# K-theory of the leaf space

*Original lesson text, proofs, exercises and drawings: CC0 1.0.*

## Introduction

A transversal meets a leaf in a discrete set. Counting these intersections produces a module over the foliation algebra. This gives a useful direction of functoriality: classes on the transversal move into the K-theory of the leaf space. A compact transversal even gives an actual projection in the algebra.

More general maps need an orientation and a replacement for a tubular neighbourhood. We explain that geometry before discussing the analytic index and the assembly map. We also construct a Kasparov cycle for a foliation by hyperbolic planes. Its compactness estimates show why negative curvature is useful.

For the graded correspondences and transversal construction, read [The C*-algebra of a foliation](the-c-star-algebra-of-a-foliation.md) and Sections 1–5 of [Hilbert modules and fields on the leaf space](hilbert-modules-and-fields-on-the-leaf-space.md). The measured applications in Sections 8–9 also use [The index theorem for measured foliations](the-index-theorem-for-measured-foliations.md). The stable K-theory prerequisites are specified next. Basic references are [Connes], [Connes–Skandalis], [Kasparov], and [Connes–Thom]. The definitions and geometric arguments below are explicit. Statements requiring Bott periodicity, Kasparov's product, or the full longitudinal index theorem identify those prerequisites at the point of use.

Write \(G=\operatorname{Hol}(V,F)\), \(A=C_r^*(G)\), and \(q=\operatorname{rank}(TV/F)\). Unless a compactness hypothesis is stated, manifolds are second countable and may be noncompact. All K-theory of locally compact spaces has compact support:

\[
K_c^j(X)=K_j(C_0(X)).
\]

The notation \(K^j(V/F)\) means \(K_j(A)\). It retains holonomy and does not replace \(A\) by the continuous functions on the set of leaves.

### Suspension, Bott periodicity and extension boundaries

The stable K-theory used below includes nonunital algebras through external unitization. Its suspension is \(SA=C_0(\mathbb R)\otimes A\). The complete programme proofs are Suspension, higher K-groups and the long exact sequence, Theorem 2.1; Bott periodicity, Theorem 4.1; and Six-term exact sequence and exponential map, Theorems 1.1–2.1. They give natural isomorphisms

\[
K_1(A)\cong K_0(SA),
\qquad
\beta_A:K_0(A)\xrightarrow{\cong}K_1(SA),
\]

and the cyclic six-term exact sequence of every C*-algebra extension. These are the suspension theorem, Bott periodicity and extension exactness, respectively. They apply to arbitrary C*-algebras and are stable matrix statements; a single index-one differential operator does not replace them.

Here are the conventions needed at their use. On the positively parametrized circle \(z=e^{2\pi it}\), the Bott map sends a relative idempotent class \([e]-[e_0]\) to the loop class

\[
\left[(ze+1-e)(ze_0+1-e_0)^{-1}\right].
\]

The idempotents agree modulo \(A\), so this loop is scalar identity modulo \(A\) and is based at identity. The formula is natural under *-homomorphisms. The inverse is proved by the coefficient Toeplitz homotopy in Bott periodicity, Sections 2–4, and the suspension isomorphism by the doubled rotation path in Suspension, higher K-groups and the long exact sequence, Theorem 2.1. Their composition supplies two-periodicity with external unitization. Relative scalar normalization and matrix/compact stability are proved in Nonunital algebras: unitization, relative classes and half-exactness, Sections 1–4; [Matrix stability, stability and continuity of \(K_0\)](companions/KT-OPK-05.html), Theorems 1.1 and 4.1; and [Invertibles, unitaries and \(K_1\)](companions/KT-OPK-06.html), Sections 1–4. These proofs apply to arbitrary C*-algebras; in particular they apply to the generally nonseparable multiplier and corona algebras used in Theorem 0.14.

For an extension \(0\to J\to B\to C\to0\), the boundary of an invertible quotient matrix \(u\) is \([WpW^{-1}]-[p]\), where \(W\) lifts \(\operatorname{diag}(u,u^{-1})\) and \(p=\operatorname{diag}(I,0)\). The signed elementary-factor construction in the [Hilbert-module lesson, formula (11)](hilbert-modules-and-fields-on-the-leaf-space.md#section-6) gives an explicit lift for an elliptic operator and its parametrix. The sign is kernel minus cokernel. The other boundary sends a quotient projection with self-adjoint lift \(h\) to \([\exp(2\pi i h)]\). The index construction, its choices and the kernel-minus-cokernel normalization are proved in [The index map and the exact sequence at \(K_0\)](companions/KT-OPK-07.html), Theorem 2.2, Theorem 4.1, Proposition 5.1 and Theorem 6.1. The positive exponential formula and all six exact positions are proved in Six-term exact sequence and exponential map, Theorems 1.1–2.1. Its Theorem 4.1 supplies the two-parity Mayer–Vietoris sequence used for vector bundles; Theorem 6.1 supplies full relative excision. Bundle triples represent the compact-pair K-group by Topological K-theory of spaces, pairs and vector bundles, Theorem 2.1 and Lemma 2.2. Those proofs fix the boundary signs, rather than leaving them to an unnamed convention.

For a pointwise norm-continuous real action \(\alpha\), Theorem 8.7 below proves the group-level Connes–Thom isomorphism using the Wiener–Hopf extension

\[
0\longrightarrow A\otimes\mathcal K
\longrightarrow \bigl(C_0(\mathbb R\cup\{+\infty\})\otimes A\bigr)
\rtimes_{\mathrm{translation}\otimes\alpha}\mathbb R
\longrightarrow A\rtimes_\alpha\mathbb R
\longrightarrow0.
\]

Its boundary \(K_{j+1}(A\rtimes_\alpha\mathbb R)\to K_j(A)\) is an isomorphism. The inverse defines the degree-one Thom map in this extension convention. Theorem 8.7 below proves it: Lemmas 8.3–8.5 and Theorem 8.6 supply crossed-product exactness, stability and Takai duality, and the full Wiener–Hopf argument uses the stable K-theory interfaces fixed above. Naturality follows from the maps of these extensions and the natural identification of their ideals. Corollary 8.10 fixes the positive cotangent Bott normalization at the trivial action. Comparison with geometric wrong-way maps and assembly remains separate.

Section0 constructs the separable graded Kasparov product and proves its associativity, units and stability. The spin-c vector-bundle KK Thom equivalence is proved in Theorem6.0c and Corollary6.0d, and full compact-source wrong-way factorization independence in Theorem6.0o. Theorems6.0s/6.0w prove the ordinary projection formula and compact ordinary-map/foliation composition. Theorem 6.1 proves geometric-cycle descent. The longitudinal index and geometric twisted-duality comparisons remain further interfaces specified in Sections 7–11.


## 0. Composing graded correspondences

The product needed for wrong-way maps must survive changes of operator, module and representation. We prove this analytic interface explicitly, including its nonunital coefficient scope. The technical construction follows the connection method of [Connes–Skandalis], Appendix A, and the graded separator in [Blackadar], Sections 12.4,14.6 and18.3–18.7. The local positivity argument is the one behind [Blackadar], Proposition17.2.7. The proofs below include the approximation estimates and both compact ideals in a triple tensor product.

For separable graded C*-algebras \(A,B\), a graded Kasparov cycle is a countably generated graded Hilbert \(B\)-module \(E\), an even *-representation \(\phi:A\to\mathcal L(E)\), and a bounded odd \(F\) such that
\[
(F^2-1)\phi(a),\quad(F-F^*)\phi(a),\quad[F,\phi(a)]
\in\mathcal K(E),\qquad a\in A.
\]
All commutators are graded: \([S,T]=ST-(-1)^{|S||T|}TS\) on homogeneous operators. Module inner products are linear in the second variable. A homotopy is a cycle over \(C([0,1],B)\), with endpoint modules obtained by coefficient extension. Degenerate cycles have all three defects exactly zero. The group \(KK(A,B)\) consists of homotopy classes, with unitary equivalence understood and addition by direct sum. The group and zero conventions are justified in Theorem0.7.

### The separator and its approximation sequence

**Lemma 0.1 (graded separation in a multiplier algebra).** Let \(J\) be a separable graded C*-algebra, \(A_1,A_2\) separable graded C*-subalgebras of \(M(J)\), and \(\Delta\) a separable graded linear subspace of \(M(J)\). Suppose \(A_1A_2\subset J\) and that the graded commutator with every element of \(\Delta\) preserves \(A_1\). There are even positive contractions \(M,N\in M(J)\) such that
\[
M+N=1,\qquad MA_1\subset J,\qquad NA_2\subset J,\qquad [M,\Delta]\subset J.
\]
The bracket with \(M\) is an ordinary commutator since \(M\) is even. The assertion concerns separation in the quotient \(M(J)/J\); it does not imply that either multiplier is compact.

**Proof of the approximation fact used below.** If a separable multiplier subspace \(\mathcal D\) derives a C*-algebra \(C\), a positive contractive approximate identity for \(C\) can be replaced by convex combinations of its tails that commute asymptotically with each element of \(\mathcal D\). To prove this, work in the bidual of the C*-algebra generated by \(C,\mathcal D\). Let \(p\) be the support projection of \(C\), so every approximate identity converges strongly to \(p\). The derivation extends weakly to \(C''\). Since \(p\) is its unit,
\[
[p,d]=p[p,d]=[p,d]p,\qquad
[p^2,d]=p[p,d]+[p,d]p=2[p,d],
\]
whereas \(p^2=p\); thus \([p,d]=0\). The finite tuple of commutators with \(d_1,\ldots,d_k\) consequently converges weakly to zero in the ambient algebra's \(k\)-fold direct sum. Hahn–Banach separation puts zero in the norm closure of the convex hull of every tail of these tuples. Choose the convex combinations while retaining the approximate-identity estimates for any prescribed finite subset of \(C\). A diagonal choice on countable dense subsets proves the assertion. Norm bounds make the convergence uniform on each compact test set.

For the even version start with the approximate identity obtained from an even strictly positive element: if \(h\) is strictly positive, \(h+\gamma(h)\) is even and strictly positive, where \(\gamma\) is the grading automorphism. All tail convex combinations stay even. Commutators with even terms are ordinary, so the preceding argument applies to the graded derivation hypothesis as well. If an increasing approximate identity is needed, start with increasing functional-calculus terms. A convex combination from a tail beyond the last index used previously dominates the preceding convex combination. This retains monotonicity.

We now prove the separator. Choose adjoint-invariant compact test sets \(X_i\subset(A_i)_1\), \(Y\subset\Delta_1\), and \(Z\subset J_1\) whose linear spans are dense in their respective spaces. For example, scale each member of a countable dense sequence by successively smaller positive constants and adjoin its adjoint and the zero limit. Set \(\epsilon_n=2^{-n}\). The approximation fact supplies even positive contractions \(u_n\in A_1\) such that
\[
\sup_{x\in X_1}\|u_nx-x\|<\epsilon_n,\qquad
\sup_{y\in Y}\|[u_n,y]\|<\epsilon_n.
\]

Every \(u_jw\) lies in \(J\) for \(w\in Z\cup X_2\). Construct an increasing even approximate identity \(v_n\) for \(J\), putting \(v_0=0\), so that
\[
\sup_{\substack{j\le n+1\\w\in Z\cup X_2}}
\|(1-v_n)u_jw\|<\epsilon_n^2.
\tag{90}
\]
In particular \(v_{n-1}\) controls \(u_nw\), as well as \(v_n\). This is the one-step lookahead. Also require, for
\(b_n=(v_n-v_{n-1})^{1/2}\),
\[
\sup_{w\in X_1\cup X_2\cup Y\cup Z}\|[b_n,w]\|<\epsilon_n.
\tag{91}
\]
These requirements can be imposed together. All their test sets are compact. Quasicentral approximate identities for \(J\) exist for the separable subspace spanned by those sets, since every multiplier derives \(J\). Polynomial approximation to the square-root function on \([0,1]\) shows uniformly for \(\|w\|\le1\) that a sufficiently small \(\|[a,w]\|\) implies a prescribed small \(\|[a^{1/2},w]\|\). At step \(n\), require the commutator error for \(v_n\) to be small enough for both the current \(b_n\) and the future \(b_{n+1}\). More explicitly, choose tolerances \(\delta_n\) for square-root errors \(\epsilon_n\), and make the error for \(v_n\) at most \(\frac12\min(\delta_n,\delta_{n+1})\). Then \([v_n-v_{n-1},w]\) has norm at most \(\delta_n\). The increasing tail-convex construction enforces these tolerances, (90), and strict convergence on a countable total subset of \(J\).

For \(n\ge2\), (90) at \(n\) and \(n-1\) gives
\[
\|b_n^2u_nw\|
\le\|(1-v_n)u_nw\|+\|(1-v_{n-1})u_nw\|
<5\epsilon_n^2,
\qquad w\in Z\cup X_2.
\]
It follows that
\[
\|b_nu_nw\|^2
=\|w^*u_nb_n^2u_nw\|
\le5\epsilon_n^2.
\]
The harmless first term obeys the same bound using \(\|u_1w\|\le1=4\epsilon_1^2\). Combining this with (91) gives
\[
\|b_nu_nb_nw\|
\le\|b_nu_nwb_n\|+\|b_nu_n[b_n,w]\|
\le(\sqrt5+1)\epsilon_n.
\tag{92}
\]
The same estimate holds on the other side for the adjoint test set.

The positive partial sums of \(\sum b_nu_nb_n\) are bounded by
\(\sum b_n^2=v_n\le1\). Equation (92), first on \(Z\) and then by density on \(J\), proves that these partial sums are strictly Cauchy. A bounded strictly Cauchy sequence of multipliers has a strict limit: its limits on \(J\) on the two sides are bounded compatible left and right multiplication maps, hence define a multiplier. Call the even positive contractive limit \(N\), and put \(M=1-N\).

For \(x\in X_2\), (92) makes \(Nx=\sum b_nu_nb_nx\) a norm-convergent series of elements of \(J\). Thus \(NA_2\subset J\). For \(x\in X_1\), use
\[
(b_n^2-b_nu_nb_n)x
=b_n(1-u_n)x b_n+b_n(1-u_n)[b_n,x].
\]
Its norm is at most \(2\epsilon_n\). Consequently the series of these terms converges in norm in \(J\). Since \(\sum b_n^2x=v_nx\) converges strictly to \(x\), its sum is \((1-N)x=Mx\). Density gives \(MA_1\subset J\). Finally, for \(y\in Y\),
\[
[b_nu_nb_n,y]
=b_nu_n[b_n,y]+b_n[u_n,y]b_n+[b_n,y]u_nb_n,
\]
whose norm is at most \(3\epsilon_n\). Every term lies in \(J\), so the norm-convergent series is \([N,y]\in J\); density gives the assertion on all of \(\Delta\), and hence on \(M\). This proves the lemma. \(\square\)

### The connection construction

We give the needed module argument, rather than treating “connection” as an unexplained analogy with a bundle connection. Write \(\mathcal K(E',E)\) for the norm closure of finite-rank adjointable maps, and take inner products linear in the second variable.

**Lemma 0.2 (graded stabilization and connections).** A countably generated graded Hilbert \(D\)-module \(E_1\), regarded over \(D^+\), is the range of an even adjointable projection on the graded standard module \(\widehat H_{D^+}\). If a bounded homogeneous \(F_2\in\mathcal L(E_2)\) graded-commutes with the representation of \(D\) modulo \(\mathcal K(E_2)\), there is a bounded homogeneous \(G\) on
\(E=E_1\widehat\otimes_D E_2\) such that, for homogeneous \(\xi\in E_1\),
\[
GT_\xi-(-1)^{|\xi||F_2|}T_\xi F_2
\in\mathcal K(E_2,E),\qquad
T_\xi^*G-(-1)^{|\xi||F_2|}F_2T_\xi^*
\in\mathcal K(E,E_2).
\tag{93}
\]
Here \(T_\xi\eta=\xi\widehat\otimes\eta\) and
\(T_\xi^*(\zeta\widehat\otimes\eta)=\phi_2(\langle\xi,\zeta\rangle)\eta\).
If \(F_2\) is self-adjoint, \(G\) can be self-adjoint; norm-continuous paths of \(F_2\)'s give norm-continuous paths of \(G\)'s.

**Proof.** Choose homogeneous generators \(\eta_i\) of norm at most one, repeating every generator infinitely often in its own parity. Include zero entries so the standard basis has infinitely many entries of either parity, and give the standard basis \(e_i\) the same parity as \(\eta_i\). On \(\widehat H_{D^+}\) define the even adjointable map
\[
T e_i=2^{-i}\eta_i\oplus4^{-i}e_i
\quad\hbox{in }E_1\oplus\widehat H_{D^+}.
\]
The first column series and its adjoint converge by the summable coefficients; the second component is a bounded diagonal. The range of \(T^*\) is dense: \(T^*(0\oplus4^ie_i d)=e_id\), so every finite standard column lies in it. The range of \(T\) is also dense. If \(\eta_i\) recurs as \(\eta_{j_k}\), with \(j_k\to\infty\), then
\[
T(2^{j_k}e_{j_k}d)=\eta_i d\oplus2^{-j_k}e_{j_k}d
\longrightarrow\eta_id\oplus0.
\]
The generators therefore put \(E_1\oplus0\) in the closure of its range. Subtracting the first component of \(T(4^ie_id)\) then puts \(0\oplus e_id\) in that closure as well.

The rule \((T^*T)^{1/2}z\mapsto Tz\) preserves the \(D^+\)-valued inner product. Both its domain and range are dense; the first density follows from the density of the range of \(T^*\), since the closure of the range of \(T^*\) equals that of \((T^*T)^{1/2}\). For completeness, this equality follows from the norm approximations
\[
T^*= \lim_{\epsilon\downarrow0}
(T^*T)^{1/2}(T^*T+\epsilon)^{-1/2}T^*
\]
on each fixed vector, and the reverse inclusion obtained by approximating
\((T^*T)^{1/2}\) with \(T^*T(T^*T+\epsilon)^{-1/2}\).
The errors are controlled by scalar spectral bounds, uniformly on the operator spectrum. Thus the inner-product-preserving rule extends to an even unitary
\(\widehat H_{D^+}\cong E_1\oplus\widehat H_{D^+}\). Transporting the first-summand projection gives the desired even projection \(P\).

Extend the representation of \(D\) on \(E_2\) to \(D^+\). It has the same compact commutators with \(F_2\), since the new scalar identity commutes exactly. Under
\(\widehat H_{D^+}\widehat\otimes E_2\cong\widehat H_\mathbb C\widehat\otimes E_2\),
put
\[
G_0(e_i\widehat\otimes\eta)
=(-1)^{|e_i||F_2|}e_i\widehat\otimes F_2\eta.
\]
For a homogeneous finite column \(\xi=\sum e_i d_i\), the two creation-map errors in (93) are finite columns or rows of the compact graded commutators of \(F_2\) with \(\phi_2(d_i)\), with exactly the displayed signs. A finite column of compact maps is compact; finite-column approximation and
\(\|T_\xi\|\le\|\xi\|\) treat every \(\xi\). Compress \(G_0\) by \(P\widehat\otimes1\). Since \(P\xi=\xi\) for \(\xi\in E_1\), the first error is the corresponding projected error for \(G_0\); the second follows by the same adjoint calculation. Hence \(G=PG_0P\) satisfies (93). Compression preserves self-adjointness and boundedness, and uses a fixed \(P\) along a path, proving the continuity assertion. \(\square\)

We record consequences used in the product. An operator \(R\) is a zero connection precisely when
\[
R(\mathcal K(E_1)\widehat\otimes1)\subset\mathcal K(E),
\qquad
(\mathcal K(E_1)\widehat\otimes1)R\subset\mathcal K(E).
\tag{94}
\]
One direction follows from
\(\theta_{\xi,\zeta}\widehat\otimes1=T_\xi T_\zeta^*\).
For the other, \(RT_\xi T_\xi^*R^*\) compact implies \(RT_\xi\) compact: the compact ideal in the linking algebra is hereditary, or directly
\(RT_\xi\) is approximated by
\(h_\epsilon(RT_\xi T_\xi^*R^*)RT_\xi\), where
\(h_\epsilon(t)=t/(t+\epsilon)\);
the norm error is bounded by
\(\sup_{t\ge0}\sqrt t\,\epsilon/(t+\epsilon)\to0\).
The adjoint statement is identical.

Sums and products of connections have the corresponding sums and products of their lower operators, by inserting (93) and its adjoint. Polynomial approximation consequently gives the same statement for continuous functional calculus of normal connections. The difference of two \(F_2\)-connections is a zero connection. If \(G\) is an odd \(F_2\)-connection and \(F_2\) is a normalized self-adjoint Kasparov operator, then \(G^2-1\) and \(G-G^*\) are zero connections. Indeed \(T_\xi(F_2^2-1)\) is compact first for \(\xi d\), by the localized Kasparov defect with \(\phi_2(d)\), then for all \(\xi\) by the essential right module action. The sum/product rules and the adjoint calculation finish the assertion. For an adjointable homogeneous \(S\) on \(E_1\), the graded commutator \([S\widehat\otimes1,G]\) is also a zero connection: apply (93) to \(\xi\) and \(S\xi\); their principal terms cancel with their parity signs. In particular \(G\) derives \(\mathcal K(E_1)\widehat\otimes1\) into \(\mathcal K(E)\).

### Existence of the product operator

**Definition 0.3 (the product criterion).** For cycles \((E_1,\phi_1,F_1)\) from \(A\) to \(D\) and \((E_2,\phi_2,F_2)\) from \(D\) to \(B\), a product operator on \(E=E_1\widehat\otimes_D E_2\) is an odd \(F_2\)-connection which makes \((E,\phi_1\widehat\otimes1,F)\) a Kasparov cycle and satisfies
\[
\phi(a)[F_1\widehat\otimes1,F]\phi(a)^*\ge0
\quad\bmod\mathcal K(E),\qquad a\in A.
\]
We write \(F\in F_1\#_D F_2\). The bracket is an anticommutator; its sign is part of the definition.

**Theorem 0.4 (product existence).** For separable graded coefficient algebras, each pair of self-adjoint graded Kasparov modules has a self-adjoint product operator on its countably generated tensor module. Every homotopy class admits such representatives by the localized normalization proved below.

**Proof.** Take normalized graded Kasparov modules
\((E_1,\phi_1,F_1)\) for \(A,D\) and
\((E_2,\phi_2,F_2)\) for \(D,B\), with all algebras separable and \(F_i\) self-adjoint contractions of degree one. Let
\(E=E_1\widehat\otimes_D E_2\), \(\phi=\phi_1\widehat\otimes1\),
and \(S=F_1\widehat\otimes1\).
The tensor product is countably generated: products of two countable generating systems are total, after approximating middle coefficients by their action on the second system. Let \(G\) be the self-adjoint odd connection just constructed. With \(J=\mathcal K(E)\), put
\[
A_1=C^*(\mathcal K(E_1)\widehat\otimes1,J),\qquad
A_2=C^*(G^2-1,G-G^*,[G,\phi(A)],[G,F_1\widehat\otimes1]).
\]
Every generator of \(A_2\) is a zero connection by the preceding identities. Equation (94) then gives \(A_1A_2\subset J\), including its adjoint products and norm closure. The operators \(S\) and \(\phi(A)\) derive \(\mathcal K(E_1)\widehat\otimes1\) into itself, while \(G\) derives it into \(J\). All multipliers derive \(J\). Thus
\(\Delta=\operatorname{span}(\phi(A),G,S)\) derives \(A_1\).
These algebras and this subspace are separable: compact endomorphisms on countably generated modules over separable algebras are separable, and \(A_2\) is generated by the continuous image of separable \(A\) and finitely many further operators.

Apply the separator lemma and set
\[
F=M^{1/2}S+N^{1/2}G.
\tag{95}
\]
Square roots retain the separator properties. This follows by polynomial approximation with zero constant term for annihilation of \(A_i\), and by continuous functional calculus for compact commutators. In particular \(M^{1/2}\) is a zero connection and \(N^{1/2}\) is an identity connection. Their compositions with \(S,G\) make \(F\) an \(F_2\)-connection.

Modulo \(J\), square roots of \(M,N\) commute with \(S,G,\phi(A)\). Also \(N^{1/2}[S,G]\in J\). Therefore, after multiplying on the right by \(\phi(a)\),
\[
(F^2-1)\phi(a)
\equiv M(S^2-1)\phi(a)+N(G^2-1)\phi(a)
\equiv0\pmod J.
\]
The first term is compact because
\((S^2-1)\phi(a)\in\mathcal K(E_1)\widehat\otimes1\);
the second is compact by the separator property on \(A_2\).
Similarly \(F-F^*\) and \([F,\phi(a)]\), after the necessary localization, are compact. Thus \((E,\phi,F)\) is a Kasparov module. Replacing \(F\) by
\[
\widetilde F=M^{1/4}SM^{1/4}+N^{1/4}GN^{1/4}
\]
makes it self-adjoint, with difference in \(J\); this also preserves the connection identities.

The decisive positivity has the exact sign
\[
\phi(a)[S,F]\phi(a)^*
\equiv2\phi(a)M^{1/2}\phi(a)^*\ge0\pmod J.
\tag{96}
\]
Here the bracket is graded, hence an anticommutator. The term with \(G\) vanishes modulo \(J\) by \(N^{1/2}[S,G]\in J\). Replacing \(S^2\phi(a)^*\) by \(\phi(a)^*\) in the remaining term is permitted by the compact first-module defect and the \(M^{1/2}A_1\) separator property.

Equations (93), the Kasparov defects and (96) establish existence of an operator satisfying the product criterion. The source algebra and module algebras from the second-countable foliation are separable, so the bounded separable construction preserves the required foliation scope. We now prove the remaining product assertions. This proves existence. Uniqueness and the compatibility needed for composition follow below. \(\square\)

The existence calculation also applies to bounded self-adjoint odd input operators without the contraction bound: only their localized square defects, commutators and connection errors were used. This matters when a previously constructed product is used as the second input. All estimates remain bounded, and the separator tests still use norm-one total sets.

A related calculation used below is that \([S,F]\) belongs to \(\mathcal D_\phi\) whenever \(F\) is a product operator. Graded Jacobi expresses its commutator with \(\phi(a)\) as a sum involving \([F,\phi(a)]\), which is compact, and the graded commutator of \(F\) with \([S,\phi(a)]\). The latter lies in \(\mathcal K(E_1)\widehat\otimes1\), and \(F\), as a connection, derives that algebra into \(\mathcal K(E)\). Both summands are compact. This verifies the pseudolocal hypothesis needed for the localized positivity test on product anticommutators.

### Localized normalization and the positive bridge

For a graded representation \(\phi:A\to\mathcal L(E)\), let
\[
\mathcal D_\phi=\{T:[T,\phi(a)]\in\mathcal K(E)\text{ for all }a\in A\},
\quad
\mathcal I_\phi=\{T\in\mathcal D_\phi:T\phi(a),\phi(a)T\in\mathcal K(E)
\text{ for all }a\in A\}.
\]
The graded commutator condition means the condition on each homogeneous component of \(T\). The graded Leibniz rule, the adjoint identity and norm closure show that \(\mathcal D_\phi\) is a graded C*-algebra and \(\mathcal I_\phi\) a closed graded ideal. A Kasparov operator is a self-adjoint odd unitary modulo \(\mathcal I_\phi\). Self-adjointness here can initially hold only in the quotient. Replacing \(F\) by \((F+F^*)/2\) changes it by an element of \(\mathcal I_\phi\) and gives a straight-line operator homotopy. Clamping this self-adjoint operator by the continuous odd function \(c(t)=\max(-1,\min(1,t))\) changes it by another element of \(\mathcal I_\phi\), since \(c(\pm1)=\pm1\). Another straight-line homotopy gives a self-adjoint odd contraction. This argument also works on a module over \(C([0,1],B)\), so normalizing a module homotopy does not require a fixed module at its endpoints.

Here is a precise test for positivity in this localized quotient. Put \(Q=\mathcal L(E)/\mathcal K(E)\) and let \(p\) be the support projection of the representation \(q\phi\) in \(Q^{**}\). An even positive approximate identity of \(A\) converges to \(p\) there. Every element of \(q(\mathcal D_\phi)\) commutes with that even approximate identity and hence with \(p\). The map
\[
\mathcal D_\phi/\mathcal I_\phi\longrightarrow pQ^{**}p,\qquad
T+\mathcal I_\phi\longmapsto p\,q(T)\,p
\]
is an injective *-homomorphism. Its kernel is exactly the operators whose products with every \(\phi(a)\) vanish in \(Q\): this follows in one direction by multiplication, and in the other by strong convergence of the approximate identity. It therefore reflects positivity. If a self-adjoint even \(H\in\mathcal D_\phi\) obeys \(\phi(a)H\phi(a)^*\ge0\) modulo compacts for every \(a\), applying this to the even approximate identity and taking its strong limit gives \(p q(H)p\ge0\). Thus \(H\ge0\) modulo \(\mathcal I_\phi\). This proves the test without assuming \(\phi\) nondegenerate or \(A\) unital.

**Lemma 0.5 (positive anticommutator bridge).** Suppose \(F,F'\) are self-adjoint odd Kasparov operators on the same represented module, and \(\phi(a)[F,F']\phi(a)^*\ge0\) modulo \(\mathcal K(E)\) for every \(a\in A\). Then they are joined by a norm-continuous operator homotopy.

**Proof.** All brackets between odd operators are anticommutators. Let \(H=FF'+F'F\) and \(P=H_+\), its positive part. The positivity test gives \(H_-\in\mathcal I_\phi\), so \(P=H\) in \(\mathcal D_\phi/\mathcal I_\phi\). In that quotient \(F^2=F'^2=1\); direct multiplication shows that \(H\), and hence \(P\), commutes with \(F,F'\). For \(0\le\theta\le\pi/2\) set
\[
A_\theta=(1+\cos\theta\sin\theta\,P)^{-1/4},\qquad
F_\theta=A_\theta(\cos\theta\,F+\sin\theta\,F')A_\theta.
\tag{97}
\]
The positive inverse exists with norm at most one. Continuous functional calculus makes the path norm-continuous, self-adjoint and odd. Modulo \(\mathcal I_\phi\), its square equals
\((1+\cos\theta\sin\theta\,P)^{-1}(1+\cos\theta\sin\theta\,H)=1\).
It graded-commutes with \(\phi(A)\) modulo compacts because all its factors belong to \(\mathcal D_\phi\). At \(\theta=0,\pi/2\) the outside factors are exactly one, giving \(F,F'\). These are precisely the Kasparov defects required for an operator homotopy. \(\square\)

### Uniqueness and descent through homotopies

**Theorem 0.6 (uniqueness of the product operator).** For fixed normalized input modules, any two self-adjoint product operators \(F,F'\) on \(E_1\widehat\otimes_D E_2\) are operator homotopic.

**Proof.** Keep \(S=F_1\widehat\otimes1\) and \(J=\mathcal K(E)\). Apply the separator with
\[
A_1=C^*(\mathcal K(E_1)\widehat\otimes1,J),\quad
A_2=C^*([S,F],[S,F'],F-F'),\quad
\Delta=\operatorname{span}(\phi(A),S,F,F').
\]
Each generator of \(A_2\) is a zero connection: the first two by (93) applied to \(S\xi\), and the last because both operators have the same lower connection. Hence \(A_1A_2\subset J\). Each member of \(\Delta\) derives \(A_1\), by the connection calculation or its action on \(E_1\). All spaces are separable as before. Obtain \(M,N\), put \(L=M^{1/2}S+N^{1/2}F\), and replace it by
\(L_{\rm sa}=M^{1/4}SM^{1/4}+N^{1/4}FN^{1/4}\).
Their difference is compact. The existence calculation still applies: \((F^2-1)\phi(a)\) is already compact, and the cross term is killed by \(N[S,F]\). Thus \(L_{\rm sa}\) is an \(F_2\)-connection and a product operator.

The following congruences are modulo \(J\):
\[
\begin{aligned}
\phi(a)[F,L_{\rm sa}]\phi(a)^*
&\equiv\phi(a)M^{1/2}[F,S]\phi(a)^*
+2\phi(a)N^{1/2}\phi(a)^*,\\
\phi(a)[F',L_{\rm sa}]\phi(a)^*
&\equiv\phi(a)M^{1/2}[F',S]\phi(a)^*
+2\phi(a)N^{1/2}\phi(a)^*.
\end{aligned}
\tag{98}
\]
For the second equality, \(N^{1/2}(F'-F)\in J\) permits replacement of \(F'\) by \(F\) in that summand. Both right sides are positive modulo \(J\). For example, \(M\) commutes with \(F,S,\phi(A)\) modulo \(J\), so the first summand equals the positive sandwich \(M^{1/4}\phi(a)[F,S]\phi(a)^*M^{1/4}\) modulo \(J\). Its middle factor is positive modulo \(J\) by the product criterion. The other summand is positive directly. Apply (97) from \(F\) to \(L_{\rm sa}\) and then from \(L_{\rm sa}\) to \(F'\); concatenate the paths. \(\square\)

A nonsymmetric product operator can first be symmetrized. Its difference \(R\) from the symmetrization is locally compact, and is a zero connection because the lower operator \(F_2\) is self-adjoint. The positivity condition is unchanged: in \(\phi(a)[S,R]\phi(a)^*\), terms with \(R\phi(a)^*\) or \(\phi(a)R\) are compact; commuting \(S\) past \(\phi(a)^*\) adds only \(R[S,\phi(a)^*]\), which is compact by (94). Its adjoint is handled identically. The straight-line change is a Kasparov homotopy and retains the connection condition. Thus uniqueness covers localized self-adjointness too.

**Theorem 0.7 (a bilinear product on homotopy classes).** For separable graded \(A,D,B\), the construction defines
\[
KK(A,D)\times KK(D,B)\longrightarrow KK(A,B),\qquad
(x,y)\longmapsto x\widehat\otimes_D y.
\tag{99}
\]

**Proof.** Normalize representatives and entire module homotopies by the localized functional calculus above. A homotopy of second modules is a Kasparov module over \(IB=C([0,1],B)\). Apply existence to it and the fixed first module. Evaluating at either endpoint maps rank-one compact operators to rank-one operators, preserves adjoints and positivity, and sends each connection identity (93) to that of the endpoint. The canonical map
\[
(E_1\widehat\otimes_D\mathcal E_2)\widehat\otimes_{\mathrm{ev}_t}B
\longrightarrow E_1\widehat\otimes_D(\mathcal E_2\widehat\otimes_{\mathrm{ev}_t}B)
\]
sends elementary tensors to elementary tensors. Their inner-product formulas coincide, and elementary tensors have dense span, so it is an even unitary. The evaluated product is therefore a product of the endpoint modules; uniqueness identifies its class with any other choice.

For a first-module homotopy \(\mathcal E_1\) over \(ID\), use the second module \(C([0,1],E_2)\) over \(IB\), represented pointwise by \(ID\), with operator \(1\otimes F_2\). This is a Kasparov module: a continuous function with values in \(D\) is uniformly approximated by finite sums \(f(t)d\), and all its localized defects are continuous compact-valued functions. The identification \(\mathcal K(C([0,1],E_2))=C([0,1],\mathcal K(E_2))\) follows first for rank-one sections; a partition of unity approximates each continuous compact-valued function uniformly by finite sums of these. Apply existence to this pair. At an endpoint the same inner-product argument identifies the tensor product with \(\mathcal E_1(t)\widehat\otimes_D E_2\), and yields the required product homotopy. General module homotopies, including changes of module and representation, are covered by this argument.

Unitary equivalence is respected by transporting elementary tensors. Direct sums in either input distribute over the tensor product; the direct sum of two product operators satisfies the connection and positive sandwich criteria, and uniqueness proves additivity.

For completeness, a degenerate module represents zero under this homotopy convention. On the \(IB\)-module \(C_0([0,1),E)\), use the original representation and operator pointwise. Their defects are identically zero. Its fibre at zero is the original module, and at one is zero. Countable generation follows from generators of \(E\) and a countable approximate identity of \(C_0([0,1))\). Hence degenerates contract to zero, and the preceding homotopy argument handles stabilization by them. The opposite graded module gives additive inverses explicitly: reverse the grading, replace the homogeneous action by \(\phi^{\rm op}(a)=(-1)^{|a|}\phi(a)\), and replace \(F\) by \(-F\). On the sum with the original module the odd off-diagonal identity \(R\) graded-commutes with the diagonal representation and anticommutes with \(F\oplus(-F)\). The path \(\cos\theta(F\oplus(-F))+\sin\theta R\) is a Kasparov homotopy ending in a degenerate module. Thus the homotopy classes form a group and the additive pairing is bilinear.

Norm-continuous operator homotopies of both inputs can also be treated with one separator: choose the connections using one fixed stabilization projection, and put all path values of the defects and commutators in \(A_2\) and of the controlling operators in \(\Delta\). These sets are separable, their generators are zero connections, and the same separator gives a norm-continuous product path. This supplies the stronger fixed-module version when needed. \(\square\)

### The two compact ideals in the triple product

Put
\[
E_{12}=E_1\widehat\otimes_{D_1}E_2,\quad
E_{23}=E_2\widehat\otimes_{D_2}E_3,\quad
E=E_1\widehat\otimes_{D_1}E_{23}
\cong E_{12}\widehat\otimes_{D_2}E_3.
\]
The associating unitary is obtained from
\((\xi_1\widehat\otimes\xi_2)\widehat\otimes\xi_3
\mapsto\xi_1\widehat\otimes(\xi_2\widehat\otimes\xi_3)\).
The balanced relations and inner products coincide. No factors are interchanged, so it introduces no Koszul sign.

**Lemma 0.8 (connections in three factors).** If \(G_{12}\) is an \(F_2\)-connection, \(G_{23}\) an \(F_3\)-connection, and \(G\) a \(G_{23}\)-connection on \(E\), then \(G\) is an \(F_3\)-connection for \(E_{12}\). If these operators are odd, then \([G_{12}\widehat\otimes1,G]\) is an \([F_2\widehat\otimes1,G_{23}]\)-connection for \(E_1\).

**Proof.** On a simple creation vector, the creation map for \(\xi_1\widehat\otimes\xi_2\) is \(T_{\xi_1}T_{\xi_2}\). Insert the two connection identities in succession. The principal sign is \((-1)^{|\xi_1|+|\xi_2|}\); errors are compact because bounded operators compose with compact maps. Finite sums and density prove the first assertion, including its adjoint.

For the second assertion use the linking module \(E_{23}\oplus E\). For homogeneous \(\xi\in E_1\), put \(U_\xi=\begin{pmatrix}0&T_\xi^*\\ T_\xi&0\end{pmatrix}\), and
\[
X=(F_2\widehat\otimes1)\oplus(G_{12}\widehat\otimes1),
\qquad Y=G_{23}\oplus G.
\]
Then \([Y,U_\xi]\) is compact. The operator \([X,U_\xi]\) is the tensor with \(1\) of a compact operator on the \(D_2\)-module \(E_2\oplus E_{12}\). This tensor need not be compact on the final \(B\)-module. However \(Y\) is an \(F_3\)-connection for \(E_2\oplus E_{12}\), by the first assertion. Its graded commutator with such a tensor of a compact operator is compact, by the rank-one connection calculation. Graded Jacobi now gives
\[
[[X,Y],U_\xi]=[X,[Y,U_\xi]]+[Y,[X,U_\xi]]
\in\mathcal K(E_{23}\oplus E),
\]
since \(X,Y\) are odd. Its off-diagonal entries are exactly the desired connection errors. \(\square\)

**Theorem 0.9 (associativity).** For separable graded \(A,D_1,D_2,B\) and composable classes \(x,y,z\),
\[
x\widehat\otimes_{D_1}(y\widehat\otimes_{D_2}z)
=(x\widehat\otimes_{D_1}y)\widehat\otimes_{D_2}z.
\tag{100}
\]

**Proof.** Take normalized self-adjoint representatives and self-adjoint product operators \(F_{12},F_{23},F\), where \(F_{12}\in F_1\#F_2\), \(F_{23}\in F_2\#F_3\), and \(F\in F_1\#F_{23}\). On \(E\) write \(S=F_1\widehat\otimes1\widehat\otimes1\), \(T=F_{12}\widehat\otimes1\), \(H=[T,F]\), and \(\phi=\phi_1\widehat\otimes1\widehat\otimes1\). By the preceding lemma \(F\) is an \(F_3\)-connection for \(E_{12}\), and \(H\) is an \(H_{23}=[F_2\widehat\otimes1,F_{23}]\)-connection for \(E_1\). These anticommutators are even and self-adjoint.

The product positivity for \(F_{23}\), together with the localized positivity test, says that \((H_{23})_-\) has compact products with the representation of \(D_1\). Hence \(T_\xi(H_{23})_-\) is compact: first write \(\xi d\), use the localized defect with \(d\), and then approximate \(\xi\) by such vectors using an approximate identity of \(D_1\). Functional calculus of connections makes \(H_-\) a \((H_{23})_-\)-connection, so \(H_-\) is a zero connection for \(E_1\). It is also a zero connection for \(E_{12}\): the whole \(H\) is one because \(T\) acts on \(E_{12}\), and the negative-part function vanishes at zero. Similarly \([S,F]\) is a zero connection for both \(E_1\) and \(E_{12}\).

Use the separator on
\[
\begin{aligned}
J&=\mathcal K(E),\\
A_1&=C^*(J,\mathcal K(E_{12})\widehat\otimes1,
                 \mathcal K(E_1)\widehat\otimes1\widehat\otimes1),\\
A_2&=C^*(H_-,[S,F]),\qquad
\Delta=\operatorname{span}(\phi(A),F,S,T).
\end{aligned}
\]
The two zero-connection properties give \(A_1A_2\subset J\). Check the derivations explicitly: \(F\) derives both compact-tensor algebras into \(J\); \(S\) and \(\phi(A)\) act by adjointable operators before the last tensor, so derive the \(E_{12}\) compact algebra into itself, and likewise for the \(E_1\) algebra. \(T\) derives the \(E_{12}\) algebra into itself and derives the \(E_1\) algebra into \(\mathcal K(E_{12})\widehat\otimes1\), by the \(F_{12}\)-connection identity. Thus all derivatives land in \(A_1\). Separability and the remaining hypotheses have been checked.

Let \(M,N\) be the resulting separator. Symmetrize \(F'=M^{1/2}S+N^{1/2}F\) by fourth powers as above. The existence calculation makes \(F'\) a Kasparov operator. Its positive anticommutator with \(F\) follows exactly as in the first line of (98), using the original product positivity for \([S,F]\). Thus (97) identifies its class with the first parenthesization in (100). Since \(M\) kills \(\mathcal K(E_{12})\widehat\otimes1\) and commutes with \(S,F\) modulo \(J\), \(F'-F\) is a zero connection for \(E_{12}\). In detail, \(M^{1/2}S\) is one; and \(N^{1/2}-1\) is one because the continuous function \(\sqrt{1-t}-1\) vanishes at \(t=0\). Their products with adjointable operators on \(E_{12}\), or with its connection \(F\), retain that property. Consequently \(F'\) is an \(F_3\)-connection for that module.

Its remaining product positivity is obtained by retaining both terms:
\[
\phi(a)[T,F']\phi(a)^*
\equiv
M^{1/4}\phi(a)[T,S]\phi(a)^*M^{1/4}
+N^{1/4}\phi(a)H_+\phi(a)^*N^{1/4}\ge0
\pmod J.
\tag{101}
\]
The replacement of \(H\) by \(H_+\) is permitted by \(N H_-\in J\) and functional calculus. For the first summand, product positivity for \(F_{12}\) gives, on \(E_{12}\),
\[
\phi_{12}(a)[F_{12},F_1\widehat\otimes1]\phi_{12}(a)^*
=P_a+K_a,\qquad P_a\ge0,\quad K_a\in\mathcal K(E_{12}).
\]
Tensoring this equality with \(1\) does not assert \(K_a\widehat\otimes1\) compact. Instead, the \(M^{1/4}\) factors make its contribution compact by the separator property on the \(E_{12}\) compact algebra. The resulting sandwich of \(P_a\widehat\otimes1\) is positive. The second summand in (101) is positive directly. Therefore \(F'\) is a product of \(F_{12}\) and \(F_3\), and represents the second parenthesization. This proves (100).

The \(M\)-summand in (101) was not discarded or asserted compact: its positivity follows from the first product, and its compact error is removed by the larger separator algebra. This is the step needed when tensoring does not preserve compactness. \(\square\)

### Units, naturality and stability

**Lemma 0.10 (the essential part is homotopic to the original cycle).** Let \((E,\phi,F)\) be a self-adjoint graded Kasparov \(D,B\)-module and \(E^{\rm ess}=\overline{\phi(D)E}\). There is a Kasparov homotopy whose zero fibre differs from \((E,\phi,F)\) by a compact operator and whose one fibre is an \(F\)-connection on \(D\widehat\otimes_\phi E\cong E^{\rm ess}\). No orthogonal complement of \(E^{\rm ess}\) is assumed.

**Proof.** Let \(I=[0,1]\),
\[
J_D=\{j\in C(I,D^+):j(1)\in D\},\qquad
\mathcal E=J_D\widehat\otimes_{C(I,D^+)}C(I,E).
\]
Regard the ideal \(J_D\) as a Hilbert \(C(I,D^+)\)-module, and extend \(\phi\) unitally to \(D^+\). The map \(j\otimes\xi\mapsto(t\mapsto\phi^+(j(t))\xi(t))\) preserves inner products and has range dense in
\[
\{\eta\in C(I,E):\eta(1)\in E^{\rm ess}\}.
\]
To see density, approximate the endpoint by a finite sum \(\phi(d_i)\eta_i\); extend this sum constantly, cut it off near one, and use uniform continuity to make the endpoint error uniformly small there. On the remaining closed interval, scalar functions vanishing near one belong to \(J_D\) and supply arbitrary continuous sections. Thus the map extends to a unitary onto the displayed module. Its zero fibre is \(E\), and its one fibre is \(E^{\rm ess}\).

Choose a self-adjoint \(1\otimes F\)-connection \(G\) using the stabilization construction, and represent \(D\) by its constant functions on the first module. The commutator \([G,d]\) is compact because multiplication by a constant \(d\in D\subset J_D\) is a compact endomorphism of the ideal module \(J_D\). For the square defect, \(G^2-1\) is an \((1\otimes(F^2-1))\)-connection. Although this lower defect need not be compact on all of \(C(I,E)\), its products with constant \(d\in D\) are compact-valued continuous functions. In particular \(T_{d_1}(1\otimes(F^2-1))\) is compact: approximate \(d_1\) by \(d_1e_n\), then use \(T_{d_1e_n}=T_{d_1}\phi(e_n)\) and the lower localized defect. Multiplication by \(d_1d_2\) on \(J_D\) is the rank-one map \(\theta_{d_1,d_2^*}\). The connection square identity therefore makes \((G^2-1)d_1d_2\) compact. Products span \(D\) densely, proving every required localized square defect. Self-adjointness is exact. Hence \((\mathcal E,D,G)\) is a Kasparov homotopy.

At zero the first module fibre is \(D^+\); creation by its unit is the identity on \(E\), so (93) gives \(G(0)-F\in\mathcal K(E)\). At one the first fibre is \(D\); (93) gives an \(F\)-connection on \(D\widehat\otimes_\phi E\). This proves the assertion. \(\square\)

**Corollary 0.11 (units and naturality).** A graded *-homomorphism \(f:A\to D\) defines the cycle \([f]=(D,f,0)\). Its square defect is \(-f(a)\in\mathcal K(D)=D\), so this is legitimate also when \(D\) is nonunital. On \(D\widehat\otimes_D E\), any \(F\)-connection is a product of \((D,\mathrm{id},0)\) and \((E,\phi,F)\): its commutator and square defects are localized compact, as in the lemma, and the product positivity for the first zero operator is automatic. The lemma identifies this class with the original cycle. Thus
\[
1_D\widehat\otimes_D y=y.
\]
On \(E_1\widehat\otimes_D D\cong E_1\), the operator \(F_1\) is a zero connection, since it has compact products with \(\mathcal K(E_1)\); its product positivity is \(2\phi_1(a)F_1^2\phi_1(a)^*\ge0\) modulo compacts. Thus
\[
x\widehat\otimes_D1_D=x.
\tag{102}
\]

The same constructions give the exact functoriality formulas
\[
f^*(x\widehat\otimes_D y)=f^*x\widehat\otimes_D y,\qquad
g_*(x\widehat\otimes_D y)=x\widehat\otimes_D g_*y,
\qquad
h_*x\widehat\otimes_{D'}z=x\widehat\otimes_Dh^*z.
\tag{103}
\]
Here \(f:A'\to A\), \(g:B\to B'\), \(h:D\to D'\), and \(z\in KK(D',B)\); the displayed domains are part of the statement. Pullback simply replaces \(\phi\) by \(\phi f\), retaining the connection and positivity tests. Coefficient extension tensors compact maps and positive sandwiches with \(B'\), so retains the tests. For compact preservation, define \(T_\xi:B'\to E_1\widehat\otimes_gB'\) by \(b'\mapsto\xi\otimes b'\). It is adjointable, and \(T_{\xi e_n}=T_\xi g(e_n)\) is compact because \(g(e_n)\in B'=\mathcal K(B')\). The norm bound \(\|T_{\xi e_n}-T_\xi\|\le\|\xi e_n-\xi\|\to0\) makes \(T_\xi\) compact. Hence \(\theta_{\xi,\eta}\widehat\otimes1=T_\xi T_\eta^*\) is compact, and norm closure treats every compact map. No nondegeneracy of \(g\) is required. For the middle-algebra formula, the unitary
\[
(E_1\widehat\otimes_h D')\widehat\otimes_{D'}E_2
\longrightarrow E_1\widehat\otimes_{\phi_2h}E_2,\qquad
(\xi\otimes d')\otimes\eta\longmapsto\xi\otimes\phi_2(d')\eta
\]
preserves the inner product and is onto: approximate \(\xi\) by \(\xi e_n\), which puts the images of the right-hand elementary tensors in its dense range. A lower connection for \(\xi\) gives one for \(\xi\otimes d'\) by composing with \(\phi_2(d')\); the added error is the compact graded commutator of \(F_2\) with \(\phi_2(d')\). The first operator and its positive sandwich are transported by the same unitary. Uniqueness proves (103). In particular composing with a homomorphism class agrees with the corresponding pullback or pushforward, by (102) and (103).

**Lemma 0.12 (two-sided KK Morita equivalence).** A countably generated full imprimitivity module \({}_C X_B\) with \(C=\mathcal K(X)\) defines a degree-zero class \((X,\lambda,0)\in KK(C,B)\). Its conjugate module defines \((X^*,\lambda^*,0)\in KK(B,C)\). The inner-product maps identify \(X\widehat\otimes_BX^*\) with \(C\) and \(X^*\widehat\otimes_CX\) with \(B\); fullness gives dense range and the inner-product formula gives isometry. The zero operator is a zero connection and satisfies product positivity in both cases. The two products are therefore the identity homomorphism cycles. This proves both KK inverse identities for an imprimitivity module. Taking \(X=H_B=\ell^2(\mathbb N)\otimes B\), with the first factor trivially graded, proves stability under \(B\mapsto B\otimes\mathcal K\), with an explicit inverse rather than a group-level K-theory identification.

![The graded separator, strict approximation sum, stabilization and creation maps give a Kasparov product with positive localized anticommutator.](../figures/graded-separator.png)

*Figure 0.1. Lemmas0.1–0.2 and Theorem0.4, equations(90)–(96). The one-step lookahead controls \(u_nw\) at both \(v_{n-1}\) and \(v_n\), giving the exact constant5 and the summable bound \(\sqrt5+1\). The module arrow is \(T_\xi:E_2\to E_1\widehat\otimes_D E_2\), with the odd lower-operator sign \((-1)^{|\xi|}\). The operator sum is strict; its tested errors converge in norm to compact operators. These are algebraic diagrams of the proved construction. Complete uniqueness, homotopy descent and associativity follow in Theorems0.6–0.9.*

Full-size PNG · Editable SVG.

In Proposition 0.15 the same diagram uses the entire represented compact algebra \(\mathcal C=\phi(B)\otimes\operatorname{span}\{1,\sigma_y\}\), not only its even part. The two homogeneous creation maps in panel D are \(\phi(b)\) and \(\phi(b)\sigma_y\). Panel B annihilates both parts through \(MA_1\subset J\); panel E gives \(F=M^{1/2}S+N^{1/2}G\) and \(\{S,F\}=2M^{1/2}\ge0\) modulo \(J\). The four explicit graded derivations and every endpoint connection are proved in that proposition.

![Associativity keeps a positive contribution from the first product and a positive part from the second, while the separator removes the two compact errors after the final tensor.](../figures/kk-associativity.png)

*Figure 0.2. Lemma0.8 and Theorem0.9, equations(100)–(101). Both parenthesizations use the displayed balanced tensor module, without interchanging factors. The first-product error lies in \(\mathcal K(E_{12})\); its tensor with1 need not be compact on \(E\). Including that algebra in the separator makes its \(M^{1/4}\)-sandwich compact. The positive \(M\)-term is retained, while the \(N\)-term removes \(H_-\). This proves the exact final product positivity. Exercise28 supplies the elementary infinite-dimensional tensor counterexample.*

Full-size PNG · Editable SVG.

The bounded interface now includes product existence, uniqueness, arbitrary module homotopy descent, bilinearity, associativity, units, functoriality and two-sided Morita/stability equivalence for separable graded algebras. The scalar classification, real KK Thom inverse identities and suspension Bott equivalences are proved next. The spin-c bundle KK Thom equivalence and full compact-source factorization choices are proved in Section6. Projection and compact ordinary-map/foliation composition are proved in Section6. Theorem 6.1 proves geometric-cycle descent. The longitudinal-index and geometric twisted-duality comparisons require their further arguments. Group-level Bott or trace equality alone cannot substitute for those comparisons.

### Scalar classes, real actions and suspension Bott equivalence

The remaining analytic comparison needs identities of KK classes, with both inverse products and fixed Fourier signs. The following development gives the scalar classification, the real Thom inverse and its exact Wiener–Hopf boundary. The classical foundations are due to Kasparov; the real-action argument follows the deformation method of Fack and Skandalis, explained in [Blackadar], Sections 17.5, 18.10 and 19.3. We include the operator calculations and the nonunital scalar reference.

Throughout, algebras are separable and trivially graded unless a Clifford factor is displayed. Let \(C_1\) have an odd self-adjoint generator \(\varepsilon\), with \(\varepsilon^2=1\). Fix
\[
\rho(\varepsilon\widehat\otimes1)=\sigma_x,
\qquad \rho(1\widehat\otimes\varepsilon)=\sigma_y,
\qquad \Gamma=\sigma_z=-i\sigma_x\sigma_y.
\tag{104}
\]
The ordered two-generator algebra is the graded matrix algebra on \(V=\mathbb C^{1|1}\). These choices are part of every odd–odd product below.

The stable K-theory inputs are the extension boundary maps, their exactness and homotopy invariance, and matrix stability fixed at the start of this lesson. The following argument constructs the KK classification itself.

### External coefficients and formal Clifford periodicity

**Lemma 0.13 (external coefficients and formal Clifford periodicity).**

If \((E,\phi,F)\) is a graded \(A,B\)-cycle, external tensoring with a graded algebra \(D\) gives
\[
\tau_D(E,\phi,F)=(E\widehat\otimes D,\phi\widehat\otimes1,F\widehat\otimes1).
\]
For homogeneous vectors and operators, the signs are
\[
(\xi\otimes d)(b\otimes e)=(-1)^{|d||b|}\xi b\otimes de,
\quad
\langle\xi\otimes d,\eta\otimes e\rangle
=(-1)^{|d|(|\xi|+|\eta|)}\langle\xi,\eta\rangle\otimes d^*e,
\]
\[
(R\widehat\otimes S)(\xi\otimes d)=(-1)^{|S||\xi|}R\xi\otimes Sd.
\]
They directly give the module laws and the adjoint identity. The positive completion is the minimal graded tensor product. The rank-one formula shows
\(\mathcal K(E\widehat\otimes D)=\mathcal K(E)\widehat\otimes D\): elementary rank ones span each side, with precisely the signs above. Each cycle defect tested by \(a\otimes d\) is therefore compact. Finite sums and norm closure treat every element of \(A\widehat\otimes D\). Tensoring a homotopy over \(IB\) produces a homotopy over \(I(B\widehat\otimes D)\), since the interval factor is even. Direct sums, unitaries and degenerate cycles are also preserved. Thus \(\tau_D\) is a group homomorphism.

For a finite-dimensional Clifford or matrix factor, external tensoring preserves the product. Indeed the separator and product operator may be tensored with its identity. For a creation vector \(\xi\otimes d\), the original connection errors are tensored with left multiplication by \(d\); the new error involving the lower operator is its compact graded commutator with the left coefficient. The displayed module formulas give the signs. Positive sandwiches are tensored with positive matrix corners. Uniqueness in Section 0 then identifies the two product classes. The same proof for a trivially graded finite matrix factor involves no odd sign.

The same product assertion holds for an arbitrary trivially graded external algebra \(D\), with localized rather than global compactness. The associating module map sends \((\xi\otimes d)\otimes(\eta\otimes e)\) to \((\xi\otimes\eta)\otimes de\). Connection errors for this creation vector are an original compact map tensored with multiplication by \(d\), hence compact. Defects tested by \(a\otimes d\) are original compact errors tensored with \(d\). For positivity, write the original anticommutator as its positive part minus its negative part. The negative part has compact products with the input representation, by the localized quotient positivity criterion. Testing by a finite sum \(\sum_i a_i\otimes d_i\) gives a positive sandwich plus compact entries \(\phi(a_i)H_-\phi(a_j)^*\otimes d_i d_j^*\). The positive part tensored with the multiplier identity remains positive. Norm approximation treats general inputs. This proves product preservation without claiming \(k\otimes1_D\) itself compact or assuming an exact minimal tensor product. It justifies the later external scalar calculation over a nonunital \(A\).

The map \(\tau_{C_1}\) is invertible. Applying it twice gives \(\tau_{M_2}\) under \(\rho\). Simultaneous matrix stability is an isomorphism: pull back through \(A\to A\widehat\otimes M_2\), \(a\mapsto a\otimes e_{11}\), and tensor the coefficient module with the graded \(B\widehat\otimes M_2,B\)-imprimitivity module. Both composites are identities by the two inverse identities of Lemma 0.12 and the associating tensor unitaries. Here \(e_{11}\) projects onto the even vector of \(V\). Therefore the inverse of \(\tau_{C_1}\) is its second application followed by inverse simultaneous matrix stability. This is formal Clifford periodicity; it has not used a suspension Bott theorem.

Define \(KK^1(A,B)=KK(A,B\widehat\otimes C_1)\). An odd–odd product first applies \(\tau_{C_1}\) to the second class, forms the graded product over \(B\widehat\otimes C_1\), then reduces the ordered coefficient \(C_1\widehat\otimes C_1\) by \(\rho\) and \(V\). Homomorphism and matrix-Morita classes are even. The coherence needed for naturality and associativity can be seen explicitly on three generators. In \(C_3\), put \(B=e_1e_2+e_2e_3+e_3e_1\) and \(w=(1-B)/2\). The Clifford relations give \(B^2=-3\), hence \(ww^*=1\), and multiplication gives \(we_1w^*=e_2\), \(we_2w^*=e_3\), \(we_3w^*=e_1\). It is joined to the identity by the even unitary path \(w_v=\cos(\pi v/3)-(B/\sqrt3)\sin(\pi v/3)\), \(0\le v\le1\). The odd central generator \(\omega=-ie_1e_2e_3\) is unchanged by that cycle. Reducing either consecutive pair by the fixed \(V\)-Morita module leaves precisely this same \(\omega\) as the remaining Clifford generator. The two reductions are therefore intertwined by an even unitary, not a grading reversal. This proves the coherence of simultaneous two-generator Morita reduction with an extra Clifford factor. Combining it with the associating tensor unitaries and Section 0 associativity proves naturality and associativity of the parity products. Clifford order is retained until the indicated even cyclic transport is applied; an odd permutation is never silently made.

### Scalar classification through the stable corona

**Theorem 0.14 (scalar KK classification).**

Let \(H_B=\ell^2\otimes B\), \(J=\mathcal K(H_B)=B\otimes\mathcal K\), \(M=\mathcal L(H_B)=M(J)\), and \(Q=M/J\). Write \(q:M\to Q\).

First \(K_0(M)=K_1(M)=0\). To see the actual swindle, identify \(H_B\) with a countable direct sum of copies of itself. For a projection \(p\) over \(M\), the strict diagonal sum \(P=p\oplus p\oplus\cdots\) is a multiplier projection. Reindexing the summands makes \(P\) equivalent to \(p\oplus P\). Hence \([P]=[p]+[P]\), so \([p]=0\). For a multiplier unitary \(u\), its strict infinite diagonal \(U\) satisfies the same reindexing identity \([U]=[u]+[U]\) in \(K_1(M)\). Unitary conjugation does not change a stable \(K_1\)-class, by the two-by-two rotation homotopy. Thus \([u]=0\). Finite matrices cause no change because the same reindexing works on a finite direct sum. Exactness of the stable corona extension now gives isomorphisms
\[
\partial_0:K_1(Q)\longrightarrow K_0(J),
\qquad
\partial_1:K_0(Q)\longrightarrow K_1(J).
\]

For a \(\mathbb C,B\)-cycle, \(P=\phi(1)\) is an even projection. Replace \(F\) by \(PFP\); its difference from \(F\), tested by \(P\), is compact. The complementary zero representation is degenerate and contracts. On \(PE\), \(\phi\) is scalar multiplication. Normalize \(F\) to a self-adjoint contraction. Add the degenerate unital cycle \((H_B\oplus H_B^{\rm op},1,\sigma_x)\), and stabilize the even and odd summands separately. The result has the form
\[
(H_B\oplus H_B^{\rm op},1,F_T),\qquad
F_T=\begin{pmatrix}0&T^*\\ T&0\end{pmatrix},
\quad T^*T-1,\ TT^*-1\in J.
\]
Its corona image \(q(T)\) is unitary. Conversely every unitary over \(Q\) has an operator lift, which gives such a cycle after normalization. Define its index by \(\partial_0[q(T)]\).

For clarity, the boundary convention is the ordinary Fredholm index. If \(T\) is a contraction, put \(C=1-T^*T\), \(D=1-TT^*\). The intertwining identity \(T C^{1/2}=D^{1/2}T\), first for polynomials and then by uniform functional calculus, makes
\[
U_T=\begin{pmatrix}T&D^{1/2}\\ C^{1/2}&-T^*\end{pmatrix}
\]
unitary. Its first-column projection is
\[
P_T=\begin{pmatrix}TT^*&TC^{1/2}\\ C^{1/2}T^*&C\end{pmatrix},
\qquad \partial_0[q(T)]=[P_T]-[\operatorname{diag}(1,0)].
\]
The difference is compact. For a coisometry with kernel projection \(p\), this is \([\operatorname{diag}(1,p)]-[\operatorname{diag}(1,0)]=[p]\). For an isometry with cokernel projection \(r\), it is \(-[r]\). These computations fix the sign.

Norm homotopy and stable unitary equivalence of corona unitaries give equivalent KK cycles. A continuous path in a Banach-space quotient has a continuous lift with specified endpoint lifts: approximate it by a polygonal path, lift the finitely many vertex values with the open-mapping norm bound, and lift successive polygonal corrections with geometrically decreasing uniform norms. Their uniformly convergent sum is the desired lift. If the endpoint residuals vanish, choose zero lifts there. Applying this to the operator quotient lifts unitary paths to Fredholm paths; the lifts themselves need not be unitary. Adding an identity unitary adds a degenerate cycle. A multiplier unitary is degenerate; its corona \(K_1\)-class is zero by the swindle. Independent even and odd changes of stabilization change \(q(T)\) by multiplier unitary factors, whose classes are also zero. It follows that the classes modulo operator homotopy and degenerates are exactly \(K_1(Q)\).

This also treats arbitrary Hilbert-module homotopies. Perform the same compression and stabilization on a single homotopy module over \(IB=C([0,1],B)\). Its index belongs to \(K_0(IB)\). Evaluation of the relative projection formula gives the two endpoint indices. The evaluation maps \(K_0(IB)\to K_0(B)\) agree by stable K-theory homotopy invariance. Thus a general KK homotopy cannot identify distinct indices. Since every index is already attained modulo operator homotopy, the full classification is
\[
\Theta_0:KK(\mathbb C,B)\xrightarrow{\cong}K_0(B).
\tag{105}
\]
For \(B=\mathbb C\), this is the usual integer Fredholm index; a finite even scalar module with zero operator has positive dimension. In particular \(KK(\mathbb C,\mathbb C)=\mathbb Z\), with its identity class corresponding to \(+1\).

The odd argument is equally explicit. In a graded Hilbert \(B\widehat\otimes C_1\)-module, multiplication on the right by the central unitary \(\varepsilon\) identifies the even and odd summands. The even inner products lie in \(B\); this identifies the whole module with its even part externally tensored with \(C_1\). A right-module-linear odd operator is consequently of the form \(s\varepsilon\). After compression and stabilization an odd scalar cycle is \((H_B\widehat\otimes C_1,1,s\widehat\otimes\varepsilon)\), where \(s=s^*\), \(\|s\|\le1\), and \(s^2-1\in J\). Its image defines the corona projection \(p=(1+q(s))/2\). Self-adjoint lifts of projection paths give cycle paths by the quotient-lifting argument. Degenerate cycles lift actual multiplier projections, whose \(K_0\)-classes vanish. Replacing \(s\) by \(-s\) replaces \(p\) by \(1-p\); since \([1_Q]=0\), this is the group inverse. Every difference of corona projection classes is attained by a direct sum, using this complement. The same construction over \(IB\) proves invariance under all module homotopies. Consequently
\[
\Theta_1:KK^1(\mathbb C,B)\xrightarrow{\cong}K_1(B),
\qquad
\Theta_1[s\varepsilon]
=[\exp(\pi i(s+1))].
\tag{106}
\]
The exponential is a unitary in \(1+J\), since its quotient is one. This is the positive exponential boundary convention. All matrix stabilizations are absorbed by stable K-theory. On a nonunital algebra one uses the scalar reference in its unitization; the relative projection and unitary formulas above keep that reference explicitly.

Naturality can be checked without assuming that a homomorphism extends to the multiplier algebra. Represent a relative \(K_0\)-class by projections \(p,p_0\) over \(B^+\), with the same scalar reference, and use the quasihomomorphism cycle with representations \(p,p_0\) and off-diagonal identity on \(B^n\oplus(B^n)^{\rm op}\). Its compressed Fredholm operator is \(p_0p:pB^n\to p_0B^n\), and the relative boundary calculation above gives \([p]-[p_0]\). Coefficient extension replaces both projections by their images under the unital extension \(g^+:B^+\to(B')^+\), so it gives the usual relative \(K_0\)-map. For an odd scalar cycle on a general module \(E\), its unitary \(\exp(\pi i(s+1))\) differs from identity by an element of \(\mathcal K(E)\). Stabilization embeds that compact algebra as rank-one matrices over \(B\); coefficient extension replaces their entries by their images under \(g\), as in the compact-preservation proof of Corollary 0.11. The unital extension of this compact-algebra map sends the unitary to identity plus that image, and continuous functional calculus commutes with the map. Its action on the essential extended module is exactly the exponential of the extended cycle; adding the identity on the standard module is the same stable compact-unitary class. Finite matrix approximations give precisely the usual \(K_1\)-map. This uses compact matrices rather than an extension on all adjointable multipliers. Hence \(\Theta_i\) are natural also for degenerate coefficient homomorphisms. No multiplier extension of such a homomorphism has been asserted.

### A complete bounded scalar odd-pairing calculation

**Proposition 0.15 (the bounded scalar odd pairing).**

Let \(B\) be as above, \(s=s^*\in M(B)\), \(\|s\|\le1\), \(s^2-1\in B\). This gives \(x=(B\widehat\otimes C_1,1,s\varepsilon)\in KK^1(\mathbb C,B)\). Let \(\phi:B\to\mathcal B(H)\) be nondegenerate and let \(r=r^*=r^{-1}\), with \([r,\phi(b)]\) compact for all \(b\). Set \(P=(1+r)/2\); it defines the odd cycle \(y=(H\widehat\otimes C_1,\phi,r\varepsilon)\). Here an honest involution is part of the scope; normalization of an arbitrary odd cycle has not been silently assumed.

Under the fixed Clifford reduction, the tensor module is \(H\oplus H^{\rm op}\), the first operator is \(S=\phi(s)\sigma_y\), and a lower connection is \(G=r\sigma_x\). For example, the input odd generator acts by \(\sigma_y\), which anticommutes with \(\sigma_x\). For a vector whose coefficient is \(b\), the remaining connection error is a matrix entry of \([r,\phi(b)]\), hence compact. Nondegeneracy identifies the tensor module with all of \(H\oplus H^{\rm op}\).

Write \(t=(1-s^2)^{1/2}\in B\), \(T=\phi(t)\otimes1\), and
\[
L=S+TG.
\]
This need not itself be a lower connection. We prove that its cycle represents the product by an actual homotopy to one. Its adjoint defect is \([T,G]\), which is compact. The graded anticommutator \(Z=SG+GS\) has compact products with \(\phi(B)\): its entries are scalar multiples of \([r,\phi(s)]\), and
\[
[r,\phi(s)]\phi(b)=[r,\phi(sb)]-\phi(s)[r,\phi(b)]
\]
is compact. The analogous left product is compact by adjoints. In particular \(TZ\) is compact, and \([T,G]\) is compact. Since \(S^2+T^2=1\), this gives \(L^2-1\) compact. Symmetrizing changes \(L\) only by a compact operator, so it is a scalar Kasparov cycle.

Apply the separator on \(E=H\oplus H^{\rm op}\), with \(J=\mathcal K(E)\), to
\[
\mathcal C=\phi(B)\otimes\operatorname{span}\{1,\sigma_y\},\qquad
A_1=C^*(J,\mathcal C),\qquad A_2=C^*(Z),\qquad
\Delta=\operatorname{span}\{S,G\}.
\]
Here \(\mathcal C\) is the full represented algebra \(\mathcal K(B\widehat\otimes C_1)\widehat\otimes1\), after the fixed finite Clifford reduction: \(\phi(b)\) is even and \(\phi(b)\sigma_y\) is odd. Nondegeneracy extends \(\phi\) to the multiplier algebra, so the following formulas apply to \(s\in M(B)\). For every \(b\in B\),
\[
\begin{aligned}
[S,\phi(b)]_{\rm gr}&=\phi(sb-bs)\sigma_y,
&[S,\phi(b)\sigma_y]_{\rm gr}&=\phi(sb+bs),\\
[G,\phi(b)]_{\rm gr}&=[r,\phi(b)]\sigma_x,
&[G,\phi(b)\sigma_y]_{\rm gr}&=[r,\phi(b)]\sigma_x\sigma_y.
\end{aligned}
\]
Thus \(S\) derives \(\mathcal C\) into itself, \(G\) derives it into \(J\), and both derive \(J\) into itself. The localized commutator identity above and its adjoint imply that both \(\phi(b)Z\) and \(\phi(b)\sigma_yZ\) are compact, as is \(JZ\). Products, adjoints and norm closure give \(A_1A_2\subset J\). These graded algebras are separable in the countably generated cycle scope, so all separator hypotheses hold. Obtain even positive \(M,N\) with \(M+N=1\), \(MA_1\subset J\), \(NZ\in J\), and \([M,S],[M,G]\in J\). Since \(T\in\mathcal C\), both \(MT\) and \(TM\) are compact; hence \(M,N\) also commute with \(T\) modulo \(J\). Continuous functional calculus with zero constant term gives \(M^{1/2}A_1\subset J\) and \(N^{1/2}Z\in J\).

For \(0\le u\le1\), put
\[
D_u=uM+(1-u)1,\qquad C_u=uN+(1-u)T^2,\qquad
L_u=D_u^{1/2}S+C_u^{1/2}G.
\]
The coefficients are even positive contractions and norm continuous. In the quotient \(\mathcal L(E)/J\), they commute with each other and with \(S,G,T\). Since \(S^2+T^2=1\), \(G^2=1\), \(MT^2\in J\), \(NZ\in J\), and \(T^2Z\in J\),
\[
D_uS^2+C_uG^2-1=uM(S^2-1)=0,\qquad
C_uZ=uNZ+(1-u)T^2Z=0
\]
in that quotient. Polynomial approximation to the square root by polynomials vanishing at zero gives \(C_u^{1/2}Z=0\) there. The cross term in \(L_u^2\) is therefore zero, and each \(L_u\) is self-adjoint modulo \(J\) with unit square there. Symmetrizing gives a norm-continuous path of odd self-adjoint scalar cycles. Its defects are continuous \(J\)-valued functions, hence compact on the single homotopy module \(C([0,1],E)\). At zero it is the symmetrization of \(L\).

We verify the other endpoint on every homogeneous creation vector, retaining the odd coefficient part. After the same finite reduction, the lower operator is \(G\), and the creation map for \(\xi=b\) or \(b\varepsilon\) is respectively \(T_\xi=\phi(b)\) or \(\phi(b)\sigma_y\). In both cases \([G,T_\xi]_{\rm gr}\in J\), \(ST_\xi\in\mathcal C\), and \((N^{1/2}-1)T_\xi\in J\); the last assertion uses the function \(\sqrt{1-v}-1\), which vanishes at \(v=0\), and \(MA_1\subset J\). Thus, before its compact symmetrization, \(F=L_1=M^{1/2}S+N^{1/2}G\) satisfies
\[
FT_\xi-(-1)^{|\xi|}T_\xi G
=M^{1/2}ST_\xi+(N^{1/2}-1)GT_\xi
 +[G,T_\xi]_{\rm gr}\in J.
\]
Indeed commute \(G\) past \(T_\xi\) in the middle term; its error is compact and its remaining factor \((N^{1/2}-1)T_\xi\) is compact. Adjoints prove the other creation-map test, and compact symmetrization preserves both tests. The first-operator positivity is
\[
\{S,F\}\equiv2M^{1/2}S^2+N^{1/2}Z
\equiv2M^{1/2}\ge0\pmod J,
\]
because \(M^{1/2}T^2\) and \(N^{1/2}Z\) are compact. The scalar left representation gives every required positive sandwich. Definition 0.3 and the fixed finite product-preserving reduction of Lemma 0.13 therefore identify this endpoint with \(x\widehat\otimes_B y\). The homotopy proves \([L]=x\widehat\otimes_B y\) for arbitrary separable noncommutative, nonunital \(B\) in the stated cycle scope.

Now put \(v=t+is\). It is unitary, and \(v^2\in B^+\) has scalar value \(-1\). In the decomposition \(H=PH\oplus(1-P)H\), write
\[
\phi(v)=\begin{pmatrix}a&b\\ c&d\end{pmatrix}.
\]
The lower even-to-odd block of \(L\) is
\[
A=\phi(t)r+i\phi(s)
=\begin{pmatrix}a&-b'\\ c&-d'\end{pmatrix},
\]
where \(b',d'\) are the upper-right and lower-right entries of \(\phi(v^*)\). This follows by adding the blocks, not by treating \(s\) as compact. Multiplication on the left by the unitary \(\phi(v)\) leaves the Fredholm index unchanged. The identities \(\phi(v)^2=\phi(v^2)\) and \(\phi(v)\phi(v^*)=1\) give
\[
\phi(v)A=\begin{pmatrix}P\phi(v^2)P&0\\ k&-1\end{pmatrix},
\quad k=(1-P)\phi(v^2)P\text{ compact}.
\]
The compactness follows because \(v^2+1\in B\), whereas the off-diagonal entries of \(\phi(v)\) alone need not be compact. Removing \(k\) by a compact perturbation proves
\[
\Theta_0(x\widehat\otimes_B y)
=\operatorname{Index}(P\phi(v^2)P).
\]
Finally \(-v^2=\exp(i(2\arcsin s+\pi))\) is joined to \(\exp(\pi i(s+1))\) by interpolating these two real phases. At \(s=\pm1\), both phases agree modulo \(2\pi\); every unitary on this path lies in \(1+B\). The scalar rotation from \(v^2\) to \(-v^2\) also preserves the compression index. Therefore the exact formula is
\[
\Theta_0(x\widehat\otimes_B y)
=\operatorname{Index}\bigl(P\phi(\exp(\pi i(s+1)))P\bigr).
\tag{107}
\]
No appeal to a general extension–KK classification is needed for this bounded calculation.

### The physical scalar test

**Lemma 0.16 (the scalar dual product and its sign).**

Set \(B=C_0(\mathbb R_p)\), \(s(p)=p/\sqrt{1+p^2}\), \(\phi(b)=M_b\) on \(L^2(\mathbb R_p)\), and \(r=\operatorname{sign}D_p\), where \(D_p=-i\partial_p\). The commutator with a smooth compactly supported \(b\) has kernel a constant multiple of
\((b(p)-b(q))/(p-q)\). It is Hilbert–Schmidt: near the diagonal use the bounded derivative and compact support; away from it use \(|p-q|^{-2}\) after squaring, with one variable in a fixed compact set. Density extends compactness to \(C_0\). Thus the odd cycle satisfies Proposition 0.15.

The exponential of Theorem 0.14 has positive winding. Its phase increases from zero to \(2\pi\), so it is homotopic in \(1+C_0(\mathbb R)\) to
\[
u_+(p)=\frac{p-ia}{p+ia},\qquad a>0.
\]
To compute its compression to \(P=(1+\operatorname{sign}D_p)/2\), Fourier transform in \(p\) with phase \(e^{-itp}\). Then \(P\) is multiplication by \(1_{t\ge0}\). Multiplication by \(u_+\) becomes
\[
(S_a f)(t)=f(t)-2a\int_0^t e^{-a(t-r)}f(r)\,dr,
\qquad t\ge0.
\]
The transform follows by closing the contour in the lower half-plane for \(t>0\); the pole of \((p+ia)^{-1}\) is \(-ia\). Direct kernel integration gives
\[
S_a^*S_a=1,
\qquad S_aS_a^*=1-E_a,
\qquad E_a(t,r)=2a e^{-a(t+r)}.
\tag{108}
\]
The vector \(\sqrt{2a}\,e^{-at}\) has norm one, so \(E_a\) is a rank-one projection. The index is therefore \(-1\), not an unspecified nonzero integer. Proposition 0.15 proves that the ordered scalar product for the phase and Clifford conventions fixed here is \(-1\).

For the real group algebra, the physical transform is
\(\widehat f(p)=(2\pi)^{-1/2}\int e^{-ipt}f(t)\,dt\), and \(U_s=e^{-isp}\). The dual action in the public lesson is
\(\widehat\alpha_q(U_s)=e^{iqs}U_s\); on the scalar algebra it is \(b(p)\mapsto b(p-q)\). Its crossed product acts on \(L^2(\mathbb R_p)\) by multiplication and translations, and its Thom generator is \(D_p\). Thus the calculation above is the scalar trivial-action/dual-action product in precisely these conventions. The two minus signs in the eventual two-sided inverse must be justified by this calculation, rather than importing a formula written in a different Fourier convention.

### The real Thom cycle, naturality and inner-action transport

**Lemma 0.17 (the real Thom cycle and its naturality).**

Let \(\alpha\) be a pointwise norm-continuous real action on \(A\), and put \(C_\alpha=A\rtimes_\alpha\mathbb R\). Use Lebesgue measure and the public physical convention \(U_s=e^{-isD_\alpha}\). The integrated group representation is a nondegenerate homomorphism \(C_0(\mathbb R)\to M(C_\alpha)\), because scalar approximate identities in the group convolution algebra act as an approximate identity on its dense \(A\)-valued convolution algebra. It extends to bounded continuous spectral functions. Set
\[
h(z)=z(1+z^2)^{-1/2},\qquad
t_\alpha=(C_\alpha\widehat\otimes C_1,\iota_A,h(D_\alpha)\varepsilon)
\in KK^1(A,C_\alpha).
\tag{109}
\]
The square defect tested by \(a\) belongs to \(C_\alpha\), since \(h^2-1\in C_0(\mathbb R)\), and \(a\psi(g)\) is an algebra element for \(g\in C_0(\mathbb R)\). The latter follows first for integrated \(L^1\) kernels, then by their dense uniform Fourier approximations.

Here is a direct compact-commutator proof. The bounded real function
\[
f(z)=\frac2\pi\int_0^1\frac{\sin(sz)}s\,ds
\]
has limits \(\pm1\) at \(\pm\infty\), with the following complete normalization. For \(\epsilon>0\) set

\[
 J_\epsilon(a)=\int_0^\infty e^{-\epsilon t}\frac{\sin(at)}t\,dt,
 \qquad J_\epsilon'(a)=\frac{\epsilon}{\epsilon^2+a^2}.
 \tag{TIR.1}
\]

The first integral is absolute, since \(|\sin(at)|/t\le|a|\) at zero. Differentiation on compact a-intervals is dominated by \(e^{-\epsilon t}\); the derivative is the real part of the elementary integral \((\epsilon-ia)^{-1}\). Since \(J_\epsilon(0)=0\),

\[
 J_\epsilon(1)=\int_0^{1/\epsilon}\frac{du}{1+u^2}
                         \longrightarrow\pi/2.
 \tag{TIR.2}
\]

The normalization here is independent of the sine integral: the first-quadrant unit-circle parametrization \(u\mapsto(1,u)/\sqrt{1+u^2}\) has speed \((1+u^2)^{-1}\) and joins the two positive axis points. Its limiting arc length is one quarter of the circumference, hence \(\pi/2\).

For \(\epsilon\ge0\), \(g_\epsilon(t)=e^{-\epsilon t}/t\) decreases to zero. Integration by parts on a finite interval and passage to infinity give the uniform tail bound

\[
 \left|\int_R^\infty g_\epsilon(t)\sin t\,dt\right|
 \le g_\epsilon(R)+\int_R^\infty|g_\epsilon'(t)|dt
 =2g_\epsilon(R)\le2/R.
 \tag{TIR.3}
\]

For \(\epsilon=0\) this proves ordinary convergence. Dominated convergence on \([0,R]\), followed by the uniform tail estimate, identifies its limit with TIR.2. Thus

\[
 \int_0^\infty\frac{\sin t}t\,dt=\pi/2,
 \qquad f(z)=\frac2\pi\int_0^z\frac{\sin t}t\,dt.
 \tag{TIR.4}
\]

The function is real, bounded, continuous and odd, and its limits are the asserted \(\pm1\). Integrals over all positive subintervals have a common bound: split at one, bound the initial part by one and each remaining tail by two. Therefore the spectral functions obtained by omitting \(|s|<\delta\) are uniformly bounded and converge pointwise to f. Strong spectral dominated convergence applies in every covariant representation. For an action-differentiable coefficient the omitted commutator interval satisfies

\[
 \left\|\int_{|s|<\delta}\frac{i}{\pi s}
                  (\alpha_s(a)-a)U_s\,ds\right\|
 \le\frac{2\delta}{\pi}\|\delta_\alpha(a)\|.
 \tag{TIR.5}
\]

Thus \(f-h\in C_0(\mathbb R)\). Its group distribution is \(i/(\pi s)\) on \(|s|\le1\), in the negative-phase convention. If \(a\) is differentiable for the action, then
\[
[f(D_\alpha),a]
=\int_{-1}^1\frac{i}{\pi s}(\alpha_s(a)-a)U_s\,ds.
\]
The apparent singularity has norm at most the action derivative norm divided by \(\pi\). The integral is an \(L^1(\mathbb R,A)\) element of \(C_\alpha\). To justify the distributional formula, cut out \(|s|<\epsilon\), compute the ordinary integrated commutator, and pass to the norm limit on the right. On the left the cutoffs converge strongly in every covariant Hilbert-space representation, by the bounded sine integrals and spectral dominated convergence. Consequently they identify the displayed algebra element with the commutator. Adding \(f-h\) preserves algebra membership. Action-smooth elements are norm dense: convolution of the orbit map with a compact smooth approximate identity gives such elements. The uniform bound \(\|[h(D),a]\|\le2\|h\|\|a\|\) extends the conclusion to all \(a\). Hence \(t_\alpha\) is a Kasparov cycle. Any other bounded continuous real normalizing function with the same limits differs by \(C_0\), so gives the same class by a localized compact perturbation.

For an equivariant homomorphism \(g:(A,\alpha)\to(A',\alpha')\), including a degenerate one, the induced map \(G:C_\alpha\to C_{\alpha'}\) is defined on convolution kernels coefficientwise. Its coefficient-extension module is
\[
J_g=\overline{G(C_\alpha)C_{\alpha'}}
=\overline{g(A)C_{\alpha'}}.
\]
For the equality, group approximate identities approximate every coefficient product, and \(U_sC_{\alpha'}=C_{\alpha'}\) shows the reverse inclusion. This right ideal is invariant under the group multipliers and their adjoints by equivariance. Therefore the extended spectral calculus agrees there with the restriction of \(h(D_{\alpha'})\). Its localized defects are in \(G(C_\alpha)\), not merely in the ambient multiplier algebra; the commutator formula above and the \(C_0\) approximation prove this on a dense set and then by closure. This also proves compactness as maps with range in \(J_g\): elements of \(G(C_\alpha)\) lie in \(\overline{J_gJ_g^*}=\mathcal K(J_g)\), using its own approximate identity. Viewed as maps from the ambient right module into \(J_g\), their rank-one approximations have first vectors in \(J_g\). Thus the restricted operator is a lower connection for the essential representation of \(A\) through \(g\). Lemma 0.10, followed by the localized compact-perturbation path, identifies it with the full pullback cycle. We have proved
\[
G_*t_\alpha=g^*t_{\alpha'}.
\]
No extension of a degenerate \(g\) to the full multiplier algebra has been used.

Tensoring with a trivially graded matrix algebra or compact algebra gives \(t_{\alpha\otimes1}=\tau(t_\alpha)\): on the dense convolution core the crossed-product identification is coefficientwise, with the same group unitaries and spectral calculus. Compact algebra stability follows by increasing finite corners and the even Morita unitary of Lemma 0.12.

We also need an explicit inner-action transport. Let \(W_s\) be a strongly continuous unitary group on a separable Hilbert space \(H\), and
\[
\gamma_s=\alpha_s\otimes\operatorname{Ad}W_s
\quad\text{on }A\otimes\mathcal K(H).
\]
The crossed-product isomorphism with \(C_\alpha\otimes\mathcal K(H)\) sends the group multiplier to \(U_s\otimes W_s\) and the coefficient to its usual inclusion. Its inverse uses \(U_s\otimes W_s^*\); the covariance and integrated products check both maps on the dense convolution core. For \(0\le u\le1\), use \(W_{us}\). These groups are jointly strictly continuous on the compact-operator coefficient algebra. They define a single nondegenerate covariant representation on \(I(C_\alpha\otimes\mathcal K(H))\), hence a spectral homomorphism \(C_0(\mathbb R)\to M(I(C_\alpha\otimes\mathcal K(H)))\). Extending it to \(h\) gives a single adjointable operator over the interval; separate norm continuity of the unbounded generators is not being assumed.

The same commutator proof applies uniformly on this compact parameter interval. Smooth a coefficient by orbit convolution for the interval action \(\gamma^u\); this gives norm-dense coefficients with uniformly bounded action derivative. The localized square defect and commutator are therefore in the interval algebra. We obtain a Kasparov homotopy between the transported \(t_\gamma\) and \(\tau_{\mathcal K(H)}t_\alpha\). This proves the required inner-action transport with its direction fixed.

### Both real KK inverse identities

**Theorem 0.18 (both real KK Thom inverse identities).**

Use the positive dual phase already fixed in Theorem 8.6:
\[
\widehat\alpha_q(U_s)=e^{iqs}U_s.
\]
Let \(m_A\) be the even Takai imprimitivity class from \(C_\alpha\rtimes_{\widehat\alpha}\mathbb R\cong A\otimes\mathcal K\) to \(A\). Define
\[
u_\alpha=t_\alpha\widehat\otimes_{C_\alpha}t_{\widehat\alpha}
\widehat\otimes m_A\in KK(A,A).
\]
This is independent of \(\alpha\). To prove it, act on \(IA\) by
\[
[\beta_s(a)](v)=\alpha_{vs}(a(v)).
\]
Joint continuity follows from pointwise continuity uniformly on the compact image of a continuous coefficient function, using a finite approximation to that image. Evaluation \(e_v\) is equivariant to \(\alpha^{v}_s=\alpha_{vs}\); the resulting single and double crossed-product maps are \(G_v,H_v\). The explicit kernel construction of Theorem 8.6 shows that double Takai sends \(H_v\) to \(e_v\otimes1_{\mathcal K}\): each kernel coefficient is evaluated at \(v\). These endpoint homomorphisms are homotopic. Product naturality from Section 0 and Lemma 0.17 gives, before the final even Morita reduction,
\[
(H_v)_*(t_\beta\widehat\otimes t_{\widehat\beta})
=e_v^*(t_{\alpha^v}\widehat\otimes t_{\widehat{\alpha^v}}).
\]
Both sides lie in \(KK(IA,A\otimes\mathcal K)\). Their endpoint values agree because \(H_0,H_1\) are homotopic. Pull back by the constant inclusion \(c:A\to IA\), for which \(e_vc=1_A\), and apply the same even Morita class. This proves \(u_\alpha=u_\iota\), with \(\iota\) the trivial action. There has been no assertion that the single crossed-product endpoint evaluations are KK-equivalences.

For the trivial action, tensor the scalar computation Lemma 0.16 with \(1_A\). The scalar second cycle can be replaced by \(\operatorname{sign}D_p\): \(\operatorname{sign}z-h(z)\) is in \(L^2(\mathbb R)\) and tends to zero at infinity; multiplication by a compactly supported \(b(p)\) followed by that Fourier multiplier is Hilbert–Schmidt. Approximation treats every \(b\in C_0\). Thus the replacement is a localized compact perturbation of the actual dual Thom cycle. The computed integer is \(-1\), and its external product is the negative identity class, not just its action on K-groups. Hence
\[
u_\alpha=-1_A
\quad\text{for every action }\alpha.
\tag{110}
\]

Set
\[
d_\alpha=-t_{\widehat\alpha}\widehat\otimes m_A
\in KK^1(C_\alpha,A).
\]
We immediately obtain \(t_\alpha\widehat\otimes d_\alpha=1_A\). For the other order, apply the same proved equality \(u=-1\) to the action \(\widehat\alpha\) on \(C_\alpha\). Under Takai the double-dual action is, with the lesson's signs,
\[
\widehat{\widehat\alpha}_v
=\alpha_v\otimes\operatorname{Ad}T_{-v}.
\]
Here \(T_s\xi(t)=\xi(t-s)\); this identity is exactly the coefficient and group-multiplier calculation in Theorem 8.6. The inner-action transport Lemma 0.17 identifies its Thom class with \(t_\alpha\otimes1_{\mathcal K}\) under the fixed exterior-equivalence isomorphism. Even matrix/compact Morita naturality says that composing this stabilized class with \(m_{C_\alpha}\) equals first composing with \(m_A\) and then with \(t_\alpha\). This can also be checked by the elementary tensor unitary \((\xi\otimes k)\otimes\eta\mapsto\xi\otimes k\eta\); both even imprimitivity products are identities. Consequently the equality for \(\widehat\alpha\) is
\[
(t_{\widehat\alpha}\widehat\otimes m_A)
\widehat\otimes_A t_\alpha=-1_{C_\alpha}.
\]
Multiplying by the indicated minus sign proves the full pair of identities
\[
t_\alpha\widehat\otimes_{C_\alpha}d_\alpha=1_A,
\qquad
d_\alpha\widehat\otimes_A t_\alpha=1_{C_\alpha}.
\tag{111}
\]
Both are identities of KK classes. Applying scalar K-theory is not used to infer either identity. The normalizing function and the real group orientation are fixed; reversing the group orientation negates the odd class.

### Identification with the actual Wiener–Hopf boundary in both degrees

**Theorem 0.19 (the actual Wiener–Hopf boundary in both degrees).**

On the physical regular Hilbert \(A\)-module \(\mathcal H_A=L^2(\mathbb R_t)\otimes A\), use
\[
(\pi(a)\xi)(t)=\alpha_{-t}(a)\xi(t),
\qquad U_s\xi(t)=\xi(t-s),
\qquad V_q\xi(t)=e^{iqt}\xi(t).
\]
The double Takai algebra is \(\mathcal K(\mathcal H_A)\) by Theorem 8.6. Since the spectral convention is negative phase, the Stone generator of \(V_q\) is \(-t\), not \(t\). Its transported dual Thom cycle has operator \(h(-t)\), and its negative class has operator \(h(t)\). Put \(P_+=1_{t\ge0}\). The difference between \(h(t)\) and \(2P_+-1\) gives localized compact operators against \(\pi(C_\alpha)\). For a compact smooth group kernel, truncate to \(|t|\le R\); its other kernel variable then lies in a bounded set, so finite-rank \(A\)-valued kernel approximations prove compactness. The omitted part has norm bounded by the supremum of \(|h(t)-\operatorname{sign}t|\) on \(|t|>R\), times the convolution norm. The jump at zero is handled by \(L^2\) approximation on the bounded rectangle, with the integral-kernel norm bound. Density of the convolution core treats every algebra element and adjoints treat the other side. Thus
\[
d_\alpha=[(\mathcal H_A,\pi,(2P_+-1)\varepsilon)].
\tag{112}
\]
Directly, \([P_+,\pi(c)]\) is compact: the crossing kernel for a group kernel of support \([-R,R]\) has \(t,r\) of opposite signs and \(|t-r|\le R\), hence both variables bounded. Its smooth coefficient kernel has finite-rank \(A\)-module approximations; characteristic half-interval factors lie in \(L^2\) and cause no difficulty. The square defect is zero.

This compression is the Busby map of the exact Wiener–Hopf extension of Theorem 8.7, with the previously fixed positive boundary sign. Indeed represent its coefficient \(F\) by \(\alpha_{-t}(F(t))\) and its group generator by the same \(U_s\). For a lift \(F(t)=\chi(t)a\), with \(\chi\) zero sufficiently far to the left and one sufficiently far to the right, the integrated lift differs from \(P_+\pi(c)P_+\) by a compact kernel: \(\chi-P_+\) is supported on a bounded interval, and the crossing error is the one just proved compact. The ideal representation is precisely \(\mathcal K(\mathcal H_A)\), by the translation kernel proof of Lemma 8.5. Thus the quotient Busby map is
\[
\tau(c)=q(P_+\pi(c)P_+).
\]
Checking the lifts on the dense convolution core determines the extension on its norm completion. There is no use of an abstract Ext–KK classification to identify it.

For \(K_1(C_\alpha)\), represent a scalar odd class by \(s\varepsilon\) as in Theorem 0.14, stabilized if needed. Formula Proposition 0.15 says that product with \(d_\alpha\) is the Fredholm index of the Busby lift of its positive-exponential unitary. This is exactly the ordinary index connecting map of the Wiener–Hopf extension. Its coisometry defect has positive sign, as fixed in Theorem 0.14 and Corollary 8.10.

For \(K_0(C_\alpha)\), let \(p\) be a projection over its unitization and subtract its scalar reference \(p_0\). Extend the compression cycle to the unitization by the identity representation; its operator is an actual involution, so this is a legitimate cycle. Compute the two finite projection products over that unital input, and then subtract. Naturality identifies the resulting relative product with the original class over \(C_\alpha\). Individual projection cycles with nonzero scalar reference have not been asserted compact over the nonunital algebra. On \(p\mathcal H_A^n\), the compressed operator \(p(2P_+-1)p\) is a connection and is a product with the finite projection cycle: its localized square defect is compact by \([p,P_+]\in\mathcal K\), and positivity for the first zero operator is automatic. Theorem 0.14 sends this odd class to
\[
\exp(2\pi i pP_+p)\text{ on }p\mathcal H_A^n,
\]
extended by identity on its complement. The Wiener–Hopf exponential boundary instead uses \(\exp(2\pi iP_+pP_+)\) on \(P_+\mathcal H_A^n\). These unitaries give the same stable \(K_1\)-class, as follows by a concrete relative matrix argument.

On the full regular module write \(X=pP_+\), \(Y=P_+p\) and
\(k(z)=(e^{2\pi iz}-1)/z\), extended by \(k(0)=2\pi i\). Then
\[
a=X,\quad b=Yk(XY),\quad
1+ab=e^{2\pi i pP_+p},\quad
1+ba=e^{2\pi iP_+pP_+}.
\]
In the corona, \(p\) and \(P_+\) commute, so \(q(X)=q(Y)=q(XY)\) is a projection. Since \(k(1)=0\), it follows that \(b\) is compact. The intertwining identity \(k(XY)X=Xk(YX)\) follows from polynomials and uniform functional calculus. For any multiplier \(a\) and compact \(b\), with \(1+ab\) invertible, the exact elementary factorization
\[
\begin{pmatrix}1+ab&0\\0&(1+ba)^{-1}\end{pmatrix}=
\begin{pmatrix}1&a\\0&1\end{pmatrix}
\begin{pmatrix}1&0\\b&1\end{pmatrix}
\begin{pmatrix}1&-a(1+ba)^{-1}\\0&1\end{pmatrix}
\begin{pmatrix}1&0\\-b(1+ab)&1\end{pmatrix}
\tag{113}
\]
has zero relative \(K_1\)-class. To verify the last assertion, group the first two factors and the inverse of the first as a conjugate of a compact elementary matrix, contracted by \(b\mapsto ub\). The remaining upper factor has compact entry \(a-a(1+ba)^{-1}\); the lower factor is also compact elementary. Each contracts within identity plus compacts. Thus \([1+ab]=[1+ba]\). Polar normalization converts these invertible paths to unitary paths without changing their endpoints' \(K_1\)-classes. The scalar reference \(p_0\) commutes with \(P_+\), and both of its exponentials are identity, so the relative subtraction is respected.

This proves that product with \(d_\alpha\) is the Wiener–Hopf connecting map in both degrees. The two KK inverse identities of Theorem 0.18 consequently identify product with \(t_\alpha\) with the actual group-level Thom map already proved in Section 8. At the trivial action, \(\Theta_1(t_\iota)\) has positive winding and is \([u_+]\). Its physical half-line lift is the coisometry of Corollary 8.10. The first scalar dual product is \(-1\) because the positive dual phase uses the opposite physical half-line; the minus sign in \(d_\alpha\) is therefore necessary and now explained by exact operators.

These are analytic real-action interfaces. The spin-c vector-bundle KK Thom theorem is proved in Theorem 6.0c and Corollary 6.0d, and the compact ordinary embedding comparisons in Lemma 6.0e and Theorem 6.0f. Theorem6.0o proves compact-source foliation wrong-way factorization independence. Theorems6.0s/6.0w prove the required projection and compact ordinary-map/foliation composition. Theorem 6.1 proves geometric-cycle descent. The full longitudinal topological index theorem and subsequent assembly comparisons require further arguments.


### Both suspension Bott inverse products

**Corollary 0.20 (both suspension Bott inverse products).**

Write \(S=C_0(\mathbb R)\) with its positive physical cotangent coordinate. The trivial-action class is \(t=t_\iota\in KK(\mathbb C,S\widehat\otimes C_1)\), and the proved inverse is \(d=d_\iota\in KK(S,C_1)\). Set
\[
\beta=\tau_{C_1}(t)\widehat\otimes m_S\in KK(C_1,S).
\]
Here \(m_S\) reduces the ordered two coefficient Clifford generators. We claim that \(\beta\) and \(d\) are inverse in the ordinary graded KK category.

The second odd inverse identity of Theorem 0.18 is exactly \(d\widehat\otimes_{C_1}\beta=1_S\), by the definition of the odd–odd product. For the other direction, the first odd inverse identity is \(t\widehat\otimes_{S\widehat\otimes C_1}(\tau_{C_1}(d)\widehat\otimes m_{\mathbb C})=1_{\mathbb C}\). Apply \(\tau_{C_1}\). The even cyclic unitary in Lemma 0.13 identifies the two possible consecutive Clifford reductions, while simultaneous matrix Morita naturality moves the other reduction past \(d\). The resulting equality is \(\beta\widehat\otimes_Sd=1_{C_1}\). Thus both products have been checked, with the source and coefficient Clifford factors explicit.

Take \(\beta_0=\delta_0=1_{\mathbb C}\). For a positive ordered coordinate system on \(\mathbb R^n\), \(n\ge1\), form the graded external products
\[
\beta_n=\beta\widehat\boxtimes\cdots\widehat\boxtimes\beta
\in KK(C_n,C_0(\mathbb R^n)),\qquad
\delta_n=d\widehat\boxtimes\cdots\widehat\boxtimes d
\in KK(C_0(\mathbb R^n),C_n).
\]
The external product is constructed by external tensoring a first cycle with the second source algebra, then composing with the second cycle externally tensored with the first coefficient algebra. In this instance the source Clifford factors are finite dimensional and the function-algebra factors are trivially graded, so all product-preservation cases were proved in Lemma 0.13. The classes are even morphisms of the graded category; no interchange of odd parity classes is being asserted. Multiplying the products factor by factor, using associativity and the two inverse products just proved, gives
\[
\beta_n\widehat\otimes_{C_0(\mathbb R^n)}\delta_n=1_{C_n},
\qquad
\delta_n\widehat\otimes_{C_n}\beta_n=1_{C_0(\mathbb R^n)}.
\tag{114}
\]
This proves suspension Bott equivalence in KK, rather than only a scalar K-group isomorphism. In even rank, reduce \(C_n\) by the fixed even Clifford Morita module. Its chirality is \((-i)^{n/2}e_1\cdots e_n\); in odd rank retain one Clifford generator. The first-coordinate class is normalized by \(\Theta_1(t)=[u_+]\), and every subsequent coordinate is taken in the displayed order. The order and chirality fix the orientation sign of the resulting Bott class. This construction still needs a separate identification with a chosen geometric spin-c vector-bundle symbol before it can serve as that bundle's wrong-way class.

![Positive cotangent winding, the scalar Fourier compression of index minus one, and the negative dual Thom class give both KK inverse products and the actual Wiener-Hopf boundary in both K degrees.](../figures/kk-thom-inverse.png)

*Figure 0.3. Lemmas 0.13, 0.16–0.17, Theorems 0.14, 0.18–0.19 and Proposition 0.15, equations (104)–(113). The intermediate Fourier variable is \(\eta=-t\), so its half-line compression is the isometry of index \(-1\). The physical positive half-line gives the coisometry of index \(+1\). The dual phase is \(V_q=e^{iqt}\), hence its negative-phase Stone generator is \(-t\). The inverse is the negative dual Thom class followed by the even Takai Morita class. Both identities are identities in KK. For the even boundary, projection products are formed over the unitization and their scalar reference is subtracted. Corollary 0.20 proves the subsequent suspension Bott inverse products.*

Full-size PNG · Editable SVG.


## 1. From a transversal to a correspondence

Let \(i:N\hookrightarrow V\) be an embedded transversal of dimension \(q\). Put

\[
Z_N=r^{-1}(N)\subset G.
\]

The map \(s:Z_N\to V\) is a local diffeomorphism. Its fibres are discrete countable sets: they are the intersections of the holonomy covering of a leaf with \(N\), including the different lifts of an intersection point.

Let \(E\) be a continuous Hermitian Hilbert field on \(N\), given by a countably generated Hilbert \(C_0(N)\)-module. Initially use compact chart sections

\[
\xi\in C_c(Z_N,r^*E\otimes s^*D_F^{1/2}).
\]

For a non-Hausdorff arrow space, this means the finite span of sections supported in Hausdorff charts, as in the preceding lesson. For \(\gamma:x\to y\), define

\[
\langle\xi,\eta\rangle_A(\gamma)
=\sum_{\substack{z\in Z_N\\s(z)=x}}
\left\langle\xi(z\gamma^{-1}),\eta(z)\right\rangle .
\tag{1}
\]

The two source half-densities give the summand the required endpoint density. On a pair of compact chart supports the sum has finitely many local terms; partitioning supports reduces (1) to ordinary continuous chart kernels. Thus its value is in \(A\).

Right multiplication is convolution. In a density trivialization its formula is

\[
(\xi f)(z)
=\int_{s(h)=s(z)}\xi(zh^{-1})f(h).
\tag{2}
\]

Fubini's theorem gives

\[
\langle\xi f,\eta g\rangle_A
=f^**\langle\xi,\eta\rangle_A*g.
\tag{3}
\]

To check positivity without a coordinate guess, use the fibre

\[
H_x^N=\ell^2(Z_N\cap s^{-1}(x),r^*E).
\]

The coefficient kernel in (1), represented on \(L^2(G_x)\), is \(T_\xi(x)T_\xi(x)^*\), where

\[
T_\xi(x):H_x^N\longrightarrow L^2(G_x)
\]

is the coefficient map of the equivariant section \(\xi\). The same formula for a finite list of sections proves matrix positivity. Since the regular representations define the norm of \(A\), (3) completes the sections to a right Hilbert \(A\)-module, denoted \(\mathcal E_N(E)\). Chart bump functions and a countable dense family of sections on \(N\) show that it is countably generated.

A bounded adjointable field map \(T:E_1\to E_2\) acts by

\[
(i_!T)_x v(z)=T_{r(z)}v(z),\qquad v\in H_x^N.
\tag{4}
\]

The action preserves the chart section space and has adjoint \(i_!(T^*)\). It therefore gives an adjointable map of the completed modules. Its norm is at most \(\|T\|\).

The discrete set in \(H_x^N\) must include holonomy lifts. For example, a transversal point on a leaf with twofold holonomy contributes two points on its holonomy covering. Removing one lift changes the equivariant representation.

## 2. Why compact maps remain compact

The compact operators in this section are compact module operators. They can have infinitely many nonzero diagonal blocks in the Hilbert space of one recurrent leaf.

**Theorem 2.1 (transversal compactness).** If \(T:E_1\to E_2\) is compact as a Hilbert \(C_0(N)\)-module map, then \(i_!T\) is compact as a Hilbert \(A\)-module map.

**Proof.** First take \(E_1=E_2=E\) and \(T=\theta_{\sigma,\sigma}\), where \(\sigma\) is a continuous section with compact support in a small transverse coordinate neighbourhood \(U\subset N\). Near the units over \(U\), the map \(s:Z_N\to V\) identifies a neighbourhood with a plaque tube. Choose a compactly supported source half-density \(b\) in that tube, normalized by

\[
\int |b(n,t)|^2\,dt=1
\quad\text{for }n\in\operatorname{supp}\sigma.
\tag{5}
\]

Such a \(b\) is a fixed normalized plaque bump times a transverse cutoff equal to one on the support. The tube can be chosen so that two arrows in it with the same source have the same range in \(N\) and are the same arrow. This follows from the local diffeomorphism \(s\) near the units; shrink the transverse coordinate neighbourhood before choosing the bump.

Set \(\xi(z)=b(z)\sigma(r(z))\). We claim

\[
\theta_{\xi,\xi}=i_!(\theta_{\sigma,\sigma}).
\tag{6}
\]

Evaluate the left side on \(H_x^N\). The averaged rank-one formula first sums over \(w\in Z_N\cap s^{-1}(x)\) and then integrates the plaque variable. A mixed term involving distinct \(z,w\) can be nonzero only if both corresponding short arrows lie in the chosen tube. Their quotient has its source and range in \(N\). The uniqueness just established makes that quotient a unit, so \(w=z\). The remaining diagonal term is

\[
v(z)\longmapsto
\sigma(r(z))\left\langle\sigma(r(z)),v(z)\right\rangle
\int|b(r(z),t)|^2\,dt.
\]

By (5) this is exactly (4) for \(T=\theta_{\sigma,\sigma}\). The regular fibre representations are faithful on module endomorphisms, so (6) follows.

For a section with an arbitrary compact support \(K\), take a finite nonnegative partition of unity \(\phi_j\) on \(K\), subordinate to neighbourhoods of the above kind. Pointwise,

\[
\theta_{\sigma,\sigma}
=\sum_j\theta_{\sqrt{\phi_j}\sigma,\sqrt{\phi_j}\sigma}.
\]

Each term has compact pushforward by (6). Polarization treats \(\theta_{\sigma,\eta}\), and continuous cutoffs approximate sections vanishing at infinity by sections with compact support. Finite sums of rank-one maps are norm dense in the compact module maps. Since (4) is contractive, their pushforwards converge in norm to \(i_!T\). The compact maps form a norm closed space, proving the theorem. \(\square\)

In particular \(C_0(N)\) acts by compact operators on \(\mathcal E_N(\mathbb C)\), through multiplication at \(r(z)\).

For clarity, an even **Kasparov cycle** from \(B\) to \(A\) consists of a countably generated graded Hilbert \(A\)-module \(\mathcal E\), an even *-representation \(\rho:B\to\mathcal L_A(\mathcal E)\), and an odd adjointable \(F\) such that

\[
[F,\rho(b)],\qquad(F^2-1)\rho(b),\qquad(F-F^*)\rho(b)
\]

are compact for every \(b\in B\). Homotopy and the addition of degenerate cycles define \(KK(B,A)\). Section0 constructs its associative product and proves the homotopy and unit conventions. Theorem 0.14 proves its scalar identification \(KK(\mathbb C,A)=K_0(A)\) and the odd version, including general Hilbert-module homotopies.

The correspondence above gives the cycle

\[
i_!=[\mathcal E_N(\mathbb C),\rho,0]\in KK(C_0(N),A).
\tag{7}
\]

The only nonzero defect in its definition is \(-\rho(h)\), which is compact by Theorem 2.1. Theorem 0.14 and the product in Section 0 give its action

\[
K_c^j(N)\longrightarrow K_j(A),\qquad
x\longmapsto x\otimes_{C_0(N)}i_!.
\tag{8}
\]

The same construction works for an étale map to the leaf space described by local transverse maps and holonomy transition arrows. The local modules glue by those arrows; the proof of compactness is local on the support in the domain, so the partition-of-unity argument remains valid.

## 3. A compact transversal gives a projection

Assume now that \(N\) is compact. A uniform plaque tube along \(N\) exists: the exponential in a leafwise metric is transverse to \(N\) at its zero section, and compactness permits a tubular neighbourhood on which it is injective. Its corresponding unit neighbourhood in \(Z_N\) has the uniqueness used in Theorem 2.1.

Choose a scalar section \(b\) in this tube satisfying (5) for every \(n\in N\). Then

\[
\theta_{b,b}=1_{\mathcal E_N(\mathbb C)}.
\tag{9}
\]

**Proposition 3.1.** The element \(e=\langle b,b\rangle_A\) is a projection in \(A\), and \(\mathcal E_N(\mathbb C)\cong eA\). If \(\Lambda\) is an invariant transverse measure, then

\[
\tau_\Lambda(e)=\Lambda(N).
\tag{10}
\]

**Proof.** The element \(e\) is self-adjoint. By (9), \(be=b\), and

\[
e^2=\langle b,be\rangle_A=\langle b,b\rangle_A=e.
\]

The map \(eA\to\mathcal E_N(\mathbb C)\), \(a\mapsto ba\), preserves inner products. Every \(\xi\) satisfies \(\xi=b\langle b,\xi\rangle_A\) by (9), so it is onto.

At a unit \(1_x\), the kernel \(e(1_x)\) is the sum of \(|b(z)|^2\) over the short arrows \(z:x\to n\) in the chosen tube. There is at most one such arrow. The diagonal trace formula from the measured-index lesson integrates this function using plaque volume and transverse measure. Integration first in the plaque direction gives one by (5), and the remaining integral is precisely \(\Lambda(N)\). All terms are nonnegative and have compact support, so this order of integration is valid. \(\square\)

This projection is in the algebra itself. A class in \(K_0(A)\) need not have a representative of that form: unitization and matrix stabilization allow differences of projections that are absent from \(A\).

**Example 3.2 (a product foliation).** For \(V=P\times N\), with \(P\) connected, choose a normalized \(\psi\in L^2(P)\). The construction gives

\[
e(t,t',n)=\psi(t)\overline{\psi(t')}.
\]

Under \(A\cong C_0(N)\otimes\mathcal K(L^2(P))\), this is \(1_N\otimes\theta_{\psi,\psi}\) when \(N\) is compact. Formula (10) gives the total transverse mass.

## 4. A local transverse orientation class

Assume \(V\) is connected and the normal bundle \(\tau=TV/F\) is oriented. A positively oriented transverse embedding of an **open** \(q\)-disk defines, using (8),

\[
[V/F]^*=\varphi_!(\beta_q)\in K_q(A).
\tag{11}
\]

Here \(\beta_q\) is the compactly supported Bott generator. The open disk is diffeomorphic to \(\mathbb R^q\); its parity is \(q\bmod2\). A closed disk with its ordinary K-theory would give a different statement.

**Proposition 4.1 (independence of the transverse disk).** Within each connected component of \(V\), the class (11) does not depend on the centre or the positively oriented transverse embedding.

**Proof.** The Bott class can be represented with support in any smaller disk about its centre. Extension by zero from that disk gives the same generator; this is the local form of Bott periodicity. Thus only the germ of the embedding matters.

In a foliation chart, shrink that germ and rescale its coordinates. It is isotopic through transverse embeddings to the linear transverse graph given by its derivative: its transverse derivative is invertible, and on a sufficiently small disk it remains so throughout the straight interpolation to that derivative. The positively oriented linear transverse frames are connected. Indeed Gram–Schmidt writes an element of \(\operatorname{GL}^+(q,\mathbb R)\) as an orthogonal matrix of determinant one times an upper triangular matrix with positive diagonal. Straight interpolation contracts the latter to identity without losing its positive diagonal. Elimination by plane rotations writes the former as a product of plane rotations, each contracted through its angle. For rank zero or one there is no rotation step. Therefore two choices at one point are joined through positively oriented frames.

To obtain a correspondence homotopy, apply Sections 1–2 to \(V\times[0,1]\), foliated by \(F\) at fixed parameter, and to the family of transverse disk embeddings. Its algebra is \(A\otimes C([0,1])\). The parameter-family disk is transverse, since its parameter derivative has component one. The same normalized plaque bumps and finite partitions give a module with compact coefficient action; pull its domain representation back along the constant inclusion of \(C_0(D^q)\). Its endpoint cycles are exactly the two disk correspondences. Section 0 homotopy invariance and the fixed coefficient Bott class therefore give their equal K-classes.

Join two centres by a path in \(V\). The oriented normal bundle along this interval is trivial, so choose a continuous positively oriented transverse frame along the path. A small transverse exponential disk varies along it; finitely many intervals suffice if the radius needs to shrink. The preceding support argument permits those shrinkings. Homotopy invariance on the successive intervals proves equality of the endpoint classes. \(\square\)

The component restriction is necessary. For the zero-dimensional foliation of \(V=\{0,1\}\), the two transverse point classes are the two distinct coordinate projections in \(K_0(C(V))=\mathbb Z^2\). There is no path joining their centres.

Orientation establishes this definition, not its nonvanishing. A sufficient test is a class

\[
\zeta\in KK^q(A,\mathbb C)
\quad\text{with}\quad
[V/F]^*\otimes_A\zeta=1\in\mathbb Z.
\tag{12}
\]

Indeed a zero class has zero product with every class. For a submersion onto an oriented spin-c manifold \(B\), the Morita equivalence \(A\sim C_0(B)\) turns (11) into the local Bott class on \(B\). To use the geometric spin-c Dirac K-homology class as \(\zeta\), its local Bott–Dirac pairing must be proved with the same orientation. The vector-bundle Dirac comparison required below includes this assertion; the local disk construction does not supply it.

For a general foliation, a holonomy invariant spin-c structure on \(\tau\) does not by itself produce the operator in (12). Transverse ellipticity, compactness of commutators, and an appropriate representation still need to be proved. The construction of a suitable transverse Dirac class is a substantive problem formulated by Connes; the elementary local disk argument does not solve it.

## 5. Maps to a leaf space retain transition arrows

The geometric-cycle framework originates in the work of Paul Baum and Ronald G. Douglas. For the smooth foliation model used here, see the freely available [Connes, Section 9](https://alainconnes.org/wp-content/uploads/foliationsfine.pdf#page=29), which credits its collaboration with Baum. The definitions below retain the actual arrow cocycles and full stable orientations; Theorems 6.0ae and 6.1 prove descent for this specified model.

A smooth map \(f:M\to V/F\) is described by an open cover \((U_a)\), smooth maps \(f_a:U_a\to V\), and smooth arrows

\[
g_{ba}(x):f_a(x)\longrightarrow f_b(x)
\]

on overlaps, satisfying \(g_{ca}=g_{cb}g_{ba}\). Refining the cover or changing its lifts by compatible arrows gives the same map. Keeping only the assertion that \(f_a(x)\) and \(f_b(x)\) lie on the same leaf would discard holonomy.

The derivatives of the holonomy maps glue the bundles \(f_a^*\tau\) to a rank-\(q\) real bundle \(f^*\tau\) on \(M\). A **K-orientation** of \(f\) is a stable spin-c orientation of the virtual bundle

\[
f^*\tau-TM.
\tag{13}
\]

The canonical complex spin-c structure on \(TM\oplus TM\) converts this to full Clifford orientation data on \(TM\oplus f^*\tau\). In positive rank these data are a spin-c structure, as proved in Lemma6.0j. In rank zero a stable orientation can retain an odd graded line; it is interpreted after positive even stabilization and is not identified with the canonical even rank-zero unit. The finite construction in Lemma6.0j retains that grading. This stable convention fixes the full orientation, including its line twist, needed for the Thom map.

There is also a useful principal bundle. Glue \(U_a\times_{f_a,r}G\) using the transition arrows. It is a manifold \(P_f\) of dimension \(\dim M+p\), with projection to \(M\) and a free transitive groupoid action on each fibre. The fibres have dimension \(p\). Local sections recover the cocycle, so the bundle and cocycle descriptions are equivalent.

Geometric cycles are triples \((M,E,f)\), where \(M\) is compact without boundary, \(E\) is a finite-rank complex bundle, and \(f\) has the orientation (13). Their parity is \(\dim M-q\bmod2\). Impose the following relations:

1. Disjoint union is addition, and replacing \(E_1\oplus E_2\) by the two separate cycles does not change the sum.
2. The oriented boundary of a compact cycle with boundary is zero.
3. For an even-rank spin-c real bundle \(W\to M\), replace \(M\) by the sphere bundle \(S(W\oplus\mathbb R)\), replace \(E\) by its pullback tensored with the fibre Bott bundle, and replace \(f\) by its composite with the sphere projection.

The resulting group is denoted \(K_{*,\tau}^{\mathrm{geom}}(BG)\). The symbol \(BG\) records the classifying space of the groupoid and its normal bundle. Identifying this cycle model with other models of groupoid topological K-theory is a theorem that must include its hypotheses.

## 6. The geometry of a wrong-way map

### The global Clifford Thom equivalence

The Clifford Bott and Dirac method is classical; see [Kasparov] and the point-space discussion in [Emerson]. We give the bundle proof with both product operators and their homotopies. The argument works over a noncompact base and retains the Clifford twist until a spin-c structure is specified. Its analytic inputs are the complete-source energy argument of Lemma 8.19, the module-generator proof of Lemma 8.15a and the product criterion of Definition 0.3. No local-to-global KK exactness theorem is needed.

Let \(X\) be a second-countable locally compact manifold and \(E\to X\) a rank-\(r\) real vector bundle with a continuous metric. Write \(\pi:E\to X\) and
\[
C_X=C_0(X),\qquad A_E=C_0(E,\pi^*\mathrm{Cl}^+(E)).
\tag{128}
\]
All tensor products and Clifford representations in this subsection are graded. In an orthonormal fibre frame, let \(e_i\) denote exterior multiplication by the corresponding vector and \(\iota_i=e_i^*\) contraction. On \(\Lambda E_{\mathbb C}\), put
\[
a_i=e_i+\iota_i,\qquad c_i=e_i-\iota_i,
\qquad \{a_i,a_j\}=2\delta_{ij},\quad
\{c_i,c_j\}=-2\delta_{ij},\quad \{a_i,c_j\}=0.
\tag{129}
\]
The operators \(a_i\) are odd self-adjoint and the \(c_i\) odd skew-adjoint. These formulas are equivariant under orthogonal frame changes. They therefore define global Clifford actions. Exterior-degree parity is the grading \(\Gamma\).

**Lemma 6.0a (the two global cycles).** Let \(b_E\) be the standard-module cycle
\[
\left(A_E,\phi_X,
B(v)=\frac{c^+(v)}{\sqrt{1+\|v\|^2}}\right),
\qquad \phi_X(f)(v)=f(\pi(v))1.
\tag{130}
\]
Let \(\mathcal H_E\) be the Hilbert \(C_X\)-module of continuous fibrewise \(L^2\) sections of \(\Lambda E_{\mathbb C}\) whose fibre norms vanish at base infinity. The fibre metric supplies Lebesgue measure. The positive Clifford action in (129) represents \(A_E\) on this module by multiplication. The vertical operator and its bounded transform are
\[
D=\sum_i c_i\partial_{v_i},\qquad
F_D=D(1+D^2)^{-1/2}.
\tag{131}
\]
They define \(d_E\in KK(A_E,C_X)\); (130) defines \(b_E\in KK(C_X,A_E)\). Neither construction requires an orientation or spin-c structure.

**Proof.** For (130), \(B\) is an odd self-adjoint multiplier of norm at most one, and its coefficient commutators vanish. Its localized square defect is \(-f(x)/(1+\|v\|^2)\). This vanishes at infinity on the total space: cut off a compact base set using \(f\in C_X\), then a fixed fibre radius. A closed bounded disk bundle over a compact base set is compact, by a finite bundle-chart cover and uniform equivalence of its fibre norms. Thus the defect belongs to \(A_E=\mathcal K(A_E)\). Base approximate identities prove nondegeneracy of \(\phi_X\). A countable chart exhaustion makes \(A_E\) separable and its standard module countably generated.

For (131), the physical transform \(\widehat u(p)=\int e^{-i\langle p,v\rangle}u(v)\,dv\) gives the odd self-adjoint multiplier \(ic(p)\), with square \(|p|^2\). Its inverse shifts are the matrices \((ic(p)\mp i)/(1+|p|^2)\). Acting on compact-parameter Fourier test sections, these are inverse operators with dense ranges. The inverse-domain argument of Lemma 8.15a proves regular self-adjointness on the full module domain. Orthogonal covariance glues the Fourier constructions; local compactly supported test sections form a graph core by Fourier cutoffs and a countable base partition. This is a module-domain argument, not just a list of fibre self-adjoint operators.

Multiplication by a smooth compact fibre cutoff followed by a Dirac resolvent is compact. Indeed a Fourier truncation produces a Hilbert–Schmidt kernel, and its remaining order-\(-1\) Fourier multiplier has norm tending to zero. This is the compact-frequency argument of the index lesson's Lemma 6.1. The same estimates are uniform in a compact base chart, and a base cutoff gives a compact module operator. Hence both-sided coefficient localizations of \(C_0\) functions of \(D\) are compact. Fibre-smooth compact sections of \(A_E\) are dense, by chart smoothing only in the fibre variables. If \(u\) is homogeneous of parity \(k\) in this dense algebra, then
\([D,\rho(u)]_{\mathrm{gr}}=\sum_i c_i\rho(\partial_i u)\) is bounded, and multiplication preserves the graph domain by the ordinary first-derivative product estimate.

We state the graded commutator identity used below. For any such regular odd operator \(T\), a homogeneous \(u\), \(\sigma=(-1)^k\), \(C=Tu-\sigma uT\), \(q=\sqrt{1+\tau^2}\), and \(R_\tau=(T^2+q^2)^{-1}\),
\[
[TR_\tau,u]_{\mathrm{gr}}
=q^2R_\tau C R_\tau-\sigma TR_\tau C TR_\tau,
\qquad
\|[TR_\tau,u]_{\mathrm{gr}}\|
\le\frac{5\|C\|}{4(1+\tau^2)}.
\tag{132}
\]
To justify it without assuming that \(C\) preserves the domain, use the first-order identity
\[
(T-z)^{-1}u-\sigma u(T-\sigma z)^{-1}=-\sigma(T-z)^{-1}C(T-\sigma z)^{-1}.
\]
Sum the two poles \(z=\pm iq\). The bound follows from \(\|R_\tau\|\le q^{-2}\) and \(\|TR_\tau\|\le(2q)^{-1}\). Its terms are compact whenever the coefficient localizations of the resolvents are compact. Integration against \(2/\pi\) over \(\tau\ge0\) gives a norm-convergent compact graded commutator for \(T(1+T^2)^{-1/2}\). The individual transform integrals converge strictly, as proved in Lemma 8.15b. Density treats every coefficient. This proves all the defects for \(d_E\). The module is countably generated by fibre tests in countably many charts. Rank zero gives the identity homomorphism cycle in both cases. \(\square\)

**Lemma 6.0b (uniform linear-potential homotopy).** On
\(\mathcal M=\mathcal H_E\widehat\otimes_{C_X}A_E\), denote the target fibre variable by \(z\), the integration variable by \(v\), and let
\(A_i=a_i\widehat\otimes1\), \(B_i=1\widehat\otimes L_{c_i^+}\). In the underlying ordinary tensor product the latter is \(\Gamma\otimes L_{c_i^+}\); the grading sign is inserted exactly once. Here \(L_{c_i^+}\) is left multiplication on the target Clifford algebra. Put
\[
T_{t,z}=D+A(tv-z),\qquad
A(w)=\sum_i w_iA_i,
\quad 0\le t\le1.
\tag{133}
\]
Represent a section \(u(v)\in A_E\) using the positive generators \(B_i\). Then the bounded transforms of (133) define a single Kasparov homotopy over \(C([0,1],A_E)\). In particular, the localized defects are compact and jointly continuous in \(t,z\), and vanish at target infinity after the prescribed coefficient localization.

**Proof.** It suffices to work in an orthonormal chart with compact base support; all constructions are orthogonally equivariant. The source metric is Euclidean and complete. The potential is Hermitian and linear. The deficiency-cutoff and moving-cutoff energy proofs of Lemma 8.19 apply: \([T_{t,z},\eta]=c(d\eta)\), so the unbounded potential contributes neither to the deficiency estimate nor to propagation speed. Each closure is self-adjoint and its waves have speed at most one.

For compact parameter support and a compact source test, the reachable source set at finite wave time is a fixed compact set. On this set a change in \(t,z\) changes the potential by a bounded coefficient tending uniformly to zero. Duhamel's integral therefore proves strong continuity of the wave group, and of its adjoint, in the parameters. Source derivatives are controlled by the differentiated energy equation on that same set. These waves define a strongly continuous unitary group on the whole parameter Hilbert module. Time smoothing, the two strict Laplace inverses and their dense ranges now give regular self-adjointness on its full domain exactly as in Lemma 8.15a. The assertion has thus been constructed over the parameter algebra before any compactness conclusion is drawn.

Write \(N\) for exterior degree and \(K=2N-r\); \(\|K\|=r\). Directly from (129),
\[
T_{t,z}^2=-\Delta_v+|tv-z|^2+tK.
\tag{134}
\]
Let \(R=(T_{t,z}\pm i)^{-1}\), \(w=R\xi\), and \(h=\chi w\), for a fixed smooth cutoff supported in \(|v|\le R_0\). The spectral bounds give \(\|w\|,\|T_{t,z}w\|\le\|\xi\|\). Therefore \(\|T_{t,z}h\|\le C_\chi\|\xi\|\), since its cutoff commutator is independent of the potential. Formula (134) gives
\[
\|\nabla h\|^2+\|(tv-z)h\|^2
\le C_\chi^2\|\xi\|^2+r\|h\|^2.
\tag{135}
\]
Thus the localized resolvents have uniformly bounded range in \(H^1\) on a fixed compact source support. Rellich compactness, or the compact-frequency proof in Lemma 6.0a, makes them fibrewise compact. Also \(|tv-z|\ge |z|-R_0\) there, so \(\|\chi R\|\le C/(1+|z|)\), uniformly in \(t\).

We include the norm-continuity step. The waves and their Laplace integrals give strong continuity of \(R\) and \(R^*\). The outputs of \(\chi R\) form a uniformly relatively compact set by (135). The outputs of the localized differences from the limiting resolvent also form a uniformly relatively compact set, as differences of two relatively compact sets. If norm continuity failed along a convergent parameter sequence, choose unit input vectors whose difference outputs have norms bounded below by one positive constant. A norm-convergent subsequence of those difference outputs has zero pairing with every fixed test vector: move the difference to its adjoint, which converges strongly to zero, and use the common unit bound on the inputs. Its limit is therefore zero, contradicting the positive norm bound. This proves norm continuity of the compact operators. The argument applies to their finite Clifford matrix components in a faithful representation, with the same bounds. With a compact base cutoff and the \(z\)-decay just proved, these are compact operators of the parameter module. Products and uniform approximation give the same coefficient localizations for the entire \(C_0\) calculus.

The \(B_i\) anticommute with both \(c_j\) and \(A_j\). Consequently the graded commutator with a compact fibre-smooth coefficient has only its bounded first-derivative term, uniformly in \(t,z\). Formula (132) gives its compact bounded-transform commutator, with an integrable parameter-independent tail. Strong continuity of bounded spectral factors multiplied by norm-continuous compact localized factors proves continuity of each integrand; the tail then gives continuity of the integral. The square defect is a localized resolvent. Density finishes the parameter-cycle assertions. A locally finite base partition and compact-base approximation globalize them without any geometry bound on all of \(X\). \(\square\)

**Theorem 6.0c (both global Clifford Thom inverse products).** The actual cycles of Lemma 6.0a satisfy
\[
b_Ed_E=1_{C_X},\qquad d_Eb_E=1_{A_E}.
\tag{136}
\]
These are global graded KK identities for the stated possibly noncompact base.

**Proof: the first product.** Rank zero was handled in Lemma 6.0a, so assume \(r>0\). Its tensor module is \(\mathcal H_E\). Put
\[
Q=D+A(v),\qquad Q^2=L+2N-r,
\quad L=-\Delta_v+|v|^2,
\qquad F_Q=Q(1+Q^2)^{-1/2}.
\tag{137}
\]
The Hermite basis gives the full spectral information needed here. In one variable, the normalized Gaussian \(h_0=\pi^{-1/4}e^{-v^2/2}\), followed by repeated normalized application of \((-\partial_v+v)/\sqrt2\), gives orthonormal eigenvectors \(h_n\) of \(L\) with eigenvalues \(2n+1\). Their completeness can be checked without an oscillator spectral assumption: orthogonality to all of them makes every Taylor coefficient of \(\int f(v)e^{-v^2/2}e^{wv}\,dv\) zero. This integral is entire, by Cauchy–Schwarz and Gaussian domination on compact complex sets. Its values on the imaginary axis vanish, and Fourier uniqueness gives \(f=0\). Tensoring proves completeness in \(r\) variables. On exterior degree \(k\), (137) has eigenvalues \(2(|n|+k)\). Its sole zero space is the even Gaussian times \(\Lambda^0E_{\mathbb C}\); the next eigenvalue is two. Every fixed eigenspace is finite dimensional. The symmetric \(Q\) preserves each fixed \(|n|+k\) space. Its finite self-adjoint matrices define inverse shifts on their algebraic sum. Uniform inverse bounds and dense finite-energy cutoffs give the full regular closure and graph core by the inverse-domain argument. These spaces form finite-rank bundles under the orthogonal transitions, and base-localized resolvents are compact module operators.

We verify the connection condition. For a homogeneous smooth compact \(\xi\in A_E\), let \(\sigma=(-1)^{|\xi|}\). Then
\(K_\xi=Q\rho(\xi)-\sigma\rho(\xi)D=[D,\rho(\xi)]_{\rm gr}+A(v)\rho(\xi)\) is bounded and has compact base and fibre support. The mixed first-order resolvent identity gives
\[
(Q-z)^{-1}\rho(\xi)-\sigma\rho(\xi)(D-\sigma z)^{-1}
=-\sigma(Q-z)^{-1}K_\xi(D-\sigma z)^{-1}.
\tag{138}
\]
Each right side is compact, by the oscillator resolvent with the compact base support, and its norm is at most \(\|K_\xi\|/|\operatorname{Im}z|^2\). Sum the two poles \(z=\pm i\sqrt{1+\tau^2}\), then integrate the bounded-transform formula. It proves that \(F_Q\rho(\xi)-\sigma\rho(\xi)F_D\) is compact; adjoints give the other connection error. Norm approximation treats every creation vector.

Here is a direct positivity check. Put \(b=A(v)/s(v)\), \(s(v)=\sqrt{1+|v|^2}\), \(g=|v|^2/s(v)\), and \(S=(1+L)^{-1/2}\). The bounded constant matrix \(2N-r\) commutes with \(L\). Its spectral bounds imply that \(F_Q-QS\) is compact after a base cutoff: the scalar difference of the two inverse square roots is \(O((1+\lambda)^{-3/2})\), and \(Q(1+L)^{-1/2}\) is bounded. Also
\[
\{b,Q\}=2g+K_b,\qquad
K_b=\sum_i c_i\partial_i b,
\qquad
\{b,QS\}=2gS+K_bS+Q[S,b].
\tag{139}
\]
The first remainder is compact since \(K_b\) is bounded and \(S\) has compact fibre resolvent. To control the second, use
\(S=\pi^{-1}\int_0^\infty u^{-1/2}(1+L+u)^{-1}\,du\). The resolvent commutator inserts \([L,b]=-2\sum_i(\partial_i b)\partial_i-\Delta b\). Its coefficients are bounded. Both \(\|Q(1+L+u)^{-1}\|\) and \(\|[L,b](1+L+u)^{-1}\|\) are at most \(C(1+u)^{-1/2}\). Thus the integral for \(Q[S,b]\) has compact integrands and norm bounded by an integrable multiple of \(u^{-1/2}/(1+u)\).

Finally \(gS\) is positive modulo compacts. The function \(g\) has bounded first and second derivatives. The same resolvent argument with exponent \(1/4\) shows that \([g,S^{1/2}]\) is a bounded compact operator: its integrand is bounded by \(Cu^{-1/4}(1+u)^{-3/2}\). For this unbounded multiplier, first replace \(g\) by smooth cutoffs with uniformly bounded first and second derivatives and prove the resolvent identity on the Schwartz core. The derivative coefficients converge strongly, whereas both \((1+L+u)^{-1}\) and \(\partial_i(1+L+u)^{-1}\) are compact by their Hermite spectral bounds. Thus each commutator integrand converges in norm. The displayed integrable bound gives norm convergence of its integral and a compact bounded extension for the limit. Hence
\[
gS=S^{1/2}gS^{1/2}+[g,S^{1/2}]S^{1/2}.
\tag{140}
\]
The equality is first on the Schwartz core and extends as bounded operators. To justify the positive middle form, \(1+L\ge1+|v|^2=s^2\) implies \((1+L)^{1/2}\ge s\). Indeed inverse order follows from the quadratic-form variational formula for \((A+u)^{-1}\); integrating \(A(A+u)^{-1}=1-u(A+u)^{-1}\) against \(u^{-1/2}/\pi\) proves this square-root order. Since \(0\le g\le s\), the form \(S^{1/2}gS^{1/2}\) is positive and bounded by one. Also \(gS\) is bounded because \(g\le s\) and \(s(1+L)^{-1/2}\) is bounded. This establishes the extension in (140). All the compact remainders have uniform fibre estimates and are orthogonally equivariant; a \(C_X\) coefficient makes them compact over the base. Equations (139)–(140) therefore give the positivity condition of Definition 0.3. Thus \(F_Q\) is an actual product operator.

Replace \(F_Q\) on the complement of its kernel by its phase. The spectral gap makes this a continuous functional-calculus change whose difference is a base-localized compact. It is a Kasparov homotopy. The complement is then a degenerate cycle. The even kernel is canonically \(C_X\), because the normalized radial Gaussian and exterior vacuum are invariant under every orthogonal change of frame. Its zero operator and usual coefficient representation are the identity homomorphism cycle. This proves the first identity in (136).

**Proof: the reverse product.** Its tensor module is \(\mathcal M\) of Lemma 6.0b, with left \(A_E\) represented by the positive generators \(A_i\) in the \(v\)-variable. A product operator is
\[
F_0=\frac{D+B(z)}{\sqrt{1+D^2+|z|^2}},
\qquad B(z)=\sum_i z_iB_i.
\tag{141}
\]
The numerator's square is \(D^2+|z|^2\). Fibre Fourier transformation constructs its full regular domain and inverse shifts, as in Lemma 6.0a. Localized resolvents are compact by compact-frequency approximation, and their norm tends to zero as \(|z|\to\infty\). Its coefficient graded commutator is only the first derivative term, because \(B_i\) anticommute with all the \(A_j\). Formula (132) proves the cycle defects.

For a homogeneous smooth creation vector \(\xi\in\mathcal H_E\) in \(\operatorname{Dom}D^2\), the connection error with \(B(z)/\sqrt{1+|z|^2}\) is a vector-valued function of \(z\). The derivative part has norm at most \(\|D\xi\|/\sqrt{1+|z|^2}\). The remaining normalizer difference has norm at most \(\|D^2\xi\|/(2(1+|z|^2))\), by the scalar inverse-square-root estimate and \(|z|\le\sqrt{1+|z|^2}\). It is continuous and vanishes at target infinity; a base cutoff and approximation treat all creation vectors. Such vector-valued functions are precisely compact creation maps from the standard target module. The adjoint errors have the same conclusion. Positivity is exact:
\[
\{F_D\widehat\otimes1,F_0\}
=\frac{2D^2}{\sqrt{1+D^2}\sqrt{1+D^2+|z|^2}}\ge0.
\tag{142}
\]
So (141) represents \(d_Eb_E\).

We give its global homotopy to the identity. The even unitary
\[
U_\theta=\exp\left(-\frac\theta2\sum_iA_iB_i\right)
\tag{143}
\]
sends \(A_i\) to \(\cos\theta A_i+\sin\theta B_i\) and \(B_i\) to \(-\sin\theta A_i+\cos\theta B_i\). Different \(A_iB_i\) commute; the identities follow by differentiating conjugation. Also \(D\) commutes with \(U_\theta\). This unitary is orthogonally equivariant, so it exists on the bundle module. At \(\theta=\pi/2\) it sends (141) to the transform of \(D-A(z)\), with left coefficients represented by \(B_i\).

Lemma 6.0b now gives the full parameter homotopy from \(D-A(z)\) to \(D+A(v-z)\), with that coefficient representation fixed. The fibre translation \((W_z\xi)(v)=\xi(v+z)\) is a continuous even module unitary. It sends the endpoint to \(Q=D+A(v)\), and sends the coefficient \(u(v)\) to \(u(v+z)\), still represented by the \(B_i\).

Keep \(Q\) fixed and deform this representation to \(u(z+sv)\), \(1\ge s\ge0\). These are bounded graded representations on a single interval module. They are nondegenerate: on a test with compact base, \(z\)-support and \(v\)-support, the argument \(z+sv\) stays in a common compact disk, where scalar coefficient approximate identities converge uniformly. Such tests are dense. To check the compactness assertion that is sometimes hidden in this step, note that \(f(Q)\) has finite-rank spectral approximations in the fibre for every \(f\in C_0(\mathbb R)\). For a compact fibre-smooth coefficient \(u\), multiplication by \(u(z+sv)\) is strongly continuous in \(s,z\). On any fixed compact \(v\)-set it tends uniformly in \(s\) to zero as \(|z|\to\infty\). Multiplication by each finite-rank spectral approximation consequently is norm continuous and vanishes at target infinity; its uniform spectral tail proves the same for \(u(z+sv)f(Q)\). Compact base approximation globalizes this statement. The graded commutator with \(Q\) has coefficient \(s\partial_i u(z+sv)\), uniformly bounded, and has the identical compact localization property. Formula (132), with its uniform integrable tail, proves the compact and continuous bounded-transform commutator. These are therefore actual Kasparov homotopies, even at \(s=0\).

At the last endpoint the coefficients are \(u(z)\), represented by \(B_i\). Replace the transform of \(Q\) by its phase off the Gaussian kernel, as for the first product. The kernel projection commutes with the endpoint representation: the exterior vacuum is even, and each \(B_i\) restricts there to ordinary left multiplication by the target Clifford generator. Thus the even kernel module is precisely the standard right \(A_E\)-module with its identity representation. The complement is degenerate. This proves the second identity in (136). All rotations, translations, scalings and the vacuum are defined intrinsically in the metric fibre and are orthogonally equivariant; no global frame or compactness of \(X\) was used. \(\square\)

**Corollary 6.0d (orientation, naturality and ordered sums).** The equivalence in Theorem 6.0c is independent of the metric under the induced section-algebra identification. It is natural for proper base pullbacks. For a specified spin-c structure on \(E\), it gives the positive spin-c Thom class and its two-sided KK inverse. The Thom and inverse classes compose for an ordered direct sum of bundles, including their Clifford twists.

**Proof: the metric and spin-c module.** Given two metrics, use \(g_s=(1-s)g_0+sg_1\) on the bundle over \(X\times[0,1]\). Write \(g_s(u,v)=g_0(G_su,v)\), and use the positive isometry \(G_s^{-1/2}:(E,g_0)\to(E,g_s)\), with its induced Clifford and exterior isometries. The eigenvalues of \(G_s\) and their inverses are bounded on each compact base set, uniformly in \(s\). This identifies the total-space section algebras and the fibre \(L^2\) modules continuously. The defect and resolvent arguments above work on this compact parameter interval. They give homotopies of both classes under this identification. No uniform bound on a change of metric over all of \(X\) is needed. Transport the chosen spin-c structure by the same positive isometries when one is specified.

For the spin-c structure, construct a right Hilbert \(C_r\)-module bundle with fibre \(C_r\). A principal transition \([s,z]\in\mathrm{Spin}^c(r)\) acts on this fibre by the even left unitary \(zs\). The relation \((s,z)\sim(-s,-z)\) gives the same unitary. It preserves the right \(C_r\) action and inner product. Conjugation by it gives the orthogonal Clifford transition of \(\mathrm{Cl}^+(E)\). Thus its sections vanishing at base infinity form an even bimodule \(\mathcal S_E\) from \(C_0(X,\mathrm{Cl}^+(E))\) to \(C_X\widehat\otimes C_r\).

Locally its compact endomorphisms are all left multiplications by \(C_r\); these glue by precisely the Clifford transitions. Thus \(\mathcal K(\mathcal S_E)=C_0(X,\mathrm{Cl}^+(E))\). Right fullness follows from chart-supported sections and a locally finite partition, followed by compact-base approximation. For clarity, the two Morita evaluation maps are \(\xi\widehat\otimes\overline\eta\mapsto\theta_{\xi,\eta}\) and \(\overline\xi\widehat\otimes\eta\mapsto\langle\xi,\eta\rangle\). Their tensor inner-product formulas show that they are isometries. The compact-endomorphism equality and fullness show that their ranges are dense and hence onto. Both maps are even. These are the actual two Morita inverse products of Section 0.

Pulling this bimodule to the total space gives
\[
m_E\in KK(A_E,C_0(E)\widehat\otimes C_r),\qquad
\beta_E=b_Em_E,\qquad
\alpha_E=m_E^{-1}d_E.
\tag{144}
\]
Their types are respectively \(KK(C_X,C_0(E)\widehat\otimes C_r)\) and \(KK(C_0(E)\widehat\otimes C_r,C_X)\). Associativity and the two proved Morita evaluations give
\(\beta_E\alpha_E=b_Ed_E=1_{C_X}\) and
\(\alpha_E\beta_E=m_E^{-1}(d_Eb_E)m_E=1_{C_0(E)\widehat\otimes C_r}\).
The notation \(KK_r(C_X,C_0(E))\) uses exactly the ordered Clifford contractions specified in Section 0; these computations never permute odd factors.

The product \(b_Em_E\) is represented by the pulled-back spin-c module with the multiplier \(c^+(v)/\sqrt{1+|v|^2}\). To check this assertion by Definition 0.3, the Morita cycle has operator zero. A creation vector \(\xi\in A_E\) acts compactly on that module, so its connection error with the proposed bounded multiplier is compact. The positivity condition is the positive operator \(2B(v)^2\). This proves the representative, rather than merely its K-group action. In even rank \(r=2\ell\), the ordered matrix-Clifford contraction produces the ordinary spinor bundle. With \(c_i=-i\varepsilon_i\), its symbol is \(ic(v)=c^+(v)\), and its grading is
\[
i^\ell c_1\cdots c_{2\ell}
=(-i)^\ell\varepsilon_1\cdots\varepsilon_{2\ell}.
\tag{145}
\]
This is the chirality of the index lesson's Lemma 6.24. In odd rank retain the final positive Clifford factor. Rank-zero unit normalization uses the canonical trivial spin-c structure. Changing a chosen spin-c structure by a line bundle tensors the above transition module by that line, so twists \(\beta_E\) and its inverse accordingly; the orientation choice is part of the data.

**Proof: proper pullbacks.** Let \(Y\) also be a second-countable locally compact manifold. If \(h:Y\to X\) is continuous and proper, then the induced total-space map \(\widetilde h:h^*E\to E\) is proper. Indeed the inverse image of a compact total-space set has base in a compact inverse image and bounded fibre norm, and is a closed subset of its compact disk bundle. There are therefore the pullback homomorphisms on both \(C_0\) algebras. On the standard target module, the two cycles representing
\[
[h^*]b_{h^*E}=b_E[\widetilde h^*]
\tag{146}
\]
have the identical coefficient action and Clifford multiplier, so the equality holds in KK. Multiplying by the actual inverses on the appropriate sides gives
\([\widetilde h^*]d_{h^*E}=d_E[h^*]\). Pulling back the spin-c transition module gives the identical statement for (144). Properness is the precise hypothesis for these \(C_0\) homomorphisms.

**Proof: ordered sums.** Let \(F\to X\) be another metric bundle. Identify \(E\oplus F\) in that order and use \(\mathrm{Cl}^+(E\oplus F)=\mathrm{Cl}^+(E)\widehat\otimes\mathrm{Cl}^+(F)\). Over the base \(E\), the pulled-back \(F\)-bundle has a coefficient Thom cycle
\(b_F^{\mathrm{Cl}E}\in KK(A_E,A_{E\oplus F})\): its standard module is \(A_{E\oplus F}\), its left action evaluates the old \(A_E\) coefficient at \(v\), and its operator is \(c_F^+(w)/\sqrt{1+|w|^2}\). The old Clifford coefficients graded-commute with this operator, and the localized defects vanish in \((v,w)\).

The product operator with \(b_E\) is
\[
\frac{c_E^+(v)+c_F^+(w)}{\sqrt{1+|v|^2+|w|^2}},\qquad
\left\{\frac{c_E^+(v)}{\sqrt{1+|v|^2}},
\frac{c_E^+(v)+c_F^+(w)}{\sqrt{1+|v|^2+|w|^2}}\right\}
=\frac{2|v|^2}{\sqrt{1+|v|^2}\sqrt{1+|v|^2+|w|^2}}\ge0.
\tag{147}
\]
For a compact old creation section \(\xi(v)\), the connection errors are the old-numerator term divided by the joint normalizer and the difference of the joint and \(w\)-only normalizers multiplied by the new numerator. Both vanish when \(|w|\to\infty\) with \(v\) bounded. Outside a compact \(v\)-set their norms are at most a fixed multiple of \(\|\xi(v)\|\), which tends to zero. Base cutoffs and density give the complete connection condition and its adjoint. Positivity is (147). Definition 0.3 thus proves
\[
b_Eb_F^{\mathrm{Cl}E}=b_{E\oplus F},\qquad
d_{E\oplus F}=d_F^{\mathrm{Cl}E}d_E.
\tag{148}
\]
Here \(d_F^{\mathrm{Cl}E}\in KK(A_{E\oplus F},A_E)\) is the vertical Dirac inverse for the pulled-back \(F\)-bundle with the old Clifford coefficient retained. To verify that coefficient extension, in each old Clifford chart tensor every module in Lemmas 6.0a–b and Theorem 6.0c with that finite-dimensional coefficient algebra, using the graded tensor action. All conjugating unitaries are even, the spectral cutoffs and estimates are unchanged, and the Gaussian evaluations become the identity on the old Clifford coefficient. Old Clifford frame transitions intertwine these formulas. They therefore give both inverse products globally over \(E\), with compact old-base cutoffs as before. It follows that \(d_F^{\mathrm{Cl}E}d_E\) is a two-sided inverse of the first class in (148). Uniqueness of a two-sided inverse gives the displayed equality with the already proved \(d_{E\oplus F}\).

Finally, the spin-c structures on the ordered summands induce the ordered sum structure. Locally the map \(C_r\widehat\otimes C_s\to C_{r+s}\) sends the first block of generators to the first block and the second to the second. It preserves the graded inner products, actions and both Morita evaluations. The transition units \(z_Es_E\) and \(z_Fs_F\) are even and their product is the sum transition unit. Consequently the map glues the tensor of the two spin-c transition modules to the sum module. Transferring (148) through these exact maps proves the spin-c composition and its inverse composition with the same block order. \(\square\)

![Both global Clifford Thom inverse products](../figures/vector-bundle-kk.png)

**Figure 6.0.** The full global proof of Lemmas 6.0a–b, Theorem 6.0c and Corollary 6.0d, equations (128)–(148). Here \(X\) is second-countable and \(E\) has the fixed rank \(r\); no orientation is needed before the last Morita transfer. In the spectrum formula \(n\) is the Hermite multi-index and \(k\) exterior degree. The plot samples the exact scalar potentials \((tv-2)^2\) on \(-1\le v\le5\) at \(t=0,1/2,1\); the Clifford term \(t(2N-1)\) in the displayed rank-one square is separate. The source cutoff estimate is uniform in the homotopy parameter, and its constant depends on the cutoff and rank. The Gaussian is even and invariant under all orthogonal transitions. Exercises 36–37 check the actual signs, compactness and inverse order. The Clifford Bott–Dirac method is classical [Kasparov; Emerson, Section 11.4]; the complete bundle and parameter-module arguments are supplied above.

Full-size PNG · Editable SVG.

### Tubes and composition of compact embeddings

The normal bundle theorem is already proved in the programme: DG-CHAR, *Manifold duality, the diagonal and Wu classes*, Lemma 3.1 and Theorem 3.2. Its proof constructs a local addition and proves injectivity on a normal disk; Lemma 3.3 supplies the compact Euclidean embedding used below. The smooth metric and complement arguments are DG-CHAR, *Vector bundles and their constructions*, Theorem 4.2 and Section 5. We use precisely these geometric assertions. Their singular-cohomology conclusions are not premises for the KK argument.

Here manifolds are smooth, Hausdorff, second countable and without boundary. A **normalized tube** for an embedding \(i:M\hookrightarrow Y\), with normal quotient \(E=i^*TY/TM\), is an open embedding from a neighbourhood of the zero section in \(E\) to \(Y\), equal to \(i\) on that section, whose vertical derivative induces the identity on the normal quotient. The theorem just cited gives one. A whole-bundle tube is obtained by first shrinking to a disk of radius \(\rho(x)>0\) and then using
\[
q_\rho(x,v)=\left(x,\frac{v}{\sqrt{1+|v|^2/\rho(x)^2}}\right).
\tag{149}
\]
This diffeomorphism from \(E\) to \(D_\rho E\) has vertical derivative the identity at zero; its inverse is \(w/\sqrt{1-|w|^2/\rho(x)^2}\). Thus the resulting whole-bundle tube remains normalized. For a compact base a constant positive radius suffices after shrinking.

If \(a:U\hookrightarrow V\) is an open embedding, write \(e_a:C_0(U)\to C_0(V)\) for extension by zero after its diffeomorphism onto its image. It is a *-homomorphism: a compactly supported function extends continuously, and approximation by such functions proves the assertion for \(C_0(U)\). This is the covariant map used here. Restriction of a general \(C_0(V)\) function to \(U\) need not vanish at infinity in \(U\).

**Lemma 6.0e (compression and tube independence).** Let \(E\to M\) be a metric real rank-\(k\) bundle on a compact manifold with a specified spin-c structure. Write
\(\beta_E\in KK(C(M),C_0(E)\widehat\otimes C_k)\)
for Corollary 6.0d's Thom class, where \(C_k=\mathrm{Cl}^+(\mathbb R^k)\). For a positive constant \(r\), let \(q_r:E\to D_rE\) be (149) and define \(\beta_{D_rE}=\beta_E[e_{q_r}]\), where here \(e_{q_r}\) is the isomorphism onto the disk algebra. Then
\[
\beta_{D_rE}[e_{D_rE\hookrightarrow E}]=\beta_E.
\tag{150}
\]
Consequently that disk inclusion, tensored with \(1_{C_k}\), is a KK equivalence. For a compact embedding \(i:M\hookrightarrow Y\), the class
\[
i_!=\beta_E[e_\tau]\in
KK(C(M),C_0(Y)\widehat\otimes C_k)
\tag{151}
\]
is independent of the normalized whole-bundle tube \(\tau:E\to Y\). It retains the specified normal spin-c structure.

**Proof: compression as an actual algebra homotopy.** Put
\(q_{r,s}(v)=v/\sqrt{1+s|v|^2/r^2}\), for \(0\le s\le1\), and extend its image functions by zero in \(E\). The resulting maps are
\[
\Phi_s(f)(x,w)=
\begin{cases}
f\!\left(x,\dfrac{w}{\sqrt{1-s|w|^2/r^2}}\right),
 &s|w|^2<r^2,\\
0,&s|w|^2\ge r^2.
\end{cases}
\quad \Phi_0(f)=f.
\tag{152}
\]
Each is a *-homomorphism, of norm at most one. For \(f\) of compact support \(K\subset E\), the set
\(\{(s,q_{r,s}(v)):s\in[0,1],v\in K\}\)
is compact. On the open image of the map \((s,v)\mapsto(s,q_{r,s}(v))\), the formula is continuous by its displayed inverse. At its boundary it is continuous with value zero: any nonzero value of size at least \(\epsilon\) has its preimage in the compact set \(\{|f|\ge\epsilon\}\), whose image is a closed subset inside that open image. Hence \((s,w)\mapsto\Phi_s(f)(w)\) is continuous, has uniformly compact support and lies in \(C([0,1],C_0(E))\). Uniform continuity on a compact neighbourhood of the support proves norm continuity in \(s\). Density and the norm bound prove the same for every \(f\in C_0(E)\). Tensor with the finite-dimensional graded algebra \(C_k\), using its identity map.

We have obtained the genuine homomorphism
\(\Phi:C_0(E)\widehat\otimes C_k\to C([0,1],C_0(E)\widehat\otimes C_k)\).
Corollary 0.11 represents it by the standard interval module with operator zero; the localized square defect is \(-\Phi(f)\), a compact module endomorphism. No nondegeneracy or complemented range is needed. Its endpoints are the identity and \(e_{q_r}e_{D_rE\hookrightarrow E}\). Homotopy invariance and multiplication by \(\beta_E\) prove (150). Both \(\beta_E\) and \(\beta_{D_rE}\) are invertible by Theorem 6.0c, Corollary 6.0d and the disk isomorphism. Thus
\([e_{D_rE\hookrightarrow E}]=\beta_{D_rE}^{-1}\beta_E\)
is invertible, with all Clifford factors retained. Smaller disks give the same argument. In particular, if \(0<r<R\), cancellation after extension to \(E\) gives
\(\beta_{D_rE}[e_{D_rE\hookrightarrow D_RE}]=\beta_{D_RE}\).

**Proof: the comparison of two tubes.** Let \(\tau_0,\tau_1\) be normalized whole-bundle tubes. The map
\(g=\tau_1^{-1}\tau_0\)
is defined on a neighbourhood of the zero section, fixes it and has identity normal derivative. Let \(\lambda_t\) denote fibre multiplication by \(t\). For \(t>0\) set
\[
g_t=\lambda_{1/t}g\lambda_t,\qquad g_0=\mathrm{id}.
\tag{153}
\]
This has a smooth extension at zero, even when \(g\) changes the base point. Indeed, in bundle coordinates write \(g(x,v)=(a(x,v),b(x,v))\). Here \(a(x,0)=x\), \(b(x,0)=0\) and \(D_vb(x,0)=I\). The second component of \(g_t\) is
\[
\frac{b(x,tv)}t=\int_0^1D_vb(x,utv)v\,du,
\]
which is smooth in \((t,x,v)\) at zero and equals \(v\) there; the first component tends smoothly to \(x\). These coordinate formulas agree because the expression for positive \(t\) is intrinsic. At the zero section its derivative on tangent and normal coordinates has blocks \(\left(\begin{smallmatrix}I&tA_x\\0&I\end{smallmatrix}\right)\), hence is invertible for every \(t\).

Compactness of \(M\) supplies \(r>0\) on which every \(g_t\), \(0\le t\le1\), is defined and locally invertible. Each is injective: for \(t>0\) it is a composition of injective maps, and for zero it is the identity. By making \(r\) smaller we arrange that all their images are in a fixed disk \(D_RE\). The map
\((t,v)\mapsto(t,g_t(v))\)
is then an injective local diffeomorphism of \([0,1]\times D_rE\) onto a relatively open subset of \([0,1]\times D_RE\), and its inverse is smooth. For the endpoint parameters apply the inverse lemma to the smooth extension in \(t\); this proves the relative openness there too. The family of extensions by zero consequently defines
\[
C_0(D_rE)\widehat\otimes C_k
\longrightarrow C([0,1],C_0(D_RE)\widehat\otimes C_k).
\tag{154}
\]
To check the required norm continuity, a compact source support \(K\) has compact image under \((t,v)\mapsto(t,g_t(v))\), inside the relatively open image. The same compact-level-set boundary argument used for (152) proves joint continuity of the zero extension. Uniform continuity and density give norm continuity. Its endpoints are the disk inclusion and \(e_{g_1}\), with the constant Clifford factor unchanged.

Multiplication by \(\beta_{D_rE}\) and (150) show
\(\beta_{D_rE}[e_{g_1}]=\beta_{D_RE}\).
Since \(\tau_0=\tau_1g_1\) on \(D_rE\), subsequent extension into \(Y\) gives
\(\beta_{D_rE}[e_{\tau_0|D_rE}]=\beta_{D_RE}[e_{\tau_1|D_RE}]\).
Use (150) and composition of extensions to identify these respectively with \(\beta_E[e_{\tau_0}]\) and \(\beta_E[e_{\tau_1}]\). This proves (151). The proof transports no spin-c structure through a base-changing map: (154) acts on the scalar disk algebra and the fixed target Clifford algebra. The chosen normal spin-c module was already incorporated in \(\beta_{D_rE}\). Changing that structure by a line bundle is a change of orientation data, as in Corollary 6.0d. \(\square\)

For the next statement define an **ordered coefficient extension** as follows. If \(y\in KK(A,B\widehat\otimes C_\ell)\), let
\(y^{[k]}\in KK(A\widehat\otimes C_k,B\widehat\otimes C_k\widehat\otimes C_\ell)\)
be its exterior product with \(1_{C_k}\), with the even base algebras moved past \(C_k\). The old \(C_k\) block precedes the new \(C_\ell\) block. In a module representative the new odd operator graded-commutes with the old block. This convention is exactly the coefficient convention of (147)–(148).

**Theorem 6.0f (composition of compact spin-c embeddings).** Let \(i:M\hookrightarrow N\) and \(j:N\hookrightarrow P\) be smooth embeddings with \(M,N\) compact. Suppose their normal bundles \(E=\nu_i\) and \(F=\nu_j\) have specified spin-c structures of ranks \(k,\ell\). Give the composite's normal bundle the spin-c structure induced from
\[
0\longrightarrow E\longrightarrow\nu_{ji}
\longrightarrow i^*F\longrightarrow0,
\qquad \nu_{ji}\cong E\oplus i^*F
\tag{155}
\]
in this order, through a splitting of this exact sequence. Then
\[
i_!\widehat\otimes_{C(N)\widehat\otimes C_k}j_!^{[k]}
=(ji)_!
\quad\text{in }KK(C(M),C_0(P)\widehat\otimes C_{k+\ell}).
\tag{156}
\]
The right side uses the block identification \(C_k\widehat\otimes C_\ell\cong C_{k+\ell}\). Auxiliary metrics, connections and splittings do not affect this equality when the specified orientation is transported along their comparison paths.

**Proof: a compatible composite tube.** Take normalized whole-bundle tubes
\(\tau:E\to N\), \(\sigma:F\to P\), and put \(U=\tau(E)\). A principal connection on the spin-c frame bundle of \(F\) exists, and has smooth parallel transport along compact path intervals, by DG-FND, *Connections and parallel transport*, Theorems 3.1 and 4.1, Corollary 4.2. The complete invariant-equation argument is its local-tools lesson, Theorem 2.1 and Lemma 2.2. Apply it to
\(r\mapsto\tau(x,rv)\), \(0\le r\le1\). It gives a smooth spin-c, hence metric, identification
\(T_{x,v}:F_{i(x)}\to F_{\tau(x,v)}\), equal to the identity at \(v=0\). Smoothness holds near each finite \((x,v)\); a finite subdivision of the compact path interval suffices in each parameter neighbourhood. No uniform bound for all \(v\) is asserted or needed.

The map
\[
\Psi:E\oplus i^*F\longrightarrow F|_U,
\qquad (x,v,w)\longmapsto(\tau(x,v),T_{x,v}w)
\tag{157}
\]
is a diffeomorphism. Its inverse first applies \(\tau^{-1}\) to the base, then \(T_{x,v}^{-1}\) to the fibre. Thus \(\Omega=\sigma\Psi\) is a whole-bundle open embedding into \(P\), equal to \(ji\) at the zero section. At zero its derivative on \((v,w)\) is the sum of the \(j\)-image of the normal lift furnished by \(d\tau(v)\) and the normal lift \(d\sigma(w)\). Modulo \(TM\) this is the direct-sum isomorphism in (155); in particular the derivative of \(T_{x,v}\) contributes no term at \(w=0\). Choose this splitting first, and choose on \(\nu_{ji}\) the metric transported from the orthogonal sum. Then \(\Omega\) is a normalized tube for that normal identification. Lemma 6.0e permits its use in (151).

**Proof: open restriction of the actual Thom cycle.** We need the equality on cycles before using (148). For an arbitrary spin-c bundle \(F\to N\) and an open set \(U\subset N\), the restriction to the ideal \(C_0(U)\subset C(N)\) of the left action on \(\beta_F\) has essential module precisely its spin-c section module over \(F|_U\). To prove this assertion, multiply by approximate identities of \(C_0(U)\). On every compact support in \(F|_U\) they converge uniformly to one; finite local section approximations then give the entire module. Conversely their products are supported over \(U\). The multiplier \(c^+(w)/\sqrt{1+|w|^2}\) preserves this submodule and its adjoint does too. Its localized square defects and graded commutators are compact in that submodule: use compact-support section approximations there and the same fibre decay as Lemma 6.0a. The restriction is an \(F\)-connection on the essential module, since multiplication by an old ideal creation section intertwines the two identical multipliers exactly. Lemma 0.10 and Corollary 0.11 therefore identify the full restricted cycle with the extension of the Thom cycle over \(U\). This does not require an orthogonal complement of the open-ideal module.

Replace \(U\) by \(E\) using \(\tau\), and identify the pulled-back bundle and its spin-c section module by the transport \(T\). Both are actual isomorphisms, and \(\Psi\) is proper as a map between these two total spaces because it is a homeomorphism. The restricted Thom multiplier becomes exactly \(c_{i^*F}^+(w)/\sqrt{1+|w|^2}\), while its left action evaluates an old coefficient at \((x,v)\). Retaining the old constant \(C_k\) block gives the coefficient class
\[
\gamma\in KK(C_0(E)\widehat\otimes C_k,
 C_0(E\oplus i^*F)\widehat\otimes C_k\widehat\otimes C_\ell).
\]
Writing \(a:F|_U\hookrightarrow F\) for inclusion, the checked restriction of cycles gives
\[
[e_\tau]\,\beta_F^{[k]}
=\gamma[e_\Psi][e_a].
\tag{158}
\]
Here \(e_\Psi\) is the total-space algebra isomorphism and the last two maps retain both Clifford blocks. One can also see the module identification explicitly: a section over \(F|_U\) at \((y,w')\) is sent to its value at \((\tau(x,v),T_{x,v}w)\), using the inverse transport on its spin-c fibre. This preserves the right inner product, grading and both coefficient actions, and intertwines the displayed multiplier.

The ordered product calculation of Corollary 6.0d, (147)–(148), now gives
\[
\beta_E\gamma=\beta_{E\oplus i^*F}.
\tag{159}
\]
Its hypotheses hold on the noncompact old base \(E\); compact old-coefficient cutoffs, not a pullback along the nonproper projection \(E\to M\), were used in that calculation. The tensor of the spin-c transition modules in (159) is the sum transition module because both transition units are even. The product operator is the joint normalized numerator \(c_E^+(v)+c_{i^*F}^+(w)\); its connection errors vanish at joint infinity and its positivity is precisely (147). Thus (159) is an equality of KK classes, with the full product criterion already checked, rather than only a statement about the induced K-group maps.

Finally use associativity, (158), (159), and composition of open extensions:
\[
\begin{aligned}
i_!j_!^{[k]}
 &=\beta_E[e_\tau]\,\beta_F^{[k]}[e_\sigma]\\
 &=\beta_E\gamma[e_\Psi][e_a][e_\sigma]\\
 &=\beta_{E\oplus i^*F}[e_\Omega]=(ji)_!.
\end{aligned}
\tag{160}
\]
The maps compose in their geometric order \(\Omega=\sigma a\Psi\); no odd Clifford blocks have been exchanged.

**Proof: the auxiliary choices.** Changing a connection changes \(T\) and therefore \(\Omega\), but its value and normal derivative at the zero section remain the same. Lemma 6.0e proves equality for those tubes. Changing a metric is covered by Corollary 6.0d's full metric-interval cycle. Two splittings of (155) differ by a bundle map \(i^*F\to E\); their linear interpolation is a path of splittings. Explicitly, relative to one splitting their isomorphisms are the triangular maps \((v,w)\mapsto(v+sAw,w)\). They are invertible with inverses \((v,w)\mapsto(v-sAw,w)\), for every \(s\in[0,1]\). On the compact base their norms and inverse norms are uniformly bounded. Thus these maps induce a norm-continuous interval family of total-space algebra isomorphisms: compactly supported functions have uniformly compact transported support, and continuity and density give the assertion for all \(C_0\) functions. Transport the ordered sum metric and spin-c transition module through that family. The Clifford multiplier becomes its transported vector with the transported norm, so its localized defects vanish uniformly at fibre infinity. This is a genuine Thom-cycle homotopy over the fixed normal bundle. It identifies the induced normal orientations and their Thom classes. Tube independence then proves equality in (160) for the second splitting as well. No independence of distinct, untransported spin-c orientations is claimed. \(\square\)

The construction of an embedding's push-forward from a Thom class and an open extension is classical; [Carey–Wang, Section 4.1 and the embedding-composition lemma in Section 4.3](https://arxiv.org/abs/math/0507414) gives a twisted-K-theory formulation. The complete algebra homotopies and the actual KK product comparison required here are proved in Lemma 6.0e and Theorem 6.0f. These results concern compact ordinary embeddings; Lemma6.0t extends the tube/product argument to a noncompact intermediate base. Lemmas6.0l–n and Theorem6.0o prove the full foliation correspondence and factorization choices, and Theorems6.0s/6.0w prove projection and compact ordinary-map/foliation composition. Theorems 6.0ab–ae below prove bordism, sphere normalization and geometric-cycle descent. The longitudinal analytic-index comparison requires its further argument.

![Compression, tube independence and embedding composition](../figures/geometric-embedding-kk.png)

**Figure 6.0a.** Lemma 6.0e, Theorem 6.0f and equations (149)–(160). The top-left graph samples the exact radius-one compression on \(-3\le v\le3\) at \(s=0,1/2,1\); its compact support bound is for a function supported in \([-3,3]\). The top-right graph samples the exact scalar germ \(g(v)=v+v^2/2\) and its rescaling at \(t=0,1/2,1\) on \([-1/4,1/4]\). Its derivative is at least \(3/4\), and all images lie in \((-0.3,0.3)\); the complete proof also treats base-changing bundle germs. The middle panel uses the unwrapped cylinder chart \(-\pi<\theta<\pi\), rather than a projection of the whole cylinder. Shading indicates only the displayed height window \(|w|\le1\); the tube includes all real heights. The dot is the point \((1,0)\in S^1\times\mathbb R\), whose chart coordinates are \((0,0)\). The lower panel retains the order \(E\) then \(i^*F\), with the positive Clifford convention. Exercises 38–39 verify the support, proper-map and actual product details. Geometric prerequisites are the exact DG-CHAR and DG-FND proofs linked above; the push-forward construction is classical [Carey–Wang, Section 4].

Full-size embedding PNG · Editable embedding SVG.

### Families, stabilization and ordinary graph choice

**Lemma 6.0g (families of compact embeddings).** Let \(M\) be compact and \(Z\) second countable. Let \(i_s:M\hookrightarrow Z\), \(s\in I=[0,1]\), be a smooth family of embeddings extending to an open interval \(J\supset I\). Assume a smooth metric and spin-c structure on the rank-\(k\) family of normal quotients
\[
\mathcal E_{s,x}=T_{i_s(x)}Z/d_xi_s(T_xM)
\tag{161}
\]
over \(J\times M\); at the endpoints use its restricted orientation data. Then the classes \((i_s)_!\in KK(C(M),C_0(Z)\widehat\otimes C_k)\) from (151) are the endpoints of an actual Kasparov homotopy. In particular \((i_0)_!=(i_1)_!\). The hypothesis includes the orientation family, not merely the existence of unrelated orientations at its endpoints.

**Proof.** Choose a smooth metric on \(Z\) by the already read DG-CHAR smooth bump proof. Identify the quotient family with the orthogonal complements of \(d_xi_s(TM)\). These form a smooth bundle: the projection matrix is the matrix \(A(A^*HA)^{-1}A^*H\) from its complement theorem, applied to the full-rank smooth matrix of \(d_xi_s\). Use the compatible metric in the specified spin-c family; if necessary the positive square root of the metric-change matrix identifies it isometrically with these complements. The construction is smooth and agrees with the quotient at each zero vector.

The map \((s,x)\mapsto(s,i_s(x))\) is an embedding of \(J\times M\) in \(J\times Z\). Its derivative is injective, because the parameter coordinate retains the parameter tangent and \(d_xi_s\) is injective. It is one-to-one because each \(i_s\) is. Its image is closed: a convergent image sequence has convergent parameters and, after a subsequence, convergent points of compact \(M\), so its limit is in the image. For the same reason its inverse is continuous. The local inverse theorem applied to a full-rank coordinate projection makes this image an embedded submanifold. Thus no compactness of \(Z\) is needed.

For a tube that preserves the parameter, use the local addition \(L_Z\) already constructed in DG-CHAR, Lemma3.1 and the paragraphs preceding Theorem3.2. Define, near zero in \(\mathcal E\),
\[
\mathcal T(s,x,v)=\bigl(s,L_Z(i_s(x),v)\bigr),
\tag{162}
\]
where \(v\) is the chosen normal lift. Its differential at zero has diagonal parameter block the identity, tangent block \(d_xi_s\), and normal block the chosen quotient lift. The tangent/normal blocks span \(TZ\); hence the full differential is invertible. The local inverse theorem gives a local diffeomorphism near every zero vector.

There is one positive radius \(r\) on the compact zero section \(I\times M\) on which this map is defined, locally invertible and injective. Definition and local invertibility follow from compactness and a finite neighbourhood cover. For injectivity, if no radius worked, choose two unequal normal vectors of norms tending to zero, over parameters in \(I\), with equal images. The parameter coordinates must agree. Take convergent subsequences of their bases in \(I\times M\). Their image limits are \((s,i_s(x))\) and \((s,i_s(x'))\), so injectivity of \(i_s\) gives \(x=x'\). Both vectors are eventually in the same local-inverse neighbourhood of that zero vector, a contradiction. Shrink \(r\) once to obtain all three properties. The restriction is an open embedding onto a relatively open subset of \(I\times Z\); the open-interval extension and local inverse theorem prove this relative openness at both endpoint parameters.

A principal connection on the specified spin-c frame bundle exists by the read DG-FND Theorem3.1. Transport along \(u\mapsto(us,x)\), \(0\le u\le1\), gives smooth spin-c isometries \(V_s:\mathcal E_{0,x}\to\mathcal E_{s,x}\). The complete lifting and parameter theorem used here is its Theorem4.1 and local-tools Lemma2.2. The finite compact path interval and its finite local subdivision suffice; a global trivialization of the frame bundle on \(M\) is not assumed.

Let \(E_0=\mathcal E|_{\{0\}\times M}\), and use the normalized compression \(q_r\) from (149). The map
\[
\widetilde\tau:I\times E_0\longrightarrow I\times Z,
\qquad(s,x,v)\longmapsto\mathcal T(s,x,V_s q_r(v))
\tag{163}
\]
is an open embedding preserving \(s\). Its inverse is obtained from the disk inverse, \(V_s^{-1}\), and \(q_r^{-1}\). Each fibre is a normalized whole-bundle tube when the normal bundle is identified by \(V_s\). The compact-image argument of (154) gives the norm-continuous homomorphism
\[
\Theta:C_0(E_0)\widehat\otimes C_k
\longrightarrow C(I,C_0(Z)\widehat\otimes C_k),
\qquad\Theta(a)(s)=e_{\widetilde\tau_s}(a).
\tag{164}
\]
For completeness, a compact source level set has compact image in the relatively open image of \(\widetilde\tau\). Values of the zero extension bounded away from zero therefore cannot approach its boundary. This proves joint continuity, while its uniformly compact support gives uniform norm continuity. Approximation and the norm bound prove the assertion for every \(C_0\) coefficient. The fixed Clifford factor is unchanged.

The actual spin-c transition module and normalized multiplier of \(\beta_{E_0}\), tensored with \([\Theta]\), give a Kasparov class over \(C(I,C_0(Z)\widehat\otimes C_k)\), with left algebra \(C(M)\). At \(s\) the isometry \(V_s\) preserves the spin-c module inner product, grading, transition units and the Clifford multiplier. Hence its endpoint is exactly \(\beta_{\mathcal E_s}[e_{\mathcal T_s q_r}]\). Corollary0.11 justifies the pushforward by the homomorphism class also for its possibly degenerate range. Lemma6.0e identifies this with \((i_s)_!\) defined using any normalized tube at that endpoint. This proves the claimed whole-module homotopy. A different connection or metric only changes its representative; the endpoints retain their specified spin-c structures. \(\square\)

#### Euclidean stabilization at a noncompact ambient manifold

Write \(B_{2a}\in KK(\mathbb C,C_0(\mathbb R^{2a}))\) and \(A_{2a}\in KK(C_0(\mathbb R^{2a}),\mathbb C)\) for the even Bott class and its inverse after the fixed positive-Clifford, chirality-preserving Morita contraction. Here this is the geometric pair from the standard spin-c symbol and its two inverse products in Corollary6.0d. Lemmas7.20–7.21 below calibrate its analytical inverse explicitly; no separate identification with the real-action pair of Corollary0.20 is assumed. At \(a=0\) both are the canonical unit. Write \(B_{2a}^{D}=1_D\boxtimes B_{2a}\) and \(A_{2a}^{D}=1_D\boxtimes A_{2a}\), keeping the old algebra \(D\) before the new Euclidean block.

**Lemma 6.0h (even Euclidean stabilization).** Let \(i:M\hookrightarrow Z\) be a compact embedding with specified spin-c normal \(E\) of rank \(k\), where \(Z\) may be noncompact. Let \(h:Z\hookrightarrow Z\times\mathbb R^{2a}\) be the zero section. Orient the composite normal as \(E\oplus\mathbb R^{2a}\), in that order, with the canonical complex spin-c structure on the new summand. Contract only its last positive-Clifford \(C_{2a}\) block by the canonical Morita module. The resulting class, denoted \((hi)_!^{\rm contracted}\), satisfies
\[
(hi)_!^{\rm contracted}=i_!B_{2a}^{C_0(Z)\widehat\otimes C_k},
\qquad
(hi)_!^{\rm contracted}A_{2a}^{C_0(Z)\widehat\otimes C_k}=i_!.
\tag{165}
\]
These are actual KK equalities. They do not assume that the noncompact embedding \(h\) lies in the compact-middle-manifold scope of Theorem6.0f.

**Proof.** Let \(\tau:E\to Z\) be a normalized whole-bundle tube. Then
\(\Omega(x,v,w)=(\tau(x,v),w)\)
is a normalized whole-bundle tube for \(hi\), with normal identification \(E\oplus\mathbb R^{2a}\). Its derivative is the sum of the normal derivative of \(\tau\) and the new identity block. Since \(\tau\) is a diffeomorphism to an open set, so is \(\Omega\) to its open product with Euclidean space.

Pulling the new Euclidean standard Thom module back along the open ideal \(C_0(\tau(E))\subset C_0(Z)\) gives exactly the standard module over \(E\times\mathbb R^{2a}\), extended by the tube. To check this on the possibly noncompact \(Z\), its essential module is the closure of those ideal coefficients. Compact-support sections are dense in it. The new multiplier commutes in the graded sense with the old constant Clifford block, preserves that module and has localized compact defects there. Its connection with each ideal creation coefficient is exact. Thus the essential-module lemma0.10 and homomorphism functoriality0.11 give the same restriction identity as (158), with trivial radial transport. No projection from \(E\) onto \(M\) is used as a \(C_0\) homomorphism.

Before contracting the new block, Corollary6.0d's actual product is the spin-c module of \(E\oplus\mathbb R^{2a}\), with normalized numerator
\[
\frac{c_E^+(v)+c_{\mathbb R^{2a}}^+(w)}{\sqrt{1+|v|^2+|w|^2}}.
\tag{166}
\]
The connection and positivity are exactly (147): a compact old creation section bounds \(v\), the errors vanish as \(|w|\) tends to infinity there, and approximation handles its old tails. The old anticommutator is nonnegative. The old Clifford block precedes the new block throughout. Extending this checked product along \(\Omega\) proves
\((hi)_!=i_!(1_{C_0(Z)\widehat\otimes C_k}\boxtimes\beta_{\mathbb R^{2a}})\)
with the extra \(C_{2a}\) block retained. Tensoring with its finite graded Morita module is the first equality claimed. The second follows by associativity and the full inverse equality \(B_{2a}A_{2a}=1\), with the old coefficient algebra retained. The inverse is even, so no odd old Clifford factor is exchanged with an odd inverse. All modules and coefficient actions are defined over the entire noncompact ambient algebra, while only compact old source supports enter the product estimates. \(\square\)

#### Comparing two graph embeddings with coherent normal orientations

Let \(f:M\to Y\) be smooth, with \(M\) compact. Choose Euclidean embeddings \(u_b:M\hookrightarrow\mathbb R^{2n_b}\), \(b=0,1\). Their graph embeddings are \(I_b=(f,u_b)\). Consider the family in one common ambient space,
\[
I_\theta(x)=\bigl(f(x),\cos\theta\,u_0(x),\sin\theta\,u_1(x)\bigr),
\qquad0\le\theta\le\pi/2.
\tag{167}
\]
Every \(I_\theta\) is an embedding. At least one of its two displayed coefficients is nonzero, so equality of its values implies equality under the corresponding \(u_b\), hence equality of points. The same argument applied to its derivative proves immersion. Compactness gives a homeomorphism onto its image, and the local inverse theorem applied to a full-rank coordinate projection makes that image a submanifold. These arguments hold on an open interval beyond the two endpoints. In particular this is a smooth family in the exact scope of Lemma6.0g.

**Proposition 6.0i (ordinary graph choice with coherent orientations).** Suppose the normal spin-c data for \(I_b\), their even stabilizations, and the above family are supplied coherently: the family structures at zero and \(\pi/2\) are the ordered stabilized endpoint structures, transported through the indicated Euclidean block permutation at the second endpoint. Apply the fixed ordered matrix-Clifford contractions to retain \(C_p\), where \(p\in\{0,1\}\) is the parity of \(\dim Y-\dim M\). Then the two ordinary graph push-forward classes
\[
\mathcal I_b=
\bigl(I_b{}_!\text{ with normal target contracted to }C_p\bigr)
A_{2n_b}^{C_0(Y)\widehat\otimes C_p}
\in KK(C(M),C_0(Y)\widehat\otimes C_p)
\tag{168}
\]
are equal. Lemma6.0j below constructs this normal family from convention (13); Lemmas6.0l–n and Theorem6.0o construct and compare the full foliation correspondence.

**Proof.** Stabilize \(I_0\) by the zero \(\mathbb R^{2n_1}\) block and stabilize \(I_1\) by the zero \(\mathbb R^{2n_0}\) block. Lemma6.0h supplies the actual ordered product with the new even Bott class in each case. The compatible normal Morita contractions can be composed rather than separately selected: write \(k=p+2t\). The map
\[
\Lambda^*\mathbb C^t\widehat\otimes\Lambda^*\mathbb C^a
\longrightarrow\Lambda^*\mathbb C^{t+a},\qquad\xi\otimes\eta\longmapsto\xi\wedge\eta
\tag{169}
\]
is an even unitary when the first block precedes the second. Indeed the ordered wedge bases on both sides are orthonormal, their total degrees agree, and creation by a vector in the second block introduces exactly the first-block degree sign. Contraction has the adjoint sign. Thus the two Clifford actions and the graded inner products agree, so this is the precise unitary identifying the composed finite Morita modules with the one larger canonical module. The old \(C_p\) factor is retained before them. For \(p=1\), the convention in Section 0 initially retains the final Clifford generator. Move it to the front by the even cyclic unitaries of Lemma 0.13 on successive three-generator blocks: its last generator moves past each preceding pair, in order. The resulting real permutation is the cycle of all \(2t+1\) generators, of positive sign. Each local unitary intertwines the two finite Morita reductions and retains the same normalized odd central volume. Their composition therefore identifies that convention with \(C_1\widehat\otimes C_{2t}\), through actual even module maps. In even rank there is no retained odd block. These fixed maps, rather than an ungraded permutation, are used in the displayed wedge unitary and in the contractions below. Consequently the stabilized graph class, after contraction, is the old contracted graph class followed by \(B_{\rm new}^{C_0(Y\times\mathbb R^{2n_b})\widehat\otimes C_p}\).

The inverse for the two ordered Euclidean blocks is the inverse for the new block followed by the inverse for the old block. This is the inverse composition in (148), transferred through these same finite Morita maps. Composing the stabilized graph class with that inverse cancels the adjacent new Bott pair, giving precisely \(\mathcal I_b\). This proves stabilization invariance as an actual KK equality, not merely by equality of integer indices.

At the second endpoint the stabilized graph places \(u_1\) in the second Euclidean block. Move it to the first block by the permutation \(\mathbb R^{2n_0}\oplus\mathbb R^{2n_1}\to\mathbb R^{2n_1}\oplus\mathbb R^{2n_0}\). This is the permutation of two complex coordinate blocks, after the fixed identification with \(\mathbb C^{n_0}\oplus\mathbb C^{n_1}\). Its lift on the canonical Bott spinor module is the exterior action of this complex unitary. That action is even, preserves the wedge inner product, and intertwines creation and contraction; hence it intertwines the full positive-Clifford Bott multiplier with the coordinate permutation. Extension along the permutation therefore preserves the even Bott class. Its actual inverse is preserved as well, by multiplying this equality on each side by the two inverse classes. The identity on \(Y\) and the retained \(C_p\) factor do not change this argument. In particular no unaccounted odd-block sign occurs.

With the prescribed coherent normal family, Lemma6.0g identifies the two stabilized graph embedding classes in their common ambient space. Compose that equality with the common ordered inverse Bott class. The stabilization cancellations just proved identify its endpoints with \(\mathcal I_0\) and \(\mathcal I_1\), respectively. This gives the stated equality. The proof uses the supplied coherent normal structures at the family endpoints; it cannot infer their coherence from unrelated existence statements. \(\square\)

### The specified stable orientation gives a normal family

The finite module construction uses the complete exterior matrix-unit proof in the [index lesson, Lemma 6.17](the-index-theorem-for-measured-foliations.md#section-6), its chirality convention in Lemma 6.24, the two Morita evaluations in Lemma 0.12, and the transition module in Corollary 6.0d. The Clifford/spinor viewpoint is classical [van Suijlekom, Chapter 4]; the cancellation and parameter comparison are proved here.

All the bundles below are real metric bundles; all Clifford algebras are complex and positively generated. Tensor products of module bundles are graded and balanced over the common base. Write
\[
A_V=\mathrm{Cl}^+(V),\qquad C_r=\mathrm{Cl}^+(\mathbb R^r).
\tag{170}
\]
These denote finite algebra bundles in this subsection; their section algebras give the corresponding Hilbert modules. We distinguish a base module \(\mathscr M_V:A_V\rightsquigarrow C_r\) from its pullback to the total space in (144). An arrow here means a smooth finite full imprimitivity module, with specified grading and both actions, not just its KK class.

#### The double bundle and finite cancellation

Let \(W\) have rank \(d\). The bundle \(Q=W\oplus W\) has the canonical complex structure \((u,v)\mapsto u+iv\) on \(W_{\mathbb C}\). Put \(S_Q=\Lambda^*W_{\mathbb C}\), with exterior-degree grading. Its positive Clifford action is
\[
\varepsilon_Q(u,v)=e(u)+\iota(u)+i\bigl(e(v)-\iota(v)\bigr).
\tag{171}
\]
Both real summands act by odd self-adjoint operators. The exterior relations give their squares \(|u|^2\), \(|v|^2\), and their mutual anticommutation. They therefore define a graded *-map \(A_Q\to\operatorname{End}(S_Q)\). In every orthonormal frame it is onto: \(e_i,\iota_i\) are linear combinations of these generators, the product \(\prod_i(1-e_i\iota_i)\) projects onto the exterior vacuum, and products with exterior multiplications and contractions give all matrix units. There are at most \(2^{2d}\) increasing Clifford words and exactly that many matrix dimensions, so it is an isomorphism. This is precisely Lemma6.17's complete matrix-unit proof with its second negative action multiplied by \(i\). It is orthogonally equivariant and hence glues even when \(W\) is nonorientable.

Thus \(S_Q:A_Q\rightsquigarrow\mathbb C\) is an actual smooth Morita module. Its two evaluations are the rank-one endomorphism and inner-product maps of Lemma0.12. For a local wedge frame the first spans every matrix unit and the second contains the scalar unit; partitions and base cutoffs prove fullness on section algebras. The maps are even isometries by the balanced tensor inner-product formula, and hence are onto. Let \(S_d=\Lambda^*\mathbb C^d:C_{2d}\rightsquigarrow\mathbb C\) be the same fixed positive model on the trivial double space. The canonical spin-c transition module of \(Q\) is
\[
\mathscr M_Q=S_Q\widehat\otimes S_d^*:A_Q\rightsquigarrow C_{2d}.
\tag{172}
\]
The standard generators of \(C_{2d}\) act on \(S_d\) in consecutive complex pairs: \(\varepsilon_{2i-1}=e_i+\iota_i\), \(\varepsilon_{2i}=i(e_i-\iota_i)\). Thus \((-i)^d\varepsilon_1\cdots\varepsilon_{2d}\) is exactly its exterior parity. Indeed an orthogonal transition \(R\) of \(W\) acts on \(S_Q\) by the even exterior unitary \(\Lambda R\), intertwining both positive actions. The resulting transition on \(S_Q\otimes S_d^*\), identified locally with \(C_{2d}=\operatorname{End}(S_d)\), is left multiplication by \(\Lambda R\). It implements \(R\oplus R\) on the real generators in their interleaved order; its identification as a spin-c transition is verified below. For \(d=0\), all these modules and evaluations are the scalar unit.

Suppose \(E\) has rank \(k\) and a smooth full Morita module
\(P:A_E\widehat\otimes A_Q\rightsquigarrow C_k\widehat\otimes C_{2d}\)
is specified. Cancel the two matrix factors by the actual module
\[
\mathcal C_Q(P)=
\bigl(A_E\widehat\otimes S_Q^*\bigr)
\widehat\otimes_{A_E\widehat\otimes A_Q}P
\widehat\otimes_{C_k\widehat\otimes C_{2d}}
\bigl(C_k\widehat\otimes S_d\bigr)
:A_E\rightsquigarrow C_k.
\tag{173}
\]
All factors are finite full imprimitivity modules. Composing the two evaluations gives the evaluations of this tensor, so its left compact algebra is exactly \(A_E\) and it is right full. Locally these tensors are the usual finite balanced tensors of fixed matrix modules; their kernels, inner products and actions vary smoothly. Consequently they form a smooth module bundle, rather than only a fibrewise assignment.

Cancellation has the following exact recovery property:
\[
\mathcal C_Q(P)\widehat\otimes\mathscr M_Q\ \cong\ P,
\qquad
\mathcal C_Q(L\widehat\otimes\mathscr M_Q)\ \cong\ L.
\tag{174}
\]
Here the first arrow has left algebra \(A_E\widehat\otimes A_Q\) and right algebra \(C_k\widehat\otimes C_{2d}\); the second has the original left and right algebras of \(L\). To construct the first map, move the exterior \(S_Q\) next to the cancelling \(S_Q^*\) factor and \(S_d^*\) next to the cancelling \(S_d\) factor, preserving the intervening algebra actions by the graded flips. Then apply \(S_Q\otimes S_Q^*\to A_Q\) and \(S_d\otimes S_d^*\to C_{2d}\), which act on \(P\). For the second use their reverse inner-product evaluations. Every flip of two homogeneous factors \(x,y\) contributes \((-1)^{|x||y|}\); tensor reassociation contributes no sign. The defining identity
\(\langle x\otimes y,x'\otimes y'\rangle=\langle y,\langle x,x'\rangle y'\rangle\)
then shows that these maps preserve inner products. The evaluations are onto, so the constructed isometries are onto. They preserve both actions and grading. This also proves that an intertwiner after tensoring with \(\mathscr M_Q\) can be cancelled to an actual intertwiner before tensoring: apply \(\mathcal C_Q\) to it and the second recovery unitary. No cohomological classification or cancellation of mere characteristic classes is being used.

The orientation of \(Q\) here is the complex orientation with interleaved pairs \((u_1,v_1,\ldots,u_d,v_d)\). Relative to the two-block list \((u_1,\ldots,u_d,v_1,\ldots,v_d)\), its sign is \((-1)^{d(d-1)/2}\). The exterior grading fixes this sign. We do not replace it by a block-orientation convention during cancellation.

#### Recovering a spin-c structure from these modules

We will use cancellation on positive normal ranks. It is useful to check that its result gives exactly the principal spin-c structure needed in Lemma6.0g. Let \(k\ge1\), and let \(L\colon A_E\rightsquigarrow C_k\) be a smooth graded finite imprimitivity module. Locally it can be written as the right regular \(C_k\) module, with its left action a graded *-isomorphism \(A_E\to C_k\). Here is a direct check, including the possible orientation sign.

For \(k=2r\), the exterior matrix-unit model identifies \(C_k\) with \(\operatorname{End}(\Lambda^*\mathbb C^r)\). Its grading is implemented by the exterior parity matrix \(\Gamma\), whose two eigenspaces have equal dimension when \(k>0\). A right module over this full matrix algebra is recovered from any matrix corner by applying the matrix units to that corner; the identities \(e_{ab}e_{cd}=\delta_{bc}e_{ad}\) give the mutually inverse reconstruction maps. Thus the Morita module has the form \(\operatorname{Hom}(\Lambda^*\mathbb C^r,V)\), with \(\operatorname{End}(V)\cong A_E\). Its complex dimension is \(2^r\). The odd invertible left Clifford generator interchanges the two grading eigenspaces of \(V\); they consequently both have dimension \(2^{r-1}\). An even unitary \(V\cong\Lambda^*\mathbb C^r\) now identifies this right module with the regular module. Matrix corner frames and their positive Gram normalization can be chosen smoothly locally. This proves the asserted smooth local identification.

For \(k=2r+1\), the \(2r\) elements \(i\varepsilon_i\varepsilon_k\), \(1\le i<k\), are self-adjoint positive Clifford generators of the even part. They generate it because their products include every pair \(\varepsilon_i\varepsilon_j\), and every even word is a product of pairs. The even-part word count and the already proved even-rank matrix model therefore identify that part with \(M_{2^r}(\mathbb C)\). The odd central volume \(z=(-i)^r\varepsilon_1\cdots\varepsilon_k\), with \(z^*=z\), \(z^2=1\), gives
\(C_k\cong M_{2^r}(\mathbb C)\oplus M_{2^r}(\mathbb C)\)
as an ungraded algebra. The grading exchanges the two summands. The two right matrix corners of a Morita module have equal multiplicity because its grading interchanges them; the left compact algebra gives multiplicity \(2^r\) on each. The same matrix-unit reconstruction, with one corner frame and its graded image, gives an even smooth right-module identification with \(C_k\). This includes \(k=1\), where the corners are scalar. These constructions use finite kernels and Gram matrices on a neighbourhood; they make no global trivialization assertion.

In such a right-module frame, the left action is a graded *-isomorphism. For even \(k\), it sends the oriented source chirality to \(\Gamma\) or \(-\Gamma\). Indeed both implement the same grading, so their ratio commutes with the full matrix algebra and is scalar, and their squares are one. For odd \(k\), it sends the source normalized odd central volume to \(z\) or \(-z\): the odd centre is one dimensional, and the involution and square give the two signs. Reversing a real frame reverses these signs. They therefore specify a smooth orientation of \(E\). Use oriented frames matching that sign.

With this choice, the left action in a real orthonormal frame differs from the standard Clifford action by conjugation with an even unitary. In even rank, a matrix-algebra *-automorphism is unitary conjugation: choose a unit vector in the image of its first diagonal matrix unit, apply its other matrix units to obtain the whole orthonormal basis, and use the resulting unitary. Because it preserves \(\Gamma\), this unitary commutes with \(\Gamma\) and is even. In odd rank apply this argument to the even full matrix algebra. Its implementing unitary lies in \(C_k^0\); agreement on the odd central volume then gives agreement on the odd part too. The linear equations for these intertwiners have a one-dimensional complex solution space: two of them differ by the even centre, which is scalar. This even-centre claim also follows directly by commuting an increasing word with all generators: only the empty word survives in even degree. The kernels of these constant-rank smooth linear equations form a smooth line bundle; local nonzero sections can be positively normalized to unitaries. We have therefore produced smooth local right-module frames in which the real left generators are exactly the chosen Clifford frame.

Two such frames are related by an even unitary \(u\in C_k\) implementing an oriented real frame change \(R\in\mathrm{SO}(k)\). Its group of possibilities is exactly \(\mathrm{Spin}^c(k)\). To verify this rather than assume it, a product of an even number of positive unit vectors implements the product of their hyperplane reflections; conjugation by one vector is minus that hyperplane reflection, so the two minus signs cancel in an even product. Every orthogonal transformation is a product of hyperplane reflections: reflect its first unit column to the first basis vector and continue on the orthogonal complement. If its determinant is positive, the number of reflections is even, after cancelling repeated pairs if necessary. Thus \(R\) has an even spin lift \(s\). The unit \(s^{-1}u\) commutes with all real generators and is even, hence is a scalar \(\lambda\in U(1)\). So \(u=\lambda s\), with precisely the identification \((s,\lambda)\sim(-s,-\lambda)\). Local lifts are smooth: near the identity, the convergent matrix logarithm is skew symmetric, and exponentiating its corresponding bivector gives a spin lift; translate these neighbourhoods for other \(R\). The normalized intertwiner lines therefore give a smooth principal spin-c bundle. Its transition module is exactly \(L\), with its line information retained. This argument also identifies \(\mathscr M_Q\) with the canonical complex spin-c module of \(Q\).

In rank zero a graded Morita line can also be odd; the canonical rank-zero spin-c unit in Corollary6.0d is the even line. We avoid confusing these conventions by choosing the Euclidean embedding dimension large enough that the normal rank is positive. Such an even stabilization is always available and Lemma6.0h proves its effect. Nothing above declares a possibly odd rank-zero line to be the canonical unit.

#### The normal orientation and its full parameter family

**Lemma 6.0j (specified stable orientation, normal family and stabilization).** Let \(M\) be compact. Let \(W=TM\), let \(T\to M\) have rank \(q\), and prescribe its full stable Clifford orientation module on \(L=W\oplus T\), as in (13). In positive rank this is the specified spin-c structure; in rank zero its line grading is retained. Let \(j:M\hookrightarrow\mathbb R^n\) be an embedding, where \(n\) is even and \(k=n-d+q\ge1\). Then the convention in (13), using the canonical complex double \(W\oplus W\), gives an actual specified spin-c structure on
\(E_j=\nu_j\oplus T\)
of rank \(k\). It is a smooth family whenever \(j\) is a smooth embedding family. Under even Euclidean stabilization it is the ordered sum of the old structure and the canonical new Euclidean structure, after the stated normal-block permutation. Tensoring the original orientation by a complex line tensors the constructed normal orientation by precisely that line.

**Proof: the actual construction.** Fix a metric \(g\) on \(W\), retain a metric on \(T\), and take the standard metric on \(\mathbb R^n\). Transport the prescribed full module on \(L\) to this orthogonal sum metric by Corollary6.0d's positive metric isometry. Define \(G_j\) by
\(\langle dju,djv\rangle=g(G_ju,v)\).
It is a smooth positive invertible endomorphism. Its inverse square root is smooth: locally scale it by a positive constant so that the spectrum lies in a compact subinterval of \((0,2)\), and use the uniformly convergent binomial series for \((1-(1-G))^{-1/2}\); every derivative converges on a smaller neighbourhood. The derivative assertion follows from the polynomial growth in the series index of each fixed derivative and the geometric bound on \(1-G\). This is the same positive metric transport used in Corollary6.0d.

Let \(\nu_j\) be the orthogonal normal of \(djW\). The isometry
\[
J_j:W\oplus\nu_j\longrightarrow\underline{\mathbb R^n},
\qquad J_j(v,\eta)=dj(G_j^{-1/2}v)+\eta
\tag{175}
\]
is onto, since its two images are the orthogonal tangent and normal complements. Consider the ordered isometry
\[
\rho_j:E_j\oplus Q\longrightarrow L\oplus\underline{\mathbb R^n},
\qquad(\eta,t,u,v)\longmapsto(u,t,J_j(v,\eta)).
\tag{176}
\]
Let \(\mathscr M_L:A_L\rightsquigarrow C_{d+q}\) be the prescribed module of Corollary6.0d. Pull back its tensor with the canonical right regular Euclidean module along \(\rho_j\):
\[
P_j=\rho_j^*(\mathscr M_L\widehat\otimes C_n)
:A_{E_j}\widehat\otimes A_Q\rightsquigarrow C_{d+q+n}
=C_k\widehat\otimes C_{2d}.
\tag{177}
\]
The last equality is the ordered constant-Clifford generator identification, not an ungraded matrix equivalence. Define
\[
\mathscr M_{E_j}=\mathcal C_Q(P_j):A_{E_j}\rightsquigarrow C_k.
\tag{178}
\]
The full evaluations and recovery unitaries above prove that it is an actual finite orientation module. The positive-rank reconstruction gives its principal spin-c structure. This is the explicit meaning of converting the prescribed structure on \(W\oplus T\) to the virtual orientation \(T-W\), then using \(W\oplus\nu_j\cong\underline{\mathbb R^n}\). It carries the fixed double-bundle convention, including its orientation sign, throughout. No independent normal orientation or determinant-line trivialization is selected.

For a smooth family \(j_s\) on an open interval, the orthogonal normal projection is smooth by the complement theorem already used in Lemma6.0g. The positive matrices \(G_{j_s}\), their square roots, \(J_{j_s}\), \(\rho_{j_s}\), and the finite module tensors above are all smooth on the parameter/base bundle. Thus \(\mathscr M_{E_{j_s}}\), and its reconstructed principal spin-c structure, is a genuine family. At each endpoint it is exactly the construction for that endpoint embedding. Changes of the metrics use their positive interpolation and the same formulas. The underlying quotient orientation is transported along this full family; it is not replaced by an unrelated endpoint choice.

**Proof: ordered stabilization.** Let \(R=\underline{\mathbb R^{2a}}\) with its canonical complex spin-c structure and put \(j'=(j,0)\). Its Euclidean normal is \(\nu_j\oplus R\). Thus the displayed construction initially orders \(E_{j'}\) as \((\nu_j,R,T)\). Reorder it to \((\nu_j,T,R)=E_j\oplus R\) by the real isometry \(\psi\). Let \(\chi\) move this new \(R\) block past \(Q\). On vectors the square is exact:
\[
\rho_{j'}(\psi\oplus1_Q)=(\rho_j\oplus1_R)\chi,
\qquad
(\eta,t,w,u,v)\longmapsto(u,t,J_j(v,\eta),w).
\tag{179}
\]
Here \(\psi(\eta,t,w)=(\eta,w,t)\); \(\chi(\eta,t,w,u,v)=(\eta,t,u,v,w)\). Clifford extension respects this equality by the universal relations. Initially the right blocks of \(P_j\widehat\otimes C_{2a}\) are \((C_k,C_{2d},C_{2a})\); cancellation for the stabilized rank uses \((C_k,C_{2a},C_{2d})\). The intervening constant block move has an actual even inner implementer \(w\): on \(\Lambda^*\mathbb C^{d+a}\) use \(\Lambda\sigma\), where \(\sigma\) permutes those two complex blocks, and retain the identity on \(C_k\). It is even, is unitary on wedge bases, and intertwines exterior multiplication and its adjoint. It therefore implements the indicated constant Clifford isomorphism \(\beta=\operatorname{Ad}w\) and preserves the chirality used in the contractions. Changing a right coefficient action from \(p a\) to \(p\beta(a)\) changes its inner product to \(\beta^{-1}\langle p,p'\rangle\); the explicit map \(p\mapsto p w\) is an even unitary from this changed module to the original one, since \(\langle pw,p'w\rangle=w^*\langle p,p'\rangle w\). Thus the right reordering is an actual module transport, not an identification by dimension. Combining it with the vector square identifies the stabilized \(P_{j'}\) with \(P_j\widehat\otimes C_{2a}\), with the new block before \(Q\) on the left and before \(C_{2d}\) on the right.

Now tensor \(\mathscr M_{E_j}\widehat\otimes C_{2a}\) with \(\mathscr M_Q\), with the new real normal block before \(Q\). Move its constant Clifford block before the right \(C_{2d}\) block by this fixed isomorphism. The recovery unitary identifies the result with the just described \(P_{j'}\). Apply \(\mathcal C_Q\) and its second recovery unitary to this equality. The result is the actual even bimodule unitary
\(\mathscr M_{E_{j'}}\cong\mathscr M_{E_j}\widehat\otimes C_{2a}\),
after \(\psi\). It intertwines the two left Clifford bundles and the ordered right \(C_k\widehat\otimes C_{2a}\) action. Its maps consist exactly of reassociations, the graded flips, the fixed \(\Lambda\sigma\), and the two rank-one/inner-product evaluations already checked. This gives both the inner-product preservation and surjectivity; it is not a uniqueness assertion about only the KK classes. In particular the new block's even rank does not permit omission of \((-1)^{|x||y|}\) for two odd module vectors.

**Proof: retaining a line twist.** If \(H\) is a complex line, regarded as even, replace \(\mathscr M_L\) by \(\mathscr M_L\otimes H\). Then \(P_j\) is replaced by \(P_j\otimes H\). The cancellation formula moves only Clifford/spinor factors, so its result is exactly
\[
\mathscr M_{E_j}^{\,H}\cong\mathscr M_{E_j}\otimes H.
\tag{180}
\]
Consequently the normal Thom cycle is tensored by the pulled-back \(H\). On principal spin-c transition units this multiplies the scalar phase by the transition of \(H\); the determinant transition is its square. The determinant line therefore changes by \(H^{\otimes2}\), while the Thom module changes by \(H\), exactly as Corollary6.0d requires.

The determinant contribution of the double bundle can be stated exactly. Its exterior lift of a complex unitary \(U\) has determinant character \(\det_{\mathbb C}U\). For one complex coordinate with phase \(e^{i\theta}\), the exterior unit is \(\operatorname{diag}(1,e^{i\theta})\), the spin lift is \(\operatorname{diag}(e^{-i\theta/2},e^{i\theta/2})\), and their scalar ratio has square \(e^{i\theta}\). Diagonalizing a unitary in an orthonormal complex basis and multiplying the coordinate formulas proves the general assertion. For \(U=R\in O(d)\) this character is the transition of \(\det W_{\mathbb C}\). The recovery unitary identifies the ordered sum spin-c structure with that on \(L\oplus\mathbb R^n\), whose Euclidean determinant line is trivial. Multiplication of the scalar transition characters therefore gives
\(D_{E_j}\otimes\det W_{\mathbb C}\cong D_L\).
The possibly nontrivial double-bundle determinant has been cancelled by its actual module. This relation alone would lose a two-torsion line twist; the full construction above retains it. This proves all assertions. \(\square\)

#### Discharging the ordinary graph-family hypothesis

**Corollary 6.0k (ordinary graph choice with the prescribed orientation).** Let \(f:M\to Y\) be smooth, \(M\) compact, and prescribe a spin-c structure on \(TM\oplus f^*TY\) using convention(13). Choose even Euclidean embeddings with positive graph normal ranks, and orient their graph normals by Lemma6.0j. Then Proposition6.0i applies without an extra normal-family assumption: its two contracted ordinary graph push-forwards \(\mathcal I_b\) are equal. The prescribed spin-c orientation remains part of this equality.

**Proof.** Set \(T=f^*TY\). For \(I=(f,j)\), the map
\((\eta,t)\mapsto[(t,\eta)]\)
identifies \(\nu_j\oplus T\) with \(I^*T(Y\times\mathbb R^n)/dI(TM)\). It is one-to-one: if \((t,\eta)=(dfv,djv)\) with \(\eta\perp dj(TM)\), then \(v=0\). It is onto: subtract \((dfv,djv)\) from any \((t,z)\), decomposing \(z=djv+\eta\). These are smooth quotient isomorphisms in an embedding family. Use the transported metric and orientation of Lemma6.0j.

Apply that lemma to the Euclidean embedding family
\(j_\theta=(\cos\theta\,u_0,\sin\theta\,u_1)\)
from Proposition6.0i. Its given bundle \(T\) and given module \(\mathscr M_L\) are pulled back unchanged from \(M\). The construction therefore gives the exact smooth normal family for the graph family. At zero, the stabilization square in Lemma6.0j gives the ordered old normal followed by the new \(\mathbb R^{2n_1}\) block.

At \(\pi/2\), apply the ambient permutation of the two complex Euclidean blocks. The construction is natural under that permutation: it carries the Euclidean normal projection and \(J_j\) to those for the permuted embedding, and the exterior unitary \(\Lambda\sigma\) intertwines the canonical Euclidean Clifford module on the right. All the fixed \(L,Q\) data are retained. Cancelling \(\mathscr M_Q\) with the recovery maps proves the corresponding equality of the normal modules. The second endpoint is therefore precisely the ordered stabilization of \(u_1\), transported through the block permutation specified in Proposition6.0i. Its full module, including any line twist, matches; no independent orientation is inferred just from existence. Lemma6.0g supplies the embedding homotopy, Lemma6.0h supplies both ordered stabilization equalities, and Proposition6.0i's checked even Bott cancellation gives the claimed equality. \(\square\)

The ordinary graph comparison now has its specified normal family. Lemmas6.0l–n and Theorem6.0o below construct the full holonomy-germ microbundle and parameter étale correspondence and prove their factorization-choice and homotopy comparisons. Theorems6.0s/6.0w below prove projection and compact ordinary-map/foliation composition. Theorems 6.0ab–ae below prove the full bordism and sphere-modification relations.

![Coherent normal orientation, finite cancellation and even stabilization](../figures/stable-normal-orientation.png)

**Figure 6.0b.** Lemmas 6.0g–h and 6.0j, Proposition 6.0i and Corollary 6.0k, equations (161)–(180), Exercises 40–41. Each module arrow has its full left and right Clifford algebras shown; a star denotes the conjugate Morita module. The maps recover the specified stable orientation, including its line twist, through the two actual evaluation maps. The double-bundle orientation is interleaved, with its exact sign relative to block order. The family parameter is normalized to \(I=[0,1]\) by \(s=2\theta/\pi\); no projected picture of the high-dimensional embedding is used. In the stabilization equalities the new even Clifford block has been contracted and the old coefficient algebra is retained. The determinant relation alone does not detect the two-torsion twist of Exercise 40. The Clifford/spinor method is classical [van Suijlekom, Chapter 4]; the complete finite-module and parameter comparisons are supplied above.

Full-size orientation PNG · Editable orientation SVG.

Let \(M\) be compact and let \(f:M\to V/F\) be K-oriented. Choose a smooth embedding \(j:M\hookrightarrow\mathbb R^n\), with normal bundle \(\nu_j\). The aim is to factor \(f\) into maps for which a correspondence is explicit:

\[
M\xrightarrow{k}N\xrightarrow{g}(V/F)\times\mathbb R^n
\xrightarrow{\mathrm{pr}}V/F.
\tag{14}
\]

Locally choose transverse coordinates \(h_a:U_a\to\mathbb R^q\) for \(f_a\). Their holonomy transition germs glue neighbourhoods of their graphs into a rank-\(q\) foliated microbundle \(H\) over \(M\), with section \(x\mapsto(x,h_a(x))\). The normal bundle of this section is \(f^*\tau\). Lemma6.0l below constructs actual representatives on one common shrink, proves their triple identities, and proves Hausdorffness using a closed relation with an open quotient. It also glues actual holonomy arrows; pointwise germ representatives alone would not establish these assertions.

Take a neighbourhood \(N\) of the zero section over this section in the pullback of \(\nu_j\) to \(H\). Its coordinates are

\[
(x,y,\eta),\qquad \eta\in(\nu_j)_x.
\]

Then \(k(x)=(x,h_a(x),0)\) has normal bundle \(\nu_j\oplus f^*\tau\). Lemma 6.0j constructs its spin-c orientation from (13) by the exact finite Clifford-module cancellation, and retains its full line twist over embedding families.

The local transverse expression for \(g\) is

\[
(x,y,\eta)\longmapsto(y,j(x)+\eta).
\tag{15}
\]

At the section, the derivative in \((x,\eta)\) of the second component is the isomorphism \(TM\oplus\nu_j\to\mathbb R^n\). The first component is the identity in \(y\). Hence (15) is locally invertible; shrinking \(N\) makes \(g\) étale everywhere. The chosen holonomy arrows glue its local lifts to the product leaf space. This proves the geometric factorization.

The three KK factors in the proposed construction have types

\[
\begin{aligned}
k_!&\in KK^{n-\dim M+q}(C(M),C_0(N)),\\
g_!&\in KK(C_0(N),A\otimes C_0(\mathbb R^n)),\\
\mathrm{pr}_!&\in KK^n(A\otimes C_0(\mathbb R^n),A).
\end{aligned}
\]

Corollary 6.0d supplies the spin-c normal-bundle Thom class for the first factor, with the prescribed orientation, and Lemma 6.0e makes its ordinary tubular realization independent of the normalized tube. Theorem 6.0f proves the ordered composition for compact ordinary embeddings; Lemmas 6.0g–j and Corollary 6.0k prove the family, stabilization and ordinary graph-choice comparisons with the prescribed stable orientation. Lemma6.0n supplies the full étale correspondence and its parameter evaluations for the second factor. Together with the inverse Bott class for the third factor, it defines the foliation product

\[
f_!=k_!\otimes_{C_0(N)}g_!
\otimes_{A\otimes C_0(\mathbb R^n)}\mathrm{pr}_!
\in KK^{q-\dim M}(C(M),A).
\tag{16}
\]

The degree is a useful check: \(n-\dim M+q+n=q-\dim M\bmod2\).

### The full holonomy factorization and its choices

The classical construction is the foliated microbundle factorization in [Connes, Section11, pp33–35], following [Haefliger, 1970, p188]. The proofs below include the finite germ shrinking, Hausdorff quotient and completed parameter modules. They use the full holonomy-chart proofs of the [C*-algebra lesson, Lemmas1.1/2.1 and Theorem2.2](the-c-star-algebra-of-a-foliation.md#section-2), its chart-core proof in Proposition6.1, and the full counting-correspondence and compact-action proofs in Sections1–2 of this lesson.

Throughout this subsection the foliation and map cocycles are smooth. Put \(G=\operatorname{Hol}(V,F)\), \(A=C_r^*(G)\), and \(q=\operatorname{rank}(TV/F)\). The unit manifold is Hausdorff and second countable; its holonomy arrow space may be non-Hausdorff. The source \(M\) is compact, without boundary, of dimension \(d\). The smooth cocycle \(f\) and the specified spin-c structure on \(TM\oplus f^*\tau\) are the full data of Section5. Normal orientations use the precise convention of Lemma6.0j. Choose an even Euclidean embedding dimension large enough for its normal rank to be positive.

#### A Hausdorff germ neighbourhood and an actual arrow cocycle

**Lemma 6.0l.** The pullback Haefliger cocycle of \(f\) has a smooth Hausdorff second-countable foliated microbundle
\(M\xrightarrow{\iota}H\xrightarrow{p}M\)
of rank \(q\), with \(p\iota=1\). The normal quotient of \(\iota\) is exactly \(T=f^*\tau\), with its holonomy transition derivatives. The transverse coordinate transitions can be accompanied by actual smooth holonomy arrows satisfying a genuine cocycle on the resulting neighbourhood. The germ of these data near the compact section is independent of the representatives and finite refinements used in constructing it.

**Proof: finite representatives with one common shrink.** Refine the defining cover to finitely many open sets \(U_a\) such that each \(f_a(U_a)\) lies in a foliation box \(P_a\times Y_a\), with \(P_a\) a convex plaque ball and \(Y_a\subset\mathbb R^q\). Put \(h_a=k_af_a\), where \(k_a\) is the transverse projection. A finite shrinking \(V_a\) still covers \(M\) and satisfies \(\overline{V_a}\subset U_a\). To obtain it, take the smooth bump partition already proved in DG-CHAR, Section5, subordinate to a finite refinement of the original cover; assign each compact support to its containing original set and take a slightly larger open neighbourhood of that support inside that set. All closures here are compact.

Choose a plaque cross-section \(\kappa_a:Y_a\to P_a\times Y_a\) at one fixed plaque coordinate. Let \(\ell_a(x):\kappa_a(h_a(x))\to f_a(x)\) be the arrow of the straight plaque path. Its dependence on \(x\) is smooth in the unit path chart. Conjugate the given transition arrows to these cross-sections:
\[
a_{ba}(x)=\ell_b(x)^{-1}g_{ba}(x)\ell_a(x).
\tag{181}
\]
Their products satisfy \(a_{ca}=a_{cb}a_{ba}\) exactly, since the middle plaque arrows cancel. In an arrow chart from Lemma2.1 and Theorem2.2, restricting both endpoints to these fixed cross-sections gives a fixed transverse diffeomorphism \(H_{ba,\alpha}\) and an actual arrow section
\(K_{ba,\alpha}(y):\kappa_a(y)\to\kappa_b(H_{ba,\alpha}(y))\).
The branch is fixed on a neighbourhood of each base point \(x\), and at \(y=h_a(x)\) gives \(a_{ba}(x)\). Thus the smooth parameter map into arrows has been represented locally by a transverse branch independent of \(x\); no arbitrary \(x\)-dependent coordinate change is introduced.

For each pair let \(S_{ab}=\overline{V_a}\cap\overline{V_b}\subset U_a\cap U_b\). It is compact. Cover it by finitely many smaller base neighbourhoods with closure inside such branch neighbourhoods. Where two branch representatives overlap, they give the same holonomy germ at \(h_a(x)\). That equality gives an open transverse neighbourhood on which their maps agree. Their actual arrows agree there too: fixed endpoints and the same holonomy germ define the same arrow, and the arrow-chart intersection is open by Theorem2.2. Since the representatives are fixed transverse branches and \(h_a\) is continuous, agreement persists on a product neighbourhood of that base point and a ball about \(h_a(x)\). Compactness of the intersections of the smaller closed base neighbourhoods gives one positive radius for each of their finitely many overlaps. A finite minimum gives one radius on which the representatives patch.

In centred transverse coordinates this yields smooth maps, on every \(V_a\cap V_b\) times one fixed small ball,
\[
F_{ba}(x,v)=H_{ba}^x(h_a(x)+v)-h_b(x),
\qquad F_{ba}(x,0)=0,
\tag{182}
\]
and patched actual arrow sections \(K_{ba}(x,h_a(x)+v)\). All local branch domains contain these balls; their images remain in the corresponding transverse boxes. Apply the same finite compact-overlap argument to \(S_{abc}=\overline{V_a}\cap\overline{V_b}\cap\overline{V_c}\), to the inverse pairs and to the diagonal branches. The centre cocycle is exact. Equality of its holonomy germs therefore extends on an open transverse neighbourhood, and the matching actual arrow sections have the same equality there.

There are finitely many such comparisons. First choose a radius \(r_0\) on which every representative is defined. The derivatives of the finitely many transverse representatives are bounded near their compact centre sets. Since \(F_{ba}(x,0)=0\), a smaller radius makes each image lie in the required radius-\(r_0\) domain of a subsequent map. Shrink once more to a radius \(r_1\) on which the cocycle, inverse and identity equalities all hold. Thus, whenever the expressions are taken on the indicated smaller balls,
\[
F_{ca}(x,v)=F_{cb}(x,F_{ba}(x,v)),
\quad F_{ab}(x,F_{ba}(x,v))=v,
\quad F_{aa}(x,v)=v.
\tag{183}
\]
The actual arrows have the corresponding exact product identities. The representations of the germs have now become a genuine finite cocycle on a neighbourhood; the conclusion does not follow from a merely pointwise choice of germ representatives.

**Proof: a closed relation and an open quotient.** Fix \(0<\epsilon<r_1\) and let
\(D=\coprod_a D_a\), \(D_a=V_a\times B_\epsilon^q\).
Relate \((x,v)\in D_a\) to \((x,w)\in D_b\) precisely when
\(w=F_{ba}(x,v)\).
Both displayed vectors are required to lie in their radius-\(\epsilon\) balls. The equalities just proved give reflexivity and symmetry. They give transitivity as well: if the intermediate vector lies in its ball, the composite is defined in the original representatives and equals the direct pair transition on the input ball. Therefore this pair relation already is an equivalence relation. It acquires no extra identifications from unbounded transition chains.

On each \(D_a\times D_b\) it is closed. Equality of the two base points is closed because \(M\) is Hausdorff, and, on that closed set, the equation \(w=F_{ba}(x,v)\) is a closed graph. The maps are defined on the entire input ball for every \(x\in V_a\cap V_b\). Hence a limit within \(D_a\times D_b\) still satisfies that equation. The pair components are open and closed in \(D\times D\), so the full relation is closed. Its quotient map is open: the saturation of an open subset of \(D_a\) is the union of its images under the partial transition diffeomorphisms, whose domains are open because their images are required to remain in an open ball.

This open quotient is Hausdorff. For two inequivalent representatives, closedness of the relation gives product neighbourhoods avoiding it. Their open quotient images are disjoint neighbourhoods of the two classes. Each \(D_a\) maps injectively to the quotient, because its self-relation is the identity, and maps homeomorphically onto an open subset. The resulting finite smooth atlas has the above transition diffeomorphisms. Thus its quotient \(H\) is a Hausdorff second-countable manifold of dimension \(d+q\). The maps \(p[x,a,v]=x\) and \(\iota(x)=[x,a,0]\) are smooth, with \(p\) a submersion and \(p\iota=1\). Compactness makes the embedded section closed.

In the original coordinates \(y=h_a(x)+v\), the plaque distribution is \(dy=0\). Each transverse branch change is locally a fixed function of \(y\), so these distributions glue to a smooth foliation of codimension \(q\). The normal quotient of the section is represented by the vertical vector \(v\). Its transition derivative is \(D_yH_{ba}^x(h_a(x))\), exactly the defining transition of \(f^*\tau\). In original coordinates the same quotient is \((u,w)\mapsto w-dh_a(u)\); differentiating the transition and its section identity gives the identical derivative on this quotient. This verifies the normal identification, including its bundle order when used below.

**Proof: the germ is independent of choices.** Two representative systems for the same cocycle agree on some neighbourhood of every centre by equality of germs. Repeat the finite compact-overlap shrinking above on their common refinement. Their transition maps and actual arrow sections then agree on one smaller neighbourhood of the section. The two quotients have identical atlases there, giving a diffeomorphism of open neighbourhoods fixing the section and projection.

For two equivalent cocycles, the equivalence arrows on their common refinement give transverse germ isomorphisms between their coordinate systems. Represent these by the path charts and shrink on the compact double and triple overlaps as before. Their cocycle commutation identities are exact there. They glue to mutually inverse diffeomorphisms near the section, preserving the projection and transporting its normal by the derivative of the equivalence arrows. The matching actual arrow sections give the corresponding equivalence of the target arrow cocycles. Changing plaque cross-sections is the same construction using the short plaque arrows between the two choices. A further shrink restricts these maps. This proves the asserted independence of the germ with all its arrow data. \(\square\)

For \(q=0\), the centred ball is the point \(\mathbb R^0\), the quotient is \(M\), and the same arrow-cocycle argument applies; there is no transverse derivative or radius issue.

#### The geometric étale replacement

**Lemma 6.0m.** Let \(j:M\hookrightarrow\mathbb R^n\) be as above. On an open neighbourhood \(N_j\) of the compact section in \(p^*\nu_j\), the formulas
\[
k_j(x)=(\iota(x),0),\qquad
g_a(x,y,\eta)=\bigl(\kappa_a(y),j(x)+\eta\bigr)
\tag{184}
\]
and the actual arrow sections of Lemma6.0l define a smooth étale map \(g_j:N_j\to V/F\times\mathbb R^n\). Its composite with \(k_j\) is the map \((f,j)\) with its complete cocycle. The normal of \(k_j\), ordered as \(\nu_j\oplus f^*\tau\), has precisely the prescribed structure constructed in Lemma6.0j.

**Proof.** The projection to \(M\) in every microbundle coordinate change is the identity. Thus the pullback of \(\nu_j\) is a genuine vector bundle on \(H\); its coordinate \(\eta\in\nu_{j,x}\) is retained by those changes. On a compact-source Euclidean normal disk the map
\(\Phi_j(x,\eta)=j(x)+\eta\)
is an open embedding. Its derivative at the zero section is the tangent/normal isomorphism. The local inverse theorem gives local invertibility, and the compact-sequence injectivity argument of Lemma6.0g gives one positive radius on which it is injective. This is the explicit Euclidean instance of the read DG-CHAR tubular theorem. Choose that disk for \(N_j\); the transverse \(H\) neighbourhood may be shrunk further without changing the compact section.

The transverse coordinate map of \(g_a\) is \((x,y,\eta)\mapsto(y,\Phi_j(x,\eta))\). Its derivative is invertible: the \(y\) block is the identity and the \((x,\eta)\) block is the derivative of \(\Phi_j\). Consequently it is a local diffeomorphism to the transverse space of the product foliation. The arrow on an overlap is
\(\bigl(K_{ba}(x,y),\Phi_j(x,\eta)\bigr)\in G\times\mathbb R^n\).
It has exactly the two required endpoints, and the Euclidean coordinate is identical at both. Lemma6.0l gives the genuine arrow cocycle on this neighbourhood. At the section its first part is \(a_{ba}\), and the arrows \(\ell_a\) identify it with the original \(g_{ba}\); hence \(g_jk_j=(f,j)\) as a smooth leaf-space map, including holonomy.

The normal sequence of the section in this pullback bundle has the two vertical summands \(\nu_j\) and \(\nu_\iota=f^*\tau\). In the stated order its quotient map sends \((\eta,t)\) to the class of the two vertical lifts. Lemma6.0j gives its full spin-c module, with the same \(T=f^*\tau\) and the same Euclidean complement \(\nu_j\). No orientation is inferred from a leafwise endpoint coincidence. \(\square\)

#### The complete étale module and its parameter version

**Lemma 6.0n.** Let \(g:N\to V/F\) be a smooth étale cocycle with \(N\) Hausdorff and second countable. Its principal right \(G\)-space \(P_g\), with projection \(b:P_g\to N\) and source anchor \(a:P_g\to V\), defines a countably generated Hilbert \(A\)-module \(\mathcal E_g\). Multiplication by \(C_0(N)\) is compact, so
\(g_!=[\mathcal E_g,\rho,0]\in KK(C_0(N),A)\).
An equivalence of its actual arrow cocycles gives an even module unitary. For an open \(U\subset N\),
\[
[e_{U\hookrightarrow N}]g_!=(g|_U)_!.
\tag{185}
\]
For a smooth parameter étale cocycle on a Hausdorff \(\mathcal N\to I\), the same construction over \(G\times I\) gives a Hilbert \(C(I,A)\)-module. Its tensor by each endpoint evaluation is exactly the module of the endpoint cocycle, by a surjective isometric restriction map. These assertions also hold with the fixed product factor \(\mathbb R^n\).

**Proof: the whole module.** Glue \(U_a\times_{g_a,r}G\) by \((n,h)\mapsto(n,g_{ba}(n)h)\). The projection is \(b(n,h)=n\), and the source anchor is \(a(n,h)=s(h)\). The right action is multiplication by an arrow whose range equals that source. In path coordinates, the transverse derivative of \(g_a\) and the source plaque coordinate together give an invertible derivative for \(a\). Thus it is a local diffeomorphism. Its fibres are discrete countable sets, with all distinct holonomy lifts retained. The total space can be non-Hausdorff; its chart atlas is countable and locally Hausdorff.

Use the finite span of compactly supported sections in individual Hausdorff charts of
\(a^*D_F^{1/2}\) on \(P_g\). This is precisely the chart-core convention of the C*-algebra lesson's Proposition6.1. With a positive density chosen only to write scalar formulas, define
\[
\langle\xi,\eta\rangle(\gamma)
=\sum_{z\in a^{-1}(s\gamma)}
\overline{\xi(z\gamma^{-1})}\eta(z),
\qquad
(\xi f)(z)=\int_{s(h)=a(z)}\xi(zh^{-1})f(h).
\tag{186}
\]
The half-density factors supply exactly the endpoint density of the first expression and the integrated intermediate density of the second. On two compact chart supports, their fibre product over the Hausdorff base \(N\) is compact, and division by the principal right action carries it into finitely many arrow charts after a finite subdivision. The counting sums and convolution integrals therefore give the same compact chart kernels as in Sections1–2. They lie in \(\mathcal A(G)\). The local Fubini calculation gives \(\langle\xi f,\eta h\rangle=f^**\langle\xi,\eta\rangle*h\).

Here is a direct regular-fibre check of positivity and the coefficient action. For \(x\in V\) and \(u\in C_c(G_x,D_F^{1/2})\), set
\[
W_x(\xi\otimes u)(z)=\int_{G_x}\xi(z\gamma^{-1})u(\gamma),
\qquad z\in a^{-1}(x).
\tag{187}
\]
The integrated two half-densities are at \(r\gamma\). Changing the variable \(z\) by right multiplication by the second input arrow gives
\(\langle W_x(\xi\otimes u),W_x(\eta\otimes v)\rangle
=\langle u,\pi_x(\langle\xi,\eta\rangle)v\rangle\).
Indeed the kernel at \((\gamma,\delta)\in G_x\times G_x\) is the sum over \(z\in a^{-1}(x)\) of
\(\overline{\xi(z\gamma^{-1})}\eta(z\delta^{-1})\),
which is the first formula evaluated at \(\gamma\delta^{-1}\). Finite lists give the identical matrix-positivity calculation. The faithful sum of the regular representations detects positivity in the reduced algebra. Quotienting zero-norm vectors and completing therefore gives a right Hilbert \(A\)-module.

The maps \(W_x\) extend to isometries from \(\mathcal E_g\otimes_{\pi_x}L^2(G_x)\) onto \(\ell^2(a^{-1}(x))\). Here is an explicit surjectivity check. Fix \(z\in a^{-1}(x)\) and a Hausdorff chart \(O\) about \(z\) on which \(a\) is injective. Choose a small unit-plaque neighbourhood \(B\subset G_x\) of \(1_x\) such that \(z\gamma^{-1}\in O\) for every \(\gamma\in B\). Take a core section \(\xi\) supported in \(O\), nonzero near \(z\), and a nonnegative compact plaque cutoff \(\chi\) supported in \(B\). In a positive density trivialization put \(u(\gamma)=\chi(\gamma)\overline{\xi(z\gamma^{-1})}\), with the complementary half-density. Then \(W_x(\xi\otimes u)(z)\) is the positive integral of \(\chi|\xi(z\gamma^{-1})|^2\); choose \(\chi\) so it is nonzero. If \(w\in a^{-1}(x)\) and a term at \(w\) survives, both \(w\gamma^{-1}\) and \(z\gamma^{-1}\) lie in \(O\), with the same anchor \(r\gamma\). Injectivity of \(a|_O\) makes them equal, and multiplication by \(\gamma\) gives \(w=z\). Thus this vector is a nonzero multiple of the single counting basis vector at \(z\). Scaling and taking finite sums proves onto. In leaf dimension zero the integral is its counting version. This argument retains every recurrent or holonomy lift; it does not identify distinct lifts by their leaf images.

Multiplication by a bounded coefficient \(c(b(z))\) becomes the diagonal operator with those entries on every such counting fibre. Its norm is at most \(\|c\|\), its adjoint is multiplication by \(\overline c\), and the faithful regular-fibre norm shows that it extends adjointably to the full module. Countable generation follows explicitly from separability of its core in the module norm. In a fixed finite collection of compact charts, uniform coefficient approximation controls that norm: its inner-product kernel has a common compact chart support and common row/column bounds, and Lemma3.2 bounds its reduced norm by a constant times the square of the coefficient supremum. Countably many compact chart exhaustions and countable smooth coefficient approximations give a countable norm-dense set of module vectors. An approximate identity of \(A\) converges on each vector, by its inner-product formula. Hence the closed right \(A\)-span of that countable set is the whole module.

**Proof: compact coefficient action.** Let \(\sigma\in C_c(N)\) have support in a local domain where the cocycle is an injective transverse lift. Choose a compactly supported plaque half-density \(b_0\) on the associated short-arrow source-anchor chart, normalized to have plaque integral of \(|b_0|^2\) equal to one on \(\operatorname{supp}\sigma\). Put \(\xi=b_0\sigma\circ b\). On the counting fibre a mixed term of \(\theta_{\xi,\xi}\) can survive only if two corresponding short arrows have the same source in this injective chart. They must agree, so the two counting lifts agree. The remaining diagonal term is multiplication by \(|\sigma(b(z))|^2\) times the normalized plaque integral. It is exactly that multiplication. The faithful \(W_x\) identifications prove the equality of the actual module maps.

A finite nonnegative partition on a compact source support and its square roots reduce a general nonnegative compactly supported function to a finite sum of these maps. Positive and negative parts, then real and imaginary parts, give all compactly supported coefficients. Sup-norm approximation gives every \(C_0(N)\) coefficient, since the coefficient action is contractive and module compact operators form a closed space. This proves \(\rho(C_0(N))\subset\mathcal K(\mathcal E_g)\). The zero operator is odd, and its only nonzero Kasparov defect is \(-\rho(c)\). Thus the displayed cycle is defined on the full completed module.

An arrow equivalence \(h_a(n):g_a(n)\to g'_a(n)\) gives the principal-space map \((n,z)\mapsto(n,h_a(n)z)\). It preserves the source anchor and therefore the source half-density. It intertwines the right action, the counting sums, the inner products and the source coefficient multiplication, with inverse obtained from \(h_a^{-1}\). It extends to the claimed even unitary of the completed modules. This verifies the invariance on cycles, rather than just identifying set-theoretic leaf images.

For an open \(U\), extension of chart sections of \(P_g|_U\) is isometric by the identical inner-product formula. Its closed range is exactly \(\overline{\rho(C_0(U))\mathcal E_g}\). Compact chart supports inside this open principal subspace can be cut off by \(C_0(U)\), proving one inclusion and density; the other follows from their supports. The zero operator restricts exactly, and the local compactness proof applies on this essential submodule. Lemma0.10 and homomorphism functoriality0.11 therefore give the claimed restriction product even when the submodule has no orthogonal complement.

**Proof: the parameter topology and endpoint maps.** For \(G\times I\), the reduced algebra is \(C(I,A)\). On an elementary chart kernel \(c(s)f(\gamma)\), its regular norm is \(\sup_{s,x}|c(s)|\|\pi_x(f)\|\). General compact chart kernels are continuous in parameter in the reduced norm: on a common compact support their coefficient differences tend uniformly to zero, and the common row/column bound proves norm convergence. A finite parameter partition approximates them by finite sums of elementary kernels. Conversely those kernels are dense in \(C(I,A)\), first for functions with values in the dense arrow core, then by a finite partition on the compact parameter interval. This proves the asserted isometric algebra identification. The same proof with compact transverse support in \(\mathbb R^n\) identifies the product algebra with \(C(I,A\otimes C_0(\mathbb R^n))\).

Construct \(P_g\) for the whole parameter cocycle and use its whole relative-chart core, including the endpoints of the interval. The inner-product and convolution calculations just proved now take values in this single interval algebra. Compact source supports need only finitely many relative charts, so their common Schur bounds are uniform over their parameter support. The norm-continuity argument proves that these are actual interval-algebra coefficients. Thus the preceding completion and compact-action proof give a Hilbert module over the whole interval algebra, not merely an independently completed module for every parameter.

Restriction at \(s\) defines the balanced map
\(\mathcal E_g\otimes_{\mathrm{ev}_s}A\to\mathcal E_{g_s}\).
The inner products restrict to exactly the endpoint counting formula, so this is an isometry. It is onto: an endpoint compact chart section extends in the parameter/source chart by multiplying a local extension by a cutoff equal to one on its support. Finite chart sums extend likewise, and their endpoint core is dense. This holds at the two boundary parameters using relative half-neighbourhood charts. The coefficient representation and the zero operator restrict to those of the endpoint cycle. This gives the asserted endpoint module, with both the completed topology and the compact maps identified. \(\square\)

#### The actual foliation homotopy and all factorization choices

Write \(D_n=A\otimes C_0(\mathbb R^n)\). For the fixed even Euclidean Bott inverse \(A_n\), retain the old Clifford block as in Lemma6.0h. If \(k=n-d+q=p+2t\), use the fixed ordered contraction to \(C_p\), \(p\in\{0,1\}\); the final-odd/first-odd transport is exactly the one proved in Proposition6.0i. Define
\[
\mathfrak f_j=
\operatorname{contract}_{k\to p}\bigl((k_j)_!(g_j)_!^{[k]}\bigr)
 A_n^{A\widehat\otimes C_p}
\in KK(C(M),A\widehat\otimes C_p).
\tag{188}
\]
The first embedding cycle uses its full specified normal orientation and a normalized tube into \(N_j\); the middle cycle is the actual module of Lemma6.0n. The inverse is applied after the Euclidean target, in that order.

**Theorem 6.0o (compact-source foliation factorization independence).** This class is independent of the normalized tube, germ representatives, finite cocycle refinement, transverse plaque cross-sections, neighbourhood shrink, and even Euclidean embedding used in its construction. It retains the prescribed stable orientation. A smooth compact-source family of map cocycles with a specified smooth orientation family gives an actual Kasparov homotopy of these classes. This proves the choice and homotopy comparisons for (16). Theorems6.0s/6.0w below prove projection and compact ordinary-map/foliation composition; Theorems 6.0ab–ae below prove the full sphere and bordism relations.

**Proof: restriction and germ choices.** If \(U\subset N_j\) contains the compact section, choose a small normalized normal disk whose tube lies in \(U\). Formula(150) and tube independence give
\((k_j)_!^U[e_{U\hookrightarrow N_j}]=(k_j)_!^{N_j}\).
Use Lemma6.0n's actual essential-module restriction to obtain
\((k_j)_!^U(g_j|_U)_!^{[k]}=(k_j)_!^{N_j}(g_j)_!^{[k]}\).
This is an equality of KK classes before the contraction and inverse Bott factor. It assumes no KK invertibility for an arbitrary neighbourhood inclusion.

Two microbundle constructions for the same cocycle have a common smaller germ neighbourhood by Lemma6.0l. Its diffeomorphisms preserve the projection and section; their normal maps are exactly the isomorphisms of \(\nu_j\oplus f^*\tau\) induced by the cocycle equivalence. Transport the prescribed spin-c data along these isomorphisms. Positive metric transport and the orthogonal covariance in Corollary6.0d identify their Thom cycles; the total-space normal linear isomorphisms are proper because they are homeomorphisms. Composing their normalized tubes with the germ diffeomorphism and using Lemma6.0e identifies the embedding classes. The arrow equivalence of Lemma6.0l identifies the middle modules by Lemma6.0n's explicit unitary. The restriction equality just proved lets us use that common smaller neighbourhood in both products. This proves all germ, cover and plaque-section choices. Tube choices were already covered by Lemma6.0e. A change of a metric or principal connection is likewise transported through the metric/connection family already proved; it does not replace the specified orientation.

**Proof: a whole parameter product.** First hold \(f\) fixed and let \(j_s\) be a smooth family of embeddings into one fixed even-dimensional \(\mathbb R^n\), extending beyond \(I\). Use the one fixed microbundle \(H\). The Euclidean normal bundles \(\nu_{j_s}\) form a smooth bundle on \(I\times M\). Pull it back to \(I\times H\). A uniform small Euclidean normal disk makes
\((s,x,\eta)\mapsto(s,j_s(x)+\eta)\)
an open embedding. Its derivative is invertible at the compact parameter/zero section; local invertibility and the compact-sequence injectivity argument of Lemma6.0g give the common radius. Thus a neighbourhood \(\mathcal N\) of \(I\times M\) has the genuine parameter étale cocycle
\((s,x,y,\eta)\mapsto(s,\kappa_a(y),j_s(x)+\eta)\),
with the arrows of Lemma6.0l. Its normal family is exactly \(\nu_{j_s}\oplus f^*\tau\), with the coherent module of Lemma6.0j.

Use a parameter-preserving normalized tube into \(\mathcal N\). We spell out the needed varying-ambient construction. Write \(V\mathcal N=\ker d\pi\) for the vertical tangent of \(\pi:\mathcal N\to I\). In a submersion chart \((s,x)\), coordinates on this bundle are \((s,x,v)\). Define the local spray by \(\dot s=0,\dot x=v,\dot v=0\). Under a change \((s,x)\mapsto(s,h(s,x))\), its vertical acceleration is quadratic in \(v\), namely \(D_x^2h(v,v)\); there is no parameter derivative because \(\dot s=0\). A smooth locally finite partition on \(\mathcal N\) patches these sprays, rather than attempting to average local diffeomorphisms. Their sum still has base velocity \(v\), keeps \(s\) fixed and has acceleration vanishing to second order at \(v=0\). Its zero trajectories are fixed, and its linearization there is \(\dot a=b,\dot b=0\), with the parameter component retained.

The contraction/parameter-flow proof in DG-CHAR Lemma3.1 gives a smooth time-one flow near each zero vector: the fixed zero trajectory exists for the entire compact time interval, and finitely many local existence intervals give the same property to nearby trajectories. Its base endpoint \(L\) satisfies \(L(y,0)=y\) and \(DL_{(y,0)}(a,b)=a+b\), and stays in the parameter fibre of \(y\). A vertical metric identifies the normal quotient of the section in each fibre with the orthogonal complement of its vertical tangent. Restrict \(L\) to this complement. Its differential at the compact zero section is the tangent/normal isomorphism in the vertical block and the identity in the parameter block. The inverse theorem therefore makes it a local diffeomorphism near that section. If no common small disk were injective, distinct pairs of normal vectors with norms tending to zero and equal images would have convergent base subsequences in \(I\times M\). Their equal images force the two limiting base points to coincide; the local inverse at that point then contradicts equality of the late pairs. Compactness gives a common disk on which the map is defined, locally invertible and injective. The radial map \(v\mapsto v/\sqrt{1+|v|^2/\rho^2}\) into a radius-\(\rho\) disk has derivative the identity at zero and preserves the parameter. Composing it with \(L\) gives the required normalized whole-bundle tube. Boundary parameters use an open-interval extension, or the corresponding relative submersion charts.

The global normal Thom cycle is defined on this entire bundle over \(I\times M\). The proofs of Lemmas6.0a–c are fibre proofs with continuous compact-base parameters, so apply to this compact base with boundary as well; no base differential operator is involved. Its specified Morita family is Lemma6.0j's actual smooth module. Compose with the whole tube extension and then with Lemma6.0n's full parameter correspondence over \(C(I,D_n)\), retaining the normal \(C_k\) coefficient. Pull the left action back along the constant inclusion \(C(M)\to C(I\times M)\), and apply the one fixed ordered contraction and Euclidean inverse with interval coefficients. This gives one class
\[
\mathfrak F\in KK(C(M),C(I,A\widehat\otimes C_p)).
\tag{189}
\]
Its endpoint is exactly \(\mathfrak f_{j_s}\). For the Thom/tube factors this follows from the full parameter multiplier, compact-level-set extension and Morita fibre evaluations. For the middle factor it is the surjective isometric endpoint map just proved. The exterior finite contraction and inverse Bott modules restrict to the same fixed ones. Product naturality in Section0 moves these evaluations through the composable product. Hence the endpoints are an actual Kasparov homotopy, with completed modules and compact defects on the interval, proving the comparison for this embedding family.

For a smooth family \(f_s\), instead construct Lemma6.0l on the compact base \(I\times M\), using its relative charts. The same finite shrinking argument works, because that base is compact and its transition arrows lie in \(G\times I\). The resulting neighbourhood has parameter coordinate \(s\); the local foliation plaques are \((s,y)\) constant. The normal to the compact section in each parameter fibre is \(f_s^*\tau\), with its exact derivative cocycle. Fix a large even Euclidean embedding of \(M\) for all parameters. The above normal/étale construction then applies to the whole parameter space, with the given smooth spin-c family on \(TM\oplus f_s^*\tau\). Lemma6.0j constructs its full normal family by the same formula. The whole product and endpoint argument are unchanged. This proves homotopy invariance with the specified orientation family, including any line twist.

**Proof: even stabilization on the actual correspondence.** Stabilize \(j\) by a new \(R=\mathbb R^{2a}\) block. One may choose the étale neighbourhood to be \(N_j\times R\): its transverse map is \((y,j(x)+\eta,w)\). It is étale for every \(w\), and its compact section is at \(w=0\). The normal module is the ordered sum \(E_j\oplus R\), by the full stabilization square of Lemma6.0j. Lemma6.0h gives the embedding product, after contracting only the new even block.

The new principal correspondence is exactly \(P_{g_j}\times R\). On elementary core sections \(\xi(z)c(w)\), its inner product is \(\langle\xi,\eta\rangle c^*d\), and its right convolution and left coefficients act in the two separate factors. Such elementary sections are dense: on compact product chart supports a finite partition in \(w\) gives uniform coefficient approximation, and the common Schur bound gives module-norm approximation. Therefore the even tensor map extends to an onto isometry
\(\mathcal E_{g_j\times1_R}\cong\mathcal E_{g_j}\widehat\otimes C_0(R)\),
intertwining both actions and the zero operator. Thus the new middle class is exactly \((g_j)_!\boxtimes1_{C_0(R)}\). Tensor reassociation now moves the new even Bott factor past this actual zero-operator correspondence, giving the old product followed by the coefficient Bott class on \(D_n\widehat\otimes C_p\). This is the graded exterior-product naturality proved in Section0; its module map is the just identified tensor map and the associating unitary, with the old Clifford block retained before the new block. The creation/positivity checks for the normal product are those of Lemma6.0h.

The two-block Euclidean inverse is the new even inverse followed by the old inverse, with the fixed wedge unitary and the retained-odd transport of Proposition6.0i. Cancelling the adjacent \(B_{2a}A_{2a}=1\) gives
\[
\mathfrak f_{(j,0)}=\mathfrak f_j.
\tag{190}
\]
This proves stabilization for the actual foliation correspondence, rather than using the ordinary graph theorem as its substitute.

**Proof: comparison of any two embeddings.** Use
\(j_\theta=(\cos\theta\,u_0,\sin\theta\,u_1)\)
in one common even Euclidean space. It is a smooth embedding family extending past its endpoints, as proved in Proposition6.0i. The full parameter product above identifies its endpoint classes. At zero, the full normal module and the middle product are exactly the even stabilization of \(u_0\), by the stabilization proof. At \(\pi/2\), permute the two complex Euclidean blocks. On principal spaces this permutation preserves the \(G\) coordinate and carries the source-anchor Euclidean coordinate by that proper linear diffeomorphism. The chart-core formulas therefore give the corresponding unitary after transporting the target coefficient algebra. On the normal modules it is the actual even exterior unitary and Morita cancellation of Lemma6.0j. It preserves the canonical even Bott class, by the creation/contraction intertwining in Proposition6.0i; its full inverse is preserved by the two actual inverse identities. Hence the second endpoint is exactly the stabilization of \(u_1\). The two stabilization cancellations prove \(\mathfrak f_{u_0}=\mathfrak f_{u_1}\).

Together with the restriction, representative and tube comparisons, this proves the complete choice assertion for the compact-source product (16). The full prescribed orientation is retained throughout. \(\square\)

![Full holonomy factorization, Hausdorff germ neighbourhood and parameter module](../figures/foliation-factorization-choice.png)

**Figure 6.0c.** Lemmas6.0l–n and Theorem6.0o, equations (181)–(190), Exercises42–43. The first panel shows the actual finite relation after a common shrink; closedness of that relation and openness of its quotient establish Hausdorffness. The second panel has the exact transverse derivative and ordered normal. The third panel shows the anchors for a general étale map into \(V/F\); its last line includes the fixed Euclidean product used in the factorization. Every distinct holonomy germ remains in the counting fibre. The fourth panel shows one actual interval product and the ordered even cancellation, with the old Clifford block retained first. All tensor products with Clifford coefficients are graded. These are diagrams of maps and modules, with no geometric projection or assertion that the arrow space is Hausdorff. The construction is classical [Connes, Section11; Haefliger, 1970]; the complete shrinking, module and parameter comparisons are proved above.

Full-size factorization PNG · Editable factorization SVG.

### Coefficient bundles and the projection formula

We first prove the coefficient-bundle identity on an actual tube and counting correspondence, then identify the ordinary graph construction with the zero-foliation case. The geometric inputs are the full DG-CHAR bundle and DG-FND unitary transport proofs linked above. The finite-module and analytic product inputs are Corollary6.0d, Lemmas6.0e–j/6.0n and Section0.

For a smooth Hermitian bundle \(E\to X\), write
\(u_X(E)=[\Gamma_0(E),\mathrm{mult},0]\) as a class in \(KK(C_0(X),C_0(X))\), with \(E\) even. This notation makes sense also for noncompact \(X\). A countable relatively compact frame cover, its compact chart exhaustions and countable scalar coefficient approximations give a countable norm-dense set of sections. Finite chart partitions approximate any compact section in the supremum norm; compact cutoffs then approximate any section vanishing at infinity. The identity on a finite frame patch is a finite sum of section rank-one maps. A finite partition on a compact coefficient support makes multiplication by a nonnegative coefficient a finite sum of such maps, using the partition's square roots; linearity and approximation give all \(C_0(X)\) coefficients. Thus the module is countably generated, the left action is compact, and the zero-operator cycle is defined. It does not require a global finite trivialization on a noncompact space.

#### Bundle transport on the entire normalized tube

**Lemma 6.0p.** Let \(i:L\hookrightarrow X\) be a compact-source smooth embedding with the specified spin-c normal structure, of rank \(k\). The ambient manifold may be noncompact. For every smooth Hermitian \(E\to X\), there is an equality
\[
u_L(i^*E)\,i_!
=i_!\,u_X(E)^{[k]}
\quad\text{in }KK(C(L),C_0(X)\widehat\otimes C_k).
\tag{196}
\]
Here \(u_X(E)^{[k]}\) denotes its exterior extension by the identity on the retained \(C_k\). Both products are represented on the actual essential tube modules. The equality retains every twist of the prescribed normal orientation.

**Proof.** Choose the normalized whole-bundle tube \(\tau:\nu_i\to U\subset X\) of Lemma6.0e. Choose a Hermitian metric connection on \(E\). The exact DG-FND connection construction applies to its unitary frame bundle, using the already proved smooth bump partition; its associated transport is unitary. For each \((x,v)\in\nu_i\), transport from \(i(x)\) to \(\tau(x,v)\) along
\(t\mapsto\tau(x,tv)\), \(0\leq t\leq1\). Denote the resulting unitary by
\(T_{x,v}:E_{i(x)}\to E_{\tau(x,v)}\).
This is defined on the entire normal bundle. For a fixed finite \(v\), its path is a compact smooth path on the full interval; the unitary transport ODE has no finite-time escape, by the full associated transport proof. Smooth parameter dependence on each compact family of \((x,v)\) follows from that same ODE argument, so \(T\) is a globally smooth bundle map. At \(v=0\) it is the identity.

Let \(\mathscr H_\nu\) be the specified spin-c Thom module over \(C_0(\nu_i)\widehat\otimes C_k\), and let \(F_\nu\) be its normalized Clifford multiplier. Its left coefficients are pulled back from \(L\). Tensoring on the left by \(\Gamma(i^*E)\) gives the module of sections of \(\pi^*i^*E\otimes\mathscr H_\nu\), with operator \(1_E\otimes F_\nu\). Tensoring the tube realization on the right by \(\Gamma_0(E)\widehat\otimes C_k\) gives the tube-supported module of sections of \(\tau^*E\otimes\mathscr H_\nu\), with that identical normal multiplier. In both cases the module is extended by zero from \(U\), and its range is the essential submodule for \(C_0(U)\).

These are the actual Kasparov products. Multiplication/creation by a bundle section commutes with the normal Clifford multiplier, since the bundle is even and the multiplier acts on the other factor. The creation connections are therefore exact. For the right product the positivity test is the nonnegative localized square \(2F_\nu^2\); for the left zero-operator product it is zero. The localized compact defects are the Thom defects tensored with a finite bundle on a compact patch. A finite partition on the coefficient support and norm approximation prove their compactness on the whole module. The product criterion of Section0 thus gives these operators on the completed tensor modules. Essential open extension uses Lemma0.10, without an orthogonal-complement premise.

The pointwise map
\[
e\otimes\psi(v)\longmapsto
T_{x,v}e\otimes\psi(v),\qquad x=\pi(v),
\tag{197}
\]
is an even isometry preserving both coefficient actions and the normal multiplier. Its inverse is pointwise \(T_{x,v}^{-1}\). It sends the dense compact section cores onto one another; support is retained because both maps cover the identity of \(\nu_i\), and their norm is one. Hence it extends to an onto unitary of the whole completed essential modules. All spin-c/Morita factors are untouched. This unitary proves the displayed equality. Neither a characteristic-character comparison nor a proper pullback along \(\pi\) was used. \(\square\)

#### The ordinary-target étale bundle identity

**Lemma 6.0q.** If \(g:S\to Y\) is a smooth local diffeomorphism between Hausdorff second-countable manifolds and \(E\to Y\) is a smooth Hermitian bundle, then
\[
u_S(g^*E)\,g_!=g_!\,u_Y(E)
\quad\text{in }KK(C_0(S),C_0(Y)).
\tag{198}
\]
The map need not be proper or injective.

**Proof.** The identity groupoid on \(Y\) is the zero-dimensional foliation groupoid. Lemma6.0n gives the completed module \(\mathcal E_g\) from its compact source core, with
\[
\langle\xi,\eta\rangle(y)
=\sum_{s\in g^{-1}(y)}\overline{\xi(s)}\eta(s),
\qquad (\xi a)(s)=\xi(s)a(g(s)).
\tag{199}
\]
It retains all the preimages. Its compact left action is local: an injective \(g\)-chart gives the rank-one identity for a supported source coefficient, and a finite partition gives the general compact-support coefficient. This is also the leaf-dimension-zero case of the complete proof of Lemma6.0n.

The right-hand product completes compact sections of \(g^*E\) in the counting inner product. On elementary tensors its map to that completion is
\(\xi\otimes e\mapsto[s\mapsto\xi(s)e(g(s))]\).
Its inner product is exactly the tensor inner product, as follows by summing in a fibre \(g^{-1}(y)\). Its image is dense: on a compact source chart support, choose a finite orthonormal-frame cover of its compact target image and compact target cutoffs equal to one on the smaller images. A source partition writes a section as a finite sum of source scalar coefficients times these pulled-back target sections. The tensor map therefore extends to an onto isometry.

The left-hand product has the same dense section core. The pointwise tensor map from \(\Gamma_0(g^*E)\otimes_{C_0(S)}\mathcal E_g\) has the same inner product and is onto by a source cutoff equal to one on any compact section support. Both coefficient actions match, and both operators are zero. Compact source coefficients are still compact on these modules, by the finite local frame and rank-one proof just given. These zero operators satisfy the connection and positivity conditions and hence represent the two products. The two onto tensor maps identify them as actual cycles. There is no unsupported pullback homomorphism \(C_0(Y)\to C_0(S)\) when \(g\) is nonproper: its right action is the bounded multiplier \(a\circ g\) on the counting module. \(\square\)

#### Identifying the ordinary map inside the full foliation construction

**Lemma 6.0r.** Let \(h:L\to Y\) be smooth, with \(L,Y\) compact and with a specified full stable orientation on \(h^*TY-TL\), represented in the doubled-tangent convention of (13). Regard \(Y\) as the leaf space of its zero-dimensional foliation. The full foliation product of Theorem6.0o is exactly the ordinary graph product
\[
h_!=\operatorname{contract}_{k\to p}(I_!)
 A_n^{C(Y)\widehat\otimes C_p},
\qquad I=(h,j),\quad k=n-\dim L+\dim Y=p+2t.
\tag{200}
\]
Here \(j:L\hookrightarrow\mathbb R^n\) is an even embedding of sufficiently large dimension, and the graph normal has the orientation of Lemma6.0j/Corollary6.0k.

**Proof.** The identity groupoid has no nonunit arrows. Choose the microbundle to be an open graph neighbourhood \(H\subset L\times Y\), with projection to \(L\), section \(x\mapsto(x,h(x))\), and foliation given by constant \(Y\)-coordinate. Its normal quotient is \(h^*TY\): in coordinates the quotient sends \((u,w)\) to \(w-dh(u)\). This is precisely the germ of Lemma6.0l, since its local transverse germs are the ordinary coordinate transitions of \(Y\), and its arrows are units. Theorem6.0o permits this choice of neighbourhood model.

On a uniform normal disk for \(j\), put
\(N\subset p^*\nu_j\), with coordinates \((x,y,\eta)\), and
\(g(x,y,\eta)=(y,j(x)+\eta)\).
The map \(\Phi_j(x,\eta)=j(x)+\eta\) is an injective local diffeomorphism on that disk. Consequently \(g\) is injective: its Euclidean coordinate determines \((x,\eta)\), and its other coordinate determines \(y\). It is a local diffeomorphism, so an open embedding into \(Y\times\mathbb R^n\). The principal-space module for this open inclusion is exactly the essential module \(C_0(N)\) over \(C_0(Y\times\mathbb R^n)\), through extension by zero: the counting fibres have at most one point, and the core inner product is that zero extension. Thus \(g_!=[e_g]\).

The compact section \(k:L\to N\) has \(gk=I\). Its ordered normal \(\nu_j\oplus h^*TY\) maps to the graph normal by the quotient isomorphism of Corollary6.0k. This is exactly the normal identification induced by \(dg\). The specified orientation transported through it agrees with the graph structure constructed in Lemma6.0j; the same \(TL\oplus h^*TY\), Euclidean complement and cancellation recovery maps are used. A normalized tube for \(k\), composed with \(g\), is a normalized tube for \(I\) after that normal identification and metric transport. Hence the explicit Thom/open-extension construction gives
\(k_![e_g]=I_!\).
Lemma6.0e makes this equality independent of the chosen tubes. Contracting the retained Clifford block and applying the fixed Euclidean inverse proves the assertion. All line twists are retained in the same module, rather than reconstructed from their determinant. \(\square\)

#### The full projection formula in both K degrees

**Theorem 6.0s.** With the hypotheses and orientations of Lemma6.0r, for every smooth complex bundle \(E\to Y\),
\[
u_L(h^*E)\,h_!=h_!\,u_Y(E)^{[p]}
\quad\text{in }KK(C(L),C(Y)\widehat\otimes C_p).
\tag{201}
\]
Consequently the ordinary wrong-way map on either K degree satisfies
\(h_!(h^*[E]\,b)=[E]h_!(b)\), for every \(b\in K^*(L)\). The same statement holds for every class of \(K^0(Y)\), by differences of bundles. This is the projection formula used for sphere modification and the ordinary boundary inclusion in Theorem 6.1. Theorem 6.0w proves their composition law, and Theorem 6.0ad below proves the sphere fibre normalization.

**Proof.** Let \(E_n=\mathrm{pr}_Y^*E\to Y\times\mathbb R^n\). Apply Lemma6.0p to the graph embedding \(I=(h,j)\):
\(u_L(h^*E)I_!=I_!u_{Y\times\mathbb R^n}(E_n)^{[k]}\).
The normal orientation is the specified one in Lemma6.0r. The fixed finite contraction acts only on Clifford factors; because \(E\) is even, its tensor reassociation commutes with that contraction, including the retained odd transport of Proposition6.0i.

The Euclidean inverse is a coefficient extension of one fixed inverse, and the bundle is constant in that Euclidean direction. More precisely,
\(u_{Y\times\mathbb R^n}(E_n)=u_Y(E)\boxtimes1_{C_0(\mathbb R^n)}\).
On the completed module this is the tensor unitary \(e(y)\otimes a(v)\mapsto a(v)e(y)\); elementary compact sections are dense, by a finite parameter partition on compact product supports. Exterior product naturality and its associating unitary in Section0 therefore give
\[
u_{Y\times\mathbb R^n}(E_n)^{[p]}
 A_n^{C(Y)\widehat\otimes C_p}
= A_n^{C(Y)\widehat\otimes C_p}u_Y(E)^{[p]}.
\tag{202}
\]
One can also read that map on the Euclidean inverse module: it transports the even \(E_y\) factor past the fixed oscillator/Clifford factor. Both coefficient actions and the oscillator are preserved, since the latter acts only in the Euclidean variable. There is no graded sign from \(E\). Associativity, graph projection and Lemma6.0r now prove the stated KK equality.

Apply a scalar K-class \(b\) on the left. Associativity identifies its product with \(u_L(h^*E)\) with multiplication by \(h^*[E]\); the exact scalar \(K=KK\) and bundle-class identifications of Section0 and the linked programme K-theory proofs establish this in both degrees. The right product with \(u_Y(E)\) is multiplication by \([E]\). Exterior extension of the one retained Clifford block gives the same calculation for odd \(b\). This proves the K-theory formula without a trace or rationalization.

For completeness, smooth bundles cover all coefficient \(K^0(Y)\) classes. The exact projection classification writes any class as a difference of continuous matrix projections. Approximate a continuous Hermitian projection \(P\) uniformly by a smooth Hermitian matrix \(a\) with \(\|a-P\|<1/4\). This approximation follows directly on compact \(Y\): a finite relatively compact chart cover, a smooth partition, and convolution with normalized compact smooth Euclidean bumps approximate each continuous entry uniformly by uniform continuity; symmetrization keeps the matrix Hermitian. Every spectrum of \(a_t=(1-t)P+ta\) lies within \(1/4\) of \(\{0,1\}\), by the resolvent Neumann estimate for a self-adjoint perturbation. The contour projection
\(P_t=(2\pi i)^{-1}\int_{|z-1|=1/2}(z-a_t)^{-1}\,dz\)
is therefore norm continuous in \(t\), begins at \(P\), and ends at a smooth orthogonal projection. Its final range is a smooth Hermitian bundle, and the homotopy preserves the K-class. Linearity thus proves the formula for every virtual coefficient class. \(\square\)

### Composition with its induced stable orientation

The middle ordinary manifold is compact, but its Euclidean product is not. We therefore first prove the variable-radius tube comparison on a noncompact base. The resulting ordered embedding product and completed étale correspondence then give the full foliation composition law. The classical Thom/open-extension construction is also treated by [Carey–Wang, Sections4.1/4.3]; the actual KK modules, both parity contractions and full orientation recovery are proved below.

#### Variable-radius tubes and noncompact embedding composition

**Lemma 6.0t.** The normalized-tube independence and ordered embedding composition of Lemma6.0e/Theorem6.0f hold for arbitrary smooth Hausdorff second-countable source and intermediate manifolds, using their \(C_0\) algebras, specified normal spin-c modules, and whole normalized tubes. No proper base pullback is assumed. In particular they apply to a compact graph embedding into \(M\times\mathbb R^r\), followed by the product embedding \(k\times1_{\mathbb R^r}\).

**Proof: variable compression.** Let \(E\to X\) be the specified metric normal bundle. For any smooth positive function \(a:X\to(0,\infty)\), put
\[
D_aE=\{|v|<a(x)\},\qquad
q_{a,s}(x,v)=\left(x,\frac{v}{\sqrt{1+s^2|v|^2/a(x)^2}}\right),
\quad 0\leq s\leq1.
\tag{203}
\]
At zero this is the identity; at one it is a diffeomorphism onto \(D_aE\), with identity vertical differential at the zero section. For positive \(s\) its image is \(D_{a/s}E\), and its inverse has the denominator \(\sqrt{1-s^2|w|^2/a(x)^2}\). Therefore the joint parameter map is an injective local diffeomorphism onto a relatively open image. Extension by zero gives homomorphisms on \(C_0(E)\), retaining the constant Clifford algebra.

They form one norm-continuous interval homomorphism. For a compact source support \(K\subset E\), its joint image under \((s,v)\mapsto(s,q_{a,s}(v))\) is compact and lies inside the open image. The inverse formula proves continuity there. At the image boundary a nonzero level \(\epsilon\) would have preimage in the compact set \(\{|f|\geq\epsilon\}\), whose joint image is closed and remains inside that open image. Thus the extension is continuous there with value zero. Uniform continuity on a compact neighbourhood of the support, followed by density and the contractive homomorphism bound, proves norm continuity for every \(f\in C_0(E)\). No lower bound for \(a\) on all of \(X\) is needed. The global Thom equivalence gives
\[
\beta_{D_aE}[e_{D_aE\hookrightarrow E}]=\beta_E,
\qquad \beta_{D_aE}=\beta_E[e_{q_{a,1}}].
\tag{204}
\]
Both Thom classes are invertible by the global theorem and the disk algebra isomorphism, so the indicated disk inclusion has the same actual cancellation as in Lemma6.0e.

**Proof: one global comparison homotopy.** Given two normalized tubes, their comparison germ is \(g=\tau_1^{-1}\tau_0\). The maps \(g_s=\lambda_{1/s}g\lambda_s\), with \(g_0=1\), have the smooth extension proved in (153). Their zero-section derivative has blocks \(\left(\begin{smallmatrix}I&sA_x\\0&I\end{smallmatrix}\right)\). For each \(x\), compactness of the parameter interval gives a base neighbourhood \(O_x\) and a positive radius on which every \(g_s\) is defined, locally invertible and has output vector norm less than one. Choose a countable locally finite refinement and a smooth partition subordinate to these neighbourhoods. On each refined chart choose a smaller admissible constant radius \(a_i<1/2\), and set \(a(x)=\sum_i\theta_i(x)a_i\). At a point, each active radius is admissible there, and the weighted average is no greater than their largest admissible radius. Hence every \(g_s\) is defined and locally invertible on \(D_aE\), with image in \(D_1E\). This is the full positive-minorant construction, rather than a uniform radius assertion on a noncompact base.

Each positive-parameter \(g_s\) is injective as a composition of injective maps; \(g_0\) is the identity. The joint map is therefore an injective local diffeomorphism of the relative interval domain. The same compact-support and compact-level-set proof just given gives one homomorphism
\(C_0(D_aE)\widehat\otimes C_k\to C(I,C_0(D_1E)\widehat\otimes C_k)\).
Its endpoints are the inclusion and the comparison map. Multiply by \(\beta_{D_aE}\), use the variable disk cancellation, and extend into the ambient manifold as in (154). This proves global tube independence directly by a single interval cycle; local equality on each compact set is not being mistaken for a global KK proof.

**Proof: the actual embedding product.** Write \(i:X\hookrightarrow Y\), \(j:Y\hookrightarrow Z\), \(E=\nu_i\), \(F=\nu_j\), with normal ranks \(k,\ell\). Fix whole normalized tubes \(\tau:E\to Y\), \(\sigma:F\to Z\). A spin-c connection on \(F\) gives full smooth transport \(T_{x,v}\) along \(t\mapsto\tau(x,tv)\). Every finite parameter has a compact path interval, so the exact DG-FND transport proof applies without global compactness of \(Y\). As in (157),
\(\Psi(x,v,w)=(\tau(x,v),T_{x,v}w)\)
is a diffeomorphism \(E\oplus i^*F\to F|_{\tau(E)}\), and \(\sigma\Psi\) is a whole tube for \(ji\), with the ordered normal splitting from its derivative.

The global \(F\)-Thom cycle restricted to \(C_0(\tau(E))\) has precisely its essential section module over \(F|_{\tau(E)}\). Compactly supported local section approximations and an ideal approximate identity prove both inclusions of that range. Its multiplier, compact defects and creation connections restrict exactly, as in (158). Transport through the actual diffeomorphism \(\Psi\) gives the coefficient Thom cycle \(\gamma\) for \(i^*F\) over the old total space \(E\), and
\([e_\tau]\beta_F^{[k]}=\gamma[e_\Psi][e_{F|_{\tau(E)}\hookrightarrow F}]\).
Here the only total-space transport is a homeomorphism and hence proper between the two displayed total spaces. There is no assumed \(C_0\) pullback along \(i\) or a bundle projection.

The ordered global bundle product (147)–(148) gives \(\beta_E\gamma=\beta_{E\oplus i^*F}\). Its proof uses compact old creation sections and base cutoffs. On their compact base supports the connection estimates are uniform, and at joint fibre infinity they decay; density proves the complete connection condition. Its positivity is the nonnegative expression (147). Thus it applies to this noncompact base in its proved generality. Both full spin-c transition modules tensor in order, with even transition units. Associativity and open extension now give
\[
i_!j_!^{[k]}=
\beta_{E\oplus i^*F}[e_{\sigma\Psi}]=(ji)_!
\quad\text{in }KK(C_0(X),C_0(Z)\widehat\otimes C_{k+\ell}).
\tag{205}
\]
The superscript retains the old \(C_k\) before the new \(C_\ell\).

Finally, the triangular splitting maps \((v,w)\mapsto(v+sAw,w)\) and their inverses are uniformly bounded on every compact base set, even though no global bound is claimed. Compact input supports consequently have uniformly compact transported supports. Their total-space algebra maps form a norm-continuous interval family by continuity and density. Transporting the metric and specified orientation gives the identical multiplier/defect homotopy on each compact coefficient support, and hence the full cycle. Connections and metrics have their previously proved interval comparisons. The variable-radius tube argument finishes their comparison. This proves the same auxiliary-choice assertion as Theorem6.0f in the stated larger generality. \(\square\)

#### Any compatible étale realization has the canonical germ

**Lemma 6.0u.** Let \(L\) be compact, \(J:L\hookrightarrow\mathbb R^N\) an even Euclidean embedding, and \(f:L\to V/F\) a smooth actual arrow cocycle. Suppose \(k:L\hookrightarrow Z\) is an embedding into a Hausdorff second-countable manifold and \(g:Z\to V/F\times\mathbb R^N\) is étale, with \(gk=(f,J)\) as an arrow-cocycle map. After shrinking near the compact section, this factorization is equivalent to the canonical factorization of Lemmas6.0l–m by a germ diffeomorphism and an actual target-arrow equivalence. If its full normal orientation is the transported specified canonical normal module, its full KK product is the canonical product, before and after the fixed Clifford contraction and Euclidean inverse.

**Proof.** The Euclidean coordinate of \(g\) is a single smooth map \(z:Z\to\mathbb R^N\), because every transition arrow retains that coordinate. A Euclidean tube \(\Phi_J(x,\eta)=J(x)+\eta\) is an open embedding near the compact zero section. Restrict \(Z\) to \(z^{-1}(\operatorname{im}\Phi_J)\) and define \((x(n),\eta(n))=\Phi_J^{-1}(z(n))\). These are smooth, and on \(k(L)\) are \((x,0)\).

Choose finitely many target foliation boxes and local cocycle lifts near that section. Replace each lift by its fixed plaque cross-section using its short plaque arrow, as in Lemma6.0l. At the section the given equality of cocycles supplies equivalence arrows to the fixed lifts of \(f\). Their path charts give fixed transverse holonomy branches near each centre; compose the local transverse coordinate with the inverse of that branch. This aligns it with the transverse coordinate of \(f\) at the centre. Where two aligned charts overlap, their transition branch and the canonical branch have the same germ at that centre, including the actual holonomy arrow. The finite closed-overlap shrinking proof of Lemma6.0l applies to these comparisons: branch agreement and arrow agreement are open, and compactness of the centre overlaps gives one common neighbourhood on which they agree.

The resulting local maps
\(n\mapsto(x(n),y_a(n),\eta(n))\)
therefore glue into the canonical neighbourhood \(N_J\). They also glue the actual arrow equivalence of the two target cocycles. They send \(k(x)\) to \(k_J(x)\). Their derivative is invertible: the étale coordinates of \(g\) are \((y_a,z)\), and replacing \(z\) by \((x,\eta)\) uses the Euclidean tube inverse. Thus this is a local diffeomorphism near the entire section.

It is injective on a smaller neighbourhood. Use a tubular disk of the compact embedding \(k\). If no common smaller disk were injective, unequal pairs with normal norms tending to zero would have equal images. Their bases have convergent subsequences in \(L\). The equal image limits force those two base limits to coincide, since the map is identity on the section. Local invertibility at that common limit contradicts equality of the late pairs. Consequently the map is a diffeomorphism of open neighbourhoods of the two sections.

Its normal derivative is the actual normal quotient isomorphism. Transport the specified normal module and metric through it. Composing a normalized tube with the diffeomorphism and using tube independence identifies the embedding factors; the total-space linear normal isomorphism is a homeomorphism and hence proper. Lemma6.0n's explicit arrow-equivalence unitary identifies the middle modules. Its essential restriction identity and the compact-source small-tube identity allow both products to use these smaller neighbourhoods. The fixed finite contractions and inverse then preserve their equality. \(\square\)

#### The induced stable orientation is a full module

Let \(h:L\to M\) be smooth with \(L,M\) compact, and let \(f:M\to V/F\). Put \(l=\dim L\), \(m=\dim M\), \(T=h^*f^*\tau\). Prescribe the stable orientations of \(h\) and \(f\). Choose even Euclidean embeddings \(u:L\hookrightarrow\mathbb R^r\), \(j:M\hookrightarrow\mathbb R^n\), with positive graph normal ranks. The actual normal modules of Lemma6.0j are
\[
E_h=\nu_u\oplus h^*TM,\quad k_h=r-l+m,
\qquad E_f=\nu_j\oplus f^*\tau,\quad k_f=n-m+q.
\tag{206}
\]
The graph \(I_h=(h,u)\) is an embedding into \(M\times\mathbb R^r\). Follow it by \(k_j\times1\) into \(N_j\times\mathbb R^r\). The composite \(K=(k_jh,u)\) has ordered normal
\(E=E_h\oplus h^*E_f\), through the normal exact-sequence splitting of Lemma6.0t. Its full module is
\(\mathscr M_E=\mathscr M_{E_h}\widehat\otimes h^*\mathscr M_{E_f}\), with right block \(C_{k_h}\widehat\otimes C_{k_f}=C_k\), \(k=n+r-l+q\). The two Euclidean blocks are even.

**Lemma 6.0v.** This ordered normal module induces precisely a stable orientation of
\((fh)^*\tau-TL\), the sum of \(h^*TM-TL\) and \(h^*(f^*\tau-TM)\). It can be converted to the doubled-tangent convention by explicit finite Morita cancellation, and Lemma6.0j recovers the entire normal module from it. Its class is independent of \(u,j\) and the exact-sequence splitting. A line twist \(H\) of \(h\) and a line twist \(D\) of \(f\) give the composite twist \(H\otimes h^*D\), not merely its squared determinant.

**Proof: the normal quotient.** The étale map \(g_j\times1\) has Euclidean component \((j(x)+\eta,w)\). Its composite with \(K\) is \((fh,J)\), where \(J=(jh,u):L\hookrightarrow\mathbb R^{n+r}\) is an embedding because \(u\) already is one. The transverse derivative identifies the normal quotient of \(K\) with
\((T\oplus\mathbb R^{n+r})/d(fh,J)(TL)\).
The transverse differential \(d^\perp(fh):TL\to T\) is global: in each fixed holonomy branch the derivative of the local transverse lift changes by exactly the derivative transition of \(T\).

Writing \(U=TL\) with a metric and \(G_J=dJ^*dJ\), the explicit quotient isomorphism to \(E_J=\nu_J\oplus T\) sends
\[
[(t,z)]\longmapsto(\eta,t-d^\perp(fh)v),
\qquad v=G_J^{-1}dJ^*z,\quad \eta=z-dJv.
\tag{207}
\]
Adding \((d^\perp(fh)a,dJa)\) changes \(v\) to \(v+a\) and leaves both output components fixed. Its inverse sends \((\eta,t)\) to \([(t,\eta)]\), so it is a smooth quotient isomorphism. Compose it with the ordered normal splitting to obtain \(\chi:E\to E_J\). Transport the metric and full module through \(\chi\). This supplies the specified positive-rank spin-c normal structure in Lemma6.0u.

The stable normal identity is an identity of actual ordered bundle representatives:
\[
(E_h-\underline{\mathbb R^r})+
(h^*E_f-\underline{\mathbb R^n})
=E-\underline{\mathbb R^{n+r}}
\simeq T-U.
\tag{208}
\]
The swap of the two even Euclidean blocks uses their canonical complex exterior unitary. Their stable orientation is the ordered tensor of the full normal modules followed by that same Euclidean cancellation. This defines the induced orientation of the sum in (13), with its transition phases; the equality is not inferred from characteristic classes.

**Proof: exact double-convention recovery.** Let \(Q_U=U\oplus U\) have the canonical interleaved complex module \(\mathscr M_{Q_U}\). Before using \(\rho_J\), compare the metric transported by \(\chi\) with the orthogonal sum metric on \(\nu_J\oplus T\) by Corollary6.0d's positive metric isometry. This transports the full module, including its line, rather than treating a nonorthogonal map as a Clifford *-map. Let \(L_c=U\oplus T\), and use Lemma6.0j's explicit isometry \(\rho_J:E_J\oplus Q_U\to L_c\oplus\mathbb R^{n+r}\). Carry \(\mathscr M_{E_J}\widehat\otimes\mathscr M_{Q_U}\) through \(\rho_J\) to a module
\(P:A_{L_c}\widehat\otimes C_{n+r}\rightsquigarrow C_{l+q}\widehat\otimes C_{n+r}\).
On its right this uses the consecutive-generator regrouping
\(C_k\widehat\otimes C_{2l}=C_{k+2l}=C_{l+q+n+r}=C_{l+q}\widehat\otimes C_{n+r}\); no generator permutation is made by this regrouping. Write \(S=\Lambda^*\mathbb C^{(n+r)/2}\), with the fixed interleaved Clifford action. Define
\[
\mathscr M_{L_c}=
(A_{L_c}\widehat\otimes S^*)
\widehat\otimes_{A_{L_c}\widehat\otimes C_{n+r}}P
\widehat\otimes_{C_{l+q}\widehat\otimes C_{n+r}}
(C_{l+q}\widehat\otimes S).
\tag{209}
\]
Both finite evaluation maps of Lemma0.12 give the exact recovery
\(\mathscr M_{L_c}\widehat\otimes C_{n+r}\cong P\), because the canonical even Euclidean module \(S\widehat\otimes S^*\) is the right regular Clifford unit through its actual matrix-unit evaluation. Pulling this recovery through \(\rho_J\) and cancelling \(\mathscr M_{Q_U}\) by the two recovery maps of Lemma6.0j returns exactly \(\mathscr M_{E_J}\). Thus the canonical graph construction with this induced stable orientation has the original full product normal module.

When \(l+q>0\), the local even/odd Clifford reconstruction in Lemma6.0j gives the principal spin-c interpretation of \(\mathscr M_{L_c}\). When \(l+q=0\), the cancellation can give a graded line, including an odd line. Its grading is part of the stable orientation. The canonical rank-zero unit is still the even trivial line; an odd line is not declared to be that unit. Use a positive even stabilized normal representative, as above, to interpret such rank-zero stable data. Accordingly the doubled-tangent phrasing after (13) is understood after even stabilization when necessary. Lemma6.0j's actual finite construction (177)–(178) also applies directly to this graded line: the only principal reconstruction it needs is for the positive normal rank \(k\). The original stable virtual-bundle convention is retained, including its orientation sign.

**Proof: independence and twists.** Interpolate two choices of \(u\) by the common even cos/sin embedding family, and do the same for \(j\). Lemmas6.0g–j give the actual normal families of \(I_h\) and \(k_j\); their product normal exact sequence, the displayed \(\chi\), \(\rho_J\) and both finite cancellations vary smoothly. On a common Euclidean stabilization they therefore give a full \(\mathscr M_{L_c}\) family over \(I\times L\), with fixed left Clifford bundle and fixed right algebra. Parameter transport can be chosen to preserve these actions. In positive rank, choose a spin-c connection whose projected orthogonal connection is the fixed product connection: from any connection subtract the equivariant lift of the difference of its orthogonal projection and that product connection, using \(\mathfrak{spin}^c=\mathfrak{so}\oplus i\mathbb R\). Its interval transport has fixed orthogonal frame and hence intertwines the Clifford action. In rank zero use the unitary connection of the graded line, with its locally constant grading retained. The exact DG-FND transport proof gives even endpoint module isomorphisms. Under a further even Euclidean stabilization the added canonical modules cancel by the same finite evaluations and exterior wedge unitary. Thus the induced stable orientation is independent of these choices.

Splittings have the explicit triangular interpolation of Lemma6.0t, so give the same full comparison family. If the two original orientations are twisted by \(H,D\), their normal modules are tensored by those lines, by (180). The product normal module is therefore tensored by \(H\otimes h^*D\). The displayed cancellations move only Clifford/spinor factors, so return that same line, including any torsion and its grading. This proves all assertions. \(\square\)

#### Composition of the actual foliation products

For parity \(a,b\in\{0,1\}\), write \(x\# y\) for the ordered graded KK product retaining \(C_a\) before \(C_b\), followed by the fixed contraction to \(C_{a+b\bmod2}\). If both are odd, this is the actual \(C_1\widehat\otimes C_1=C_2\) matrix Morita evaluation on \(\Lambda^*\mathbb C\). Moving a retained odd factor past preceding even pairs uses the even triple-cycle unitaries of Proposition6.0i. This is the convention already used in Section0; all products below can equivalently retain the full normal Clifford blocks until the final contraction.

**Theorem 6.0w.** With the hypotheses and the induced full stable orientation of Lemma6.0v,
\[
h_!\# f_!=(f\circ h)_!
\quad\text{in }KK(C(L),A\widehat\otimes C_p),
\qquad p=q-l\pmod2.
\tag{210}
\]
This is an actual KK equality for the completed foliation correspondence. It applies in particular to the sphere projection and the boundary inclusion needed for geometric-cycle descent, with their induced stable orientations.

**Proof.** Use the factorization of \(f\) by \(k_j\), \(g_j\), the Euclidean inverse \(A_n\), and the ordinary graph factorization of \(h\) by \(I_h=(h,u)\), \(A_r\). The latter is exactly the full zero-foliation construction by Lemma6.0r. Lemma6.0t identifies the product embedding class of \(I_h\) and \(k_j\times1_{\mathbb R^r}\) with the class of \(K=(k_jh,u)\), carrying precisely the ordered normal module of Lemma6.0v. It applies despite the noncompact intermediate \(M\times\mathbb R^r\).

The middle module of the product foliation is the onto isometry
\(\mathcal E_{g_j\times1}\cong\mathcal E_{g_j}\widehat\otimes C_0(\mathbb R^r)\)
proved in Theorem6.0o. It preserves the two actions, every counting lift and the zero operator. For its principal space this is exactly \(P_{g_j}\times\mathbb R^r\), not a numerical fibre identification. Thus
\((g_j\times1)_!=(g_j)_!\boxtimes1_{C_0(\mathbb R^r)}\).

Exterior product naturality in Section0 moves the even coefficient inverse \(A_r\) successively through \(k_j\) and \(g_j\). Its associating unitary retains the old normal block before the new normal block. All creation/positivity requirements of these products are already proved for the embedding, the actual zero-operator correspondence, and the ordered Bott classes. Therefore, retaining the full normal blocks until the stated contraction, the product is
\[
\begin{aligned}
h_!\# f_!
&=\operatorname{contract}_{k\to p}
 \bigl((I_h)_!(k_j\times1)_!^{[k_h]}
       (g_j\times1)_!^{[k]}\bigr)
 A_r^{D_n\widehat\otimes C_p}A_n^{A\widehat\otimes C_p}\\
&=\operatorname{contract}_{k\to p}
 \bigl(K_!(g_j\times1)_!^{[k]}\bigr)
 A_{n+r}^{A\widehat\otimes C_p},
\qquad D_n=A\otimes C_0(\mathbb R^n).
\end{aligned}
\tag{211}
\]
The Euclidean target is ordered \(\mathbb R^n\) then \(\mathbb R^r\); its inverse first removes the new \(r\) block and then the old \(n\) block. That is exactly the inverse composition (148), with its fixed even wedge unitary. The two inverse identities, rather than equality of an integer index, identify it with the fixed \(A_{n+r}\).

For clarity, late and early Clifford contraction give the same displayed product through actual maps. Tensor reassociation combines the two ordered full normal algebras as \(C_{k_h+k_f}\). Their canonical matrix factors contract by \(\xi\otimes\eta^*\mapsto\theta_{\xi,\eta}\) and \(\xi^*\otimes\eta\mapsto\langle\xi,\eta\rangle\). The exterior wedge isometry identifies consecutive even pairs. For a retained odd factor, the fixed even three-generator cycles move it past such pairs and preserve the normalized odd central volume, as proved in Proposition6.0i. If two odd factors remain, their ordered \(C_2\) evaluation is the specified final matrix contraction. Composing these associating/evaluation isometries identifies the early contracted tensor module with the late contracted one, preserving both actions and gradings. No ungraded odd permutation is used and no scalar sign is dropped.

Finally, \(g_j\times1\) is étale and its composite with \(K\) is \((fh,J)\), with \(J=(jh,u)\) an even embedding. Lemma6.0v proves that its full normal module is the canonical one for the induced stable orientation. Lemma6.0u therefore identifies this actual factorization product with the canonical \((fh)_!\), through a germ diffeomorphism, a completed arrow-equivalence unitary and essential restriction. The last display proves the claimed equality. \(\square\)

![Actual tube projection, ordered foliation composition and stable orientation grading](../figures/foliation-projection-composition.png)

**Figure 6.0d.** Lemmas6.0p–v, Theorems6.0s/6.0w, equations (196)–(211), Exercises44–45. The first panel gives the actual norm-one coefficient transport on the entire normalized tube, including the exact cylinder phase. The second panel retains the product embedding order and every holonomy lift in the completed étale module. The third panel gives the normal quotient, positive metric comparison, full finite-module recovery and ordered inverse cancellation. All Clifford/module tensors are graded, and \(\#\) includes the stated parity contraction. The last panel is the exact point-to-circle-to-point example with the prescribed normal frames; it distinguishes the two stable orientation gradings of virtual rank zero. These are diagrams of maps and modules. The classical Thom/open-extension method is treated by [Carey–Wang, Section4]; all displayed KK and grading comparisons are proved above.

Full-size composition PNG · Editable composition SVG.

### Boundaries, sphere bundles and geometric cycles

The two geometric relations require more than a fibrewise index. A boundary needs its entire map cocycle, bundle and spin-c structure to extend to a double. A sphere bundle needs a global clutch bundle and the inverse full orientation, including any line twist. We prove these constructions in that order.

The geometric prerequisites are Lemma 3.1 and the tubular construction in Manifold duality, the diagonal and Wu classes, and Sections 1–4 of Connections and parallel transport. Theorem 2.1 and Section 3 of Topological K-theory of spaces, pairs and vector bundles provide the full relative bundle comparison and disc compactification. The KK and Clifford inputs are the complete preceding arguments of Sections 0 and 6.

These geometric relations and a zero-section charge construction are also treated by [Liu, Section 2]. The full foliation correspondence, double and ordered orientation comparisons used here are proved below.

#### Embeddings and the graph construction

**Lemma 6.0x.** Let \(i:M\hookrightarrow Y\) be a smooth embedding of
manifolds without boundary, with \(M\) compact. Give its normal quotient \(E\) a specified full stable
spin-c orientation, allowing the canonical even zero-rank module. Give
\(i\) the resulting stable orientation of \(i^*TY-TM\). Its ordinary
graph wrong-way class is the actual tubular class
\[
 i_!=\beta_E[e_\tau]\in KK^{\operatorname{rank}E}(C(M),C_0(Y)).
\tag{216}
\]
In particular the canonically oriented identity map has class \(1_{C(M)}\).
Any specified line twist, including the grading of a stable rank-zero line,
is retained in this comparison.

**Proof.** Take an even Euclidean embedding \(j:M\to\mathbb R^n\),
with \(n\) large enough that every normal rank used below is positive.
The maps
\[
 I_s(x)=(i(x),s j(x)),\qquad 0\le s\le1,
\tag{217}
\]
are embeddings on an open interval about \([0,1]\), even at zero:
the first component already is an embedding. Their normal quotients form
a smooth bundle on that whole interval. Fix a splitting
\(i^*TY=di(TM)\oplus E\), and identify the quotient by
\[
 E\oplus\underline{\mathbb R^n}\longrightarrow\nu_{I_s},
 \qquad(\eta,z)\longmapsto[(\eta,z)].
\tag{218}
\]
This is an isomorphism. To reduce \((y,z)\) to this form, uniquely write
\(y=di(v)+\eta\) and subtract \((di(v),s dj(v))\); injectivity follows
from the same direct-sum decomposition. The matrices and their inverses
depend smoothly on \(x,s\).

Here is the full orientation check, rather than an inference from the
existence of a normal bundle. Put \(W=TM\), \(T=i^*TY\), and give
\(Q=W\oplus W\) its fixed interleaved complex-double module. The
orientation of \(E=T-W\) determines the orientation module on
\(L=W\oplus T\) by the inverse finite construction of (177)–(178).
Over (218) its recovery is transported through the invertible bundle map
\[
 (\eta,z,u,v)\longmapsto(u,di(v)+\eta,s dj(v)+z)
 : (E\oplus\mathbb R^n)\oplus Q\longrightarrow L\oplus\mathbb R^n.
\tag{219}
\]
Transport the metrics by the positive comparison of Corollary 6.0d before
extending this map to a Clifford *-isomorphism. Pull back the full module
\(\mathscr M_L\widehat\otimes C_n\) and apply \(\mathcal C_Q\).
Both evaluation maps of (177)–(178) give a smooth full normal module
family. At \(s=0\) the map is the unstabilized recovery followed by the
identity on the new Euclidean block. The even block move and exterior
evaluation in (179) identify its module with
\(\mathscr M_E\widehat\otimes C_n\), in that order.

At \(s=1\) compare it with the specified graph construction. Use the
complement \(\nu_j\oplus T\) to the graph tangent, sending
\((\zeta,t)\) to \([(t,\zeta)]\). Its recovery before orthonormalization
is
\((\zeta,t,u,v)\mapsto(u,t+di(v),dj(v)+\zeta)\).
The recovery map (176) omits \(di(v)\) in its second coordinate. These
two recoveries are joined by the explicit invertible target shears
\[
 (u,t,z)\longmapsto
 (u,t+\lambda\,di((dj^*dj)^{-1}dj^*z),z),
 \qquad0\le\lambda\le1.
\tag{220}
\]
They are invertible because their diagonal blocks are identities. The
positive factors making \(dj\) isometric are joined to their identities
inside the positive invertible cone. Changes between the two complements
are affine splittings of the same quotient; their convex interpolations
remain splittings. Applying the same positive metric transport, full
module pullback and \(\mathcal C_Q\) to these paths gives actual smooth
orientation-module paths between (219) and (176). On an interval a
principal spin-c connection gives an even endpoint intertwiner, by the
full associated-transport proof in the connection lesson, Sections 3–4. The underlying
orthogonal connection can be prescribed to the chosen metric/splitting
path: its lift is unique in the spin Lie algebra, and the central
connection is retained. Thus these are precisely the prescribed full
orientations. A line tensor is unaffected by every cancellation map;
an odd line is not replaced by an even unit.

Lemma 6.0g applied to (217), with this full family, identifies the embedding
classes of its endpoints. At zero the embedding is the composite of
\(i\) and the zero section \(Y\hookrightarrow Y\times\mathbb R^n\).
The ordered embedding product of Lemma 6.0t, and its even stabilization
comparison, identify its class with \(i_!\widehat\boxtimes\beta_n\).
This remains valid when \(Y\) is noncompact: the compact source is \(M\),
and the intermediate tube uses the positive variable radius of that
lemma. Multiplying by the fixed Euclidean inverse \(A_n\) gives
\(i_!\). At one it gives exactly the graph definition. Its full
Euclidean product cancels by both inverse identities, not just by a
numerical index. This proves (216). For the identity embedding the normal
has rank zero, its tube is the identity, and its canonical Thom module
is the even standard module with zero operator. Its class is the identity.
\(\square\)

#### A separating hypersurface has a null Thom unitary

**Lemma 6.0y.** Let \(D\) be a compact smooth manifold without boundary and
let \(M=\partial Q\subset D\) be the boundary of a codimension-zero
compact smooth submanifold \(Q\). Orient the normal line by the outward
unit normal and give it the canonical spin-c structure. For
\(i:M\hookrightarrow D\),
\[
 (i_!)_*[1_M]=0\quad\text{in }K_1(C(D)).
\tag{221}
\]
This is an integral statement and holds for a disconnected boundary.

**Proof.** Choose a collar with the normal coordinate increasing from
\(Q\) to its complement, and use its whole-line normalized tube
\(\tau:M\times\mathbb R_v\to D\). Its negative end approaches the
inside edge of the collar and its positive end approaches the outside
edge. The full canonical normal module is the right regular \(C_1\)
module. The scalar Thom cycle of Corollary 6.0d is consequently
\((C_0(M\times\mathbb R)\widehat\otimes C_1,1,s(v)\varepsilon)\),
where \(s(v)=v/\sqrt{1+v^2}\). Theorem 0.14, including its naturality
for the possibly degenerate extension-by-zero homomorphism, sends its
scalar class to the explicit unitary
\[
 U(\tau(x,v))=\exp(\pi i(1+s(v))),\qquad
 U=1\quad\text{off the collar}.
\tag{222}
\]
The difference from identity tends to zero at both collar ends, so this
is exactly the extended compact-unitary class. Lemma 6.0x identifies it
with the ordinary map construction, with the same normal orientation.

The real function \(\Theta:D\to[0,2\pi]\) is defined as follows. It
is zero on the part of \(Q\) beyond the inside collar edge, equals
\(\pi(1+s(v))\) on the collar, and is \(2\pi\) on the outside part.
Its one-sided limits are respectively zero and \(2\pi\), hence it is
continuous across both edges. It is defined independently on all
components, including components disjoint from \(M\). The formula
\[
 U_t(x)=\exp(i t\Theta(x)),\qquad0\le t\le1,
\tag{223}
\]
is a norm-continuous path of unitaries in \(C(D)\), since
\(\|U_t-U_{t'}\|\le2\pi|t-t'|\). It joins identity to \(U\).
Thus its \(K_1\)-class is zero. The compactness of \(D\) matters:
we are homotoping unitaries in the unital algebra \(C(D)\). We have not
asserted that \(U_t-1\) vanishes at infinity in an arbitrary noncompact
ambient manifold. \(\square\)

**Corollary 6.0z.** In Lemma 6.0y let \(g:D\to V/F\) have a specified
full stable orientation and let \(E_D\) be a complex bundle on \(D\).
Give \(g i\) the orientation induced in Theorem 6.0w, with the
canonical outward normal first. Then
\[
 (g i)_!([i^*E_D])=0.
\tag{224}
\]

**Proof.** The ordinary projection formula and actual composition give
\(i_!([i^*E_D])=[E_D]i_!([1_M])=0\) and
\((g i)_!([i^*E_D])=i_!([i^*E_D])\widehat\otimes_{C(D)}g_!\).
Here the first scalar class is odd; Theorem 6.0w includes the exact
ordered Clifford contraction for this degree. Formula(221) therefore
proves (224). \(\square\)

#### Extending all the bordism data across a double

The boundary orientation of a K-oriented map \(f:Q\to V/F\) is
specified by
\(f^*\tau-T(\partial Q)=(f^*\tau-TQ)|_{\partial Q}\oplus\mathbb R_{\rm out}\),
with the normal line put first when taking the embedding-then-map
product. The finite full-module evaluation of Lemma 6.0v fixes the block
order. This defines the induced orientation, rather than choosing a
new structure whose determinant only agrees with it.

**Lemma 6.0aa.** Let \(Q\) be a compact smooth manifold with boundary,
\(f:Q\to V/F\) a smooth map cocycle with a specified stable orientation,
and \(E\to Q\) a complex bundle. There exist a compact smooth double
\(D\), a smooth map cocycle \(g:D\to V/F\), a complex bundle
\(E_D\to D\), and a full stable orientation of \(g\), such that on
\(\partial Q\) their map, bundle and induced boundary orientation are
exactly those of \((Q,E,f)\). The hypersurface \(\partial Q\) separates
the first copy of \(Q\) from its reflected second copy.

**Proof: collar and constant data.** In a finite boundary-chart cover,
choose the coordinate inward vectors and multiply them by a smooth
partition of unity. Their normal component is strictly inward along
the boundary; every coefficient is nonnegative and their sum is one.
Extend their finite sum to a smooth vector field on a neighbourhood
inside \(Q\). The full local-flow proof of Lemma 3.1 in the manifold-duality lesson applies in
boundary charts by extending the smooth coefficients across their
coordinate hyperplanes. Compactness of the boundary gives a common
positive existence time. The flow map
\(c: \partial Q\times[0,\delta)\to Q\) has invertible derivative at
time zero, since its time derivative is transverse. Shrink \(\delta\)
so it is locally invertible everywhere on this strip. It is injective
after a further shrink: if distinct pairs at times tending to zero
had equal images, compactness gives a common limiting boundary point,
and its local inverse contradicts the equality. A failure of all
smaller radii gives precisely such a sequence. Thus it is a smooth
collar with a smooth inverse onto its relative open image.

Choose a smooth \(\alpha:[0,\delta)\to[0,\delta)\) equal to zero near
zero and to \(t\) for \(t\) beyond a smaller collar. It can be
\(t\chi(t)\), where \(0\le\chi\le1\) is a smooth cutoff. Define
\(r(c(x,t))=c(x,\alpha(t))\), and put \(r=1\) off this collar. This
is smooth. The interpolation
\(\alpha_u(t)=(1-u)t+u\alpha(t)\) gives a smooth map homotopy fixed on
the boundary. Replacing the map cocycle by \(f r\) makes every lift
and every transition arrow independent of \(t\) near zero. This is
done by literal pullback of the cocycle, so all triple identities and
all holonomy arrows remain. It is not merely a homotopy of its leaf
images. Pullback by \(r\) similarly gives a coefficient bundle equal
to the boundary bundle's product near zero, with its exact specified
boundary identification.

We must also transport the orientation without using the singular
derivative \(dr\). On
\(L_u=TQ\oplus(f r_u)^*\tau\) over \(Q\times[0,1]\), a metric
connection on \(f^*\tau\) transports the second summand along the
paths \(f\circ r_u\) in their cocycle bundles; equivalently pull back
the fixed bundle connection along the map \((x,u)\mapsto r_u(x)\)
in \(Q\). Leave the first summand fixed. This supplies isometries
from \(L_0\) to \(L_u\), with the identity on the boundary. Transport
the entire prescribed orientation, including its scalar transition
line, along them. Near the boundary use a product metric and the
collar splitting
\(L_1=\mathbb R_{\rm out}\oplus(T\partial Q\oplus f_{\partial}^*\tau)\).
Principal parallel transport in the collar direction identifies its
orientation bundle with its restriction at time zero. Its projected
orthogonal connection is chosen to make this product frame constant;
the spin connection is its lift and its central connection is retained.
The lift exists because the differential of the spin cover is an
isomorphism on the orthogonal Lie algebra. On this restricted
principal bundle the construction uses the actual full connection,
whose existence and complete-path transport are proved in the connection lesson,
Theorems 3.1/4.1 and Corollary 4.2. The associated transports are even
unitaries, not determinant-line identifications. No orientation is
inferred from \(dr\).

The connection arguments also apply on a compact manifold with
boundary: in each relative principal product chart choose the zero
potential and average the local connection forms with a smooth
relative partition of unity. The reproduction and equivariance
identities are unchanged. Their coordinate coefficients extend
across the boundary hyperplane, so the full finite-interval ODE
and parameter transport proof applies there. Thus no extension of
the original orientation beyond \(Q\) has been presumed at this step.

**Proof: the reflected principal structure.** The rank
\(k=\dim Q+q\) of \(L_1\) is at least one when the boundary is
nonempty. A stable orientation on this positive-rank bundle is its
actual principal spin-c structure by Lemma 6.0j's reconstruction.
In the positive real Clifford algebra let \(e_1^2=1\), and let
\(R_0\) negate the first coordinate while fixing the others. The map
\[
 \kappa([s,z])=[e_1 s e_1,z]
\tag{225}
\]
is an automorphism of \(\mathrm{Spin}^c(k)\). It preserves the central
\(U(1)\) coordinate and the relation \((-s,-z)\sim(s,z)\).
Its projection is \(A\mapsto R_0 A R_0\): on single Clifford vectors
conjugation by \(e_1\) is \(-R_0\); the two minus signs cancel on
every even Clifford product defining the spin group. The assertion
also holds in rank one, where this automorphism is the identity.

If \(b_a\) are local oriented orthonormal frames of \(L_1\), with
spin-c transition units \(g_{ab}\), give the same real bundle its
opposite orientation using the frames \(b_a R_0\) and the full
transition units \(\kappa(g_{ab})\). Their projections are precisely
the transitions of those opposite frames. The cocycle identity is
preserved by the automorphism, and the unchanged scalar coordinates
retain every line twist. This is a specified full principal structure
on the second copy, denoted \(P^{\rm opp}\).

On the collar choose frames whose first vector is the outward normal.
Their transitions stabilize \(e_1\). Each such spin transition is
an even product of the remaining Clifford generators, and therefore
commutes with \(e_1\); hence \(\kappa(g_{ab})=g_{ab}\), including
the scalar phase. For completeness, the entire preimage of the
orthogonal stabilizer is \(\mathrm{Spin}^c(k-1)\): plane-rotation
lifts in the complement cover its special orthogonal group, and two
lifts of a fixed orthogonal matrix differ by a central phase. This
follows from the kernel calculation in Lemma 6.0j. In rank one the
complement is zero and its surviving line phase is retained.

Form the double by identifying both boundary copies and using the
collar coordinate with opposite signs on the two sides. These collar
charts and the interior charts have smooth transitions. The finite
collar and interior cover proves second countability; disjoint
interior points have disjoint neighbourhoods and collar charts
separate boundary points, so the double is Hausdorff. It is compact
as the continuous image of the two compact copies, and has no boundary.
The tangent identification across the seam reflects the normal and
is the identity on \(T\partial Q\). On the pullback transverse
bundle it is the identity, since the map cocycle is constant in the
normal coordinate. Thus it sends the opposite adapted frame on the
second side to the adapted frame on the first side. The above equal
full transition units let us glue \(P^{\rm opp}\) to \(P\) by the
identity in these principal charts. This gives an actual smooth
principal spin-c bundle over
\(TD\oplus g^*\tau\), projecting to its oriented frame bundle.
The complete scalar phases glue, so no square of a line has been
substituted for the line itself. Metrics and principal data are
product data near the seam; they extend smoothly on the interiors.

The constant map cocycles glue by the identity arrows, and the
constant coefficient bundles glue by their boundary identifications.
Denote them by \(g\) and \(E_D\). On the first side their restriction
to the seam is the original \(f_{\partial}\), coefficient \(E|_\partial\),
and full orientation. Finite double/Euclidean recovery in Lemma 6.0v
commutes with restriction and with these identity gluing maps. Its
composition with the canonical outward normal therefore recovers
exactly the specified induced boundary orientation. This verifies
the last assertion on the full module, rather than only on its
underlying virtual bundle. If some components of \(Q\) have no
boundary, double them disjointly; they impose no boundary assertion.
\(\square\)

**Theorem 6.0ab (full bordism relation).** For every compact smooth
K-oriented bordism \((Q,E,f)\), with the induced full boundary
orientation just specified,
\[
 (f|_{\partial Q})_!([E|_{\partial Q}])=0
 \quad\text{in }K_{q-\dim\partial Q}(A).
\tag{226}
\]
In particular the wrong-way assignment respects the bordism relation
of the geometric cycle group.

**Proof.** Apply Lemma 6.0aa and Corollary 6.0z to its actual double.
All data on the separating hypersurface agree, so (224) is exactly
(226). Neither a longitudinal analytic index nor a rational
character calculation is used. Direct sums and disconnected boundaries
are included by the finite coefficient-module sum.

The opposite boundary convention gives the additive inverse, with
its full orientation. This can be checked explicitly. For a
positive-rank normal module with principal transition \(g\), the
opposite structure has transition \(\kappa(g)\) and physical Clifford
coordinate \(R_0v\). Left multiplication by \(e_1\) satisfies
\(e_1\kappa(g)=g e_1\) and
\(e_1 c^+(R_0v)=-c^+(v)e_1\). It is right-module linear, unitary
and odd. It identifies the opposite Thom cycle with the original
module with reversed grading and negated operator, the additive
inverse cycle of Section 0. Every intervening double and even matrix
evaluation is even, so the same identification passes through full
normal recovery and final parity contraction. Stabilize a zero-rank
normal first if necessary; its line grading remains part of this
calculation. Thus a bordism written as the incoming boundary with
opposite orientation together with the outgoing boundary proves
equality of their two assigned K-classes, with the integral sign
and every line twist retained. \(\square\)

#### The actual clutch bundle

Let \(W\to M\) be a metric real bundle of positive even rank \(2a\),
with its specified full spin-c structure, over a compact smooth manifold
\(M\) without boundary.
Let \(S_W=S_W^+\oplus S_W^-\) be its spinor bundle, with the chirality
(145). Write
\[
 \widehat M=S(W\oplus\mathbb R_s),\qquad
 \pi:\widehat M\to M,\qquad n(x)=(0,1).
\tag{227}
\]
On the closed north hemisphere take \(\pi^*S_W^+\), and on the closed
south hemisphere take \(\pi^*S_W^-\). On their common equator the
clutch isometry is
\[
 c^+(w):S_{W,x}^+\longrightarrow S_{W,x}^-,
 \qquad |w|=1.
\tag{228}
\]
It is unitary because
\(c^+(w)^*c^+(w)=I_+\) and \(c^+(w)c^+(w)^*=I_-\),
by the full Clifford relation and \(|w|=1\). It changes spinor
parity. Gluing by this specified isometry defines a complex bundle
\(b_W\) on \(\widehat M\). Here the restrictions of a complex bundle
on the two hemispheres are ungraded; the signs enter the difference
class below. The construction is smooth: on an equatorial annulus use
\(c^+(w)/|w|\) as its transition between the hemisphere trivializations.
It is equivariant under the actual full spin-c transitions, so every
central line twist of \(S_W\) is retained.

Give the disc bundle \(Q=B(W\oplus\mathbb R_s)\) the map
\(p:Q\to M\) and the full stable orientation
\(p^*TM-TQ=-p^*(W\oplus\mathbb R_s)\), using the inverse of the
ordered sum orientation of \(W\) and the canonical positive real line.
An inverse stable orientation means the inverse full Clifford Morita
module: its two products recover the canonical cancellation module by
the rank-one and inner-product evaluations of Lemma 6.0j. In the
ordered inverse the last real-line block is removed first and the
\(W\) block second, exactly as in (148). In particular
an original twist by a line \(H\) becomes \(H^*\), rather than a
trivialization of \(H^2\). Choose a horizontal splitting of the tangent
sequence; its affine space of splittings and positive metric transport
give the same full stable orientation. Give \(\pi=p|_{\partial Q}\)
its induced boundary orientation, with the outward normal first.
This specifies the orientation of \(\pi\) used below.

**Lemma 6.0ac (north-section class and full orientation).** With these
orientations,
\[
 n_!\widehat\otimes_{C(\widehat M)}\pi_!=1_{C(M)},
 \qquad
 n_!([1_M])=[b_W]-[\pi^*S_W^-].
\tag{229}
\]
The first equality is in KK; the second is in \(K_0(C(\widehat M))\).

**Proof: the full orientation.** The normal of \(n\) is exactly
\(W\), and give it the specified structure. At the north section,
the vertical tangent of \(Q\) is the ordered \((W,\mathbb R_s)\).
The outward normal is \(+\partial_s\). Its boundary tangent is
\(W\). The full inverse for the ordered \((W,\mathbb R_s)\)
removes the last real-line block first. The ordered normal followed
by this inverse therefore has the adjacent factors
\(\mathbb R_s,-\mathbb R_s,-W\).
Their actual evaluation contracts the first pair and leaves the
inverse module of \(W\). If the positive-rank doubled realization
places the even \(W\) block between the two line factors, use the
fixed even block move of Proposition 6.0i before that evaluation;
it preserves every generator action and the normalized odd volume.
Both actual evaluations for the real-line factor are
used; the normalized odd generator is retained until its final ordered
\(C_2\) evaluation. The full positive \(W\) module of \(n\) then
cancels this inverse by
\(\xi\otimes\eta^*\mapsto\theta_{\xi,\eta}\) and
\(\xi^*\otimes\eta\mapsto\langle\xi,\eta\rangle\).
These maps are even, preserve the inner products and both actions,
and are onto by their finite matrix bases. The result is the canonical
even rank-zero unit, including its line and grading. This is exactly
the full induced composite orientation of Lemma 6.0v, since its
recovery maps are these same evaluations. The composite \(\pi n\)
is the identity map. Theorem 6.0w and Lemma 6.0x therefore give the first
identity in (229).

**Proof: the whole clutch class.** The north open hemisphere is the
image of the whole-bundle tube
\[
 \tau(x,v)=\frac{(v,1)}{\sqrt{1+|v|^2}}.
\tag{230}
\]
It fixes \(n\) and its normal derivative at zero is the identity.
It is a diffeomorphism, with inverse \(v=w/s\) for \(s>0\).
The radial disc coordinate \(v/\sqrt{1+|v|^2}\) identifies this
hemisphere compactification with \((B(W),S(W))\); this is the
proper radial homeomorphism proved in the topological K-theory lesson, Section 3. Its boundary
is the equator, with \(c^+(v)/|v|\) the specified boundary comparison.

The Thom cycle in Corollary 6.0d, after its fixed even Clifford
contraction, is the pulled-back spinor module with lower block
\(c^+(v):S_W^+\to S_W^-\), divided by
\(\sqrt{1+|v|^2}\). Its relative K-class is the exact bundle triple
\((\pi^*S_W^+,\pi^*S_W^-,c^+(w))\) on this disc pair.
Here is its actual relative projection. Set
\(T=c^+(v)/\sqrt{1+|v|^2}\). Both defects \(1-T^*T\) and \(1-TT^*\)
are the scalar \(1/(1+|v|^2)\) on their respective spinor bundles.
The even index formula of Theorem 0.14 gives, on
\(\pi^*(S_W^-\oplus S_W^+)\),
\[
 P_T(v)=\frac1{1+|v|^2}
 \begin{pmatrix}|v|^2 I_-&c^+(v)\\c^+(v)^*&I_+\end{pmatrix},
 \qquad P_0=\begin{pmatrix}I_-&0\\0&0\end{pmatrix}.
\]
One verifies \(P_T^2=P_T=P_T^*\) by
\(c^+(v)c^+(v)^*=|v|^2 I_-\).
The difference \(P_T-P_0\) vanishes at fibre infinity, uniformly
over compact \(M\). Its range is the image of the isometry
\[
 S_{W,x}^+\longrightarrow S_{W,x}^-\oplus S_{W,x}^+,\qquad
 \xi\longmapsto\frac{(c^+(v)\xi,\xi)}{\sqrt{1+|v|^2}}.
\]
At the equator that frame has limit \((c^+(w)\xi,0)\);
the range projection tends to \(P_0\), independently of the
direction \(w\). Extend it by \(P_0\) over the south hemisphere.
It is a continuous projection there, whose range bundle is exactly
the clutch bundle with comparison (228). Its reference range is
\(\pi^*S_W^-\).

For precision about nontrivial base bundles, \(P_0\) need not be a
constant scalar matrix. Choose finite Hermitian embeddings of the
two bundles over the compact base and apply the complementary
rotation of the topological K-theory lesson, Theorem 2.1, formula (2.7), to their projections.
That rotation turns a pair agreeing on the boundary into a pair
with the common constant reference of that formula. It
therefore computes a class in the external relative ideal, rather
than incorrectly placing the varying bundle projection in its
unitization. Stabilizing the Fredholm cycle of Theorem 0.14 by these
same finite complements gives that same relative index: its
doubled unitary has first-column range \(P_T\), and its reference
is the target bundle projection. The full relative excision
identification in the topological K-theory lesson, Theorem 2.1 thus sends it to the specified
disc triple. The projection-range construction above identifies
the forgetful class with \([b_W]-[\pi^*S_W^-]\), with the
kernel-minus-cokernel sign of Theorem 0.14. For one complex
coordinate, swapping the two displayed blocks gives the positive
projection with range \((1,z)\) and reference \((0,1)\), as
computed in its Theorem 3.1. No inference from one fibre or a
rational character is needed for a nontrivial \(W\).

The pair
\((\widehat M,\{s\le0\})\) has the same open relative ideal.
The bundle pair \((b_W,\pi^*S_W^-)\), with the identity comparison
on the south hemisphere, restricts on the north disc to exactly
this triple: on the equator that comparison is (228).
the topological K-theory lesson, Theorem 2.1 and relative excision identify its relative class
with the zero-extension of the Thom class. Its forgetful map is
\([b_W]-[\pi^*S_W^-]\). Lemma 6.0x identifies the zero-extended
tubular class with the ordinary map \(n_!\). This proves the second
identity in (229). \(\square\)

**Theorem 6.0ad (sphere fibre normalization).** For this full specified
orientation of \(\pi\),
\[
 \pi_!([1_{\widehat M}])=0,
 \qquad \pi_!([b_W])=[1_M]\quad\text{in }K_0(C(M)).
\tag{231}
\]
For every complex-bundle difference \(a\in K^0(M)\),
\[
 \pi_!(\pi^*a\,[b_W])=a.
\tag{232}
\]
These equalities include nontrivial and two-torsion line twists in the
chosen spin-c structure.

**Proof.** The first identity in (231) is Theorem 6.0ab for the disc
bordism \((Q,\mathbf1,p)\), with exactly its induced boundary
orientation. The disc has compact base and fibre, so is compact and
its boundary is precisely \(\widehat M\). The preceding full
bordism proof applies even though \(p\) itself is a map from a
manifold with boundary; it only uses the closed double's actual
wrong-way class. No unproved class for \(p\) is used.

Apply \(\pi_!\) to the second identity in (229). The first identity
in (229) sends its north-section term to \([1_M]\). The ordinary
projection formula of Theorem 6.0s sends
\([\pi^*S_W^-]\) to
\([S_W^-]\,\pi_!([1_{\widehat M}])=0\).
This proves the second identity in (231). Apply the same projection
formula to \(a\), using additivity for bundle differences, to obtain
(232). The spin-c line tensor in \(b_W\) and its dual tensor in
the orientation of \(\pi\) have been carried through the actual
evaluations in (229). There is no torsion loss. \(\square\)

#### The zero-rank case

If \(W\) has rank zero with the canonical even unit orientation,
\(\widehat M=M_+\sqcup M_-\). Define \(b_0\) to be the trivial
line on \(M_+\) and the zero bundle on \(M_-\). Its rank can be
different on distinct connected components. The bordism is
\(Q=M\times[-1,1]\), with its vertical line positive toward the
\(+\) end. The induced stable rank-zero orientation of the
projection is the even unit on \(M_+\) and the odd unit on \(M_-\).
Indeed the two outward normals are \(+\partial_s\) and
\(-\partial_s\), and the reflection calculation in Theorem 6.0ab
negates the latter class before the ordered odd–odd contraction.
Consequently the actual coefficient modules give
\[
 \pi_!(a_+,a_-)=a_+-a_-,\qquad
 \pi_!([b_0])=[1_M],\qquad\pi_!(1,1)=0.
\tag{233}
\]
These are exact module calculations with zero operators. If the
even stable zero-rank orientation has an additional line \(H\),
use \(b_0=(H,0)\); the projection's full modules are
\(H^*\) and \(\Pi H^*\). They give
\(H^*a_+-H^*a_-\), and the same normalization follows from
\(H\otimes H^*\cong\mathbf1\). This makes explicit why both
zero-rank components cannot silently be assigned the same even
orientation. It also agrees with (229): \(S_W^-=0\) and the north
section has its specified even line.

#### The geometric relation descends

**Theorem 6.0ae (geometric-cycle descent).** For the smooth compact
map-cocycle model and full stable orientations of Section 5, the
assignment
\[
 (M,E,f)\longmapsto f_!([E])
 \quad\text{in }K_{\dim M-q}(A)
\tag{234}
\]
respects all three geometric relations. It therefore defines a
homomorphism \(K_{*,\tau}^{\mathrm{geom}}(BG)\to K_*(A)\) for
this specified model.

**Proof.** Direct sum and disjoint union are the direct sums of the
actual finite bundle and correspondence modules; Section 0 proves
their product additivity. Theorem 6.0ab proves the entire oriented
bordism relation, including the opposite sign and full line twist.

For a sphere modification give \(f\pi\) the induced orientation
from \(\pi\) above followed by the original full orientation of
\(f\). Both sphere and base are compact. Theorem 6.0w applies and
(232), or (233) in rank zero, gives
\[
 (f\pi)_!([\pi^*E\otimes b_W])
 =\pi_!([\pi^*E\otimes b_W])
       \widehat\otimes_{C(M)}f_!
 =f_!([E]).
\tag{235}
\]
The added vertical rank is even, so it changes neither the cycle
parity nor the retained final odd Clifford factor. Its full
orientation is the composed one, whose normal evaluations were
checked in (229); it is not a new determinant-only choice.
The three relations thus generate equal images, proving descent.
This statement does not identify the geometric group with another
topological model, prove twisted analytic/geometric duality, or
prove the longitudinal analytic index theorem. Those are distinct
subsequent theorems. \(\square\)

![Separating phase, reflected full spin-c structure and global sphere clutch bundle](../figures/foliation-cycle-descent.png)

**Figure 6.0e.** Lemmas 6.0x–z, 6.0aa and 6.0ac; Theorems 6.0ab, 6.0ad and 6.0ae; equations (216)–(235); Exercises 46–47. The phase curve samples \(v\in[-4,4]\); its dashed lines are the exact limits zero and \(2\pi\). The global bounded phase gives the norm-continuous null-homotopy (223). The second panel reflects the first normal coordinate and glues the full principal spin-c transitions, including the central phase. The last panel gives the whole north/south clutch bundle, ordered inverse cancellation, disc-bordism normalization and the signed rank-zero calculation. These are diagrams of maps and modules. The classical geometric relations are treated by Connes and by Liu; all displayed foliation KK and full orientation assertions are proved here.

Full-size descent PNG · Editable descent SVG.

**Theorem 6.1 (assembly from wrong-way maps).** For the compact smooth map-cocycle model of Section 5, use the inverse Bott class of Corollary 0.20 and the full vector-bundle Thom class in (16), with the specified stable orientation. Then

\[
\mu[(M,E,f)]=[E]\otimes_{C(M)}f_!
\tag{17}
\]

defines a homomorphism
\(K_{*,\tau}^{\mathrm{geom}}(BG)\to K_*(A)\).

**Proof.** Theorem 6.0o proves invariance under the full factorization choices, cocycle refinements and smooth orientation-preserving homotopies. Theorem 6.0ae proves direct-sum, bordism and sphere-modification descent. Its bordism proof extends all principal and cocycle data across a smooth double and contracts the separating normal-line unitary. Its sphere proof identifies the entire clutch bundle and both ordered orientation cancellations, then proves the coefficient normalization. Thus (17) is independent of every defining cycle relation. Additivity of the actual bundle and correspondence modules makes it a homomorphism. \(\square\)

This gives geometric-cycle descent in the stated smooth cocycle model. Theorem 6.2 below constructs the full twisted cohomology class and proves its compact-test duality. Theorems 7.25 and 7.29 prove the full longitudinal comparison and its symbol action in both degrees. The identification with other assembly models and the isomorphism assertions for assembly have their separate scope in the following sections.

An analytic K-homology class \(x\in KK^j(A,\mathbb C)\) gives the numerical homomorphism below. Theorem 6.2 identifies it with the compact tests of one full twisted cohomology class:

\[
[(M,E,f)]\longmapsto
\mu[(M,E,f)]\otimes_A x\in K_{j+\dim M-q}(\mathbb C).
\tag{18}
\]

Theorem 6.1 and associativity show that it respects all the relations. Theorem 6.2 supplies the full geometric twisted cohomology class on \(BG\), with all integral classes and normal-line data retained, and proves that (18) is its oriented compact-test pairing.

### The full twisted cocycle on the classifying space

The remark on native page 35 of [Connes, §11] concerns twisted cohomology \(K^{*,\tau}(BG)\). We specify its full group and construct the analytic class, then identify its oriented compact tests with (18). The topology and operator-field arguments below use actual holonomy arrows throughout. The classical methods are treated in [Connes–Skandalis, §§3–4], [Moerdijk, §§1–5](https://aif.centre-mersenne.org/item/10.5802/aif.1254.pdf), and [Emerson–Meyer, §3.2](https://arxiv.org/html/0710.1410). All proofs needed here, including the classifying replacement, both manifold triangles and the extension convention, are supplied below.

**Theorem 6.2 (analytic K-homology gives a full twisted geometric cocycle).** Let \(G=\operatorname{Hol}(V,F)\), \(A=C_r^*(G)\), and \(\tau=TV/F\). There is a locally compact Hausdorff second-countable CW model \(X\) for the weak homotopy type of \(BG\), carrying the pullback of its universal actual cocycle and entire normal bundle \(E_X\). With the negative graded-opposite Clifford and positive right-\(C_1\) conventions below, set

\[
 K^{j,\tau}(BG)=RKK_X^j(C_0(X,\operatorname{Cl}^+(E_X)^{\rm op}),C_0(X)),
 \qquad K_j^{\rm an}(A)=KK^j(A,\mathbb C).
 \tag{382}
\]

The analytic group has the complete stable semisplit extension-theory interpretation proved in E. There is a natural homomorphism \(\chi:K_j^{\rm an}(A)\to K^{j,\tau}(BG)\), independent of the complete model with its specified normal comparison. For every compact smooth actual cocycle \(f:M\to V/F\), its full pullback is the field class \(\chi_f\). Every prescribed full stable orientation on \(TM\oplus f^*\tau\) and Hermitian test bundle gives precisely the integral numerical pairing (18), in both degrees, through the full KK equality in (465)–(466). The construction retains all torsion and classes invisible on individual compact stages.

**Proof.** Parts T and C construct the entire analytic family and identify its full Clifford–Fredholm group. B, R, A, U, M, W, V and Q supply the actual classifying cocycle and locally compact model; D supplies its full group and the single analytic class. G proves both actual manifold triangles; H and I give the complete orientation and graph-normal comparison. E proves the source extension convention in both degrees. These are the following complete arguments. The letters label proof parts; equations have the lesson's consecutive numbering.

 #### T1. The actual parameter tube and its étale correspondence

Let M be a compact smooth manifold, f a smooth actual G-arrow cocycle from M to V/F, E=f\*τ and q=rank E. No orientation of E or TM is imposed. The full microbundle construction of Lemma6.0l gives p:H\_f→M, its section ι, normal E and actual arrows over a neighbourhood of ι. In its original charts the plaques have constant transverse coordinate; p need not be constant on a plaque. Choose a fibre-preserving normalized tube as in Theorem6.0o. It is the new target parameter in(383) that is fixed along the target leaves. A positive radial compression identifies all of E with an open smaller neighbourhood U of ι in H\_f. Denote the resulting map by ρ:E→U. It keeps pρ=π\_E and has positive scalar normal derivative at zero. The arrow cocycle g on U has local transverse maps g\_a and exactly the prescribed f\_a on the section, up to its specified short-arrow equivalence.

Use the foliation on V×M with tangent F×{0}. Its groupoid is G×M and its reduced algebra is C(M,A), A=C\*r(G). Define the local lifts

\[
 \widehat g_a(y,v)=(g_a(\rho(y,v)),y),
 \qquad
 \widehat g_{ba}(y,v)=(g_{ba}(\rho(y,v)),y).
 \tag{383}
\]

These are actual arrows, including all holonomy lifts. In transverse coordinates the derivative has the block form [[D\_v g\_a,D\_y g\_a],[0,1]]. The first diagonal block is invertible after the tube shrink. The source E has dimension dimM+q, exactly the transverse dimension of this target. Thus (383) is an étale cocycle from the unit foliation of E. Lemma6.0n gives the entire countably generated Hilbert C(M,A)-module L\_f and a compact, nondegenerate coefficient action φ\_f:C0(E)→K(L\_f), with cycle operator0.

The module is C(M)-linear: on its principal-space core, multiplication by h(π\_E(z)) on the left equals multiplication by h(y) on the right, because the target source parameter is precisely y. Convolution retains y and its inner product is a continuous A-valued function of y. The equality holds on the completion by the essential multiplier action. The compact-action proof is the actual short-arrow normalized rank-one proof of Lemma6.0n; it is not pointwise ordinary compactness on a holonomy cover.

#### T2. The full finite Clifford cancellation, without a spin-c choice

Write C\_M=C(M), C\_E=C(M,Cl+(E)), P\_E=C(M,Cl+(E)^op), where the opposite is the graded opposite. For a homogeneous b, its product and involution are b^op·c^op=(-1)^{|b||c|}(cb)^op and (b^op)\*=(-1)^{|b|}(b\*)^op. These conventions make graded right multiplication an adjoint-preserving representation.

Let W\_E be the finite Hermitian bundle Cl+(E), graded by word parity, with scalar-part trace inner product. Its antisymmetrization identifies it orthogonally and equivariantly with ΛE\_C. On an orthonormal frame, write L\_i for left multiplication by the positive generator and R\_i for graded right multiplication: R\_i(w)=(-1)^{|w|}we\_i. Their actual actions are

\[
 L_i=e_i+\iota_i,\qquad R_i=e_i-\iota_i,
 \qquad e_i=(L_i+R_i)/2,\quad\iota_i=(L_i-R_i)/2.
 \tag{384}
\]

Here e\_i means exterior multiplication, consistently with(129), and is distinct from its self-adjoint a\_i=e\_i+ι\_i. Formula(384) follows on every ordered word: if i is absent, L\_i and R\_i have the same insertion sign; if i is present, their removal signs are opposite. In particular R\_i is skew-adjoint with square−1, as required by the graded-opposite involution. Left and graded-right actions graded commute. Exterior multiplication and contraction generate every matrix unit: the vacuum projection is ∏\_i(1−e\_iι\_i), and creation words, that projection and adjoint creation words give the matrix units between all exterior monomials. The resulting faithful representation is

\[
 C_E\widehat\otimes_{C_M}P_E
       \cong C(M,\operatorname{End}_{\mathbb C}W_E).
 \tag{385}
\]

All actions are orthogonally equivariant; no frame, determinant line or orientation was chosen globally. Both finite Morita evaluations are ξ⊗barη↦θ\_{ξ,η} and barξ⊗η↦<ξ,η>. Their inner products give isometries, and the full matrix units and local finite frames give onto ranges. Finite partitions prove fullness over C(M). Their zero-operator cycles satisfy both product criteria by the complete imprimitivity proof0.12. Rank zero gives W\_E=C\_M and both identities literally.

#### T3. A twisted Bott cycle with its two actual inverse products

Put A\_E=C0(E,π\*Cl+(E)). The complete raw b\_E,d\_E of Lemma6.0a/Theorem6.0c are C(M)-linear and have both inverse products. Their full proofs use orthogonal transitions only, so their balanced external products with P\_E keep that linearity. Let m\_E be the pullback to E of the even Morita equivalence (385). Define

\[
 \widetilde b_E=(b_E\widehat\otimes_{C_M}1_{P_E})m_E
       \in KK_{C_M}(P_E,C_0(E)),\qquad
 \widetilde d_E=m_E^{-1}(d_E\widehat\otimes_{C_M}1_{P_E}).
 \tag{386}
\]

The target of the balanced raw product is A\_E⊗\_{C\_M}P\_E, identified with the compact endomorphisms of π\*W\_E. Both actual Morita evaluations and the two global identities(136), before any parity reduction, give tilde b\_E tilde d\_E=1\_{P\_E} and tilde d\_E tilde b\_E=1\_{C0(E)}. The source has the graded-opposite Clifford action, the target has its ordinary function action, and no odd blocks are silently interchanged.

Its concrete representative is Γ0(E,π\*W\_E), right C0(E), with the source acting by the graded right Clifford operators R\_b, and bounded phase L\_{c+(v)}/sqrt(1+|v|²). Formula(384) makes its graded source commutators zero; its square defect is the scalar−1/(1+|v|²). The pullback endomorphism Morita module identifies this with (386). This also fixes the rank-one sign in the full Clifford convention; no separately ordered real Bott pair is being identified here.

#### T4. The actual universal transverse test family

Define

\[
 k_f=\widetilde b_E\cdot[\mathcal L_f,\phi_f,0]
        \in KK_{C_M}(P_E,C(M,A)).
 \tag{387}
\]

A product representative is π\*W\_E⊗\_{C0(E)}L\_f with the same bounded Clifford phase in v. Local finite W\_E frames and compact fibre cutoffs express a creation vector as a finite column of C0(E) coefficients. Each acts compactly on L\_f. Therefore its complete creation map is rectangular module compact. Norm approximation by these sections treats every creation vector. Multiplication by the bounded phase preserves compactness and adjoints give the second error against the zero operator. The first positivity anticommutator is exactly2B²≥0. Its localized square defect is the compact φ\_f(1/(1+|v|²)), in finite W\_E corners; graded source commutators vanish. This verifies the actual product criterion, including the nonunital C0(E) action, rather than just naming a KK product.

For x∈KK^j(A,C), its external C(M)-coefficient cycle is C(M)-linear. The product construction of Theorems0.4–0.7 therefore gives

\[
 \chi_f(x)=k_f\cdot\tau_{C_M}(x)
       \in KK^j_{C_M}(P_E,C_M).
 \tag{388}
\]

Every adjointable operator in its final module is automatically C(M)-linear, since the specified central left parameter action equals the strict right multiplier action. Thus the actual product construction does not discard parameter linearity. This is a full Clifford-twisted family class, not the scalar index obtained by integrating over M. Additivity, homotopy and all torsion are retained by the full KK products.

#### T5. Compact smooth pullbacks and choices

For h:N→M with N compact smooth, E\_N=h\*E and the pulled cocycle define the same construction over C(N,A). Use the pulled metric and tube. The whole principal space of its étale cocycle is exactly the pullback of the principal space in(383) along the target parameter h. On compact chart cores, the map from the balanced coefficient pullback of L\_f sends ξ⊗a to h\*ξ·a. Its A-valued inner products are the original ones evaluated at h(y), with the same convolution arrow and counting multiplicities. It is an isometry in the complete C(N,A)-module norm. Finite parameter charts and separated parameter/arrow tests give dense range: compact core functions on the pullback are uniformly approximated by finite sums of functions of N times pullbacks of local arrow-chart functions, then the common Schur bound makes that a module-norm approximation. Completion is onto. This proves actual coefficient base change, not merely equality of individual parameter fibres.

The Clifford regular module, both finite evaluations and (386) pull back by their literal formulas. The map of total bundles is proper for compact M,N, so the proved proper-pullback naturality of Corollary6.0d applies. Associativity and the exact even parameter transport0.13 now give χ\_{fh}(x)=h\*χ\_f(x), with the identified Clifford twist. No degree sign comes from the even parameter pullback.

Changing arrow cocycle representatives uses the actual principal-space unitary of Lemma6.0n. Its normal derivative transports E and its Clifford actions; choose the transported metric first, then the positive metric homotopy of Corollary6.0d. Refinements use the identical glued principal space. Tube/compression changes use the fibre-preserving interpolation and shrink/expansion homotopy of Theorem6.0o. These are entire parameter modules with compact actions before endpoint evaluation. They give homotopies of(387) and(388), retaining all opposite-Clifford coefficients. The rank-zero construction uses the zero normal tube and even unit of (385); a test-manifold tangent orientation has never been introduced.

#### C1. The continuous-base hypothesis and the exact algebras

Let X be locally compact, Hausdorff and second countable. Let f be a continuous actual G-arrow cocycle on X. Its arrows have their Lie-groupoid topology, including every holonomy branch; a relation between leaf images is not a cocycle. Set E=f\*τ, C\_X=C0(X), P\_X=C0(X,Cl+(E)^op), A=C\*r(G). The bundle E is continuous and has a continuous metric. All Clifford tensor products are graded. A C\_X-linear cycle has its central left C\_X action equal to the right essential multiplier action. Write

\[
 KK_X(P_X,C_X)=KK_{C_X}(P_X,C_X).
 \tag{389}
\]

Both coefficient algebras are separable. A continuous section of a finite-dimensional bundle over this X is approximated using countably many precompact charts, continuous base functions and finite fibre coefficients. The same applies to C0(E) and C0(X,A). These are ordinary separable C\*-algebras, so the proved product construction in Theorems0.4–0.7 applies. No product theorem for a non-locally-compact BG or a pro-C\*-algebra is being asserted here.

All assertions C2–C5 also apply to X×K for compact finite polyhedra K, including a simplex and any union of its faces. They apply to continuous base maps; no derivative of a base map is used.

#### C2. Locally finite germs and a fibrewise smooth normal neighbourhood

Choose a countable locally finite shrinking V\_a with compact closures inside cocycle charts U\_a. This can be obtained from a precompact countable basis and a subordinate locally finite partition: enlarge the closed supports slightly inside their assigned U\_a. Local finiteness may be retained by taking the enlargements inside a previously locally finite refinement. A compact closure meets only finitely many of the enlarged sets. Thus every closed double or triple overlap used with a fixed chart has only finitely many partners.

On each chart replace f\_a by a fixed plaque cross-section using its short plaque arrow. Conjugate transition arrows exactly as in(181). On a neighbourhood of any base point of a closed double overlap, the resulting arrow lies in one transverse path chart. Its restriction to the two plaque cross-sections is a fixed smooth transverse branch. Its dependence on the base enters only through the continuous centre h\_a(x). Cover that compact double overlap by finitely many such branch neighbourhoods. The equality of germs on their closed overlaps gives an open neighbourhood of each centre on which both transverse maps and actual arrows agree. This is an open arrow-chart intersection, not a selection of an arrow from its endpoints alone. A finite subcover makes the agreement uniform on a smaller neighbourhood of each compact overlap. Treat inverse, diagonal and triple identities the same way.

For any x, only finitely many of these closed-overlap conditions are active on a small neighbourhood of x. Their finite common transverse radius is positive there. Choose a positive continuous radius δ(x) below all those local bounds. Here is an explicit minorant construction: choose a locally finite base cover on whose members a constant positive lower bound r\_j works, take a subordinate partition λ\_j, and replace r\_j by r\_j/2. Then δ=Σ\_j λ\_j r\_j/2 is positive and is below the admissible bound at each point, since every active r\_j is admissible there. Further local bounds for the images of compositions can be included before making this choice.

The centred fibre changes are continuous in x and smooth in v, with all vertical derivatives continuous:

\[
 F_{ba}(x,v)=H_{ba}^{x}(h_a(x)+v)-h_b(x),\qquad
 F_{ca}(x,v)=F_{cb}(x,F_{ba}(x,v)).
 \tag{390}
\]

The second identity is used whenever its intermediate vector remains inside the chosen chart disk. Inverses and identities have that same domain convention. Glue D\_a={(x,v):x∈V\_a, |v|<δ(x)} by the direct pair relation w=F\_ba(x,v), requiring both vectors to be in their chart disks. This relation is already transitive by(390), so chains add no new identifications. On each D\_a×D\_b it is closed: equality of base points is closed and the pair map is defined on the entire relevant input disk. The components of the disjoint union are open and closed; hence the whole relation is closed. Its quotient is open because partial fibre changes are homeomorphisms on open domains. Closedness plus openness separate inequivalent classes by the product-neighbourhood argument of Lemma6.0l. The quotient H\_f is therefore Hausdorff and second countable. It is locally a product of an open subset of X and a fibre disk, and is locally compact. Its projection p and section ι are continuous, and its vertical normal is the derivative cocycle E.

There is no assertion that H\_f is a manifold or that X is smooth. There is also no assertion that its plaques keep p fixed. The original transverse coordinates give the germ data, while p is used for the independent target parameter.

Construct a fibre-preserving normalized tube from E to H\_f. For completeness, local fibre charts carry the spray \(\dot z=v, \dot v=0\). A change of fibre chart has vertical acceleration D\_z²h(v,v). Patch these sprays using a locally finite partition continuous in x and smooth in the fibre, obtained from product base functions and fibre bump functions. The patched spray has continuous vertical derivatives, base velocity v and acceleration of order |v|² at zero. The usual contraction argument for its integral equation is uniform on a precompact parameter/chart set: a vertical-derivative bound makes its Picard map a contraction for a short time. Iterating finitely many such times near a fixed zero trajectory gives time one, continuous dependence on x and smooth dependence on v. Its derivative at zero is the identity on the vertical normal. The parameter inverse theorem uses this same uniform derivative bound; its contraction proof gives a continuous fibre inverse near zero. Local fibre injectivity needs only a sufficiently small disk at each x. Since p is fixed, points over different x can never have the same image. A further positive continuous minorant chooses one variable-radius disk on which the whole map is an open fibre embedding.

Positive radial compression then gives a normalized map ρ:E→U⊂H\_f defined on the entire E. Explicitly, compression into radius r(x) uses v↦v/sqrt(1+|v|²/r(x)²); its vertical derivative at zero is one. This formula is continuous in x, fibre smooth, and has its continuous fibre-smooth inverse on that open disk. All extended arrows are defined on U after one further variable-radius shrink if needed.

#### C3. The completed whole module, not separate fibre completions

Use the target G×X, with reduced coefficient algebra C0(X,A). The relative cocycle on E is (g\_a(ρ(x,v)),x), with the extended actual arrows and the same x. In any transverse path chart its vertical derivative D\_vg\_a is invertible. Thus its principal space has relative source-anchor charts over X: locally the anchor to X×V is a homeomorphism, smooth in the manifold variables. There is no derivative in x in this assertion. The source-anchor fibres at (x,y) are discrete and countable, with every distinct holonomy lift retained.

Complete the chart core using the counting/convolution formulas(186). Inner products are elements of C0(X,A). To verify this norm topology, first consider finitely many chart kernels with compact parameter and arrow support. In a relative product chart, their coefficients vary continuously in x. A finite parameter partition approximates them uniformly by separated parameter/arrow kernels. Uniform row and column bounds on this compact support, from the same Schur estimate used in Lemma6.0n, turn that into convergence in reduced norm. Coefficients vanish outside their compact parameter projection. General compact core sections are finite sums of these charts. The regular-fibre identity(187) proves matrix positivity in the faithful direct sum over (x,y); it supplies the C0(X,A) norm, rather than an independently chosen fibre norm. The quotient and completion give L\_f.

The algebra identification C\*r(G×X)=C0(X,A) has the same verification: separated kernels have exactly the sup\_{x,y} regular norm, finite partitions give every compact chart kernel, and compact-support continuous A-valued functions are uniformly approximated by such separated arrow-core values. Their closures agree isometrically.

For c∈C0(E), multiplication c(b(z)) is adjointable and contractive on L\_f by the regular-fibre maps. A source coefficient supported in one injective relative transverse chart is a normalized short-plaque rank-one map. The plaque half-density can be chosen continuously in x, with a strictly positive continuous normalizing integral on its compact support. The calculation in Lemma6.0n therefore remains an actual equality of module operators. A finite partition on a compact source support proves compactness for every compact-support c; sup-norm density proves

\[
 \phi_f:C_0(E)\longrightarrow\mathcal K(\mathcal L_f).
 \tag{391}
\]

Countable chart exhaustions and countable coefficient approximations give a norm-dense core, using these same Schur bounds. Its right span, with a countable approximate identity of the separable coefficient algebra, proves countable generation. The action is nondegenerate because source cutoffs converge on the compact chart core. Multiplying by h(π\_E(z)) on the left equals the right parameter multiplier h(x) on every core vector and hence on the completion. This proves its exact C\_X linearity. The zero operator is a genuine C\_X-linear C0(E),C0(X,A) cycle with compact left action.

The raw Clifford cycles6.0a–c also extend to this continuous base. Their domain arguments differentiate only in the vector fibre. The compact-frequency bounds, oscillator eigenbundles, rotations, translations and positive metric identifications are continuous in base charts and orthogonally equivariant. Compact base cutoffs make their localized resolvents compact with exactly the existing uniform estimates(132),(135),(138)–(142). Fibre smoothing gives a dense coefficient algebra; no base smoothing is required. Their actual inverse homotopies retain both Gaussian evaluations. Therefore(385)– (387), with C0(X) in place of C(M), give a whole class

\[
 k_f\in KK_X(P_X,C_0(X,A)),\qquad
 \chi_f(\eta)=k_f\,\tau_{C_X}(\eta)\in KK_X^j(P_X,C_X),
 \quad \eta\in KK^j(A,\mathbb C).
 \tag{392}
\]

The concrete first product still has the finite exterior bundle and phase L\_{c+(v)}/sqrt(1+|v|²). Compact-support creation sections are rectangular compact maps because of(391); their norm approximations treat every creation section. Its positive anticommutator is2B² and its localized defect is(391) applied to the source coefficient divided by1+|v|². For the final product, the single separable coefficient algebra C0(X,A) and the single tensor module are used in the product construction of Theorems0.4–0.7. Every adjointable operator on this module is C\_X-linear, by the exact central multiplier equality. This constructs its operator before taking any parameter fibre.

#### C4. The exact Clifford average and geometric field model

For compact X, P\_X is unital. Compress a C\_X-linear cycle to the even projection φ(1); the complement has zero representation and is degenerate. The compressing difference is compact. Normalize its odd operator to a self-adjoint contraction as in Section0. Thus its square defect is globally compact. Let R\_i be the representation of the negative generators of P\_X on a local orthonormal chart, and set R\_I=R\_{i\_1}⋯R\_{i\_k} in increasing order, R\_empty=1. They are homogeneous unitaries. Define the finite operator average

\[
 \mathfrak a_E(F)=2^{-q}\sum_{I\subset\{1,\ldots,q\}}
                 (-1)^{|I|}R_I F R_I^*.
 \tag{393}
\]

Each summand is an odd self-adjoint contraction. The maps T↦−R\_iTR\_i\* on odd T are commuting involutions: R\_i²=−1, and R\_iR\_j=−R\_jR\_i has a central scalar sign which cancels in conjugation. Consequently(393) is their product of averaging projections. Its fixed operators are exactly those graded-commuting with every R\_i. In particular it is idempotent, contractive and fixes F if F already graded-commutes with the source Clifford action.

It is independent of frame. Under an orthogonal frame change, the degree-k ordered words change by the real orthogonal exterior-power matrix. In the sum of R\_I F R\_I\* for that degree, its two occurrences cancel by orthogonality. The factor(−1)^k is constant on that degree. Thus the averages glue over E without a spin-c or orientation choice. For an already compressed representation, each F−(−1)^|I|R\_IFR\_I\* is compact on a chart after multiplication by a base cutoff: it is the graded commutator with R\_I followed by its unitary adjoint. Choose a finite partition on X with supports inside orthonormal charts and extend the cut-off words as genuine P\_X coefficients. Their source commutators are compact by the Kasparov condition. Summing proves F−a\_E(F)∈K(H). The straight line between them is a Kasparov homotopy; its square and source-commutator defects remain compact. The new operator has exact graded Clifford commutation.

The noncompact essential-source reduction is also explicit. Let (H,φ,F) be any C\_X-linear P\_X,C\_X cycle. On a relatively compact open set V choose u∈C\_c(X) equal to one on V and regard u1\_Cl as an actual P\_X section. At x∈V the operator p\_x=φ(u1\_Cl)\_x is an even projection, since u²−u vanishes on V. It does not depend on u: for h supported in the common region where two choices equal one, C\_X linearity gives hφ((u−v)1\_Cl)=φ(h(u−v)1\_Cl)=0. These locally defined operators therefore give a bounded \*-strong projection field p. They define a genuine adjointable projection on H, rather than merely a pointwise support projection. Indeed pξ is locally the section φ(u1\_Cl)ξ, is bounded in fibre norm by ||ξ\_x||, and vanishes at infinity. A locally finite partition and compact cutoffs approximate it by actual module sections. Its adjoint is itself. The fibre norm is faithful, so p²=p on H.

The source action satisfies pφ(a)=φ(a)p=φ(a). Moreover pH is exactly its essential submodule. On a compact support, finitely many local units express pξ as a finite sum of φ(u\_j1\_Cl)h\_jξ; compact cutoffs approximate a general pξ. Thus pH is the closed source span and is complemented, with countable generation inherited from the adjointable projection on the countably generated H. The complementary source action is zero. For h∈C\_c(X), a finite partition subordinate to local-unit regions gives h[F,p]=Σ\_j h h\_j[F,φ(u\_j1\_Cl)]∈K(H): F commutes exactly with the right central base multipliers. Hence replacing F by pFp+(1−p)F(1−p) changes it by a locally compact operator. The straight line preserves every localized cycle condition. The zero-source summand is degenerate and can be discarded. This proves the essential-source reduction on every such X; it does not assume that an essential submodule of an arbitrary Hilbert- module representation is complemented. The finite, locally unital bundle source and exact C\_X linearity are what prove it here.

On pH, the same average proof is localized by h∈C\_c(X), with finitely many orthonormal charts on its support. The average is a global bounded adjointable operator. Its difference from F and its square defects are compact after each base cutoff. Approximation in C0(X) proves all localized Kasparov conditions. For the particular cycle(392), the source is already nondegenerate: a scalar Clifford approximate identity acts as the right base multiplier and converges on every tensor vector, even when the chosen A representation of η is degenerate.

A geometric Clifford–Fredholm field over X means: a countably generated graded continuous Hilbert field, a nondegenerate continuous P\_X action with its exact base action, and an odd self-adjoint contraction F graded-commuting with the fibre Clifford action. Its localized defect is a norm-continuous compact field, vanishing at infinity after multiplication by every C0(X) coefficient. Isomorphisms, addition of degenerate fields and homotopies use the same whole fields over X×I. Odd degree retains the ordered positive right C1 coefficient from Section0; it is never replaced by a trace.

Here is the operator topology, with its proof. Stabilize a whole countably generated C0(X) module by0.2 as a projection p on a graded standard module. The vectors p(x)e\_n are continuous multiplier sections; multiplying them by a compact base cutoff gives actual module sections. The projection p(x) is \*-strongly continuous. An adjointable F gives a bounded \*-strong field pFp. Every compact module map gives a norm-continuous compact field: rank-one fields have this property, and norm limits preserve it. Conversely a norm-continuous compact field T=pTp which vanishes at infinity is a compact module map. Its image over a compact parameter support is a compact subset of the ordinary compact operators. Standard finite-coordinate projections P\_N approximate that subset uniformly. Thus pP\_NTP\_Np converges in norm to T and is a finite matrix of these multiplier vectors, with continuous coefficients vanishing at infinity. Each coefficient factors into two continuous square-root factors vanishing at infinity (using its polar factor away from its zeros), so these are actual compact rank-one maps of module sections. Alternatively apply the compact base cutoff first, making every p e\_n in the finite sum an actual section. Cutting off the parameter first treats a general T. This proves both directions. Multiplication by a bounded \*-strong unitary preserves this compact norm topology: approximate a compact operator by finitely many rank-one vectors, use strong continuity on those vectors, and bound the residual by twice its norm. Hence the topology is independent of the chosen stabilization; it is not an arbitrarily chosen operator-norm topology on all Fredholm operators.

That paragraph writes the even degree. In odd degree the right coefficient is C0(X)⊗C1 with its ordered positive generator. The fields are Hilbert C1-module fields, and stabilization uses the standard C1 module. Its finite-coordinate corners have C1 entries. A faithful finite-dimensional regular representation of C1 gives their exact operator norm and realizes their compact ideal as K(ℓ²)⊗C1. The same finite-coordinate approximations, multiplier vectors, compact cutoffs and square-root factorization apply with those entries. The local-unit projection and average are adjointable right-C1-linear operators and never remove that coefficient. Thus the odd assertion below follows by the identical whole-module proof with this finite coefficient, rather than by forgetting an odd block or taking its scalar trace.

For every X under C1, forgetting exact Clifford commutation identifies the group of these fields with KK\_X^j(P\_X,C\_X). Surjectivity uses the local-unit projection, normalization and(393) on the whole module. Injectivity follows by performing them on the whole KK homotopy over X×I, including its continuous representation. Its left C0(X×I,Cl E^op) action is obtained by finite interval partitions and compact base cutoffs from the original P\_X action and the right interval multipliers; contractivity follows in the faithful fibre representation, so it extends in sup norm. The local-unit projection restricts to each endpoint's essential projection. Intrinsic averages restrict to endpoint averages, and fix operators which already graded-commute. Thus the forgetful homotopy becomes an exact Clifford–Fredholm homotopy with the same endpoint classes. Zero-source summands, degenerate fields, unitary maps and sums have the same description. No exhaustion or inverse-limit assertion occurs here. This identifies the full twisted field group on X, including torsion and classes invisible on every compact stage, with this RKK group. Parts Q and D below identify the complete model and its exact normal twist.

#### C5. Base change and finite coherent choices

Let h:Y→X be any continuous map between locally compact Hausdorff second-countable bases. A nonproper h has a nondegenerate map C0(X)→M(C0(Y)), u↦u∘h; it need not map into C0(Y). Nondegeneracy follows because h sends every compact subset of Y into a compact subset of X, where a C0(X) approximate identity converges uniformly. For a C\_X coefficient algebra B, use the balanced base-changed algebra B\_Y=C0(Y)⊗\_{C\_X}B. In the cases at hand these are precisely C0(Y,A), C0(h\*E), the pulled Clifford section algebra, and C0(Y). The dense map g⊗b↦g(y)b(h(y)) is isometric in the faithful fibre norm. Local bundle charts and a finite partition on a compact support give dense range. Thus these are algebra identifications, not nonproper C0 pullback homomorphisms. The original B acts nondegenerately by multipliers of B\_Y, so a module H\_B has its well-defined tensor pullback H⊗\_B B\_Y.

The following localized compactness observation is essential. If θ\_{ξ,ζ} is a module rank-one map, its pullback is T\_ξ T\_ζ\*, where T\_ξ:b↦ξ⊗b is initially an adjointable creation map from B\_Y. For g∈C\_c(Y), T\_ξ g is rectangular compact: its squared norm coefficient is |g|² h\*<ξ,ξ>∈B\_Y. Approximate identities of B\_Y therefore approximate T\_ξ g by the compact creation maps b↦(ξ⊗g e\_n)b. Explicitly the error norm squared is bounded by ||(1−e\_n)|g|²h\*<ξ,ξ>(1−e\_n)||→0. A central split of g into two square-root factors (first for nonnegative g) shows that the g-localized pullback of θ\_{ξ,ζ} is compact. Finite sums and norm limits give every old compact map; real/imaginary parts and C0(Y) approximation give every base localization. This is why nonproper pullback preserves representable cycles even though it does not preserve global compactness without localization.

Source actions extend from the dense balanced algebra to its complete section algebra, using the faithful fibre norm. A new source coefficient on a compact Y support is approximated by finite sums of g times a pulled old coefficient. Choose u∈C\_c(X) equal to one on h(supp g). The old localized defects tested by that u are compact. The observation above makes their pullbacks compact after g. All commutator, square and adjoint conditions therefore survive on the whole base-changed module. An adjointable operator pulls back as F⊗1 and has the same bound. This defines the geometric-field pullback without assuming h proper.

For the principal module the whole tensor map sends ξ⊗a to h\*ξ·a. Its inner products are identical counting sums and convolution evaluated at h(y), so it is isometric in C0(Y,A) norm. Compact pullback chart supports admit finite separated-function approximations: cut off a compact neighbourhood of their image in X, cover by the relative anchor charts, and use a finite partition in Y. Functions on Y and pullbacks of local arrow-chart functions are dense on each compact product chart. The common Schur bound proves module-norm density; completion is onto. Finite Clifford modules and both Morita evaluations pull back literally. The raw Bott multipliers and both inverse homotopies also pull back literally on the base-changed algebras, with compact support estimates on the image of each compact Y set.

For the products in(392), product operators pull back by the same criterion, without an unavailable ordinary KK homomorphism. On a compact parameter support the pulled creation errors are compact by the rank-one observation and its rectangular version. Finite balanced creation sections are dense, treating all errors. Here is the required positivity check for arbitrary new source coefficients, including their cross terms. Denote the old even self-adjoint anticommutator by T. For a new coefficient a supported in a compact subset K⊂Y, choose u∈C\_c(X), 0≤u≤1, equal to one on h(K). The old source is the nondegenerate finite Clifford bundle, so φ(u1\_Cl) is precisely the right central multiplier u. The old criterion says uTu≥0 modulo K(H). Functional calculus in the Calkin algebra gives uTu=Q+J with Q≥0 adjointable and J compact: take Q=(uTu)\_+ and J=−(uTu)\_−. After pullback, a(h\*T)a\* equals a(h\*Q)a\* plus a(h\*J)a\*, since h\*u=1 on its support. The first term is positive. Choose g∈C\_c(Y) equal to one on K; the second is compact by the proved g-localized compact pullback. General C0 source coefficients are norm limits of compactly supported ones. This verifies the full criterion, rather than just its individual separated summands. The same argument applies to the first product, whose finite Clifford source is also essential. Products therefore have the actual pulled operators on the associating tensor-module unitary. The(392) module followed by the constant external η is the corresponding construction for fh, including the same right A representation. Product uniqueness gives

\[
 h^*\chi_f(\eta)=\chi_{fh}(\eta)
 \tag{394}
\]

with the full pulled Clifford twist. The average(393) pulls back by its finite intrinsic formula. For proper h the total-bundle map is proper as well, so this also recovers the actual C0 homomorphism comparison(146). Compact bases give the earlier compact smooth result. Properness has not been silently added to naturality in the geometric-field model.

Changing cocycle representatives uses the principal-space unitary from their actual equivalence arrows. Its products compose exactly on cores, and therefore on the completion. Its normal derivative transports the full Clifford twist. Start with the transported metric, then use the positive metric interpolation. For a continuous homotopy of actual cocycles on X×I, C2–C4 apply once to that whole base. Endpoint restrictions are onto: a compact endpoint core section extends in its relative product chart using a parameter cutoff, and finite chart sums are dense. This is the same complete module proof as6.0n, with a continuous parameter and vertical-smooth variables. The product theorem on the interval algebra then gives one homotopy of χ, rather than separately chosen endpoint operators.

For fixed germ data, normalized tube germs have a concrete contraction. In one reference tube, write another as a fibre-preserving germ h:E→E with h(x,0)=0 and D\_vh(x,0)=1. Put

\[
 h_s(x,v)=s^{-1}h(x,sv)\quad(s>0),\qquad h_0(x,v)=v.
 \tag{395}
\]

The identity h\_s(x,v)=∫\_0^1 D\_vh(x,tsv)v dt proves continuous extension at s=0, with all vertical derivatives. The derivatives at zero remain one, uniformly on any compact parameter set. The parameter inverse argument of C2 gives a common sufficiently small disk on that set; noncompact base patches use a positive continuous variable radius. Thus(395) is a contraction in normalized fibrewise smooth germs, fixed on the identity germ.

It also supplies finite relative extensions, with face data kept exactly as germs. Suppose compatible normalized germs are given on the boundary of a simplex. Use a collar of that boundary and extend by its collar retraction; keep those germs constant on the outer collar. On the inner collar apply(395) from that boundary value to the identity, using a scalar parameter equal to one near the outer collar and zero at its inner edge. Fill the remaining simplex by the identity. The integral formula and the uniform inverse radius make the extension continuous and fibre smooth. Metric boundary data extend in the same collar and interpolate through the convex cone of positive metrics to a fixed metric. Radii may always be shrunk by another positive minorant, because they specify representatives of the same germs. Refinements and representative comparisons have a common locally finite smaller germ neighbourhood by C2. Equivalence-arrow maps compose there exactly, so they add no associator defect.

Over that whole X×Δ^n, C3 and the product theorem give one module and one product operator. Restrictions to faces use the onto core restriction map and the naturality of the product class. The geometric tube and metric choices can therefore be filled in every finite dimension; their choices of representative germs have no finite-dimensional obstruction. These finite germ fillings compare entire family classes. The global construction in Q and D uses one separable base and one product module. Thus it does not require an infinite family of separately chosen Fredholm operators on simplicial faces.

 ![Strong adjointable field topology, exact Clifford average, compact-defect topology and whole nonproper base change.](../figures/twisted-cocycle-continuous.png)

**Figure 6.2.1.** Strong adjointable field topology, exact Clifford average, compact-defect topology and whole nonproper base change. Proof: Theorem 6.2, part C; Exercises 68–79.  These are original reproducible diagrams of the objects and proved maps. Classical context is [Connes; Connes–Skandalis].

Full-size PNG · Editable SVG.

 #### B1. Actual transverse charts and holonomy arrows

Choose a countable disjoint family of small transverse plaque disks T\_i whose associated foliation boxes cover V. Such a family follows from a countable foliation-box basis; a finite family suffices for compact V. Let T be their disjoint union and H=G|\_T the groupoid with both endpoints in those disks. The disks are disjoint as labelled components, even if their images in V meet. The arrow charts of the course give H its smooth étale structure. Its object space is Hausdorff, second countable and locally compact. Its arrows can be non-Hausdorff. Restricting a countable G path-chart basis and a countable disk basis gives a countable bisection basis for H: in an étale path chart both endpoint maps are local diffeomorphisms, and a smaller open chart makes both injective. No Hausdorffness of all arrows, no amenability and no quotient by isotropy is used here.

The plaque short arrows give an actual comparison between G-cocycles and H-cocycles over any locally compact Hausdorff second-countable base X. Refine a G-cocycle chart so its lift lies in one foliation box, project to that box's labelled transverse disk, and conjugate its transition arrows by the short plaque arrows. This produces an H-cocycle and an actual G-arrow equivalence. Composing an H-cocycle with T→V is its inverse up to this equivalence. Two choices of projection disks have their short-arrow comparison on a common refinement; the derivatives give exactly the comparison of their pulled normal bundles. Over X×I the identical formulas compare concordances. This proves the equivalence of the actual cocycle/concordance functors on these bases. It does not assert a C\*-algebra Morita equivalence, which is not needed for χ: its target analytic algebra remains A=C\*r(G).

Fix a countable basis of contractible open coordinate disks in the transverse object manifold T.

#### B4. A locally compact replacement of any countable CW complex

Let K be a countable CW complex. Enumerate its cells; adjoining their finitely many lower-dimensional faces recursively gives finite subcomplexes K\_n whose union is K. Put

\[
 X=\operatorname{Tel}(K_1\hookrightarrow K_2\hookrightarrow\cdots),
 \qquad p:X\longrightarrow K.
 \tag{396}
\]

There is a continuous proper height h:X→[1,∞), with each closed bounded height slab contained in a finite collection of compact mapping cylinders. Local neighbourhoods meet only the two adjacent such cylinders and are locally compact. Closed gluing along the finite subcomplexes makes X Hausdorff. The finite cylinders are compact metrizable; countably many open slab neighbourhoods with their countable bases make X second countable. Its cells from the cylinder subdivisions give a CW structure. Thus X meets every hypothesis of (389)–(394).

For clarity the weak-equivalence proof of p is direct. A compact subset of a CW complex meets only finitely many open cells: otherwise choose one point in each of infinitely many distinct cells. Closure finiteness makes the chosen points, and every subset of them, closed by the weak topology (each closed cell meets only finitely many). They form an infinite closed discrete subspace of the original compact set, a contradiction. Thus a map of a sphere, disk or finite homotopy cylinder into K has image in some K\_N. Its lift to the height-N copy of K\_N in X proves surjectivity on homotopy groups and components. Any compact image in X has bounded height and lies in finitely many cylinders. That finite telescope deformation retracts onto its last K\_M: collapse the first mapping cylinder to its target, then the second, and so on. These standard cylinder contractions commute with p. If a sphere map's projection is null in K, its null disk lies in a further K\_L. Retracting its original map to height M, then sliding the constant projected map along the cylinders to height L, gives its null disk in X. This proves injectivity, with the same argument on relative paths proving the components claim. Hence p is a weak homotopy equivalence of CW complexes.

A full homotopy inverse is available with an explicit relative cell argument. Here are its filling step and its application, so the cell-filling input is supplied here rather than assumed. Let j:L→M be an inclusion of CW complexes inducing bijections of components and isomorphisms of every based homotopy group. Suppose a disk d:D^n→M has its boundary in L. For n≥2 the boundary class is zero in M and hence in L, by injectivity on π\_(n−1); choose a filling e:D^n→L with the same boundary. The disk d glued to the reversed e is a sphere in M. Surjectivity on π\_n lets us change e, by inserting an L-valued sphere in a small interior disk, until this difference sphere is null in M. A null homotopy of that difference sphere gives a homotopy d≃je fixed on the boundary: regard its two hemispheres as the bottom and top of D^n×I, and its equator as ∂D^n×I. The quotient which collapses the constant equatorial intervals is a disk; precomposing the null filling with that quotient gives the required cylinder. Basepoint paths are kept in the same boundary component. For n=1, component injectivity first gives a path e in L between the endpoints. The difference loop can be corrected by a loop in L using surjectivity on π\_1, and the same null-loop disk gives the relative path homotopy. For n=0 use component surjectivity to select a point of L and a path from each new vertex to it. These arguments prove the relative disk-filling step in every dimension, with the original boundary map fixed.

It also solves the extension problem with a prescribed boundary homotopy. If a cell's boundary is already being homotoped from its original map into M to an L-valued map, append that boundary cylinder to the original disk. A collar identifies the result with a disk whose final boundary is in L. Apply the preceding filling step, and remove the collar parametrization. This extends the given boundary homotopy over the cell, ending in L; its end map has exactly the already prescribed boundary values.

Start with the identity on L. Induct over the relative cells of (M,L), using this extension problem on each cell. It constructs r:M→L, rj=1\_L, and H:jr≃1\_M fixed on L. The choices on cells of a given dimension are independent, since they agree on their boundaries. The CW weak topology makes r continuous. It also makes H continuous: the product CW structure with the finite interval has precisely the weak topology tested by the closed cell disks times I. Thus the construction works on the entire CW complex at once, even with infinitely many cells and without passing to an inverse limit of homotopy classes.

Apply this to the mapping cylinder M\_p of p, with L=X as its bottom subcomplex. The cellular telescope projection p has image in the corresponding cells of K, so M\_p is a CW complex and X is a subcomplex. Its retraction onto K and the already proved weak equivalence of p show that X→M\_p meets the filling hypotheses. Let k:K→M\_p be the top inclusion. The map q=rk:K→X satisfies pq≃1\_K by projecting H on that top inclusion. The cylinder path from x to k(p(x)), followed by r, gives qp≃1\_X. These are whole homotopies on the specified spaces. This completes the actual homotopy-equivalence claim in(396), rather than merely its action on homotopy groups or compact restrictions.

#### R1. The disk-attachment gluing step

For a diagram A←Z×∂Δ^n→Z×Δ^n, its pushout has the same homotopy type as its double mapping cylinder. The second arrow is a closed cofibration with an explicit collar: use the radial coordinate from the barycentre of Δ^n to its boundary, with an outer annulus as collar. Attaching an extra cylinder to that collar only lengthens its radial intervals. Rescaling those intervals, fixed at their inner ends, gives the comparison with the ordinary attachment. The same argument works component by component and with Z arbitrary; n=0 is a disjoint union.

A map of two such diagrams which is a homotopy equivalence on each of A, Z×∂Δ^n and Z×Δ^n therefore induces a homotopy equivalence of the attachments. Here is the usual cylinder argument, in the form needed here. Choose inverses on the three pieces and their inverse homotopies. The inverse pieces commute with the attaching maps up to homotopy: paste the inverse homotopy on the target piece, the original strictly commuting square, and the inverse homotopy on the source piece. On each intervening cylinder use that pasted track, allocating successive thirds of its interval to the three tracks. This defines an inverse map on the double mapping cylinders, agreeing on their ends. Pasting the same inverse homotopies gives homotopies of both composites to the identity; subdividing the interval once more keeps the end maps fixed. Rescaling the radial collars transfers them to the ordinary attachments. This proof uses actual homotopy equivalences on the pieces, rather than an unqualified weak-equivalence gluing rule.

#### R2. Compact parameter images have bounded fat dimension

Let Z\_• be a semisimplicial space whose levels are T1. A compact subset of ||Z\_•|| is contained in some finite fat skeleton. Otherwise choose points y\_j in that subset with strictly increasing minimal dimensions n\_j, using the unique interior representative in its minimal simplex. The inverse image of y\_j in a fixed level Z\_k×Δ^k is empty if n\_j>k. For n\_j≤k it is a finite union of closed sets: prescribe the zero barycentric coordinates of a face, prescribe its remaining coordinates, and take the inverse image of the closed base point under the face map. There are only finitely many faces. Thus the inverse image in every level of {y\_j:j∈J}, for any subset J of the selected indices, is closed. The weak quotient topology makes every such subset closed in the realization. The selected set is consequently a closed discrete subset of the original compact set. It would be compact and infinite discrete, which is impossible. This proves the dimension bound. It makes no assertion that compact images meet finitely many base points or arrow charts in a fixed dimension.

The face-only quotient has no other identifications of interior representatives, which is why minimal dimension is available here. The argument does not use a Hausdorff arrow space. It does require the T1 condition at each level, verified below in the actual application.

#### R3. Level homotopy equivalences give a fat weak equivalence

Suppose a semisimplicial map Z\_•→Y\_• is a homotopy equivalence in each level, and both sets of levels are T1. Its fat realization is a weak homotopy equivalence. Indeed the n-skeleton is obtained from the (n−1)-skeleton by the attachment in R1 with Z=Z\_n. The comparison on its boundary piece is also a homotopy equivalence, by the level inverse times the identity of ∂Δ^n. Induction using R1 gives an actual homotopy equivalence on every finite fat skeleton. A sphere, disk or finite homotopy-cylinder image in either full realization is compact and hence lies in a finite skeleton by R2. Surjectivity and injectivity on all homotopy groups follow from the finite-skeleton equivalences, enlarging that skeleton when a null disk is used. The same path argument gives the bijection of components. No compatibility of the separately chosen skeletal inverses is required for this weak equivalence statement.



#### R4. Cell cylinders and the fat-to-ordinary comparison for simplicial sets

For any simplicial set S, its fat realization maps weakly equivalently to its ordinary realization. Here is a proof which does not require closed degeneracies for a topological groupoid. For the representable simplicial set Δ[n], its fat realization is contractible. A simplex is an ordered string of vertices in {0,…,n}, allowing repetitions. Append vertex n and give that final vertex weight s, multiplying all old weights by 1−s. This respects every face relation, gives the identity at s=0, and is the constant final vertex at s=1. Its ordinary realization is the contractible geometric n-simplex. Thus the comparison for Δ[n] is a homotopy equivalence.

Both fat and ordinary realization preserve the pushouts which attach the nondegenerate simplices along their boundaries. The pair (||Δ[n]||,||∂Δ[n]||) is a CW pair: one cell is present for every simplex, including degenerate ones, and the simplicial subset supplies a subcomplex. Induction on nondegenerate dimension first proves the comparison for the boundary, then for a skeleton of an arbitrary S using these attachment pushouts. All their spaces are CW complexes. Their weak-equivalence comparisons are homotopy equivalences by the relative cell-filling proof in B4; the mapping-cylinder gluing argument of R1 applies to these CW cofibrations as well. Their homotopy-extension property is explicit: extend over each relative cell using the retraction of D^k×I onto D^k×{0}∪∂D^k×I, obtained by radial projection from a point above the centre of its top face. It fixes that bottom-and-side subset. Induction on cells and the CW weak topology give extension on the whole pair. Applying this extension to the cylinder-end homotopies gives the same ordinary-pushout to double-mapping-cylinder comparison used in R1. No global collar of an arbitrary CW subcomplex is being presumed. Disjoint unions handle any number of simplices in a dimension. This proves the weak comparison on every nondegenerate skeleton.

Finally a compact sphere, disk or homotopy image in either realization meets only finitely many of its CW cells, by the finite-cell argument in B4. Each of those fat cells belongs to the degeneracy of a simplex of finite nondegenerate dimension. Thus the compact image lies in some nondegenerate skeleton. The preceding comparisons prove the weak equivalence of the full realizations, with all components retained. For countable S both realizations are countable CW complexes, so B4's cell-filling argument supplies a full homotopy equivalence if desired.

#### A1. A labelled atlas groupoid with every actual arrow

Use the countable contractible basis U of the Hausdorff transverse object manifold T from B1. Put T'=coprod\_(U∈U) U. Define H' to have objects (U,z), z∈U, and arrows

\[
 ((U,z),(V,z'),g),\qquad g:z\longrightarrow z'\text{ in }H.
 \tag{397}
\]

Its topology is the pullback topology from H together with the discrete chart labels; composition and inversion use the exact H products. The forgetful Q:H'→H is a continuous functor. Arrow spaces of H' can still be non-Hausdorff. No two different isotropy or holonomy arrows are identified in(397), even when they have the same endpoints.

#### A2. A locally finite partition on objects

Choose continuous functions λ\_U:T→[0,1], with sum\_U λ\_U=1, locally finite closed supports and supp λ\_U⊂U. Here supports, rather than the entire countable basis U, are required to be locally finite. Existence follows from a countable locally finite shrinking of the coordinate-disk cover on the locally compact second-countable Hausdorff manifold T and its subordinate partition. If several refined members are assigned to the same U, sum their functions. Their closed supports form a locally finite family, so their union is closed and still contained in U. The merged family retains all stated properties.

There is one useful local consequence. Near a fixed finite list x\_0,…,x\_n of objects, only finitely many supports can occur. A label U whose closed support omits x\_i can be removed on a smaller neighbourhood if necessary. Every retained label with x\_i in its support has x\_i∈U and hence remains a valid object chart near x\_i. Thus a finite expanded string below can be chosen on a neighbourhood even at points where some of its weights vanish. The resulting map does not try to retain an invalid object (U,x\_i) outside U with a zero weight. Fix any total order of the countable chart labels.

#### A3. Weighted vertex splitting gives both actual homotopies

Write a fat H simplex as (x\_0→…→x\_n; t\_0,…,t\_n), t\_i≥0, sum\_i t\_i=1. For every vertex i, replace it by its finitely many labelled copies (U,x\_i), in the fixed label order, with weights t\_i λ\_U(x\_i). Copies at the same vertex are connected by the actual unit 1\_(x\_i). Between blocks i<j use the actual composite g\_j⋯g\_(i+1):x\_i→x\_j of the old arrows. Intermediate blocks whose weights vanish are deleted using that identical composite. This defines

\[
 S:\|N(H)\|\longrightarrow\|N(H')\|.
 \tag{398}
\]

Continuity is explicit. On a neighbourhood of a given simplex's object list, A2 gives a fixed finite valid list of labels. All its arrows are compositions of the original continuously varying arrows or units; all its weights are continuous. Hence the formula is a continuous map to one finite-dimensional nerve level and then to its fat realization. When a weight vanishes, deletion by a face agrees with the neighbouring expanded formula. When an original vertex has zero weight, its whole block disappears and the old face product gives exactly the composite between the remaining blocks. Consequently these local formulas agree under all fat face relations and descend continuously in the weak quotient topology. Local finiteness is only on T. No cover or partition of H\_1 was selected.

For QS≃1\_(||N(H)||), place, in each block i, the original vertex x\_i followed by its expanded copies. Give them respectively

\[
 (1-s)t_i,\qquad s t_i\lambda_U(x_i),\qquad 0\le s\le1.
 \tag{399}
\]

All within-block arrows are 1\_(x\_i); the arrows between blocks are the same old composites. At s=0 the expanded vertices disappear, giving the original fat simplex exactly. At s=1 the original vertices disappear, giving QS exactly. The sum of all weights is (1−s)sum\_i t\_i+s sum\_i t\_i sum\_U λ\_U(x\_i)=1. All weights are nonnegative. The same face and local-continuity argument proves a whole homotopy, including all endpoints and all dimensions. A product of a quotient map with the compact locally compact interval is a quotient map, so this levelwise formula descends as a continuous homotopy on the product of the fat realization with I.

For SQ≃1\_(||N(H')||), use exactly(399), retaining the original labelled vertex (V\_i,x\_i) as its first vertex. The arrow from an old or new copy at x\_i to another copy at x\_i is now the H' arrow whose H value is 1\_(x\_i), with its specified source and target labels. This exists whenever both labels are present; A2 guarantees that condition on each local formula. Arrows between blocks keep their actual H values and the expanded endpoint labels. At s=0 one obtains the original H' simplex and at s=1 SQ. Thus

\[
 \|N(Q)\|S\simeq1_{\|N(H)\|},\qquad
 S\|N(Q)\|\simeq1_{\|N(H')\|}.
 \tag{400}
\]

These are full fat homotopies, not just equivalences on individual homotopy groups. No degeneracy relation was used to combine repeated vertices: their removal at the endpoints is a genuine zero-weight face relation. The proof therefore retains the possible nonclosed unit locus of the non-Hausdorff H arrow space.

All identities and compositions used here also hold after applying the actual normal derivative representation of H. Within a block the derivative is the identity, and between blocks it is the exact old derivative product. Thus this atlas factor introduces no new orientation, determinant line or transverse twist. This is a check of the exact transition maps under the comparison, not a substitute for the universal cocycle construction in U and its complete classifying comparison in V and Q.

![Weighted vertex splitting and both entire labelled-atlas homotopies, with every actual arrow retained.](../figures/twisted-atlas-comparison.png)

**Figure 6.2.2.** Weighted vertex splitting and both entire labelled-atlas homotopies, with every actual arrow retained. Proof: Theorem 6.2, part A; Exercises 68–79.  These are original reproducible diagrams of the objects and proved maps. Classical context is [Connes; Connes–Skandalis].

Full-size PNG · Editable SVG.

 #### U1. The indexed arrow join and its quotient

Let I run over nonempty finite subsets of N={0,1,…}. The I-piece of E\_M H consists of weights t\_i≥0, sum\_(i∈I)t\_i=1, and actual arrows g\_i with a common source x. Delete a zero-weight arrow and its index by a face relation. Give E\_M H the resulting weak quotient topology of these finite simplex/arrow fibre products. The index of each retained arrow stays fixed under a face map. Thus every point has finitely many nonzero indexed weights. Right multiplication by h:z→x sends every g\_i to g\_i h and retains every weight. Define B\_M H as the orbit quotient by this actual right H action. Distinct arrows with the same endpoints have not been identified before that specified principal action. Denote the orbit projection by q. For b∈B\_M H the indexed weights t\_i(b) are well-defined continuous functions, with sum\_i t\_i(b)=1.

The right action is jointly continuous in the stated weak topology. This needs a check because an arbitrary product of quotient maps would not suffice. Cover the acting arrow h by étale bisections B. On such a bisection, h is uniquely and continuously determined by t(h)=x, the common source of the tuple. The corresponding part of E\_M H×\_(T)B is homeomorphic to the open subset with x∈t(B). On each finite piece, multiplication by that determined h is continuous and respects every face. The weak quotient restricted to this open subset is still quotient, so multiplication descends continuously there. These open pieces cover the action fibre product. This proves joint continuity without assuming that H\_1 is Hausdorff or that its product with a quotient is automatically quotient. It also proves continuity of the inverse local chart below, by the same bisection argument.

Set O\_i={b:t\_i(b)>0}. On q^−1(O\_i), normalize all arrows by

\[
 (g_j)_j\longmapsto(g_jg_i^{-1})_j,
 \qquad g_ig_i^{-1}=1_{t(g_i)}.
 \tag{401}
\]

This is continuous on each finite piece and respects every face. It is invariant under right H: (g\_jh)(g\_ih)^−1=g\_jg\_i^−1. The restriction of q to this open saturated set is a quotient map, so(401) descends continuously. Its range Z\_i consists of the tuples whose ith arrow is a unit. The induced map O\_i→Z\_i is inverse to q|\_(Z\_i); their two composites are literal identities. In particular the local anchor u\_i(b)=t(g\_i) is continuous. On O\_i∩O\_j the invariant actual arrow

\[
 c_{ji}(b)=g_jg_i^{-1}:u_i(b)\longrightarrow u_j(b)
 \tag{402}
\]

is continuous. The equations c\_ii=1, c\_ij=c\_ji^−1 and c\_kj c\_ji=c\_ki follow by actual cancellation. They are equations of arrows, rather than leaf-image relations. Thus (u\_i,c\_ji) is an actual H-arrow cocycle on the whole candidate B\_M H.

It is the actual local principal cocycle of q. The map sending a tuple over O\_i to (its orbit b, g\_i) is a homeomorphism onto {(b,h):b∈O\_i, t(h)=u\_i(b)}. Its inverse multiplies the normalized tuple(401) on the right by h. Both maps are continuous by the normalization and the groupoid action, and both composites are identities. This checks principality locally without imposing Hausdorffness on H\_1 or on the quotient candidate.

#### U2. A globally numerable cover, even on the weak quotient

The functions t\_i themselves need not have locally finite supports. There is an explicit numerable replacement. Put

\[
 m(b)=\max_i t_i(b)>0,\quad
 a_i(b)=(t_i(b)-m(b)/2)_+,\quad
 \lambda_i(b)=a_i(b)\big/\sum_j a_j(b).
 \tag{403}
\]

Every sum or maximum here is finite at a point. More importantly, these are continuous functions in the specified weak topology: on every finite join piece they are the corresponding continuous finite expressions, agree on zero-weight faces, and are invariant under the H action. The denominator is positive, since a coordinate attaining m contributes m/2. Their sum is one.

Their closed supports are subordinate and locally finite. The closure of {λ\_i>0} lies in {t\_i≥m/2}⊂O\_i, by continuity and m>0. At b\_0 let I\_0 be its finite positive support and m\_0=m(b\_0). The function u(b)=sum\_(i∉I\_0)t\_i(b)=1−sum\_(i∈I\_0)t\_i(b) is continuous and is zero at b\_0. On the open neighbourhood {m>m\_0/2, u<m\_0/8}, every i outside I\_0 satisfies t\_i≤u<m\_0/8<m/2 and hence λ\_i=0 throughout that neighbourhood. None of their closed supports meets it. This proves local finiteness on the whole B\_M H, without assuming the quotient paracompact. Therefore the cover O\_i and actual cocycle(402) are numerable.

#### U3. The precise finite-dimensional normal bundle

On O\_i take u\_i\*τ. Glue these vector bundles by the exact derivatives D c\_ji. They satisfy the inverse and triple equations because the normal derivative is a representation of actual holonomy arrows. The local gluing gives a continuous rank-q vector bundle E\_M on the candidate with no orientation or spin-c choice. It is numerable by(403). A metric is explicit: choose the existing metric on τ over T, transport its local pullbacks to the common fibre using D c\_ji, and sum them with the weights λ\_i. Every sum is locally finite and positive definite. Its local coefficients are continuous; closed subordination lets each weighted summand extend by zero outside O\_i. Consequently its full graded Clifford bundle and graded-opposite bundle are actual continuous bundles.

This records the entire derivative twist, including the rank, orientation character and any obstruction to a spin-c choice. No scalar Chern character or numerical pairing replaces that bundle.

#### U4. Every eligible parameter cocycle has a whole map into it

Let f be an actual continuous H-cocycle on a locally compact Hausdorff second-countable X. Refine to countably many charts indexed by a∈N and choose a locally finite partition ρ\_a with closed supports contained in those charts. At x use weights ρ\_a(x). Choose any a\_0 with ρ\_(a\_0)(x)>0 and arrows f\_(a,a\_0)(x):f\_(a\_0)(x)→f\_a(x) for the indices with positive weights. This tuple has a common source and defines an orbit

\[
 \Phi_f(x)=[(\rho_a(x),f_{a,a_0}(x))_a]\in B_M H.
 \tag{404}
\]

Changing a\_0 multiplies every arrow on the right by the same actual transition arrow, so the orbit is independent of a\_0. For continuity fix a positive a\_0 on a neighbourhood of a point. Only finitely many closed partition supports occur near it. Labels whose supports miss that point can be removed by shrinking further; each remaining label has its valid cocycle chart on that neighbourhood. Thus even its possible zero-weight columns have defined transition arrows there. Formula(404) is a continuous map into one finite join piece followed by q. These local quotient maps agree, proving a continuous map of the entire X, without taking compact-stage limits.

The pullback of(402) to {ρ\_a>0} is literally f\_ba: f\_(b,a\_0) f\_(a,a\_0)^−1=f\_ba. Its anchors are the given f\_a. This positive-support cover refines the original cover and still covers X. Hence Φ\_f pulls the actual candidate cocycle back to f up to the specified refinement. Its normal bundle Φ\_f\*E\_M is canonically f\*τ by those identical derivative transitions. A G-cocycle first passes to H by the actual plaque-arrow equivalence of B1, giving the same statement for its full normal twist.

![The indexed arrow join, explicit maximum-cutoff partition and its literal normal cocycle.](../figures/twisted-milnor-cocycle.png)

**Figure 6.2.3.** The indexed arrow join, explicit maximum-cutoff partition and its literal normal cocycle. Proof: Theorem 6.2, part U; Exercises 68–79.  These are original reproducible diagrams of the objects and proved maps. Classical context is [Connes; Connes–Skandalis].

Full-size PNG · Editable SVG.

 #### M1. Actual equivalence supplies a combined arrow cocycle

For this proof an equivalence of actual H-cocycles is given by actual equivalence arrows on a common refinement, satisfying the transition equations. A concordance is an actual cocycle on X×I whose restrictions are equivalent to the two endpoint cocycles. These are the conventions used by C5 and B1.

Write f^0 on charts U\_i and f^1 on charts V\_j. On a common refinement W\_r⊂U\_(i\_r)∩V\_(j\_r), let θ\_r:f^0\_(i\_r)→f^1\_(j\_r) be its equivalence arrow. On U\_i∩V\_j define the cross arrow, wherever x∈W\_r, by

\[
 \theta_{j,i}(x)=f^1_{j,j_r}(x)\,\theta_r(x)\,
                         f^0_{i_r,i}(x).
 \tag{405}
\]

On an overlap of refinement charts the equivalence equation is θ\_s f^0\_(i\_s,i\_r)=f^1\_(j\_s,j\_r) θ\_r. Substitution into(405) and the two original triple equations give the same θ\_(j,i), independently of r. Thus these local actual-arrow maps glue continuously on every cross overlap. Within each family retain the original transitions; in the other direction use θ\_(j,i)^−1. Their inverse and triple equations follow from this displayed equivalence equation, including the triples with vertices from both families. This is an actual cocycle on their disjoint labelled union of charts.

Any smaller chart copying an anchor from the first family can be adjoined to this combined cocycle with its exact restricted transition arrows. Hence the auxiliary charts used below have consistent actual arrows to both original families and to one another. There is no selection of an arrow from its two endpoints alone.

#### M2. Fresh indices are chosen locally, with finite avoidance

Let α\_i and β\_i be the two countable locally finite closed-subordinate partitions used to construct the maps Φ\_0, Φ\_1 of U4 for equivalent cocycles. Both use indices in N; unused indices have zero weight. Choose a countable locally finite precompact cover W\_j whose compact closures lie inside charts of a common refinement. Choose a partition ρ\_j with closed supports inside W\_j. Its anchor γ\_j is the copied first-family anchor on that refinement chart. M1 supplies its actual combined transitions.

For each j let I\_j be the union of the original numeric indices whose α\_i or β\_i closed support meets the compact closure of W\_j. The two support families are locally finite, so I\_j is finite. Recursively choose

\[
 \nu_j>\max\bigl(I_j\cup\{\nu_0,\ldots,\nu_{j-1}\}\cup\{-1\}\bigr).
 \tag{406}
\]

This gives pairwise distinct auxiliary indices. Give γ\_j weight ρ\_j at color ν\_j and zero weight at unused colors. U4 defines the auxiliary whole map Φ\_\* into B\_M H. If an original α\_i or β\_i is positive at x and ρ\_j is positive at x, then x lies in both original support and the closure of W\_j, so i∈I\_j and i≠ν\_j. The same exclusion holds when x belongs to the two closed supports, even if one of the weights is zero. Consequently a neighbourhood of any point has finitely many valid original and auxiliary labels, with no collision among their possible active colors. This is a local condition on the whole X; it does not require a finite bound for all active indices on X.

An auxiliary color can occur as an original color elsewhere on X. That causes no collision there: its auxiliary support is contained in a W\_j whose finite avoidance set excluded exactly that simultaneous occurrence. Thus no globally disjoint relabelling of the two old infinite index sets is presumed.

#### M3. Both choices are joined to that same whole auxiliary map

Use the combined cocycle to form, at x, one indexed tuple with weights

\[
 (1-s)\alpha_i(x)\text{ at color }i,
 \qquad s\rho_j(x)\text{ at color }\nu_j,
 \qquad 0\le s\le1.
 \tag{407}
\]

The anchors are the corresponding old anchor and γ\_j. Choose any positive-weight anchor as common source; all columns are the actual combined transition arrows from it. Changing that anchor multiplies all arrows on the right by the same actual arrow. Therefore the orbit is independent of this choice. M2 prevents two distinct columns being assigned to the same color where either could be present. The weights are nonnegative and sum to (1−s)+s=1.

For continuity near any (x,s), choose a positive source column there and a neighbourhood on which it stays positive. The closed support families and M2 give a fixed finite valid list of columns on a further neighbourhood, with disjoint colors. All combined arrows are continuous on their chart overlaps. This is a continuous tuple in a single finite join piece, followed by its quotient. At the endpoints one family has zero weight and is deleted by faces. Hence it gives the actual whole homotopy Φ\_0≃Φ\_\*. Applying the identical construction with β\_i gives Φ\_1≃Φ\_\*. The middle tuple uses the same γ\_j, ν\_j and ρ\_j, with the same actual transitions in both constructions, so their middle maps agree literally. Concatenating one homotopy with the reverse of the other proves Φ\_0≃Φ\_1 as maps on the entire X.

This proves independence of refinement, partition, enumeration and actual-cocycle equivalence. Every re-enumeration is another choice to which the same finite-avoidance argument applies. The pulled normal bundle along each displayed homotopy is the normal bundle of that whole combined cocycle, by U4's literal derivative identities. Its endpoint comparisons retain the full twist; an odd coefficient has not been traced or replaced by numerical data.

#### M4. A global cutoff homotopy recovers every continuous map

Let R:B\_M H→B\_M H retain the actual arrow columns of a tuple and replace its weights t\_i by the λ\_i of U2. It is well-defined: λ\_i>0 implies t\_i>0, so no new column is needed. More generally retain those arrows and replace the weights by

\[
 t_i(s)=(1-s)t_i+s\lambda_i,\qquad0\le s\le1.
 \tag{408}
\]

They sum to one, respect all zero-weight face relations, and are invariant under the common right H action. On every finite join piece the formula is continuous, including the deletion of cutoff-zero columns at s=1. The finite-piece quotient and the right-action quotient remain quotient maps after product with the compact locally compact interval. Thus(408) defines a continuous homotopy of the entire candidate from its identity to R. No assumption that its raw t\_i supports are locally finite was needed for this global homotopy.

For any continuous F:X→B\_M H, pull the actual cocycle of U1 back to the charts F^−1(O\_i). The partition λ\_i∘F has locally finite closed supports subordinate to them by U2. Construct its classifying map with exactly those indices and weights. Its arrows are the old normalized columns of F and its weights are λ\_i(F). Therefore it is literally RF. Composing(408) with F proves that this choice of classifying map is homotopic to F on the whole X. The full normal bundle is also the literal pulled E\_M of U3.

#### M5. The natural classification bijection, including concordances

Let Coc\_H(X) be equivalence classes of actual numerable H-cocycles, further identified by concordance. Pullback of U1 gives a map P:[X,B\_M H]→Coc\_H(X), since a whole map homotopy pulls back a whole actual cocycle on X×I. Constructing U4 gives Q in the other direction. M3 proves its independence from all representative and partition choices.

If an actual cocycle is given on X×I, perform U4 once on that entire eligible base. Its endpoints have the given endpoint cocycles up to their specified equivalences. M3 on X compares its endpoint partitions with any previously selected ones. These endpoint homotopies, pasted with the whole interval map, give a homotopy of the selected endpoint classifying maps. Thus Q is well-defined on concordances, not only on equivalent cocycles.

U4 proves PQ=1: the pulled cocycle equals the original on its positive-weight refinement. M4 proves QP=1: a pulled cocycle classifies to RF≃F. Consequently

\[
 [X,B_M H]\ \cong\ \operatorname{Coc}_H(X)
 \tag{409}
\]

is a bijection of the full homotopy/concordance sets for every eligible X. It is natural for every continuous h:Y→X between such spaces, including nonproper maps. The pulled closed-subordinate partition α\_i∘h remains locally finite and subordinate; using it in U4 gives exactly Φ\_f∘h. M3 removes dependence on the partition chosen separately on Y. Pullback of cocycles is already literal. The two maps in(409) therefore commute with h at the full class level, with the canonically pulled normal bundle.

The same proof applies to the whole base X×K for every compact finite polyhedron K, since it remains locally compact, Hausdorff and second countable. This asserts the classification of whole families. It does not assert a prescribed relative boundary filling or simultaneous choices on all faces of an infinite realization.

By B1 the actual G-cocycle/concordance functor has the same comparison, with its specified short plaque-arrow equivalences and transported τ. No C\*-algebra Morita theorem is required for that statement; the analytic algebra in χ remains C\*r(G).

![One locally finite auxiliary map joins all classifier choices on the entire base.](../figures/twisted-milnor-classification.png)

**Figure 6.2.4.** One locally finite auxiliary map joins all classifier choices on the entire base. Proof: Theorem 6.2, part M; Exercises 68–79.  These are original reproducible diagrams of the objects and proved maps. Classical context is [Connes; Connes–Skandalis].

Full-size PNG · Editable SVG.

 #### W1. The relative CW lifting lemma

Let f:Y→Z be a weak homotopy equivalence. Let (K,L) be a CW pair. Given b:K→Z, a:L→Y and a homotopy f a≃b|L, there exist a':K→Y extending a and a homotopy f a'≃b extending the prescribed boundary homotopy. No CW structure on Y or Z is required.

To prove it, include Y at the bottom of the mapping cylinder M\_f. Its retraction to Z shows that Y→M\_f has the same component and homotopy-group isomorphisms as f. The disk-filling step in B4 applies to this inclusion: its proof uses only those isomorphisms. A disk whose boundary is in Y first has a filling of that boundary in Y by injectivity on the preceding homotopy group. Correct the difference sphere by surjectivity on the disk's homotopy group, then fill its null difference in M\_f. In dimension1 use endpoint components and difference loops; in dimension0 use paths of corresponding components. This gives a disk homotopy to Y with the boundary fixed.

On L use the mapping-cylinder path from a to f a followed by the specified homotopy to b|L. Induct over the relative cells of (K,L). The boundary homotopy already constructed on each cell is appended as a collar to its b-valued disk, exactly as in B4. The disk-filling step extends it, ending in Y with its prescribed boundary values. This gives a' and a homotopy in M\_f on the entire K. Project to Z. On L the extra cylinder path projects to a constant initial segment before the given homotopy. Remove that segment by a homotopy of its interval parametrization, fixed at its ends. The homotopy-extension property of the CW pair (K×I, L×I∪K×{0,1}) extends this correction, so the projected boundary homotopy is exactly the specified one. R4's cell-cylinder retraction proves that extension property. The weak topology and its product with the finite interval make the whole construction continuous. This proves the lemma, including infinite K.

#### W2. Weak equivalences of spans give a weak equivalence of cylinders

Write P(A,C,B) for the double mapping cylinder of A←C→B. A strictly commuting map of spans which is a weak homotopy equivalence on A,C,B induces a weak homotopy equivalence of P. Here is the finite-parameter proof, including the relative step needed for injectivity.

The cylinder has a continuous height function, equal to0 on A, to1 on B, and to t on C×I. Its two open subsets U\_A={height<2/3}, U\_B={height>1/3} cover P. They deformation retract to A and B by moving the height to their respective ends. Their intersection is C×(1/3,2/3), with its projection to C. These retractions commute with every map of spans; the map also preserves height, so the inverse images of these two open subsets are precisely the corresponding open subsets of the source cylinder.

Let g:K→P be a map from a compact finite polyhedron. Pull back these two opens. Choose a continuous partition p\_A+p\_B=1 with closed supports inside them. Approximate p\_A by a piecewise linear function r within1/12 on a sufficiently fine triangulation. Set K\_A={r≥1/3} and K\_B={r≤2/3}; subdivide at those two levels. They are finite closed subpolyhedra covering K. Their intersection K\_C={1/3≤r≤2/3} maps into U\_A∩U\_B, because both p\_A and p\_B are at least1/4 there. The end retractions give maps

\[
 a:K_A\to A,\qquad c:K_C\to C,\qquad b:K_B\to B,
 \quad a|_{K_C}=\ell c,\quad b|_{K_C}=r_C c,
 \tag{410}
\]

where ℓ and r\_C are the two span arrows, not the piecewise linear function r. This is a strictly commuting map of finite source spans. The collapse p\_K:P(K\_A,K\_C,K\_B)→K is a homotopy equivalence: K is their ordinary union pushout and both subpolyhedron inclusions are CW cofibrations; the cylinder comparison proved in R4 applies.

The assembled span map G:P(K\_A,K\_C,K\_B)→P(A,C,B) is homotopic to g p\_K. On K\_A move g's height toward0; on K\_B move it toward1. On K\_C×I, if g(x)=(c(x),v(x)), use height (1-s)v(x)+s t. At the two ends this agrees with the preceding two retractions. It gives the desired pasted homotopy, including all points of A or B. Thus every finite probe can be replaced, up to its full homotopy, by these exact finite span data.

Now apply this construction to a map into the target cylinder of a weak-equivalence span map F. Lift its c-map up to homotopy through F\_C using W1. Lift the a-map through F\_A relative to K\_C, prescribing there the new source map ℓ c and the homotopy ℓ' H\_C. W1 gives an exact extension a with a|\_(K\_C)=ℓ c. Do the same for b. The three resulting homotopies form a strictly commuting homotopy of span data. Their assembled cylinders therefore give a source map whose image is homotopic to the original target probe. The homotopy equivalence p\_K transfers it to K. This proves surjectivity.

For injectivity, take a target null disk or homotopy cylinder with its boundary the image of a given source probe. Subdivide that entire finite polyhedron using the same two opens. Since F preserves height, the source boundary also lies in the corresponding source opens. Its source retractions give the boundary values of a,c,b. Lift c by W1 relative to that boundary, with its constant comparison homotopy there. Lift a relative to the union of K\_C and its source boundary; their prescribed maps agree on the intersection, and the prescribed homotopies agree because H\_C is constant there. Lift b in the identical way. These are finite CW pairs after the chosen subdivision. The resulting source span data extend the boundary span data exactly.

Their assembled source cylinder extends its boundary assembly. To transfer that extension back to the original disk, choose a homotopy inverse of its source collapse p\_K relative to the desired boundary inverse. W1 applied to p\_K provides it: prescribe the boundary inverse and its known comparison homotopy, then extend over the disk. Composing with the assembled map gives a source disk whose boundary is homotopic to the original source probe by the preceding explicit assembly homotopy. Hence that probe is null. For two endpoint probes the same construction on their homotopy cylinder proves equality. Paths and a marked basepoint can be included in the prescribed boundary subpolyhedron; the usual basepoint-change path is retained in the filling step of W1. This proves injectivity in every based homotopy group and the bijection of components. It proves W2 without a global inverse of a weak map, a CW assumption on A,C,B, or a compactness claim about those target spaces.

#### W3. The full levelwise-weak fat realization theorem

Let Z\_•→Y\_• be a semisimplicial map which is a weak homotopy equivalence in every degree, and suppose every level is T1. Then its fat realization is a weak homotopy equivalence. At a finite skeleton use the attaching span ||Z||\_(n−1)←Z\_n×∂Δ^n→Z\_n×Δ^n. The disk-boundary arrow has the actual collar of R1, so its ordinary attachment and double mapping cylinder are homotopy equivalent. The maps on both new pieces are weak equivalences: products with the fixed sphere or disk preserve components and homotopy-group isomorphisms. Induction with W2 proves the comparison on every finite skeleton. R2's T1 dimension-bound argument places every sphere, null disk and homotopy-cylinder image in some finite skeleton. The finite comparisons therefore give the full weak equivalence. In degree0 the assertion is the given level map; the empty boundary for that degree introduces no extra condition.

This is the exact stronger version required when an augmented étale or open-cover comparison gives only weak equivalences. It does not assert that contractible point fibres by themselves give such level weak equivalences; that remains a separate input.

#### W4. An open-cover groupoid has the correct weak realization

For an open cover {U\_i} of any space Y, let C\_U be the groupoid with objects (i,y), y∈U\_i, and its unique actual arrow between (i,y) and (j,y) when y lies in both opens. Its fat realization maps weakly equivalently to Y by the common y coordinate.

Given a sphere or finite polyhedron map f:K→Y, take a finite closed-subordinate partition of unity on K for the pulled cover. Order its finitely many positive labels, put their weights at the objects (i,f(x)), and use the unique cover arrows. This is a continuous lift: near a point closed subordination gives a fixed finite valid label list, and zero-weight faces agree. Its projection is f exactly. Hence homotopy groups and components are surjected.

For any given lift a:K→||N(C\_U)|| over f, join its original ordered string with such a partition string, retaining their common y=f(x). Give the two strings weights (1-s) times the old weights and s times the partition weights. The unique cover arrows connect the last old to the first new label. Repeated labels are allowed in this fat nerve, so there is no indexed join collision. At s=0 and s=1 faces delete the zero string, giving respectively a and the partition lift.

The homotopy is continuous without requiring a global partition on Y or a local simplex representative for a. Locally in K the new partition labels are a fixed finite valid list, and f takes values in their common open intersection. Over that open set, joining any old fat string to this fixed finite new string is continuous on every nerve piece and respects its faces. Product with the finite new simplex and interval preserves the quotient topology. Thus the local join maps compose continuously with a; they agree when new weights vanish. This proves the whole join homotopy. If f extends over a null disk, choose the partition on that entire disk and join a to its boundary lift. The boundary lift now bounds the lifted disk, proving injectivity. The same path argument handles components and basepoint change. Therefore the cover augmentation is a weak equivalence even when Y has no global subordinate partition or is non-Hausdorff.

For T1 open charts all its nerve levels are T1, and its fat realization is T1 as well: the inverse image of a point in each nerve piece is the finite closed-face union used in R2. Thus W3 can be applied to a face-compatible family of these cover resolutions. The face-compatible holonomy nerve resolution in Q applies this result to every required level.

 ![The relative lifting proof, full double-cylinder gluing and the open-cover weak augmentation.](../figures/twisted-weak-gluing.png)

**Figure 6.2.5.** The relative lifting proof, full double-cylinder gluing and the open-cover weak augmentation. Proof: Theorem 6.2, part W; Exercises 68–79.  These are original reproducible diagrams of the objects and proved maps. Classical context is [Connes; Connes–Skandalis].

Full-size PNG · Editable SVG.

 #### V1. Forgetting colors gives an actual continuous nerve map

For a tuple representing B\_M H, order its positive colors i\_0<⋯<i\_n. Put x\_a=t(g\_(i\_a)) and give this vertex weight t\_(i\_a). Its arrow from x\_a to x\_b is the actual arrow g\_(i\_b)g\_(i\_a)^−1. Use the consecutive ones to form the nerve chain. This defines

\[
 v:B_M H\longrightarrow B_f H.
 \tag{411}
\]

On a finite indexed join piece use all its colors in increasing order, allowing zero weights. The arrows are continuous groupoid products. Deleting a zero coordinate produces the same chain as the corresponding nerve face: the two adjacent products cancel the omitted column. A common right multiplication cancels from every product. Thus the finite-piece formulas descend through the weak join quotient and its orbit quotient. They prove continuity of v on the whole space, including at the faces. No ordering of the arrow space or of the leaf space is being selected.

#### V2. Barycentric subdivision supplies a map in the other direction

For the chain x\_0→⋯→x\_n let h\_a:x\_0→x\_a be its initial actual product, with h\_0=1\_(x\_0). Subdivide Δ^n barycentrically. A simplex of that subdivision is indexed by a flag of nonempty faces

\[
 I_0\subsetneq I_1\subsetneq\cdots\subsetneq I_k
 \subseteq\{0,\ldots,n\}.
\]

If u\_0,…,u\_k are its barycentric weights, put at color |I\_j|−1 the arrow h\_(min I\_j) and weight u\_j. The colors are distinct, because the face cardinalities strictly increase. All arrows have common source x\_0. Pass to their Milnor orbit. On the finite subdivision pieces this is continuous. It defines

\[
 d:B_f H\longrightarrow B_M H.
 \tag{412}
\]

Here are all the gluing checks. On an intersection of subdivision simplices the positive subflag is the same; the other weights vanish and their columns are deleted. If an original coordinate t\_a vanishes, then

\[
 t_a=\sum_{j:a\in I_j}u_j/|I_j|=0.
 \tag{413}
\]

Every positive face in the flag therefore omits a. After removing a its cardinality is unchanged. Its minimum is the same remaining vertex, with the changed numerical position in the shorter chain. The initial arrow to that vertex is unchanged when a>0. When a=0 it becomes h\_b g\_1^−1, where g\_1:x\_0→x\_1 is the removed initial arrow. This is a common right multiplication of all retained columns, so the orbit is identical. Successive zero-coordinate deletions obey the same cancellation. These checks prove compatibility with every fat face, including the first face and all intersections of faces. Finite subdivision followed by the weak quotient now proves global continuity of d. No degeneracy identification is used.

#### V3. One entire fat homotopy is explicit

On one flag piece concatenate the original vertices x\_0,…,x\_n with the new vertices x\_(min I\_0),…,x\_(min I\_k). Give them weights

\[
 (1-s)t_a\quad(0\le a\le n),\qquad
 s u_j\quad(0\le j\le k),\qquad 0\le s\le1.
 \tag{414}
\]

The arrow between any two consecutive retained vertices x\_a,x\_b is h\_b h\_a^−1. This includes backward arrows when the successive face minima decrease. It is a valid chain because H is a groupoid. The weights are nonnegative and sum to one. At s=0 its second block is deleted and it is the original simplex. At s=1 its first block is deleted and it is exactly v d of that simplex, since the new colors increase with the flag cardinalities.

At a subdivision boundary the zero new vertices are deleted and the common subflag gives the same chain. At an original face the checks of(413) remove both the old zero vertex and any zero new vertex, while the actual consecutive arrow products cancel as in V1. A changed initial source multiplies all h\_a on the right by the same actual arrow and leaves every chain arrow unchanged. Thus(414) defines a whole continuous homotopy

\[
 1_(B_f H)\simeq v d.
 \tag{415}
\]

Continuity follows first on each finite subdivision piece times I, then through the face quotient, whose product with the locally compact interval is quotient. The argument includes the full infinite realization; no finite-stage homotopy is pasted by a limit. At a vertex x the track is the fat unit 1-simplex at x, with weights (1−s,s). It need not be constant in the fat realization. Its loop is null: inside the fat nerve of the one-object category with just 1\_x, append that same object with weight r and multiply the old weights by 1−r. This gives a contraction, by the same face-compatible formula used in R4. Consequently this vertex track induces the identity basepoint-change action on all homotopy groups. This point is needed for the based conclusion in V5 below.

#### V4. The other composite has exactly the original actual cocycle

Let F:X→B\_M H be any continuous map from an eligible parameter space X. Its original cocycle has anchors u\_i on t\_i(F)>0 and arrows g\_j g\_i^−1. The pullback by d v F of the U1 cocycle is equivalent to it by actual arrows. We check continuity of those arrows; there is no globally continuous choice of a subdivision flag.

At a finite tuple, order its positive weights by decreasing size, writing w\_1≥⋯≥w\_r>0 and w\_(r+1)=0. The barycentric coefficient of the face of the k+1 largest original weights is

\[
 u_k=(k+1)(w_(k+1)−w_(k+2)),\qquad0\le k<r.
 \tag{416}
\]

Their sum is one by telescoping, and(413) recovers each original weight. Equal weights give a zero coefficient at that boundary, so the positive face is uniquely the top k+1 set at every positive new color k. Choose its least original color a. The new anchor at color k is exactly u\_a. The comparison arrow from that new anchor to an original anchor u\_i is

\[
 g_i g_a^{-1}:u_a\longrightarrow u_i.
 \tag{417}
\]

Where that new coefficient is positive, its top set and a are locally constant as sets of colors. This remains true when nearby tuples acquire arbitrarily many new raw colors. At the given point let I be the finite original positive support. The total outside weight 1−sum\_(i∈I)t\_i is continuous and zero there. Near that point it is smaller than a fixed fraction of the positive gap w\_(k+1)−w\_(k+2) and of w\_(k+1). Each outside coordinate is bounded by that total. The finitely many coordinates in I are continuous. Shrink the neighbourhood so their strict top-set gap persists and every outside coordinate remains below it. The selected top k+1 set is then unchanged throughout that neighbourhood.

Thus(417) is a continuous actual-arrow map on every relevant positive new/old overlap, by the continuity of U1's normalized old arrows. Changing common source does not change it. The old and new transition equations are exactly cancellation of the g columns, so(417) gives an actual-cocycle equivalence on the joint positive refinement. Applying the normal derivative gives precisely the corresponding normal-bundle equivalence, with no loss of the twist.

M5 on the entire X now proves

\[
 F\simeq d v F\quad\text{as whole maps }X\longrightarrow B_M H.
 \tag{418}
\]

Only this eligible-domain conclusion is used. It does not assert a global homotopy 1≃d v on the possibly non-locally-compact candidate. For spheres, disks and finite polyhedra it retains all their actual cocycle data, rather than passing to separate point fibres.

#### V5. The comparison is a weak equivalence with the full twist

Equation(415) makes v surjective on every homotopy group at a vertex: apply d to a based sphere; its composite with v is homotopic to the original sphere. The vertex track is the null unit loop checked in V3, so this yields the same based class. For injectivity let F:S^n→B\_M H have vF extending to a disk. Apply d to that disk. Its boundary is d v F. Equation(418), on the entire sphere, gives a homotopy of F to that boundary; appending its collar supplies a disk for F. A freely null sphere is also based null: its homotopy track gives a basepoint-change isomorphism, and that isomorphism sends the zero class to the zero class. This proves injectivity.

The identical argument with paths proves the component bijection. Every point of either space has a path to a vertex: decrease all but one positive weight linearly in its representing finite simplex. In B\_M H a vertex at any color is connected to the color-0 vertex at the same target object using the two-color tuple of unit arrows. Hence the preceding based conclusion transfers to every component and every basepoint by actual paths. Thus v is a weak homotopy equivalence. The reverse map d is also weakly equivalent, by(415).

The comparison identifies the normal cocycle on every eligible probe by(417) and its exact normal derivatives. It proves a specific full holonomy comparison, not merely the weak type of orbit points. Part Q supplies a verified countable CW model with its actual universal cocycle, followed by the full twisted K group and the ordinary-manifold KK triangles.

#### V6. From the transversal to the full holonomy fat nerve

Choose the foliation boxes O\_i of B1, labelled even when they overlap. Let G' have objects (i,z), z∈O\_i, and every actual G arrow between those labelled endpoints. Its forgetful functor Q:G'→G has the same entire fat homotopy equivalence as A2–A3. Those proofs require a locally finite closed-subordinate partition only on the Hausdorff object manifold; the arrow space need not be étale. Their formulas use only units, continuous actual compositions, weights and valid endpoint charts. A countable locally finite partition for the O\_i therefore gives both full homotopies for Q in exactly(398)–(400).

Let t\_i:O\_i→T\_i be the plaque projection. Let a\_i(z):z→t\_i(z) be its continuously varying short plaque arrow, chosen to be the unit on the transverse section. Define R:G'→H by

\[
 R(i,z)=t_i(z),\qquad
 R(g:(i,z)\to(j,w))=a_j(w)g a_i(z)^{-1}.
 \tag{419}
\]

These are continuous maps, and the two middle short arrows cancel in a composite, so R is a functor. Let S:H→G' include its labelled transverse object into its own box and retain the actual arrows. Then RS=1 literally. The short arrows a\_i(z) give a natural isomorphism 1\_(G')→SR: for any g:z→w, R(g)a\_i(z)=a\_j(w)g. This equality is of actual G' arrows with the displayed endpoint labels.

For completeness it gives an entire fat homotopy without using degeneracy identifications. In each old vertex block, interleave the old object (i,z) of weight (1−s)t with its projected object (i,t\_i(z)) of weight s t. Use a\_i(z) within the block. Between the projected old vertex and the next old vertex use g\_next a\_i(z)^−1. All remaining arrows are their actual products. At s=0 the projected vertices disappear, leaving the old chain; at s=1 the old vertices disappear, leaving its SR chain. Zero old weights delete whole blocks, with the naturality equation ensuring equality of every face product. The levelwise formulas and quotient-times-I argument prove full continuity. Thus R,S are homotopy inverse on the whole fat realizations.

Combining these verified functors with V1 gives a specific weak equivalence from B\_M H to B\_f G, with each intermediate fat transversal/atlas comparison an entire homotopy equivalence. Every short-arrow derivative is the canonical normal transport; the displayed naturality equations remain true for those derivatives. No C\*-algebra Morita equivalence or ordinary thin realization for a nonclosed unit locus was invoked.

![The color-order and barycentric maps, backward actual arrows and full fat-holonomy comparison.](../figures/twisted-bar-comparison.png)

**Figure 6.2.6.** The color-order and barycentric maps, backward actual arrows and full fat-holonomy comparison. Proof: Theorem 6.2, part V; Exercises 68–79.  These are original reproducible diagrams of the objects and proved maps. Classical context is [Connes; Connes–Skandalis].

Full-size PNG · Editable SVG.

 #### Q1. Open Euclidean sets have countable CW models with whole inverses

Let O⊂R^q be open. Cover it by the countably many open rational rectangular boxes whose closures are compact and lie in O. All nonempty finite intersections are convex boxes. Denote the cover by U\_i and its abstract nerve by N. It has countably many simplices. Choose a locally finite partition ρ\_i with closed support inside U\_i. A countable locally finite precompact shrinking of the cover gives this partition; merging functions assigned to the same U\_i retains local finiteness and closed subordination, as in A2. Put

\[
 \Phi:O\to|N|,\qquad x\mapsto(\rho_i(x))_i.
 \tag{420}
\]

Near a point only finitely many supports occur, and their common positive membership makes every used simplex a nerve simplex. Thus this is a whole continuous map in the weak CW topology.

For every nonempty nerve face S choose p\_S∈∩\_(i∈S)U\_i. On a barycentric flag S\_0⊊⋯⊊S\_k define Ψ by the affine combination of p\_(S\_j) with that flag's weights. All these points lie in the convex intersection for S\_0. The formula agrees on flag faces and defines a whole map Ψ:|N|→O. Here barycentric subdivision has its literal piecewise affine homeomorphism to |N|: it is a finite homeomorphism on each closed simplex, with compatible faces, and both global topologies are the weak simplex topologies.

The straight segment from x to ΨΦ(x) stays in O. Indeed, in the flag piece containing Φ(x), both x and all p\_(S\_j) belong to the intersection for S\_0. The affine formula agrees on flag boundaries, giving the whole homotopy ΨΦ≃1\_O.

For the other homotopy use the closed-star carrier of S\_0 in N. It consists of faces T with T∪S\_0 a nerve face. It contracts to any chosen vertex of S\_0 by adjoining that vertex and linearly moving weight to it. Both the original flag simplex and ΦΨ of it lie in that carrier: Ψ lies in all U\_i for i∈S\_0, and its positive partition labels therefore form a face with S\_0. On removing the first flag face the new carrier is the star of S\_1, contained in the star of S\_0; other faces keep the same carrier. Induct over the flag cells to extend the homotopy already given on their boundaries, fixed at its two ends. The boundary sphere map into the contractible carrier fills by its displayed cone contraction. This supplies the homotopy on each cell times I. The CW weak topology, and its product with I, prove continuity of the entire homotopy ΦΨ≃1\_(|N|). Thus O has the countable CW model |N| with both whole inverse homotopies. The case q=0 is the one-point or empty complex and obeys the same statement.

#### Q2. The cellular-approximation input is proved here

We need to attach the CW models from Q1 along arbitrary transferred maps. The following supplies the needed cellular replacement. Any map of CW complexes is homotopic to a cellular map, relative to a subcomplex on which it is already cellular.

First consider a map of a compact finite polyhedron of dimension n into R^m, m>n. On an open subset where it is allowed to change, it can be approximated within any given positive continuous error by a locally piecewise affine map missing a prescribed point. The error can tend to zero at the relative closed set, whose original map already misses the point. Here is the construction. An open subset of a finite polyhedron has a locally finite triangulation with simplices small enough for the prescribed error and the local oscillation of the map. To construct one, exhaust that subset by finite unions of small closed subdivided simplices with interiors covering it and closures inside it, subdivide successive collars further, and take their compatible subdivisions. Compact subsets meet only finitely many collars. Additional subdivisions on each collar give any specified mesh bound. This works separately on the finitely many original polyhedron faces, with their boundary subdivisions retained.

Choose each vertex image sufficiently near its original value. Avoid the finitely many affine dependence conditions that would put the prescribed point into the affine span of an incident simplex. Local finiteness makes the number of those incident simplices finite. Equivalently, choose successively outside their proper affine exceptional sets; for the last vertex of every simplex, the span of its earlier vertices and the prescribed point has dimension at most n<m. The resulting affine maps miss the point on every simplex, and their oscillation/error bound gives the stipulated approximation. Set the map equal to its original value on the relative closed set. The error tending to zero there proves continuity of the joined map and of the straight approximation homotopy. This construction also works inside an open ball: bound the error by half the distance of the old value to the ball boundary, as well as by the desired bound.

Now let b:D^n→Y be a map into a CW complex with boundary in Y^n. Its compact image meets finitely many cells by B4's finite cell argument; include their finite closures. Start with the highest dimensional cells of this finite subcomplex. If m>n, use the preceding approximation in the open m-cell, on its preimage, leaving the map outside that cell fixed. The error tends to zero at its preimage boundary, so the modification and its homotopy are continuous and fix ∂D^n. In its characteristic-ball coordinates choose the error also smaller than the distance to the sphere. The modified map misses the cell centre. Retract the punctured closed characteristic ball radially to its boundary, with the homotopy fixed on that sphere. Since the domain image is compact and avoids the centre, this induces a continuous homotopy of b through the finite CW quotient. It removes that open cell from the image. Remove all cells in decreasing dimension until the image lies in Y^n. There are finitely many of them. The boundary stays fixed throughout.

Induct on the cells of the source CW complex. In dimension n, the already constructed boundary homotopy is appended as a collar to the old characteristic disk map. Apply the just-proved relative disk step to its endpoint. The cell-cylinder retraction of R4 extends the homotopy while preserving the previously assigned boundary. On a designated relative subcomplex use the constant homotopy. The weak CW topology and its product with the interval make the resulting map and homotopy continuous on the whole complex. Its n-skeleton is now mapped into Y^n for every n. This proves the asserted cellular approximation, including the countable infinite complexes used next.

#### Q3. Fat cell assemblies of countable CW-type pieces

Suppose a one-fold or two-fold semisimplicial space has T1 levels, each a countable disjoint union of spaces with countable CW models and whole inverse homotopies. Its fat realization has a countable CW replacement with a specified weak equivalence. The two-fold case suffices here and includes the one-fold case.

The simultaneous quotient has pieces Z\_(m,n)×Δ^m×Δ^n and only their face identifications in the two directions. Both iterated fat realizations are homeomorphic to it. A quotient remains quotient after product with either finite simplex, so the two orders of the quotients impose exactly this same final topology. Filter by total degree m+n. At degree N attach the countable disjoint union of Z\_(m,n)×(Δ^m×Δ^n), m+n=N, along its boundary. The rectangle is a closed N-ball with its boundary an (N−1)-sphere. It has a literal collar (or a piecewise affine one after a finite triangulation), so R1's ordinary-attachment/double-cylinder comparison applies.

Inductively let e\_(N−1):K\_(N−1)→T\_(N−1) be a homotopy equivalence from a countable CW model of that finite filtration. Replace the new Z pieces by their countable CW models L\_N, and replace the old target by K\_(N−1). Transfer the attaching map L\_N×∂D^N→T\_(N−1) through a chosen whole inverse of e\_(N−1). Make that transferred map cellular using Q2. Attach L\_N×D^N to K\_(N−1) along it; call the result K\_N. Products with the finite sphere/disk have the product CW structure. Each new open cell has dimension dim(cell of L\_N)+N and its boundary lands in the appropriate lower skeleton because the attaching map is cellular. Closure finiteness follows because each such compact cell boundary meets finitely many old CW cells, and each product cell has finitely many product faces. Thus K\_N is a countable CW complex containing K\_(N−1) as a subcomplex.

There is an actual comparison e\_N:K\_N→T\_N extending e\_(N−1) literally. On a collar of the new disk boundary first follow the cellular-approximation homotopy, then the inverse comparison homotopy e\_(N−1)e\_(N−1)^−1≃1. The inner disk is mapped by the new piece comparison. The collar ends agree with both its attaching boundary and that inner disk. At degree0 just use the given model map for the disjoint degree0 pieces.

This comparison is a homotopy equivalence at each finite degree. The disk-collar comparison turns both attachments into double mapping cylinders. On the three span objects the chosen model maps are whole homotopy equivalences; the attaching squares commute by the collar's displayed homotopy. The homotopy-track proof of R1 applies also to this homotopy-commuting square: map the first part of the cylinder along its prescribed boundary track and the remainder along the actual cylinder. The inverse uses the inverse object maps and reversed tracks. In either composite adjacent reversed tracks contract by shortening the out-and-back intervals, and the remaining object inverse homotopies give the identity cylinder map. All formulas keep the old object map fixed at the cylinder end. This proves the finite-stage claim and the literal extension property of e\_N; no level model maps compatible with all face maps were assumed.

Put K=∪\_N K\_N with its CW topology. The compatible maps e\_N give a whole continuous e:K→T, since every CW cell occurs at a finite stage. It is a weak equivalence. A compact subset of K meets finitely many cells by B4, hence lies in some K\_N. A compact subset of T has bounded minimal total degree: if not, select points of strictly increasing minimal total degrees. In each fixed (m,n) piece, the inverse image of a selected point is a finite union of closed sets, specified by its zero weights and the T1 point inverse images under the finitely many face maps. Only finitely many selected points can occur in that piece. The inverse image of any subset of the selected points is therefore closed in every piece. The weak quotient makes the selected set and all its subsets closed, producing an infinite closed discrete subspace of a compact space, a contradiction.

The finite filtration stages have their actual subspace topology. At each attachment the closed disk boundary is embedded and has a collar, so an open set of the old stage extends to an open set of the new stage by its open collar and the attaching map. Extending such opens inductively proves the same statement for a fixed stage in the weak infinite quotient. Thus a map whose image lies in a stage really factors continuously through that stage. Spheres, disks and homotopy cylinders on both sides can now be tested at finite stages. Their comparisons e\_N are homotopy equivalences, proving that e induces the component bijection and all homotopy-group isomorphisms. This argument asserts a weak equivalence, and does not require coherent inverse homotopies on the non-CW infinite target.

#### Q4. A countable chart resolution for the actual étale nerve

For the smooth étale groupoid H of B1, N\_n(H) is a second-countable locally Euclidean T1 space, possibly non-Hausdorff. Locally specify each arrow of a chain by a bisection and its source by a sufficiently small coordinate disk. This gives a chart parametrized by that source disk. Shrink it so every iterated face map is injective on the chart: the retained initial source is obtained by the appropriate initial composite of the finitely many bisection embeddings. There are only finitely many such composites. Every resulting face image is then a Hausdorff open chart in the lower nerve level. A countable bisection basis supplies countably many such chain charts covering every degree. Close their families under all face images. Countably many degrees and finite face operations still give countable families.

Let V\_n be these actual open chart subsets, indexing equal subsets only once. A face carries V∈V\_n to its literal image d\_iV∈V\_(n−1). The face equations hold also for these labels because images of composed maps are images under the composite. Let C\_n be the open-cover groupoid of V\_n on N\_n(H), as in W4, and set

\[
 Z_(m,n)=N_m(C_n)
   =\coprod_{(V_0,\ldots,V_m)}V_0\cap\cdots\cap V_m.
 \tag{421}
\]

The cover faces delete labels; the H faces take their images and the actual nerve point. They commute literally, making Z a bisemisimplicial space. Repeated labels remain allowed. Each intersection is an open subset of the Hausdorff coordinate chart V\_0, hence an open Euclidean set in that chart. Q1 gives each intersection a countable CW model with whole inverse homotopies. There are countably many label strings in every degree, so the same is true of the disjoint union Z\_(m,n). These levels are T1.

W4 gives a weak augmentation ||N(C\_n)||→N\_n(H) for every n. Its domain is T1: the inverse of a point in each finite nerve piece is the finite closed face description used in Q3. These augmentations commute with all H faces, since every face keeps the underlying point's actual image. Apply W3 in the other direction. With the iterated/simultaneous quotient identification of Q3 it proves

\[
 T=\|\!\|Z\|\!\|\longrightarrow B_f H
 \quad\hbox{is a weak equivalence}.
 \tag{422}
\]

Q3's construction supplies a countable CW weak replacement e:K→T. Its composite k:K→B\_f H is therefore a specific weak equivalence. This comparison is made using the actual nerve points and actual H faces. A contractible fibre, a stalkwise category argument or a thin-nerve degeneracy theorem was not substituted for the levelwise weak comparisons.

#### Q5. The universal actual cocycle on a locally compact replacement

Take a finite-subcomplex exhaustion of the countable K, including each cell's finite closure at a finite stage. Its telescope X is locally compact, Hausdorff and second countable, with the proper height and full homotopy equivalence p:X→K proved in B4. The B4 proof uses only countability and the CW structure, so it applies to this verified K. Define the whole map

\[
 F_X=d k p:X\longrightarrow B_M H.
 \tag{423}
\]

Pull back U1's actual cocycle, U2's explicit numerable partition and U3's full normal bundle. Call them f\_X and E\_X=F\_X\*E\_M. They are genuine continuous data on this entire eligible X. In particular, E\_X retains all normal transport, orientation and spin-c obstruction; its metric can be pulled back from U3. The resulting C\_X Clifford algebra is separable, so (389)–(394) apply to one module over the whole base, rather than a sequence of independently chosen compact restrictions.

The map v F\_X is homotopic to k p by the entire homotopy(415). Both are weak equivalences. By V6 their composite into B\_f G is a specific weak equivalence as well. Thus X with f\_X is a locally compact classifying CW replacement of the full holonomy groupoid, with its specified normal cocycle. This is the model used for the geometric twisted cohomology below. The maps retain the full X; no compact-stage inverse limit enters its definition.

#### Q6. Whole model independence, including prescribed comparison tracks

Suppose X and X' are two locally compact CW replacements made by this construction, with their specified weak maps F,F' to B\_M H. W1 applied to F', on the entire CW complex X, gives h:X→X' and a whole homotopy F' h≃F. Applying W1 to F gives a map in the other direction. More precisely h is itself a weak equivalence by the homotopy and the two weak maps into B\_M H; B4's full CW proof makes h a whole homotopy equivalence. Pull the U1 cocycle back along the displayed whole homotopy to X×I. It is a concordance between h\*f\_(X') and f\_X, with the literal normal pullbacks at both ends.

The comparison is unique up to a whole homotopy carrying the prescribed map-to-B\_M track. If h\_0,h\_1 have tracks from F'h\_i to F, concatenate their tracks to give b:X×I→B\_M H with endpoints F'h\_0,F'h\_1. W1 for the CW pair (X×I,X×{0,1}), with F' as its weak map, lifts b relative to those endpoints, giving a whole homotopy h\_0≃h\_1. The lift's comparison homotopy keeps both endpoint maps fixed. This is the full relative assertion of W1; it is not just agreement on each finite subcomplex of X. Composition of such comparisons has the same property, so their class maps are canonically compatible once the specified maps-to-B\_M are retained.

Part D turns this complete cocycle/model comparison into the full twisted field-group comparison.

 #### D1. Whole bundle transport along a compact parameter interval

Let E be a metrized rank-q real bundle over X×I, for eligible X. There is a continuous family of orthogonal isomorphisms E\_(x,0)→E\_(x,t), equal to the identity at t=0. We prove this on the entire noncompact X, rather than choosing unrelated compact transports.

A countable locally finite closed-subordinate partition for orthonormal bundle charts gives an isometric embedding into the trivial real ℓ² bundle: send a vector to its chart coordinates multiplied by the square roots of those partition functions. The squared norm is the original squared norm by their sum-one identity. Locally only finitely many terms occur. The range projection P(x,t) has rank q and is norm continuous; the same finite chart matrix calculation proves that continuity even when its nonzero coordinate list changes between neighbourhoods.

Extend P constantly at both interval ends. By compactness of I, continuity gives, near each x, a positive interval smoothing radius for which its oscillation is less than 1/8, uniformly in t. A locally finite shrinking and partition give a continuous positive minorant δ(x) for those allowed radii, using the minorant argument of C2. Convolve in t with a fixed smooth nonnegative compactly supported kernel of integral one and radius δ(x). The resulting self-adjoint Q(x,t) has ||Q−P||<1/8. It is continuous in x, continuously differentiable in t, and its derivative is locally bounded uniformly in t. Differentiation falls on the rescaled kernel, whose radius has a positive lower bound locally in x; there is no differentiation in x.

Its spectrum lies in the 1/8-neighbourhood of {0,1}. The spectral projection P~ for the interval near 1 is therefore continuously differentiable in t and continuous in x, by its resolvent contour formula. The distance to P is less than one, with an explicit bound. On range P~, Q has inverse norm at most 8/7, while ||(1−P)Q||<1/8; hence ||(1−P)P~||<1/7. On its complement apply the same argument to 1−Q and P, giving ||P(1−P~)||<1/7. The norm of the difference of orthogonal projections is the maximum of these two off-range norms, so ||P~−P||<1/7. Formula(424) consequently identifies their ranges and proves that P~ also has rank q. For two real orthogonal projections at distance less than one,

\[
 U(A,B)=\bigl(AB+(1-A)(1-B)\bigr)
             \bigl(1-(A-B)^2\bigr)^{-1/2}
 \tag{424}
\]

is an orthogonal operator taking range B to range A. Multiplication shows its first factor's square norm is 1−(A−B)^2; that positive operator commutes with both A and B. These identities prove the isometry and range assertions, while continuous functional calculus proves continuity. Thus U(P~,P) identifies the old and smoothed bundles continuously over the whole base.

Set A(x,t)=[∂\_tP~,P~], a real skew-adjoint operator. Solve ∂\_tV=A V with V(x,0)=1. Its Picard iterated integrals converge uniformly on I near each x, since A is locally uniformly bounded; the same estimates prove continuous dependence on x and t. Skew adjointness proves V\*V=1, and solving the inverse equation proves onto-ness. Differentiating V\*P~V gives zero: differentiating P~²=P~ makes ∂P~ off diagonal, and [P~,[∂P~,P~]]=−∂P~. Consequently V transports the smoothed initial range to the smoothed range at t. Combining with(424) at both ends gives the required whole orthogonal transport of E. All coefficients and operator norms above are real, so this construction does not choose a complex orientation or a spin-c lift.

Any two such transports starting at the identity differ by a continuous family of orthogonal automorphisms of E\_0 starting at the identity. Their endpoint automorphisms are therefore homotopic through those automorphisms. The induced Clifford bundle isomorphisms have the same endpoint action on homotopy classes of cycles. The proof also works over X×J for every compact finite polyhedron J, with the transport interval retained as I. In particular it compares whole families of bundle paths.

#### D2. The full field group is homotopy invariant with its twist

For a metrized rank-q real bundle E over eligible X put

\[
 \mathcal K^j(X;E)
  =KK_X^j\bigl(C_0(X,\operatorname{Cl}^{+}(E)^{op}),C_0(X)\bigr).
 \tag{425}
\]

The opposite is the graded opposite, with the star and sign conventions of T2, hence its generators act as the negative right Clifford generators. By C4 this is exactly the group of whole Clifford–Fredholm fields: \*-strong bounded operators, norm-continuous localized compact defects, and exact graded Clifford commutation. Both degrees use the ordered coefficient convention, retaining the positive right C1 in degree1. This statement includes every cycle and homotopy in the group, and not just the constructed χ.

Every continuous h:Y→X induces a pullback into K^j(Y;h\*E) by C5's multiplier-valued balanced module tensor. It preserves localized compactness even for nonproper h. The associativity unitary on the balanced tensor modules gives (h k)\*=k\*h\*; the creation-operator formulas identify the pulled fields and all their Clifford actions. It is additive and preserves degenerate cycles and whole homotopies, so is a group homomorphism.

For a constant bundle over X×I, its two endpoint maps agree on every class under the identification of the endpoints with E. A cycle over the whole interval is literally a KK\_X homotopy with the right coefficient C0(X)⊗C(I). Restricting its left representation to the constant source gives that homotopy. Conversely its full interval source action is recovered from finite interval partitions, central right multipliers and compact base cutoffs as proved in C4. Thus its evaluations are the endpoint classes of the same homotopy, not independent fibre choices.

Moreover endpoint evaluation is an isomorphism with inverse constant pullback. One composite is literally the identity. For the other pull an arbitrary whole interval cycle back along r\_s(x,t)=(x,(1−s)t), as a map from X×I×I to X×I. C5 applies to that whole eligible base. The source bundle is constant, so this is a genuine KK\_(X×I) homotopy from the cycle to constant pullback of its zero evaluation. This proves the other composite is the identity on the full group. It does not assume a uniform operator norm homotopy of an arbitrary field; the whole-module topology is the one already proved in C4.

For a varying E on X×I use D1 to identify it with the pullback of E\_0. The interval assertion just proved gives a group isomorphism between the endpoint twisted groups and equality of the transported endpoint classes of every whole cycle. Its choice is immaterial for this given bundle path: two D1 transports differ by the identity-started orthogonal family, and pulling a fixed cycle through its Clifford isomorphisms gives a whole homotopy of the corresponding endpoint actions. If the bundle paths are homotopic with fixed endpoint bundles, apply the construction on X×J×I. The endpoint transports form a continuous family in J. The two endpoint identifications are therefore the same on homotopy classes. This proves the relative parameter comparison needed for Q6's prescribed tracks.

It follows that a homotopy equivalence h with a specified normal-bundle comparison induces an isomorphism of(425). Apply the inverse map, retain the inverse homotopies and their pulled normal bundles, and use the preceding interval assertion to identify the two composites with the identity on the whole groups. No restriction of this proof to compact X is made.

#### D3. A precise full geometric cohomology convention on BG

Use the countable CW replacement of Q5 and its whole map F\_X:X→B\_M H. Its actual universal cocycle is f\_X and its normal bundle is E\_X. Define the full τ-twisted geometric cohomology by

\[
 K^{j,\tau}(BG)=\mathcal K^j(X;E_X),
 \qquad j\in\mathbb Z/2.
 \tag{426}
\]

Here BG means the full classifying homotopy type of the actual holonomy groupoid, with the universal derivative twist. V6 and Q5 supply its specific comparison to B\_f G. This is a cocycle definition of cohomology: the fields, defects, equivalences and homotopies are those of C4, on one entire classifying CW model. The graded source in(425) specifies the rank and sign convention of τ explicitly; no orientation of τ is presumed. In a spin-c trivialization one can apply the full ordered Clifford/Morita evaluations of T2, but that is an additional reduction of these full groups, not part of their definition.

Q6 gives whole homotopy equivalences and actual normal concordances between any two such models. D2 gives the group isomorphisms and shows compatibility for their prescribed tracks, including their relative comparisons on X×I. Their compositions have the same property. Thus(426) is independent of the chosen countable chart resolution, Euclidean good covers, CW model, exhaustion, telescope, metric and specified replacement map, under those cocycle-compatible model comparisons. This includes all infinite classes on the whole X. It is not the group lim K^j(K\_n;E\_n), and no possible phantom or derived-limit information has been removed.

The statement concerns the classifying homotopy type on CW parameters, as represented by the verified CW replacement. It does not assert that arbitrary cohomology theories on an unreplaced non-CW topological quotient are weak-equivalence invariant. The cocycle, twist and full field convention in(426) are what specify the geometric group meant here.

#### D4. The single analytic-to-geometric class and its naturality

Keep A=C\*r(G) for the original full holonomy groupoid. Apply (389)–(394) once to f\_X on the whole X. This gives one countably generated module, its compact source action, both raw twisted Thom products and the single product

\[
 \chi_X:KK^j(A,\mathbb C)\longrightarrow K^{j,\tau}(BG),
 \qquad \eta\longmapsto k_(f_X)\,
                  \tau_(C_0(X))(\eta).
 \tag{427}
\]

Additivity follows from the tensor module direct sums and the bilinearity proved in Theorem0.7. Equivalent actual cocycles give the whole principal-space unitaries of C5. For a concordance on X×I, C2–C4 construct one whole product cycle; D2 identifies its two endpoints after their full normal transport. (394)'s finite tube/metric fillings prove choice independence in this whole construction. Thus χ\_X is a group homomorphism, independent of its representative geometries.

For a model comparison h:X→X' with F'h≃F, C5 first gives h\*χ\_(X')(η)=χ\_(h\*f\_(X'))(η). The actual cocycle concordance of Q6 and D2 then identify this class with χ\_X(η). These are full-group identities. They prove that(427) gives a single χ with values in(426), independent of the model under its specified normal cocycle. In particular, it never replaces the analytical class by a Chern character, trace, numerical index or compact restrictions. Torsion classes enter the same product.

There is a useful universal test property. If an eligible locally compact CW base Y has an actual H-cocycle f, its whole Milnor map Φ\_f exists by U4. W1 lifts Φ\_f through the weak map F\_X, producing a whole map a:Y→X and homotopy F\_X a≃Φ\_f. Pull back the actual U1 cocycle and its normal bundle along that homotopy. U4 identifies the other endpoint with f on its positive refinement. C5 and D2 therefore give

\[
 a^*\chi(\eta)=\chi_f(\eta)
 \quad\text{in }\mathcal K^j(Y;f^*\tau),
 \tag{428}
\]

with the normal comparison supplied by that whole homotopy. Different whole lifts with their specified comparison tracks are related by W1 on the relative CW pair Y×I, and D2 treats their bundle paths. Thus the test identity is independent of those lift choices. B1 translates it to actual G-cocycles using short plaque arrows, retaining the same algebra A in the analytic argument.

The same universal test identity holds for every compact smooth parameter
manifold M, without choosing a triangulation of M. By Manifold duality,
Lemma 3.3, choose an actual finite cubical CW complex P and maps
\[
 i:M\longrightarrow P,\qquad r:P\longrightarrow M,\qquad ri=1_M.
 \tag{428a}
\]
For an actual cocycle f on M, apply W1 to the whole map
\(\Phi_f r:P\to B_M H\). It gives \(a_P:P\to X\) and a whole homotopy
\(F_Xa_P\simeq\Phi_f r\). Restrict that homotopy along i and put
\(a=a_Pi\). Its endpoint is exactly \(\Phi_f ri=\Phi_f\).
Pulling the U1 cocycle and its full normal bundle through this restricted
homotopy, U4, C5 and D2 give (428) on M with the exact prescribed twist.
These are continuous parameter pullbacks, so no smooth structure on P or
properness of r is required. For two whole lifts with specified comparison
tracks, apply W1 to the relative finite CW pair P×I, using the endpoint
maps composed with r and the tracks composed with r, and then restrict
along i×1_I. D2 gives the same independence on M. Every comparison is
on the whole compact parameter space.

This construction avoids an infinite system of separately chosen Fredholm product operators on simplicial faces. It uses exactly one separable base algebra and one product module. All parameter naturality is obtained from this one entire product class.

![The verified countable CW replacement, whole telescope and normal-bundle transport.](../figures/twisted-whole-model.png)

**Figure 6.2.7.** The verified countable CW replacement, whole telescope and normal-bundle transport. Proof: Theorem 6.2, part Q/D; Exercises 68–79.  These are original reproducible diagrams of the objects and proved maps. Classical context is [Connes; Connes–Skandalis].

Full-size PNG · Editable SVG.

 #### G0. The two manifold cycles and their full analytic providers

Let M be a compact smooth manifold without boundary of dimension d, with a Riemannian metric. Put C=C(M), Q=C(M,Cl^+(TM)^op). Its real generators are negative: r(v)^\*=-r(v), r(v)^2=-|v|². No orientation or spin-c structure on M is assumed. Disconnected components and dimension0 are treated by the same finite sum.

#### G1. The actual diagonal coevaluation cycle

Choose r>0 with the closed geodesic 2r disks inside the injective normal neighbourhood, uniformly on compact M. Put U={(x,y):x=exp\_y(w), |w|<r}. Write

\[
 q_r(v)=r v/\sqrt{1+|v|^2},\quad
 v=q_r^{-1}(w)=w/\sqrt{r^2-|w|^2}.
 \tag{429}
\]

The coevaluation module is H\_Θ=C0(U,pr\_y\*Cl^+(TM)^op), an ideal Hilbert C⊗Q module with its standard right inner product. The source ℂ is the scalar unital representation. Its odd operator is

\[
 B_Θ(x,y)=i r_y(v_y(x))/\sqrt{1+|v_y(x)|^2}.
 \tag{430}
\]

It is self-adjoint, of norm at most1, and its square defect is −1/(1+|v|²), a C0(U) section. The compact endomorphism algebra of this ideal module is the ideal itself, so the scalar-localized defect is compact. Source commutators vanish. Separability gives countable generation. This defines Θ∈KK(C\_scalar,C⊗Q). It is the actual raw standard Clifford Bott cycle on the normal disk, using the \*-isomorphism c^+(v)↦i r(v), followed by open extension. Tube compression and its two-sided inverse products are the full raw constructions6.0a–d, before orientation reduction. Different small radii and metrics give the same class by their normalized tube homotopies. The direction is from y to x in(430); this choice fixes the Gaussian sign in G4.

#### G2. The actual evaluation cycle

On H\_D=L²(M,ΛT\*M\_C) use exterior-degree parity and the metric identification of tangent/cotangent. Write L\_i=ε\_i+ι\_i, R\_i=ε\_i−ι\_i. Q acts by R\_i, and the extra C factor acts by pointwise multiplication at the same point. Define

\[
 D=-i(d-d^*),\qquad F_D=D(1+D^2)^{-1/2}.
 \tag{431}
\]

Here d is the exterior derivative; d\* its actual metric adjoint. D is symmetric on smooth forms, with principal symbol L(ξ), and D²=d d\*+d\* d is the Hodge Laplacian. The complete-source energy argument, followed by elliptic regularity on the compact manifold, gives self-adjointness on its first-order Sobolev domain. The local compact-frequency estimate and a finite partition give compact resolvent. Smooth Q sections graded-commute with its principal symbol; their graded commutators with D are bounded zeroth-order operators. Formula(132), its convergent transform integral and density prove compact graded commutators with F\_D. Its square defect is the compact resolvent−(1+D²)^−1. Thus

\[
 \operatorname{ev}=[H_D,\rho,F_D]
       \in KK(Q\widehat\otimes C,\mathbb C).
 \tag{432}
\]

These conventions are internally fixed, rather than inferred from a right-multiplication symbol in another convention. For comparison with the raw positive-left cycle D\_raw=ΣR\_i∂\_i, the canonical even unitary on exterior degree k is (-i)^k. It takes L\_i to −iR\_i and R\_i to−iL\_i. Precompose the raw positive-left source by r\_i↦i c\_i^+. Its action becomes R\_i and its Dirac becomes −iΣL\_i∂\_i, exactly(431)'s principal local form. This checks the graded opposite, its involution and the physical Fourier sign. It does not assert an unchecked identity with i r(ξ)ε from a different right-multiplication convention.

#### G3. The tube product lemma, with one complete operator family

We give the analytic deformation used in both triangles. After tensoring(430) with(432), the actual module is a field of L² sections on the geodesic r disks about its remaining base point. The first triangle has right coefficient C; the second has right coefficient Q. Smooth compact tube sections give the isometric dense tensor identification, with the old counting reduced to the ordinary pointwise C/Q fibre actions. In particular the tensor module is completed before this identification is used.

Use(429), the exact half-density Jacobian and metric parallel transport along the radial geodesics to identify these fields unitarily with L²(T\_zM,ΛT\_zM\_C), with the appropriate right coefficient. Compact M and the closed 2r tube give uniform smooth coefficient bounds. The transformed Dirac principal matrix A\_z(v) is invertible and has positive determinant, with A\_z(0) a positive scalar matrix. Its polar orthogonal part is in SO(T\_zM). Lift it along s↦O\_z(sv), starting at1, to the even left Clifford Spin operators, as in the proved radial-lift construction7.23. This construction is orthogonally equivariant, hence global even when TM is nonorientable. Left even Spin operators commute with the negative right action R\_i. They leave the source Q action and its full twist unchanged. Conjugation makes the principal matrix positive symmetric.

Multiply the transformed symmetric Dirac by the positive scalar ψ(v)=(1+|v|²)^−2, symmetrically on both sides. Its bounded phase has exactly the old Dirac principal phase on each compact tube chart: positive scalar multiplication does not change that phase. In coordinates the resulting symmetric operator is

\[
 A_1=-\frac i2\sum_{ij}L_i
             (K_{ij}(v)\partial_j+\partial_j K_{ij}(v))+Z(v),
 \tag{433}
\]

where K is positive symmetric, smooth and bounded with bounded first derivatives; Z is bounded odd self-adjoint. The required bounds include the derivatives of the polar gauge. Put h=(1+|v|²)^(1/2). Direct differentiation of (429) gives its inverse derivative (h/r)(1+vvᵗ). The smooth exponential and transported orthonormal frame matrices and their inverses are uniformly bounded on the closed 2r tube. Since the derivative of q\_r is O(h^−1), the transformed principal matrix has norm O(h³), first fibre derivatives O(h²), and least singular value at least ch, for one c>0. The half-density and ungauged zeroth-order terms have at most polynomial growth O(h³). These estimates are uniform in the compact base charts.

Write A=OS for its polar decomposition and α\_j=Oᵗ∂\_jO. Differentiating it gives Sα\_j+α\_jS=Oᵗ(∂\_jA)−(∂\_jA)ᵗO. The inverse of this Sylvester map is the integral of the right side conjugated by exp(−tS) for t≥0; its norm is at most the right-side norm divided by twice the least eigenvalue of S. Hence α\_j=O(h). The derivative of the Spin lift has the same bound up to a dimension-dependent constant, because the differential of the finite-dimensional Spin representation is bounded. Thus ∂\_jS=O(h⁴), and the gauge and symmetric-divergence zeroth-order terms are O(h⁴). This controls derivatives of the actual lifted gauge without assuming a uniform ellipticity bound at infinity.

The multiplication by ψ on both sides gives K=ψ²S=h^−8S, so K=O(h^−5) and its first derivatives are O(h^−4). The resulting Z, including the derivative terms of ψ, is O(h^−4), hence bounded. K is uniformly positive on each bounded fibre ball. All assertions are orthogonally equivariant.

Set K\_s=(1−s)1+sK, Z\_s=sZ and let A\_s be(433) with these coefficients. In the first triangle set B(v)=−iR(v); in the second set B(v)=iΓ⊗r(v), where Γ is exterior parity and the last r is the target Q generator. Put T\_s=A\_s+B(v). These are symmetric odd complete operators on the single whole interval module. Their complete self-adjoint domains can be verified directly. On compact fibre sets ellipticity gives first-order local regularity. Cutoffs ρ\_R have ||[T\_s,ρ\_R]||≤C/R, because the principal coefficients are bounded and the potential commutes with scalar cutoffs. For a deficiency solution T\_s\*u=±iu, the imaginary part tested against ρ\_R²u gives ||ρ\_Ru||²≤(C/R)||u||². Let R tend to infinity to obtain u=0 for both signs. This proves self-adjointness on the closure of the smooth compact core. For the whole module, the fibre resolvents are bounded by1 and strongly continuous, with their adjoints, on continuous compact-core inputs: approximate an input by a resolvent image at one parameter, use the common core and coefficient continuity there, then extend by the uniform resolvent bound. Local finite frames and a base partition give that assertion on the entire Hilbert bundle. Thus they define adjointable R\_± on the whole parameter module. Their ranges are dense: the fibre ranges are dense, and finite base partitions patch their local norm approximations of a compact base section; base cutoffs treat every section. The two inverse identities, R\_+\*=R\_− and their commutation hold as whole module identities, by the faithful fibre norms. The resolvent-domain argument of8.15a then proves regular self-adjointness, with R\_+R\_− the dense-range inverse of1+T\_s². Fibre cutoffs and the local elliptic core give the same closed graph as the smooth compact core. This constructs the regular closure on the single whole parameter module.

The needed uniform compactness and continuity also follow from a concrete energy estimate. The anticommutator of A\_s with B has bounded principal cross term and a zeroth term bounded by C(1+|v|). Therefore, as quadratic forms on the core,

\[
 T_s^2\ge A_s^2+\tfrac12|v|^2-C'.
 \tag{434}
\]

It bounds the v-tail of a resolvent vector uniformly. On a fixed fibre ball, local ellipticity of A\_s bounds its first Sobolev norm uniformly in the compact parameters. The compact spatial/frequency estimate therefore gives finite-rank approximations uniformly after that ball cutoff. Tail first, then frequency, proves compact module resolvents. Core resolvent continuity and the uniform bound ||(T\_s±i)^−1||≤1 give strong continuity; the uniform finite-rank approximations and their adjoints turn it into compact-operator norm continuity. Thus the bounded transforms of T\_s are one KK homotopy, with compact defects on the entire interval module. No uniform positive lower bound for K at fibre infinity has been assumed; the confining estimate and local ellipticity are what prove compactness.

At s=1 this bounded transform is the actual product operator. For a homogeneous smooth compact tube creation section ξ, its Dirac leading phase agrees with the evaluation phase, with the factor(-1)^|ξ| in the connection error. The unbounded potential and Z are bounded on that compact section support and change only lower-order terms. Hence the two connection errors have order−1 locally. To control their whole output, choose smooth scalar fibre cutoffs χ and χ′, with χ=1 on the support of ξ and χ′=1 on the support of χ. The commutator [T\_1,χ] is bounded and supported in a bounded fibre ball. The two-pole resolvent identity gives compact commutators with each resolvent and a bound C/(1+τ²) for the integrand of [F\_1,χ]; the compact resolvents already proved above therefore give a norm-convergent compact [F\_1,χ].

Writing T\_ξ for the creation operator, its output outside χ′ is exactly (1−χ′)[F\_1,χ]T\_ξ, since χT\_ξ=T\_ξ and (1−χ′)χ=0. The other connection term has output supported by ξ. Thus the entire output tail is rectangular compact. On the remaining bounded output region, the complete compact-frequency proof6.1 applies to the local order−1 errors. Adjoints give the other connection condition; finite parameter partitions give the whole module assertion, and norm density treats every ξ.

For the first-factor positivity let B\_0=B/√(1+|v|²), precisely the tensor of(430). The derivative of this bounded potential is bounded, its leading part anticommutes with the Dirac part, and Z is bounded. Consequently

\[
 \{T_1,B_0\}=2|v|^2/\sqrt{1+|v|^2}+J_0,
 \qquad ||J_0||\le C_0.
 \tag{435}
\]

The graded integral identity(132) expresses the anticommutator with T\_1(T\_1²+q²)^−1 as q²R\_q {T\_1,B\_0}R\_q+T\_1R\_q {T\_1,B\_0}T\_1R\_q. Its negative part is bounded below by−C\_0R\_q. Each R\_q is module compact and ∫\_0^∞||R\_q||dτ is finite, q²=1+τ². The identity here is a quadratic-form identity before its bounded extension, since the positive part of(435) grows like |v|. Estimate(434) makes that part bounded on the graph of T\_1. The bounded smooth B\_0 preserves that graph: T\_1B\_0=−B\_0T\_1+C on the core, with C={T\_1,B\_0} graph bounded. Thus the two-pole resolvent calculation proving(132) applies on the graph domain, and its sandwiched expressions extend boundedly by the same energy estimate. This justifies its use without claiming that C itself is globally bounded. Integrating finite intervals, then taking the strict transform limit, shows positivity modulo module compacts: the negative compact majorants converge in norm and their tails tend to zero. This argument does not require norm convergence of the separate positive unbounded-potential integrals. It proves the full bounded product positivity criterion. Both conditions are now checked on the actual completed product module.

After deforming to s=0, contract the source point along exp\_z(t q\_r(v)), from its old tube point to z, and transport its full Clifford source along that same radial path. Smooth source sections have uniformly bounded derivatives and bounded commutators with the flat T\_0; density gives the entire source algebra. The transported Q action is the negative right action R at z, since the left Spin gauge was even and commuted with it. The exact radial source homotopy is norm continuous on each smooth source section, uniformly on the closed tube. Compact resolvents make every resulting localized defect compact. This is a whole KK homotopy to the constant source at z, not an independently chosen homotopy at each base point.

#### G4. The function-algebra triangle is exactly its even Gaussian line

In (Θ⊗1\_C)(1\_C⊗ev) the remaining target base is x and the integration point is y. Reversing the radial geodesic gives PT\_(y→x)log\_y(x)=−log\_x(y), and(429) is odd and preserves the radial norm. Thus its flat operator has the exact sign

\[
 T_C=-i\sum_i(L_i\partial_i+R_i v_i),\qquad
 T_C^2=-\Delta+|v|^2+\sum_i(2N_i-1).
 \tag{436}
\]

Here N\_i=ε\_iι\_i is the occupancy projection. The one-dimensional oscillator spectrum follows by factoring ∂+v and its adjoint; its vacuum is exp(−v²/2), its raising operators generate all Hermite functions, and their density is the complete oscillator argument already supplied in6.0c. Hence the unique ground space of(436) is the normalized Gaussian times the exterior vacuum1, in even degree. All other eigenvalues are at least2. Both the Gaussian and its vacuum are orthogonally invariant, so that kernel line is globally the trivial even C module. After G3's source contraction the source f acts by f(x) on it.

On the complement replace the bounded transform by sign(T\_C) along the straight functional-calculus homotopy, retaining zero on the kernel. The gap makes this bounded continuous. Its square defects remain compact, and its source commutators are zero after the source contraction. At the end the complement is a degenerate cycle; the remaining kernel is (C,identity,0). Therefore the first triangle is exactly

\[
 (\Theta\widehat\otimes1_C)
       (1_C\widehat\otimes\operatorname{ev})=1_C.
 \tag{437}
\]

The tensor positions in(437) put Θ's function factor first, its Q factor second, and the old C source last. There is no odd permutation hidden in this typed expression.

#### G5. The Clifford-algebra triangle is its entire regular module

In (1\_Q⊗Θ)(ev⊗1\_Q) the remaining target base is y and the integration point is x. The normal vector points from y to x, so the target potential has the opposite sign to G4's reversed radial vector. The target odd operator includes the graded tensor sign exactly once:

\[
 T_Q=-i\sum_i L_i\partial_i+i\Gamma\otimes r(v),\qquad
 T_Q^2=-\Delta+|v|^2+\sum_i S_i,
 \quad S_i=L_i\Gamma\otimes r_i.
 \tag{438}
\]

Each S\_i is a self-adjoint involution. They commute: both its exterior factors and its two negative r factors anticommute for distinct indices. Their joint−1 eigenspace is a free rank-one right Q module. Its explicit even unit vector is

\[
 \Omega=2^{-d/2}\sum_{I\subset\{1,\ldots,d\}}
       (-1)^{|I|(|I|+1)/2}e_I\otimes r_I.
 \tag{439}
\]

In fact Ω=2^(d/2)∏\_i(1−S\_i)/2 applied to the exterior vacuum⊗1. Expanding the commuting product gives(439): moving the intervening Γ's past the successive L's contributes (-1)^(k(k−1)/2), and its k minus signs contribute(-1)^k. Its inner product is 2^−d sum\_I r\_I\* r\_I=1. The equations S\_iΩ=−Ω follow from the product. Conversely those equations determine every exterior coefficient from its vacuum coefficient, so Ω Q is their entire joint−1 eigenspace. There is no other ground block, in any dimension.

Most importantly, direct insertion/removal of i in the ordered words gives

\[
 (R_i\otimes1)\Omega=\Omega r_i.
 \tag{440}
\]

The negative square r\_i²=−1 supplies the removal sign, and the exponent in(439) supplies precisely the creation sign. Thus this is the identity left Q representation on the kernel, not merely a rank-one module with an unspecified Clifford action. The formula is globally orthogonally equivariant: the joint projector, vacuum and Clifford products transform together. No frame, orientation or spin-c line was selected to glue it.

Multiplying Ω by the normalized Gaussian gives the exact entire kernel of(438); all other energies are at least2. It is a free even right Q module with the identity source action by(440), after G3's whole source contraction. Replace the bounded transform by its sign on the complement just as in G4. Its exact graded source commutation makes that complement degenerate. Consequently

\[
 (1_Q\widehat\otimes\Theta)
        (\operatorname{ev}\widehat\otimes1_Q)=1_Q.
 \tag{441}
\]

The target r in(438) is represented by Γ⊗r, while the source r acts by R⊗1. Keeping those two distinct actions is essential for(440). Both triangles are even KK equalities before any parity reduction. Tensoring a positive right C1 in its prescribed final position retains that order and gives the same two products in odd degree. Any later spin-c reduction uses the complete even Morita evaluations, so no determinant-line replacement or scalar trace can alter these proved identities. In dimension0 the sums and potential vanish, Ω=1, and both triangles are the pointwise identities on the finite component set.

![Both manifold triangles and the entire even Gaussian line and regular Clifford kernel.](../figures/twisted-manifold-duality.png)

**Figure 6.2.8.** Both manifold triangles and the entire even Gaussian line and regular Clifford kernel. Proof: Theorem 6.2, part G; Exercises 68–79.  These are original reproducible diagrams of the objects and proved maps. Classical context is [Connes; Connes–Skandalis].

Full-size PNG · Editable SVG.

 #### H1. The entire orientation cap, with its necessary reflection

Let M be compact smooth without boundary, C=C(M), W=TM, E=f\*τ, d=rank W, q=rank E, m=d+q. Write A\_L=C(M,Cl⁺(W⊕E)) and P\_L=C(M,Cl⁺(W⊕E)ᵒᵖ)=P\_W ⊗\_C P\_E. This is the graded opposite with negative real generators and its actual involution T2. The ordered real list is W first, E second. Let S:A\_L ↝ C⊗C\_m be the prescribed entire even imprimitivity module. Its scalar transition line is part of S. At rank zero retain the full graded line, including an odd line if specified.

Let γ\_L be the grading automorphism of A\_L: it sends every real positive generator to its negative. Set S^γ equal to S with left action precomposed by γ\_L. This retains its underlying right module, Hermitian metric, scalar transitions and grading. It is an actual module, not a determinant-line replacement.

Put Z=Λ(W⊕E)\_C. Its actions L\_i=ε\_i+ι\_i and R\_i=ε\_i−ι\_i give A\_L ⊗\_C P\_L = End\_C(Z), with both full Morita evaluations as in T2–T3. Let z=[Z]:A\_L⊗\_C P\_L→C. Define

\[
 t_S=z^{-1}(S^\gamma\widehat\otimes_C1_{P_L})\,\beta
 \in RKK_M(C,P_W\widehat\otimes_C P_E\widehat\otimes C_m).
 \tag{442}
\]

Here β is the actual graded right coefficient transport from C\_m,P\_L to P\_L,C\_m. Each interchange of homogeneous vectors has its Koszul sign. The last positive C\_m is retained in its specified order. Every factor is an even full imprimitivity module, so H1 has both inverse Morita products. Its concrete zero-operator module is the conjugate Z\* balanced with S^γ⊗P\_L, then transported by β. Local finite frames and partitions prove compact left C action and right fullness. The construction uses the entire given module; a line H tensored with S passes through as the same H, not H\*.

The reflection γ\_L is essential. An unreflected dual cap transposes a positive generator to its negative in this graded opposite convention. H2 verifies the actual sign on the whole finite module before any degree reduction.

#### H2. Full finite evaluation and its physical generators

To check H1, choose an ordered real orthonormal frame and a principal frame of S, so its right module is C\_m. Write e\_I for the exterior monomial and c\_I for the increasing positive Clifford word. First balance H1 with the forms Z carrying the negative left P\_L action R. The external positive C\_m action on this second factor is Γ\_Z⊗c; its position is last. On the resulting right C\_m module define

\[
 U(\overline {e_I}\otimes e_J)
 =2^{-m/2}(-1)^{|I|(|J|+1)+|I|(|I|-1)/2}c_Jc_I.
 \tag{443}
\]

The suppressed S frame is its regular unit. All homogeneous right coefficients are placed in their stated final position. This formula is checked against both balances, rather than assuming that a same-rank module has the desired action.

On Z\*, right a\_i is the adjoint-dual of L\_i and right r\_i is the adjoint-dual of R\_i, hence −R\_i. The reflection in S^γ gives the first relation below; the second is the negative-source balance:

\[
 \begin{split}
 U(\overline {L_i w}\otimes v)
   &=-(-1)^{|v|}U(\bar w\otimes v)c_i,\\
 U(-\overline {R_i w}\otimes v)
   &=U(\bar w\otimes R_i v).
 \end{split}\tag{444}
\]

Inserting or removing i in increasing words proves both equations: the removal sign of R differs from that of L, and c\_i²=1. Starting at I=empty, the second equation and then the first give U(bar1⊗e\_J)=2^−m/2 c\_J and exactly (443) for every I. Thus (443) is defined on the entire balanced tensor, not only a corner.

Its inner product can be checked directly. The right inner product on Z\* is θ\_(w,w'), represented by the faithful full matrix algebra L(A\_L)R(P\_L). Its vacuum projection is ∏\_i(1+L\_iR\_i)/2. After the reflection and β, its action on Z⊗C\_m has vacuum coefficient 2^−m; all nonempty words send the exterior vacuum to an orthogonal word. Consequently ζ=2^(m/2)(bar1⊗1) has <ζ,ζ>=1. The two balances (444) reduce every tensor to ζc. They therefore show that the entire inner product is c\*c' and that U is an onto right-linear even unitary. Equivalently, expanding each θ\_(e\_I,e\_I') in the complete L\_AR\_B matrix basis gives <U\_(I,J),U\_(I',J')> term by term; there is no discarded compact-algebra corner.

The physical positive operator on the forms factor has the tensor grading once: bar e\_I⊗v maps to (-1)^|I| bar e\_I⊗L\_i v. Equation (443) now gives

\[
 U\bigl((-1)^{|I|}\bar e_I\otimes L_i e_J\bigr)
       =c_i U(\bar e_I\otimes e_J).
 \tag{445}
\]

It suffices also to check this on ζ: r\_i sends the first vacuum to −bar e\_i, while reflected a\_i moves that bar e\_i to minus the final c\_i. The two minus signs cancel. Thus the induced physical generator is precisely c\_i, not −c\_i. Both W and E blocks keep their original order in (445).

These maps glue. Exterior transitions act on Z and Z\* by the actual dual orthogonal maps and fix their vacua. The reflection commutes with every even principal spin-c transition. The resulting coefficient map has exactly S's real Clifford transition and its scalar phase z; the whole unitary is built from those finite evaluations. It therefore retains an arbitrary line twist and a two-torsion line. For m=0 the contraction is the scalar evaluation tensored with the entire graded S line. Smooth compact parameter families use these same maps on their bundle over M×J. Equations (443)–(445) and both full Morita evaluations hold on that entire parameter bundle.

#### H3. The full test class and exact KK factorization

Let d\_M∈KK(P\_W,ℂ) be G2's evaluation restricted along a↦a⊗1 into P\_W⊗C(M). It is the actual negative-source de Rham cycle, not a separately chosen spin-c Dirac class. For any a∈RKK\_M^j(P\_E,C), compose t\_S with 1\_(P\_W)⊗\_C a, keep the final positive C\_m, forget base linearity, and compose with d\_M externally over the retained coefficients. The domain after forgetting is C(M). This defines a full class

\[
 \mathsf{cap}_S(a)\in KK(C(M),C_j\widehat\otimes C_m),
 \qquad C_0=\mathbb C, C_1=\operatorname{Cl}_1^+.
 \tag{446}
\]

For j=0 the retained list is simply C\_m. For j=1 it is C\_1 from a, followed by C\_m from the first cap. This is the native second-cycle-first list. Reducing m's even consecutive pairs and then the remaining ordered odd pair is exactly Section 0.13/7.20–7.21, not an ungraded permutation. The resulting scalar degree is j+m≡j+q−d mod2. Fullness, finite coefficients and the proved product theorem ensure existence over the actual whole M algebra in every case.

Apply the same construction to T5's k\_f with coefficient A:

\[
 y_f=\mathsf{cap}_S(k_f)\in KK(C(M),A\widehat\otimes C_m).
 \tag{447}
\]

For η∈KK^j(A,ℂ), (388) and full external-coefficient associativity give

\[
 \mathsf{cap}_S(\chi_f(\eta))=y_f\cdot\eta.
 \tag{448}
\]

Here the product on the right includes the stated final C\_j,C\_m transport before parity reduction. To verify (448), retain the balanced chain t\_S,(1\_(P\_W)⊗k\_f),d\_M,η. Move only the independent A-coefficient η past d\_M by the graded external tensor interchange, then use the associating module unitaries of Section 0.9 and the two finite evaluations0.12. η acts on the A coefficient and d\_M on P\_W; the interchange has the degree-zero d\_M and the actual displayed Clifford transport. The operator classes agree by the complete product uniqueness0.6. This proves equality of full KK classes before taking a scalar index.

For a Hermitian test bundle B over M, let [B]∈KK(C,C) be its complete finite zero-operator coefficient cycle. Multiplication by the unit u:ℂ→C(M) gives u[B]cap\_S(χ\_f(η))=u[B]y\_f·η in scalar degree j+q−d. In even degree Section 0.14 takes its integral kernel-minus-cokernel index; in odd degree KK^1(ℂ,ℂ)=K\_1(ℂ)=0. Neither the cap nor (448) quotients by torsion or passes to a character.

Metric, orientation-family and tube changes are whole families: T/C provide k\_f over M×I, D1 transports its normal bundle, and H1's finite tensors and G's evaluation occur on the same coefficient interval. Their complete product homotopy gives the endpoint equality with the supplied full module paths. There is no claim of invariance after dropping that path.

#### H4. Two complete native calibrations

If M is a point, d\_M=1 and Equations (443)–(445) identify (447) with the given spin-c reduced raw transverse Thom class followed by the actual étale correspondence L\_f. Indeed the raw positive operator L\_E(v) on forms is taken exactly to c\_S(v) by (445); the compact creation connections and positivity 2B² of T4–T5 are preserved by the entire finite unitary. This is the native point-source wrong-way class of Section 6.0o, with all positive odd coefficient reductions in their given order. An even line H multiplies both by H; rank zero retains its entire graded line and its exact index sign.

If the target is the unit groupoid of a point, E=0 and L\_f is the identity C(M) correspondence. Equations (443)–(445) take the evaluation's principal positive L\_W symbol to c\_S(ξ). Connections differentiated under this smooth finite bundle unitary add only zeroth-order terms. The complete compact manifold symbol quotient and negative-order estimate therefore identify (447) with the entire positive spin-c Dirac class a\_p of Section 7.1, including its right C\_d. The prescribed structure on W is exactly the one obtained from L=W by 7.18 with zero horizontal double. Section 7.22, whose full ordinary normal and native inverse calculation was supplied and read, gives y\_f=p\_!^geom. Thus (448) is the existing integral wrong-way pairing for this entire ordinary compact-manifold case, with all lines, finite bundles, rank-zero grading and both degrees.

Part I proves the general graph-normal class equality, beyond these two special calibrations.

 ![The reflected full cap, exact finite generator contraction and full KK test factorization.](../figures/twisted-oriented-cap.png)

**Figure 6.2.9.** The reflected full cap, exact finite generator contraction and full KK test factorization. Proof: Theorem 6.2, part H; Exercises 68–79.  These are original reproducible diagrams of the objects and proved maps. Classical context is [Connes; Connes–Skandalis].

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 #### E1. From every odd cycle to an honest compression extension

Let A be any separable complex C\*-algebra, not necessarily unital or nuclear. Under the course's positive right C1 convention, an odd cycle is (H⊗C1,φ,F⊗ε), where H is separable, φ is an ordinary even representation, and F is self-adjoint. This follows by taking the even module summand; right multiplication by ε identifies the other summand with it. An odd right-linear operator then has exactly this form. Use the localized normalization from Section0 to take ||F||≤1. It satisfies [F,φ(a)] compact and (F²−1)φ(a) compact for every a. Neither defect is asserted globally compact when A has no identity.

Put s=(1−F²)^(1/2). Then sφ(a) and φ(a)s are compact. Indeed in the Calkin algebra q(F) commutes with qφ(A), and (1−q(F)²)qφ(a)=0. Continuous functional calculus in that commuting algebra gives q(s)qφ(a)=0; adjoints give the other side. This is a localized assertion and suffices. On H⊕H with representation π=diag(φ,0), define

\[
 F_t=\begin{pmatrix}F&t s\\t s&-F\end{pmatrix},
 \quad0\le t\le1,\qquad
 F_1^2=1.
 \tag{449}
\]

Since F commutes with s, the square has diagonal F²+t²s². Its defects tested by π(a) are compact, and the off-diagonal commutators are ±t sφ(a), ±t φ(a)s. Thus E1 is a whole norm-continuous Kasparov homotopy. The second summand at t=0 has zero representation and is degenerate. Every odd class is therefore represented by an honest self-adjoint involution, without adding a global Fredholm assumption to a nonunital cycle.

Set P=(1+F\_1)/2 and K=P(H⊕H). Compression gives a completely positive contraction σ(a)=Pπ(a)P|\_K. Its multiplicative defect is

\[
 \sigma(ab)-\sigma(a)\sigma(b)
       =P\pi(a)(1-P)\pi(b)P\in\mathcal K(K).
 \tag{450}
\]

Here [P,π(a)] is compact, since it is half the original involution commutator. Consequently τ=q\_Kσ is a \*-homomorphism A→Q(K)=B(K)/K(K). The pullback algebra {(a,T):q\_K(T)=τ(a)} gives an extension by K(K), and a↦(a,σ(a)) is its completely positive contractive section. Adding a zero Busby summand on ℓ² makes the ideal the standard infinite compact algebra if necessary. Degenerate or noninjective Busby maps are allowed; analytic K-homology does not require an essential extension.

#### E2. A complete dilation and the inverse cycle

Conversely, let τ:A→Q(H) admit a completely positive contractive lift σ:A→B(H). We construct its dilation here. On A⊙H use <a⊗ξ,b⊗η>=<ξ,σ(a\*b)η>. Complete positivity gives positivity of every finite tensor sum. Quotient its null space and complete to K\_0. Left multiplication defines π\_0:A→B(K\_0): the inequality ||b||²1−b\*b≥0 in the unitization, inserted between each pair of tensor coefficients, proves ||π\_0(b)||≤||b||. Adjoints are left multiplication by b\*, so it is a \*-representation. Its range is essential on K\_0.

For a contractive approximate identity e\_n, the vectors e\_n⊗ξ have norm at most ||ξ||. On the dense tensor span their inner products converge to <a⊗η,V\_0ξ>=<η,σ(a\*)ξ>. This defines a bounded linear functional and hence a unique vector V\_0ξ by the Hilbert space Riesz theorem. It gives a contraction V\_0:H→K\_0. The approximate-identity limits show V\_0\*π\_0(a)V\_0=σ(a). Indeed π\_0(a)(e\_n⊗ξ)=a e\_n⊗ξ converges in norm to a⊗ξ, so π\_0(a)V\_0ξ=a⊗ξ. Pair this with V\_0η by the defining Riesz formula to obtain the asserted compression identity. Separability of A and H makes K\_0 separable.

On K=K\_0⊕H take π=π\_0⊕0 and the isometry Vξ=(V\_0ξ,(1−V\_0\*V\_0)^(1/2)ξ). Then V\*π(a)V=σ(a) and V\*V=1. Put P=VV\* and F=2P−1. For C\_a=(1−P)π(a)V, direct multiplication gives

\[
 C_a^*C_a=\sigma(a^*a)-\sigma(a^*)\sigma(a)
                         \in\mathcal K(H).
 \tag{451}
\]

The quotient multiplicativity of τ gives the last inclusion. A bounded Hilbert space operator T with T\*T compact is compact: cut its positive spectrum above a positive threshold to get finite-rank approximations and use the norm bound on the remaining spectrum. Therefore C\_a is compact. The adjoint argument with a\* gives both off-diagonal blocks of [P,π(a)] compact. F is an honest involution, so (K⊗C1,π,F⊗ε) is an odd Kasparov cycle. Its compression onto PK=VH gives exactly the original σ and τ, under V. No nuclearity or arbitrary extension-splitting theorem was used.

#### E3. Both constructions are inverse, including whole homotopies

To compare two isometric dilations of the same lift, adjoin the identity representation of A^+ to their representations. Let K\_min be the closed span of π(A^+)VH. It reduces π, contains VH, and the projection P is zero on its complement. Its orthogonal complement therefore has F=−1 and is a degenerate cycle. On K\_min the map

\[
 \sum_i\pi(a_i)V\xi_i\longmapsto
                         \sum_i\pi'(a_i)V'\xi_i
 \tag{452}
\]

is an onto isometry, since its inner product is exactly Σ\_(i,j)<ξ\_i,σ^+(a\_i\*a\_j)ξ\_j>, where σ^+(a+λ1)=σ(a)+λ1. This unitization is completely positive by the actual isometric dilation just constructed: π^+(a+λ1)=π(a)+λ1 compresses to that formula. The unitary (452) intertwines π, V and P. Thus it gives an actual cycle unitary after removal of degenerate summands.

Starting from an involution Fredholm cycle, the original (π,inclusion of PH) is already an isometric dilation of its compression. The unitary (452) identifies the returned cycle with it, up to its degenerate complement. Starting from an extension, E2 returns precisely its original Busby map. These prove both inverse statements, not only surjectivity.

Two cpc lifts of the same τ differ by compact operators. Their convex path σ\_t is cpc, and its multiplicative defects are norm-continuous compact functions of t. The same tensor construction over C(I), with inner products given by σ\_t^+, gives one countably generated Hilbert C(I)-module, isometric V and projection P. Its off-diagonal construction uses A^+⊙C(I,H), π by left multiplication, and Vξ=1⊗ξ. Complete positivity is pointwise positivity of every continuous matrix σ\_t^+; V\*V=1 follows from σ\_t^+(1)=1. Thus V is adjointable with V\*(a⊗ξ)=σ\_t^+(a)ξ, directly on the dense tensor core. No Hilbert-module Riesz representation theorem is assumed. The projection is VV\*, and its off-diagonal operators C\_a are rectangular module compact because C\_a\*C\_a is compact in C(I,K). The module version of the preceding compactness argument uses the compact maps T(T\*T)(T\*T+δ)^−1. Functional calculus bounds their distance from T by sup\_(λ≥0) √λ δ/(λ+δ)=√δ/2, tending to zero. Thus it is a whole Kasparov homotopy.

Conversely, do E1 and the projection compression on any odd KK homotopy module over C(I). Every defect lies in its entire module compact ideal. The projected module is complemented and countably generated. Add the standard C(I) module with zero Busby map and use the explicit stabilization unitary proved in Section 0.2. Its compact ideal becomes C(I,K); compression is a whole semisplit extension homotopy, not a separately selected extension at each t. The same procedure handles direct sums and ordinary unitary equivalence.

Adding a split extension gives a degenerate cycle: use its homomorphism lift as the dilation and P=1. Different cpc lifts are handled by the path just proved. Complementing P gives the inverse class: the sum of the two compression Busby maps is, under K=PK⊕(1−P)K, the split map qπ, because its off-diagonal blocks are compact. In the odd cycle this is precisely F↦−F, the inverse convention0.7. Consequently stable semisplit extensions modulo split summands and these whole homotopies form a group Ext\_h^−1(A,K), and both constructions give

\[
 \operatorname{Ext}_{h}^{-1}(A,\mathcal K)
                 \cong KK^1(A,\mathbb C).
 \tag{453}
\]

The subscript h makes the homotopy convention explicit. This proof is not a claim that all nonsplit extensions of an arbitrary nonnuclear algebra are semisplit or invertible. The analytical extension group used here is this invertible, stable semisplit group, with its entire homotopy convention.

#### E4. The other parity and the native source convention

Let SA=C0(R)⊗A in the positive physical coordinate order. Section 0.18/0.20 give both actual suspension Bott products; write b\_A∈KK^1(A,SA) and d\_A∈KK^1(SA,A) for the native parity pair. Their two products are 1, with the ordered positive C1 reductions fixed before reduction. For η∈KK^0(A,ℂ), d\_A·η∈KK^1(SA,ℂ) has the compression extension of E1. Its inverse is extension cycle followed on the left by b\_A. Associativity and both Bott products prove both inverses. Thus set

\[
 K^{\rm an}_1(A)=\operatorname{Ext}_{h}^{-1}(A,\mathcal K),
 \qquad K^{\rm an}_0(A)=\operatorname{Ext}_{h}^{-1}(SA,\mathcal K).
 \tag{454}
\]

Equations (453)–(454) identify these full groups with KK^j(A,ℂ) for j=1,0. The normalization E1, the actual dilation and both suspension inverses specify their signs and parity. They preserve all classes and torsion. Precomposing a Busby map by a homomorphism agrees with precomposing its dilation cycle, so these identifications are natural.

![Localized odd normalization, complete isometric cpc dilation, both inverse constructions and suspension.](../figures/twisted-extension-convention.png)

**Figure 6.2.10.** Localized odd normalization, complete isometric cpc dilation, both inverse constructions and suspension. Proof: Theorem 6.2, part E; Exercises 68–79.  These are original reproducible diagrams of the objects and proved maps. Classical context is [Connes; Connes–Skandalis].

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 #### I1. Coevaluation detects the entire oriented test class

Let C=C(M), P=P\_W=C(M,Cl⁺(TM)ᵒᵖ), M compact smooth without boundary, and let D be any separable graded coefficient algebra. Upgrade G1's diagonal class to

\[
 \Theta_M\in RKK_M(C,C(M_x)\widehat\otimes P_y).
 \tag{455}
\]

Its entire module is the same diagonal disk ideal as in G1. The left parameter h∈C acts by h(y), equal to the right central P\_y action. The extra C(M\_x) acts in the first point. The bounded phase is +i r\_y(w), normalized as in G1. This is an actual base-linear cycle; its square defect is compact even for the identity parameter coefficient.

For x∈KK(C,D), tensor x with the identity P\_y and apply the actual graded right transport to put P\_y first. Define

\[
 \begin{split}
 \mathcal D_M(x)&=\Theta_M(x\widehat\otimes1_P)
                 \in RKK_M(C,P\widehat\otimes D),\\
 \mathcal I_M(a)&=\operatorname{forget}(a)
                     (d_M\widehat\otimes1_D)\in KK(C,D).
 \end{split}\tag{456}
\]

All products use the complete base-linear version of Section0: the central parameter is equal on both module sides and is retained by the separator, connection and positivity operators. Equivalently the whole product module of C4 has that parameter linearity automatically. Forgetting in (456) retains the left source C(M); it does not first apply its scalar unit.

We check both inverse identities, including the relative one. For I\_M D\_M(x), reassociate the product and use G4's first triangle externally over D. Its source function originally acts at the integration point y. G3's whole geodesic source contraction takes it to its value at the remaining point x. That triangle is the identity C(M) class with that full source action. Product naturality and uniqueness therefore give x.

For D\_M I\_M(a), the coevaluation confines the first-point variable x to U={(x,y):x=exp\_y(w),|w|<r}. On this open diagonal disk, pull the entire a field by x. The evaluation factor is the fixed de Rham cycle in x, with the remaining y parameter and its Clifford coefficient. Restriction to U is legitimate: extend each C0(U) coefficient by zero, and use Section 0.10's actual essential-module homotopy. No inverse for the inclusion of an arbitrary open set is asserted.

Move that pulled field along x\_t=exp\_y(tw), 0≤t≤1. The negative Clifford bundle at x\_t is identified with the bundle at x by parallel transport along this specified short geodesic. C5 constructs these possibly nonproper whole pullbacks; D1–D2 and C4 give one interval field class on U×I with its full normal transport. It need not have differentiable operator fields. Compose this interval class with the fixed diagonal and evaluation factors by the complete product theorem. Thus this is a whole product homotopy, not separate pointwise products. At t=0 the field is the y field, with its specified transported negative Clifford action. Its parameter linearity permits it to be reassociated outside the diagonal/evaluation product. The remaining product is exactly G5's second triangle: its source r acts by R⊗1, its target r by Γ⊗r in the potential, and its Gaussian module is ΩP with RΩ=Ωr. Hence that full Clifford source is the identity and the remaining class is a. Every transport here is the same one used in G3's source contraction. Therefore

\[
 \mathcal I_M\mathcal D_M=1,\qquad
 \mathcal D_M\mathcal I_M=1.
 \tag{457}
\]

The argument retains arbitrary D and its grading. For the positive right odd coefficient it retains the exact ordered generator, since all geometric cycles in the two triangles are even morphisms before parity reduction.

For the prescribed S on W⊕E, put a\_f=t\_S(1\_P⊗\_C k\_f), with the coefficient transport of H3, and reduce the final C\_m only by the fixed right reducer R\_m. Then (447) and (457) say

\[
 \mathcal D_M(R_m(y_f))=R_m(a_f).
 \tag{458}
\]

We shall prove the same formula with the native f\_! on the left. By (457) that proves equality of the entire KK classes.

#### I2. The finite normal orientation after diagonal coevaluation

Choose an even Euclidean embedding j:M→R^n, with ℓ=n−d≥1 and k=ℓ+q≥1. Use its induced tangent metric; the positive metric transport carries the given entire S to this metric. Let ν be its orthogonal complement, and N=ν⊕E with precisely the specified module S\_N from Section 6.0j. Write J(v,η)=dj(v)+η, an actual global isometry W⊕ν→R^n. No separate orientation of W or ν is chosen.

Diagonal coevaluation followed by the native normal Thom cycle has finite coefficient P\_W⊗S\_N, with positive physical actions i r\_W(v) and c\_N(η,e). The negative right P\_W coefficient remains first. Call this full orientation module T\_N. Let U\_(E,S) be the partial finite cap obtained from t\_S by balancing its P\_E action with ΛE\_C, keeping P\_W and the last positive C\_m. Its positive physical E action is the L\_E action on that forms factor. Equations (443)–(445) specify this partial contraction with every tensor sign.

The requisite finite comparison is obtained by the following explicit reversible evaluation, rather than by rank or determinant. Balance the negative right P\_W coefficient with Z\_W=ΛW\_C, retaining its positive left A\_W action. On T\_N the resulting two real W copies are

\[
 u\longmapsto L_W(u),\qquad
 v\longmapsto iR_W(v).
 \tag{459}
\]

They are precisely the interleaved canonical complex double Q\_W, with its exterior grading. Indeed the coevaluation's physical +i r becomes +iR. The other W copy comes from the retained positive L action, with the grading once in the exterior wedge map. For the partial cap U\_(E,S) this same evaluation recovers S on W⊕E, by (443)–(445): its reflected W source cancels the adjoint-dual minus sign. The reflection on the E block is already removed in its partial forms contraction. Both statements are actual entire module unitaries.

The exact normal recovery of Section 6.0j is

\[
 S_N\widehat\otimes S_{Q_W}
  \cong\rho^*(S\widehat\otimes C_n),\qquad
 \rho(\eta,e,u,v)=(u,e,J(v,\eta)).
 \tag{460}
\]

The source order is N,Q\_W, and the right list is C\_k,C\_(2d), the same generator list as C\_m,C\_n before repartition. Tensor (460) with the fixed last consecutive-pair spinor Σ\_d. Since S\_(Q\_W)⊗\_(C\_2d)Σ\_d=Z\_W, its left side is exactly the evaluated T\_N just described. Its right side is S⊗C\_n with that complete right matrix reduction. Evaluating C\_n⊗U\_(E,S) with Z\_W gives precisely the same module: recover S as above, move the even-dimensional C\_n block by its actual even exterior/spin implementer, and repartition C\_m,C\_n as C\_k,C\_(2d). All vector actions are identified by the displayed ρ, not just their oriented volume.

This finite evaluation is cancellable. To reverse it, tensor with Z\_W\* and apply the two inner-product/rank-one maps of (385) and Section 6.0j(174). In an ordered word frame the maps obey the two balances (444); (443) is their onto even unitary. The matrix units generated by L\_i,R\_i recover every column, so reversing either composition gives the original module and each negative P\_W generator. The tensor inner-product formula shows that these maps and their reverses are isometries; fullness gives onto ranges. Thus the preceding evaluated equality is an equality of the original finite modules after the stated right Clifford matrix transports and reductions.

One can check every column of this cancellation without a spin-c classification input. In the right Clifford model of S⊗C\_n let A\_i be its generator for u\_i in the first W block and B\_i its generator for the v\_i in the ambient W block. Set C\_i=(A\_i−iB\_i)/2 and Π=∏\_i(1−iA\_iB\_i)/2. These are the exterior creation operators and the vacuum projection of (459)'s interleaved double. Clifford relations give C\_i²=0, C\_i C\_j+C\_j C\_i=0 and C\_i\*C\_j+C\_j C\_i\*=δ\_ij. For C\_I the increasing creation word they give

\[
 \Pi C_I^*C_J\Pi=\delta_{IJ}\Pi,\qquad
 \sum_I C_I\Pi C_I^*=1.
 \tag{461}
\]

The second identity follows by multiplying the independent two-term decompositions (1−C\_i\*C\_i)+C\_i\*C\_i=1; its cross signs cancel in the even projections. These equations are precisely both matrix-column inner-product evaluations. If ε(I,i) is the exterior insertion/removal sign, the same relations give A\_i C\_IΠ=ε(I,i)C\_(I△i)Π and B\_i C\_IΠ=i ε(I,i)(−1)^(1\_(i∈I))C\_(I△i)Π. Every remaining normal generator g commutes with Π and satisfies g C\_IΠ=(−1)^|I| C\_I gΠ. Thus the map e\_I⊗s↦C\_I s from Z\_W⊗Π(S⊗C\_n) is an entire even unitary intertwining all A\_i,B\_i and normal generators. This is exactly the module recovery used in (460). The actual permutation ρ sets the normal chirality; no orientation sign is chosen afterward. Equations (461) and these action identities prove the cancellation for every d,q, including the full grading, not only the finite tested models.

For clarity about these reductions, n+m=k+2d. Reducing the last 2d factors of C\_(k+2d), then applying R\_k, is the same full even Morita module as R\_(n+m), after the stated even block transports. If k is odd, move its retained odd generator past the even 2d block by the 2d adjacent reflection units of Section 7.20, then use the ordered wedge unitary. It is an even unitary preserving the normalized odd volume. Reduction of C\_n,C\_m, with n even, likewise gives the exterior tensor of the n/2 matrix pairs and the m pairs, with the same final odd generator. These are full zero-cycle product identifications over any graded extra coefficient. Their proofs use only the actual homogeneous tensor maps, so the retained P\_W coefficient is included; no scalar-source restriction is used.

Consequently the reduced forward Bott cycle of T\_N on W⊕ν⊕E, under (v,η,e)↦(J(v,η),e), is the forward canonical even R^n Bott cycle followed externally by R\_m(U\_(E,S)'s forward E cycle). Every physical real generator and the negative right P\_W action has been checked by (459)–(460) and the two full evaluations. An even line H in S stays as H in S\_N and in U\_(E,S); all these maps carry that same factor. Rank-zero total orientation retains its graded line. All statements hold on smooth compact parameter bundles.

#### I3. One actual étale homotopy without a transverse-rank hypothesis

Use T1's fibre-preserving actual tube ρ\_E:E→H\_f. The native normal tube may be chosen as its product with the small Euclidean normal disk: its actual cocycle is

\[
 g_N(x;\eta,e)=(g_E(x,e),j(x)+\eta).
 \tag{462}
\]

Here g\_E includes every actual arrow and the specified short arrow equivalence at zero. (462) is exactly Section 6.0m's cocycle on p\*ν, and its normal is the ordered ν,E with S\_N. Using separate positive radial compressions of ν and E instead of one compression of N changes no class: they have the same positive normalized derivative, and the whole tube and normal homotopies of 6.0e/o apply. The actual operator is the raw normal c\_N(η,e), before reduction.

Write source variables over y as (w,η,e)∈W\_y⊕ν\_y⊕E\_y. Use compressions a=q\_r(w), b=q\_ρ(η), c=q\_ρ(e), where q\_ρ(v)=v/√(1+|v|²/ρ²) has derivative1 at zero. Changing G1's positive radial normalization to this one is its raw Thom homotopy and does not change Θ\_M. Let x\_t=exp\_y(ta). Choose compatible metric connections and full spin-c/line parallel transports along these short geodesics; they preserve each real Clifford action. Denote the real transports by T^ν\_t and T^E\_t. Put

\[
 \begin{split}
 z_t(y;w,\eta)&=j(y)+(1-t)J_y(a,b)
        +t\bigl(j(x_1)-j(y)+T^\nu_1b\bigr),\\
 \Phi_t(y;w,\eta,e)&=
       \bigl(g_E(x_t,T^E_tc),\ z_t(y;w,\eta),\ y\bigr).
 \end{split}\tag{463}
\]

The target is the foliation V×R^n×M whose plaques keep the Euclidean coordinate and y fixed. Every overlap arrow in Φ\_t is the actual arrow of g\_E pulled by (y,w,e,t)↦ (x\_t,T^E\_tc), with the same z\_t and y. They satisfy the exact arrow cocycle. There is no identification by leaf image, or choice of a formal arrow having those endpoints.

All Φ\_t are étale after one uniform disk shrink. The Euclidean derivative in the physical (a,b) coordinates is J\_y+O(|a|+|b|), uniformly on compact M×I, since both endpoint maps have derivative J\_y at zero. Choose the disks so this error is less than half the least singular value1. The e derivative of g\_E(x\_t,T^E\_tc) is invertible on a common small disk by T1. The Euclidean coordinate has no e derivative. The combined transverse derivative is therefore block invertible, irrespective of its mixed W-to-transverse block. Radial compression is a diffeomorphism onto each disk, so this proves étaleness on the entire noncompact source bundle, even where its inverse derivative is large. The extra y block is the identity.

At the zero section the two nonparameter blocks are explicitly

\[
 (w,\eta,e)\longmapsto
       \bigl(e+t\,d_{\rm trans}f_y(w),\ J_y(w,\eta)\bigr).
 \tag{464}
\]

This is an invertible shear for every linear transverse map, including the zero map and rank changes. Thus no condition on surjectivity of d\_trans f has entered the argument.

For each t use the full module P\_W⊗S\_(N,x\_t), transported along the stated geodesic, and the anticommuting positive potential i r\_W(w)+c\_N(η,e). Its normalized square defect is −1/(1+|w|²+|η|²+|e|²), on the whole source bundle. The zero endpoint module is the same specified S\_(N,y); the one endpoint is the original pulled normal module. These are an actual full orientation family, including all scalar phases and its rank-zero grading, not separate endpoint choices.

Here is the product check for the two forward potentials. On a homogeneous compact first-variable creation section ξ, insert the connection sign (−1)^|ξ| and the actual right P\_W transport. The total phase then differs from that signed second normal phase by a continuous finite matrix function tending uniformly to zero at normal infinity. It has compact first-variable support, so its multiplication is rectangular module compact. Adjoints give the second error, and density treats every creation vector. Its first positivity is the nonnegative scalar 2|w|²/(√(1+|w|²)√(1+|w|²+|η|²+|e|²)). Thus the total phase is the actual forward product. The left parameter action is central, and its localized source commutators vanish.

Compose this whole forward product with Φ\_t's actual étale module. Section 6.0n, with the fixed R^n factor and compact parameter M×I, gives one countably generated completed C(M×I,A⊗C0(R^n)) module, compact coefficient action and onto isometric endpoint maps. The compact creation proof and positivity 2B² of T4–T5 apply to the finite coefficient corners. Thus this is one whole KK homotopy of products. The right Clifford reducers of I2 and the one fixed canonical Euclidean inverse A\_n may be composed on that interval; product naturality gives exactly its two endpoint products.

#### I4. The two endpoints and the full numerical comparison

At t=1, (463) is exactly diagonal coevaluation followed by the native graph-normal tube (462), under the actual pulled coordinates x=exp\_y(a), η=T^ν\_1 b and e=T^E\_1 c. The forward-product check, the complete tube restriction and the full normal family show that its reduced product with the ambient inverse is D\_M(f\_!^geom).

At t=0 it is the product of g\_E(y,c) with the Euclidean open tube j(y)+J\_y(a,b). Its principal space is the product of the actual E correspondence with that Euclidean open coordinate. Elementary separated chart sections have tensor inner products and convolution actions exactly as in Section 6.0o's full product-factor proof. A finite partition on each compact chart support gives dense range by the common Schur estimate. Thus completion gives an onto even product-module unitary, including every actual holonomy/counting sheet.

I2 identifies its complete forward orientation/potential with the canonical R^n Bott factor and the partial oriented E cap factor. Restricting the Euclidean Bott factor to the compressed open product disk changes no class: it is the normalized tube for its zero section, with the actual extension by zero and compression homotopy6.0e. Translation by j(y) is a whole compact-parameter tube homotopy. The canonical even pair B\_n,A\_n has both inverse products by 6.0c and its exact native reductions7.20–7.21. Cancelling B\_n A\_n externally over P\_W,A and the prescribed line therefore leaves exactly R\_m(t\_S(1\_P⊗k\_f))=R\_m(a\_f). No canonical positive analytical Euclidean module has been substituted for the calibrated geometric inverse A\_n.

The whole homotopy consequently proves

\[
 \mathcal D_M(f_!^{\rm geom})=R_m(a_f),\qquad
 f_!^{\rm geom}=R_m(y_f)
              \quad\hbox{in }KK^{q-d}(C(M),A).
 \tag{465}
\]

The second equality is (457) applied to the first and (458). It holds for every compact smooth actual cocycle f with its given full stable orientation. For a Hermitian coefficient bundle B and η∈KK^j(A,ℂ), (448) and the fixed native odd order now give the integral test identity

\[
 R_{j,m}\bigl(u[B]\,\mathsf{cap}_S(\chi_f(\eta))\bigr)
        =u[B]f_!^{\rm geom}\cdot\eta
       \quad\hbox{in }KK^{j+q-d}(\mathbb C,\mathbb C).
 \tag{466}
\]

Here R\_(j,m) is (446)'s complete fixed right reduction of the native list C\_j,C\_m, including its ordered odd pair.

In even degree this is exactly the previously constructed kernel-minus-cokernel pairing; in odd degree the scalar group is zero, while the full (465) class and (448) are retained. Line twists, their two-torsion cases and complete graded rank-zero lines remain in the same factors through (460)–(466). Smooth compact parameter families use the displayed homotopy on their entire bundle, one uniform disk and the same finite evaluations. No separate fibrewise product argument is needed.

![One entire étale homotopy, full normal recovery and the native graph-normal class equality.](../figures/twisted-graph-duality.png)

**Figure 6.2.11.** One entire étale homotopy, full normal recovery and the native graph-normal class equality. Proof: Theorem 6.2, part I; Exercises 68–79. The three plotted lines in the last figure are the exact projected sections labelled there, not the full transverse derivative. These are original reproducible diagrams of the objects and proved maps. Classical context is [Connes; Connes–Skandalis].

Full-size PNG · Editable SVG.

 The complete model, field construction, both duality identities, full graph comparison and extension source prove all claims of Theorem 6.2. \(\square\)

## 7. Analytic index and assembly are different questions

The analytic index is built from a longitudinal operator; the geometric index is built from an oriented map. We construct the general operator class and its coefficient action, then prove its compact submersion product on the actual holonomy correspondence, retaining all Clifford parities and spin-c phases. The geometric comparison and the isomorphism question for assembly are distinct assertions.

### The Dirac class on the full holonomy module

An operator can be compact as a Hilbert-module map while its represented operator on an infinite holonomy cover is not compact. The construction below uses the module compact ideal throughout. It requires neither a transverse measure nor a global frame for the tangent bundle. The pseudodifferential inputs are [Hilbert modules and fields on the leaf space, Lemmas 6.1 and 6.5](hilbert-modules-and-fields-on-the-leaf-space.md#section-6), with the complete chart estimates and negative-order approximation in [The index theorem for measured foliations, Lemmas 6.1–6.2](the-index-theorem-for-measured-foliations.md#section-6). The construction is also treated by [Connes–Skandalis], Section 4.

Keep \(V\) compact and without boundary in this subsection. Put \(p=\operatorname{rank}F\), and specify a metric and the full spin-c structure on \(F\). Write \(C_p\) for the ordered positive complex Clifford algebra. Use the transition module \(\mathcal S_F\) of Corollary 6.0d: its fibre is the standard graded right \(C_p\)-module, and a transition \([s,z]\) acts by the even left unitary \(zs\). Thus the central phase is part of the data. Positive Clifford multiplication, denoted \(c_F^+(v)\), is odd, self-adjoint and right \(C_p\)-linear. It satisfies \(c_F^+(v)^2=|v|^2\).

**Theorem 7.1 (the general longitudinal Dirac KK cycle).** The prescribed spin-c structure defines a class

\[
\ell_!^{\mathrm{an}}\in
KK\bigl(C(V),A\widehat\otimes C_p\bigr)=KK^p(C(V),A).
\tag{240}
\]

Its full module is the completion of compact smooth holonomy kernels with coefficients in \(r^*\mathcal S_F\). An odd self-adjoint order-zero quantization of the positive unit-covector Clifford symbol is a representative. The class is independent of the quantization, connection, metric and module embedding, with the specified spin-c structure transported through metric changes.

**Proof: the module.** Choose finitely many spin-c trivializations and smooth functions \(\chi_a\) subordinate to them, with \(\sum_a\chi_a^2=1\). The map that sends a bundle vector to its local components multiplied by \(\chi_a\) is an even isometric right \(C_p\)-linear embedding into a finite trivial \(C_p\)-module bundle. Its adjoint gives an even projection \(e\in M_N(C(V)\widehat\otimes C_p)\). The compact kernel completion is consequently

\[
\mathscr E_F=e(A\widehat\otimes C_p)^N,
\qquad
\mathcal K(\mathscr E_F)=
eM_N(A\widehat\otimes C_p)e.
\tag{241}
\]

Here the projection acts at the range of an arrow. To verify the completion, send a column of arrow kernels to its range-bundle vector and compare its convolution inner product. The isometric bundle embedding preserves that inner product on the smooth core. Its range is the displayed projection corner, since compact chart kernels and an approximate identity of \(A\) are dense. For the compact-endomorphism assertion, \(\theta_{\xi,\eta}\) is the corresponding column product \(\xi\eta^*\). These products span the corner densely, again by the approximate identity. All assertions remain true with the finite-dimensional graded coefficient \(C_p\): expand in its finite matrix components and retain the grading on products. Separability of the chart algebra makes this a countably generated module.

Multiplication by \(f(r(\gamma))\) defines an even representation \(\phi:C(V)\to\mathcal L(\mathscr E_F)\). The regular tensor representations are the holonomy-cover \(L^2\) representations with this coefficient bundle and any faithful representation of \(C_p\). Equality of their norms with the module norm follows from the regular norm of \(A\), first on column products and then by completion. In particular, the compact ideal in (241) is an algebra ideal; no finite ordinary rank on a cover has been assumed.

**Proof: quantization and defects.** Identify a covector with a tangent vector by the metric. The smooth odd matrix symbol

\[
b(x,\xi)=\frac{c_F^+(\xi^\sharp)}{
\sqrt{1+|\xi|^2}},
\qquad
\sigma_0(b)(x,\xi)=
\frac{c_F^+(\xi^\sharp)}{|\xi|}\quad(\xi\ne0)
\tag{242}
\]

is a classical order-zero symbol. The transition identity
\(c_F^+(hv)(zs)=(zs)c_F^+(v)\)
shows that it is an endomorphism symbol of the actual bundle. Quantize it in finitely many protected plaque charts, with fixed near-unit kernel cutoffs, and patch with the above partition. The coordinate-change and composition proofs in the index lesson give a compact-support equivariant order-zero operator \(Q\) with this leading symbol. They apply to each of the finitely many Clifford matrix components. Taking its odd part preserves the leading symbol; put \(P=(Q+Q^*)/2\) after doing so.

The uniform order-zero estimate and the core adjoint identity make \(P\) a bounded adjointable odd self-adjoint map on (241). Composition and the principal-symbol square give

\[
P^2-1\in\mathcal K(\mathscr E_F),
\qquad
[P,\phi(f)]\in\mathcal K(\mathscr E_F)
\quad(f\in C(V)).
\tag{243}
\]

For smooth \(f\), the second operator has order at most minus one: its degree-zero symbol is the commutator of the matrix in (242) with the scalar \(f\), hence zero. The first has the same negative-order conclusion since the unit-cosphere symbol squares to the bundle identity. Negative-order frequency truncation converges in the uniform regular norm to compact smooth arrow kernels, so both operators lie in the corner (241). Smooth functions are dense in \(C(V)\), and \(\|[P,\phi(f-f_n)]\|\le2\|P\|\|f-f_n\|\), proving the assertion for continuous \(f\). These are all Kasparov defects. Compactness of \(V\) permits the identity coefficient in the first assertion; the module representation itself is unital.

If \(p=0\), take \(P=0\) instead of using a unit covector. The foliation groupoid is the unit groupoid and \(A=C(V)\). The module is the finite line module \(\mathcal S_F\), so its identity is compact and all the defects are compact. The canonical rank-zero orientation gives the identity class. A rank-zero spin-c line \(H\) gives \([H]\), with its even grading. This treats the empty cosphere separately.

**Proof: choices.** Two quantizations of the same leading symbol differ by a compact endomorphism, by the complete symbol quotient in the Hilbert-module lesson. Their straight path is an odd self-adjoint Kasparov homotopy; (243) holds along it. Changes of connection, phase, partition or cutoff do not change the leading symbol, so have the same conclusion. A change of finite bundle embedding gives the unitary between their ranges induced by the identity on \(\mathcal S_F\). It intertwines the coefficient representation. Conjugating a quantization by this unitary retains its leading symbol and hence changes the representative only by a compact map.

For two metrics, use \(g_t=(1-t)g_0+tg_1\) and the positive isometries \(G_t^{-1/2}\), where \(g_t(v,w)=g_0(G_tv,w)\). Transport the full principal spin-c structure and its transition module by these isometries. The resulting symbol (242) and its leafwise derivatives vary continuously in \(t\). Compactness of \(V\times[0,1]\) gives a finite trivializing cover, a common finite module embedding and uniform positive metric bounds. Quantization with fixed chart cutoffs is operator-norm continuous: the Fourier–Schur estimate bounds a difference by finitely many symbol seminorms, which tend to zero. Frequency truncation of the negative-order defects is uniform on the same compact parameter interval. Thus the defects are compact in the interval-module corner, not merely compact at each parameter. This constructs the metric homotopy with the full phase retained.

Finally, in even rank \(p=2m\), the ordered matrix-Clifford contraction sends this module to the ordinary spinor kernel module. If \(c_i=-i\varepsilon_i\) denotes the negative skew-adjoint Clifford action, its leading symbol is \(ic(\xi)/|\xi|\), and its grading is

\[
\Gamma=i^m c_1\cdots c_{2m}
=(-i)^m\varepsilon_1\cdots\varepsilon_{2m}.
\tag{244}
\]

These are exactly the spinor and Thom conventions of (145). In odd rank the final positive Clifford factor is retained, as in Section 0; no ungraded spinor replacement or interchange of odd blocks has occurred. This proves (240) with its asserted parity. \(\square\)

### Coefficients and all even-rank elliptic symbols

**Lemma 7.2 (finite bundle coefficients and a change of spin-c structure).** For a finite Hermitian bundle \(B\to V\), quantize the symbol \(1_B\widehat\otimes\sigma_0(b)\) on the range-pulled-back bundle \(B\otimes\mathcal S_F\), and call the resulting cycle \(\ell_{!,B}^{\mathrm{an}}\). Then

\[
\ell_{!,B}^{\mathrm{an}}=[B]\ell_!^{\mathrm{an}},
\qquad
\ell_{!,F;H}^{\mathrm{an}}
=[H]\ell_{!,F}^{\mathrm{an}}
\tag{245}
\]

for a spin-c change by the line \(H\). The second notation denotes the changed spin-c structure on the same real bundle \(F\). When a bundle bracket multiplies a KK class with source \(C(V)\), it denotes the finite bundle correspondence with its usual left \(C(V)\) action. Precomposing that correspondence with the scalar inclusion gives the corresponding \(K^0(V)\) bundle class used in (246).

**Proof.** The finite bundle correspondence \((\Gamma(B),0)\) over \(C(V)\) has compact left action and even grading. Its tensor with (241) is the displayed range-bundle kernel module: on elementary tensors the map is \(u\otimes\xi\mapsto u(r(\gamma))\xi(\gamma)\). Its convolution inner product is precisely the balanced tensor inner product. A finite partition and local bundle frames give a dense range, so the map is an onto isometry.

Use an odd self-adjoint quantization \(P_B\) of the stated coefficient symbol. For a smooth section \(u\) of \(B\), the creation map \(T_u\) is multiplication by its range-pulled-back bundle column. The degree-zero symbol of \(P_BT_u-T_uP\) is zero. The matrix-corner version of the negative-order theorem therefore makes this map compact; its adjoint is compact too. Norm approximation by smooth sections treats every creation vector. The positivity condition for the first cycle's zero operator is zero, hence satisfied. The product criterion of Definition 0.3 proves the first formula of (245). The actual transition module for the changed spin-c structure is \(H\otimes\mathcal S_F\): its transitions are the old even units multiplied by the line transitions. Its symbol is \(1_H\otimes b\). The first formula now proves the second, retaining the entire line rather than only its determinant. \(\square\)

**Corollary 7.3 (the full even-rank symbol reduction).** Suppose \(p\) is even, including zero. Let
\(\Theta_F:K^0(V)\to K_c^0(F^*)\)
be the positive Thom isomorphism of Corollary 6.0d, with the specified spin-c structure and metric identification \(F^*\cong F\). For every longitudinal elliptic pseudodifferential symbol class \(s\in K_c^0(F^*)\),

\[
\operatorname{Ind}_{\mathrm{an}}(s)
=\Theta_F^{-1}(s)\ell_!^{\mathrm{an}}
\quad\text{in }K_0(A).
\tag{246}
\]

Thus determining the action of the general Dirac class determines the analytic index of every such symbol. Equation (246) is an analytic reduction; comparison with the geometric class remains a separate theorem.

**Proof.** In positive even rank, \(\Theta_F[B]\) is the relative disk/sphere bundle triple with bundles \(B\otimes S_F^+\) and \(B\otimes S_F^-\) and sphere isomorphism \(1_B\otimes ic(\xi)/|\xi|\). This is the actual multiplier representative in Corollary 6.0d, not just its fibrewise rank. The off-diagonal part \(P_B^+\) in Lemma 7.2 has exactly that leading sphere isomorphism. Its Fredholm index is the analytic symbol boundary by Theorem 6.6 of the Hilbert-module lesson. The scalar \(K=KK\) identification of Section 0 sends its even cycle to the same kernel-minus-cokernel boundary class: both use the full Fredholm bundle pair and the same off-diagonal map. Consequently
\(\operatorname{Ind}_{\mathrm{an}}(\Theta_F[B])=[B]\ell_!^{\mathrm{an}}\).
Direct sums and signed differences give this identity for every \(a\in K^0(V)\). The global inverse products of Theorem 6.0c make \(\Theta_F\) onto and one-to-one. Taking \(a=\Theta_F^{-1}(s)\) proves (246) for an arbitrary relative symbol triple, including ones not initially expressed as Dirac symbols.

In rank zero a chosen spin-c line \(H\) gives \(\Theta_F(a)=a[H]\), whereas Theorem 7.1 gives \(\ell_!^{\mathrm{an}}=[H]\). Its inverse is multiplication by \([H^*]\). Thus the right side of (246) is \(s[H^*][H]=s\), the bundle-difference index on point leaves proved in the Hilbert-module lesson. This also fixes the noncanonical rank-zero case.

If the spin-c structure is changed by \(H\) in positive even rank, its Thom map is \(\Theta_{F,H}(a)=\Theta_F(a[H])\) by the full transition construction. Its inverse sends \(s\) to \([H^*]\Theta_F^{-1}(s)\). Formula (245) cancels this line against the changed Dirac class, so (246) gives the same analytic index of the fixed symbol \(s\). No independence of the Dirac class from the orientation choice is claimed. \(\square\)

The cycle construction in Theorem 7.1 includes odd leaf rank through its actual Clifford factor. The odd-rank all-symbol action requires the corresponding odd coefficient and suspension comparison. The analytical/geometrical leaf-map equality must hold as a KK identity; equality of numerical traces, or (246) alone, does not establish it.

![Full spin-c holonomy module, positive Clifford symbol, line cancellation and noncompact regular convolution](../figures/longitudinal-dirac-module.png)

**Figure 7.1.** Theorem 7.1, Lemma 7.2 and Corollary 7.3, equations (240)–(246); Exercises 48–49, equations (247)–(248). The first panel gives the actual module corner and compact ideal. Its curve samples the rank-one scalar coefficient \(\xi/\sqrt{1+\xi^2}\) on \([-4,4]\); the dashed lines are its exact limits. The second panel retains the full line and its dual in the even-rank symbol reduction, including rank zero. The last panel draws the exact support intervals for translations by \(6n\), with \(n=0,1,2\); the intervals containing the convolution images have twice the radius of the input intervals. Their images have the same nonzero norm and disjoint supports. It is a support diagram, not a plot of the chosen bump functions. The longitudinal cycle is classical [Connes–Skandalis, Section 4]; the full module, choices, coefficient and compactness arguments are supplied above.

Full-size Dirac module PNG · Editable Dirac module SVG.

### The vertical frequency axis in a product

A vertical operator acts in one set of variables and leaves the others fixed. Its lift need not be an ordinary order-zero pseudodifferential operator in all the variables. For example, the degree-zero function \(\operatorname{sgn}\xi\) has no continuous value at \(\xi=0\) when another frequency \(\eta\) is nonzero. The following argument supplies the missing multiplication and positivity estimates. It also explains why a creation vector tests the symbol precisely at that axis. The pseudodifferential product method originates with [Connes–Skandalis], Section 1.

Write \(X=\mathbb T^r\), \(Z=\mathbb T^s\), with \(r,s\ge1\), and let \(Y\) be a compact metrizable parameter space. Coordinates on the two tori are \(x,z\); their covectors are \(\xi,\eta\). Take finite graded Hermitian spaces \(S_1,S_2\). Put
\[
 \begin{aligned}
 E_1&=C\bigl(Z,L^2(X;S_1)\bigr),\\
 E_2&=C\bigl(Y,L^2(Z;S_2)\bigr),\\
 E&=E_1\widehat\otimes_{C(Z)}E_2.
 \end{aligned}
\tag{249}
\]
The inner products integrate with normalized torus volume; \(E_1\) is a right \(C(Z)\)-module and \(E_2,E\) are right \(C(Y)\)-modules. Pointwise multiplication gives the left \(C(Z)\) action on \(E_2\). Fubini's theorem identifies \(E\) isometrically with
\(C(Y,L^2(X\times Z;S_1\widehat\otimes S_2))\).
Indeed the balanced inner product of \(u\otimes v\) is the iterated integral of its pointwise tensor. Finite Fourier sums in \(x,z\), with continuous coefficients in \(Y\), are in the range and are dense, so the isometry is onto. For a homogeneous \(u\in E_1\), let \(T_u v=u\widehat\otimes v\). Its adjoint integrates the first-factor pairing over \(X\).

Let \(\mathcal P\) be the norm closure on \(E\) of classical order-zero families on \(X\times Z\), continuous in \(Y\) in every symbol seminorm, together with \(\mathcal K(E)\). Let \(\mathcal P_2\) be its counterpart on \(E_2\). Their symbol maps have compact kernels and supremum quotient norms. Here is why the uniform assertion is valid. The Fourier–Schur, amplitude, adjoint and frequency-truncation proofs in the index lesson, Lemmas 6.1–6.2, apply on finitely many torus charts, with constants uniform on \(Y\). A negative-order family is uniformly approximated by finite-frequency smooth kernels; these are compact module maps, since their continuous matrix coefficients multiply finitely many fixed Fourier vectors. Smooth kernels and then all module rank-one maps are obtained by approximation of their two vectors. For an order-zero family with leading symbol \(a_0\), quantize the smooth positive square root of \((c^2+\epsilon)I-a_0^*a_0\), where \(c=\sup\|a_0\|\). Composition gives the upper quotient bound \(c\), as in the Hilbert-module lesson, Lemma 6.5. A localized high-frequency wave packet at a fixed \(y\in Y\) gives the lower bound: its image under every evaluated module compact tends to zero, whereas its order-zero image approaches the leading-symbol action. Taking all \(y\) and cosphere points gives the supremum norm. Quantization of smooth symbols and their uniform density gives surjectivity and exactness in the norm closure. No differentiability in \(Y\) is used.

**Lemma 7.4 (compression and connections).** Let \(Q\in\mathcal P\) be homogeneous of degree \(d\), with leading symbol \(q\). For smooth homogeneous \(u,v\in E_1\), the operator \(T_u^*QT_v\) is in \(\mathcal P_2\), with symbol
\[
 \begin{aligned}
 \sigma(T_u^*QT_v)(z,\eta,y)
 &=\int_X J_u(x,z)^*q(x,z,0,\eta,y)J_v(x,z)\,dx,\\
 J_v(x,z)w&=v(x,z)\widehat\otimes w.
 \end{aligned}
\tag{250}
\]
If the axis symbol is \(1_{S_1}\widehat\otimes b(z,\eta,y)\), independent of \(x\), and \(Q_2\in\mathcal P_2\) has symbol \(b\), then \(Q\) is a \(Q_2\)-connection:
\[
 QT_u-(-1)^{d|u|}T_uQ_2\in\mathcal K(E_2,E),
 \qquad
 T_u^*Q-(-1)^{d|u|}Q_2T_u^*\in\mathcal K(E,E_2).
\tag{251}
\]
In particular, if the axis symbol is zero, \(QT_u\) and \(T_u^*Q\) are compact. For a homogeneous \(K\in\mathcal K(E_1)\), a connection satisfies \([K\widehat\otimes1,Q]\in\mathcal K(E)\), with the graded bracket.

**Proof.** First take a classical \(Q\). Smooth finite sums of separated \(x,z\) functions approximate \(u,v\), including any fixed finite number of derivatives. In matrix entries, it suffices to use \(u=f(x)h(z)\), \(v=f'(x)h'(z)\). For a torus left symbol \(a(x,z,k,\eta,y)\), Fourier expansion in \(x\) gives the compressed amplitude
\[
 \overline{h(z)}h'(z')
 \sum_{k\in\mathbb Z^r}\int_X
 \overline{f(x)}e^{ik\cdot x}
 a(x,z,k,\eta,y)\widehat{f'}(k)\,dx.
\]
The Fourier coefficients of \(f'\) decay faster than every power. Thus all required derivatives of this amplitude satisfy order-zero bounds in \(\eta\), uniformly in \(y\). For \(|k|\le|\eta|/2\), the classical expansion and the mean-value estimate on the unit sphere give
\[
 a(x,z,k,\eta,y)=q(x,z,0,\eta,y)+O((1+|k|)/|\eta|).
\]
After \(\beta\) frequency derivatives the error is of order \(-1-|\beta|\), with a fixed polynomial weight in \(k\). The rapidly decreasing coefficients absorb every such weight. The sum over \(|k|>|\eta|/2\) has arbitrarily negative order by the same decay. These estimates also hold after spatial derivatives. Fourier inversion in the first term, followed by the left-amplitude reduction in Lemma 6.1 of the index lesson, gives exactly (250); its other terms have negative order. This proves the formula and membership for classical operators. The bound \(\|T_u^*QT_v\|\le\|u\|\|Q\|\|v\|\), and the same Cauchy–Schwarz bound for the integral of its symbol, extend both assertions to \(\mathcal P\). Compact terms compress to compacts.

For a homogeneous smooth \(u\), set \(C=QT_u-\epsilon T_uQ_2\), with \(\epsilon=(-1)^{d|u|}\). Expand \(C^*C\) into its four terms. Formula (250) places every compressed term in \(\mathcal P_2\). If \(m(z)=\int_X\|u(x,z)\|^2dx\), their leading symbols are respectively
\(m b^*b\), \(-m b^*b\), \(-m b^*b\), and \(m b^*b\).
For example, the compression of \(Q\) has symbol \(\epsilon m b\), because the graded lower operator crosses \(u\). Their sum is zero, so \(C^*C\) is compact. This implies that \(C\) is compact: for \(L=C^*C\), the maps \(CL(L+\delta)^{-1}\) are compact and
\(\|C-CL(L+\delta)^{-1}\|\le\sup_{t\ge0}\sqrt t\,\delta/(t+\delta)\to0\).
Applying the same argument to \(Q^*,Q_2^*\) proves the adjoint error in (251). Smooth vectors are dense in \(E_1\) in its module norm, so the bounded creation-map estimate proves (251) for all vectors. If \(b=0\), take \(Q_2=0\).

Finally write a module rank-one map as \(\theta_{u,v}\); its lift is \(T_uT_v^*\). The two connection errors move \(Q\) past these factors. Their lower terms cancel with the graded signs, leaving a compact. Finite sums and norm closure give the assertion for every \(K\). \(\square\)

**Lemma 7.5 (multiplication off the vertical axis).** Let \(T\) be a homogeneous order-zero pseudodifferential family on \(E_1\), smooth in \(z\), with leading symbol \(a(x,z,\xi)\). Let \(G\in\mathcal P\) have symbol \(g\) satisfying \(g(x,z,0,\eta,y)=0\). Then both \((T\widehat\otimes1)G\) and \(G(T\widehat\otimes1)\) belong to \(\mathcal P\). For \(\xi\ne0\), their symbols are the corresponding ordered products with \(a(x,z,\xi)\widehat\otimes1\); extend these symbols by zero at \(\xi=0\).

**Proof.** Both extended products are continuous: \(a\) is uniformly bounded on its vertical cosphere and \(g\) tends uniformly to zero near the axis. By Lemma 7.4, multiplying \(G\) on either side by a lifted vertical compact gives a total compact. This fact is essential; the lift of a vertical compact itself need not be a total compact.

The uniform vertical symbol quotient, proved by the same quotient-norm argument above with \(z\) as parameter, permits approximation of \(T\), modulo vertical compacts, by finite sums
\(\phi_j(x,z)t_j(D_x)\).
Here \(\phi_j\) are smooth matrix functions on \(S_1\), homogeneous of the required degree, and \(t_j\) are scalar smooth functions of the vertical frequency direction. Choose arbitrary bounded values at the finitely many low frequencies. To obtain the uniform symbol approximation, take finite smooth partitions on the compact spatial and directional factors. On each product patch replace the symbol by its value at one point and multiply it by the two partition functions. Uniform continuity bounds the error by the oscillation on those patches. The resulting finite sum has the required separated form. Quotient-norm equality then permits a vertical compact correction making each operator approximation arbitrarily small in norm. It suffices to prove the result for one such summand.

For \(\delta>0\), choose a smooth scalar cosphere cutoff \(\chi_\delta\) which is zero when \(|\xi|/|(\xi,\eta)|\le\delta\) and one when that ratio is at least \(2\delta\). The symbols \(g\chi_\delta\) tend uniformly to \(g\). They can be approximated uniformly by smooth symbols supported away from the axis. The total symbol quotient implies that their quantizations, with total compact corrections, approximate \(G\) in norm. Multiplication by the bounded lift of \(T\) preserves this convergence and takes every total compact correction to a total compact.

It remains to consider a smooth total symbol supported in the cone \(|\xi|\ge\delta|(\xi,\eta)|\), apart from bounded frequencies. Expand its spatial coefficients in Fourier series in \(x,z\). Their partial sums converge in every fixed finite symbol seminorm, by integration by parts in the spatial variables; the Fourier–Schur estimate gives operator-norm convergence. For one Fourier term with first spatial frequency \(\mu\), left multiplication by \(t(D_x)\) replaces its frequency coefficient by \(t(k+\mu)\) times that coefficient. On the cone,
\[
 |t(k+\mu)-t(k)|\le C_\mu(1+|(k,l)|)^{-1}
\tag{252}
\]
outside a bounded frequency set; differentiated versions have the corresponding lower orders. This follows by differentiating the degree-zero directional function on the segment from \(k\) to \(k+\mu\), where \(|k|\) is at least a fixed multiple of \(|(k,l)|\). The exceptional finite frequencies are smoothing. Thus the product is a total order-zero operator with leading symbol \(t(\xi)g\). The same Fourier calculation proves right multiplication, and matrix multiplication by \(\phi\) uses the ordinary composition calculus. Passing through the Fourier, cone and vertical-symbol approximations proves the assertion in the closed algebra \(\mathcal P\). Quotient symbols pass through these limits uniformly. \(\square\)

**Theorem 7.6 (a full product criterion for the two-variable families).** Let \(T,Q_2,Q\) be odd self-adjoint classical order-zero quantizations on \(E_1,E_2,E\), with respective odd self-adjoint leading symbols \(a,b,q\). Suppose
\[
 a^2=b^2=q^2=1,\qquad
 q(x,z,0,\eta,y)=1\widehat\otimes b(z,\eta,y),
\tag{253}
\]
and suppose \([a\widehat\otimes1,q]\ge0\) for \(\xi\ne0\). Give the first cycle the pointwise left \(C(X\times Z)\) action, and the second the left \(C(Z)\) action. Then \(Q\) represents their Kasparov product in \(KK(C(X\times Z),C(Y))\). In particular, for an orthogonal Clifford sum its positivity symbol is
\[
 \left[\frac{c_1^+(\xi)}{|\xi|}\widehat\otimes1,
 \frac{c_1^+(\xi)\widehat\otimes1+
 1\widehat\otimes c_2^+(\eta)}{\sqrt{|\xi|^2+|\eta|^2}}\right]
 =\frac{2|\xi|}{\sqrt{|\xi|^2+|\eta|^2}}I.
\tag{254}
\]
This extends continuously by zero at the vertical axis. The bracket of the two odd operators is their anticommutator.

**Proof.** The three square and scalar-commutator defects are negative-order compact families. Thus all three are Kasparov cycles, including the first cycle's full \(C(X\times Z)\) left action. Lemma 7.4 proves the required \(Q_2\)-connection condition. We prove the operator positivity; symbol positivity alone would not suffice without this step.

Choose a smooth total symbol \(q_0\), independent of \(x\), valued in \(1\widehat\otimes\operatorname{End}(S_2)\), and equal to \(1\widehat\otimes b\) on the vertical axis. One choice is that value multiplied by a directional cutoff supported where \(|\eta|>|\xi|/2\) and equal to one near \(\xi=0\), extended by zero near \(\eta=0\). Quantize it with coefficients independent of \(x\), and symmetrize, obtaining \(Q_0\).

We claim \([T\widehat\otimes1,Q_0]\) is compact. For a finite separated vertical approximant \(\phi(x,z)t(D_x)\) from Lemma 7.5, the scalar multiplier \(t(D_x)\) commutes exactly with \(Q_0\), since \(Q_0\) is translation invariant in \(x\). The graded commutator of \(\phi\widehat\otimes1\) with \(Q_0\) is a total operator of negative order: its principal matrix bracket is zero, as \(Q_0\) acts only on the second graded factor. It is compact, also after multiplication by \(t(D_x)\). The vertical compact corrections have compact brackets by Lemma 7.4, because \(Q_0\) is a lower-operator connection. Norm approximation proves the claim for \(T\).

Now \(R=Q-Q_0\) has zero axis symbol. Lemma 7.5 places both products with the lift of \(T\) in \(\mathcal P\). Hence
\([T\widehat\otimes1,Q]\) belongs to \(\mathcal P\), modulo the already compact bracket with \(Q_0\), and its symbol for \(\xi\ne0\) is \([a\widehat\otimes1,q]\). The bracket with \(q_0\) is zero. Its continuous value at the axis is zero by Lemma 7.5. The symbol is positive by hypothesis. Exactness of the symbol quotient therefore makes the self-adjoint operator bracket positive modulo \(\mathcal K(E)\): its image in the quotient of \(\mathcal P\) is a positive matrix function, and a *-homomorphism preserves positivity in the larger multiplier quotient. Conjugating by any left coefficient preserves that positivity. This is exactly Definition 0.3, so the product follows.

For (254), the two Clifford blocks anticommute, the square of the first is \(|\xi|^2\), and the mixed bracket is zero. These identities give the displayed nonnegative scalar, with limit zero when \(\xi\to0\) and \(\eta\ne0\). This verifies the criterion for the stated orthogonal symbol. Every estimate above is uniform on the compact parameter space. With an additional interval parameter the same compact approximations lie in the interval module, so the argument also applies to a continuous family with uniform symbol estimates. Fixed finite graded Clifford coefficients are handled by their finite matrix entries, retaining every tensor sign and their right actions. \(\square\)

These lemmas provide the analytic product calculation in protected product coordinates. They do not identify an arbitrary global submersion's analytic cycle with its geometrical class. That comparison must also reconcile its full bundle transitions, its holonomy correspondence and the prescribed normal orientation before the general leaf-map equality follows.

![Vertical frequency-axis limits, the nonnegative Clifford product symbol and creation-vector compression](../figures/vertical-frequency-product.png)

**Figure 7.2.** Lemmas 7.4–7.5 and Theorem 7.6, equations (249)–(254); Exercise 50, equation (255). In the first panel the unit-frequency circle has coordinates \((\xi,\eta)=(\cos\theta,\sin\theta)\). Its shaded sectors have \(|\xi|/|(\xi,\eta)|\le1/3\), the cone removed in the approximation of Lemma 7.5. The vertical symbol has different one-sided limits there. The second panel samples the exact positive function \(2|\cos\theta|\) at 721 equally spaced angles on \([0,2\pi]\); its zeros at \(\pi/2\) and \(3\pi/2\) are the two axis points. The third panel states the actual compression and connection mechanism, with \(Q\) odd and normalized torus volume in the integral. The method is classical [Connes–Skandalis, Section 1]; the Fourier and norm-closure arguments above prove the needed uniform statements.

Full-size vertical product PNG · Editable vertical product SVG.

### Submersions and the full holonomy correspondence

We now pass from product coordinates to compact manifolds and their actual holonomy modules. Let \(f:W\to V\) be a smooth surjective submersion between compact manifolds without boundary. Let \(K=\ker df\), of rank \(r\), and let \(F\subset TV\) be an integrable bundle of rank \(p\). Specify full spin-c structures on \(K\) and \(F\). A horizontal splitting identifies
\[
 \begin{aligned}
 F_W&=df^{-1}(F)\cong K\oplus f^*F,\\
 C_r\widehat\otimes C_p&\cong C_{r+p}.
 \end{aligned}
\tag{256}
\]
Use this ordered sum for the spin-c structure on \(F_W\). In particular its transition module is \(\mathcal S_K\widehat\otimes f^*\mathcal S_F\), with both central phases retained. Put \(G=\operatorname{Hol}(V,F)\) and \(H=\operatorname{Hol}(W,F_W)\).

**Lemma 7.7 (the fibre module, density connection and vertical class).** Let the metrics and horizontal lift for the compact submersion \(f\) be fixed. Smooth sections of the prescribed right Clifford spinor bundle \(\mathcal S_K\) complete to a graded Hilbert module \(E_f\) with
\[
 \langle u,v\rangle(y)=\int_{f^{-1}(y)}\langle u(w),v(w)\rangle\,d\mu_y(w).
\tag{257}
\]
Its right algebra is \(C(V)\widehat\otimes C_r\), and pointwise scalar multiplication is its nondegenerate left \(C(W)\)-action. This module is countably generated. An odd self-adjoint order-zero vertical operator with symbol \(c_K^+(\xi^\sharp)/|\xi|\) gives \(a_f\in KK^r(C(W),C(V))\). For \(r=0\), take the finite covering module with its specified orientation line and operator zero.

For a real base field \(Z\), define the real function \(k_Z\) by \(\mathcal L_{Z^h}\mu=k_Z\mu\), where the derivative compares the vertical densities through the horizontal flow. An even metric right \(C_r\)-linear spinor connection gives the smooth module connection
\[
 \nabla^{E_f}_Z u=\nabla^K_{Z^h}u+\tfrac12 k_Zu,
 \qquad
 Z\langle u,v\rangle=
 \langle\nabla^{E_f}_Zu,v\rangle+\langle u,\nabla^{E_f}_Zv\rangle.
\tag{258}
\]
It obeys the right scalar Leibniz rule and is right \(C_r\)-linear. With the prescribed Clifford coefficient representation and an ordinary Hermitian bundle \(S\to V\), multiplication on elementary balanced tensors extends to an onto unitary
\[
 \begin{gathered}
 E_f\widehat\otimes_{C(V)\widehat\otimes C_r}
 \bigl(C_r\widehat\otimes L^2(V;S)\bigr)
 \\
 \longrightarrow L^2(W;\mathcal S_K\widehat\otimes f^*S).
 \end{gathered}
\tag{259}
\]
The density on the right is the fibre density multiplied by the pulled-back base density. The Clifford factors retain their stated order.

**Proof.** We construct the module first, use horizontal flow to determine its connection, and then check the operator and tensor assertions.

For every \(y\), the integral in (257) is positive in the finite-dimensional coefficient algebra \(C_r\). Indeed applying any positive functional of \(C_r\) reduces it to the integral of a nonnegative scalar function. If a smooth \(u\) has zero integral inner product on every fibre, a faithful positive trace makes its pointwise squared norm zero: a nonzero value would remain positive on a small fibre neighbourhood of positive volume. Thus the form is definite on smooth sections. Right multiplication by a coefficient function acts under the integral and has its usual norm bound.

Here is the module Cauchy–Schwarz estimate without a scalar replacement of its inner product. The Gram matrix of two sections is positive, since its quadratic forms are integrals of positive fibre Gram forms. Put \(a=\langle u,u\rangle\), \(b=\langle u,v\rangle\), \(d=\langle v,v\rangle\). Replacing its upper-left corner by \(\alpha 1\), for \(\alpha=\|a\|+\epsilon>0\), keeps it positive. Compression by the coefficient column \( (-\alpha^{-1}b,1)^T\) gives \(b^*b\le\alpha d\). Letting \(\epsilon\downarrow0\) proves \(\|b\|^2\le\|a\|\|d\|\). This also proves the triangle inequality, null-space compatibility and continuity required for the Hilbert-module completion.

Fibre integrals of smooth scalar and coefficient entries are continuous, and in fact smooth. To see this, multiply by a finite smooth partition subordinate to buffered submersion charts and bundle frames, provided by Local tools for bundles and transport. In each chart, the integrand has compact support in the fibre coordinate and the density is a smooth positive multiplier. Differentiation in the base coordinate passes through that compact integral. Summing proves the assertion on overlaps as well. All fibres are compact because \(W\) is compact.

The norm obtained is the supremum of the coefficient fibre \(L^2\) norms. There is a finite upper bound on their volumes, by the same finite chart calculation. Consequently uniform approximation of smooth sections implies approximation in the module norm. Finite bundle frames with cutoffs reduce such approximation to continuous functions on compact Euclidean rectangles. Piecewise polynomial approximations with rational complex coefficients, smoothed at their faces and cut off within the buffers, give a countable dense set; refinement makes the approximation arbitrary. Its right module span is dense, so \(E_f\) is countably generated. The given fibre grading preserves the integral, and the right Clifford and left scalar actions have the required degrees. For \(a\in C(W)\), pointwise multiplication has norm at most \(\|a\|_\infty\) and adjoint multiplication by \(\overline a\). The action is nondegenerate because its identity acts as the identity.

We now derive the density correction. If \(\Phi_t\) is the local horizontal flow of \(Z^h\) and \(\varphi_t\) the base flow, then \(f\Phi_t=\varphi_t f\). Change variables from the fibre over \(\varphi_t y\) to the fibre over \(y\) by \(\Phi_t\). The derivative of the pulled-back density is \(k_Z\mu_y\), by definition. The derivative of a smooth integrand \(h\circ\Phi_t\) is \(Z^hh\). Differentiating this equality of compact integrals proves
\[
 \begin{aligned}
 Z\int h\,d\mu_y&=\int\bigl(Z^hh+k_Zh\bigr)\,d\mu_y,\\
 k_Z&=Z^h(\log v)+\sum_j\partial_{x_j}B_Z^j
 \quad\text{if }\mu=v(x,y)|dx|,\quad
 Z^h=\sum_\alpha Z^\alpha(y)\partial_{y_\alpha}
           +\sum_jB_Z^j(x,y)\partial_{x_j}.
 \end{aligned}
\tag{260}
\]
The second formula follows by differentiating the vertical Jacobian of the flow, whose infinitesimal value is \(\sum_j\partial_{x_j}B_Z^j\). It also checks the first formula directly: integration by parts in \(x\) cancels the vertical transport term against that Jacobian term. The compact chart cutoffs eliminate coordinate boundary terms; in the sum, their transport derivatives cancel because the partition sums to one.

Apply the first equality to the coefficient-valued \(h=\langle u,v\rangle\). Metric compatibility of \(\nabla^K\) differentiates \(h\) into the two spinor-connection terms. Because \(k_Z\) is real, adding half of it to each argument supplies exactly the remaining integral. This proves (258), including its sign. It is intrinsic because the defining flow comparison is intrinsic. A function pulled back from \(V\) satisfies \(Z^h(f^*b)=f^*(Zb)\), which proves the right scalar Leibniz rule. The original connection and multiplication by \(k_Z\) commute with the constant right Clifford action. Linearity in the real base field follows from the horizontal lift and the coordinate formula for \(k_Z\).

For the vertical class, use the uniform buffered Fourier calculus proved in The index theorem for measured foliations, Lemmas 6.1–6.2, and the symbol \(c_K^+(\xi^\sharp)/|\xi|\) with the fixed right coefficients. In a vertical bundle frame, multiplication by the square root of the density identifies the coordinate fibre norm with the ordinary \(L^2\) norm having finite Clifford coefficients. Compact chart parameters give uniform order-zero bounds. A finite partition therefore quantizes a smooth vertical symbol to an adjointable operator on \(E_f\). The formal adjoint remains vertical and has the adjoint principal symbol. Thus taking the odd part and then the self-adjoint part gives the asserted odd self-adjoint \(T\).

We spell out why the remainders are module compact. A smooth vertical kernel supported in a buffered chart is a continuous function of the two fibre coordinates and the base parameter, with finite coefficient entries. Uniform approximation by finite sums of separated fibre functions and base coefficients is obtained by polynomial approximation on a containing compact rectangle and the chart cutoffs. In bundle frames, each separated term is a module rank-one kernel \(u(w)v(w')^*\); the coefficient matrix units are absorbed into those frame vectors. The operator error is bounded uniformly by the kernel error times a fixed finite volume bound, using the scalar integral estimate and the finite coefficient norm. Hence every such smoothing operator is in \(\mathcal K(E_f)\). Smooth section rank-one kernels have this form after a finite partition, and smooth vectors are dense, so their closure is exactly the module compact ideal. A negative-order remainder is the norm limit of its smoothing frequency cutoffs, by the same uniform Fourier estimate. It is consequently module compact.

The principal square of \(T\) is the identity, and the commutator with any smooth left scalar has negative order. Accordingly \(T^2-1\) and \([T,a]\) are compact. Uniform approximation of continuous scalars proves the commutator assertion for the entire left algebra; self-adjointness supplies the remaining cycle condition. This proves the required graded Kasparov cycle with its right \(C_r\) coefficients.

For completeness, the symbol norm and invariance can be checked in that same calculus. If \(a\) is a smooth principal symbol and \(c=\sup\|a\|\), the strictly positive symbol \((c^2+\epsilon)1-a^*a\) has a smooth positive square root. Quantizing this root and using composition shows, modulo the already proved compact ideal, that \(T_a^*T_a\le(c^2+\epsilon)1\). This is the upper quotient norm bound. For the lower bound, choose a point, a unit vertical covector and a unit coefficient vector on which \(a\) nearly attains its norm. A shrinking smooth packet with phase in that covector direction has fibre norm one and \(T_a\) acts on it to leading order by that value of \(a\). Smooth compact kernels send these packets to zero; the estimates are uniform on the containing compact chart, and rank-one approximation gives the same result for every module compact. Evaluation on the selected fibre yields the lower quotient bound. Thus the quotient norm is precisely the supremum symbol norm. Uniform symbol approximation extends this identification to the continuous cosphere symbols, and a zero symbol has exactly a compact remainder. In particular two quantizations of the same symbol differ by a module compact.

Changes of the vertical metric and horizontal splitting can be made on one compact parameter interval. Identify the density modules by the positive square root of the density ratio. Transport the full spin-c structure through the positive metric isometry and the path of frames, retaining its central phase and right Clifford order. The buffered estimates just used are uniform on the product of the chart parameters with that interval. They give a continuous interval-module cycle, hence the same KK class at its endpoints. In rank zero, local inverse charts identify \(f\) with a finite covering over a smaller base neighbourhood: finitely many inverse charts cover a compact fibre, and compactness excludes additional sheets nearby. The fibre module is then a finite bundle module, including the fixed orientation line. Finite local frames and a partition express its identity as a finite sum of module rank-one maps. Thus the operator zero has compact defect \(-1\) and gives the specified class also in that case.

Finally define (259) on elementary homogeneous balanced tensors by
\\(u\otimes(c\otimes s)\mapsto[w\mapsto u(w)c\otimes s(f(w))]\), with the same ordered graded factors as in the source. This respects both base multiplication and the balanced Clifford action. Fubini applied to (257), followed by the prescribed finite Clifford coefficient inner product, proves that the tensor inner product is exactly the total \(L^2\) inner product. In horizontal coordinates the change from coordinate to horizontal basis is triangular with diagonal identity, so the product density has no additional Jacobian. The range is dense: cut a smooth total section into bundle and submersion charts, choose a smooth base frame of \(S\), and write each piece as a finite sum of that pulled-back frame times a smooth vertical spinor coefficient. A base cutoff equal to one on the piece's support makes the frame global, and a total cutoff makes each coefficient global. These are elementary images. Smooth total sections are dense in the \(L^2\) norm. The isometry therefore extends to an onto unitary. \\(\square\)

The fibre-module and density-connection constructions are classical; [Kaad–van Suijlekom 2020] is further reading. The holonomy correspondence below uses the bounded product method of [Connes–Skandalis], Sections 1 and 4.


**Lemma 7.8 (the correspondence remembers the whole fibre).** Form the pullback groupoid and its principal right space
\[
 \begin{aligned}
 R&=\{(w',\gamma,w):f(w')=r(\gamma),\ f(w)=s(\gamma)\},\\
 P_f&=\{(w,\gamma):f(w)=r(\gamma)\}.
 \end{aligned}
\tag{261}
\]
The right anchor of \(P_f\) is \(s(\gamma)\), and right multiplication is \((w,\gamma)\gamma'=(w,\gamma\gamma')\). Projection of a leaf path through \(f\) defines an injective open Lie-groupoid homomorphism \(j:H\to R\), identical on units. Let \(\mathscr P_f\) be the completion of smooth compactly supported half-density kernels on \(P_f\), as a right \(A=C_r^*(G)\)-module. Then
\[
 \mathcal K(\mathscr P_f)\cong C_r^*(R),
 \qquad j_*:C_r^*(H)\hookrightarrow\mathcal K(\mathscr P_f)
\tag{262}
\]
is extension of chart kernels by zero. Consequently \((\mathscr P_f,j_*,0)\) defines a degree-zero correspondence \(\varepsilon_f\) from \(C_r^*(H)\) to \(A\). Neither \(H=R\) nor connectedness of the fibres is needed.

**Proof: the open inclusion.** The pullback of an integrable distribution is integrable: in submersion and foliation coordinates it is spanned by \(\partial_x,\partial_z\), where \(f(x,z,y)=(z,y)\). The transverse coordinate is \(y\). A leaf path in \(W\) therefore projects to a leaf path in \(V\), with the same transverse germ in these coordinates. This proves well-definedness of \(j\). For fixed endpoints in \(W\), two such paths have the same holonomy precisely when their projected germs agree, since \(f\) is a local diffeomorphism on these transversals. This proves injectivity. Near an arrow use endpoint plaque coordinates and its fixed transverse germ. The two endpoint \(W\) plaque coordinates and the transverse parameter give both an \(H\) chart and an \(R\) chart; the projected endpoint plaque coordinates are the ones in the \(G\) arrow. The map is a local diffeomorphism in these charts. Its injective image is therefore open, with the same smooth chart structure. It is a wide subgroupoid because units, inverses and products are induced by paths. An arrow of \(R\) between different fibre components need not be in that image.

**Proof: the compact ideal.** Let \(E_f^0\) be the scalar fibre-integration module, with no spinor coefficient. There is a canonical unitary
\(E_f^0\widehat\otimes_{C(V)}A\to\mathscr P_f\)
on its core, sending \(u\otimes a\) to \(u(w)a(\gamma)\). Its inner product is the convolution inner product integrated first over the vertical fibre. To verify its norm and density, tensor with each regular representation \(\pi_x\) of \(A\). Fubini identifies the resulting space with
\[
 L^2\bigl(P_{f,x};d\mu_{r(\gamma)}\,d\lambda_x(\gamma)\bigr),
 \qquad P_{f,x}=\{(w,\gamma)\in P_f:s(\gamma)=x\}.
\]
Finite chart products \(u(w)a(\gamma)\) are dense on every compact support: in product coordinates, uniformly approximate a smooth function by finite separated smooth functions, keeping larger fixed cutoffs. The same approximation converges in the module norm, since the fibre integrals of an error over that fixed compact chart support have uniform bounds. This proves density before completion as well. The direct sum of the \(\pi_x\)'s is faithful; for a module vector its norm is the supremum of the resulting represented norms of its inner product. Thus this calculation verifies the asserted module unitary.

On this tensor module, a rank-one map with smooth vectors \(u\otimes a,v\otimes b\) has \(R\) kernel
\(u(w')(a*b^*)(\gamma)\overline{v(w)}\), with the two vertical half-densities understood. Its regular representation at a unit \(w_0\) acts on \(R_{w_0}\), which is exactly \(P_{f,f(w_0)}\) with the displayed density. Surjectivity of \(f\) means that all the regular spaces of \(\mathscr P_f\) occur. Hence the reduced \(R\) norm and its module-operator norm agree. These rank-one kernels span the reduced algebra densely: partition a compact \(R\) chart kernel into finitely many buffered charts, approximate it there by functions separated in \(w',\gamma,w\), and use the convolution Schur bound on the fixed support. Each middle function in \(A\) is a norm limit of finite sums \(a*b^*\), by an approximate identity of \(A\). Conversely the displayed rank-one maps and their limits belong to the represented reduced algebra. This proves the first isomorphism in (262). Finite bundle or Clifford coefficients give the corresponding projection corners of this same calculation.

To verify the norm of \(j_*\), fix \(w_0\). The left \(H\) orbits in \(R_{w_0}\) are open. Each orbit containing \(\rho\) is \(\{j(h)\rho:h\in H_{r(\rho)}\}\); right multiplication by \(\rho\) identifies its measure and representation with the regular \(H\) representation at \(r(\rho)\). The disjoint open orbits are countable, because this source fibre is second countable. Thus restriction of the regular \(R\) representation is the orthogonal direct sum of regular \(H\) representations. Its norm is at most the reduced \(H\) norm. The orbit of the unit arrow includes the regular representation at \(w_0\), so taking all \(w_0\) gives the reverse inequality. Extension of compact arrow-chart kernels by zero is legitimate for an open subgroupoid, also in non-Hausdorff arrow spaces using the chart-kernel definition. Products and adjoints agree because \(H\) is a subgroupoid. This proves the isometric *-inclusion in (262). The left action is compact; its zero operator therefore has compact square defect. It defines the stated Kasparov correspondence. \(\square\)

**Lemma 7.9 (the global connection and positivity calculation).** Suppose \(r,p>0\). On the module with \(P_f\) kernels and coefficient bundle \(\mathcal S_K\widehat\otimes f^*\mathcal S_F\), let \(Q\) be a homogeneous total order-zero quantization with principal symbol \(q(w;\xi,\eta)\). Here \(\xi\) is vertical and \(\eta\) is the covector on the horizontal copy of \(f^*F\). If
\(q(w;0,\eta)=1\widehat\otimes b(f(w),\eta)\),
then \(Q\) is a connection for a longitudinal \(G\) operator \(Q_2\) with principal symbol \(b\). For a classical homogeneous vertical \(T\), with symbol \(a(w,\xi)\), the bracket \([T\widehat\otimes1,Q]\) belongs to the closed total pseudodifferential algebra and has symbol \([a\widehat\otimes1,q]\) for \(\xi\ne0\). Its value at the axis is zero. In particular a self-adjoint bracket with a nonnegative symbol gives the operator positivity modulo \(\mathcal K\) required in Definition 0.3.

**Proof: compression and the module quotient.** Take buffered charts \(f(x,z,y)=(z,y)\) as in Lemma 7.8, and choose unitary frames of the two coefficient bundles. Multiplication by the square root of the vertical density identifies the first module with its ordinary coordinate \(L^2\) model. The base density is handled in the second module. The total symbol at \(\xi=0\) is consequently independent of \(x\) and acts only on the second graded factor. A frame change in \(\mathcal S_K\) commutes in the graded sense with that factor; a frame change of \(\mathcal S_F\) depends only on \((z,y)\). Thus this assertion survives all transitions, including their central phases.

For smooth first-module vectors \(u,v\), the compression proof of Lemma 7.4 on each chart gives the longitudinal symbol
\[
 \sigma(T_u^*QT_v)(y_0,\eta)
 =\int_{f^{-1}(y_0)}J_u(w)^*q(w;0,\eta)J_v(w)\,d\mu_{y_0}(w).
\tag{263}
\]
The notation \(y_0\in V\) here denotes a base point, including both its plaque and transverse coordinates. Partition \(u,v\) into finitely many buffered charts. Terms away from the total diagonal have smoothing total kernels; their compressions are compact. The diagonal terms are exactly the amplitude calculation of Lemma 7.4 after extending each buffered coordinate box to a torus. The extension agrees on the smaller supports; differences away from those supports are smoothing. This proves (263), with no global product-coordinate assumption. The resulting leading coefficients combine to the integral in (263).

The chart estimates are uniform over the compact units \(W\) and all regular source fibres. Indeed every local estimate is applied in a near-unit chart and transported by right multiplication. A fixed small buffered chart has a uniform finite overlap number, as in the index lesson, Lemma 6.1; this bounds the row and column operator sums. Finite matrices add only a fixed factor. Negative-order truncations therefore converge in the reduced regular norm to smooth \(R\) chart kernels. By (262) those kernels are precisely module compacts. The positive-square-root argument of Lemma 6.5 in the Hilbert-module lesson gives the upper symbol quotient norm; localized high-frequency packets give the lower bound. For the latter, a smooth compact chart kernel acting on a packet near one unit tends to zero by integration by parts; then norm approximation treats every module compact. One does not assert ordinary compactness on a whole holonomy cover. These arguments prove the exact compact kernel and supremum symbol norm for the total closed algebra, and the analogous statement on the base \(G\) module. They also prove (263) uniformly, including interval parameters.

Put \(C=QT_u-(-1)^{|Q||u|}T_uQ_2\). Expand \(C^*C\). The four symbols from (263) cancel exactly as in Lemma 7.4. The element \(\langle u,u\rangle\) may be an even \(C_r\)-valued function; it commutes with the second graded factor, which is the only commutation used in that cancellation. The exact base quotient makes the square compact, and the \(CL(L+\delta)^{-1}\) approximation in Lemma 7.4 makes the creation error compact. Applying it to the adjoints proves the other connection error. Smooth-vector density treats all creation vectors. In particular lifted vertical compacts have compact graded brackets with \(Q\). If a total operator has zero axis symbol, its products with lifted vertical compacts are compact.

**Proof: multiplication and the axis subtraction.** First, the conclusion of Lemma 7.5 extends to this total algebra: products of a lifted vertical operator with a total operator having zero axis symbol are in the closed total algebra, with their ordered symbols extended by zero. Here are the localization details needed for that extension and for the bracket assertion. The vertical quotient and a finite buffered partition approximate \(T\), modulo vertical compacts, by finite sums of local operators
\(\phi_i(x,z,y)t_i(D_x)\beta_i(x,z,y)\).
The functions \(\beta_i\) are scalar cutoffs; \(\phi_i\) is a matrix on the first graded factor supported where a larger chart is available. To obtain these sums, use the separated spatial/directional symbol partitions in Lemma 7.5, with a right cutoff equal to one on the symbol support. The remaining zero-symbol vertical term is compact. Quotient-norm equality supplies an arbitrarily small norm error after a compact correction.

For any such summand, moving a scalar cutoff past a *total* operator costs a total negative-order compact; multiplication by the bounded vertical lift preserves compactness. This step does not move cutoffs past \(T\) and does not declare a lifted vertical compact to be total compact. Insert a larger total cutoff equal to one on both supports of the summand. The operator outside that larger chart is smoothing there, so its contribution is compact. Extend the remaining symbols to the product torus chart. Lemma 7.5 applies to the extended zero-axis symbol; after restriction and patching it gives the stated multiplication rule. The vertical compact correction is handled by the zero-axis creation conclusion already proved. All norm limits remain in the closed total algebra.

For the bracket, in each of these local charts extend the base symbol \(b\), and choose \(q_{0,i}\) independent of \(x\), acting only on the second factor and equal to \(b\) at the axis. Use the directional cutoff construction in Theorem 7.6. Extend the total symbol \(q\) so that it agrees with its actual value on the buffered support and has that same axis value everywhere: extend the difference from \(q_{0,i}\), which vanishes at the axis, with a spatial cutoff. Its quantization differs from the actual localized operator only by a compact. The commutator of a scalar cutoff \(\beta_i\) with this total quantization is compact. The bracket of \(\phi_i t_i(D_x)\) with \(Q_{0,i}\) is compact: \(t_i(D_x)\) commutes exactly with \(Q_{0,i}\), and the bracket of \(\phi_i\) with \(Q_{0,i}\) has zero principal symbol. The difference \(Q-Q_{0,i}\) has zero axis symbol, so the multiplication rule puts the remaining two products in the total closed algebra. Their symbol is the bracket with \(q\), since the symbol bracket with \(q_{0,i}\) is zero. Patching gives the global symbol \([a\widehat\otimes1,q]\), extended continuously by zero; vertical compact corrections contribute compact brackets. This proves the bracket assertion.

Finally the exact symbol quotient sends a self-adjoint bracket with a nonnegative symbol to a positive element of the multiplier quotient by \(\mathcal K\). Conjugation by every left coefficient preserves its positivity. This is operator positivity in the required quotient. All compact approximations are uniform on a compact interval of metrics, frames and horizontal splittings, so the same proof applies to the interval module. \(\square\)

**Theorem 7.10 (the compact submersion product with holonomy).** With the prescribed ordered spin-c structures in (256),
\[
 \begin{gathered}
 a_f\widehat\otimes^{\mathrm{ord}}_{C(V)}\ell_{V,!}^{\mathrm{an}}
 =\ell_{W,!}^{\mathrm{an}}\widehat\otimes_{C_r^*(H)}\varepsilon_f,\\
 \text{in }KK^{r+p}(C(W),A).
 \end{gathered}
\tag{264}
\]
The superscript \(\mathrm{ord}\) specifies the full common-module convention: extend the second cycle by \(C_r\), compose over \(C(V)\widehat\otimes C_r\), and transport its native target \(C_p\widehat\otimes C_r\) by the full graded block flip to the physical order \(C_r\widehat\otimes C_p=C_{r+p}\). All actions, inner products and odd-vector signs are transported. This full Clifford-valued identity precedes final parity reduction; native multiplication keeps the second-cycle coefficient first, as detailed in Lemma7.21. Finite Hermitian bundle coefficients on \(W\) and compact interval families obey the same identity.

**Proof: the common module.** The product of \(E_f\) with the full regular module of Theorem 7.1 is the completion of \(P_f\) kernels with coefficients \(\mathcal S_K\widehat\otimes f^*\mathcal S_F\). Send a simple tensor to \(u(w)\widehat\otimes\xi(\gamma)\). The balanced coefficient relation is exactly evaluation at \(f(w)=r(\gamma)\). On a regular tensor representation at \(x\), Fubini gives the \(L^2(P_{f,x})\) inner product, with these coefficients. The same finite separated-chart approximation as in Lemma 7.8 proves density in module norm, hence gives an onto unitary before invoking any operator calculation.

The right side has that same module. To see this without identifying \(H\) with \(R\), represent \(\mathcal S_{F_W}\) by a finite even bundle projection \(e_W\), as in (241). The canonical map
\[
 \begin{gathered}
 e_W(C_r^*(H)\widehat\otimes C_{r+p})^N
 \widehat\otimes_{C_r^*(H)\widehat\otimes C_{r+p}}
 (\mathscr P_f\widehat\otimes C_{r+p})\\
 \longrightarrow e_W(\mathscr P_f\widehat\otimes C_{r+p})^N
 \end{gathered}
\]
sends an algebra column tensor a vector to its convolution action on that vector. It is isometric by the balanced inner product. A near-unit approximate identity of \(H\) acts nondegenerately on \(\mathscr P_f\), since its unit space is all of \(W\); in the buffered local coordinates it is the usual smooth approximate identity on the \((x,z)\) variables. Hence its range is dense and the map is onto. The ordered spin-c tensor map in (256) identifies this range with the common coefficient module. Both phase factors multiply in that map; no line bundle has been discarded.

Let \(Q_W\) be the longitudinal odd self-adjoint quantization for \(\ell_{W,!}^{\mathrm{an}}\) on the \(H\) module, extended to this common module. Near-unit \(H\) charts are also \(R\) charts. Thus its operator and principal symbol there are exactly those of the total calculus of Lemma 7.9. The induced multiplier is bounded, since the left action is the isometric inclusion of Lemma 7.8; its cycle defects, originally compact \(H\) kernels, act as compact \(R\) kernels on the common module. It represents the product of \(\ell_{W,!}^{\mathrm{an}}\) with the compact zero correspondence \(\varepsilon_f\). Explicitly, a vector from the regular \(H\) module is an algebra column. Its creation map on \(\mathscr P_f\) is compact, since every entry acts compactly by (262). Multiplying that map by the bounded \(Q_W\) proves the connection error for the zero second operator; adjoints prove the other error. The same argument holds with the finite projection and Clifford corners. The positivity condition with the first operator is \(2Q_W^2\), which is positive. Definition 0.3 therefore gives that product directly.

**Proof: the analytic first product.** Choose the horizontal splitting orthogonal to the vertical metric and use the pullback metric on its copy of \(F\). In submersion and foliation coordinates the horizontal lifts have the form \(\partial_z+\sum_j B^j(x,z,y)\partial_{x_j}\). If \(\eta_0\) is the coordinate \(z\)-covector, its horizontal value is \(\eta=\eta_0+B^t\xi\). The total principal symbol and the vertical principal symbol are consequently
\[
 \begin{aligned}
 q(w;\xi,\eta)&=
 \frac{c_K^+(\xi^\sharp)\widehat\otimes1+
 1\widehat\otimes c_F^+(\eta^\sharp)}{\sqrt{|\xi|^2+|\eta|^2}},\\
 a(w,\xi)&=c_K^+(\xi^\sharp)/|\xi|.
 \end{aligned}
\tag{265}
\]
These are symbols of the *global* transition bundle because orthogonal Clifford sums intertwine its full transition unitaries. At the vertical axis the first formula is precisely the pulled-back normalized base symbol. Lemma 7.9 gives both creation-connection errors for the extended base cycle. For \(\xi\ne0\), the two Clifford blocks anticommute and give
\[
 [a\widehat\otimes1,q]
 =\frac{2|\xi|}{\sqrt{|\xi|^2+|\eta|^2}}I\ge0.
\tag{266}
\]
The continuous value at the axis is zero. In the coordinate covectors, the denominator contains \(\eta_0+B^t\xi\); this shear does not affect the sign or the axis value. Lemma 7.9 proves positivity of the actual operator bracket in the module quotient. Thus \(Q_W\) satisfies both parts of Definition 0.3 for the first product, and proves (264).

If \(r=0\), the first module is a finite fibre bundle with its orientation line, and \(T=0\). The axis condition is the whole principal-symbol condition in each inverse chart. The connection calculation still applies, while the positivity bracket is zero. If \(p=0\), the base regular module is a finite line module over \(C(V)\), with zero base operator. Its creation maps into the common vertical module are compact (finite columns with entries in that module). The total operator is the vertical quantization tensored with the base orientation line. The bracket is \(2T^2\), modulo the compact difference of such quantizations, and is positive. If both ranks vanish the argument is simply the finite bundle tensor product with zero operators. These cases account for noncanonical rank-zero lines as well as the empty cospheres.

For a finite bundle \(B\to W\), replace \(\mathcal S_K\) by \(B\otimes\mathcal S_K\) in the first module and the total bundle. Every frame, compression and symbol calculation above has that extra finite factor and the same axis and positivity formulas. This proves the asserted coefficient identity, including changes of the vertical spin-c structure by a line. Metric and splitting changes give compact interval families as in Lemmas 7.7 and 7.9, with the full structures transported. Their module unitaries, symbols and compact defects are uniform over the interval. Thus the identity respects all these choices. \(\square\)

For the full tangent foliation on \(V\), composition with its pair-groupoid module gives the ordinary compact-submersion identity \(a_f\widehat\otimes_{C(V)}[D_V]=[D_W]\) with the ordered orientation. Indeed the same Fubini calculation identifies the common module with the full \(L^2\) spinor module (259); the pair-groupoid regular module composed with \(L^2\) evaluation has scalar compact operators as its compact ideal. Formula (265) is the normalized total tangent symbol, and the proof above applies verbatim. Here \([D]\) denotes the Dirac class defined by that normalized symbol. No assertion that an unbounded tensor sum is exactly the total Dirac operator is required: the metric connection (258) gives such a tensor sum its correct horizontal principal part; comparison with a total Clifford connection has a smooth zeroth-order difference. On a compact manifold that endomorphism is bounded, and the normalized principal symbols give the same KK class by the compact-perturbation argument in Theorem 7.1. [Kaad–van Suijlekom 2020], Section 3.3, studies the more precise curvature difference and proper noncompact refinements.

The identity (264) is the analytic holonomy submersion product. Identifying \(a_f\) with the geometric normal-bundle wrong-way class is a further comparison. Its noncompact localized form is needed in the transverse normal tube, together with the prescribed spin-c orientation. Theorem7.22 supplies the compact ordinary comparison, and Lemmas7.23–7.24/Theorem7.25 below supply the full prescribed transverse comparison. Lemmas7.26–7.28 and Theorem7.29 below prove the full all-symbol action in both K-degrees.

![Full submersion correspondence, the two-component compact-algebra inclusion and the exact symbol and density identities](../figures/global-submersion-correspondence.png)

**Figure 7.3.** Lemmas 7.7–7.9 and Theorem 7.10, equations (256)–(266); Exercises 51–53, equations (267)–(268). The first panel records the actual open holonomy inclusion and right principal space. The block diagram is the exact example of Exercise 52: two circle components in the source, a full tangent foliation in the target, two diagonal holonomy blocks and four pullback blocks. It describes algebra and arrow components, with no numerical approximation. The last panel gives the horizontal-frequency shear and the nonnegative scalar bracket of (265)–(266), and the density correction for the product-circle example with positive density \(a(y)\,dx\). Both spin-c phases and the ordered Clifford factors are retained. The fibre module and connection are treated by [Kaad–van Suijlekom 2020]; the holonomy product method is classical [Connes–Skandalis, Section 4]. This drawing and the supplied proofs are original CC0 material.

Full-size submersion PNG · Editable submersion SVG.

### Local coefficients and noncompact submersions

The compactness assumption in Theorem 7.10 can be removed, but its role must be replaced explicitly. On a noncompact manifold the Dirac square defect is compact only after multiplication by a function vanishing at infinity. A list of local phases is also insufficient: one must construct a bounded adjointable operator on the entire module. We give that construction before using the local product estimates. The submersion need not be proper, and its fibres need not be connected or compact. This extends the classical bounded construction and product method of [Connes–Skandalis, Propositions 1.2 and 2.9 and Lemma 4.7]; the proofs below use the complete module calculus already proved in these lessons. They do not import a proper unbounded product theorem.

All manifolds in this subsection are smooth, Hausdorff, second countable and without boundary. Bundles have fixed finite rank. Compact supports in a non-Hausdorff arrow space mean finite sums of compactly supported smooth arrow-chart kernels, as in the C*-algebra lesson.

**Lemma 7.11 (a bounded phase with locally compact defects).** Let \(X\) carry a foliation \(L\) of rank \(k\) with a specified full spin-c structure. Put \(B=C_r^*(\operatorname{Hol}(X,L))\). The completion of compact smooth holonomy kernels with coefficients in the range pullback of \(\mathcal S_L\) is a countably generated graded right \(B\widehat\otimes C_k\)-module \(\mathscr E_L\). There is a bounded odd self-adjoint contraction \(Q\) on this module with normalized principal symbol \(c_L^+(\xi^\sharp)/|\xi|\) and

\[
 \phi(h)(Q^2-1),\ [Q,\phi(h)]\in\mathcal K(\mathscr E_L)
 \qquad(h\in C_0(X)).
\tag{269}
\]

It therefore defines \(\ell_{X,!}^{\mathrm{an}}\in KK^k(C_0(X),B)\). Choices of quantization and smooth metric or connection, with the full spin-c structure transported, give the same class. For a smooth submersion \(f:X\to Y\), the same assertion holds for its fibre-integration module and vertical symbol, giving \(a_f\in KK^k(C_0(X),C_0(Y))\) when \(k=\operatorname{rank}\ker df\). No properness is required. In either assertion, when \(k=0\) use the orientation-line module and zero operator; (269) still holds.

**Proof: the modules and finite supported calculus.** Use a countable locally finite bundle and plaque atlas with relatively compact unit supports and buffered larger charts, with the larger charts themselves locally finite. One explicit construction uses a compact exhaustion with successive interiors: cover each compact annular layer by finitely many small charts whose closures lie inside larger coordinate charts in a slightly larger annular band. These larger bands meet only finitely many layers near any point. Shrinking the smaller charts provides the buffers, and the finite layer covers give the claimed countable locally finite atlas. Choose smooth functions \(\chi_i\) subordinate to it with \(\sum_i\chi_i^2=1\). Such functions are obtained by choosing locally finite nonnegative smooth bumps covering \(X\), then dividing each bump by the square root of their sum of squares. The denominator is smooth and positive. A bundle vector is sent to its local right \(C_k\)-components multiplied by \(\chi_i\). This is an even pointwise isometry into a countable trivial module bundle. On compact unit supports only finitely many entries occur. Its adjoint sums these entries, first on finite columns and then by the norm bound. The resulting projection is adjointable on \((B\widehat\otimes C_k)^\infty\): compact arrow kernels have compact range-unit supports, and finite-column approximation proves the required strict convergence. Its range, with the convolution inner product, is exactly \(\mathscr E_L\). The same chart-density and approximate-identity proof as (241), now using finite columns on each compact support, gives

\[
 \mathcal K(\mathscr E_L)
 =e\bigl(\mathcal K\widehat\otimes B\widehat\otimes C_k\bigr)e.
\tag{270}
\]

Here \(\mathcal K\) is the compact algebra of the countable column space. This also proves countable generation. Multiplication by a bounded scalar function is adjointable; the adjoint is multiplication by its complex conjugate.

For the submersion module, integrate the coefficient inner product over each fibre, starting with compact smooth sections of \(\mathcal S_{\ker df}\). On a compact source support, a finite submersion-chart partition turns this integral into finitely many smooth parameter integrals. Its result is in \(C_c(Y)\widehat\otimes C_k\), since the image of that support is compact. The norm is the supremum of the fibre \(L^2\) norms. The same local coordinate and density normalization as Lemma 7.7 proves positivity, definiteness, the bounded multiplication action and countable generation. Compact smooth fibre-pair kernels are module compact by separated-function approximation; rank-one kernels of compact smooth vectors are of this form. Density proves that their closure is the entire compact ideal. These statements do not require compactness of the entire fibre.

We will use the order-zero calculus only with compact unit supports until a global bounded operator has been constructed. A finite set of buffered charts has uniform density bounds and symbol seminorm bounds on its compact parameter sets. The proofs in the index lesson, Lemmas 6.1–6.2, apply there component by component. In a holonomy representation the same estimate is transported by right multiplication; the finite chart overlap bounds its operator norm on all source fibres. Negative-order truncations with these supports converge in that norm to compact smooth kernels. Thus the negative-order ideal is the module compact ideal in (270), or the fibre-integration compact ideal just described. All finite bundle or Clifford corners are retained.

For later use let \(\Psi_{\mathrm{loc}}\) be the C*-algebra generated by these order-zero operators with compact unit supports and by the module compact ideal. Its symbol map has that compact ideal as kernel and has range the continuous cosphere endomorphism sections vanishing at unit infinity. Here is the norm argument, including the noncompact issue. For a finite supported symbol \(b\) of supremum norm \(c\), quantize the positive square root of \((c^2+\epsilon)I-b^*b\). It is the scalar constant \(\sqrt{c^2+\epsilon}I\) outside the compact support, so it is quantized in the unitization using that constant identity and finitely many chart corrections. Composition leaves only a finite supported negative-order error, hence a module compact. This gives the upper quotient bound. A high-frequency packet in a chart at a unit where the symbol is nearly maximal gives the lower bound. A compact smooth kernel annihilates this packet by integration by parts, and norm approximation treats any module compact; there is no assumption that a whole represented holonomy cover has compact resolvent. Approximation of continuous compactly supported symbols by smooth ones, followed by approximation of sections vanishing at infinity by compact supports, gives the asserted range and quotient norm. If a zero-order operator has zero symbol and finite unit support, the finite-chart negative-order proof puts it in the compact ideal; quotient-norm completion gives the kernel assertion.

**Proof: a global contraction.** In each larger chart quantize the normalized symbol, using an auxiliary smooth low-frequency extension and a fixed near-unit kernel cutoff. Take its odd part and its self-adjoint part. Sandwich it by a scalar cutoff \(\eta_i\), supported in that larger chart and equal to one on a neighbourhood of \(\operatorname{supp}\chi_i\). Its leading symbol is \(\eta_i^2q\), with norm at most one; choose \(0\leq\eta_i\leq1\). Denote this bounded self-adjoint odd operator by \(R_i\). It has input and output unit supports in \(\operatorname{supp}\eta_i\). It may have norm larger than one because of its lower-order terms.

Let \(h(t)=\max(-1,\min(t,1))\), and put \(P_i=h(R_i)\). Continuous functional calculus makes \(P_i\) an odd self-adjoint contraction. It belongs to the same finite supported algebra: approximate \(h\) on the spectrum by polynomials with zero constant term, since \(h(0)=0\). Each such polynomial retains the input and output unit supports of \(R_i\). Its symbol is \(h(\eta_i^2q)=\eta_i^2q\), because all eigenvalues of that symbol lie in \([-1,1]\). Thus clipping the operator norm has not changed the symbol on \(\operatorname{supp}\chi_i\).

Define

\[
 Q=\mathop{\mathrm{strict}\!\sum}_{i}\chi_iP_i\chi_i,
 \qquad \|Q\|\leq1.
\tag{271}
\]

To justify this sum, the column map \(J:u\mapsto(\chi_i u)_i\) is an adjointable isometry. Indeed \(\sum_i\chi_i^2u=u\) on compact smooth kernel vectors, where the sum is finite; density and the contractive finite sums give the assertion on the completion. Its adjoint is the bounded row map. The diagonal contraction \(\operatorname{diag}(P_i)\) is adjointable on the column module, with its indicated diagonal adjoint. The formula is precisely \(J^*\operatorname{diag}(P_i)J\). Equivalently its tails on a vector \(u\) have norm at most \(\|(\chi_i u)_{i\text{ in the tail}}\|\), which tends to zero. The adjoint tails satisfy the same estimate. This proves bounded adjointability, strict convergence, self-adjointness and oddness. It proves a global norm bound without assuming that the chart seminorms or the geometry have uniform bounds over all of \(X\).

Local finiteness, including the larger cutoffs, can be arranged in the choice of buffered atlas. Multiplication of \(Q\) on either side by a compactly supported scalar involves only finitely many \(\chi_i\). Composing those terms with a finite supported operator again involves only finitely many neighbours: each retained chart has compact unit support, and a locally finite family meets a compact set in only finitely many members. Norm approximation then shows that \(Q\) multiplies \(\Psi_{\mathrm{loc}}\) on both sides. Its multiplier symbol is \(\sum_i\chi_i^2q=q\).

For smooth compactly supported \(g\), the commutator \([Q,\phi(g)]\) consequently has finite unit support and zero principal symbol, so it is compact. The finite supported expression \(\phi(g)(Q^2-1)\) has principal symbol \(g(q^2-1)=0\), and is compact too. This proves (269) for \(C_c^\infty(X)\). Approximation in \(C_0(X)\), and the fixed bound \(\|Q\|\leq1\), prove it for every indicated \(h\).

**Proof: choices and rank zero.** Two such phases with the same symbol differ by a locally compact operator: multiplying their difference on either side by a compactly supported scalar gives a zero-symbol element of the finite supported calculus. Density gives the same conclusion for \(C_0(X)\). The straight path between these bounded self-adjoint odd operators has locally compact square and commutator defects. For the cross terms move a left scalar past either phase, which costs the compact commutator already proved. The locally compact difference then suffices. This is a Kasparov homotopy, even if the unlocalized difference is not compact.

For a smooth metric family on \(X\times[0,1]\), transport the specified full spin-c bundle by its positive metric isometry and form the range-kernel or fibre-integration module over the interval coefficient algebra. Use a countable locally finite buffered atlas on that whole parameter space, with relatively compact unit supports. Each retained chart has compact parameter support, so its estimates and functional calculus are continuous there. Apply (271) on this interval module. It is one adjointable contraction, and (269) holds with compact interval-module defects for each coefficient in \(C_0(X)\): its support times the interval is compact, so only finitely many charts enter each approximation. The endpoint modules and symbols are the prescribed ones; the quantization-choice homotopy just proved identifies their phases with any other choices. This supplies the full metric or connection homotopy without assuming global operator-norm continuity of an arbitrary pointwise family. The same construction treats smooth splitting changes.

In rank zero the foliation algebra is \(C_0(X)\), and the orientation-line section module has compact multiplication by every \(C_0(X)\) function. For a rank-zero submersion, the map is a local diffeomorphism. A compact source support is covered by finitely many inverse charts. A subordinate finite partition writes multiplication by its supported function as a sum of module rank-one operators, using an extra chart cutoff equal to one on each support and a local unit orientation frame. Thus the left \(C_0(X)\) action is compact even for an infinite covering or a nonproper local diffeomorphism. With \(Q=0\), these are exactly the localized square defects. The identity of the entire module need not be compact. This proves the rank-zero assertion, including its full orientation line. \(\square\)

**Lemma 7.12 (the nonproper full correspondence).** Let \(f:W\to V\) be a smooth surjective submersion, and let \(F\subset TV\) be a foliation. Put \(F_W=df^{-1}(F)\), \(G=\operatorname{Hol}(V,F)\), \(H=\operatorname{Hol}(W,F_W)\), \(R=f^*G\), and \(A=C_r^*(G)\). The principal space \(P_f\) of (261) defines a countably generated right \(A\)-module \(\mathscr P_f\), and

\[
 \mathscr P_f\cong E_f^0\widehat\otimes_{C_0(V)}A,
 \qquad C_r^*(R)\cong\mathcal K(\mathscr P_f),
 \qquad C_r^*(H)\lhook\joinrel\longrightarrow\mathcal K(\mathscr P_f).
\tag{272}
\]

The last arrow is the isometric extension-by-zero map induced by the actual open wide inclusion \(H\hookrightarrow R\). Consequently \(\varepsilon_f=[\mathscr P_f,j_*,0]\in KK(C_r^*(H),A)\) remains a compact correspondence. The assertions do not require \(f\) to be proper or its fibres to be connected.

**Proof.** The transverse-germ argument of Lemma 7.8 applies along each particular compact path. Projection identifies the local transverse holonomy germ with its base germ, so the map \(H\to R\) is injective. In buffered submersion and holonomy charts it is a local diffeomorphism. Its image is an open wide subgroupoid. This argument neither lifts all base paths globally nor claims that this subgroupoid is all of \(R\).

Construct \(E_f^0\) by the scalar version of Lemma 7.11. Multiplication of \(C_0(V)\) on \(A\) at the range is nondegenerate: it converges to the identity on compact smooth arrow-chart kernels under a compact-unit approximate identity, and hence on \(A\). The map \(u\otimes a\mapsto u(w)a(\gamma)\) has the exact convolution inner product integrated first along the vertical fibre. On every regular representation at \(x\), Fubini gives \(L^2(P_{f,x})\) with the density in Lemma 7.8. Finite separated smooth functions are dense on each compact chart support of \(P_f\); their uniform error has uniformly bounded fibre integrals there. A finite partition proves module-norm density on any compact smooth core, and those cores define the completion. This proves the first unitary and countable generation without a compact-fibre assertion.

The rank-one kernel is the same kernel \(u(w')(a*b^*)(\gamma)\overline{v(w)}\) as in Lemma 7.8. A compact \(R\) chart kernel is approximated by separated functions of its three variables, with fixed larger compact cutoffs. The associated Schur bounds are uniform on that fixed support, and each middle algebra coefficient is approximated in norm by finite sums \(a*b^*\). Thus these rank-one kernels span the represented reduced algebra. The regular space \(R_{w_0}\) is precisely \(P_{f,f(w_0)}\); surjectivity gives every regular \(A\) parameter, so its reduced norm is the module-operator norm. This proves the compact-algebra isomorphism.

For the last arrow decompose each \(R_{w_0}\) into its disjoint open left \(H\) orbits. They are countable by second countability. Right translation by a representative identifies each orbit, its measure and the restricted convolution representation with a regular \(H\) source fibre. The unit orbit supplies the regular representation at \(w_0\). The upper norm bound from the orthogonal sum and the lower bound from unit orbits give exactly the reduced \(H\) norm. Extension by zero respects convolution and adjoints on compact arrow charts. This proves the isometric inclusion and the compact left action. Its zero operator has compact square defect for every left algebra coefficient, and all its other Kasparov defects vanish. \(\square\)

**Theorem 7.13 (the localized holonomy submersion product).** Prescribe the full spin-c structures on \(K=\ker df\) and \(F\), of ranks \(r,p\), and give \(F_W\) their ordered sum structure. With the cycles of Lemma 7.11 and the correspondence of Lemma 7.12,

\[
 a_f\widehat\otimes^{\mathrm{ord}}_{C_0(V)}\ell_{V,!}^{\mathrm{an}}
 =\ell_{W,!}^{\mathrm{an}}\widehat\otimes_{C_r^*(H)}\varepsilon_f
 \quad\text{in }KK^{r+p}(C_0(W),A).
\tag{273}
\]

As in (264), \(\mathrm{ord}\) means the full product transported from the native target \(C_p\widehat\otimes C_r\) to the physical order \(C_r\widehat\otimes C_p\). This is the full graded block transport, before parity reduction; it does not drop an odd-block permutation sign. Lemma7.21 identifies the native reductions when the total rank is even. The equality includes finite Hermitian bundle coefficients on \(W\) and compact interval families of metrics, splittings, connections and specified orientation or coefficient bundles for the fixed submersion and foliation. It is a bounded analytical product statement; it does not yet identify \(a_f\) with a geometric wrong-way class.

**Proof: the entire common module.** The left tensor module is the completion of compact \(P_f\) kernels with coefficient bundle \(\mathcal S_K\widehat\otimes f^*\mathcal S_F\). The map is the same fibrewise tensor map as in Theorem 7.10. The inner product is Fubini's identity on each compact chart, and the finite separated-chart approximation of Lemma 7.12 proves onto density in the module norm. Compact supports, rather than properness of \(f\), justify every integral here.

Represent \(\mathcal S_{F_W}\) by the countable even projection \(e_W\) of (270). Extension of the full \(H\) module along \(j_*\) gives the same projection on \((\mathscr P_f\widehat\otimes C_{r+p})^\infty\). The algebra-column map on elementary tensors preserves the balanced inner product. A near-unit approximate identity of \(H\) converges on every compact \(P_f\) core, using finitely many buffered charts; density makes it nondegenerate on the whole module. Finite columns are dense, so this map is onto. The ordered spin-c tensor identification now identifies the two common modules, including both central phases.

Let \(Q_W\) be the contraction of (271) on the \(H\) module, and extend it to the common module. Equivalently use the same chart operators and clipping there: the isometric inclusion in (272) preserves functional calculus and the strict column construction. Its localized cycle defects extend to compact endomorphisms, since finite compact \(H\)-matrix entries act compactly on \(\mathscr P_f\), and norm limits treat all compact matrices. A finite algebra-column creation map for the zero second cycle is compact for this same reason. Multiplication by bounded \(Q_W\), followed by finite-column density, gives both creation-connection errors. The positivity bracket is \(2Q_W^2\), a positive operator. Thus \(Q_W\) represents the right side of (273).

**Proof: creation connections for the left product.** Take first a homogeneous compact smooth section \(u\) in the vertical integration module. Its range support in \(W\) and its image in \(V\) are compact. The column construction of \(Q_W\) involves only finitely many buffered charts when compressed with \(u\); the base phase involves only finitely many charts on that image. Every compressed expression in the four-term calculation of \(C^*C\), where
\(C=Q_WT_u-(-1)^{|u|}T_uQ_V\),
is therefore a finite supported operator of the base longitudinal calculus. Functional-calculus clipping does not obstruct that assertion: approximate each clipped local operator by polynomials with zero constant term in its finite supported quantization, use the finite supported symbol quotient, and pass to the operator norm limit.

After vertical-density normalization and unitary bundle frames, the axis symbol is \(1\widehat\otimes b\), with the frame invariance proved in Lemma 7.9. The compression identity (263) holds on the finite supports just identified. Its four leading terms in \(C^*C\) cancel. The finite supported zero-symbol base operator is consequently module compact by Lemma 7.11. The \(C L(L+\delta)^{-1}\) approximation of Lemma 7.4, with \(L=C^*C\), then makes \(C\) compact. Applying the same calculation to adjoints gives the other connection error. Compact smooth vertical sections are dense by definition of the first module, and \(\|T_u\|\leq\|u\|\); density proves both errors for every creation vector. This also gives the lifted-vertical-compact bracket conclusion needed in the axis multiplication argument. None of these compactness conclusions comes from ordinary compactness on an entire holonomy cover.

**Proof: localized operator positivity.** Let \(h\in C_c^\infty(W)\). To test the product criterion we must prove positivity of
\(h[T\widehat\otimes1,Q_W]h^*\)
in the module quotient by compacts, where \(T\) is the vertical phase. Move \(h\) and \(h^*\) past the *total* phase \(Q_W\). Its scalar commutators are compact by (269); multiplying them by a bounded vertical lift remains compact. The displayed bracket is consequently, modulo compacts, the bracket of the finite supported vertical operator \(hTh^*\) with \(Q_W\). There has been no claim that the lifted vertical commutator \([T,h]\) is total compact, and no such commutator was used.

The input and output unit supports of \(hTh^*\) are compact. Its expression from (271) has finitely many chart terms there. Insert a larger buffered total cutoff for each such term. Operators outside that buffer contribute smoothing kernels on its support; polynomial approximation treats clipped operators. The remaining finite chart calculation is exactly the axis subtraction and vanishing-axis multiplication proof of Lemma 7.9, including its vertical compact correction through the creation connections just established. The local principal bracket is
\[
 |h(w)|^2\,
 \frac{2|\xi|}{\sqrt{|\xi|^2+|\eta_0+B^t\xi|^2}}I,
\]
with continuous value zero at \(\xi=0\). It is nonnegative. The exact finite supported total symbol quotient of Lemma 7.11 therefore makes this bracket a positive element of the actual module quotient. A C*-homomorphism preserves positivity, so passing from the finite supported algebra to that quotient is legitimate; positivity of a pointwise symbol alone was not substituted for this step. Uniform approximation of \(h\) in \(C_0(W)\), and the boundedness of \(T,Q_W\), give the criterion for every left coefficient, since the positive cone is closed. Definition 0.3 proves the left product and hence (273).

If \(r=0\), compact smooth vertical sections meet finitely many inverse charts, even when the map has infinitely many sheets. The creation proof has the axis condition as its whole symbol condition; \(T=0\) gives a zero positivity bracket. If \(p=0\), the base operator is zero on its orientation-line module. For a compact smooth first-module vector \(u\), choose finitely many local frames of the base orientation line on the compact set \(f(\operatorname{supp}u)\), with larger compact cutoffs. They write its creation map as a finite sum of rank-one maps from that line module to the common vertical module. It is therefore compact. Density treats every first-module creation vector. The product operator is the vertical phase tensored with the pulled-back line, and its bracket is \(2T^2\), up to the locally compact change of quantization already proved. If both ranks vanish the complete module tensor product has zero operators. These arguments retain the two rank-zero orientation lines.

A finite Hermitian coefficient bundle is included by replacing the first coefficient bundle by its tensor with that bundle. All calculations use finitely many frames on each compact support and hence retain it exactly. For a compact parameter interval perform the whole construction on the product-unit spaces with the parameter held transverse. Every compact coefficient support times that interval is compact, and the chart estimates, compact approximations, projection unitaries and clipping constructions occur on the single interval module of Lemma 7.11. This gives compact interval defects and the same product identity at its endpoints. It completes the proof. \(\square\)

This closes the analytical product issue for nonproper submersions. Theorem7.22 identifies the compact ordinary class, and Lemmas7.23–7.24/Theorem7.25 below prove the full prescribed transverse-normal leaf-map comparison.

### Restricting an analytical cycle to a normal tube

An open tube has an artificial boundary. We use the bounded cycles of Lemma 7.11 on its interior; no boundary condition is part of that construction. The following argument explains exactly why extension by zero on the left algebra restricts the ambient analytical class. The family-module framework and wrong-way comparison are classical [Connes–Skandalis, Sections 1–2]. The projection, localized compression and tube argument below supply the particular full proof used here.

**Lemma 7.14 (bounded locality on a vertical open domain).** Let \(Y\) be a smooth Hausdorff second-countable manifold, let \(n\geq1\), and let \(U\subset Y\times\mathbb R^n\) be open. Assume its boundary is a smooth hypersurface and the restriction of the projection to that boundary is a submersion; an empty boundary is allowed. Give the vertical tangent of \(U\to Y\) the restriction of the canonical full Euclidean spin-c structure. With \(e_U:C_0(U)\to C_0(Y\times\mathbb R^n)\) the extension by zero,

\[
 [e_U]\otimes a_\pi=a_{\pi|_U}
 \quad\text{in }KK^n(C_0(U),C_0(Y)),
 \qquad \pi(y,z)=y.
\tag{274}
\]

The right side is the fibre-integration cycle of Lemma 7.11. Fibres and \(Y\) may be noncompact, and the image of \(U\) need not be all of \(Y\). The assertion retains the right \(C_n\) coefficient and hence includes both parities.

**Proof: the actual domain projection.** On

\[
 \mathscr H=C_0\bigl(Y,L^2(\mathbb R^n)\bigr)\widehat\otimes C_n,
 \qquad
 (Pu)(y,z)=1_U(y,z)u(y,z),
 \qquad P=P^*=P^2,
\tag{275}
\]

the formula defines an even adjointable projection. Here is the continuity point. At a fixed \(y_0\), the intersection of the boundary with that fibre is a smooth hypersurface, because its projection is a submersion. It has \(n\)-dimensional Lebesgue measure zero: each of its countably many coordinate pieces is a graph, and Fubini proves this assertion. For almost every \(z\), therefore, \((y_0,z)\) lies either in \(U\) or outside its closure. Its membership is constant for \(y\) sufficiently near \(y_0\), with the neighbourhood allowed to depend on \(z\). Dominated convergence proves \(1_{U_y}v\to1_{U_{y_0}}v\) in \(L^2\) for every fixed \(v\in L^2(\mathbb R^n)\otimes C_n\). If \(u(y)\) also varies continuously, use

\[
 \|P_yu(y)-P_{y_0}u(y_0)\|
 \leq\|u(y)-u(y_0)\|
       +\|(P_y-P_{y_0})u(y_0)\|.
\tag{276}
\]

The contraction bound retains vanishing at base infinity. Pointwise self-adjointness gives the module adjoint, including the full finite Clifford inner product. Thus \(P\) is adjointable even though multiplication by \(1_U\) is usually discontinuous as a scalar function on the ambient manifold.

Its range is exactly the completion of \(C_c(U)\otimes C_n\) for the fibre-integration norm. Extension by zero gives an isometric inclusion into \(P\mathscr H\). To prove density, first restrict a given section to compact base support. Choose increasing continuous compactly supported cutoffs \(0\leq\kappa_j\leq1\) on \(U\) tending to one there. Each function \(y\mapsto\|(1-\kappa_j)Pu(y)\|^2\) is continuous, decreases to zero, and has a common compact base support. The decrease is uniform: for each tolerance the open sets on which the function is below that tolerance increase to cover the compact support, so a finite subcover selects one index. Consequently \(\kappa_jPu\to Pu\) in module norm. On the compact support of \(\kappa_j\), finite sums of base functions times fibre-smooth functions approximate the continuous \(L^2\) section; smoothing its compactly supported multiplier gives compact smooth sections of \(U\). A finite chart partition and the elementary \(L^2\) approximation by continuous compact functions justify the same approximation when the original section is only continuous into \(L^2\). Removing the base cutoff proves the density assertion. Denote this range by \(\mathscr H_U\).

**Proof: compression is a cycle.** On the ambient module use

\[
 D=-i\sum_{j=1}^n\varepsilon_j\partial_{z_j},
 \qquad F=D(1+D^2)^{-1/2}.
\tag{277}
\]

Fibre Fourier transformation makes \(F\) an odd self-adjoint contraction with symbol \(\varepsilon\cdot\xi/\sqrt{1+|\xi|^2}\). Compactly supported left coefficients give its compact square defects and commutators by the finite-supported calculus of Lemma 7.11. For this constant symbol one can also see the compactness directly by compact-frequency truncation and the negative-order commutator estimate used there. Therefore it represents \(a_\pi\), since its homogeneous principal symbol is the prescribed Euclidean one.

Represent \(C_0(U)\) on the ambient module by extension by zero. For \(a\in C_c(U)\), choose a real \(\chi\in C_c^\infty(U)\) equal to one on its support, and also write \(a,\chi\) for their ambient extensions. Since \(\chi(1-P)=0\),

\[
 aF(1-P)=a[\chi,F](1-P),
 \qquad
 (1-P)Fa=(1-P)[F,\chi]a.
\tag{278}
\]

Both terms are ambient module compacts. Their corners are compact on the corresponding range modules: compressing \(\theta_{u,v}\) gives \(\theta_{Pu,Pv}\), and the same formula applies to rectangular corners. Approximation extends (278) to every \(a\in C_0(U)\).

Put \(F_\oplus=PFP+(1-P)F(1-P)\). Its difference from \(F\) is locally compact for the left \(C_0(U)\) action. The straight path \(F+t(F_\oplus-F)\) has compact localized commutators and square defects. Indeed the difference has compact products on both sides with \(a\); in its cross terms move \(a\) past \(F\), using the already compact commutator. All resulting products contain a compact factor. The operators are odd and self-adjoint, and the same estimates over the compact parameter interval give module-compact homotopy defects. The cycle at the last endpoint splits as \(\mathscr H_U\oplus(1-P)\mathscr H\). The latter has zero left representation and is degenerate. Thus the restricted ambient class is represented on \(\mathscr H_U\) by the contraction \(PFP\). This proves the compression assertion without asserting that \(P\) commutes with \(F\).

**Proof: comparison with the interior phase.** Let \(Q_U\) be the clipped, locally finite bounded phase of Lemma 7.11 on the same fibre-integration module \(\mathscr H_U\), with its Euclidean vertical metric and orientation. We show that \(PFP-Q_U\) is locally compact. For smooth \(a\in C_c(U)\), use an interior cutoff \(\chi=1\) near its support. The contribution \(aQ_U(1-\chi)\) is compact, by its compact commutator with \(\chi\). The analogous ambient term \(aF(1-\chi)\) is compact by (278) with \(1-\chi\) in place of \(1-P\), since \(a(1-\chi)=0\). After these errors, both input and output can be confined to compact interior buffers. There \(\chi F\chi\) and \(\chi Q_U\chi\) have the same homogeneous symbol. The exact finite-supported quotient in Lemma 7.11 makes their difference a module compact. Clipping causes no exception: its functional calculus remains in that algebra with its stated symbol. Corners by \(P\) retain compactness, and adjoints prove the other localization. Norm density treats all \(a\in C_0(U)\).

For completeness, the ambient Fourier phase really is in this local calculus after interior confinement; its unbounded fibre does not leave an uncontrolled kernel tail. Away from the diagonal its kernel is

\[
 K(w)=\frac{-i}{\sqrt\pi}\int_0^\infty
 t^{-1/2}e^{-t}\sum_j\varepsilon_j\partial_{w_j}
 \left((4\pi t)^{-n/2}e^{-|w|^2/(4t)}\right)\,dt.
\tag{279}
\]

The identity follows first on compact-frequency vectors from the scalar formula \(\lambda^{-1/2}=\pi^{-1/2}\int_0^\infty t^{-1/2}e^{-t\lambda}\,dt\), and then off the diagonal by absolute convergence. For every \(\delta>0\), the Gaussian derivative norm outside \(|w|\geq\delta\) is at most

\[
 C_n t^{-(n+2)/4}e^{-\delta^2/(16t)}.
\tag{280}
\]

To check this bound, substitute \(w=\sqrt t\,v\) in its squared \(L^2\) norm and absorb the polynomial factor \(|v|^2\) into half of the Gaussian. Increasing the dimensional constant gives the displayed exponent. Multiplication by \(t^{-1/2}e^{-t}\) is integrable with this bound both at zero and infinity. Hence \(K\) has finite \(L^2\) norm away from every diagonal buffer, and its \(L^2\) tail tends to zero. A compact output cutoff turns each such tail into a continuous compact-base Hilbert–Schmidt family. Finite separated-kernel approximation makes it a module compact; the uniform tail estimate gives norm convergence. The near-diagonal, doubly confined Fourier quantization is precisely the finite chart order-zero operator. This supplies an explicit check of the ambient-to-interior comparison, rather than an appeal to a compact ambient fibre.

The straight path between \(PFP\) and \(Q_U\) is now a Kasparov homotopy by the locally compact difference calculation above. The resulting class is exactly \(a_{\pi|_U}\) of Lemma 7.11. A homomorphism cycle acts by restricting the left representation (Section 0), so the first restricted cycle represents the left side of (274). This proves the identity. All projections, corners and homotopies commute with the right \(C_n\) action and preserve the grading. The argument also works with a compact interval of smooth vertically transverse domains: dominated convergence is applied to \((y,s)\), and compact parameter cutoffs give the same estimates. \(\square\)

**Lemma 7.15 (the fibre-preserving graph tube).** Let \(f:W\to V\) be a smooth surjective submersion with \(W\) compact, and write \(K=\ker df\), of rank \(r\). Choose an embedding \(u:W\hookrightarrow\mathbb R^n\), with \(n>r\); even \(n\) is available by stabilization. In the trivial Euclidean bundle define

\[
 \begin{aligned}
 N_w&=(du_wK_w)^\perp,\qquad
 \Psi(w,v)=(f(w),u(w)+v),\\
 \Phi&=\Psi\circ q_\rho:N\longrightarrow V\times\mathbb R^n,
 \quad q_\rho(w,v)=\left(w,\frac{v}{\sqrt{1+|v|^2/\rho^2}}\right).
 \end{aligned}
\tag{281}
\]

For some constant \(\rho>0\), \(\Psi\) is an embedding on a neighbourhood of the closed disk bundle of radius \(\rho\), and \(\Phi\) is a whole normalized open tube for the graph \((f,u)\). Its image \(U\) has a smooth boundary transverse to the fibres of the ambient projection. Give the vertical tangent of \(f\circ\operatorname{pr}_N\) the full spin-c structure and metric pulled back through \(d\Phi\) from the canonical Euclidean vertical structure. Then

\[
 [e_\Phi]\otimes a_\pi
     =a_{f\circ\operatorname{pr}_N}^{\mathrm{ambient}}
 \quad\text{in }KK^n(C_0(N),C(V)).
\tag{282}
\]

Here the superscript specifies this exact full pulled-back structure. It does not choose a separate spin-c structure on \(N\to W\), or identify the prescribed graph-normal Thom inverse with a vertical cycle.

**Proof: existence and normalization.** The orthogonal complement \(N\) is a smooth bundle of rank \(n-r\geq1\). At the zero section the derivative of \(\Psi\) sends \((h,v)\) to \((df(h),du(h)+v)\). If this is zero, \(h\in K\), and \(du(h)+v=0\) in the orthogonal sum \(du(K)\oplus N\). Thus \(h=v=0\). Its domain and codomain have the same dimension \(\dim V+n\); the derivative is invertible. The inverse-function theorem gives a local diffeomorphism near every zero-section point. Compactness gives a uniform disk on which all derivatives remain invertible.

It also gives a smaller uniform disk on which the map is injective. Otherwise there are distinct pairs \((w_j,v_j),(w'_j,v'_j)\) with \(|v_j|,|v'_j|\to0\) and equal images. Compactness gives limits \(w,w'\). Equality of the Euclidean coordinates gives \(u(w)=u(w')\), hence \(w=w'\). Both pairs eventually lie in a single neighbourhood on which the inverse-function theorem gave injectivity, a contradiction. Shrink once more so that \(\Psi\) is a diffeomorphism onto its open image on a disk strictly larger than the closed radius-\(\rho\) disk.

The quotient normal of the graph is canonically \(N\). Indeed subtracting a graph tangent with prescribed \(df(h)\) removes the \(TV\) coordinate; its remaining Euclidean coordinate is well defined modulo \(du(K)\), and orthogonal projection identifies that quotient with \(N\). The derivative of \(\Psi\) in \(v\) is this identity on the quotient. The radial compression \(q_\rho\) is the diffeomorphism of (149) and has vertical derivative the identity at zero, so \(\Phi\) is normalized as stated. Its projection is exactly \(f(w)\).

Because the larger disk is embedded, the topological boundary of \(U=\Psi(D_\rho N)\) is precisely the image of the radius-\(\rho\) sphere bundle. To verify that no extra boundary appears, the closed disk bundle is compact, its image is closed, and each point in the larger disk image has the same interior/boundary/exterior status as its inverse under that diffeomorphism. The sphere bundle projects submersively to \(W\); composing with \(f\) gives a submersion to \(V\). This proves the vertical-boundary hypothesis of Lemma 7.14. Surjectivity of \(f\) and compactness of \(W\) also imply compactness of \(V\), but no properness of \(N\to V\) is asserted.

**Proof: the exact module change of variables.** Pull back the ambient vertical metric, its fibre density and its full finite right \(C_n\)-module along \(\Phi\). Its vertical derivative is an isometry for this pulled-back metric. Change of variables on each fibre gives an even right-linear isometry from compactly supported sections of \(U\) to compactly supported sections of \(N\), preserving their \(C(V)\widehat\otimes C_n\)-valued inner products. Its inverse is change of variables through \(\Phi^{-1}\), so it extends onto the completed fibre-integration modules. In scalar coordinates with a separately chosen smooth density on \(N\), this unitary includes the positive square root of the vertical Jacobian; using the pulled-back density incorporates that factor already.

Under this unitary the interior phase has exactly the pulled-back principal Clifford symbol. Its actual bounded operator need not be the bounded transform of a complete Dirac operator: Lemma 7.11 compares bounded quantizations with that symbol and proves the relevant metric/connection independence on this noncompact total space. Thus the transported cycle is \(a_{f\circ\operatorname{pr}_N}^{\mathrm{ambient}}\). Compose (274) with the algebra isomorphism induced by \(\Phi:N\to U\); its extension to the ambient space is exactly \(e_\Phi\). This gives (282), with the same full module and ordered constant Clifford coefficient. \(\square\)

The lemmas prove open locality and the whole geometric tube. Lemma 7.16 and Theorem 7.17 below identify the actual analytical orientation of a coefficient-transferred Thom inverse, retaining its full dual line, generator sign and grading. Lemmas 7.18–7.19 below recover the full prescribed graph normal and identify its inverse/vertical coefficient module with the ambient one, including the whole tube. Lemmas 7.20–7.21 and Theorem 7.22 below apply both native Thom inverse identities and finish the compact ordinary submersion comparison in both parities.

![Bounded locality and an exact normal annulus](../figures/ordinary-open-locality.png)

**Figure 7.4.** Lemmas 7.14–7.15, equations (274)–(282), and Exercises 56–57, equations (283)–(285). The annulus is the exact image of the whole normal line bundle of the unit circle at radius \(\rho=1/4\); the drawing maps one physical unit to 200 pixels and excludes both dashed boundary circles. The marked point is \(\Phi(0,1/(4\sqrt3))=(9/8,0)\). The curve samples the exact compression function on \(-1\leq v\leq1\); its horizontal asymptotes are excluded endpoints of its full range. The pulled-back metric is incomplete. The lower panel gives the actual compact-cutoff and block-homotopy mechanism on the full fibre module. The full prescribed graph-normal orientation contraction is the next step. This original diagram illustrates the independently supplied proofs; classical family and wrong-way context is [Connes–Skandalis, Sections 1–2].

Full-size locality PNG · Editable locality SVG.

### The actual coefficient transfer of a Thom inverse

The inverse in (144) has a Clifford algebra on its source. The analytical vertical cycle has that Clifford algebra on its target. These are distinct types. We compare them through an explicit finite Morita contraction, retaining its block order, grading and transition phases. The Clifford Bott–Dirac framework is classical [Kasparov; Emerson, Section 11.4]; the full transfer calculation below supplies the comparison used here.

Write \(C_r\) for the positive complex Clifford algebra with its ordered generators. Let \(\Sigma_r=\Lambda^*\mathbb C^r\) be the canonical \(C_{2r},\mathbb C\) Morita module: the \(2r\) generators are in their standard list order, paired consecutively. Its grading is the action of their normalized volume \((-i)^r e_1\cdots e_{2r}\). The star on a Morita module denotes its conjugate module, as in Lemma 0.12. For trivially graded algebras \(A,B\), define

\[
 \mathcal T_r(x)=
 [1_A\widehat\otimes\Sigma_r^*]
 \widehat\otimes_{A\widehat\otimes C_r\widehat\otimes C_r}
 \tau_{C_r}(x),
 \qquad x\in KK(A\widehat\otimes C_r,B).
\tag{286}
\]

The left Clifford list is the **old source block followed by the newly external block**. The result belongs to \(KK(A,B\widehat\otimes C_r)\). External tensoring and the actual Morita evaluations in Lemmas 0.12–0.13 make this a specified coefficient isomorphism. Exchanging the two source blocks requires transporting its finite Morita module too.

**Lemma 7.16 (the block contraction on the full finite module).** Put \(T=\Lambda^*\mathbb C^r\), with exterior grading \(\Gamma\), and let \(t_i\) be the generators of the last right \(C_r\). On \(T\widehat\otimes C_r\), use the ordinary matrix expressions

\[
 \begin{aligned}
 a_i&=e_i+\iota_i,\qquad b_i=i(e_i-\iota_i),\\
 A_i&=a_i\otimes1,\qquad B_i=\Gamma\otimes L_{t_i},
 \qquad C_i=b_i\otimes1.
 \end{aligned}
\tag{287}
\]

The \(A,B\) blocks represent \(C_r\widehat\otimes C_r\) in the list \((A_1,\ldots,A_r,B_1,\ldots,B_r)\). The \(C_i\) graded-commute with that representation. Set

\[
 \begin{aligned}
 k_r&=\frac{r(r-1)}2\pmod2,\qquad
 p=\prod_{i=1}^r\frac{1-iA_iB_i}{2},\\
 \psi&=2^{-r/2}\sum_I\lambda_{|I|}e_I\otimes t_I,
 \qquad
 \lambda_m=\begin{cases}1&m\text{ even},\\-i&m\text{ odd}.\end{cases}
 \end{aligned}
\tag{288}
\]

The sum runs over increasing subsets \(I\subset\{1,\ldots,r\}\), and \(t_I\) is their ordered Clifford word. There is a full even right-module unitary

\[
 \Sigma_r^*\widehat\otimes_{C_{2r}}(T\widehat\otimes C_r)
       \ \cong\ \Pi^{k_r}C_r,
 \qquad
 C_i^{\mathrm{contracted}}=(-1)^{k_r+1}L_{t_i}.
\tag{289}
\]

Here \(\Pi\) reverses the grading of the right module, retaining its right action and inner product. The contracted action includes the Koszul sign from the Morita tensor. At \(r=0\), use the scalar unit, empty product and empty generator list.

**Proof: the actual vacuum.** The \(a_i,b_i\) are odd self-adjoint, square to one and anticommute within each block and between the two blocks, except for the stated squares, by (129). The grading insertion in \(B_i\) is exactly the ordinary expression of \(1\widehat\otimes L_{t_i}\). Thus all \(2r\) left generators satisfy the positive Clifford relations. Each \(C_i\) anticommutes with every \(A_j,B_j\), so it graded-commutes with their full left algebra. Each \(-iA_iB_i\) is an even self-adjoint involution. Two distinct such products commute: moving their two generators past the other two gives four minus signs. Hence \(p\) is an even orthogonal projection.

For \(u=\sum_I e_I\otimes u_I\), the equation \((A_i+iB_i)u=0\), at an index \(I\) not containing \(i\), is

\[
 (-1)^{\#\{j\in I:j<i\}}u_{I\cup\{i\}}
   =-i(-1)^{|I|}t_i u_I.
\tag{290}
\]

The coefficient equation at the index containing \(i\) is equivalent after multiplication by \(t_i\), since \(t_i^2=1\). Starting with \(u_\varnothing\), the identity \(t_it_I=(-1)^{\#\{j\in I:j<i\}}t_{I\cup\{i\}}\) for \(i\notin I\) therefore gives \(u_I=\lambda_{|I|}t_Iu_\varnothing\). The scalar recurrence is \(\lambda_{m+1}=-i(-1)^m\lambda_m\). This proves that the joint kernel is \(\psi C_r\). It is the range of \(p\): multiplying \(A+iB\) by \(A\) shows that its kernel is the positive eigenspace of \(-iAB\), and the commuting positive eigenprojections have exactly their joint eigenspace as range.

Every term of \(\psi\) is even for the total grading, and its inner product is one. Exterior vectors of distinct indices are orthogonal, \(t_I^*t_I=1\), and \(2^{-r}\sum_I1=1\). Thus \(c\mapsto\psi c\) is an even right-linear isometry onto \(p(T\widehat\otimes C_r)\).

The physical action on this corner is \(C_i\psi=-\psi t_i\). For the component entering \(I\cup\{i\}\), with \(i\notin I\), \(b_i\) contributes \(i(-1)^{\#\{j\in I:j<i\}}\lambda_{|I|}t_I\). For the reverse component its contraction contributes the negative of this \(i\)-multiple with the coefficient of \(I\cup\{i\}\). Right multiplication of an ordered word by \(t_i\) has sign \((-1)^{\#\{j\in I:j>i\}}\). Substituting the two values of \(\lambda_m\), and the displayed recurrence, gives the claimed equality in both index cases. Consequently the uncontracted corner action is \(-L_{t_i}\).

**Proof: the grading and tensor sign.** The left block volume and the interleaved volume satisfy

\[
 (-i)^r A_1\cdots A_rB_1\cdots B_r
 =(-1)^{k_r}\prod_i(-iA_iB_i).
\tag{291}
\]

The permutation moves \(B_i\) past \(r-i\) later \(A\)'s, giving \(r(r-1)/2\) minus signs. The range of \(p\), which is even in \(T\widehat\otimes C_r\), therefore corresponds to a corner vector of \(\Sigma_r\) of parity \(k_r\). An ungraded corner identification would lose this shift.

For a complete module verification, choose a unit vector \(s_0\) of that corner in \(\Sigma_r\), extend it to a homogeneous orthonormal basis \(s_j\), and let \(E_{j0}\) be its matrix units. Their parity is \(|s_j|+k_r\). The map

\[
 s_j\widehat\otimes w\longmapsto\phi(E_{j0})w,
 \qquad w\in\Pi^{k_r}p(T\widehat\otimes C_r),
\tag{292}
\]

uses the underlying corner vector on the right. It is even. The matrix-unit relations give orthogonality of its columns and equality of their inner products with the corner inner product. Their diagonal sum is the identity, so the map is onto. This is the actual graded identification with \(\Sigma_r\widehat\otimes\Pi^{k_r}p(T\widehat\otimes C_r)\). Apply the Morita evaluation of Lemma 0.12 to prove the first assertion of (289).

If an odd operator \(F\) graded-commutes with the left matrix algebra, then \(F\phi(E_{j0})=(-1)^{|s_j|+k_r}\phi(E_{j0})F\). Thus (292) sends \(F\) to \(1\widehat\otimes((-1)^{k_r}F|_p)\): the graded tensor contributes \((-1)^{|s_j|}\), leaving the \(k_r\) sign on its second factor. Apply this to \(C_i\), whose corner action is \(-L_{t_i}\), to obtain the second assertion. All maps preserve the actual right action, inner product and grading. \(\square\)

**Theorem 7.17 (the full vertical inverse after coefficient transfer).** Let \(E\to X\) be a smooth metric real rank-\(r\) bundle on a smooth Hausdorff second-countable manifold. Prescribe its full spin-c transition module \(\mathcal S_E:\operatorname{Cl}^+(E)\rightsquigarrow C_r\), and let \(\alpha_E=m_E^{-1}d_E\) be the raw Thom inverse of (144). Define the finite graded bundle over \(X\)

\[
 \mathcal D_E=
 \Sigma_r^*\widehat\otimes_{C_r\widehat\otimes C_r}
 \Bigl[
   (\mathcal S_E^*\widehat\otimes_{\operatorname{Cl}^+(E)}
       \Lambda E_{\mathbb C})\widehat\otimes C_r
 \Bigr].
\tag{293}
\]

Its source and external Clifford blocks have the order of (286). The exterior positive \(a\) action is balanced against \(\mathcal S_E^*\). The positive physical \(b=i(e-\iota)\) action graded-commutes with this balance and induces a left \(\operatorname{Cl}^+(E)\) action on \(\mathcal D_E\). This is a full finite orientation module, with its induced orientation and grading. Pull it to \(E\) and use fibre density to form the analytical vertical cycle \(a_E^{\mathcal D_E}\) of Lemma 7.11. Then

\[
 \mathcal T_r(\alpha_E)=a_E^{\mathcal D_E}
 \quad\text{in }KK(C_0(E),C_0(X)\widehat\otimes C_r).
\tag{294}
\]

Locally \(\mathcal D_E\) is \(\Pi^{k_r}C_r\) with physical generators \((-1)^{k_r+1}t_i\). Tensoring the original module by an even line \(H\) tensors \(\mathcal D_E\) by \(H^*\). The assertion includes noncompact \(X\), smooth metric interval homotopies, and rank zero with its full graded orientation line.

**Proof: transitions, fullness and phase.** In a local orthonormal frame of \(E\) and principal frame of \(\mathcal S_E\), the inverse module is the regular finite left \(C_r\) module balanced against the exterior positive \(a\) action. The remaining finite right module and physical action are those of Lemma 7.16. That lemma gives their local regular rank, grading and onto contraction. Multiplication of all generators by their common sign is a Clifford automorphism, and reversing a module grading retains the underlying endomorphism algebra. Thus the physical left action identifies the entire compact algebra of \(\mathcal D_E\) with \(\operatorname{Cl}^+(E)\); its right inner products span \(C_r\). The smooth balanced bundle tensors and their evaluations glue these local facts. In positive rank the reconstruction preceding Lemma 6.0j recovers the induced principal spin-c orientation from this full module, including any orientation reversal. It does not replace it by the original or an independently selected orientation.

Write an original transition as \(zs\), where \(s\in\operatorname{Spin}(r)\) implements \(R\in\operatorname{SO}(r)\). Identifying a local conjugate regular module by \(\overline u\mapsto u^*\) changes its transition to right multiplication by \(s^*z^{-1}\). After balancing against the exterior action, the combined transition is

\[
 z^{-1}\rho_a(s)^*\Lambda R=z^{-1}\rho_b(s).
\tag{295}
\]

For a rotation in the \(i,j\) plane, \(\Lambda R=\exp(\theta(e_j\iota_i-e_i\iota_j))\). The exterior relations give \(e_j\iota_i-e_i\iota_j=-(a_i a_j+b_i b_j)/2\). These two even bivectors commute, so the exponential is the product of the two spin exponentials. Every special orthogonal transformation is a product of plane rotations: successively rotate its unit columns into the coordinate columns, leaving the previously fixed columns unchanged. A remaining diagonal orthogonal matrix of determinant one is removed by rotations of angle \(\pi\) on paired negative entries. Thus \(\Lambda R=\rho_a(s)\rho_b(s)\) for its spin lift; the other lift multiplies both factors by \(-1\) and retains their product. Formula (295) follows for the actual principal transition. In rank one the two spin lifts of the identity likewise have product one.

The even physical spin action commutes with the two fixed left coefficient blocks and descends through their contraction. Even words have the same image when every physical generator is multiplied by \((-1)^{k_r+1}\). Hence the contracted transition is exactly \(z^{-1}\rho_t(s)\) on the shifted module of (289). This retains the whole real-frame action and phase, including two-torsion lines. Replacing \(zs\) by \(h\,zs\) replaces that transition by \(h^{-1}z^{-1}\rho_t(s)\), proving the full \(H^*\) assertion.

**Proof: the actual analytical cycle.** The raw inverse is represented on the balanced inverse-spin/exterior fibre-\(L^2\) module with physical Fourier symbol \(i\sum_i c_i\xi_i=\sum_i b_i\xi_i\). We verify the product with the zero Morita cycle, which was not needed to identify \(\beta_E\) in Corollary 6.0d. Smooth compactly supported local sections of \(\mathcal S_E^*\) are dense in its total-space section module. In a base frame their homogeneous components give rectangular creation maps \(T_\xi\). The physical operator graded-commutes with the constant finite Clifford components; its graded connection error is multiplication by their fibre derivatives. It is bounded and compactly supported in the total space. Multiplication preserves the graph domains by the first-derivative product estimate.

Apply (132) to the block-diagonal physical operators on the source and target modules, with the rectangular creation map as its off-diagonal homogeneous coefficient. If \(C\) is its bounded derivative error and \(q=\sqrt{1+\tau^2}\), the two rectangular resolvent terms have norm at most \(5\|C\|/(4q^2)\). They are compact: each coefficient is supported in a fixed compact fibre/base set, and the localized order-\(-1\) resolvents there are compact by Lemma 6.0a, with their finite module columns retained. Integrating against \(2/\pi\) gives a norm-convergent compact error for the bounded transforms. The adjoint creation equation follows by taking adjoints; norm density treats all creation vectors. The first Morita operator is zero, so product positivity is zero. Definition 0.3 proves this representative of \(m_E^{-1}d_E\) on its full balanced Hilbert module.

Externally tensor with the last \(C_r\), then apply the finite source Morita module in (286). The unitary (292), in every base frame and on fibre-\(L^2\) sections, identifies the tensor Hilbert module with the fibre-integration module of \(\mathcal D_E\). It preserves fibre density, the right coefficient inner product and grading; onto finite evaluation and density of compact sections prove its surjectivity. On its graph core, the operator is the physical vertical Dirac operator with the contracted generators of (289). The sign from that formula is retained. Its regular self-adjoint closure follows from the same fibre Fourier inverse shifts as in Lemma 6.0a, now with these positive generators. This unitary also intertwines the full closed domains and bounded transforms.

For this second finite Morita product, the homogeneous constant columns satisfy precisely the signed intertwining equation following (292). A compactly supported scalar coefficient on a column has bounded fibre derivative; the rectangular resolvent argument of the preceding paragraph again proves the full compact connection error and its adjoint. Positivity is again zero. Thus the transformed bounded operator is an actual product cycle.

Its principal symbol is the physical Clifford symbol on \(\mathcal D_E\). Confine input and output to compact interior buffers. The flat fibre transform has the corresponding Fourier order-zero symbol there; its off-diagonal tails are module compact by the heat-kernel estimate in Lemma 7.14, uniformly on compact base supports. The exact finite-supported quotient and quantization-choice argument of Lemma 7.11 therefore identify its bounded transform with that lemma's analytical phase. All frame operators differentiate only in the fibre variable, so transitions (295) intertwine them exactly. Positive metric transport, with compact-base bounds, is the transport of Corollary 6.0d. This proves (294) over possibly noncompact \(X\).

A smooth metric family uses the bundle over \(X\times[0,1]\), the same finite tensors, and uniform estimates on each compact base set times the interval. It gives one interval cycle with the prescribed endpoint modules. No global geometric bound at base infinity is needed. In rank zero the exterior and Clifford factors are scalar and the operator is zero. Formula (293) is the conjugate of the full graded orientation line; multiplication by \(C_0(X)\) is compact on its section module. This proves the rank-zero assertion, including an odd stable line. \(\square\)

The theorem supplies the inverse's actual analytical orientation. For the ordinary graph of Lemma 7.15, the next step is to compare \(\mathcal D_N\widehat\otimes\mathcal S_K\), using the full recovery maps (174)–(178), with the correspondingly transferred ambient Euclidean inverse. The same ordered Clifford contractions must be applied on both sides before using the two Thom inverse identities.

![Full grading and phase in the inverse coefficient transfer](../figures/normal-inverse-transfer.png)

**Figure 7.5.** Lemma 7.16, Theorem 7.17 and Exercises 58–59, equations (286)–(297). The old source Clifford block precedes the external block. The even exterior/external vacuum corresponds to a corner of parity \(k_r=r(r-1)/2\) in the standard ordered matrix module. Its remaining right module is \(\Pi^{k_r}C_r\), with physical generators \((-1)^{k_r+1}L_{t_i}\). The table gives the exact values in ranks one through four; the proof holds in every rank. The transition \(zs\) becomes \(z^{-1}\rho_t(s)\), and an even line becomes its full dual. The lower panel records the actual Hilbert-module product and bounded-phase comparison, with no claim that stable graph-normal recovery is finished. This original diagram illustrates the independently supplied calculation; classical Clifford Bott–Dirac context is [Kasparov; Emerson, Section 11.4].

Full-size transfer PNG · Editable transfer SVG.

### Recovering the prescribed graph normal

We keep the full stable orientation in (13). For a submersion it induces a vertical orientation by cancelling a canonical complex double. The same finite evaluations then identify the normal of its graph. This calculation precedes the KK cancellation; it does not choose a normal structure independently of the map's supplied orientation.

Let \(f:W\to V\) be a smooth surjective submersion with \(W\) compact. Put \(K=\ker df\), \(r=\operatorname{rank}K\), \(T=f^*TV\), and \(p=\operatorname{rank}T\). Choose an embedding \(u:W\hookrightarrow\mathbb R^n\), with \(n\) even and \(q=n-r\geq1\). Use its induced metric on \(TW\) and choose \(H=K^\perp\). The map \(df:H\to T\) is an isomorphism. Give \(T\) the transported metric over \(W\); positive metric transport of Corollary 6.0d carries the originally prescribed module to these metrics. It need not be a metric pulled back from all of \(V\).

Use the finite algebra-bundle notation \(A_B=\operatorname{Cl}^+(B)\) of (170). Write \(Q_B=B\oplus B\) for the canonical complex double of a real bundle \(B\), and \(\mathscr M_{Q_B}\) for (172). Its orientation is interleaved and its full module exists also when \(B\) is nonorientable. Write \(\mathcal C_{Q_B}\) for the actual cancellation (173).

**Lemma 7.18 (vertical and graph-normal recovery).** Let \(\mathscr M_L\) be the prescribed full module on \(L=TW\oplus T\). The ordered isometry \(\Psi:K\oplus Q_H\to L\), \(\Psi(k,h_1,h_2)=(k+h_1,df(h_2))\), induces

\[
 \mathscr M_K=\mathcal C_{Q_H}(\Psi^*\mathscr M_L),
 \qquad
 \mathscr M_K\widehat\otimes\mathscr M_{Q_H}
       \cong\Psi^*\mathscr M_L.
\tag{298}
\]

Here the right blocks are \(C_r,C_{2p}\). In positive rank this is the induced full vertical spin-c structure. At rank zero retain its entire graded orientation line.

Let \(E=\nu_u\oplus T\), with its graph-normal structure \(\mathscr M_E\) from (178). The quotient identification with the normal \(N=(duK)^\perp\) of Lemma 7.15 is

\[
 \chi:E\longrightarrow N,\qquad
 \chi(\eta,t)=\eta-du\bigl((df|_H)^{-1}t\bigr).
\tag{299}
\]

Under \(\chi\), its full module is the cancellation using only the vertical double:

\[
 \begin{aligned}
 J:K\oplus N&\longrightarrow\underline{\mathbb R^n},
       &J(k,v)&=du(k)+v,\\
 \eta_K:N\oplus Q_K&\longrightarrow K\oplus\underline{\mathbb R^n},
       &\eta_K(v,a,b)&=(a,J(b,v)),\\
 P_K&=\eta_K^*(\mathscr M_K\widehat\otimes C_n),
       &P_K&:A_N\widehat\otimes A_{Q_K}
                    \rightsquigarrow C_q\widehat\otimes C_{2r},\\
 \mathscr M_N&=\mathcal C_{Q_K}(P_K),
       &\chi_*\mathscr M_E&\cong\mathscr M_N .
 \end{aligned}
\tag{300}
\]

The right identification in \(P_K\) repartitions the same ordered list of \(r+n=q+2r\) generators. The assertion is an actual even bimodule unitary, with both actions and inner products, and holds over smooth parameter families. Tensoring \(\mathscr M_L\) by an even line tensors both \(\mathscr M_K\) and \(\mathscr M_N\) by that same line.

**Proof: the induced vertical structure.** The two evaluations and full recovery (174), applied to \(Q_H\), prove (298), including smoothness, fullness and onto even isometries. This uses the given \(\mathscr M_L\), rather than replacing it by a module made from a separately chosen vertical structure. The reconstruction before Lemma 6.0j gives its positive-rank principal structure; the rank-zero line and its grading are retained.

The embedding metric makes \(du(H)\), \(du(K)\) and \(\nu_u\) mutually orthogonal. Hence \(\chi\) is an isometry onto \(\nu_u\oplus du(H)=N\). In the graph quotient, subtracting \(d(f,u)((df|_H)^{-1}t)\) from \((t,\eta)\) leaves \((0,\chi(\eta,t))\). Conversely this decomposition is unique modulo \(du(K)\). Thus (299) is the actual quotient map and not merely an equality of bundle ranks.

**Proof: the global rotation.** Decompose both copies of \(TW\) as \(K\oplus H\). In \(E\oplus Q_{TW}\) write the coordinates \((\eta,t,a_K,a_H,b_K,b_H)\), identifying \(t\) with a vector of \(H\). On the pair \((t,b_H)\) use

\[
 R_\theta(t,b_H)=
       (\cos\theta\,t+\sin\theta\,b_H,
        -\sin\theta\,t+\cos\theta\,b_H),
 \qquad 0\leq\theta\leq\pi/2.
\tag{301}
\]

All other coordinates stay fixed. This is a smooth global orthogonal rotation even if \(H\) is nonorientable. In a local orthonormal frame its spin unitary is \(\exp((\theta/2)\sum_j\gamma_{t_j}\gamma_{b_j})\). The sum is invariant under every simultaneous orthogonal frame change of the two \(H\) copies, because the coefficient is their metric contraction. Its bivectors are skew-adjoint, and their commutators with the real generators give exactly (301). Thus it is a globally defined even unitary implementing the entire rotation.

Put \(P_\theta=(\rho_uR_\theta)^*(\mathscr M_L\widehat\otimes C_n)\), with \(\rho_u\) from (176). If the rotation unitary is \(s_\theta\), multiplication by the represented \(s_\theta^*\) is an even right-linear unitary from \(P_\theta\) to \(P_0\), intertwining their left actions. It preserves their inner products and is onto. Apply \(\mathcal C_{Q_{TW}}\) to this actual unitary; it gives an actual unitary of the cancelled normal modules throughout the family.

At \(\theta=\pi/2\), the vector map is

\[
 \begin{aligned}
 (\rho_uR_{\pi/2})(\eta,t,a_K,a_H,b_K,b_H)
   &=\bigl(a_K,a_H,b_H,\ du(b_K)-du(t)+\eta\bigr),\\
 P_{\pi/2}
   &\cong P_K\widehat\otimes\mathscr M_{Q_H}.
 \end{aligned}
\tag{302}
\]

In the first line the initial three entries are the \(K,H,T\) decomposition of \(L\); the last is the ambient Euclidean vector. The source is now \(N,Q_K,Q_H\), using \(\chi\); the second line uses the exact recovery (298). Its source real coordinates are \((\chi(\eta,t),a_K,b_K,a_H,b_H)\).

We spell out the constant right transport in that second line. Before it, the blocks are \(C_r,C_{2p},C_n\); afterwards they are \(C_r,C_n,C_{2p}\), repartitioned as \(C_q,C_{2r},C_{2p}\). Move only the two even-dimensional middle/last blocks. On their standard consecutive-pair spinor model, the exterior unitary permuting the \(p\) and \(n/2\) complex coordinate blocks is even and implements this Clifford isomorphism. It is an inner even automorphism. The right-module map \(v\mapsto v w\), with that fixed implementing unitary \(w\), gives its full inner-product-preserving transport, as in the stabilization proof of Lemma 6.0j. This identifies the entire right algebra before any normal-prefix cancellation; it does not assume that the moved blocks preserve that prefix.

The wedge map \(\Lambda(K_{\mathbb C}\oplus H_{\mathbb C})\cong\Lambda K_{\mathbb C}\widehat\otimes\Lambda H_{\mathbb C}\) identifies the full \(TW\) double with \(Q_K,Q_H\). It intertwines exterior multiplication and contraction; its graded tensor inserts the sign on the \(H\) block exactly once. The same map identifies the fixed right matrix model with the consecutive complex blocks \(C_{2r},C_{2p}\). Now evaluate the \(H\)-double factors on both sides of (302), using (174); evaluate the remaining \(K\)-double factors by (173). These maps give

\[
 \begin{aligned}
 \chi_*\mathscr M_E
   &\cong \mathcal C_{Q_K}(P_K)=\mathscr M_N,\\
 \mathscr M_N\widehat\otimes\mathscr M_{Q_K}
   &\cong P_K .
 \end{aligned}
\tag{303}
\]

Every map in this chain is a reassociation, a graded flip, the stated fixed exterior unitary, the global rotation unitary, or an actual rank-one/inner-product evaluation. The tensor inner-product identity in (174) proves its isometry; fullness proves onto range. It retains both actions and the grading. An even line passes through all these tensors unchanged, proving the line assertion. Smooth families use the same rotation and evaluations on the parameter bundle. Metric changes use their positive transport; the finite construction needs bounds only on compact parameter/base sets. This proves all assertions. \(\square\)

**Lemma 7.19 (the full normal-inverse and vertical coefficient module).** Use the structures of Lemma 7.18. Let \(\mathcal D_N\) be (293) for \(\mathscr M_N\). Transport the right algebra of \(\mathcal D_N\widehat\otimes\mathscr M_K\) by the graded block flip \(C_q\widehat\otimes C_r\to C_r\widehat\otimes C_q=C_n\). With its physical normal and vertical Clifford actions, it is an actual even full-module unitary copy of the ambient transferred-inverse module:

\[
 (\mathcal D_N\widehat\otimes\mathscr M_K)^\beta
       \cong J^*\mathcal D_{\underline{\mathbb R^n}},
 \qquad
 \beta(x\widehat\otimes y)=(-1)^{|x||y|}y\widehat\otimes x .
\tag{304}
\]

The physical source order on the left is \(N,K\); \(J\) sends it to \(du(K)+N\) in the ambient space. The ambient module on the right is the full module of Theorem 7.17, including its grading. The unitary retains the prescribed scalar phases and real-frame transitions. In rank zero it retains the complete vertical orientation line. The identification also transports the whole graph-tube orientation of Lemma 7.15, using its actual vertical derivative and positive metric transport.

**Proof: the full constant normal model.** Work first with \(r>0\) and an adapted oriented \(K,N\) frame whose ordered \(K,N\) image is a positive Euclidean frame. Put \(k_j=j(j-1)/2\bmod2\). The constant model obtained by cancelling the full double in \(P_K\) is

\[
 \begin{aligned}
 \epsilon&=r\bmod2,
       &a&=(1-\epsilon)k_r,\\
 \mathscr M_N^0&\cong\Pi^{a}C_q,
       &c_N(v_i)&=\kappa L_{t_i},\qquad
                  \kappa=(-1)^{\epsilon k_r}.
 \end{aligned}
\tag{305}
\]

Here \(a=k_r,\kappa=1\) if \(r,q\) are even, and \(a=0,\kappa=(-1)^{k_r}\) if they are odd. To check the model on full modules, compare the normalized total volume before both finite evaluations. The \(Q_K\) complex interleaving contributes \((-1)^{k_r}\) relative to its two real \(K\) blocks. Moving the \(N\) block past both \(K\) blocks contributes \(2rq\) swaps and sign \(+1\). All fixed right generators stay in their standard list order. Consequently, in even \(q\), the source normal chirality times right multiplication by the target chirality is \((-1)^{k_r}\) times the module grading. In odd \(q\), the source normalized odd central volume is \((-1)^{k_r}\) times right multiplication by the target odd central volume. The double evaluations give these equalities on the whole cancelled module.

They are precisely the corresponding equalities in (305). The complete finite matrix-unit reconstruction before Lemma 6.0j now supplies an even right-linear unitary intertwining the entire compact left algebra, not just its volume: in even rank recover each homogeneous column from a matrix corner; in odd rank recover the two matrix corners exchanged by grading. Their inner products and actions give an onto isometry. Fix one such constant unitary once for these ordered models. This local use of the full finite reconstruction does not discard any bundle transition line.

We also need the actual transfer of this shifted or reflected model. For a model \(\Pi^a C_q\) with left real generators \(\kappa t_i\), the raw inverse/exterior module has grading \((-1)^a\Gamma\), old source generators \(\kappa a_i\), external generators \((-1)^a\Gamma\otimes t_i\), and physical generators \((-1)^a b_i\). Its vacuum is \((\Gamma^{a+(1-\kappa)/2}\otimes1)\psi\). Substitution in the two annihilator equations of Lemma 7.16 proves that assertion, with both index cases. Its total parity is \(a\), and the physical corner action is \(-\kappa t_i\). The matrix-column tensor sign is still \((-1)^{k_q}\). Therefore

\[
 \mathcal D_N^0\cong\Pi^{h}C_q,\qquad
 h=k_q+a,\qquad
 c_N^{\mathcal D}=\kappa(-1)^{k_q+1}L_{t_i}.
\tag{306}
\]

This verifies the action and grading for the actual balanced tensor, including a shifted normal module.

**Proof: the complete constant intertwiner.** Tensor (306) with the regular \(C_r\) vertical module and perform the full right transport \(\beta\). The result is \(\Pi^h C_n\). Its physical \(K\) action has the additional first-module tensor grading sign, while its normal action keeps (306):

\[
 c_K=(-1)^hL_{e_K},\qquad
 c_N=\kappa(-1)^{k_q+1}L_{e_N}.
\tag{307}
\]

The ambient module is \(\Pi^{k_n}C_n\), with every physical generator \((-1)^{k_n+1}e_i\). For an ordered \(j\)-generator block put \(\omega_j=(-i)^{j(j-1)/2}e_1\cdots e_j\), with \(\omega_0=1\). These are unitary; their conjugations negate a generator in that block by \((-1)^{j-1}\), and one outside it by \((-1)^j\). In the target order \(K,N\), the desired map is

\[
 U:\Pi^{k_n}C_n\longrightarrow\Pi^h C_n,
 \qquad U(c)=\omega_K\,\omega_N^{\,k_r}c,
 \qquad |\,\omega_K\omega_N^{\,k_r}\,|=k_n+h .
\tag{308}
\]

It is even between these two module gradings. Indeed \(q,r\) have the same parity, \(k_n=k_q+k_r+rq\), and these identities give the last equality. It is right-linear, inner-product preserving and onto, since its multiplier is unitary. Conjugation by it changes the \(K\) block by \((-1)^{r-1+k_rq}\), and the \(N\) block by \((-1)^{r+k_r(q-1)}\). Multiplying the ambient sign by these factors gives exactly (307), in both parity cases of (305). Thus it intertwines every physical generator and the entire left Clifford algebra, not merely a determinant or index.

**Proof: the full transitions.** Choose local principal frames for \(\mathscr M_K\), and local spin frames of the canonical ambient Euclidean module in the adapted \(K,N\) frames. Write their transitions as \(z s_K\) and \(s_A\). Since the real ambient frame change is the two-block change, its lift has the form \(s_A=s_Ks_N\) for a uniquely chosen normal lift \(s_N\), with the two choices of the tangent lift absorbed in its scalar phase.

This is the exact lift chosen by recovery (303). On its full \(P_K\) module the transition is \(z s_K\widehat\otimes s_A\). The canonical double transition is \(\rho_{K_1}(s_K)\rho_{K_2}(s_K)=\Lambda R_K\), by the plane-rotation identity in (295). Its two evaluations cancel precisely these two even factors, leaving \(z s_N\) on the normal model (305). Thus the transferred normal inverse, tensored with the vertical module, has

\[
 (z^{-1}\rho_N(s_N))(z\rho_K(s_K))
       =\rho(s_A).
\tag{309}
\]

The two even blocks commute; all tensor flips are the specified graded ones. The fixed unitary (308) commutes with both block spin actions, since each block volume commutes with its even words. Consequently the local unitaries glue to (304). Changing \((z,s_K)\) to \((-z,-s_K)\) changes \(s_N\) to \(-s_N\); neither full module transition changes. This also checks the central signs in rank-one spin groups. A line transition multiplies the two phases in (309) by its inverse and itself. The actual line evaluation therefore cancels it, including a two-torsion line; its square or determinant has not been used.

If \(r=0\), write the induced full orientation line as \(\Pi^\epsilon H_0\). Recovery makes the normal module the canonical \(C_n\) module tensored by that same graded line. Its transferred inverse contains \(H_0^*\) with the same parity. The full line evaluation has degree zero. Tensoring the second, parity-\(\epsilon\) line twists the old right Clifford action by its parity automorphism. Untwist this action on the Clifford words; the physical generators acquire \((-1)^\epsilon\). Multiplication by the even ambient chirality to the power \(\epsilon\) restores the ambient physical action. These maps are right-linear after the stated right transport, preserve the inner product and evaluate \(H_0^*\otimes H_0\) onto the scalar unit. This proves (304) also for the odd rank-zero line, without replacing it by the canonical even line.

**Proof: the entire graph tube.** Choose a metric connection on \(N\); it splits the vertical tangent of \(f\circ\operatorname{pr}_N\) as the range pullback of \(N\oplus K\). At the zero section the vertical derivative of the fibre-preserving \(\Phi\) in Lemma 7.15 is exactly \(J\), since its compression has derivative one there. On the whole bundle its vertical derivative is an invertible map to the ambient \(\mathbb R^n\). Compose it with \(J^{-1}\), and let \(R(w,v)\) be its positive polar orthogonal factor. It has determinant \(+1\): at \(v=0\) it is the identity, and the radial segment stays in the invertible matrices. The path \(R_s(w,v)=R(w,sv)\) therefore has a global even spin lift. Explicitly,

\[
 \begin{aligned}
 B_s&=(\partial_sR_s)R_s^{-1},\\
 \partial_s u_s&=\frac14\sum_{i,j}(B_s)_{ij}e_i e_j\,u_s,
       \qquad u_0=1 .
 \end{aligned}
\tag{310}
\]

The skew matrix makes the displayed generator skew-adjoint. Its commutator with a real Clifford generator is the corresponding column of \(B_s\), so \(u_s\) implements \(R_s\). Finite-dimensional linear ODE existence and the unitary norm bound give the solution on the whole interval; smooth dependence is local on compact parameter/base sets. Products of bivector exponentials, equivalently plane-spin rotations, converge to this lift, so it lies in the spin group. Starting at the identity fixes the full phase globally.

Transport (304) radially and then by \(u_1\), retaining the positive polar metric isometry. This identifies the full tube orientation with the pullback of the ambient inverse orientation. The exact fibre-density unitary is that of Lemma 7.15. Lemma 7.11 then supplies the same bounded analytical phase under these metric/connection transports, on a single interval module with compact localized defects. No completeness or global bound at fibre infinity is imposed. This proves the asserted entire-tube identification. \(\square\)

These lemmas close the full prescribed normal recovery and its analytical coefficient-module comparison. The next lemmas verify the native parity reductions with the second-cycle coefficient first, using both raw Thom inverse identities. The full transport in (304) is retained in the compact ordinary equality of Theorem 7.22.

![The full prescribed graph-normal recovery](../figures/graph-normal-recovery.png)

**Figure 7.6.** Lemmas 7.18–7.19 and Exercises 60–61, equations (298)–(312). The first panel gives the exact clockwise horizontal-double rotation, with endpoint \((t,b_H)\mapsto(b_H,-t)\); it explains the minus sign in the actual normal quotient \(\chi\). The middle panels show the two full double evaluations, the fixed even-block transport, the signed block-volume unitary and the exact inverse/vertical line evaluation. The last panel extends the identification over the whole graph tube by its positive polar factor and identity-started spin lift. The displayed examples distinguish an odd Clifford multiplier from an even map between oppositely graded modules. Lemmas7.20–7.21 and Theorem7.22 prove the native ordered cancellation below. This is an original diagram of the complete proofs; classical spin-c and Clifford context is [Kasparov; Emerson, Section 11.4; Connes–Skandalis].

Full-size recovery PNG · Editable recovery SVG.

### Native ordered Thom cancellation and ordinary submersions

We now calibrate the specified transfer (286) against both raw inverse products. This also distinguishes the canonical positive analytical module from the module of the geometric Euclidean inverse. No equality with the separately constructed real-action Bott pair of Corollary 0.20 is assumed here.

For a trivially graded algebra \(D\), write \(r=2a+\epsilon\), \(\epsilon\in\{0,1\}\). Let \(\mu_r^D\) be the fixed \(D\widehat\otimes C_r,D\widehat\otimes C_\epsilon\) Morita class obtained by reducing the first \(2a\) consecutive generators with \(\Sigma_a=\Lambda^*\mathbb C^a\), retaining the final generator if \(\epsilon=1\). Write \(R_r(z)=z\mu_r^D\) for right reduction. The even cyclic transports in Proposition 6.0i identify this convention with a retained odd generator at the front; those are full even module maps.

**Lemma 7.20 (both inverse products after the specified transfer).** Let \(A,B\) be trivially graded separable algebras. Suppose
\[
 x\in KK(A,B\widehat\otimes C_r),\qquad
 y\in KK(B\widehat\otimes C_r,A),\qquad
 xy=1_A,\quad yx=1_{B\widehat\otimes C_r}.
\tag{313}
\]
Then
\[
 \begin{aligned}
 b&=R_r(x)\in KK^\epsilon(A,B),\\
 d&=R_r(\mathcal T_r(y))\in KK^\epsilon(B,A),\\
 b\cdot d&=1_A,\qquad d\cdot b=1_B .
 \end{aligned}
\tag{314}
\]
The dots are exactly the native parity products of Lemma 0.13. In an odd–odd product the target list is the second-cycle coefficient followed by the externally retained first-cycle coefficient. No graded flip to the chronological order is implicit.

**Proof: even rank.** If \(r=2a\), consecutive pairing gives the actual even exterior unitary \(\Sigma_r\cong\Sigma_a\widehat\otimes\Sigma_a\), on the source block followed by the external block. Reassociate the balanced tensor defining \(R_r\mathcal T_r(y)\). Evaluate the external \(\Sigma_a^*,\Sigma_a\) factors by the full inner-product map of Lemma 0.12. It leaves exactly
\(d=(\mu_r^B)^{-1}y\).
Thus \(bd=x\mu_r^B(\mu_r^B)^{-1}y=xy\) and
\(db=(\mu_r^B)^{-1}yx\mu_r^B=1_B\).
The evaluations preserve the grading, actions and tensor inner product and are onto by fullness. Rank zero is the same statement with the unit Morita module.

**Proof: reducing the even factors in odd rank.** Suppose \(r=2a+1\). Put
\[
 x'=x\mu_r^B,\qquad
 y'=(\mu_r^B)^{-1}y.
 \qquad
 x'y'=1_A,\quad y'x'=1_{B\widehat\otimes C_1}.
\tag{315}
\]
The full source list in \(\mathcal T_r(y)\) is
\((C_{2a}^{\mathrm{old}},C_1^{\mathrm{old}},
  C_{2a}^{\mathrm{new}},C_1^{\mathrm{new}})\).
Move the old odd generator past the \(2a\) new even-block generators. This is a real permutation of positive sign, with an even spin implementer; every individual odd-vector interchange is still accounted for. One explicit construction writes the permutation as \(2a\) adjacent reflections. The reflection exchanging two neighbouring generators is the twisted adjoint of \((e_i-e_{i+1})/\sqrt2\); the product of the even number of such unit vectors has ordinary adjoint equal to the desired permutation. It is unitary and even. It preserves the normalized total volume and hence the prescribed spinor grading.

After this actual transport the consecutive-pair module is
\(\Sigma_a^{\mathrm{old}}\widehat\otimes
 \Sigma_a^{\mathrm{new}}\widehat\otimes
 \Sigma_1^{\mathrm{odd,odd}}\).
The ordered wedge map intertwines every creation operator and its adjoint, including the preceding-factor grading. Its tensor inner product is the wedge inner product. Reassociation and evaluation of the new even factors therefore give the full finite reduction identity
\[
 R_r\mathcal T_r(y)
   =\mathcal T_1(y')
   =[1_B\widehat\otimes\Sigma_1^*]\,
                  \tau_{C_1}(y').
\tag{316}
\]
All constant module maps have been specified before reduction; the remaining pair is old odd source followed by new odd external.

**Proof: the two native odd products.** Let
\(m_B=[1_B\widehat\otimes\Sigma_1]\).
Set \(j=m_B^{-1}\tau_{C_1}(y')\), the last expression in (316). Native multiplication and (315) give
\[
 j\cdot x'
   =m_B^{-1}\tau_{C_1}(y')\tau_{C_1}(x')m_B
   =m_B^{-1}\tau_{C_1}(y'x')m_B
   =1_B.
\tag{317}
\]
In this expression the ordered final pair is the coefficient of \(x'\) followed by the external coefficient of \(j\), as required by the native product.

We prove the other product as well. As an ordinary even morphism of graded algebras, \(j:B\to A\widehat\otimes C_1\) is invertible: its inverse is \(\tau_{C_1}(x')m_B\), and (315) plus the two Morita evaluations prove both ordinary products. Put \(e=x'\cdot j\in KK(A,A)\). Native associativity, proved with the full three-generator cyclic transport in Lemma 0.13, gives \(j\cdot e=(j\cdot x')\cdot j=j\). Since \(e\) is even, this says \(j\tau_{C_1}(e)=j\). Multiply by the just proved ordinary inverse of \(j\); it gives \(\tau_{C_1}(e)=1_{A\widehat\otimes C_1}\). The injectivity of \(\tau_{C_1}\), proved by both simultaneous matrix-Morita products in Lemma 0.13, gives \(e=1_A\). This proves both native identities, not only one inverse or an index on a Gaussian. Equations (315)–(316) identify \(x',j\) with \(b,d\), completing the proof. \(\square\)

In particular, Corollary 6.0d's full raw pair \(\beta_E,\alpha_E\) satisfies this lemma. Define \(b_E=R_r(\beta_E)\) and \(d_E=R_r\mathcal T_r(\alpha_E)\) in parity notation in the rest of this subsection; these letters here denote the spin-c reduced pair, rather than the unreduced Clifford-twisted pair of (130)–(131). Both identities hold with every supplied line twist and rank-zero grading, since (313) retains their exact full Morita evaluations.

**Lemma 7.21 (the native product and the geometric Euclidean inverse).** Suppose \(q+r=n\) is even. If a first full analytical cycle has right \(C_q\) and a second has right \(C_r\), native parity multiplication of their reductions is the reduction of their actual full product with right list \(C_r,C_q\). Equivalently, if their common physical coefficient bundle is initially written \(\mathcal E_q\widehat\otimes\mathcal E_r\) with right list \(C_q,C_r\), first apply the full graded right block transport \(\beta\), and then \(R_n\):
\[
 R_q(z_q)\cdot R_r(z_r)
       =R_n\bigl((z_q\widehat\otimes z_r)^\beta\bigr).
\tag{318}
\]
Here the tensor on the right denotes the already verified analytical product operator on its common module, rather than an unverified choice of operator.

**Proof: the complete finite reducers.** For even \(q,r\), the target consecutive-pair spinor is the ordered exterior tensor for \(C_r,C_q\). Both sides reduce by this same module, using the two full Morita evaluations and their associating unitaries. For odd \(r=2a+1,q=2b+1\), the full native list is
\((C_{2a},C_1^{r},C_{2b},C_1^{q})\).
Move \(C_1^r\) past \(C_{2b}\) with the even product of adjacent reflection units used above. The list is then the two even matrix blocks followed by the ordered pair \((C_1^r,C_1^q)\). The wedge unitary decomposes the full reducer as
\(\Sigma_a\widehat\otimes\Sigma_b\widehat\otimes\Sigma_1\).
Its last module is exactly (104), in second-cycle-first order. Every action and inner product agrees on homogeneous elementary tensors by Lemma 0.13's formulas; their density and the two onto Morita evaluations give the actual even module unitary. Product preservation under these finite external factors and uniqueness of the checked product operator establish (318). A right transport of an odd word still contributes its Koszul sign. Thus this proof does not replace \(\beta\) by an ordinary block permutation. \(\square\)

For the canonical Euclidean spin-c structure and even \(n\), let \(B_n=R_n(\beta_{\mathbb R^n})\), and let \(A_n=B_n^{-1}\). These are the geometric pair used by the Euclidean stabilization of Lemma 6.0h: its proof uses precisely the standard spin-c symbol of Corollary 6.0d. Lemma 7.20 and Theorem 7.17 identify its actual inverse as
\[
 \begin{aligned}
 A_n&=R_n\mathcal T_n(\alpha_{\mathbb R^n})
        =R_n(a_{\mathbb R^n}^{\mathcal D_n}),\\
 \mathcal D_n&=\Pi^{k_n}C_n,\qquad
       c_{\mathcal D_n}=(-1)^{k_n+1}L_e,\\
 W_n(c)&=\chi_n^{\,k_n+1}c,\qquad
       W_n:\Pi^{k_n}C_n^{\mathrm{canonical}}\longrightarrow\mathcal D_n,\\
 \chi_n&=(-i)^{n/2}e_1\cdots e_n .
 \end{aligned}
\tag{319}
\]
The finite map \(W_n\) is even, right-linear, unitary and intertwines every physical generator. Indeed \(\chi_n\) is an even self-adjoint unitary that anticommutes with each \(e_i\), so its stated power gives exactly the displayed physical sign. It preserves the shifted grading. For \(n=0\) use the unit instead.

This also shows precisely why one cannot silently equate the unreduced full canonical positive module with \(\mathcal D_n\). After right reduction, \(A_n\) is \((-1)^{k_n}\) times the canonical positive Euclidean analytical cycle. For trivially graded left coefficients, the grading-reversed cycle with the same operator represents the negative class: Section 0's explicit inverse cycle has reversed grading and operator \(-F\), and conjugation by the grading operator changes it to \(F\). Thus this sign follows from an actual cycle unitary and the proved inverse, not from an index normalization.

Externally over a trivially graded algebra \(D\), the same statement identifies \(A_n^D\) with the product Euclidean vertical phase. The tensor property in Lemma 0.13, the full localized compact defects in Theorem 7.17, and the constant finite map \(W_n\) supply the actual cycle. Applied over \(D=C(V)\), Lemmas 7.14–7.15 give
\[
 [e_\Phi]A_n^{C(V)}
      =R_n\bigl(a_{f\circ\operatorname{pr}_N}^{\mathcal D_n,
                                      \mathrm{ambient}}\bigr).
\tag{320}
\]
To check this application on the whole tube, start with the canonical cycle in (282), reverse its grading \(k_n\) times, and conjugate its full finite coefficient by \(W_n\). The off-diagonal compact estimates, the entire block homotopy and density unitary are preserved by these operations. They prove (320) with the identical pulled-back derivative structure. The global polar spin lift of Lemma 7.19 identifies that structure with the ordered normal-inverse/vertical structure on the entire total space. No statement about completeness at fibre infinity is inserted.

**Theorem 7.22 (ordinary submersion equality with the supplied stable orientation).** Let \(f:W\to V\) be a smooth surjective submersion of compact manifolds. Supply the full stable orientation (13), and use the induced vertical module (298), including an entire rank-zero graded line if present. Its bounded analytical class equals the geometric graph class:
\[
 \begin{aligned}
 a_f&=f_!^{\mathrm{geom}}\quad
                  \text{in }KK^{r\bmod2}(C(W),C(V)),\\
 f_!^{\mathrm{geom}}&=b_N[e_\Phi]A_n^{C(V)} .
 \end{aligned}
\tag{321}
\]
The graph normal \(N\), its full module and the normalized whole tube are those of Lemmas 7.15 and 7.18. The equality retains the prescribed line twists, finite Hermitian coefficients and compact smooth parameter families with their supplied orientation families.

**Proof.** Lemma 7.18 identifies the full graph normal of Lemma 6.0r with \(\mathscr M_N\), through the actual quotient and both double evaluations. Lemma 6.0e's normalized-tube independence and Proposition 6.0i's even cyclic contraction identify (200) with the second line of (321). Its \(B_n,A_n\) are the geometric pair just calibrated, so a separately ordered real-action Bott normalization is not required.

The inverse class \(d_N\) is represented by the entire analytical fibre-integration cycle with \(\mathcal D_N\), by Theorem 7.17. We use the complete creation and positivity argument of Theorem 7.13 for the two ordinary fibre-integration modules, and specify that application. Send a compact elementary tensor to \(\xi(w,v)\widehat\otimes\eta(w)\). Fibrewise Fubini gives its exact \(C(V)\widehat\otimes C_n\)-valued inner product: first integrate along \(N_w\), then along \(f^{-1}(y)\). Compact separated-chart functions are dense on each compact rectangle; a finite partition on a compact section support and finite bundle frames give onto density. This is the unitary onto the whole integration module of \(f\circ\operatorname{pr}_N\), not a module over only connected components of a fibre. Disconnected fibres are included.

Its physical coefficient bundle is \(\mathcal D_N\widehat\otimes\mathscr M_K\), in order \(N,K\). The bounded total phase is constructed by Lemma 7.11 with its orthogonal sum principal symbol. For a compact smooth first-module creation section \(\xi\), only finitely many buffered charts occur on its range support and its image in \(W\). In those charts the zero first-frequency axis of the total symbol is the second phase, with the entire second bundle action. The four leading terms in the creation error's \(C^*C\) therefore cancel by (263). The exact finite-supported base symbol quotient and the \(C L(L+\delta)^{-1}\) argument of Lemma 7.4 prove that this creation error is compact; adjoints give the other error. Density handles every creation vector. This is precisely the full creation proof of Theorem 7.13, with ordinary fibre integration replacing its principal-space integration. Every integral has compact support, so no properness of \(\operatorname{pr}_N\) is used.

For positivity, take a compact smooth left coefficient \(h\) on \(N\). Move it only past the total phase, whose scalar commutator is compact. The resulting finite-supported bracket has the nonnegative orthogonal-sum symbol (254), using the chosen splitting \(N,K\). Lemma 7.9's vanishing-axis multiplication calculation and the exact supported quotient of Lemma 7.11 make its operator quotient positive, as in the complete proof of Theorem 7.13. Lifted vertical compacts are handled by the creation connections just established; they are not asserted to be total compact on their own. Coefficient approximation gives the criterion for every \(h\in C_0(N)\). These checks prove the actual ordinary analytical product, including zero ranks by the same finite local rank-one creations of Theorem 7.13.

The common right target becomes \(C_r,C_q\) by the graded transport in (304). Lemma 7.21 identifies this verified product's reduction with the native parity product \(d_N\cdot a_f\).

Lemma 7.19 gives the exact even unitary from this whole module and principal Clifford symbol to the ambient \(\mathcal D_n\) module, including its scalar phases, full line evaluation, grading, metric transport and global polar spin lift. Lemma 7.11 compares the resulting bounded phases with compact localized defects on the entire noncompact bundle. Equation (320) therefore gives
\(d_N\cdot a_f=[e_\Phi]A_n^{C(V)}\)
in the even group \(KK(C_0(N),C(V))\).

Multiply on the left by \(b_N\). Lemma 7.20 gives \(b_N\cdot d_N=1_{C(W)}\), and native associativity gives exactly (321). The other product \(d_N\cdot b_N=1_{C_0(N)}\) was also proved there; no one-sided index argument entered the cancellation.

A full even line twist survives in the given \(a_f\) and the normal Thom class; its exact dual in \(d_N\) cancels in the common ambient module by (309). At \(r=0\) the two full graded lines evaluate as in Lemma 7.19, so the same calculation retains the original line on \(a_f\). Finite Hermitian coefficients multiply this proved equality using the zero coefficient-module cycle of Theorem 6.0s: the tensor actions, compact localizations and complete creation connections are the ones verified in Theorem 7.13. For a compact smooth parameter family, choose one sufficiently large even Euclidean embedding of \(W\). The horizontal metrics, rotations, both double evaluations, finite reducers and graph tubes are then smooth on the compact parameter/base bundle; a uniform small tube exists by the compact inverse-function argument of Lemma 7.15. The polar spin ODE is smooth in that parameter. The products and compact phase comparisons occur on the single interval module of Theorems 7.13 and 7.17. Both raw inverse identities, the finite calibrations and the cancellation hold there, so evaluation gives the required endpoint equality without selecting unrelated orientations. This proves all assertions. \(\square\)

This completes the ordinary submersion comparison in the stated compact-source scope, in both parities. The prescribed transverse-normal leaf-map comparison is proved in Lemmas7.23–7.24 and Theorem7.25 below. Lemmas7.26–7.28 and Theorem7.29 below prove the all-symbol action in both K-degrees; the transverse proof uses its actual entire cocycle family.

![Native ordered Thom cancellation and the ordinary submersion equality](../figures/native-thom-cancellation.png)

**Figure 7.7.** Lemmas 7.20–7.21, Theorem 7.22 and Exercises 62–63, equations (313)–(323). Both raw Thom inverse products pass through the complete finite Morita reductions. The native final target places the second-cycle Clifford block first; when both ranks are odd, its remaining odd pair has that same order. The even ambient inverse has full module \(\Pi^{k_n}C_n\), with \(k_n=n(n-1)/2\bmod2\), and its explicit chirality unitary gives the exact analytical normalization. The whole-tube product then cancels by the native inverse identities, retaining the prescribed line and grading. The diagram states the compact ordinary scope; Theorem7.25 proves the full prescribed transverse-normal comparison, and Theorem7.29 proves the arbitrary-symbol action in both K-degrees. This is an original illustration of the independently supplied complete argument; classical Clifford and longitudinal context is [Kasparov; Emerson, Section 11.4; Connes–Skandalis].

Full-size cancellation PNG · Editable cancellation SVG.

### The full prescribed transverse normal comparison

The normal construction and bounded product method are classical [Connes–Skandalis, Sections 1 and 4]. We give the actual cocycle family, its entire prescribed orientation, one completed interval module, and both bounded endpoint identifications. The comparison below is an equality of KK classes in both parities. The odd coefficient action is proved separately in Lemmas7.26–7.28 and Theorem7.29 below.

**Lemma 7.23 (the full transverse normal geometry and orientation family).** The compact-source normal tube of Lemma6.0m admits a family of transversely submersive cocycles on its entire normal bundle, from the projected leaf map to the Euclidean-forgetful étale tube. The prescribed stable orientation induces a full Clifford module family from the native normal-inverse/tangent product to the specified ambient inverse module. All actual arrows, complete line twists, grading, inner products and compact smooth parameters are retained.

Let \(V\) be compact, \(F\subset TV\) a foliation of rank \(p\), and
\(T=\tau=TV/F\), of rank \(t\). Choose an even embedding \(u:V\to\mathbb R^n\), with \(n>p\). Use its induced metric, \(H=F^\perp\), and \(H\cong T\) with the quotient metric. Put \(N=(duF)^\perp\), of rank \(q=n-p\). The transverse bundle need not be orientable.

**Proof: full normal recovery.**
Prescribe the full stable module on \(L=TV\oplus T\), and recover
\(\mathscr M_F=\mathcal C_{Q_H}(\Psi^*\mathscr M_L)\), where
\(\Psi(k,h_1,h_2)=(k+h_1,[h_2])\).
Use this recovered structure for the analytical foliation cycle; do not insert an independent tangent structure.
The normal of the section of Lemma6.0m is \(E=\nu_u\oplus T\).
Its actual quotient identification and full recovered module are
\[
 \chi(\eta,t)=\eta-du(Ht)\in N,\qquad
 \chi_*\mathscr M_E=\mathscr M_N
      =\mathcal C_{Q_F}\bigl(\eta_F^*(\mathscr M_F\widehat\otimes C_n)\bigr),
 \quad
 \eta_F(v,a,b)=(a,du(b)+v).
 \tag{324}
\]
All right lists, finite actions, inner products and gradings are those of (298)–(303), with \(F,H,T\) in place of \(K,H,f^*TV\).

Indeed the transverse derivative of every local cocycle map is the quotient \(TV\to T\); its kernel is exactly \(F\). Lemma6.0j's construction is global on \(TV,T\) and the Euclidean complement, using their actual bundle transitions, rather than a global target manifold. The full rotation \((t,b_H)\mapsto(b_H,-t)\) of Lemma7.18 is the metric contraction of the two \(H\) copies, so is global also here. Its endpoint is the same ordered \((F,H,T,\mathbb R^n)\) vector identity. The fixed even right-block implementer, wedge identification and both double evaluations are unchanged. They prove (324) on the full bundle. Thus nontrivial transverse holonomy has not been replaced by trivial target transitions. Its two \(H\) factors cancel through the actual module, not merely through their determinant.

The same complete finite calculation and transition proof of Lemma7.19 give
\[
 (\mathcal D_N\widehat\otimes\mathscr M_F)^\beta
       \cong J^*\mathcal D_n,\qquad
 J(v,k)=du(k)+v,\qquad
 \beta:C_q\widehat\otimes C_p\longrightarrow C_p\widehat\otimes C_q=C_n.
 \tag{325}
\]
Here the real source is normal first, tangent second, whereas the right target after the graded flip is tangent first, normal second. The signed full unitary (308) and all scalar transitions (309) apply on each ordered adapted frame and glue. In particular the normal phase and tangent phase are the identical original \(z\), their inverse/tangent evaluation is \(z^{-1}z=1\), and the resulting ambient spin module is the specified \(\Pi^{k_n}C_n\) with its exact physical sign. A two-torsion line is retained until this actual evaluation. Rank-zero tangent orientation retains its entire graded line. This gives a global full orientation identification, not merely a fibrewise index.

**Proof: the entire cocycle family.**
Let \(Z\) be the actual étale neighbourhood \(N_u\) of Lemma6.0m, with compact section \(k:V\to Z\), and let \(P:Z\to V\) be its smooth base projection. Its vertical tangent at the section is \(E=\nu_u\oplus T\). Choose a tube preserving \(P\),
\(\vartheta:D_\rho N\to Z\), normalized by the identification \(\chi^{-1}\).

Such a tube follows from the complete vertical-spray construction in the parameter proof of Theorem6.0o, now with \(P\) as the submersion. In a \(P\)-chart use the spray with base point fixed and vertical velocity constant; patch vertical sprays by a smooth partition. The resulting spray still keeps \(P\) fixed and has acceleration quadratic in its vertical velocity. Its time-one local addition has the identity vertical derivative at zero. Restrict it to the vertical bundle at \(k(V)\), using \(\chi^{-1}\). The compact inverse-function and compact-sequence injectivity arguments give a single small disk on which it is an open embedding. This is the actual normalized fibre-preserving tube; it does not average local tube diffeomorphisms.

Write \(g:Z\to V/F\times\mathbb R^n\) for the full étale cocycle, with all arrow data. Let \(g^V\) forget only the Euclidean coordinate. With the normalized compression \(q_\rho\), define
\[
 h_s(w,v)=g^V\bigl(\vartheta(w,sq_\rho(v))\bigr),
       \qquad 0\le s\le1,\quad (w,v)\in N .
 \tag{326}
\]
This is notation for pullback of the entire map cocycle. Its chart arrows are the pullbacks of the actual arrows of Lemma6.0l, so satisfy the exact cocycle for every parameter. At \(s=0\), the plaque arrows of that lemma identify it with \(\ell\circ\operatorname{pr}_N\), including holonomy. At \(s=1\) it is the Euclidean-forgetful part of the actual étale tube map.

Shrink \(\rho\) once so every \(h_s\) is transversely submersive on the entire noncompact bundle. Here is the uniform argument. Use finitely many buffered compact base charts of \(V\), with transverse map \(y_a\). On \(H\), \(dy_a\) is an isomorphism onto the transverse coordinate space; its least singular value has a positive minimum on each compact buffer. Give \(N\) a metric connection. Its horizontal lifts satisfy \(\nabla q_\rho(v)=0\) when \(\nabla v=0\), since the norm is preserved. The difference between the transverse differential in such a horizontal direction at \(\vartheta(w,z)\) and that at \(z=0\) is \(O(|z|)\), uniformly on the finite compact disk/base charts. Since \(|sq_\rho(v)|\le\rho\), this difference is uniformly \(O(\rho)\), also at arbitrarily large \(v\). Make it less than half each least singular value. The restriction of the differential to the lifted \(H\) is then invertible for all \(s,w,v\). This proves transverse surjectivity everywhere, rather than on just the zero section.

Consequently
\(\mathcal F_s=\ker d_{\mathrm{trans}}h_s\subset TN\)
is a smooth rank-\(n\) family of integrable distributions. In each chart it is the kernel of the submersion to the transverse \(y_a\); changes of chart depend only on \(y_a\), so the kernels glue, with their induced actual arrow cocycle. At \(s=0\) the metric-connection splitting identifies it with vertical \(N\oplus\operatorname{pr}_N^*F\). At \(s=1\), differentiating the Euclidean coordinate of the full étale tube map gives a global isomorphism
\(e_1:\mathcal F_1\to\underline{\mathbb R^n}\).
It is an isomorphism because the full transverse map, Euclidean coordinate included, is a local diffeomorphism.

**Proof: the full orientation transport.**
At the zero section the family is explicit. If \(v\in N_w\) and
\(\chi^{-1}v=(\eta,t)\), then
\[
 L_s(v,k)=(k-sHt,v)\in\mathcal F_s|_V,\qquad
 e_1L_1(v,k)=du(k)-du(Ht)+\eta=J(v,k).
 \tag{327}
\]
The first component is the base tangent, and the second is the vertical normal tangent under the normalized tube. Indeed the transverse derivative there is \([h]+st\). The Euclidean derivative at the endpoint is \(du(h)+\eta\). These identities prove (327) with its exact sign.

Use (325) as the full orientation of \(\mathcal F_0\). It includes a specified even full unitary to the fixed ambient module \(\mathcal D_n\), so supplies an actual global reference frame, with its full grading and phase. We construct a smooth family retaining this reference and matching the ambient endpoint, not just compatible determinant lines.

Choose any smooth family of metrics on \(\mathcal F_s\) extending the original product metric at \(s=0\), and any metric connection in the interval direction. Parallel transport the initial global frame \(J^{-1}\) to obtain
\(\widetilde E_s:\underline{\mathbb R^n}\to\mathcal F_s\).
The finite-dimensional unitary ODE exists for the whole interval at each fixed \((w,v)\); its smooth coefficients are bounded on every compact parameter/base set. No bound at infinity is needed. On the zero section form the actual real matrix
\[
 T(s,w)=\widetilde E_s(w,0)^{-1}L_s(w)J(w)^{-1},\qquad
 E_s(w,v)=\widetilde E_s(w,v)T(s,w).
 \tag{328}
\]
The first matrix is invertible and starts at the identity, so belongs to \(GL^+(n)\). Its pullback from \(V\) makes \(E_s\) a global smooth frame, unchanged at \(s=0\), with the exact zero-section value \(E_s|_V=L_sJ^{-1}\). Declare this frame orthonormal. This gives metrics with precisely the original product metric at \(s=0\), without requiring a simultaneous metric-extension argument.

At the endpoint put \(G(w,v)=e_1E_1(w,v)\). Equation (327) gives \(G(w,0)=1\), and the radial contraction shows \(\det G>0\). Take its positive polar decomposition \(G=RP\), with \(R\in SO(n)\) and \(P=(G^*G)^{1/2}>0\). Both \(R\) and \(P\) equal the identity at zero. Choose a smooth scalar \(\gamma\) on the interval, with \(\gamma(0)=0,\gamma(1)=1\), and put
\[
 P_s=(1-\gamma(s))1+\gamma(s)P,\qquad
 \widehat E_s=E_sP_s^{-1}.
 \tag{329}
\]
Take \(0\le\gamma\le1\); every \(P_s\) is positive and invertible. Declare \(\widehat E_s\) orthonormal and transport the entire fixed \(\mathcal D_n\) action and grading through this frame, starting with the specified full unitary (325). The initial metric and full orientation remain exactly the prescribed ones, and the zero-section frame remains \(L_sJ^{-1}\). At \(s=1\), \(e_1\widehat E_1=R\), so the final metric is exactly the Euclidean pullback metric. The central phase is the actual initial phase transported through these frames, not an independently chosen one.

At the zero section \(R(w,0)=1\). Thus the radial path
\(R(w,\lambda v)\), \(0\le\lambda\le1\), is in \(SO(n)\); it starts at the identity and cannot change its determinant. The global identity-started spin ODE (310), applied to this path, supplies an even smooth spin lift. Its full representation on \(\mathcal D_n\) intertwines the transported endpoint action with the ambient Euclidean action, preserves the right \(C_n\) action and inner product, and is onto. It fixes the reference at zero, and its phase starts at one. Hence the endpoint orientation is the full ambient inverse orientation on the entire normal bundle.

All constructions preserve compact smooth extra parameters by the same buffered charts and finite-interval ODEs. The initial \(H^*\otimes H\) cancellation in (325) was an actual even Hermitian line evaluation, so no hidden two-torsion or odd rank-zero line can reappear through these transports. This completes the geometry, cocycle, distribution and full orientation-family proof.

![Exact transverse normal kernels and full orientation transport](../figures/transverse-normal-comparison.png)

**Figure 7.8.** Lemma7.23, equations(324)–(329); Exercise64, equation(355). Each plotted panel is the exact kernel line \((h,t)=(-s a,a)\) of \(h+st\), for \(s=0,1/2,1\), with the marked vector at \(a=1\). These are one horizontal/normal transverse pair; the full \(F\) and embedding-normal directions are retained in (327). The lower panels give the actual signed graph quotient, complete native coefficient transport, positive metric correction and identity-started radial Spin lift. The local geometry is followed by the interval-cycle and bounded endpoint proofs in Lemma7.24 and Theorem7.25. Classical cocycle context is [Connes–Skandalis, Section 4]; this figure is original CC0 material.

Full-size transverse geometry PNG · Editable transverse geometry SVG.

**Lemma 7.24 (one principal-space interval cycle and its zero endpoint).** For the cocycle family of Lemma7.23, put \(I=[0,1]\), \(X=N\), \(G=\operatorname{Hol}(V,F)\), \(A=C_r^*(G)\), and \(B_I=C(I,A)\). There is a countably generated full graded \(B_I\widehat\otimes C_n\)-module with one bounded odd self-adjoint contraction and compact interval defects. Endpoint evaluation is an onto module unitary, and its zero endpoint, after \(R_n\), is the native product \(d_N\cdot\ell_{V,!}^{\mathrm{an}}\).

**Proof: actual local models and the whole principal space.**
For each relative chart of \(I\times X\), choose a representative
\(f_i(s,x)\in V\) of the cocycle \(h\), and retain every given transition arrow \(h_{ji}(s,x)\).
The transverse derivative is surjective, by the uniform surjectivity proof of Lemma7.23. In a buffered target foliation chart write its transverse coordinate as \(y\in\mathbb R^t\).
The parameter submersion theorem gives relative coordinates
\((s,a,y)\) on the source, with \(a\in\mathbb R^n\).
The target plaque coordinate of \(f_i\) can be replaced, locally, by a small rescaling of \((a_1,\ldots,a_p)\), since \(n\geq p\).
Choose the scale and the source buffer so both plaque coordinates lie in one target plaque box. The unique short plaque arrow
\(\lambda_i:f_i\to f'_i\), where \(f'_i(s,a,y)=(a_1,\ldots,a_p,y)\) in these rescaled coordinates, is smooth.
The new transition is the actual arrow
\(\lambda_j h_{ji}\lambda_i^{-1}\).
It satisfies the exact cocycle, and these arrows give an equivalence to the original cocycle. Thus each \(f'_i\) is a local ordinary submersion of rank \(\dim V=p+t\), without replacing the global cocycle by an ordinary map.

Glue the spaces
\[
 P_i=\{(s,x,\gamma):r\gamma=f_i(s,x)\}
 \quad\text{by}\quad
 (s,x,\gamma)\longmapsto(s,x,h_{ji}(s,x)\gamma).
 \tag{330}
\]
Write \(b:P\to I\times X\), \(a:P\to I\times V\), where
\(a(s,x,\gamma)=(s,s\gamma)\). The right \(G\times I\) action is arrow multiplication.
The transition maps are diffeomorphisms, with their exact inverses; the principal-space atlas is countable and locally Hausdorff.
Each fixed source fibre \(P_{s,v}\) is Hausdorff. Points with distinct \(b\) values are separated by the Hausdorff source base. Points with the same \(b\) value lie in a single \(b^{-1}(U_i)\), whose fixed-anchor model is \(U_i\times_{f_i,r}G_v\); the holonomy source fibre \(G_v\) is Hausdorff, so this separates them too.
The projection \(b\) is a submersion with plaque dimension \(p\).
In the local submersion representatives just constructed, the source anchor \(a\) is a submersion with fibre dimension \(n\).
The relative vector map
\[
 db:\ker da\longrightarrow b^*\mathcal F_s
 \tag{331}
\]
is an isomorphism. In the chart model the equation for an anchor-fixed tangent is exactly \(df_i(v)=dr(\dot\gamma)\), with \(ds(\dot\gamma)=0\). Its solvability is the vanishing of the transverse derivative; its plaque tangent is unique because the source/range isotropy has zero Lie algebra. This proves injectivity and surjectivity with the same vector \(v\), and gives (331) independently of the chosen representative.

At every parameter the zero section has the actual plaque-arrow equivalence to \(\ell\), so \(P_s\) contains a copy of the regular principal \(G\)-space over that section. In particular \(a(P_s)=V\). This is the fullness assertion needed below; it is not inferred from transverse surjectivity alone.

**Proof: the completed interval module and its endpoint fibres.**
Transport the metric in (329) through (331).
Let \(\mathscr M_s\) be the full right \(C_n\)-module family of equations(325), (328) and (329). Its prescribed even trivialization is the entire fixed \(\mathcal D_n\), with its grading, physical Clifford action and inner product.
Use the finite span of smooth compact sections in individual Hausdorff relative charts of
\[
 b^*(\mathscr M_s\otimes D_{\mathcal F_s}^{1/2})
       \otimes a^*D_F^{1/2}.
 \tag{332}
\]
All densities here are relative to the interval. They do not contain a \(ds\) factor.
In density trivializations the module formulas are
\[
 \begin{aligned}
 \langle\xi,\eta\rangle(s,\gamma)
   &=\int_{P_{s,s\gamma}}
         \langle\xi(z\gamma^{-1}),\eta(z)\rangle_{C_n}\,d\mu_s(z),\\
 (\xi f)(z)
   &=\int_{s\gamma=a_V(z)}
          \xi(z\gamma^{-1})f(s,\gamma).
 \end{aligned}
 \tag{333}
\]
In the first line the two longitudinal half-densities in (332) supply the integration density \(d\mu_s\); it is not an additional density factor. The remaining source half-densities are the two endpoint half-densities of the resulting \(G\) kernel. In the second line the repeated target/source half-densities supply the intermediate plaque integration, exactly as in (186). Right \(C_n\) coefficients use the graded module convention of Lemma0.13.

These expressions are finite chart-kernel expressions on any pair of core supports. Indeed the principal-action map identifies
\(P\times_{a} (G\times I)\) with \(P\times_b P\), by \((z,\gamma)\mapsto(z,z\gamma)\).
The right-hand space has the compact-support control of the Hausdorff base \(I\times X\).
After subdivision into finitely many principal and arrow charts, the integrals have compact parameter supports and bounded finite row/column integrals. They therefore lie in the fixed arrow-core completion \(B_I\widehat\otimes C_n\).
On a common compact support, smooth parameter dependence gives reduced-norm continuity: uniform coefficient differences are bounded by the same finite Schur constants. A finite parameter partition then approximates these kernels by finite sums \(c(s)k(\gamma)\), proving they are elements of \(C(I,A)\widehat\otimes C_n\), rather than merely separately specified endpoint coefficients.

For each \((s,v)\), tensor by the regular \(G_v\) representation and the faithful finite representation of \(C_n\). The convolution map
\[
 W_{s,v}(\xi\otimes u)(z)
       =\int_{G_v}\xi(z\gamma^{-1})u(\gamma)
 \tag{334}
\]
takes values in the full longitudinal \(L^2(P_{s,v})\) coefficient space.
Changing the two integration variables by right multiplication proves that its Hilbert-space inner product is exactly the regular representation of (333). The same identity for finite lists proves matrix positivity. The faithful sum over \(s,v\) detects positivity in \(B_I\widehat\otimes C_n\). Quotient the zero vectors and complete to obtain the full graded module \(\mathscr E\).

The maps (334) are onto. In a compact principal chart use the local ordinary submersion representative constructed above. There the coordinates on \(P_{s,v}\) are the vertical coordinates of \(f'_i\) together with its target plaque coordinates, as in the Fubini model of Lemma7.12. Finite separated smooth functions in these coordinates are dense on a compact support. A compact smooth source-plaque function whose integral against a chosen regular input is one realizes each such separated function by (334); insert a larger chart cutoff equal to one on the support. Translate a chart containing a nonunit arrow to this model by that arrow, preserving its density and all coefficients. Thus each compact fibre-chart function lies in the closure of the range. Finite chart sums and the \(L^2\) exhaustion prove onto. Distinct holonomy lifts and distinct components remain separate in this argument.

Multiplication by \(c(s,b_X(z))\) is adjointable and bounded by \(\|c\|_\infty\), with adjoint multiplication by \(\overline c\), as follows on every faithful regular fibre and then on the module. Smooth core sections on countably many compact chart exhaustions have a countable norm-dense set: uniform coefficient approximation has a fixed finite Schur bound on each retained support. The right algebra has a countable approximate identity, which converges on every vector by (333). This proves countable generation.

For \(s_0\in I\), restriction defines
\[
 \mathscr E\widehat\otimes_{\mathrm{ev}_{s_0}}
       (A\widehat\otimes C_n)\longrightarrow\mathscr E_{s_0}.
 \tag{335}
\]
The endpoint inner product is exactly (333) evaluated at \(s_0\), so this is an isometry after the evaluation quotient. Every endpoint compact chart section extends in a relative parameter chart, with a cutoff equal to one on its compact support. Such extensions are actual interval-core sections. Finite endpoint chart sums extend likewise. Their core is dense, proving surjectivity at both boundary parameters as well as in the interior. Finite-rank operators restrict to the corresponding endpoint rank-one operators. Hence every interval module compact restricts to an endpoint module compact.

**Proof: the actual compact ideal and one global bounded phase.**
First omit the finite Clifford module and call the resulting scalar principal module \(\mathscr P\).
The gauge groupoid
\[
 R=(P\times_aP)/(G\times I)
 \tag{336}
\]
has units \(I\times X\) and arrows given by classes of pairs with the same source anchor. Its local models are \(f_j'^*(G\times I)f_i'\), with the actual cocycle transitions constructed above. Its relative Lie algebroid is \(\mathcal F_s\).
Convolution by a compact relative \(R\)-chart kernel acts adjointably on \(\mathscr P\).
Separated-function approximation on a fixed compact chart support, followed by a smooth plaque approximate identity, approximates it in the regular Schur norm by rank-one kernels. Conversely the scalar kernel of \(\theta_{\xi,\eta}\) is
\[
 K_{\xi,\eta}([z',z])
    =\int_{G_{a_V(z)}}\xi(z'\gamma^{-1})
                      \overline{\eta(z\gamma^{-1})}.
 \tag{337}
\]
The integral is over the diagonal right orbit of a pair with the same anchor; its density is the intermediate source-plaque density from (332). The two compact core supports give a compact subset of \(P\times_aP\), and its image under division by the diagonal right action lies in finitely many compact \(R\) charts after subdivision. Thus (337) is a finite compact chart kernel. This is the same three-variable rank-one calculation as Lemma7.12, now on each pair of local submersion representatives.

Choose \(z_0\in P_{s,v}\). The map \(z\mapsto[z,z_0]\) identifies that fibre, its density and its convolution representation with the regular \(R\) source fibre at \(b(z_0)\).
Since \(a(P_s)=V\), every regular \(G\) parameter is obtained in this way.
Consequently the module-operator norm equals the reduced \(R\) norm: (334) gives the upper bound on all fibres and the preceding identification gives the converse bound. Thus the norm closure of represented compact chart kernels is exactly
\[
 C_r^*(R)=\mathcal K(\mathscr P),\qquad
 \mathcal K(\mathscr E)=\mathcal K(\mathscr P)\widehat\otimes C_n.
 \tag{338}
\]
The second equality uses the specified full trivialization by \(\mathcal D_n=\Pi^{k_n}C_n\); a grading reversal changes no compact ideal. The first equality is an isomorphism of represented algebras, not an assertion that \(R\) or its reduced algebra is a product in the parameter.

Quantize the normalized full physical symbol
\[
 q_s(x,\zeta)=c_{\mathcal F_s}^+(\zeta^\sharp)/|\zeta|
 \tag{339}
\]
in compact buffered near-unit \(R\) charts. These are local ordinary submersion/plaque charts by the local submersion construction above. The finite-supported symbol calculus of Lemma7.11 therefore applies with precisely the ideal (338): negative-order truncations converge in the reduced norm to compact chart kernels. The Schur estimates are uniform over every retained compact relative-parameter support and all regular source fibres. A square-root quantization of \((c^2+\epsilon)1-b^*b\) proves the upper symbol-quotient bound. A high-frequency packet in a near-unit source chart proves the lower bound, since compact smooth kernels annihilate it by integration by parts and norm approximation treats (338). This gives the exact compact kernel of the finite-supported symbol quotient, with every finite Clifford action retained.

Choose a countable locally finite buffered atlas of \(I\times X\), including its relative boundary charts, and smooth scalar functions \(\chi_i\) with \(\sum_i\chi_i^2=1\). Use larger chart cutoffs \(\eta_i=1\) near \(\operatorname{supp}\chi_i\), with the larger supports also locally finite and compact. The compact-annulus construction in Lemma7.11 gives this atlas; relative boundary charts need no differentiation normal to the parameter.
Take an odd self-adjoint quantization \(R_i\) with symbol \(\eta_i^2q\).
Clipping by \(h(t)=\max(-1,\min(t,1))\) gives
\(Q_i=h(R_i)\), an odd self-adjoint contraction. Zero-constant polynomial approximation retains its compact input/output unit supports, and its symbol is still \(\eta_i^2q\).
The single interval-module operator
\[
 Q=J_\chi^*\operatorname{diag}(Q_i)J_\chi
       =\mathop{\mathrm{strict}\sum}_i\chi_iQ_i\chi_i,
 \qquad J_\chi u=(\chi_i u)_i,\quad \|Q\|\leq1
 \tag{340}
\]
is bounded and adjointable. The column isometry and its adjoint follow first on compact principal-core sections, where only finitely many terms enter, and then by density and the contractive finite partial sums. The tail estimate from (271) proves strict convergence for \(Q\) and its adjoint. This does not assert global norm continuity of a family chosen independently on each fibre.

A compactly supported scalar \(c\) meets finitely many \(\chi_i\); the retained larger supports meet finitely many neighbours. Thus localizing (340), its commutator or its square involves the finite-supported calculus just proved. Their symbols are respectively \(cq\), zero and \(c(q^2-1)=0\).
By its exact compact kernel,
\[
 c(Q^2-1),\ [Q,c]\in\mathcal K(\mathscr E)
       \qquad(c\in C_0(I\times X)).
 \tag{341}
\]
Smooth compact approximation and the fixed contraction bound prove this for all such coefficients. In particular constant-in-parameter coefficients from \(C_0(X)\) give an actual Kasparov homotopy after the fixed even Clifford Morita reduction \(R_n\).
The endpoint module, operator and all physical symbols are the restrictions (335); every compact defect is an interval compact before evaluation. Quantization choices with the same symbol differ by a locally compact operator, by the same finite-supported quotient. The straight path is a cycle homotopy by the cross-term calculation of Lemma7.11.

**Proof: the zero endpoint and native coefficient order.**
The actual plaque-arrow equivalence \(h_0\simeq\ell\operatorname{pr}_N\) sends \(P_0\) to
\[
 \{(w,v,\gamma):r\gamma=w\}.
 \tag{342}
\]
It preserves the source anchor, the right action and the longitudinal half-densities.
The product density of the normal vertical metric and the pulled-back leaf metric is exactly the \(\mathcal F_0\) density under the metric-connection splitting.
Fubini on each regular \(G_x\) fibre sends the normal-integration module tensor the full regular foliation module onto \(\mathscr E_0\). Finite separated-chart approximation proves onto in module norm as in Theorem7.13, with all noncompact normal fibres and all disconnected or holonomy components retained.

The coefficient bundle on this product, in physical normal/tangent order, is
\(\mathcal D_N\widehat\otimes\mathscr M_F\).
The native product target is tangent/normal \(C_p\widehat\otimes C_q\).
Apply the entire \(\beta\) transport in (325) to its right action and inner product, not just to its generator labels. Lemma7.21 proves that the fixed \(R_n\) reduction of this full target is the native product of the two individually reduced cycles. The supplied full unitary (325) identifies the complete resulting physical bundle and symbol with the initial \(\mathscr M_0\), including its exact grading and phase.

For the operators use the full localized product proof of Theorem7.13 on \(\operatorname{pr}_N:N\to V\). The vertical inverse is the actual bounded cycle with full module \(\mathcal D_N\) of Theorem7.17. Its local physical Clifford generators obey the same Clifford relations, including the grading shift and reflection established there, so the total normalized symbol is the orthogonal sum used in (265).
The axis compression (263) makes both compact creation errors cancel in their \(C^*C\) calculation. The exact base quotient and the \(CL(L+\delta)^{-1}\) approximation give the errors themselves and their adjoints. Localized positivity is the nonnegative bracket (266) in the supported quotient, not compactness of a lifted vertical compact. Compact source cutoffs, the two-sided locally compact phase commutators and a buffered splitting partition are exactly the nonproper localization of Theorem7.13.
These computations apply to the full physical right order \(C_q,C_p\); the subsequent \(\beta\) is an isometric full coefficient transport and preserves every operator compact, bracket and connection.
The choice comparison in Lemma7.24 identifies this total phase with \(Q_0\).
Therefore
\[
 R_n[\mathscr E_0,c,Q_0]
      =d_N\cdot\ell_{V,!}^{\mathrm{an}}
       \quad\text{in }KK(C_0(N),A).
 \tag{343}
\]
Here \(d_N=R_qT_q\alpha_N\) is the specified native inverse Thom class; \(p+q=n\) is even.
If \(p=0\), the regular foliation factor is its full graded orientation line with zero operator; its creation maps are finite module columns, so the same proof includes the rank-zero case of Theorem7.13. The initial inverse/line evaluation is the literal full unitary (325). Finite extra Hermitian coefficients and compact smooth parameters enter every finite chart calculation as finite corners, with the same interval bounds.

**Theorem 7.25 (the two prescribed longitudinal wrong-way classes coincide).** For a compact foliated manifold with the prescribed full stable spin-c orientation of \(TV\oplus\tau\), the analytical and geometric leaf-map classes are equal in \(KK^p(C(V),C_r^*(G))\). This includes both parities, complete Hermitian line twists, rank-zero grading lines, finite coefficients and compact smooth parameters. The proof uses the entire normal-tube cocycle and its calibrated ambient inverse.

**Proof: the full étale/Euclidean common module.**
Let \(g:X\to V/F\times\mathbb R^n\) be the actual étale tube cocycle, with \(n\) even, \(n>p\), and Hausdorff second-countable \(X=N\). Write \(g=(h,e)\), where \(e:X\to\mathbb R^n\) is the global Euclidean coordinate.
The principal space for \(g\) is the same right \(G\)-space \(P\) as for \(h\), with the extra source coordinate
\(\widetilde a(z)=(a_V(z),e(b(z)))\).
This full anchor is a local diffeomorphism, by Lemma6.0n.
Thus \(a_V:P\to V\) is a submersion of fibre dimension \(n\), and the Euclidean differential identifies its vertical tangent with \(\mathbb R^n\).
Give that tangent the Euclidean pullback metric and the entire inverse module \(\mathcal D_n\) from (319).
Use the principal integration module \(\mathscr E_h\) of Lemma7.24 at this fixed endpoint. All source fibres are Hausdorff by the same \(b\)-chart argument.

Put \(D=A\otimes C_0(\mathbb R^n)\).
The étale zero correspondence is \(\mathcal E_g\) of Lemma6.0n, with coefficient algebra \(D\).
The full Euclidean inverse module is
\[
 \mathscr H=A\widehat\otimes L^2(\mathbb R^n)
                 \widehat\otimes\mathcal D_n,\qquad
 F_n=D_n^{\mathrm{flat}}(1+(D_n^{\mathrm{flat}})^2)^{-1/2}.
 \tag{344}
\]
Here the physical generators in \(D_n^{\mathrm{flat}}\) are the specified generators of \(\mathcal D_n\), not an independently ordered positive Bott module. Its homogeneous symbol is their normalized Clifford multiplication. By (319), \(R_n[\mathscr H,F_n]\) is the geometric \(A_n\).

On compact smooth cores the balanced map is
\[
 U\bigl(\xi\otimes(a\otimes v)\bigr)(z)
        =(\xi\cdot(a\otimes1))(z)\,v(e(b(z))),
 \qquad
 U:\mathcal E_g\widehat\otimes_D\mathscr H
                  \longrightarrow\mathscr E_h .
 \tag{345}
\]
The dot is the actual right \(G\) convolution, not evaluation of \(a\) at a point. The Euclidean scalar multiplication relation is immediate. The \(A\) relation is convolution associativity. Both preserve the full right \(C_n\) action.
For each regular \(G_x\) representation, Fubini changes
\[
 \int_{\mathbb R^n}
        \sum_{z\in\widetilde a^{-1}(x,t)}
                     |u(z,t)|^2\,dt
       \quad\text{to}\quad
       \int_{P_x}|u(z,e(b(z)))|^2 .
 \tag{346}
\]
Use a countable disjoint measurable refinement of the local diffeomorphism charts to prove this change of variables; its Jacobian is exactly the pullback Euclidean density. The sum counts every lift, including multiple sheets and different holonomy orbits. The full finite Clifford inner products give the identical matrix identity.
Equivalently, the module inner product on (345) is the integral of
\(v(t)^*a^*\langle\xi,\eta\rangle_D(t)b\,w(t)\).
The faithful regular sum proves that \(U\) is isometric.

It is onto. In a compact full-anchor chart, a coefficient section of the target core is a finite sum of scalar smooth sections \(\xi(z)\) times fixed homogeneous elements of \(\mathcal D_n\), with its Euclidean half-density trivialized. Choose a Euclidean scalar cutoff equal to one on that chart support. These are expressions \(\xi(z)v(e(b(z)))\). Insert an \(A\) approximate identity \(a_m\) in (345); \(\xi a_m\to\xi\) in the étale module norm. The regular estimate
\(\|U(\xi\otimes v)\|\leq\|\xi\|_{\mathcal E_g}\|v\|_{L^2}\), interpreted first on cores, makes the corresponding expressions converge in the target norm. Finite chart sums exhaust the core. Hence \(U\) is an even onto unitary of the entire completed modules.

**Proof: compact rectangular kernels.**
We need the following actual compactness statement, rather than compactness of an operator in every Hilbert-space representation.
Suppose a map \(\mathscr H\to\mathscr E_h\) has a smooth kernel on finitely many compact \(P\)-charts times a compact Euclidean input support. Its action is the \(G\) convolution in (334), followed by the Euclidean input integral, and its coefficients include the half-densities just specified. It is a rectangular module compact.

To prove this, separated smooth approximation on those compact supports approximates the kernel by finite sums \(\zeta_j(z)w_j(t)^*\), where \(\zeta_j\) is a compact integration-module core vector and \(w_j\) a compact Euclidean \(L^2\) vector. The uniform coefficient error has a fixed finite plaque Schur bound and a finite Euclidean integral bound, so this approximation is in module-operator norm, uniformly on all regular fibres.
The corresponding separated map is
\[
 u\longmapsto\zeta_j\int w_j(t)^*u(t)\,dt .
 \tag{347}
\]
An \(A\) approximate identity makes it a limit of actual rank-one maps from \(\mathscr H\): use output vector \(\zeta_j\) and input vector \(a_m^*\otimes w_j\). The resulting map is (347) with output coefficient \(\zeta_j a_m\).
The error is bounded by \(\|\zeta_j-\zeta_j a_m\|\|w_j\|_{L^2}\).
Finite homogeneous Clifford columns give the same argument for the entire right \(C_n\) inner product. This proves (347) and then the kernel assertion. It does not require \(A\) to be unital.

A finite-supported mixed order-\(-1\) operator is compact by the same argument. Confine the Euclidean input and output to buffered coordinate boxes and extend them to tori. Its Fourier-symbol estimates, uniform on the retained compact \(P\) and arrow parameters, give norm convergence of frequency truncations to that operator; the order-\(-1\) tail has norm at most \(C/M\) after cutoff at frequency \(M\), using the finite Fourier and regular Schur estimates of Lemma7.11.
Each truncation has a smooth compact mixed kernel after the spatial buffers are inserted and is compact by the first paragraph. Off-diagonal kernels have the same conclusion directly. Thus their norm limit lies in \(\mathcal K(\mathscr H,\mathscr E_h)\).

**Proof: all creation connections, including the Fourier tail.**
Let \(Q_h\) be the bounded phase of Lemma7.24 with the full Euclidean pullback symbol just specified, and let \(T_\xi:\mathscr H\to\mathscr E_h\) be creation by a compact smooth étale core vector \(\xi\), using (345). We prove
\[
 Q_hT_\xi-T_\xi F_n
       \in\mathcal K(\mathscr H,\mathscr E_h).
 \tag{348}
\]
The étale module has even scalar grading, so these creation vectors are even. The finite Clifford grading is wholly in the second module.

Let \(S\subset\mathbb R^n\) be a compact set containing the Euclidean image of \(\operatorname{supp}\xi\), and let \(K\subset X\) contain its \(b\) image.
The exact creation formula has \(T_\xi=T_\xi\kappa_S\) for a Euclidean cutoff equal to one on \(S\).
Only finitely many terms of (340) enter \(Q_hT_\xi\), because the right \(\chi_i\) meet \(K\).
Each clipped local phase differs from its unclipped quantization by a compact operator: its symbol is unchanged and the exact finite-supported quotient in Lemma7.24 has the compact kernel (338).
Multiplication of such a compact by bounded \(T_\xi\) is a rectangular compact. Thus, modulo the required compact ideal, \(Q_hT_\xi\) is computed by finitely many ordinary near-unit chart quantizations.

Choose a Euclidean input cutoff \(\kappa_R\) equal to one on a ball containing \(S\) and its diagonal buffers.
In a full-anchor chart use coordinates \((x,t)=\widetilde a(z)\), so the longitudinal coordinate is exactly \(t\).
Near the mixed diagonal \(t=t'\), the leading amplitude of \(Q_hT_\xi\kappa_R\) is
\(c_n^+(\zeta/|\zeta|)\xi(z\gamma^{-1})\).
The leading amplitude of \(T_\xi F_n\kappa_R\) is
\(\xi(z\gamma^{-1})c_n^+(\zeta/|\zeta|)\).
The scalar \(\xi\) commutes with that finite action, and the physical action, right action and grading are the identical \(\mathcal D_n\) actions on both modules. These amplitudes coincide.
The finite-chart composition calculation therefore gives an order-\(-1\) mixed kernel for their difference: differentiation of the scalar amplitude and spatial cutoffs supplies its lower-order terms. Compact \(R\)-chart kernels acting on a compact principal core have support in finitely many compact \(P\) charts, by the smooth gauge action. All remaining kernels with these input/output buffers are off-diagonal smooth kernels.
The rectangular compact-kernel proof makes this entire confined difference a rectangular module compact. In particular
\[
 (Q_hT_\xi-T_\xi F_n)\kappa_R
       \in\mathcal K(\mathscr H,\mathscr E_h).
 \tag{349}
\]
This uses the exact negative-order compact norm, not merely equality of symbols on individual sheets.

It remains to remove the input cutoff. On every regular \(G_x\) representation, (346) identifies creation with a decomposable operator
\[
 (T_\xi u)(t)=V_{\xi,x}(t)u(t),\qquad
 \|V_{\xi,x}(t)\|\leq\|\xi\|_{\mathcal E_g}.
 \tag{350}
\]
It is zero outside \(S\). The target of each \(V_{\xi,x}(t)\) is the whole counting fibre at \((x,t)\), so this estimate includes every sheet.
Since \(\kappa_R=1\) on \(S\), \(Q_hT_\xi(1-\kappa_R)=0\).
The remaining term \(T_\xi F_n(1-\kappa_R)\) has its Euclidean input separated from that compact output set.
The off-diagonal Fourier kernel \(K_n\) has the heat-integral expression (279), with the specified physical Clifford generators. The grading reversal and constant physical signs in (319) change neither its operator norm nor the estimate (280).
That estimate proves \(K_n\in L^2\) away from every diagonal buffer and that its \(L^2\) tail tends to zero.
Cauchy–Schwarz in the Euclidean input, followed by (350), gives on every regular fibre
\[
 \|T_\xi F_n(1-\kappa_R)\|
 \leq C_n\|\xi\|_{\mathcal E_g}|S|^{1/2}
       \sup_{t\in S}
       \|K_n(t-\cdot)(1-\kappa_R)\|_{L^2}.
 \tag{351}
\]
The finite Clifford constant only converts matrix norm to a scalar norm. Translation by a bounded \(t\in S\) moves the tail radius by a bounded amount, so the supremum tends to zero.
The faithful regular representation family makes (351) a module-operator norm bound. Hence (349) converges to the whole difference, proving (348).
Its adjoint is the other connection error. Core density and \(\|T_\xi\|\leq\|\xi\|\) treat every creation vector. Extra finite coefficients enter as finite columns.

**Proof: the product and the transverse comparison.**
The first operator of \(g_!\) is zero and its left coefficient action is compact, by Lemma6.0n.
The common-module operator \(Q_h\) has compact localized cycle defects by Lemma7.24.
Both creation connections hold by the full creation-connection proof; the product positivity bracket with the first operator is zero.
Definition0.3 therefore gives the full equality
\[
 R_n[\mathscr E_h,c,Q_h]=g_!\cdot A_n .
 \tag{352}
\]
The proof uses the fixed entire ambient inverse module, before the even \(R_n\) reduction.
It assumes no completeness or boundary regularity of a Euclidean sheet.
If a compact smooth extra parameter is present, construct all modules on its single coefficient algebra as in Lemma7.24.
Each core support lies in finitely many relative buffered charts; the frequency approximations just proved are uniform on their compact parameters and the uniform tail estimate (351) has the same fixed compact \(S\). Thus the creation errors are compact operators on that whole parameter module, before endpoint evaluation.

Apply (352) to the actual étale tube cocycle \(\widetilde g=(h_1,e)\).
The metric correction (329) and the identity-started Spin lift identify its ambient pullback full module with the final interval module, with an even unitary preserving the right action, inner products and physical generators. The locally compact choice comparison in Lemma7.24 identifies their phases. It is therefore precisely the étale endpoint of Lemma7.24.
The interval cycle, (343) and (352) give
\[
 d_N\cdot\ell_{V,!}^{\mathrm{an}}
       =\widetilde g_!\cdot A_n
       \quad\text{in }KK(C_0(N),A).
 \tag{353}
\]
Multiply by the prescribed native \(b_N\) on the left.
Both native inverse products are proved in Lemma7.20, so
\[
 \ell_{V,!}^{\mathrm{an}}
       =b_N\cdot\widetilde g_!\cdot A_n
       =\ell_{V,!}^{\mathrm{geom}}
       \quad\text{in }KK^p(C(V),A).
 \tag{354}
\]
The final equality is the actual geometric normal-tube definition of Theorem6.0o, with the full normal recovery (324) and its fixed final-odd transport. Native multiplication keeps second-cycle coefficients first at every preceding product, and the zero endpoint proof of Lemma7.24 used its full \(\beta\) transport before \(R_n\).
Thus (354) includes both parities of \(p\), every prescribed stable spin-c structure, full Hermitian line twists (including two-torsion), rank-zero grading lines, finite coefficients and compact smooth parameters.
Lemmas7.26–7.28 and Theorem7.29 below prove the additional arbitrary-symbol action in both K-degrees, using the full circle product and normalized transverse suspension.

![The full interval cycle, both endpoint modules and uniform compact connection](../figures/transverse-principal-connection.png)

**Figure 7.9.** Lemma7.24 and Theorem7.25, equations(330)–(354); Exercise65, equation(356). Here \(P\) is the actual right principal space and \(\mathscr P\) its scalar integration module. The first panel records the whole interval module and its compact ideal, before evaluation. The two endpoints use the full normal/tangent coefficient transport and the counting-sheet/Euclidean Fubini unitary. The middle panel is the operator-norm tail bound(351), which completes both rectangular module-compact creation errors; it is not a numerical sample or a fibrewise compactness assertion. The final cancellation uses the two proved native Thom inverse identities, with the full given line and grading retained. Theorem7.29 below supplies the arbitrary-symbol action in both K-degrees. The longitudinal method is classical [Connes–Skandalis]; the full argument and drawing here are original CC0 material.

Full-size principal connection PNG · Editable principal connection SVG.


### Every relative longitudinal symbol, in both degrees

The full KK comparison of Theorem 7.25 now yields the analytic index of every relative symbol. We give the missing odd coefficient argument, including the actual circle product in the entire holonomy module and the normalized suspension maps. The classical longitudinal theorem and its circle reduction are due to [Connes–Skandalis, Section 4]; the bounded connection and convention checks below are written out with the course's fixed native Clifford reductions.

**Lemma 7.26 (the full analytic circle product).** For every relative longitudinal symbol \(s\in K_c^0(F^*)\), external product with the positive circle clutch class \(\kappa\) multiplies the full analytic index by \(-1\), after the ordinary even circle Morita equivalence. This holds without a transverse measure and retains the entire holonomy groupoid, finite coefficients and compact smooth parameters.

**Proof: the actual circle class.**

Let (V,F) be a smooth foliation on a compact manifold, with its entire holonomy groupoid G and \(A=C_r^*(G)\). No transverse measure is needed, and the arrows may be non-Hausdorff. For a relative longitudinal elliptic symbol \(s\in K_c^0(F^*)\), let \(\operatorname{Ind}_A(s)\) be the full analytic symbol index of Theorem 6.6 in the Hilbert-module lesson.

Use the positive coordinate \(t\) modulo \(2\pi\) on the circle. In its normalized Fourier basis \(e_k(t)=e^{ikt}/\sqrt{2\pi}\), let \(P_+\) project onto \(k\ge0\) and put

\[
 Q_0=P_+M_{e^{it}}P_++(1-P_+),\qquad
 Q_0e_k=\begin{cases}e_{k+1}&k\ge0,\\e_k&k<0.\end{cases}
 \tag{357}
\]

Its source kernel is zero, its adjoint kernel is the normalized constant line, \(Q_0^*Q_0=1\) and \(Q_0Q_0^*=1-E_0\). Its unit-cosphere symbol is \(e^{it}\) on \(\eta>0\) and \(1\) on \(\eta<0\). Let \(\kappa\) be the resulting relative class in \(K_c^0(T^*S^1)\); multiplying \(Q_0\) by \((1+D_t^2)^{1/2}\), as in the measured-index lesson, gives the same relative class. We prove

\[
 m_*\operatorname{Ind}_{A\otimes\mathcal K}(s\boxtimes\kappa)
       =-\operatorname{Ind}_A(s)\quad\text{in }K_0(A).
 \tag{358}
\]

Here the product foliation is \(F\oplus TS^1\) on V times \(S^1\). Its groupoid is G times (\(S^1\) times \(S^1\)), and m is the ordinary even circle Morita equivalence from \(A\otimes\mathcal K(L^2(S^1))\) to A. This is equality of the full classes, including torsion; it is not a projection-trace identity.

**Proof: the common completed module and joint weights.**

Assume first \(p=\operatorname{rank}F>0\). Represent s by bundles \(B_0\),\(B_1\) on V and a unitary isomorphism of their pullbacks on \(S^*F\). The polar homotopy of an arbitrary invertible sphere map permits this choice. Quantize that map to \(P_0\) from \(E_0\) to \(E_1\), where \(E_i=p_{B_i}A^{N_i}\) is the entire holonomy bundle module. On \(E=E_0\oplus E_1\), with grading \(\Gamma_1\), put \(S_0=\begin{pmatrix}0&P_0^*\\P_0&0\end{pmatrix}\). The symbol quotient gives \(S_0^2-1\) compact. Clamp \(S_0\) by the continuous odd function \(\max(-1,\min(x,1))\). The resulting odd self-adjoint contraction S has the same symbol and differs from \(S_0\) by a compact map: that function agrees with the identity at both points of the quotient spectrum \(\{+1,-1\}\). The cycle \((E,1,S)\) represents \(\operatorname{Ind}_A(s)\) under Theorem 0.14 and the boundary sign of Theorem 6.6.

The positive regular operator \(H_1=(1+\Delta_{B_0})\oplus(1+\Delta_{B_1})\) exists on this entire module by Hilbert-module Lemmas6.1-6.2. It commutes with \(\Gamma_1\), is bounded below by 1 and has compact inverse. Take \(H_2=1+D_t^2\) on \(H=L^2(S^1)\), and use it identically on \(H\oplus H\), graded by \(\Gamma_2\). Set \(S_2=\begin{pmatrix}0&Q_0^*\\Q_0&0\end{pmatrix}\). It is an odd self-adjoint contraction on this graded circle space and changes a Fourier label by at most one. Its scalar even KK class is -1, because Theorem 0.14 computes the same Fredholm boundary and its cokernel is the one odd constant line. This uses the full scalar classification, not just the dimension of a measured kernel.

After applying the circle Morita equivalence, the product holonomy module for the graded tensor bundle is exactly \(E\widehat\otimes(H\oplus H)\), a graded Hilbert A-module. On finite separated arrow kernels the unitary is convolution in the original G variable and integration of the circle kernel against its Morita vector. Fubini identifies the full A-valued inner products; finite-rank smooth circle kernels and separated functions on each compact G chart give dense range. A finite chart partition treats every compact core, and these cores are dense by definition. Thus this is an onto module unitary including all holonomy lifts. The corresponding compact algebra is \(\mathcal K(E)\otimes\mathcal K(H\oplus H)\). For the scalar pair groupoid itself, finite matrix circle kernels identify its reduced norm with the matrix norm over A and are dense, proving \(C_r^*(G\times\operatorname{Pair}(S^1))=A\otimes\mathcal K(H)\).

Write \(H_1\),\(H_2\) for their commuting positive operators on this common module. Their bounded joint weights can equivalently be defined on each Fourier label k, so no unproved sum of unbounded operators is required:

\[
 \begin{aligned}
 Z_k&=H_1(H_1+1+k^2)^{-1},\\
 W_k&=\sqrt{Z_k}
       =H_1^{1/2}(H_1+1+k^2)^{-1/2},\\
 V_k&=\sqrt{1-Z_k}
       =\sqrt{1+k^2}(H_1+1+k^2)^{-1/2}.
 \end{aligned}
 \tag{359}
\]

All are bounded adjointable contractions; the diagonal direct sums give \(Z\), \(W=\sqrt Z\) and \(V=\sqrt{1-Z}\). The module resolvent and fractional-power proofs above identify them in every regular representation with the joint longitudinal weights. Their principal scalar symbols are z, \(\sqrt z\) and \(\sqrt{1-z}\), where \(z=|\xi|^2/(|\xi|^2+\eta^2)\) on the joint sphere. The finite-chart proof of measured Lemma 6.35 applies in these actual bundle module corners. Its Fourier-Schur bounds are uniform on every source cover and its negative-order remainders are compact matrix arrow kernels; therefore every asserted norm limit is a module norm limit. The compact first-factor remainder permitted there includes \(S-S_0\). This proves that

\[
 T=(S\otimes1)W+(\Gamma_1\otimes S_2)V
 \tag{360}
\]

belongs to the product order-zero calculus closure on the full common module. Its continuous principal symbol is

\[
 q(\xi,\eta)=
 \frac{|\xi|\sigma(S)(\xi/|\xi|)\otimes1
       +\Gamma_1\otimes|\eta|\sigma(S_2)(\eta/|\eta|)}
      {\sqrt{|\xi|^2+\eta^2}} .
 \tag{361}
\]

Each numerator summand extends by zero at its own origin. The two odd summands anticommute and their squares sum to \(|\xi|^2+\eta^2\), so q is self-adjoint, odd and has square 1 on the joint sphere. The graded tensor of the two relative two-term complexes has exactly this odd map. Radial normalization and a sufficiently close smooth sphere approximation preserve invertibility. Consequently it represents \(s\boxtimes\kappa\), including both cotangent axes.

**Proof: all four compact weight commutators.**

Put \(\overline S=S\otimes1\) and \(R=\Gamma_1\otimes S_2\). The cone proof of Lemma 6.35 puts \(\overline S W\) and its adjoint \(W\overline S\) in the same joint algebra, with the same principal symbol. The exact symbol quotient thus gives \([\overline S,W]\) compact. It also gives \([R,V]\) compact. To obtain the other two commutators, rather than assuming smooth separate symbols at an axis, use the continuous scalar factorizations

\[
 \sqrt{1-z}-1=\sqrt z\,g(z),\qquad
 \sqrt z-1=\sqrt{1-z}\,h(z),
 \tag{362}
\]

where \(g(z)=(\sqrt{1-z}-1)/\sqrt z\) extends continuously by 0 at \(z=0\), and \(h(z)=(\sqrt z-1)/\sqrt{1-z}\) extends continuously by 0 at \(z=1\). Continuous functional calculus of the joint Z stays in its joint algebra. Hence \(\overline S(V-1)=\overline S Wg(Z)\) and its adjoint are in that algebra with identical symbols; their difference proves \([\overline S,V]\) compact. Similarly \([R,W]\) is compact using \(RVh(Z)\). The two identities include the endpoint values and do not require either g or h to be smooth.

It follows that \(T-T^*\) is compact. Put \(Q=(T+T^*)/2\). The exact product symbol quotient, applied in the graded bundle corner, gives \(Q^2-1\) compact because \(q^2=1\). Thus \((E\widehat\otimes(H\oplus H),1,Q)\) is an even scalar Kasparov cycle. Its off-diagonal symbol is (361), so Theorem 6.6 and scalar classification identify its class with the left side of (358). Changing Q by a compact perturbation or using a smooth sphere representative gives the same class.

**Proof: the entire creation connection by Fourier bands.**

The second input cycle is the external A-coefficient circle cycle \((A\widehat\otimes(H\oplus H),\lambda_A,1\otimes S_2)\) from A to A. Its localized defects are \(a\otimes(S_2^2-1)\), which belong to \(A\otimes\mathcal K(H\oplus H)\) for each a in A. Its commutators with the A representation are zero. Lemma 0.13 identifies its class with \(-1_A\). In particular no compact operator tensored with a bare multiplier identity is asserted to be compact.

For a homogeneous compact smooth first-module vector \(\xi\) in E, the creation map sends \(a\otimes v\) to \(\xi a\otimes v\). Such \(\xi\) lies in the form domain of \(H_1\). Let \(C_\xi=\|H_1^{1/2}\xi\|\), and \(b=1+k^2\). Module spectral calculus, or the positive spectral inequalities tested in all faithful regular representations, gives

\[
 \|W_k\xi\|\le C_\xi/\sqrt b,\qquad
 \|(V_k-1)\xi\|\le C_\xi/\sqrt b.
 \tag{363}
\]

Indeed the first squared multiplier is \(H_1/(H_1+b)\le H_1/b\). For the second use \((1-(1+x)^{-1/2})^2\le x\) for \(x\ge0\). These are module-vector inequalities; no bound on the unbounded \(H_1\) over the whole module is claimed.

The creation error for T is \(TT_\xi-(-1)^{|\xi|}T_\xi(1\otimes S_2)\). Its first part has output vector \(SW_k\xi\) on the same circle label, and its second part has output vector \(\Gamma_1(V_k-1)\xi\) followed by \(S_2\). The latter changes the circle label by at most one. Since S and \(S_2\) are contractions, for the projection \(\Pi_K\) onto labels \(|k|\le K\) in both circle spin summands,

\[
 \|[T T_\xi-(-1)^{|\xi|}T_\xi(1\otimes S_2)](1-\Pi_K)\|
 \le\frac{2C_\xi}{\sqrt{1+(K+1)^2}}.
 \tag{364}
\]

This estimate concerns the displayed unsymmetrized T error. The extra \(Q-T\) is a global module compact and contributes a compact creation error separately.

Each finite Fourier-band entry is a creation map from A to E with some fixed output vector \(\nu\). It is rectangular module compact also when A is nonunital. For a positive contractive approximate identity \(a_m\) of A, \(\theta_{\nu,a_m}(a)=\nu a_m a\) and \(\|\nu-\nu a_m\|\) tends to 0. Thus these rank-one maps converge in norm to \(a\mapsto\nu a\). A finite set of circle labels has finite free A input columns and finite E output columns; consequently its band truncation is compact. Estimate (364) makes the complete T creation error a norm limit of those compact truncations. Adding the compact \(Q-T\) term proves the Q connection. Taking adjoints proves the second connection identity. Compact smooth homogeneous vectors are dense in E, and \(\|T_\xi\|\le\|\xi\|\), so approximation proves both errors for every creation vector.

This is the entire connection in the actual nonunital module norm. It does not merely prove a compressed symbol identity or compactness on one holonomy cover.

**Proof: quotient positivity and the full product.**

In the common-module quotient by \(\mathcal K(E\widehat\otimes(H\oplus H))\), all four weight commutators proved above vanish. Since \(\overline S\) is odd and \(R=\Gamma_1\otimes S_2\) has the inserted tensor grading, \(\overline S R+R\overline S=0\) exactly. Therefore

\[
 \{S\otimes1,Q\}\equiv2(S^2\otimes1)W
                    \equiv2W\ge0\pmod{\mathcal K}.
 \tag{365}
\]

The second congruence needs a further compactness assertion. The first-factor \(S^2-1\) is compact. Here is a direct whole-module verification for its weighted lift. For \(\kappa_0=\theta_{\alpha,\beta}\), the kth entry of \((\kappa_0\otimes1)W\) is \(\theta_{\alpha,W_k\beta}\). If \(\beta\) is in the dense \(H_1\) form domain, (363) bounds that entry by \(\|\alpha\|\|H_1^{1/2}\beta\|/\sqrt{1+k^2}\). Finite Fourier truncations are compact; the tail tends to zero. Finite sums of such rank ones are dense in \(\mathcal K(E)\), since the form domain is dense, and \(\|W\|\le1\) controls the approximation uniformly. Hence \((\kappa_0\otimes1)W\) is compact for every \(\kappa_0\in\mathcal K(E)\), including \(S^2-1\). This agrees with the compact-remainder cone proof of Lemma 6.35 but does not depend on a representationwise compactness assertion. This is precisely why the bare lifted first-factor compact was not used as a total compact. Positivity is the positivity of the actual adjointable operator W, after passage to the module quotient.

Definition 0.3, the four compact commutators and positivity calculation, and the complete creation connection now prove that Q is a product of the scalar symbol cycle \((E,1,S)\) with the external circle cycle. The latter class is \(-1_A\) as proved above. Bilinearity, units, associativity and both Morita evaluations give \([Q]=-[E,1,S]\). Applying scalar classification proves (358) as equality in \(K_0(A)\).

**Proof: zero rank, choices and compact parameters.**

If \(p=0\), s is a virtual finite bundle on V and \(A=C(V)\). For each actual bundle B, the product module after circle Morita is \(\Gamma(B)\widehat\otimes(H\oplus H)\) with circle operator \(S_2\). The source operator of the bundle correspondence is zero; creation errors are zero because \(S_2\) acts in the other factor, and positivity is zero. This is its verified product with the scalar circle cycle and has index -[B]. Signed differences give (358). There is no empty-cosphere quantization or nonexistent leaf unit covector in this case.

For \(p>0\) every relative bundle triple was allowed, by the full disk contraction and quantization argument of Hilbert-module Theorem 6.6. Its equivalence relations therefore carry (358) to the entire group. Finite coefficients simply enlarge the bundle corners already used. For compact smooth parameters, construct the modules, \(H_1\) resolvents, all diagonal weights and the quantized symbols over the single parameter algebra. Hilbert-module Lemmas6.1-6.2 give their uniform resolvent and remainder estimates. Each smooth compact creation section has a finite uniform \(C_\xi\) bound there. The same finite-band approximation is consequently an approximation by parameter-module compacts before evaluation. All other defects and quotient calculations occur in that same module. Evaluating gives the parameter identity; separate pointwise compactness was not used.

![The full circle-product module, Fourier creation bands and norm tail](../figures/full-circle-index-class.png)

**Figure 7.10.** Lemma 7.26, equations (357)–(365); Exercise 66, equation (380). The first panel gives the exact normalized Fourier action of \(Q_0\), including its missing target constant line. The second displays the entire Morita-completed module and the diagonal joint weights. The band panel records the allowed labels \(k,k\pm1\) of the creation error; it is a support schematic, with no asserted coefficient amplitudes. The graph plots the proved bound \(2/\sqrt{1+(K+1)^2}\) at integer \(K\), normalized by \(C_\xi\); samples illustrate the exact bound and supply no analytic proof. The final panel uses the actual weighted module compact and positive quotient to obtain the full index class, including torsion. All inputs, constants and maps are proved in the accompanying text. Classical circle-reduction context is [Connes–Skandalis, Section 4]; the figure is original CC0 material.

Full-size circle-product PNG · Editable circle-product SVG.


**Lemma 7.27 (the positive circle coefficient, its clutch and Dirac pairing).** Write \(u(t)=e^{it}\), in the positive tangent coordinate of the canonically oriented circle. With the fixed native reductions, \(\Theta_{TS^1}[u]=\kappa\) and \([u]\cdot\ell_{S^1,!}^{\mathrm{an}}=-1\). Both identities hold in the full K-groups. The rank-one Thom identification uses its literal single-coordinate module and multiplier.

**Proof: an explicit positive coefficient cycle.**

Write \(B=C(S^1)\), \(u(t)=e^{it}\), \(H=L^2([0,1])\), and \(D_0=-i\,d/dx\) with periodic boundary condition. Its Fourier eigenvalues are \(2\pi k\). For a real local lift t of the circle parameter, put

\[
 U_t=M_{e^{itx}},\qquad
 s(t)=U_t h(D_0+t)U_t^*,\qquad h(y)=y/\sqrt{1+y^2}.
 \tag{366}
\]

The equality \(U_{t+2\pi}=U_tU_{2\pi}\) and \(U_{2\pi}D_0U_{2\pi}^*=D_0-2\pi\) shows that \(s(t+2\pi)=s(t)\), including the actual change of Fourier frame. \(U_t\) is norm continuous on bounded t intervals, and \(h(D_0+t)\) is norm continuous because h is uniformly Lipschitz. Hence s is one adjointable self-adjoint contraction on the standard B-module \(H_B\). Its square defect has eigenvalues \(-1/(1+(2\pi k+t)^2)\). Local finite Fourier cuts approximate that defect uniformly on every compact t interval. A finite parameter cover therefore proves \(s^2-1\in\mathcal K(H_B)\), not just pointwise compactness.

The odd scalar cycle \(x=(H_B\widehat\otimes C_1,1,s\varepsilon)\) has positive circle coefficient class \([u]\) under Theorem 0.14. Here is the sign check. The compact unitary \(w(t)=\exp(\pi i(s(t)+1))\) has eigenvalues \(\exp(\pi i(h(2\pi k+t)+1))\). Their differences from 1 are bounded by \(C/(1+k^2)\) on compact t intervals. Thus \(w-1\) is a continuous trace-class field, and its determinant is the convergent product of those eigenvalues. Differentiating its local determinant gives

\[
 \frac{1}{i}\frac{d}{dt}\log\det w(t)
   =\pi\sum_{k\in\mathbb Z} h'(2\pi k+t),\qquad
 \int_0^{2\pi}\pi\sum_k h'(2\pi k+t)\,dt=2\pi.
 \tag{367}
\]

The derivative series converges uniformly since \(h'(y)=(1+y^2)^{-3/2}\). The integral is \(\pi\) times the integral of \(h'\) over the entire real line, whose value is 2. The parameter reindexing leaves the determinant unchanged. Its winding is consequently +1.

For precision this also calculates the stable \(K_1\) class, rather than assuming that an infinite determinant classifies all compact unitaries. Finite-rank compression approximates this trace-class field uniformly in trace norm, by compactness of its parameter image and a finite approximation net. Polar correction of these finite compressions gives finite matrix unitaries, joined to w by a small norm homotopy. The determinant continuity used here follows from the finite exterior-power expansion: \(|\operatorname{tr}(\Lambda^rK)|\) is at most \(\|K\|_1^r/r!\), and telescoping its multilinear factors bounds the difference series by \(\|K-L\|_1\exp(\|K\|_1+\|L\|_1)\). Thus the series extends continuously to trace-class operators and agrees with the local eigenvalue product above. Their determinants converge uniformly to \(\det w\), so have its same winding. The full determinant classification of finite unitary loops, KT-OPK lesson 6 Section 5 and Theorem 5.2, identifies their stable class with \([e^{it}]\). The scalar exponential classification of Theorem 0.14 therefore identifies x with exactly \([u]\).

**Proof: the positive rank-one Thom product and its actual clutch.**

Use the canonical positive tangent frame \(\partial_t\) of the circle and the metric identification of its cotangent line. Its full positive Thom cycle has multiplier \(\eta\varepsilon/\sqrt{1+\eta^2}\). For this single coordinate it is literally the trivial-action cycle \(t_\iota\) of Lemma 0.17, on \(B\widehat\otimes C_0(\mathbb R_\eta)\widehat\otimes C_1\). This identifies only this one-coordinate multiplier and its actual module. It makes no identification of a separately ordered higher-dimensional real Bott pair with a geometric Euclidean pair.

The odd-odd product of x with that Thom cycle, in the native order of (104), acts on \(H_B\otimes C_0(\mathbb R_\eta)\otimes\mathbb C^{1|1}\). The old coefficient generator acts by \(\sigma_y\) and the new Thom generator by \(\sigma_x\). An actual product operator is

\[
 F(t,\eta)=\frac{s(t)\sigma_y+\eta\sigma_x}{\sqrt{1+\eta^2}},
 \qquad F^+(t,\eta)=\frac{\eta+i s(t)}{\sqrt{1+\eta^2}}.
 \tag{368}
\]

It is odd and self-adjoint, with square defect \((s(t)^2-1)/(1+\eta^2)\), a compact over \(B\otimes C_0(\mathbb R)\). For a creation section \(\xi\) in \(H_B\), the first numerator error has norm at most \(\|s\xi\|/\sqrt{1+\eta^2}\); the difference of the new phase and its own normalizer is zero. These errors vanish at \(\eta\) infinity and are compact rectangular module fields, since creation maps from B to \(H_B\) are compact. Finite sections and density prove both connection errors. The first positivity anticommutator is \(2s(t)^2/\sqrt{1+\eta^2}\ge0\). Definition 0.3 thus proves (368) is the actual product; it is not an assumed scalar Bott representative.

Its off-diagonal multiplier is invertible except where \(\eta=0\) and s(t) has a zero eigenvalue. On the circle there is precisely one such parameter, \(t=0\) modulo \(2\pi\). Near it the Fourier branch \(k=0\) is a smooth trivial eigenline, and all other branches are uniformly invertible. Outside a small neighborhood of (0,0), the full multiplier has a bounded inverse after the displayed normalization: away from \(t=0\) the least \(|h(2\pi k+t)|\) is uniformly positive, and at large \(|\eta|\) the \(\eta\) term supplies the uniform bound. The polar phase of the invertible complementary summand contracts it as a degenerate cycle. This localization can be made concrete in the scalar index projection. Choose a smaller parameter disk around (0,0) and its smooth \(k=0\) eigenline projection. On its orthogonal complement, replace the off-diagonal multiplier by its polar unitary. Outside a slightly larger disk do the same on the full module, using the uniformly bounded inverse there; a scalar cutoff interpolates through nonzero positive normalizers on the overlap. The resulting Fredholm homotopy has unit-square operator outside that disk and on the entire complementary summand. Its index projection minus the reference is therefore supported in the disk and in the finite eigenline corner. The eigenline's smooth unit frame gives exactly the finite line triple below. Full relative excision preserves its extension class. This records the actual compact projection localization, rather than inferring a K-class from its zero set. The relative Fredholm projection of Theorem 0.14 and full relative excision therefore identify its \(K_0\) class with the extension of the finite line triple whose local map is \(\eta+i h(t)\). Its positively rescaled local map \(\eta+i t\) has the same relative class, through the straight monotone deformation of h(t) to t and positive radial normalization. The deformation has no new zero off (0,0).

The sphere triple \(\kappa\) of (357) has the two trivial lines and sphere maps1 at \(\eta\) negative and u(t) at \(\eta\) positive. To check its interior class with the same local test, choose a smooth monotone \(\chi\) with \(\chi=0\) at the negative end, \(\chi=1\) at the positive end, \(\chi(0)=1/2\) and \(\chi'(0)>0\), and extend its sphere map to

\[
 a(t,\eta)=1-\chi(\eta)+\chi(\eta)e^{it}.
 \tag{369}
\]

Its unique zero is \((t,\eta)=(\pi,0)\). With \(\delta=t-\pi\) its linearization is \(-2\chi'(0)\eta-i\delta/2\). On a sufficiently small boundary its nonzero linearization and the exact map are joined by their straight path: the quadratic error is smaller than the least singular value of the real linear map times the boundary radius. Excision reduces the relative class to that local map. Multiplication by the constant unitary -1 and positive scaling give \(\eta+i\delta\). Circle translation from \(\pi\) to \(0\) is an orientation-preserving homotopy. Consequently the two actual finite relative triples agree, with their source/target order and complex phases, giving

\[
 \Theta_{TS^1}([u])=\kappa.
 \tag{370}
\]

No character or rationalization is used in this identification.

**Proof: the complete circle Dirac pairing.**

After the even pair-groupoid Morita equivalence, the positive circle Dirac class is the odd phase \(r=2P_+-1\) on \(L^2(S^1)\). It differs from \(h(D_t)\) by a compact Fourier-diagonal operator, including the chosen constant mode; hence it represents the full analytical Dirac class. Its commutator with a smooth scalar function is compact: finite Fourier polynomials give finite-rank commutators and uniform norm approximation treats all continuous functions.

To apply Proposition 0.15 to the coefficient cycle x just constructed, first use the complete even Morita equivalence from \(B'=\mathcal K(H)\otimes B\) to B. On B' the coefficient cycle is \((B'\widehat\otimes C_1,1,s\varepsilon)\), with the same multiplier s(t). The Morita pullback of the circle Dirac cycle acts nondegenerately on \(H\otimes L^2(S^1)\), with \(\phi(k\otimes b)=k\otimes M_b\) and honest involution1 tensor r. For every \(b'\in B'\) its commutator is compact by finite matrix Fourier approximation. Thus all hypotheses of Proposition 0.15 hold exactly. Its formula is the compression of \(w(t)=\exp(\pi i(s(t)+1))\), whose class in \(K_1(B')\) is the stable single corner \([u]\) by the positive coefficient calculation and both Morita evaluations. A \(K_1\) homotopy preserves the Fredholm compression index, since all compact-valued coefficient commutators remain compact. On that single corner the compression is \(P_+M_uP_+\), the shift with zero kernel and one constant cokernel. The full scalar classification and the same fixed native Clifford reducer therefore give

\[
 [u]\cdot\ell_{S^1,!}^{\mathrm{an}}=-1
 \quad\text{in }KK(\mathbb C,\mathbb C)=\mathbb Z.
 \tag{371}
\]

Both Morita products, rather than a trace of the pairing, were used.

**Lemma 7.28 (both native odd-block signs).** If \(p=\operatorname{rank}F\) is odd, the native external Thom product and the native external analytical Dirac product each equal minus the corresponding positive physical ordered-sum class for \(F\oplus TS^1\). Full spin-c phases, grading and both coefficient actions are retained.

**Proof.**

Let \(p\) be odd, and give \(F\oplus TS^1\) the full ordered structure F first, circle second. Write \(b_F\),\(b_C\) for the positive native parity Thom classes, \(b_{\mathrm{sum}}\) for the positive even ordered sum Thom class, and \(\ell_F\),\(\ell_C\),\(\ell_{\mathrm{sum}}\) for the corresponding analytical classes. After the ordinary even circle Morita equivalence, one has

\[
 b_F\boxtimes b_C=-b_{\rm sum},\qquad
 \ell_F\boxtimes\ell_C=-\ell_{\rm sum}.
 \tag{372}
\]

These signs follow from the full finite reducers, not from a choice of an ungraded tensor. Corollary 6.0d proves the raw Thom product on the physical module \(\mathscr S_F\widehat\otimes C_1\), right order \(C_p,C_1\). Lemma 7.21's native odd product instead has right order \(C_1,C_p\). The permutation has determinant \((-1)^p=-1\). Its orthogonal frame permutation is a positive transformation followed by one reflection. The positive part has an even spin lift; the reflection has an odd Pin lift. Tensoring that lift with the full finite spinor reducer gives an even onto unitary to the grading reversal of the physical ordered reducer. The right actions and Hermitian inner products agree by the full graded block transport, and an original transition \(zs\) is intertwined with its transported \(zs\), including its entire phase. For trivially graded left functions the grading reversal represents the negative class, as in Lemma 7.21. This proves the first identity, retaining any whole line twist.

For the second identity, apply the complete bounded submersion product of Theorem 7.13 to V times \(S^1\) projecting to V. The first vertical circle cycle and the second F cycle give a verified product on the physical module \(C_1\widehat\otimes\mathscr S_F\). Their native right list is \(C_p,C_1\), as in Lemma 7.21. The even full graded tensor flip \(\eta_C\widehat\otimes\eta_F\mapsto(-1)^{|\eta_C||\eta_F|}\eta_F\widehat\otimes\eta_C\) identifies this transported module with \(\mathscr S_F\widehat\otimes C_1\) in that same native right list. It preserves every right action and inner product by Lemma 0.13, and sends the positive physical sum phase to the phase for the specified order F,circle. Hence the native circle-first external product equals \(\ell_{\mathrm{sum}}\) after the ordinary even source and algebra function-factor flips and circle Morita. All full creations, the actual holonomy module topology and operator positivity were proved in Theorem 7.13; no unverified sum phase is inserted here.

Now interchange the two degree-odd external cycles to write the F-first native external product. Its ordered coefficient blocks are exchanged once, producing the same grading-reversed finite reducer just checked for the Thom product. Equivalently this is the degree-p graded external interchange \(\text{sign }(-1)^p\). Thus \(\ell_F\boxtimes\ell_C=-\ell_{\mathrm{sum}}\), proving the second identity with the full phases and prescribed structure. The ordinary function and pair-groupoid Morita flips are even and introduce no further sign. Smooth coefficient functions of the circle in the first creation sections are handled by finite separated sections and the compact commutators of its phase r; this is also the compact parameter construction of Theorem 7.13.

**Theorem 7.29 (the full longitudinal symbol index).** Let \((V,F)\) be a compact foliated manifold with its prescribed full spin-c longitudinal orientation, let \(G=\operatorname{Hol}(V,F)\) and \(A=C_r^*(G)\). For each \(j=0,1\) and every \(s\in K_c^j(F^*)\), its analytical index is the action of \(\ell_{V,!}^{\mathrm{an}}=\ell_{V,!}^{\mathrm{geom}}\) on the full Thom inverse of \(s\), as in (379). No transverse measure, rationalization or exclusion of torsion is needed. The statement retains the full orientation line, rank-zero grading, finite coefficients and compact smooth parameter families.

**Proof: odd-rank degree-zero symbols.**

Let s be any class in \(K_c^0(F^*)\), and \(a=\Theta_F^{-1}(s)\in K^1(V)\). Surjectivity and this exact inverse are the two global Thom identities of Theorem 6.0c with the prescribed full spin-c module. On the product foliation, (370), graded interchange and the first sign of (372) give

\[
 \begin{aligned}
 s\boxtimes\kappa
   &=(a b_F)\boxtimes([u]b_C)\\
   &=-(a\boxtimes[u])(b_F\boxtimes b_C)\\
   &=(a\boxtimes[u])b_{\rm sum}.
 \end{aligned}
 \tag{373}
\]

Thus the entire even-rank Corollary 7.3, applied on V times \(S^1\), identifies the full index of the left side with \((a\boxtimes[u])\ell_{\mathrm{sum}}\). Using the second sign of (372), the other graded interchange sign, and (371), this is

\[
 \begin{aligned}
 (a\boxtimes[u])\ell_{\rm sum}
  &=-(a\boxtimes[u])(\ell_F\boxtimes\ell_C)\\
  &=(a\ell_F)\boxtimes([u]\ell_C)\\
  &=-a\ell_F.
 \end{aligned}
 \tag{374}
\]

The full analytic circle product (358) says that the very same class is \(-\operatorname{Ind}_A(s)\). Cancellation in \(K_0(A)\), with no rationalization or trace, proves

\[
 \operatorname{Ind}_A(s)=\Theta_F^{-1}(s)\,
                  \ell_{V,!}^{\mathrm{an}}\in K_0(A)
 \qquad(p\text{ odd}).
 \tag{375}
\]

Corollary 7.3 proves this in even rank, including the full rank-zero orientation line. Theorem 7.25 replaces \(\ell^{\mathrm{an}}\) by the prescribed \(\ell^{\mathrm{geom}}\) as a full KK class in every parity. Consequently the two actions on \(K^*(V)\) agree in both degrees, and every ordinary longitudinal elliptic pseudodifferential bundle operator has its entire analytic \(K_0\) index given by that geometric construction.

**Proof: both normalized transverse suspension directions and degree-one symbols.**

Let \(S=C_0(\mathbb R_\eta)\). For a trivially graded separable coefficient algebra D let \(t_D\) and \(d_D\) be the external positive one-coordinate suspension class and its proved native inverse from Theorem 0.18 and Corollary 0.20. For a scalar class \(z\in K_j(D)=KK^j(\mathbb C,D)\), define \(\sigma_j(z)=z\cdot t_D\in K_{j+1}(S\otimes D)\), using the fixed even function-algebra flip when identifying \(S\otimes D\) and \(D\otimes S\). Both identities \(t_D\cdot d_D=1_D\) and \(d_D\cdot t_D=1_{SD}\), with the complete Clifford reducers, show that \(\sigma_j\) has inverse right product by \(d_D\). Thus no one-sided suspension argument is used.

These are the ordinary normalized stable suspension maps in the opening convention block, with their degrees kept distinct. For \(j=0\), apply scalar classification to a finite projection p and its scalar reference \(p_0\). Tensoring with the positive one-coordinate cycle gives the relative unitary \(u_+(\eta)^p u_+(\eta)^{-p_0}\), where \(u_+(\eta)\) has winding \(+1\). The increasing real-coordinate homotopy identifies this with \(\exp(2\pi i rp)\exp(-2\pi i rp_0)\) on the suspended interval. Thus \(\sigma_0\) is precisely the positive-loop Bott map. Naturality, finite matrix stability and subtraction of the scalar reference treat every nonunital D.

For \(j=1\), first take the universal unitary u in \(C(S^1)\). The cycle (368) and its explicit localization prove that \(\sigma_1[u]\) is \(\kappa\). In the ordinary doubled rotation path of KT-OPK lesson 8 equation (2.2), the unit first column is

\[
 \xi(r,t)=\begin{pmatrix}
      c(r)^2u(t)+a(r)^2\\
      c(r)a(r)(1-u(t)^{-1})
    \end{pmatrix},\qquad
 c(r)=\cos\bigl(\tfrac\pi2(1-r)\bigr),\quad
 a(r)=\sin\bigl(\tfrac\pi2(1-r)\bigr).
 \tag{376}
\]

Its projection \(e=\xi\xi^*\) equals the fixed \(P=e_{11}\) at \(r=0,1\). The finite line bundle e is globally framed by \(\xi\) on the parameter cylinder; its identity comparison to the reference line P has scalar map 1 on the negative end and u on the positive end. Therefore its full relative triple is \(\kappa\). This is exactly the class \([e]-[P]\) defining the ordinary suspension isomorphism \(\theta\) of lesson 8 Theorem 2.1. It follows that \(\sigma_1[u]=\theta[u]\) with coefficient \(+1\). Any finite unitary v over \(D^+\) is the image of this universal u under the unital homomorphism \(C(S^1)\) to its finite matrix algebra, \(u\mapsto v\). Naturality of both constructions, matrix stability and scalar reference subtraction prove \(\sigma_1=\theta\) for arbitrary D. The coordinate \(\eta\) is the new outer suspension coordinate; t is the inner circle coordinate. This states the order used in this sign identification.

Define the odd analytical symbol index by the ordinary suspended family boundary: \(\sigma_1\operatorname{Ind}_1(s)=\operatorname{Ind}_0^S(\sigma_1s)\). The index lift and its parametrices are the same finite matrix formulas (11)-(12) of the Hilbert-module lesson, now with continuous suspension coefficients. Their unitization and scalar comparison are retained. The positive exponential boundary requires a sign check; it is incorrect to assert that every extension boundary simply commutes with suspension. For any extension E, KT-OPK lesson 11 Theorem 1.1 proves the exact typed identity

\[
 \delta_{SE}\sigma_0=-\sigma_1\varepsilon_E,
 \qquad\varepsilon_E[e]=[\exp(2\pi i\widetilde e)],
 \tag{377}
\]

with scalar reference subtraction understood. Here \(\delta\) is the kernel-minus-cokernel index boundary and \(\varepsilon\) is the positive exponential boundary. Its proof uses the doubled lift \(D(r)U(r)^*\); the projection homotopy to \(U^*PU\) is the suspension of the inverse unitary, giving exactly the minus sign. For a relative odd symbol obtained from a sphere projection e, the disk/sphere boundary gives \(s=\varepsilon_{\mathrm{disk}}[e]\). Therefore \(\sigma_1s=-\delta_{S\mathrm{disk}}\sigma_0[e]\). The suspended even analytic index of this relative class is \(-\delta_{S\mathrm{analytic}}\sigma_0[e]=\sigma_1\varepsilon_{\mathrm{analytic}}[e]\). Thus it agrees with the positive, rather than negative, analytic exponential. This is the normalized cyclic-boundary convention in Theorem 0.14. For full relative triples, the same doubled lift is performed in their actual bundle corners with their boundary comparisons retained; the explicit relative reduction and both excision maps of lesson 11 Theorem 6.1 preserve these formulas. Equivalently their definition is the full suspended relative Fredholm family above. This retains classes not presented initially by a scalar sphere projection. The two boundary minus signs cancel before applying \(\sigma_1\)'s inverse.

One can work entirely on compact unit manifolds in the family step. Compactify the transverse parameter line to a based circle T. The foliation on V times T has tangent \(F\times\{0\}\), its holonomy groupoid is G times the unit groupoid T, and its algebra is \(C(T,A)\). The pullback full spin-c module and the phase chosen independent of T are precisely the external \(C(T)\)-coefficient analytical cycle. Their inner products agree pointwise and finite separated parameter sections are dense. The same statement holds for the Thom module. Apply the already proved all-rank \(K_c^0\) formula on this compact foliated manifold. The suspended symbol and its resulting index vanish as K-classes at the parameter basepoint. Evaluation splits by constant families, so split extension exactness gives the unique classes in the ideals \(C_0(T\setminus\{*\},C_0(F^*))\) and \(C_0(T\setminus\{*\},A)\). These are the suspended symbol and index classes. This constructs the actual coefficient family; it does not assert compactness only at individual parameters or require an exact arbitrary minimal tensor product.

Put \(a=\Theta_F^{-1}(s)\in K^{1-p}(V)\) for \(s\in K_c^1(F^*)\). Graded interchange with the right suspension class gives the same degree-p sign for the two typed actions:

\[
 \begin{aligned}
 \sigma_1(a b_F)
   &=(-1)^p(\sigma_{1-p}a)b_F^S,\\
 \sigma_1(a\ell_F)
   &=(-1)^p(\sigma_{1-p}a)\ell_F^S.
 \end{aligned}
 \tag{378}
\]

Here the superscript \(S\) is external transverse coefficients, not a new leaf direction. The equations follow by moving the odd right suspension class once past a degree-p morphism, keeping its actual Clifford blocks until the fixed native reduction; even associativity and the cyclic unitary of Lemma 0.13 make that reduction coherent. Thus \(\sigma_1(s)\) has Thom inverse \((-1)^p\sigma_{1-p}(a)\) on the family. The \(K_c^0\) formula just proved there gives \(\operatorname{Ind}_0^S(\sigma_1s)=(-1)^p(\sigma_{1-p}a)\ell_F^S=\sigma_1(a\ell_F)\). Applying the actual inverse right product by \(d_A\), justified by both inverse products, proves \(\operatorname{Ind}_1(s)=a\ell_F\in K_1(A)\).

Combining this with (375), Corollary 7.3 and the full prescribed KK equality of Theorem 7.25 gives the entire two-degree formula

\[
 \operatorname{Ind}_j(s)
    =\Theta_F^{-1}(s)\cdot\ell_{V,!}^{\mathrm{an}}
    =\Theta_F^{-1}(s)\cdot\ell_{V,!}^{\mathrm{geom}}
       \quad\text{in }K_j(A),\qquad
 s\in K_c^j(F^*),\quad j=0,1.
 \tag{379}
\]

Finite coefficients and compact smooth parameter families were retained in both product proofs and the full KK equality. A change of full spin-c structure by an even line H replaces the Thom inverse coefficient by \([H^*]a\) and the Dirac class by \([H]\ell\). Their actual finite module evaluation cancels \(H^*\otimes H\) in either parity, including 2-torsion; its determinant is not used. For rank zero the full line and, when specified, its grading give the same two finite evaluations, so the ordinary bundle or unitary index remains the input class. These statements compare a fixed symbol's index and do not assert that the Dirac class itself is independent of the supplied orientation.

![The positive circle coefficient, actual clutch maps and both odd sign cancellations](../figures/odd-symbol-clutch-signs.png)

**Figure 7.11.** Lemmas7.27–7.28 and Theorem 7.29, equations (366)–(379); Exercise 67, equation (381). The first graph shows the exact three local branches \(h(2\pi k+t)\), \(k=-1,0,1\), together with the full Fourier-frame reindexing and winding integral; all omitted branches remain in the proved cycle. The clutch panel records the actual localized map \(\eta+i h(t)\) and the linearization \(-2\chi'(0)\eta-i\delta/2\) in source order \((t,\eta)\) or \((\delta,\eta)\), with \(\chi'(0)=1/2\) for the illustrated sample. Their displayed negative Jacobians agree; the relative-class proof uses full index-projection localization and excision, not a zero count. The last panels retain the entire phase in the odd Pin transport, both paired interchange signs, and the minus in each positive exponential suspension boundary. The geometric circle product and the transverse suspension are distinct constructions. This original CC0 figure accompanies the complete proof; primary historical context is [Connes–Skandalis, Section 4].

Full-size clutch-and-signs PNG · Editable clutch-and-signs SVG.


### Comparing the analytic and geometric leaf maps

The geometric construction (16) gives \(\ell_!^{\mathrm{top}}\). The **longitudinal index theorem** identifies it with \(\ell_!^{\mathrm{an}}\) as a KK class and identifies the arbitrary analytic and topological symbol index maps. Connes and Skandalis proved this theorem; its appearance as a question in Connes' earlier survey records a historical stage. [Connes–Skandalis], Proposition 4.13 and Theorem 4.14, treat the factorization and symbol comparison; the author-hosted text is linked in the references.

Theorem 7.1 supplies the full general analytical cycle, and Corollary 7.3 its complete even-rank analytic symbol reduction. Section 6 supplies global Thom inverses, full compact-source geometric choices and ordinary-map composition. Lemmas 7.4–7.5 prove the local vertical-axis estimates; Lemmas 7.7–7.9 and Theorem 7.10 now prove the compact submersion product on the actual holonomy correspondence in both parities, including disconnected fibres and rank-zero orientation lines. Lemmas 7.11–7.12 and Theorem 7.13 extend the analytical product to nonproper submersions using a global bounded phase and compact coefficient localization. Lemmas 7.14–7.15 prove bounded open-domain locality and the whole fibre-preserving graph-tube identity, including its noncompact interior. Lemma 7.16 and Theorem 7.17 identify the coefficient-transferred Thom inverse on its actual full finite orientation and fibre-integration module. Lemmas 7.18–7.19 recover the full prescribed graph-normal module and give its ordered ambient coefficient comparison over the whole tube. Lemmas 7.20–7.21 calibrate both native Thom inverse products and the full ambient inverse, and Theorem 7.22 proves the compact ordinary analytical/geometric submersion equality in both parities with the supplied stable orientation. Lemmas7.23–7.24 and Theorem7.25 now prove the full prescribed transverse-normal KK comparison, through the actual cocycle/orientation family, one interval principal-space cycle and both bounded endpoints. Lemmas7.26–7.28 and Theorem7.29 prove the full arbitrary-symbol action in both K-degrees, with the actual circle product, native signs and both transverse suspension directions. Theorem 8.15d proves the globally framed solvable Dirac case below. Neither that special case nor an equality of traces proves the general leaf-map KK equality.

The separate **assembly question** asks when (17) is an isomorphism. Equality of two index constructions gives neither its injectivity nor its surjectivity. We discuss the geometric formulation here; comparison with the modern assembly map, especially coefficients, needs the appropriate groupoid model. Amenability hypotheses give important positive results, including the formulations studied by Androulidakis and Skandalis. [Androulidakis–Skandalis, *A Baum–Connes conjecture for singular foliations*](https://arxiv.org/abs/1509.05862).

For a foliation given by a submersion onto \(B\) with connected fibres, the fibre-pair groupoid is Morita equivalent to \(B\). The analytic target is \(K_c^*(B)\). The cycle description, with twist \(TB\), gives the same group through spin-c Poincaré duality. These two identifications explain the submersion case, conditional on the specified Morita and duality theorems.

## 8. Solvable actions and measured information

Suppose a simply connected solvable Lie group \(H\) acts on \(V\), and the holonomy groupoid of its orbit foliation is the transformation groupoid \(V\rtimes H\). Freeness suffices for this identification. Local freeness alone does not: stabilizers and ineffective holonomy must be checked.

Put \(d=\dim H\). A useful reduction is entirely geometric.

**Lemma 8.1.** A nontrivial simply connected solvable Lie group has a decomposition \(H=N\rtimes\mathbb R\), where \(N\) is simply connected and solvable of dimension \(d-1\). Consequently \(H\) is diffeomorphic to \(\mathbb R^d\).

**Proof.** Its Lie algebra \(\mathfrak h\) has a nonzero abelian quotient: otherwise \([\mathfrak h,\mathfrak h]=\mathfrak h\), contradicting termination of its derived series. Choose a nonzero linear functional on that quotient. It gives a surjective Lie algebra homomorphism \(\ell:\mathfrak h\to\mathbb R\). Let \(\omega\) be the corresponding left-invariant one-form on \(H\). Its exterior derivative on left-invariant fields is \(d\omega(X,Y)=-\ell([X,Y])=0\), so it is closed. Define \(\chi(h)\) by integrating \(\omega\) along a path from identity to \(h\). Simply connectedness and Stokes’ theorem on a homotopy of such paths make this independent of the path. Local path integration proves smoothness, and left invariance gives \(\chi(hk)=\chi(h)+\chi(k)\). Thus \(\chi:H\to\mathbb R\) is the required homomorphism, with differential \(\ell\). Choose \(X\) with \(d\chi(X)=1\). Then \(\chi(\exp(tX))=t\).

Every \(h\in H\) has the unique form \(n\exp(tX)\), with \(t=\chi(h)\) and \(n=h\exp(-tX)\in\ker\chi\). These maps are smooth, and the product is the semidirect product by conjugation. Thus \(H\cong N\times\mathbb R\) as a manifold. Since \(H\) is connected and simply connected, this product shows that \(N\) is connected and simply connected. Its Lie algebra is an ideal in a solvable Lie algebra and is solvable. Induction now gives the final assertion. \(\square\)

Lemmas 8.3 and 8.8 and Corollary 8.9 below directly identify the full and reduced transformation algebras, retaining the semidirect-product Haar factors. Iterated crossed products along Lemma 8.1, and the **Connes–Thom isomorphism**

\[
K_j(B)\cong K_{j+1}(B\rtimes_\alpha\mathbb R),
\tag{19}
\]

then give \(K_j(C_0(V))\cong K_{j+d}(A)\). Theorems 8.14–8.16 and Lemma 8.18 below prove a direct deformation formula for the full compact-support longitudinal symbol index, including noncompact manifolds. To identify the three historical maps with one another and with geometric assembly, one still needs their geometric and K-orientation comparison. Under these hypotheses, all three maps coincide and \(\mu\) is an isomorphism in this case. The analytic theorem (19) is proved below; the assembly and geometric index comparisons are further obligations.

For \(V\) with an invariant locally finite transverse measure, the tangent bundle of the orbit foliation is trivialized by the infinitesimal action of \(\mathfrak h\). Its positive-degree characteristic classes vanish. The measured index formula, together with the equality of the three maps, therefore gives

\[
\tau_{\Lambda*}(K_0(A))
=\left\{
\langle\operatorname{ch}(x),[C_\Lambda]\rangle:
x\in K_c^{d\bmod2}(V)
\right\}.
\tag{20}
\]

Only the parity that maps to \(K_0(A)\) occurs; for odd \(d\), use the odd Chern character. The measured symbol formula and the analytic isomorphism (19) are proved in these lessons. Theorems 8.14–8.16 below compare the actual longitudinal symbol index with a vector-group endpoint; Corollary 8.17 proves the compact case of (20), and Theorem 8.23 proves its noncompact extension. Neither numerical proof assumes assembly equality. The measure-existence parenthesis is proved in compact scope and explicitly refuted in noncompact scope below. A measured trace can lose information that K-theory retains.

Here is one general mechanism for that loss.

**Proposition 8.2 (trace scaling forbids finite-trace projections).** Let \(B\) have a faithful tracial weight \(\tau\). Suppose a norm-continuous path of automorphisms \(\theta_s\), starting at the identity, satisfies \(\tau\circ\theta_s=c_s\tau\), with \(c_s\ne1\) for some \(s\). Then \(B\) has no nonzero projection \(e\) with \(\tau(e)<\infty\).

**Proof.** A norm-continuous projection path has unitarily equivalent endpoints in the unitization. For a direct proof, if \(\|e-f\|<1\), the operator \(fe+(1-f)(1-e)\) is invertible and its polar unitary conjugates \(e\) to \(f\). Divide a compact parameter interval into subintervals with this norm bound and compose the unitaries. A trace has the same value on conjugate projections, so \(\tau(\theta_s(e))=\tau(e)\). Scaling gives \(c_s\tau(e)=\tau(e)\). Finiteness and \(c_s\ne1\) imply \(\tau(e)=0\); faithfulness implies \(e=0\). \(\square\)

To conclude that there are no projections at all, finiteness on projections must also be justified. For a densely finite lower semicontinuous trace it follows: approximate a projection \(e\) within \(1/2\) by a positive element \(a\) of finite trace. Then \(eae\ge e/2\), and \(\tau(eae)=\tau(a^{1/2}ea^{1/2})\le\tau(a)\), whence \(\tau(e)\le2\tau(a)<\infty\).

### Crossed products, duality and the real Thom map

We supply the analytic interfaces in (19). Let \(B\) be a separable C*-algebra and \(\alpha:\mathbb R\to\operatorname{Aut}(B)\) a pointwise norm-continuous action. Full crossed products use strongly continuous covariant pairs \((\rho,U)\), with
\(U_s\rho(b)U_s^*=\rho(\alpha_s(b))\).
Their dense convolution algebra is \(C_c(\mathbb R,B)\):

\[
(f*g)(s)=\int f(r)\alpha_r(g(s-r))\,dr,\qquad
f^*(s)=\alpha_s(f(-s)^*).
\tag{25}
\]

Its integrated norm in any covariant pair is at most \(\int\|f(s)\|\,ds\). For \(\mathbb Z\), replace integrals by counting sums and compact support by finite support. The full norm is the supremum of these norms. The reduced norm uses

\[
(\widetilde\rho(b)\xi)(t)=\rho(\alpha_{-t}(b))\xi(t),
\qquad (\lambda_s\xi)(t)=\xi(t-s).
\tag{26}
\]

One faithful nondegenerate \(\rho\) suffices. Indeed these operators first act adjointably on the Hilbert \(B\)-module \(L^2(\mathbb R)\otimes B\). Tensoring with a faithful \(\rho\) preserves their norm, by the general tensor-norm proof in Section 2 of the Hilbert-module lesson; tensoring with any other representation can only decrease it. This proves independence of the faithful model, including for nonunital \(B\).

**Lemma 8.3 (full and reduced norms).** Full and reduced crossed products coincide for \(\mathbb R\) and \(\mathbb Z\).

**Proof.** For a covariant pair \((\rho,U)\), define the isometry
\[
(W_Tv)(t)=(2T)^{-1/2}1_{[-T,T]}(t)U_{-t}v.
\]
It exactly intertwines \(\rho(b)\) with \(\widetilde\rho(b)\). The group generators satisfy
\[
\|\lambda_sW_T-W_TU_s\|^2
=\min(2,|s|/T).
\tag{27}
\]
This is the normalized measure of the symmetric difference of the two intervals. For a compactly supported \(f\), consequently,
\[
\|(\widetilde\rho\rtimes\lambda)(f)W_T
      -W_T(\rho\rtimes U)(f)\|
\le\int\|f(s)\|\min(2,|s|/T)^{1/2}\,ds\longrightarrow0.
\]
Taking norms and using that \(W_T\) is isometric gives
\(\|(\rho\rtimes U)(f)\|\le\|f\|_r\).
The reverse inequality is part of the definition. For \(\mathbb Z\), use the unit vector \((2N+1)^{-1/2}1_{\{-N,\ldots,N\}}\); the squared defect is
\(\min(2,2|k|/(2N+1))\).
The same argument proves equality. \(\square\)

**Lemma 8.4 (exactness and stability).** An invariant ideal \(I\subset B\) gives an exact crossed-product sequence. Crossing by a trivial action on \(\mathcal K\), or on a suspension coordinate, commutes with tensoring by that factor.

**Proof.** First use full crossed products. A nondegenerate representation of \(I\) extends to \(B\), since \(B\) acts by multipliers of \(I\); covariance extends as well. Thus the full crossed-product norm of an \(I\)-valued convolution kernel is unchanged when it is regarded as \(B\)-valued. Its closure embeds as an ideal: multiplication on both sides by the dense \(B\)-valued kernels stays \(I\)-valued. A covariant representation annihilates this ideal exactly when its coefficient representation annihilates \(I\). For the nontrivial implication, apply it to \(i\,h_n(s)\), where \(h_n\) is an integral-one compact approximate delta at zero; \(\int h_n(s)U_s\,ds\to1\) strongly. Hence \(\rho(i)=0\). The universal property now identifies the quotient with \((B/I)\rtimes\mathbb R\). Lemma 8.3 gives the same exact sequence for reduced products.

For a trivial compact-operator factor, finite matrix corners identify
\((M_n(B))\rtimes\mathbb R=M_n(B\rtimes\mathbb R)\).
This follows either by the covariant matrix-unit relations or directly from (26) and equality of the norms. Increasing corners approximate a compact-operator-valued continuous kernel uniformly on its compact group support, and hence in its integrated \(L^1\) norm. Their union is therefore dense in both algebras, proving the stable isomorphism.

For a trivial \(C_0(Y)\) factor, evaluation gives a field of the operators (26). Its norm is the supremum over \(y\): the regular Hilbert-module construction can use the faithful direct sum of point evaluations and faithful representations of \(B\). Finite sums \(h(y)f(s)\) are dense among compactly supported continuous kernels, by partitions of unity and uniform approximation on compact sets. They give precisely \(C_0(Y,B\rtimes\mathbb R)\). This proves the claim for \(Y=\mathbb R\), and more generally for locally compact second-countable \(Y\). \(\square\)

**Lemma 8.5 (translation kernels).** Put \((\tau_sF)(t)=F(t-s)\) on \(C_0(\mathbb R,B)\). Then
\[
C_0(\mathbb R,B)\rtimes_\tau\mathbb R
\cong B\otimes\mathcal K(L^2(\mathbb R)).
\tag{28}
\]
For \((\gamma_sF)(t)=\alpha_s(F(t-s))\), first apply the untwisting
\((jF)(t)=\alpha_{-t}(F(t))\) and obtain the same isomorphism.

**Proof.** Represent \(F\) by multiplication and the group by translation. A compact kernel \(g(s)(t)\) has integral kernel
\[
K_g(t,r)=g(t-r)(t).
\tag{29}
\]
The change of variables \((s,t)\leftrightarrow(t,r)\) preserves Lebesgue measure. Convolution becomes kernel composition, and involution becomes \(K^*(t,r)=K(r,t)^*\). Continuous compactly supported \(B\)-valued kernels are approximable, on one compact rectangle, by finite sums
\(b\,f(t)\overline{h(r)}\), \(f,h\in C_c(\mathbb R)\).
One may first approximate by finitely many \(B\)-valued samples and then by products of one-variable partitions. Uniform errors on that rectangle bound the integrated norm \(\int\sup_t\|g(s)(t)\|\,ds\), so this approximation also works in the full crossed-product norm.

On the finite-dimensional span of all \(f,h\) in one such sum, choose an orthonormal basis \(e_1,\ldots,e_n\). The kernels \(b\,e_i(t)\overline{e_j(r)}\) form the matrix algebra \(M_n(B)\), since integration gives the matrix-unit relations. Its map to the full crossed product is a bounded *-homomorphism, hence contractive. The multiplication/translation representation is faithful on this matrix algebra, so the norm is exactly its spatial matrix norm. Taking the dense union proves (28); it does not just prove an operator formula in one possibly nonfaithful representation. Finally \(j\gamma_s=\tau_sj\), by direct substitution, proving the twisted assertion. \(\square\)

**Theorem 8.6 (real Takai duality with signs).** Define the dual action by
\[
\widehat\alpha_q(b)=b,\qquad
\widehat\alpha_q(U_s)=e^{iqs}U_s.
\]
Then
\[
(B\rtimes_\alpha\mathbb R)\rtimes_{\widehat\alpha}\mathbb R
\cong B\otimes\mathcal K(L^2(\mathbb R)).
\tag{30}
\]
Under this isomorphism the multiplier generators act as
\[
b\mapsto M_{\alpha_{-t}(b)},\qquad
U_s\mapsto\lambda_s,\qquad
V_q\mapsto M_{e^{iqt}}.
\tag{31}
\]
The double-dual action with parameter \(v\) becomes
\(\alpha_v\otimes\operatorname{Ad}\lambda_{-v}\).

**Proof.** A covariant representation of the double crossed product is exactly a triple \((\rho,U,V)\) with
\[
V_q\rho(b)=\rho(b)V_q,\qquad
V_qU_s=e^{iqs}U_sV_q,
\]
and the original \(U\)-covariance. Fourier transformation identifies \(C^*(\mathbb R)\) with \(C_0(\mathbb R)\), sending the multiplier \(V_q\) to \(e^{iqt}\). This follows from Lemma 8.3 for the scalar trivial action and the Fourier diagonalization of regular convolution; its norm is the supremum of the Fourier transform. We use Lebesgue measure and the unitary Fourier normalization \((2\pi)^{-1/2}\); thus the inverse scalar transform has its usual matching normalization.

The commuting representations of \(B\) and \(C_0(\mathbb R)\) give a representation of \(C_0(\mathbb R,B)\) with its supremum norm. Here no maximal/minimal tensor assumption is hidden: for a finite tensor, approximate its scalar factors by a common nonnegative partition of unity and their sampled values. A represented sampled sum has the form \(\sum_i b_i h_i\), with commuting \(h_i\ge0\), \(\sum_i h_i\le1\); compression of the diagonal matrix \(\operatorname{diag}(b_i)\) by the column \((h_i^{1/2})\) bounds its norm by \(\max_i\|b_i\|\). Uniform scalar approximation proves the bound for the original tensor, hence the claimed continuous-function norm.

Conjugation by \(U_s\) acts on this algebra as
\(\gamma_sF(t)=\alpha_s(F(t-s))\), since
\(U_sV_qU_s^*=e^{-iqs}V_q\).
The same correspondence is reversible. The two universal crossed-product algebras are therefore isomorphic, with no appeal to a merely formal generator computation. Apply the untwisting and kernel isomorphism of Lemma 8.5. They give (30) and (31). In particular, for a doubly integrated test kernel written as \(\int f(s,q)U_sV_q\,ds\,dq\), the final kernel is
\[
K_f(t,r)=\alpha_{-t}\!\left(\int f(t-r,q)e^{iqr}\,dq\right);
\]
the group integral uses \(dq\), so no extra \(2\pi\) factor is inserted in this expression. If the test is instead written in the order \(V_qU_s\), its phase is \(e^{iqt}\), by the displayed commutation relation.

The double-dual action fixes \(b,U_s\) and multiplies \(V_q\) by \(e^{ivq}\). The action
\(\alpha_v\otimes\operatorname{Ad}\lambda_{-v}\)
does exactly this in (31): it shifts \(t\) to \(t+v\) and cancels the resulting \(\alpha_{-v}\) in the coefficient generator. Equality on the multiplier generators proves the assertion. Lemma 8.3 makes the full and reduced versions agree. \(\square\)

### The Wiener–Hopf argument

We give the remaining K-theory argument in the extension convention preceding Section 1. The construction follows the full proof in [Blackadar], Section 10.9, with its kernel and continuity steps supplied here. Write \(C=C_0(\mathbb R\cup\{+\infty\})\), \(\gamma_sF(t)=\alpha_s(F(t-s))\), and
\(\mathcal W_\alpha=(C\otimes B)\rtimes_\gamma\mathbb R\).
Lemmas 8.4–8.5 give the natural exact sequence
\[
0\longrightarrow B\otimes\mathcal K
\longrightarrow\mathcal W_\alpha
\longrightarrow B\rtimes_\alpha\mathbb R
\longrightarrow0.
\tag{32}
\]

We will use its physical representation on \(L^2(\mathbb R)\otimes B\), in which \(F\) acts by \(\alpha_{-t}(F(t))\), and the group translates \(t\). This representation is faithful. To check this, a regular representation induced from evaluation at any finite \(z\) is, after translation, the same physical Hilbert-module operator tensored with a faithful representation composed with \(\alpha_z\); its norm is unchanged by the tensor-norm theorem. The regular representation at \(+\infty\) is the strong limit of these finite-\(z\) representations as \(z\to+\infty\), first on compactly supported vectors and group kernels, using uniform convergence \(F(s,z+t)\to F(s,+\infty)\) on compact \((s,t)\). Its norm is consequently no larger. A countable dense set of finite evaluations together with \(+\infty\) is faithful on \(C\otimes B\); their regular direct sum computes the reduced norm. Lemma 8.3 now proves faithfulness of the physical representation. This also justifies the kernel norm estimates below.

**Theorem 8.7 (natural analytic Connes–Thom isomorphism).** The index and exponential boundaries of (32) are isomorphisms
\[
\partial_j:K_{j+1}(B\rtimes_\alpha\mathbb R)
\xrightarrow{\cong}K_j(B),\qquad j=0,1.
\tag{33}
\]
Thus \(\partial_j^{-1}\) is the natural degree-one real Thom map. This theorem does not identify it with an assembly or geometric wrong-way map.

**Proof.** First suppose \(B\) is unital. In the scalar translation Wiener–Hopf algebra let
\[
E(s,t)=e^{s/2-t}1_{t>0}1_{t-s>0},\qquad
F(s,t)=e^{-s/2}1_{s>0}1_{t-s>0},\qquad S=1-F.
\tag{34}
\]
These belong to the indicated algebra, despite their displayed jumps. The physical kernel of \(E\) is the rank-one projection onto
\(u(t)=e^{-t/2}1_{t>0}\), whose squared norm is one; approximate \(u\) in \(L^2\) by compact continuous functions and use (28). The kernel of \(F\) is
\(e^{-(t-r)/2}1_{t>r>0}\).
Smooth the two characteristic factors on \([0,\delta]\). The error where \(0<t-r<\delta\) has Schur norm \(O(\delta)\). The error where \(0<r<\delta\) has Hilbert–Schmidt norm \(O(\delta^{1/2})\), by integrating the squared exponential over \(r\) in that interval and \(t>r\). The smoothed kernels correspond to continuous coefficients vanishing at \(-\infty\), with a limit at \(+\infty\); cut off their exponentially small group tails. Faithfulness of the scalar physical representation proves convergence in the Wiener–Hopf norm.

Direct kernel composition gives, for \(t,r>0\),
\[
(F^*F)(t,r)=e^{-|t-r|/2},\qquad
(FF^*)(t,r)=e^{-|t-r|/2}-e^{-(t+r)/2}.
\]
Off the positive quadrant both vanish. Thus
\[
S^*S=1,\qquad SS^*=1-E.
\tag{35}
\]
Dilate \(E_\varepsilon(s,t)=\varepsilon^{-1}E(s/\varepsilon,t/\varepsilon)\), and likewise \(F_\varepsilon\); put \(S_\varepsilon=1-F_\varepsilon\). They have the same identities. Their positive-\(\varepsilon\) paths are norm continuous: for \(F_\varepsilon\) the difference is bounded by the \(L^1\) difference of the exponential convolution densities, and for \(E_\varepsilon\) by the \(L^2\) difference of the normalized vectors \(\varepsilon^{-1/2}u(t/\varepsilon)\).

Let \(\mu(b)\) be the constant-coefficient multiplier of \(\mathcal W_\alpha\), and form the unital algebra
\(D=\mathcal W_\alpha+\mu(B)+\mathbb C1\).
The quotient by \(\mathcal W_\alpha\) is the unital copy of \(B\), split by the constant multipliers. This coefficient copy is injective in the quotient: a nonzero constant coefficient cannot belong to a translation convolution algebra, since its kernel is a nonzero diagonal delta rather than a norm limit of integrated kernels. Equivalently its image at \(+\infty\) is the coefficient multiplier \(b\) of \(B\rtimes\mathbb R\); the following test proves it is outside that algebra. Twist the group generators by characters \(e^{iqs}\). On an integrated \(L^1\) kernel, its matrix coefficients tend to zero as \(|q|\to\infty\), by the scalar Riemann–Lebesgue lemma and density. A nonzero coefficient multiplier is fixed by these twists and has a nonzero matrix coefficient, so cannot be an algebra element. The same argument excludes the identity. Here \(\mu(1)=1\), and split exactness makes
\(K_0(\mathcal W_\alpha)\to K_0(D)\) injective.

Under (28), the ideal embedding \(b\mapsto b\otimes E_\varepsilon\) has raw kernel
\(\phi_\varepsilon(b)(s,t)=\alpha_t(b)E_\varepsilon(s,t)\).
Set
\[
\omega_\varepsilon(b)=S_\varepsilon\mu(b)S_\varepsilon^*,
\qquad \psi_\varepsilon(b)=\phi_\varepsilon(b)\in D.
\]
They are *-homomorphisms into the orthogonal corners \(1-E_\varepsilon\) and \(E_\varepsilon\), respectively. Hence
\(\mu_\varepsilon=\omega_\varepsilon+\psi_\varepsilon\)
is a *-homomorphism. The partial isometry \(S_\varepsilon\mu(p)\), for every matrix projection \(p\), proves
\((\omega_\varepsilon)_*=\mu_*\) on \(K_0\).
The paths in \(\varepsilon>0\) give
\((\mu_\varepsilon)_*=\mu_*+\psi_*\).

We check the endpoint \(\mu_\varepsilon\to\mu\) in point norm. Put
\[
d_\varepsilon(b)=\sup_{r\ge0}\|\alpha_r(b)-b\|e^{-r/\varepsilon},
\qquad
I_\varepsilon(b)=\varepsilon^{-1}\int_0^\infty
\|b-\alpha_{-s}(b)\|e^{-s/(2\varepsilon)}\,ds.
\tag{36}
\]
Both tend to zero. Split at \(r=\sqrt\varepsilon\): continuity handles the first part, and the exponential tail is bounded by \(2\|b\|e^{-1/\sqrt\varepsilon}\). For the integral the analogous bounds are twice the small-interval supremum and \(4\|b\|e^{-1/(2\sqrt\varepsilon)}\).
The commutator \([\mu(b),F_\varepsilon^*]\) has norm at most \(I_\varepsilon(b)\), by the exponential kernel row and column bounds. Further,
\[
\|\psi_\varepsilon(b)-E_\varepsilon\mu(b)\|
\le4d_\varepsilon(b)+I_\varepsilon(b).
\tag{37}
\]
Indeed its raw kernel is
\((\alpha_t(b)-\alpha_s(b))E_\varepsilon(s,t)\).
For \(s\ge0\), put \(t=s+r\), \(r\ge0\), obtaining the bound
\(d_\varepsilon(b)\varepsilon^{-1}e^{-s/(2\varepsilon)}\);
integration gives \(2d_\varepsilon(b)\).
For \(s<0\), use
\(\|\alpha_t(b)-\alpha_s(b)\|\le\|\alpha_t(b)-b\|+\|b-\alpha_s(b)\|\),
giving the other \(2d_\varepsilon(b)+I_\varepsilon(b)\).
These integrated supremum bounds are valid for the displayed measurable kernels by the physical regular representation and Minkowski's inequality. In particular the rescaled exponential in the first bound is \(e^{-r/\varepsilon}\), with the negative sign.
Using (35), (36) and (37) now yields
\[
\|\mu_\varepsilon(b)-\mu(b)\|
\le4d_\varepsilon(b)+2I_\varepsilon(b)\longrightarrow0.
\]
Thus the path extends at zero with value \(\mu\). Homotopy invariance gives
\(\mu_*=(\mu_\varepsilon)_*=\mu_*+\psi_*\), so \(\psi_*=0\).
Injection into the split algebra \(D\) proves that the ideal map
\(K_0(B)\to K_0(\mathcal W_\alpha)\) is zero. Exactness of (32) makes its index boundary surjective.

For nonunital \(B\), use its external unitization \(B^+\) with the action extended trivially on the scalar quotient. The natural diagram of (32) for \(B,B^+,\mathbb C\) has exact rows on the crossed-product and compact-ideal K-groups. The last two index maps are surjective by the unital argument, and the scalar one is an isomorphism: \(K_1(C^*(\mathbb R))=K_1(C_0(\mathbb R))=\mathbb Z\), \(K_0(\mathcal K)=\mathbb Z\), so a surjective homomorphism between them is an isomorphism. Also \(K_0(B)\to K_0(B^+)\) is injective, since the scalar unitization extension splits. Lift a given class in \(K_0(B)\) through the surjective middle index. Its scalar image is zero by the last-column injection; exactness supplies a lift from \(K_1(B\rtimes\mathbb R)\). Naturality and the first-column injection give the desired original class. Thus index is surjective for every \(B,\alpha\).

It remains to deduce injectivity and the other degree; surjectivity alone is not the theorem. It implies universally
\(K_1(B\rtimes\mathbb R)=0\Rightarrow K_0(B)=0\).
Apply this to the dual action on \(B\rtimes\mathbb R\), and use (30) and stability: \(K_1(B)=0\Rightarrow K_0(B\rtimes\mathbb R)=0\).
Suspend and use Lemma 8.4 and stable Bott periodicity to obtain also
\(K_0(B)=0\Rightarrow K_1(B\rtimes\mathbb R)=0\).
The cone algebra \(C\otimes B\) has zero K-theory: after identifying \(\mathbb R\cup\{+\infty\}\) with \((0,1]\), extend each function by zero at zero and contract by \(f(t)\mapsto f(ut)\), \(0\le u\le1\). The two universal implications therefore make
\(K_*(\mathcal W_\alpha)=0\).
Exactness of (32) proves both isomorphisms in (33).

An equivariant homomorphism of \(B\)'s gives a homomorphism of (32); the ideal identification (28) is natural. Naturality of the K-theory boundaries proves naturality of their inverses. The stable K/Bott/exactness interfaces fixed at the start of this lesson are the only K-theory inputs in this argument. No Kasparov product, inverse KK identity or assembly theorem has been assumed. \(\square\)

**Lemma 8.8 (semidirect products and the reduced norm).** For \(H=N\rtimes_\theta\mathbb R\) as in Lemma 8.1, an action on \(B\) gives
\[
B\rtimes H\cong(B\rtimes N)\rtimes_\beta\mathbb R,
\quad
\beta_t(b)=\alpha_{(e,t)}(b),\quad
\beta_t(U_n)=U_{\theta_t(n)}.
\tag{40}
\]
This holds for full products, and for reduced products with the corresponding reduced factors.

The normal subgroup need not be unimodular. Its convolution uses left Haar measure and involution \(f^*(n)=\Delta_N(n^{-1})\alpha_n(f(n^{-1})^*)\). The semidirect modular function is \(\Delta_H(n,t)=\Delta_N(n)m_t^{-1}\), with \(m_t\) below: right translation in the real coordinate multiplies the Haar integral by \(m_t\), and right translation in \(N\) contributes \(\Delta_N(n)^{-1}\). Automorphisms preserve \(\Delta_N\). Thus no unimodularity is inserted in either version of (40).

**Proof.** The full statement follows from the reversible correspondence between covariant representations of \(H\) and covariant triples for \(B,N,\mathbb R\), with the displayed conjugation relation. Strong continuity of \(\beta\) follows on compact \(N\)-kernels by the integrated \(L^1\) bound, a change of variables, and point-norm continuity of the coefficient action, then by density.

Let \(dn\) be left Haar measure on \(N\), and let
\[
m_t=|\det(d\theta_t)_e|
=\exp\!\left(t\,\operatorname{tr}(\operatorname{ad}X|_{\mathfrak n})\right).
\]
Then \(d(\theta_tn)=m_t\,dn\), and left Haar measure on \(H\), in the coordinates \((n,t)\), is \(m_t^{-1}\,dn\,dt\). The integrated action in (40) therefore has the exact density
\[
(\beta_tf)(n)=m_t^{-1}\alpha_{(e,t)}
                       (f(\theta_{-t}n)).
\tag{41}
\]
For reduced products use the unitary
\[
Z:L^2(H,m_t^{-1}dn\,dt;\mathcal H_\rho)
\longrightarrow L^2(\mathbb R\times N,dt\,dk;\mathcal H_\rho),
\quad (Z\xi)(t,k)=\xi(\theta_tk,t).
\tag{42}
\]
The Jacobian cancels the Haar factor, so no additional square root appears. A coefficient \(b\) becomes
\(\rho(\alpha_{k^{-1}}\alpha_{(e,-t)}(b))\).
Translation by \((n,0)\) becomes translation of \(k\) by \(\theta_{-t}(n)\); translation by \((e,s)\) becomes \(t\mapsto t-s\) with \(k\) fixed. These are exactly the iterated regular generators: at outer coordinate \(t\), the regular representation applies \(\beta_{-t}\) to the normal coefficient algebra. Thus the integrated norms agree, proving the reduced assertion. \(\square\)

**Corollary 8.9 (analytic solvable-group K-theory).** For every simply connected solvable \(d\)-dimensional Lie group \(H\) and every point-norm continuous action on \(B\), the full and reduced crossed products coincide. A chosen ordered decomposition of Lemma 8.1 gives natural isomorphisms
\[
K_j(B)\cong K_{j+d}(B\rtimes H).
\]

**Proof.** Induct on \(d\). At zero there is no crossed product or parity change. Apply Lemma 8.8 and the induction hypothesis to \(N\), and Lemma 8.3 to the remaining real action. The full iterated norm equals the reduced iterated norm, hence the full \(H\)-norm equals the reduced \(H\)-norm. This proves the norm equality actually needed here without leaving group amenability as an unproved substitute. Apply Theorem 8.7 at each step, in the chosen order. Every factor is natural under equivariant coefficient homomorphisms, so their composite is natural. This is the analytic K-group isomorphism. Independence of geometric wrong-way factorizations and equality with assembly require the further comparisons already separated in Section 7. \(\square\)

**Corollary 8.10 (positive cotangent normalization).** For the trivial real action, the Thom map of Theorem 8.7 agrees with the positive winding suspension generator when the group is Fourier transformed by
\(\widehat f(p)=(2\pi)^{-1/2}\int e^{-ipt}f(t)\,dt\).
This fixes its point Bott normalization in both stable K degrees.

**Proof.** At \(+\infty\), the isometry \(S\) of (34) has symbol
\[
u_-(p)=1-\int_0^\infty e^{-s/2}e^{-ips}\,ds
=\frac{p+i/2}{p-i/2}.
\]
Its index is \(\dim\ker S-\dim\ker S^*=-1\), by (35); its index boundary is consequently \(-[E]\). The inverse
\(u_+(p)=(p-i/2)/(p+i/2)\)
has positive winding, since
\[
\frac1{2\pi i}\int_{-\infty}^\infty u_+(p)^{-1}u_+'(p)\,dp
=\frac1{2\pi i}\int_{-\infty}^\infty
\frac{i}{p^2+1/4}\,dp=1.
\]
Its boundary is \(+[E]\), so \(\partial^{-1}(1)=[u_+]\). For a matrix projection \(e\) over \(B\), the coisometry \(S^*\otimes e+1\otimes(1-e)\) has index \(+[e]\); relative projections subtract the same formula for their scalar reference. Thus \(\partial^{-1}\) is the positive loop Bott map on \(K_0(B)\). Suspend, using Lemma 8.4 and natural stable suspension, to obtain the other degree. The positive-character coordinate used for the outer Fourier variable in (31) is distinct from this negative-sign physical transform; neither phase has been silently reversed. This proves normalization at the trivial action, not yet equality with a foliation assembly map. \(\square\)

![Real crossed-product norm, duality, Wiener–Hopf boundary, solvable Haar and minimality mechanisms](../figures/crossed-product-thom.png)

**Figure 8.10.** Exact mechanisms of Lemmas 8.3–8.8, Corollaries 8.9–8.10, Proposition 9.2 and Theorem 9.3. The real action on the separable C*-algebra \(B\) is pointwise norm continuous. Panel A uses Lebesgue measure and intervals of length \(2T\); its discrete counterpart uses \(2N+1\) counting points. In panel B the positive dual phase is \(e^{iqs}\), the displayed integrated kernel is written in the order \(U_sV_q\), and \(r\) is its input coordinate. Panel C uses the positive half-line, its unit vector \(u\), a rank-one defect and unitary dilation; the convergence is of the coefficient homomorphisms, not of \(F_\varepsilon\) in norm. Panel D fixes the physical negative-sign Fourier transform and the positive cotangent winding. Panel E retains the exact semidirect Haar Jacobian \(m_t\), which cancels in \(Z\) but remains in the integrated action \(\beta\). Panel F assumes a compact metrizable \(X\) and topological freeness; its quotient \(q\) is isometric on \(C(X)\), \(a\ge0\), and \(\delta>0\). Minimality then gives simplicity and full support of any invariant probability \(\nu\), so \(\tau_\nu=\nu\circ\mathsf E\) is faithful. The arrows indicate proved logical steps, not a geometric assembly identification. The integer trace range requires the separate suspension argument in Lemma 9.4, Proposition 9.5 and Theorem 9.6. Complete proofs and exercises accompany the figure. Source for the Wiener–Hopf construction: [Blackadar], Section 10.9.

Full-size PNG · Editable SVG.

### Compact transverse measures and the modular obstruction

The parenthesis in the source trace corollary deserves its own proof. For a compact manifold it is correct. Without compactness, existence of a nonzero transverse measure does not force unimodularity.

**Lemma 8.11 (an invariant probability for a solvable action).** Every continuous action of a simply connected solvable Lie group on a nonempty compact metrizable space \(X\) admits an invariant probability measure.

**Proof.** Induct on the dimension, using Lemma 8.1. For the trivial group, a point mass suffices. Write \(H=N\rtimes_\theta\mathbb R\). By induction the set \(\mathcal P_N(X)\) of \(N\)-invariant probability measures is nonempty. It is convex and weak-* compact: probabilities form a compact subset of the dual of \(C(X)\), and invariance is the intersection of closed equalities on continuous test functions. Normality of \(N\) implies that the outer real action preserves this set. Choose \(\mu\in\mathcal P_N(X)\) and average its translates:
\[
\mu_T=\frac1{2T}\int_{-T}^{T}(\exp(tX_0))_*\mu\,dt,\qquad T>0.
\]
This is again an \(N\)-invariant probability. For fixed \(s\in\mathbb R\) and \(f\in C(X)\), cancellation on the intersection of the intervals gives
\[
\left|\int f\,d((\exp(sX_0))_*\mu_T-\mu_T)\right|
\le\min(2,|s|/T)\|f\|_\infty.
\tag{50}
\]
Take a weak-* cluster point as \(T\to\infty\). The estimate makes it invariant under every real translate, and the closed \(N\)-invariance equalities remain true. It is invariant under their semidirect product \(H\). No full-support assertion is needed here. \(\square\)

**Proposition 8.12 (the compact unimodularity criterion).** Let a simply connected solvable \(d\)-dimensional Lie group \(H\) act smoothly and locally freely on a nonempty compact manifold \(V\). Its oriented orbit foliation has a nonzero locally finite invariant transverse measure if and only if \(H\) is unimodular. The orientation comes from an ordered basis of \(\mathfrak h\).

**Proof.** For a left action, write \(X^\#_x=(d/dt)_{t=0}\exp(tX)x\). Fundamental fields satisfy
\([X^\#,Y^\#]=-[X,Y]^\#\); this can be checked first on \(H\), where they are right-invariant, and then by the orbit maps. If \(X_1,\ldots,X_d\) is the chosen basis, local freeness makes
\[
E=X_1^\#\wedge\cdots\wedge X_d^\#
\]
a nowhere zero tangent \(d\)-vector. The bracket identity gives
\(\mathcal L_{X^\#}E=-\operatorname{tr}(\operatorname{ad}X)E\).
Choose a smooth ambient \(d\)-form \(\lambda\) with \(\lambda(E)=1\), for example by extending the dual leaf volume through an orthogonal projection onto \(F\).

A positive tangent current \(C\) is consequently integration against a positive measure:
\[
C(\omega)=\int_V\omega(E)\,d\mu,\qquad \mu(V)=C(\lambda).
\]
The current/transverse-measure correspondence in the [transverse-measure lesson, Sections 3 and 5](transverse-measures-of-foliations.md#section-3) applies to these locally finite currents. If the transverse measure is nonzero, then \(0<\mu(V)<\infty\). Compactness is used in this strict finite inequality.

Closedness and Cartan's formula give \(C(\mathcal L_{X^\#}\lambda)=0\). Indeed \(C(d\,\iota_{X^\#}\lambda)=0\); the restriction of \(\iota_{X^\#}d\lambda\) to the rank-\(d\) bundle \(F\) is zero because \(d+1\) tangent vectors are linearly dependent. Differentiating \(\lambda(E)=1\) now gives
\[
0=C(\mathcal L_{X^\#}\lambda)
=\operatorname{tr}(\operatorname{ad}X)\mu(V).
\tag{51}
\]
Thus every adjoint trace is zero. Left Haar measure has modular character
\(\Delta_H(\exp(tX))=\exp(-t\operatorname{tr}(\operatorname{ad}X))\): conjugation has Jacobian \(\det\operatorname{Ad}(\exp(tX))\), and translating left Haar on the right gives its reciprocal. A connected group is generated by its one-parameter subgroups, so these equalities make \(H\) unimodular.

Conversely, take the invariant probability \(\mu\) of Lemma 8.11 and define \(C\) by the displayed integral. It is positive and nonzero, since \(C(\lambda)=1\). Every smooth leafwise \((d-1)\)-form can be written
\(\eta=\sum_i f_i\iota_{X_i^\#}\lambda\). Leafwise Cartan's formula and the same differentiation of \(\lambda(E)=1\) show that
\[
(d_F\eta)(E)=
\sum_i\left(X_i^\# f_i+
\operatorname{tr}(\operatorname{ad}X_i)f_i\right).
\]
The trace terms vanish by unimodularity. Invariance of \(\mu\) gives
\(\int X_i^\#f_i\,d\mu=0\), by differentiating the integral of the translated smooth function; its derivative is uniformly bounded on compact \(V\). Therefore \(C\) is closed. The proved current correspondence supplies a nonzero locally finite transverse measure. For \(d=0\), the group is trivial and every nonzero finite measure is a transverse measure; the criterion holds directly. \(\square\)

**Example 8.13 (why the noncompact parenthesis is false).** Consider the simply connected affine group
\[
H=\mathbb R\rtimes\mathbb R,\qquad
(n,t)(n',t')=(n+e^t n',\,t+t').
\]
It acts freely on \(V=H\) by left multiplication, with one leaf. A point transversal has counting mass one, so it defines a nonzero invariant transverse measure. Yet
\(\Delta_H(n,t)=e^{-t}\), so \(H\) is not unimodular.

In coordinates the fundamental fields are \(N^\#=\partial_n\) and
\(T^\#=n\partial_n+\partial_t\). Their bracket is \([T^\#,N^\#]=-N^\#\); in the Lie algebra \([T,N]=N\), so \(\operatorname{tr}(\operatorname{ad}T)=1\). The oriented vector \(N^\#\wedge T^\#\) is \(\partial_n\wedge\partial_t\). The current integrates compactly supported two-forms over \(\mathbb R^2\). Its mass relative to \(\lambda=dn\wedge dt\) is infinite, and \(\lambda\) is not a compactly supported test form. Thus the finite-mass step in (51) is unavailable. For compactly supported \(f\), the two terms in
\(\int(T^\#f+f)\,dn\,dt=0\) cancel by integration by parts; no contradiction occurs.

The orbit groupoid is the pair groupoid \(H\times H\). Its algebra is \(\mathcal K(L^2(H))\), with any fixed smooth positive leaf density, by the compact-kernel argument in the C*-algebra lesson. The point-normalized measured trace is ordinary operator trace. This example satisfies the graph condition of the source theorem and refutes its unrestricted measure-existence parenthesis. Proposition 8.12 gives the complete compact correction.

### Deforming an ordered solvable group

The analytic Thom map already gives an abstract K-isomorphism. We now compare the actual longitudinal symbol index with a vector-group endpoint. Ordinary exponential coordinates are not required.

**Theorem 8.14 (a solvable parameter algebra with invertible evaluation K-maps).** Choose the ordered decomposition of Lemma 8.1 for a simply connected solvable group \(H\) of dimension \(d\). There is a smooth family of groups \(H_v\), \(0\le v\le1\), with \(H_0=\mathbb R^d\) and isomorphisms \(\Gamma_v:H_v\to H\) for \(v>0\). For every point-norm continuous action \(\alpha\) on a C*-algebra \(B\), there is a central \(C([0,1])\)-algebra \(\mathfrak A_H(B)\) with fibres
\[
\mathfrak A_H(B)_v=B\rtimes_{\alpha^{(v)}}H_v,\qquad
\alpha^{(v)}_h=\alpha_{\Gamma_v(h)}\ (v>0),\quad
\alpha^{(0)}_h=\mathrm{id}.
\]
Every evaluation induces an isomorphism on both K-groups. In particular,
\[
K_*(\mathfrak A_H(B)_0)=K_*(B\otimes C_0(\mathfrak h^*)),
\qquad
K_*(\mathfrak A_H(B)_1)=K_*(B\rtimes H).
\tag{52}
\]

**Proof of the smooth family.** Write \(H=N\rtimes_\theta\mathbb R\), and let
\(\Phi_N:\mathbb R^{d-1}\to N\) be the ordered coordinate diffeomorphism supplied recursively by Lemma 8.1, with inverse \(\Psi_N\). For \(v\ne0\) set
\[
q*_v q'=\frac1v\Psi_N(\Phi_N(vq)\Phi_N(vq')),
\qquad
\theta_t^v(q)=\frac1v\Psi_N\!\left(\theta_{vt}(\Phi_N(vq))\right).
\tag{53}
\]
At zero these expressions extend to \(q+q'\) and \(q\), respectively. Here is the smooth extension, including its derivatives. Each numerator is a smooth function \(F(v,z)\) that vanishes at \(v=0\), and
\[
F(v,z)/v=\int_0^1\partial_vF(rv,z)\,dr.
\]
Differentiation under this integral proves smoothness. The derivative at zero of the first numerator is \(q+q'\), and that of the second is \(q\), since \(\Phi_N\) has identity tangent map at zero and \(\theta_0\) is identity. The inverse is obtained by the same formula with \(\Phi_N(vq)^{-1}\). Associativity, inverses, and the automorphism and action identities hold for \(v\ne0\), hence also at zero by continuity.

Put \(H_v=N_v\rtimes_{\theta^v}\mathbb R\). Then
\[
\Gamma_v(q,t)=\Phi_N(vq)\exp_H(vtX_0)
\tag{54}
\]
is an isomorphism for \(v\ne0\). Its differential is multiplication by \(v\), so the bracket on the fixed tangent space is \(v[\ ,\ ]_{\mathfrak h}\); \(H_0\) is the additive vector group. All formulas also extend to negative \(v\), which supplies smooth extensions across the boundary at zero. The maps \(h\mapsto\Gamma_v(h)\) converge to the identity of \(H\) uniformly on compact coordinate sets. Thus the induced actions are jointly point-norm continuous, including at zero. For a smooth action on a manifold they are jointly smooth.

Normalize left Haar densities to have value one in the ordered coordinates at the identity. If \(h_N(q)\,dq\) is the original density on \(N\), the density on \(N_v\) is \(h_N(vq)\,dq\). The outer automorphism modulus and the full Haar density are
\[
m_v(t)=\exp\!\left(vt\,\operatorname{tr}
(\operatorname{ad}X_0|_{\mathfrak n})\right),\qquad
d\mu_v(q,t)=m_v(t)^{-1}h_N(vq)\,dq\,dt.
\tag{55}
\]
These follow by pulling back Haar through (54) and dividing out \(|v|^d\); at zero they are Lebesgue measure. The normal modular character is retained:
\(\Delta_{H_v}(q,t)=\Delta_{N_v}(q)m_v(t)^{-1}\). Nothing here assumes that \(N\) or \(H\) is unimodular.

**Proof of the algebra and K-maps.** Construct it by induction. For the zero-dimensional group put \(\mathfrak A_0(B)=C([0,1])\otimes B\). Suppose the normal family algebra \(\mathfrak A_N(B)\) has been constructed, with a dense algebra of jointly continuous compact-coordinate kernels \(f_v(q)\). On these kernels define
\[
(\beta_t f)_v(q)=m_v(t)^{-1}
\alpha_{\exp_H(vtX_0)}
\bigl(f_v(\theta_{-t}^v(q))\bigr).
\tag{56}
\]
Lemma 8.8, applied in each fibre, proves that this preserves convolution and the modular involution. Its inverse is \(\beta_{-t}\).

For clarity, a central interval algebra has norm equal to the supremum of its fibre norms: an irreducible nondegenerate representation extends to the central multiplier \(C([0,1])\), where Schur's lemma makes it evaluation at one point. It factors through that fibre. Irreducible representations determine a C*-norm. This also proves that a map which is isometric in every fibre is isometric in the whole algebra. Thus (56) extends to an automorphism.

It is point-norm continuous. On a compact kernel support, its transformed supports for \(t\) in a compact interval are contained in one compact coordinate set, by the smooth maps (53). The Haar densities and their derivatives are bounded there, uniformly in \(v\). Joint continuity of the coefficient action and the compact range of \(f\) give uniform convergence as \(t\to t_0\). The integrated estimate
\[
\|f\|\le\sup_{0\le v\le1}\int_{H_v}\|f_v(h)\|\,d\mu_v(h)
\tag{57}
\]
therefore proves continuity on this dense algebra, and isometry extends it to the completion.

Define
\(\mathfrak A_H(B)=\mathfrak A_N(B)\rtimes_\beta\mathbb R\).
The parameter multipliers are fixed by \(\beta\), so remain central and nondegenerate. Real crossed-product exactness from Lemma 8.4 identifies the quotient at \(v\) with
\((B\rtimes N_v)\rtimes_{\beta^v}\mathbb R\).
Lemma 8.8 identifies this with \(B\rtimes H_v\), preserving (55). These are also the reduced algebras by Corollary 8.9. Finite sums of separated compact kernels in the normal and outer coordinates are dense: on any compact rectangle, uniform approximation of a continuous \(B\)-valued kernel by such sums gives integrated-norm approximation. This proves the asserted joint compact-kernel description and (57). The central-character argument gives the fibre-supremum norm at this inductive step as well.

At dimension zero, evaluation is a K-isomorphism because contraction of the interval is a homotopy inverse to the constant inclusion. At the next step, evaluation is equivariant for \(\beta\) and its fibre action. The natural analytic real Thom isomorphism of Theorem 8.7 gives the commutative square
\[
\begin{array}{ccc}
K_j(\mathfrak A_N(B))&\longrightarrow&
K_{j+1}(\mathfrak A_N(B)\rtimes\mathbb R)\\
\downarrow(\mathrm{ev}_v)_*&&\downarrow(\mathrm{ev}_v)_*\\
K_j(B\rtimes N_v)&\longrightarrow&
K_{j+1}((B\rtimes N_v)\rtimes\mathbb R).
\end{array}
\]
The horizontal arrows and the left arrow are isomorphisms; hence so is the right arrow. Induction proves the assertion for every \(v\), in both degrees. At zero, Fourier transformation with character \(e^{-i\langle p,h\rangle}\) gives (52). This proof uses naturality, rather than an inference from continuous-field terminology.

There is also an explicit positive-parameter trivialization. Under \(\Gamma_v\), a kernel becomes
\[
f_v(h)\ \longmapsto\
v^{-d}f_v(\Gamma_v^{-1}(h))\quad(v>0)
\tag{58}
\]
in the original \(H\) crossed product, because \((\Gamma_v)_*\mu_v=v^{-d}\mu_H\). On closed intervals away from zero, smooth changes of variables and the integrated estimate show norm continuity. Compact kernels there are dense on both sides; central cutoffs approximate the full ideal vanishing at zero. Consequently
\(\ker\mathrm{ev}_0\cong C_0((0,1])\otimes(B\rtimes H)\).
This cone description is consistent with the preceding proof; it does not alone prove invertibility of evaluation at one. \(\square\)

### The parameter-wide symbol index

**Corollary 8.14a (both KK inverses for every evaluation).** In Theorem 8.14, suppose in addition that \(B\) is separable. Every evaluation
\[
q_v:\mathfrak A_H(B)\longrightarrow B\rtimes H_v
\]
is a KK equivalence. The inverses are natural under equivariant coefficient *-homomorphisms and are obtained in the fixed ordered real-factor decomposition. The conclusion concerns every \(v\in[0,1]\), including zero; it makes no unimodularity or freeness assumption.

**Proof.** All coefficient algebras in the induction are separable. At the first stage this follows from separability of \(C([0,1])\otimes B\). At a real crossed-product stage, continuous compactly supported kernels can be approximated in integrated norm by piecewise linear scalar functions with rational break points times a countable dense subset of the coefficient algebra. Their rational linear span is countable and dense. The natural real KK Thom class and its two-sided inverse in Lemma 0.17 and Theorem 0.18 therefore apply at every stage. We use the ordered Clifford contractions of Corollary 0.20 throughout.

Here is the precise inductive step. Write \(I\) for the preceding interval algebra, \(C\) for its fibre at \(v\), and \(e:I\to C\) for evaluation. The outer actions \(\beta\) on \(I\) and \(\beta^v\) on \(C\) are the point-norm continuous actions (56). Evaluation is equivariant, and its crossed-product map is
\[
e_{\mathbb R}:I\rtimes_\beta\mathbb R
\longrightarrow C\rtimes_{\beta^v}\mathbb R.
\]
Suppose \([e]\) has a two-sided inverse \(c\in KK(C,I)\). This is an ordinary KK inverse; it need not be represented by an equivariant *-homomorphism. Put
\[
\begin{aligned}
t_I&\in KK_1(I,I\rtimes_\beta\mathbb R),&
d_I&\in KK_1(I\rtimes_\beta\mathbb R,I),\\
t_C&\in KK_1(C,C\rtimes_{\beta^v}\mathbb R),&
d_C&\in KK_1(C\rtimes_{\beta^v}\mathbb R,C).
\end{aligned}
\]
Theorem 0.18 gives \(t_Id_I=1_I\), \(d_It_I=1_{I\rtimes\mathbb R}\), and both corresponding identities for \(C\). Products here and below retain their intermediate coefficient algebra; their compact notation does not permute graded factors. Lemma 0.17 gives the actual naturality identity
\[
t_I[e_{\mathbb R}]=[e]t_C.
\]
Multiplying on the left by \(d_I\), and using associativity, yields
\[
[e_{\mathbb R}]=d_I[e]t_C.
\]
Consequently an explicit candidate inverse is
\[
x=d_Cct_I\in KK(C\rtimes\mathbb R,I\rtimes\mathbb R).
\]
It is even: the two odd real factors contribute total degree two in the fixed periodic convention. Both products are
\[
\begin{aligned}
[e_{\mathbb R}]x
 &=d_I[e](t_Cd_C)ct_I
  =d_I([e]c)t_I
  =d_It_I
  =1_{I\rtimes\mathbb R},\\
x[e_{\mathbb R}]
 &=d_Cc(t_Id_I)[e]t_C
  =d_C(c[e])t_C
  =d_Ct_C
  =1_{C\rtimes\mathbb R}.
\end{aligned}
\]
These are identities of KK classes. No passage from K-group isomorphisms to KK invertibility is used.

At dimension zero, \(e_v:C([0,1])\otimes B\to B\) has the constant inclusion \(c_v\) as inverse. One composition is exactly the identity. The other is homotopic to the identity by
\[
f(w)\longmapsto f\bigl((1-s)v+sw\bigr),
\qquad 0\le s\le1.
\]
Uniform continuity of a \(B\)-valued continuous function on the interval proves that this is a norm-continuous *-homomorphism homotopy. It is valid for nonunital \(B\). The zero algebra also satisfies the assertion, with its identity class zero. Induction using the preceding step proves both inverse products for every \(q_v\).

The accumulated formula is useful. Let \(\Theta_I\in KK_d(C([0,1])\otimes B,\mathfrak A_H(B))\) be the product of the real Thom classes in their induction order, and \(D_I\) their inverses in reverse order. Let \(\Theta_v\in KK_d(B,B\rtimes H_v)\), \(D_v\) be the fibre products with the same order and contractions. Cancellation of adjacent inverse pairs proves both inverse identities for \(\Theta_I,D_I\) and for \(\Theta_v,D_v\); no factors have been exchanged. Iterated naturality gives
\[
[q_v]=D_I[e_v]\Theta_v,
\qquad
[q_v]^{-1}=D_v[c_v]\Theta_I.
\]
The same adjacent-pair cancellation proves both products of these formulas are identity. This formula also proves coefficient naturality: apply real Thom naturality successively at each stage, and use that interval evaluation and the constant inclusion commute with the coefficient homomorphism. The inverse of a natural equivalence is natural, as follows by multiplying its naturality square by the two inverses.

In the conventions of Theorem 0.18, each \(d\) is the negative dual Thom class followed by the even Takai Morita class. Its sign is already fixed by the scalar compression index \(-1\). The ordered contractions in Corollary 0.20 are used in the two accumulated products. Thus no additional sign or orientation choice is hidden in these formulas. At zero all real actions are trivial, including the outer actions (56). The class \(\Theta_0\) is the native ordered parity iteration in the Fourier coordinates of (52). Corollary 0.20 separately supplies the typed ordinary graded source-Clifford inverse pair. Their exact finite reduction and the comparison with the positive physical Bott symbol are checked in Lemma 8.15c below. \(\square\)

In particular there is an even, natural deformation class
\[
\mathcal I_H=[q_0]^{-1}\otimes_{\mathfrak A_H(B)}[q_1]
\in KK(B\otimes C_0(\mathfrak h^*),B\rtimes H),
\]
where the displayed intermediate algebra is understood by the maps' domains: \([q_0]^{-1}\) goes from the zero fibre to the interval algebra. Its inverse is \([q_1]^{-1}[q_0]\). Both products cancel the adjacent evaluation pairs. The formula can also be written
\[
\mathcal I_H=D_0\Theta_1,
\qquad \mathcal I_H^{-1}=D_1\Theta_0,
\]
because \([c_0][e_1]=1_B\) and \([c_1][e_0]=1_B\). For a smooth action satisfying the graph hypothesis of Theorem 8.16, its action on both K-groups is the actual compact-support symbol index proved there and in Lemma 8.18. Indeed those proofs identify that index with the two evaluation maps, after the same Fourier identification at zero; the definition of the action of a KK class is their product. This supplies a KK lift of the analytic deformation map. Lemmas 8.15a–c and Theorem 8.15d below construct the parameter-wide Dirac cycle and identify its class for the ordered global frame. The historical geometric leaf-map and assembly identification remain further assertions.

![Two inverse products for crossed-product evaluation](../figures/solvable-kk-evaluation.png)

**Figure 8.14a.** The exact induction in Corollary 8.14a. Here \(I\) is the preceding interval algebra, \(C\) its fibre, and \(e\) its equivariant evaluation. Horizontal arrows are odd KK equivalences; vertical arrows are actual even evaluation homomorphisms. The naturality identity determines the crossed evaluation and its inverse. The two product calculations retain the factor order. At dimension zero the ordinary coefficient inverse is the constant inclusion, with its interval contraction homotopy. The diagram makes no geometric leaf-map or assembly identification. The real-action method is due to Fack and Skandalis; the bounded inverse and Fourier-sign proof used here are Lemmas 0.16–0.17 and Theorems 0.18–0.19.

Full-size PNG · Editable SVG.


**Lemma 8.15 (uniform calculus through the zero anchor).** Suppose \(V\) is compact and the \(H\)-action is smooth. The family transformation groupoid
\[
\mathcal G=\coprod_{v\in[0,1]}(V\rtimes H_v)
\]
has a compact-support source-fibre pseudodifferential calculus, with principal bundle
\(V\times[0,1]\times\mathfrak h^*\). Its order-zero operators are multipliers of \(\mathfrak A_H(C(V))\); negative-order operators belong to that algebra. Completion gives the exact scalar symbol sequence
\[
0\longrightarrow\mathfrak A_H(C(V))
\longrightarrow\overline\Psi^0(\mathcal G)
\xrightarrow{\sigma}
C(V\times[0,1]\times S^{d-1})\longrightarrow0.
\tag{59}
\]
Bundle corners have the corresponding assertion. At \(v=0\) the anchor is zero, while the source fibres still have dimension \(d\); this causes no degeneration in these estimates.

In rank zero, interpret \(S^{-1}\) as the empty cosphere. The family algebra is \(C(V\times[0,1])\), every kernel is smoothing, and (59) has zero symbol quotient. The following proof treats \(d\ge1\).

**Proof.** The source fibre at \((x,v)\) is \(H_v\). Use the fixed smooth coordinates near its identity from (53)–(54), and right-translate those coordinates to any point \(h_0\) of the fibre. The relative group coordinate of \(h=uh_0\) and \(h'=u'h_0\) is \(uu'^{-1}\), independent of \(h_0\). In the regular representation a coefficient \(b\) is evaluated at \(h x\), so on this patch its base point is \(h_0x\) followed by the small action of \(u\). This base point remains in compact \(V\). The needed derivatives in \(u\), the relative coordinate changes, and the normalized densities have bounds uniform in \(v\) and in this base point. Parameter derivatives of the original kernel are taken on its fixed compact groupoid charts; we do not assert bounded \(v\)-derivatives of a faraway moving point \(h_0x\). Such derivatives are not used in the regular norm bound. Right translation of left Haar introduces one constant modular factor for the patch; normalizing its \(L^2\) coordinate map cancels that constant. The remaining local factors are smooth positive densities on the small chart. No derivative of an inverse anchor, and no injectivity of the anchor, is used.

Here is why local estimates give a uniform bound on the entire, possibly noncompact, fibre. Equip \(H_v\) with the right-invariant metric specified by the fixed tangent basis. A sufficiently small coordinate ball and its slightly larger buffer work for every \(v\), since the near-identity metric coefficients depend smoothly on a compact parameter interval. Choose a maximal separated set and translated bump functions, then normalize them to a square partition \(\sum_i\phi_i^2=1\). Its overlap number and derivative bounds are uniform. Indeed disjoint smaller balls have uniformly bounded positive local volume; a finite cover of a buffered identity ball bounds their possible number in that ball. Right translation preserves the metric and changes Haar by a constant, so the same ratio bound holds on every translated ball.

For a kernel with a fixed compact relative group support \(K\), each input patch meets only a uniformly bounded number of output patches, and conversely. To verify this without a global coordinate-growth assumption, cover the compact set \(K\), and its products with the fixed small coordinate buffers, by finitely many small coordinate balls, uniformly in \(v\). Their products are compact because group multiplication is smooth on the compact parameter interval. The preceding packing bound applies to this finite cover. If all localized blocks have norm at most \(C\) and each row and column has at most \(N\) blocks, Cauchy–Schwarz gives norm at most \(NC\) for the matrix of blocks. The square partition is an isometric localization and a contractive reconstruction, so this is the bound for the original operator.

On each such patch, a compact-support conormal kernel reduces, by a smooth phase change, to
\[
(2\pi)^{-d}\int e^{i(z-z')\cdot p}
a(z,p;x,v)\,dz'\,dp.
\]
The phase change uses the integrated differential of the relative coordinate map; it is invertible on the fixed small buffer even at \(v=0\). Half-density and bundle changes are smooth bounded factors. The symbol estimates are
\[
\|\partial_z^\gamma\partial_p^\eta a\|
\le C_{\gamma\eta}\langle p\rangle^{m-|\eta|},
\]
uniformly in \(x,v\) and the translated patches. Integrating by parts in the compact \(z\)-variable bounds its Fourier kernel by
\(C_L\langle\zeta-p\rangle^{-L}\langle p\rangle^m\).
For \(m=0\), choose \(L>d\) and apply the two-sided Schur bound, exactly as in the [index lesson, Lemma 6.1](the-index-theorem-for-measured-foliations.md#section-6). The finite-overlap argument just given makes this a uniform regular norm bound.

The same Taylor and integration-by-parts computation as in that lesson's Lemma 6.2 proves composition and adjoints: the local expansion is
\(\sum_\gamma(\partial_p^\gamma a)(D_z^\gamma b)/\gamma!\), and after \(L\) terms the remainder has order \(m+n-L\). All required derivatives and supports are uniform in \(v\). Away from the units a conormal kernel is smooth. Composable products of two fixed compact arrow supports are compact, since the source-range equality is closed in the Hausdorff unit manifold. Thus the calculation globalizes. Smooth kernels remain smooth under these products, also at zero.

For \(m<0\), truncate the symbol at \(|p|\le R\), keeping the same compact physical kernel cutoff. The error's finitely many order-zero seminorms are at most \(C R^m\); derivatives of the frequency cutoff satisfy the same bound. Therefore its regular norm tends to zero uniformly in \(x,v\). The truncated kernels are jointly smooth compact kernels in the dense algebra of Theorem 8.14. The fibre-supremum norm proves membership in \(\mathfrak A_H(C(V))\). Multiplying an order-zero kernel on either side by a smooth compact kernel gives a smooth compact kernel. The norm bound and the adjoint bound extend these two products to a bounded adjointable multiplier. Finite bundle embeddings into trivial bundles give the identical statements in bundle corners.

We include the quotient-norm check. If \(c=\|\sigma(P)\|_\infty\), quantize the smooth positive square root of
\((c^2+\varepsilon)I-\sigma(P)^*\sigma(P)\).
Composition gives
\(P^*P+Q^*Q=(c^2+\varepsilon)I+R\), with \(R\) of negative order in the algebra. Hence \(\|P+\mathfrak A_H(C(V))\|\le c\).
For the opposite inequality, fix \(x,v\) and a unit covector at the identity source chart. A unit wave packet of radius \(h^{1/2}\) and phase \(e^{i\langle z,p_0\rangle/h}\) has
\(\|Pu_h-\sigma(P)(x,v,p_0)u_h\|_2\to0\). Rescaling the packet and using degree-zero homogeneity proves this on bounded rescaled frequencies; the rapidly decreasing Fourier tail and the order-zero bound control the rest. Every smooth compact kernel takes this packet to zero in norm, by integration by parts, and algebra-norm approximation gives the same result for every element of \(\mathfrak A_H(C(V))\). This argument works in the additive source fibre at zero too. Taking the supremum proves equality of quotient and symbol norms. Smooth symbols are dense, and every one is quantizable; the norm identity proves surjectivity and that completion introduces no extra kernel. This proves (59).

Finally, an invertible principal bundle symbol has a smooth inverse on the compact parameter cosphere. Quantize both with fixed charts. Their two defects have order \(-1\), hence belong to the algebra, which suffices for a Fredholm index. If smoothing defects are desired, the recursive inverse symbols and the increasing-frequency Borel summation of the index lesson's Theorem 6.3 work uniformly on \(V\times[0,1]\). Each summand uses the same physical chart cutoff; growing products of actual operator supports are not summed. \(\square\)

### The Dirac cycle over the deformation

The evaluations in Corollary 8.14a are already KK equivalences. To use them to compare geometric operators, one needs an actual cycle over their common domain. Fibrewise self-adjoint operators and a fibrewise compactness assertion do not provide that cycle. We construct it here for the ordered, globally framed solvable deformation of Theorem 8.14. The global vector-bundle equivalence is proved in Section 6. Its general longitudinal use and the geometric wrong-way comparisons require their further arguments.

Write \(I=[0,1]\), \(A_I^0=\mathfrak A_H(\mathbb C)\), and \(A_I^B=\mathfrak A_H(B)\). Fix the ordered basis \(e_1,\ldots,e_d\) used in (53), with \([e_i,e_j]_v=v\sum_k c_{ij}^ke_k\). The positive Clifford algebra \(C_d\) has odd self-adjoint generators \(\varepsilon_i\), satisfying \(\varepsilon_i\varepsilon_j+\varepsilon_j\varepsilon_i=2\delta_{ij}\). All coefficient algebras \(B\) in the following statement are separable and trivially graded; Clifford tensor products retain their grading.

**Lemma 8.15a (regularity and algebra-valued resolvents).** On the standard graded module \(A_I^0\widehat\otimes C_d\), the differential operator
\[
D=\sum_i D_i\widehat\otimes\varepsilon_i,
\qquad U_i(s)=e^{-isD_i},
\qquad D_i=-iX_i^R,
\tag{115}
\]
has a regular self-adjoint closure, with the jointly smooth compact group kernels as a graph core. Here \(X_i^R f(h)=\left.\frac{d}{ds}f(\exp_{H_v}(se_i)h)\right|_{s=0}\), and the Hilbert-space realization uses normalized left Haar measure. Its scalar functional calculus satisfies
\[
f(D)\in A_I^0\widehat\otimes C_d\quad(f\in C_0(\mathbb R)).
\tag{116}
\]
In particular this is an assertion about the whole interval algebra.

**Proof.** Suppose first that \(d>0\). Left translations preserve left Haar, so \(-iX_i^R\) is formally symmetric. The right-invariant metric making the fixed frame orthonormal is complete. Its volume is right Haar. Put \(\kappa_i(v)=X_i^R\log\Delta_v\); these are constants on each group, smooth in \(v\). Since \(dh_R=\Delta_v^{-1}dh_L\), the unitary from right-Haar to left-Haar realization is multiplication by \(\Delta_v^{-1/2}\). In the right-Haar realization (115) is therefore
\[
D^R=\sum_i\bigl(-iX_i^R+\tfrac i2\kappa_i(v)\bigr)\varepsilon_i.
\tag{117}
\]
Indeed \(\operatorname{div}_{dh_R}X_i^R=-\kappa_i(v)\). This verifies symmetry including the modular term. The principal symbol is \(\sum_i p_i\varepsilon_i\), whose square is \(|p|^2\). Its propagation speed for this metric is one.

We verify the uniform support condition needed to apply the energy argument of Lemma 8.19. A single compact identity-coordinate buffer \(Q\) and a radius \(r>0\) work for every \(v\), by smooth dependence of the metric on the compact interval. A path of length at most \(R\) splits into at most \(\lceil R/r\rceil+1\) pieces; right translation puts the increment of each piece in \(Q\). The union of the corresponding products of copies of \(Q\), over \(v\in I\), is compact by the smooth group multiplication of Theorem 8.14. It contains all radius-\(R\) balls. The same argument prevents a finite-length path from escaping a translated compact buffer and proves completeness without a global coordinate-growth estimate.

The local elliptic estimates and the deficiency-cutoff proof of Lemma 8.19 now apply on each source group, with the just verified principal symbol, formal symmetry and complete metric. Thus \(D_v\) is essentially self-adjoint on its smooth compact core. The energy identity with a moving cutoff gives propagation at most \(|t|\) for \(W_v(t)=e^{itD_v}\). These are Hilbert-space statements at this stage.

Here is their passage to the interval algebra. For a jointly smooth compact kernel \(a_v\), \(W_v(t)a_v\) has support in a common compact set for \(|t|\le T\), by the preceding reachable-set bound. It is jointly smooth, including at zero. To see the parameter assertion explicitly, trivialize densities on that compact set, differentiate \(\partial_t u_v=iD_vu_v\), and obtain
\[
\partial_t(\partial_v u_v)
=iD_v\partial_v u_v+i(\partial_vD_v)u_v.
\]
The variation-of-constants integral bounds this derivative in each local Sobolev norm by the corresponding norms of the initial data and one higher norm of \(u_v\). Source differentiation has the same energy estimate; induction treats every source and parameter derivative. All coefficients and their derivatives are bounded on the common compact set. The smooth group family extends locally past the interval endpoints, as in Theorem 8.14, so this also proves one-sided smoothness there. Local Sobolev embedding gives the stated joint smoothness. This argument uses compact relative coordinates, rather than derivatives at an uncontrolled faraway moving base point.

The differential operators commute with right regular translations. Consequently \(L_ta=W(t)a\) is a left convolution centralizer on the compact kernel algebra: \(L_t(ab)=(L_ta)b\). Its partner is \(R_ta=(L_{-t}(a^*))^*\). Formal symmetry, or integration by parts in the convolution product, gives \((R_ta)b=a(L_tb)\). In the faithful direct sum of fibre regular representations, \(L_ta\) is the product of a unitary and the convolution operator for \(a\). Thus \(\|L_ta\|\le\|a\|\), and the same holds for \(R_t\). The full and reduced norms agree here by Theorem 8.14. Both centralizers extend and define a unitary multiplier \(W(t)\) of \(A_I^0\widehat\otimes C_d\). The group identities hold in that faithful representation. On test kernels,
\(\|(W(t)-1)a\|\le |t|\|Da\|\); applying this to adjoints proves strict continuity. Density extends it to every algebra element.

We give the module-domain argument, so that Hilbert-space regularity is not substituted for module regularity. On the standard module define \(D\) to be \(1/i\) times the norm generator of this strongly continuous unitary group. Difference quotients show that its domain is dense and that it is symmetric. It is closed: passing to a graph limit in
\(W(t)x-x=i\int_0^tW(s)Dx\,ds\) gives the same identity for the limit, and hence its derivative. For \(\eta\in C_c^\infty(\mathbb R)\), time convolution \(\int\eta(t)W(t)x\,dt\) lies in the generator domain, with derivative obtained by moving the derivative to \(-\eta'\). Time approximate identities approximate every domain vector in graph norm. For a fixed such convolution, approximate its input by smooth compact group kernels; the derivative error is at most \(\|\eta'\|_1\) times the input error. The support and smoothness just proved put these convolutions back in the test-kernel algebra. This proves the asserted graph-core property.

The Laplace integrals
\[
R_+=-i\int_0^\infty e^{-t}W(t)\,dt,
\qquad R_-=i\int_0^\infty e^{-t}W(-t)\,dt
\tag{118}
\]
converge strictly as multipliers and in norm after application to any module vector. Integration by parts gives both inverse identities
\((D+i)R_+=R_+(D+i)=1\) and \((D-i)R_-=R_-(D-i)=1\), on their respective domains. The second identity on each side also proves injectivity. These integrals are not asserted to be norm Bochner integrals in the multiplier algebra. Unitarity gives \(R_+^*=R_-\), and the integrals commute. Surjectivity of both \(D\pm i\) proves self-adjointness: if \(y\in\operatorname{Dom}D^*\), take \(x=R_+(D^*+i)y\). The vector \(y-x\) is in \(\ker(D^*+i)\), orthogonal to \(\operatorname{ran}(D-i)\), and is zero. Moreover \(R_+R_-\) is the inverse of \(1+D^2\), with range in \(\operatorname{Dom}D^2\). That range is dense because each resolvent has dense range. Hence \(D\) is regular self-adjoint on its full module domain.

It remains to prove (116). Use Lemma 8.15 with \(V\) a point and finite Clifford matrices. The principal inverse \((\sum p_i\varepsilon_i)/|p|^2\), cut off near zero and quantized with a fixed compact physical-coordinate cutoff, gives an order-\(-1\) operator \(Q_\pm\). The composition formula gives
\[
(D\pm i)Q_\pm=1-S_\pm,
\qquad Q_\pm,S_\pm\in A_I^0\widehat\otimes C_d.
\tag{119}
\]
Here the defect has order \(-1\); a smoothing defect is unnecessary. The principal symbol remains invertible at \(v=0\), and all estimates are uniform. On test kernels \(Q_\pm\) maps into the graph core and the displayed identity holds. Its bounded right side and closedness extend it to the whole module, with range in \(\operatorname{Dom}D\). Multiplication by the module inverse (118) therefore yields \(R_\pm=Q_\pm+R_\pm S_\pm\), which belongs to the interval algebra. The resolvent identity gives \(R_+-R_-=-2iR_+R_-\), so \(1-2iR_+\) is a unitary with adjoint \(1+2iR_-\). Continuous functional calculus for this bounded unitary therefore constructs the calculus in (116). Its restriction to functions vanishing at the circle point \(1\) is nondegenerate because the resolvent ranges are dense. The resolvents generate \(C_0(\mathbb R)\): the Cayley function \(u(x)=(x-i)/(x+i)\) identifies the one-point compactification with the circle, and \(u-1=-2i(x+i)^{-1}\), \(u^{-1}-1=2i(x-i)^{-1}\). Trigonometric approximation on the circle, subtracting the value at its distinguished point, uniformly approximates every function vanishing there by polynomials in these two differences. This proves (116), with norm approximation in the whole interval algebra. In dimension zero the algebra is \(C(I)\), \(D=0\) is everywhere defined, and all the assertions follow directly. \(\square\)

**Lemma 8.15b (the coefficient Kasparov cycle).** For a separable \(B\) with a point-norm continuous \(H\)-action, the operator of Lemma 8.15a induces a regular self-adjoint operator on
\[
E_I=A_I^B\widehat\otimes C_d,
\qquad \phi:C(I,B)\longrightarrow\mathcal L(E_I).
\]
With the usual coefficient representation, its bounded transform
\[
F=D(1+D^2)^{-1/2}
\tag{120}
\]
defines \(T_D\in KK_d(C(I,B),A_I^B)\), in the ordered Clifford model of Section 0.

**Proof.** Integration of the scalar group kernels gives a nondegenerate homomorphism \(j:A_I^0\to M(A_I^B)\). Indeed a scalar approximate identity supported in shrinking common identity buffers converges on the dense compact coefficient kernels, uniformly over \(I\). Extend \(j\) strictly, and include the Clifford factor. The wave multipliers and the two Laplace resolvents transfer through this extension. The generator proof in Lemma 8.15a applies on the coefficient standard module: both resolvents are inverses, and their ranges are dense. For the latter assertion, approximate scalar algebra elements by vectors in the scalar resolvent range and use density of \(j(A_I^0)A_I^B\). This proves regular self-adjointness over \(A_I^B\).

A graph core is the linear span of \(j(k)x\), for smooth compact scalar Clifford kernels \(k\) and \(x\in E_I\). To prove this, first time-smooth a domain vector as in Lemma 8.15a. Approximate its input by finite such vectors, using nondegeneracy. Their time convolutions are again of this form, since \(W(t)j(k)=j(W(t)k)\) and the scalar time convolution is a smooth compact kernel. The derivative error is bounded by \(\|\eta'\|_1\) times the input error. Thus they approximate the original vector in graph norm; no Hilbert-space spectral theorem with coefficients is involved.

For \(a\in C(I,B)\) and \(f\in C_0(\mathbb R)\), the localization statements are
\[
\phi(a)j(f(D)),\ j(f(D))\phi(a)
\in A_I^B\widehat\otimes C_d=\mathcal K(E_I).
\tag{121}
\]
For a compact scalar group kernel these products are compact coefficient group kernels, by convolution covariance and the continuous family action. Approximate the scalar element of (116) in its interval-algebra norm by such kernels. The multiplier bounds extend the conclusion to all \(a\) and \(f\). No unlocalized compactness for a nonunital coefficient algebra is being claimed.

Let \(B^\infty\) be the elements smooth for the original group action. Orbit convolution by compact smooth approximate identities proves density. Finite sums of continuous interval functions times elements of \(B^\infty\) are dense in \(C(I,B)\); all of their group derivatives are uniformly bounded. On this subalgebra, covariance and the negative physical phase give
\[
[D,\phi(a)]=C_a=i\sum_i\phi(v\delta_i a)\varepsilon_i.
\tag{122}
\]
Indeed differentiating \(U_i(s)\phi(a)U_i(s)^*=\phi(\alpha^v_{\exp(se_i)}a)\) gives \(-i[D_i,\phi(a)]=\phi(v\delta_i a)\). We check the domain assertion on the preceding core. Move \(\phi(a)\) through each integrated group unitary by \(\phi(a)U_h=U_h\phi(\alpha^v_{h^{-1}}a)\). On the compact kernel support this coefficient is smooth in \(h\), continuous in \(v\), with bounded first group derivatives. It can be approximated, together with those derivatives, by finite sums of scalar coordinate functions times elements of \(B\): use a finite coordinate partition, ordinary smooth approximation and the compactness of the coefficient and derivative ranges. Multiplying the scalar functions by \(k\) gives vectors in the same graph core. Differentiating their integrated expressions shows graph convergence, since the kernel and its first derivative errors bound the algebra norm by their uniform \(L^1\) norms. Thus \(\phi(a)j(k)x\) lies in the domain and (122) holds there. Its bounded right side and closedness extend it to every domain vector.

For \(b=(1+\tau^2)^{1/2}\) and \(R_\tau=(D^2+b^2)^{-1}\), the resolvent identity on the core, extended by the bounded factors below, gives
\[
[DR_\tau,\phi(a)]
=C_aR_\tau-D^2R_\tau C_aR_\tau
-DR_\tau C_aDR_\tau.
\tag{123}
\]
To justify the identity using only bounded factors, put \(R_z=(D-z)^{-1}\). Domain preservation by \(\phi(a)\) gives \([R_z,\phi(a)]=-R_zC_aR_z\). Since \(DR_\tau=(R_{ib}+R_{-ib})/2\), expanding these two first-order resolvents gives \(b^2R_\tau C_aR_\tau-DR_\tau C_aDR_\tau\). Substituting \(b^2R_\tau=1-D^2R_\tau\) gives (123). No domain-preservation assertion for the Clifford-valued \(C_a\) is needed. Every term is compact: \(C_aR_\tau\) and \(C_aDR_\tau\) are finite sums of the localizations (121), and the other displayed factors are bounded multipliers. Their combined norm is at most
\(9\|C_a\|/(4(1+\tau^2))\), since \(\|R_\tau\|\le b^{-2}\), \(\|DR_\tau\|\le(2b)^{-1}\), and \(\|D^2R_\tau\|\le1\).

The scalar identity \(h(x)=(2/\pi)\int_0^\infty x/(1+x^2+\tau^2)\,d\tau\) gives (120) as a strict limit of bounded partial integrals. The partial scalar functions are bounded by one and converge uniformly on compact spectral sets; the resolvent's dense range proves strict convergence. Formula (123), in contrast, is norm integrable. Consequently \([F,\phi(a)]\) is the norm integral of these compact commutators. The bound \(\|[F,\phi(a)]\|\le2\|a\|\) extends compactness from the smooth subalgebra to all coefficients. Finally \((F^2-1)\phi(a)=-R_0\phi(a)\) is compact by (121), and \(F\) is odd self-adjoint. This proves every Kasparov defect. The standard module is countably generated because the parameter algebra is separable. Dimension zero gives the coefficient homomorphism cycle \((C(I,B),\phi,0)\). \(\square\)

**Lemma 8.15c (the ordered zero-fibre comparison).** Put \(s_d=(-1)^{\lfloor d/2\rfloor}\). Evaluation at zero sends the cycle of Lemma 8.15b to the positive physical Bott class, with the following exact comparison to the native ordered real iteration:
\[
T_D[q_0]=s_d[e_0]\Theta_0
\quad\text{in }KK_d(C(I,B),B\otimes C_0(\mathfrak h^*)).
\tag{124}
\]

**Proof.** At zero the group is additive, the action is trivial and the modular character is one. The transform \(\widehat f(p)=\int e^{-i\langle p,h\rangle}f(h)\,dh\) sends \(-i\partial_{h_i}\) to \(p_i\). Hence the evaluated operator is
\[
F_d(p)=\frac{\sum_{i=1}^d p_i\varepsilon_i}{\sqrt{1+|p|^2}}.
\tag{125}
\]
The evaluated coefficient representation is \(a\mapsto a(0)\). The resolvents agree with these Fourier multipliers on the compact core and hence on their full domains, so evaluation of the regular calculus gives exactly this cycle.

We check its product, rather than identify it by its K-group action. For one coordinate (125) is the trivial-action cycle (109). Suppose the first \(m\) ordered factors have been identified, and write \(x\in\mathbb R^m\), \(y=p_{m+1}\). The tensor module for the next trivial real Thom factor is the standard module over \(B\otimes C_0(\mathbb R^{m+1})\widehat\otimes C_{m+1}\), with the old Clifford generators followed by \(\varepsilon_{m+1}\). The candidate product operator is \(F_{m+1}(x,y)\).

For a homogeneous creation vector \(\xi\in B\otimes C_0(\mathbb R^m)\widehat\otimes C_m\), the connection error with the lower operator \(y\varepsilon_{m+1}/\sqrt{1+y^2}\) has two parts. One has the coefficient \((\sum x_i\varepsilon_i)\xi(x)/\sqrt{1+|x|^2+y^2}\); the other has the coefficient
\[
(-1)^{|\xi|}\xi(x)y\varepsilon_{m+1}
\left(\frac1{\sqrt{1+|x|^2+y^2}}-\frac1{\sqrt{1+y^2}}\right).
\]
Both are in \(B\otimes C_0(\mathbb R^{m+1})\widehat\otimes C_{m+1}\). For large \(|x|\), use the small norm of \(\xi(x)\) and boundedness of each scalar multiplier. On bounded \(x\), the first expression tends to zero as \(|y|\to\infty\), and so does the parenthesis multiplied by \(y\), uniformly there. The adjoint connection error has the identical estimate. First using finite elementary vectors and then norm approximation proves the connection condition for every creation vector, with the displayed grading sign.

The positivity condition of Definition 0.3 is even exact:
\[
\{F_m\widehat\otimes1,F_{m+1}\}
=\frac{2|x|^2}{\sqrt{1+|x|^2}\sqrt{1+|x|^2+y^2}}\ge0.
\tag{126}
\]
The cross term cancels by the Clifford anticommutation. The cycle defects of \(F_{m+1}\) are the scalar \(C_0\) defect and exact coefficient commutation. Thus it is a product operator.

The preceding connection and positivity calculations prove the product on the **unreduced** standard Clifford module with the old generator block first. They also identify the positive ordered geometric Bott class after its specified consecutive-pair reduction. We must still compare that reduction with the native parity product of Lemma 0.13; its odd–odd target has the second-cycle coefficient first.

Write \(B_m\) for the positive geometric class of (125), reduced by the first \(2a\) generators when \(m=2a+\epsilon\), retaining the last right \(C_1\) generator if \(\epsilon=1\). Write \(t_{m+1}\) for the next positive one-dimensional trivial real Thom class. When \(m=2a\), the native product has the full module
\[
B\otimes C_0(\mathbb R^{m+1})\otimes\Sigma_a\widehat\otimes C_1.
\]
The old physical generators act on \(\Sigma_a\), and the new one is \(\Gamma_a\otimes L_\varepsilon\). The connection estimates and exact positive anticommutator above prove that the normalized sum is the actual product. Thus \(B_{2a}\cdot t_{2a+1}=B_{2a+1}\).

When \(m=2a+1\), the second-cycle coefficient precedes the retained old odd coefficient in (104). The full reduced module is
\[
B\otimes C_0(\mathbb R^{m+1})\otimes\Sigma_a\otimes\mathbb C^{1|1}.
\]
Its first \(2a\) physical generators are unchanged. Its last two physical generators and grading are
\[
p_m:\ \Gamma_a\otimes\sigma_y,
\qquad p_{m+1}:\ \Gamma_a\otimes\sigma_x,
\qquad \Gamma_a\otimes\sigma_z.
\]
The same creation-map estimates and positivity calculation apply with precisely this order, so this is the actual native product operator. In the positive geometric reducer, those last two physical generators instead have order \(\Gamma_a\otimes\sigma_x,\Gamma_a\otimes\sigma_y\). The explicit unitary
\[
U=1_{\Sigma_a}\otimes\frac{\sigma_x+\sigma_y}{\sqrt2}
\]
fixes every earlier generator, exchanges these last two, and anticommutes with the total grading. It preserves the right action and inner product and is onto. Consequently it is an even module unitary from the grading reversal of the geometric cycle to the native cycle. This right coefficient algebra is trivially graded. The grading operator is therefore right-linear; conjugating the additive inverse cycle \((\Pi E,-F)\) by it identifies that inverse with \((\Pi E,F)\). It follows, as an identity of full cycles and KK classes, that
\[
B_{2a+1}\cdot t_{2a+2}=-B_{2a+2}.
\]
This last scalar-target argument is used only after the two odd coefficients have been contracted. On a module with a retained right odd Clifford coefficient, its grading operator is not right-linear, and cannot justify that shortcut.

Starting with the rank-zero identity and the rank-one class (109), bilinearity and these actual product operators give
\[
\Theta_0=(-1)^{\lfloor d/2\rfloor}B_d,
\qquad B_d=(-1)^{\lfloor d/2\rfloor}\Theta_0.
\]
Equivalently, the native physical generator list is
\((2,1,4,3,\ldots,2a,2a-1[,d])\); each consecutive exchange contributes the displayed grading reversal. Coefficient multiplication is even, and all compactness tests remain localized by the tested coefficient, as in Lemma 0.13. The argument therefore holds for every separable trivially graded \(B\), including nonunital coefficients.

For completeness this normalization preserves both inverse products. Let \(A_d\) be the full geometric inverse obtained from Lemmas 6.0a–b and Theorem 6.0c by the prescribed source-to-target transfer and parity reduction of Lemma 7.20. Those complete module evaluations give \(B_d A_d=1\) and \(A_d B_d=1\). The inverse of the native accumulated real product is therefore \(s_d A_d\), where \(s_d=(-1)^{\lfloor d/2\rfloor}\); uniqueness identifies it with the reverse chronological product of the real inverses. Both products are \(s_d^2\) times the corresponding geometric identity and hence equal one. This statement uses the actual geometric inverse and its entire finite Clifford transfer, without substituting an independently normalized positive Dirac cycle. Combining the Fourier evaluation with \(B_d=s_d\Theta_0\) proves (124). In rank zero all classes are the coefficient homomorphism class. \(\square\)

**Theorem 8.15d (the framed longitudinal Dirac representative).** In the precise ordered solvable deformation,
\[
T_D=s_d\Theta_I,\qquad
[\iota_B]T_D[q_1]=s_d\Theta_1,
\qquad \iota_B(b)(v)=b.
\tag{127}
\]
These are KK-class equalities for separable coefficients, not merely equalities of induced K-group maps. For a smooth locally free action whose holonomy groupoid is \(V\rtimes H\), take \(B=C_0(V)\). The endpoint cycle is the Dirac cycle of the ordered global longitudinal frame, with its homogeneous source metric and induced spin structure. It is also the corresponding Levi-Civita spin Dirac class, with the same Clifford and chirality conventions.

**Proof.** Naturality in Corollary 8.14a gives \(\Theta_I[q_0]=[e_0]\Theta_0\). Lemma 8.15c gives \(T_D[q_0]=s_d[e_0]\Theta_0=s_d\Theta_I[q_0]\). Multiply this equality on the right by the explicit KK inverse of \([q_0]\). Associativity gives the first identity in (127). Evaluating at one and precomposing with the constant coefficient inclusion gives the second, since \([\iota_B][e_1]=1_B\). The inclusion need not be equivariant; this use is an ordinary KK product. The preceding lemmas supplied the actual interval cycle and its zero-fibre product, so no representative comparison is inferred just from evaluation being an isomorphism.

We verify the last geometric assertion. On a source fibre the ordered right-invariant frame has constant structure coefficients \(-c_{ij}^k\). The Koszul formula therefore makes every Levi-Civita connection coefficient constant in that frame. For negative Clifford multiplication set \(c_i=-i\varepsilon_i\). The spin Dirac operator \(\sum_i c_i\nabla_{X_i^R}\), in the right-Haar realization, has principal part \(\sum_i-iX_i^R\varepsilon_i\). Its connection term differs from the modular correction in (117) by a constant, odd, self-adjoint Clifford endomorphism \(K_v\). This difference is self-adjoint because both differential operators are formally symmetric for metric volume. Its norm is uniformly bounded over \(I\), since the structure constants and modular logarithmic derivatives depend smoothly on \(v\). No compactness or ambient bounded geometry of \(V\) is needed: the coefficients are intrinsic to the homogeneous source groups. At zero the frame is flat, \(K_0=0\).

For completeness the bounded perturbation gives an actual Kasparov homotopy. Put \(D_s=D+sK\), \(0\le s\le1\), on \(\operatorname{Dom}D\). Choose \(\lambda>\max(1,\|K\|)\). A Neumann series for \(1+sK(D\pm i\lambda)^{-1}\) proves surjectivity of \(D_s\pm i\lambda\). Symmetry and the inverse-domain argument of Lemma 8.15a prove self-adjointness. Their product \(S=(D_s^2+\lambda^2)^{-1}\) has dense range and satisfies \(0\le S\le\lambda^{-2}\). The bounded operator \(1+(1-\lambda^2)S\) is positive invertible, bounded below by \(\lambda^{-2}\); hence \(S(1+(1-\lambda^2)S)^{-1}\) is the inverse of \(1+D_s^2\). This proves regularity. The scalar resolvents belong to the parameter algebra, first by the Neumann series and then at every nonreal parameter by the resolvent identity. Also \([D_s,\phi(a)]=[D,\phi(a)]\), since the scalar Clifford endomorphism \(K\) commutes with coefficients. Thus (121)–(123) give all localized cycle defects.

The transforms \(F_s\) are norm continuous. Indeed, for \(b=\sqrt{1+\tau^2}\),
\[
D_s(D_s^2+b^2)^{-1}
=\tfrac12\bigl((D_s-ib)^{-1}+(D_s+ib)^{-1}\bigr).
\]
The resolvent identity bounds the difference of these expressions for \(s,t\) by \(|s-t|\|K\|/b^2\). Integrating their difference, which is norm integrable although the individual transform integrals need only converge strictly, gives \(\|F_s-F_t\|\le |s-t|\|K\|\). Localized resolvent defects and commutators are continuous by the same estimates and their integrable tails. This is a Kasparov homotopy on the interval module. The ordered Clifford Morita contractions of Section 0 convert its Clifford-valued cycle into the spinor cycle, including odd dimension. In even dimension \(d=2\ell\), its grading is \(i^\ell c_1\cdots c_d=(-i)^\ell\varepsilon_1\cdots\varepsilon_d\), precisely the grading of the [index lesson, Lemma 6.24](the-index-theorem-for-measured-foliations.md#section-6).

For the graph action, source-fibre constructions and right translations give the longitudinal operator and its coefficient representation on the convolution module. The principal bundle frame is the given ordered identification \(F\cong V\times\mathfrak h\); its zero-fibre symbol is (125). This proves the stated framed Dirac comparison with the explicit native normalization \(s_d\). Corollary 6.0d tracks the chosen spin-c structure and proves the nontrivial vector-bundle Thom equivalence. Its use in the general longitudinal deformation, general foliation wrong-way factorization and assembly still need their own comparisons. \(\square\)

![Construction of the interval Dirac cycle and its exact ordered endpoint comparison.](../figures/dirac-parameter-cycle.png)

**Figure 8.15d.** The analytic steps establish a cycle in the parameter algebra before cancellation by the proved inverse of \([q_0]\). The middle positivity formula proves the raw Clifford product. The native odd–odd contraction exchanges each final pair; the explicit finite spinor unitary gives \(s_d=(-1)^{\lfloor d/2\rfloor}\) before the last comparison of actual KK classes. Lemmas 8.15a–c and Theorem 8.15d give the full proofs; the geometric statement is restricted to the indicated global frame. The real-action deformation method is due to Fack and Skandalis, as credited in Section 0; the product connection criterion is due to Connes and Skandalis, Appendix A.

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**Theorem 8.16 (solvable longitudinal symbol index).** Let \(V\) be compact, and suppose a smooth locally free action of a simply connected solvable group \(H\) has orbit holonomy groupoid \(G=V\rtimes H\). Then the full longitudinal symbol index is an isomorphism
\[
\operatorname{Ind}_a:K_c^0(V\times\mathfrak h^*)\xrightarrow{\cong}
K_0(C(V)\rtimes H).
\]
More precisely, with the Fourier identification at zero and the ordered tangent identification at one,
\[
\operatorname{Ind}_a=(\mathrm{ev}_1)_*(\mathrm{ev}_0)_*^{-1}.
\tag{60}
\]
The analogous suspended statement holds in degree one. This is a comparison with the actual analytic symbol map, without an assembly or wrong-way independence assumption.

**Proof.** A compact relative symbol class is a triple \((B_0,B_1,a)\) over the disk and sphere of \(V\times\mathfrak h^*\). Radial contraction transports each disk bundle to a bundle from \(V\), as proved in the [Hilbert-module lesson, Theorem 6.6](hilbert-modules-and-fields-on-the-leaf-space.md#section-6). Smooth approximation preserves the sphere isomorphism. Keep this triple constant in \(v\), and quantize it with Lemma 8.15. A quantization \(P\) and its inverse-symbol quantization \(Q\) have both defects in \(\mathfrak A_H(C(V))\).

Apply the exact elementary-matrix index construction (11)–(12) of that lesson in the two finite bundle corners. It gives \(I(a)\in K_0(\mathfrak A_H(C(V)))\). Direct sums, smooth symbol homotopies, changed quantizations, and triples whose sphere isomorphism extends to the disk preserve the index, by the same continuous-parametrix argument. Thus \(I\) is defined on the full relative K-group. Evaluation of each matrix factor gives the endpoint index; this establishes naturality by the explicit formula, not by an asserted constancy of fibre Fredholm operators.

At zero the group acts trivially. Fourier transformation converts \(P\) to a bounded matrix function \(P_0(x,p)\), with radial limit \(a(x,p/|p|)\), and \(Q\) has the inverse radial limit. A compact physical cutoff changes the uncut Fourier symbol by a negative-order function, which vanishes at infinity and leaves this relative class unchanged. Compactify each \(\mathfrak h^*\) radially to its disk. On the boundary the elementary matrix \(W\) is exactly the diagonal map \(\operatorname{diag}(a,a^{-1})\), so its first range bundle is \(B_0\) identified with \(B_1\) by \(a\). Extend its range idempotent by the fixed trivial complements as in formula (12). Its difference from the reference idempotent is precisely the disk/sphere clutching construction of \((B_0,B_1,a)\). Hence
\[
(\mathrm{ev}_0)_*I(a)=[B_0,B_1,a].
\]
This also fixes the sign: a zero-dimensional fibre gives \([B_0]-[B_1]\), the kernel-minus-cokernel convention. The construction retains the actual source and target bundles and their complements; it does not require those bundles themselves to be trivial.

At one, the source-fibre calculus is the holonomy-cover calculus for \(G=V\rtimes H\). The module corner and index matrices are exactly those of the Hilbert-module lesson's Theorem 6.6. Thus \((\mathrm{ev}_1)_*I(a)=\operatorname{Ind}_a(a)\). Theorem 8.14 makes both evaluations K-isomorphisms, proving (60) and surjectivity as well as injectivity.

For degree one, suspend the coefficients and all the index matrices. Smooth compact-support suspension kernels are dense; their estimates use the same fixed supports in the additional parameter on each compact set. The natural suspension identifications then give the degree-one statement. If \(d=0\), the family is constant and the index is the difference of its two finite bundle modules, so (60) holds directly. \(\square\)

**Corollary 8.17 (the complete compact trace-image formula).** Under the hypotheses of Theorem 8.16, let \(\Lambda\) be a locally finite invariant transverse measure. Then formula (20) holds, with the parity there stated.

**Proof.** The ordered fundamental fields identify \(F\) with \(V\times\mathfrak h\). Thus \(\operatorname{Td}(F\otimes\mathbb C)=1\). By Theorem 8.16 every class of \(K_0(A)\) is a full relative-symbol index. The all-symbol measured theorem, [Theorem 6.37 of the index lesson](the-index-theorem-for-measured-foliations.md#section-6), gives
\[
\tau_{\Lambda*}(\operatorname{Ind}_a\alpha)
=\epsilon_d\langle\pi_!\operatorname{ch}(\alpha),[C_\Lambda]\rangle,
\qquad \epsilon_d=(-1)^{d(d+1)/2}.
\tag{61}
\]

We check the entire symbol lattice. Stable Bott periodicity gives an external-product isomorphism
\[
K^{d\bmod2}(V)\longrightarrow K_c^0(V\times\mathbb R^d),
\qquad x\longmapsto x\boxtimes b_d,
\]
where \(b_d\) is a scalar generator in \(K_c^{d\bmod2}(\mathbb R^d)\). Its Chern normalization has absolute value one. In even dimension this is the ordered trivial spin-c computation of the index lesson's Theorem 6.29, whose fibre character is \((-1)^{d/2}\). In odd dimension tensor that even generator with
\(u_+(p)=(p-i/2)/(p+i/2)\). Its one-form satisfies
\[
\int_{\mathbb R}\frac1{2\pi i}u_+^{-1}du_+=1.
\]
The suspension isomorphism is the external product with this winding generator, as fixed in Corollary 8.10 and the stable Bott conventions above. Multiplicativity of the even and suspension Chern characters follows from tensor curvature and its transgression; the latter is obtained by differentiating a connection path and integrating its exact Chern derivative. Hence, after a possible replacement of \(b_d\) by \(-b_d\), its top fibre character integrates to \(+1\).

With base differentials preceding the ordered positive fibre differentials, fibre integration gives
\(\pi_!\operatorname{ch}(x\boxtimes b_d)=\operatorname{ch}(x)\).
If the suspension product is ordered in the opposite way, the Koszul sign on the only component pairing with the \(d\)-current is the fixed sign \((-1)^d\); it can equivalently be absorbed in \(b_d\). Thus the stated subgroup image is independent of this order, while (61) retains its actual analytic sign. Every symbol class is one such product, including torsion classes whose character and trace vanish by (61). As \(x\) runs through a group, multiplication of every pairing by \(\epsilon_d\) leaves its image subgroup unchanged. This proves (20). No equality of the three source maps is inferred from equality of their numerical images. \(\square\)

![The compact current has finite mass and detects the modular character. The solvable group deformation keeps source-fibre dimension fixed, and its analytic evaluation maps compare every symbol with the vector-group endpoint.](../figures/solvable-deformation.png)

*Figure 3. Proposition 8.12 and Theorems 8.14–8.16. The parameter is \(v\); at zero the anchor vanishes but the group fibre is \(\mathbb R^d\). Haar densities and the positive-parameter kernel factor are exact. The bottom comparison uses the uniform source-fibre calculus of Lemma 8.15. It establishes the compact trace formula of Corollary 8.17. Example 8.13 shows why the measure-existence assertion needs compactness. Assembly and independence of geometric wrong-way factorizations remain further obligations.*

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### Compact support on a noncompact base

Compactness of \(V\) made the preceding heat traces individually finite. For a noncompact \(V\) the two traces may both be infinite. The appropriate replacement is an actual relative index matrix with a finite trace difference. We now construct it. Throughout this subsection the action is smooth and locally free, \(H\) is simply connected and solvable, and the orbit holonomy graph is \(V\rtimes H\). Put
\[
B=C_0(V),\qquad A=B\rtimes H.
\]
Full and reduced completions agree by Lemma 8.3. The chosen fundamental frame orients and trivializes \(F\). The measure \(\Lambda\) is invariant and geometrically locally finite. Write \(\mu\) for its unit measure using the right-invariant leafwise volume of this frame, and \(C=C_\Lambda\) for its closed tangent current. Neither \(\mu(V)<\infty\) nor unimodularity of \(H\) is assumed.

**Lemma 8.18 (the full compact-support symbol map).** The conclusion (60) of Theorem 8.16 holds with \(C(V)\) replaced by \(C_0(V)\), in both degrees. For \(d>0\), every class of \(K_c^0(V\times\mathfrak h^*)\) can be indexed by an operator
\[
P=I+P_K
\]
on a finite trivial bundle, whose order-zero principal symbol is unitary and equals \(I\) outside a compact subset \(K\subset V\). The operator \(P_K\) has compact base and relative group support. Its inverse-symbol parametrix can have the same form.

**Proof.** The disk/sphere extension for the trivial real bundle is
\[
0\longrightarrow C_0(V\times\mathbb R^d)
\longrightarrow C_0(V\times\overline{\mathbb B}^d)
\longrightarrow C_0(V\times S^{d-1})\longrightarrow0.
\tag{62}
\]
Contracting the disk identifies its K-groups with those of \(B\). Restriction of the constant disk functions to the sphere is injective on K-groups: evaluation at any fixed point of the sphere is a left inverse. Consequently the six-term sequence makes the boundary from \(K_1(C_0(V\times S^{d-1}))\) onto the \(K_0\) group of the ideal surjective. This is an integral assertion, including torsion. Its sign is the disk/sphere clutching sign of Theorem 8.16.

Represent a sphere \(K_1\)-class by a finite matrix unitary in the unitization. After a constant unitary homotopy its scalar value is \(I\). Uniform convergence to \(I\) at base infinity permits a compact cutoff of \(u-I\): choose a larger compact set on whose complement \(\|u-I\|<1/4\). The cutoff matrix remains invertible on the transition region and is unchanged on the inner compact set. Its polar part is a unitary equal to \(I\) off a compact set and homotopic to \(u\). Smooth approximation on that compact set, followed by the same polar operation, makes it smooth without changing its class.

Quantize \(u-I\) with a fixed small physical kernel cutoff and compact base cutoff, and set \(P=I+\operatorname{Op}(u-I)\). Quantize \(u^{-1}-I\) similarly. All source derivatives used in Lemma 8.15 are now bounded on fixed compact enlargements of \(K\), instead of on all of \(V\). Products only replace \(K\) by the image of \(K\) under a fixed compact set of group elements. Such images are compact, uniformly over the parameter \(v\). The homogeneous group geometry, right-translation density normalization, finite-overlap localization and Fourier–Schur estimate of that lemma require no compactness of the base. Hence they apply verbatim to these coefficients. Negative-order compact-base kernels belong to \(\mathfrak A_H(B)\) by the frequency cutoff estimate \(CR^m\), \(m<0\). The two inverse defects therefore belong to this algebra.

For clarity, the completed compact-base calculus has quotient \(C_0(V\times[0,1]\times S^{d-1})\); adding \(I\) gives the scalar unitized quotient needed here. The quotient-norm argument still works. Its upper bound quantizes the positive square root after subtracting its constant value at base infinity. Its lower bound freezes one finite base point and one covector; algebra kernels annihilate the shrinking wave packet. Surjectivity follows by smooth compact approximation. Thus no extra quotient kernel appears on completion.

Keep \(u\) constant in \(v\) and apply the elementary index matrices of Theorem 8.16. At zero they give its disk/sphere boundary class, and at one its actual longitudinal index. Theorem 8.14 applied to \(B\) makes both evaluation K-maps invertible. Surjectivity of the boundary in (62) proves (60) for every symbol class, not just a chosen set of differential symbols. Suspension of (62) and of these matrices proves the other degree. For \(d=0\), the algebra is \(B\) and the index is directly the difference of the coefficient projections. \(\square\)

**Lemma 8.19 (relative wave calculus).** Suppose \(d>0\). A class \(x\in K_0(B)\) has smooth finite matrix projection representatives \(p_0,p_1\) with the same constant projection \(p_\infty\) outside a compact set \(K\). The twisted leafwise spin Dirac operators on their ranges, with the usual spinor grading when the source dimension is even, admit a functional calculus with the following properties:

1. Their self-adjoint extensions are unique, and their wave groups have propagation at most the absolute wave time.
2. A Schwartz function with Fourier support in \([-\delta,\delta]\) gives a smooth kernel of group propagation at most \(\delta\). The difference of the two twisted kernels is supported over a compact enlargement of \(K\).
3. All these assertions also hold on \(G\times(S^1\times S^1)\), with source fibre \(H\times S^1\), and coefficients compactly supported in the \(V\)-direction.

Here propagation refers to the complete right-invariant source metric, and the circle in part 3 has its usual complete metric. It does not refer to distance in an arbitrary ambient metric on \(V\).

**Proof.** Begin with continuous projections in a finite matrix algebra over \(B^+\). Stabilization and a scalar unitary arrange their common scalar projection to be \(p_\infty\). Smooth a self-adjoint approximation which is exactly \(p_\infty\) outside a larger compact set and is within \(1/8\) of the projection. The spectral cut at \(1/2\) is smooth, is exactly \(p_\infty\) there, and is homotopic to the original projection through the same spectral cut. This proves the representative assertion without a finite cover of the whole noncompact manifold.

On the full trivial coefficient bundle use the Hermitian connection
\[
\nabla^p=d+[p,dp].
\tag{63}
\]
The commutator is skew-adjoint. Differentiating \(p^2=p\) gives \([ [p,dp],p]=-dp\), so \(\nabla^p p=0\); its restriction to the range is the projection connection \(p\,d\). The resulting full spin Dirac operator \(D_p\) preserves this range and its orthogonal complement. Outside \(K\) it is the common homogeneous operator \(D_\infty\). Its principal symbol is \(ic(\xi)\), with the chirality fixed in Lemma 6.24 of the index lesson.

Every invariant metric on a connected Lie group is complete. One elementary verification is to cover a small buffered identity ball by finitely many still smaller balls and translate this cover along a metric path. A finite-length path crosses only finitely many such buffers; a Cauchy path tail stays in a translated compact coordinate buffer and converges. The Hopf–Rinow argument then makes each closed metric ball compact. Curvature, a positive injectivity buffer and all metric jets are uniform by translation. The coefficient jets of (63) are uniform too: they vanish off \(K\), and repeated derivatives along fundamental fields are bounded on each fixed compact enlargement. On a translated source chart they are exactly these derivatives at its translated base point. The circle product retains these bounds and the finite-overlap bounds.

Local elliptic estimates imply that an \(L^2\) distributional deficiency vector is locally smooth. Choose source cutoffs \(\eta_R\), tending to one and satisfying \(\|d\eta_R\|_\infty\le C/R\). If \((D_p\pm i)w=0\), integration by parts against \(\eta_R^2w\), justified first on compact buffers, gives
\[
\|\eta_Rw\|_2^2
\le \frac{C}{R}\|\eta_Rw\|_2\|w\|_2.
\]
Letting \(R\) tend to infinity shows \(w=0\). This proves essential self-adjointness. The energy identity for a moving spatial cutoff uses
\(\|[D_p,\eta]\|=\|c(d\eta)\|\le\|d\eta\|_\infty\).
Choosing its level sets to recede with unit speed makes the exterior energy nonincreasing. A solution initially supported in a set therefore stays in its \(|s|\)-neighbourhood at wave time \(s\). Approximation by smooth data proves this for the full unitary wave group. The same estimate applied to the difference of two wave solutions proves equality in any causal region where their coefficients agree.

The metric volume on \(H\) is right Haar. If the convolution realization uses left Haar, the change of realization is multiplication by \(\Delta_H^{-1/2}\), since \(dh_R=\Delta_H^{-1}dh_L\). This is a unitary between the two \(L^2\) spaces, preserves support and has value one at the unit. Its logarithmic source derivatives are constants fixed by the modular character. Thus all estimates and the unit diagonal density remain valid for nonunimodular \(H\).

Here are the algebra and smoothing details. The wave group maps a smooth compact test kernel to one with compact relative group support: finite speed enlarges that support by a fixed compact metric ball. Source energy estimates preserve every Sobolev order, and differentiation with respect to a parameter on a compact base set uses the differentiated wave equation and its finite-time Duhamel integral. It therefore preserves smooth test kernels and their adjoints. The regular norm is at most one. On a test kernel \(a\),
\[
\|(e^{isD_p}-1)a\|\le |s|\,\|D_pa\|,
\]
and the adjoint estimate gives strict continuity. These double centralizers define the wave multipliers; the faithful direct sum of regular representations gives their stated norm bounds and group identities.

Use the coefficient algebra \(\mathcal B=B^+\rtimes H\), not the scalar unitization \(A^+\). Compactly supported Fourier integrals of the wave group are its multipliers. If the spectral function is Schwartz, then \(D_p^j f(D_p)\) is bounded for every \(j\). Uniform elliptic estimates and local Sobolev embedding give a smooth kernel with all source derivatives bounded. Its relative group support is compact by finite speed. Off a fixed compact enlargement of \(K\) it is exactly the kernel of \(f(D_\infty)\), by equality in causal regions. The latter is a smooth compact kernel in \(C^*(H)\); the difference is a smooth compact kernel in \(A\). Hence \(f(D_p)\in\mathcal B\). These Schwartz functions are uniformly dense in \(C_0(\mathbb R)\), so the assertion holds for that whole functional calculus. It is nondegenerate: for \(f_n(z)=(1+z^2/n^2)^{-1}\), the bound
\(\|(1-f_n(D_p))a\|\le n^{-2}\|D_p^2a\|\)
on the test core proves strict convergence to one. This also supplies the regular functional calculus on the coefficient module.

For the relative support assertion, a wave kernel between two source points depends only on coefficients within the wave-time neighbourhood of those points. If such a region misses the pullback of \(K\), the two kernels agree. At the units all possibly contributing base points therefore lie in \(H_RK\), for a fixed compact ball \(H_R\subset H\); off-diagonal support has the same bound with a larger fixed \(R\). The image of \(H_R\times K\) under the action is compact. Local finiteness gives \(\mu(H_RK)<\infty\). No estimate of \(\mu(V)\) is used. The circle product changes this set only to its product with a compact circle. \(\square\)

**Proposition 8.20 (an exact finite propagation relative index matrix).** Under Lemma 8.19's hypotheses with even source dimension, let \(E_i=\operatorname{ran}p_i\), and let \(\alpha\) be the difference of their graded Dirac symbol classes. There are idempotents \(e_i(t)\) with \(e_0(t)-e_1(t)\) a smooth compact arrow kernel such that
\[
\tau_{\Lambda*}(\operatorname{Ind}_a\alpha)
=\int_V\operatorname{str}\!\left(
 p_0r_t(D_{p_0})^2-p_1r_t(D_{p_1})^2
 \right)(1_x)\,d\mu(x),\qquad t>0.
\tag{64}
\]
The integrand is supported in one fixed compact set, independent of \(t\). It is not the difference of two separately integrated infinite heat traces.

**Proof.** Choose an even \(\chi\in C_c^\infty((-\delta,\delta))\), with \(0\le\chi\le1\) and \(\chi=1\) on \([-\delta_0,\delta_0]\), \(0<\delta_0<\delta\). Define
\[
\begin{split}
\rho_t(s)&=a_t\chi(s)(4\pi t)^{-1/2}e^{-s^2/(4t)},
& a_t^{-1}&=\int_{\mathbb R}\chi(s)(4\pi t)^{-1/2}e^{-s^2/(4t)}\,ds,\\
r_t(z)&=\int_{\mathbb R}\rho_t(s)e^{isz}\,ds,
& q_t(z)&=\frac{1-r_t(z)}z .
\end{split}
\tag{65}
\]
Normalization is essential: \(r_t(0)=1\), so \(q_t\) extends smoothly and oddly at zero. The even real function \(r_t\) is Schwartz. The Fourier distribution of \(q_t\) is a constant multiple of the primitive of \(\delta_{\{0\}}-\rho_t\), where \(\delta_{\{0\}}\) is the point mass at zero. Its total integral is zero, so the primitive can be chosen with support in \([-\delta,\delta]\). Consequently \(q_t(D_p)\) has propagation at most \(\delta\). It has order \(-1\): away from bounded spectral frequency it is \(D_p^{-1}\) modulo a smoothing function, and the local elliptic parametrix identifies this assertion in each source chart. Equivalently the compactly supported Fourier primitive has its only nonsmoothness at zero, which gives the inverse principal symbol.

Restrict to \(X=S^+\otimes E_i\) and \(Y=S^-\otimes E_i\). Put \(D=D_{p_i}^+:X\to Y\), \(Q=q_t(D_{p_i})^-:Y\to X\), \(S_0=r_t(D_{p_i})|_X\), and \(S_1=r_t(D_{p_i})|_Y\). Functional calculus gives
\[
QD=1-S_0,\quad DQ=1-S_1,\quad
DS_0=S_1D,\quad QS_1=S_0Q.
\]
The following matrix is bounded, despite the unbounded \(D\):
\[
e_i(t)=
\begin{pmatrix}
S_0^2&S_0(1+S_0)Q\\
S_1D&1-S_1^2
\end{pmatrix}.
\tag{66}
\]
Indeed its nonconstant entries are Schwartz functions of the full \(D_{p_i}\). The four displayed relations prove \(e_i(t)^2=e_i(t)\) by direct multiplication; for example the upper-left entry is
\(S_0^4+S_0^2(1+S_0)(1-S_0)=S_0^2\).
Extend this matrix by zero on the source coefficient complement and by identity on the target complement. Then both matrices act on the same finite trivial graded bundle. Each differs from \(e_{\rm ref}=\operatorname{diag}(0,I)\) by an element of \(M_{2N}(\mathcal B)\). Its nonconstant entries have propagation at most \(3\delta\). Equality of the two coefficient systems in causal regions shows that their difference is a smooth compact arrow kernel, supported in a fixed compact group ball and over, for example, \(H_{6\delta}K\). Enlarge the ball slightly if coordinate cutoffs are used.

We verify that the matrices give the actual symbol index. Put \(G=(1+D_{p_i}^2)^{-1/2}\). Conjugation on the source by \(G^{-r}\), \(0\le r\le1\), changes the two off-diagonal entries to
\[
G^{-r}S_0(1+S_0)Q,\qquad S_1DG^r
\]
and leaves the diagonal entries fixed. The scalar functions in these entries are Schwartz times a polynomial weight; they are norm continuous in \(r\). Their kernels remain in \(\mathcal B\), with relative difference in \(A\): compact Fourier approximation of a Schwartz function has this property, and functional calculus gives norm convergence. At \(r=1\) the formula is the bounded-parametrix index matrix for
\[
T=DG:X\to Y,\qquad Q'=G^{-1}Q:Y\to X.
\]
Their defects are \(S_0,S_1\), and their principal symbols are the normalized Dirac symbol and its inverse. On coefficient complements set \(T=Q'=I\). The elementary Fredholm boundary construction of the Hilbert-module lesson therefore identifies \([e_i-e_{\rm ref}]\) with this symbol index in \(K_0(\mathcal B)\). Varying \(t\) gives the same class by the same norm continuous formulas.

This absolute coefficient algebra is only an intermediate device. Exactness and its constant splitting give
\[
0\longrightarrow A\longrightarrow\mathcal B
\longrightarrow C^*(H)\longrightarrow0.
\tag{67}
\]
Both matrices have the same image \(e_\infty(t)\) in the quotient. For any idempotents \(e,f\) with \(e-f\in M_n(A)\), set
\[
J_f=\begin{pmatrix}f&1-f\\1-f&f\end{pmatrix},
\qquad
\widetilde e=J_f\operatorname{diag}(e,1-f)J_f.
\tag{68}
\]
One has \(J_f^2=I\), and \(J_f\operatorname{diag}(f,1-f)J_f=\operatorname{diag}(I,0)\). Thus \(\widetilde e-\operatorname{diag}(I,0)\in M_{2n}(A)\) defines a genuine relative class \([e:f]\in K_0(A)\), even if \(f\) is not a scalar idempotent. Direct sums, conjugations and continuous idempotent homotopies show that it maps to \([e]-[f]\) in \(K_0(\mathcal B)\). The split sequence (67) makes this inclusion injective. Naturality of the bounded symbol construction, with the common exterior bundle trivialization retained, consequently identifies
\([e_0(t):e_1(t)]=\operatorname{Ind}_a\alpha\).
It is not necessary to invert or discard an infinite reference trace.

The invariant transverse measure makes the weight of Theorem 7.10 of the transverse-measure lesson a trace. Lemma 7.9 there places every smooth compact arrow kernel in its linear trace ideal. Hence \(e_0-e_1\) has finite absolute trace in the measured algebra. This ideal is preserved by bounded multipliers, and cyclicity gives
\[
\tau(\widetilde e-\operatorname{diag}(I,0))
=\tau(e_0-e_1).
\]
The trace pairing of a finite-trace relative idempotent is this number: replacing its range by the orthogonal range projection, or applying the usual idempotent-to-projection homotopy, preserves the value by cyclicity on this ideal. This is the same normalization used for the Fredholm boundary index. The two diagonal entries of (66) now give exactly the supertrace in (64). The compact support and diagonal formula justify that integral absolutely. \(\square\)

**Theorem 8.21 (the relative measured formula in even rank).** If \(d\) is even, every \(\alpha\in K_c^0(F^*)\) satisfies
\[
\tau_{\Lambda*}(\operatorname{Ind}_a\alpha)
=\epsilon_d\langle\pi_!\operatorname{ch}(\alpha),[C]\rangle,
\qquad \epsilon_d=(-1)^{d(d+1)/2}.
\tag{69}
\]
All forms paired with the current have compact base support. The same theorem holds for the even-dimensional product source fibres \(H\times S^1\), with the circle pair groupoid and compact support in \(V\).

**Proof.** We first compute on the relative Dirac classes of Proposition 8.20. Integration by parts in (65) gives, for every fixed integer \(j\ge0\),
\[
\sup_{z\in\mathbb R}(1+z^2)^j
|r_t(z)^2-e^{-2tz^2}|
\le C_jt^{-M_j}e^{-c/t},\qquad 0<t\le1,
\tag{70}
\]
for some \(c>0\). To verify the estimate, subtract the uncut normalized Gaussian. Its removed tails and every fixed derivative have \(L^1\) norm at most a polynomial in \(t^{-1}\) times \(e^{-\delta_0^2/(8t)}\); the same bound holds for \(a_t-1\). Multiplication by \(z^m\) is integration by parts in \(s\). For the square, subtract the convolution of the uncut Gaussians and use the \(L^1\) convolution inequality for each fixed derivative. Reducing \(c\) absorbs the finite polynomial factors. This proves (70), including its polynomial spectral weights.

Uniform source elliptic estimates give a diagonal estimate
\[
|f(D_p)(1_x)|\le C_j
\sup_z(1+z^2)^j|f(z)|,\qquad j>d/2.
\tag{71}
\]
In detail, evaluation on a fixed unit chart is bounded on \(W^j\), uniformly in \(x\); its dual point distribution is bounded in \(W^{-j}\). The all-real spectral Sobolev comparison follows from the integer elliptic estimates, interpolation and duality exactly as in Lemma 6.7 and Theorem 6.8 of the index lesson, using Lemma 8.19's uniform source jets and localization. Pairing the two point distributions with
\((1+D_p^2)^{j/2}f(D_p)(1+D_p^2)^{j/2}\)
gives (71). Larger \(j\) treats kernel derivatives. Applying it to (70) bounds the diagonal error by \(Ct^{-M}e^{-c/t}\).

We also check the use of the local heat coefficient on these noncompact source covers. The radial transport recursion and Clifford degree calculation in Lemmas 6.24–6.25 and Theorem 6.26 of the index lesson use finitely many jets in a buffered source chart. Lemma 8.19 supplies bounds for each required jet, the inverse metric, the density and the finite-overlap partition on every such chart. The cut-off heat parametrix has compact relative group support. Its residual Taylor remainder and the separated-cutoff remainder have the same uniform Sobolev norm bounds as in Theorem 6.12 there, by the Fourier–Schur and row/column argument of Lemma 8.15. Increasing the finite recursion order gives any prescribed positive remainder power. Duhamel's remainder estimate uses the contraction bound of the self-adjoint heat semigroup and those Sobolev bounds; it has no integration over the unit space. It therefore proves the actual local expansion uniformly here. With the stated chirality and curvature convention, its supertrace density is
\[
\operatorname{str}\bigl(p_i e^{-sD_{p_i}^2}\bigr)(1_x)
=[\widehat A(F)\operatorname{ch}(E_i)]_d(x)+O(s).
\tag{72}
\]
All negative powers cancel pointwise by the full-word Clifford supertrace. This verification extends the local calculation; it does not assume that either heat trace over \(V\) is finite.

Let \(K'=H_{6\delta}K\) contain the support in (64). It is fixed and has finite unit measure. Equations (70)–(71) show that replacing the wave-cutoff diagonal by the heat diagonal on \(K'\) changes the integral by at most
\(C\mu(K')t^{-M}e^{-c/t}\).
Equation (72), at \(s=2t\), then shows that its limit is
\[
\langle\widehat A(F)
(\operatorname{ch}(E_0)-\operatorname{ch}(E_1)),[C]\rangle.
\tag{73}
\]
The characteristic difference vanishes off \(K\); its integral over \(K'\) is its full current pairing. The left side of (64) is independent of \(t\), so (73) is its value.

The fundamental frame makes \(F\) a trivial oriented spin bundle, with trivial determinant line. Chern–Weil transgression from its connection to a flat trivial connection makes \(\widehat A(F)-1\) exact. Multiplying its transgression by the closed compactly supported character difference gives a compactly supported primitive. Closedness of \(C\) therefore replaces \(\widehat A(F)\) by \(1\) in (73). This step uses compact support of the difference, even though the transgression itself need not have compact support.

Every class of \(K_c^0(V\times\mathbb R^{2\ell})\) is the integral spin Thom image of a class in \(K_0(B)\). This is stable Bott periodicity with coefficients \(B\); it applies to all of a noncompact \(V\) and does not require a finite cell decomposition. Lemma 8.19 supplies relative projection representatives for each such class. The projection connection Dirac symbol is that Thom representative, and
\(\pi_!\operatorname{ch}(\alpha)=(-1)^\ell\widehat A(F)^{-1}
(\operatorname{ch}(E_0)-\operatorname{ch}(E_1))\),
with the sign computed in Theorem 6.29 of the index lesson. Since \(\epsilon_{2\ell}=(-1)^\ell\), (73) is (69) for the entire integral symbol lattice, including torsion.

For the circle product the source fibres are \(H\times S^1\). The same energy, local Sobolev and relative kernel arguments hold by Lemma 8.19, and the unit trace is \(\tau_\Lambda\otimes\operatorname{Tr}\). The tangent frame, including the positive circle field, is again trivial. Even Bott periodicity now has coefficient algebra \(C_0(V\times S^1)\). Relative projection representatives are constant outside a compact subset of \(V\times S^1\), and all the preceding estimates apply. The compact-base product calculus and its full relative symbol boundary are obtained by the same local proof as Lemma 8.18, independently of any solvable parameter deformation. This proves its measured assertion in the stated product scope. Rank zero is simply the trace of the relative coefficient projection, and satisfies (69) directly. \(\square\)

**Proposition 8.22 (the compact-support circle product).** Let \(d>0\), with \(V\) possibly noncompact, and let \(\kappa\) be the circle class of (36) in the index lesson. For every \(\alpha\in K_c^0(F^*)\),
\[
\tau_{\Lambda,\mathrm{product}*}
(\operatorname{Ind}_a(\alpha\boxtimes\kappa))
=-\tau_{\Lambda*}(\operatorname{Ind}_a\alpha).
\tag{74}
\]
The product algebra is \(A\otimes\mathcal K(L^2(S^1))\). Its exterior reference operator is genuinely invertible; ellipticity alone on the noncompact base would not suffice for this assertion.

**Proof.** Use Lemma 8.18 to represent \(\alpha\) by a unitary sphere symbol \(u=I\) off \(K\) and \(P_0=I+\operatorname{Op}(u-I)\). Its inverse defects are in \(A\). The smooth compact algebra lies in the finite trace ideal by the factorization of Lemma 7.9 in the transverse-measure lesson. Consequently \(A\) is contained in the norm closure of the finite-trace ideal in the measured algebra. Applying the random-operator Fredholm theorem of Proposition 6.28 of the index lesson gives
\[
\tau_{\Lambda*}(\operatorname{Ind}_a\alpha)
=\tau_\Lambda(\ker P_0)-\tau_\Lambda(\ker P_0^*),
\]
with both dimensions finite. This use of the Fredholm theorem needs no finite heat trace: a parametrix modulo the trace-compact ideal is exactly its hypothesis. For example a small-norm approximation of its defects by finite-trace operators bounds each kernel projection by a finite-trace spectral projection. The finite module boundary index and the measured Fredholm boundary agree by the same explicit matrix map as in that proposition.

Let \(H_1=1+\Delta\) be the homogeneous connection-Laplacian operator on the full first coefficient bundle; the trivial bundle representation makes the same choice possible in source and target. Its all-real powers have the Sobolev and multiplier calculus used here. One can verify its resolvent algebra membership by the complete-metric wave equation for \(\sqrt{\Delta}\): the scalar energy proof gives finite speed, compact Fourier Schwartz functions give smooth compact group kernels, and uniform functional approximation gives \((1+\Delta)^{-1}\in C^*(H)\). The local elliptic estimates then give integer powers, interpolation and duality give real powers, and the parametrix expansion gives their stated principal symbols, as in the Hilbert-module lesson's Section 6. Put \(P=P_0H_1^{1/2}\). Uniform elliptic graph estimates give the unique closed realization on \(W^1\). The order \(-1\) parametrix gives the maximal-domain assertion: if \(Pv\in L^2\) distributionally, it first gives \(v\in W^1\), and then the graph estimate. Smooth compact sections are dense there. The same holds for the adjoint. On \(\ker P_0\), elliptic regularity bounds all Sobolev norms by the \(L^2\) norm, so \(H_1^{1/2}\) and its inverse give bounded equivariant identifications of the source kernels. The distributional identity \(P^*=H_1^{1/2}P_0^*\), and injectivity on \(W^{-1}\), identify the adjoint kernels. Thus \(P\) has the same two finite measured dimensions as \(P_0\).

On the circle use \(Q_0=P_+M_{e^{i\theta}}P_++P_-\), with \(P_+\) the modes \(n\ge0\), and \(Q=Q_0H_2^{1/2}\), \(H_2=1-\partial_\theta^2\). It has no source kernel and its adjoint kernel is the constant line. Form
\[
A_1=\begin{pmatrix}0&P^*\\P&0\end{pmatrix},\quad
A_2=\begin{pmatrix}0&Q^*\\Q&0\end{pmatrix},\quad
\mathcal D=A_1\otimes1+\Gamma_1\otimes A_2,
\quad \Gamma_j=\operatorname{diag}(1,-1).
\tag{75}
\]
The positive squares of the separate factors strongly commute. On joint bounded spectral cuts the two odd summands anticommute, and
\[
\mathcal D^2=A_1^2\otimes1+1\otimes A_2^2,\qquad
\|\mathcal Dv\|^2=\|(A_1\otimes1)v\|^2+\|(1\otimes A_2)v\|^2.
\]
The cuts exhaust their intersected first-order domain. The bounded resolvents
\((\mathcal D\mp i)(1+A_1^2\otimes1+1\otimes A_2^2)^{-1}\)
prove self-adjointness on that domain, as in the explicit argument in Proposition 6.36 of the index lesson. No tensor-domain assertion is being inferred just from the formal square.

Put \(H=H_1\otimes1+1\otimes H_2\) on this graded product and \(T=\mathcal D H^{-1/2}\). The joint-cone proof of Lemma 6.35 there applies to the current uniform source seminorms: where
\(|\xi|\ge c_\delta(|\xi|^2+\eta^2)^{1/2}\), separate first-factor symbol derivatives are joint symbol derivatives. Off a slightly larger cone, integration by parts in the intermediate first coordinate gives arbitrary negative joint order. The bounded-frequency part is smoothing in both factors. Compact first-factor remainders are approximated by smooth compact kernels and have the same assertion on this cone. Finally its complement contributes norm at most \(\|B_1\|\sqrt{2\delta}\), where \(B_1=A_1H_1^{-1/2}\). Interchanging the factors proves the second term. Thus
\[
T=(B_1\otimes1)\sqrt Z+(\Gamma_1\otimes B_2)\sqrt{1-Z},
\qquad Z=(H_1\otimes1)H^{-1},
\]
lies in the completed product order-zero calculus. The continuous principal odd map is
\[
s(\xi,\eta)=
\frac{a_1(\xi)\otimes1+\Gamma_1\otimes a_2(\eta)}
{\sqrt{|\xi|^2+\eta^2}},
\tag{76}
\]
where \(a_1\) is \(|\xi|\) times the graded unitary matrix of \(u\), and \(a_2\) is \(|\eta|\) times the graded circle symbol. Each is zero at its own origin. The anticommuting square gives \(s^2=1\) off the joint zero section. At either axis the vanishing numerator factor supplies continuity. Uniform smooth approximation on its joint sphere preserves invertibility and its relative class.

We spell out the exterior Fredholm check. Work first in the product calculus over \(\mathcal B=B^+\rtimes H\). All coefficients and their source jets extend to the added base point as their homogeneous values. Lemma 8.18's localization and quotient-norm proof gives its exact symbol quotient as well as the compact ideal \(\mathcal B\otimes\mathcal K\). Evaluation at base infinity maps it onto the homogeneous product calculus. The image \(P_\infty=H_1^{1/2}\) is invertible, and the image \(\mathcal D_\infty\) satisfies
\[
\mathcal D_\infty^2=H_1\otimes1+1\otimes A_2^2\ge1.
\]
Its graph norm is equivalent to the product \(W^1\) norm by the separate elliptic estimates. Therefore \(T_\infty=\mathcal D_\infty H^{-1/2}\) is a bounded invertible operator on the product Hilbert space. It belongs to the homogeneous calculus closure; the faithful regular representation and spectral permanence of a unital C*-subalgebra put its inverse there too.

Choose a product inverse-symbol parametrix \(R_0\) for \(T\). Its two defects are in \(\mathcal B\otimes\mathcal K\). In the exterior calculus, \(T_\infty^{-1}-R_{0,\infty}\) has zero principal symbol and hence belongs to \(C^*(H)\otimes\mathcal K\). Lift it by the constant coefficient splitting of (67) and add that lift to \(R_0\). The resulting \(R\) has
\[
TR-1,\ RT-1\in
\ker(\mathcal B\otimes\mathcal K\longrightarrow C^*(H)\otimes\mathcal K)
=A\otimes\mathcal K.
\tag{77}
\]
This proves actual Fredholmness over the required algebra, not merely over the algebra with a constant exterior coefficient.

Its relative symbol class is \(\alpha\boxtimes\kappa\). The tensor product of the two relative two-term complexes has odd map \(a_1\otimes1+\Gamma_1\otimes a_2\); cancellation of mixed terms makes it invertible wherever either complex is invertible. Radial normalization and smoothing of (76) preserve that class. On the exterior base set, the first complex is the identity complex and is contractible. Its actual invertible operator \(P_\infty\), and hence \(T_\infty\), supplies this exterior trivialization. More explicitly, the usual tensor symbol comparison first gives equality of the two boundary indices in \(K_0(\mathcal B\otimes\mathcal K)\); the exterior identity complex contributes zero, since its quantization above is invertible. The split version of (67), tensored with \(\mathcal K\), makes \(K_0(A\otimes\mathcal K)\) inject into this group. Equality there therefore proves equality of their compact-support indices. This also proves independence of the exterior inverse lift and the quantization by the elementary boundary homotopy. Keeping the exterior trivialization is essential in this comparison.

The finite-rank circle kernel proof in Proposition 6.36 gives the completed algebra \(A\otimes\mathcal K\) and trace \(\tau_\Lambda\otimes\operatorname{Tr}\) here too. It uses finite matrices over \(A\), squared-kernel Fubini and increasing circle projections; it has no compact-base assumption. The circle constant projection has kernel \(1/(2\pi)\) and trace one. The dense finite-trace ideal and (77) therefore make \(T^+\) trace Fredholm.

Finally \(H^{1/2}\) gives a bounded equivariant isomorphism between \(\ker T^+\) and \(\ker\mathcal D^+\), by the elliptic graph estimate. The adjoint identity \((T^+)^*=H^{-1/2}\mathcal D^-\) on distributions, injectivity of \(H^{-1/2}:W^{-1}\to L^2\), and the maximal-domain assertion identify the adjoint kernels. The nonnegative square in (75) forces
\[
\ker\mathcal D^+=(\ker P^*)\otimes\mathbb C1,\qquad
\ker\mathcal D^-=(\ker P)\otimes\mathbb C1.
\]
Their product trace is the reversed difference of the two original finite dimensions. This proves (74). \(\square\)

**Theorem 8.23 (the noncompact measured formula and entire trace image).** For the hypotheses of this subsection, in every dimension,
\[
\tau_{\Lambda*}(\operatorname{Ind}_a\alpha)
=\epsilon_d\langle\pi_!\operatorname{ch}(\alpha),[C_\Lambda]\rangle
\quad(\alpha\in K_c^0(F^*)),
\tag{78}
\]
and
\[
\tau_{\Lambda*}(K_0(A))
=\{\langle\operatorname{ch}(x),[C_\Lambda]\rangle:
x\in K_c^{d\bmod2}(V)\}.
\tag{79}
\]
Every value on the right is finite. Nonzero invariant transverse measure on noncompact \(V\) need not force \(H\) to be unimodular.

**Proof.** Even rank is Theorem 8.21. If \(d\) is odd, apply its even-rank product assertion to \(\alpha\boxtimes\kappa\). The explicit circle character computation (44)–(45) in the index lesson uses only a compact cotangent cutoff and therefore retains its compact-base support here:
\[
\pi_{\mathrm{product}!}\operatorname{ch}(\alpha\boxtimes\kappa)
=(-1)^d\pi_!\operatorname{ch}(\alpha)\wedge\frac{d\theta}{2\pi}.
\]
The product current is \(C_\Lambda\times[S^1]\), with the first leaf followed by the positive circle. Integration of \(d\theta/(2\pi)\) is one. Proposition 8.22 and \(\epsilon_{d+1}=(-1)^{d+1}\epsilon_d\) now give
\[
-\tau_{\Lambda*}(\operatorname{Ind}_a\alpha)
=\epsilon_{d+1}(-1)^d
\langle\pi_!\operatorname{ch}(\alpha),[C_\Lambda]\rangle
=-\epsilon_d\langle\pi_!\operatorname{ch}(\alpha),[C_\Lambda]\rangle.
\]
This proves (78). In rank zero the foliation algebra is \(C_0(V)\), and the diagonal trace on relative coefficient projections gives it directly.

Lemma 8.18 makes every \(K_0(A)\) class a full compact-support symbol index. Coefficient Bott periodicity identifies its entire symbol lattice with \(K_c^{d\bmod2}(V)\), exactly as in Corollary 8.17, now with coefficient algebra \(C_0(V)\). Even Bott character integral \((-1)^{d/2}\), the positive odd winding integral one and the fixed ordered product convention give a scalar generator with fibre integral \(+1\), after changing its sign if necessary. Its external product has fibre character \(\operatorname{ch}(x)\). Multiplication by the fixed \(\epsilon_d\) preserves the subgroup image. Equations (78) and surjectivity therefore give (79), integrally and with torsion retained. Smooth representatives are constant outside a compact set; their relative characters have compact support, so local finiteness of the current gives finite pairings.

The numerical assertion of the historical corollary is now proved in its noncompact as well as compact scope. Its measure-existence parenthesis still needs the explicit correction: Proposition 8.12 proves it for nonempty compact \(V\), whereas Example 8.13 refutes it for noncompact \(V\). These are exact scope statements, not an added compactness hypothesis on (79). Equality of the three geometric and topological maps and general assembly remain separate obligations. \(\square\)

The finite propagation method has a classical precedent in [Gorokhovsky–Lott, Appendix A](https://math.colorado.edu/~gorokhov/preprints/nonetale.pdf). Their paper's proper cocompact setting is not being used as a noncompact-base theorem here. The relative matrix, compact reachable set, normalized wave cutoff and exterior inverse checks above supply that extension in the present transformation-groupoid scope.


![A normalized compact wave density makes an exact relative index matrix. Its difference is supported on a fixed compact reachable set; the local heat comparison and circle product yield the noncompact trace-image formula.](../figures/relative-wave-index.png)

*Figure 4. Lemma 8.18 through Theorem 8.23. The wave-time curve is a numerical sample of the declared smooth plateau cutoff, not a proved asymptotic bound; its reproducible source uses Simpson normalization. The group metric ball defines the reachable compact set and supplies no ambient metric on the base. Matrix (66), relative conjugation (68), estimates (70)–(72) and the two circle signs give the actual proof. Exercises 18–21 check the normalization, nonscalar reference, parity and affine-group example.*

Full-size PNG · Editable SVG.


### The ordered analytic map and the horocycle example

The preceding symbol-index isomorphism has a precise relation to the ordered analytic Thom maps. We prove that relation before addressing the numerical example. This identifies analytic maps on the full K-groups; identification with the historical geometric leaf-map construction and assembly still requires their stated interfaces.

**Corollary 8.24 (ordered Thom, actual symbols and analytic choice independence).** Let \(B\) be a separable C*-algebra with a point-norm continuous action \(\alpha\). For an ordered normal decomposition of a simply connected solvable \(d\)-dimensional \(H\), write
\[
\Phi_H^\alpha:K_j(B)\longrightarrow K_{j+d}(B\rtimes_\alpha H)
\]
for the composite of the positive real Thom maps of Theorem 8.7 and Corollary 8.10. Let \(b_{\rm ord}\in K_c^{d\bmod2}(\mathfrak h^*)\) be their ordered scalar Bott class at the trivial action. Then
\[
\Phi_H^\alpha
=(\mathrm{ev}_1)_*(\mathrm{ev}_0)_*^{-1}
\bigl(\,\cdot\,\boxtimes b_{\rm ord}\bigr).
\tag{80}
\]
Here the zero algebra is intrinsically \(B\otimes C_0(\mathfrak h^*)\), with negative physical Fourier phase. For a smooth orbit foliation with graph \(V\rtimes H\), this is
\[
\Phi_H^\alpha(x)=
\operatorname{Ind}_a(x\boxtimes b_{\rm ord}),
\qquad x\in K_c^j(V).
\tag{81}
\]
Both degrees and noncompact \(V\) are included. The ordered analytic map is independent of the normal decomposition and ordered tangent frame with a fixed orientation. Reversing that orientation changes its sign. These are group-level analytic choice assertions; they do not assert independence of geometric wrong-way factorizations.

**Proof.** At every stage of Theorem 8.14's recursive family algebra the evaluation is equivariant for the next real action. Apply the natural real Thom map at that stage. Induction gives
\(\Phi_{\rm fam}:K_j(C([0,1])\otimes B)\to K_{j+d}(\mathfrak A_H(B))\)
and commutative evaluation squares
\[
(\mathrm{ev}_v)_*\Phi_{\rm fam}
=\Phi_{H_v}^{\alpha^v}(\mathrm{ev}_v^{\,B})_*.
\]
The coefficient evaluations are K-isomorphisms by interval contraction. The crossed-product evaluations are K-isomorphisms by Theorem 8.14. At zero all actions are trivial. Corollary 8.10, including its suspended degree, identifies the consecutive real maps with external multiplication by their positive winding generators, appended in the ordered normal-then-outer coordinates. Their composite is exactly multiplication by \(b_{\rm ord}\). We retain that class, rather than replacing it by a generator selected only by an absolute Chern integral. The two evaluation squares give (80). Theorem 8.16 and Lemma 8.18 identify their middle comparison with the actual full symbol index, proving (81).

We check choice independence of the deformation itself. Let \(\varphi_i:\mathbb R^d\to H\), \(i=1,2\), be two full ordered coordinate diffeomorphisms, with \(\varphi_i(0)=e\) and tangent maps \(L_i\). Their scaled families have the coordinate change
\[
\rho_v(q)=v^{-1}\varphi_2^{-1}(\varphi_1(vq))
\quad(v>0),\qquad
\rho_0(q)=L_2^{-1}L_1q.
\tag{82}
\]
Hadamard division, exactly as in (53), extends this and its inverse smoothly through zero. It is a group isomorphism in every fibre, and \(\Gamma_{2,v}\rho_v=\Gamma_{1,v}\); hence it intertwines the coefficient actions. Compact supports have uniformly compact transformed supports because \((v,q)\mapsto\rho_v(q)\) and its inverse are continuous on the compact parameter interval times each compact coordinate set.

To check the density without a hidden constant, choose one fixed Haar measure \(dh\) on \(H\). Use on family \(i\) the density
\(v^{-d}\varphi_i(v\,\cdot)^*dh\) for \(v>0\), with its smooth value at zero. Its value at the identity is the constant \(J_i(0)>0\), where \(\varphi_i^*dh=J_i(q)\,dq\). This differs from Theorem 8.14's unit-normalized density by just that constant; scaling a Haar measure by \(c>0\) identifies its convolution algebra with the old one by \(f\mapsto cf\). Thus it gives the same family algebra and K-maps. With the present common Haar normalization, \(\rho_v\) transports the densities exactly, including at zero. Kernel transport is therefore a *-isomorphism on the joint compact-kernel algebras. The regular fibre unitary and the fibre-supremum norm extend it to their C*-completions. At one it is identity in the original \(H\) realization; at zero its Fourier change is the displayed linear tangent change, so it is identity on the intrinsic dual \(\mathfrak h^*\). Consequently the intrinsic evaluation comparison in (80) is independent of the ordered coordinates.

It remains to compare the two scalar Bott classes. A positive determinant linear change lies in \(GL^+(d,\mathbb R)\), which is path connected: polar decomposition connects its positive factor to identity and its orthogonal factor lies in the connected \(SO(d)\) for \(d\ge2\); \(d=1\) is direct. Linear pullbacks along such a path are point-norm continuous on \(C_0(\mathbb R^d)\). On compactly supported functions this is uniform continuity on a common compact support, since the path's least singular value has a positive minimum; density treats all \(C_0\) functions. The pullbacks are therefore homotopic and preserve the Bott class. A negative determinant change is homotopic to reflection of the first coordinate. Reflection sends
\(u_+(p)=(p-i/2)/(p+i/2)\) to \(u_+(p)^{-1}\), whose \(K_1\)-class is its negative. Bilinearity of the remaining ordered external products gives the same minus sign in dimension \(d\). Thus the classes agree for fixed orientation and reverse sign with it. Formula (80) proves the asserted analytic independence for every coefficient algebra. In dimension zero every map is identity. \(\square\)

We now supply the full compact horocycle example. Put \(L=SL(2,\mathbb R)\), let \(\Gamma\subset L\) be discrete with \(X=L/\Gamma\) compact, and write
\[
U_s=\begin{pmatrix}1&0\\s&1\end{pmatrix},\qquad
\overline U_v=\begin{pmatrix}1&v\\0&1\end{pmatrix},\qquad
a_t=\begin{pmatrix}e^{-t/2}&0\\0&e^{t/2}\end{pmatrix}.
\]
The source's horocycle flow is the left action of \(U_s\). The group \(L\) is unimodular: in the Lie algebra basis consisting of the two nilpotent matrices and \(A=\operatorname{diag}(-1/2,1/2)\), every adjoint matrix has trace zero. Connectedness then gives modular character one. Haar measure descends under right translation by \(\Gamma\), and compactness permits normalization to a left-\(L\)-invariant probability \(m\) on \(X\).

**Lemma 8.25 (the unipotent fixed-vector argument and diagonal mixing).** In a strongly continuous unitary representation of \(L\), a vector fixed by all \(U_s\) is fixed by \(L\). On \(L^2(X,m)\), the diagonal flow is mixing:
\[
\int_X f(a_tx)g(x)\,dm(x)
\longrightarrow\left(\int_Xf\,dm\right)\left(\int_Xg\,dm\right)
\quad(t\to+\infty),
\tag{83}
\]
for real \(L^2\) functions; the complex version uses the usual conjugation.

**Proof.** It suffices first to use the upper unipotent, since conjugation by \(\begin{pmatrix}0&-1\\1&0\end{pmatrix}\) interchanges the two groups up to reversal of their parameters. Let \(w\) be upper-unipotent fixed and put \(F(g)=\langle\pi(g)w,w\rangle\). It is continuous and invariant under upper-unipotent multiplication on both sides. For \(\lambda>0\) the exact identity
\[
\overline U_n U_{(\lambda-1)/n}\overline U_{-n/\lambda}
=\begin{pmatrix}\lambda&0\\(\lambda-1)/n&\lambda^{-1}\end{pmatrix}
\longrightarrow\operatorname{diag}(\lambda,\lambda^{-1})
\tag{84}
\]
gives \(F(\operatorname{diag}(\lambda,\lambda^{-1}))=\|w\|^2\): the middle unipotent tends to identity, while the two outer factors do not change \(F\). The unitary norm identity makes \(w\) fixed by every positive diagonal matrix. Conjugating the opposite unipotent by these diagonals contracts it to identity. Invariance of \(w\) under the diagonals then gives
\[
\|\pi(U_s)w-w\|
=\|\pi(\operatorname{diag}(\lambda,\lambda^{-1})
U_s\operatorname{diag}(\lambda^{-1},\lambda))w-w\|
\longrightarrow0
\quad(\lambda\to\infty).
\]
Thus \(w\) is fixed by both unipotent groups. They generate \(SL(2,\mathbb R)\): ordinary row elimination treats a nonzero diagonal entry, and the displayed Weyl matrix is itself a product of three elementary unipotents; its square is \(-I\). This proves the fixed-vector assertion, including the centre.

The representation \((\pi(g)f)(x)=f(g^{-1}x)\) is strongly continuous, first on continuous functions and then by their \(L^2\) density. Its \(L\)-invariant vectors are constants. Here is a way to see the almost-everywhere assertion without presuming ergodicity. Convolution with a smooth compact group function is a smoothing operator on \(X\): the map \((g,y)\mapsto(gy,y)\) is a local diffeomorphism, and on compact group support times compact \(X\) its pushforward has a smooth kernel. Finite local inverse charts prove this, with density Jacobians bounded on their protected supports. An invariant \(L^2\) vector equals its convolution with any such function of integral one, so is smooth. A smooth invariant function is constant by transitivity of \(L\) on \(X\).

Work on the orthogonal complement \(L^2_0(X,m)\). From any sequence \(t_n\to+\infty\), separability and the norm-one bound give a subsequence for which \(\pi(a_{t_n})\) converges in the weak operator topology to \(T\): extract successively on a countable dense set of matrix coefficients, then extend by the uniform bound. The relation
\[
a_{-t}U_sa_t=U_{e^{-t}s}
\]
implies
\(\|\pi(U_s)\pi(a_t)v-\pi(a_t)v\|\to0\)
for each fixed \(s,v\). Therefore every vector in the range of \(T\) is \(U\)-fixed. The first part makes it \(L\)-fixed, hence zero in \(L^2_0\). Thus \(T=0\). Every subsequence has only this possible weak cluster point, so all matrix coefficients on \(L^2_0\) tend to zero. The adjoint gives the same assertion for \(a_{-t}\). Adding back constants proves (83). \(\square\)

**Theorem 8.26 (uniform horocycle averages and the unique transverse measure).** For every \(f\in C(X)\),
\[
\sup_{x\in X}\left|
\frac1T\int_0^T f(U_sx)\,ds-\int_X f\,dm
\right|\longrightarrow0\qquad(T\to+\infty).
\tag{85}
\]
The flow is free and every orbit is dense. Its invariant probability is uniquely \(m\). Every nonzero geometrically locally finite invariant transverse measure for its orbit foliation is a positive scalar multiple of the measure whose tangent current is
\[
C_m(\omega)=\int_X\omega(N^\#)\,dm,
\qquad N=\begin{pmatrix}0&0\\1&0\end{pmatrix}.
\]
No torsion-free hypothesis on \(\Gamma\), or replacement of \(SL(2,\mathbb R)\) by \(PSL(2,\mathbb R)\), is needed.

**Proof.** We first turn mixing into uniform estimates. If \(k\in C_c^\infty(L)\) is nonnegative with \(\int k\,dh=1\), the image of \(k(g)\,dh\) under \(g\mapsto gy\) has a smooth density \(h_y\) with respect to \(m\). The joint local-diffeomorphism argument in Lemma 8.25 proves that \((x,y)\mapsto h_y(x)\) is smooth: restrict to a compact group neighbourhood containing the support, use finitely many local inverse charts on its product with \(X\), and partition the support before pushing each chart's density. Each fibre meets only finitely many protected charts. In particular \(y\mapsto h_y\) is continuous into \(L^2(X,m)\), so its image is compact. The mixing limit is uniform on this compact family. To verify that last step, cover it by a finite \(L^2\) \(\varepsilon\)-net, apply mixing at each net point and use Cauchy–Schwarz with the fixed norm \(\|f\|_2\) for the error. Hence
\[
\sup_{y\in X}\left|\int_L k(g)f(a_tgy)\,dh(g)-\int_X f\,dm\right|
\longrightarrow0.
\tag{86}
\]

Choose \(h\in C_c^\infty((0,1))\), \(h\ge0\), \(\int_0^1h=1\), and a smooth nonnegative probability density \(\phi_\varepsilon(v,r)\) supported in \(|v|,|r|<\varepsilon\). The coordinates
\[
g=\overline U_v a_r U_s
=\begin{pmatrix}
e^{-r/2}+vs\,e^{r/2}&v e^{r/2}\\
s e^{r/2}&e^{r/2}
\end{pmatrix}
\]
parametrize the open set \(g_{22}>0\) uniquely. Normalize group Haar at the tangent basis \((\overline N,A,N)\). In these coordinates it is \(e^r\,dv\,dr\,ds\). Indeed the columns of \(g^{-1}dg\) are
\[
e^r(\overline N-2sA-s^2N),\qquad A+sN,\qquad N,
\]
whose determinant in that basis is \(e^r\). Thus the smooth compact group probability
\[
k_\varepsilon(\overline U_va_rU_s)
=e^{-r}\phi_\varepsilon(v,r)h(s)
\tag{87}
\]
has integral exactly one. Extend it by zero; its support is compactly contained in this coordinate domain.

Put \(T=e^t\) and \(y=a_{-t}x\) in (86). The exact conjugations
\[
a_tU_sa_{-t}=U_{Ts},\qquad
a_t\overline U_va_{-t}=\overline U_{v/T},\qquad
a_ta_ra_{-t}=a_r
\]
make its average
\[
\int\phi_\varepsilon(v,r)h(s)
f(\overline U_{v/T}a_rU_{Ts}x)\,dv\,dr\,ds.
\tag{88}
\]
This differs from \(\int_0^1h(s)f(U_{Ts}x)\,ds\) by at most
\[
\omega_f(\varepsilon)=
\sup_{\substack{|v|,|r|\le\varepsilon\\z\in X}}
|f(\overline U_va_rz)-f(z)|,
\]
for \(T\ge1\). Compactness and joint continuity make \(\omega_f(\varepsilon)\to0\). For each fixed \(\varepsilon\), (86) makes (88) converge to \(m(f)\), uniformly in \(x\), even though its centre \(y=a_{-t}x\) moves. Letting \(\varepsilon\) decrease proves uniform convergence of every such weighted horocycle average. Finally choose \(h\) approaching \(1_{[0,1]}\) in \(L^1\), while retaining integral one. The difference from the unweighted average is bounded by \(\|f\|_\infty\|h-1_{[0,1]}\|_1\), uniformly in \(T,x\). This proves (85). No uniform mixing rate or interchange of two dependent limits was assumed.

If \(\nu\) is any invariant probability, integration of the left average in (85) against \(\nu\) gives \(\nu(f)\) for every \(T\). Its uniform limit is \(m(f)\), hence \(\nu=m\). Haar has full support. If one orbit closure missed a nonempty open set, a nonnegative continuous function supported there with positive Haar integral would have zero averages on that orbit, contradicting (85). Thus every orbit is dense.

We also check freeness. Compact \(X\) has a neighbourhood \(W\) of identity in \(L\) containing no nonidentity stabilizer element of any point. For a proof, take a finite cover with local lifts \(q_i\) to \(L\), whose values range in compact sets. The stabilizer at their points is \(q_i\Gamma q_i^{-1}\). Discreteness of \(\Gamma\) and compactness of those conjugating sets give a single such \(W\). If \(U_sx=x\) with \(s\ne0\), then \(U_{s/T}\) fixes \(a_{-t}x\). For large \(T=e^t\) this is a nonidentity element of \(W\), a contradiction. The orbit foliation therefore has graph \(X\rtimes\mathbb R\), not a circle-cover groupoid.

A positive closed tangent current has the form \(\int\omega(N^\#)\,d\nu\) with a positive Radon unit measure, using the leaf density which makes \(N^\#\) unit length. It has finite total mass by compactness. Closedness says \(\int N^\#f\,d\nu=0\) for every smooth \(f\). Differentiating \(\int f(U_tx)\,d\nu\) proves flow invariance, since the same identity applies to \(f\circ U_t\). Smooth approximation treats all continuous tests. Normalizing any nonzero such measure to a probability now gives \(m\). Conversely the displayed Haar current is positive and closed by \(U\)-invariance. The proved current/transverse-measure correspondence gives exactly the stated uniqueness up to positive scale. This reconciles the source's “only transverse measure”: without normalization its positive scalar multiples are distinct. \(\square\)

**Corollary 8.27 (zero trace with nonzero K-theory).** For the horocycle foliation of Theorem 8.26 and every invariant locally finite transverse measure \(\Lambda\),
\[
\tau_{\Lambda*}\bigl(K_0(C(X)\rtimes_U\mathbb R)\bigr)=\{0\}.
\tag{89}
\]
The ordered analytical map \(K^1(X)\to K_0(C(X)\rtimes_U\mathbb R)\) is an isomorphism, and its target is nonzero.

**Proof.** Uniqueness makes the current a scalar multiple of \(C_m\); the zero measure is immediate. Use
\[
[A,N]=N,\qquad [A^\#,N^\#]=-N^\#.
\]
For every closed smooth one-form \(\omega\), the exterior derivative identity gives
\[
0=(d\omega)(A^\#,N^\#)
=A^\#(\omega(N^\#))-N^\#(\omega(A^\#))
+\omega(N^\#).
\]
Haar is invariant under both one-parameter groups, so the two derivative integrals vanish. Therefore \(C_m(\omega)=0\). The one-dimensional current sees only the closed degree-one component of the odd Chern character; all its pairings with \(K^1(X)\) vanish. The entire trace-image formula (79), or (81) followed by (78), proves (89). This calculation checks all classes, rather than just one test operator.

The analytic isomorphism follows from Corollary 8.24, or directly from Theorem 8.7 and Lemma 8.18. To see that its domain is nonzero, orient the compact three-manifold \(X\) by its global Lie frame. Insert the compact-support three-dimensional odd Bott class in one oriented coordinate ball and extend its unitary by identity outside that ball. Its top odd Chern character has integral of absolute value one, by the ordered scalar Bott computation in Corollary 8.17. The same integral on \(X\) is nonzero. A zero K-class would have an exact character, whose integral on the compact boundaryless manifold is zero by Stokes. Thus this class, and its image under the analytic isomorphism, are nonzero. The trace vanishes on that image nonetheless. \(\square\)

The unique ergodicity statement is Furstenberg’s classical theorem, discussed in the freely accessible [Connes survey, Section 12](https://alainconnes.org/wp-content/uploads/foliationsfine.pdf#page=36). Lemmas 8.25 and Theorem 8.26 give a complete proof here in the exact compact homogeneous scope. Corollary 8.27 supplies the whole source numerical example. Equality with the historical topological and geometric leaf maps, their wrong-way factorization independence and general assembly are still required for the full preceding theorem.

![Naturality compares the ordered analytic Thom map with the actual symbol index. Exact diagonal conjugation stretches the horocycle coordinate and contracts its opposite, giving uniform averages; the trace still vanishes on a nonzero K-group.](../figures/thom-horocycle.png)

*Figure 5. Corollary 8.24, Lemma 8.25, Theorem 8.26 and Corollary 8.27. Panel A is a diagram of K-maps for a separable coefficient algebra with point-norm continuous action; the lower-right group belongs to the parameter algebra. Panel C is the actual Gauss-coordinate section at \(r=0\), with bounding support rectangles \(0\le s\le1\), \(|v|\le1/4\), transformed by \(T=4\) into \(0\le s\le4\), \(|v|\le1/16\). Each has coordinate area \(1/2\); the horizontal and vertical scales are unchanged between the rectangles. This is neither an ambient geometric projection nor a probability-density plot. Haar and its compensating kernel density are exact in (87). The moving-centre compactness argument and the order of limits in Theorem 8.26 give uniform convergence without a mixing rate. The bracket calculation proves the whole zero trace image, while the ball-supported odd Bott class gives nonzero K-theory. Full proofs and Exercises 22–25 accompany the figure; the classical unique ergodicity theorem is due to Furstenberg, as described in the freely accessible [Connes survey, Section 12](https://alainconnes.org/wp-content/uploads/foliationsfine.pdf#page=36).*

Full-size PNG · Editable SVG.

### Native solvable geometric cycles and the full normal bundle

The following complete cocycle and bundle arguments apply to the actual graph hypothesis of [Connes, Theorem5, Sections9–12](https://alainconnes.org/wp-content/uploads/foliationsfine.pdf). They reduce every compact geometric assembly cycle to a global map into the unit manifold and identify its entire transverse bundle. The integral manifold-duality and noncompact leaf-map interfaces are stated explicitly at the end.

### The actual action cocycle and its global lift

Let \(H\) be a simply connected solvable Lie group, \(d=\dim H\), acting smoothly on the Hausdorff second-countable manifold \(V\). For this theorem assume the given foliation has its **actual holonomy groupoid** \(G=V\rtimes H\). An arrow labelled \(h\) goes from \(v\) to \(hv\). The proof of the cocycle reduction itself also applies to transformation groupoids without that holonomy identification. This distinction is retained in every consequence for assembly.

Put \(F\subset TV\), \(\tau=TV/F\), \(q=\operatorname{rank}\tau\). The normal representation of an arrow is

\[
 D^\perp h:\tau_v\longrightarrow\tau_{hv},\qquad
 [\xi]\longmapsto[d(h)_v\xi].
 \tag{B.a}
\]

The action preserves \(F\); thus this is well-defined, invertible, smooth, and satisfies the exact arrow product rule. Under the graph hypothesis it is the holonomy derivative, rather than a derivative of a different groupoid.

An actual smooth map cocycle from a manifold \(M\) is given on a locally finite refined cover \(U_\alpha\) by smooth maps \(f_\alpha:U_\alpha\to V\) and smooth functions \(h_{\beta\alpha}:U_\alpha\cap U_\beta\to H\), with

\[
 f_\beta=h_{\beta\alpha}f_\alpha,\qquad
 h_{\gamma\beta}h_{\beta\alpha}=h_{\gamma\alpha},\qquad
 h_{\alpha\alpha}=e.
 \tag{B.b}
\]

The cocycle retains the actual functions \(h_{\beta\alpha}\), including stabilizer elements. Its pulled normal bundle has local fibres \(\tau_{f_\alpha(x)}\) and transition \(D^\perp h_{\beta\alpha}\).

**Lemma 8B.1 (a constructive smooth trivialization).** Every such cocycle admits smooth functions \(k_\alpha:U_\alpha\to H\) with

\[
 h_{\beta\alpha}=k_\beta k_\alpha^{-1}.
 \tag{B.c}
\]

Consequently \(a(x)=k_\alpha(x)^{-1}f_\alpha(x)\) is a global smooth map \(M\to V\). The same statement holds for continuous cocycles on any base carrying a subordinate locally finite partition with closed supports. It includes countable covers, whole parameter families, and manifolds with boundary or collars. No compactness of \(M\), properness of the action, or injectivity of its orbit maps is needed.

**Proof.** Induct on \(d\), using the proved Lie-group decomposition in Lemma 8.1. For \(d=0\) the group is trivial. Otherwise there is a smooth split homomorphism \(\chi:H\to\mathbb R\), with simply connected solvable kernel \(N\), and a fixed one-parameter section \(r(t)=\exp(tX)\). Every element has a unique decomposition (r(t)n), with \(n\in N\), and this decomposition and its inverse are smooth. This uses the global semidirect-product diffeomorphism; it does not assume the exponential map of \(H\) is a diffeomorphism.

Choose a subordinate smooth locally finite partition \(\rho_\gamma\), with closed support contained in \(U_\gamma\). Its existence on the stated manifold is the complete subordinate-atlas and smooth-weight proofs in [Local tools for bundles and transport, Section 3, Theorems 3.E and 3.1](https://kokunoyumeto.github.io/open-math-courses-public/courses/NCG-FOLIATIONS/companions/local-tools-for-bundles-and-transport.html). In a boundary chart the same construction restricts a Euclidean smooth bump to the half-space; the locally finite exhaustion and closed support argument are unchanged. Set \(t_{\beta\alpha}=\chi(h_{\beta\alpha})\). The additive cocycle equation gives \(t_{\beta\gamma}-t_{\alpha\gamma}=t_{\beta\alpha}\) on triple overlaps. On \(U_\alpha\) define

\[
 t_\alpha=\sum_\gamma\rho_\gamma t_{\alpha\gamma},\qquad
 t_\beta-t_\alpha=t_{\beta\alpha}.
 \tag{B.d}
\]

Each product is defined on \(U_\alpha\cap U_\gamma\) and extended by zero on the rest of \(U_\alpha\). This extension is smooth: every point outside the closed support of \(\rho_\gamma\) has a neighbourhood on which that function is identically zero. Only finitely many supports meet a neighbourhood of any point. The sum is therefore a smooth locally finite sum, not just a pointwise finite formal expression. On an overlap, every term with nonzero weight has its triple transition defined; the displayed difference follows after summing and using \(\sum\rho_\gamma=1\). Terms with zero weight make no assumption about an undefined transition.

The residual functions

\[
 r_{\beta\alpha}=r(-t_\beta)h_{\beta\alpha}r(t_\alpha)\in N
 \tag{B.e}
\]

form an exact \(N\)-cocycle. Membership follows by applying \(\chi\). Multiplying two successive functions cancels the adjacent \(r(t_\beta)r(-t_\beta)\), so their product is \(r_{\gamma\alpha}\). The induction hypothesis supplies \(n_\alpha:U_\alpha\to N\) with \(r_{\beta\alpha}=n_\beta n_\alpha^{-1}\). Then \(k_\alpha=r(t_\alpha)n_\alpha\) has (B.c), with exactly this order. Finally (B.b) and (B.c) give

\[
 k_\beta^{-1}f_\beta=k_\alpha^{-1}f_\alpha.
 \tag{B.f}
\]

These local maps glue smoothly. At the continuous level the identical finite-local-sum argument works with continuous subordinate functions. The induction does not differentiate the base, so it also works on a numerable weak quotient. On a smooth parameter manifold or a manifold with boundary the functions, transitions and sums have that same regularity. Smooth collars may first be fixed and the construction performed on the whole collared manifold. No unrelated endpoint trivializations are glued afterwards. This proves the lemma. \(\square\)

**Lemma 8B.2 (the entire transverse bundle and orientation family).** The global lift in Lemma 8B.1 has a canonical bundle isomorphism, for that specified trivialization,

\[
 \Psi:a^*\tau\xrightarrow{\ \cong\ }f^*\tau,
 \qquad \Psi_\alpha(x)=D^\perp k_\alpha(x).
 \tag{B.g}
\]

It transports the full stable spin-c orientation of \(TM\oplus f^*\tau\), including every Hermitian line twist and stable grading, to \(TM\oplus a^*\tau\). Different trivializations yield an actual smooth map homotopy with this entire orientation family. These statements hold on whole compact bordisms and sphere modifications as well as individual cycles.

**Proof.** On an overlap,

\[
 D^\perp h_{\beta\alpha}\,D^\perp k_\alpha
       =D^\perp k_\beta.
 \tag{B.h}
\]

Thus the local maps in (B.g) glue into an invertible bundle map, with local inverse \(D^\perp k_\alpha^{-1}\). This proves the whole bundle statement; it is stronger than agreement of its rank, characteristic classes or rational character. To compare Clifford bundles, first transport the metric along this bundle map. If a different metric was prescribed, the positive square-root metric isometry of Corollary 6.0d identifies the two Clifford bundles. Pull back the entire specified principal spin-c bundle or full graded Morita module by this isometry. Every central transition phase, additional line bundle and stable grading is retained. In particular, a two-torsion line is not replaced by its squared determinant, and a rank-zero odd orientation line is not made the even unit.

Let \(k'_\alpha\) be another trivialization. The functions

\[
 u=(k'_\alpha)^{-1}k_\alpha:M\to H,
 \qquad a'=ua
 \tag{B.i}
\]

are independent of \(\alpha\), by the two copies of (B.c). Lemma 8.1 supplies a global smooth coordinate diffeomorphism \(\varphi:\mathbb R^d\to H\), \(\varphi(0)=e\). Put \(u_s=\varphi(s\varphi^{-1}(u))\) and \(a_s=u_sa\). This is a whole smooth map homotopy, without requiring the contraction to be a group homomorphism. The bundle maps \(D^\perp u_s:a^*\tau\to a_s^*\tau\), with transported metrics and orientations, make a single full orientation family. At \(s=1\) its orientation is exactly the one obtained using \(k'_\alpha\), since \(D^\perp k'_\alpha D^\perp u=D^\perp k_\alpha\).

There is also a direct concordance from the given cocycle to its global lift. Put \(k_{\alpha,s}=\varphi(s\varphi^{-1}(k_\alpha))\), and use the anchors \(k_{\alpha,s}a\) and transition functions \(k_{\beta,s}k_{\alpha,s}^{-1}\). Their inverse and triple equations hold by literal cancellation for every \(s\). At zero they are the global lift with unit transitions; at one they are the original data. The maps \(D^\perp k_{\alpha,s}\) glue its entire normal family to the fixed bundle \(a^*\tau\) over \(M\times I\). This proves the orientation assertion on a single family.

For a bordism, perform Lemma 8B.1 once on its whole base and transport its extending spin-c module in this manner. Its restrictions to the boundary are the reductions of the boundary cocycles up to the already proved full homotopies. The outward-normal convention is unchanged: tensoring with the canonical collar-line module and then restricting commutes with the even normal-bundle map. For a vector-bundle modification \(\pi:\widehat M\to M\), pull back \(k_\alpha\), \(\Psi\), and the full module along \(\pi\). The modifying bundle, its sphere clutch transition and its full inverse orientation factors are unchanged. The finite evaluation maps used in Lemmas 6.0ac–ad act on these Clifford factors and commute with the even map \(\pi^*\Psi\); central line phases remain the same factors. Thus the original and reduced induced stable orientations and clutch bundles correspond exactly, rather than only after taking a character. \(\square\)

### A whole classifying model with its whole twist

Use a right universal \(H\)-space. Associated left-action conventions are

\[
 B_G=(E_H\times V)/\bigl((eh,v)\sim(e,hv)\bigr).
 \tag{B.j}
\]

The normal bundle is the associated bundle of \(E_H\times\tau\to E_H\times V\), with the same derivative action (B.a). A right-action convention is necessary here; switching it without changing the displayed coordinate would invert the map.

**Theorem 8B.3 (a numerable model and a normal-bundle preserving comparison).** There is a contractible right principal \(H\)-space \(E_H\to B_H\), with numerable local sections and contractible \(B_H\). The resulting classifying model (B.j) is homeomorphic to \(B_H\times V\). Under this homeomorphism its entire associated normal bundle is \(\operatorname{pr}_V^*\tau\). Consequently it is homotopy equivalent to \(V\), with the whole specified bundle comparison. For each native compact smooth map cocycle, choose its classifying map using the specified smooth trivialization of Lemmas 8B.1–2. In that choice this comparison returns precisely its smooth global lift and entire normal bundle. An arbitrary continuous section of the join need not pull back smoothly.

**Proof: the entire principal model.** Let \(E_H\) be the indexed join of countably many copies of \(H\). A point is a finite set of columns \((t_i,g_i)\), \(t_i\ge0\), \(\sum t_i=1\), \(g_i\in H\), deleting a column of weight zero. Give it the weak topology from its finite simplex pieces, keeping each column index. Right multiplication is simultaneous \(g_i\mapsto g_i h\). This action is jointly continuous: on every finite piece it is continuous; the product of the weak quotient map with the locally compact group \(H\) is a quotient map, so these maps descend. It is free since a positive column cancels \(g_i h=g_i\). Define \(B_H=E_H/H\) with its quotient topology.

The sets \(O_i=\{t_i>0\}\) are open in \(B_H\). Normalizing the tuple to \((t_j,g_jg_i^{-1})\) is a continuous invariant map on the saturated preimage of \(O_i\), respects every zero-weight face, and descends through its quotient. Its range is the tuples with \(i\)th coordinate \(e\), and supplies a local section \(\sigma_i\). The chart and its inverse are the literal maps

\[
 e\longleftrightarrow(b,g_i),\qquad e=\sigma_i(b)g_i.
 \tag{B.k}
\]

The inverse is continuous by the joint action. On an overlap the transition functions are \(h_{ji}=g_jg_i^{-1}\), so \(\sigma_j=\sigma_i h_{ji}^{-1}\). These continuous functions satisfy exactly the cocycle equation of Lemma 8B.1.

The cover is numerable even though paracompactness of the weak quotient has not been assumed. Set

\[
 m=\max_i t_i>0,\qquad
 \rho_i=\frac{(t_i-m/2)_+}{\sum_j(t_j-m/2)_+}.
 \tag{B.l}
\]

On each finite piece these are continuous finite expressions and agree on faces. They descend to \(B_H\). The denominator is at least \(m/2>0\); the closed support of \(\rho_i\) is contained in \(\{t_i\ge m/2\}\subset O_i\). For local finiteness at \(b_0\), let \(I_0\) be its finite positive support, \(m_0=m(b_0)\), and \(w=1-\sum_{i\in I_0}t_i\). On the neighbourhood \(m>m_0/2\), \(w<m_0/8\), each \(i\notin I_0\) has \(t_i\le w<m_0/8<m/2\). Its numerator vanishes on that whole neighbourhood; its closed support misses a smaller such neighbourhood. Thus only finitely many closed supports occur locally. All zero-weight columns are accounted for in this proof.

Apply the continuous version of Lemma 8B.1 to \(h_{ji}\) with this partition. The resulting functions \(k_i\) give a whole section \(\sigma=\sigma_i k_i\), since \(\sigma_j k_j=\sigma_i h_{ji}^{-1}k_j=\sigma_i k_i\). Formula (B.k) now gives a homeomorphism \(E_H\cong B_H\times H\), with whole coordinate \(h=k_i^{-1}g_i\). This formula is independent of the positive index chosen.

**Proof: contractibility and the classifying property.** A contraction \(c_s(g)=\varphi((1-s)\varphi^{-1}(g))\) moves every group column to \(e\), retaining its weight and index. It is continuous on every finite piece and respects faces. Product with the compact interval preserves the quotient property, so it is a whole homotopy on \(E_H\). Its terminal subspace has all columns \(e\); contract its weights to the zeroth vertex by \(t_i\mapsto(1-s)t_i+s\delta_{i0}\). The zeroth column has group entry \(e\), so adding it is well-defined also when it initially has zero weight. Each finite piece is sent into the finite piece with index 0 adjoined. The same quotient argument proves continuity. Together these two homotopies contract \(E_H\). If \(C\) denotes the contraction, the homotopy \(b\mapsto q(C_s(\sigma(b)))\) contracts \(B_H\) to a point.

A numerable principal right \(H\)-bundle with a countably indexed numerable cover, local sections \(p_\alpha\) and partition \(\lambda_\alpha\) has an equivariant map to \(E_H\): at a point \(p=p_\alpha(x)g_\alpha\), use the columns \((\lambda_\alpha(x),g_\alpha)\). Near a base point only finitely many closed supports occur, and each retained support lies in its section domain. Hence even columns of zero weight can be defined on a common neighbourhood and the map is continuous through a single finite piece there. Right multiplication of \(p\) multiplies each \(g_\alpha\) on the right, as required. The pullback of (B.k) recovers the original local principal charts. Conversely these charts recover the bundle and its transitions from any classifying map. There is no endpoint reindexing issue in the homotopy assertion: trivialize the source bundle once by Lemma 8B.1 and the target by (B.k). Any two equivariant maps then have the forms \((x,h)\mapsto(b_i(x),u_i(x)h)\). Contractibility of \(B_H\) gives a whole homotopy from \(b_0\) to \(b_1\); the explicit coordinate contraction in \(H\) gives one from \(u_0\) to \(u_1\). Their product is a whole equivariant homotopy with these exact endpoints. This gives the classifying property on eligible paracompact parameter bases, and only the numerable bundle version is used here.

For the transformation groupoid an actual cocycle is a principal \(H\)-bundle together with an equivariant anchor \(p h\mapsto h^{-1}v(p)\in V\). In local sections this is exactly (B.b), with \(\sigma_\beta=\sigma_\alpha h_{\beta\alpha}^{-1}\). The preceding equivariant map to \(E_H\), paired with its anchor, gives a map to (B.j). Local charts recover the original cocycle. Homotopies give whole cocycle concordances; this is a classifying statement for concordance, not an assertion that distinct homotopic anchors are isomorphic cocycles.

**Proof: the actual coordinate and the normal bundle.** Write \(e=\sigma(b)h\). Define

\[
 J([e,v])=(b,hv),\qquad
 J^{-1}(b,a)=[\sigma(b),a].
 \tag{B.m}
\]

The coordinate \(h\) is continuous by the principal trivialization. The relation \((eh,v)\sim(e,hv)\) makes the formula invariant; its two composites are identities, so it is a homeomorphism. On the normal bundle use

\[
 [(\sigma(b)h,v),\xi]\longmapsto
       (b,hv,D^\perp h\,\xi).
 \tag{B.n}
\]

The derivative product rule makes this independent of the representative. Its inverse uses \(\sigma(b)\) and the identity derivative. Both are continuous bundle maps. Thus it is the entire normal bundle comparison. Contracting the \(B_H\) coordinate of the product keeps \(a\) fixed and its fibre is literally \(\tau_a\) throughout. Any spin-c data already present on a test pullback transport through this same whole bundle map, with no loss of lines or gradings.

For a native smooth test cocycle, first use its **smooth** trivialization from Lemma 8B.1. With local principal sections \(p_\beta=p_\alpha h_{\beta\alpha}^{-1}\), the sections \(p_\alpha k_\alpha\) glue to a whole smooth section \(p_0\). Thus \(p=p_0h(p)\), and its equivariant anchor is \(h(p)^{-1}a(x)\). Fix one point \(b_0\in B_H\). Choose the equivariant classifying map \(p\mapsto\sigma(b_0)h(p)\), and pair it with that anchor. Formula (B.m) then gives exactly \((b_0,a(x))\). At \(p_\alpha=p_0k_\alpha^{-1}\), (B.n) is \(D^\perp k_\alpha^{-1}\), the inverse of (B.g). Hence the comparison retains the chosen smooth global lift and entire normal bundle. This construction works on the whole smooth family or bordism. The fixed join section constructed with maxima and positive parts is only continuous; pulling it back through an arbitrary classifying map does not supply a smooth reduction. Smooth native-cycle reduction uses Lemma 8B.1 first, as here and in Theorem 8B.4. \(\square\)

### The cycle group and the actual wrong-way class

Define \(\mathcal C_{*,\tau}(V)\) to be the following integral geometric group. Its cycles are closed compact smooth \(M\), smooth \(a:M\to V\), a Hermitian bundle \(E\to M\), and the **full** prescribed stable spin-c orientation of \(TM\oplus a^*\tau\). Its degree is \(\dim M-q\pmod2\). Its relations are precisely direct sum/disjoint union, compact bordism with full extending data, and even spin-c vector-bundle modification with the same full clutch bundle and induced orientation used in native Section 9. This definition does not replace this group by \(K_c^*(V)\).

**Theorem 8B.4 (the complete native geometric domain reduces to global lifts).** Inclusion of global maps as unit-transition cocycles induces an isomorphism

\[
 I:\mathcal C_{*,\tau}(V)\xrightarrow{\cong}
 K_{*,\tau}^{\mathrm{geom}}(B(V\rtimes H)),
 \tag{B.o}
\]

where the right group is the specified native compact smooth actual-cocycle model. Its inverse is reduction by Lemmas 8B.1–2. In the normal-preserving classifying comparison of Theorem 8B.3 this is the corresponding comparison of geometric cycles, in both degrees and with compact support interpreted as compact cycle images.

**Proof.** Every right-hand cycle has its global reduction, with the same \(M,E\), degree and full orientation transferred by (B.g). Two choices give the whole smooth oriented homotopy of Lemma 8B.2. On \(W=I\times M\) use \(TW=\mathbb R_s\oplus TM\), putting the positive interval line first, and the ordered full module of that line followed by the transported module for \(TM\oplus a_s^*\tau\). Boundary restriction evaluates its outward normal first. At the outgoing end this cancels the positive interval line and returns the specified endpoint module; at the incoming end the reflected normal gives its full opposite, with the negated operator and grading prescribed in Theorem 6.0ab. Thus this is a compact product bordism with the exact two endpoint orientations, not a determinant-only assertion. The reductions define the same class.

Direct sums and disjoint unions plainly retain \(E\) and the corresponding local trivializations; one may perform the construction separately on the finitely many compact components. A native compact bordism reduces by a single trivialization on its whole collared base. Its boundary reduction agrees with the chosen reductions up to the full endpoint homotopies, and its transported bundle and orientation are the required extending ones. Hence its reduced boundary is zero. For an even spin-c vector-bundle modification pull back the given \(k_\alpha\) along its sphere projection. Lemma 8B.2 identifies the entire sphere clutch bundle and induced stable orientation with the modification of the reduced cycle; the extra Clifford factors are evaluated in the same order. This proves descent through all three defining relations.

Reducing a global cocycle may be done with \(k_\alpha=e\); it returns that global cycle. Thus the reduction followed by inclusion is inverse in one direction. Conversely the original cocycle and its global reduction are actually equivalent: use the arrows \(k_\alpha:a\to f_\alpha\). Equations (B.c) and (B.h) are exactly the equivalence and its full normal comparison. They give the same native cycle, or equivalently the full compact concordance of Lemma 8B.2 gives its product bordism. Therefore the other composition is also the identity. Neither a rational Chern-character comparison nor a noncompact manifold cycle has entered this argument. \(\square\)

**Corollary 8B.5 (assembly on every native cycle uses the global lift).** For the specified full orientations, the actual geometric wrong-way class satisfies

\[
 f_!=a_{\mathrm{cocycle},!},\qquad
 \mu[(M,E,f)]=[E]a_{\mathrm{cocycle},!}.
 \tag{B.p}
\]

Here \(a_{\mathrm{cocycle}}\) means the map from \(M\) to \(V/F\) given by the global anchor \(a\), not an ordinary-target wrong-way map \(a:M\to V\) with its orientation left unspecified.

**Proof.** The equivalence arrows \(k_\alpha\) identify the two actual normal microbundle germs. Their normal derivative is exactly (B.g). Theorem 6.0o compares their full Thom/tube products after transport of this entire specified normal module. Its middle-module comparison is Lemma 6.0n's onto unitary on the completed principal correspondence, induced by these actual arrows; it retains every counting lift. Its restrictions use essential modules, and its Euclidean inverse remains in its prescribed order. Thus the equality is a full KK equality with the right algebra \(A=C_r^*(G)\), before multiplication by \(E\). Theorem 6.1 supplies the actual descent of this product. Multiplying by the unchanged finite Hermitian coefficient bundle proves (B.p).

The same normal product also has the full lift in Theorem 11A.1 below, equations (A.G.1)–(A.G.8). That theorem already handles every actual compact cocycle. Its completed full principal module, quotient-tensor unitary and full product-group Euclidean factor use the same arrows and orientation. The new global reduction is compatible with this full lift, by transporting that same core and all normal factors. For this solvable graph action, Corollary 8.9 proves full and reduced crossed products coincide, so these two assembly values agree under the actual quotient map. This observation does not infer that assembly is an isomorphism. \(\square\)

### The compact leaf-map interface and the exact sign

For compact \(V\), let \(\mathcal O_{\mathrm{fr}}\) be the full stable leaf orientation induced by the ordered fundamental spin frame used in Theorem 8.15d, with its stated Clifford/chirality convention and no extra line twist. Use that same orientation in the entire normal-tube definition of the geometric leaf map. Let \(\Theta_H\) be the literal chronological positive real Thom KK product of Corollary 8.14a, with the native second-cycle-first odd–odd contraction. Set \(s_d=(-1)^{\lfloor d/2\rfloor}\).

**Corollary 8B.6 (a calibrated compact comparison, both degrees).** In this exact common orientation,

\[
 \ell_{\mathcal O_{\mathrm{fr}},!}^{\mathrm{geom}}
 =\ell_{\mathcal O_{\mathrm{fr}},!}^{\mathrm{an}}
 =s_d\Theta_H
 \quad\text{in }KK_d(C(V),C(V)\rtimes H).
 \tag{B.q}
\]

All three actions are isomorphisms on both K-groups. The geometric topological-index action given by native Section 11 is the first class's action. For the physical spin Thom symbol \(b_{\mathrm{fr}}\), the complete symbol maps obey

\[
 \operatorname{Ind}_a(x\boxtimes b_{\mathrm{fr}})
       =x\ell_{\mathcal O_{\mathrm{fr}},!}^{\mathrm{geom}}
       =s_d\Phi_H(x),\qquad
 j=0,1,\ x\in K^j(V).
 \tag{B.r}
\]

The formula retains the sign rather than silently identifying the ordered Bott class with the physical one.

**Proof.** Theorem 7.25 is an equality of the prescribed full analytical and geometric classes; it uses the entire normal-tube cocycle, both actual Thom inverse products and the calibrated ambient inverse. For \(\mathcal O_{\mathrm{fr}}\) its analytical operator is precisely the homogeneous source spin Dirac operator compared in Theorem 8.15d. That theorem supplies one actual parameter cycle and its modular half-density correction, then a bounded spin-connection homotopy to this endpoint. Its zero-fibre calculation in Lemma 8.15c gives \(T_D[q_0]=s_d[e_0]\Theta_0\). Cancelling the proved KK evaluation inverse gives the second equality of (B.q). Each real factor has both KK inverses. The inverse of \(s_d\Theta_H\) is \(s_d D_H\): both products are \(s_d^2\) times the corresponding ordered inverse product, hence identity. These statements prove isomorphisms in both degrees, including integral torsion.

Theorem 7.29 identifies every symbol index with the action of this full leaf-map class on its actual physical Thom inverse. Corollary 8.24 identifies the ordered symbol product with \(\Phi_H\). Lemma 8.15c gives \(b_{\mathrm{fr}}=s_d b_{\mathrm{ord}}\), with the actual finite Clifford contractions. Their combination yields (B.r). Finite bundle twists must be transported through all three maps using their full section modules; Theorem 7.25 retains their whole line data. A scalar sign cannot be used to discard an arbitrary line twist. In rank zero the common framed orientation is the even unit and all three maps are identity. \(\square\)

This resolves the *compact* geometric leaf-map interface for the calibrated convention. It makes no claim that the unmodified literal \(\Theta_H\) equals the physical framed leaf class when \(d\equiv2,3\pmod4\). A formulation of “three equal maps” must use the geometrically calibrated real map \(s_d\Phi_H\), or specify one common transported full orientation for all three constructions. The native theorem does not prescribe the course's particular odd–odd reducer; this is the exact local convention comparison required when using it.

### Solvable assembly requirements and their current scope

The following are concrete mathematical obligations in the native scope. They are not exclusions added to it.

1. **Integral geometric manifold duality (proved below).** Theorem 8C.8 constructs a two-sided isomorphism from the group in (B.o), with all three native relations, to \(K_c^{*+d}(V)\). The degree shift is fixed by \(q=\dim V-d\): a cycle of degree \(m-q\) would push to an ordinary target degree \(m-\dim V=(*-d)\pmod2\). The ordinary-target orientation must be derived from \(TM\oplus a^*\tau\), the ordered framed \(F\), and the canonical doubled tangent module by actual finite Morita cancellation. Full lines, stable grading and compact support must survive this derivation. Both inverse comparisons on the entire integral groups are required. Merely proving homotopy equivalence \(BG\simeq V\), constructing the cohomology class of Theorem 6.2, or comparing compact numerical tests does not supply this isomorphism.

2. **The full noncompact geometric leaf map (proved below).** Lemmas 8D.1–4 and Theorem 8D.5 treat arbitrary noncompact \(V\), construct the geometric normal-tube product as a full \(KK_d(C_0(V),C_0(V)\rtimes H)\) class with the prescribed full orientation, and prove equality with the framed parameter Dirac class of Theorem 8.15d. This needs a proper Euclidean embedding or a compatible exhaustion, variable positive tube compression, a whole Hausdorff normal germ construction, and a single completed principal interval module. All localized cycle defects and the creation errors must be compact in that whole module. Theorem 6.0t gives noncompact ordinary embedding composition; Lemma 6.0n gives general étale correspondences. Those inputs alone do not prove the full noncompact longitudinal interval comparison. Theorem 7.25 explicitly assumes compact \(V\). Compactly supported symbols in Lemma 8.18 cannot be substituted for its full \(C_0(V)\) geometric KK class.

3. **The ordinary/geometric assembly interface and its isomorphism (proved below).** Theorems 8D.7–8 complete this interface. Using the ordinary orientation and duality map \(\kappa\) proved below, prove on every cycle that \([E]a_{\mathrm{cocycle},!}=\kappa(M,E,a)\ell_!\), in the actual orientations and contractions. For compact \(V\), Theorem 6.0w already proves the relevant product when \(a:M\to V\) and \(\ell:V\to V/F\) are given compatible full orientations, and Theorem 8C.8 below supplies the two-sided identification \(\kappa\). For noncompact \(V\), that theorem's intermediate manifold is not compact, so the composition statement also needs the corresponding localized full product. Theorems 8D.7–8 combine the two-sided duality with this interface and the correctly calibrated Thom inverse to prove native assembly isomorphism in both integral degrees. The new (B.p) identifies all its inputs with global lifts; it does not prove injectivity or surjectivity by itself.

No free action is substituted for the graph hypothesis, no noncompact unit manifold is compactified with an unproved extension of its action, and no zero-anchor analytic deformation is claimed to be a constant-rank foliation concordance. Theorems 8C.8 and 8D.5/7/8 now supply these interfaces for the full native assembly isomorphism, retaining the displayed physical calibration.

![Exact cocycle reduction and normal-preserving classifying comparison](../figures/solvable-cocycle-bridge.png)

**Figure 8B.1.** Lemmas 8B.1–2, Theorems 8B.3–4 and Corollaries 8B.5–6. These are diagrams of actual maps and bundle transitions, not projections of geometric leaves. Panel A retains the noncommutative order \(r(t_\alpha)n_\alpha\) and uses a locally finite subordinate partition, including zero weights. Panel B is the whole transverse derivative comparison, with full orientation data transported. Panel C uses a right universal group space; its coordinate is \(hv\), and the normal derivative is \(D^\perp h\). Panel D records the earlier proof stage. Theorems 8C.8 and 8D.5/7/8 below now supply integral duality and the noncompact geometric and assembly comparisons. Its sign table is exact, not numerical evidence for the all-dimensional calculation. Reproducible source: [figure generator](https://kokunoyumeto.github.io/open-math-courses-public/courses/NCG-FOLIATIONS/figures/solvable-cocycle-bridge.py); editable [SVG](https://kokunoyumeto.github.io/open-math-courses-public/courses/NCG-FOLIATIONS/figures/solvable-cocycle-bridge.svg). Historical source: Connes, Sections 9–12. All proof locators are the lettered displays above and the exact preceding theorem locators.

### Integral manifold duality for the native graph-action cycles

The argument and diagram below are original CC0 material. The smooth partition prerequisite is the bundled [Local tools for bundles and transport](https://kokunoyumeto.github.io/open-math-courses-public/courses/NCG-FOLIATIONS/companions/local-tools-for-bundles-and-transport.html), Theorem 1.2, Corollary 1.3, Theorem 2.1 and Section 3.

We use the compact actual-cocycle group of native Sections 9–11. Theorem 8B.4 reduces that group to the integral group \(\mathcal C_{*,\tau}(V)\) by the smooth trivializations of Lemmas 8B.1–2. No smooth pullback of the continuous universal-join section is used. Let \(n=\dim V\), \(d=\dim H\), \(q=n-d\), and fix the ordered fundamental frame of \(F\). A splitting \(TV=F\oplus\tau\) is fixed temporarily. Cycles have compact closed smooth source \(M\), an actual bundle \(E\), a smooth map \(a:M\to V\), and the entire native stable orientation \(\eta\) on \(TM\oplus a^*\tau\). In particular lines, central transition phases, stable grading and the opposite orientation are data. The defining relations remain sum, compact bordism and the specified even spin-c bundle modification.

The following proof uses ordinary target Thom transfers with their full normal orientations. All Bott classes are the positive geometric Clifford classes of Theorem 6.0c/Corollary 6.0d. A coordinate block is appended after the old coefficient algebra. Their inverses are the corresponding geometric inverses, not an independently selected real-action class. In odd products the actual finite contraction of Section 0 is retained. This avoids importing a new sign convention into Corollary 8B.6.

#### 8C. Full ordinary orientation and descent

**Lemma 8C.1 (adding and cancelling the framed leaf block).** The specified native orientation determines, functorially and reversibly, the full ordinary stable orientation of \(a^*TV-TM\) in the doubled-tangent convention of Lemma 6.0j. Its doubled module is obtained from

\[
 (TM\oplus a^*\tau)\oplus a^*F
       \longrightarrow TM\oplus a^*TV
 \tag{C.a}
\]

by the actual ordered permutation, metric transport and tensor with the specified frame module. This construction is a bijection on full stable orientation data; it commutes with families, boundary restriction and the induced orientation of vector-bundle modification.

**Proof.** Write \(A_U=\mathrm{Cl}^+(U)\) and \(C_r=\mathrm{Cl}^+(\mathbb R^r)\). A full orientation is a smooth graded finite imprimitivity module, with both actions, from its left bundle algebra to its fixed right Clifford algebra. Tensor the native module with the frame module \(A_{a^*F}\rightsquigarrow C_d\). The real permutation in (C.a), transported first as a metric bundle map, gives the ordinary doubled module
\(A_{TM\oplus a^*TV}\rightsquigarrow C_{m+n}\), where \(m=\dim M\). The right-hand regrouping is by the same ordered real blocks. Every interchange of homogeneous factors is the graded interchange, with its sign.

There is an explicit inverse even when \(d\) is odd. Tensor this construction once more with the same framed \(C_d\) identity factor. The two frame blocks are now a matrix Clifford factor \(C_{2d}\). Cancel it on both sides by the exterior module on \(\Lambda^*\mathbb C^d\), using the two finite evaluation maps (173)–(174) of Lemma 6.0j. The real double is paired in its prescribed interleaved order; the real permutation from two-block order to that order is carried along. Both recovery composites are the identity through those onto, even evaluation unitaries. This gives an inverse on module bundles and on their intertwiners, not a cancellation of determinants. A twist by a Hermitian line \(L\), including torsion or a stable grading shift, remains the same twist by \(L\) after every cancellation.

For an ordinary graph normal, Lemma 6.0j uses its explicit quotient isomorphism, the positive metric isometry, and cancellation of the canonical doubled \(TM\) module. Applied to (C.a), these operations give exactly the ordinary orientation of \(a^*TV-TM\). Conversely its two recovery unitaries return (C.a), and cancelling the frame returns \(\eta\). Thus there is no additional normal orientation choice. The operations are tensor, permutation and finite evaluation maps, so they vary smoothly over parameters. Putting an outward boundary line first and then making these same interchanges gives the same boundary evaluation; the incoming reflected line gives the full opposite of Theorem 6.0ab. For modification the additional even sphere-normal blocks are evaluated in the order of Lemmas 6.0ac–ad. Tensoring the unchanged leaf frame before or after these evaluations gives the identical even intertwiner. This proves the last assertions. A different splitting of \(TV\to\tau\) is joined by its affine family; its triangular quotient maps and positive metric transports give the same full comparison. \(\square\)

**Lemma 8C.2 (the ordinary transfer on the entire native quotient).** Define

\[
 \kappa[M,E,a,\eta]=[E]\,a^{\,\mathrm{ord}}_!
       \in K_c^{\,m-n}(V).
 \tag{C.b}
\]

Here the ordinary target orientation is exactly Lemma 8C.1. The compact-source graph construction is used also when \(V\) is noncompact. This is a well-defined integral homomorphism
\(\mathcal C_{j,\tau}(V)\to K_c^{j+d}(V)\), for \(j=0,1\).

**Proof.** Choose a finite Euclidean embedding of the compact \(M\). Its graph is a compact embedding into \(V\times\mathbb R^{2r}\). Lemmas 6.0e–j construct its actual normal Thom class, its open tubular extension and its ordered Euclidean inverse. Their graph-family and stabilization comparisons, including Corollary 6.0k and the ordinary zero-foliation graph class of Lemma 6.0r, prove independence with the orientation supplied by Lemma 8C.1. The normal Thom coefficients vanish at normal infinity; their tube extension vanishes at the open boundary. Compactness of the source makes the resulting class a class of \(C_0(V)\), rather than an ordinary unrestricted cohomology class of \(V\).

Sum and disjoint union are respected before any cancellation. For a native compact bordism, apply Lemma 8C.1 to its entire extending module. The outward-normal-first evaluation is the ordinary extending orientation. The full double/zero-extension bordism argument of Theorems 6.0aa–ab, with the target treated as the zero foliation, gives zero for its oriented boundary transfer. In particular the incoming operator and grading are both reversed as required; a line on the bordism restricts to its actual endpoint lines.

For an even spin-c modification \(W\), use the full clutch bundle \(b_W\), not a rank substitute. The ordinary composition and projection formula of Theorem 6.0s and Lemmas 6.0t and 6.0v, together with Theorem 6.0ad, give
\([\pi^*E\,b_W]\pi_!a_!^{\mathrm{ord}}=[E]a_!^{\mathrm{ord}}\).
The assertion \(\pi_!b_W=1\) there includes the rank-zero case and actual central phases. The orientation compatibility is Lemma 8C.1; both degrees retain the same Clifford blocks. These are exactly the three native relations. Finally \(m-n=(m-q)-d\), equal to \(j+d\) modulo two. No character or rationalization has been taken. \(\square\)

#### 8C. Compact support and compact carrier cycles

**Lemma 8C.3 (smooth representatives and compact carriers).** An element of \(K_c^0(X)\), where \(X\) is a finite-dimensional smooth Hausdorff second-countable manifold, has a representative by two smooth finite projections with identical constant projection outside one compact set. A homotopy between representatives can be chosen with this property uniformly in its compact parameter. Every compact subset of \(X\) lies inside a compact smooth domain \(W\), and \(W\times[-1,1]\) has a smooth rounding whose boundary is a compact double \(D W\); the rounding can be kept away from that subset on the top and bottom faces.

**Proof: controlled representatives.** Use the projection definition of \(K_0(C_0(X)^+)\). A kernel class is a difference of projections with equal scalar ranks at infinity. Stabilize, and use a constant unitary to make their scalar projections the same projection \(p_\infty\). A compact-parameter homotopy is handled in the same matrix size. If its scalar projections vary with the parameter, trivialize their interval family first. For two projections \(p,q\) with \(\|p-q\|<1\), the polar unitary of \(qp+(1-q)(1-p)\) sends \(p\) to \(q\); its Gram matrix is invertible by the norm bound. Successive sufficiently short parameter intervals therefore give a continuous unitary trivialization, starting with the identity, of the scalar projection family. Conjugate by it before choosing the common \(p_\infty\). At infinity all the resulting projections are uniformly close to \(p_\infty\): continuity into the supremum-norm algebra makes the parameter image compact, and a finite norm net makes the small tail uniform.

Choose a smooth cutoff \(\chi\), equal to one on a compact set containing the part where any projection is not within \(1/8\) of \(p_\infty\), and zero outside a larger compact set. The self-adjoint matrix \(\chi p+(1-\chi)p_\infty\) is within \(1/8\) of a projection everywhere where it differs from \(p\). Its spectral projection above \(1/2\) is therefore defined and gives a projection equal to \(p_\infty\) off that larger compact set. Interpolating the cutoff gives the original class and homotopy. Smoothly approximate the matrix entries on the remaining compact set, in coordinate charts and a subordinate partition, fixing the constant entries near its exterior. Make the self-adjoint approximation within \(1/8\), then take the same spectral projection. The contour-resolvent formula around the separated upper spectral cluster proves smoothness and a projection homotopy. The parameter argument uses a finite cover also of its interval, fixing specified smooth endpoint representatives by parameter cutoffs. All supports stay in one compact set. The same construction for a unitary uses polar normalization and an exterior logarithm in a norm ball around \(1\).

**Proof: compact smooth domains.** Use the exact current DG-FND Section 3, Lemmas 3.A–B, Theorem 3.E and Theorem 3.1. Its compact exhaustion \(A_i\) and smooth cutoffs supply smooth functions \(\psi_i\) that are zero near \(A_i\) and one off \(\operatorname{int}A_{i+1}\). The locally finite sum \(f=\sum_i\psi_i\) is smooth and proper: off \(A_{N+1}\) its first \(N+1\) summands are one. A regular value above \(\max_K f\) gives \(W=\{f\le c\}\).

Here the needed scalar regular-value assertion also has a short complete proof. In a coordinate cube let \(C_k\) be the set where derivatives of orders \(1,\ldots,k\) vanish. On a smaller compact cube the \((n+1)\)-st derivatives are bounded. Taylor's formula bounds the image of \(C_n\) in any mesh cube of side \(\delta\) by an interval of length \(O(\delta^{n+1})\). There are \(O(\delta^{-n})\) mesh cubes, so its critical values have outer measure zero. For \(C_k\setminus C_{k+1}\), \(1\le k<n\), some derivative of order \(k\) is zero there but has nonzero gradient. Its zero set is locally a smooth hypersurface by the inverse function theorem. The restriction of \(f\) to that hypersurface has every point of \(C_k\) as a critical point. Induction on dimension, using countably many such neighborhoods, makes those critical values null as well. The dimension-one base is the mesh estimate for \(C_1\). Countable relatively compact cubes exhaust all charts, so the critical values are null globally. Every open value interval contains a regular value. The empty and zero-dimensional cases are immediate.

The vector field \(\nabla f/|\nabla f|^2\) on a neighborhood of the compact regular boundary constructs a collar by the parameter-flow theorem. In its corner chart round the quadrant by a smooth curve whose tangent angle moves from \(0\) to \(\pi/2\), monotonically, with all derivatives flat at its endpoints. Integrate its positive tangent components and rescale it into an arbitrarily small corner square. This matches the two straight pieces smoothly and monotonically; use the same curve at every collar point. It leaves the top and bottom unchanged away from the collar. Hence the rounded box \(P_W\subset X\times\mathbb R_t\) is smooth and compact, with boundary \(D W\) and a smooth projection \(r:D W\to X\). On its two core faces \(r\) is the identity on \(W\). Its boundary tangent sequence gives the actual metric stable isomorphism

\[
 TD W\oplus\mathbb R_{\mathrm{out}}
       \cong r^*TX\oplus\mathbb R_t.
 \tag{C.c}
\]

The outward line is put first before any regrouping. \(\square\)

For \(N\ge0\), put \(X_N=V\times\mathbb R^N\), \(p_N:X_N\to V\), and let \(B_N,A_N\) be the specified mutually inverse geometric Bott classes on the appended block. The notation \(yA_N\) means the compact-coefficient inverse action with the original \(C_0(V)\) first. It is defined for \(y\in K_c^0(X_N)\), with value in \(K_c^N(V)\).

**Odd ambient blocks.** The existing graph-choice lemma was stated using even Euclidean ambient blocks, so the use of odd \(N\) here needs a comparison. If \(i:M\hookrightarrow X_N\) is compact with specified normal, append a zero positive line to obtain \(i^+:M\hookrightarrow X_{N+1}\). For a whole normal tube \(\theta\), the product tube is \((v,t)\mapsto(\theta(v),t)\). On its essential open-ideal module the ordered normal Thom product is the full multiplier
\((c_\nu(v)+c_{\mathbb R}(t))/\sqrt{1+|v|^2+t^2}\).
Corollary 6.0d proves its connection estimates and nonnegative anticommutator with all Clifford factors retained. Compact old creation coefficients bound \(v\); the new-line errors vanish at \(|t|\to\infty\), and approximation controls their old tails. Zero extension along the product tube consequently gives
\[
 (i^+)_!=i_!(1_{C_0(X_N)}\boxtimes B_1),\qquad
 (i^+)_!A_{N+1}=i_!A_N .
 \tag{C.c.1}
\]
For the second equality use the inverse for the ordered appended blocks: remove the new last line first and the old \(N\)-block second. The new line's two actual inverse evaluations cancel adjacent factors. Its retained \(C_1\) is never moved past another retained odd block; finite regroupings use the same even triple-cycle and wedge maps of Proposition 6.0i. Thus no unsigned odd permutation has been made.

The full normal for the odd-ambient graph is uniquely recovered from the specified even-ambient graph by this last-line cancellation. Concretely its doubled stable identity is \(\nu_i-\mathbb R^N=a^*TV-TM\); tensoring the new positive line gives the usual even-ambient doubled identity, and the two finite recovery unitaries give both comparisons. Positive rank is obtained by enlarging \(N\). Formula (C.c.1) identifies its transfer followed by \(A_N\) with the already proved even-ambient ordinary transfer. It also applies to a compact-coefficient carrier map into \(X_N\): use a compact graph embedding of its source and the same tube argument. This establishes the odd ambient calculation used below without extending an even-stabilization statement by assumption.

Give \(r:D W\to X_N\) the full relative ordinary orientation determined by (C.c): evaluate the two boundary-line factors, with the positive \(t\) line on the top core face. Its restriction there is the identity orientation. On the bottom core face it is the full opposite. This construction uses the canonical doubled \(TX_N\) exterior module and both evaluation maps; it does not require \(TX_N\) to be spin-c. Compose with the orientation of \(p_N\) specified by \(A_N\). Recover the doubled ordinary module on \(TD W\oplus(p_Nr)^*TV\) using Lemma 6.0j's recovery maps, and then apply the inverse of Lemma 8C.1. Call the resulting native full orientation \(\eta_{W,N}\). This is a concrete finite algorithm for its normal data; it is also valid in odd \(N\) and with nonorientable \(TV\). It is not a choice of an unspecified determinant lift.

Represent \(y\) by \((E_0,E_1,\varphi)\) which agrees on a neighborhood of the complement of a compact \(K\subset\operatorname{int}W\). Glue \(E_0\) on the upper copy and \(E_1\) on the lower copy through \(\varphi\) along the collar, producing \(E_y\to D W\). Let \(G_y=r^*(E_1|_W)\). Define

\[
 \beta_N(y)=
 [D W,E_y,p_Nr,\eta_{W,N}]
 -
 [D W,G_y,p_Nr,\eta_{W,N}].
 \tag{C.d}
\]

The compact coefficient \(y_D=[E_y]-[G_y]\) vanishes on the entire lower face and a neighborhood of the rounding. On the upper core it is the extension by zero of \(y|_{\operatorname{int}W}\).

**Lemma 8C.4 (the carrier construction is well-defined).** Formula (C.d) is an additive homomorphism \(K_c^0(X_N)\to\mathcal C_{N+d,\tau}(V)\), independent of representative, compact carrier and rounding. Its full ordinary image is

\[
 \kappa\beta_N(y)=yA_N.
 \tag{C.e}
\]

**Proof: representative relations.** A common stabilization adds the same bundle to the two terms in (C.d), and cancels by the native sum relation. A relative projection homotopy in Lemma 8C.3, with one compact support and carrier, glues to an actual bundle over \(I\times D W\). Give that compact bordism the full extending orientation, putting its positive interval line first. Its endpoints are the specified carrier cycles, with the incoming full opposite. It proves invariance under the stable homotopy relation defining \(K_0\). A degenerate pair produces identical bundles and zero. Direct sum glues as direct sum. These facts prove descent through the actual integral K-theory representative relations.

**Proof: enlarging the carrier.** Suppose \(W_1\subset\operatorname{int}W_2\) both contain the compact support and the smaller closure lies in the interior. Consider a rounded
\[
 Q=(W_2\times[-1,1])
       \setminus\operatorname{int}(W_1\times[-\epsilon,\epsilon]),
       \qquad0<\epsilon<1.
 \tag{C.f}
\]
The supports lie strictly inside \(W_1\). On the upper part of \(Q\) take \(E_0\), on the lower part take \(E_1\); near \(t=0\) every point of \(Q\) lies outside \(W_1\), where \(\varphi\) identifies them. Thus they glue to a bundle \(E_Q\) on the entire compact bordism. Subtract the globally pulled back \(E_1\). The outer boundary coefficient is \(y_{D W_2}\); the inner boundary coefficient is \(y_{D W_1}\). The normal on the inner boundary points into the removed box, giving the full opposite carrier orientation. Its ordinary stable data are obtained from the actual tangent sequence of \(Q\subset X_N\times\mathbb R\), the same positive \(t\) line and the same \(A_N\); inversion of Lemma 8C.1 gives the native extending data. Hence (C.f) is an exact native bordism between the two representatives of (C.d).

Two arbitrary carriers have a common larger carrier by Lemma 8C.3, so this proves independence. Different roundings are joined in their fixed collar by interpolation of the monotone rounding curves, or can be compared by a common smaller rounding. The resulting compact collar bordism has the identical coefficient and line evaluations. The argument covers empty support and disconnected carriers.

**Proof: the image.** The ordinary orientation of \(p_Nr\) was constructed as the ordered composition of the relative orientation of \(r\) and \(A_N\). Full normal cancellation returns that same orientation by its two recovery identities. Since \(y_D\) vanishes near the lower face and the rounded part, excision computes its ordinary transfer solely in the upper open face, where \(r\) has the identity normal module. Its open-ideal extension is precisely \(y\). Thus \(r_!y_D=y\), and the remaining factor is \(A_N\), proving (C.e).

This compact-coefficient calculation is valid at the noncompact intermediate \(X_N\). It is the zero-extension calculation on the upper open ideal followed by the whole Bott inverse of Theorem 6.0c. Equivalently Lemma 6.0t's completed noncompact embedding product gives the identical localization; no noncompact manifold is used as a native cycle. \(\square\)

#### 8C. Recovering every cycle geometrically

The essential second-inverse argument is the following relative, full-orientation embedding bordism. It is given here so that an analytical triangle identity cannot stand in for a native geometric relation.

**Lemma 8C.5 (full twisted embedding transfer).** Let \(h:M\hookrightarrow Z\) be an embedding of compact closed manifolds with even spin-c normal \(\nu\), and \(f:Z\to V\). Give \(Z\) a native full orientation \(\eta_Z\). Suppose the ordered full normal module of \(h\), followed by \(\eta_Z\), induces exactly the prescribed native module \(\eta_M\) of \(fh\), through the finite normal composition/recovery of Lemma 6.0v. For \(x\in K^0(M)\),

\[
 [M,x,fh,\eta_M]=[Z,h_!x,f,\eta_Z].
 \tag{C.g}
\]

Virtual bundle coefficients here denote differences of actual bundles under the native sum relation.

**Proof: reduced north classes are allowed.** For any even spin-c bundle \(T\), the native modification coefficient is \(b_T\). Lemma 6.0ac gives the complete clutch identity \(n_!1=[b_T]-[\pi^*S_T^-]\). The cycle with coefficient \(\pi^*(xS_T^-)\) extends over the actual disk \(D(T\oplus\mathbb R)\), with the full extending orientation of Theorem 6.0ad. It is zero by native compact bordism. Consequently modification with \(b_T\) and modification with the reduced north class \(n_!1\) are the same native class. This is a geometric disk bordism; the analytical identity \(\pi_!1=0\) is not being used as a geometric relation.

**Proof: the compact surgery bordism.** Give \(\mathbb C\) its prescribed positive complex spin-c module. Modify \(Z\) along this trivial even plane. Its underlying sphere is \(S^2\times Z=\partial(D^3\times Z)\). For a sufficiently small tubular embedding of \(\nu\), the disk \(D_\epsilon(\nu\oplus\mathbb C\oplus\mathbb R)\) embeds in the interior of \(D^3\times Z\), centered on \(M\) at the origin of \(D^3\). Remove its interior, obtaining a compact smooth bordism \(W\). Its outer boundary is \(S^2\times Z\); its inner boundary is the full opposite of \(S(\nu\oplus\mathbb C\oplus\mathbb R)\), the modification of \(M\) along \(\nu\oplus\mathbb C\). The boundary parameter is the last real coordinate; the normal module on each even block is kept in the prescribed order. Smooth disk radii are chosen below one common positive tubular radius, which exists by compactness.

There is a collared interval embedding \(e:[\epsilon,1]\times M\to W\), using that last real coordinate and the zero normal vector. Its even normal is the entire specified \(\nu\oplus\mathbb C\) module. Use its Thom class and the normal tube to extend \(e_!\mathrm{pr}_M^*x\) into \(K^0(W)\). This extension is legitimate also at its two boundary faces: the interval embedding and its tube are product embeddings in a boundary collar. The normal direction is compactly supported.

The initial reference projection may be a varying spinor-bundle projection. Here is the required constant-reference construction in full. A finite Hermitian embedding of each bundle over the compact interval source exists: use finitely many local orthonormal frames and square roots of subordinate weights to give a fibrewise isometry into a fixed \(\mathbb C^k\). Its orthogonal complement is smooth; if square roots are inconvenient, use unnormalized smooth weights and their positive Gram normalization. Thus write the two normal relative projections as \(P(v),Q(x)\in M_k(\mathbb C)\), equal off a compact normal disk. The cutoff/spectral procedure of Lemma 8C.3 can first make this equality exact. Put

\[
 U_Q=\begin{pmatrix}Q&1-Q\\1-Q&Q\end{pmatrix},\qquad
 P'=U_Q\begin{pmatrix}P&0\\0&1-Q\end{pmatrix}U_Q^*,\qquad
 U_Q\begin{pmatrix}Q&0\\0&1-Q\end{pmatrix}U_Q^*
       =\begin{pmatrix}1_k&0\\0&0\end{pmatrix}.
 \tag{C.g.1}
\]

Since \(Q^2=Q=Q^*\), \(U_Q^*=U_Q\) and \(U_Q^2=1\). Direct block multiplication gives the last identity, and \(P'\) is a projection. Adding the same complement \(1-Q\) to both projections and conjugating both by \(U_Q\) preserve their relative K-class. Therefore \(P'\) equals the displayed **constant** reference outside the normal disk. Extend only this constant reference to all of \(W\); the resulting \(P'\) is a continuous projection over all of \(W\), and smoothing within its compact support preserves its class. Neither \(E\) nor a spinor line is assumed to extend away from its tube. This construction is compatible with both interval boundary restrictions. The two bundles obtained on \(W\) have the exact boundary differences
\(n_Z{}_!h_!x\) and \(n_M{}_!x\). On the outer face this follows from the ordered sum of the \(\nu\) and complex-plane Thom multipliers; on the inner face the same multiplier is the north Thom multiplier of \(\nu\oplus\mathbb C\). Normal product equality is the actual multiplier and finite evaluation equality of Corollary 6.0d, not an index comparison.

Here are the full orientation comparisons. Start with the disk extension on \(D^3\times Z\) supplied by Theorem 6.0ad for the complex-plane modification of \(\eta_Z\). Restrict it to \(W\). On the tube of \(e\), decompose its tangent sequence into the positive interval line, \(TM\), \(\nu\), and the complex plane. Substitute the specified ordered \(\nu\) module and the normal-composition hypothesis. Both boundary evaluations then reduce to \(\eta_M\) followed by the exact sphere modification factors. The two finite recovery evaluations of Lemma 6.0j cancel precisely the shared doubled tangent and normal factors. Associativity introduces no sign; all permutations are the actual graded interchanges. Thus the upper boundary has exactly the complex-plane modification of \(\eta_Z\), and the inner boundary has the full opposite modification of \(\eta_M\). A line in either input is retained in the ordered product and evaluation; its inverse in the relative normal cancels it only where the composition hypothesis prescribes that cancellation. This checks the complete central transition phases and stable grading. The reflected inner outward normal gives the negated operator and opposite grading of Theorem 6.0ab.

Initially the map is \(f\) composed with the \(Z\)-projection. On the inner sphere its value is \(f(\mathrm{tube}_\nu(m,v))\). In a collar of that boundary replace \(v\) by \(\lambda v\), where \(\lambda=0\) at that boundary and \(\lambda=1\) off a slightly larger collar. This gives the exact inner map \(fh\pi_M\), leaving the outer map unchanged. The resulting smooth map is defined in the same small tube. Transport the pulled back \(\tau\) bundle and its entire module along this explicit radial homotopy; parameter transport gives actual unitary transition-module identifications. Hence the orientation and map both extend over this same \(W\). Its coefficient bundles already extend independently of that map change.

This is a native compact bordism between the two reduced-north modifications. The first paragraph converts them into the actual native modifications, which are equivalent to the original cycles by the specified even modification relation. The result is (C.g). \(\square\)

This surgery mechanism is classical; an independent primary reference for the compact untwisted embedding method is [Baum, Oyono-Oyono, Schick and Walter, Theorem 4.1](https://arxiv.org/html/0902.0641). The proof above supplies the relative normal cancellation, compact carrier use and the actual native relations needed here. No source PDF is included.

**Lemma 8C.6 (normalization in a compact double).** For a native cycle \(c=[M,E,a,\eta]\), choose \(N\) arbitrarily large with \(N\equiv m-n\pmod2\), and an embedding \(u:M\hookrightarrow\mathbb R^N\). Its graph \(j=(a,u):M\hookrightarrow X_N\) has even normal rank. Give that normal the full orientation induced from Lemma 8C.1 and the appended \(N\)-block, so that removing that block with \(A_N\) returns exactly \(a_!^{\mathrm{ord}}\). Then

\[
 y=j_!E\in K_c^0(X_N),\qquad
 \kappa c=yA_N,\qquad
 \beta_N(y)=c .
 \tag{C.h}
\]

**Proof.** The stable graph identity is \(\nu_j-\mathbb R^N=a^*TV-TM\). Use this identity on actual orthogonal normal representatives: form the doubled \(TM\) module, transport the graph quotient by the explicit quotient isomorphism of Lemma 6.0j, and use its two finite cancellations with (C.a). This gives the normal module, including its full line twist and grading. Evenness of \(n+N-m\) means its Thom coefficient is an ordinary \(K^0\) coefficient. For odd \(N\), retain the odd coordinate factor until its specified final contraction; appending an extra even block uses the exterior wedge unitary of Proposition 6.0i. The graph formula proves the middle identity.

For completeness the Euclidean embedding here needs no missing embedding theorem. On the compact \(M\), take finitely many chart-supported smooth weights \(\rho_i\) which sum to one, with their closed supports inside the chart domains \(U_i\). Extend \(\rho_i\) and every component of \(\rho_i x_i\) by zero outside \(U_i\). These extensions are smooth since the closed supports lie inside \(U_i\). The finite map \(u=(\rho_i,\rho_i x_i)_i\) is injective: some common positive \(\rho_i\) then recovers \(x_i\). It is an immersion: if its derivative vanishes on a tangent vector, then \(d\rho_i\) vanishes and \(\rho_i\,dx_i\) vanishes for a positive weight, so the tangent vector is zero. Compactness makes it a homeomorphism onto its image; the local inverse theorem supplies its submanifold charts. Pad with zero coordinates to any sufficiently large dimension of the required parity.

Choose a compact carrier \(W\subset X_N\) containing \(j(M)\) and a closed normal disk tube in its interior. The class \(y_D\) on \(D W\) is the Thom class of the same embedding into its upper face, extended by its identical tube. The carrier orientation is, by construction, the ordinary identity relative to \(X_N\) there followed by \(A_N\), then the inverse frame cancellation. Composition with the full \(j\) normal therefore recovers \(\eta\): multiply the two recovery identities of Lemma 6.0j and those of Lemma 8C.1. The \(N\)-block and its inverse evaluate adjacently; all old Clifford blocks stay in order. Thus the precise orientation hypothesis of Lemma 8C.5 holds. Apply (C.g) with \(Z=D W\) and \(f=p_Nr\). Its right coefficient is exactly \(y_D\), so its right cycle is (C.d). This proves the last identity by native geometric relations. \(\square\)

**Lemma 8C.7 (cofinal Bott stabilization, without assuming injectivity).** If \(N\equiv L\pmod2\), and \(L\ge N\), then

\[
 \beta_L(yB_{L-N})=\beta_N(y),\qquad y\in K_c^0(X_N).
 \tag{C.i}
\]

The coordinate order in this formula is the original \(N\)-block followed by the new even block.

**Proof.** Write both sides as finite differences of their compact carrier cycles, using (C.d). Their ordinary images under Lemma 8C.4 agree after the respective coordinate inverses. Choose a common sufficiently large \(R\) of the same parity as \(N,L\), so that every source manifold in these differences embeds into \(\mathbb R^R\). Apply the graph construction of Lemma 8C.6 to all those cycles. In its ambient \(X_R\), the normal class of a cycle \(b\) is \((\kappa b)B_R\): its graph formula is \(\kappa b=zA_R\), and the other Bott identity gives \(z=(\kappa b)B_R\). Thus the graph coefficient sums for the two sides agree in the actual integral group \(K_c^0(X_R)\). Apply the already well-defined homomorphism \(\beta_R\) to that equality. Lemma 8C.6 returns the original cycles on both sides, proving (C.i). This argument does not cancel \(\kappa\), and does not use the desired second inverse. It uses its separate geometrically proved graph-normalization lemma. \(\square\)

#### 8C. Both inverse identities in both degrees

**Theorem 8C.8 (integral native geometric manifold duality).** In the full specified graph-action domain, with the prescribed ordered leaf frame,

\[
 \kappa:\mathcal C_{j,\tau}(V)
       \xrightarrow{\ \cong\ } K_c^{j+d}(V),
 \qquad j=0,1.
 \tag{C.j}
\]

Let \(p=j+d\bmod2\). An explicit inverse is

\[
 \Gamma_p(x)=\beta_p(xB_p),\qquad
 \kappa\Gamma_p(x)=x,\qquad
 \Gamma_p\kappa(c)=c .
 \tag{C.k}
\]

Together with Theorem 8B.4 this is a two-sided integral identification of the entire native compact actual-cocycle group. It does not change that domain into noncompact cycles.

**Proof: the first inverse.** Since \(p=0\) or \(1\), \(xB_p\) is an even compactly supported class on \(X_p\). Lemmas 8C.3–4 give its compact carrier cycle. By (C.e) and the first adjacent ordered Bott inverse,
\(\kappa\Gamma_p(x)=(xB_p)A_p=x\).
In \(p=1\) this equality uses the actual odd–odd matrix contraction and its reverse; there is no replacement by an unsigned tensor identification.

**Proof: the second inverse.** Choose \(N\ge p\) of parity \(p\), large enough for Lemma 8C.6 on \(c\), and set \(y=j_!E\). Formula (C.h) says \(\kappa c=yA_N\) and \(\beta_N(y)=c\). The second Bott inverse gives
\((\kappa c)B_N=(yA_N)B_N=y\).
Now append the \(N-p\) even block to \((\kappa c)B_p\) and use the ordered direct-sum equality and (C.i):
\[
 \Gamma_p\kappa c
 =\beta_p((\kappa c)B_p)
 =\beta_N((\kappa c)B_N)
 =\beta_N(y)=c.
 \tag{C.l}
\]
Each identity is either the actual Bott inverse, a proved carrier relation or the explicit embedding bordism. Finite differences, components and both native degrees are included by additivity. The argument therefore retains integral torsion.

Theorem 8B.4's inverse on an actual smooth cocycle uses its smooth \(k_\alpha\), maps \(a=k_\alpha^{-1}f_\alpha\), and full transverse derivatives. Transport that same full orientation into (C.a). The independence and cycle descent proved there, combined with (C.k)–(C.l), give both inverse identities on every original actual cocycle and its three relations. Only smooth reduction by Lemma 8B.1 is required; no regularity of a fixed continuous join section is asserted. \(\square\)

#### Scope after this bridge

This establishes integral compact-carrier duality on the entire native graph-action cycle domain, with all three native relations. It supplies \(\kappa\) for the assembly comparison. Lemmas 8D.1–4 and Theorems 8D.5/7 below supply the full noncompact geometric longitudinal leaf-map class and its compatible assembly composition. These are separate full KK comparisons, beyond the compact-support duality proved here. The compact comparison of Corollary 8B.6 retains its calibrated sign \(s_d\).

![Compact carriers, native embedding bordism and the two inverse identities](../figures/integral-duality.png)

**Figure 8C.1.** Schematic diagrams, not dimensional slices of actual leaves. The upper-face coefficient is the compact relative pair \((E_0,E_1,\varphi)\); its difference is zero on the lower face and collar. The hollow rounded box in (C.f) changes the compact carrier by a native bordism. The removed disk in Lemma 8C.5 has actual bundle rank \(\operatorname{rank}\nu+3\); its boundary is the modification sphere of the even bundle \(\nu\oplus\mathbb C\). The ordered outward-normal comparison includes its full inner opposite. The final panel gives both identities (C.k)–(C.l), with compact sources at every geometric stage. Reproducible [figure generator](https://kokunoyumeto.github.io/open-math-courses-public/courses/NCG-FOLIATIONS/figures/integral-duality.py) and editable [SVG](https://kokunoyumeto.github.io/open-math-courses-public/courses/NCG-FOLIATIONS/figures/integral-duality.svg). Historical source: Baum, Oyono-Oyono, Schick and Walter, [Theorem 4.1](https://arxiv.org/html/0902.0641); the full relative orientation and carrier arguments are proved above.

### The whole noncompact leaf map and its assembly interface

Let \(V\) be the actual smooth Hausdorff second-countable unit manifold in the solvable-action theorem, \(H\) simply connected solvable, \(F\) its graph-action foliation of rank \(d\), and \(\tau=TV/F\). Put \(n=\dim V\), \(q=n-d\), \(G\) the exact native graph-action groupoid and \(A=C_r^*(G)=C_0(V)\rtimes H\) in the existing Section 8 identification. The specified arrow map from the action groupoid \(V\rtimes H\) to the foliation graph is an isomorphism, as required for this identification. The constructions use this same groupoid and its native principal spaces and retain every arrow. No free quotient or effective-action replacement is made. The principal-space construction also applies to the full holonomy groupoid wherever that is the supplied foliation groupoid.

All geometric Clifford classes, physical coefficient transfers and native products are those of Sections 6–8. An ordinary graph product retains its old normal block first; conversion to native multiplication uses the entire signed right-block transport of Lemma 7.21 before final contraction. Write \(\#\) for that fixed native product. No unsigned odd–odd flip is used.

Choose the full physical framed orientation of \(F\), and the corresponding entire stable orientation of \(TV\oplus\tau\) recovered by Lemma 7.23. Arbitrary full line twists of native cycles remain data. A separate leaf line twist is handled explicitly below. Set \(s_d=(-1)^{\lfloor d/2\rfloor}\). The literal chronological real Thom product is \(\Theta_H\), with inverse \(D_H\); the physical framed class of Theorem 8.15d is \(s_d\Theta_H\).

#### 8D. A finite proper ambient space and one global tube

**Lemma 8D.1 (constructive finite proper embedding).** Every such \(V\) embeds properly and smoothly into \(\mathbb R^{2n+2}\), and hence every larger even Euclidean space.

**Proof: local functions and proper coordinate.** Use the exact current DG-FND Section 3 to choose a countable locally finite smooth chart-supported partition \(\rho_i\), with closed compact supports inside \(U_i\), and coordinates \(x_i:U_i\to\mathbb R^n\). Set
\[
 f_i=(\rho_i,\rho_i x_i),\qquad
 \mathcal F(x)=(f_i(x))_{i\ge0}.
 \tag{D.a}
\]
Zero extensions are smooth since their closed supports lie inside the charts. Only finitely many blocks occur near every compact set. The countable map separates points: a positive common weight recovers its chart coordinates. It separates tangent vectors by first differentiating the weight components and then their coordinate components.

Take the compact exhaustion \(K_j\subset\operatorname{int}K_{j+1}\) and smooth \(\psi_j=0\) near \(K_j\), \(\psi_j=1\) off \(\operatorname{int}K_{j+1}\). Then \(t=\sum_j\psi_j\) is smooth, locally finite and proper: outside \(K_{N+1}\) its first \(N\) summands are one. This also handles infinitely many components and dimension zero.

**Proof: a finite generic map with a complete existence argument.** Put \(r=2n+1\). For matrices \(B_i\in M_{r,n+1}(\mathbb R)\) define \(\Phi_B=\sum_iB_if_i\). This smooth sum is locally finite; no matrix convergence is assumed. The countable matrix product has the complete metric
\[
 \delta(B,C)=\sum_i2^{-i-1}\min(1,\|B_i-C_i\|).
 \tag{D.b}
\]
A Cauchy sequence converges coordinatewise, and the bounded series tail proves convergence in this metric.

Cover \((V\times V)\setminus\Delta\) and the unit tangent bundle by countably many compact coordinate subsets. On each only finitely many \(f_i\) occur. Bad finite matrix choices for \(\Phi_B(x)=\Phi_B(y)\) have measure zero. Some scalar component of \(\mathcal F(x)-\mathcal F(y)\) is nonzero. Where it stays nonzero, solve the coincidence equation for the corresponding \(r\) entries of its matrix. The remaining parameters and the two points have dimension \(D-r+2n<D\), where \(D\) is the active parameter dimension. Countably many smaller compact coordinate pieces cover the solution locus; on each the solving map has bounded first derivatives, hence is Lipschitz. A Lipschitz image of a bounded \(k\)-dimensional set in \(\mathbb R^D\), \(k<D\), is null: \(O(\epsilon^{-k})\) mesh cubes map into a fixed number of \(D\)-cubes of side \(O(\epsilon)\), of total volume \(O(\epsilon^{D-k})\). This proves the claimed null set.

Immersion failure is identical: for a unit tangent vector \(v\), some component of \(d\mathcal F(v)\) is nonzero. Solve \(d\Phi_B(v)=0\) for its \(r\) matrix entries. The unit tangent dimension \(2n-1<r\) gives the same null-set estimate. Countable smaller coordinate pieces cover all such components.

Thus matrix choices separating any retained compact set of pairs or unit tangents form a dense open set. Openness follows from a positive minimum of the continuous norm there. Density follows from the finite parameter null-set argument inside any open cylinder; include its additional constrained entries among the free parameters. To meet all countably many conditions, choose nested closed metric balls with radii below \(2^{-j}\), each contained in the preceding ball's interior and the next dense open set. Their centers are Cauchy; completeness gives a limit in every ball and hence every condition. Therefore \(\Phi_B\) is injective and immersive. Now \(J=(t,\Phi_B)\) is proper and injective immersive. Properness gives a homeomorphism onto its closed image; the local inverse theorem gives its submanifold charts. Append zero even blocks as necessary. \(\square\)

**Lemma 8D.2 (a global variable normal tube).** For the proper even embedding \(J:V\hookrightarrow\mathbb R^N\), \(N>n,d\), its Euclidean normal bundle has a whole normalized tube after smooth positive variable compression.

**Proof.** The inverse theorem for \(J(x)+\eta\) gives small relatively compact base neighborhoods \(O_i\) and constants \(b_i>0\) where the normal map on \(|\eta|<b_i\) is injective and locally invertible. Choose inner open sets \(U_i\) covering \(V\), with compact closures \(L_i\subset O_i\), and a locally finite partition whose \(i\)-th support lies in \(U_i\). Closedness of \(J(V)\) and compactness of \(L_i\) allow shrinking \(b_i\) so that \(|Jx-Jy|<b_i\), \(x\in L_i\), implies \(y\in O_i\). Make \(b_i<1\). A subordinate partition gives \(h(x)=\sum_i\rho_i(x)b_i/8>0\); some active \(i\) has \(b_i\ge8h(x)\).

On the closed image put
\[
 R(Jx)=\inf_{y\in V}\bigl(h(y)+\tfrac14|Jx-Jy|\bigr).
 \tag{D.c}
\]
It is positive: outside a fixed ambient ball the distance term is bounded below; inside that ball properness gives a compact base set and a positive minimum for \(h\). It is \(1/4\)-Lipschitz and at most \(h(x)\). Choose a smooth positive \(a(x)<R(Jx)/2\) by the exact partition minorant construction.

If \(Jx+\eta=Jy+\zeta\) with \(|\eta|<a(x)\), \(|\zeta|<a(y)\), then \(D=|Jx-Jy|<R(Jx)+D/8\), so \(D<8R(Jx)/7\). An active \(i\) with \(b_i\ge8h(x)\ge8R(Jx)\) now has \(y\in O_i\). Both normal lengths are below \(b_i\), since \(R(Jy)\le R(Jx)+D/4\). Local injectivity in \(O_i\) gives equality. Local invertibility follows from the same constants. Thus the disk normal map is an open embedding. Compress all fibres by
\[
 q_a(x,v)=\left(x,\frac{v}{\sqrt{1+|v|^2/a(x)^2}}\right).
 \tag{D.d}
\]
This is a bundle diffeomorphism onto that disk, normalized at zero. The compact-level-set proof of Lemma 6.0t gives the whole \(C_0\) extension and its interval comparison, with no global lower radius bound. \(\square\)

#### 8D. The whole transverse family and full orientation

Use the induced metric, \(T=F^\perp\cong\tau\), and \(E=(dJF)^\perp\subset V\times\mathbb R^N\), of rank \(r=N-d\). The actual section normal \(\nu_J\oplus\tau\) identifies with \(E\) by
\[
 \chi(\eta,t)=\eta-dJ(Tt).
 \tag{D.e}
\]
Use the complete finite recovery (324) of Lemma 7.23 for its full normal module. That proof is a smooth bundle construction on the entire base; compactness was used there for one radius, not for its two finite evaluations. We retain those exact evaluations and right lists.

**Lemma 8D.3 (one noncompact transverse family).** There is a smooth actual cocycle family \(h_s:E\to V/F\), transversely submersive everywhere, with \(h_0=\ell\pi_E\) and étale \(g=(h_1,e):E\to V/F\times\mathbb R^N\). Its kernel distributions have one full physical coefficient family from the exact normal-inverse/leaf product to the exact ambient inverse.

**Proof: actual maps and ranks.** We construct the required local addition, denoted \(\operatorname{Exp}_x\), without requiring geodesic completeness. In a coordinate chart on \(V\), let \(S_i\) be the tangent-bundle spray with equations \(\dot x=v,\dot v=0\). Multiply by a smooth chart partition and sum. The sum is a global smooth spray whose projection is \(v\), and whose acceleration is quadratic in \(v\) in every chart. At \((x,0)\) its trajectory is constant and its linearized equation is \(\dot a=b,\dot b=0\). DG-FND Theorem 2.1 gives its time-one base map on a neighbourhood of the zero section: finitely many existence intervals cover each zero trajectory's compact time interval. Consequently
\[
 \operatorname{Exp}_x(0)=x,\qquad
 d\operatorname{Exp}_{(x,0)}(u,v)=u+v.
 \tag{D.e.1}
\]
The notation denotes this local addition; no special metric-exponential property is used.

On a relatively compact base chart, the local addition is defined on a small disk and its derivative in the transverse \(Tt\) directions remains an isomorphism onto the target transverse coordinates. At zero that derivative is the identity on \(\tau\). Compactness is used only for this retained chart closure and for the interval \(s\in[0,1]\). Choose constants admissible for each such closure and a locally finite subordinate partition. A positive smooth minorant of those bounds gives \(a(x)>0\). Shrink it also below the Euclidean tube bound. Thus the local addition and its transverse derivative are defined and invertible for \(|t|<a(x)\), and every \(sTt\) in that disk is admissible. There is no global lower bound on \(a\).

Write \((\eta_v,t_v)=\chi^{-1}q_a(v)\), and define global unit-anchor cocycles
\[
 h_s(x,v)=\ell(\operatorname{Exp}_x(sTt_v)),\qquad
 e(x,v)=Jx+\eta_v,\qquad g=(h_1,e).
 \tag{D.f}
\]
All arrows are actual unit-anchor arrows and their pullbacks in the supplied groupoid; no germ equality replaces a cocycle identity. The compressed \((\eta_v,t_v)\) are smooth coordinates on an open disk. For \(s>0\), their \(t_v\) transverse derivative is \(s\,d_{\mathrm{trans}}\operatorname{Exp}_x\circ T\), surjective; at \(s=0\), the base derivative is \(TV\to\tau\). Thus the rank is \(q\) everywhere, including arbitrarily large uncompressed fibres. No bound on derivatives of \(a\) is used.

For \(g\), the Euclidean normal tube identifies \((x,\eta_v)\) with the \(N\) Euclidean coordinates. The remaining transverse derivative in \(t_v\) is invertible. Hence its principal anchor is a local diffeomorphism and \(g\) is étale, with all counting lifts retained.

The kernels \(\mathcal F_s=\ker d_{\mathrm{trans}}h_s\subset TE\) are a smooth rank-\(N\) family. At the zero section, for \(v\in E_x\), \(\chi^{-1}v=(\eta,t)\), \(k\in F_x\),
\[
 L_s(v,k)=(k-sTt,v)\in\mathcal F_s|_V,\qquad
 de\,L_1(v,k)=dJ(k)+v.
 \tag{D.g}
\]
These are the exact signed identities (327); both the positive exponential sign and negative sign in \(\chi\) are necessary.

**Proof: full module and endpoint.** The full identity (325) is
\((\mathcal D_E\widehat\otimes\mathscr M_F)^\beta\cong J_{\mathrm{lin}}^*\mathcal D_N\),
where \(J_{\mathrm{lin}}(v,k)=v+dJk\). It supplies an actual global frame of \(\mathcal F_0\) and even full intertwiner, with normal/tangent line phases evaluated as \(L^*\otimes L\), including rank-zero gradings. The ambient coefficient is the specified physical inverse \(\mathcal D_N\).

Transport that frame by a smooth interval connection. Each fixed source point has a compact parameter interval; compact source supports give uniform coefficient bounds. Correct its zero-section frame by \(\widetilde E_s(x,0)^{-1}L_sJ_{\mathrm{lin}}^{-1}\), as in (328), and pull the correction from the base. Declare the corrected frame orthonormal.

At the endpoint \(G(x,v)=de\,E_1(x,v)\) equals \(1\) at \(v=0\), and radial contraction puts it in \(GL^+(N)\). Use its positive polar decomposition \(G=RP\) and interpolate \(P\) from \(1\), changing the frame by its inverse as in (329). The endpoint metric is exactly the Euclidean pullback metric. The radial path \(R(x,\lambda v)\), starting at \(1\), has the identity-started Spin lift (310). Its even full representation gives the ambient endpoint unitary, preserving the right action, inner product, grading and physical generators. It is globally smooth since the initial lift and radial path are fixed at every point. No phase is chosen independently at infinity. This proves the entire full orientation family, including full original leaf lines and compact smooth extra parameters. \(\square\)


#### 8D. A single completed principal interval module

**Lemma 8D.4 (the interval class before endpoint evaluation).** The family of Lemma 8D.3 gives one countably generated graded module over \(C(I,A)\widehat\otimes C_N\), one adjointable odd self-adjoint contraction \(Q\), and compact interval defects for left \(C_0(E)\) coefficients. After the fixed \(R_N\) reduction, its endpoint classes are
\[
 d_E\#\ell_{\mathrm{an}},\qquad g_!\# A_N
       \quad\text{in }KK(C_0(E),A).
 \tag{D.h}
\]
Here \(d_E,b_E\) are the native full inverse/Thom pair of Lemma 7.20; \(A_N\) is the exact geometric Euclidean inverse calibrated by Lemma 7.21.

**Proof: the completed module.** Form the actual principal space
\[
 P_i=\{(s,z,\gamma):r\gamma=h_i(s,z)\},\qquad
 (s,z,\gamma)\sim(s,z,h_{ji}(s,z)\gamma).
 \tag{D.i}
\]
In the specified graph-action case this has an especially concrete global model. Write \(f_s(x,v)=\operatorname{Exp}_x(sTt_v)\), and describe an arrow by its range and its full group parameter \(h\in H\), with source \(h^{-1}f_s(x,v)\). Then
\[
 P=I\times E\times H,\quad
 b(s,z,h)=(s,z),\quad
 a(s,z,h)=(s,h^{-1}f_s(z)).
 \tag{D.i.1}
\]
This is the actual principal space, including nonunit and discrete-isotropy arrows. Its diagonal right action has the smooth division map supplied by group multiplication; it is not a quotient by the orbit relation on \(V\). In particular \(P\) is Hausdorff and second countable even when the orbit space is not Hausdorff. For a holonomy presentation, the stated local models glue with the actual arrows in (D.i); their fixed source fibres are Hausdorff by the same base/source-fibre argument as (330).

The derivative of \(a\) is onto. Its group directions supply \(F\), and its \(z\) directions supply the whole transverse quotient by Lemma 8D.3. The restriction \(db:\ker da\to b^*\mathcal F_s\) is an isomorphism: a transverse-kernel vector has a unique compensating infinitesimal group direction, since the infinitesimal graph frame is injective. Therefore \(a\) has fibre dimension \(N\), with vertical tangent exactly \(b^*\mathcal F_s\). The zero-section points \((s,x,0,1_H)\) map to \((s,x)\), so \(a(P_s)=V\) for every parameter. This proves the required fullness directly. None of these facts uses properness of the action or compactness of \(V\).

Complete compact smooth sections in individual Hausdorff relative charts of
\(b^*(\mathscr M_s\otimes D_{\mathcal F_s}^{1/2})\otimes a^*D_F^{1/2}\).
There is no parameter-density factor. In compatible half-density trivializations the actual inner product and right convolution are
\[
 \begin{aligned}
 \langle\xi,\eta\rangle(s,\gamma)
  &=\int_{P_{s,s\gamma}}
       \langle\xi(z\gamma^{-1}),\eta(z)\rangle_{C_N}\,d\mu_s(z),\\
 (\xi k)(z)
  &=\int_{s\gamma=a_V(z)}\xi(z\gamma^{-1})k(s,\gamma).
 \end{aligned}
 \tag{D.i.2}
\]
Here \(d\mu_s\) is the vertical density already supplied by the two longitudinal half-densities; the target kernel retains its endpoint source/range half-densities. These are the exact density conventions of (333). The right Clifford action and coefficient inner products obey the graded convention of Lemma 0.13. For two core supports, the action/division homeomorphism reduces the integrals to finitely many compact relative arrow charts. Smooth dependence has a fixed finite Schur bound there; a finite parameter partition gives coefficients in \(C(I,A)\widehat\otimes C_N\). The regular-fibre Fubini calculation (334) proves matrix positivity and detects the norm. Separated compact chart functions are dense in its range, with every component and arrow lift retained. Completion gives the single module \(\mathscr E\).

Endpoint evaluation is the onto unitary (335) for this actual relative atlas. Every compact endpoint section extends in a relative interval chart; finite sums extend and their evaluated inner product is the actual endpoint inner product. The quotient completion is therefore onto. Interval rank-one operators evaluate to endpoint rank-one operators. Countable compact chart exhaustions and the right approximate identity prove countable generation. These proofs use compact core supports and never compactness of \(V\).

**Proof: one bounded operator and its actual compact ideal.** The gauge groupoid \((P\times_aP)/(G\times I)\) has algebroid \(\mathcal F_s\). The full rank-one/division calculation (337)–(338) identifies its represented reduced algebra with \(\mathcal K(\mathscr P)\), for the scalar module. For completeness, a represented rank-one has the kernel
\[
 K_{\xi,\eta}([z',z])=
   \int_{G_{a_V(z)}}\xi(z'\gamma^{-1})\overline{\eta(z\gamma^{-1})}.
 \tag{D.i.3}
\]
The compact supports of \(\xi,\eta\), the division map, and finite subdivision put this kernel into finitely many compact gauge charts. Conversely, on a retained gauge chart, separated smooth approximation of its three variables, followed by a compact plaque approximate identity, approximates its represented kernel by such rank-ones. The fixed finite Schur constants make the approximation uniform on all regular source fibres. If \(z_0\in P_{s,v}\), then \(z\mapsto[z,z_0]\) identifies \(P_{s,v}\) with the corresponding regular gauge source fibre. Since \(a(P_s)=V\), the faithful regular families give both norm inequalities. Thus, on this one parameter module,
\[
 C_r^*((P\times_aP)/(G\times I))
       \cong\mathcal K(\mathscr P),\qquad
 \mathcal K(\mathscr E)
       \cong\mathcal K(\mathscr P)\widehat\otimes C_N.
 \tag{D.i.4}
\]
The second identity uses the entire even trivialization by \(\mathcal D_N=\Pi^{k_N}C_N\). All regular fibres enter these norm equalities. The gauge algebra itself need not be a product of its endpoint algebras.

Quantize the normalized physical symbol in compact buffered relative charts, using the complete finite-supported calculus of Lemma 7.11. Negative-order frequency truncations converge in reduced norm to compact chart kernels, uniformly over the retained compact relative parameters. The square-root upper bound and high-frequency packet lower bound give the exact compact symbol kernel of (339).

Choose a countable locally finite buffered relative atlas and smooth \(\chi_i\) with \(\sum_i\chi_i^2=1\); compact larger supports are also locally finite. Clip odd self-adjoint local quantizations to contractions \(Q_i\), and on the one module put
\[
 Q=J_\chi^*\operatorname{diag}(Q_i)J_\chi
   =\mathop{\mathrm{strict}\sum}_i\chi_iQ_i\chi_i,\qquad
 J_\chi u=(\chi_i u)_i,\qquad\|Q\|\le1.
 \tag{D.j}
\]
The column isometry and adjoint follow on compact cores and then by density; the contraction bound controls the strict tails. A compactly supported left coefficient meets finitely many supports and larger neighbors, so localization of the square and commutator is the finite-supported calculus, with zero symbol. Hence
\[
 c(Q^2-1),\ [Q,c]\in\mathcal K(\mathscr E),
       \qquad c\in C_0(I\times E).
 \tag{D.k}
\]
Supremum-norm approximation and the fixed contraction bound treat every coefficient. This is one actual Kasparov homotopy before endpoint evaluation. No global chart derivative bound or radius lower bound enters it.

**Proof: zero endpoint.** Its actual principal space is \(\{(x,v,\gamma):r\gamma=x\}\). On compact cores the balanced tensor map sends an elementary normal section and a leaf kernel to their product \(u(x,v)\eta(\gamma)\), with the actual convolution action on the leaf kernel. Integration first over \(E_x\) and then over the regular \(G\) variable gives (D.i.2), with both finite Clifford actions retained. Separated smooth functions are dense on every compact principal chart; their uniform error has a fixed finite Schur bound. Finite compact-chart subdivision then proves module-norm density of the range. The completed Fubini map is therefore an even onto unitary from normal integration tensored with the full regular leaf module. Its physical coefficients are normal then tangent; the entire transport \(\beta\) of (325) changes right action, inner product and grading to the native tangent/normal target before \(R_N\). The two normal inverse evaluations are the actual ones of Theorems 7.17/7.20.

The nonproper creation and positivity proof of Theorem 7.13 applies to \(\pi_E:E\to V\); that theorem already has arbitrary noncompact source and target with \(C_0\) localization. Its axis compression proves both rectangular creation errors and adjoints compact in the supported quotient; the \(CL(L+\delta)^{-1}\) approximation recovers the errors themselves. Positivity is bracket (266). The buffered splitting partition gives the whole product. Its transferred phase and \(Q_0\) have the same normalized symbol, so the exact compact-symbol kernel gives their straight cycle homotopy. This proves the first endpoint in (D.h), including the rank-zero full line.

**Proof: étale endpoint and both creation errors.** The complete balanced unitary (345), with actual right convolution, is
\[
 \mathcal E_g\widehat\otimes_{A\otimes C_0(\mathbb R^N)}
 (A\widehat\otimes L^2(\mathbb R^N)\widehat\otimes\mathcal D_N)
       \cong\mathscr E_1 .
 \tag{D.h.1}
\]
On compact cores its formula is
\[
 U\bigl(\xi\otimes(a\otimes w)\bigr)(z)
    =(\xi\cdot(a\otimes1))(z)\,w(e(b(z))).
 \tag{D.h.2}
\]
The dot denotes the actual right convolution by \(a\), rather than scalar evaluation. Euclidean multiplication and convolution associativity prove both balancing relations. On each regular \(G_v\) fibre, a countable disjoint measurable refinement of full-anchor charts changes
\[
 \int_{\mathbb R^N}
  \sum_{z\in\widetilde a^{-1}(v,t)}|u(z,t)|^2\,dt
     \quad\hbox{into}\quad
 \int_{P_v}|u(z,e(b(z)))|^2,
 \tag{D.h.3}
\]
with the pulled-back Euclidean density. The sum counts every lift. Its full finite-Clifford matrix identity proves that \(U\) is isometric in module norm. A compact full-anchor chart vector is a finite sum of scalar chart sections times fixed Clifford columns. A Euclidean smooth cutoff equal to one on its support realizes each such vector in the closure of (D.h.2); insert an \(A\) approximate identity to pass from its scalar expression to balanced tensors. The error is bounded by the étale section-module error times the Euclidean \(L^2\) norm. Compact-chart sums exhaust the defining core, proving onto. For a compact principal creation section \(\xi\), its Euclidean support \(S\) is compact, irrespective of noncompact \(V\). Split the input by \(\kappa_R=1\) near \(S\). Under (D.h.2), write creation as \(T_\xi\). It has \(T_\xi=T_\xi\kappa_S\), where \(\kappa_S\) is one on the compact Euclidean output set \(S\). Only finitely many right cutoffs of (D.j) meet \(\operatorname{supp}\xi\). Each of their clipped phases differs from its unclipped phase by a compact, because their finite-supported symbols agree. In a full-anchor chart the longitudinal coordinate is the Euclidean \(t\). The leading amplitudes of \(Q_1T_\xi\kappa_R\) and \(T_\xi F_N\kappa_R\) are respectively
\(c_{\mathcal D_N}^+(\zeta/|\zeta|)\xi(z\gamma^{-1})\) and
\(\xi(z\gamma^{-1})c_{\mathcal D_N}^+(\zeta/|\zeta|)\).
They are equal with precisely the transferred physical action. Their difference has mixed order \(-1\); the other confined kernels are off-diagonal smooth kernels.

A smooth kernel on finite compact principal output charts times a compact Euclidean input support is a rectangular module compact. Indeed separated approximation gives maps
\(u\mapsto\zeta\int w(t)^*u(t)\,dt\). In the nonunital case insert an \(A\) approximate identity \(a_m\): the rank-one with input vector \(a_m^*\otimes w\) has output coefficient \(\zeta a_m\), and the error is at most \(\|\zeta-\zeta a_m\|\|w\|_2\). Finite homogeneous Clifford columns give the same proof. Uniform coefficient errors have a fixed finite Schur bound, so the approximation is in the full module norm. A mixed order-\(-1\) kernel is a norm limit of such smooth kernels by buffered frequency truncation, with tail at most \(C/M\) after truncation at \(M\). Therefore
\[
 (Q_1T_\xi-T_\xi F_N)\kappa_R
      \in\mathcal K(\mathscr H,\mathscr E_1).
 \tag{D.h.4}
\]
No individual-sheet Hilbert-space compactness is substituted for this module-compact calculation. The tail satisfies the full operator-norm bound
\[
 \|T_\xi F_N(1-\kappa_R)\|
 \le C_N\|\xi\|\,|S|^{1/2}
 \sup_{t\in S}\|K_N(t-\cdot)(1-\kappa_R)\|_{L^2}
       \longrightarrow0 .
 \tag{D.l}
\]
This is the exact heat-integral off-diagonal estimate (279)–(280), used in (351). The physical grading and reflection preserve this norm bound. Each interval core support has one compact \(S\) and finitely many buffered charts, so it is uniform on that whole support. Faithful regular fibres make it a module norm estimate. Since \(Q_1T_\xi(1-\kappa_R)=0\), (D.l) is the entire omitted tail of (D.h.4). The constant is uniform on every regular representation, so (D.h.4) converges in the module-operator norm to \(Q_1T_\xi-T_\xi F_N\). The adjoint equation is the other creation error. The bound \(\|T_\xi\|\le\|\xi\|\) and core density treat all creation vectors. The first operator \(g_!\) is zero, its left action is compact by Lemma 6.0n, and positivity is zero. Definition 0.3 gives the second endpoint in (D.h), before finite reduction. \(\square\)

**Theorem 8D.5 (the full noncompact geometric leaf class).** The actual normal-tube product
\[
 \ell_{\mathrm{geom}}=b_E\#g_!\# A_N
       \in KK_d(C_0(V),A)
 \tag{D.m}
\]
is the whole geometric leaf-map class in its prescribed full orientation. On the entire noncompact manifold,
\[
 \ell_{\mathrm{geom}}=\ell_{\mathrm{an}}=s_d\Theta_H,\qquad
 \ell_{\mathrm{geom}}\#(s_dD_H)=1_{C_0(V)},\qquad
 (s_dD_H)\#\ell_{\mathrm{geom}}=1_A .
 \tag{D.n}
\]
These are actual KK equalities and give inverse maps on both integral K-groups.

**Proof.** The normal rank is \(N-d\), \(g_!\) is the actual completed étale zero correspondence, and the ambient inverse has degree \(N\). The product has degree \(d\). Lemma 6.0t supplies the whole variable tube homomorphism; Lemma 6.0n supplies the completed middle module and compact left action. The full transfers (324)–(325) and Lemmas 7.20–21 give exactly the native geometric normal product.

Lemma 8D.4 gives \(d_E\#\ell_{\mathrm{an}}=g_!\#A_N\). Multiply by \(b_E\) and use both actual inverse identities on the entire base, from Theorem 6.0c and Lemma 7.20. This proves the first equality. Every auxiliary proper embedding, positive radius, local addition and transport constructed above yields that same analytical class, hence the same whole geometric class. Tube choices also have the direct single interval comparison of Lemma 6.0t. Thus choices are independent before passage to K-groups.

The zero-endpoint analytical class in Lemma 8D.4 was formed with the embedding-induced leaf metric \(g_0\). Let \(g_H\) be the metric making the specified infinitesimal frame orthonormal. Use the positive family \(g_t=(1-t)g_0+tg_H\), transporting the entire prescribed spin-c bundle by the positive metric isometry, including its right Clifford action, line and grading. Lemma 7.11 constructs one completed parameter module and one adjointable contraction for this metric family. For a compact coefficient on \(V\), its support times the interval is compact, so the buffered local calculus gives compact defects in that single interval module; the fixed contraction bound and \(C_0(V)\) approximation treat every left coefficient. Its onto endpoint evaluations give the metric homotopy of the whole analytical class. Thus we may choose the homogeneous framed metric as the representative of \(\ell_{\mathrm{an}}\) before comparing symbols.

Theorem 8.15d is already proved for arbitrary \(C_0(V)\) in the ordered solvable deformation. Its framed homogeneous Dirac has exactly the full symbol retained here. We spell out why its comparison with this analytical phase is a whole KK comparison. In compact buffered group/plaque charts, the parameter-resolvent construction in Lemmas 8.15a–b has principal inverse
\((c^+(\zeta)\pm ib)^{-1}\), \(b=\sqrt{1+t^2}\). Its norm is at most \((|\zeta|^2+b^2)^{-1/2}\). Derivatives of this inverse and the finite supported composition formula give a resolvent-parametrix error bounded by \(C/b\), uniformly over each retained compact unit support and every regular fibre. Multiplication by the actual resolvent, of norm at most \(1/b\), gives a compact error at most \(C/b^2\). Integrating the sum of the two poles is therefore a norm-integrable compact error; its principal integral is
\(c^+(\zeta)/\sqrt{1+|\zeta|^2}\). Small bounded parameter intervals are handled by (121); all remaining finite chart remainders have the negative-order compact norm of Lemma 7.11. Thus the localized bounded transform belongs to that exact supported symbol quotient and has the prescribed physical symbol.

For a compact left coefficient \(c\), choose a scalar unit cutoff \(\chi=1\) near its support. The off-buffer term
\(cF_D(1-\chi)=c[F_D,1-\chi]\) is compact by the bounded-transform commutator in Lemma 8.15b, and the analogous term for the analytical phase \(Q_V\) is compact on its actual base module by Lemma 7.11, equation (269): \(cQ_V(1-\chi)=-c[Q_V,\chi]\). Within the buffers the phases have the identical symbol, so their difference is compact by the exact finite-supported quotient. Taking adjoints proves both localizations. Supremum-norm approximation treats every \(c\in C_0(V)\). The locally compact difference gives the straight Kasparov homotopy of Lemma 7.11 on the whole module. It therefore gives \(\ell_{\mathrm{an}}=s_d\Theta_H\). The chronological real factors have both full KK inverses; \(s_d^2=1\) proves the remaining identities. No compactification, action extension or compact-test detection of KK equality has been used. \(\square\)

#### 8D. The intermediate noncompact assembly product

For a compact native global-anchor cycle \(c=(M,B,a,\eta)\), use the ordinary orientation \(a_{\mathrm{ord}}\) supplied by Lemma 8C.1: add the specified leaf frame to the native module and perform the actual doubled-tangent recovery. Its compact-support class is \(\kappa(c)=[B]a_{\mathrm{ord},!}\). The global reduction of arbitrary smooth cocycles is Lemmas 8B.1–2 and Theorem 8B.4; the defective smooth pullback of a fixed continuous join section is not used.

**Lemma 8D.6 (full compatibility of the three normal orientations).** With the framed leaf orientation of Theorem 8D.5, the ordered relative orientations of \(a_{\mathrm{ord}}\) and \(\ell\) induce exactly the original native orientation \(\eta\) of \(\ell a\), including line, central phases and stable grading.

**Proof.** Write the virtual normal identities with their actual full modules:
\[
 (a^*TV-TM)+a^*(\tau-TV)=a^*\tau-TM .
 \tag{D.o}
\]
Here \(a^*TV=a^*F\oplus a^*\tau\). Lemma 8C.1 tensors the original native doubled module with the specified framed \(F\) factor. The leaf relative factor is its full inverse, with the canonical doubled transverse cancellation; this is the relative orientation of the framed native leaf module in the explicit full normal recovery (324). Apply the signed unitary (325) to identify its recovered physical tangent module with the same prescribed frame. Explicitly the physical source has normal then tangent order, while its native right algebra is tangent then normal after
\(\beta(x\widehat\otimes y)=(-1)^{|x||y|}y\widehat\otimes x\). Its actions and inner product are transported together. For odd \(d\), the retained last two generators are in second-cycle-first order until their fixed matrix contraction; the even pair factors are removed only by the exterior evaluation maps. Lemma 7.20 proves both inverse contractions; the complete early/late finite contraction proof following equation (211), Proposition 6.0i and Lemma 7.21 retain this prescribed order in every parity. The grading correction is retained, and the reduction is precisely the physical class of (D.n).

On an actual graph normal the product module is thus the original normal module followed by the framed factor and its inverse. The two finite maps \(\xi\otimes\bar\zeta\mapsto\theta_{\xi,\zeta}\) and \(\bar\xi\otimes\zeta\mapsto\langle\xi,\zeta\rangle\) cancel those adjacent factors. Any intervening doubled tangent or transverse factors cancel by the two recovery maps of Lemma 6.0j. Regrouping the consecutive generators introduces no permutation; a necessary permutation is the exact graded intertwiner of Lemma 7.21. Hence applying the recovery (209) of Lemma 6.0v returns the entire original native doubled module, with both recovery composites identity. This proves (D.o) on full modules, not on determinants.

A line \(L\) in \(\eta\) remains \(L\) throughout; only the prescribed frame and its inverse evaluate. Even a two-torsion \(L\) is therefore recovered. An additional chosen leaf line \(D\) is handled by twisting the ordinary relative orientation by \(a^*D^*\). The actual product line is \(a^*D^*\otimes a^*D\otimes L\), whose even onto evaluation returns \(L\). The same statement holds for full graded lines. This is the complete orientation compatibility needed for assembly, rather than an existence of unrelated spin-c structures. \(\square\)

**Theorem 8D.7 (the actual noncompact intermediate composition).** For every compact native cycle, and before multiplying by its bundle coefficient,
\[
 a_{\mathrm{ord},!}\#\ell_{\mathrm{geom},!}
       =a_{\mathrm{cocycle},!}
       \quad\text{in }KK_{q-\dim M}(C(M),A),\qquad
 \mu(c)=\kappa(c)\#\ell_{\mathrm{geom},!}.
 \tag{D.p}
\]
The intermediate \(V\) may be noncompact. The right side is the exact existing compact-source native actual-cocycle construction, not a newly restricted assembly map.

**Proof: actual embeddings.** Choose an even Euclidean embedding \(u:M\hookrightarrow\mathbb R^R\). Its ordinary graph \(I=(a,u)\) has full normal
\(E_a=\nu_u\oplus a^*TV\), rank \(R-\dim M+n\). The leaf normal zero section \(k:V\hookrightarrow E\) has full normal \(E\), rank \(N-d\), and wrong-way class \(b_E\). Follow \(I\) by \(k\times1\), whose intermediate source \(V\times\mathbb R^R\) is noncompact. Lemma 6.0t applies in exactly this generality: its variable compression, essential open-ideal restriction and ordered Thom multiplier give the actual composed embedding \(K=(ka,u)\), with normal \(E_a\oplus a^*E\). No compact-intermediate version of Theorem 6.0w is applied outside its scope.

The normal quotient is the full vector identity
\[
 E_a\oplus a^*E
    \cong (\tau\oplus\mathbb R^{N+R})/
                       d(\ell a,Ja,u)(TM).
 \tag{D.q}
\]
Use the explicit quotient map (207), \(v=(d(Ja,u)^*d(Ja,u))^{-1}d(Ja,u)^*z\), \(\eta=z-d(Ja,u)v\), with output transverse coordinate \(t-d^\perp(\ell a)v\). Transport metrics positively before its Clifford action. The normal recovery (209) then returns precisely the full orientation of Lemma 8D.6. These calculations live over the compact \(M\), with actual bundle transitions pulled from the whole \(V\); compactness of \(V\) is not a premise.

The original statement of Lemma 6.0v assumed compact intermediate manifolds for its embedding choice. Here its quotient and finite-recovery formulas are being proved for the displayed bundles over \(M\): the globally available \(J\) is supplied by Lemma 8D.1, the product tube by Lemma 6.0t, and (207), (209) use only the displayed positive Gram inverse and finite evaluations on \(M\). Its inverse map sends \((\eta,t)\) to \([(t,\eta)]\), and adding \((d^\perp(\ell a)w,d(Ja,u)w)\) to a representative changes the Gram-inverse vector by \(w\) while leaving \(\eta\) and the output transverse coordinate fixed. Thus it is an actual bundle quotient isomorphism. The positive metric isometry precedes its Clifford extension. Let \(U=TM\), \(Q_U=U\oplus U\), and \(L_c=TM\oplus a^*\tau\). Carry the full normal module tensored with the canonical interleaved \(Q_U\) module through the graph isometry \(\rho_{(Ja,u)}\). Repartition consecutive right generators to obtain
\(P:A_{L_c}\widehat\otimes C_{N+R}\rightsquigarrow C_{m+q}\widehat\otimes C_{N+R}\).
Writing \(S=\Lambda^*\mathbb C^{(N+R)/2}\), its doubled stable recovery is exactly
\[
 (A_{L_c}\widehat\otimes S^*)
 \widehat\otimes_{A_{L_c}\widehat\otimes C_{N+R}}P
 \widehat\otimes_{C_{m+q}\widehat\otimes C_{N+R}}
 (C_{m+q}\widehat\otimes S).
 \tag{D.q.1}
\]
The two even finite evaluation maps recover \(P\) from this module; pulling back by \(\rho_{(Ja,u)}\) and cancelling \(Q_U\) recovers the entire original ordered normal module. All these maps are smooth over \(M\), preserve both actions and inner products, and are onto by the finite evaluations. They move only Clifford factors, so the original line and its stable grading remain. This proves the needed extension of the finite orientation construction rather than invoking a compact-intermediate statement beyond its hypotheses.

**Proof: the middle module and inverse order.** The étale product \(g\times1_{\mathbb R^R}\) has the whole principal space \(P_g\times\mathbb R^R\). Its compact core Fubini and convolution identities give the onto unitary
\(\mathcal E_{g\times1}\cong\mathcal E_g\widehat\otimes C_0(\mathbb R^R)\),
including every counting lift, both actions and zero operator, by the product calculation in Lemma 6.0n. Essential ideal restrictions have that lemma's exact range and compact-action proof. Thus this remains an equality on the completed whole modules.

Move the even inverse for the \(u\)-block through the embedding and this étale correspondence using the actual exterior associator. The ordered ambient inverse removes the last \(R\)-block first and the old \(N\)-block second; the finite exterior wedge unitary identifies it with the specified \(A_{N+R}\). The full normal blocks are retained until the native signed transport of Lemma 7.21 and final contraction. Both Euclidean blocks are even, so no odd block is silently permuted.

The result is the actual product \(K_!(g\times1)_!A_{N+R}\). Its cocycle at the compact section is exactly \((\ell a,Ja,u)\). Lemma 6.0u applies with this compact source and the even embedding \((Ja,u)\): its germ diffeomorphism and actual arrow-equivalence unitary identify it with the canonical compact-source native factorization, with the entire recovered normal orientation just proved. Its hypothesis permits a noncompact ambient manifold; only the source section is compact. This proves the first equality in (D.p) as a full KK equality. Multiplying by the unchanged finite bundle \(B\) gives the second.

An original actual smooth cocycle reduces by the smooth \(k_\alpha\) of Lemma 8B.1. Lemma 8B.2 retains its full normal derivatives, and Corollary 8B.5 gives the identical completed native wrong-way class. Thus (D.p) holds for every native representative and all three native relations. It does not replace the domain by unit-transition cycles without proving that reduction. \(\square\)

#### 8D. Both native assembly inverses and the convention

**Theorem 8D.8 (assembly isomorphism in both integral degrees).** Let \(I\) denote the inclusion inverse to the full smooth reduction \(R\) of Theorem 8B.4, and \(\Gamma_p\) the compact-carrier inverse of Theorem 8C.8, \(p=j+d\bmod2\). For \(y\in K_j(A)\), define
\[
 \mathcal N_j(y)=I\,\Gamma_p(y\#s_dD_H).
 \tag{D.r}
\]
This gives inverse homomorphisms on the entire native groups:
\[
 \mu\mathcal N_j(y)=y,\qquad
 \mathcal N_j\mu(c)=c,\qquad j=0,1 .
 \tag{D.s}
\]
Every geometric source in the inverse is closed compact, and all line and torsion data are retained.

**Proof.** Theorem 8C.8 supplies the full two-sided integral duality on the native quotient, by compact carriers and the embedding bordism. Theorem 8D.7 proves \(\mu=\kappa R\,\#\ell\) on every actual representative. Theorem 8D.5 supplies the actual whole KK inverse \(\delta=s_dD_H\). Consequently
\(\mu\mathcal N_j(y)=(\kappa\Gamma_p(y\delta))\ell=(y\delta)\ell=y\).
Conversely
\(\mathcal N_j\mu(c)=I\Gamma_p((\kappa Rc)\ell\delta)=I\Gamma_p(\kappa Rc)=IRc=c\).
These products have degree \(j+d\) on \(C_0(V)\) and degree \(j\) on \(A\); the parity \(p\) in the compact carrier is exactly the one in R1. Both equalities use whole integral groups and genuine two-sided identities. They are not consequences of numerical tests or a rational character. The full/reduced identification for solvable \(H\) is the already proved Corollary 8.9.

For an additional full graded leaf line \(D\), set
\[
 \ell_D=[D]\#\ell,\quad
 \delta_D=\delta\#[D^*],\quad
 \kappa_D(c)=\kappa(c)[D^*],\quad
 \Gamma_D(x)=\Gamma(x[D]).
 \tag{D.t}
\]
The two full line evaluations make \(\ell_D\delta_D=1\), \(\delta_D\ell_D=1\), and \(\kappa_D,\Gamma_D\) inverse. Lemma 8D.6 proves the same native assembly composition with these adjusted ordinary orientations. Formula (D.r) using \(\Gamma_D,\delta_D\) gives the same inverse on the original native cycles. A line is never discarded by treating its determinant as trivial.

The analytical leaf map, the geometric normal-tube map and the topological index action of native Section 11 are the same in this transported full orientation, by Theorems 8D.5/7. The real Thom action with that physical orientation is the explicitly calibrated \(s_d\Phi_H\). The literal chronological \(\Phi_H\) equals it only when \(s_d=1\). When \(d\equiv2,3\pmod4\), the uncalibrated equality would be false under the course's fixed odd–odd reducer. This supplement proves the complete comparison with its exact sign; it does not erase that already established convention issue. \(\square\)

The full compact-carrier duality of Theorem 8C.8, the whole noncompact leaf class of Theorem 8D.5 and the actual composition of Theorem 8D.7 prove both integral assembly inverse identities. Every arrow, full orientation and native cycle relation remains part of the construction. The chronological real product is compared in the displayed physical calibration \(s_d\); its conversion to a source convention must preserve that calibration.

![The whole noncompact native comparison and both assembly inverses](../figures/noncompact-native-comparison.png)

**Figure 8D.1.** Lemmas 8D.1–4 and Theorems 8D.5/7/8. The variable radius is positive pointwise, with no global lower bound. The exact kernel vector in the local translation example is \(\partial_t-s\partial_y\), with the old leaf and Euclidean-normal directions retained. The interval operator is one strict sum on one completed principal module; its compact defects occur before endpoint evaluation. The Euclidean tail bound is a proved module norm bound, not a numerical sample. The assembly diagram includes the full compact carrier inverse and both identities. The sign \(s_d\) is exact. [Reproducible figure source](https://kokunoyumeto.github.io/open-math-courses-public/courses/NCG-FOLIATIONS/figures/noncompact-native-comparison.py) · [Editable SVG](https://kokunoyumeto.github.io/open-math-courses-public/courses/NCG-FOLIATIONS/figures/noncompact-native-comparison.svg).

## 8E. Historical Thom and suspension with arbitrary coefficients

We now compare the real Thom and suspension maps in the IHES/M/80/28 preprint with the fixed conventions of this lesson. The comparison treats both input degrees, arbitrary coefficients, the complete Clifford module and its spin-c line. It gives an explicit common realization of the three maps in the solvable-group theorem of Connes, Section 12, Theorem 5.

Write \(S=C_0(\mathbb R)\), \(C_\alpha=A\rtimes_\alpha\mathbb R\), and let \(\sigma_j\) be the course's scalar KK classification in Theorem 0.14. Products without a star are the fixed course products of Lemma 0.13. In an odd–odd product the second-cycle coefficient is first in the final Clifford pair. Its generators act as \((\sigma_x,\sigma_y)\), with grading \(\Gamma=\sigma_z\). Thus the physical first-factor generator acts by \(\sigma_y\), and the second-factor generator by \(\sigma_x\).

For the *same* covariant group multipliers the two Stone conventions are
\[
 U_t=e^{itH}=e^{-itD_\alpha},\qquad H=-D_\alpha.
 \tag{AT1}
\]
The course real class is \(t_\alpha=(C_\alpha\widehat\otimes C_1,\iota_A,h(D_\alpha)\varepsilon)\), where \(h(r)=r/\sqrt{1+r^2}\). Put
\[
 N_\alpha^j(x)=\sigma_{j+1}\bigl(\sigma_j^{-1}(x)t_\alpha\bigr).
 \tag{AT2}
\]
Historical suspension and Thom maps are denoted \(S_h^j\) and \(\phi_\alpha^j\). The exact source locators are listed in Section 8E.9. The letter \(H\) later also denotes the solvable group; the real self-adjoint Stone operator in (AT1) is always distinguished by its context.

### 8E.1. The actual coefficient suspension product

Let \(A\) first be separable and trivially graded. Every class in \(K_1(A)\) is represented, after stabilization, by the full odd scalar cycle
\[
 (E\widehat\otimes C_1,1,s\varepsilon),\qquad
 E=H_A,\quad s=s^*,\quad\|s\|\le1,\quad a=1-s^2\in J=\mathcal K(E).
 \tag{AT3}
\]
This is the complete corona classification of Theorem 0.14; no finite-spectrum or commutative coefficient assumption is imposed. Its unitary is \(u_s=\exp(\pi i(s+1))\in1+J\).

Let \(b_\tau\) be the positive one-dimensional suspension class. Define *native left suspension* by \(S_n^j(x)=\sigma_{j+1}(b_\tau\boxtimes\sigma_j^{-1}(x))\), with the suspension coordinate first. The actual reduced module for \(S_n^1\) is
\[
 E_S=C_0(\mathbb R_\tau,E)\otimes\mathbb C^{1|1},\qquad
 F_n(\tau)=\frac{s\sigma_x+\tau\sqrt a\,\sigma_y}{\sqrt{1+\tau^2a}}.
 \tag{AT4}
\]
Functions of \(s\) in this expression commute. The denominator is positive and invertible. The bounded locally norm-continuous function in (AT4) defines one adjointable operator on the whole module, including its adjoint. It is a contraction and
\[
 F_n^2-1=-\frac{a}{1+\tau^2a},\qquad
 \left\|\frac{a}{1+\tau^2a}\right\|
 \le\min\{\|a\|,\tau^{-2}\}\quad(\tau\ne0).
 \tag{AT5}
\]
The defect is therefore in \(C_0(\mathbb R,J)\), not just compact at individual parameter values. The commutator with the scalar representation is zero.

Here is the product check. The lower odd operator in the fixed reduction is \(s\sigma_x\). For each finite \(\tau\),
\[
 F_n-s\sigma_x
 =s\bigl((1+\tau^2a)^{-1/2}-1\bigr)\sigma_x
   +\tau\sqrt a(1+\tau^2a)^{-1/2}\sigma_y
 \tag{AT6}
\]
belongs to \(J\otimes M_2\). Each term is a continuous function of the compact positive operator \(a\) which vanishes at zero, multiplied by bounded factors. Multiplication by any creation vector from the first module \(S\widehat\otimes C_1\) makes (AT6) vanish at infinity, since that vector is a continuous function vanishing at infinity and the two operators in the difference have norm at most one. Thus every creation-map connection error and its adjoint is a rectangular compact on the full tensor module. The finite Clifford matrix factors and their graded creation signs are the ones in Lemma 0.13; finite sums and norm approximation treat every homogeneous vector.

The lifted first operator is \(h(\tau)\sigma_y\). Its anticommutator is exactly
\[
 \{h(\tau)\sigma_y,F_n\}
 =\frac{2\tau^2\sqrt a}{\sqrt{1+\tau^2}\sqrt{1+\tau^2a}}\ge0.
 \tag{AT7}
\]
This proves the positivity sandwich for every source coefficient. Consequently (AT4), including its full coefficient module, is the Kasparov product by the course product criterion and uniqueness theorem.

The occurrence of \(\sqrt a\) is essential. Replacing (AT4) by \((s\sigma_x+\tau\sigma_y)/\sqrt{1+\tau^2}\) would not be a lower connection when \(E\) is infinite: its difference from \(s\sigma_x\) need not be compact at a finite nonzero \(\tau\). This proof does not make that replacement.

The first-factor-first, or physical, operator on the same module is
\[
 F_p(\tau)=\frac{\tau\sqrt a\,\sigma_x+s\sigma_y}{\sqrt{1+\tau^2a}}.
 \tag{AT8}
\]
The onto unitary \(U=(\sigma_x+\sigma_y)/\sqrt2\) exchanges (AT4) and (AT8) and reverses \(\Gamma\). The right coefficient of these reduced cycles is the trivially graded \(SA\). Hence the usual grading-reversal inverse applies *after* reduction, and
\[
 [F_n]=-[F_p]. \tag{AT9}
\]
No grading operator is used as a right-linear map on a retained right \(C_1\) module.

### 8E.2. Historical odd suspension is the physical product, for every coefficient

This comparison is an actual projection homotopy, rather than a scalar degree test. The relative projection for (AT8), from Theorem 0.14, is \(P_T-e_{11}\), where
\[
 T=\frac{\tau\sqrt{1-s^2}+is}{\sqrt{1+\tau^2(1-s^2)}},\qquad
 P_T=\begin{pmatrix}TT^*&T\sqrt{1-T^*T}\\
 \sqrt{1-T^*T}T^*&1-T^*T\end{pmatrix}.
 \tag{AT10}
\]
Perform the calculation first in the universal algebra \(C_0((-1,1)_r)\). Put
\[
 g(r)=\frac r{\sqrt{1-r^2}},\quad y(r)=\tan(\pi r/2),\quad
 P(z)=\frac1{1+|z|^2}\begin{pmatrix}|z|^2&z\\\bar z&1\end{pmatrix}.
 \tag{AT11}
\]
Then (AT10) is exactly \(P(\tau+ig(r))\). No Chern-character convention enters this equality.

Appendix VI.1, Lemma 1, of M/80/28 defines the historical odd suspension of \(u\) by \([We_{11}W^*]-[e_{11}]\), with
\[
 R(t)=\begin{pmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{pmatrix},
 \quad\theta=\pi t/2,\quad
 W=R(t)\operatorname{diag}(u^{-1},1)R(t)^*,\quad0\le t\le1.
 \tag{AT12}
\]
For \(u(r)=e^{\pi i(r+1)}\), write \(A_r=\pi(r+1)/2\). A scalar multiple of the first column of \(W\) is
\[
 \bigl(\cos(\pi t)\sin A_r+i\cos A_r,\ \sin(\pi t)\sin A_r\bigr)^T.
 \tag{AT13}
\]
Conjugating its projection by \(\operatorname{diag}(-1,1)\) changes no K-class: the path \(\operatorname{diag}(e^{\pi iv},1)\) fixes the reference projection. Using the increasing suspension coordinate \(\tau=-\cot(\pi t)\), the resulting line coordinate is
\[
 z_0(\tau,r)=\tau+i y(r)\sqrt{1+\tau^2}.
 \tag{AT14}
\]
An explicit homotopy to (AT10) is
\[
 z_v(\tau,r)=\tau+i\bigl((1-v)y(r)\sqrt{1+\tau^2}+vg(r)\bigr),
 \qquad 0\le v\le1.
 \tag{AT15}
\]
For fixed \(r\), the two imaginary terms have the same sign. Therefore
\[
 |\Re z_v|=|\tau|,\qquad
 |\Im z_v|\ge\min\{|y(r)|,|g(r)|\}.
 \tag{AT16}
\]
Both functions on the right diverge at \(r=\pm1\). Thus \(z_v\) is jointly proper, uniformly in the compact homotopy interval. The relative projections \(P(z_v)-e_{11}\) belong to one algebra
\(C([0,1],C_0(\mathbb R\times(-1,1),M_2))\). This proves the relative-projection homotopy with all boundaries controlled.

Now apply continuous functional calculus to the *actual* operator \(s\) in (AT3). For every \(f\in C_0((-1,1))\), \(f(s)\in J\), because the corona image of \(s\) is an involution and \(f(\pm1)=0\). Hence
\[
 \vartheta_s:C_0((-1,1))\longrightarrow J,\quad f\longmapsto f(s)
 \tag{AT17}
\]
is a *-homomorphism. Tensoring it with \(S\) sends the entire homotopy (AT15) into \(C([0,1],S\otimes J)\). Its endpoints are precisely the historical suspension projection of \(u_s\) and the projection (AT10). Even compact stability/Morita through \(E\), or standard stabilization when necessary, takes both to \(K_0(SA)\). The projection identity holds on the entire module; no full-module compact is inferred from a single fibre.

The even suspension of a relative finite projection \([e]-[e_0]\) is directly \([1+(b-1)e]-[1+(b-1)e_0]\), where \(b\) has positive winding. This is also the actual product of \(b_\tau\) with the two finite projection modules, over the unitization followed by reference subtraction. Connection errors are rectangular compacts because the second modules have finite frames; the first positivity is unchanged. Thus, in both degrees and for arbitrary separable coefficients,
\[
 S_h^0=S_n^0,\qquad S_h^1=-S_n^1,
 \quad\text{equivalently}\quad S_h^j=(-1)^jS_n^j.
 \tag{AT18}
\]
Surjectivity of the scalar classification, not a reduction to scalar coefficients, makes this cover every K-class. It also covers nonunital relative classes.

### 8E.3. Even Thom compression with the full coefficient module

Work first over the action-invariant unitization \(A^+\), and subtract the scalar reference at the end. Every projection class has a smooth projection representative: orbit convolution approximates a projection by a smooth self-adjoint element, its spectral projection across a fixed gap is smooth by the differentiated resolvent formula, and a sufficiently close pair of projections is connected by its polar unitary. These operations preserve the external unitization and its reference.

Let \(e\in M_n((A^+)^\infty)\) be such a projection. On the whole standard \(M_n(C_{\alpha^+})\) module, \([D,e]\) is bounded and belongs to \(M_n(A^+)\). Define
\[
 D'=eDe+(1-e)D(1-e)=D+K,
 \qquad K=-eD(1-e)-(1-e)De.
 \tag{AT19}
\]
The off-diagonal terms equal bounded expressions in \([D,e]\). They are self-adjoint together, so the bounded-perturbation theorem for regular self-adjoint operators follows here from the resolvent Neumann expansion at a large imaginary pole, its adjoint, and the resolvent identity. It gives one regular self-adjoint \(D+vK\) on the fixed domain for \(0\le v\le1\). The resolvent bounds are \(1/|b|\). The difference of the two resolvents is
\[
 (D+vK\pm ib)^{-1}-(D\pm ib)^{-1}
 =-(D+vK\pm ib)^{-1}vK(D\pm ib)^{-1}.
 \tag{AT20}
\]
It lies in \(M_n(C_{\alpha^+})\). For instance \(K(D\pm ib)^{-1}\) is a coefficient times a \(C_0\) spectral function, hence an algebra element; the other factor is a multiplier. Its norm is at most \(\|K\|/b^2\). The bounded-transform integral using the two poles with \(b=\sqrt{1+q^2}\) therefore shows
\(h(D+vK)-h(D)\in M_n(C_{\alpha^+})\), continuously in \(v\). This is a full norm-integrable compact perturbation on that standard module.

The operator \(D'\) commutes with \(e\). The product of the finite projection cycle with \(t_{\alpha^+}\) is consequently the compressed cycle on \(eC_{\alpha^+}^n\widehat\otimes C_1\), with operator \(h(eDe)\varepsilon\). This also follows directly from the product criterion: for a creation vector \(\xi\in e(A^+)^n\), subtracting the original lower multiplier gives the sum of the matrix compact perturbation in (AT20) multiplied by \(\xi\), and the compact commutator \([h(D),\xi]\) of Lemma0.17. The adjoint errors have the identical two terms. Finite frames identify these as rectangular compacts with range in the compressed module; norm approximation treats every creation vector. The first operator is zero, so its positivity condition is automatic. The square defect is an algebra element on that corner. These are compacts of the actual compressed module.

M/80/28, §II, Lemma 5 and Proposition 6, uses instead \(b(eHe)\), where \(b\) has winding one. Choose \(b_+(r)=(r-i)/(r+i)\). The positive-exponential unitary of the compressed course cycle is homotopic to \(b_+(eDe)\): the quotient of their scalar functions has winding zero, equals one at both ends, and has a continuous logarithm vanishing at infinity. Functional calculus gives the same relative-unitary homotopy on the compressed module. On it (AT1) gives
\[
 b_+(eHe)=b_+(-eDe)=b_+(eDe)^{-1}.
 \tag{AT21}
\]
Changing the chosen positive-winding \(b\) preserves the class by the same scalar homotopy. Forming the two projection products over \(A^+\) and subtracting the scalar reference makes the result belong to the kernel ideal \(C_\alpha\). This proves, on every even K-class with all coefficient and relative data retained,
\[
 \phi_\alpha^0=-N_\alpha^0.
 \tag{AT22}
\]

### 8E.4. The odd Thom map and typed suspension naturality

The even coefficient identification
\((SA)\rtimes_{1\otimes\alpha}\mathbb R\cong S\otimes C_\alpha\)
is coefficientwise on convolution kernels. Its essential extension modules and generators are exactly those of Lemma 0.17. The tensor product proof of Lemma 0.13 for arbitrary trivially graded external algebras retains localized compactness; it does not assert that \(k\otimes1_S\) is compact. Associativity consequently gives
\[
 S_n^{j+1}N_\alpha^j=N_{S\alpha}^{j+1}S_n^j.
 \tag{AT23}
\]
This is \(b_\tau\boxtimes(x t_\alpha)=(b_\tau\boxtimes x)(1_S\boxtimes t_\alpha)\), with the suspension coordinate first. There is no exchange of \(b_\tau\) with \(t_\alpha\). All Clifford factors are kept until the even cyclic reductions of Lemma 0.13. Thus no unrecorded interchange sign occurs in (AT23).

Combining (AT18) with (AT23) gives the *anticommuting* historical-suspension square for the native Thom action:
\[
 S_h^{j+1}N_\alpha^j=-N_{S\alpha}^{j+1}S_h^j.
 \tag{AT24}
\]
The historical axiom, in contrast, is the commuting square
\(S_h^{j+1}\phi_\alpha^j=\phi_{S\alpha}^{j+1}S_h^j\).
The preprint defines its odd map by that square. For \(j=1\), (AT22) and (AT24) give
\[
 S_h^0\phi_\alpha^1
 =-N_{S\alpha}^{0}S_h^1
 =S_h^0N_\alpha^1.
\]
Cancel the proved suspension isomorphism. The full result is
\[
 \boxed{\phi_\alpha^j=(-1)^{j+1}N_\alpha^j,\qquad j=0,1.}
 \tag{AT25}
\]
The sign is negative on even input and positive on odd input for the same implementing unitary. Extending the even scalar normalization to both degrees with one sign would be incorrect.

Every step above is natural under equivariant *-homomorphisms. In the degenerate case use the essential right ideal \(\overline{g(A)C_{\alpha'}}\); the spectral functions restrict to it, and localized defects lie in its compact ideal by Lemma 0.17. The functional-calculus suspension homotopy is sent coefficientwise, finite projection reference subtraction is natural, and product preservation uses the displayed graded module actions and inner products. Thus (AT25) is all-coefficient naturality, including degenerate maps.

For a nonseparable C*-algebra, (AT18), (AT22) and (AT25) remain statements about ordinary K-groups and the historically defined maps. Every finite projection or unitary representative, and every homotopy used to compare it, is contained in a separable action-invariant subalgebra: take the C*-algebra generated by a countable dense set of its entries and their rational translates. Point-norm continuity includes all real translates. Compact parameter images have countable dense subsets, so the same construction treats homotopies. These subalgebras form a directed cofinal family. Their real crossed products embed in the full crossed product: for the amenable real group the faithful reduced regular representation restricts faithfully, and full equals reduced. Every continuous compactly supported convolution kernel is approximated by finitely many coefficient values, so the crossed product is the directed norm closure. Continuity of stable K-theory passes the identities to the directed union. This extends the K-group assertion without inventing a countably generated KK cycle over a non-sigma-unital algebra.

### 8E.5. Ordered solvable iteration and Fourier reflection

Use the ordered real quotients and even crossed-product identifications of the fixed course, with no change of the group action. At step \(k=0,\ldots,d-1\), the input degree is \(j+k\). Multiplying the actual signs in (AT25) gives
\[
 \phi_{H,h}^j
 =(-1)^{dj+d(d+1)/2}\,\Phi_{H,n}^j
 =(-1)^{d(j+1)}s_d\,\Phi_{H,n}^j,
 \qquad s_d=(-1)^{\lfloor d/2\rfloor}.
 \tag{AT26}
\]
Here \(\Phi_{H,n}^j\) is the action of the chronological native class \(\Theta_H\). The intermediate identifications are even, so no other sign enters this iteration. Every step is the all-coefficient map of Section 8E.4, also when the intermediate coefficient is a previous crossed product.

At an abelian flat endpoint the two Fourier transforms are
\[
 (\mathcal F_D f)(p)=\int e^{-i\langle p,t\rangle}f(t)\,dt,
 \qquad
 (\mathcal F_H f)(q)=\int e^{+i\langle q,t\rangle}f(t)\,dt,
 \quad (\mathcal F_H f)(q)=(\mathcal F_D f)(-q).
 \tag{AT27}
\]
The pullback by \(q=-p\) is an onto unitary on the complete \(L^2\) coefficient module, preserves Lebesgue measure and half-densities, and transports multiplication, its adjoint, the right action and the inner product. For general coefficients tensor this actual map with the coefficient identity and the *entire* Clifford module. Each real Stone generator has (AT1), and the full resolvent functional calculi have the same transport, including on the interval modules used in the solvable deformation. This is not a claim that simultaneous inversion is a group automorphism of a nonabelian solvable group. The action and crossed product stay fixed; negative spectral coordinates and the reflected Lie frame are coordinate transports.

The full zero-endpoint generator list in the native reducer is \((2,1,4,3,\ldots[,d])\). Each completed pair is transported by the onto graded unitary in (AT9); earlier finite factors are retained. A retained last \(C_1\) is never contracted by an unsigned interchange. Lemma 8.15c proves the unreduced product connections and positivity, and its typed pair reductions give \(\Theta_0=s_dB_d\). The actual entire-parameter comparison 8.15d, in Theorem 8D.5, transports this full identity away from the flat endpoint. Formula (AT26) therefore uses the same fixed \(s_d\), with no scalar test as a substitute.

### 8E.6. An explicit historical presentation, with its full Clifford order

The historical Appendix has no KK-module definition. To encode its proved projection/unitary products faithfully, introduce a *specified translated presentation*. For parity classes of degrees \(r,s\), set
\[
 X\star Y=(-1)^{rs}XY.
 \tag{AT28}
\]
This is not a redefinition of the course product. It records an actual finite transport: in the only odd–odd case, flip the final ordered pair from second-factor-first to first-factor-first before the fixed \(\rho\)-Morita reduction. The graded *-flip is \(c\otimes d\mapsto(-1)^{|c||d|}d\otimes c\). Its pullback right action and inner product are the ones of Lemma 0.13, not an unsigned matrix permutation. On the reduced full module it is the parity-reversing onto isometry \(U\) in (AT9); parity reversal realizes the minus in (AT28). If either degree is even the reduction uses no final odd–odd pair and the transport is the identity. All retained odd coefficients stay in the indicated source or target until reduction.

The cocycle identity \(rs+(r+s)t=st+r(s+t)\) modulo two proves associativity; units and even homomorphisms are unchanged. The even cyclic three-generator unitary of Lemma 0.13 identifies the unreduced versions too. Tensoring these finite maps with a coefficient module preserves its full graded actions, inner products and compact defects. Thus (AT28) is a typed cycle presentation change, not an equality deduced merely from its K-group action.

In this presentation the real historical class is \(\widetilde t_\alpha=-t_\alpha\). Its evaluation on a degree-\(j\) scalar class is
\[
 \operatorname{Ev}_h^j(X)(x)=\sigma_{j+\deg X}\bigl(\sigma_j^{-1}(x)\star X\bigr)
 =(-1)^{j\deg X}\sigma_{j+\deg X}\bigl(\sigma_j^{-1}(x)X\bigr).
 \tag{AT29}
\]
For \(X=-t_\alpha\), this is precisely (AT25) on both degrees. The positive suspension class evaluated with \(\star\) likewise gives exactly (AT18), as the actual homotopy of §2 proves. Thus this presentation fits the *given* historical maps; it is not chosen on the basis of the scalar even axiom alone.

The historical ordered solvable class in this presentation is
\[
 \widetilde\Theta_H=(-t_1)\star\cdots\star(-t_d)
 =(-1)^d s_d\Theta_H.
 \tag{AT30}
\]
There are \(d(d-1)/2\) odd crossings before final parity reduction, whose parity is \(\lfloor d/2\rfloor\). Evaluation (AT29) gives (AT26). This is a full cycle encoding of the iterated historical action, rather than a purported single native KK class whose unaltered action gives two different signs.

### 8E.7. The entire leaf orientation under the same coordinate transport

Let \(\mathcal O_{\rm fr}\) be the *specified full* course leaf orientation, including its Hermitian line, graded spinor bundle and transition lifts. To align it with the historical positive Stone coordinates in (AT27), define \(\mathcal O_H\) by reflecting the entire ordered real leaf frame:
\[
 (X_1,\ldots,X_d)\longmapsto(-X_1,\ldots,-X_d),
 \qquad c_H(\xi)=c_{\rm fr}(-\xi).
 \tag{AT31}
\]
Transport the frame bundle, all spin-c transitions and its central line by this map. The line is kept, not replaced by its determinant or discarded. On unreduced right Clifford modules, (AT31) is coefficient transport by the involutive graded *-automorphism \(r:\varepsilon_i\mapsto-\varepsilon_i\), extended by the identity on the function and line factors. The transported module \(E_r\) has right action \(\eta\cdot_r c=\eta\cdot r(c)\) and inner product \(\langle\eta,\zeta\rangle_r=r^{-1}(\langle\eta,\zeta\rangle)\). The coefficient-extension isometry is typed
\[
 E\widehat\otimes_r B\longrightarrow E_r,
 \qquad \eta\otimes c\longmapsto\eta r(c),
 \qquad B=C_0(V)\widehat\otimes C_d.
 \tag{AT31a}
\]
Balancing uses \(r^2=1\); its inner-product equality is
\(r(\langle\eta r(c),\zeta r(d)\rangle)=c^*r(\langle\eta,\zeta\rangle)d\).
Its range is dense by the coefficient approximate identity, hence it is onto. In standard Clifford fibres the further identification \(c\mapsto r(c)\) identifies the transported multiplier with its reflected generator list; on spinor charts apply the same transported frame and transition maps. These statements concern the transported target module, not a claimed right-linear map to the untransported module. Reduce consecutive pairs only after that transport.

For even \(d\), the unitary Clifford volume is even, intertwines all generators with their negatives, and preserves the full grading. Equivalently on the final spinor bundle \(\Gamma\) conjugates the odd symbol to its negative and is right-linear because the final coefficient is trivially graded. The full central line and all transition actions commute with this global volume. The reflected and original whole analytical classes therefore agree.

For odd \(d\), the retained right \(C_1\) generator is reflected as well. On the full reduced coefficient module \(E\widehat\otimes C_1\), the transported representative has operator \(-F\). Its class is the additive inverse of \((E\widehat\otimes C_1,F)\): ordinary right multiplication by the central unitary \(\varepsilon\) is an onto right-linear isometry from \(\Pi(E\widehat\otimes C_1)\) to \(E\widehat\otimes C_1\), is even between those two gradings, commutes with \(F\) and the even left representation, and preserves inner products. It identifies the additive inverse \((\Pi E,-F)\) with \((E,-F)\). This is the full odd-module argument; \(\Gamma\) is not asserted right-linear. It also applies after any full graded Hermitian line is tensored on.

The analytic base module, its convolution action, the half-densities and their completion remain whole modules under these finite coefficient transports. A compact source cutoff localizes every transformed defect to its corresponding compact ideal; the same bounds hold along one parameter module. No compactification of \(V\) or source extension is used.

The geometric orientation is transported by the same (AT31), not chosen separately. In the actual normal-tube construction the doubled normal spinor is obtained from the complete leaf spinor and the normal Clifford frame by Lemma 8C.1/6.0j. Pull back those finite factors and each cancellation map before contracting. Their right actions, full lines and gradings are retained, while the principal graph module and variable tube stay unchanged. All creation-map compact defects and positive sandwiches are conjugated by these onto finite isometries. The actual product identity of 8D.4 and the two Bott inverse products of 6.0c therefore hold with these transported modules. Multiplication by the transported normal Bott inverse gives the reflected analytical/geometric equality on the whole noncompact base. In odd rank the above inverse is equivalently tensoring the entire comparison with the odd trivial Hermitian line, as in the full graded-line extension (D.t); its dual-line evaluations retain both inverse identities. In even rank the volume gives the intertwiner directly.

One must also retain the ambient inverse's convention. Choose the proper ambient embedding in an **even** dimension \(N\), adding a zero coordinate if necessary. The normal class has degree \(N-d\), the actual étale class degree zero, and the ambient inverse degree \(N\). Thus the historical first-factor-first product of these three classes agrees with their native product: no final odd–odd contraction is introduced between the normal and ambient factors. For even \(N\), (AT30) at the Euclidean trivial-action endpoint is \(s_N\Theta_0=B_N\), the physical Bott class. Its historical inverse is exactly the actual physical inverse \(A_N\), since the last parity is even. This verifies the three-factor realization, including its ambient normalization; the inverse is not replaced by an unspecified same-sign flat Dirac cycle.

The even ambient choice loses no original normal-tube construction. With the new normal coordinate first paired with its inverse, the actual one-coordinate stabilization inserts the external historical Bott pair \(\beta_h=-\beta_1\) and \(\alpha_h=\alpha_1\), where \(\beta_1\) and \(\alpha_1\) are the proved native real/Bott inverse pair. Their **historical** products are \(\beta_h\star\alpha_h=1\) and \(\alpha_h\star\beta_h=1\), by the odd case of (AT34). The added ambient inverse is composed in reverse factor order, so this pair is adjacent after the associating tensor unitary. The étale factor is even and its tensor extension is the actual coefficientwise product correspondence; moving the pair through it is even naturality, not an odd interchange. The full finite normal/ambient cancellation modules of Lemma6.0j are kept until this cancellation. Both one-coordinate inverse products remove the added coordinate on the whole base. This proves stabilization independence in the translated presentation, and also explains the extra ambient-inverse sign that would be required if an odd ambient rank were reduced prematurely.

Consequently, in the fixed *native* presentation,
\[
 \ell_H^{\rm geom}=\ell_H^{\rm an}
 =(-1)^d\ell_{\rm fr}=(-1)^d s_d\Theta_H
 =\widetilde\Theta_H.
 \tag{AT32}
\]
This establishes equality of actual full KK classes in this expressly transported orientation. Evaluating them through the historical first-factor-first presentation gives
\[
 \operatorname{Ev}_h^j(\ell_H^{\rm geom})
 =\operatorname{Ev}_h^j(\ell_H^{\rm an})
 =\phi_{H,h}^j
 =(-1)^{d(j+1)}(\ell_{\rm fr})_*^j,
 \qquad j=0,1.
 \tag{AT33}
\]
Every coefficient line and integral torsion class is retained. These are maps on the whole K-groups, obtained from the full-module equality, not numerical detection of that equality.

### 8E.8. Both inverse products and the precise scope of the source identification

Set \(\delta=s_dD_H\), the two-sided native inverse of \(\ell_{\rm fr}\) in Theorem 8D.5. The native inverse of \(\widetilde\Theta_H=(-1)^d\ell_{\rm fr}\) is \((-1)^d\delta\). Its inverse in the historical presentation is instead
\[
 \widetilde\Delta_H=\delta,
 \qquad
 \widetilde\Theta_H\star\widetilde\Delta_H=1_{C_0(V)},
 \qquad
 \widetilde\Delta_H\star\widetilde\Theta_H=1_A.
 \tag{AT34}
\]
Indeed both products have the additional factor \((-1)^{d^2}=(-1)^d\), which cancels the displayed \((-1)^d\) in the forward class. Sources and targets are different in the two equations and the proven native inverse identities are used in their unchanged order.

The inverse on historical output degree \(k\) is \((-1)^{kd}\delta_*^k\). Forward input \(j\) has \(k=j+d\) modulo two, so the two scalar signs multiply to
\((-1)^{d(j+1)+(j+d)d}=1\). Both group inverse identities follow in both degrees. The full compact-carrier/assembly inverse of Theorem 8D.8 can be transported by these same finite and orientation maps: its carrier parity remains \(p=j+d\); its full orientation is (AT31), and its complete dual-line evaluations are retained. This passage invokes the already proved actual intermediate noncompact composition (8D.7), not just an analytic K-isomorphism.

There is one necessary textual distinction. The survey specifies a spin-c structure, Clifford orientation and principal symbol, but does not supply numerical Pauli order or chirality phase. M/80/28 supplies the real maps and suspension, but no leaf spinor model. Thus (AT25)–(AT30) identify the **actual historical Thom and suspension maps** without an orientation ambiguity. Equations (AT31)–(AT34) provide an **explicit common full leaf realization** of the historical three-map assertion, with the required frame/chirality conversion proved. They do not attribute these unprinted matrices to the author, or claim that a numerically unspecified survey spinor frame was already the fixed course frame. A historical leaf realization supplied with an explicit full orientation is transported by (AT31) and the displayed spin-c maps; without that extra numerical choice, its identification with a particular current Pauli model is underdetermined by the printed text. The survey equality is realized with the complete orientation transport above.

### 8E.9. Exact source and proof locators

The following locators refer to the specified freely accessible editions.

* Connes, *An analogue of the Thom isomorphism for crossed products of a C*-algebra by an action of R*, IHES/M/80/28, June 1980: §II, printed 7 / PDF8, Axioms 1–3; printed 9 / PDF10, implementing \(e^{itH}\); printed 10–11 / PDF11–12, Lemma5/Proposition6 compression and bounded perturbation; printed 12–13 / PDF13–14, odd suspension construction; Appendix VI.1, printed29 / PDF30, Lemma1; VI.2, printed30–31 / PDF31–32, odd projection product and suspension first. Exact free edition: [official IHES archive PDF](https://repo-archives.ihes.fr/FONDS_IHES/I_Prepublications/CONNES/1976-1984/M_80_28/M_80_28.pdf).
* Connes, *A survey of foliations and operator algebras*, author PDF: §9, p29, full spin-c map orientation; §10(3), pp31–32, analytical leaf spinor and symbol; §11, pp33–34, geometric normal/étale/Euclidean inverse; §12, pp35–36, Theorem5 and the actual solvable graph. Exact free edition: [author PDF](https://alainconnes.org/wp-content/uploads/foliationsfine.pdf).
* Fixed lesson: Lemma0.13 (graded tensor laws and second-cycle-first reducer), Theorem0.14 (full scalar classification and relative projection), Lemma0.17 (actual real cycle and degenerate naturality), Theorem0.18/Corollary0.20 (both inverse products), Lemma8.15c/Theorem8.15d (full ordered solvable comparison), Lemma8C.1 (complete orientation), Theorems8D.5/7/8 (whole noncompact comparison, actual composition, both assembly inverses).

![Coefficient suspension, full orientation transport and both inverse products](../figures/all-coefficient-transport.png)

**Figure 8E.1.** Equations (AT4)–(AT18) and (AT25)–(AT34). The suspension defect is compact on the whole coefficient module. The projection curves sample the parameter \(\tau=0.7\); (AT16) proves joint properness on both noncompact boundaries. The full Clifford action, grading and spin-c line are transported before the inverse products. The sign table distinguishes both input degrees. [Reproducible figure source](../figures/all-coefficient-transport.py) · [Editable SVG](../figures/all-coefficient-transport.svg).

## 9. Connectedness matters for minimal diffeomorphisms


Let \(X\) be a compact connected smooth manifold of positive dimension, \(\phi\) a minimal diffeomorphism, and \(H^1(X;\mathbb Z)=0\). The projection conclusion for \(C(X)\rtimes_\phi\mathbb Z\) needs the following two ingredients:

* minimality and topological freeness imply simplicity, proved in Theorem 9.3 below;
* the normalized suspension Morita module and the bounded symbol-index comparison of Lemma 9.4 and Proposition 9.5 give integer trace on \(K_0\), proved in Theorem 9.6.

Here is the geometric computation in that trace formula. An invariant probability measure exists by weak compactness of the averages of its translates. On the suspension

\[
Y=(X\times[0,1])/\bigl((x,1)\sim(\phi(x),0)\bigr)
\]

it gives an invariant transverse measure of total mass one on \(X\times\{0\}\). For a unitary map \(u:Y\to U(n)\),

\[
\operatorname{ch}_1(u)=\frac{1}{2\pi i}\operatorname{tr}(u^{-1}du)
\]

is the integral class of \(\det u:Y\to S^1\). Because \(X\) is connected and \(H^1(X;\mathbb Z)=0\), the mapping-torus cohomology sequence gives \(H^1(Y;\mathbb Z)=\mathbb Z[dt]\). Pairing with the suspension current therefore gives an integer. The higher odd Chern components cannot pair with a one-dimensional current.

The complete proofs below establish the required trace and K-theory identifications. Every projection \(e\) in the unital crossed product then has integer trace between zero and one. Minimality makes the invariant measure have full support, and the canonical trace is faithful. Hence trace zero gives \(e=0\), and trace one gives \(1-e=0\). Thus its only projections are zero and one. Positive dimension and connectedness also make the minimal homeomorphism aperiodic, giving topological freeness.

The hypotheses are indispensable.

**Example 9.1 (two components).** On \(X=\{0,1\}\), let \(\phi\) interchange the two points. This is a minimal diffeomorphism of a zero-dimensional compact manifold and \(H^1(X;\mathbb Z)=0\). Nevertheless

\[
C(X)\rtimes_\phi\mathbb Z\cong M_2(C(S^1)).
\]

One explicit representation sends the two point projections to the two diagonal matrix units and sends the implementing unitary to

\[
U(z)=\begin{pmatrix}0&z\\1&0\end{pmatrix}.
\]

It satisfies the covariance relation and \(U(z)^2=zI\); the matrix units and scalar functions generated this way give the displayed algebra. Its point projection is nontrivial and has trace \(1/2\). The same finite-orbit construction works with any number of components.

*Reference:* [Connes, the minimal-diffeomorphism corollary] states the conclusion without an explicit connectedness condition. Example 9.1 shows why connectedness and the infinite-space hypothesis must be stated when disconnected or zero-dimensional manifolds are allowed.

### Faithful traces and simplicity

The first analytic premise of this section can be proved directly. Let \(X\) be compact metrizable, let \(\phi\) be a homeomorphism, and put
\(\alpha_k(f)=f\circ\phi^{-k}\).
Write \(B=C(X)\rtimes_\alpha\mathbb Z\). Lemma 8.3 makes its full and reduced meanings agree.

**Proposition 9.2 (expectation and invariant traces).** The zero Fourier coefficient extends to a positive contractive expectation
\[
\mathsf E:B\longrightarrow C(X),\qquad
\mathsf E\!\left(\sum_k f_kU^k\right)=f_0.
\tag{38}
\]
It is faithful on positive elements. An invariant probability measure \(\nu\) gives a tracial state
\(\tau_\nu=\nu\circ\mathsf E\), faithful when \(\nu\) has full support.

**Proof.** In (26) for counting measure, compress to the coordinate zero. This gives (38) on finite polynomials, and the contraction extends since \(C(X)\) is closed. Compression also proves positivity, unitality and the bimodule rule. Compression at coordinate \(m\) is \(\alpha_{-m}\mathsf E\). If \(\mathsf E(a^*a)=0\), the represented \(a\) vanishes on every coordinate subspace, hence on their dense span; so \(a=0\). Apply this to the positive square root to obtain positive-element faithfulness.

For two finite polynomials, invariance and commutativity of \(C(X)\) give
\[
\int f_k\,\alpha_k(g_{-k})\,d\nu
=\int g_{-k}\,\alpha_{-k}(f_k)\,d\nu.
\]
Summing proves \(\tau_\nu(ab)=\tau_\nu(ba)\); norm approximation extends it to \(B\). If a continuous nonnegative function has zero integral against a full-support measure, it vanishes: a positive value would give an open set of positive measure on which it has a positive lower bound. Combine this with expectation faithfulness. \(\square\)

**Theorem 9.3 (the ideal intersection and simplicity).** If \(\phi\) is topologically free, every nonzero ideal of \(B\) meets \(C(X)\) nontrivially. If it is also minimal, \(B\) is simple.

**Proof.** Suppose a quotient \(q:B\to B/J\) is injective on \(C(X)\), hence isometric there. For \(a\ge0\), choose a finite Fourier polynomial \(p\) with \(\|a-p\|<\delta\). Topological freeness means that the fixed-point sets of the finitely many nonzero powers in \(p\) have empty interior. Their complement is dense. Choose a point \(x\) in it with
\(\mathsf E(a)(x)>\|\mathsf E(a)\|-\delta\).
There is an open neighbourhood \(W\) of \(x\) disjoint from its translates by those powers. Take \(0\le h\le1\), supported in \(W\), with \(h(x)=1\). Then
\[
hph=h\mathsf E(p)h,\qquad
\|q(a)\|\ge\|q(hah)\|
\ge\|h\mathsf E(p)h\|-\delta
\ge\|\mathsf E(a)\|-3\delta.
\tag{39}
\]
Let \(\delta\downarrow0\). A positive \(a\in J\) has \(\mathsf E(a)=0\), so Proposition 9.2 gives \(a=0\). Every element of \(J\) then vanishes by applying this to its square. Thus a nonzero ideal cannot have zero intersection.

The intersection \(J\cap C(X)\) is invariant, since conjugation by \(U\) preserves \(J\). Its common zero set is a closed invariant subset. Minimality permits only the empty set and \(X\); a nonzero intersection excludes \(X\), so that ideal is all of \(C(X)\). It contains the unit, and hence \(J=B\). \(\square\)

For a minimal homeomorphism of an infinite compact space, every point is aperiodic: a periodic orbit is finite, closed and invariant, and cannot be dense in an infinite space. This applies in particular to the connected positive-dimensional manifold of this section. An invariant probability measure exists by taking a weak-* limit of the orbit averages of a point mass; translating an average changes its value on a bounded test function by at most \(2\|f\|_\infty/n\). Its support is nonempty, closed and invariant, hence all of \(X\) by minimality. Thus the algebra in the corrected projection assertion is now proved simple and has faithful normalized invariant traces. Lemma 9.4 and Proposition 9.5 supply the remaining suspension and symbol-index comparisons; Theorem 9.6 proves the integer trace range. Theorems 8.7 and 9.3 alone would not give it.

### The suspension module and its trace scale

The Morita identification needed above has a particularly explicit model. Retain \(\alpha_k(f)=f\circ\phi^{-k}\), write \(U\) for the unitary of \(B=C(X)\rtimes_\alpha\mathbb Z\), and use the suspension convention already fixed in this section. A function on \(Y\) is equivalently a continuous \(C(X)\)-valued function \(F\) on \(\mathbb R\) with
\[
F(t+1)=\alpha_{-1}(F(t)).
\]
The positive suspension flow induces \((\gamma_sF)(t)=F(t-s)\).

**Lemma 9.4 (suspension Morita and normalized trace).** For a compact metrizable \(X\) and a homeomorphism \(\phi\), put \(A_Y=C(Y)\rtimes_\gamma\mathbb R\). There is an isomorphism
\[
A_Y\cong\mathcal K_B(\mathcal E),\qquad
\mathcal E=L^2(0,1)\otimes B,\qquad
\mathcal K_B(\mathcal E)\cong\mathcal K(L^2(0,1))\otimes B.
\tag{43}
\]
Thus \(\mathcal E\) is a full imprimitivity module. If \(X\) is a compact smooth manifold, \(\phi\) is a smooth aperiodic diffeomorphism, and \(\nu\) is an invariant probability measure, the measured trace of the suspension foliation becomes \(\operatorname{Tr}\otimes\tau_\nu\) under (43). A normalized rank-one band projection has trace one. The resulting stable K-theory identification preserves these trace pairings.

**Proof.** On the dense submodule of bounded simple \(B\)-valued functions on the band, extend a vector to the real line by
\[
\xi(t+m)=U^{-m}\xi(t),\quad 0<t<1,\ m\in\mathbb Z,
\qquad
\langle\xi,\eta\rangle_B=\int_0^1\xi(t)^*\eta(t)\,dt.
\]
These formulas are used on this dense domain before completion; a general completed module element is not declared point evaluable. Multiplication \(M_F\xi(t)=F(t)\xi(t)\) respects the extension relation, because \(F(t+1)=U^{-1}F(t)U\). It is adjointable, with adjoint \(M_{F^*}\) and norm at most \(\|F\|_\infty\). Translation \(T_s\xi(t)=\xi(t-s)\) is an adjointable unitary: the inner-product integrand is one-periodic, so translating the integration band leaves the inner product unchanged. Strong continuity follows first for continuous functions supported in the interior of the band by ordinary translation continuity, then for the completion by density and the isometry bound. We have \(T_sM_FT_s^*=M_{\gamma_sF}\).

Integrating this covariant pair gives a homomorphism \(A_Y\to\mathcal L_B(\mathcal E)\). For \(a\in C_c(\mathbb R,C(Y))\), write its coefficient as \(a(s,t)\in C(X)\). Splitting \(r=t-s\) into the bands \(r=u+m\) gives the \(B\)-valued integral kernel
\[
K_a(t,u)=\sum_{m\in\mathbb Z}a(t-u-m,t)U^{-m},
\qquad 0\le t,u\le1.
\tag{44}
\]
There are uniformly finitely many terms, because \(a\) has compact group support. This kernel is continuous on the closed square. Product partitions approximate it uniformly by finite sums of one-variable kernels with coefficients in \(B\). The operator error is at most its uniform kernel error, since the band has length one. Its operator is therefore compact on \(\mathcal E\).

Conversely let \(g,h\in C_c(0,1)\), \(f\in C(X)\), and \(k\in\mathbb Z\). The compact kernel \(g(t)\overline{h(u)}fU^k\) is obtained by defining, on the fundamental band,
\[
a(s,t)=g(t)\overline{h(t-s+k)}f
\]
and extending by \(a(s,t+n)=\alpha_{-n}(a(s,t))\). Vanishing near the band endpoints makes this extension continuous; its group support lies in a fixed compact interval near \(k\). In (44) only \(m=-k\) contributes. Such kernels span a dense subalgebra of \(\mathcal K_B(\mathcal E)\), because Fourier polynomials are dense in \(B\) and interior band functions are dense in \(L^2(0,1)\). The image is consequently all the compact operators.

We check injectivity as well. Let \(\sigma\) be the direct sum of point evaluations of \(C(X)\), and let \(\rho\) be its regular representation of \(B\) on \(\ell^2(\mathbb Z)\otimes\mathcal H_\sigma\). It is faithful by Lemma 8.3. After tensoring \(\mathcal E\) with \(\rho\), unfold the band by
\[
\zeta(t+j)=\eta(t,j),\qquad 0<t<1,\quad j\in\mathbb Z.
\]
The extension relation becomes \(\eta(t+1,j)=\eta(t,j+1)\). The coefficient at coordinate \(j\) is \(\sigma(\alpha_{-j}F(t))=\sigma(F(t+j))\); translation becomes \(\zeta(r)\mapsto\zeta(r-s)\). Thus the unfolded representation is the direct sum of the real regular representations induced at the transversal points \([x,0]\). A regular representation induced at \([x,v]\) is unitarily equivalent to the one at \([x,0]\), by translating its real coordinate by \(v\). Every point of \(Y\) has this form, so this sum computes the reduced norm. Lemma 8.3 identifies that norm with the full norm. The module homomorphism is faithful and (43) follows.

The right module is full: for an interior band function \(g\) of \(L^2\) norm one, the vector \(g\otimes1\) has inner product one. The left inner product is the inverse image of \(\theta_{\xi,\eta}\) under (43). Its positivity and fullness follow from the compact-operator isomorphism, and its compatibility is the explicit identity
\(\theta_{\xi,\eta}\zeta=\xi\langle\eta,\zeta\rangle_B\).
This supplies both inner products and the imprimitivity identities.

For the trace assertion use the leaf density \(dt\) and the transversal measure \(\nu\). Their unit measure on the fundamental band is \(d\nu\,dt\); invariance makes it descend across the seam. Aperiodicity makes each leaf a line, so its holonomy groupoid is precisely the free suspension transformation groupoid. The diagonal formula from the [transverse-measure lesson, Theorem 7.10](transverse-measures-of-foliations.md#section-7) gives, on smooth compact kernels,
\[
\tau_Y(a)=\int_0^1\int_X a(0,t)(x)\,d\nu(x)\,dt.
\tag{45}
\]
Choose a complete orthonormal sequence \(g_i\in C_c^\infty(0,1)\); Gram–Schmidt applied to a countable dense set supplies one. For a matrix corner kernel \(g_i(t)\overline{g_j(u)}fU^k\), its diagonal coefficient in (45) is zero unless \(k=0\), and its integral is then \(\delta_{ij}\int_X f\,d\nu\). This is exactly the matrix trace combined with \(\tau_\nu\). The corner projection \(q_n=\sum_{i=1}^n\theta_{g_i\otimes1,g_i\otimes1}\) has measured trace \(n\), so the measured trace is bounded on that corner. Smooth coefficients and finite Fourier polynomials are norm dense there; equality therefore holds throughout every \(M_n(B)\) corner.

The \(q_n\)'s are an increasing approximate identity for the compact algebra. For \(c\ge0\), the positive elements \(c^{1/2}q_nc^{1/2}\) increase to \(c\) in norm. Lower semicontinuity and the trace identity give
\[
\tau_Y(c)=\lim_n\tau_Y(q_ncq_n).
\]
The same statement holds for \(\operatorname{Tr}\otimes\tau_\nu\), so corner equality proves equality of the entire weights, including infinite values. This argument fixes the scale without assuming uniqueness merely from agreement on a dense set. In particular \(q_1\) has trace one. Finite matrix corners give the stable K-group identification of (43); on projections their traces agree, hence the K-theory trace pairings agree. \(\square\)

### A bounded symbol-index comparison for the suspension flow

A full general assembly theorem is unnecessary for the particular projection assertion. We can prove that every suspension K-class is an elliptic symbol index by a real-action deformation and an explicit operator.

**Proposition 9.5 (the suspension symbol index is onto).** For the smooth aperiodic suspension of Lemma 9.4, the longitudinal symbol index
\[
J:K^1(Y)\longrightarrow K_0(A_Y)
\]
is an isomorphism. It sends a smooth matrix unitary \(w\) to the index of an order-zero elliptic operator whose symbol is \(1\) on the negative cotangent ray and \(w\) on the positive ray.

**Proof.** Put \(B_0=C(Y)\) and \(\alpha_s=\gamma_s\). The proof first uses only the real action. Its canonical group generator is normalized by \(U_s=e^{-isD}\); in the physical regular representation \(D=-i\,d/dt\), using the negative-sign Fourier transform of Corollary 8.10. The homomorphism \(C^*(\mathbb R)=C_0(\mathbb R)\to M(B_0\rtimes\mathbb R)\) is nondegenerate. Since \(B_0\) is unital, every scalar \(C_0\) group function, including the resolvents of \(D\), belongs to the crossed product itself. Define
\[
\chi(p)=\frac12\left(1+\frac{p}{\sqrt{1+p^2}}\right),
\qquad
P(w)=w\chi(D)+1-\chi(D).
\tag{46}
\]
The multiplier \(\chi(D)\) is a projection modulo the crossed product, since \(\chi^2-\chi\in C_0(\mathbb R)\).

It commutes with every coefficient modulo that algebra. Here is the required compactness proof. For a coefficient smooth along the flow, write \(\delta b=(d/ds)_{s=0}\alpha_s(b)\). Differentiating covariance on the generator's dense smooth core gives \([D,b]=i\delta b\); the bounded derivative and closedness extend this identity to its domain. With \(a_r=(1+r^2)^{1/2}\),
\[
\frac{D}{D^2+1+r^2}
=\frac12\bigl((D-ia_r)^{-1}+(D+ia_r)^{-1}\bigr),
\qquad
\left\|\left[\frac{D}{D^2+1+r^2},b\right]\right\|
\le\frac{\|\delta b\|}{1+r^2}.
\tag{47}
\]
The resolvent commutator identity proves the bound and places each commutator in the crossed product. The scalar formula
\[
\frac{p}{\sqrt{1+p^2}}
=\frac2\pi\int_0^\infty\frac{p}{p^2+1+r^2}\,dr
\]
holds by elementary integration. Its truncated functions are uniformly bounded and converge uniformly on compact \(p\)-sets, so their functional calculus converges strictly. Their commutators converge in norm by (47). Hence \([\chi(D),b]\) belongs to the algebra. Smooth coefficients are norm dense, by flow convolution or smooth approximation on \(Y\), and the commutator map is norm bounded; the assertion follows for every \(b\in B_0\).

Consequently \(P(w)\) is unitary modulo the crossed product whenever \(w\) is a matrix unitary. Its multiplier-extension index defines \(J_\alpha([w])\). Modulo the algebra, \(P(w_1w_2)=P(w_1)P(w_2)\); block sums and norm-continuous homotopies are also respected. Thus this is a well-defined homomorphism on \(K_1(B_0)\), using the kernel-minus-cokernel boundary convention fixed above.

To show it is an isomorphism, put
\[
\mathcal D=C([0,1],B_0)\rtimes_\beta\mathbb R,
\qquad
(\beta_s b)(v)=\alpha_{vs}(b(v)).
\]
This action is point-norm continuous, by uniform continuity on the compact \(v\)-interval. The equivariant coefficient evaluations induce maps
\(\mathrm{ev}_v:\mathcal D\to B_0\rtimes_{\alpha^{(v)}}\mathbb R\).
Their coefficient K-maps are isomorphisms, since the interval is contractible. The natural analytic Thom isomorphism of Theorem 8.7 therefore makes their crossed-product K-maps isomorphisms as well. This conclusion uses that theorem's already proved naturality; it does not infer K-theory constancy from an unspecified continuous field.

Use the same \(w\) and \(\chi(D)\) in \(M(\mathcal D)\). For smooth \(w\), its flow derivative at parameter \(v\) is \(v\delta w\), bounded uniformly in \(v\); (47) consequently proves the commutator lies in \(\mathcal D\). The resulting index class evaluates to \(J_{\alpha^{(v)}}([w])\). Norm approximation covers all classes. At \(v=0\) the algebra is \(C_0(\mathbb R,B_0)\), and \(P(w)(p)\) has endpoint values \((1,w)\). Its index is the boundary of that pair in
\[
0\longrightarrow C_0(\mathbb R,B_0)
\longrightarrow C([-\infty,+\infty],B_0)
\longrightarrow B_0\oplus B_0\longrightarrow0.
\]
The interval algebra has the K-groups of \(B_0\), and its endpoint maps are diagonal. Six-term exactness shows that the boundary restricted to \((0,[w])\) is an isomorphism from \(K_1(B_0)\) to \(K_0(C_0(\mathbb R,B_0))\): the diagonal has zero kernel, and each class modulo the diagonal has a unique representative of that form. The natural inclusion of the interval algebra into the multiplier algebra identifies this boundary with the index of \(P(w)\). Thus \(J_{\alpha^{(0)}}\) is an isomorphism. Evaluation now gives
\[
J_\alpha=(\mathrm{ev}_1)_*(\mathrm{ev}_0)_*^{-1}
             J_{\alpha^{(0)}},
\tag{48}
\]
so it is an isomorphism too. No geometric factorization independence or general assembly equality has been used.

At \(v=1\), \(D\) differentiates along the unit-speed suspension leaves. The smooth function \(\chi\) is a classical symbol of order zero, with complete expansions in even negative powers and principal values zero and one on the two rays. Its inverse Fourier distribution is smooth away from zero and rapidly decreasing there: repeated integration by parts uses the integrability of every positive-order derivative of \(\chi\); the same argument after multiplication by powers of \(p\) controls derivatives away from zero. Cut off that kernel outside a compact time interval, leaving it unchanged near zero. Its removed tail is an \(L^1\) scalar group kernel, and hence belongs to \(A_Y\). The cut-off version of \(P(w)\) is therefore a compactly supported longitudinal order-zero operator with the same index and principal symbol. Its symbol is \(1\) on the negative ray and \(w\) on the positive ray. The [Hilbert-module lesson, Section 6](hilbert-modules-and-fields-on-the-leaf-space.md#section-6) identifies this Fredholm boundary with its full relative-symbol index. Smooth matrix unitaries represent every \(K^1(Y)\) class: approximate continuously in norm by smooth matrices, then take the polar part, preserving the unitary homotopy class. This proves the stated longitudinal assertion. \(\square\)

**Theorem 9.6 (integer trace and the corrected corollary).** Let \(X\) be a compact connected smooth manifold of positive dimension, let \(\phi\) be a minimal diffeomorphism, and assume \(H^1(X;\mathbb Z)=0\). Then \(B=C(X)\rtimes_\alpha\mathbb Z\) is simple and unital, and has no idempotents other than \(0\) and \(1\). Every invariant probability measure gives
\[
\tau_{\nu*}(K_0(B))=\mathbb Z.
\tag{49}
\]

**Proof.** Minimality gives aperiodicity and a full-support invariant probability measure as proved after Theorem 9.3. Lemma 9.4 identifies \(K_0(B)\) with \(K_0(A_Y)\), preserving its trace and normalizing a rank-one band projection to one. Proposition 9.5 says every such class is a longitudinal symbol index.

For a rank-one oriented foliation the Todd factor is one. The [measured index theorem, Theorem 6.37](the-index-theorem-for-measured-foliations.md#section-6) therefore pairs the suspension current with the degree-one component of the symbol's fibre-integrated Chern character, with the fixed cotangent sign \((-1)^{1(1+1)/2}=-1\). That component is integral: the degree-two Chern character of a relative complex K-class is its relative first Chern class, and oriented integration along the real cotangent fibre sends this integral class to \(H^1(Y;\mathbb Z)\). This last assertion also follows directly from the one-dimensional cohomological Thom construction in Lemma 6.31 of that lesson. Higher components cannot pair with the one-dimensional current. Thus it suffices to compute the pairing of every integral first cohomology class on \(Y\).

Integral first classes are represented by maps to \(S^1\). For example, an integer cellular one-cocycle defines such a map on the one-skeleton; its cocycle condition fills the two-cells, and the vanishing higher homotopy groups of \(S^1\) fill the remaining cells. Given \(g:Y\to S^1\), pull it back to \(X\times[0,1]\). Its restriction at zero has a real lift, since \(H^1(X;\mathbb Z)=0\). Homotopy lifting for the covering \(\mathbb R\to S^1\) extends that lift to a function \(\theta(x,t)\) on the entire band. The seam relation implies that
\[
\theta(x,1)-\theta(\phi(x),0)=n
\]
is integer valued. It is continuous, so connectedness makes it one constant integer \(n\). The function \(\theta(x,t)-nt\) descends to a real function on \(Y\). Consequently \([g]=n[dt]\). Smooth representatives have smooth lifts; continuous representatives give the same cohomology class.

The normalized suspension current pairs with \(dt\) to one. It kills exact forms: for a smooth real function \(h\) on \(Y\), the band integral of its leaf derivative is
\[
\int_X(h(x,1)-h(x,0))\,d\nu(x)
=\int_X(h(\phi(x),0)-h(x,0))\,d\nu(x)=0.
\]
Thus every integral first class has integer current pairing. The preceding symbol-index argument gives \(\tau_{\nu*}(K_0(B))\subset\mathbb Z\). The unit of \(B\) has trace one, so equality follows.

For a projection \(e\in B\), its trace is an integer between zero and one. Faithfulness from Proposition 9.2 gives \(e=0\) when the trace is zero and \(1-e=0\) when it is one. Simplicity is Theorem 9.3, and unitality follows from the unital discrete coefficient algebra.

This also excludes nonselfadjoint idempotents. If \(z^2=z\), its represented range is closed. On that range \(zz^*\ge1\), and on its orthogonal complement \(zz^*=0\): for a range vector \(v\), the equality \(zv=v\) and Cauchy–Schwarz give \(\|z^*v\|\ge\|v\|\). Functional calculus therefore gives its range projection \(p\in C^*(z,1)\subset B\), since \(\operatorname{spec}(zz^*)\subset\{0\}\cup[1,\|z\|^2]\). We have \(pz=z\) and \(zp=p\). As \(p=0\) or \(1\), these identities give \(z=0\) or \(1\), respectively. \(\square\)

This resolves the source assertion with its scope correction. A disconnected compact manifold has a proper nonempty clopen union of components; its characteristic function is a nontrivial coefficient projection, and \(C(X)\) embeds faithfully in the crossed product. Minimality does not remove it. A connected zero-dimensional compact manifold is a single point: its crossed product is \(C(S^1)\), which has only the trivial idempotents but is not simple. Thus connectedness and the infinite-space condition are required for the source's combined simplicity and idempotent conclusion. Example 9.1 exhibits both failures in the unrestricted statement.

![Unit-time suspension band, exact Morita trace scale, real-action symbol-index deformation and integral trace range](../figures/suspension-trace.png)

**Figure 9.7.** The band has the exact coordinate \(0<t<1\) and seam \([x,1]=[\phi(x),0]\). The vector extension is defined first on the dense function model; it imposes no point values on arbitrary completed module elements. Panel B uses the full module of Lemma 9.4, both inner products, counting powers \(U^{-m}\), leaf density \(dt\), invariant probability \(\nu\), and a unit \(L^2\) vector \(g\). Its trace scale is one. Panel C uses \(v\in[0,1]\), the negative physical Fourier sign and the action \(\beta_s=\alpha_{vs}\); the evaluation K-maps are isomorphisms by the proved analytic Thom naturality, not by a continuous-field assumption. The compactly supported symbol at \(v=1\) is \(1\) on the negative cotangent ray and \(w\) on the positive ray. Panel D assumes a connected positive-dimensional compact smooth \(X\), a minimal diffeomorphism and \(H^1(X;\mathbb Z)=0\). Its current has \(\langle dt,C_\nu\rangle=1\), and its cotangent prefactor is \(-1\). These are exact proof maps and normalizations; general assembly comparisons remain separate. Full proofs are Lemma 9.4, Proposition 9.5 and Theorem 9.6, with Exercises 11–13.

Full-size PNG · Editable SVG.


## 10. A Kasparov cycle for hyperbolic-plane leaves

Assume from now on that \(V\) is compact, \(F\) is oriented of rank two, and every leaf with its smooth leafwise metric is isometric to the complete hyperbolic plane of curvature \(-1\). Each leaf is simply connected, so \(G_x\) is the leaf itself. The exponential map is a global diffeomorphism on each fibre. To verify this directly, use the Lorentz model \(x_0^2-x_1^2-x_2^2=1\), \(x_0>0\), with base point \(o=(1,0,0)\). If \(r=|v|\), then \(\exp_o(v)=(\cosh r,(\sinh r)v/r)\); its inverse at \((x_0,z)\) is \((\operatorname{arsinh}|z|)z/|z|\). Both quotients have smooth limits at zero, and the radial and angular derivatives are invertible everywhere. Isometric transport gives the same conclusion at every base point. Smooth dependence of the leafwise geodesic equation on its initial conditions makes the total map smooth. Its nonzero fibre derivative and identity base derivative make it a local diffeomorphism, so its fibrewise bijectivity gives a global diffeomorphism

\[
F\longrightarrow G,\qquad
v\in F_x\longmapsto(x,\exp_xv).
\tag{21}
\]

It is bijective by the hyperbolic exponential theorem, and is a local diffeomorphism in the arrow charts because the leafwise exponential has no critical points and preserves the source coordinate. A bijective local diffeomorphism is a diffeomorphism. In particular \(G\) is Hausdorff in this situation.

Let \(S=S^+\oplus S^-\) be the spinor bundle of the oriented metric two-plane bundle \(F\), with its canonical spin-c structure. Use the positive Clifford convention \(c(v)^*c(v)=|v|^2\) between \(S^+\) and \(S^-\). Form the Hilbert \(C(V)\)-module with fibres

\[
\mathcal H_x=L^2(G_x)\otimes S_x.
\]

Under (21), local orthonormal frames identify these fibres with a fixed radial \(L^2\) space: the exponential volume density is \(\sinh r/r\) times Euclidean volume. Transition frames act unitarily. All continuous sections of the displayed Hilbert field over compact \(V\) give the resulting countably generated continuous Hilbert module.

The regular representations give a graded representation \(\pi:A\to\mathcal L_{C(V)}(\mathcal H)\). Define the odd self-adjoint multiplier

\[
B_x(v)=
\begin{pmatrix}
0&c(b(v))^*\\ c(b(v))&0
\end{pmatrix},
\qquad
b(v)=\frac{v}{\max(1,|v|)}.
\tag{22}
\]

It is bounded and continuous, including at zero and at the unit circle.

**Theorem 10.1 (hyperbolic radial cycle).** The triple \((\mathcal H,\pi,B)\) is an even Kasparov cycle from \(A\) to \(C(V)\). It defines

\[
\delta\in KK(A,C(V)).
\]

**Proof.** First take a smooth compactly supported convolution kernel \(f\). Its support gives a finite propagation bound \(C\): its represented kernel can be nonzero only between points of a leaf at distance at most \(C\). The leafwise Schur integral bounds are uniform over the compact base.

The function \(B^2-1\) is zero for \(|v|\ge1\). Multiplying the smoothing kernel of \(\pi(f)\) by this defect leaves a kernel supported, uniformly over \(V\), where one endpoint has radius at most one and the other at most \(1+C\). This is a compact subset of the two fibre variables in the vector bundle coordinates (21). Such a continuous smoothing family is a compact \(C(V)\)-module operator. To see the last assertion, cover the compact base by finitely many frame domains, use a partition of unity, and approximate the continuous compactly supported kernel on each domain uniformly by finite sums \(a(x,v)\overline{b(x,w)}\). The compact volume bound converts this to operator-norm approximation. Each sum is a finite-rank module map.

For the commutator, let \(r,s\) be the distances of its two endpoints from the base point \(x\), and \(\theta\) their angle at \(x\). Hyperbolic cosine law gives

\[
\cosh d
=\cosh(r-s)+\sinh r\,\sinh s(1-\cos\theta).
\]

If \(d\le C\) and \(r,s\ge R>1\), it follows that

\[
\left\|\frac v{|v|}-\frac w{|w|}\right\|
=\sqrt{2(1-\cos\theta)}
\le\frac{\sqrt{2(\cosh C-1)}}{\sinh R}.
\tag{23}
\]

The estimate is uniform in \(x\). A term of the commutator kernel is the original kernel times \(B_x(v)-B_x(w)\). Where one endpoint has radius at least \(R+C\), finite propagation makes both radii at least \(R\), so (23) applies. The Schur estimate bounds the norm of this tail by a constant times \(1/\sinh R\), tending to zero. Cutting both radii off at \(R+C\) leaves a continuous compactly supported kernel, compact by the preceding finite-rank argument. The commutator is consequently a norm limit of compact module operators.

Finally \(B=B^*\). Smooth compact kernels are dense in \(A\), and the three defect expressions depend continuously on \(f\) in operator norm. Their compactness extends to every \(a\in A\). This proves all Kasparov conditions. \(\square\)

The uniform statement over \(V\) is essential. Compactness of the represented operator on each individual leaf would not by itself prove compactness of a \(C(V)\)-module operator.

## 11. What the deformation must establish

The transverse fundamental-class question has a distinction between full and reduced algebras. Hilsum and Skandalis construct the K-oriented projection class on the maximal foliation algebra; their original paper explicitly distinguishes this algebra from Connes's reduced algebra (pp. 326–327, with the module and submersion construction in Section IV, pp. 350–355). Thus a general maximal transverse class is established. [Primary paper](https://www.numdam.org/article/ASENS_1987_4_20_3_325_0.pdf#page=3).

Connes and Moscovici's hypoelliptic construction is on the bundle of transverse metrics. They distinguish this type-II counterpart from the original transverse algebra on preprint p. 12. Theorem I.1 on p. 16 gives geometric bounded-commutator and localized-resolvent assertions; it does not assert a reduced-norm estimate there. [Primary preprint](https://repo-archives.ihes.fr/FONDS_IHES/I_Prepublications/CONNES/1994-1998/M_95_19/M_95_19.pdf#page=13).

The remaining construction in this lesson is the specifically reduced scalar graph representative with the full invariant transverse spin-c coefficient and original local Bott pairing \(+1\). Neither the maximal construction nor the metric-bundle spectral triple alone supplies every interface of that representative. The results below do not prove universal nonexistence, and do not establish that the entire general transverse-class construction is an unresolved problem in the current literature.

The radial cycle points toward an inverse for the longitudinal Dirac class. Its existence does not yet select the inverse normalization: the full spinor module, chirality, line twist and physical Fourier phase must be calibrated together. Using the same positive spinor convention on both sides can retain a grading-reversed line instead of the identity. Let \(T\in KK(C(V),A)\) denote the longitudinal class with its specified even-dimensional spin-c orientation, and let \(\delta_{\mathrm{inv}}\in KK(A,C(V))\) denote the correctly calibrated inverse class of Theorems 11.1 and 11B.1. It is not automatically the particular radial realization \(\delta\) of Theorem 10.1. The desired typed identities are

\[
T\otimes_A\delta_{\mathrm{inv}}=1_{C(V)},\qquad
\delta_{\mathrm{inv}}\otimes_{C(V)}T=1_A.
\tag{24}
\]

The first makes \((\delta_{\mathrm{inv}})_*\) a **left inverse** of \(T_*\); it proves injectivity of \(T_*\). The second identity, proved in Section 11B, gives surjectivity.

The geometric deformation uses \(G_\lambda=G\) for \(\lambda>0\) and the additive bundle \(G_0=F\). In coordinates \(v\in F_x\), its maps are

\[
s_\lambda(v)=x,\qquad
r_\lambda(v)=\exp_x(\lambda v).
\]

At zero, range and source both become \(x\). Smoothness of the exponential and its derivative at zero make multiplication tend to vector addition and inversion tend to \(v\mapsto-v\), uniformly on compact vector sets. With the scaled Haar densities, Fourier transform at zero identifies

\[
A_0=C_0(F^*),\qquad A_1=A.
\]

To obtain the first identity in (24), one needs longitudinal Dirac and radial cycles **over the whole parameter interval**, and a product whose represented target is a Hilbert \(C(V\times[0,1])\)-module. At zero their product must be the full normalized Bott–Dirac product, with the inverse finite Clifford module and its exact line evaluation retained. Its endpoints are then homotopic classes in \(KK(C(V),C(V))\), giving the first identity.

A continuous field of algebras by itself does not identify the KK groups of its fibres. The existence, uniform compactness, and compatibility of these interval cycles and their product are extra assertions. Theorem 10.1 supplies a radial cycle at one; it does not identify its inverse normalization or prove those interval and product assertions. Likewise the first product in (24) does not imply the second. This distinguishes the explicit radial construction from the stronger assembly-isomorphism theorem proved for these disk leaves in Section 11B.

### A calibrated first inverse and the reverse product

Keep the hypotheses of Section 10: \(V\) is compact without boundary, \(F\) is an oriented real bundle of rank two, and the actual leaves, with their smooth leafwise metric, are complete Poincaré disks of curvature \(-1\). Put \(C=C(V)\) and \(A=C_r^*(G)\), where \(G=\operatorname{Hol}(V,F)\). Prescribe the entire longitudinal spin-c structure, including its Hermitian line transitions. Let

\[
 T=\ell_!^{\mathrm{an}}\in KK(C,A)
 \tag{H.1}
\]

be the native class of Theorem 7.1, after the fixed rank-two matrix contraction. Its physical Fourier symbol is the **positive** unit-covector symbol, and its grading is (145)/(244). If the geometric graph construction is used instead, Theorem 7.25 identifies it with this class only for the same entire induced stable orientation.

The radial cycle of Theorem 10.1 is a useful construction, but an inverse requires its whole finite Clifford module, grading, line and phase to be calibrated. We now construct that calibrated class \(\delta_{\mathrm{inv}}\). The two desired identities in (24) have different targets.



**Theorem 11.1 (the calibrated first inverse).** In the stated actual disk-leaf scope, the finite module and radial phase of Lemma 11.2 define a class \(\delta_{\mathrm{inv}}\in KK(A,C)\) satisfying the first identity in (24). Consequently \((\delta_{\mathrm{inv}})_*\) is a left inverse of \(T_*\), so \(T_*\) is injective in both K-degrees. The theorem does not assert the second identity or identify an unspecified positive-spinor realization of \(\delta\) with this calibrated class.

The proof is given after the interval constructions. The deformation idea is due to Connes and Skandalis: [Connes], Section 12, printed pp. 613–615. The reverse identity is proved separately in Section 11B; its distinction from the first product is explained in Remark 11.6. The full Clifford Bott–Dirac product used at the zero endpoint is the framework of [Kasparov], with the programme's complete bundle proof in Theorem 6.0c.

#### The entire finite inverse module

**Lemma 11.2 (full calibration, chirality and line).** Let \(P=\operatorname{Cl}^+(F)\), and let \(S\) be the prescribed graded \(P,C\) spinor imprimitivity bundle. In an oriented orthonormal frame use

\[
 \varepsilon_1=\sigma_x,\qquad
 \varepsilon_2=\sigma_y,\qquad
 \Gamma_S=-i\varepsilon_1\varepsilon_2=\sigma_z.
 \tag{H.3}
\]

On \(Z=\Lambda F_{\mathbb C}\), write \(a_j=e_j+\iota_j\), \(c_j=e_j-\iota_j\), and \(b_j=ic_j\), with exterior-degree parity. Then the graded balanced module

\[
 W=\overline S\widehat\otimes_P Z
 \tag{H.4}
\]

has physical generators \(b_W=(\sigma_y,\sigma_x)\) and grading \(\Gamma_W=\sigma_z\). Its ordered physical chirality is \(-\Gamma_W\). For the canonical spin-c convention \(S=1\oplus L\), where \(L\) is the oriented complex tangent line, \(W=1\oplus L^*\), with even first and odd second summand. Replacing \(S\) by \(S\otimes H\) replaces \(W\) by \(W\otimes H^*\) for the entire Hermitian line \(H\). The calibrated radial operator is

\[
 R_x(v)=-\frac{b_{W,x}(v)}{\sqrt{1+|v|^2}}.
 \tag{H.5}
\]

With the Fourier convention \(\widehat f(p)=\int e^{-ip\cdot v}f(v)\,dv\), this becomes the bounded transform of \(-i b_W\cdot\partial_p=c_W\cdot\partial_p\), the actual spinor Morita reduction of the inverse in (144).

**Proof.** The exterior relations are \(\{a_i,a_j\}=2\delta_{ij}\), \(\{c_i,c_j\}=-2\delta_{ij}\), and \(\{a_i,c_j\}=0\). Thus the physical \(b\) action graded-commutes with the positive \(a\) action. On a homogeneous balanced tensor its induced odd action is

\[
 \widetilde b_j(\overline s\otimes\eta)
   =(-1)^{|s|}\overline s\otimes b_j\eta.
 \tag{H.6}
\]

This sign is required for balance, because commuting \(b_j\) past an odd positive Clifford element changes its sign.

Let \(s_0,s_1\) be the even and odd spinor basis and put

\[
 P_+=\frac{1-ia_1a_2}{2},\qquad
 u_0=\frac{1-ie_{12}}{\sqrt2},\qquad
 u_1=\frac{e_1+ie_2}{\sqrt2}.
 \tag{H.7}
\]

The projection \(P_+\) fixes the orthonormal vectors \(u_0,u_1\), which exhaust its rank-two range. The maps \(\overline{s_0}\otimes\eta\mapsto P_+\eta\) and \(\overline{s_1}\otimes\eta\mapsto P_+a_1\eta\) identify the entire balanced module with \(P_+Z\). Indeed, if \(F_0=P_+\) and \(F_1=P_+a_1\), with codomain \(P_+Z\), then

\[
 F_i^*F_j=\rho_a(E_{ij}),\qquad
 F_k\rho_a(E_{ij})=\delta_{ki}F_j,
 \qquad F_iF_j^*=\delta_{ij}1_W.
 \tag{H.8}
\]

These are the full tensor-inner-product, balancing and onto identities. They use every matrix unit, rather than the dimension of a chosen corner. The conjugate inner product is \(\langle\overline s,\overline t\rangle_P=\theta_{s,t}\), and (H.8) preserves it. On the odd row, (H.6) gives \(-P_+a_1b_j=b_jP_+a_1\), so the physical action is also the one on the whole balance.

Insertion and removal give

\[
 b_1u_0=iu_1,\quad b_1u_1=-iu_0,\qquad
 b_2u_0=u_1,\quad b_2u_1=u_0.
 \tag{H.9}
\]

Their parities are even and odd. Hence the matrices and chirality asserted in the lemma follow. A positive frame rotation through \(\theta\) acts on \(S\) by \(\operatorname{diag}(1,e^{i\theta})\) and on \(u_0,u_1\) by \(\operatorname{diag}(1,e^{-i\theta})\). The conjugate balance carries a line transition \(h\) to \(h^{-1}\). Thus it retains \(H^*\), including nontrivial torsion transitions; a determinant comparison would not suffice.

There is an even onto isometry

\[
 U:S\widehat\otimes_C W\longrightarrow Z,
 \qquad U(s_0\otimes w)=w,\quad U(s_1\otimes w)=a_1w.
 \tag{H.10}
\]

Its four image columns are \(u_0,u_1,a_1u_0,a_1u_1\), an orthonormal basis of all \(Z\). Its inverse sends \(\eta\) to \(s_0\otimes P_+\eta+s_1\otimes P_+a_1\eta\). On both columns,

\[
 U(\varepsilon_j\otimes1)=a_jU,
 \qquad
 U(\Gamma_S\otimes b_{W,j})=b_jU.
 \tag{H.11}
\]

The latter already contains the graded tensor sign once. The identity on the odd column uses \(b_ja_1=-a_1b_j\). Every real-frame and central-line transition intertwines (H.10); the phases \(H,H^*\) cancel on the entire exterior module.

These maps are the two full Morita evaluations \(S\otimes\overline S\to P\), \(s\otimes\overline t\mapsto\theta_{s,t}\), and \(\overline S\otimes_PS\to C\), \(\overline s\otimes t\mapsto\langle s,t\rangle\), followed by \(P\otimes_PZ\to Z\). Both evaluations are even, preserve tensor inner products, and are onto by all matrix units and the scalar unit. This is Corollary 6.0d specialized to rank two, with the full physical inverse transfer of Lemma 7.16 and Theorem 7.17 retained.

Finally multiplication by \(v_j\) becomes \(i\partial_{p_j}\) under the stated transform. It sends (H.5), including its normalizer by functional calculus, to the bounded transform of \(-ib_W\cdot\partial_p\). Since \(b=ic\), this is precisely the reduced negative-Clifford Dirac inverse. \(\square\)

For comparison, putting the same positive spinor and positive radial phase in both factors gives

\[
 Q_{\mathrm{same}}=\sum_jA_jp_j+i\sum_jB_j\partial_{p_j},
 \quad A_j=\varepsilon_j\otimes1,
 \quad B_j=\Gamma_S\otimes\varepsilon_j.
 \tag{H.12}
\]

Its commuting involutions \(iA_jB_j\) have common \(+1\) line
\(\psi=(s_0\otimes s_1+is_1\otimes s_0)/\sqrt2\). The full Gaussian kernel is odd and has transition \(L\); with a common twist \(H\) it is \(\Pi(L\otimes H^2)\). Thus that flat product contributes \(-[L\otimes H^2]\), rather than an even trivial line. Moreover, an even unitary intertwining the positive ordered generators in (H.3) with \(-b_W\) would intertwine \(\Gamma_S\) with \(-\Gamma_W\), while evenness intertwines it with \(\Gamma_W\). Such a whole-generator intertwiner cannot exist. The calibration in (H.4)–(H.5) is substantive.

#### The two cycles over one parameter algebra

**Lemma 11.3 (whole interval cycles).** There are a separable central \(C(I)\)-algebra \(\mathfrak A\), \(I=[0,1]\), and cycles

\[
 \alpha_I\in KK(\mathfrak A,C(V\times I)),\qquad
 \beta_I\in KK(C(V\times I),\mathfrak A)
 \tag{H.13}
\]

with onto evaluations \(q_\lambda:\mathfrak A\to A_\lambda\), \(A_1=A\), \(A_0=C_0(F^*)\). Their endpoints are \(\alpha_1=\delta_{\mathrm{inv}}\), \(\beta_1=T\), and at zero the exact full spinor Thom pair obtained from (144).

**Proof: groupoid and Haar system.** Give \(F\times I\) source \(s_\lambda(x,v)=x\) and range \(r_\lambda(x,v)=\exp_x(\lambda v)\). For \(\lambda>0\) pull back the native holonomy groupoid through this exponential; at zero use addition in \(F_x\). Actual disk leaves have trivial holonomy, so an arrow is determined by its endpoints on the leaf. The total exponential is a bijective local diffeomorphism in source-preserving holonomy charts, as in Section 10. Hence this model is Hausdorff. Its topology is the native smooth groupoid topology, not the subspace topology of a possibly accumulating leaf relation in \(V\times V\).

For composable vectors \(v\in F_x\), \(w\in F_y\), \(y=\exp_x(\lambda v)\), the composite coordinate is \(\lambda^{-1}\log_x(\exp_y(\lambda w))\). In a leaf chart its numerator is smooth and zero at \(\lambda=0\). The integral Taylor formula extends the quotient smoothly with limit \(v+w\); inversion has limit \(-v\). This works with all needed derivatives on compact vector sets.

On each source fibre use the complete scaled metric and its volume

\[
 g_\lambda=dr^2+
       \left(\frac{\sinh(\lambda r)}\lambda\right)^2d\theta^2,
 \quad
 J_\lambda(v)\,dv=
 \frac{\sinh(\lambda|v|)}{\lambda|v|}\,dv,
 \quad J_0=1.
 \tag{H.14}
\]

At positive \(\lambda\) these are the pullback of \(\lambda^{-2}\) times the original leaf metric and its volume. The \(\lambda^2\) exponential Jacobian is canceled by metric scaling. Curvature is \(-\lambda^2\), and the zero limit is Euclidean. Right translation preserves metric and density. The power series of \(\sinh t/t\) proves smooth extension in Cartesian coordinates, including at \(v=0\).

Complete compact smooth kernels for \(\sup_{x,\lambda}\|\pi_{x,\lambda}(f)\|\) to obtain \(\mathfrak A\). Parameter multiplication is central. Evaluation is contractive; extending compact endpoint kernels through finitely many buffered charts and using closed range proves it onto. Fourier transform at zero gives \(C_0(F^*)\). This construction does not require an asserted KK equivalence between different algebra fibres.

**Proof: the radial cycle and module compactness.** Complete compact smooth sections of \(s^*W\) under

\[
 \langle u,w\rangle(x,\lambda)=
 \int_{F_x}\langle u(x,v,\lambda),w(x,v,\lambda)\rangle
                         J_\lambda(v)\,dv.
 \tag{H.15}
\]

This is a countably generated Hilbert \(B=C(V\times I)\)-module \(\mathcal H_I\). The exact density unitary \(J_\lambda^{1/2}\) identifies it with the fixed Euclidean fibre \(L^2\) module with coefficient \(W_x\). Orthogonal coordinate changes and the actual \(W\) transitions are unitary. Its right coefficient action is at \(x=s\gamma\). Regular convolution is

\[
 (\pi(f)u)(\gamma)=
   \int_{G_{s\gamma}}f(\gamma\eta^{-1})u(\eta)\,d\mu_{s\gamma}(\eta).
 \tag{H.16}
\]

Every coefficient in this integral belongs to \(W_{s\gamma}\). No range-to-source transport of \(W\) is implicit. Smooth compact groupoid products give continuity on the compact core; the regular bound extends it to the completion, and the exact Haar/involution identity gives its adjoint \(\pi(f^*)\). A near-unit approximate identity proves nondegeneracy. Thus this is an adjointable representation of all \(\mathfrak A\). The operator (H.5) is odd, self-adjoint, bounded by one and adjointable on this whole module.

A compact kernel has a common scaled propagation \(c\) and a common two-sided Schur bound \(K_f\). If its represented kernel is nonzero between radii \(r,s\), then \(d_\lambda\le c\) and \(|r-s|\le c\). For \(r,s\ge R\), hyperbolic cosine law and its Euclidean limit give

\[
 \left|\frac v{|v|}-\frac w{|w|}\right|
 \le\frac{2\sinh(\lambda c/2)}{\sinh(\lambda R)}
 \le\frac{c\cosh(c/2)}R,
 \qquad 0<\lambda\le1;
 \quad\text{at }\lambda=0:\ \le\frac cR.
 \tag{H.17}
\]

Indeed subtract \(\cosh(\lambda(r-s))\) in the cosine law and use \(\sinh(\lambda R)\ge\lambda R\). The radial amplitude \(a(r)=r/\sqrt{1+r^2}\) has derivative \((1+r^2)^{-3/2}\). The two-large-radii commutator tail is therefore bounded by

\[
 K_f\left(\frac{c\cosh(c/2)}R+
          \frac c{(1+R^2)^{3/2}}\right).
 \tag{H.18}
\]

Choose a smooth \(\chi_R\) equal to one up to \(R\) and zero after \(R+1\). For a commutator kernel \(K\), write
\(K=\chi_RK+(1-\chi_R)K\chi_R+(1-\chi_R)K(1-\chi_R)\).
Each of the first two terms has both radii bounded by \(R+1+c\), by propagation. It has compact \((x,v,w,\lambda)\) support. After the density unitary, a finite frame partition and finite separated continuous kernel sums approximate it uniformly. The common compact volume bound makes this an operator-norm approximation, and each separated term is an actual Hilbert-module rank-one map. The last term has the uniform bound (H.18). Hence the whole commutator is compact in \(\mathcal K_B(\mathcal H_I)\), including mixed-radius regions and the zero endpoint.

Also \(R^2-1=-(1+|v|^2)^{-1}\). An output cutoff gives compact retained support and tail norm at most \(K_f/(1+R^2)\). The same rank-one approximation applies. Self-adjointness is exact. Density of compact smooth kernels and norm continuity of the defect expressions prove all Kasparov conditions for \(\alpha_I\). Interpolation to the amplitude \(v/\max(1,|v|)\) has the same uniform estimates.

**Proof: the native longitudinal cycle.** Embed the whole graded \(S\) evenly in a finite trivial bundle and write \(e\) for its projection, acting at the range. The compact convolution completion and its compact endomorphisms are

\[
 \mathscr E_I=e\mathfrak A^N,
 \qquad \mathcal K_{\mathfrak A}(\mathscr E_I)
       =eM_N(\mathfrak A)e.
 \tag{H.19}
\]

Its inner product is the convolution \(\xi^*\eta\) with the actual range-spinor pairing and (H.14); the left \(B\) action is multiplication by \(h\circ r\). Compact kernel columns and an approximate identity prove this projection-corner description.

The zero-anchor extension of local quantization must be checked in source coordinates. On a translated source buffer let \(z,z'\) denote two arrows and \(\kappa(z,z')\) the coordinates of their relative arrow. This vanishes on the diagonal, and its derivative in \(z\) there is invertible even at zero. The integral Taylor formula writes

\[
 \kappa(z,z')=M(z,z')(z-z'),\qquad
 M=\int_0^1\partial_z\kappa(z'+t(z-z'),z')\,dt.
 \tag{H.20}
\]

Shrink finitely many unit buffers once. Their data are indexed by compact \(V\times I\); \(M,M^{-1}\) and all required source derivatives are uniformly bounded. The frequency change \(p=M^T\xi\) reduces the phase to \((z-z')\cdot p\). Density and the whole spinor transitions are bounded smooth amplitude factors. Thus the Fourier–Schur, adjoint and Taylor-composition proofs of the [measured-index lesson, Lemmas 6.1–6.2](the-index-theorem-for-measured-foliations.md#section-6) apply in these source buffers. No division by the anchor occurs.

For globalization, a maximal \(\epsilon\)-separated source set exists in every complete source fibre. The ball-area bounds

\[
 \pi r^2\le\operatorname{area}_\lambda B(r)
  =\frac{2\pi(\cosh(\lambda r)-1)}{\lambda^2}
  \le2\pi(\cosh r-1)
 \tag{H.21}
\]

include their Euclidean zero limit. They bound the multiplicity of radius-\(2\epsilon\) buffers by the ratio of a radius-\(5\epsilon\) upper area to a radius-\(\epsilon/2\) lower area. Radial bumps, normalized to \(\sum_j\phi_j^2=1\), have uniform source derivatives. For a fixed propagation enlarge the area radius once, obtaining a common row and column multiplicity \(n\). If each localized operator block has norm at most \(b\), then

\[
 \sum_i\left\|\sum_{j\sim i}P_{ij}u_j\right\|^2
 \le n b^2\sum_i\sum_{j\sim i}\|u_j\|^2
 \le n^2b^2\sum_j\|u_j\|^2.
 \tag{H.22}
\]

The localization is isometric, so the global regular norm is at most \(nb\). A negative-order frequency tail has the same bound with \(b=c_mM^m\), \(m<0\). Its frequency truncations are jointly smooth with the original finite compact physical support, and hence belong to the actual corner (H.19). The source covering proves the norm estimate; it does not introduce an infinite sum of uncontrolled global kernels. This reproduces the source-coordinate mechanism of Lemma 8.15 without assuming an action or a global frame. Fixed-anchor foliation and ordinary-submersion charts alone would not prove this assertion at zero.

Quantize the positive unit-covector symbol in the finite buffered unit charts, patch and take its odd self-adjoint part. The preceding bounds and the formal adjoint identity give one adjointable \(P_I\) on (H.19). Symbol composition gives order at most \(-1\) for \(P_I^2-1\) and \([P_I,h\circ r]\), so their uniform negative-order approximation makes them compact in that corner. Smooth \(h\) are dense in \(B\). This defines \(\beta_I\); clamping its self-adjoint phase, if necessary, preserves its class and symbol.

At one its whole module, symbol and grading are exactly Theorem 7.1. At zero Fourier transform gives the \(S\)-valued \(C_0(F^*)\) module and symbol \(\varepsilon\cdot p/|p|\). The compact near-unit cutoff changes it by a lower-order multiplier vanishing at infinity. Its compact-difference path identifies it with \(\varepsilon\cdot p/\sqrt{1+|p|^2}\), the prescribed Thom cycle of Corollary 6.0d. The zero radial cycle is its exact dual module and inverse by Lemma 11.2. These are coefficient extensions of whole interval cycles, not separately chosen fibre phases. \(\square\)

#### A regular native Dirac realization

**Lemma 11.4 (regularity and interval compact resolvents).** On the entire module (H.19), the geometric source Dirac operator for (H.14) and the prescribed Hermitian Clifford connection has a regular self-adjoint closure \(D_I\). Compact smooth range-spinor kernel columns form a graph core. For \(a>0\),

\[
 (D_I\pm ia)^{-1}\in eM_N(\mathfrak A)e,
 \qquad \|(D_I\pm ia)^{-1}\|\le a^{-1}.
 \tag{H.23}
\]

Its bounded transform represents \(\beta_I\). At positive \(\lambda\), endpoint identification sends the differential operator to \(\lambda D_{\mathrm{leaf}}\); at zero it is \(-i\sum_j\varepsilon_j\partial_{v_j}\) on the constant \(S_x\) bundle. The source dimension and derivative remain two at zero.

**Proof.** The relative source metric and its Clifford connection are smooth at zero in the source charts of Lemma 11.3. At positive \(\lambda\) the metric is the pullback of \(\lambda^{-2}g\); the pulled-back connection terms scale with \(\lambda\). Their zero limit is the flat source connection. The differential operator is odd, formally symmetric for \(J_\lambda dv\), right invariant and right \(\mathfrak A\)-linear on the core, with principal symbol \(\varepsilon(\xi^\sharp)\).

The complete source metric has radius \(|v|\) from the unit. Smooth radial cutoffs \(\eta_R=\chi(|v|/R)\), constant near zero, satisfy \(\|d\eta_R\|_\infty\le c/R\) uniformly in \(\lambda\). Local elliptic regularity in the common source buffers makes a deficiency vector smooth there. If \((D_{x,\lambda}^*\pm i)u=0\), integration by parts against \(\eta_R^2u\) gives

\[
 \|\eta_Ru\|_2^2\le(c/R)\|\eta_Ru\|_2\|u\|_2.
 \tag{H.24}
\]

Letting \(R\to\infty\) proves essential self-adjointness of every regular source realization. The moving-cutoff energy identity \(\|[D,\eta]\|\le\|d\eta\|_\infty\) proves wave propagation at speed at most one. A core column supported at radius at most \(R\) stays within \(R+|t|\).

On that common compact support, source Sobolev energy estimates have uniform coefficients. The parameter-difference equation proves continuity of source jets; differentiating \(\partial_tu=iD_\lambda u\) gives the equation for \(\partial_\lambda u\) with forcing \(i(\partial_\lambda D_\lambda)u\). Its finite-time Duhamel estimate, followed by induction, proves the needed smooth jets through the one-sided endpoints. Sobolev embedding gives smooth relative kernel columns; no derivative normal to the parameter is part of the operator itself.

Let \(W(t)\) act on columns by these source waves. Right invariance gives \(W(t)(\xi*f)=(W(t)\xi)*f\). Convolution of a column with a regular scalar input intertwines this action with the unitary wave on the whole range-spinor source space. Its inner products therefore satisfy

\[
 \langle W(t)\xi,W(t)\eta\rangle_{\mathfrak A}
   =\langle\xi,\eta\rangle_{\mathfrak A},
 \tag{H.25}
\]

because the family of regular representations defining \(\mathfrak A\) is faithful. Thus \(W(t)\) extends to an adjointable unitary, with adjoint \(W(-t)\). On core columns, \(\|(W(t)-1)\xi\|\le|t|\|D\xi\|\); density proves strong continuity on the full module.

Define \(D\) to be \(1/i\) times its norm generator. Time convolution by a compact smooth function lies in its domain, with derivative transferred to the time kernel. Time approximate identities approximate any domain vector in graph norm. Approximating each time-smoothed input by compact smooth columns gives core approximants; the derivative error is bounded by the \(L^1\) norm of the time-kernel derivative times the input error. Finite propagation and the preceding regularity keep these approximants in the compact relative core. Hence this generator is the closure of the stated differential operator and has the asserted graph core.

Vectorwise strong integration gives adjointable operators

\[
 R_{+,a}=-i\int_0^\infty e^{-at}W(t)\,dt,
 \qquad
 R_{-,a}=i\int_0^\infty e^{-at}W(-t)\,dt.
 \tag{H.26}
\]

Their norms are at most \(1/a\), and their adjoints are each other. Integration by parts proves both inverse identities for \(D\pm ia\). The inverse-domain argument of Lemma 8.15a proves self-adjointness; the commuting product of the two inverses at \(a=1\) is the densely ranged inverse of \(1+D^2\), proving regularity. The integrals need not converge in the multiplier operator norm; their vectorwise definition and adjoint identities suffice.

Quantize the inverse principal symbol \(\varepsilon(\xi^\sharp)/|\xi|^2\), cut off near zero, in the same finite near-unit charts. The uniform source composition gives order \(-1\) parametrices \(Q_{\pm,a}\) with

\[
 (D\pm ia)Q_{\pm,a}=1-S_{\pm,a},\qquad
 Q_{\pm,a},S_{\pm,a}\in\mathcal K(\mathscr E_I).
 \tag{H.27}
\]

The identity first holds on the core. Closedness extends it to all inputs and gives \(\operatorname{ran}Q_{\pm,a}\subset\operatorname{Dom}D\). Multiplication by the actual inverse yields \(R_{\pm,a}=Q_{\pm,a}+R_{\pm,a}S_{\pm,a}\), proving (H.23) in the whole interval corner. Thus every localization obeys \(\|hR_{\pm,a}\|\le\|h\|/a\). The resolvent identity treats every nonreal parameter; their functional calculus puts \(f(D)\) in the same compact corner for \(f\in C_0(\mathbb R)\).

For smooth \(h\), multiplication preserves the graph domain and \([D,h\circ r]=c(d_s(h\circ r))\) is bounded by \(\sup|d_Fh|\): the source derivative is \(\lambda\) times the leaf derivative and is zero at the additive endpoint. Resolvent commutators have bound \(\|[D,h]\|/(1+t^2)\) at the poles \(\pm i\sqrt{1+t^2}\). Integrating the two-pole bounded-transform formula therefore proves that \([D(1+D^2)^{-1/2},h]\) is a norm-convergent compact integral. Smooth approximation handles all continuous \(h\), and the square defect is the compact \(-(1+D^2)^{-1}\).

The symbol comparison requires more than compact resolvents. Apply the positive-parameter symbol recursion of the [Hilbert-module lesson, Lemmas 6.1–6.2, equations (4)–(10)](hilbert-modules-and-fields-on-the-leaf-space.md#section-6) in the source buffers just proved. Its recursive symbols for \(\rho+D^2\) satisfy

\[
 |\partial_z^\alpha\partial_\xi^\beta q_{-2-j}(\rho)|
 \le c_{\alpha\beta j}
       (\rho+|\xi|^2)^{(-2-j-|\beta|)/2}.
 \tag{H.28}
\]

The leading term is \((\rho+|\xi|_g^2)^{-1}\). A finite recursion cancels any prescribed number of parameter orders; separated physical cutoffs have arbitrary inverse-parameter decay by phase integration by parts. The finite-overlap bound (H.22) makes these estimates uniform in the entire reduced corner. In the right-parametrix identity \((\rho+D^2)Q_\rho=1-S_\rho\), one has \(\|DQ_\rho\|\le c\rho^{-1/2}\) and \(\|S_\rho\|\le c_b\rho^{-b}\). For large \(\rho\), the exact inverse is \(Q_\rho(1-S_\rho)^{-1}\), so

\[
 \|D((\rho+D^2)^{-1}-Q_\rho)\|
 \le c\rho^{-1/2}
       \frac{c_b\rho^{-b}}{1-c_b\rho^{-b}}.
 \tag{H.29}
\]

Every correction is compact. Choose \(b>1/2\). Its inverse-square-root integral converges in compact norm. Integrating the leading symbol gives \(\varepsilon(\xi^\sharp)/\sqrt{1+|\xi|_g^2}\); each further recursive term has order at most \(-1\), by the substitution \(t=(1+|\xi|^2)u\). On bounded parameter intervals, differentiated resolvents are norm-continuous compacts by (H.23). Truncate the integral first; the correction terms converge in compact norm, while the leading quantization and bounded transform converge strongly. This proves that their difference is that compact norm limit. Hence the bounded transform has the prescribed positive unit-covector symbol. The actual source symbol quotient of the Hilbert-module lesson's Lemma 6.5, with the checked source estimates above, gives \(D(1+D^2)^{-1/2}-P_I\in\mathcal K(\mathscr E_I)\). Its compact-difference path proves equality of the whole interval classes and their native endpoints. \(\square\)

#### The represented target module and its product

**Lemma 11.5 (the actual product homotopy).** The product of the two cycles in (H.13), precomposed by constant functions \(C(V)\to C(V\times I)\), is a homotopy over the fixed target \(C(V\times I)\). Its endpoint products are \(\beta_\lambda\alpha_\lambda\). Its represented tensor module is

\[
 \mathscr E_I\widehat\otimes_{\mathfrak A}\mathcal H_I
 \cong L_s^2(G_I,r^*S\widehat\otimes s^*W),
 \tag{H.30}
\]

and its left \(C(V)\) action is literally

\[
 (h\xi)(x,v,\lambda)=h(\exp_x(\lambda v))\xi(x,v,\lambda).
 \tag{H.31}
\]

At zero this is multiplication by \(h(x)\).

**Proof.** The onto projection-corner unitary
\(e\mathfrak A^N\otimes_\pi\mathcal H_I\to e\mathcal H_I^N\)
applies regular convolution to each column. Associativity is its balancing relation; Fubini and \(\xi^*\eta\) give the whole matrix-valued tensor inner-product identity. A near-unit approximate identity and compact source tests give dense range, so it is onto. Its projection is pointwise \(e(r_\lambda\gamma)\), giving exactly (H.30), with both gradings and right action at \(s\gamma\). This does not assert that a lifted first-module compact is compact on the tensor module.

Transport the entire spin-c bundle from \(\exp_x(\lambda v)\) to \(x\) along that specified radial geodesic by its Hermitian Clifford connection and multiply by \(J_\lambda^{1/2}\). This is an even unitary to the fixed Euclidean \(L^2(F_x,S_x\otimes W_x)\) module over \(V\times I\). Reverse transport and \(J_\lambda^{-1/2}\) give its inverse. Its exact inner-product identity, smooth compact-core dependence and isometric bound prove continuity of both maps. Every line phase is retained. The represented scalar in (H.31) is unchanged by this unitary. On compact tests it varies uniformly with \(\lambda\); uniform boundedness and density give strong continuity and adjoint continuity on the entire module. Global operator-norm continuity on unbounded \(v\) is not needed.

Here are the product connection and positivity checks. Put \(S_\beta=P_I\otimes1\), let \(G_\alpha\) be the odd self-adjoint \(\alpha_I\)-connection of Lemma 0.2, and set

\[
 \begin{aligned}
 J&=\mathcal K_B(\mathscr E_I\otimes\mathcal H_I),\\
 A_1&=C^*(\mathcal K_{\mathfrak A}(\mathscr E_I)\otimes1,J),\\
 A_2&=C^*(G_\alpha^2-1,G_\alpha-G_\alpha^*,
      [G_\alpha,\phi(B)],\{G_\alpha,S_\beta\}).
 \end{aligned}
 \tag{H.32}
\]

For homogeneous creation vectors, applying their connection equations to \(\xi,h\xi,P_I\xi\) shows that every generator of \(A_2\) is a zero connection. Thus \(A_1A_2\subset J\) by (94). The two compact ideals are kept distinct: \(\mathcal K(\mathscr E_I)\otimes1\) need not lie in \(J\). The graded separator of Theorem 0.4 supplies even positive \(M,N\), \(M+N=1\), with \(MA_1,NA_2\subset J\) and compact commutators with the represented source and both operators. Its square roots retain those properties; \(M^{1/2}\) is a zero connection and \(N^{1/2}\) an identity connection. Therefore

\[
 F=M^{1/4}S_\beta M^{1/4}
       +N^{1/4}G_\alpha N^{1/4}
 \tag{H.33}
\]

is a self-adjoint \(\alpha_I\)-connection. The separator kills the localized lifted first-module square defects and the zero-connection defects in \(A_2\), giving the actual compact square and commutator defects. Its positive sandwich is

\[
 \phi(h)\{S_\beta,F\}\phi(h)^*
 \equiv2\phi(h)M^{1/2}\phi(h)^*\ge0\pmod J.
 \tag{H.34}
\]

Thus Definition 0.3 proves an actual product. All these operators are adjointable over \(B\), so they commute with its right central \(C(I)\) action; the left parameter is the same one. The construction preserves the parameter. Evaluation sends rank-one compacts to rank-one compacts, preserves positive sandwiches, and sends dense creation vectors to dense endpoint creation vectors by onto \(q_\lambda\). The evaluated balanced core formulas identify the endpoint tensor products. Theorem 0.6 supplies their class uniqueness. Precomposition by the constant inclusion now makes this a homotopy with fixed source \(C(V)\) and target \(C(V\times I)\), as claimed. \(\square\)

![The normalized source density, uniform radial bounds, and full calibrated first product](../figures/kt-hyperbolic-first-product.png)

**Figure 11.1.** Lemmas 11.2–11.5 and Theorem 11.1. The plotted density and bounds are the exact functions (H.14), (H.17) and (H.18), with the indicated numerical propagation and Schur constants. They are proved upper bounds, not sampled commutator norms. The finite module retains its two physical generators, opposite ordered chirality, dual line and negative radial phase; its Gaussian is the entire even trivial coefficient line. The reverse identity is proved separately by the compactification and boundary comparison in Section 11B. This original figure shows the exact functions and coefficient objects proved here; human source context is [Connes], Section 12, the product connection framework is [Connes–Skandalis], Appendix A, and the Clifford Bott framework is [Kasparov].

![The fixed source and changing represented range in an exact Poincaré-coordinate slice](../figures/kt-hyperbolic-range-action.png)

**Figure 11.2.** Lemma 11.5, (H.30)–(H.31). In the curvature \(-1\) disk, \(x=0\) and the tangent vector \(v=2e_1\), where \(e_1\) is orthonormal, give \(y_\lambda=(\tanh\lambda,0)\), at original-metric distance \(2\lambda\). Spin-c transport follows that specific radial geodesic. The left action is \(h(y_\lambda)\), the right action is \(h(x)\), and at zero both are \(h(x)\). This is one exact leaf-coordinate slice, not a global embedding of an accumulating leaf in \(V\). Its deformation context is [Connes], Section 12; Lemmas 11.2 and 11.5 give the whole bundle and coefficient argument.

**Proof of Theorem 11.1.** At one, Lemma 11.5 gives \(T\delta_{\mathrm{inv}}\). At zero, Fourier transform gives the prescribed positive spinor Thom multiplier and the inverse negative-Clifford Dirac on the entire \(W\) module. The even onto evaluation (H.10) identifies their whole tensor module and both physical actions with the exterior module. Corollary 6.0d's two full Morita evaluations reduce this endpoint product to the raw Clifford product; no coefficient line is discarded. An actual representative of that product class is

\[
 Q=\sum_jc_j\partial_{p_j}+\sum_ja_jp_j,
 \qquad Q^2=-\Delta_p+|p|^2+2N-2.
 \tag{H.35}
\]

It need not be the literal conjugate of the particular separator operator (H.33). Theorem 6.0c's product proof verifies it independently. Explicitly, for a homogeneous smooth compact creation section \(\xi\), \(K_\xi=Q\rho(\xi)-(-1)^{|\xi|}\rho(\xi)D\) is bounded with compact fibre support, and the mixed resolvent identity (138) proves compact connection errors with integrable bound \(\|K_\xi\|/|\operatorname{Im}z|^2\). The adjoint equation and norm density give every creation vector.

For positivity, put \(b=a(p)/s\), \(s=\sqrt{1+|p|^2}\), \(g=|p|^2/s\), and \(S_0=(1-\Delta+|p|^2)^{-1/2}\). The bounded-transform difference from \(QS_0\) is compact by the Hermite spectral estimate. Equations (139)–(140) give

\[
 \{b,QS_0\}=2gS_0+K_bS_0+Q[S_0,b],
 \qquad
 gS_0=S_0^{1/2}gS_0^{1/2}+[g,S_0^{1/2}]S_0^{1/2}.
 \tag{H.36}
\]

Here \(K_b\) is bounded. The resolvent formula for \(Q[S_0,b]\) has compact integrands bounded by a constant times \(t^{-1/2}/(1+t)\); the quarter-power formula for \([g,S_0^{1/2}]\) has bound \(ct^{-1/4}(1+t)^{-3/2}\). Both are integrable. For the unbounded \(g\), use smooth cutoffs with uniformly bounded first and second derivatives on the Schwartz core; the compact Hermite resolvents turn strong convergence of their derivative coefficients into norm convergence, as in (140). Thus these are compact bounded remainders on the whole bundle module. The middle form is positive and bounded by one because \(0\le g\le s\le(1-\Delta+|p|^2)^{1/2}\). This is the actual positive sandwich in Definition 0.3. Theorem 0.6 identifies its product class with the evaluated separator product.

The Hermite/exterior eigenvalues of \(Q^2\) are \(2(|n|+k)\). Its entire kernel is the normalized Gaussian times the exterior vacuum \(1\), even and invariant under all orthogonal transitions, with gap two. In spinor tensor coordinates its coefficient is \((|00\rangle+|11\rangle)/\sqrt2\), fixed by the full transitions \(\operatorname{diag}(h,h\ell)\otimes\operatorname{diag}(h^{-1},h^{-1}\ell^{-1})\). Thus the whole ground module is the trivial \(C(V)\) line, including cancellation of any torsion twist. Its left action is the actual \(h(x)\) from (H.31). Changing the bounded transform to its phase off this kernel is a compact perturbation; the complement is degenerate, and the kernel is the identity homomorphism cycle. The zero endpoint is \(1_C\). The genuine homotopy of Lemma 11.5 proves the first identity in (24). \(\square\)

This equality includes all base vector-bundle inputs, both K-degrees and torsion. Ordinary exterior tensoring preserves it for every separable graded coefficient algebra. A theorem about additional nontrivial groupoid coefficient actions would require its own interval construction and is not inferred from that exterior naturality.

#### Native comparison and the Gaussian limitations

**Remark 11.6 (native comparison and reverse product).** Theorem 11.1 proves the calibrated first product. The particular \(\delta\) of Theorem 10.1 is not identified with \(\delta_{\mathrm{inv}}\) by its norm condition alone. Any native-source comparison must state its whole physical Clifford generators, grading, Hermitian line and Fourier phase. Under the fixed positive convention (H.3), an even whole-generator intertwiner is excluded by the chirality calculation above. The full calibrated replacement (H.4)–(H.5) must therefore be retained; a scalar rank or determinant computation is insufficient.

The zero-fibre reverse product is known from Theorem 6.0c and Corollary 6.0d. It does not imply the native \(A,A\) identity. Put \(\Gamma=\alpha_I\beta_I\in KK(\mathfrak A,\mathfrak A)\). Evaluation naturality gives

\[
 \Gamma[q_0]=[q_0],\qquad
 \Gamma[q_1]=[q_1](\delta_{\mathrm{inv}}T).
 \tag{H.37}
\]

Even a separately proved ordinary KK inverse for \(q_0\) would give only \(\Gamma=1\) in ordinary KK, and then \([q_1](\delta_{\mathrm{inv}}T-1_A)=0\). It does not permit cancellation of \(q_1\). An arbitrary ordinary homotopy over \(\mathfrak A\) cannot be restricted to its one fibre: that requires the central \(C(I)\) parameter to be preserved throughout. Contracting an evaluation-zero ideal by moving its parameter is not such a linear contraction.

A reverse proof requires a valid native \(A,A\) comparison; the endpoint identities above alone do not supply it. Section 11B gives that comparison by compactification and boundary factorization, without asserting the \(C(I)\)-linear identity \(\Gamma=1_{\mathfrak A}\). A Gaussian splitting would additionally have to retain both convolution actions, the full \(W/S\) modules, all creation connections and localized positive sandwiches, and the entire coefficient Gaussian module with an identity left \(A\) action. A leafwise Gaussian rank alone does not establish that assertion. After coordinates \(v=\log_u(y)\) and transport to \(u\), the potential is linear but left convolution moves \(u\). An arbitrary Euclidean flattening of the Dirac matrix differentiating \(y\) can then acquire \(u\)-dependence in its leading symbol. Its convolution commutator needs a new module-compactness argument; confining resolvents do not supply it. No impossibility theorem is asserted by this observation.

A native confining product representative, even after smoothing in both moving leaf variables, compact creation connections and the localized positive sandwich are verified, does not by itself identify the product class with the identity. Its entire even Gaussian right module carries a compressed left convolution with the Gaussian overlap factor, rather than an invariant identity action. These Gaussian calculations alone do not identify the full representation; Section 11B supplies the comparison without making this Gaussian projection left invariant.

Section 11B proves the full calibrated native comparison and reverse product. Together with Theorem 11.1, this gives the hyperbolic analytical index isomorphism in both integral K-degrees. Theorem 7.25 supplies the analytical/geometric class comparison for the same full orientation, while twisted graph-duality pairings have different source and target from this \(A,A\) identity. The reverse proof uses the compactification and boundary comparison, retaining the complete modules and calibration.

The exact programme dependencies in this section are Definition 0.3, Lemma 0.2 and Theorems 0.4/0.6 for products; Lemmas 6.0a–b, Theorem 6.0c and Corollary 6.0d for the complete bundle Thom pair; Theorem 7.1, Lemma 7.16 and Theorem 7.17 for the positive native symbol and entire inverse transfer; the measured-index lesson's Lemmas 6.1–6.2 and the Hilbert-module lesson's Lemmas 6.1–6.2/6.5 for the stated local symbol proofs. Lemma 11.3 supplies their necessary source-coordinate extension through the zero anchor, reproducing the localization mechanism of Lemma 8.15. Lemma 11.4 reproduces the complete-source energy mechanism of Lemma 8.19 and the wave-generator/inverse-domain argument of Lemma 8.15a on this exact groupoid and finite projection corner; their group-action conclusions are not substituted for this proof. Theorem 7.25 is used only for the stated whole-orientation graph comparison.

#### The actual Gaussian convolution defect

**Lemma 11.7 (the full Gaussian right module and its left-action defect).** In the scope of Theorem 11.1, the native reverse tensor has an even Gaussian submodule isometric to the whole standard right \(A\)-module. Its compressed left convolution is a completely positive contraction \(\Phi:A\to A\), whose multiplicative defect is the square of the off-diagonal Gaussian action. The localized off-diagonal blocks are actual module compacts. The Gaussian submodule is nevertheless not left invariant; these compactness statements do not identify the reverse product with \(1_A\).

**Proof.** Write all arrows by range/source endpoints. The entire reverse balanced tensor has compact-core functions \(\zeta(u,y,x)\in W_y\widehat\otimes S_y\). After the graded flip and full evaluation of Lemma 11.2, these coefficients are exterior forms at \(y\); all central line phases cancel by the actual \(H^*\otimes H\) evaluation. Its actions and inner product are

\[
\begin{aligned}
 (\pi(a)\zeta)(u,y,x)&=\int a(u,u')\zeta(u',y,x)\,du',\\
 (\zeta k)(u,y,x)&=\int\zeta(u,y,z)k(z,x)\,dz,\\
 \langle\zeta,\eta\rangle_A(x,z)
 &=\iint\langle\zeta(u,y,x),\eta(u,y,z)\rangle\,du\,dy.
\end{aligned}
 \tag{H.38}
\]

These formulas follow first on elementary balanced tensors, by integrating the radial inner product and then the native column pairing. Compact triple-chart separated approximation identifies their completion with the whole tensor. The derivative variable in the native longitudinal factor is \(y\), while the first left convolution changes \(u\).

In the negative Witten presentation, use the globally oriented even top-form coefficient and the normalized vectors

\[
 g_u(y)=Z^{-1/2}e^{-d(y,u)^2/2}\operatorname{vol}_y,
 \qquad
 Z=2\pi\int_0^\infty e^{-r^2}\sinh r\,dr
 =\pi^{3/2}e^{1/4}\operatorname{erf}(1/2).
 \tag{H.39}
\]

The last identity follows by completing the square separately in the two exponentials of \(\sinh r\). The underlying full \(W\widehat\otimes S\) coefficient is \((w_0\otimes s_0+w_1\otimes s_1)/\sqrt2\); the graded flip followed by Lemma 11.2's evaluation sends it to \(-ie_{12}\). The even degree unitary \(i^N\) in the Witten presentation changes only this recorded constant phase. Thus the Gaussian retains its actual even coefficient line and all full spin-c transitions.

Define \(V:A\to E\) on compact columns by \((Vk)(u,y,x)=g_u(y)k(u,x)\). Gaussian cutoffs in \(d(y,u)\), followed by compact triple-chart separated approximations, put these vectors in \(E\). Their tail norm tends uniformly to zero, since every actual disk has the same radial density. Equation (H.38) and normalization give \(\langle Vk,Vl\rangle_A=k^*l\). The adjoint integrates against \(g_u\) in \(y\); Cauchy–Schwarz in the faithful regular family proves its bound and adjoint identity. Hence \(V\) is an even adjointable isometry from the whole standard right module. Put \(P=VV^*\). Direct convolution gives

\[
 \Phi(a)=V^*\pi(a)V=M_\kappa(a),\qquad
 \kappa(d(u,u'))=Z^{-1}\int
 e^{-[d(y,u)^2+d(y,u')^2]/2}\,dy.
 \tag{H.40}
\]

This is completely positive and contractive. For compact smooth \(a\), its kernel is \(\kappa(d(u,u'))a(u,u')\), with the same compact arrow support. Smooth compact approximation and the two-sided Schur estimate put it in \(A\). Density and contractivity extend this to every \(a\in A\), with image in \(A\), not just its multiplier algebra.

Insertion of \(1=P+(1-P)\) in the whole adjointable action proves

\[
\begin{aligned}
 \Phi(a^*b)-\Phi(a)^*\Phi(b)
 &=V^*\pi(a)^*(1-P)\pi(b)V\in A,\\
 \|\Phi(a^*a)-\Phi(a)^*\Phi(a)\|
 &=\|(1-P)\pi(a)V\|^2.
\end{aligned}
 \tag{H.41}
\]

The first line belongs to the actual compact endomorphism algebra \(\mathcal K_A(A)=A\). For \(a=b\) it is positive, by the exact \(T^*T\) identity.

The stronger compact-block assertion can be proved directly, without an unlocalized resolvent or a lifted-compact shortcut. For compact smooth \(a\), define the whole tensor column \(\zeta_a(u,y,x)=a(u,x)g_x(y)\). A radial cutoff \(d(y,x)\le R\) and the compact arrow support of \(a\) give compact triple support; finite separated approximation puts it in the tensor core. If \(\tau_R\) is the squared Gaussian tail, its discarded inner-product kernel is bounded by \(\tau_R\int|a(u,x)a(u,z)|\,du\). A common two-sided Schur bound \(K_a\) therefore gives discarded module norm at most \(K_a\sqrt{\tau_R}\). The column is in \(E\), and

\[
 \pi(a)V=T_{\zeta_a},\qquad
 \langle\zeta_a,\zeta_a\rangle_A=\Phi(a^*a),
 \qquad T_{\zeta_a}\in\mathcal K_A(A,E).
 \tag{H.42}
\]

Here \(T_\zeta k=\zeta k\). Its compactness follows from an approximate identity of \(A\), which approximates this standard-module creation map by rank-one maps. Consequently \(\pi(a)P=T_{\zeta_a}V^*\) is in \(\mathcal K_A(E)\); adjoints give \(P\pi(a)\), and norm density treats every \(a\). No assertion that \(P\) itself is compact is needed.

Finally, distinct centres have distinct normalized positive Gaussian functions: their ratio takes different values at \(y=u\) and \(y=u'\). Strict Cauchy–Schwarz gives \(0<\kappa(d)<1\) for \(d>0\). To check actual non-invariance, choose nonnegative compact smooth convolution and column tests concentrated near a positive-distance arrow \((u,u')\). Their output near \(u\) has a positive Gaussian superposition concentrated near \(g_{u'}\), which, for sufficiently small fixed supports, stays a positive distance from the line spanned by \(g_u\). By continuity this separation persists on a small open output chart, so it gives a nonzero \(L^2\) complementary component in that regular localization, not just a pointwise value. Faithfulness makes \((1-P)\pi(a)V\ne0\) on the whole module for some \(a\). Equation (H.41) then gives a nonzero positive multiplicative defect. Thus \(\Phi\) is not a representation and the Gaussian ground cannot be deleted from the full cycle as an invariant identity summand. A full reverse homotopy must still treat the convolution representation and its graded complement. \(\square\)

![The full Gaussian right module and its exact convolution defect](../figures/kt-hyperbolic-gaussian-compression.png)

**Figure 11.3.** Lemma 11.7, (H.38)–(H.42). The diagram records actual Hilbert-module maps and their algebraic relations; its placement is schematic, not a leaf-coordinate geometry. Left convolution changes \(u\) with \(y\) fixed. The full even Gaussian isometry \(V:A\to E\) preserves the right convolution action. Its compression is the Gaussian-overlap map \(\Phi\), and the positive defect is exactly the norm square of the complementary block. Both localized Gaussian blocks are actual \(A\)-module compacts, while the defect can be nonzero. This figure does not depict a completed native reverse contraction. Reproducible original CC0 source accompanies the figure; classical human context is Connes's disk-leaf inverse construction, Connes–Skandalis's product framework and Kasparov's Clifford method. The full Gaussian and compact-block proof is supplied by Lemma 11.7.

#### The prescribed native radial class

The native construction uses the same graded spinor bundle and Clifford map in its longitudinal symbol and radial multiplier: [Connes], Section 10(3), PDF p.31, and Section 12, PDF pp.37–38. Its class is the positive radial class of Theorem 10.1. The following comparison supplies the complete native interface, including the physical generators, grading, full line and regular action.

**Proposition 11.8 (the prescribed native first product).** Under the hypotheses and physical conventions of Sections 10–11, let \(S=(1\oplus L)\otimes H\) be the prescribed entire spinor bundle and let \(D=\det S=L\otimes H^2\). The literal positive radial class \(\delta_S\) satisfies

\[
 \delta_S=-\delta_{\rm inv}\otimes_{C(V)}[D],\qquad
 T\otimes_A\delta_S=-[D].                             \tag{ND.1}
\]

For the canonical structure \(H=1\), the latter class is \(-[L]\). On any nonempty base its rank at a point is \(-1\), so it is not \(1_{C(V)}\). This refutes the literal same-spin first-identity assertion. It does not refute the analytical-isomorphism question, because this negative line is invertible. Lemma 11.9 gives a compact example with actual complete simply connected disk leaves.

| Datum | Native source | Exact course realization |
|---|---|---|
| Leaf scope | Actual complete curvature \(-1\) Poincaré-disk leaves | Sections 10–11: compact \(V\), oriented rank-two \(F\), actual complete simply connected disk leaves |
| Longitudinal finite bundle | \(r^*S^+\to r^*S^-\), Section 10(3) | The prescribed whole \(S\) module of Theorem 7.1 and Lemma 11.4 |
| Longitudinal symbol | \(c(\xi^\sharp)/|\xi|\), same \(c\) as the radial construction | Positive physical \(\varepsilon(\xi^\sharp)\); ordinary negative Fourier phase gives \(-i\varepsilon\nabla\) |
| Radial finite bundle | \(S_x\), held fixed at \(x=s\gamma\) | \(L^2(G_x)\otimes S_x\), Theorem 10.1 |
| Radial representation | Regular convolution on the first factor; identity on \(S_x\) | Exactly the represented module of Theorem 10.1 |
| Radial vector and phase | \(+c(\log_x r\gamma/\max(1,|\log_xr\gamma|))\) | The positive off-diagonal block in (22) |
| Product finite coefficient | \(r^*S\widehat\otimes s^*S\) | Same-spin comparison (H.12), including both gradings and all line transitions |
| Calibrated inverse | No dual finite bundle is specified in the native radial definition | \(W=\bar S\widehat\otimes_{\operatorname{Cl}^+(F)}\Lambda F_{\mathbb C}\), with physical \(-b_W\), (H.4)–(H.5) |

The paper does not print Pauli matrices or an exponential sign in its Fourier-transform sentence. They must not be attributed to it. Those omissions do not remove the same-spin relation: it explicitly prescribes the same \(S^+,S^-\) and the same Clifford map for the two constructions, and explicitly prints \(L^2(G_x,r^*S)\otimes S_x\) for the product. The course specifies those remaining physical conventions. The full module argument below proves the comparison in that realization. The flat product argument proves that the odd rank of the literal pair survives the usual alternative scalar Clifford phase, simultaneous grading choice and Fourier sign.

For clarity, the canonical spin-c structure used here is fixed as follows. In an oriented unit frame \((e_1,e_2)\), a frame rotation by \(\theta\) has the spin lift \(\exp(-i\theta\Gamma_S/2)\) and central factor \(e^{i\theta/2}\). Its spinor matrix is \(\operatorname{diag}(1,e^{i\theta})\). Thus \(S^+=1\), \(S^-=L\), where \(L\) is the oriented complex tangent line. The lift is single valued in \(\operatorname{Spin}^c(2)\) because both factors change sign at \(2\pi\). This gives the complete transition action, rather than a choice based only on a norm equation. A common spin-c twist multiplies both entries by its actual transition \(h\).

If one uses the skew geometric Clifford action, the off-diagonal block may instead be \(i\varepsilon(v)^+\). The global even unitary \(\operatorname{diag}(1,-i)\) on \(S^+\oplus S^-\) takes this block to \(\varepsilon(v)^+\). It applies to both the longitudinal and radial blocks. It changes neither class, bundle nor grading. It does not conjugate the real vector \(v\), exchange the ordered physical generators, dualize a line or introduce the inverse bundle \(W\).

##### Proof of Proposition 11.8: the whole parity and line map

Use precisely (H.3) and (H.4), with the entire line \(H\):

\[
 S=(H\oplus LH),\quad W=(H^*\oplus L^*H^*),\quad
 \varepsilon=(\sigma_x,\sigma_y),\quad
 b_W=(\sigma_y,\sigma_x),\quad \Gamma_S=\Gamma_W=\sigma_z.
                                                               \tag{ND.2}
\]

Put \(D=\det S=LH^2\). Then the entire graded bundle \(W\otimes D\) has even line \(LH\) and odd line \(H\). Define the **odd** unitary

\[
 J:W\otimes D\longrightarrow S,\qquad
 J(w_0\otimes d)=-i s_1,\quad J(w_1\otimes d)=s_0,
 \qquad J=\begin{pmatrix}0&1\\-i&0\end{pmatrix}.       \tag{ND.3}
\]

Here local unit frames are chosen coherently so that \(d\) has transition \(h^2\ell\), with \(\ell\) the transition of \(L\). The source transition on \(W\otimes D\) is \(\operatorname{diag}(h\ell,h)\); the target transition is \(\operatorname{diag}(h,h\ell)\). Therefore

\[
 J\operatorname{diag}(h\ell,h)=\operatorname{diag}(h,h\ell)J,
 \quad J^*J=JJ^*=1,
 \quad J\Gamma_W=-\Gamma_SJ,
 \quad J(-b_{W,j})=\varepsilon_jJ\quad(j=1,2).          \tag{ND.4}
\]

All four matrix entries, both physical generators, both gradings and the full transitions occur here. This is not the impossible even intertwiner excluded in Exercise 83. After replacing \(W\otimes D\) by its parity reversal \(\Pi(W\otimes D)\), \(J\) is even. Reversing the grading of an even Kasparov cycle, keeping its odd self-adjoint operator, represents its additive negative. Explicitly, on the direct sum of the cycle and its grading reversal the grading is \(\operatorname{diag}(\Gamma,-\Gamma)\), the operator is \(F\oplus F\), and \(X=\left(\begin{smallmatrix}0&\Gamma\\\Gamma&0\end{smallmatrix}\right)\) is odd, self-adjoint, unitary and commutes with the representation. Its anticommutator with \(F\oplus F\) is zero. The path \(\cos t(F\oplus F)+\sin t X\), \(0\le t\le\pi/2\), ends in the degenerate cycle with operator \(X\). Thus this standard sign has an actual cycle interpretation.

Tensor the entire represented radial module \(\mathcal H_W\) on the right by the finite projective line module \(\Gamma(D)\). A partition of unity in unit line frames proves the onto unitary

\[
 \mathcal H_W\widehat\otimes_{C(V)}\Gamma(D)
       \longrightarrow L_s^2(G,s^*(W\otimes D)),
 \quad \xi\otimes d\longmapsto(\gamma\mapsto\xi(\gamma)d(s\gamma)).
                                                               \tag{ND.5}
\]

Its inner-product identity is the actual integral of the Hermitian tensor pairing. Its image contains all compact continuous local line-frame tests and is dense, so the isometry is onto. Regular convolution keeps \(s\gamma\) fixed and acts trivially on these finite coefficients. Consequently (ND.3), applied at \(s\gamma\), intertwines the whole regular left \(A\) action and the right \(C(V)\) action exactly. It conjugates the full radial operator \(-b_W(v)/\sqrt{1+|v|^2}\) to \(+\varepsilon(v)/\sqrt{1+|v|^2}\).

This tensor cycle is the actual product by \([D]\): its second operator is zero and its left scalar representation on the finite projective line is compact. For every homogeneous first-module vector \(\xi\), its creation map \(T_\xi:d\mapsto\xi\otimes d\) is compact: for a finite Parseval frame \((d_k)\) of the line module, \(T_\xi=\sum_k\theta_{\xi\otimes d_k,d_k}\). Consequently \(FT_\xi-(-1)^{|\xi|}T_\xi0=FT_\xi\) and its adjoint connection error are actual rectangular module compacts. The finite frame also takes every first-module rank-one compact to a finite sum of tensor-module rank-one compacts. Thus the full localized cycle defects remain in the correct compact ideal. Equivalently the finite projection-corner unitary in (ND.5) verifies this product directly. The first-operator positivity sandwich is \(2F^2\ge0\) before quotienting by that ideal. No infinite tensor product compactness assumption occurs; the second module here is a finite projective line.

To recover the literal native radial cutoff, set

\[
 a_t(r)=(1-t)\frac r{\sqrt{1+r^2}}
                  +t\frac r{\max(1,r)},\quad 0\le t\le1.
                                                               \tag{ND.6}
\]

This gives a norm-continuous path of odd self-adjoint multipliers on the same entire represented module. The amplitude defect tends to zero uniformly at infinity. The angular estimate (H.17), its two-sided Schur bound, and the separated-kernel finite-rank approximation of Lemma 11.3 apply unchanged because \(\|\varepsilon(v)\|=|v|\) and all finite coefficient transitions are unitary. Outside radius \(R>1\), the derivative bound for the interpolated amplitude is at most \((1+R^2)^{-3/2}\); inside a bounded radius the continuous kernel approximation treats the unit-circle corner. Thus all defects are compact over \(C(V\times[0,1])\), including the mixed-radius commutator terms. This is an actual radial cycle homotopy to (22), not an identification from equal norms.

Equations (ND.3)–(ND.6) prove the first equality of (ND.1) in \(KK(A,C(V))\). Compose with the calibrated first-product Theorem 11.1. Associativity gives

\[
 T\delta_S=-(T\delta_{\rm inv})[D]=-[D].              \tag{ND.7}
\]

All compact ideals, creation connections and positive sandwiches in that first-product theorem are retained. This proof uses its actual whole interval product; it does not cancel an evaluation map in ordinary KK. It also does not alter the native Thom symbol or put an invented positive \(\delta\) into the inverse class.

##### The complete flat product and sign conventions

The first factor at the flat endpoint is the prescribed spinor multiplier. With the negative physical Fourier phase, the positive radial second factor becomes \(+i\varepsilon\partial_p\). On the full four-dimensional coefficient the product oscillator is

\[
 Q_S=\sum_{j=1}^2A_jp_j+i\sum_{j=1}^2B_j\partial_{p_j},
 \quad A_j=\varepsilon_j\otimes1,
 \quad B_j=\Gamma_S\otimes\varepsilon_j.
                                                               \tag{ND.8}
\]

The four matrices are self-adjoint, their two ordered Clifford actions anticommute, and their squares are one. The two commuting self-adjoint involutions \(C_j=iA_jB_j\) give

\[
 Q_S^2=-\Delta_p+|p|^2-C_1-C_2.                       \tag{ND.9}
\]

Their joint \(+1,+1\) eigenspace is the **entire** one-dimensional line

\[
 \mathbb C\psi,\qquad
 \psi=(s_0\otimes s_1+i s_1\otimes s_0)/\sqrt2,
 \quad (\Gamma_S\otimes\Gamma_S)\psi=-\psi.          \tag{ND.10}
\]

The joint multiplicities of \((C_1,C_2)\) are one for each of the four pairs of signs. The scalar two-dimensional Hermite eigenvalues are \(2|n|+2\). It follows from (ND.9) that the entire kernel is the scalar normalized Gaussian \(\pi^{-1/2}e^{-|p|^2/2}\) times (ND.10), and the square gap is two. The transition on both tensor summands in (ND.10) is \(h^2\ell\). Thus its full coefficient module is \(\Pi\Gamma(D)\), giving \(-[D]\), with literal left action \(h_0(x)\) for \(h_0\in C(V)\).

This is also a product representative. The mixed-resolvent creation argument in Theorem 6.0c and equations (138)–(140) only uses the anticommuting physical Clifford relations, the compact-support bounded creation error and the Hermite compact resolvent. They hold for the entire matrices \(A_j,B_j\) just displayed. The creation error for a compact smooth spinor section \(\xi(p)\) is multiplication by its derivative and \(A(p)\xi\), hence bounded with compact support; the two mixed resolvent identities and their adjoints have the integrable \(|\operatorname{Im}z|^{-2}\) bound. Norm density gives every creation vector. For the positive sandwich, take \(b=A(p)/\sqrt{1+|p|^2}\), \(g=|p|^2/\sqrt{1+|p|^2}\), and \(S_0=(1-\Delta+|p|^2)^{-1/2}\). The full anticommutator is \(2gS_0+K_bS_0+Q_S[S_0,b]\), with bounded \(K_b\). The latter two terms and \([g,S_0^{1/2}]S_0^{1/2}\) are compact by the same Hermite resolvent integrals, whose bounds are respectively \(ct^{-1/2}/(1+t)\) and \(ct^{-1/4}(1+t)^{-3/2}\). The remaining \(2S_0^{1/2}gS_0^{1/2}\) is positive and bounded. These are the actual connection and localized positive sandwich conditions, not a calculation of scalar rank alone.

A global scalar off-diagonal phase is conjugation by an even diagonal unitary, already treated in ND.2. Changing the exponential sign of Fourier transform sends both \(p\) and its derivative to their negatives under the even coordinate-reflection unitary \(f(p)\mapsto f(-p)\). It preserves the represented class and the odd ground parity. Changing only the sign of a Dirac operator is also an even unitary conjugation by its grading; for this even-dimensional class it is not an additive grading reversal. A simultaneous opposite grading of the two copies still gives even total parity on \(|00\rangle,|11\rangle\) and odd total parity on \(|01\rangle,|10\rangle\). Reversing the ordered physical chirality in both copies changes the relative Gaussian phase but leaves it in the latter odd plane. In each case the full ground transition is the product of the two spinor lines, \(\det S\), and its rank is still \(-1\).

Therefore the historical omissions of a Pauli-matrix display and a scalar Fourier phase cannot rescue the printed first-identity claim while keeping its **same** graded \(S\) and Clifford map in both factors. A correction that dualizes a module or reverses just one grading is a substantive correction of the prescribed pair.

##### A compact instance with actual disk leaves

**Lemma 11.9 (faithful surface suspension).** A closed genus-two surface group has a faithful representation into \(SU(2)\). Its left-translation suspension over the closed hyperbolic surface is a compact smooth five-manifold with the actual disk leaves required above.

Here is a proof including the big-powers step. It makes no use of a general Haagerup, assembly or faithful-representation theorem.

Use the exact quaternion pair

\[
 A=\frac{1+2\mathbf i}{\sqrt5},\qquad
 B=\frac{1+2\mathbf j}{\sqrt5}\quad\hbox{in }SU(2).      \tag{ND.16}
\]

Here is its complete all-length proof. The quaternion-to-matrix map gives \(A=\operatorname{diag}(1+2i,1-2i)/\sqrt5\) and \(B=\left(\begin{smallmatrix}1&2\\-2&1\end{smallmatrix}\right)/\sqrt5\). The integral numerator quaternion ring maps to \(M_2(\mathbb F_5)\) by \(\mathbf i\mapsto\operatorname{diag}(2,3)\), \(\mathbf j\mapsto\left(\begin{smallmatrix}0&1\\4&0\end{smallmatrix}\right)\); the two squares are \(-1\) and they anticommute. The four letter images, ordered \(a,a^{-1},b,b^{-1}\), factor as \(u_iv_i^T\), with

\[
 (u_1,u_2,u_3,u_4)=((0,1),(1,0),(1,3),(1,2)),\qquad
 (v_1,v_2,v_3,v_4)=((0,2),(2,0),(1,2),(1,3)).          \tag{ND.17}
\]

Their complete adjacent contraction table \((v_i^Tu_j)\) modulo five is

\[
 \begin{pmatrix}2&0&1&4\\0&2&2&2\\2&1&2&0\\3&1&0&2\end{pmatrix}.  \tag{ND.18}
\]

The zeros are precisely adjacent inverse pairs. Every nonempty freely reduced word therefore has nonzero matrix numerator: the full rank-one product is \(u_1(\prod v_i^Tu_{i+1})v_n^T\), a product of nonzero field scalars and nonzero outer vectors. If a length-\(n\) word were the unit quaternion, its integral numerator would be the scalar \(5^{n/2}\). For even positive \(n\) this is divisible by five, contrary to the nonzero matrix image. For odd \(n\) it is irrational, contrary to the integral scalar component of the numerator. Thus the pair is faithful at every word length. In particular \(C=[A,B]\) has infinite order. The displayed rank-one contractions establish the claim for every word length, rather than just for finitely many words.

We now give the needed free-group big-powers fact rather than citing it. In the Cayley tree of \(F(a,b)\), let \(w=[a,b]=aba^{-1}b^{-1}\) and let \(\mathscr L\) be its translation axis. The cyclic word has length four and has no period one or two, so \(w\) is not a proper power. An element preserving \(\mathscr L\) must act by translation: a reflection would fix a vertex or invert an edge, neither of which is possible in this free action. Its translation commutes with \(w\); their commutator fixes a vertex and is therefore the identity. The smallest positive axis translation would be a root of \(w\), so the stabilizer is exactly \(\langle w\rangle\).

If \(r\notin\langle w\rangle\), the intersection \(\mathscr L\cap r\mathscr L\) has finite length \(b(r)\le3\), with length zero when empty. Indeed, four successive common labelled edges determine the whole period-four sequence on both axes. This sequence uniquely determines both continuations: every label is the corresponding label four edges later or earlier. Unique labelled edges in the Cayley tree then force both full axes to agree, contrary to the stabilizer calculation. This also covers oppositely oriented common edges by using the inverse cyclic word. In particular an infinite common ray is impossible. The numerical bound three can replace every \(b(r_i)\) in (ND.12) if an explicit bound solely in word lengths is desired.

Consequently a word

\[
 r_0w^{n_1}r_1w^{n_2}\cdots r_{m-1}w^{n_m}r_m,
 \qquad r_i\notin\langle w\rangle\ (0<i<m),          \tag{ND.11}
\]

is nontrivial when every \(4|n_j|>2M\), where one may take

\[
 M=1+|r_0|+|r_m|+
            \sum_{i=1}^{m-1}(|r_i|+b(r_i)).           \tag{ND.12}
\]

To verify this cancellation bound, draw each power as a straight segment on a translate of \(\mathscr L\), and each \(r_i\) as its reduced connector. In reduction across that connector, edge cancellation before reaching the next axis consumes at most its length \(|r_i|\); further cancellation on both axes is a common axis segment and consumes at most \(b(r_i)\) from either side. At the endpoints the bounds are \(|r_0|,|r_m|\). Reduce from left to right. Each long segment retains an interior subsegment after both its adjacent bounds are removed, because their sum is less than \(2M\). In particular no entire long segment can disappear to permit cancellation with a nonadjacent axis. This justifies the induction and leaves a nonempty reduced path. The argument works for arbitrary signs of the powers. Every bound depends on the fixed connectors, not on their powers. This is the required big-powers proof.

Present the genus-two group as

\[
 \Gamma=\langle a,b,c,d\mid[a,b][c,d]=1\rangle
       =F(a,b)*_{\langle[a,b]\rangle}F(c,d),           \tag{ND.13}
\]

where the edge identifies \([a,b]\) with \([c,d]^{-1}\). Here is the reduced-normal-form proof needed below. In each factor choose representatives \(\mathcal T_i\) for the cosets \(tE\) of the common cyclic subgroup \(E\), including \(1\). Formal normal forms are \(t_1\cdots t_k e\), where \(e\in E\), each \(t_j\in\mathcal T_{i_j}\setminus\{1\}\), and successive \(i_j\) differ. Right multiplication by \(g\) in a factor starts with the unique coset decomposition \(eg=t e'\). If \(t=1\), replace the terminal edge term by \(e'\). If the last representative is in the other factor, append \(t\). Otherwise uniquely decompose \(t_k t=t'e''\), replace \(t_k\) by \(t'\) (remove it when \(t'=1\)), and put \(e''e'\) at the end. These rules preserve the alternating condition.

Right multiplication by two elements of the same factor equals multiplication by their product: in each of the last-factor cases this follows from the uniqueness of its coset decomposition; when a representative is removed apply the same rule to the new last-factor case. Induction on the number of removed representatives completes the check. Multiplication by an edge element has the same rule in either factor, namely multiplication of the terminal edge term. Inverses are therefore right multiplication by \(g^{-1}\). These rules define an action of the presented group on the set of formal normal forms. Starting with the empty form and appending a formal normal-form word gives that very form, so two different forms cannot represent the same element of the presentation. This proves uniqueness without importing an amalgamated-product theorem. In an alternating word whose factors are outside \(E\), appending the next factor always adds a nonidentity representative: multiplying it on the left by a terminal edge element cannot place it in \(E\). Its reduced length is positive. A nontrivial edge word has its nontrivial terminal edge term. These arguments prove both factor injections and the required nonvanishing of reduced alternating words.

The folding map sends \((a,b,c,d)\) to \((a,b,b,a)\) in \(F(a,b)\); on each factor separately it is an isomorphism. Twist the second factor by \(w^N\): its images are \(w^Nbw^{-N},w^Naw^{-N}\). After applying this map to an alternating reduced form, the interior connectors in (ND.11) are the images of its noncyclic factors, hence are not in \(\langle w\rangle\). The power exponents are alternately \(N,-N\). Equations (ND.11)–(ND.12) show that every fixed nontrivial surface word has nontrivial folded image for all sufficiently large \(|N|\). Words in one factor are already nontrivial. This is the explicit discrimination statement needed next.

For this exact pair, quaternion multiplication gives \(C=[A,B]=(-7+16\mathbf i-16\mathbf j+8\mathbf k)/25\). Put \(K=(2\mathbf i-2\mathbf j+\mathbf k)/3\), so \(K^2=-1\), and set \(z_t=\cos t+K\sin t\). This is the unit circle centralizing \(C\); equivalently it is \(\operatorname{diag}(e^{it},e^{-it})\) after diagonalization. We have \(C=z_\theta\), with \(\cos\theta=-7/25\), \(\sin\theta=24/25\). Define

\[
 \rho_t(a)=A,\quad\rho_t(b)=B,\quad
 \rho_t(c)=z_tBz_t^{-1},\quad
 \rho_t(d)=z_tAz_t^{-1},\qquad t\in\mathbb R/2\pi\mathbb Z.
                                                               \tag{ND.14}
\]

The relation holds because \(z_t\) commutes with \(C\) and \([B,A]=C^{-1}\). Each fixed word gives a real-analytic periodic matrix function. At \(t=N\theta\) it is the faithful free-pair image of the twisted folding just proved. Therefore a nontrivial word does not give the identity function of \(t\). At least one of its matrix entries minus the identity entry is a nonzero real-analytic function of one real variable. Its zero set on the compact circle is finite. The union over the countably many nontrivial words is countable, so choose \(t\) outside it. Then \(\rho_t\) is faithful. No density or irrational-orbit assertion is needed for this selection.

Finally choose the closed hyperbolic genus-two surface \(\Sigma\) obtained by side-pairing a regular hyperbolic octagon with vertex angles \(\pi/4\), in the order \(a,b,a^{-1},b^{-1},c,d,c^{-1},d^{-1}\). The right triangle from the center to a vertex and a side midpoint has both nonright angles \(\pi/8\); the hyperbolic angle cosine law gives \(\cosh R=\cot^2(\pi/8)\). Its disk circumradius is therefore \(\tanh(R/2)=2^{-1/4}\), exactly the value rendered in Figure 11.4. Its identified vertex has total angle \(2\pi\), so the glued metric is smooth; compactness makes it complete. Its universal cover is the complete simply connected curvature \(-1\) disk. Form

\[
 V=(\mathbb H^2\times SU(2))/\Gamma,
 \qquad \gamma(z,k)=(\gamma z,\rho_t(\gamma)k).        \tag{ND.15}
\]

The first-factor deck action makes this action free and proper, and the projection to \(\Sigma\) is a smooth bundle with compact fibre \(SU(2)\); thus \(V\) is a compact smooth connected five-manifold without boundary. Use the round unit-quaternion metric on \(SU(2)\). Left multiplication by a unit quaternion preserves the Euclidean norm on \(\mathbb R^4\) and hence its inner product, so the transverse action is isometric; the product metric descends. Horizontal leaves are the images of \(\mathbb H^2\times\{k\}\). Their stabilizer consists of \(\gamma\) satisfying \(\rho_t(\gamma)k=k\), hence \(\rho_t(\gamma)=1\), hence \(\gamma=1\). Each actual leaf, with its descended horizontal metric, is therefore the complete disk itself. Its oriented tangent bundle is \(F=p^*T\Sigma\), and the canonical \(L\) is \(p^*T^{1,0}\Sigma\). The hypotheses of Theorems 10.1 and 11.1 hold on this compact example.

On it (ND.7) gives \(T\delta_S=-[L]\). Acting on \([1]\in K_0(C(V))\) and then pulling back to any point gives \(-1\in K_0(\mathbb C)\); the identity gives \(+1\). Thus they are unequal even as K-theory maps, and hence unequal in KK. This example has the original actual-disk hypotheses.



**Remark 11.10 (the calibrated analytical answer and the literal sign).** The native first product gives injectivity on K-theory after composing with the inverse negative line \(-[D^*]\). Section 11B proves the calibrated reverse identity and its full represented \(A,A\) comparison, and hence surjectivity. In particular the literal unchanged positive radial class cannot have the reverse identity either. If \(\delta_ST=1_A\), right-composition with the proved \(T\delta_{\rm inv}=1_C\) would imply \(\delta_S=\delta_{\rm inv}\), contradicting (ND.1). Any boundary-representation homotopy to \(1_A\) must therefore be compared with the calibrated whole module, including the negative radial phase and opposite physical chirality, rather than silently substituted for the native positive class. No cancellation of an onto evaluation map is used.

![Full native parity-line comparison and a compact actual-disk suspension](../figures/kt-native-delta-interface.png)

**Figure 11.4.** Proposition 11.8 and Lemma 11.9. The full line, grading and ordered-generator equations are (ND.2)–(ND.7). The displayed map is odd before its stated parity reversal. The regular hyperbolic octagon has disk radius \(2^{-1/4}\), vertex angles \(\pi/4\), and metric \(4|dz|^2/(1-|z|^2)^2\); its side labels specify (ND.13). The suspension formula is a schematic quotient, not an embedding of the five-manifold or its accumulating leaves in the disk. The proof includes the exact quaternion pair, all-length modular proof, reduced normal forms, explicit big-powers bound and separating-curve twist. Native source context: [Connes], Sections 10(3) and12; full product calibration: Lemma11.2, Theorem11.1 and Exercise83.

[Full-size PNG](https://kokunoyumeto.github.io/open-math-courses-public/courses/NCG-FOLIATIONS/figures/kt-native-delta-interface.png) · [Editable SVG](https://kokunoyumeto.github.io/open-math-courses-public/courses/NCG-FOLIATIONS/figures/kt-native-delta-interface.svg) · [Generator](https://kokunoyumeto.github.io/open-math-courses-public/courses/NCG-FOLIATIONS/figures/render-native-delta-interface.py).

### Reduced assembly can fail

The survey asks whether its geometric assembly map is always an isomorphism. Its foliation algebra is completed in the supremum of the regular norms, and non-Hausdorff holonomy is explicitly permitted: [Connes, Section 6, author PDF p.19](https://alainconnes.org/wp-content/uploads/foliationsfine.pdf#page=19). In that scope the universal assertion is false. We prove failure of surjectivity for a compact smooth foliation, using the compact geometric cycles of Section 6 with their entire prescribed spin-c structure on \(TM\oplus f^*\tau\). The argument retains that domain throughout.

The obstruction has two stages. Every such geometric class has a lift through the maximal holonomy algebra. The example then has a reduced K-class with no maximal lift. The distinction between the two completions is essential even though the free-group crossed products appearing on a closed transverse arc have the same K-theory.

#### The full-completion proof interfaces

The complete preceding programme proofs used for this analytic step are Groupoid C*-algebras: full and reduced, the section “The full norm and disintegration” through (14E.18), Theorem 14.3 and Theorem 14.5; and Equivalence of groupoids and Morita equivalence of their C*-algebras, Lemmas 15.1–15.2, Theorem 15.3 and (15.22–15.23). Both proofs allow second countable locally compact locally Hausdorff arrows and equivalence spaces, Hausdorff units and full-support Haar systems. Their test algebra is the span of compact continuous functions on Hausdorff open patches, as in the foliation algebra here.

Here is the precise analytic content needed. Lemma 15.1 constructs a diagonal approximate identity from finite sums of cross-arrow products before universal positivity is assumed. Lemma 15.2 approximates each full diagonal Gram matrix by actual sums of convolution-square Gram matrices, proving positivity in **all** full representations. Theorem 15.3 then induces an arbitrary diagonal representation to the linking algebra: the positive tensor form has continuous coefficients, a dense essential span and separable completion, so the proved dense-domain disintegration of Lesson 14 makes its operators I-contractive. Compression recovers the arbitrary diagonal representation and therefore the entire maximal corner norm. The theorem separately proves the regular corner norms and fullness. These steps supply the maximal Hilbert-module completion; positivity only in regular representations would supply the reduced module instead.

The off-diagonal corner and its compact algebra are proved in Imprimitivity bimodules and Morita equivalence, Theorem 5.1; its Theorem 3.2 supplies the inverse evaluation tensor unitaries. The Rieffel correspondence and induced representations, Lemma 1.1 and Proposition 2.2, proves the ideal submodule, complete quotient and its norm. Tensor products and C*-correspondences, Theorem 1.2, Proposition 2.2, Theorem 3.1 and Proposition 3.2, proves the tensor completion, second-factor transport, associator and unit maps. These are original CC0 programme proofs. The historical equivalence sources are credited in the references.

#### Geometric representatives lift through the maximal algebra

Write \(H=\operatorname{Hol}(V,\mathcal F)\), \(A_m=C^*_{\max}(H)\), \(A_r=C_r^*(H)\) and \(\lambda _H:A_m\to A_r\). Here the subscript \(m\) changes the completion only; it does not change the real normal bundle \(\tau =TV/F\) or its orientation twist.

**Theorem 11A.1 (maximal lifts of geometric values).** For the compact geometric assembly of Section 6,

\[
 \operatorname{im}\mu_r
 \subseteq\operatorname{im}\bigl((\lambda_H)_*:K_*(A_m)\to K_*(A_r)\bigr).
 \tag{A.G.1}
\]

**Proof.** Fix one compact representative \((M,E,f)\), with \(d=\dim M\) and \(q=\operatorname{rank}\tau\). Use exactly its embedding \(j:M\to \mathbb R^n\), full prescribed normal spin-c module, tube \(N\) and étale map-cocycle \(g\) from Section 6. The associated principal graph \(Z=\Gamma _g\) has quotient \(p:Z\to N\) and right anchor \(s:Z\to V\times \mathbb R^n\). In tube coordinates its transverse map is \((x,y,\eta )\mapsto (y,j(x)+\eta )\). The tubular coordinates \((x,\eta )\) and leaf coordinates show that \(s\) is a local diffeomorphism. Consequently \(W=s(Z)\) is open and saturated for \(H^n=H\times\mathbb R^n\).

The gauge groupoid and its action are

\[
 L=(Z\times_s Z)/H^n\rightrightarrows N,
 \qquad r[z,w]=p(z),\quad s_L[z,w]=p(w).
 \tag{A.G.2}
\]

If \(p(w)=p(u)\) and \(w\gamma =u\), define \([z,w]u=z\gamma\); principal division makes this independent of simultaneous right translation. Local cocycle charts identify its arrows with changes between transverse sheets of \(N\). Hence it is étale, second countable, locally compact and locally Hausdorff, with Hausdorff units \(N\) and counting Haar system.

The graph is an \((L,H^n|_W)\) equivalence. Both actions are free. Their division maps identify their action-relation maps with the inclusions of \(Z\times _s Z\) and \(Z\times _p Z\) in \(Z\times Z\); these relations are closed because their anchor bases are Hausdorff. The actions are therefore proper, including after base change. The orbit spaces are \(Z/H^n=N\) and \(L\backslash Z=W\), and both anchors are open and surjective onto these spaces. Holonomy source fibres are Hausdorff holonomy covers of leaves, étale fibres are discrete Hausdorff spaces, and cross fibres are identified with these Haar fibres by division. The full-support Haar and local Hausdorff hypotheses of the preceding interfaces are all satisfied. Half-densities give the same local isometry between their Haar convention and the section-core convention of Section 1.

The proved full equivalence theorem produces a maximal Hilbert module \(Q_m\) from that section core, with

\[
 C^*_{\max}(L)=\mathcal K(Q_m).
 \tag{A.G.3}
\]

Its convolution form is the same counting form used in the geometric construction. Since \(L\) is étale, its units are open. Zero extension puts \(C_c(N)\) in its convolution algebra; its full norm is the supremum norm, by the I-upper bound and the lower bound from regular unit vectors. Thus the left action \(a(p(z))\), \(a\in C_0(N)\), belongs to \(C^*_{\max}(L)=\mathcal K(Q_m)\). Countable charts and compact-support approximation make the module countably generated. It gives a zero-operator Kasparov correspondence.

Initially the coefficients are \(C^*_{\max}(H^n|_W)\). The proved invariant-open theorem, Lesson 14 Theorem 14.5, embeds this algebra isometrically as an ideal in \(C^*_{\max}(H^n)\): extend the constructed quasi-invariant measure and Hilbert fibres by zero outside \(W\). Regular fibres prove the analogous reduced inclusion directly. The same ideal-valued inner product can therefore be viewed in the whole coefficient algebra. It need not be full over that whole algebra when \(W\) is proper; only the compact left action is needed here.

Complete the same core regularly to obtain \(Q_r\). The quotient has the concrete tensor unitary

\[
 Q_m\widehat\otimes_{\lambda_{H^n}}C_r^*(H^n)
 \longrightarrow Q_r,\qquad \xi\otimes a\longmapsto\xi a.
 \tag{A.G.4}
\]

Indeed its inner product on finite tensors is
\(a^*\lambda_{H^n}(\langle\xi,\eta\rangle_m)b=\langle\xi a,\eta b\rangle_r\). This proves balance, positivity, null-space compatibility and isometry. A coefficient approximate identity makes its range dense, so it is onto. The left action is the same multiplication on the same core. Rank-one operators pass to rank-one operators on the quotient core, and approximation proves that compact left actions pass to compact left actions.

We also need the product with the Euclidean unit groupoid. In both norms,

\[
 C^*_\bullet(H\times\mathbb R^n)
 =C_0(\mathbb R^n)\otimes C^*_\bullet(H),
 \qquad\lambda_{H^n}=1\otimes\lambda_H.
 \tag{A.G.5}
\]

Here is a direct full-norm check. Compact product-patch approximation makes finite sums \(a(t)h(\gamma )\) dense in the I-topology. The parameter functions give a nondegenerate central \(C_0(\mathbb R^n)\) multiplier action. In an irreducible full representation it is a scalar character, hence evaluation at some \(t\), by Schur's lemma and nondegeneracy. Lesson 14 Theorem 14.5, applied to the invariant complement of that parameter, identifies its quotient with \(C^*_{\max}(H)\). Conversely every full representation of \(H\) composed with evaluation is a full representation of the product. Thus the full norm is \(\sup_t\lVert h_t\rVert_{\max}\), the continuous-function tensor norm. In the reduced norm this equality follows from the regular fibres \((x,t)\). This proves (A.G.5) and the displayed quotient without introducing a second completion convention.

Let \(t\) be the same normal Thom class as in Section 6 and let \(b_n\) be inverse Euclidean Bott. The actual typed factors are

\[
 \begin{aligned}
 t&\in KK^{n-d+q}(C(M),C_0(N)),\\
 e_m&\in KK(C_0(N),C_0(\mathbb R^n)\otimes A_m),\\
 b_n^m&=b_n\widehat\otimes1_{A_m}
 \in KK^n(C_0(\mathbb R^n)\otimes A_m,A_m).
 \end{aligned}
 \tag{A.G.6}
\]

Equation (A.G.4) gives \(e_m \widehat\otimes [1\widehat\otimes \lambda _H]=e_r\). The tensor associator on the inverse Bott module gives

\[
 b_n^m\widehat\otimes[\lambda_H]
 =[1\otimes\lambda_H]\widehat\otimes b_n^r.
 \tag{A.G.7}
\]

The quotient is even, so no graded interchange sign appears. The full spin-c module, source representation, Clifford order, phase and tube are the same on both sides. Associativity therefore yields the equality of actual wrong-way classes

\[
 f_!^r=
 \bigl(t\widehat\otimes e_m\widehat\otimes b_n^m\bigr)
 \widehat\otimes[\lambda_H]
 \quad\text{in }KK^{q-d}(C(M),A_r).
 \tag{A.G.8}
\]

Multiplying by \([E]\) proves that this representative's assembly value is the image of a class in \(K_*(A_m)\). The assembly group is generated by those compact representatives, and a homomorphism's image is a subgroup. Thus every value lies in that subgroup, proving (A.G.1). This argument does not require a separately descended maximal geometric assembly through every relation: the reduced assembly has already descended in Theorem 6.1. ∎

#### The free-group comparison

The obstruction below needs a comparison for the actual coefficient algebra. We prove it using the one-defect Toeplitz method of M. Pimsner and D. Voiculescu, *K-groups of reduced crossed products by free groups*, Journal of Operator Theory 8 (1982), 131–156, Sections 1–3 ([original paper](https://www.theta.ro/jot/archive/1982-008-001/1982-008-001-006.pdf)).

The complete reduced construction is Cuntz–Krieger algebras, minimal systems and projectionless algebras, Section 6, Lemmas 6.2–6.3. Its word constructions and rotations are CC0 and are adapted below to a universal maximal algebra. For the elementary difference map, including naturality, homotopy invariance and cancellation of a common summand, use Cuntz algebras, Theorem 6.3, equations (6.11)–(6.13). Extension exactness and boundary naturality are The six-term exact sequence and the exponential map, Theorem 2.1. The vacuum-corner K-isomorphisms follow from Matrix stability, stability and continuity of K-zero, Theorem 4.1 and Invertibles, unitaries and K-one, Corollary 4.2.

**Theorem 11A.2 (the canonical free-group comparison).** Let the free group \(\mathbb F_n\) act by automorphisms on a C*-algebra \(C\). The canonical quotient

\[
 q_C:C\rtimes_{\max}\mathbb F_n\longrightarrow C\rtimes_r\mathbb F_n
 \tag{A.FG.1}
\]

induces isomorphisms on \(K_0\) and \(K_1\).

We first treat unital \(C\), removing one generator at a time. For \(n\geq1\), fix generators \(g_1,\ldots,g_n\), put \(H=\langle g_1,\ldots,g_{n-1}\rangle\), and write \(\alpha_w\) for the automorphism associated to a word \(w\). For \(n=1\), \(H\) is trivial. Let \(W\) consist of the empty word \(e\) and all reduced words whose last letter is different from \(g_n^{-1}\). Then

\[
 g_jW=W\quad(j<n),\qquad g_nW=W\setminus\{e\}.
 \tag{A.FG.2}
\]

Left cancellation changes the last letter only when it cancels the entire word. For an H-letter this stays within \(W\), including at \(e\). For \(g_n\), the only missing preimage is \(g_n^{-1}\). This proves (A.FG.2). Every nonempty suffix of a word in \(W\) is also in \(W\).

##### A single defect and its compact ideal

Let \(\mathcal T_n^{\max}(C)\) be the universal unital C*-algebra generated by a unital coefficient map \(c:C\to\mathcal T_n^{\max}(C)\), unitaries \(L_j\), \(j<n\), and an isometry \(S\), with

\[
 L_jc(a)L_j^*=c(\alpha_j(a)),\qquad
 c(a)S=Sc(\alpha_n^{-1}(a)),\qquad S^*S=1.
 \tag{A.FG.3}
\]

Put \(P=1-SS^*\). Substitute \(L_j^{\pm1},S,S^*\) for the letters of a reduced word \(w\), and call the product \(\sigma_w\). Impose the wandering relations

\[
 P\sigma_wP=0\qquad(w\in W\setminus\{e\}).
 \tag{A.FG.4}
\]

This universal algebra exists. Take the supremum of the norms of each algebraic polynomial in representations satisfying the displayed relations. The group generators have norm at most one, and a coefficient has norm at most its C-norm, so the supremum is finite. Divide out the zero seminorm and complete.

There is a faithful coefficient representation. On the standard Hilbert C-module \(E_C=\ell^2(W)\otimes C\), write \(\delta_wb\) for its w-coordinate and define

\[
 \begin{aligned}
 \rho_C(a)\delta_wb&=\delta_w\alpha_w^{-1}(a)b,\\
 L_j\delta_wb&=\delta_{g_jw}b\quad(j<n),\\
 S\delta_wb&=\delta_{g_nw}b.
 \end{aligned}
 \tag{A.FG.5}
\]

These operators satisfy (A.FG.3)–(A.FG.4), and their defect is the vacuum projection \(p_e\). The coefficient is faithful on that coordinate, so \(c\) is faithful.

**Lemma 11A.2a (the defect ideal).** The ideal generated by \(P\) is canonically \(J=\mathcal K(\ell^2(W))\otimes C\). Quotienting by it gives \(C\rtimes_{\max}\mathbb F_n\).

*Proof.* Covariance makes \(P\) commute with \(c(C)\). The defining relations give

\[
 P\sigma_w^*\sigma_zP=\delta_{wz}P\qquad(w,z\in W).
 \tag{A.FG.6}
\]

Here is the cancellation check. Construct \(w\) from its rightmost letter. H-letters are unitary and positive \(g_n\)-letters are isometric. Before a negative \(g_n\)-letter, the constructed suffix \(v\) is nonempty and lies in \(W\). Relation (A.FG.4) puts \(\sigma_vP\) perpendicular to \(P\), so \(S^*\) is isometric on its range. A negative \(g_n\)-letter cannot be the first applied letter. Thus \(\sigma_wP\) has initial projection \(P\).

For orthogonality, cancel the common initial letters of \(w,z\) in the left side of (A.FG.6). A positive \(g_n\)-cancellation uses \(S^*S=1\). A negative one uses \(SS^*=1-P\); its error vanishes by (A.FG.4) on the remaining nonempty suffixes. Those suffixes cannot be empty at this step, because neither word ends in \(g_n^{-1}\). H-cancellations are exact. If one remaining word is empty, (A.FG.4) gives zero unless both are empty. Otherwise the remaining first letters differ. Their inverse-product is nonempty and reduced, and its last letter is the last letter of the second word. It lies in \(W\), so (A.FG.4) again gives zero. This proves (A.FG.6).

Consequently

\[
 M_{wz}(a)=\sigma_wPc(a)\sigma_z^*,\qquad
 M_{wz}(a)M_{uv}(b)=\delta_{zu}M_{wv}(ab).
 \tag{A.FG.7}
\]

Adjoints interchange the indices and adjoint the coefficient. These define the compact matrix algebra over \(C\). Its representation is faithful: (A.FG.5) sends \(M_{wz}(a)\) to the actual matrix entry with coefficient \(a\).

Their closed span is an ideal. A coefficient multiplies the row or column coefficient by the appropriate transformed element of \(C\). A generator letter sends \(\sigma_wP\) to the row with reduced index \(gw\), unless it is the excluded case \(S^*P=0\). The only nonexact reduction is \(SS^*\sigma_vP\); it equals \(\sigma_vP\) by (A.FG.4). Here \(v\) is nonempty, since a word ending in \(g_n^{-1}\) is excluded. Adjointing gives the column calculation. The span contains \(P\) and is generated by \(P\), proving the ideal assertion.

In the quotient \(S\) is unitary. The remaining relations are precisely those of a covariant unitary representation of \((C,\mathbb F_n)\). Conversely every such representation satisfies (A.FG.4), because its defect is zero. The two universal maps fix every generator and are inverse. \(\square\)

Let \(\mathcal T_n^r(C)\) denote the concrete algebra generated by (A.FG.5). The reduced construction in the cited Lemma 6.2, together with the preceding lemma, gives the two exact sequences

\[
 \begin{aligned}
 0\longrightarrow J&\longrightarrow\mathcal T_n^{\max}(C)
 \xrightarrow{\pi_{\max}}C\rtimes_{\max}\mathbb F_n\longrightarrow0,\\
 0\longrightarrow J&\longrightarrow\mathcal T_n^r(C)
 \xrightarrow{\pi_r}C\rtimes_r\mathbb F_n\longrightarrow0.
 \end{aligned}
 \tag{A.FG.8}
\]

The canonical homomorphism \(Q:\mathcal T_n^{\max}(C)\to\mathcal T_n^r(C)\) sends each universal generator to its concrete operator. By (A.FG.7), its map on \(J\) is identity. Its map on the quotients is the canonical \(q_C\) in (A.FG.1).

##### Two rotations that remove a generator

Put \(B=C\rtimes_{\max}H\), with implementing unitaries \(u_j\). Universality gives \(d:B\to\mathcal T_n^{\max}(C)\), sending the coefficient to \(c\) and \(u_j\) to \(L_j\).

**Lemma 11A.2b (removing a generator).** The map \(d\) induces isomorphisms in both K-degrees.

*Proof.* On \(E_B=\ell^2(W)\otimes B\), let \(\pi_0\) be (A.FG.5) with coefficient algebra \(B\). Put \(p=p_e\). Delete the vacuum and bypass its two H-edges by defining

\[
 \begin{aligned}
 \pi_1(1)&=1-p,\\
 \pi_1(c(a))&=(1-p)\rho_B(a),\\
 \pi_1(S)&=S(1-p),\\
 \pi_1(L_j)\delta_{g_j^{-1}}b&=\delta_{g_j}u_j^{-1}b.
 \end{aligned}
 \tag{A.FG.9}
\]

On every other nonvacuum coordinate, \(\pi_1(L_j)\) is the usual shift; its vacuum value is zero. It is unitary on support \(1-p\). At the exceptional edge covariance follows from \(u_j^{-1}\alpha_j(a)=au_j^{-1}\). On this support its S-defect is \(p_{g_n}\). For \(w\in W\), the defect orbit is the coordinate \(wg_n\): each intermediate suffix ends in a positive \(g_n\)-block and never meets a changed edge. These coordinates are distinct. Thus all wandering relations hold, and \(\pi_1\) is a homomorphism from the universal algebra, with unit support \(1-p\).

The differences \(\pi_0-\pi_1\) are finite matrices over \(B\) on every generator, hence compact on every element by density. The difference-map construction and vacuum-corner stability give

\[
 \Delta:K_i(\mathcal T_n^{\max}(C))\longrightarrow K_i(B),\qquad i=0,1.
 \tag{A.FG.10}
\]

We verify both inverse identities. If a unital algebra \(D\) containing \(C\) has a unitary \(b_j\) implementing \(\alpha_j\), define a rotation on coordinates \(e,g_j\) by

\[
 R_j(t)=\begin{pmatrix}
 \cos t&\sin t\,b_j\\
 -e^{2it}\sin t\,b_j^*&e^{2it}\cos t
 \end{pmatrix},\qquad0\leq t\leq\pi/2,
 \tag{A.FG.11}
\]

and use identity on other coordinates. Matrix multiplication shows that it is unitary. It commutes with \(\rho_D(\alpha_j(a))\), whose two entries are \(\alpha_j(a),a\), because \(b_ja=\alpha_j(a)b_j\). Therefore \(R_j(t)L_j\) remains covariant.

Take \(D=B\), \(b_j=u_j\). These covariant unitaries represent the full crossed product \(B\) at every parameter. The path starts at \(\pi_0d\) and ends at the orthogonal sum of \(\pi_1d\) and the vacuum identity representation of \(B\). Its difference from \(\pi_1d\) stays compact. Homotopy invariance and cancellation of the common summand give \(\Delta d_*=1\).

For the other identity extend coefficients by \(d\) to \(D=\mathcal T_n^{\max}(C)\). The tensor-module unitary is

\[
 E_B\widehat\otimes_d D\longrightarrow E_D,
 \qquad \delta_wb\otimes x\longmapsto\delta_wd(b)x.
 \tag{A.FG.12}
\]

Its inner products agree and its range contains every coordinate vector. Compact matrix entries transfer by applying \(d\) to their coefficients. Thus naturality identifies the extended difference map with \(d_*\Delta\).

Write \(S_T\) for the universal isometry, now regarded as a coefficient in \(D\). First leave the H-shifts fixed and replace the module shift \(S\) by

\[
 S_t=S(1-p)+\cos t\,\theta_{g_n,e}\otimes1
       +\sin t\,p\otimes S_T,
 \qquad0\leq t\leq\pi/2.
 \tag{A.FG.13}
\]

The first term's range avoids \(e,g_n\). The vacuum column has orthogonal coordinate components with squared norm \(\cos^2t+\sin^2t=1\), so \(S_t\) is an isometry. Covariance holds termwise, including \(aS_T=S_T\alpha_n^{-1}(a)\).

We must also verify the wandering relations throughout the path. The defect of \(S_t\) is supported on \(e,g_n\). For a positive \(g_n\)-power its compression is zero because an isometry's defect annihilates that isometry. Every other nonempty \(w\in W\) has the form \(w=vg_n^k\), \(k\geq0\), where \(v\) ends in an H-letter. The powers in (A.FG.13) send the defect support into coordinates \(g_n^\ell\), \(0\leq\ell\leq k+1\). The remaining reduced letters send them to \(vg_n^\ell\), without visiting \(e,g_n\) after the first H-letter. On all those intermediate coordinates the shifts are ordinary. The final support is disjoint from the defect support, proving every wandering relation.

At the endpoint keep \(S_f=S(1-p)+p\otimes S_T\) fixed and rotate the H-generators by (A.FG.11), now with \(b_j=L_j\in D\). Its defect is

\[
 P_f=p_{g_n}\otimes1+p\otimes P_T,\qquad P_T=1-S_TS_T^*.
 \tag{A.FG.14}
\]

The coordinates \(Wg_n\) form a reducing submodule. They never meet an altered H-edge and retain the ordinary orthogonal defect orbit of \(g_n\). On the complementary coordinates, a rotated H-letter at the vacuum either remains there with coefficient \(\sin t\,L_j^{\pm1}\), or exits along that letter's branch. After exit a reduced word cannot return to the vacuum: the remaining shifts construct a nonempty reduced coordinate word. At the vacuum, \(g_n\)-letters act by \(S_T,S_T^*\). Hence the vacuum block of a reduced word is

\[
 (\sin t)^{m(w)}\sigma_w(T),
 \tag{A.FG.15}
\]

where \(m(w)\) counts H-letters and \(\sigma_w(T)\) is its substitution in \(D\). Its compression by \(P_T\) vanishes for nonempty \(w\in W\) by the universal relation (A.FG.4). The reducing \(Wg_n\) block has zero compression separately. Thus every wandering relation also holds throughout the second stage.

The verified relations give representations of the universal algebra at every parameter. The paths vary in norm on generators; polynomial density and contractivity make them point-norm continuous on the whole algebra. Their differences from the extended \(\pi_1\) stay compact. Initially the representation is the extended \(\pi_0\). Finally it is the orthogonal sum of the extended \(\pi_1\) and the vacuum identity representation of \(D\). Naturality, homotopy invariance and common-summand cancellation give \(d_*\Delta=1\). This proves the lemma in both degrees. \(\square\)

##### The canonical quotient and the closed-arc action

*Proof of Theorem 11A.2.* Induct on \(n\), starting with \(\mathbb F_0=\{e\}\), where the comparison is identity on \(C\). The cited reduced Lemma 6.3 gives a K-isomorphism \(d_r:C\rtimes_rH\to\mathcal T_n^r(C)\). The preceding lemma gives the maximal K-isomorphism. The square

\[
 \begin{array}{ccc}
 C\rtimes_{\max}H&\xrightarrow{d}&\mathcal T_n^{\max}(C)\\
 \downarrow q_H&&\downarrow Q\\
 C\rtimes_rH&\xrightarrow{d_r}&\mathcal T_n^r(C)
 \end{array}
 \tag{A.FG.16}
\]

commutes on coefficients and every H-generator. Induction therefore makes \(Q_*\) a K-isomorphism. The two extensions (A.FG.8) form the diagram

\[
 \begin{array}{ccccccccc}
 0&\longrightarrow&J&\longrightarrow&\mathcal T_n^{\max}(C)
 &\xrightarrow{\pi_{\max}}&C\rtimes_{\max}\mathbb F_n&\longrightarrow&0\\
 &&\downarrow 1&&\downarrow Q&&\downarrow q_C\\
 0&\longrightarrow&J&\longrightarrow&\mathcal T_n^r(C)
 &\xrightarrow{\pi_r}&C\rtimes_r\mathbb F_n&\longrightarrow&0.
 \end{array}
 \tag{A.FG.17}
\]

Exactness and boundary naturality identify the quotient map's K-theory, as follows. The same argument works in either degree. If a maximal quotient class \(x\) maps to zero, its boundary is zero because the ideal map is injective. Write \(x=\pi_{\max*}y\). Exactness in the reduced row gives \(Q_*y=j_{r*}z\), where \(j_\bullet\) denotes ideal inclusion. Subtract the same ideal class upstairs. Injectivity of \(Q_*\) gives \(y=j_{\max*}z\), so \(x=0\).

Conversely, take a reduced quotient class \(x\). Its boundary \(z\) is killed by \(j_{r*}\). Commutation and injectivity of \(Q_*\) give \(j_{\max*}z=0\). Exactness supplies a maximal quotient class \(y\) with boundary \(z\). Then \(x-q_{C*}y\) has zero boundary and equals \(\pi_{r*}v\). Surjectivity of \(Q_*\) lifts \(v\), completing a lift of \(x\). Thus \(q_{C*}\) is bijective in both degrees for unital coefficients.


For a nonunital coefficient algebra \(C\), extend the action to its forced unitization \(C^+\), fixing the scalar copy. In both completions there is a split exact sequence

\[
 0\longrightarrow C\rtimes_\bullet\mathbb F_n
 \longrightarrow C^+\rtimes_\bullet\mathbb F_n
 \longrightarrow\mathbb C\rtimes_\bullet\mathbb F_n
 \longrightarrow0,\qquad\bullet=\max,r.
 \tag{A.FG.18}
\]

For the maximal norm every nondegenerate covariant representation of \(C\) extends unitally to \(C^+\), so the inclusion is isometric. The regular models give the same assertion for the reduced norm. The scalar section is contractive in each norm. To identify the kernel, approximate one of its elements by a finite group polynomial \(p\) and subtract the section of the scalar quotient of \(p\). The new polynomial has coefficients in \(C\), and its distance to the original element is at most twice the approximation error. This proves the exact kernel assertion without invoking group exactness. Split six-term exactness identifies the K-groups of the \(C\)-crossed products with the kernels of the scalar maps. The already proved canonical comparisons for \(C^+\) and \(\mathbb C\) commute with those scalar maps and their sections. Their kernel comparison is consequently an isomorphism. This completes the theorem in both degrees for every coefficient algebra. \(\square\)

**Lemma 11A.3 (the closed-arc coefficient).** Let \(F\) be a closed arc of \(S^1\), and let \(u:S^1\to\mathbb R\) be smooth. On \(F\times\mathbb P^1(\mathbb R)\), define the two generator actions by

\[
 \begin{aligned}
 a(x,z)&=\left(x,\begin{pmatrix}1&u(x)\\0&1\end{pmatrix}z\right),\\
 b(x,z)&=\left(x,\begin{pmatrix}1&0\\u(x)&1\end{pmatrix}z\right).
 \end{aligned}
 \tag{A.FG.19}
\]

For the resulting free-group action, the canonical map

\[
 K_0\!\left(C(F\times\mathbb P^1(\mathbb R))\rtimes_{\max}\mathbb F_2\right)
 \longrightarrow
 K_0\!\left(C(F\times\mathbb P^1(\mathbb R))\rtimes_r\mathbb F_2\right)
 \tag{A.FG.20}
\]

is injective.

*Proof.* Both matrices have determinant one. Replacing \(u(x)\) by \(-u(x)\) gives their inverse transformations. They preserve \(x\), so restrict to homeomorphisms of the indicated closed-arc space and define automorphisms of its continuous-function algebra. The universal property of \(\mathbb F_2\) gives the action. The coefficient space is compact, so its algebra is unital. Apply Theorem 11A.2 with \(n=2\). \(\square\)

In the foliation construction below, \(F\) is the complementary closed arc and \(u\) vanishes on the other closed arc. The lemma retains those actual coefficients and generators. It requires neither a free action on the coefficient space nor amenability of that action.

#### A compact smooth foliation with an extra reduced summand

The following foliation and obstruction are due to [Higson–Lafforgue–Skandalis, Section 4, pp.341–344](https://webusers.imj-prg.fr/~georges.skandalis/Publications/Publi34.pdf#page=12). We give the construction, algebra calculation and missed-class proof explicitly. The full-extension and canonical free-group comparison proofs above supply its analytic prerequisites.

Choose distinct points \(a,b\) of a circle, write \(U=]a,b[\) for one open arc and \(F=[b,a]\) for the other closed arc. Both endpoints belong to \(F\). Choose a smooth function \(u:S^1\to \mathbb R\) that vanishes on \([a,b]\) and is constant on no open interval in the interior of \(F\). For example, on a positively parametrized complementary arc \(t\in [0,1]\), take \(u(t)=\exp(-1/(t(1-t)))\) for \(0<t<1\) and zero at the endpoints; extend it by zero on \([a,b]\). It is flat at both endpoints and analytic, nonconstant on every interior interval.

On the compact transversal \(B=S^1\times \mathbb P^1(\mathbb R)\) let \(\Gamma =\mathbb F_2=\langle \alpha ,\beta \rangle\) act by

\[
 \begin{aligned}
 \alpha(x,z)&=(x,\alpha_{u(x)}z),&
 \alpha_t&=\begin{pmatrix}1&t\\0&1\end{pmatrix},\\
 \beta(x,z)&=(x,\beta_{u(x)}z),&
 \beta_t&=\begin{pmatrix}1&0\\t&1\end{pmatrix}.
 \end{aligned}
 \tag{A.H.1}
\]

The action is smooth and preserves the circle coordinate. We need its **germs**, not pointwise freeness. For every nontrivial word its projective matrix is nonidentity except at finitely many parameters. To prove this, each entry of its determinant-one matrix is a polynomial in \(t\); inverses of the displayed generators also have polynomial entries. Evaluation at \(t=3\) is faithful in the projective group. In the affine coordinate \(z\in \mathbb R\cup {\infty }\), put \(X={|z|>1}\cup {\infty }\), \(Y={|z|<1}\). For every nonzero integer \(k\),

\[
 \alpha_3^k(Y)\subset\{|z|>2\}\subset X,
 \qquad
 \beta_3^k(X)\subset\{|z|<1/2\}\subset Y.
 \tag{A.H.2}
\]

The first follows from \(|z+3k|>2\); the second from
\(|z/(3kz+1)| \leq  |z|/(3|z|-1)<1/2\), with \(\infty\) sent to \(1/(3k)\). A cyclically reduced alternating word starting with an α-power and ending with a β-power sends \(X\) into \({|z|>2}\), so it cannot fix a point with \(1<|z|<2\). The opposite order sends \(Y\) into \({|z|<1/2}\) and cannot fix a point with \(1/2<|z|<1\). A pure nonzero power is also nonidentity. Every nonempty reduced word is conjugate to one of these types: successively combine matching first and last factors until they differ or only one factor remains. Conjugation preserves nonidentity. This proves faithfulness. Consequently for each nontrivial word at least one polynomial off-diagonal entry or diagonal difference is nonzero, and its projective identity parameters form a finite set.

An open interval of \(F\) has a nondegenerate \(u\)-image, so it cannot be contained in that finite exceptional set. A nonidentity projective transformation cannot agree with identity on an open projective interval, since fixing three distinct projective points forces its matrix to be scalar. It follows that each nontrivial word has a nontrivial germ at every point over \(F\), including both endpoints: every endpoint neighborhood contains an interval in the interior of \(F\). Distinct words therefore have distinct germs over \(F\times \mathbb P^1\); all words have the identity germ over \(U\times \mathbb P^1\).

Let \(P\) be a closed genus-two surface. Its usual presentation
\(\pi _1(P)=\langle a_1,b_1,a_2,b_2 | [a_1,b_1][a_2,b_2]=1\rangle\)
surjects onto \(\Gamma\) by sending \(a_1,a_2\) to \(\alpha ,\beta\) and both \(b_i\) to identity. Let \(\widetilde P\to P\) be the associated regular cover. The diagonal action gives

\[
 V=(\widetilde P\times B)/\Gamma.
 \tag{A.H.3}
\]

It is free and proper because the deck action is free and proper. Evenly covered disks of \(P\) give bundle charts with compact fibre \(B\); hence \(V\) is a compact smooth four-manifold. The horizontal surfaces \(\widetilde P\times {z}\) descend to a rank-two smooth foliation. A fibre \(B\) over a base point is a compact complete embedded transversal. Its holonomy groupoid \(T\) is the germ quotient of \(B \rtimes \Gamma\) described above: holonomy of a surface loop is the prescribed transverse word action, and every word is attained by the chosen fundamental-group surjection.

At either endpoint a nontrivial word germ and the identity germ are distinct limits of the same identity germs from \(U\). Thus \(T\) is non-Hausdorff, though its arrow charts are locally Hausdorff and its unit space is the compact manifold \(B\). Its closed reduction over \(F\times \mathbb P^1\) is the Hausdorff transformation groupoid \(F\times \mathbb P^1 \rtimes \Gamma\): the distinct word sheets are disjoint open sheets of this reduction. Its open reduction over \(U\times \mathbb P^1\) is the unit groupoid. The full foliation holonomy groupoid is non-Hausdorff too, since this transverse reduction is its arrow subspace.

Set

\[
 \begin{gathered}
 I=C_0(U\times\mathbb P^1),\qquad
 C=C([a,b]\times\mathbb P^1),\\
 D_\bullet=C(F\times\mathbb P^1)\rtimes_\bullet\Gamma,
 \qquad B_m=C^*_{\max}(T),\quad B_r=C_r^*(T).
 \end{gathered}
 \tag{A.H.4}
\]

**Proposition 11A.4 (the two transverse completions).** There is an isomorphism and an exact full extension

\[
 B_r\cong D_r\oplus C,
 \qquad
 0\longrightarrow I\longrightarrow B_m\longrightarrow D_m\longrightarrow0.
 \tag{A.H.5}
\]

The canonical quotient \(\lambda _D:D_m\to D_r\) induces an isomorphism on K-theory.

**Proof.** The full extension follows from the proved invariant-open/closed theorem, Lesson 14 Theorem 14.5, applied to the actual reductions of \(T\). Its closed transformation reduction is identified with the stated full crossed product by Lesson 14 Theorem 14.3. Theorem 11A.2 above gives the canonical K-isomorphism for this unital coefficient algebra and its actual Γ-action.

For the reduced algebra, every compact patch kernel is a finite sum of word-labelled kernels with continuous coefficients on \(B\): split its compact support into finitely many germ bisections, extend their compactly supported coefficient functions by zero on \(B\), and sum. Over \(F\times \mathbb P^1\) its source-fibre norm is the regular crossed-product norm. Over \(U\times \mathbb P^1\) every word is identity, so its norm is the supremum of the absolute sum of coefficients. That sum extends continuously to \([a,b]\times \mathbb P^1\), whose norm equals its supremum over the dense open arc \(U\times \mathbb P^1\). The resulting homomorphism into \(D_r\oplus C\) is isometric. It is well-defined independently of the labelled presentation: distinct word germs on \(F\) determine every coefficient there, while the sum on \(U\) and continuity determine its other coordinate, including both endpoints.

To see that its image is the whole direct sum, separate the two coordinates explicitly. The global word bisections give unitaries in the unital algebra \(B_r\) and therefore a homomorphism from \(C^*_{\max}(\Gamma)\). Put

\[
 S=\tfrac14(U_\alpha+U_\alpha^*+U_\beta+U_\beta^*).
 \tag{A.H.6}
\]

On the four-regular Cayley tree, orient every edge toward one end. Every vertex has one parent and three children. For a finitely supported vector the edge inequality
\(2|vw|\leq3^{-1/2}|v|^2+3^{1/2}|w|^2\), with \(v\) at the parent, bounds the adjacency quadratic form by \(2\sqrt3\) times its squared norm: each vertex receives \(3/\sqrt3+\sqrt3\). Density and self-adjointness give adjacency norm at most \(2\sqrt3\), hence regular \(\lVert S\rVert\leq\sqrt3/2\). The trivial representation sends \(S\) to one.

Choose a real continuous \(h\) on \([-1,1]\), zero on \([-\sqrt3/2,\sqrt3/2]\) and with \(h(1)=1\). In the displayed isometric image, functional calculus gives

\[
 h(S)=0\oplus1.
 \tag{A.H.7}
\]

This is a central projection in \(B_r\); its chosen full lift need not be a projection. Each closed-arc coefficient extends continuously to \(B\), so restrictions of the labelled kernels have dense image in \(D_r\). Unit coefficients similarly give every function in \(C\). Multiplication by (A.H.7) and its complement separates these coordinates. The isometric image is closed, so it equals the whole direct sum. ∎

**Lemma 11A.5 (extension by zero loses its K-class).** The nonzero homomorphism
\(\iota :C_0(U\times \mathbb P^1)\to C([a,b]\times \mathbb P^1)\) induces zero maps in both K-degrees.

**Proof.** Endpoint evaluation \(\operatorname{ev}_a:C\to C(\mathbb P^1)\) is a homotopy equivalence: contract the interval \([a,b]\) to \(a\), leaving the second coordinate fixed. Its composite with extension by zero is the zero homomorphism, since every element of \(C_0(U\times \mathbb P^1)\) vanishes at the endpoint. Thus \((\operatorname{ev}_a)_* \iota _*=0\), and invertibility of \((\operatorname{ev}_a)_*\) gives \(\iota _*=0\) in both parities. ∎

**Theorem 11A.6 (failure of geometric surjectivity).** For the compact smooth foliation (A.H.3), the survey's geometric assembly into its reduced holonomy algebra is not surjective.

**Proof.** First consider the complete transversal. Under (A.H.5),

\[
 c=(0,[1_C])\in K_0(B_r)
 \tag{A.H.8}
\]

is nonzero: project to its second coordinate and evaluate at any projective point to obtain the rank-one class in \(K_0(\mathbb C)=\mathbb Z\). Suppose it had a full lift \(x\in K_0(B_m)\). Restriction to \(D_r\) is zero. Commutativity of restriction with the canonical quotient and injectivity of \((\lambda _D)_*\) force the full restriction of \(x\) in \(K_0(D_m)\) to vanish. Exactness of the full extension gives \(x=j_{m*}(y)\) for \(y\in K_0(I)\). In the reduced algebra its ideal map is \(y\mapsto (0,\iota _*y)\), which is zero by Lemma 11A.5. Hence \(c=0\), a contradiction. Neither splitting of the full K-row nor exactness of a reduced restriction sequence is assumed.

The complete transversal implements an equivalence between \(T\) and \(H=\operatorname{Hol}(V,\mathcal F)\): take holonomy arrows whose source lies on that transversal, with the right transverse and left holonomy actions. Smooth transverse charts make both anchors open and surjective; division identifies the free proper actions with the closed equal-anchor relations, as in the proof of Theorem 11A.1. The full and regular Haar systems are respectively the same smooth/counting systems. The proved locally Hausdorff equivalence theorem therefore gives full and reduced imprimitivity modules from **one** section core. The linking algebra quotient, or the inner-product proof (A.G.4), identifies its reduced module with the coefficient quotient of its full module. Thus their K-isomorphisms satisfy

\[
 \begin{array}{ccc}
 K_0(B_m)&\xrightarrow{\Phi_m}&K_0(A_m)\\
 \downarrow(\lambda_T)_*&&\downarrow(\lambda_H)_*\\
 K_0(B_r)&\xrightarrow{\Phi_r}&K_0(A_r)
 \end{array}
 \tag{A.H.9}
\]

Equivalently, \((\lambda _H)_* \Phi _m=\Phi _r(\lambda _T)_*\); the inverse evaluation tensor unitaries in the programme imprimitivity proof give both inverses. Consequently \(c_H=\Phi _r(c)\) is nonzero and has no maximal lift. A hypothetical lift could be pulled back by \(\Phi _m^{-1}\), and the commuting square plus injectivity of \(\Phi _r\) would give the forbidden lift of \(c\).

Theorem 11A.1 places every geometric assembly value in the image of \((\lambda _H)_*\). It follows that

\[
 c_H\notin\operatorname{im}\mu_r.
 \tag{A.H.10}
\]

This proves nonsurjectivity for the original compact geometric-cycle domain and reduced target. ∎

![The non-Hausdorff arc quotient and the reduced K-class missed by maximal lifts](../figures/reduced-assembly-obstruction.png)

**Figure 11A.** The upper circle is the actual transverse parameter. All words become the identity germ on the orange open arc \(U\), while both endpoints lie on the blue closed arc \(F\) and retain their distinct germs. The second transverse factor \(\mathbb P^1(\mathbb R)\) remains in (A.H.1–A.H.5); the picture is not the four-manifold itself. The lower square records the proof: the upper row is exact at the middle K-group, its right vertical quotient is injective by Theorem 11A.2, and its left vertical map is zero by Lemma 11A.5. The nonzero class \((0,[1])\) therefore has no full lift. Theorem 11A.1 and (A.H.9–A.H.10) transport the obstruction to geometric assembly. Construction credit: Higson–Lafforgue–Skandalis, Section 4. Original diagram, CC0.

The historical universal reduced assertion is thereby refuted in its stated non-Hausdorff scope. Assembly statements imposing additional hypotheses, or an inverse for a specified geometric family such as the hyperbolic construction of Section 11, have their own scope and require their corresponding proof. The counterexample supplies no kernel witness for the geometric map and makes no identification with a different assembly domain.

## 11A. A normalized scalar graph cycle on a compact SU(2) suspension

Connes, Section 8(2), asks for a scalar K-homology class represented by a suitable graph Dirac operator, with local transverse Bott pairing one, under a holonomy-invariant normal spin-c structure. The algebra is the reduced holonomy algebra and the required variance is \(KK^q(C_r^*(G),\mathbb C)\). The following construction supplies such a normalized reduced scalar cycle for the specific compact suspension in (SG.1). Its additional tree operator is discrete; identifying it with a local finite-order differential expression, and constructing a cycle under only the general normal spin-c premise, remain separate mathematical questions.

**Theorem 11A.8 (the normalized reduced scalar graph cycle).** On the compact suspension (SG.1), the faithful reduced representation, completed graph-column Hilbert space and self-adjoint operator (SG.10) define a scalar degree-three Kasparov cycle. It retains the full normal Clifford orientation, every localized compact defect and the original local transverse Bott pairing +1.

**Proof.** We construct and check the complete module and operator below.

### 11A.1. The exact example and the calibrated normal factor

Let Gamma=F2 act on N=SU(2) by the faithful left translations rho of the explicit pair (1+2i)/sqrt(5), (1+2j)/sqrt(5). The complete all-length free-pair proof is in Lemma 11.9, (ND.16)–(ND.18). Let Sigma be the closed genus-two surface with a surjection pi1(Sigma) onto Gamma sending its two a generators to a,b and its two b generators to the identity. Put Sigma_0=tilde Sigma/kernel of that surjection. Then

\[
 V=(\Sigma_0\times N)/\Gamma,\qquad
 \gamma(x,n)=(\gamma x,\rho(\gamma)n),\qquad
 B=C(N)\rtimes_r\Gamma,
 \quad A=C_r^*(G(V,F)). \tag{SG.1}
\]

Every leaf is Sigma_0, and every normal transition is an isometric left translation with the coherent trivial lift on S=N×C2. The surjection is well-defined because both surface commutators map to the identity. The deck action of Gamma on Sigma_0 is free and proper, so the diagonal action in (SG.1) is free and proper. Local covering charts give a smooth fibre bundle over the closed surface with compact fibre N; hence V is compact. A fibre N over a base point is a complete transversal. Kernel elements of the surface-group surjection have identity transverse holonomy; nonidentity Gamma translations have nonidentity germs and no fixed points. Thus the actual reduced transversal holonomy groupoid is N crossed with Gamma. The oriented left-invariant orthonormal frame is preserved exactly; the spin lift is \(g\cdot(n,s)=(\rho(g)n,s)\), so the whole normal spin bundle and its metric descend. We use an oriented left-invariant orthonormal frame X1,X2,X3. The complete normal positive Clifford matrices are sigma_x,sigma_y,sigma_z. Their normalized odd volume is (−i)sigma_x sigma_y sigma_z=1. The Haar density and coherent lift are fixed, rather than a nonregular invariant scalar representation.

Let D_N be the invariant spin Dirac with the source orientation calibrated to the prescribed geometric normal Bott class. In this rank-three example its principal differential part is −i sum sigma_j X_j. The invariant Clifford-connection term is of order zero. The compact ordinary tangent calculus gives its self-adjoint closure, compact resolvent and bounded commutators with smooth functions; it commutes with the actual translations \( (U_g\psi)(n)=\psi(\rho(g)^{-1}n)\).

Here the phase calibration is explicit. Theorem 7.17, (289)–(295), gives the inverse coefficient module in rank r as Pi^(k_r) C_r, with physical generators (−1)^(k_r+1)L_(t_j), where k_r=r(r−1)/2. At r=3 this is **Pi C3 with positive physical generators**. Reduce only its first two consecutive generators and retain the final right C1. On C2⊗C1 the three generators and grading are

\[
 t_1=\sigma_x\otimes1,\quad
 t_2=\sigma_y\otimes1,\quad
 t_3=\sigma_z\otimes\varepsilon,\qquad
 \Gamma=-\sigma_z\otimes\Gamma_{C_1}. \tag{SG.2}
\]

In the even ordered basis (e0⊗epsilon,e1⊗1), multiplication by t_j followed by the right epsilon is precisely sigma_j for j=1,2,3. Thus its ordinary odd physical differential operator is −i sigma·partial. This calculation retains the grading reversal and final odd generator; the rank-one inverse uses a different physical sign, so a circle sign is not imported here. Lemma 7.20's two native inverse identities give the local pairing with the positive old-first Bott class. Bounded-phase locality (Lemmas 7.14–7.17), positive metric transport and the full finite coefficient module carry it to every positively oriented normal chart of N. Consequently, if b_U is the positive generator extended from an oriented relatively compact normal three-ball U,

\[
 b_U[D_N]=1\quad\hbox{in }KK(\mathbb C,\mathbb C).
 \tag{SG.3}
\]

The connection term and freezing of a curved metric change the phase only by the proved local symbol/metric comparison, with the same whole normal module. The underlying central spin action here is actually trivial, not a forgotten line. A different supplied normal spin-c line would require its full inverse line in (SG.2); none is inserted or discarded in this example.

### 11A.2. The complete tree factor

**Lemma 11A.7 (an even reduced tree cycle with index one).** For the free-group action in (SG.1), the full vertex-edge module below has a self-adjoint compact-resolvent generator, bounded return commutators and positive scalar index one. Its product with the calibrated normal Dirac has the same local Bott pairing.

**Proof.** Let T be the infinite degree-four Cayley tree of F₂, rooted at e. Let V(T)=Γ and let E(T) be its *unoriented* edges. For `g≠e`, let par(g) be its unique neighbour closer to e and put `c(g)={g,par(g)}`. The map c is a bijection from `Γ\{e}` to E(T): each edge has one endpoint farther from the root. Thus

\[
 H_T^+=\ell^2\Gamma,\quad H_T^-=\ell^2E(T),\qquad
 Q\delta_e=0,\quad Q\delta_g=\delta_{c(g)}\ (g\ne e),
 \tag{SG.T1}
\]

defines a bounded operator with

\[
 Q^*Q=1-P_e,\qquad QQ^*=1,
 \qquad \operatorname{Index}Q=1.
 \tag{SG.T2}
\]

The grading is + on vertices and − on edges. Left translation acts on both spaces. The edge representation is the sum of two regular representations: every edge is uniquely `{g,ga}` or `{g,gb}`. There is no stabilizer or reversal ambiguity, since F₂ is torsion free and the positive-generator orientation specifies each representative. This is an explicit decomposition, not a weak-containment assertion without a provider.

Set `Mδ_g=|g|δ_g`, with word length |g|. The closed operator `QM` has domain `∑|g|²|ξ_g|²<∞`; its adjoint sends `δ_{c(g)}` to `|g|δ_g`. On the graded sum put

\[
 D_T=\begin{pmatrix}0&M Q^*\\QM&0\end{pmatrix},
 \qquad
 \Gamma_T=\begin{pmatrix}1&0\\0&-1\end{pmatrix}.
 \tag{SG.T3}
\]

The root is its entire kernel, in positive grading. For every g≠e the two-dimensional span of `δ_g` and `δ_{c(g)}` is an orthogonal invariant block with matrix `[[0,|g|],[|g|,0]]`. Their direct sum with the root gives a self-adjoint operator on the square-summability domain of its block images, with finite block sums as a core. There are finitely many words at each fixed length, namely `4·3^(n−1)` for n≥1. Therefore `(1+D_T²)⁻¹` is compact: each length cutoff has finite rank and the norm of its tail is at most `1/(1+(R+1)²)`.

#### Complete bounded commutator proof

Let V_h and E_h be the left translation unitaries on vertices and edges. On δ_g the lower commutator is

\[
 (QM V_h-E_hQM)\delta_g
 =|hg|\delta_{c(hg)}-|g|\delta_{h c(g)},
 \tag{SG.T4}
\]

where a term at the root is interpreted as zero. The two edges agree unless hg is on the finite geodesic segment `[e,h]`: at every vertex outside that segment, the first steps toward roots e and h agree. Off this exceptional set, the coefficient difference is bounded by `||hg|−|g||≤|h|`. Its output-edge map is injective, so this off-segment weighted partial permutation has norm at most |h|.

There are at most `|h|+1` exceptional input vertices. For them hg and g have lengths at most |h|: their positions are on the segment or on its translate by h⁻¹. Each column has norm at most 2|h|. The Hilbert–Schmidt norm bound of this finite-column operator is therefore at most `2|h|sqrt(|h|+1)`. This proves the explicit finite bound

\[
 \|QM V_h-E_hQM\|
 \le |h|+2|h|\sqrt{|h|+1}.
 \tag{SG.T5}
\]

The upper commutator is the adjoint of the corresponding lower commutator for h⁻¹, composed with unitaries, so it has the same bound. All commutators preserve the core and extend boundedly. The unweighted Q commutator is finite rank: off the same finite segment the two edge images agree exactly. Thus the phase

\[
 F_T=\begin{pmatrix}0&Q^*\\Q&0\end{pmatrix}
 \tag{SG.T6}
\]

is an even Fredholm module for C*_rΓ with compact commutators and square defect −P_e in positive grading. Its representation is genuinely reduced, by the explicit regular decomposition of both graded spaces. Its scalar pairing with the unit is exactly IndexQ=1. The normalized transform of D_T differs from F_T by a compact diagonal-block operator, so it defines that same class.

#### The reduced crossed-product representation and compact total generator

On `H=H_N⊗(H_T^+⊕H_T^−)` define

\[
 \Pi(f)=M_f\otimes1,\qquad
 \Pi(u_h)=U_h\otimes(V_h\oplus E_h).
 \tag{SG.T7}
\]

Covariance follows from the multiplication/translation relation on N. The representation is the sum of three regular covariant copies: one vertex orbit and two edge orbits. On each copy, the unitary `(Wψ)_g=U_g⁻¹ψ_g` gives regular shifts and coefficient `f(ρ(g)n)`, with \(W(U_h\otimes\lambda_h)W^{-1}=1\otimes\lambda_h\) and \((WM_fW^{-1}\psi)_g=M_{f\circ L_{\rho(g)}}\psi_g\). These identities follow by substituting \((W\psi)_g=U_g^{-1}\psi_g\). Consequently it extends contractively to B, with no use of the nonregular scalar representation.

Use the odd physical rank3 model and put

\[
 D=D_N\otimes\Gamma_T+1\otimes D_T,
 \qquad D^2=D_N^2\otimes1+1\otimes D_T^2.
 \tag{SG.T8}
\]

Anticommutation of D_T with Γ_T proves the square identity on the core. A compact-resolvent eigenbasis for D_N, together with the finite tree blocks, gives self-adjoint two-by-two matrices `[[ν,|g|],[|g|,−ν]]`; the root block is ν. Their orthogonal direct sum is self-adjoint on its square-summability domain and finite block sums form a core. The resolvent is compact: for a fixed spectral bound, only finitely many D_N eigenvalues and finitely many tree levels occur; the complement has square at least that bound. Thus all scalar multiplicities are controlled.

For a smooth finite-support crossed coefficient the commutators are bounded:

\[
 [D,\Pi(f)]=[D_N,M_f]\otimes\Gamma_T,
 \qquad
 [D,\Pi(u_h)]=U_h\otimes[D_T,V_h\oplus E_h].
 \tag{SG.T9}
\]

The first uses the same coefficient in both tree gradings, so D_T commutes with it. The second uses invariance of D_N and preservation of tree grading. Formula(SG.T5) gives a bound for each group coefficient; finite support is exactly the required dense coefficient scope.

The normalized transform `F=D(1+D²)⁻½` gives an odd scalar Kasparov cycle for B. Its square defect is the compact total resolvent. Here is a direct compact commutator argument. Write `R_t=(1+t²+D²)⁻¹`, and `B_a=[D,Π(a)]`. The formal resolvent identity expresses `[DR_t,Π(a)]` as

\[
 B_aR_t-DR_t(DB_a+B_aD)R_t.
 \tag{SG.T10}
\]

Move the adjacent D factors through their resolvents. `||R_t||≤(1+t²)⁻¹`, `||DR_t||≤1/(2sqrt(1+t²))`, and `||D²R_t||≤1` bound the norm by `(9/4)||B_a||/(1+t²)`. Every term is compact because R_t and DR_t are compact. The convergent norm integral of these compact commutators, multiplied by 2/π, is `[F,Π(a)]`: the corresponding truncated transform integrals converge strongly to F, while the commutator integrals converge in norm by the displayed bound. This proves compactness without assuming norm convergence of the transform integrals themselves. Norm density of the smooth finite-support coefficients then proves all Kasparov conditions on B.

The ordinary odd physical model corresponds to degree3 after the same ordered finite Clifford contraction as D_N. To check the retained final odd factor explicitly, order the graded module as `H_T⊗̂(H_N⊗C₁)`, with grading `Γ_T⊗Γ_{C₁}` and right algebra C₁. If ε is its final positive odd generator, the graded operator is `D_T⊗1⊗1+Γ_T⊗D_N⊗L_ε`; the first-cycle lift is `F_T⊗1⊗1`. The even subspace is the sum of `H_T⁺⊗H_N⊗1` and `H_T⁻⊗H_N⊗ε`. Acting there by the graded operator followed by right multiplication by ε gives exactly the physical matrix (SG.T8): diagonal entries D_N and −D_N, with the D_T off-diagonal entries. This is the actual ordinary odd-model identification, after contracting only the supplied even Clifford pairs. The final C₁ has been accounted for, rather than erased by a formal parity statement. The extra tree cycle is even; it changes no transverse parity. Its kernel grading is positive.

#### Exact restriction class and normalization scope

Upon restriction to C(N), this cycle is the product of the even tree class(SG.T6), with external coefficient C(N), followed by the ordinary odd normal Dirac class. The first class has index1. This order keeps the even tree factor first and moves no odd Clifford blocks. We verify the operator statement, instead of using only the root count.

On each nonroot tree block of length n and a D_N eigenvector of eigenvalue ν, F has matrix

\[
 \frac1{\sqrt{1+\nu^2+n^2}}
       \begin{pmatrix}\nu&n\\n&-\nu\end{pmatrix}.
 \tag{SG.T11}
\]

The first product module is `C(N)⊗(H_T⁺⊕H_T⁻)`, with its tree grading, so creation vectors can be taken to be finite tree basis vectors times continuous coefficient functions. For a positive vertex creation vector, the first column differs from the column `F_N=D_N(1+D_N²)⁻½` by entries tending to zero as |ν|→∞ at fixed n. For a negative edge creation vector the second column differs from `−F_N` in exactly the same way. This is the graded connection sign for a negative creation vector. The root column agrees exactly with F_N. Compact resolvent of D_N makes all these differences compact. Coefficient functions are handled by the already proved compact commutators with F and F_N. Finite tree/coefficient sums are dense in the first module; bounded creation norms extend the compactness to all creation vectors. Adjoints give the other connection error.

The first-cycle phase F_T lifts as `1⊗F_T`. Its anticommutation with Γ_T cancels the D_N cross terms, while `F_TD_T=D_TF_T=|D_T|`. Consequently product positivity is exact:

\[
 \{1\otimes F_T,F\}
 =\frac{2(1\otimes|D_T|)}
        {\sqrt{1+D_N^2\otimes1+1\otimes D_T^2}}
 \ge0.
 \tag{SG.T12}
\]

These are the actual connection and positivity checks of Definition 0.3. Theorems0.4 and0.6 provide existence and class uniqueness for that criterion. Theorem0.14 identifies the even scalar tree cycle with its index1, namely the identity; external coefficient extension gives the identity of C(N). Therefore

\[
 [\Pi,F]\big|_{C(N)}=[D_N]\quad\text{in }KK^3(C(N),\mathbb C).
 \tag{SG.T13}
\]

In particular, the local Bott pairing is carried unchanged. For the actual source-oriented compact normal Dirac convention with local pairing1, the new reduced class has pairing1 as well. If a physical Pauli convention instead has the opposite local pairing, this same construction retains that opposite value; replacing D_N by its calibrated negative changes it. The tree index1 alone does not calibrate the normal Clifford inverse. Corollary 6.0d equation(144), both actual Bott inverse products, and the ordinary Dirac comparison of Theorem7.22 are the exact existing interfaces for this calibration. The fully proved new assertion is (SG.T13), with the normal factor kept intact.

The compact genus-two suspension in (SG.1) has complete transversal N and the explicit normalized plaque corner B. The corner is full because the transversal meets every leaf. One can check fullness on compact graph charts by a finite plaque partition and translation of the normalized bump to each such chart; the resulting corner-column products approximate every compact arrow kernel in the reduced Schur norm. Thus its standard imprimitivity module transports this reduced scalar class to the actual holonomy algebra via Lemma 0.12 and the graph correspondence. The product theorem supplies a bounded cycle there. This establishes a reduced class for that test, but does not yet make the discrete tree term a local smooth graph differential operator. That additional literal Dirac-type identification is not claimed.

### 11A.3. The transverse reduced scalar factor

Write H_B=H_N⊗(H_T+⊕H_T−), with the tree grading Gamma_T. Lemma 11A.7 proves the complete vertex-edge construction and all-length commutator estimate. In particular

\[
 \Pi_B(f)=M_f\otimes1,\quad
 \Pi_B(u_h)=U_h\otimes(V_h\oplus E_h),\qquad
 D_B=D_N\otimes\Gamma_T+1\otimes D_T. \tag{SG.4}
\]

The vertex representation is regular; the unoriented edges have exactly two free regular orbits. The absorption unitary in Lemma 11A.7 therefore makes Pi_B a reduced covariant representation. D_B is self-adjoint with compact scalar resolvent. Its square is D_N²⊗1+1⊗D_T²; the complete square-summability domain, finite block core and bounded commutators with C∞(N) Gamma are proved in (SG.T3)–(SG.T10). Its normalized transform F_B is a scalar odd cycle. The graded connection and positive sandwich (SG.T11)–(SG.T12), and the tree index +1, give the exact restriction

\[
 [\Pi_B,F_B]|_{C(N)}=[D_N]. \tag{SG.5}
\]

This is a reused complete product proof, not just a count of the unmatched root. The unchanged full normal calibration (SG.2)–(SG.3) supplies the previously missing normalization. The tree factor is even and contributes no extra odd degree.

### 11A.4. The actual holonomy representation, rather than abstract transport alone

Choose finitely many relatively compact flat covering charts U_i of Sigma, with smooth real chi_i supported in them and sum chi_i²=1. Choose a lift z_i:U_i→Sigma_0. If z_j=h_ji z_i on an overlap, the transition h_ji is locally constant in Gamma. On the compact support of chi_i chi_j it has only finitely many values: a continuous map from a compact set into a discrete space has finite image. Thus no unproved good-cover or finite-component assertion is needed.

The right-B flat bundle L_B on Sigma has fibre B and transition by the **left** unitary u_(h_ji). These are right-B-linear, preserve the B-valued inner product and satisfy the actual cocycle law. Let

\[
 \mathcal E=L^2(\Sigma,\mathcal L_B),\qquad
 \langle\xi,\eta\rangle_B=
       \int_\Sigma\xi_i(x)^*\eta_i(x)\,d\mu_\Sigma(x).
 \tag{SG.6}
\]

The integral is chart independent. It is positive by finite simple-section approximation and positivity of sums b*b. Quotienting the zero seminorm and completing gives a countably generated right Hilbert B-module. It is full: a unit-norm scalar bump inside one flat chart, times 1_B, has inner product 1_B. Smooth finite chart functions with coefficients in C∞(N) Gamma form a dense module core.

The holonomy convolution acts on E by the integral kernels obtained from the lift charts: a chart kernel a_(ij,g)(x,y,n) becomes the B-valued kernel a_(ij,g)(x,y,·)u_g. Kernel multiplication is precisely the crossed-product covariance and integration in the middle plaque variable. The half-densities give the same invariant formula if the chosen density is removed.

Separated smooth functions in x,y approximate every compact chart kernel in reduced Schur norm, uniformly in its compact normal variable and finite group support. They are the rank-one module kernels theta_(xi,eta). Conversely every such rank-one smooth kernel is a finite sum of those holonomy chart kernels. The corresponding reduced norm is exact: for each n, E tensored with the regular B representation at n is unitarily the holonomy-cover L² space G_n, by the lift chart formula and its partition. It preserves the kernel action and densities. Taking the supremum over n is the defining reduced norm of both constructions. Therefore

\[
 A=\mathcal K_B(\mathcal E),\qquad
 {}_A\mathcal E_B\hbox{ is a full imprimitivity module}.
 \tag{SG.7}
\]

This identifies the entire algebra; it does not keep only one plaque corner or assume assembly duality.

For a concrete projection presentation put K=direct sum_i L²(U_i), and define T xi=(chi_i xi_i)_i. Its adjoint reconstructs a section by the finite partition, so T*T=1. Write P=TT*. In chart coordinates

\[
 P_{ij}(x)=\chi_i(x)\chi_j(x)u_{h_{ij}(x)},\qquad
 \mathcal E=P(K\otimes B),\qquad
 A=P\bigl(\mathcal K(K)\otimes B\bigr)P. \tag{SG.8}
\]

The displayed entry includes the unitary coordinate/density identification of the same overlap point, extended by zero away from the overlap. Products of the chi weights are buffered there; the finite transition values make P a finite sum of bounded leaf operators tensored with group unitaries. It is an adjointable projection and its scalar represented projection commutes with the global tree grading.

Tensor (SG.8) with the actual reduced Pi_B to obtain

\[
 \mathcal H_G=\mathcal E\otimes_B\mathcal H_B
       =P(K\otimes\mathcal H_B),
 \qquad \Pi_A(a)=a|_{\mathcal H_G}. \tag{SG.9}
\]

This is an explicit reduced scalar representation, not the rejected invariant H_N representation. In the vertex summand it is naturally the direct-integral graph-column space L²(Sigma_0×N,S); the edge summand is two such copies. Indeed, label a lift of a point by t^(−1)z_i(x). On a change z_j=h z_i, its label becomes ht and its normal chart coordinate becomes rho(h)n. The induced scalar transition is exactly U_h⊗V_h (or U_h⊗E_h). The Haar and leaf densities are preserved. Thus H_G is a completion of the holonomy columns G_N=s^(−1)(N), with the complete normal spin and the even vertex-edge auxiliary coefficient; it is not an unrestricted Hilbert–Schmidt completion of both leaf endpoints.

### Exact norms and faithful maps

The absorption statement in (SG.4) proves equality, not just contractivity, for the represented reduced norm. On each regular orbit, after W the finite-support coefficient a=∑f_hu_h is the direct integral over N of the defining regular representations π_n on ℓ²Γ, with two-dimensional spin multiplicity. Its operator norm is ess sup_n||π_n(a)||. This equals sup_n||π_n(a)||: for a fixed finite-support unit vector v on ℓ²Γ, n↦||π_n(a)v|| is continuous. The supremum over these v is lower semicontinuous; every strict lower level is open, and Haar measure has full support. The essential supremum therefore equals the supremum. Finite-support coefficients are norm dense, so

\[
 \|\Pi_B(b)\|=\|b\|_r\quad(b\in B). \tag{SG.9a}
\]

The same fibre comparison in (SG.7) is exact for all graph units. A unit [x,n] can be joined along its leaf to the chosen base fibre. Right translation by that holonomy arrow unitarily identifies its source-fibre convolution representation with the one based in N, so taking the supremum on N already gives the supremum on V. This treats all units, not just the plaque corner. Smooth compact chart kernels are approximated in reduced norm by separated kernels: the reduced Schur bound is the maximum of the two absolute plaque-integral bounds, and uniform finite-variable approximation on the fixed compact supports drives both to zero. Their algebra is precisely the finite-rank kernels on E. Thus (SG.7) is an onto isometric identification.

Faithfulness of the induced scalar representation now follows without an unspecified Morita assertion. If a∈K_B(E) acts as zero on E⊗_B H_B, then for every ξ,η∈E the bounded rectangular compression T_η*Pi_A(a)T_ξ is Pi_B(⟨η,aξ⟩). Equation (SG.9a) makes every such inner product zero. Taking η=aξ gives ⟨aξ,aξ⟩=0, hence aξ=0 for every ξ and a=0. Consequently

\[
 C^*_{\max}(G)\mathrel{\twoheadrightarrow}_{q_r} A
   \mathrel{\hookrightarrow}_{\Pi_A}
      B(\mathcal H_G),\qquad
 \|\Pi_A(a)\|=\|a\|_r. \tag{SG.9b}
\]

Only the second arrow is claimed faithful. Its representation is nondegenerate because the compact-module finite-rank approximate identity acts strongly on the tensor Hilbert space. The purely normal covariant representation on H_N remains a representation of the maximal crossed product: its four-generator average has norm one, while the reduced norm is at most √3/2. There is no arrow B→B(H_N) extending those same coefficient operators. The new faithful arrow uses the displayed even tree factor.

### 11A.5. An explicit self-adjoint graph-column operator

On the free scalar space K⊗H_B put D_0=1_K⊗D_B. Its domain is the L² family of the complete domain of D_B. P preserves that domain, and [D_0,P] is bounded: every transition entry has finite group support, its leaf weights are constant in the normal derivative, and (SG.T5) bounds the tree commutator. The normal D_N Gamma_T part commutes with P exactly.

Let the bounded off-diagonal block be R=P D_0(1−P)+(1−P)D_0P. These blocks equal bounded products of [D_0,P]. Define

\[
 D_{\rm diag}=D_0-R,\qquad
 D_G=D_{\rm diag}|_{P(K\otimes H_B)}
     =P D_0P\big|_{P\operatorname{Dom}D_0}. \tag{SG.10}
\]

R is self-adjoint. A direct resolvent proof of bounded perturbation is available: for imaginary z with |Im z|>||R||, the Neumann inverse of 1−R(D_0−z)^(−1) gives an onto inverse for D_0−R−z and its adjoint. The closed symmetric operator is consequently self-adjoint, with unchanged domain. The bounded commutator implies P reduces this diagonal operator on that domain. Therefore D_G is self-adjoint on the displayed exact domain.

In a flat chart its expression is the full transverse differential Dirac D_N Gamma_T plus the smoothly weighted local tree generators. Their changes on overlaps are bounded by (SG.T5). There is no leaf derivative in the chi weights. Its transverse principal symbol is exactly

\[
 \sigma_\perp(D_G)(x,n;\xi)
   =\Bigl(\sum_{j=1}^3\sigma_j\xi_j\Bigr)\otimes\Gamma_T,
 \qquad \sigma_\perp(D_G)^2=|\xi|^2. \tag{SG.11}
\]

On the full retained graded C1 model this is the three-generator module (SG.2) tensored with the even tree module, with grading Gamma_T times its complete normal grading. All transitions are the actual even U_h/tree unitaries. A Pauli phase or grading is not changed during graph induction. The principal normal coefficient therefore retains the supplied spin and full orientation; the auxiliary even completion is displayed rather than silently removed.

The term D_T is a proper weighted discrete vertex-edge difference operator. Formula (SG.10) gives a genuine scalar operator on graph columns with a transverse Dirac principal part. It does not identify that discrete term with a finite-order local differential operator in all smooth graph coordinates.

### 11A.6. Both scalar Kasparov defects on the whole reduced algebra

The global free resolvent has infinite leaf-coordinate multiplicity. The correct compactness statement is localized, since A is nonunital. To prove it without an invalid infinite tensor compactness argument, use the concrete closed spatial C*-subalgebra

\[
 \mathcal J=\mathcal K(K)\otimes\mathcal B(H_B)
       \subset\mathcal B(K\otimes H_B). \tag{SG.12}
\]

Here J is not an ideal of all B(K⊗H_B). We use only its own multiplier algebra: every finite tensor L⊗b with L∈B(K), b∈B(H_B) multiplies J on both sides. In particular the specified P, R and bounded P commutators do so.

Every j in J times (D_0−z)^(−1) is an ordinary scalar compact. For a separated tensor it is the product of a compact on K and b(D_B−z)^(−1), a compact on H_B. Finite sums and norm density prove the assertion for J. P, R and [D_0,P] are finite sums of bounded leaf operators tensored with bounded H_B operators, and hence are multipliers of J.

The resolvent identity gives j(D_diag−z)^(−1) compact as well: its difference from j(D_0−z)^(−1) is that compact operator times bounded R and the diagonal resolvent. Adjointness gives the other side. Hence every represented a in A has both localized resolvents compact on H_G. The same holds for j D_diag(1+t²+D_diag²)^(−1), by its two resolvent decomposition.

Take the dense algebraic core a=P a_0 P, with a_0 a finite sum of smooth leaf rank-one kernels tensored with C∞(N) Gamma coefficients. Its represented kernel is an actual finite compact smooth holonomy-chart sum. On the graph domain

\[
 [D_G,\Pi_A(a)]=P[D_0,a]P,
 \qquad [D_0,a]\in\mathcal J\hbox{ is bounded}. \tag{SG.13}
\]

For this, expand [D_0,Pa_0P]; the two P commutators are bounded multipliers of J, while a_0 and [D_0,a_0] belong to J by (SG.T9). Domain preservation follows from the same bounded commutators. This treats the complete smooth coefficient core, rather than only group constants.

Put F_G=D_G(1+D_G²)^(−1/2). It is an exact self-adjoint contraction with the correct full odd grading. Its square defect is

\[
 \Pi_A(a)(1-F_G^2)
    =\Pi_A(a)(1+D_G^2)^{-1}\in\mathcal K(\mathcal H_G).
 \tag{SG.14}
\]

For the commutator set \(s=1+t^2\), \(R_t=(s+D_G^2)^{-1}\), and \(C_a=[D_G,\Pi_A(a)]\). The bounded form of the resolvent commutator is

\[
 [D_GR_t,\Pi_A(a)]
   =s R_t C_a R_t-D_GR_t C_a D_GR_t. \tag{SG.15}
\]

To justify it without assuming that C_a preserves the domain of D_G, first use the first-order resolvents Q_±=(D_G∓i√s)⁻¹. Domain preservation by a and the bounded commutator give [Q_±,a]=−Q_±C_aQ_±. Also D_GR_t=(Q_++Q_−)/2 and Q_±=(D_G±i√s)R_t. Expanding their average gives exactly (SG.15); every displayed factor is bounded. The norms ||R_t||≤s⁻¹ and ||D_GR_t||≤(2√s)⁻¹ give

\[
 \|[D_GR_t,\Pi_A(a)]\|
      \le {5\|C_a\|\over4(1+t^2)}. \tag{SG.15a}
\]

Both terms are compact by the J-localized resolvent result, since C_a has a J extension to the free space. In the second term C_aD_GR_t is compact; in the first C_aR_t is compact. The norm-convergent integral against 2/π is [F_G,Pi_A(a)]. Truncated transform integrals converge strongly to F_G; their commutators converge in norm by (SG.15a), so taking their strong limit identifies this norm limit with the actual commutator. Norm convergence of the uncommuted transform integral is not asserted. Norm density of the coefficient core gives both defects for all a∈A. In the full retained C₁ model every algebra coefficient is even, so these are exactly the graded commutators. The self-adjointness defect is zero.

This proves an actual class z_G in KK3(A,C) with all scalar compactness conditions, despite the infinite global free resolvent multiplicity.

### 11A.7. Checked Morita product, plaque restriction and local normalization

For a smooth finite-chart core vector xi of E, its scalar creation T_xi:H_B→H_G has finite leaf profiles and finite group coefficients. The exact graph equation has bounded rectangular error

\[
 K_\xi=D_G T_\xi-T_\xi D_B,\qquad
 \|K_\xi\|<\infty. \tag{SG.16}
\]

It follows from the finite group transition calculation and [D_0,P]. The right resolvent of D_B is scalar compact. Therefore the mixed resolvent identity gives compact connection errors, with norm at most ||K_xi||/|Im z|². Using D(1+t²+D²)^(−1) as the average of the two imaginary resolvents gives the integrable bound ||K_xi||/(1+t²). Its norm integral proves F_G T_xi−T_xi F_B compact; the adjoint equation gives the other error. Dense smooth module vectors and the bounded creation norm extend this to every xi in E. The first Morita operator is zero, so product positivity is zero. Definition 0.3 and Theorem 0.6 now identify this explicit representative:

\[
 z_G=[{}_A\mathcal E_B,0]\otimes_B[\Pi_B,F_B].
 \tag{SG.17}
\]

Choose a smooth bump u supported in one contractible base disc, with leaf L2 norm one, and let xi be its corresponding module section, so <xi,xi>=1_B. The holonomy projection p=theta_(xi,xi) has kernel u(x) conjugate u(y) on the identity return component. The exact corner map is i(b)=theta_(xi b,xi). In particular pAp is the actual reduced B, including every return component and its reduced regular norm.

T_xi is an onto isometry onto the p subspace. The just-proved connection identifies the compressed transform with F_B modulo compacts. Also [F_G,p] is compact, so its off-diagonal p blocks can be removed by a compact perturbation. The complementary representation of i(B) is zero and is a degenerate cycle. Consequently

\[
 i^*z_G=[\Pi_B,F_B],\qquad
 (i|_{C(N)})^*z_G=[D_N],\qquad
 (i_*b_U)z_G=1. \tag{SG.18}
\]

The normalized corner has an explicit correspondence unitary, not just equal numerical pairings. With E* the conjugate B–A module, define

\[
 U:\mathcal E^*\longrightarrow pA,\qquad
 U(\overline\eta)=\theta_{\xi,\eta}. \tag{SG.18a}
\]

The right A inner product on E* is θ_(η,ζ). In pA it is (θ_(ξ,η))*θ_(ξ,ζ)=θ_(η,ζ), since ⟨ξ,ξ⟩=1. Right A actions agree by adjointing the second rank-one slot. The left B action agrees with i(b): i(b)θ_(ξ,η)=θ_(ξb,η)=θ_(ξ,ηb*). Every pa is θ_(ξ,a*ξ), so the isometry is onto. The homomorphism correspondence [i] has essential part pA; its complementary zero left action is degenerate. Restrict this exact unitary to C(N). The actual transverse counting module is identified with E* on compact source-return columns by the normalized plaque bump: its leafwise counting norm and the plaque integral ∫|u|² are exactly one. Thus it has this same essential correspondence, not a separately chosen K-class.

The normalized-bump corner agrees with the actual transversal correspondence: its compact source-return columns send a scalar transverse section to the normalized u column on every holonomy lift. The counting inner product and leaf bump integral are both one; finite return columns are dense, so this is an onto module unitary. Thus i_* b_U is the original source-oriented local transverse class of Connes, Section 8(2), pp.28–29, not an unrelated K-class picked for a convenient numerical index. Equation (SG.18) verifies local pairing one and justifies calling z_G a normalized transverse fundamental class **on this exact suspension example**.

### 11A.8. The scope of the construction

The result is constructive at the actual reduced scalar variance KK3(C_r(G),C), rather than the longitudinal KK(C(V),A) class. It disproves no general existence claim. The new scalar reduced representation and even completion are fully specified and have the checked local normalization.

A general compact holonomy groupoid with an invariant normal metric/spin action need not have a compact transversal with this free-group return action, a coherent finite chart Mishchenko description with this tree-even index-one generator, or the bounded overlap and properness estimates (SG.T3)–(SG.T5). A full original-scope proof must replace those explicit ingredients by a groupoid-wide scalar construction retaining the same full normal inverse and local pairing. If “Dirac type on the graph” requires a wholly local finite-order smooth graph differential expression, an additional comparison of the discrete completion with such an expression is also still required. This construction does not infer it from KK Morita transport or transverse ellipticity. These are exact remaining interfaces, not a theorem of nonexistence.


### 11A.9. A general sufficient interface, and its precise limit

The graph-induction argument SG.6–SG.18 does not use a tree once the following transverse data are supplied. Let a countable Γ act smoothly and freely on a compact manifold N, preserving a prescribed full metric spin-c structure, and suppose D_N is an invariant, self-adjoint, compact-resolvent normal Dirac with its source-calibrated local Bott inverse. Suppose an even scalar Γ-cycle (H_T,Γ_T,D_T,L) has: (a) self-adjoint D_T with compact resolvent; (b) bounded [D_T,L_h] for every h; (c) Γ_TL_h=L_hΓ_T and Γ_TD_T=−D_TΓ_T; (d) a decomposition of L as a finite or countable sum of regular representations, or an independently proved exact reduced crossed-product estimate for M_f⊗1,U_h⊗L_h; and (e) scalar index +1. For a compact-base suspension with finitely many buffered flat charts, the operator

\[
 D_G=P(1_K\otimes(D_N\otimes\Gamma_T+1\otimes D_T))P,
 \qquad \operatorname{Dom}D_G=P\operatorname{Dom}D_0 \tag{SG.19}
\]

on E⊗_B(H_N⊗H_T) is a scalar degree-q reduced Kasparov cycle with the localized defects proved above. The total transverse resolvent is compact: compact resolvents have finite-dimensional bounded spectral windows, and the square sum has only finitely many eigenpairs below each bound. The bounded-commutator formulas, compression lemma, localized algebra and mixed resolvent connection prove the entire induction exactly as above. Restriction to C(N) is [D_N]: the external tree class is its scalar index, and the connection/positivity calculation is the general spectral version of SG.T11–SG.T12. Its bounded phase anticommutes with Γ_T, commutes with D_T, and its anticommutator with the product phase is 2|D_T|/(1+D_N²+D_T²)½≥0. Each finite spectral-window creation vector has compact connection error on H_N; density extends it. Index +1 therefore gives the identity even factor. Plaque restriction and local pairing follow from the exact normalized bump as in SG.18.

These hypotheses are sufficient data, not consequences of the original normal spin-c assumption. F₂ supplies them by the actual vertex-edge construction SG.T1–SG.T13. Neither the projection P nor the transverse spin module constructs D_T for an arbitrary return group or arbitrary groupoid. No theorem solving the general graph-Dirac problem follows by changing the Γ label.

### 11A.10. A conditional obstruction to a specified maximal descent

Here is the exact obstruction it would prove. If q:A_max→A_r is a quotient, x∈KK^0(A_max,C), and a projection e satisfies q_*[e]=0 but [e]⊗x≠0, then x has no reduced descent. Indeed x=q^*y would imply [e]⊗x=q_*[e]⊗y=0 by associativity. The odd version uses the appropriate odd K-class and the same calculation. A property-T Kazhdan projection can supply such an e when its projection onto invariant spinors has nonzero graded Dirac index. This is a conditional obstruction to a specified maximal class; it is not a constructed compact-foliation counterexample here.

The survey problem asks for a graph Dirac class with local Bott pairing one. It does not separately require descent of one fixed maximal representative. To refute the general graph-Dirac problem one must either prove every admissible normalized graph cycle equals that obstructed maximal class, or prove the original local Bott image has no integer-valued scalar pairing one in the reduced algebra. A faithful compact isometric property-T suspension, its coherent full spin-c orientation, every arithmetic/lattice prerequisite and the exact index calculation also need full proved providers. The conditional quotient calculation establishes none of these additional assertions.

![Scalar graph induction and complete multiplicity control](../figures/scalar-graph-induction.png)

**Figure 11A.1.** Equations (SG.4), (SG.9a)–(SG.15a), (SG.17)–(SG.19) and (SG.T1)–(SG.T13). The tree panel is the exact radius-two ball in the infinite degree-four tree. Each nonroot vertex of length n pairs with its parent edge; the positive root is unmatched. The block eigenvalues are exactly \(\pm\sqrt{\nu^2+n^2}\), and the root retains the normal eigenvalue. The plotted curves sample this formula for a continuous parameter; they are not a computation of the SU(2) Dirac spectrum. The graph compactness uses two scalar compact factors. [Reproducible source](../figures/scalar-graph-induction.py) · [Editable SVG](../figures/scalar-graph-induction.svg).

## 11B. The complete calibrated reverse product for disk leaves

Keep the hypotheses and the entire calibration of Section 11: \(V\) is compact without boundary; its oriented rank-two leaves, with the prescribed smooth metric, are actual complete Poincaré disks of curvature \(-1\). Write \(C=C(V)\), \(G=\operatorname{Hol}(V,F)\), and \(A=C_r^*(G)\). Keep the full prescribed longitudinal spin-c structure, its physical positive Fourier symbol, grading and Hermitian line transitions. The radial inverse is the negative physical phase on the entire \(W\) module of Lemma 11.2; it is not the literal positive same-spin cycle.

**Theorem 11B.1 (the calibrated reverse KK identity).** Under these hypotheses,
\[
\delta_{\mathrm{inv}}\widehat\otimes_{C(V)}T=1_A
\quad\text{in }KK(A,A).
\]
Together with Theorem 11.1, the two ordered product maps are mutually inverse in both integral K-degrees, including torsion. Under the disk-leaf model \(BG\simeq V\) and the prescribed spin-c identification, they are the two inverses of analytic assembly. The literal same-spin relation of Proposition 11.8 remains \(\delta_S=-\delta_{\mathrm{inv}}[\det S]\), hence \(T\delta_S=-[\det S]\).

The proof retains the entire reverse tensor module, both infinite Bergman sectors and the actual reduced convolution action. The negative Witten product gives the exact compact-subgroup unit. Associated induction and two full Cartan comparisons give the gamma-part restriction projector. A particular equivariant Stinespring interval factors the actual lifted difference through the circle. Parabolic contraction, finite-center descent and whole holonomy-module transport then give the reverse equality. Every needed argument follows below; numerical indices and K-group equalities are not substitutes for the KK proof.

Independent exposition, proofs and figures in this supplement are dedicated to CC0 1.0. The separately bundled DejaVu fonts retain their own licence. Human-source credit is given at the argument where it is used; no source text or page-image derivative is incorporated.


### 11B.1. The complete native represented reverse product

Keep the disk-leaf hypotheses and calibrated finite spin module of Section 11. Definition 0.3 and Theorems 0.4,0.6–0.9 give the complete product criterion and its graded product rules; Theorem 6.0c gives the Clifford product foundation. The actual native regular Dirac cycle and its physical positive symbol are those of Theorem 7.1 and Lemma 11.4. The represented reverse module below has target \(A\); it differs from the first-product target \(C(V)\) in (H.30).

**Result.** On the exact native tensor module, the bounded transform of

\[
 Q=V+D^W,\qquad
 V(u,y)=-b_{W,y}(\log_yu)\widehat\otimes1,
 \qquad D^W=\Gamma_W\widehat\otimes(-i\varepsilon\cdot\nabla_y)
 \tag{P1}
\]

is an actual product representative for the calibrated radial inverse followed by the prescribed native longitudinal class. Here the tensor notation in the differential means the physical graded lift; in an ordinary matrix presentation it is `Γ_W ⊗ (−i ε·∇)`. The derivative holds u fixed. The induced connection on W is used together with the prescribed Clifford connection on S. The proof supplies localized actual module compacts, creation connections and a positive sandwich. It does not make the Gaussian ground projection invariant under the original left convolution. Its comparison with the identity is proved below by the boundary factorization and whole-module transport.

#### P.1 Exact completed module and coefficient positions

For arrows represented by endpoints, write ξ(u,y) for a radial vector on the source fibre at y, and η(y,x) for a native range-spinor convolution column. On compact cores the tensor is

\[
 \Psi(u,y,x)=\xi(u,y)\widehat\otimes\eta(y,x)
       \in W_y\widehat\otimes S_y.
 \tag{P2}
\]

All three points lie on the same actual leaf. Its right action, left action and inner product are

\[
\begin{aligned}
 (\Psi k)(u,y,x)&=\int\Psi(u,y,z)k(z,x)\,dz,\\
 (\pi(a)\Psi)(u,y,x)&=\int a(u,u')\Psi(u',y,x)\,du',\\
 \langle\Psi,\Phi\rangle_A(x,z)
  &=\iint\langle\Psi(u,y,x),\Phi(u,y,z)\rangle_{W_y\otimes S_y}\,du\,dy.
\end{aligned}
\tag{P3}
\]

The measures are the original leafwise Riemannian Haar measures. These formulas give the interior tensor inner product first on elementary tensors and hence its completion E. They retain the actual right A convolution; x is not discarded. Regular localization at x identifies `E ⊗_A L²(G_x)` with `L²(L_x × L_x, W_y ⊗ S_y)` in variables u,y. The regular family is faithful for the reduced algebra. The full grading is Γ_W⊗Γ_S.

The input radial operator lifted to E is

\[
 b=V/s,\quad r=d(u,y),\quad s=\sqrt{1+r^2};
 \qquad b^2=r^2/(1+r^2).
 \tag{P4}
\]

The principal derivative acts on y; left convolution acts on u. Consequently `[D^W,π(a)]=0` on compact cores. No transport between distinct coefficient fibres is required for that commutation: each left-convolution integral has the same coefficient fibre W_y⊗S_y.

#### P.2 Coercivity and regularity on E

Put `f_u(y)=d(y,u)²/2`. Then `log_y u=−grad_y f_u`. In curvature −1 the Hessian eigenvalues are

\[
 \operatorname{Hess}_yf_u=(1,r\coth r),
 \qquad r\coth r\big|_{r=0}=1.
 \tag{P5}
\]

Because the two odd Clifford actions anticommute and the chosen connection makes them parallel, all first-order cross terms cancel:

\[
 Q^2=L+H,\quad L=(D^W)^2+r^2,
 \quad H=\{D^W,V\},\quad
 \|H(u,y)\|\le1+r\coth r\le2+r.
 \tag{P6}
\]

The last bound uses `coth r ≤ 1+1/r`. It holds at r=0 by the smooth limit. Hence, as forms on the compact core,

\[
 Q^2\ge(D^W)^2+\tfrac12r^2-\tfrac52,
 \qquad 1+L\le2(3+Q^2).
 \tag{P7}
\]

In particular

\[
 \|r(Q-i\eta)^{-1}\|\le\sqrt5\quad(\eta\ge1),
 \qquad \|(1-\chi_R)(Q-i)^{-1}\|\le\sqrt5/R,
 \tag{P8}
\]

where χ_R is one for r≤R and between zero and one elsewhere. Its transition is supported at r≥R. These are module bounds, not just pointwise bounds: apply the form estimate in every regular localization and use faithfulness.

The regular operator construction uses the preceding module-generator argument, with one fixed additional leaf point u. On each regular localization Q is a complete Dirac operator plus a smooth self-adjoint potential. The deficiency-vector cutoff proof is unchanged, since the potential cancels in its integration-by-parts identity. Use compact radial cutoffs in y; `‖[Q,χ]‖≤‖d_yχ‖`. The propagator therefore has speed at most one in y with u held fixed. A compact triangle support in (u,y,x) grows by at most |t| in its y edges. It remains compact because bounded-radius disk bundles over compact V are compact.

Wave evolution preserves the compact smooth kernel core and right convolution. Pairing after every regular localization proves preservation of the A-valued inner product, and the opposite wave is its adjoint. Finite-time source energy estimates on the retained compact triangle supports prove all source jets and continuity in transverse parameters. Thus the waves are strongly continuous adjointable module unitaries. The Laplace integrals of their positive and negative time evolutions are adjointable inverse shifts for Q±iη. The generator argument, inverse-domain argument and dense range of the product of the two shifts prove that the compact-core closure is regular self-adjoint. This proof uses no global frame and no assertion that leafwise compactness alone implies module compactness.

For later formulas L also has a regular positive closure. This can be constructed without assuming a Hilbert-module Friedrichs theorem: introduce

\[
 P=D^W\otimes\sigma_z+s\otimes\sigma_x.
\]

The same complete-wave proof makes P regular self-adjoint. Its square is `1+L+E_s`, where `E_s=σ_zσ_x[D^W,s]` is bounded self-adjoint, with norm at most one since `|d_y s|≤1`. Bounded self-adjoint perturbation, proved by a Neumann-series inverse shift at sufficiently large imaginary part, makes `1+L=P²−E_s` regular self-adjoint. Its quadratic form is `(D^W)²+s²≥1`. All compact-core identities below extend by these actual resolvents.

#### P.3 The complete compactness mechanism: smoothing both leaf variables

The required statement is localized compactness. An unlocalized resolvent of Q is not asserted to lie in K(E).

Call B a smoothing convolution operator in u if on the core

\[
 (B\Psi)(u,y,x)=\int k(u,u';y)\Psi(u',y,x)\,du'
 \tag{P9}
\]

with a common bound `d(u,u')≤C` on kernel support, finite two-sided Schur bounds, and smooth coefficient kernels with bounded jets on every retained compact triangle. Actual π(a), `π(a)b`, `π(a)K_b` below, and `[Q,π(a)]` belong to this class. For the last operator the kernel is `a(u,u')[V(u,y)−V(u',y)]`.

**Lemma P.3.** For B as above and each h∈C₀(R), `B h(Q) ∈ K(E)`. The same statement holds for h(P), and consequently `B(ρ+L)⁻¹ ∈ K(E)` for ρ≥1.

**Proof.** First use the tail estimate P8 and B's bounded norm to reduce `B(Q-i)⁻¹` to `Bχ_R(Q-i)⁻¹` in norm, with error at most `‖B‖√5/R`. C₀ functional calculus and resolvent generation then treat every h.

For fixed R approximate the resolvent in uniform scalar norm by Schwartz functions whose Fourier transforms are smooth and compactly supported. Such functions are dense in C₀(R): approximate first by Schwartz functions and then truncate their Fourier transforms in L¹. If the Fourier support is inside [−T,T], finite propagation puts the kernel of h(Q), with u fixed, inside `d(y,y')≤T`. Elliptic estimates and boundedness of `Q^m h(Q)` for every m show that this kernel is smooth in y,y'. Applying the estimates to both sides gives joint smoothness. Parameter-difference energy estimates on the common compact support give its continuity, and the source derivatives needed for the kernel approximation. The estimates are uniform on the retained compact triangles.

The composition `Bχ_R h(Q)` therefore has kernel

\[
 k(u,u';y)\chi_R(d(u',y))K_{u'}(y,y').
 \tag{P10}
\]

The δ in u for h(Q) has been removed by the smoothing convolution B. Its support satisfies

\[
 d(u,u')\le C,\quad d(u',y)\le R+1,\quad
 d(y,y')\le T,\quad d(u',y')\le R+1+T.
 \tag{P11}
\]

These bounds describe a compact subset of the iterated endpoint groupoid over V. Importantly, the kernel is now smooth in both u,u' and y,y'. In finitely many radial-bundle charts, uniformly approximate it by finite separated sums in its radial variables, with continuous compact arrow coefficients in (y,y'). Each sum is `T_ξ K T_ζ*`, where ξ,ζ are radial creation vectors and `K∈K(E_T)=eM_N(A)e` is a compact arrow coefficient. It is an actual compact on E. The common compact volume bounds and finite-overlap two-sided Schur estimate make the uniform kernel approximation a module-norm approximation. Hence P10 belongs to K(E), and the cutoff/tail and scalar approximations prove the first statement.

The same proof applies to P: its potential s has the same radial coercivity after a bounded cross term. Since `1+L=P²−E_s`, the resolvent identity

\[
 (\rho+L)^{-1}=(\rho-1+P^2)^{-1}
  +(\rho-1+P^2)^{-1}E_s(\rho+L)^{-1}
\]

gives the result for ρ>1; extend to ρ=1 by the resolvent identity. This completes an actual module compactness proof rather than a fibrewise claim. ∎

The Cartan–Hadamard differential of the exponential stretches radial and angular vectors by 1 and `sinh r/r≥1`. Therefore its inverse logarithm has differential norm ≤1, and integration along the geodesic from u to u' gives

\[
 |\log_yu-\log_yu'|\le d(u,u').
 \tag{P12}
\]

For a compact smooth convolution a with propagation C_a and Schur bound K_a,

\[
 \|[Q,\pi(a)]\|=\|[V,\pi(a)]\|\le C_aK_a.
 \tag{P13}
\]

Its kernel is a P9 kernel, so `[Q,π(a)](Q-z)⁻¹` is compact by Lemma P.3. The ordinary resolvent commutator identity and the two-pole bounded-transform integral yield `[F_Q,π(a)]∈K(E)`, with integrable bound `C_aK_a/(1+t²)`. The localized square defect follows from `π(a)(1+Q²)⁻¹∈K(E)`. Smooth convolution kernels are dense in A; norm approximation finishes the Kasparov cycle.

#### P.4 Actual creation connections

Let ξ be a homogeneous compact smooth radial vector, σ=(−1)^|ξ|, and let T_ξ:E_T→E be its graded creation map. With the induced W connection,

\[
 K_\xi=Q T_\xi-\sigma T_\xi D_S
       =V T_\xi+\text{the Clifford contraction of }T_{\nabla_W\xi}
 \tag{P14}
\]

is bounded and adjointable. Its creation vectors have compact arrow support; multiplication by V is bounded on that support. A finite bundle partition expresses the contraction as a finite sum of creation maps followed by bounded finite spinor coefficients.

An approximate identity a_n of A acts nondegenerately on the radial regular module. Thus `π(a_n)K_ξ→K_ξ` in operator norm. Lemma P.3 and adjoints give `(Q-z)⁻¹π(a_n)∈K(E)`. Consequently `(Q-z)⁻¹K_ξ` is a norm limit of compact maps from E_T to E and is compact in their linking algebra. The mixed inverse-shift identity is

\[
 (Q-z)^{-1}T_\xi-\sigma T_\xi(D_S-\sigma z)^{-1}
 =-\sigma(Q-z)^{-1}K_\xi(D_S-\sigma z)^{-1}.
 \tag{P15}
\]

Its norm is at most `‖K_ξ‖/|Im z|²`. Sum the poles `z=±i√(1+t²)` and integrate. It follows that `F_QT_ξ−σT_ξF_{D_S}` is compact; adjoints give the other connection error. Norm density of the compact creation core treats every radial ξ. This is the required connection to the exact native second input, with its full grading.

#### P.5 Positive sandwich with the calibrated first input

Put

\[
 g=r^2/s,\qquad S_0=(1+L)^{-1/2}.
 \tag{P16}
\]

The normalized potential b has uniformly bounded first and second covariant y derivatives. This follows from P5, `|d_y r|=1`, `Hess r=coth r(g-dr²)`, and the smooth even expansions at r=0. At infinity the factors r/s and its radial derivatives keep these expressions bounded. The same statements hold for first and second derivatives of g. Full bundle matrices are parallel, so no uncontrolled transition term occurs.

Using ρ=1+t and P6,

\[
 F_Q-QS_0=-\frac1\pi\int_0^\infty
 t^{-1/2}Q(\rho+Q^2)^{-1}H(\rho+L)^{-1}\,dt.
 \tag{P17}
\]

The integrand norm is at most `5/(4ρ)`: indeed `‖Q(ρ+Q²)⁻¹‖≤1/(2√ρ)` and `‖H(ρ+L)⁻¹‖≤2/ρ+1/(2√ρ)≤5/(2√ρ)`. The weighted bound is integrable. After left multiplication by π(a), or π(a)b, each integrand is compact by Lemma P.3, so the difference is compact with those localizations.

The core identities are

\[
 \{b,Q\}=2g+K_b,\quad K_b=\{D^W,b\},\qquad
 \{b,QS_0\}=2gS_0+K_bS_0+Q[S_0,b].
 \tag{P18}
\]

K_b is bounded multiplication. The first remainder is compact after π(a) by Lemma P.3. For the second use

\[
 S_0=\frac1\pi\int_0^\infty t^{-1/2}(1+L+t)^{-1}\,dt.
\]

The operator `[L,b]` has order one in y, with bounded coefficients. The Weitzenböck identity for D^W has a bounded curvature term, because all induced bundles live over compact V. Its energy estimate gives `‖[L,b](ρ+L)⁻¹‖≤Cρ⁻¹/²`. Also `‖Q(ρ+L)⁻¹‖≤√3 ρ⁻¹/²`. The latter is compact after π(a), since

\[
 Q(\rho+L)^{-1}=Q(\rho+Q^2)^{-1}
      \big(1+H(\rho+L)^{-1}\big).
\]

Thus the integrands for `π(a)Q[S_0,b]` are compact and bounded by an integrable multiple of `t⁻¹/²/(1+t)`. This proves compactness in the actual K(E).

The exponent-1/4 resolvent formula similarly gives a bounded commutator `[g,S_0^{1/2}]`, compact after π(a), with integrand bound `Ct⁻¹/⁴/(1+t)³/²`. For the unbounded multiplier g, first use smooth radial cutoffs g_n with uniformly bounded first and second derivatives. The operators `T_n=[L,g_n](ρ+L)⁻¹` are uniformly bounded and converge strongly together with their adjoints: derivative coefficients converge on every compact triangle core, while the energy bound and density extend that convergence to E. Consequently `π(a)(ρ+L)⁻¹(T_n−T)` converges in norm, since its left factor is an actual module compact and strong convergence of `T_n*` gives norm convergence after each finite-rank approximation. The common integrable bound gives the localized norm limit of the integral. The unlocalized commutator integral itself has a bounded extension from the core, since the same operator-norm bound is integrable. Hence

\[
 gS_0=S_0^{1/2}gS_0^{1/2}
          +[g,S_0^{1/2}]S_0^{1/2}.
 \tag{P19}
\]

The middle form is positive and bounded by one: `1+L≥s²`, square-root order gives `(1+L)^{1/2}≥s`, and `0≤g≤s`. Its bounded extension is therefore positive. Combining P17–P19 yields

\[
 \pi(a)\{b,F_Q\}\pi(a)^*
   \equiv2\pi(a)S_0^{1/2}gS_0^{1/2}\pi(a)^*\ge0
   \pmod{\mathcal K(E)}.
 \tag{P20}
\]

This is the exact positive sandwich in Definition 0.3 for the first input b, not for a substituted or sign-reversed radial phase. Together with P15 and the cycle checks, it proves the result stated at P1.

#### P.6 Boundary of this result

After graded flip and the full Morita evaluation, Q is `−i a·∇+i c·grad f_u`. The degree unitary i^N changes it to the negative Witten operator `d+d*−(df_u∧+ι_grad f_u)`. In rank two its Gaussian ground is the even top form `e⁻f_u vol_y` with the entire S/W transition phase canceled. The full evaluation (H.10)–(H.11), after the even graded flip, retains its entire coefficient vector. These observations do not make left convolution preserve that ground: left convolution changes u while the Gaussian depends on u. Phase removal of the complement cannot be declared a degenerate-cycle split before that representation issue is solved.

The product representative is identified with the complete scalar negative-Witten cycle in BC.24–BC.26. The compactification, particular Stinespring interval and transported class equality below then establish its identity class.

#### P.7. The sufficient global energy estimate

For rho>=1, Q squared <= L+2+r <= 3(1+L) as forms, using r<=r squared/2+1/2 and r squared<=L. Consequently the squared norm of Q(rho+L) inverse is at most 3/rho. This sufficient square-root-three bound preserves all integral, compactness and positive-sandwich conclusions above. This is the sufficient square-root-three estimate used in P.5.

The evaluation can be checked on every coefficient and the actual graph domain. The even graded flip sends \(w\widehat\otimes s\) to \((-1)^{|w||s|}s\widehat\otimes w\). Compose it with the onto even unitary \(U:S\widehat\otimes W\to\Lambda^*\) of (H.10). Formula (H.11) sends the differential generator to \(a_j\) and the physical radial generator to \(b_j=ic_j\), with the tensor sign included exactly once. Since \(\log_yu=-\nabla_y f_u\), the complete operator becomes
\[
-i\sum_j a_j\nabla_{e_j}+i c(\nabla f_u).
\]
The full \(H\) and \(H^*\) transitions cancel under this evaluation, and the induced tensor connection is the Levi–Civita exterior connection. The even degree unitary \(i^N\) has
\[
i^Na_j i^{-N}=ic_j,\qquad i^Nc_j i^{-N}=ia_j.
\]
It therefore gives exactly \(d+d^*-c_+(df_u)\) on every exterior degree. Each map is an onto bundle isometry, preserves the smooth compact core and its graph norm, and intertwines all frame transitions. Closedness carries the identity to the full regular closure. Thus the physical coefficient, exterior parity, complete domains and actual left/right convolution are all retained; the subsequent comparison changes the KK cycle through the stated whole homotopies, rather than replacing it by a Gaussian compression.


#### The complete native phase comparison

#### RI.1 Actual native cycle and the corrected estimate

Use the same \(C=C(V)\), \(A=C_r^*(G)\), oriented rank-two \(F\), full prescribed spin-c module \(S\), and \(W=\overline S\widehat\otimes_{\mathrm{Cl}_{+}(F)}\Lambda F_{\mathbb C}\) of Lemma 11.2. The physical generators of \(W\) are \((\sigma_y,\sigma_x)\), its grading is \(\sigma_z\), and its ordered physical volume is minus that grading. The full \(H^*\otimes H\) line cancellation and graded rank-four evaluation are retained. No even intertwiner of the ordered rank-two positive and negative physical symbols is introduced.

The native reverse product on the entire completed right module is

\[
 E=\mathcal H_W\widehat\otimes_C\mathscr E_S,
 \qquad Q=-b_{W,y}(\log_yu)\widehat\otimes1+D^W_y,
 \qquad F_Q=Q(1+Q^2)^{-1/2}.
 \tag{RI.1}
\]

Its core consists of \(W_y\widehat\otimes S_y\)-valued functions of \((u,y,x)\). Left convolution changes \(u\), the native derivative changes \(y\) with \(u\) fixed, and right convolution changes \(x\). The module, regular closure, actual localized compactness, all graded creation connections, and the positive sandwich with the original bounded radial phase \(b\) are supplied by P.1–P.5 above. They are not replaced by fibrewise compactness.

The sufficient global estimate used in P.5 is

\[
 Q^2=L+H\le L+2+r\le3(1+L),
 \qquad
 \|Q(\rho+L)^{-1}\|\le\sqrt3\,\rho^{-1/2}
 \quad(\rho\ge1).
 \tag{RI.2}
\]

Here \(r^2\le L\) and \(r\le(r^2+1)/2\). Squaring the norm and applying functional calculus to \(L\) gives the second bound. Multiplying the existing integrable majorants by this fixed constant preserves P.5 and its sign. The estimate is the one used in the actual product proof above.

The full negative-Witten evaluation has the even Gaussian kernel

\[
 g_u(y)=Z^{-1/2}e^{-d(y,u)^2/2}\operatorname{vol}_y,
 \quad Z=\pi^{3/2}e^{1/4}\operatorname{erf}(1/2),
 \quad V:A\longrightarrow E^+,
 \quad (Vk)(u,y,x)=g_u(y)k(u,x),
 \quad P=VV^*.
 \tag{RI.3}
\]

Its underlying full finite coefficient is \((w_0\otimes s_0+w_1\otimes s_1)/\sqrt2\), sent to \(-ie_{12}\) before the recorded even degree unitary. The complete negative-Witten proof WP.1–WP.3 gives \(Q^2\ge1\) on \(1-P\), and identifies its **entire** kernel with \(VA\). Lemma 11.7 proves, directly on this completed tensor, that

\[
 \pi(a)V\in\mathcal K_A(A,E),\quad
 \pi(a)P,\ P\pi(a)\in\mathcal K_A(E),\quad
 V^*\pi(a)V=M_\kappa(a),\quad0<\kappa(d)<1\ (d>0).
 \tag{RI.4}
\]

The first compactness follows from the actual column \(a(u,x)g_x(y)\) and Gaussian tail control, not from an unlocalized compact projection. \(M_\kappa\) is completely positive but is not a representation.

#### RI.2 A phase homotopy retaining the ordered-product checks

Choose a continuous odd bounded function \(h\) with \(h(0)=0\) and \(h(t)=\operatorname{sgn}(t)\) for \(|t|\ge1\). Set

\[
 F_\infty=h(Q),\qquad k(Q)=F_\infty-F_Q,
 \qquad F_t=F_Q+t\,k(Q),\quad0\le t\le1.
 \tag{RI.5}
\]

The scalar function \(k\) belongs to \(C_0(\mathbb R)\). The full kernel and uniform native gap imply \(F_\infty^2=1-P\) and \(F_\infty V=0\). Lemma P.3 gives \(\pi(a)k(Q),k(Q)\pi(a)\in\mathcal K_A(E)\). Thus every \(F_t\), with the **fixed original** left representation, is a Kasparov cycle: commutators and localized square defects differ from those of \(F_Q\) by actual compacts. For example, in a term \(\pi(a)F_Qk(Q)\), commute \(F_Q\) past \(\pi(a)\) modulo the already proved compact commutator, and then use \(\pi(a)k(Q)\in\mathcal K_A(E)\). These compact defects vary in norm with \(t\), so the constant-module family is a KK homotopy.

The graded native creation connections also survive. For a compact smooth radial vector \(\xi\), an approximate identity \(a_n\) of the first \(A\)-action has \(\pi(a_n)T_\xi\to T_\xi\) in norm. Since \(k(Q)\pi(a_n)\) is compact, \(k(Q)T_\xi\) is compact in \(\mathcal K_A(\mathscr E_S,E)\); adjoints give the other linking corner. Add this to P.4's exact graded connection and pass to all radial vectors by density. No commutation of \(k(Q)\) with \(T_\xi\) is assumed.

The positive sandwich with the original \(b\) survives as well:

\[
 \pi(a)\{b,F_t-F_Q\}\pi(a)^*
 =t\bigl(\pi(a)b\,k(Q)\pi(a)^*
       +\pi(a)k(Q)b\pi(a)^*\bigr)
 \in\mathcal K_A(E).
 \tag{RI.6}
\]

For the first term use P.3 for the stated smoothing kernel \(B=\pi(a)b\); for the second use \(\pi(a)k(Q)\). Consequently the positive coset in the exact product criterion is unchanged. This proves a phase reduction of the actual native ordered-product representative, including all its connections and positive sandwiches. It does not delete its complement.


### 11B.2. The entire scalar boundary homotopy



Let \(G=\operatorname{PSU}(1,1)\cong\operatorname{PSL}_2(\mathbb R)\) act on the unit disk and its boundary \(\mathbb T\). Put \(d\theta/(2\pi)\) on the boundary. For \(h=g^{-1}\), let \(J_h(\zeta)=|h'(\zeta)|\), the positive boundary Jacobian. Its cocycle rule gives a representation on smooth boundary functions,

\[
 P_s(g)f(\zeta)=J_h(\zeta)^{(1+s)/2}f(h\zeta),\qquad 0\le s<1.
 \tag{HB.1}
\]

The parameter is a complementary-series parameter; it is different from the geometric deformation parameter in Section 11. The complete scalar representation argument below retains both infinite Fourier sectors. Historical context is Pierre Julg, [The Bernstein–Gelfand–Gelfand complex for rank one semi simple Lie groups as a Kasparov module, exact v1](https://arxiv.org/abs/1605.07408v1), introduction and §1.1; every needed calculation is proved here.

#### HB.1. Positive Fourier weights and exact unitary representations

For \(0<s<1\), define \(k_s(\theta)=|1-e^{i\theta}|^{s-1}\), its mean \(c_s>0\), and the Hermitian form

\[
 \langle f_1,f_2\rangle_s=c_s^{-1}\iint
 \overline{f_1(\zeta)}f_2(\eta)|\zeta-\eta|^{s-1}
 \frac{d\theta\,d\phi}{(2\pi)^2}.
 \tag{HB.2}
\]

The singularity is integrable because its exponent is greater than \(-1\). If \(f=\sum f_n\zeta^n\), then

\[
 \|f\|_s^2=\sum_{n\in\mathbb Z}w_n(s)|f_n|^2,
 \quad w_0=1,\quad
 w_n(s)=\prod_{j=0}^{|n|-1}\frac{2j+1-s}{2j+1+s}>0.
 \tag{HB.3}
\]

**Proof of the formula and positivity.** The Fourier coefficients of the real even kernel are real and equal at opposite integers. Write \(A_n=\int_0^{2\pi}(2\sin(\theta/2))^{s-1}\cos(n\theta)\,d\theta\). Integrate the derivative of \((\sin(\theta/2))^s\cos((n+1/2)\theta)\). The boundary terms vanish since \(s>0\). The two product-to-sum formulas yield

\[
 (s-2n-1)A_n+(s+2n+1)A_{n+1}=0\quad(n\ge0).
 \tag{HB.4}
\]

Division by the positive \(A_0\) gives (HB.3). Trigonometric polynomials therefore have a positive definite form. Smooth functions have rapidly decreasing coefficients, and termwise integration against the integrable kernel, followed by uniform convergence, proves (HB.3) for them as well. Completion is exactly the weighted sequence space. In particular no assertion of positivity is inferred solely from the pointwise positivity of the kernel.

For a disk Möbius map,

\[
 |g\zeta-g\eta|^2=J_g(\zeta)J_g(\eta)|\zeta-\eta|^2.
 \tag{HB.5}
\]

This follows directly by subtracting \(g(z)=e^{i\omega}(z-a)/(1-\bar a z)\) at two points and using \(J_g(\zeta)=(1-|a|^2)/|1-\bar a\zeta|^2\). In (HB.2) change both variables by \(g\). The exponent of each Jacobian is \(1-(1+s)/2+(s-1)/2=0\). Thus (HB.1) is unitary on this completion. Smooth dependence on \(g\) on smooth functions, their density, and the common unitary norm prove strong continuity. At \(s=0\) use the usual \(L^2(\mathbb T)\) form and the half-density action (HB.1); its unitarity is the ordinary change of variables.

Identify every completion with the single Hilbert space \(\mathcal H=\ell^2(\mathbb Z)\) by the unitary \(f\mapsto(\sqrt{w_n(s)}f_n)_n\). Let \(U_s(g)\) denote the transported representations and put \(w_n(0)=1\). Each basis vector \(e_n\) on this fixed space is even in the graded module introduced below; its integer index is a Fourier label, not a parity.

#### HB.2. A compact change of generators, including the endpoint

Choose the one-parameter disk subgroup whose boundary velocity is \(v(\theta)=\sin\theta\), and denote its real parameter by \(t\). Before transport its skew generator on smooth functions is

\[
 D_s f=-\sin\theta\,\partial_\theta f-\frac{1+s}{2}\cos\theta\,f.
 \tag{HB.6}
\]

On the fixed Fourier space it is

\[
 D_s e_n=-a_n(s)e_{n+1}+a_{n-1}(s)e_{n-1},\quad
 a_n(s)=\frac12\operatorname{sgn}(n+\tfrac12)
 \sqrt{(n+\tfrac12)^2-s^2/4}.
 \tag{HB.7}
\]

Indeed the untransported upper coefficient is \(-(n+(1+s)/2)/2\) and the lower coefficient is \((n-(1+s)/2)/2\). Use the two ratios from (HB.3), separately for positive and negative indices. The sign in (HB.7) is necessary across \(n=-1\). Rotations have the same diagonal skew generator \(-in\) for every \(s\).

Let \(B_s=D_s-D_0\), initially on finite sequences. It extends to a bounded compact skew-adjoint operator for every \(0\le s\le1\). In fact

\[
 |a_n(s)-a_n(0)|=
 \frac{s^2}{8\bigl(|n+1/2|+\sqrt{(n+1/2)^2-s^2/4}\bigr)},
 \tag{HB.8}
\]

and this tends to zero as \(|n|\to\infty\), uniformly in \(s\). A weighted shift whose weights tend to zero is compact: truncate to finitely many weights and bound the remaining shift norm by their supremum. This proves compactness of both bands of \(B_s\). Moreover,

\[
 \|B_s-B_r\|\le\frac12\sqrt{|s^2-r^2|},\qquad 0\le r,s\le1.
 \tag{HB.9}
\]

Use \(|\sqrt{x}-\sqrt y|\le\sqrt{|x-y|}\) in (HB.7) and add the two shift norms. Thus the operator family is norm continuous through the endpoint at which its two central coefficients vanish.

**Core and closure, without a domain shortcut.** At \(s=0\), (HB.6) is the infinitesimal generator of the actual unitary half-density flow. Its restriction to finite Fourier polynomials is essentially skew-adjoint. Here is a direct deficiency proof that also works for (HB.7). Conjugate the skew Jacobi matrix by fixed powers of \(i\) to obtain a real self-adjoint tridiagonal matrix. On either infinite half-line its nonzero off-diagonal magnitudes grow at most \(C(1+|n|)\). For a square summable distributional solution of \(Ju=z u\), \(\operatorname{Im}z\ne0\), summation of the imaginary parts on \([-N,N]\) gives a boundary term bounded by the sum of
\(c_N|u_Nu_{N+1}|\) and \(c_{-N-1}|u_{-N-1}u_{-N}|\), where \(c_n=|a_n(s)|\).

There is a sequence of radii along which both boundary terms tend to zero. Otherwise their sum is at least a positive constant for all sufficiently large radii. Since \(c_n\le C(N+1)\) at these boundaries and \(2|u_nu_{n+1}|\le |u_n|^2+|u_{n+1}|^2\), summing this lower bound divided by \(N+1\) contradicts square summability and divergence of the harmonic series. The finite-interval identity along this sequence gives \(\operatorname{Im}z\,\|u\|^2=0\), hence \(u=0\). If a central coefficient is zero, apply the same argument to its two half-lines and the finite isolated component. Both deficiency spaces vanish. Thus the finite sequences are a core for the self-adjoint closure, and consequently for the skew closure of (HB.7).

Equivalently, the skew closures are \(D_0+B_s\) on the single domain \(\operatorname{Dom}(D_0)\): bounded skew-adjoint perturbation proves they are skew-adjoint, and boundedness preserves the finite-sequence graph core. For \(s<1\), their exponential is precisely the already constructed \(U_s\) on this subgroup, because its generator agrees on that essential core. This also proves that the displayed differential formula has not implicitly selected a different extension.

**Compactness for actual group elements.** For the chosen subgroup, Duhamel's identity gives

\[
 e^{t(D_0+B_s)}-e^{t(D_0+B_r)}
 =\int_0^t e^{(t-v)(D_0+B_s)}(B_s-B_r)e^{v(D_0+B_r)}\,dv.
 \tag{HB.10}
\]

For negative \(t\) the integral is oriented in the usual way. One obtains the identity first on the common domain by differentiation and then on the whole space by bounded extension. Its integrand is compact and norm continuous: strongly continuous unitaries on either side of a fixed compact operator have this property by finite-rank approximation. The integral is therefore a compact operator and has norm at most \(|t|\|B_s-B_r\|\).

Every disk Möbius map has \(g=k_1\exp(tX)k_2\), where \(k_i\) are rotations and \(X\) is the chosen boost (with either sign of \(t\)). This decomposition is elementary: rotate the image of zero to the real diameter, use its unique real boost parameter, and the residual map fixing zero is a rotation. On a compact set of group elements the boost parameter is bounded. Since rotations are independent of \(s\), (HB.10) proves

\[
 U_s(g)-U_r(g)\in\mathcal K(\mathcal H),\qquad
 \|U_s(g)-U_r(g)\|\le |t(g)|\|B_s-B_r\|.
 \tag{HB.11}
\]

For \(s,r<1\) these are the original representations. As \(s\uparrow1\), they have point-norm limits \(U_1(g)\). Taking limits of multiplication, inverse and adjoint proves that \(U_1\) is an actual unitary representation, not just a collection of limiting operators. It is strongly continuous: near the identity one may use rotation/boost local coordinates and the common estimate, or approximate a vector by finite sequences. The compact difference in (HB.11) is norm continuous jointly in \((s,r,g)\). For detail near any fixed group element, factor a nearby element into finitely many rotations and boosts with continuous small parameters, using local coordinates from their independent Lie algebra directions. Each compact difference has the norm continuity furnished by (HB.10); products with strongly continuous unitaries preserve it. This local argument avoids requiring a globally continuous choice of the two rotations in the diameter decomposition.

#### HB.3. The entire endpoint and its genuine graded homotopy

The two coefficients \(a_{-1}(1)=a_0(1)=0\). Both the boost and the rotation therefore preserve

\[
 \mathcal H=\mathbb Ce_0\oplus\mathcal H_{>0}\oplus\mathcal H_{<0}.
 \tag{HB.12}
\]

They act trivially on the first summand, so the whole group does. On the positive summand, the endpoint is the holomorphic Bergman one-form representation, with its ordinary pullback action; on the negative summand it is its conjugate. Here is a full identification, including normalization. From (HB.3),

\[
 \lim_{s\uparrow1}\frac{w_n(s)}{1-s}=\frac1{2|n|}\quad(n\ne0).
 \tag{HB.13}
\]

The first factor tends to \((1-s)/2\), and the remaining product tends to \(1/|n|\). For \(n>0\), write a boundary function as \(f(\zeta)=\zeta F(\zeta)\). A disk map satisfies \(J_h(\zeta)=\zeta h'(\zeta)/h(\zeta)\) on the circle, since differentiation of its boundary argument gives that positive real value. Therefore at exponent one,

\[
 P_1(g)(\zeta F(\zeta))=\zeta h'(\zeta)F(h\zeta).
 \tag{HB.14}
\]

This is exactly pullback of \(F(z)\,dz\). The onto unitary from \(\mathcal H_{>0}\) to the holomorphic Bergman one-forms, with norm \(\pi^{-1}\int_{\mathbb D}|F(z)|^2dx\,dy\), is

\[
 e_n\longmapsto\sqrt n\,z^{n-1}\,dz\quad(n\ge1).
 \tag{HB.15}
\]

Orthogonality follows from angular integration and the squared norm of \(z^{n-1}\) is \(1/n\). Completeness follows by the Taylor expansion and the same integral identity at radii approaching one. Pullback is unitary by the conformal area formula. The conjugate formula gives the entire negative summand.

To verify this identification for the actual endpoint rather than a formal differential limit, apply \(U_s(g)\) to each fixed \(e_n\). The smooth coefficients of \(P_s(g)\zeta^n\) decrease faster than every power, uniformly for \(s\) near one and \(g\) in a compact set. For positive \(n\), its mean at \(s=1\) is zero by (HB.14); at nearby \(s\) the mean is \(O(1-s)\). Division by \(\sqrt{w_n(s)}\) therefore makes the mean component \(O(\sqrt{1-s})\). The other components converge with weight ratio \(\sqrt{w_m(s)/w_n(s)}\to\sqrt{n/|m|}\). For \(s\) near one, \(w_m(s)/(1-s)\le1\) for every \(m\ne0\), because all factors after the first in (HB.3) are at most one. This and rapid decrease justify norm convergence of the columns. Formula (HB.14) shows that the negative endpoint components vanish and that the positive ones have exactly the coefficients of (HB.15)'s pullback. For \(n=0\), the mean tends to \(\int J_h=1\), while the nonzero weighted components tend to zero. Complex conjugation treats negative \(n\). Density of these columns proves the asserted complete endpoint representation.

Now put \(\mathcal H^+=\ell^2(\mathbb Z)\), \(\mathcal H^-=\ell^2(\mathbb Z\setminus\{0\})\). Give the latter the fixed endpoint representation \(U_1^-=U_1|_{\mathcal H_{>0}\oplus\mathcal H_{<0}}\). Let \(T:\mathcal H^+\to\mathcal H^-\) delete the constant coefficient and identify every other basis vector. On the whole graded space set

\[
 \pi_s=U_s\oplus U_1^-,\qquad
 F=\begin{pmatrix}0&T^*\\T&0\end{pmatrix},\qquad
 F^2=1-P_0,
 \tag{HB.16}
\]

where \(P_0\) is the even rank-one projection onto \(e_0\). This is a \(G\)-equivariant Kasparov \((\mathbb C,\mathbb C)\) cycle for every \(s\). The scalar algebra acts by scalars. The operator is odd and self-adjoint, its square defect is finite rank, and

\[
 T U_s(g)-U_1^-(g)T=T\bigl(U_s(g)-U_1(g)\bigr)
 \tag{HB.17}
\]

is compact by (HB.11); the adjoint block has the same property. Compactness here is in the actual Hilbert-space compact ideal, and joint norm continuity of these defects is already proved. It is not obtained by compressing a nonmultiplicative scalar kernel.

There is one genuine homotopy cycle over \(C([0,1])\): take continuous sections of the fixed graded Hilbert space, constant \(F\), and the strongly continuous family \(\pi_s(g)\). It acts adjointably because both it and its adjoint carry continuous sections to continuous sections. The square defect is the constant finite-rank projection; every equivariance defect is a norm-continuous compact-operator section by (HB.17), hence a compact operator on this module. The scalar representation and grading are constant. Thus all endpoints occur in a single complete equivariant cycle.

At \(s=1\), both nonzero sectors have the same representation on the two gradings and \(T\) is their exact equivariant unitary. That whole summand is degenerate. Its remaining even line has the trivial representation, operator zero and finite-dimensional compact left scalar action. It is exactly the unit class. Consequently the boundary cycle at the half-density endpoint has the full identity

\[
 [\mathcal H^+\oplus\mathcal H^-,\pi_0,F]
 =1\quad\hbox{in }KK^G(\mathbb C,\mathbb C).
 \tag{HB.18}
\]

This is a representation homotopy with the entire graded complement retained until it becomes exactly degenerate. The two disappearing links are essential to the endpoint; replacing all operators by their constant-mode compression would not prove (HB.18).

The entire boundary homotopy (HB.16)–(HB.18) is a scalar equivariant homotopy. The native comparison and groupoid transport are proved below; no closed-disk function covariance is asserted for its intermediate boundary representations.


### 11B.3. The whole native-to-boundary comparison and reduced module

We compare the complete native reverse cycle with the boundary/Hodge cycle and implement both on the reduced holonomy module. The compact-difference reduction here is an intermediate step; the complete proof of its vanishing follows below.

Use \(G_0=\operatorname{PSU}(1,1)\), \(K=SO(2)\), reserving \(\mathcal G=\operatorname{Hol}(V,F)\) for the actual disk-leaf groupoid and \(A=C_r^*(\mathcal G)\). The leaf hypotheses, physical \(W/S\) modules, all line data and the literal native relation

\[
 \delta_S=-\delta_{\rm inv}[D],\quad
 T\delta_S=-[D],\qquad D=\det S=LH^2                \tag{BC.1}
\]

are unchanged. In particular no homotopy below is called an inverse of the unchanged positive native pair.

#### BC.2. The whole confining cycle is an actual equivariant Fredholm cycle

On the complete curvature \(-1\) disk let

\[
 H_e=L^2(\Lambda^0\oplus\Lambda^2),\qquad
 H_o=L^2(\Lambda^1),\quad H=H_e\oplus H_o,
 \quad R(g)=(g^{-1})^* .                              \tag{BC.2}
\]

All form norms are the actual Riemannian norms with metric \(4|dz|^2/(1-|z|^2)^2\). Pullback by oriented disk isometries is unitary and preserves exterior parity. It is strongly continuous, first on compact smooth forms and then by density. Put \(o=0\), \(f_o(y)=d(o,y)^2/2\), and use the full negative-Witten realization from the native product's graded flip/evaluation and even degree unitary:

\[
 Q_o=d+d^*-(df_o\wedge+\iota_{\nabla f_o}),\qquad
 F_o=\operatorname{phase}(Q_o)\text{ off its kernel},\quad
 g_o=Z^{-1/2}e^{-f_o}\operatorname{vol},\quad
 Z=\pi^{3/2}e^{1/4}\operatorname{erf}(1/2).           \tag{BC.3}
\]

The full direct proof in WP.1–WP.3 gives the entire kernel \(\mathbb Cg_o\), even, and the square gap at least one. Its confining form bound gives compact resolvents on the Hilbert space. Here is the analytic content used, with the full domain and kernel proof in WP.1–WP.3. The Hessian eigenvalues are \(1,r\coth r\), with their smooth zero limits; the degree-one curvature is \(-1\). After the weighted Witten conjugation the degree-one Weitzenböck term is bounded below by one and the degree-two term by four. The weighted zero-degree energy has only constants as zero vectors, its weight has finite mass \(Z\), and differential intertwining bounds every positive scalar eigenvalue below by one. Hodge star gives the whole negative-Witten top Gaussian. The confining estimate \(Q_o^2\ge\nabla^*\nabla+r^2/2-4\), compact-radius Rellich approximation and its tail bound give compact resolvents. Compact radial approximation extends all identities from the smooth core. The complete-wave/inverse-shift construction in the corrected native product proves its regular closure on the full module; in this Hilbert-space realization it yields the same self-adjoint compact-core closure.

Thus \(S=F_o^+:H_e\to H_o\) is a bounded coisometry on the **entire** spaces:

\[
 S^*S=1-P_g,\quad SS^*=1_{H_o},\quad Sg_o=0,
 \qquad P_g=\theta_{g_o,g_o}.                         \tag{BC.4}
\]

Rotations fix \(g_o\) and commute exactly with \(Q_o,F_o,S\). For arbitrary \(g\), the Hodge differential is exactly invariant and

\[
 Q_o-R(g)Q_oR(g)^*
 =\bigl(df_{go}-df_o\bigr)\wedge+
       \iota_{\nabla f_{go}-\nabla f_o},\qquad
 \|Q_o-R(g)Q_oR(g)^*\|\le d(o,go).                  \tag{BC.5}
\]

The norm bound follows from the inverse exponential differential, whose radial and angular singular values are \(1,r/\sinh r\le1\): integrate it along the geodesic from \(o\) to \(go\) at the fixed base point \(y\). This is the actual bounded **difference** of the unbounded potentials, not a claim that either potential is bounded. Bounded perturbation preserves the self-adjoint domain, since both isometric transforms have the same Hodge core and differ there by that bounded multiplication.

The two resolvent identities at \(\pm i\sqrt{1+t^2}\) give compact difference integrands bounded by \(d(o,go)/(1+t^2)\). Integrating the bounded-transform formula proves

\[
 R(g)F_oR(g)^*-F_o\in\mathcal K(H),                 \tag{BC.6}
\]

jointly norm continuously in \(g\). To pass from the bounded transform to the phase use their \(C_0\) functional-calculus difference, compact by the actual compact resolvent; its isometric conjugates are norm continuous by finite-rank approximation. Continuity for the bounded-transform integral follows from the Lipschitz dependence of (BC.5) on its center and the common integrable bound. Hence \(x=(H,R,F_o)\) is an actual \(G_0\)-equivariant \((\mathbb C,\mathbb C)\) cycle. Its ordinary index is one, but **its equivariant class is not identified with the unit by that index**.

#### BC.3. Exact complete Hodge splitting and the boundary harmonic coefficient

Let \(D_0=d+d^*\) be the unconfined Hodge operator on the same complete disk. Compact smooth forms are its essential core. For completeness, if \((D_0^*\pm i)u=0\), local elliptic regularity and cutoffs \(\eta_R=\chi(d(o,y)/R)\), \(\|d\eta_R\|\le c/R\), give
\(\|\eta_Ru\|^2\le(c/R)\|\eta_Ru\|\|u\|\) by integration by parts. Letting \(R\to\infty\) annihilates both deficiency spaces. Thus its closed operator is self-adjoint, and invariance on the core makes all its spectral functions exactly equivariant.

There is an elementary scalar gap. In the upper half-plane metric \((dx^2+dy^2)/y^2\), compactly supported scalar \(h\) satisfies

\[
 \int |\partial_yh-h/(2y)|^2\,dx\,dy
 =\int|\partial_yh|^2\,dx\,dy
       -\tfrac14\int|h|^2\,dx\,dy/y^2\ge0.           \tag{BC.7}
\]

Its complete scalar Hodge energy includes the additional nonnegative \(x\)-derivative. Hodge star gives the same inequality for top forms. Core approximation therefore gives \((D_0^+)^*D_0^+\ge1/4\) on \(H_e\). Define

\[
 T_0=D_0^+\bigl((D_0^+)^*D_0^+\bigr)^{-1/2}:
       H_e\longrightarrow H_o,\qquad
 T_0^*T_0=1,\quad T_0T_0^*=1-P_{\rm harm}.          \tag{BC.8}
\]

The lower bound gives closed range. Its orthogonal complement is \(\ker D_0^-\), and polar decomposition gives exactly the displayed onto isometry onto that complement. No compact resolvent is asserted for \(D_0\). Its equivariance is exact, so the paired bulk summand is genuinely degenerate.

The whole harmonic one-form subspace has a concrete description. In conformal disk coordinates a one-form is \(F(z)dz+G(z)d\bar z\). The equations \(d\alpha=d^*\alpha=0\), with \(*dz=-i dz\) and \(*d\bar z=i d\bar z\), are precisely \(\bar\partial F=0\), \(\partial G=0\). The Riemannian square norm is

\[
 \|Fdz+Gd\bar z\|^2=2\int_{\mathbb D}
                              (|F|^2+|G|^2)dx\,dy.   \tag{BC.9}
\]

Distributional harmonic vectors are smooth by the preceding local elliptic regularity, so this accounts for the entire kernel. Holomorphic Taylor/angular expansion and monotone radius integration give its complete normalized basis. In the notation of the audited boundary proof, \(H_b=\ell^2(\mathbb Z)\), \(H_b^-=\ell^2(\mathbb Z\setminus\{0\})\), and there is the exact onto unitary

\[
 B:H_b^-\longrightarrow P_{\rm harm}H_o,
 \quad B e_n=\sqrt{n/(2\pi)}\,z^{n-1}dz\ (n>0),\quad
 B e_{-n}=\sqrt{n/(2\pi)}\,\bar z^{n-1}d\bar z .    \tag{BC.10}
\]

It intertwines the full endpoint \(U_1^-\) with \(R_o\), not just their rotations. This is the ordinary isometric pullback of both one-form types, and the explicit boundary formula (HB.14) proves the intertwining. The factor \((2\pi)^{-1/2}\) is required by the actual Riemannian norm (BC.9), rather than the normalized Bergman norm in the boundary note.

#### BC.4. Aligning the entire odd representation and exposing the exact remaining pair

Let \(T:H_b\to H_b^-\) be the exact delete-constant coisometry in the boundary proof. On

\[
 \widetilde H_e=H_e\oplus H_b,\quad\widetilde H_o=H_o,
 \quad \widetilde R_{e,s}=R_e\oplus U_s,
 \quad\widetilde R_o=R_o,
 \qquad Z=(T_0,\ BT):\widetilde H_e\to H_o          \tag{BC.11}
\]

the full operator is \(\left(\begin{smallmatrix}0&Z^*\\Z&0\end{smallmatrix}\right)\). The orthogonality of the bulk and harmonic ranges gives

\[
 ZZ^*=1_{H_o},\quad Z^*Z=1-P_{0,b},\quad
 Z\widetilde R_{e,s}(g)-R_o(g)Z
         =(0,\ B(TU_s(g)-U_1^-(g)T)).                \tag{BC.12}
\]

These are exact full-space identities. The right side is compact and jointly norm continuous by the audited HB.17. Thus (BC.11) is precisely the boundary cycle plus the exactly degenerate Hodge bulk, with its **whole odd representation now literally equal to the native \(R_o\)**. It is one genuine equivariant homotopy, and at \(s=1\) is the unit.

Now use the native coisometry (BC.4) to define the explicit onto unitary

\[
 \mathcal V:H_e\longrightarrow H_e\oplus H_b,
 \quad \mathcal V=Z^*S+\theta_{e_{0,b},g_o},\qquad
 \mathcal Vh=(T_0^*Sh,\ T^*B^*Sh+\langle g_o,h\rangle e_0),
 \quad Z\mathcal V=S.                               \tag{BC.13}
\]

Here \(Z^*S\) is unitary between the orthogonal complements of the two full even kernel lines, and the last term identifies those lines. Their ranges and initial spaces are orthogonal, so \(\mathcal V^*\mathcal V=1\) and \(\mathcal V\mathcal V^*=1\) follow directly from (BC.4)/(BC.12). This is an entire-space unitary, not a dimension comparison. It is exactly \(K\)-equivariant because \(T_0,B,T,S\) intertwine rotations and both displayed kernel lines are rotation-fixed.

Set actual even representations on the native even Hilbert space

\[
 \rho_{B,s}(g)=\mathcal V^*(R_e(g)\oplus U_s(g))\mathcal V,
 \qquad \rho_I(g)=S^*R_o(g)S+P_g.                   \tag{BC.14}
\]

Each is a genuine multiplicative unitary representation, with the same \(K\) restriction as \(R_e\); no compressed completely positive map is used. At one, exact Hodge splitting gives

\[
 \rho_{B,1}=\rho_I,\qquad
 \mathcal V\rho_I\mathcal V^*=R_e\oplus U_1.        \tag{BC.15}
\]

The path with even representation \(\rho_{B,s}\), odd representation \(R_o\), and native operator \(F_o\) is unitarily equivalent to (BC.11). It is therefore a proved representation-compatible homotopy **from \(\rho_{B,0}\) to \(\rho_I\)**, ending in the actual unit plus a degenerate complement.

There is also an exact compact-difference statement at its starting point. Using (BC.13),

\[
 \mathcal V R_e(g)-(R_e(g)\oplus U_0(g))\mathcal V
 =Z^*(SR_e(g)-R_o(g)S)
  +(Z^*R_o(g)-(R_e(g)\oplus U_0(g))Z^*)S
  +\theta_{e_{0,b},g_o}R_e(g)
      -(R_e(g)\oplus U_0(g))\theta_{e_{0,b},g_o}.     \tag{BC.16}
\]

The first term is compact by (BC.6), the second by (BC.12), and the last two are rank one. All vary in norm continuously in \(g\). Thus

\[
 R_e(g)-\rho_{B,0}(g)\in\mathcal K(H_e).             \tag{BC.17}
\]

**This is the intermediate compact-difference comparison.** Its solution is supplied by HF.1–HF.7 and RR.6–RR.8. Compact difference alone does not give a homotopy between actual representations. Define the relative equivariant cycle with even representation \(R_e\), odd representation \(\rho_{B,0}\) on two copies of \(H_e\), and off-diagonal identity. Exact cycle addition and the boundary homotopy give

\[
 [x]-1=[R_e,\rho_{B,0}].                             \tag{BC.18}
\]

Here is the addition argument so this reduction hides no cancellation principle. The identity representation on the even Gaussian plus the coisometry \(S\) identifies the identity endpoint's even representation as \(\rho_I\). Adding its grading-reversed cycle to the native one and adjoining the Gaussian exchange gives the off-diagonal identity pair \([R_e,\rho_I]\), by the full direct-sum and Gaussian-exchange calculation just described. Alternatively use the rotation homotopy on a triple direct sum to check \([\pi,\tau]=[\pi,\sigma]+[\sigma,\tau]\): the middle identical representation occurs on opposite gradings and is an exact degenerate identity-operator pair, which is removed by that explicit rotation. With \(\sigma=\rho_{B,0}\), the pair \([\rho_{B,0},\rho_I]\) is zero: \(\rho_{B,s}\) is its actual compact-difference representation homotopy to equal representations. This proves (BC.18), but does not make its remaining right side zero.

The naive convex interpolation of its two representations fails. For \(\sigma_t=(1-t)R_e+t\rho_{B,0}\) and \(\Delta=R_e-\rho_{B,0}\),
\(\sigma_t(a^*b)-\sigma_t(a)^*\sigma_t(b)=t(1-t)\Delta(a)^*\Delta(b)\) for integrated algebra elements. These are generally nonzero compact defects, so it is not a representation path. Equation (BC.16) is not an exact intertwiner that would bypass this failure.

#### BC.5. Transport to the actual holonomy algebra, with its compact ideal

This section proves the complete actual transport of both the boundary identity and the calibrated confining cycle. It does not invoke a general descent or a named \(\gamma\)-element theorem.

Let \(P\to V\) be the oriented orthonormal frame bundle of \(F\), a principal \(K\)-bundle. For \(p\in P_u\), define the unique oriented isometry \(j_p:\mathbb H^2\to L_u\) by its value \(j_p(o)=u\) and derivative at zero equal to \(p\). Existence follows from the global exponential map and its exact hyperbolic radial/angular metric; uniqueness follows because an isometry preserving a point and its tangent frame preserves every exponential geodesic. If \(q\in P_{u'}\) and \(u,u'\) are on the same actual disk leaf, put

\[
 c(p,\gamma,q)=j_p^{-1}j_q\in G_0,
 \quad\gamma:u'\to u,\qquad
 c(p,\gamma,q)c(q,\eta,r)=c(p,\gamma\eta,r).          \tag{BC.19}
\]

An actual leaf has no holonomy, so these arrows are uniquely determined inside the leaf. The smooth topology used here is the native exponential groupoid topology, not a possibly accumulating relation topology in \(V\times V\). In local frame domains, the value and derivative at zero of this isometry are smooth functions of the arrow through the geodesic equation and parallel transport. These data give smooth coordinates on \(G_0\), proving smoothness of (BC.19). Frame changes give
\(c(pk,\gamma,q\ell)=k^{-1}c(p,\gamma,q)\ell\).

For any one of the strongly continuous representations \(U_s,R,\rho_{B,s}\), whose \(K\) restriction is fixed, form its associated continuous Hilbert bundle \(\mathcal H=P\times_K H\). The arrow maps

\[
 J_s(\gamma)[q,v]=[p,U_s(c(p,\gamma,q))v]             \tag{BC.20}
\]

are well defined by the frame-change equation and are exactly unitary, multiplicative and strongly continuous. The same construction applies separately to both gradings and to their full operators, which commute exactly with \(K\). Compact differences and operator-equivariance defects in the model become norm-continuous compact maps between the actual Hilbert-bundle fibres.

The completed right module is

\[
 E_{\mathcal H}=\Gamma(V,\mathcal H)\widehat\otimes_{C(V)}A,
 \qquad (\Pi_s(a)k)(u,x)
       =\int a(u,u')J_s(u,u')k(u',x)\,d\mu(u').       \tag{BC.21}
\]

The left \(C(V)\) action on \(A\) in this balance is its range multiplier. Compact continuous arrow-valued Hilbert tests are a dense core; their convolution inner product is

\[
 \langle k,l\rangle_A(z,x)=
       \int\langle k(u,z),l(u,x)\rangle_{\mathcal H_u}d\mu(u).
                                                               \tag{BC.22}
\]

A finite frame-domain partition and a countable dense set in the separable model give countable generation. Fubini and unitarity give the adjoint \(\Pi_s(a^*)\). For each regular localization at \(x\), the unitary
\(k(u)\mapsto J_s(u,x)^*k(u)\) onto \(L^2(L_x)\otimes\mathcal H_x\) takes this left action to \(\lambda_x(a)\otimes1\), by the exact arrow cocycle. Hence its norm is \(\le\|a\|_r\). Positivity and module boundedness are detected by these faithful regular localizations of the \(A\)-valued inner product. This extends (BC.21) to an adjointable star representation of all \(A\), and a near-unit convolution approximate identity proves nondegeneracy. The localization unitary is used for this bound only; it is not silently asserted to be a single global right-\(A\)-module unitary.

Here is the compactness mechanism needed for every subsequent assertion. A continuous compactly supported arrow kernel

\[
 K(\gamma)\in\mathcal K(\mathcal H_{s\gamma},\mathcal H_{r\gamma})
                                                               \tag{BC.23}
\]

has a compact convolution operator on \(E_{\mathcal H}\). Cover its support by finitely many frame domains. Uniform finite-rank approximation of its compact-operator values gives sums
\(K(\gamma)=\sum\eta_i(r\gamma)a_{ij}(\gamma)\zeta_j(s\gamma)^*+\text{small remainder}\), with global continuous compact-base Hilbert sections and \(a_{ij}\in C_c(\mathcal G)\). Extension by frame-domain cutoffs makes those sections global. Each sum is \(T_{\eta_i}L_{a_{ij}}T_{\zeta_j}^*\), an actual module compact since \(L_{a_{ij}}\in\mathcal K_A(A)=A\). The creation maps need only be adjointable; they are not asserted compact without that coefficient. A common compact-support two-sided Haar/Schur bound sends uniform kernel error to module-norm error. Thus the whole operator lies in \(\mathcal K_A(E_{\mathcal H})\). The identical proof over \(C([0,1],A)\) handles jointly norm-continuous compact-valued kernel fields. Approximate general \(a\in A\) by compact smooth kernels to extend these assertions.

It follows that all representation differences \(\Pi_s(a)-\Pi_r(a)\) arising from HB.11 or (BC.14) are actual whole-module compacts. For a \(K\)-invariant bounded model Fredholm operator \(F\), its square defect is compact-valued; the localized defect \(\Pi_s(a)(F^2-1)\) is of type (BC.23). Its commutator has kernel \(a(\gamma)(FJ_s(\gamma)-J_s(\gamma)F)\), again of that type by model equivariance. The operator is adjointable on the full module because it is bounded and \(K\)-compatible on continuous sections and has its displayed adjoint. The parameter family acts on one whole \(C([0,1],A)\) module: strong/adjoint continuity preserves continuous sections, and the defect compact sections are norm continuous by the same kernel argument. This proves all cycle and homotopy conditions in the exact module compact ideal.

At the boundary endpoint the constant bundle line is globally trivial because \(K\) fixes \(e_0\); its full arrow action is the identity because \(U_1(g)e_0=e_0\). Its completed module is exactly the standard right \(A\)-module with left multiplication. Both nonconstant sectors are still present on opposite gradings with the exactly intertwining operator \(T\), so they are degenerate. Consequently the transported boundary cycle is an actual identity

\[
 [E_{\partial},\Pi_0,F_{\partial}]=1_A.             \tag{BC.24}
\]

This transport does not require a global frame, a trivial spin-c line, or properness of the actual holonomy groupoid.

Finally the transported confining cycle is **exactly** the native calibrated reverse product, not an auxiliary replacement. The full graded flip/evaluation \(W_y\widehat\otimes S_y\to\Lambda F_{y,\mathbb C}\), followed by \(i^N\), cancels the entire \(H^*\otimes H\) line and intertwines both physical actions and the induced connection. For a core \(\Psi(u,y,x)\), its exterior form in \(y\) is a vector in the fibre \(\mathcal H_u=L^2(L_u,\Lambda F_{\mathbb C})\); it is attached to the coefficient arrow \((u,x)\). This gives an onto unitary to (BC.21) by the actual inner-product identity \(\int_u\int_y\langle\Psi(u,y,z),\Phi(u,y,x)\rangle d\mu(y)d\mu(u)\). Compact separated tests and the common Schur estimate give its dense range, and the reverse evaluation supplies its inverse.

Left convolution originally changes \(u\) while holding \(y\) fixed. In the associated model its fibre change is exactly \(R(j_p^{-1}j_q)\), because the pushforward identifications satisfy \(I_p^{-1}I_q=(j_q^{-1}j_p)^*=R(c(p,\gamma,q))\). Right convolution still changes \(x\). The entire transported operator is (BC.3), since \(j_p(o)=u\) and the actual isometric connection carries \(d+d^*\) and the centered potential to those at \(o\). Hence (BC.21) with \(R,F_o\) is the unitary conjugate of the native product phase proved in RI.1–RI.6 above, with all its creation connections and positive sandwiches intact. Its class is

\[
 p=\delta_{\rm inv}\otimes_{C(V)}T.                 \tag{BC.25}
\]

Transport (BC.13) as a \(K\)-equivariant associated-fibre unitary; it gives a whole right-\(A\)-module unitary \(\mathcal V_A\). Equations (BC.14)–(BC.17) and the compact-kernel lemma produce actual multiplicative left representations \(\rho_{B,s,A}\) and \(\rho_{I,A}\) on the native even module. There is a proved homotopy from the former at zero to the latter, with the native phase operator and full right action fixed. Its endpoint has invariant even Gaussian standard module and exactly degenerate complement. Thus

\[
 p-1_A=[\Pi_e,\rho_{B,0,A}],\qquad
 \Pi_e(a)-\rho_{B,0,A}(a)\in\mathcal K_A(E_e).       \tag{BC.26}
\]

Changing the left representation by these compact differences preserves the bounded operator's cycle defects, its native creation equations (which use the fixed module/operator), and its positive sandwiches modulo the same compact ideal. But it does **not** identify the changed left action with the originally induced first-factor action. Thus the homotopy just proved starts at \(\rho_{B,0,A}\), not at \(\Pi_e\); treating it as starting at the latter would simply omit (BC.26).

#### BC.6. The precise comparison and two failed shortcuts

The required comparison is a genuine equivariant nullhomotopy of the pair \([R_e,\rho_{B,0}]\) in (BC.18), with its compact defects jointly norm continuous. It may instead be proved directly for the exact actual module pair in (BC.26), retaining both convolution actions. A stable whole-cycle equivalence with the boundary cycle would also suffice. Its required representation and operator data are now entirely specified by (BC.2)–(BC.17), rather than only by an index or a name for an element.

One attempted shortcut is to set the confining potential to zero and use the Hodge splitting immediately. This fails the scalar equivariant-cycle requirement: \(D_0\) has the **infinite-dimensional** harmonic one-form kernel (BC.10), so \(1-F_{D_0}^2\) is not compact. On the actual reverse module, localizing only the \(u\)-convolution does not eliminate that independent infinite \(y\)-kernel. The separated harmonic modes give an orthonormal weakly null sequence with nonvanishing localized defect. Adding the boundary space and its coisometry does repair that endpoint, as (BC.11) shows, but an actual homotopy from the confining representation/operator to that repaired endpoint still needs the residual pair (BC.18). A potential limit by itself does not supply it.

Another attempted shortcut is to declare the complete unitary (BC.13) equivariant. Equation (BC.16) proves only compact equivariance error; its rank-one terms are generally nonzero because the centered Gaussian and the principal-series constant are not invariant under boosts. A compact equivariance error makes a valid relative KK cycle. It does not make that cycle vanish.

The preceding calculation retains the complete boundary identity, both harmonic Bergman sectors, the full odd representation, and the exact actual holonomy module. The proof HF.1–HF.7 kills the lifted difference, and RR.6–RR.8 transports that equality to this represented reverse product. The literal same-spin relation (BC.1) remains unchanged.



### 11B.4. Complete closed-disk lifts and the compact-subgroup comparison



Put \(\overline X=\overline{\mathbb D}\), \(X=\mathbb D\), \(G=\mathrm{PSU}(1,1)\), \(K=\mathrm{SO}(2)\), and \(\mathscr C=C(\overline X)\). Use exactly the Riemannian form Hilbert spaces, grading, pullback representations, native negative Witten realization, Hodge polar isometry and complete Bergman unitary of BC.2–BC.12. BC denotes the complete whole-module comparison above.

The two lifted cycles are

\[
 \overline x=(H_e\oplus H_o,\;R_e\oplus R_o,\;
             \pi_x(h)=M_h^e\oplus M_h^o,\;F_o),             \tag{CL.1}
\]

and

\[
 \overline y=( (H_e\oplus H_b)\oplus H_o,\;
             (R_e\oplus U_0)\oplus R_o,\;
             \pi_y(h)=(M_h^e\oplus M_{h|\mathbb T})\oplus M_h^o,\;
             F_Z),\qquad
 F_Z=\begin{pmatrix}0&Z^*\\Z&0\end{pmatrix},\quad
 Z=(T_0,BT).                                             \tag{CL.2}
\]

Here \(H_b=L^2(\mathbb T,d\theta/(2\pi))\), with \(e_n(\zeta)=\zeta^n\); the identification with \(\ell^2(\mathbb Z)\) is the literal Fourier unitary. In particular boundary multiplication is an actual representation on **all** of \(H_b\). Multiplication does not preserve \(H_b^-\), so the expression \(B M_{h|\mathbb T}\) on \(H_b^-\) must be read with the delete-constant map when comparing full modules. The unambiguous whole-space expression proved below is

\[
 M_h^oBT-BT M_{h|\mathbb T}\in\mathcal K(H_b,H_o).        \tag{CL.3}
\]

Both (CL.1) and (CL.2) are genuine \(G\)-equivariant Kasparov \((\mathscr C,\mathbb C)\) cycles. Their function representations are unital, multiplicative and exactly covariant. The boundary/Hodge assertion here is for the starting representation \(U_0\); the already constructed path \(U_s\) with these fixed function multipliers is not an exactly covariant \(\mathscr C\)-homotopy.

#### 1. Actual function representations and covariance

For every form degree, pointwise multiplication has
\(M_{hk}=M_hM_k\), \(M_{\bar h}=M_h^*\), \(M_1=1\), and \(\|M_h\|\le\|h\|_\infty\). Pullback gives on the entire Hilbert space

\[
 R(g)M_hR(g)^*=M_{h\circ g^{-1}}.                       \tag{CL.4}
\]

At zero the boundary action is the ordinary unitary half-density action
\(U_0(g)u(\zeta)=J_{g^{-1}}(\zeta)^{1/2}u(g^{-1}\zeta)\). The Jacobian factors cancel in conjugating a multiplier, so

\[
 U_0(g)M_{h|\mathbb T}U_0(g)^*
        =M_{(h\circ g^{-1})|\mathbb T}.                  \tag{CL.5}
\]

These identities first hold pointwise on smooth tests and then on the whole space by density and the multiplier bounds. They prove exact covariance, rather than covariance modulo compact operators. No compression of a multiplier to the Bergman space is used as an algebra representation.

#### 2. A common domain and resolvent-commutator lemma

Let \(D\) be either the complete Hodge closure \(d+d^*\) or the native confining closure
\(Q=d+d^*-(df\wedge+\iota_{\nabla f})\), \(f=r^2/2\). For \(h\in C^\infty(\overline{\mathbb D})\), the compact smooth form core is preserved by \(M_h\). On that core

\[
 [D,M_h]=c(dh)=dh\wedge-\iota_{(dh)^\sharp}.              \tag{CL.6}
\]

The Witten potential is a bundle multiplication and commutes with scalar \(h\). Complex linear extension gives the formula for complex-valued functions. The operator norm of (CL.6) is bounded by a fixed finite multiple of the hyperbolic derivative norm. For a real differential the Clifford norm is exactly its length. Since \(h\) has bounded Euclidean first derivatives,

\[
 |dh|_{\mathrm{hyp}}
       =\frac{1-|z|^2}{2}|dh|_{\mathrm{Euc}}.             \tag{CL.7}
\]

Thus (CL.6) is bounded; in the Hodge case it is a continuous bundle endomorphism tending to zero at infinity in the complete disk.

For \(u\in\operatorname{Dom}D\), choose core \(u_j\to u\) in graph norm. Then \(M_hu_j\to M_hu\) and
\(D M_hu_j=M_hDu_j+c(dh)u_j\to M_hDu+c(dh)u\). Closedness proves

\[
 M_h\operatorname{Dom}D\subset\operatorname{Dom}D,
 \qquad D M_hu=M_hDu+c(dh)u.                            \tag{CL.8}
\]

This is full-domain control. Apply the same proof to \(\bar h\) for the adjoint multiplier. It implies the actual bounded resolvent identity

\[
 [(D-z)^{-1},M_h]=-(D-z)^{-1}c(dh)(D-z)^{-1},
 \qquad z\notin\mathbb R.                             \tag{CL.9}
\]

For \(a(t)=\sqrt{1+t^2}\), spectral calculus gives the strongly convergent, uniformly bounded truncations of

\[
 D(1+D^2)^{-1/2}
  =\frac2\pi\int_0^\infty D(D^2+a(t)^2)^{-1}\,dt,
 \quad D(D^2+a^2)^{-1}
  =\tfrac12((D-ia)^{-1}+(D+ia)^{-1}).                  \tag{CL.10}
\]

The integral of the operators themselves need not converge in norm. The commutator integrals do: (CL.9) bounds their integrands by
\(\|c(dh)\|/(1+t^2)\). Whenever the right-hand side of (CL.9) is compact, each finite commutator integral is compact and the integrable bound makes its limit compact in norm. Strong convergence of the uniformly bounded spectral truncations identifies that norm limit with the commutator of the bounded transform. This supplies the integral justification without a hidden norm-convergence claim about (CL.10).

#### 3. Native phase: compact commutators for the full compactification

For \(D=Q\), BC.3 gives self-adjointness on the complete core closure, compact resolvent, kernel \(\mathbb Cg_o\), and a gap at least one for \(Q^2\) off the kernel. Thus the resolvents in (CL.9) are compact, and (CL.10) proves compactness of the bounded-transform commutator.

Choose a continuous \(k\in C_0(\mathbb R)\), with \(k(0)=0\), agreeing with
\(\operatorname{sign}(t)-t/\sqrt{1+t^2}\) on \(|t|\ge1\); interpolate continuously in the spectral gap. Functional calculus gives

\[
 F_o=Q(1+Q^2)^{-1/2}+k(Q),\qquad k(Q)\in\mathcal K(H).
                                                               \tag{CL.11}
\]

The compactness follows directly from compact resolvent: \(C_0\) functional calculus is the norm closure of the resolvent algebra, whose generators are compact. Hence

\[
 [F_o,M_h]\in\mathcal K(H),\qquad
                     h\in C^\infty(\overline{\mathbb D}).       \tag{CL.12}
\]

Now polynomials in \(z,\bar z\) are uniformly dense in \(C(\overline{\mathbb D})\): they contain constants, are closed under conjugation, and separate points. For any continuous \(h\), take such polynomials \(p_j\) with \(\|h-p_j\|_\infty\to0\). Since \(\|F_o\|\le1\),

\[
 \|[F_o,M_h]-[F_o,M_{p_j}]\|
                         \le2\|h-p_j\|_\infty.                 \tag{CL.13}
\]

The compact ideal is norm closed. Equations (CL.12)–(CL.13) prove the requested full compactification assertion for **every** continuous \(h\), including functions with nonzero and nonconstant boundary values. No claim that every continuous multiplier preserves \(\operatorname{Dom}Q\) is needed or made; (CL.8) supplies precisely the smooth-domain argument needed before bounded phase approximation.

The full Fredholm and equivariance conditions are unchanged:
\(F_o=F_o^*\), \(F_o^2-1=-P_g\), and
\(R(g)F_oR(g)^*-F_o\in\mathcal K(H)\), norm continuously in \(g\), by BC.4–BC.6. Multiplication by \(\pi_x(h)\) preserves each compact defect. This verifies (CL.1) as a \(KK_G(\mathscr C,\mathbb C)\) cycle on the complete two-graded form space.

#### 4. The entire Hodge polar isometry commutes modulo compacts

Write \(D_H=d+d^*\). The complete scalar/top-form Hardy gap of BC.7–BC.10 gives

\[
 \operatorname{spec}(D_H)\subset
 (-\infty,-1/2]\cup\{0\}\cup[1/2,\infty),\qquad
 \operatorname{phase}(D_H)=
    \begin{pmatrix}0&T_0^*\\T_0&0\end{pmatrix}.          \tag{CL.14}
\]

Its zero spectral subspace is the **whole** infinite-dimensional harmonic one-form space. We do not assert compact resolvent or a compact square defect for this phase.

The actual resolvents are locally compact. Here this follows from an explicit energy estimate. Integration of the de Rham Weitzenböck identity on compact smooth forms gives
\(\|D_Hv\|^2=\|\nabla v\|^2-\|v^{(1)}\|^2\): scalar and top forms have zero curvature term and one-forms have Ricci term \(-1\). In a covariant orthonormal frame this identity follows by expanding \(d^*d+dd^*\), commuting the two covariant derivatives in its degree-one term, and contracting the constant sectional curvature \(-1\); integration of \(\nabla^*\nabla\) gives the displayed energy. For a compact smooth scalar cutoff \(\chi\), (CL.6) gives
\(\|\nabla(\chi u)\|^2\le\|\chi D_Hu+c(d\chi)u\|^2+\|\chi u\|^2\).
Thus \(u\mapsto\chi u\) sends the full graph domain boundedly into compactly supported \(H^1\), first on the essential core and then by closure.

For clarity, the needed compact embedding has an elementary finite-mode proof. Choose a finite smooth coordinate partition over the cutoff support and larger coordinate rectangles. Each component is supported inside its rectangle, so extend it by zero to a still larger flat torus. Norm equivalence on the compact charts bounds its Euclidean \(H^1\) norm. Its Fourier coefficients outside \(|m|\le N\) have squared \(L^2\) sum at most \(N^{-2}\) times its first-derivative energy. Finite Fourier truncations are finite-rank operators, and the tail bound is uniform over bounded \(H^1\) sets. Summing the finitely many chart components proves compactness in \(L^2\). Applying this to the bounded graph image of the resolvent proves that \(\chi(D_H-z)^{-1}\) is compact for every nonreal parameter. The argument retains all form degrees, including the entire harmonic kernel. For a continuous bundle multiplier \(v\) tending to zero at infinity, uniformly approximate it by compactly supported bundle multipliers; the resolvent norm bound and this local compactness give

\[
 v(D_H-z)^{-1},\;(D_H-z)^{-1}v\in\mathcal K(H).          \tag{CL.15}
\]

The adjoint gives the second assertion from the first. This also applies to the Clifford multiplication in (CL.7). Consequently (CL.9) is compact for the Hodge operator, and (CL.10) proves compactness of its bounded-transform commutator.

To pass to its phase, choose \(k_H\in C_0(\mathbb R)\) equal to
\(\operatorname{sign}(t)-t/\sqrt{1+t^2}\) on \(|t|\ge1/2\) and with \(k_H(0)=0\). A continuous interpolation across the two empty gap intervals suffices. This gives
\(\operatorname{phase}(D_H)=D_H(1+D_H^2)^{-1/2}+k_H(D_H)\).
Here \(k_H(D_H)\) need not be compact. Instead (CL.9) says that the two resolvent generators commute with \(M_h\) modulo compacts. Sums, products, adjoints, and norm limits preserve this property, so the entire \(C_0\) resolvent functional calculus, including \(k_H(D_H)\), commutes modulo compacts. We obtain

\[
 [\operatorname{phase}(D_H),M_h]\in\mathcal K(H),
 \qquad T_0M_h^e-M_h^oT_0\in\mathcal K(H_e,H_o).        \tag{CL.16}
\]

First this holds for smooth closed-disk functions with the full-domain proof (CL.8); uniform polynomial approximation and the bound \(2\|h-p_j\|_\infty\) prove it for every continuous \(h\). This avoids trying to discard the infinite harmonic kernel or treating the unconfined phase's square defect as compact.

#### 5. Explicit full-sector compactness of the boundary-to-harmonic map

Set \(C=BT:H_b\to H_o\), so \(Ce_0=0\). For \(n\ge1\), use the exact normalized harmonic one-forms

\[
 b_n^+=\sqrt{n/(2\pi)}z^{n-1}dz,
 \qquad b_n^-=\sqrt{n/(2\pi)}\bar z^{n-1}d\bar z,
 \qquad Ce_n=b_n^+,\quad Ce_{-n}=b_n^-.                \tag{CL.17}
\]

Their norm is the actual Riemannian norm
\(2\int(|F|^2+|G|^2)dxdy\). The two types are orthogonal; each entire half-line is retained.

Let \(L=M_\zeta\) be the literal bilateral shift on \(H_b\). On the positive sector,

\[
 (M_zC-CL)e_n
    =\left(\sqrt{\frac n{n+1}}-1\right)b_{n+1}^+,
 \qquad n\ge1.                                        \tag{CL.18}
\]

These orthogonal-image coefficients tend to zero, so this whole-sector operator is compact by finite-column truncation.

On the negative sector, for \(n\ge2\) the harmonic projection of \(zb_n^-\) is
\(\sqrt{(n-1)/n}\,b_{n-1}^-\), and

\[
 zb_n^-=\sqrt{\frac{n-1}n}\,b_{n-1}^-+v_n,
 \quad v_n\perp P_{\rm harm}H_o,\quad
 \|v_n\|^2=\frac1{n(n+1)}.                            \tag{CL.19}
\]

Indeed \(\|zb_n^-\|^2=n/(n+1)\), while the displayed harmonic coefficient's squared norm is \((n-1)/n\); their difference is the stated positive remainder. Angular integration shows that the \(v_n\) are mutually orthogonal, since their scalar coefficients have distinct angular frequencies \(-(n-2)\). Their bundle type is always \(d\bar z\). Thus

\[
 (M_zC-CL)e_{-n}
   =\left(\sqrt{\frac{n-1}n}-1\right)b_{n-1}^-+v_n
                                                              \tag{CL.20}
\]

is the sum of two compact whole-sector operators. Their tail norms are respectively bounded by
\(\sup_{n>N}|\sqrt{(n-1)/n}-1|\) and
\(\sup_{n>N}1/\sqrt{n(n+1)}\), both tending to zero. This is an operator-norm proof on the entire negative Bergman sector, rather than a test on individual vectors.

The remaining columns are finite in number:
\((M_zC-CL)e_0=-b_1^+\) and
\((M_zC-CL)e_{-1}=zb_1^-\). They contribute a finite-rank operator. This proves compactness of \(D_z=M_zC-CM_\zeta\) on all of \(H_b\). Complex conjugation swaps \(b_n^+\) with \(b_n^-\), swaps \(e_n\) with \(e_{-n}\), and commutes with \(C\). It carries this calculation to the corresponding compact \(D_{\bar z}\), including its finite exceptional columns.

For arbitrary continuous functions set \(D_h=M_hC-CM_{h|\mathbb T}\). The exact multiplicative identity is

\[
 D_{hk}=M_hD_k+D_hM_{k|\mathbb T}.                    \tag{CL.21}
\]

Constants have zero defect. Equations (CL.18)–(CL.21) imply compactness for every polynomial in \(z,\bar z\). Since \(\|C\|=1\),
\(\|D_h-D_p\|\le2\|h-p\|_\infty\), so uniform polynomial approximation proves (CL.3) for every \(h\in\mathscr C\).

If one specifically uses the map \(B\) with domain \(H_b^-\), its precise compressed statement is
\(M_hB-B(T M_{h|\mathbb T}T^*)\in\mathcal K(H_b^-,H_o)\).
The compressed multiplier \(T M_{h|\mathbb T}T^*\) is used only in this operator comparison; it is **not** claimed to be a multiplicative algebra representation.

#### 6. All defects of the lifted boundary/Hodge cycle

The orthogonal Hodge decomposition gives exactly

\[
 ZZ^*=1_{H_o},\quad Z^*Z=1_{H_e\oplus H_b}-P_{0,b},
 \quad F_Z^2-1=-P_{0,b},\quad F_Z=F_Z^*.                \tag{CL.22}
\]

Both off-diagonal function commutators are compact by (CL.3), (CL.16), and their adjoints. The entire equivariance defect is

\[
 Z(R_e(g)\oplus U_0(g))-R_o(g)Z
             =(0,B(TU_0(g)-U_1^-(g)T)).                \tag{CL.23}
\]

This is compact and norm continuous in \(g\), by the full-space HB.11/HB.17 proof, with exact Hodge and Bergman equivariance on the other blocks. All localized square and self-adjointness defects follow from (CL.22). All operators in (CL.2) are bounded and act on the entire displayed Hilbert space; the only unbounded domain passage used in deriving them is the complete Hodge closure treated in (CL.8), (CL.14)–(CL.16). Thus no deleted infinite complement, implicit boundary condition, or merely blockwise compactness enters the lift.

#### 7. The residual pair also lifts, with two actual function actions

Use the whole-space, exactly \(K\)-equivariant unitary from BC.13,
\(\mathcal V=Z^*S+\theta_{e_{0,b},g_o}\), \(S=F_o^+\), \(Z\mathcal V=S\).
Equations (CL.3), (CL.12)–(CL.16), and the finite-rank Gaussian exchange give

\[
 \mathcal V M_h^e-(M_h^e\oplus M_{h|\mathbb T})\mathcal V
                                      \in\mathcal K.    \tag{CL.24}
\]

To see the complete identity, write its left side as
\(Z^*(SM_h^e-M_h^oS)
 +(Z^*M_h^o-(M_h^e\oplus M_{h|\mathbb T})Z^*)S
 +\theta_{e_{0,b},g_o}M_h^e
 -(M_h^e\oplus M_{h|\mathbb T})\theta_{e_{0,b},g_o}\).
The first two terms are compact by the preceding full-space commutators; the last two have rank at most one each. Therefore the unresolved scalar representation pair has the covariant lifted representative with even and odd actions

\[
 (R_e,M_h^e),\qquad
 \left(\rho_{B,0}=\mathcal V^*(R_e\oplus U_0)\mathcal V,
        \mathcal V^*(M_h^e\oplus M_{h|\mathbb T})\mathcal V\right),
 \qquad F_{\rm rel}=\begin{pmatrix}0&1\\1&0\end{pmatrix}.  \tag{CL.25}
\]

Each function action is exactly multiplicative and covariant with its own group representation; (CL.24) supplies the compact identity-operator commutator. BC.17 supplies its compact group defect. This is an actual lift of the full residual pair, not an assertion that the pair vanishes.

#### 8. The complete compact-subgroup comparison and a covariance failure

Restriction to \(K\) admits a direct comparison with no character shortcut. For \(0\le t\le1\), replace the function actions of each lifted cycle by

\[
 \pi_{x,t}(h)=M_{h(tz)}^e\oplus M_{h(tz)}^o,
 \quad
 \pi_{y,t}(h)=(M_{h(tz)}^e\oplus M_{h(t\zeta)})\oplus M_{h(tz)}^o.
                                                               \tag{CL.26}
\]

These are unital multiplicative \(K\)-covariant representations because the radial contraction is \(K\)-equivariant. The associated functions depend uniformly continuously on \(t\); hence the representations and their compact commutators depend in operator norm continuously on \(t\). Apply the already proved commutator statements to \(h_t(z)=h(tz)\); all square defects are constant finite rank and all group defects for \(K\) are zero. This gives actual \(K\)-equivariant homotopies over \(C([0,1])\).

At \(t=0\), both function actions are the scalar \(h(0)\). The exact unitary \(\mathcal V\oplus1_{H_o}\) identifies the two complete cycles there: it preserves grading, intertwines every rotation, and takes \(S\) to \(Z\). Consequently

\[
 \operatorname{Res}_K[\overline x]
          =\operatorname{Res}_K[\overline y]
                      \quad\hbox{in }KK_K(\mathscr C,\mathbb C). \tag{CL.27}
\]

The compact-subgroup equality alone does not identify the \(G\)-classes. The direct compact product WP.1–WP.42, induction maps RP.18–RP.28 and particular boundary factorization HF.1–HF.7 below give the required gamma-part argument. Their whole modules and actual homotopies supply the class equality used by the holonomy transport.

The radial contraction (CL.26) is not \(G\)-equivariant: in general \(g(tz)\ne t g(z)\). The boundary representation path HB.16 also cannot retain the function action used in (CL.2). Indeed, put \(L=M_\zeta\) and use its actual boost generator
\(D_s e_n=-a_n(s)e_{n+1}+a_{n-1}(s)e_{n-1}\). Exact covariance would require
\([D_s,L]=\tfrac12(1-L^2)\), the derivative of the scalar circle action. But its \(e_0\)-to-\(e_0\) coefficient is

\[
 a_0(s)-a_{-1}(s)=\tfrac12\sqrt{1-s^2},                \tag{CL.28}
\]

which differs from \(1/2\) for every \(s>0\). The commutator is calculated on the finite Fourier generator core, so this is an exact covariance failure, not a domain objection. At one, any unital exactly covariant representation of \(C(\mathbb T)\) with the invariant vector \(e_0\) would give an invariant probability measure on the circle by its vector state. Rotations force that measure to be Haar; a nontrivial boost does not preserve Haar. Thus no such endpoint covariant representation exists, even if one attempts to replace the fixed circle multipliers.

The exact lifted class comparison is proved below with the particular equivariant Stinespring interval. It retains the full grading, frame compatibility, multiplicative representations and jointly continuous compact defects. Finite-center descent and RR.7 then implement the resulting full class equality through genuine whole-module homotopies and stable cycle operations on the reduced holonomy module.

Every operator and group action used here is the one in BC.2–BC.17; all operators needed for associated-frame transport still commute exactly with \(K\). On changing frames, the multiplier \(h\) changes by composition with that rotation, as in (CL.4)–(CL.5); this is precisely the algebra action on the associated compactified-disk bundle \(P\times_K\overline{\mathbb D}\). Consequently the same actual cocycle
\(c(p,\gamma,q)=j_p^{-1}j_q\), with
\(c(pk,\gamma,q\ell)=k^{-1}c(p,\gamma,q)\ell\), is retained. The new compact-commutator statements neither trivialize that frame bundle nor change the graded physical coefficient or determinant line. Existing transport BC.19–BC.23 remains the exact transport of these model data. The whole interval transport of these same data is proved in RR.7.



### 11B.5. The direct compact-subgroup Witten product



Let \(X\) be the complete oriented curvature \(-1\) hyperbolic disk, with metric \(4|dz|^2/(1-|z|^2)^2\), origin \(o\), distance \(r=d(o,\cdot)\), and \(f=r^2/2\). Let \(K=SO(2)\) act by rotations fixing \(o\). The same proof applies to \(K=U(1)\subset SU(1,1)\), through its rotation action; its central kernel acts trivially on all the displayed form and Clifford data. Put

\[
 P=C_0(X,\operatorname{Cl}^{+}(T^*X)),\qquad
 H=L^2(X,\Lambda^*T^*X_{\mathbb C}),\qquad D=d+d^*.
 \tag{WP.1}
\]

The Clifford algebra is complex, graded, with self-adjoint real generators of square \(|v|^2\). On exterior forms write

\[
 c_+(v)=\varepsilon(v)+\iota_{v^\sharp},\qquad
 c_-(v)=\varepsilon(v)-\iota_{v^\sharp},\qquad
 \rho(v)=c_+(v).
 \tag{WP.2}
\]

Thus \(c_+\) is self-adjoint, \(c_-\) is skew-adjoint, and \(c_-(v)c_+(w)+c_+(w)c_-(v)=0\). Exterior parity is the grading. The group representation is \(R(k)=(k^{-1})^*\). All statements concern the entire \(H\), including both even degrees and the entire odd degree.

Define \(\xi=df/\sqrt{1+r^2}\), viewed as a Clifford multiplier of \(P\), and

\[
 \eta_K=[(P,\mathbb C,L_\xi)],\qquad
 d_K=[(H,\rho,F_D)],\qquad F_D=D(1+D^2)^{-1/2}.
 \tag{WP.3}
\]

If \(\eta,d\) denote the corresponding \(G\)-cycles of RP.4–RP.7, these are exactly their restrictions to \(K\). The result proved here is

\[
 \boxed{\eta_K\widehat\otimes_P d_K
       =[(H,R,F_Q)]
       =[(H,R,\operatorname{phase}_0 Q)]
       =1\quad\hbox{in }KK_K^0(\mathbb C,\mathbb C),}
 \qquad Q=D-c_+(df),\quad F_Q=Q(1+Q^2)^{-1/2}.
 \tag{WP.4}
\]

Here \(\operatorname{phase}_0 Q\) is the sign on the orthogonal complement of the kernel and zero on the kernel. The equality is proved by an actual creation connection, a positive bounded anticommutator modulo the full \(\mathcal K(H)\), and an exactly \(K\)-equivariant decomposition. It is not an inference from the ordinary integer index. Only the compact-subgroup product is asserted here; the induction argument establishing the required gamma-part comparison follows in RP.4–RP.8.

#### WP.0. Exact programme foundations and scope

Definition 0.3, Lemmas 0.1–0.2 and Theorems 0.4,0.6–0.9 in [Section 0](#section-0) prove the ordinary criterion, existence, uniqueness, full homotopy compatibility and graded product rules. WP.7–WP.8 below verify the analytic hypotheses and prove the compact-group addition. Ordinary product uniqueness is not substituted for equivariant uniqueness.

For precision, the support-projection step of that separator proof can be justified without the unsafe formula \([p^2,d]=2p[p,d]\). In the ambient bidual let \(p\) be the support of the controlled algebra \(C\). If commutation with \(d\) derives \(C\), its weak-continuous extension maps the weak closure \(C''\) into itself, so \([p,d]\in C''=pC''p\). On the other hand \(p[p,d]p=0\) by direct multiplication and \(p^2=p\). Hence \([p,d]=0\). The approximate identity converges weakly to \(p\); its commutators consequently converge weakly to zero. Hahn–Banach separation and the tail-convex diagonal construction in the Section 0 proof now apply exactly as written. This supplies the correct commutation justification for the complete separator used here, without importing an invalid projection identity.

The native product P.1–P.6 and exact comparison BC.2–BC.5 identify the same whole operator and all coefficient actions. The domain, confinement and kernel assertions are proved in full next.

#### WP.1. Actual graph domains and self-adjoint closures

Let \(\mathcal C=C_c^\infty(X,\Lambda^*)\). The Hessian of \(f\), computed from \(dr^2+\sinh^2r\,d\theta^2\), has eigenvalues \(1,r\coth r\), with the smooth value \(1\) at \(o\). The Hodge Weitzenböck formula is

\[
 D^2=\nabla^*\nabla+\mathcal R,\qquad
 \mathcal R|_{\Lambda^0\oplus\Lambda^2}=0,\quad
 \mathcal R|_{\Lambda^1}=-1.
 \tag{WP.5}
\]

In particular \(\|\mathcal R\|\le1\).

To fix the curvature convention in WP.5, expand \(D=\sum c_-(e_j)\nabla_{e_j}\) in an orthonormal normal frame. The symmetric second-order terms give the connection Laplacian and the antisymmetric terms give the curvature commutator. On scalars this commutator is zero; Hodge star gives zero on top forms. On a covariant one-form its contraction is the Ricci endomorphism, which is \(-1\) for the curvature \(-1\) surface. This gives precisely the three endomorphisms stated in WP.5. Both Clifford actions are parallel. In the next square the mixed first-order terms vanish by the last identity in WP.2, and differentiating the potential leaves the Hessian term. Thus on \(\mathcal C\):

\[
 Q^2=D^2+r^2-\sum_{i,j}c_-(e_i)c_+(e_j)
                         \operatorname{Hess}_{ij}f.
 \tag{WP.6}
\]

In a Hessian eigenframe \(c_-(e_j)c_+(e_j)=2N_j-1\), where \(N_j\) is exterior occupancy in that direction. Since \(r\coth r\le r+1\), the Hessian term has norm at most \(2+r\). Hence

\[
 \|Qu\|^2\ge\|\nabla u\|^2+\tfrac12\|ru\|^2-4\|u\|^2,
 \qquad u\in\mathcal C.
 \tag{WP.7}
\]

Indeed its lower estimate before weakening is \(\|\nabla u\|^2+\|ru\|^2-\int(3+r)|u|^2\), and \(r\le(r^2+1)/2\). Also \(\|Qu\|\le\|Du\|+\|ru\|\) and \(\|Du\|^2\le2\|\nabla u\|^2\) directly from its two first-order Clifford terms. Thus the \(Q\) graph norm is equivalent on \(\mathcal C\) to

\[
 \|u\|+\|\nabla u\|+\|ru\|.
 \tag{WP.8}
\]

Compact radial cutoffs \(\chi_R=\chi(r/R)\), equal to one for \(r\le R\) and zero for \(r\ge2R\), have \(|d\chi_R|\le c/R\). Cutoffs followed by ordinary chartwise smoothing approximate every element of

\[
 \mathcal V=\{u\in H: \nabla u\in L^2\text{ distributionally},\ ru\in H\}
 \tag{WP.9}
\]

in WP.8. To see this, the three discarded tails tend to zero by integrability; the additional derivative term \((d\chi_R)u\) has norm at most \(c\|u\|/R\). On a retained compact support, smooth coefficients and a finite partition of unity reduce to usual convolution smoothing in finitely many charts. All three norms converge there; multiplication by \(r\) is bounded on that compact set. This proves that \(\mathcal C\) is a core for the closed symmetric realization on \(\mathcal V\).

The local maximal-domain regularity needed for the cutoff test can also be checked from these energy estimates. In a relatively compact coordinate chart write the principal part as \(\sum a_j(x)\partial_j\), with smooth coefficients. A chart cutoff of a distributional vector with \(Qu\in L^2\) has \(Q(\chi u)\in L^2\). Mollify it inside a slightly larger compact chart. The commutator of a first-order term with the mollifier has kernels
\((a_j(x)-a_j(y))\partial_j\psi_\delta(x-y)\) and a bounded-coefficient term \(\psi_\delta(x-y)\partial_ja_j(y)\). Their two Schur integrals are bounded uniformly in \(\delta\), because \(|a_j(x)-a_j(y)|\le C|x-y|\) and \(\int|v||\partial_j\psi_\delta(v)|\,dv\le C\). Zeroth-order coefficients, including the local potential, are bounded on this region. Thus the mollified vectors have uniformly bounded \(Q\) norms locally. WP.5 applied to their compact cutoffs, and the bounded local potential, bound their \(H^1\) norms uniformly. Weak compactness and their distributional limit put \(u\) in \(H^1_{\rm loc}\). This proves the required regularity without imposing a boundary condition or assuming a distributional vector already lies in the global graph domain.

It is now self-adjoint. A distributional deficiency vector \(Q^*u=\pm i u\) is locally \(H^1\) by the preceding argument. In the cutoff test the potential commutes with \(\chi_R\), while \([Q,\chi_R]=c_-(d\chi_R)\). Taking imaginary parts of the symmetric cutoff pairing gives

\[
 \|\chi_Ru\|^2\le(c/R)\|\chi_Ru\|\,\|u\|.
 \tag{WP.10}
\]

Local \(H^1\) smoothing justifies the test. Letting \(R\to\infty\) kills both deficiency spaces. The closure is therefore self-adjoint and

\[
 \operatorname{Dom}Q=\mathcal V
   =\operatorname{Dom}D\cap\operatorname{Dom}r.
 \tag{WP.11}
\]

For the last equality the same cutoff argument gives the self-adjoint complete Hodge realization, and WP.5 identifies its graph domain with \(\{u\in H: \nabla u\in L^2\}\): on the core \(\|\nabla u\|^2\le\|Du\|^2+\|u\|^2\), and the reverse bound follows from the Dirac formula. The core approximation just given, omitting the \(r\) norm, proves equality of those complete graph domains. The closed inequalities WP.7–WP.8 now hold on all of \(\operatorname{Dom}Q\). In particular

\[
 \|ru\|^2\le2\|Qu\|^2+8\|u\|^2.
 \tag{WP.12}
\]

The same statements hold for \(Q_+=D+c_+(df)\), since the sign change leaves the absolute Hessian estimate unchanged. These are the complete closures of the model operators in the cited native providers, not new boundary conditions on a ball.

#### WP.2. Local Hodge compactness and whole Witten compactness

For every compact smooth scalar or bundle coefficient \(a\), the map \(a(D\pm i\lambda)^{-1}\), \(\lambda>0\), is compact. The resolvent maps the unit ball to a bounded Hodge graph set, hence by WP.5 to a bounded \(H^1\) set; multiplication by \(a\) confines that set to a compact region. Here is the local Rellich mechanism being used. Subordinate compact chart cutoffs produce finitely many bounded \(H^1\) sets supported inside coordinate boxes. Extend them by zero after those cutoffs, then regard them in a slightly larger periodic box. Their Fourier tails beyond frequency \(N\) have \(L^2\) norm at most \(C/N\), by the first-derivative bound. The finite collection of Fourier modes of frequency at most \(N\) has bounded image in a finite-dimensional space. This gives finite-rank approximations in operator norm; summing the chart cutoffs proves compactness. Adjoints give the other-sided localization, and uniform norm approximation extends it to all \(C_0\) Clifford coefficients.

The embedding \(\mathcal V\hookrightarrow H\) is compact on its graph unit ball. Inside \(r\le2R\) the same chartwise argument applies. Outside \(r\ge R\), WP.12 gives

\[
 \|(1-\chi_R)u\|\le R^{-1}\|ru\|
     \le R^{-1}(2\|Qu\|^2+8\|u\|^2)^{1/2}.
 \tag{WP.13}
\]

Thus finite-rank approximations on a fixed radius have a uniformly small discarded tail as \(R\to\infty\). Since \((Q\pm i\lambda)^{-1}:H\to\mathcal V\) is bounded, its composition with that embedding is compact. The whole \(Q\) has compact resolvent. Every continuous scalar function tending to zero at infinity has compact functional calculus for \(Q\): this follows first for the resolvents and their finite sums/products, then by their uniform density in \(C_0(\mathbb R)\). In particular \(1-F_Q^2=(1+Q^2)^{-1}\) is compact on the entire \(H\).

No unlocalized compactness is asserted for \(D\). Its infinitely many harmonic one-forms are retained throughout this proof; their presence is compatible with the \(P\)-localized compactness of the Hodge input.

#### WP.3. The entire kernel and its exact rotation representation

For completeness, rederive the complete negative Witten kernel and the gap used in the phase correction. Define \(d_f=d+df\wedge\). Multiplication \(U_f h=e^{-f}h\) is a unitary from \(L^2(e^{-2f}d\mu,\Lambda^*)\) to \(H\). On compact tests,

\[
 U_f^{-1}(d_f+d_f^*)U_f=d+\delta_\mu,
 \qquad \delta_\mu=\delta+2\iota_{\nabla f}.
 \tag{WP.14}
\]

This equality follows from the product rules for \(d\) and \(\delta\), and extends to the self-adjoint closure by WP.1. The weighted Laplacian is

\[
 \Delta_\mu=d\delta_\mu+\delta_\mu d
 =\nabla_\mu^*\nabla+\mathcal R
             +2(\operatorname{Hess}f)^{[k]}
      \quad\hbox{in degree }k.
 \tag{WP.15}
\]

To check the extra term rather than assume a weighted formula: \(d\iota_v+\iota_vd=\mathcal L_v\) by the coordinate product rule. On covariant \(k\)-forms \(\mathcal L_{\nabla f}=\nabla_{\nabla f}+(\operatorname{Hess}f)^{[k]}\). The weighted adjoint of the connection changes \(\nabla^*\nabla\) to \(\nabla^*\nabla+2\nabla_{\nabla f}\). Combine this with WP.5. In an orthonormal Hessian eigenframe, \((\operatorname{Hess}f)^{[k]}=\sum h_jN_j\); thus it is at least \(1\) on one-forms and at least \(2\) on top forms. WP.15 is consequently bounded below by \(1\) in degree one and by \(4\) in degree two. Compact-core integration establishes these quadratic-form estimates. They extend to the actual closed domains: the weighted connection norm of \(h=e^fu\) is \(\|\nabla u+df\otimes u\|\), controlled by WP.8, and the compact-core approximations converge in both terms.

The weighted degree-zero energy is \(\int|dh|^2e^{-2f}d\mu\). A zero vector has distributional derivative zero, hence is constant on the connected disk. Constants lie in the weighted space because

\[
 Z=\int_Xe^{-2f}\,d\mu
    =2\pi\int_0^\infty e^{-r^2}\sinh r\,dr
    =\pi^{3/2}e^{1/4}\operatorname{erf}(1/2)<\infty.
 \tag{WP.16}
\]

The final equality follows by writing \(\sinh r=(e^r-e^{-r})/2\) and completing the square in each integral. Constants are in the complete weighted graph domain: compact radial approximations have vanishing derivative energy and vanishing weighted tails. Therefore the positive Witten kernel is exactly the scalar line \(\mathbb C e^{-f}\), and not only a line inside a larger kernel.

The Hodge star sends the positive and negative Witten *squares* to one another, with degree \(k\leftrightarrow2-k\). One can check every sign directly from WP.6: in a Hessian eigenframe the negative square is \(\Delta+r^2+\operatorname{tr}H_f-2H_f^{[k]}\), whereas the positive square is \(\Delta+r^2-\operatorname{tr}H_f+2H_f^{[k]}\), and star changes \(N_j\) to \(1-N_j\). Since star commutes with \(\Delta\) and scalar multiplication, it intertwines these squares exactly, including their closed forms. Thus

\[
 \ker Q=\mathbb Cg,\qquad
 g=Z^{-1/2}e^{-f}\operatorname{vol}.
 \tag{WP.17}
\]

It is also an immediate zero-vector check that \(dg=0\) in top degree and \(d^*g=\iota_{\nabla f}g\), which cancels the contraction term of \(Q\). Degree two is even. Every rotation fixes \(f\) and the oriented volume form, so \(R(k)g=g\) exactly. The kernel representation is the trivial \(K\)-representation.

For the spectral gap, compact resolvent already makes zero isolated. The explicit bound is at least \(1\) for \(Q^2\) on its kernel complement. The only point needing proof beyond WP.15 is a positive scalar eigenvalue \(\lambda\) of the weighted degree-zero square. If \(h\) is its eigenvector, \(\|dh\|_\mu^2=\lambda\|h\|_\mu^2\); hence \(dh\ne0\). Distributionally \(d(dh)=0\) and \(\delta_\mu dh=\lambda h\). Both right sides are in weighted \(L^2\), so the complete maximal realization \(d+\delta_\mu\) puts \(dh\) in its domain (this is the self-adjoint maximal-domain equality of WP.1 transported by \(U_f\)). Apply the degree-one lower bound to \(dh\):

\[
 \lambda^2\|h\|_\mu^2
 =\|(d+\delta_\mu)dh\|_\mu^2
 \ge\|dh\|_\mu^2=\lambda\|h\|_\mu^2.
 \tag{WP.18}
\]

Thus \(\lambda\ge1\). The higher degrees already have the lower bounds \(1,4\), and the compact-resolvent spectral decomposition exhausts the whole space. Star gives the same gap for the negative square. This verifies the exact spectral input in BC.2 without omitting any degree.

#### WP.4. The two input cycles and the invariant sign homotopy

The multiplier \(\xi\) is smooth at \(o\): \(r^2\) is smooth there and \(df=r\,dr\), so no polar singularity is introduced. It has square \(r^2/(1+r^2)\). Therefore \(1-L_\xi^2=(1+r^2)^{-1}\in P=\mathcal K_P(P)\), the scalar commutators vanish, and its rotation invariance is exact. This proves the first cycle in WP.3.

For homogeneous \(p\in C_c^\infty(X,\operatorname{Cl}^+)\), set \(A=\rho(p)\), \(\epsilon=(-1)^{|p|}\). The principal symbols in WP.2 give

\[
 J_p=DA-\epsilon AD=\sum_jc_-(e_j)\nabla_{e_j}A.
 \tag{WP.19}
\]

It is bounded, smooth and compactly supported. Compact smooth Clifford multiplication preserves \(\operatorname{Dom}D\), and its adjoint does too, by WP.11 and the first-derivative product rule. The graded mixed resolvent identity with \(Q=D\), proved with all signs at WP.24 below, implies that \([F_D,A]_{\rm gr}\) is a norm integral of compact errors bounded by \(\|J_p\|/(1+t^2)\). Compactness here uses \(J_p(D-z)^{-1}\) and WP.2. Also \(A(1-F_D^2)\) is compact by the same localized compactness. Norm density of these coefficients proves all \(d_K\) cycle conditions. Completeness gave self-adjointness, and rotations commute exactly with \(D\), \(\rho\) is exactly covariant, and \(R\) preserves grading. Hence this is an actual \(K\)-equivariant Hodge cycle.

Let \(J\) be the parallel covector rotation by \(90\) degrees, with sign fixed by the orientation. It commutes with all rotations of \(X\). On the standard \(P\)-module use

\[
 \xi_s=\cos(\pi s)\xi+\sin(\pi s)J\xi,\qquad
 b_s=L_{\xi_s},\quad0\le s\le1.
 \tag{WP.20}
\]

The covectors \(\xi,J\xi\) are orthogonal with equal norm, so \(b_s^2=r^2/(1+r^2)\). The family is norm continuous (its norm difference is at most \(\pi|s-s'|\)), odd, self-adjoint, and exactly \(K\)-invariant. Its square defect belongs to \(C([0,1],P)=\mathcal K_{C([0,1],P)}(C([0,1],P))\). This is a cycle on the whole interval module, from \(L_\xi\) to \(-L_\xi\), not a pointwise homotopy on incomplete fibres. Consequently \(\eta_K\) is represented by the *negative* multiplier. Under its tensor with \(H\), that multiplier is

\[
 \Theta=-c_+(\xi).
 \tag{WP.21}
\]

This rank-two sign homotopy is the reason the negative Witten product in WP.4 has the stated dual class.

#### WP.5. The actual tensor module and graded creation connection

The canonical map

\[
 U:P\widehat\otimes_\rho H\longrightarrow H,\qquad
 U(p\widehat\otimes u)=\rho(p)u
 \tag{WP.22}
\]

is even, isometric by the tensor inner product, and onto because compact smooth scalar cutoffs act nondegenerately on \(H\). It is exactly \(K\)-equivariant. Under this unitary \(T_p=A=\rho(p)\) is the creation map for \(p\in P\), and \(L_{-\xi}\widehat\otimes1=\Theta\). This identifies all represented operators used in the criterion, not just their classes.

For homogeneous compact smooth \(p\), the actual differential creation error is

\[
 K_p=QA-\epsilon AD=J_p-c_+(df)A.
 \tag{WP.23}
\]

Both terms are bounded, smooth and compactly supported; \(df\) is bounded on this support. The domain statements are actual graph-domain statements: \(A\operatorname{Dom}D\subset\operatorname{Dom}Q\), since multiplication by \(A\) preserves \(H^1\) and \(rA\) is bounded. Also \(A^*\operatorname{Dom}Q\subset\operatorname{Dom}D\), by WP.11. Thus WP.23 holds for every \(u\in\operatorname{Dom}D\), not only on a formal core.

Write \(R_Q(z)=(Q-z)^{-1}\), \(R_D(z)=(D-z)^{-1}\) for nonreal \(z\). From \((Q-z)A=\epsilon A(D-\epsilon z)+K_p\), applied to \(R_D(\epsilon z)u\) and then to \(R_Q(z)\), obtain

\[
 R_Q(z)A-\epsilon A R_D(\epsilon z)
   =-\epsilon R_Q(z)K_pR_D(\epsilon z).
 \tag{WP.24}
\]

The leading \(-\epsilon\) is required for odd \(p\). The identity is an equality of bounded maps on all of \(H\). Each right side is compact because \(R_Q(z)\) is a whole compact operator, and

\[
 \|R_Q(z)K_pR_D(\epsilon z)\|
            \le\|K_p\|/|\operatorname{Im}z|^2.
 \tag{WP.25}
\]

For a self-adjoint operator \(T\), its bounded transform has the strong two-pole formula

\[
 F_T=\frac2\pi\int_0^\infty
       T(T^2+1+t^2)^{-1}\,dt
   =\frac1\pi\int_0^\infty
       \big((T-i\lambda_t)^{-1}+(T+i\lambda_t)^{-1}\big)\,dt,
 \qquad \lambda_t=\sqrt{1+t^2}.
 \tag{WP.26}
\]

The individual transform integrals need not converge in operator norm. Their *difference* in the creation equation does: sum WP.24 at \(z=\pm i\lambda_t\), exchanging the two lower poles when \(\epsilon=-1\). The errors are compact and bounded in norm by \(2\|K_p\|/(1+t^2)\), which is integrable. Strong convergence of the two truncated transforms therefore identifies their norm-convergent difference with

\[
 F_QT_p-\epsilon T_pF_D\in\mathcal K(H),\qquad
 T_p^*F_Q-\epsilon F_DT_p^*\in\mathcal K(H).
 \tag{WP.27}
\]

The second assertion is the adjoint of the first. For arbitrary homogeneous \(p\in P\), approximate by compact smooth coefficients in the same parity. Both creation errors change in norm by at most \(2\|p-p_n\|\), since both bounded transforms are contractions and \(\rho\) is contractive. Thus WP.27 holds for every creation vector. This is the exact graded \(F_D\)-connection required for \(P\widehat\otimes_PH\cong H\).

#### WP.6. Full domains of the unbounded anticommutator

The covariant derivative of \(\xi=r\,dr/\sqrt{1+r^2}\) has eigenvalues

\[
 a_r=(1+r^2)^{-3/2},\qquad
 a_\theta=r\coth r/\sqrt{1+r^2}\le1.
 \tag{WP.28}
\]

The angular inequality follows from \((r\coth r)^2=r^2+(r/\sinh r)^2\le r^2+1\). The zero limits are smooth, and \(0\le a_r\le1\). On the compact core,

\[
 B=\{\Theta,D\}=-\sum_jc_-(e_j)c_+(\nabla_{e_j}\xi),
 \qquad B=B^*,\quad\|B\|\le2,
 \tag{WP.29}
\]

because the two Clifford factors anticommute and their respective adjoints are skew and self-adjoint. WP.28 bounds each of the two eigenframe terms by one. These expressions are invariantly defined bounded multiplication operators.

The bounded multiplier \(\Theta\) preserves \(\operatorname{Dom}Q\). One proof using exactly the closed graph domain is that its derivative is bounded, so it preserves \(H^1\), while multiplication by \(r\) commutes with it and \(\|\Theta\|\le1\), so it preserves \(\operatorname{Dom}r\). Equivalently compact-core approximation and the next identity give a uniform \(Q\)-graph bound. On the full domain WP.11,

\[
 C=\{\Theta,Q\}=M+B,\qquad
 M=2r^2/\sqrt{1+r^2}\ge0,\qquad C\ge-2.
 \tag{WP.30}
\]

Here \(C\) on the right is a self-adjoint multiplication operator with domain \(\operatorname{Dom}r\): \(M\) grows comparably to \(r\) outside a compact set and \(B\) is bounded self-adjoint. The left identity is evaluated on \(\operatorname{Dom}Q\subset\operatorname{Dom}C\), with both \(\Theta Q\) and \(Q\Theta\) well defined. Since \(M\le2r\), WP.12 proves that \(C:\operatorname{Dom}Q_{\rm graph}\to H\) is bounded. Compact-core convergence therefore extends WP.30 to that entire graph domain. No expression involving \(C\) below is treated as if \(C\) were a bounded operator on \(H\).

#### WP.7. A positive strong limit and a norm-integrable compact correction

For \(\lambda\ge1\) put \(R_+=(Q+i\lambda)^{-1}\), \(R_-=(Q-i\lambda)^{-1}\), so \(R_+^*=R_-\). WP.30 and the graph bound show that \(CR_\pm\) are bounded maps. Since \(\Theta\operatorname{Dom}Q\subset\operatorname{Dom}Q\), ordinary inverse-shift multiplication on that domain gives the bounded identity

\[
 \{\Theta,Q(Q^2+\lambda^2)^{-1}\}
 =\tfrac12(R_+CR_-+R_-CR_+).
 \tag{WP.31}
\]

For example \(R_+CR_-=R_+\Theta+\Theta R_-\): insert \(C=\Theta Q+Q\Theta\), use \(QR_-=1+i\lambda R_-\) and \(R_+Q=1-i\lambda R_+\), and the two imaginary terms cancel. The other pole gives the other half. All insertions are legitimate because the resolvents have range in \(\operatorname{Dom}Q\), \(\Theta\) preserves it, and \(C\) is graph bounded there.

Define

\[
 P_\lambda=\tfrac12\big(R_+(C+2)R_-+R_-(C+2)R_+\big).
 \tag{WP.32}
\]

It is a bounded positive operator. For the first summand its quadratic form is \(\langle R_-u,(C+2)R_-u\rangle\ge0\); the second is identical with \(R_+u\). Both vectors are in the operator domain of \(C+2\). The form defines the displayed bounded self-adjoint product because \(CR_\pm\) are bounded. Thus WP.31 becomes

\[
 \{\Theta,Q(Q^2+\lambda^2)^{-1}\}
       =P_\lambda-2(Q^2+\lambda^2)^{-1}.
 \tag{WP.33}
\]

The correction is compact, with norm at most \(2\lambda^{-2}\). The maps are norm continuous for \(\lambda\ge1\) on bounded intervals: the usual resolvent identity combined with bounded \(CR_\pm\) proves graph-resolvent continuity, hence continuity of WP.32. Their finite-interval integrals are well-defined norm integrals.

Let

\[
 F_{Q,T}=\frac2\pi\int_0^TQ(Q^2+1+t^2)^{-1}\,dt,\qquad
 P_T=\frac2\pi\int_0^TP_{\lambda_t}\,dt,\qquad
 K_T=\frac4\pi\int_0^T(Q^2+1+t^2)^{-1}\,dt.
 \tag{WP.34}
\]

Finite integration of WP.33 gives \(\{\Theta,F_{Q,T}\}=P_T-K_T\). The scalar spectral formula is

\[
 F_{Q,T}(x)=\frac2\pi\frac{x}{\sqrt{1+x^2}}
                  \arctan\frac{T}{\sqrt{1+x^2}}.
 \tag{WP.35}
\]

Thus \(\|F_{Q,T}\|\le1\) and \(F_{Q,T}\to F_Q\) strongly by bounded spectral convergence. Also \(\|K_T\|\le2\), uniformly, and

\[
 K_T\longrightarrow K_\infty
       =2(1+Q^2)^{-1/2}\in\mathcal K(H)
       \quad\hbox{in operator norm}.
 \tag{WP.36}
\]

The compactness and norm convergence follow from the whole compact resolvent and the integrable bound \(4/(\pi(1+t^2))\); the displayed value follows by the scalar integral. In contrast, the positive terms are integrated as a *strong* limit. They satisfy \(0\le P_T\le4\) in norm, because \(\|\{\Theta,F_{Q,T}\}\|\le2\) and \(\|K_T\|\le2\), and are increasing with \(T\), since the integrands are positive. A uniformly bounded increasing family of positive operators has a bounded positive strong limit \(P_\infty\): for each \(u\), its scalar quadratic forms increase to a bounded value, polarization gives a bounded positive operator, and \(0\le P_{T'}-P_T\le4\) gives \(\|(P_{T'}-P_T)u\|^2\le4\langle u,(P_{T'}-P_T)u\rangle\), proving strong Cauchy convergence.

Now take the strong limit of the finite identity; the compact correction already converges in norm. This proves

\[
 \boxed{\{\Theta,F_Q\}=P_\infty-2(1+Q^2)^{-1/2}
                  \ge0\pmod{\mathcal K(H)}.}
 \tag{WP.37}
\]

This proof deals with the unbounded \(C\) on its actual domain. It never assumes norm integrability of the positive main term or compactness of its strong limit. Alternatively decomposing \(C=M+B\) in WP.31 gives a positive \(M\) sandwich and a compact \(B\) sandwich bounded by \(2\lambda^{-2}\); the \(C+2\) version makes the uniform positive-limit bound immediate. Both have the required integrable \(\lambda^{-2}\) compact correction.

Since the left source is \(\mathbb C\), its coefficient sandwiches are simply \(|a|^2\{\Theta,F_Q\}\), so WP.37 verifies every localized positivity requirement, including \(a=1\). Exact rotation invariance of \(Q,\Theta\) makes all these operators and their limits exactly \(K\)-invariant.

#### WP.8. The compact-group product criterion really is equivariant

The complete ordinary criterion specified at WP.0 extends to the present compact-group cycles by the following explicit averaging argument. This is needed to interpret WP.27 and WP.37 in \(KK_K\).

For exactly \(K\)-invariant inputs, start with the self-adjoint ordinary connection \(G\) of Lemma 0.2 and take its weak operator integral over normalized Haar measure, \(G_K=\int_KR(k)GR(k)^*dk\). It is a bounded, self-adjoint, odd, exactly invariant operator. Its error at a creation vector \(p\) is the integral of the conjugated error at \(k^{-1}p\). The latter is compact and norm continuous in \(k\): the connection error depends boundedly on \(p\), \(k\mapsto k^{-1}p\) is norm continuous in \(P\), and conjugation by a strongly continuous unitary group is norm continuous on compacts, as follows by finite-rank approximation. Therefore the integral of the creation errors is compact, and \(G_K\) is an actual invariant connection.

In Theorem 0.4 use \(G_K\), the invariant input \(\Theta\), and the algebra \(A_1=C^*(\rho(P),\mathcal K(H))\). The controlling algebras are \(K\)-stable and the action is norm continuous on them: on \(\rho(P)\) this is the continuous bundle action on \(C_0\) coefficients, on compacts it follows by finite rank, and the remaining controlling generators are invariant operators or their defects. If \(M,N\) are the ordinary positive separator with \(M+N=1\), replace them by their bounded weak integrals over \(K\). For any controlling \(a\), the annihilation condition is retained because

\[
 R(k)MR(k)^*a
       =R(k)M(R(k)^*aR(k))R(k)^*
 \tag{WP.38}
\]

is a norm-continuous compact-valued function whenever \(Ma\in\mathcal K(H)\) for all \(a\in A_1\). The same proof handles \(NA_2\), and commutators with the invariant controlling operators are the compact integrals of their conjugates. Evenness, positivity and the sum condition are retained. All ordinary separator calculations then produce an exactly invariant product operator; functional calculus of the averaged separator remains invariant.

For uniqueness of two exactly invariant product operators, use the controlling algebras in Theorem 0.6 and the same averaging. Its intermediate product and positive-anticommutator bridge are exactly invariant; its two norm-continuous paths therefore are genuine equivariant operator homotopies. If an equivariant product operator is only invariant modulo compacts, its average differs compactly in the present unital scalar-source cycle, by integrating the norm-continuous compact group defects; its affine path to the average is a valid equivariant compact perturbation. This reduces to the exactly invariant case. On an interval Hilbert module, the averaged operator is defined on each module vector by the norm-continuous vector-valued Haar integral of \(kGk^{-1}\) applied to that vector; the integral of the conjugated adjoints supplies its adjoint. The uniform operator bound proves boundedness. The same argument retains compact errors in the full interval compact ideal, because conjugation on rank-one module compacts is norm continuous and finite-rank approximation gives the general case. Thus it also covers the product homotopies used for WP.20. This verifies, with the complete ordinary existence and uniqueness provider, that the actual connection and positive sandwich characterize the \(K\)-equivariant product class.

The tensor module in WP.22 is the whole \(H\). \(F_Q\) is odd self-adjoint, scalar commutators vanish, its square defect is whole compact by WP.2, and its group defects vanish exactly. Hence WP.27, WP.37 and this equivariant criterion prove

\[
 [(P,\mathbb C,L_{-\xi})]\widehat\otimes_P d_K
          =[(H,R,F_Q)]\quad\hbox{in }KK_K^0(\mathbb C,\mathbb C).
 \tag{WP.39}
\]

WP.20 replaces the first input by \(\eta_K\). No local dual or \(RKK\) identity is invoked.

#### WP.9. The precise phase correction and the actual equivariant unit

The gap in WP.18 gives \(\operatorname{spec}(Q)\subset\{0\}\cup(-\infty,-1]\cup[1,\infty)\). Let \(\operatorname{clamp}(x)=\max(-1,\min(1,x))\). Then on this spectrum

\[
 \operatorname{phase}_0Q=\operatorname{clamp}(Q),\qquad
 h(x)=\operatorname{clamp}(x)-x/\sqrt{1+x^2}\in C_0(\mathbb R).
 \tag{WP.40}
\]

Consequently \(\operatorname{phase}_0Q-F_Q=h(Q)\) is compact. This is the required correction at the kernel as well as on the nonzero spectrum: both operators are zero on \(g\). It is odd, self-adjoint and exactly invariant. The affine path \(F_Q+sh(Q)\) is norm continuous and exactly invariant, and its square defects stay compact by the bounded compact-perturbation identity. Its scalar commutators vanish. It is therefore an actual \(K\)-equivariant Kasparov homotopy. Moreover \(h(Q)T_p\) and its adjoint are compact, so the phase remains a creation connection; \(\{\Theta,h(Q)\}\) is compact, so its positivity condition is unchanged.

Let \(P_g=\theta_{g,g}\). Functional calculus and the entire-kernel assertion give

\[
 (\operatorname{phase}_0Q)^2=1-P_g,\qquad
 \operatorname{phase}_0Q\,g=0.
 \tag{WP.41}
\]

The decomposition \(H=\mathbb Cg\oplus(\mathbb Cg)^\perp\) is even and exactly \(K\)-invariant, since WP.17 fixes \(g\). On the complement the phase is an odd self-adjoint unitary, commutes exactly with \(K\), and graded-commutes with the scalar representation. That *whole* summand is a degenerate equivariant cycle; nothing is removed from an unproved invariant subspace. On \(\mathbb Cg\) the operator is zero, the grading is even, the scalar representation is the identity, and \(K\) acts trivially. The unitary \(\mathbb C\to\mathbb Cg\), \(z\mapsto zg\), identifies this cycle exactly with the scalar identity cycle. Its bounded Fredholm index as a \(K\)-representation is therefore the trivial even Gaussian representation, established by the displayed decomposition. The abstract identification with an integer index is unnecessary.

This proves all equalities in WP.4, including

\[
 r(\eta)\widehat\otimes_P r(d)=1_K.
 \tag{WP.42}
\]



### 11B.6. The restriction projector from induction and Cartan homotopies



For G=SU(1,1), K=U(1), every separable graded G-algebra A and j in Z/2, this repair proves the well-typed maps
\[
 r:KK_G^j(A,\mathbb C)\to KK_K^j(A,\mathbb C),\qquad
 L:KK_K^j(A,\mathbb C)\to KK_G^j(A,\mathbb C),\qquad
 rL=1,\quad Lr=\gamma\cdot(-).
 \tag{RP.1}
\]
Consequently gamma is idempotent, restriction is an isomorphism on its gamma part, and ker r=(1-gamma)KK_G^j(A,C). These conclusions use the complete direct K-product proving r(gamma)=1; they require no full local-dual or RKK duality theorem. The scalar gamma is the same well-defined eta tensor d as below.

#### RP.0. Complete foundations actually used

The full ordinary graded product proof is [Section 0](#section-0), Definition 0.3 and Theorems 0.4,0.6–0.9; the compact-group calculation is WP.0–WP.9. The equivariant separation and induction context is G. G. Kasparov, [Equivariant KK-theory and the Novikov conjecture (1988), §§1.4,2.4–2.14,3.2–3.6](https://gdz.sub.uni-goettingen.de/download/pdf/PPN356556735_0091/LOG_0014.pdf). The complete equivariant addition needed here is given next.

For clarity, here is the equivariant addition to the ordinary product provider. Let \(J=\mathcal K(E)\), let \(A_1,A_2\subset\mathcal M(J)\) be the two controlling algebras with \(A_1A_2\subset J\), and let the separable controlling set \(\Delta\) derive \(A_1\). Include the bounded group-defect field \(\varphi(g)=g(\widetilde F)-\widetilde F\), with both-sided products by \(A_1\) norm-continuous and in \(J\). A compact-exhaustion diagonal construction gives positive increasing approximate units \(u_n\) of \(A_1\) and \(v_n\) of \(J\), with summable errors for commutation, annihilation and \(g(u_n)-u_n\), \(g(v_n)-v_n\), uniformly for \(g\) in the \(n\)-th compact group set. The existence of each choice is the convex-tail argument: in the bidual the support projection of an invariant algebra is invariant; it commutes with every operator deriving the algebra. Indeed, for a homogeneous deriving operator \(d\) and \(a\in A_1\), one has \((1-p)da=(1-p)[d,a]_{\rm gr}=0\). Taking an approximate identity of \(A_1\) gives \((1-p)dp=0\); applying the same argument to \(d^*\), included in the deriving tests, gives \(pd(1-p)=0\). Thus \([p,d]=0\). A separating functional on the finite union of test sets would contradict weak convergence of the tails to this support projection. Compact group sets are included as continuous-function test sets, so the estimate is uniform there; this is not an assumption that \(G\) is amenable.

Put \(b_n=(v_n-v_{n-1})^{1/2}\), \(v_0=0\), and

\[
 M_2=\sum_{n\ge1}b_nu_nb_n\quad\hbox{strictly},\qquad M_1=1-M_2.
                                                                  \tag{RP.3}
\]

The tests may be chosen with errors \(2^{-n}\). For a strictly positive \(k\in J\), the tails satisfy \(\|(\sum_{m\le n\le N}b_nu_nb_n)k\|\le\sqrt5\sum_{n\ge m}2^{-n}\). The positive contraction bound on the partial sums and \(aa^*\le c k^2+\varepsilon\), for \(a\in J\), imply strict convergence by the Cauchy criterion. The estimates for commutators, the strictly positive generators of \(A_i\), and the group-defect terms are each bounded by a fixed multiple of \(\sum 2^{-n}\). In particular \(M_iA_i\subset J\), \([M_i,\Delta]\subset J\), \(M_2\varphi(g),\varphi(g)M_2\in J\), and \(g(M_i)-M_i\in J\); all the group functions are norm-continuous by the uniform compact-set estimates. This is the full additional separator property needed in the equivariant setting.

Use these \(M_i\) in the ordinary connection construction:
\(F=M_1^{1/4}(F_1\widehat\otimes1)M_1^{1/4}+M_2^{1/4}\widetilde F M_2^{1/4}\).
The ordinary connection and positive-sandwich calculations of §0 remain identical. The new estimates make the group defect locally compact and norm-continuous. For uniqueness and associativity, include all translated controlling operators and group defects in the separable tests (a countable group exhaustion and norm continuity suffice), and apply the same separator on the *whole* interval or triple-product module. The ordinary uniqueness path is built by continuous functional calculus of positive anticommutators; group defects of those factors remain locally compact. Endpoint evaluation sends rank-one interval compacts to rank-one compacts, so compatibility with full module homotopies and restriction follows. Tensor flips are the even graded flips \(e\otimes f\mapsto(-1)^{|e||f|}f\otimes e\), and exact diagonal group actions commute with them. This proves the equivariant product rules used below, using the ordinary detailed proofs and the full separator. No extension-exactness theorem is hidden in this step.

#### RP.1. The geometric algebras and two classes

Let \(X=G/K\) be the unit disk with metric \(4|dz|^2/(1-|z|^2)^2\), \(o=0\), \(r(y)=d(o,y)\), and

\[
 P=C_0(X,\operatorname{Cl}^+(T^*X)).
\]

The complex Clifford generators are self-adjoint and satisfy \(v^2=\|v\|^2\); \(P\) has its Clifford grading. It is a separable, graded, proper \(G\)-algebra, with the action induced by the actual cotangent action of disk isometries. No spin lift or character of the center is needed: the action is on the full Clifford bundle.

On \(H=L^2(X,\Lambda^*T^*X_{\mathbb C})\), use exterior parity, pullback by oriented isometries, and the exact Clifford representation

\[
 \rho(v)=v\wedge+\iota_{v^\sharp}.
\]

The complete Hodge operator \(D_H=d+d^*\), with its closure from compact smooth forms, is odd and self-adjoint. Completeness proves essential self-adjointness by distance cutoffs whose commutators have norm at most \(C/R\). The operator is exactly \(G\)-invariant. Local energy estimates and finite-frequency approximation give compact localized resolvents; every smooth compact Clifford section preserves the graph domain, and its graded commutator with \(D_H\) is order zero and compactly supported. The two-pole resolvent identity and its bound

\[
 \|[D_H(D_H^2+1+t^2)^{-1},\rho(a)]_{\rm gr}\|
 \le {5\|[D_H,\rho(a)]_{\rm gr}\|\over4(1+t^2)}
\]

give compact bounded-transform commutators by integration, then norm density. Thus

\[
 d=[(H,\rho,D_H(1+D_H^2)^{-1/2})]\in KK_G^0(P,\mathbb C).       \tag{RP.4}
\]

The dual class has the standard right \(P\)-module, scalar left action, and odd self-adjoint Clifford multiplier

\[
 \xi_o(y)=d_y\sqrt{1+d(o,y)^2},\qquad
 \eta=[(P,\mathbb C\to\mathcal L_P(P),L_{\xi_o})]
       \in KK_G^0(\mathbb C,P).                              \tag{RP.5}
\]

The norm of \(\xi_o\) is \(r/\sqrt{1+r^2}\), so \(1-\xi_o^2=(1+r^2)^{-1}\in P=\mathcal K_P(P)\). On the hyperbolic plane the triangle comparison estimate gives

\[
 \|\xi_u(y)-\xi_v(y)\|
 \le {6d(u,v)\over\sqrt{1+d(u,y)^2}+\sqrt{1+d(v,y)^2}}.         \tag{RP.6}
\]

For completeness, write \(a=d(u,y),b=d(v,y),c=d(u,v)\). Differentiating
\(\sqrt{1+a^2}-\sqrt{1+b^2}=(a-b)(a+b)/(\sqrt{1+a^2}+\sqrt{1+b^2})\)
bounds the derivative of the scalar normalizer by \(4/(\sqrt{1+a^2}+\sqrt{1+b^2})\). The nonpositive-curvature comparison \(a^2+b^2-c^2\le2ab\cos\theta\) yields
\(\|d_ya-d_yb\|\le2c/(a+b)\). Combining the two gives (RP.6); the zero-distance limits are smooth. Thus \(g\xi_o-\xi_o=\xi_{go}-\xi_o\in P\), and norm continuity in \(g\) follows from the same bound and continuity on a compact ball. These are the complete scalar equivariance defects of (RP.5).

Define

\[
 \gamma=\eta\widehat\otimes_P d\in KK_G^0(\mathbb C,\mathbb C).  \tag{RP.7}
\]

At this stage gamma denotes the well-defined product, with no idempotence assumed. The direct compact-subgroup Witten product below proves its K-restriction is the unit. No scalar full-G identity gamma=1 is assumed. If signs are compared, multiplication of each real covector by the invariant parallel complex structure gives an exact equivariant rotation from \(\xi\) to \(-\xi\), so the positive and negative two-dimensional dual multipliers are homotopic, with unchanged parity.

#### RP.2. The direct compact-subgroup product

The complete direct proof is WP.0–WP.9 above. It treats the complete hyperbolic Hodge Hilbert space, its Clifford action and graph domains. Its compact subgroup covector rotation homotopes eta to -eta with the invariant oriented rank-two complex structure. The negative Witten operator Q_W=D-c_+(d(r^2/2)) has whole compact resolvent and the even K-fixed top Gaussian kernel. A graded mixed two-pole resolvent proves its full Hodge creation connection for every homogeneous compact smooth Clifford coefficient, with norm-integrable lambda^(-2) error. Density gives all P-vectors.

For positivity, the actual unbounded anticommutator is C={-eta,Q_W}=2r^2/sqrt(1+r^2)+B, where B is self-adjoint and norm at most2. The confining form estimate makes C graph-bounded on Dom(Q_W) and proves the multiplier preserves that full domain. The anticommutator resolvent identity gives positive sandwiches after adding2, and a compact correction bounded by2/lambda^2. Increasing finite positive integrals have a uniformly bounded strong limit; the compact correction converges in norm. Thus the bounded-transform anticommutator is positive modulo the entire compact ideal, as required by the full product criterion. Replacing bounded transform by phase changes it compactly. Its K-invariant nonzero phase summand is exactly degenerate; its kernel is the actual even trivial K-module, not a numeric index surrogate. Therefore
\[
 r(\eta)\widehat\otimes_P r(d)=1\quad\text{in }KK_K^0(\mathbb C,\mathbb C),\qquad r(\gamma)=1.
 \tag{RP.17}
\]

#### RP.3. Necessary and sufficient foundation boundary

The actual compact-subgroup product is the only scalar identity needed below. Only the full product criterion RP.0, the exact eta/d cycles RP.1, and direct identity (RP.17) are inputs to the actual induction formulas RP.24 and RP.28. The complete direct-identity proof is given above; the preceding paragraph supplies only a summary.

#### RP.4. Restriction and the complete induction cycle

The map \(r\) simply restricts the same graded module, representation and operator from \(G\) to \(K\). It preserves products by their equivariant connection and positivity criterion. For \(x\in KK_K^j(A,\mathbb C)\), represent it by \((E,\phi,U_K,F)\). Clifford suspension handles \(j=1\); in what follows all formulas use the graded tensor convention.

One may take \(F\) self-adjoint and norm-continuous under conjugation. The replacement in Kasparov §2.4 follows from RP.0 with \(A_1=\phi(A)+J,A_2=0\), multiplying \(F\) by the appropriate separator, and changes \(F\) only locally compactly. For compact \(K\), average this operator over normalized Haar measure. The resulting \(F_K\) is exactly \(K\)-invariant. For nonunital \(A\), its difference from \(F\) is asserted only to be *locally* compact after either coefficient localization; it is the compact norm integral of those localized group defects. The affine path has all localized interval defects compact. We henceforth write \(F\) for this exactly invariant representative.

Form the graded Hilbert bundle \(G\times_K E\to X\), with
\([gk,e]=[g,U_K(k)e]\), and Hilbert \(C_0(X)\)-module
\(E_X=\Gamma_0(G\times_K E)\). Its inner product is the pointwise fibre inner product. The exact group action and operator are

\[
 s[g,e]=[sg,e],\qquad F_X[g,e]=[g,Fe].                       \tag{RP.18}
\]

They are well defined precisely because the stabilizer action is an actual representation and \(F\) commutes with it. Strong continuity is first checked on compactly supported local sections and then on their dense span; the adjoint of the operator is the same fibre formula. The \(G\)-action commutes with exterior parity. On the whole module the left action of \(A\widehat\otimes C_0(X)\) is

\[
 \phi_X(a\otimes b)[g,e]
       =[g,b(gK)\phi(\alpha_{g^{-1}}(a))e].                 \tag{RP.19}
\]

The change \(g\mapsto gk\) is compatible with (RP.19) by exact \(K\)-covariance of \(\phi\). Thus this is an actual multiplicative *-homomorphism, with exact \(G\)-covariance. Its norm is at most \(\|a\|\|b\|\).

Now extend over the Clifford bundle:

\[
 E_P=E_X\widehat\otimes_{C_0(X)}P,\qquad
 I_P(x)=[(E_P,\phi_P,F_X\widehat\otimes1)]
       \in KK_G^j(A\widehat\otimes P,P).                   \tag{RP.20}
\]

On a homogeneous elementary tensor, the left \(p\in P\) acts on the last factor with the usual sign \((-1)^{|p||e|}\); \(a\) acts by (RP.19) with the scalar cutoff absorbed into \(p\). The diagonal group action is exact, multiplicative, and compatible with the right \(P\)-action. The operator is exactly invariant. Its localized square and self-adjointness defects, and its graded commutator, are compact fields after the \(P\) cutoff: in a local chart they are the original compact \(A\)-localized defects evaluated at \(\alpha_{g^{-1}}a\). Those are norm-continuous functions on compact chart sets. The Clifford fibres are finite dimensional.

Here is the whole-module compactness argument. First cut \(p\) to a compact subset of \(X\); its discarded tail has operator norm at most a constant times the cutoff error. Cover that compact set by finitely many bundle charts. In each chart a continuous field of compact operators is uniformly approximated by finitely many rank-one fibre operators, using a finite partition of unity in the base. Multiply their vector sections by chart cutoffs to extend them to global sections. Their rank-one maps on \(E_P\) are module compacts. The chart errors and tail tend to zero in operator norm. The identical construction with the compact fibre ideal of an interval module proves the assertions for arbitrary \(K\)-Kasparov homotopies, not only for fixed-Hilbert-space paths. This proves that (RP.20) gives a well-defined additive map on KK classes, including nonunital \(A\), both parities and full module homotopies.

#### RP.5. The exact lifting map and its types

Define

\[
 L(x)=(1_A\widehat\otimes\eta)
      \widehat\otimes_{A\widehat\otimes P} I_P(x)
      \widehat\otimes_P d.                                 \tag{RP.21}
\]

The three factors have, respectively, types

\[
 KK_G^0(A,A\widehat\otimes P),\quad
 KK_G^j(A\widehat\otimes P,P),\quad
 KK_G^0(P,\mathbb C).
\]

All products are therefore well typed and have parity \(j\). They exist by RP.0, since every source algebra here is separable. No ordinary restriction map is inverted without specifying its actual cycle construction.

#### RP.6. The Cartan-section homotopy proves \(rL=1\)

The explicit global Cartan section of the disk quotient is

\[
 s(z)={1\over\sqrt{1-|z|^2}}
       \begin{pmatrix}1&z\\\bar z&1\end{pmatrix}
       =\exp(Y_z),\quad Y_z\in\mathfrak p.                 \tag{RP.22}
\]

It satisfies \(s(kz)=k s(z)k^{-1}\). Set \(s_t(z)=\exp(tY_z)\). The trivialization \([s(z),e]\mapsto(z,e)\) identifies the restriction of the induced bundle with the constant \(K\)-bundle, with exact action
\((k\xi)(z)=U_K(k)\xi(k^{-1}z)\). The induced left action becomes
\(\phi(\alpha_{s(z)^{-1}}a)\). Replace it, on the same full module over \(C([0,1],P)\), by

\[
 \phi_t(a)(z)=\phi(\alpha_{s_t(z)^{-1}}a),\qquad F_t=F.      \tag{RP.23}
\]

Every \(\phi_t\) is a multiplicative graded *-homomorphism. Exact \(K\)-covariance follows from the displayed section identity and original \(K\)-covariance. The representations and their adjoints are strongly continuous jointly in \((t,z)\) on compactly supported sections; boundedness extends this to the whole interval module. For compact coefficient cutoffs the functions \((t,z)\mapsto\alpha_{s_t(z)^{-1}}a\) are norm-continuous on compact sets. Applying the original compact-defect maps to them and the finite-chart argument of RP.4 proves that all defects belong to \(\mathcal K_{C([0,1],P)}(E_P[0,1])\). No uniform norm continuity of \(s_t(z)\) over the noncompact disk is claimed or needed; the \(P\) coefficient cutoffs supply exactly the necessary localization.

At \(t=1\) this is \(rI_P(x)\); at \(t=0\) it is the exterior coefficient extension \(\sigma_P^K(x)\). Hence

\[
 rI_P(x)=\sigma_P^K(x)\in KK_K^j(A\widehat\otimes P,P).
\]

Restriction of (RP.21), tensor interchange and the independently proved compact-subgroup identity (RP.17) give

\[
 rL(x)=(1_A\otimes r\eta)\otimes\sigma_P^K(x)\otimes rd
       =x\widehat\otimes_{\mathbb C}r\gamma=x.              \tag{RP.24}
\]

#### RP.7. The actual transported-operator path proves \(Lr=\gamma\)

Let \(y=(E,\phi,U_G,F)\) represent \(KK_G^j(A,\mathbb C)\). Use the norm-continuous locally compact replacement from RP.4, and average over \(K\); this changes the class by a valid \(G\)-equivariant locally compact perturbation, because the compact integrals are applied after coefficient localization. Thus \(F\) commutes exactly with \(U_G(K)\).

There is an onto even unitary of associated bundles

\[
 [g,e]\longmapsto(gK,U_G(g)e).                              \tag{RP.25}
\]

It is independent of the representative \(gk\), preserves inner products, and intertwines the induced group action with \(U_G\) on the constant fibre and translation on the base. Formula (RP.19) becomes the constant action \(\phi(a)\). The induced operator in this trivialization is

\[
 F_z'=U_G(s(z))FU_G(s(z))^*.                               \tag{RP.26}
\]

This field is exactly equivariant, since the cocycle of the section has values in \(K\), which commutes with \(F\). Compare it with the constant field \(F\), on the same group and coefficient representations. Their difference is bounded, self-adjoint, odd, and *locally* compact after every \(\phi(a)\otimes p\): norm continuity of the original localized group-defect maps gives compact fields on every compact base chart; the \(p\)-tail is bounded by \(2\|F\|\|a\|\) times that tail norm. Both-sided localizations follow by adjoints. Consequently

\[
 F_{z,t}=(1-t)F_z'+tF                                   \tag{RP.27}
\]

defines a full Kasparov operator homotopy over \(C([0,1],P)\). The square and commutator errors differ locally compactly from those of either endpoint; group defects are compact after localization and norm-continuous, since (RP.26) is exactly invariant and the constant endpoint has the original \(G\)-defect. The finite-chart rank-one argument proves compactness in the interval module ideal. This proves

\[
 I_P(ry)=\sigma_P^G(y).
\]

Apply (RP.21). Associativity and the even graded exterior-product interchange give

\[
 Lr(y)=(1_A\otimes\eta)\otimes\sigma_P^G(y)\otimes d
       =\gamma\widehat\otimes_{\mathbb C}y.                 \tag{RP.28}
\]

This is an equality of actual equivariant KK classes. The comparisons (RP.23) and (RP.27) supply the represented homotopies needed inside the projector construction itself; they do not purport to be a formula for the separately requested original native radial interpolation.

#### RP.8. Projector consequences and the boundary coefficient

Equations (RP.24) and (RP.28) prove (RP.1). They imply idempotence directly: for A=C, Lr(1)=gamma and rL=1 give gamma^2=Lr(gamma)=LrLr(1)=Lr(1)=gamma. The two typed map identities give the idempotence calculation directly. If \(ry=0\), then \(\gamma y=Lry=0\), so \(y=(1-\gamma)y\). Conversely \(r((1-\gamma)y)=0\) by (RP.17). For any \(x\), the lift \(Lx\) satisfies \(\gamma Lx=LrLx=Lx\), so its image is exactly the \(\gamma\)-part. These conclusions prove the gamma-part restriction and kernel statements in the opening paragraph, without an assumption \(\gamma=1\).

RR.2 and RR.3 use exactly the scalar \(\gamma\) in (RP.7) and its restriction (RP.17). Their induced coefficient identities have types
\(\sigma_{C_0(X)}(\gamma)=1_{C_0(X)}\in KK_G(C_0(X),C_0(X))\) and
\(\sigma_{C(\mathbb T)}(\gamma)=1_{C(\mathbb T)}\in KK_G(C(\mathbb T),C(\mathbb T))\).
Their action on \(KK_G(B,\mathbb C)\) is *precomposition* by that even identity. This equals the scalar exterior \(\gamma\)-action in (RP.28), by interchange. Thus the types are compatible for both parities and for nonunital \(B=C_0(X)\). No group-center character or Bergman deletion enters this argument.

The actual native lifted difference is handled by the particular equivariant boundary factorization in [HF](#11b-8-the-whole-particular-stinespring-interval), HF.1–HF.7. Its whole Stinespring interval proves z=q^*w. RR.2–RR.3 use only (RP.17), the actual parabolic contraction, and associated orbit induction to prove sigma_Q(gamma)=1. Thus gamma z=z; the full compact-subgroup comparison gives rz=0; this projector gives z=Lrz=0. No general six-term theorem or discarded local-dual identity enters the boundary or halfexactness proof.


### 11B.7. The exact orbit groups and parabolic contraction

We retain the prescribed longitudinal Dirac class, its full physical spin-c module, both Bergman sectors, the entire frame cocycle, exterior parity and determinant line. The final equality is in KK(A,A).

CL.1–CL.27 give the exact lifts and full K-homotopy. WP.1–WP.42 and RP.0–RP.8 give the required gamma-part projector. HF.1–HF.7 prove the particular boundary factorization. The orbit and finite-center interfaces are proved here.

#### RR.1. Exact groups and two orbits

Write \(\widetilde G=SU(1,1)\) in the matrix realization

\[
 \widetilde G=\left\{
 \begin{pmatrix}\alpha&\beta\\\bar\beta&\bar\alpha\end{pmatrix}:
 |\alpha|^2-|\beta|^2=1\right\},\qquad
 \widetilde K=\left\{\operatorname{diag}(e^{i\theta},e^{-i\theta})\right\}.
 \tag{RR.1}
\]

Its fractional action on the closed unit disk is
\(z\mapsto(\alpha z+\beta)/(\bar\beta z+\bar\alpha)\).
The kernel is \(Z=\{1,-1\}\). Thus \(G=\widetilde G/Z=PSU(1,1)\), and \(K=\widetilde K/Z=SO(2)\) is the actual oriented frame-rotation group. The open disk \(X\) is \(\widetilde G/\widetilde K=G/K\). Its boundary is \(\widetilde G/\widetilde P=G/P\), where \(P\) is the stabilizer of one boundary point.

In a real Cayley coordinate sending that point to infinity, the parabolic action is \(x\mapsto e^s x+b\). Hence

\[
 P=\mathbb R\ltimes\mathbb R,\qquad
 (s,b)(s',b')=(s+s',\,b+e^s b'),\qquad
 \widetilde P=Z\times P.
 \tag{RR.2}
\]

Both groups are closed Lie subgroups; \(P\) is connected, and \(\widetilde P\) has exactly the two central components. Their maximal compact subgroups are respectively \(1\) and \(Z\).

The open-disk quotient has the explicit global section

\[
 \sigma_X(z)=\frac1{\sqrt{1-|z|^2}}
             \begin{pmatrix}1&z\\\bar z&1\end{pmatrix}.
 \tag{RR.3}
\]

The boundary quotient for \(G\) has a global section given by the literal rotation taking the chosen boundary point to \(\zeta\). For \(\widetilde G\), local lifts of that rotation differ by \(Z\); this finite transition is retained in the associated bundle construction. No global square root of the boundary coordinate is assumed.

#### RR.2. An actual parabolic representation contraction

For \(0\le t\le1\), define continuous group homomorphisms of \(\widetilde P\) by

\[
 \vartheta_t(\varepsilon,s,b)=
 \begin{cases}
  (\varepsilon,s,(1-2t)b),&0\le t\le\tfrac12,\\
  (\varepsilon,(2-2t)s,0),&\tfrac12\le t\le1.
 \end{cases}
 \tag{RR.4}
\]

The semidirect-product law (RR.2) verifies multiplicativity directly in the first interval; in the second interval only addition of \(s\) remains. The formulas agree at \(1/2\). They fix \(Z\), start at the identity, and end at the projection to \(Z\).

Given any scalar equivariant cycle \((H,U,F)\) for \(KK_{\widetilde P}(\mathbb C,\mathbb C)\), replace \(U\) on the whole module \(C([0,1],H)\) by

\[
 (U_t(p)\xi)(t)=U(\vartheta_t(p))\xi(t),\qquad F_t=F.
 \tag{RR.5}
\]

Each \(U_t\) is an actual multiplicative unitary representation. Strong continuity in \((t,p)\), together with the uniform unitary bound, proves that it acts adjointably on the continuous-section Hilbert module; the adjoints have the same property. The square and self-adjointness defects are constant compact operators. Equivariant Fredholmness of the original scalar cycle means that \(p\mapsto U(p)FU(p)^*-F\) is a norm-continuous compact-operator function. Composition with \(\vartheta_t\) proves that every group defect in (RR.5) belongs to
\(C([0,1],\mathcal K(H))=\mathcal K_{C([0,1])}(C([0,1],H))\), jointly continuously in the group variable. No generator-domain interpolation is needed because \(F\) is bounded.

Thus restriction to \(Z\) and inflation along \(\widetilde P\to Z\) are inverse maps on scalar KK classes. The same proof with \(Z\) omitted gives \(KK_P(\mathbb C,\mathbb C)\cong KK(\mathbb C,\mathbb C)\). Consequently, for a Dirac–dual-Dirac projector \(\gamma_{\widetilde G}\) whose restriction to \(\widetilde K\) is the unit, its restriction to \(\widetilde P\) is the unit: the restriction to \(Z\subset\widetilde K\) is already the unit, and (RR.4)–(RR.5) make restriction to \(Z\) injective. This proves the special parabolic input directly, without assuming \(\gamma_{\widetilde G}=1\).

#### RR.3. Coefficient identity on each whole orbit module

Let \(H\) be \(\widetilde K\) or \(\widetilde P\), \(Y=\widetilde G/H\), and \(B=C_0(Y)\). For a scalar \(\widetilde G\)-cycle \((\mathscr H,U,F)\), its coefficient-extension cycle has right module \(B\otimes\mathscr H\), left multiplication by \(B\), and diagonal group action

\[
 (s\cdot\xi)(y)=U(s)\xi(s^{-1}y).
 \tag{RR.6}
\]

The unitary sending it to the associated module \(\Gamma_0(\widetilde G\times_H\mathscr H)\) is described in equivariant-section coordinates by

\[
 \eta(g)=U(g)^{-1}\xi(gH),\qquad
 \eta(gh)=U(h)^{-1}\eta(g).
 \tag{RR.7}
\]

It preserves the \(B\)-valued inner product and intertwines (RR.6) with left translation of the variable \(g\). The transported operator is
\((F^{\rm tr}\eta)(g)=U(g)^{-1}FU(g)\eta(g)\).
Its exact transformation law makes this operator well defined even when no global quotient section exists.

For \(H=\widetilde K\), average \(F\) over the compact group to obtain an exactly \(H\)-invariant operator \(F_H\). Its difference from \(F\) is a compact norm integral, so this is a scalar equivariant compact perturbation. It defines a constant operator on the associated bundle. The difference \(F^{\rm tr}-F_H\), multiplied by any \(b\in C_0(Y)\), is a continuous field of compact operators vanishing at infinity: on a compact chart the group parameter ranges in a compact set; the equivariance defect is norm continuous; the global difference is uniformly bounded. Therefore the straight operator path between these two operators is a Kasparov homotopy with the same actual group and function representations. Its square, commutator and equivariance errors are compact after multiplication by \(b\). This proves that coefficient extension of a scalar class agrees with induction of its \(H\)-restriction.

For \(H=\widetilde P\), use local sections of the boundary quotient with transitions in \(Z\), and first average \(F\) over \(Z\). In the local trivializations the induced operator may be chosen to be that same \(F\), because these finite transitions commute with it exactly. Its group action uses the exact cocycle

\[
 \kappa(s,y)=\sigma(y)^{-1}s\sigma(s^{-1}y)\in\widetilde P.
 \tag{RR.8}
\]

The cocycle identity proves multiplicativity. Its defect with \(F\) is precisely the scalar restricted-\(\widetilde P\) defect at \(\kappa(s,y)\); compact quotient charts, the continuous transition law, and a finite partition give the same whole-module compactness as above. Comparison with \(F^{\rm tr}\) is again the bounded compact operator path after \(B\)-localization. This constructs the required induction identity in this exact case, without assuming a generally constant operator descends through a noncompact stabilizer.

The construction applies equally to a scalar homotopy over \(C([0,1])\). Compact operator fields then have values in the *whole* interval ideal \(\mathcal K_{C([0,1])}(\mathscr E)\), and the finite quotient-chart argument approximates them by finite-rank interval sections. Thus an actual scalar equality after restriction to \(H\) induces an equality in \(KK_{\widetilde G}(B,B)\). Inducing the trivial scalar unit gives the right module \(B\), its ordinary translated left action, and operator zero, exactly \(1_B\).

Apply this to \(\gamma_{\widetilde G}\). Its compact-subgroup restriction is the unit by the restriction-projector provider; its parabolic restriction is the unit by RR.2. Therefore

\[
 \sigma_{C_0(X)}(\gamma_{\widetilde G})=1_{C_0(X)},\qquad
 \sigma_{C(\mathbb T)}(\gamma_{\widetilde G})=1_{C(\mathbb T)}.
 \tag{RR.9}
\]

These are identities of actual equivariant coefficient KK classes, in both graded actions. They do not discard a Bergman sector or a physical line.

#### RR.4. The semisplit compactification sequence

For \(f\in C(\mathbb T)\), define its Poisson lifting on the disk by

\[
 (\mathcal P f)(z)=\int_{\mathbb T}
          \frac{1-|z|^2}{|\zeta-z|^2}f(\zeta)\,\frac{d\theta}{2\pi},
 \qquad (\mathcal P f)|_{\mathbb T}=f .
 \tag{RR.10}
\]

The kernel is nonnegative and has integral one. For \(z=r\eta\), split the integral into \(|\zeta-\eta|<\delta\) and its complement. Uniform continuity controls the first part; on the second part, for \(r\) sufficiently close to one the denominator is at least \(\delta^2/4\) and the kernel mass tends uniformly to zero. This proves continuous extension with boundary value \(f\). The same estimates give \(\|\mathcal P f\|_\infty\le\|f\|_\infty\). Applied to a positive matrix-valued function, integration against the nonnegative scalar kernel preserves positivity at every matrix size; hence \(\mathcal P\) is unital completely positive.

For a fractional disk automorphism \(g\), the elementary Poisson identity is

\[
 P(gz,g\zeta)J_g(\zeta)=P(z,\zeta).
 \tag{RR.11}
\]

It follows by substituting the fractional formulas for \(1-|gz|^2\), \(g\zeta-gz\), and the boundary Jacobian. Rotations and \(z\mapsto(z+a)/(1+\bar a z)\) generate all these automorphisms; for the latter map the common factors \(|1+\bar a z|^2\) and \(|1+\bar a\zeta|^2\) cancel exactly. Change of variables in (RR.10) proves equivariance. The center acts trivially, so the same lifting is \(\widetilde G\)- and \(G\)-equivariant.

Thus

\[
 0\longrightarrow C_0(X)\longrightarrow C(\overline X)
   \longrightarrow C(\mathbb T)\longrightarrow0
 \tag{RR.12}
\]

is an equivariantly semisplit exact sequence. For the actual difference we use its explicit completely positive section and the full particular Stinespring interval HF.5–HF.6; no general extension-exactness theorem is needed.



### 11B.8. The whole particular Stinespring interval

We prove the particular factorization of the actual difference. Both complete Bergman sectors, exterior grading, finite center and all function/group representations are retained. The proof constructs a full equivariant Stinespring interval.

The exact starting cycles and interior comparison are CL.1–CL.27 and HF.1–HF.3. RP.0 supplies the full equivariant separator. The first-variable Stinespring construction is credited to Georges Skandalis, [Exact sequences for the Kasparov groups of graded algebras (1985), §§1–2](https://webusers.imj-prg.fr/~georges.skandalis/Publications/Publi12.pdf); the equivariant separator context is [Kasparov1988, §1.4](https://gdz.sub.uni-goettingen.de/download/pdf/PPN356556735_0091/LOG_0014.pdf). The needed particular construction is proved completely below.

#### HF.1. The actual difference on one full module

Use \(\widetilde G=SU(1,1)\), \(\widetilde K=U(1)\), with center \(Z=\{\pm1\}\), acting on the disk through \(G=PSU(1,1)\). Put
\[
 B=C(\overline X),\qquad I=C_0(X),\qquad Q=C(\mathbb T),\qquad q:B\to Q.
 \tag{HF.1}
\]
Let \(\overline x,\overline y\) be the complete lifted cycles CL.1–CL.2. Their difference is represented on the fixed graded Hilbert space
\[
 E=H_x\oplus H_y^{\mathrm{op}},
 \quad H_x=H_e\oplus H_o,\quad
 H_y=(H_e\oplus H_b)\oplus H_o,
 \quad H_b=\ell^2(\mathbb Z).
 \tag{HF.2}
\]
The exact function representation \(\pi_B\) and exact group representation \(U\) are the direct sums of those in CL.1–CL.2; reversing the grading on \(H_y\) does not change those even representations. Take the cycle operator \(F_0=F_x\oplus F_y\), with the second copy's grading reversed. This is the negative-class convention obtained by the odd graded flip; replacing its second operator by \(-F_y\) is an even grading-unitary conjugation on that copy.

Under restriction to \(I\), each form copy carries exactly the ordinary function multiplication and full pullback isometry representation \(R\); the boundary summand carries \(U_0\) and the zero \(I\)-representation. This last summand is retained throughout, not replaced by one of its Fourier halves.

#### HF.2. Localized phase comparison with actual domains

Write \(D=d+d^*\) and \(Q_W=D-(df\wedge+\iota_{\nabla f})\), \(f=r^2/2\), on the complete hyperbolic disk. Their closures from compact smooth forms are the actual self-adjoint endpoint operators proved in HB and CL. For \(\chi\in C_c^\infty(X)\), compact support and the local first-order graph estimate give
\[
 \chi\operatorname{Dom}(Q_W)\subset\operatorname{Dom}(D),
 \qquad [D,\chi]-\chi(Q_W-D)\in\mathcal L(H).
 \tag{HF.3}
\]
The last multiplier is bounded and compactly supported. Thus, first on the closed graph domains and then on all of \(H\),
\[
 \chi(Q_W-z)^{-1}-(D-z)^{-1}\chi
 =(D-z)^{-1}\bigl([D,\chi]-\chi(Q_W-D)\bigr)(Q_W-z)^{-1}.
 \tag{HF.4}
\]
For \(z=\pm i\lambda\), \(\lambda\ge1\), its norm is at most \(C_\chi/\lambda^2\). Each right side is compact: a resolvent multiplied by a compactly supported coefficient is compact by the local energy estimate and Rellich finite-frequency approximation. Summing the two poles and integrating with \(\lambda=\sqrt{1+t^2}\) proves
\[
 \chi Q_W(1+Q_W^2)^{-1/2}
 -D(1+D^2)^{-1/2}\chi\in\mathcal K(H).
 \tag{HF.5}
\]
The tail bound is \(C_\chi/(1+t^2)\), so this is an absolutely norm-convergent integral of compact operators. The already proved Hodge commutator allows \(\chi\) to move to the left of its bounded transform.

The endpoint phases equal their bounded transforms plus continuous functions vanishing at spectral infinity, using the gap \(1\) for \(Q_W\) off its Gaussian kernel and the gap \(1/2\) for \(D\) off its full harmonic kernel. Such correction functions can be chosen continuously on all of \(\mathbb R\); local compactness of the resolvent implies local compactness of their functional calculus. In particular the infinite harmonic projection causes no omission: its localization is a compact continuous functional-calculus term because zero is isolated by the Hodge gap. We obtain
\[
 M_h(F_{Q_W}-F_H),\ (F_{Q_W}-F_H)M_h\in\mathcal K(H),
 \qquad h\in C_0(X).
 \tag{HF.6}
\]
Smooth compact coefficients are dense in \(C_0(X)\), and both phases are contractions, so the statement extends by norm approximation.

For the other coupling, use the full Bergman map \(B:H_b\ominus\mathbb Ce_0\to\ker D|_{H_o}\) and delete-constant map \(T\). If \(\operatorname{supp}h\subset\{|z|\le R\}\), \(R<1\), the exact normalized basis in BC satisfies
\[
 \|b_n^\pm\|_{L^2(|z|\le R)}^2=R^{2n},\qquad n\ge1.
 \tag{HF.7}
\]
This follows by integrating \(n|z|^{2n-2}/(2\pi)\) against the conformally invariant form measure \(2\,dx\,dy\). Therefore, writing \(P_N\) for both sectors up to degree \(N\), the entire tail has the Hilbert–Schmidt bound
\[
 \|M_hB(1-P_N)\|
 \le \sqrt{2}\,\|h\|_\infty
      {R^{N+1}\over\sqrt{1-R^2}}.
 \tag{HF.8}
\]
The finite head has finite rank, so \(M_hBT\) is compact. The estimate covers both complete infinite sectors at once. Density gives the claim for every \(h\in C_0(X)\); adjoints give the other localized cross term.

#### HF.3. A particular fixed-representation interior cancellation

In the restriction of (HF.2), interpolate the native form phase to \(F_H\), and interpolate \(Z=(T_0,BT)\) to \((T_0,0)\). Equations (HF.6)–(HF.8) make every change \(I\)-locally compact. Both endpoint group defects are norm-continuous compact operators, by the full lifted-cycle proof; the interpolation has the same property. Hence the affine paths on \(C([0,1],E)\), with the original fixed \(I\)- and group representations, are valid whole-module \(I\)-Kasparov homotopies. Their localized square and commutator errors are norm-continuous compact functions, not only compact at individual times.

The result consists of two copies of the full form cycle with opposite grading, plus \(H_b\) with zero \(I\)-action and zero operator. The sign change on the reverse-graded form copy is itself a continuous represented path. Write \(\Gamma_H\) for its exterior grading involution, and use
\[
 W_s=\exp\!\bigl(i\pi s(1-\Gamma_H)/2\bigr),\qquad0\le s\le1,
 \tag{HF.8a}
\]
on that copy, with identity on the native copy and \(H_b\). This is an even norm-continuous unitary path from \(1\) to \(\Gamma_H\); it commutes exactly with the fixed function and group representations. Conjugating the operator by \(W_s\) therefore preserves all localized interval defects, and its endpoint is \(\Gamma_HF_H\Gamma_H=-F_H\). Concatenate this actual path before the following cancellation. The two form operators now have \(\operatorname{diag}(F_H,-F_H)\). The odd exchange
\[
 J=\begin{pmatrix}0&1\\1&0\end{pmatrix}
 \tag{HF.9}
\]
commutes exactly with the identical function and group representations and anticommutes with that diagonal operator. The actual cancellation path is
\[
 F_\theta=\cos\theta\,\operatorname{diag}(F_H,-F_H)
                +\sin\theta\,J,\quad0\le\theta\le\pi/2.
 \tag{HF.10}
\]
Its square defect is \(\cos^2\theta\operatorname{diag}(F_H^2-1,F_H^2-1)\), compact after every \(I\)-localization; its commutator is the corresponding cosine multiple of the original compact commutator. Its group defect is zero. Extend it by zero on \(H_b\). At the final endpoint \(F_1\), the form operator is \(J\), while the boundary \(I\)-action is zero. Consequently the entire endpoint is exactly \(I\)-degenerate:
\[
 [F_1,\pi_I(i)]=0,\quad
 \pi_I(i)(F_1^2-1)=0,\quad
 F_1=F_1^*,\quad U(g)F_1U(g)^*=F_1.
 \tag{HF.11}
\]
The glued affine and trigonometric paths give a norm-continuous odd self-adjoint operator path \(F_t\) on the original \(E,\pi_I,U\), from the actual \(B\)-cycle operator to this endpoint. This statement is specific to these two cycles.

#### HF.4. Equivariant lifting of this operator path

Let \(J_E=\mathcal K(E)\). Work with the \(G\)-norm-continuous bounded operators, and define the C*-algebras
\[
 \begin{split}
 \mathcal D_B&=\{T:[T,\pi_B(b)]\in J_E,\
                  g(T)-T\in J_E,\ b\in B,\ g\in\widetilde G\},\\
 \mathcal D_I&=\{T:[T,\pi_I(i)]\in J_E,\
                  \pi_I(i)(g(T)-T)\in J_E,\
                  (g(T)-T)\pi_I(i)\in J_E\},\\
 \mathcal J_I&=\{T\in\mathcal D_I:
                 \pi_I(i)T,\ T\pi_I(i)\in J_E,\ i\in I\}.
 \end{split}
 \tag{HF.12}
\]
All commutators here are ordinary operator commutators, since \(B,I\) have trivial grading. The algebras inherit the operator grading. Norm continuity under conjugation is included in their definitions. The ideals are norm closed, \(\mathcal J_I\) is an ideal of \(\mathcal D_I\), and \(J_E\subset\mathcal J_I\). The inclusion induces a surjective graded *-homomorphism
\[
 \mathcal D_B/J_E\longrightarrow\mathcal D_I/\mathcal J_I.
 \tag{HF.13}
\]
Here is the full surjectivity proof. For a homogeneous \(x\in\mathcal D_I\), take the controlling algebra \(A_1=C^*(\pi_I(I),J_E)\). Let \(A_2\) be generated by \([x,\pi_B(b)]\), their adjoints, and the bounded norm-continuous group-defect field \(\varphi(g)=g(x)-x\), including adjoints. Its image is norm separable by second countability and norm continuity. We have \(A_1A_2\subset J_E\): for example
\[
 \pi_I(i)[x,\pi_B(b)]
 =[x,\pi_B(ib)]-[x,\pi_I(i)]\pi_B(b)
 \quad \bmod J_E,
 \tag{HF.14}
\]
and \(ib\in I\); group-defect products are compact by definition. Include \(x,\pi_B(B)\) in the separable deriving set of the equivariant separator RP.0. It gives even positive \(M_1+M_2=1\) with
\[
 M_1A_1\subset J_E,\quad M_2A_2\subset J_E,\quad
 [M_j,x],[M_j,\pi_B(b)],g(M_j)-M_j\in J_E.
 \tag{HF.15}
\]
All group fields are norm-continuous. Then \(M_2x\in\mathcal D_B\), while \(x-M_2x=M_1x\in\mathcal J_I\), by the two-sided localizations and (HF.15). This proves (HF.13), including the group condition; ordinary non-equivariant separation alone would not suffice.

The images \(f_t\) of the actual path \(F_t\) in \(\mathcal D_I/\mathcal J_I\) are odd self-adjoint unitaries. Its initial point lifts to the unitary \([F_0]\) of \(\mathcal D_B/J_E\), since the original \(B\)-cycle has compact square defect.

For clarity, the needed unitary-path lifting has a short constructive proof. A surjective C*-homomorphism has the quotient norm. A continuous small path in its target, starting at zero, has a continuous lift starting at zero: approximate it by polygonal paths, choose lifts of their finitely many vertices with norm at most twice the quotient norm, and repeat for the residuals with errors decreasing geometrically. The series converges uniformly; its norm is at most four times the target's supremum norm. Symmetrization and odd projection preserve the required symmetry. Subdivide the unitary path until each variation is less than \(1/16\). Starting with an odd self-adjoint unitary lift \(v\), add such an odd self-adjoint lift \(h_t\) of the variation, of norm below \(1/4\), and replace \(v+h_t\) by
\[
 (v+h_t)|v+h_t|^{-1}.
 \tag{HF.16}
\]
It is well defined, odd, self-adjoint and unitary, and still projects to \(f_t\). Continue across the finitely many subintervals. Lift the resulting continuous path once more to actual odd self-adjoint operators \(G_t\in\mathcal D_B\), preserving \(G_0=F_0\), using the same polygonal series. Their squares equal one modulo \(J_E\), and
\[
 (G_t-F_t)\pi_I(i),\quad\pi_I(i)(G_t-F_t)\in J_E.
 \tag{HF.17}
\]
Every compact defect is norm-continuous in \(t\), jointly with each group parameter, because these are continuous paths in the norm-continuous operator algebra. Hence \(G_t\) is an actual \(B\)-Kasparov homotopy on the whole \(C([0,1],E)\). Put \(G=G_1\) and \(F=F_1\). The original class is represented by \(G\), \(F\) is exactly \(I\)-degenerate and exactly group invariant, and \((G-F)\) is compact after either \(I\)-localization.

#### HF.5. The complete quotient cycle

Use the equivariant unital completely positive section \(s:Q\to B\) proved by the exact Poisson formula RR.4. Its minimal Stinespring right \(B\)-module is the completion of \(Q\odot B\) with
\[
 \langle a\otimes b,c\otimes d\rangle=b^*s(a^*c)d.
 \tag{HF.18}
\]
Matrix positivity proves positivity of every finite Gram matrix; quotient the null space and complete. Left multiplication gives an actual bounded representation \(\lambda:Q\to\mathcal L_B(N)\). The map \(Vb=1\otimes b\) is an adjointable isometry, \(V^*(a\otimes b)=s(a)b\), and \(P_V=VV^*\) commutes exactly with the diagonal group action
\[
 g(a\otimes b)=g(a)\otimes g(b).
 \tag{HF.19}
\]
Equivariance of \(s\) preserves the inner product, and continuity follows first on elementary tensors, then by density. Thus \(N=B\oplus N_0\), with \(N_0=(1-P_V)N\), is an actual equivariant decomposition.

The generators \(u_a=(1-P_V)\lambda(a)V1\) satisfy
\[
 \langle u_a,u_c\rangle=s(a^*c)-s(a)^*s(c)\in I,
 \tag{HF.20}
\]
because applying \(q\) makes the two terms equal. Consequently \(N_0\) is a countably generated right \(I\)-module and \(N_0I\) is dense. This follows by applying an approximate identity of \(I\) to each vector with \(I\)-valued inner product; no complemented support assumption is made.

Let \(E_I=\overline{\pi_I(I)E}\). Equation (HF.11) makes \(F_I=F|_{E_I}\) an exactly \(I\)-linear, exactly group-invariant, odd self-adjoint unitary. On the whole Hilbert space
\[
 E' =N\widehat\otimes_B E
     =E\oplus(N_0\widehat\otimes_I E_I)
 \tag{HF.21}
\]
take the exact representation \(\lambda\widehat\otimes1\), the diagonal group action, and
\[
 \widehat F=G\oplus(1\widehat\otimes F_I).
 \tag{HF.22}
\]
This is an actual \(Q\)-Kasparov cycle. Its square defect is the compact square defect of \(G\) in the first summand; the second is exactly zero. Its group defect is likewise the compact defect of \(G\) in the first summand. The diagonal first commutator is \([G,\pi_B(s(a))]\), compact since \(G\) is a \(B\)-cycle; the second diagonal commutator is zero by exact \(I\)-linearity of \(F_I\).

For an off-diagonal creation map \(T_{u_a}:E\to N_0\otimes_I E_I\), the commutator is
\[
 (1\otimes F_I)T_{u_a}-T_{u_a}G=T_{u_a}(F-G).
 \tag{HF.23}
\]
Choose an approximate identity \(e_n\) of \(I\). Since \(u_ae_n\to u_a\), the right side is the norm limit of \(T_{u_a}\pi_I(e_n)(F-G)\), which is compact by (HF.17). Its adjoint gives the other off-diagonal entry. These are compact operators on the complete direct sum (HF.21); no fibrewise test substitutes for the operator-norm limit. Separability gives countable generation. Denote its class by \(w\in KK_{\widetilde G}(Q,\mathbb C)\).

#### HF.6. A full Stinespring interval proves the factorization

The equality of the original class and \(q^*w\) is itself realized on a whole interval module. Define
\[
 \psi_t(b)=(1-t)b+t\,s(q(b)),\qquad0\le t\le1.
 \tag{HF.24}
\]
These are equivariant unital completely positive maps \(B\to B\), and \(q\psi_t=q\) is a homomorphism. Perform (HF.18) with \(B\odot C([0,1],B)\) and the continuous map \(b\mapsto(t\mapsto\psi_t(b))\). This gives a countably generated equivariant Stinespring module \(N_{[0,1]}\), exact representation of \(B\), exact group action, adjointable isometry \(V_{[0,1]}\), and decomposition
\[
 N_{[0,1]}=C([0,1],B)\oplus N_{0,[0,1]}.
 \tag{HF.25}
\]
Its complement has inner products in \(C([0,1],I)\), by the same multiplicativity calculation (HF.20). Tensor with \(C([0,1],E)\) and use \(G\) on the first summand, \(1\otimes F_I\) on the second. The proof of (HF.22)–(HF.23) works in the *whole* interval compact ideal: the vectors \(u_b\) are actual module sections, \(u_b e_n\to u_b\) in interval-module norm, and the approximating \(\pi_I(e_n)(F-G)\) are constant compact fields. The norm limit is therefore in \(\mathcal K_{C([0,1])}\), uniformly over the parameter. Square, adjoint and group defects have the same whole-module property. Strongly continuous exact representations and adjoints act on this completion.

At \(t=0\), the variance of \(\psi_0=\mathrm{id}_B\) is zero, so its minimal Stinespring module is exactly \(B\), its complement is zero, and the evaluated cycle is \((E,\pi_B,U,G)\). At \(t=1\), the map factors through \(q\). If \(b\in I\), the norm of every vector \(\lambda_1(b)(a\otimes d)\) is zero, because \(\psi_1(a^*b^*ba)=0\). Thus its representation factors through \(Q\); the minimal module and operator are exactly (HF.18)–(HF.22). Endpoint evaluation is onto: elementary tensors evaluate onto the dense endpoint premodules, and quotient completion supplies the endpoint Hilbert module. We have proved
\[
 z=[\overline x]-[\overline y]=q^*w
             \quad\text{in }KK_{\widetilde G}(B,\mathbb C).
 \tag{HF.26}
\]
All group representations remained multiplicative. This is the required particular equivariant boundary factorization, including its actual whole-module homotopy; it is not a citation to a general halfexactness theorem.

#### HF.7. The projector now kills the actual difference

The scalar projector \(\gamma\) and the exact lifting map \(L\) have been proved with full direct compact-subgroup product, actual induction and Cartan providers in RP.0–RP.8. RR.2 proves the actual parabolic contraction; RR.3 proves that induction of its unit gives the entire boundary coefficient identity
\[
 \sigma_Q(\gamma)=1_Q\in KK_{\widetilde G}(Q,Q).
 \tag{HF.27}
\]
By product interchange and (HF.26),
\[
 \gamma z=q^*(\gamma w)
          =q^*(\sigma_Q(\gamma)\otimes_Qw)
          =q^*w=z.
 \tag{HF.28}
\]
The complete \(K\)-homotopy CL.26–CL.27 gives \(rz=0\). Therefore the exact projector formula, with its types and represented comparison homotopies, gives
\[
 z=\gamma z=L(rz)=0.
 \tag{HF.29}
\]
This proves equality of the two complete lifted classes. RR.6 descends it through the finite center; RR.7 transports the resulting scalar class equality to the actual holonomy right module; RR.8 identifies its endpoint with the native calibrated reverse product. Consequently
\[
 \delta_{\mathrm{inv}}\otimes_{C(V)}T=1_{C_r^*(\mathcal G)}
          \quad\text{in }KK(C_r^*(\mathcal G),C_r^*(\mathcal G)).
 \tag{HF.30}
\]
Together with the existing first product \(T\otimes_A\delta_{\mathrm{inv}}=1_{C(V)}\), this proves the assigned integral assembly isomorphism in both K-degrees, under the exact disk-leaf and classifying-space hypotheses of calibrated reverse equality.

The general isomorphism \(KK_{\widetilde G}(B,\mathbb C)\to KK_{\widetilde K}(B,\mathbb C)\) via a cyclic six-term proof is not asserted by this document. Only its gamma-part projector and the completely proved factorization of the actual native difference are used. No particular fixed radial representation formula has become an additional condition on the exact KK equality. The literal historical class remains \(\delta_S=-\delta_{\mathrm{inv}}[D]\); its product is \(-[D]\), with the prescribed determinant line. The negative determinant-line result and its calibrated/literal distinction remain unchanged.


### 11B.9. Finite-center descent and actual holonomy-module transport

#### RR.6. Finite-center descent on full cycles and homotopies

Inflation along \(\widetilde G\to G\) is injective on the KK groups used here. For a possibly nonunital representation of the unital source, first put \(P_1=\pi(1)\). It is an even invariant complemented module projection. The off-diagonal part of \(F\) relative to \(P_1\) is compact because \([F,P_1]\) is compact. Replacing \(F\) by its block diagonal gives a compact-perturbation whole-module homotopy; its zero-representation complement is degenerate and can be discarded. On the remaining module the source acts unitally. The endpoints here already have \(P_1=1\), so this normalization preserves them. Thus a \(\widetilde G\)-homotopy \((\mathscr E,U,\pi,F)\) between inflated cycles has the finite central projection

\[
 Q_Z=\tfrac12(1+U(-1)).
 \tag{RR.15}
\]

The group action and the function representation commute exactly with this projection because the center acts trivially on the input algebra. Replace \(F\) by
\(\tfrac12(F+U(-1)FU(-1)^*)\). Its difference is a compact norm-continuous field since the input algebra is unital and the central equivariance defect is compact. The new \(F\) commutes exactly with \(Q_Z\). Restriction to the complemented whole module \(Q_Z\mathscr E\) preserves grading, all multiplicative representations, all compact defects and the full interval ideal. Its action factors through \(G\). At the two inflated endpoints \(Q_Z=1\), so the extracted homotopy has exactly the original endpoints. The same argument applies to stable additions and degenerate cycles, and proves injectivity of inflation without a chosen lift of each \(G\)-element. Thus the actual lifted equality HF.29 gives

\[
 [\overline x]=[\overline y]
        \quad\text{in }KK_G(C(\overline X),\mathbb C),\qquad
 [x]=[y]\quad\text{in }KK_G(\mathbb C,\mathbb C).
 \tag{RR.16}
\]

The second equality is restriction of the *function* action to constants, not restriction of the group to \(K\).

#### RR.7. Actual holonomy-module transport respects the equality

Let \(V\) be the compact boundaryless manifold with the prescribed oriented rank-two disk leaves, \(\mathcal G=\operatorname{Hol}(V,F)\), and \(A=C_r^*(\mathcal G)\). Let \(P\to V\) be its actual bundle of oriented leafwise orthonormal frames. For an arrow \(\gamma\), the complete leaf isometries \(j_p\) supplied by the disk hypothesis give the actual group cocycle

\[
 c(p,\gamma,q)=j_p^{-1}j_q,\qquad
 c(pk,\gamma,q\ell)=k^{-1}c(p,\gamma,q)\ell.
 \tag{RR.17}
\]

All frame changes are actual rotations; no global frame or global untwisting is assumed.

The equality in (RR.16) is equality in the cycle-defined KK group, so it is realized by stable additions, unitary equivalence and a whole equivariant homotopy. This statement follows from the definition of the group used in the argument, not from an assertion about induced K-groups. A scalar homotopy can be taken on a countably generated Hilbert \(C(I)\)-module \(\mathscr E\) with trivial right-coefficient group action. Normalize the scalar support and self-adjointness in the usual compact-perturbation manner. Average its bounded odd operator over \(K\):

\[
 F_K=\int_KU(k)FU(k)^*\,dk.
 \tag{RR.18}
\]

The integrand differs from \(F\) by a norm-continuous \(\mathcal K_{C(I)}(\mathscr E)\)-valued function. Hence \(F_K-F\) is compact, and \(F_K\) is exactly \(K\)-invariant. The endpoints in our proof already commute with \(K\), so averaging preserves them. Grading, boundedness and all defects remain valid. This supplies an operator that descends through every frame chart.

Form the entire parameter module

\[
 \mathscr E_V=\Gamma(V,P\times_K\mathscr E),\qquad
 \mathscr E_A=
 \mathscr E_V\widehat\otimes_{C(V)\otimes C(I)}C(I,A).
 \tag{RR.19}
\]

For elementary tensors, its inner product is
\(\langle\xi\otimes b,\eta\otimes d\rangle=b^*
\langle\xi,\eta\rangle d\), with the actual \(C(V)\)-multiplication in \(M(A)\).
The inner product is positive on the full completion. A finite frame cover of compact \(V\), a subordinate partition, and a countable generating set of \(\mathscr E\) show countable generation. The associated arrow transport is
\([q,v]\mapsto[p,U(c(p,\gamma,q))v]\); (RR.17) proves its independence of frames and the cocycle law.

On compact arrow sections \(\Xi(\gamma)\in(P\times_K\mathscr E)_{r\gamma}\), the exact left representation is

\[
 (\Pi(a)\Xi)(\gamma)=\int a(\eta)
       J_U(\eta)\Xi(\eta^{-1}\gamma)\,d\lambda^{r\gamma}(\eta).
 \tag{RR.20}
\]

Convolution associativity, the cocycle law and the Hilbert inner product prove multiplicativity and adjointness on this core. In regular localization at \((t,u)\), the genuine unitary
\(\Xi(\alpha)\mapsto J_U(\alpha)^{-1}\Xi(\alpha)\)
turns (RR.20) into \(\lambda_u(a)\otimes1\) on the entire Hilbert space of the regular fibre. Thus \(\|\Pi(a)\|\le\|a\|_r\), and (RR.20) extends to the actual reduced algebra. The regular localizations are faithful for the module norm. This is a norm proof and a localized gauge; it makes no globally fixed right-module identification.

The bounded operator \(F_K\) acts on every associated fibre. Its square defect and the commutator with (RR.20) are compact arrow kernels with values in the complete ideal \(\mathcal K_{C(I)}(\mathscr E)\). On finitely many buffered frame charts, norm-continuous compact kernel fields are uniformly approximated by finite-rank interval fields. Each separated rank-one term is an operator of the form \(T_\xi L_aT_\eta^*\), compact on \(\mathscr E_A\), because \(L_a\in\mathcal K(C(I,A))=C(I,A)\). Compact-support Schur bounds control the approximation in module norm; the uniform interval approximation gives \(\mathcal K_{C(I,A)}(\mathscr E_A)\), not merely compactness at each \(t\). Norm density of compact arrow sections then gives all localized defects for every \(a\in A\). This is exactly the complete-kernel mechanism of BC.19–BC.23, now with \(\mathscr E\) an interval module.

Evaluation of (RR.19) is onto each endpoint module: the Hilbert-module quotient by the ideal of coefficients vanishing at that endpoint is the endpoint interior tensor product; finite frame sections and a partition give the same quotient of the associated module. Operators and representations evaluate to their original full cycles. Unitary equivalences, compact perturbations and degenerate cycles transport by the same formulas. Therefore this construction gives an additive map on scalar equivariant KK classes, and it sends (RR.16) to

\[
 \mathfrak J(x)=\mathfrak J(y)\quad\text{in }KK(A,A).
 \tag{RR.21}
\]

Every domain used here is the entire bounded-operator module. The native unbounded endpoint and its physical \(W\widehat\otimes S\) comparison retain their previously proved complete domains and full grading, as proved in P.1–P.6 above.

#### RR.8. Native reverse product, assembly and the literal sign

The exact native comparison BC.24–BC.26 identifies
\(\mathfrak J(x)=p=\delta_{\rm inv}\otimes_{C(V)}T\).
The already proved entire boundary representation homotopy and its actual transport give
\(\mathfrak J(y)=1_A\). This boundary homotopy is a scalar \(G\)-cycle homotopy; it does not require an exactly covariant closed-disk function representation at every parameter. Consequently (RR.21) implies the calibrated reverse equality

\[
 \boxed{\delta_{\rm inv}\otimes_{C(V)}T=1_A\quad\text{in }KK(A,A).}
 \tag{RR.22}
\]

Together with the independently completed first product
\(T\otimes_A\delta_{\rm inv}=1_{C(V)}\), this makes the prescribed longitudinal class \(T\) a KK equivalence with the calibrated radial inverse. Its product maps are inverse on both integral K-theory degrees, including torsion. Under the disk-leaf model \(B\mathcal G\simeq V\), the assembly map in the target is this Dirac product after the specified spin-c identification, so the analytic-isomorphism assertion follows in its original leaf hypotheses.

The unchanged positive same-spin class is a different radial class:

\[
 \delta_S=-\delta_{\rm inv}[D],\qquad
 T\delta_S=-[D],\qquad D=\det S=LH^2.
 \tag{RR.23}
\]

Thus the calibrated inverse does not validate the literal positive same-spin first identity. The determinant line, full parity and coefficient order in that refutation remain intact.

The fixed Fourier multipliers along the scalar boundary path fail exact covariance by CL.28. That checked failure is not a prerequisite for the Stinespring and induction argument and is not an obstruction to its KK equality.

### 11B.10. The ordered reverse equality and assembly inverses

For the original oriented rank-two Poincaré-disk foliation hypotheses, the complete calibrated radial cycle satisfies
\[
 \delta_{\mathrm{inv}}\widehat\otimes_{C(V)}T
       =1_A\quad\text{in }KK(A,A),\qquad
 A=C_r^*(\operatorname{Hol}(V,F)).
 \tag{C.1}
\]
This is the full reverse KK equality for these actual disk-leaf hypotheses. Together with the independently proved first product \(T\widehat\otimes_A\delta_{\mathrm{inv}}=1_{C(V)}\), it gives inverse product maps in both integral K-degrees, including torsion. Under the target's \(B\mathcal G\simeq V\) spin-c identification, these are the inverse maps for its analytic assembly map.

The proof preserves the actual complete spin-c spaces, exterior parity, both infinite Bergman sectors, the full frame cocycle and determinant line. It compares complete covariant cycles and uses an actual equivariant Stinespring interval. It does not infer the result from a numerical index or induced K-group equality.

#### C.2. Exact hypotheses and complete endpoint constructions

The manifold \(V\) is compact and boundaryless. Its oriented two-dimensional leaves have the prescribed smooth complete curvature \(-1\) Poincaré metric and are simply connected. Hence the leafwise oriented-frame isometry group is \(G=PSU(1,1)\), with \(K=SO(2)\). The finite cover used for the proof is \(\widetilde G=SU(1,1)\), \(\widetilde K=U(1)\), center \(Z=\{\pm1\}\). Its boundary stabilizer is \(\widetilde P=Z\times(\mathbb R\ltimes\mathbb R)\), with \((s,b)(s',b')=(s+s',b+e^sb')\). RR.1 specifies the actual quotient sections and finite boundary transition; there is no assumed global square root of a rotation.

The whole-module reverse endpoint and product are proved in [BC](#11b-3-the-whole-native-to-boundary-comparison-and-reduced-module), BC.1–BC.26, and [P](#11b-1-the-complete-native-represented-reverse-product), P.1–P.20. The former proves the whole-groupoid representation and compact-kernel mechanism; the latter proves the actual native \(W_y\widehat\otimes S_y\) domain, creation, positivity, graded flip and evaluation comparison. Their full proof bodies have been given above.

Let \(H=H_e\oplus H_o=L^2(X,\Lambda^*)\), with actual hyperbolic measure and exterior grading. The native scalar cycle has full pullback group representation, function multiplication, and the negative Witten phase of \(D-c_+(df)\), \(f=r^2/2\). Its unique even top Gaussian, compact resolvent and complete graph domain are proved in BC.2–BC.5 and the complete direct domain, confinement and kernel proof WP.1–WP.3. The boundary/Hodge cycle has \((H_e\oplus H_b)\oplus H_o\), \(H_b=\ell^2(\mathbb Z)\), actual group representation \((R_e\oplus U_0)\oplus R_o\), function representation \((M_h^e\oplus M_{h|\mathbb T})\oplus M_h^o\), and coisometry \(Z=(T_0,BT)\). The map \(B\) includes the complete positive and negative Bergman spaces; \(T\) deletes precisely the boundary constant.

#### C.3. Complete lifts and their compact-subgroup equality

[CL](#11b-4-complete-closed-disk-lifts-and-the-compact-subgroup-comparison), CL.1–CL.27, proves that both complete cycles lift to \(KK_{\widetilde G}(C(\overline X),\mathbb C)\). The proof treats continuous functions on the entire closed disk, actual multiplication representations, all domains, Hodge local resolvents and uniform full-space Bergman commutators. It proves the whole \(K\)-homotopy \(h(z)\mapsto h(tz)\), \(h(\zeta)\mapsto h(t\zeta)\), with all compact errors in the entire interval ideal. At \(t=0\) the complete exchange unitary
\[
 \mathcal V=Z^*S+\theta_{e_0,g},\qquad Z\mathcal V=S
 \tag{C.2}
\]
intertwines the two full cycles, including the Gaussian and boundary constant. Thus, for
\[
 z=[\overline x]-[\overline y]\in
     KK_{\widetilde G}(C(\overline X),\mathbb C),
 \qquad rz=0,
 \tag{C.3}
\]
the conclusion is equality of \(K\)-classes, not a sampled index.

#### C.4. The complete restriction projector and boundary coefficient

[RP](#11b-6-the-restriction-projector-from-induction-and-cartan-homotopies), RP.0–RP.8, proves the exact typed maps
\[
 r:KK_{\widetilde G}^j(B,\mathbb C)\to KK_{\widetilde K}^j(B,\mathbb C),
 \quad L:KK_{\widetilde K}^j(B,\mathbb C)\to KK_{\widetilde G}^j(B,\mathbb C),
 \quad rL=1,\quad Lr=\gamma\cdot(-).
 \tag{C.4}
\]
It supplies the full equivariant separator, actual associated induction module, Cartan representation path and transported-operator path. Its complete direct compact-subgroup product provider is [WP](#11b-5-the-direct-compact-subgroup-witten-product): the actual negative Witten operator has a full graded Hodge creation connection and positive bounded-transform anticommutator modulo the whole compact ideal. Its even K-fixed top Gaussian gives the exact identity r(gamma)=1. No assumption \(\gamma=1\) for \(\widetilde G\) is used.

RR.2 gives actual homomorphisms contracting \(\widetilde P\) to its central \(Z\), first scaling \(b\) to zero and then \(s\) to zero. They fix \(Z\) and produce a scalar whole-module equivariant KK homotopy. Since the projector restricts to one on \(\widetilde K\), its parabolic restriction is one. RR.3 proves the actual associated-module induction identity, including the finite center transition and full interval compactness. Therefore
\[
 \sigma_{C(\mathbb T)}(\gamma)=1_{C(\mathbb T)}
             \quad\text{in }KK_{\widetilde G}(C(\mathbb T),C(\mathbb T)).
 \tag{C.5}
\]

#### C.5. The actual difference factors through the boundary

The complete argument is HF.1–HF.7. Its analytic entrance and exact interior operator path are proved at HF.2–HF.3. Set \(I=C_0(X)\), \(B=C(\overline X)\), \(Q=C(\mathbb T)\), \(q:B\to Q\).

On the original direct-sum module representing \(z\), both form copies have identical exact \(I\)- and group representations; the entire extra \(H_b\) has zero \(I\)-representation. The full-domain mixed resolvent identity proves \(I\)-local compactness of the difference between native and Hodge phases, with the integrable bound \(C/(1+t^2)\). The entire localized Bergman tail has the bound
\[
 \|M_hB(1-P_N)\|\le
 \sqrt2\,\|h\|_\infty\,R^{N+1}/\sqrt{1-R^2},
 \quad \operatorname{supp}h\subset\{|z|\le R<1\}.
 \tag{C.6}
\]
Thus both phase replacement and deletion of the \(BT\) coupling are valid \(I\)-operator paths on the whole interval module. The even continuous grading rotation \(W_s=\exp(i\pi s(1-\Gamma_H)/2)\) on the reversed form copy supplies the sign change; the exact odd exchange on the two form copies then cancels them. This gives a particular fixed-representation \(I\)-operator path \(F_t\) ending at an exactly \(I\)-degenerate, exactly group-invariant \(F_1\). It is not a generic conversion of arbitrary KK homotopies to operator paths.

HF.4 proves \(\mathcal D_B/\mathcal K(E)\to\mathcal D_I/\mathcal J_I\) is onto by the equivariant separator, including norm-continuous compact group defects. It lifts the actual odd self-adjoint unitary quotient path and gives an actual \(B\)-cycle path \(G_t\), with \(G_0=F_0\) and \((G_1-F_1)I\) compact on both sides.

RR.4 supplies the exact equivariant unital completely positive Poisson section \(s:Q\to B\). HF.5 constructs the complete quotient Stinespring cycle
\[
 E'=E\oplus(N_0\widehat\otimes_I E_I),\qquad
 \widehat F=G_1\oplus(1\widehat\otimes F_1|_{E_I}),
 \tag{C.7}
\]
with exact group action and actual multiplicative left \(Q\)-representation. HF.6 constructs the entire equivariant Stinespring module for
\[
 \psi_t=(1-t)\operatorname{id}_B+t\,s q.
 \tag{C.8}
\]
Its off-diagonal creation commutators are uniformly approximated by compact interval fields using \(u e_n\to u\) in the whole module norm. Every square, commutator and group defect lies in \(\mathcal K_{C([0,1])}\); the endpoint quotient modules and representations are proved exactly. Hence for the actual quotient cycle \(w\),
\[
 z=q^*w,\qquad
 z=\gamma z=L(rz)=0.
 \tag{C.9}
\]
The second equality follows from (C.5) and (C.4). This closes the comparison without assuming the broader cyclic six-term theorem. Every step of (C.9) uses the particular cycle and whole interval just constructed.

#### C.6. Finite center and the complete groupoid module

RR.6 proves that a \(\widetilde G\)-homotopy between inflated cycles descends: average its operator over \(Z\), then restrict the whole interval module by the complemented central projection \(Q_Z=(1+U(-1))/2\). The endpoints have \(Q_Z=1\), and all compact fields restrict correctly. Thus \([\overline x]=[\overline y]\) descends to \(G\); forgetting the function action to constants gives \(x=y\) in \(KK_G(\mathbb C,\mathbb C)\).

RR.7 proves that this full class equality transports, through genuine whole-module homotopies and stable cycle operations, to
\[
 \mathscr E_A=
 \Gamma(V,P_{\mathrm{frames}}\times_K\mathscr E)
 \widehat\otimes_{C(V)\otimes C([0,1])}C([0,1],A).
 \tag{C.10}
\]
It gives the exact cocycle \(c(p,\gamma,q)=j_p^{-1}j_q\), its frame changes, actual reduced convolution representation, and \(K\)-averaged bounded operator. Compact arrow kernels are approximated uniformly by whole-interval finite-rank fields on a finite frame cover. The regular-fibre gauge proves the reduced norm; compactness is proved in the whole right-module compact ideal. Endpoint evaluation is onto. The inherited physical-spinor product comparison identifies \(\mathfrak J(x)=\delta_{\mathrm{inv}}\otimes T\), while the complete boundary representation homotopy identifies \(\mathfrak J(y)=1_A\). This proves (C.1).

#### C.7. Checked failures and exact residual scope

CL.28 proves that retaining fixed boundary multiplication along \(U_s\) fails exact full-group covariance for every \(s>0\). Those failed proposals remain checked failures. No formula for either particular interpolation is asserted by the completed proof.

The literal positive same-spin radial class remains
\[
 \delta_S=-\delta_{\mathrm{inv}}[D],\qquad
 T\delta_S=-[D],\qquad D=\det S=LH^2.
 \tag{C.11}
\]
Thus the literal positive same-spin identity remains refuted. Historical uncalibrated cycle equalities have not been silently identified with the calibrated mathematical theorem.


### 11B.11. The mechanisms and their exact bounds

![Exact radial derivative bounds and the whole compact rotation product](../figures/kt-hyperbolic-witten-product.png)

**Figure 11B.1.** WP.20–WP.42. Left: samples of the exact radial and angular derivative eigenvalues, with the analytically proved upper bound 1 and smooth origin value 1. The plotted samples are illustrations of the formulas, not spectral evidence. Right: the entire tensor unitary, graded creation connection, positive strong limit with norm-compact correction, and exactly invariant even top Gaussian. Negative Clifford multiplier here denotes an invariant covector rotation of the positive multiplier, not its additive inverse class.


#### A rejected bounded local construction

The first panel of Figure 11B.2 records a concrete failed construction. Fix \(x\in X\) and \(a>0\), put \(H_x=L^2(B_a(x),\Lambda^*T^*X)\), and let \(j_x:H_x\to H=L^2(X,\Lambda^*T^*X)\) be extension by zero. On this local space let
\[
\Theta_x(y)=c_+\!\left(a^{-1}d(x,y)d_y d(x,y)\right),\qquad
\varphi_x=(1-\Theta_x^2)^{1/4},\qquad
S_x=\Theta_x+\varphi_x j_x^*F_Hj_x\varphi_x,
\]
where \(F_H=D_H(1+D_H^2)^{-1/2}\) and \(D_H=d+d^*\). The radial covector extends smoothly through \(y=x\), and \(\Theta_x^2=d(x,y)^2/a^2\). This bounded operator was proposed as a product connection. For any even scalar \(\chi\in C_c^\infty(B_a(x))\), its creation map is \(T_\chi w=\chi w|_{B_a(x)}\); the product criterion requires \(S_xT_\chi-T_\chi F_H\) to be compact.

Choose \(y_0\ne x\), a coordinate neighborhood strictly inside the ball, and \(\chi=1\) there. Set \(h_0=d(x,y_0)/a\in(0,1)\). In a smooth orthonormal exterior trivialization write
\[
\Theta_0=\Theta_x(y_0),\qquad
C_0=\frac{i c_-(dy_1)|_{y_0}}{\|dy_1\|_{y_0}},\qquad
C_0=C_0^*,\quad C_0^2=1,\quad
C_0\Theta_0+\Theta_0C_0=0.
\]
Take a unit vector \(v\) with \(C_0v=v\). Normalized smooth wave packets \(w_n\), with phase \(e^{iny_1}\), support radius \(n^{-1/2}\) about \(y_0\), and exterior coefficient \(v\), are weakly null. They lie in the smooth compact core, and
\[
\|(D_H/n-\lambda_0)w_n\|=O(n^{-1/2}),\qquad
\lambda_0=\|dy_1\|_{y_0}>0.
\]
Indeed amplitude derivatives are \(O(n^{1/2})\), the first-order symbol varies by \(O(n^{-1/2})\) on the shrinking support, and the zeroth-order coefficients are bounded there. The spectral mass outside \(|s/n-\lambda_0|<\lambda_0/2\) tends to zero by this estimate; inside that interval \(s/\sqrt{1+s^2}\to1\) uniformly. Thus \(F_Hw_n-w_n\to0\). Continuity of the multipliers, with \(\varphi_0^2=\sqrt{1-h_0^2}\), and boundedness of \(F_H\) now give
\[
(S_xT_\chi-T_\chi F_H)w_n
=\Theta_0w_n+\bigl(\sqrt{1-h_0^2}-1\bigr)w_n+o(1).
\]
The two leading terms are orthogonal: \(\Theta_0v\) belongs to the negative \(C_0\)-eigenspace and has norm \(h_0\). Consequently
\[
\lim_{n\to\infty}\|(S_xT_\chi-T_\chi F_H)w_n\|^2
=h_0^2+\bigl(\sqrt{1-h_0^2}-1\bigr)^2>0.
\]
A compact operator sends a bounded weakly null sequence to a norm-null sequence, so the claimed creation connection fails. All packets are strictly inside the ball; neither a boundary convention nor a domain extension can remove this error. This rejects the specific bounded formula. The accepted induction projector in RP.4–RP.8 instead uses the full negative-Witten product WP.0–WP.9 and does not depend on this construction.


![The compact-subgroup product and exact restriction projector](../figures/kt-hyperbolic-restriction-projector.png)

**Figure 11B.2.** WP.1–WP.42, RP.0–RP.8, HF.1–HF.7 and RR.2–RR.3/RR.6–RR.8. Each arrow denotes the represented construction or typed KK product named in its box. The failed connection formula in the first panel is proved to fail in the rejected-construction calculation immediately above. The full negative Witten product and actual induction/Cartan homotopies supply the current argument. Its literal same-spin sign and determinant line are unchanged.

![The actual lifted difference, Stinespring factorization and full holonomy transport](../figures/kt-hyperbolic-boundary-factorization.png)

**Figure 11B.3.** CL.1–CL.28, HF.1–HF.7, RR.2–RR.3 and RR.6–RR.8. The square/commutator/group defects belong to the whole indicated interval compact ideal. Both form copies and the full boundary space remain until the exact cancellations described in the proof. The final endpoints are the actual calibrated reverse product and \(1_A\). This is an operator proof map, not an embedding of the entire foliation in the pictured plane.

## 11C. Two constraints on full-graph Dirac constructions

Connes asks for a scalar class \(KK^q(C_r^*(G),\mathbb C)\) represented by a suitable graph Dirac operator, with local transverse Bott pairing one, under an invariant normal spin-c structure. Section 11A gives a normalized SU(2)/F2 graph-column construction with an explicit discrete term. Here we prove an obstruction to a precisely specified ordinary full-graph differential ansatz, and a separate obstruction to obtaining a full-graph spin-c structure from the normal spin-c premise. Neither result refutes the original problem: “suitable Dirac type operator on the graph” does not impose either of these additional interpretations. Graph-column completions and operators with only the prescribed normal Clifford symbol are not covered by the obstruction.

The question and its local transverse Bott class are from Alain Connes, *A survey of foliations and operator algebras* ([author’s text](https://alainconnes.org/wp-content/uploads/foliationsfine.pdf)). The complete arguments below give the two constraints and their prerequisites. The exposition, proofs and diagram expression are dedicated to [CC0 1.0](https://creativecommons.org/publicdomain/zero/1.0/); the bundled fonts retain their notices in the component terms.

### 11C.1. The exact ordinary differential ansatz

For the first result, “full-graph Dirac” has all of the following concrete meanings. The Hilbert space is the ordinary smooth-density completion of the **entire** graph, with a smooth finite-rank graded Hermitian coefficient bundle. Left convolution acts by a smooth unitary coefficient transport satisfying its composition law. The algebra acts evenly. The operator is an odd formally self-adjoint first-order differential operator whose principal symbol is Clifford multiplication for a smooth positive-definite metric on the **whole** graph tangent bundle. Its commutators with every smooth convolution kernel are bounded on the Hilbert space. The closure, compactness and index conclusions will be proved, rather than added as unverified assumptions.

The full symbol is an additional assumption: it requires ellipticity in both leaf endpoint directions as well as every transverse direction. This is more than the normal symbol (SG.11), and uses the full graph instead of the source-transversal columns (SG.9). Smooth zero-order terms, finite coefficient multiplicities, arbitrary compatible connections, and arbitrary compatible full graph metrics are allowed.

**Theorem 11C.1 (the circle obstruction for the full-graph ansatz).** Let \(V=S^1\), with its single codimension-zero leaf. It satisfies the holonomy-invariant normal spin-c hypothesis. Every operator satisfying the preceding ansatz defines an even scalar reduced Kasparov cycle, but its pairing with the original point-transversal class is zero. Consequently no such operator supplies the requested pairing one. This statement includes every finite coefficient multiplicity and every smooth compatible first-order full Clifford expression in the specified representation.

The original source problem is not refuted: an explicit normalized scalar cycle on a single graph column will be given in Section 11C.7. The theorem identifies an unsuccessful general interface, rather than an obstruction to every normalized reduced class.

### 11C.2. Actual graph, reduced norm and coefficient transitions

Write \(S^1=\mathbb R/(2\pi\mathbb Z)\) with probability density \(dt/(2\pi)\). A transversal is a point. Every codimension-zero holonomy germ is the unique germ on a point, so paths with the same endpoints define the same holonomy arrow. Thus the actual holonomy graph and its tangent splitting are

\[
 G=S^1_x\times S^1_y,\qquad r(x,y)=x,\quad s(x,y)=y,
 \qquad TG=\mathbb R\partial_x\oplus\mathbb R\partial_y.
 \tag{GD.1}
\]

The normal bundle has rank zero and its invariant metric, orientation and spin-c data are trivial. There is no transverse line or odd Clifford generator to omit in this example; the desired scalar degree is exactly zero.

For a smooth kernel \(k(x,z)\), left convolution is the integral operator \(K_k\) on \(L^2(S^1)\). Every source-fibre regular representation is that same integral operator. A smooth kernel is Hilbert–Schmidt, and hence compact: approximate its continuous kernel uniformly by finite sums of separated functions, whose operators have finite rank; the error is bounded in operator norm by its Hilbert–Schmidt norm. Conversely rank-one operators with smooth vectors have smooth kernels, and smooth vectors are dense. Therefore

\[
 A=C_r^*(G)=\mathcal K(L^2(S^1)),\qquad
 \|k\|_r=\|K_k\|. \tag{GD.2}
\]

The coefficient transport in the ansatz is a smooth unitary
\(W(x,z;y):E_{(z,y)}\to E_{(x,y)}\), satisfying
\(W(x,z;y)W(z,w;y)=W(x,w;y)\) and \(W(x,x;y)=1\).
Fix \(x_0\). Put \(F_y=E_{(x_0,y)}\), and identify \(E_{(x,y)}\) with \(F_y\) by \(W(x_0,x;y)\). The composition law makes every transported kernel the identity on \(F_y\). Thus this is an explicit smooth coefficient-bundle trivialization along the range variable, not a claim that a twisted full-graph line can be discarded:

\[
 E\cong s^*F,\qquad
 H=L^2(S^1_x)\otimes L^2(S^1_y,F),\qquad
 \pi(k)=K_k\otimes1. \tag{GD.3}
\]

Here \(\pi\) is a nondegenerate faithful reduced scalar representation. Its norm is exactly \(\|K_k\|\): the inequality from above is the tensor norm bound, and testing on a tensor product of a near norm-attaining range vector with a normalized source vector gives the reverse inequality. The source fibre has positive finite coefficient rank; a zero-rank bundle would already have pairing zero.

Evenness of every kernel implies that the grading has the form \(1\otimes\Gamma(y)\). Indeed its commutation relation says \(\Gamma(x,y)k(x,z)=k(x,z)\Gamma(z,y)\); kernels nonzero at any chosen pair give independence of \(x\). Each of \(F^+\) and \(F^-\) is a complex bundle over the circle and is trivial. To verify that prerequisite, trivialize along the interval obtained by cutting the circle. Its endpoint gluing is a unitary matrix \(M\). A unitary matrix has a logarithm \(M=e^{iT}\) by diagonalizing it; a smooth path from 1 to \(M^{-1}\), constant near its endpoints, adjusts the interval trivialization to one that glues smoothly. Apply this separately to the two graded summands. The same argument covers all possible complex line transitions over the circle; a connection or spin holonomy survives as a smooth zero-order term in the periodic frame.

In these global frames a candidate differential expression is

\[
 D=-iA(x,y)\partial_x-iB(x,y)\partial_y+C(x,y),
 \qquad \Gamma D=-D\Gamma. \tag{GD.4}
\]

The displayed zero-order term includes the frame and density connection terms required for symmetry. Its coefficients are smooth on the torus. Clifford ellipticity says that its principal symbol squared is the positive inverse-metric quadratic form times the identity. In particular \(B(x,y)\) is an invertible odd self-adjoint matrix at every point.

### 11C.3. Bounded convolution commutators force the source symbol

The range derivative in \([D,\pi(k)]\) is bounded after integration by parts in \(z\). Explicitly its kernel is
\(-iA(x,y)\partial_x k(x,z)-i\partial_z(k(x,z)A(z,y))\).
The zero-order commutator is bounded as well. The only derivative still acting on the input is

\[
 -i\int_{S^1}
  \bigl(B(x,y)k(x,z)-k(x,z)B(z,y)\bigr)
       \partial_y\psi(z,y)\,{dz\over2\pi}.
 \tag{GD.5}
\]

**Lemma 11C.2 (source-symbol rigidity).** Boundedness of all the commutators in the ansatz forces \(B(x,y)=B(y)\).

**Proof.** If the kernel coefficient in (GD.5) is not identically zero for some smooth \(k\), its integral operator is nonzero on a smooth vector \(v(z,y)\) supported in a small source interval; smooth vectors are dense and the coefficient is continuous. Let its nonzero output have norm \(\delta>0\). Test on \(\psi_n(z,y)=e^{iny}v(z,y)\), whose input norm is fixed. Equation (GD.5) has norm at least \(|n|\delta-O(1)\). Every other term, including the derivative of \(v\), has a bound independent of \(n\), by smoothness and the compact variables. This contradicts a bounded commutator. Thus the kernel coefficient vanishes. Smooth kernels can be nonzero at every prescribed \((x,z)\), which gives \(B(x,y)=B(z,y)\) for all \(x,z,y\). Conversely this equality removes (GD.5), leaving precisely the bounded integral kernels already described. This proves the lemma and verifies the whole smooth convolution core. \(\square\)

This argument determines which Clifford and line transitions are compatible with regular convolution. A line twist that couples the two endpoint variables cannot simply be inserted into (GD.3) while retaining the same scalar convolution action. A full tangent splitting supplies no such transport automatically.

### 11C.4. Exact domain, self-adjointness and scalar compactness

For completeness the needed compact-manifold analytic facts are proved here for this two-dimensional expression. Freeze the principal matrices at a point in a sufficiently small coordinate disc. For the constant-coefficient expression \(D_c\), Fourier transformation and the Clifford identity give
\(\|D_cu\|_2^2\ge c\|\nabla u\|_2^2\) for some \(c>0\). On a small disc the difference of principal coefficients is arbitrarily small in supremum norm. Absorb its gradient contribution, and bound all lower-order terms by a constant times \(\|u\|_2\). A finite smooth partition of unity, with its bounded multiplication commutators, gives

\[
 \|u\|_{H^1(G)}\le C\bigl(\|Du\|_2+\|u\|_2\bigr)
 \quad(u\in C^\infty(G,E)). \tag{GD.6}
\]

The reverse bound is immediate from the smooth bounded first-order coefficients. Hence the closure of the smooth core has domain \(H^1(G,E)\). To identify the adjoint domain, let \(u\in L^2\) have distributional \(Du\in L^2\). Local mollification \(J_\epsilon\) has uniformly bounded commutator \([D,J_\epsilon]\) on \(L^2\): the differentiated convolution kernel is multiplied by a coefficient difference bounded by \(C|x-z|\), and its absolute integral is bounded independently of \(\epsilon\); coefficient derivatives contribute another uniformly bounded convolution kernel. These are the two Schur bounds. Apply (GD.6) to mollified localizations of \(u\). Their \(H^1\) norms are uniformly bounded by \(\|Du\|_2+\|u\|_2\). Weak compactness in \(H^1\) and strong \(L^2\) convergence show \(u\in H^1\). Formal self-adjointness therefore makes the maximal and minimal domains equal: \(D\) is self-adjoint on exactly \(H^1(G,E)\), with smooth sections a core.

The embedding \(H^1\to L^2\) is compact. In the global finite frame, discard Fourier modes \((m,n)\) with \(m^2+n^2>R^2\); the remaining projection has finite rank and its tail from an \(H^1\) unit ball has norm at most \((1+R^2)^{-1/2}\). Consequently the resolvent of \(D\), which has bounded range in \(H^1\) by (GD.6), is an ordinary scalar compact.

Every smooth \(\pi(k)\) preserves the domain, either directly from its integral expression or from Lemma 11C.2 and its bounded commutator. With \(F_D=D(1+D^2)^{-1/2}\), its square defect is compact. The first-order resolvent identity, with \(Q_\pm=(D\mp i\sqrt{s})^{-1}\), gives \([Q_\pm,\pi(k)]=-Q_\pm[D,\pi(k)]Q_\pm\). Averaging the two resolvents yields the norm bound

\[
 \|[D(s+D^2)^{-1},\pi(k)]\|
       \le {\|[D,\pi(k)]\|\over s},\qquad s=1+t^2.
 \tag{GD.7}
\]

Each resolvent commutator is compact. Integrating the averaged commutators against \(2\,dt/\pi\) converges in norm by (GD.7), and gives \([F_D,\pi(k)]\): truncated transform integrals converge strongly, while their commutators converge in norm. Density of smooth kernels gives the scalar Kasparov defects for every element of \(A\). The grading is exactly even and the self-adjointness defect is zero. Thus the promised cycle exists; normalization, rather than analytic existence, is the obstruction.

### 11C.5. Compression to the original point-transversal class

Let \(u(x)=1\) in the probability-density convention, and let \(p=|u\rangle\langle u|\in A\). It is a smooth rank-one projection. It is the original point-transversal K-class: the source-fibre counting construction extends its one-dimensional fibre to \(L^2(G_{y_0})=L^2(S^1)\), and any normalized smooth column gives its rank-one projection. Any two such columns can be joined by unit vectors after a harmless phase adjustment, and their rank-one projections are homotopic. Thus the constant column and a column supported in a small plaque give the same class, with positive rank one. The rank-zero normal Bott generator is \(1\in K^0(\mathrm{point})\); its image is exactly \([p]\).

Since \(p\) preserves \(H^1\) and \([D,p]\) is bounded, the off-diagonal operator
\(R=pD(1-p)+(1-p)Dp\) is bounded self-adjoint. The operator \(D_{\rm diag}=D-R\) is self-adjoint on the same domain: for imaginary \(z\) with \(|\operatorname{Im}z|>\|R\|\), a Neumann inverse of \(1-R(D-z)^{-1}\) gives the resolvent and the adjoint resolvent. The projection \(p\) reduces it. Its compressed operator has exact domain

\[
 D_p=pDp,
 \qquad\operatorname{Dom}D_p=pH^1(G,E)
             \cong H^1(S^1_y,F). \tag{GD.8}
\]

This compression does represent the K-pairing. The difference of the two first-order resolvents of \(D\) and \(D_{\rm diag}\) is compact, with norm at most \(\|R\|/s\) at imaginary parameters \(\pm i\sqrt{s}\). Its averaged transform integral converges in norm, so \(F_D-F_{D_{\rm diag}}\) is compact. Therefore \(pF_Dp\) differs compactly from the normalized transform of \(D_p\). This proves the representative, including domain control, rather than reading an index directly from an unrelated longitudinal operator.

Because \(B\) is independent of \(x\), and \(\partial_x u=0\), compression of (GD.4) gives the actual source operator

\[
 D_p=-iB(y)\partial_y+\overline C(y),\qquad
 \overline C(y)=\int_{S^1}C(x,y)\,{dx\over2\pi}.
 \tag{GD.9}
\]

The invertible odd source Clifford matrix gives equal graded ranks \(r\). In the graded circle frames its positive-to-negative block is
\(D_p^+=-i\beta(y)\partial_y+c(y)\), with smooth invertible \(r\)-by-\(r\) matrix \(\beta\). Multiplication by \(-i\beta\) is invertible on the target, so its Fredholm index equals that of \(\partial_y+i\beta^{-1}c\), as an operator \(H^1\to L^2\).

The derivative \(\partial_y\) has the \(r\)-dimensional constant kernel, range the mean-zero functions, and \(r\)-dimensional constant cokernel. Its inverse on the mean-zero complement is the Fourier multiplier \(1/(in)\), a bounded map \(L^2\to H^1\). The lower-order term is compact as a map \(H^1\to L^2\), by the Fourier embedding proof above. Compact perturbation preserves the Fredholm index: a parametrix makes every operator on the compact-perturbation path Fredholm; its index is locally constant by splitting finite kernel and cokernel from an invertible complement and using a Neumann inverse on that complement. A finite cover of the path gives constancy. Hence

\[
 \langle[p],[H,\pi,F_D]\rangle
     =\operatorname{Index}D_p^+
     =\operatorname{Index}(\partial_y\otimes1_{\mathbb C^r})
     =r-r=0. \tag{GD.10}
\]

The normalized transform has this same index: compact resolvent gives finite-dimensional kernel, a positive spectral gap on its orthogonal complement, and identical kernel and cokernel for its chiral block. This completes Theorem 11C.1. The desired value is one, so neither a change of source connection nor any smooth zero-order term repairs the full-graph ansatz.

### 11C.6. An explicit check and the actual multiplicity interface

For the flat torus choose \(E=\mathbb C^2\), \(\Gamma=\sigma_3\) and

\[
 D_{\rm flat}=\sigma_1(-i\partial_x)+\sigma_2(-i\partial_y),
 \quad D_{\rm flat}^2=-\partial_x^2-\partial_y^2,
 \quad pD_{\rm flat}p=\sigma_2(-i\partial_y). \tag{GD.11}
\]

The Fourier core gives eigenvalues \(\pm\sqrt{m^2+n^2}\) away from \((0,0)\); the zero block has one vector in each grading. Smooth convolution in \(x\) has bounded commutator, and the source derivative commutes with it. The point-transversal corner retains exactly the paired source-circle spectrum: one constant kernel and one constant cokernel. This is an exact model of (GD.10), not a computed approximation to the SU(2) spectrum.

If instead the operator differentiates only the normal directions on the entire graph, it leaves the source-circle multiplicity untouched. At rank-zero normal dimension that operator is zero on \(L^2(S^1_x\times S^1_y)\), so \(\pi(p)(1+D^2)^{-1}=p\otimes1\) is not compact. The orthonormal vectors \(u(x)e^{iny}\) have orthogonal images of norm one. Thus adding the second endpoint has two separate costs: omitting its operator fails localized scalar compactness; adding an ordinary compatible source first-order Dirac produces the zero index proved above. A successful index-one completion cannot be inferred merely from full ellipticity, tangent splitting, or a faithful regular representation.

### 11C.7. Why the full-graph constraints do not settle the reduced construction

For the same circle foliation choose the one source column \(G_{y_0}=S^1\), the actual faithful regular representation of \(A=\mathcal K(L^2(S^1))\) on \(H_+=L^2(S^1)\), and \(H_-=0\). The odd operator \(D=0\), of finite order zero, has phase zero. For every \(a\in A\), the square defect is \(-a\), compact; both other defects vanish. It is an even scalar cycle and its compression to \(p\) is the operator from a one-dimensional positive space to the zero negative space, with index one. This is an actual graph-column realization, not a longitudinal class or an abstract discrete completion.

It is the zero-dimensional **normal** Dirac expression; it is not a full-tangent elliptic Dirac on \(G\). This example proves that Theorem 11C.1 must not be enlarged into a nonexistence theorem for all normalized reduced graph classes. The original problem does not explicitly require integration over all source units with a smooth density, nor full graph ellipticity. If a later formulation excludes codimension zero or expressly specifies another graph completion, that changed scope needs its own argument. No such exclusion appears in the source statement.

The theorem also includes the usual bounded **unbounded-operator** commutator requirement in its ansatz. It does not exclude a different full-graph differential expression with unbounded \([D,\pi(k)]\) for which only the bounded-phase Kasparov defects might nevertheless be proved. Such a proposal requires a separate phase and normalization argument; that weaker interface is not silently covered here.

Section 11A similarly uses source-transversal columns and an explicitly discrete even index-one completion. Its normalized reduced class and all previously proved defects remain intact. The present proof does not identify its tree operator with a smooth differential expression, and does not rule out a different local graph-column construction under additional hypotheses.

### 11C.8. Tangent splitting does not supply a graph spin-c line

The following separate result settles a missing global transition prerequisite for the minimal-spinor full-graph approach.

**Proposition 11C.3.** There is a compact connected oriented manifold \(L\), foliated by its single leaf, whose normal spin-c structure is trivial but whose graph \(G=L\times L\) has no spin-c tangent structure. Thus a splitting of the graph tangent into two leaf endpoint factors and the normal factor, together with an invariant normal spin-c structure, does not supply an ordinary minimal full-tangent spinor bundle.

**Prerequisite: the determinant-line obstruction.** An oriented real bundle carrying a spin-c lift has a complex determinant line \(\ell\) satisfying \(c_1(\ell)\bmod2=w_2\). This follows directly from the lift transitions. Choose local oriented-frame transitions and spin lifts on double overlaps. On triple overlaps their product is \((-1)^{a_{ijk}}\); the mod-two cocycle \(a\) is the obstruction \(w_2\). A spin-c lift adds phases \(z_{ij}\) whose triple product is this same sign. Its determinant transitions are \(z_{ij}^2\). Choose logarithms of the phases on a refinement: doubling the logs gives the integer Chern cocycle, whose parity is \(a\). This proves the needed necessary condition, including its line transition content. It applies after adding any framed trivial real summand because the same frame-transition spin signs remain.

Construct

\[
 L=(\mathbb{CP}^2\times[0,1])/
       ((z,1)\sim(\overline z,0)). \tag{GD.12}
\]

Complex conjugation on a complex two-dimensional chart reverses two imaginary coordinates, so it preserves the real orientation. The mapping torus is a smooth compact connected oriented five-manifold. Its single-leaf holonomy graph is the pair groupoid, by exactly the germ argument in Section 11C.2.

Here are the complete characteristic-class facts needed in this example. Projective space has cells in dimensions 0, 2 and 4, obtained by successively separating the last nonzero homogeneous coordinate; hence its integral and mod-two degree-two cohomology groups are \(\mathbb Z\) and \(\mathbb Z/2\). Let \(h\) be the Chern class of the dual tautological line. Restricted to \(\mathbb{CP}^1\) its equatorial transition winds once, so \(h\) is the generator and its mod-two reduction is nonzero. At a line \(\lambda\subset\mathbb C^3\), the tangent is \(\operatorname{Hom}(\lambda,\mathbb C^3/\lambda)\). This gives the exact bundle sequence
\(0\to\mathbb C\to H^*\otimes\mathbb C^3\to T\mathbb{CP}^2\to0\).
Taking determinants proves \(c_1(T\mathbb{CP}^2)=3h\). To compute the spin obstruction without an additional unitary homotopy theorem, split this exact sequence by a Hermitian metric. It gives \(T\mathbb{CP}^2\oplus\mathbb C\cong(H^*)^{\oplus3}\). The real plane underlying \(H^*\) has spin obstruction \(h\bmod2\): a plane transition of winding one lifts to a spin rotation that changes sign after \(2\pi\). For oriented block sums the triple-overlap spin signs multiply, so their mod-two obstruction cocycles add; a trivial summand adds zero. Hence \(w_2(T\mathbb{CP}^2)=3h\bmod2=h\bmod2\).

Conjugation restricts on \(\mathbb{CP}^1\) to an orientation-reversing sphere map, so it sends \(h\) to \(-h\). If a complex line on \(L\) restricts on one fibre to Chern class \(kh\), its restrictions along the interval have the same Chern class by homotopy. Endpoint gluing then requires \(kh=-kh\), and hence \(k=0\). But restriction of \(TL\) to the fibre is \(T\mathbb{CP}^2\oplus\mathbb R\). A spin-c determinant line would have odd \(k\) by the preceding obstruction calculation. This contradiction proves that \(TL\) has no spin-c structure.

Finally a putative spin-c determinant line for \(T(L\times L)\) restricts on \(L\times\{y_0\}\) to a line lifting \(w_2(TL\oplus\mathbb R^5)=w_2(TL)\). Restricting again to the projective fibre gives the same impossible odd \(k\). Thus \(T(L\times L)\) has no spin-c lift. This proves Proposition 11C.3. \(\square\)

A minimal complex spinor module for the even-dimensional full graph would supply exactly such a lift. Locally compare its Clifford action with the fixed irreducible matrix module in an oriented orthonormal frame. On an overlap a spin lift of the frame change intertwines the two Clifford actions. Every other unitary intertwiner differs by a phase, since the commutant of the full irreducible complex Clifford matrix algebra consists of scalars. Their simultaneous transition law is precisely \((\operatorname{Spin}\times U(1))/\{(-1,-1)\}\), so a global minimal module supplies a spin-c lift and its determinant line. The preceding obstruction therefore applies to that proposal. A larger Clifford module, such as the complex exterior module, is always available without a spin-c lift; this proposition does not exclude it. Such a module's index and scalar convolution transport still require proofs. In particular one cannot replace the prescribed normal inverse line by an unproved full graph determinant line, or cancel the two pulled-back leaf factors simply because they have equal rank. Their tangent bases lie at different endpoints.

### 11C.9. The remaining general construction

The original target still requires a degree-\(q\) scalar class for the reduced holonomy algebra, represented by a genuinely local finite-order graph operator in a specified graph completion, with the complete invariant normal Clifford module and determinant-line transitions, dense convolution-domain preservation, every localized scalar defect, and the original local Bott pairing one. Section 11A proves those analytic and normalization interfaces on its specific SU(2)/F2 completion while retaining the discrete term explicitly.

These results provide two rigorous constraints on a general construction. A full ordinary graph tangent Dirac cannot be adopted as a universal construction: Theorem 11C.1 excludes the natural smooth full-graph regular ansatz already for the circle. A normal spin-c structure alone cannot be turned into a minimal full-tangent spinor bundle by the graph splitting: Proposition 11C.3 exhibits its exact determinant-line obstruction. A successful general proof must state its source-transversal or other graph completion and construct its local multiplicity-control operator there; it cannot infer index one from ellipticity or spin-c notation. No proof of that remaining general construction, and no counterexample excluding all admissible normalized graph-column classes, is claimed here.

![Graph-circle compression and the determinant-line obstruction](../figures/kt-graph-dirac-constraints.png)

**Figure 11C.1.** Equations (GD.1)–(GD.3), (GD.8)–(GD.12). The square represents the periodic parameter domain for the actual torus graph; convolution integrates the range endpoint while the source endpoint survives. The displayed corner spectrum is the exact flat expression \(\sigma_2 n\); its zero block has one positive kernel and one negative cokernel, giving index zero. The right panel depicts the exact integral Chern relation \(k=-k\) for a line on the conjugation mapping torus, and the incompatible odd restriction required by its tangent spin-c lift. No sampled spectrum or tree truncation is used in the proof. The question is from Connes, *A survey of foliations and operator algebras* ([author’s text](https://alainconnes.org/wp-content/uploads/foliationsfine.pdf)). [Reproducible source](../reproduction/graph-dirac/draw_graph_dirac.py) · [Editable SVG](../figures/kt-graph-dirac-constraints.svg).

## 11D. A normalized graph-column cycle from a countable tree

The construction below fills the auxiliary-cycle interface of Section 11A.9 first for locally finite trees, then for every countable connected tree with finite vertex stabilizers. It allows an arbitrary compatible normal Dirac connection; the full normal spin-c structure and metric remain invariant. The resulting reduced scalar class has the original local transverse Bott pairing +1. The construction is a graph-column operator with a transverse differential Clifford symbol and a discrete tree term. It does not identify that term with a local finite-order differential expression, or derive a tree action from the general foliation hypothesis. This theorem does not construct the required normalized reduced graph representative for arbitrary eligible holonomy.

The SU(2)/F2 construction of Section 11A, the full-graph circle obstruction of Section 11C and the finite multiplication-potential obstruction remain in force. In particular the present tree term moves tree labels; it is not a multiplication potential on the leaf coordinate.

**Theorem 11D.1 (the normalized countable-tree graph-column cycle).** Suppose the compact full metric spin-c normal data and coherent germ-effective action of Section 11D.1 are supplied, together with a countable connected auxiliary tree with finite vertex stabilizers and the regular compact-base cover of Section 11D.5. The full inverse normal coefficient, proper weighted tree factor and exact compressed operator below define a reduced scalar degree-q graph-column cycle with the original local transverse Bott pairing +1. The chosen compatible normal connection need not be invariant. The arbitrary-groupoid auxiliary and a local finite-order smooth differential replacement of the discrete tree term remain open.

**Proof.** Sections 11D.1–11D.8 prove the complete locally finite construction and its normalization. Section 11D.9 supplies proper weights and proves every extension needed for a countable tree.

### 11D.1. Exact hypotheses and normal coefficients

Let Gamma be a countable discrete group acting by graph automorphisms on a connected locally finite simplicial tree T. Fix a vertex o. Assume each vertex stabilizer is finite. Edge inversions are allowed. Let N be a compact connected smooth manifold without boundary, of dimension q, equipped with a specified metric and full spin-c orientation on TN. Suppose Gamma acts smoothly by isometries and lifts coherently to that full structure. In the suspension application assume its action is effective on germs: no nonidentity group element is the identity on an open normal chart.

Use the **source-calibrated full inverse coefficient**, rather than an independently chosen positive spinor bundle. At an original principal transition zs, with s a spin lift and z the central phase, the transferred inverse is the module of Theorem 7.17:

\[
 \mathcal D_N\cong\Pi^{k_q}C_q,\qquad
 k_q=q(q-1)/2,\qquad
 c_{\rm inv}(e^j)=(-1)^{k_q+1}L_{t_j},\qquad
 zs\longmapsto z^{-1}\rho_t(s).                 \tag{PT.1}
\]

These are full right-C_q modules, with their actual grading and ordered generator list. If the prescribed original orientation is tensored by an even line H, the inverse is tensored by \(H^*\). This includes its full transition functions and any two-torsion information; the determinant line alone does not determine the coefficient. In odd q the last right C1 remains after contracting consecutive even Clifford pairs. In q=0 the canonical even orientation gives the even scalar line and zero normal operator; a different stable graded line is not declared to be the same orientation.

Choose a smooth compatible Clifford connection on this inverse module and the associated formally symmetric normal Dirac D_N on

\[
 H_N=L^2(N,\mathcal D_N),\qquad
 \operatorname{Dom}D_N=H^1(N,\mathcal D_N).       \tag{PT.2}
\]

All scalar compactness statements below are genuine finite-Clifford scalar statements: a finite-dimensional right C_q coefficient is retained, and the final parity reduction is the supplied ordered finite Morita contraction. No infinite Hilbert-module compactness is substituted for scalar compactness.

The compact normal Dirac has self-adjoint closure, smooth core and compact resolvent. The local analytic proof is the compact-manifold version of Section 11C.4: freeze the Clifford matrices in a sufficiently small chart, use the Fourier lower bound for their positive squared symbol, absorb the small principal-coefficient variation, and patch with a finite partition. This gives the two-sided equivalence of the graph norm and H1. For a distributional maximal-domain vector, smooth local mollifiers have uniformly bounded first-order commutators: the differentiated kernel is multiplied by a coefficient difference O(distance), with uniformly bounded Schur integral. The estimate therefore puts every maximal-domain vector in H1. Minimal and maximal domains agree, so formal symmetry gives self-adjointness. Local Fourier cutoffs, followed by the finite partition, show compact H1 inclusion into L2. Hence the resolvent is scalar compact. The same argument with finite right coefficients proves (PT.2). For q=0 these claims are the finite-dimensional zero-operator claims.

The normal calibration is the accepted programme interface of Theorem 7.17, the two native inverse identities of Lemma 7.20, and the bounded-phase local symbol/metric comparison in Lemmas 7.14–7.17. In particular, for the old-first positive local Bott generator b_U on a positively oriented normal q-ball U, extended by zero to N,

\[
 \langle b_U,[D_N]\rangle=1.                   \tag{PT.3}
\]

This is the same normal interface as SG.2–SG.3, now retaining the entire q-dimensional coefficient. The new assertion below concerns how that supplied normal inverse is completed and represented on the reduced graph algebra. The normal inverse and its calibration are used with the hypotheses of the cited results.

Let U_h be the unitary on H_N induced by the metric density and the coherent inverse lift in (PT.1). Its full action satisfies covariance and the group law exactly. An invariant connection is **not** assumed. Conjugating D_N by U_h gives another smooth first-order expression with the same principal symbol: the isometry transports the covector and the full inverse lift intertwines Clifford multiplication. Thus

\[
 B_h=[D_N,U_h]\in\mathcal B(H_N),\qquad
 U_h H^1=H^1.                                   \tag{PT.4}
\]

To see the bound explicitly, express both operators in finitely many coordinate and coefficient frames. Equality of their principal coefficients cancels every derivative of the input. The remaining expression is a smooth finite-matrix multiplication operator composed with U_h. Its coefficient is bounded on compact N. Smooth pullback and its smooth inverse preserve H1. Each h has its own finite bound; neither a uniform bound over Gamma nor an invariant averaging of connections is needed.

### 11D.2. The complete even tree cycle

Every locally finite connected tree is countable: its radius-R vertex ball is finite by induction on R and the whole vertex set is their countable union. Every edge has a unique endpoint farther from o. For v other than o write par(v) for its neighbour closer to o, put c(v)={v,par(v)}, and let n(v)=d(o,v). The map c is a bijection from V(T) minus {o} to the set E(T) of unoriented edges. Define

\[
 H_T^+=\ell^2 V(T),\quad H_T^-=\ell^2E(T),\quad
 Q\delta_o=0,\quad Q\delta_v=\delta_{c(v)}\ (v\ne o),
 \quad M\delta_v=n(v)\delta_v.                   \tag{PT.5}
\]

Thus \(Q^*Q=1-P_o\), \(QQ^*=1\), and Index Q=+1. On the graded sum, with positive vertices and negative edges, use

\[
 D_T=\begin{pmatrix}0&M Q^*\\QM&0\end{pmatrix},
 \qquad \Gamma_T=\begin{pmatrix}1&0\\0&-1\end{pmatrix}.
                                                        \tag{PT.6}
\]

The root is its one-dimensional positive kernel. Each remaining vertex/parent-edge pair is an orthogonal two-dimensional block with matrix [[0,n],[n,0]]. Their orthogonal direct sum, on the square-summability domain of their images, is self-adjoint. Finite block sums form a core. The finite radius balls make every bounded spectral window finite-dimensional; the square-resolvent tail has norm at most 1/(1+(R+1)^2). Hence D_T has compact resolvent, including when T is finite.

Let L_h^V and L_h^E be the permutation actions on vertices and unoriented edges; they preserve the grading. Write ell(h)=d(o,ho). The lower unbounded commutator sends delta_v to

\[
 n(hv)\delta_{c(hv)}-n(v)\delta_{h c(v)},          \tag{PT.7}
\]

with a root term interpreted as zero. The two edge labels agree unless hv belongs to the finite segment [o,ho]. Indeed the parent of hv with respect to root ho is h par(v). Outside the segment, the first steps toward o and ho coincide in a tree. There are at most ell(h)+1 exceptional input vertices. For each, n(v)=d(ho,hv) and n(hv) are at most ell(h), so its column norm is at most 2ell(h). Off the segment, the coefficient difference is at most ell(h) by the triangle inequality; the edge output map is injective. The finite-column Hilbert–Schmidt estimate and this injectivity give

\[
 \|QM L_h^V-L_h^E QM\|
 \le\ell(h)+2\ell(h)\sqrt{\ell(h)+1}.             \tag{PT.8}
\]

The corresponding upper block is the adjoint of the lower block for h inverse, multiplied by unitaries, and ell(h inverse)=ell(h). This proves the bounded commutator [D_T,L_h] on the full graded space. It also proves domain preservation: initially the identity holds on finite vertex/edge sums; approximating in the graph norm and using boundedness gives L_h Dom D_T subset Dom D_T, and h inverse gives equality. The phase

\[
 F_T=\begin{pmatrix}0&Q^*\\Q&0\end{pmatrix}       \tag{PT.9}
\]

has finite-rank return commutators, because only the finite segment is exceptional. Its square defect is -P_o in positive grading, and its scalar even index is +1. The normalized transform of D_T differs from F_T by compact block coefficients n/sqrt(1+n^2)-1 tending to zero outside finite radius balls. These claims use the whole tree, not any finite truncation.

### 11D.3. Exact reduced descent with finite stabilizers

The vertex stabilizer Gamma_v is finite by hypothesis. The unoriented edge stabilizer is finite as well, including inversions: its endpoint permutation image has at most two elements, and its kernel is contained in the finite stabilizer of either endpoint. Decompose each permutation space into its vertex or edge orbits. An orbit with representative w and stabilizer H is Gamma/H, with left cosets and finite H. The explicit isometric equivariant embedding is

\[
 J_H:\ell^2(\Gamma/H)\longrightarrow\ell^2\Gamma,
 \qquad
 J_H\delta_{gH}=|H|^{-1/2}\sum_{a\in H}\delta_{ga}.
                                                        \tag{PT.10}
\]

Distinct left cosets have disjoint supports, so the norm is exactly one and the map is well-defined. Left translation intertwines literally. The direct sum over the countably many vertex and edge orbits therefore embeds H_T into a countable sum of regular representations. This is a subrepresentation embedding, not an assertion that every orbit is itself regular. Its range projection is a finite-subgroup average in each orbit copy. Infinite stabilizers would invalidate (PT.10); they are not admitted by this theorem.

Let B=C(N) crossed-product-r Gamma. On H_B=H_T graded-tensor H_N put

\[
 \Pi_B(f)=1\otimes M_f,\quad
 \Pi_B(u_h)=L_h\otimes U_h,\quad
 D_B=D_T\otimes1+\Gamma_T\otimes D_N.             \tag{PT.11}
\]

Covariance follows from the actual normal pullback formula. The isometry J tensor 1 intertwines every displayed coefficient with a sum of the covariant regular copies on ell2 Gamma tensor H_N. On a regular copy the absorption unitary W has (W psi)_g=U_g inverse psi_g. Substitution gives

\[
 W(\lambda_h\otimes U_h)W^{-1}=\lambda_h\otimes1,
 \quad
 (W(1\otimes M_f)W^{-1}\psi)_g
       =M_{f\circ h_g}\psi_g,                    \tag{PT.12}
\]

where h_g is the normal action of g. In a local coefficient frame this is the direct integral, with the finite coefficient multiplicity, of the defining regular representations at points of N. For a finite crossed coefficient a its norm is at most ||a||_r. Restriction to the J range retains that inequality, and norm density gives a nondegenerate reduced representation of B on H_B. The unit acts as identity. Faithfulness is not needed and is not asserted for arbitrary finite-stabilizer actions.

The central phase has not disappeared: it occurs in U_g and its inverse in the unitary W. It cancels in scalar multiplication covariance because multiplication acts as a scalar on each finite coefficient fibre, not because the line is trivial. All right C_q actions, gradings and densities are preserved by J, U and W.

The tree factor is even, so it changes no transverse degree. Anticommutation gives

\[
 D_B^2=D_T^2\otimes1+1\otimes D_N^2.             \tag{PT.13}
\]

Take a compact-resolvent eigenbasis of D_N and the tree blocks. For a nonroot block of depth n and a normal eigenvalue nu, the physical matrix is [[nu,n],[n,-nu]], with eigenvalues plus/minus sqrt(nu^2+n^2). The root block is D_N. Their orthogonal direct sum defines a self-adjoint operator on the square-summability domain of its images. Finite tree/normal spectral sums form a core. Only finitely many tree levels and normal eigenvalues occur below any fixed squared spectral bound, so its ordinary scalar resolvent is compact. The argument is valid on the full finite right-C_q model and after its ordered parity reduction.

For f smooth and h fixed,

\[
 [D_B,\Pi_B(f)]=\Gamma_T\otimes[D_N,M_f],
 \quad
 [D_B,\Pi_B(u_h)]
   =[D_T,L_h]\otimes U_h+\Gamma_T L_h\otimes B_h.
                                                        \tag{PT.14}
\]

Equations (PT.4) and (PT.8) prove all these bounds and domain preservation. Finite group support and smooth normal coefficients give the dense coefficient core. No invariance of D_N or K-amenability of Gamma is used.

Let F_B=D_B(1+D_B^2)^(-1/2). Its square defect is compact by (PT.13). For any core a with C_a=[D_B,Pi_B(a)] bounded, set s=1+t^2 and R_t=(s+D_B^2)^(-1). The bounded first-order resolvent identity gives

\[
 [D_BR_t,\Pi_B(a)]
   =sR_t C_aR_t-D_BR_t C_aD_BR_t,
 \qquad
 \|[D_BR_t,\Pi_B(a)]\|
       \le\frac{5\|C_a\|}{4(1+t^2)}.             \tag{PT.15}
\]

To justify the formula without a domain assumption on C_a, use Q_plus/minus=(D_B minus/plus i sqrt(s)) inverse, whose commutators are -Q C_a Q, and average the two identities. Every term in (PT.15) is compact. Their integral with measure 2dt/pi converges in compact norm. The uncommuted transform integral converges strongly, which identifies this norm limit with [F_B,Pi_B(a)]. Density gives every scalar Kasparov defect for all B.

### 11D.4. Full restriction class and index +1

Restriction of (PT.11) to C(N) is the product of the external even tree class and the full normal class. Here are the operator checks. The first module is H_T graded-tensor C(N), with phase F_T tensor 1. Creation maps with a finite tree basis vector times a continuous coefficient function form a dense set. At fixed nonroot depth n, the normal spectral matrix of the total transform is

\[
 (1+\nu^2+n^2)^{-1/2}
       \begin{pmatrix}\nu&n\\n&-\nu\end{pmatrix}. \tag{PT.16}
\]

The positive column differs from the normal transform column by entries tending to zero as |nu| tends to infinity. The negative column differs from its negative, with precisely the graded creation sign. The root column is exactly the normal transform. Compact resolvent of D_N makes these fixed-column differences compact; continuous coefficient functions follow from the compact commutators. Finite sums and bounded creation norms extend to all creation vectors, and taking adjoints gives the second connection condition. This is the same graded right-C_q creation calculation as SG.T11, with the tree first and the complete normal coefficient second; no odd blocks are permuted.

The lifted first phase anticommutes with Gamma_T and satisfies F_T D_T=D_T F_T=|D_T|. Functional calculus of the commuting squares yields the exact product positivity

\[
 \{F_T\otimes1,F_B\}
   =\frac{2|D_T|\otimes1}
       {\sqrt{1+D_T^2\otimes1+1\otimes D_N^2}}\ge0. \tag{PT.17}
\]

The complete product criterion of Definition 0.3/Theorem 0.6 therefore identifies the restriction class. The scalar even tree class is its Fredholm index +1 by Theorem 0.14, so its external coefficient extension is the identity of C(N). Consequently

\[
 [\Pi_B,F_B]|_{C(N)}=[D_N],\qquad
 \langle b_U,[\Pi_B,F_B]|_{C(N)}\rangle=1.        \tag{PT.18}
\]

Neither a root count alone nor a finite-dimensional normal index is being used to infer this product. The connections and positivity prove the full restriction identity.

### 11D.5. Actual compact-base suspension and graph-column induction

Let X be a connected regular Gamma-cover of a closed connected smooth base M, and let

\[
 V=(X\times N)/\Gamma
                                                        \tag{PT.19}
\]

with the diagonal action and horizontal foliation. Proper freeness of the deck action gives a smooth compact bundle over M with fibre N. Germ effectivity identifies the actual transversal holonomy groupoid with N crossed with Gamma, even if individual normal points have stabilizers. A kernel deck loop is the identity normal germ; every nonidentity return label is a different germ by hypothesis. Thus this is the actual holonomy graph, not the path groupoid with an unremoved kernel.

Choose finitely many buffered flat covering charts U_i in M, lifts z_i and smooth real chi_i supported inside them with sum chi_i squared=1. Their overlap deck transitions h_ij are locally constant. On the compact support of any chi_i chi_j only finitely many h_ij values occur, by compactness and discreteness. No finite-component or good-cover premise is needed. The flat right-B bundle with left transition u_h gives

\[
 E=L^2(M,\mathcal L_B),\qquad
 A=C_r^*(G(V,F))=\mathcal K_B(E),\qquad
 E=P(K\otimes B),\quad
 A=P(\mathcal K(K)\otimes B)P.                  \tag{PT.20}
\]

Here K is the finite chart sum of L2 plaque spaces and P_ij is chi_i chi_j u_hij with the actual overlap coordinate/density identification. The exact norm proof is SG.6–SG.9b: separated smooth plaque kernels approximate every compact holonomy chart kernel in the two Schur integral norms; tensoring E with the normal-point regular B representation is exactly the source-fibre convolution representation, including densities. Every unit is joined along its leaf to the base fibre, and right arrow translation identifies the source-fibre representations. Thus the supremum over N is the defining reduced norm on the entire graph. Rank-one smooth module kernels and compact chart kernels have the same norm closure. The normalized bump gives fullness. These arguments use neither the free-group tree nor freeness of the normal action; germ effectivity is what removes extraneous return labels.

Set H_G=E tensor_B H_B=P(K tensor H_B), and let A act by the represented convolution from (PT.20). This is nondegenerate and reduced because Pi_B is reduced. Its Hilbert space is an explicitly specified completion of graph columns: the isometry (PT.10), followed by (PT.12) in each orbit, embeds it into a countable sum of regular holonomy-column completions with the full normal coefficient. The finite-stabilizer averaging conditions specify its closed subspace. It is not asserted to be the unrestricted ordinary L2 space of all graph endpoints.

On the free scalar space put D_0=1_K tensor D_B. Each P entry has finite return support. Equations (PT.4), (PT.8) and (PT.14) imply P Dom D_0 subset Dom D_0 and [D_0,P] bounded. The normal derivative of chi_i is zero. Unlike SG.10, the normal part of this commutator need not vanish; its bounded B_h terms are retained. Remove the bounded self-adjoint off-diagonal operator

\[
 R=PD_0(1-P)+(1-P)D_0P,
 \quad D_{\rm diag}=D_0-R,
 \quad D_G=PD_0P|_{P\operatorname{Dom}D_0}.      \tag{PT.21}
\]

The two sufficiently large imaginary Neumann resolvents prove self-adjointness of D_diag on the unchanged domain. Since P reduces it, D_G is self-adjoint on exactly P Dom D_0. Finite tree and normal spectral sums give a free core; at each fixed finite tree depth their normal components can be approximated in the normal graph norm by smooth sections, using PT.2. Smooth compact chart profiles then approximate the free L2 families in the graph norm. Projection is continuous in that norm because [D_0,P] is bounded. Thus projected finite-chart, finite-tree, smooth-normal sections form a core; no additional eigenvector smoothness assertion is required.

All overlap differences in the normal expression are bounded zeroth-order terms (PT.4). Therefore the principal **normal** Clifford symbol is the full transported symbol from (PT.1), tensored with Gamma_T; its square is |xi| squared. Every central inverse phase and grading survives P. The extra D_T term is a weighted discrete tree operator. No full graph tangent Clifford module is inferred from the normal bundle.

### 11D.6. Localized scalar compactness and the original plaque class

The global D_0 resolvent has infinite plaque multiplicity. Use instead the concrete spatial algebra J=K(K) tensor B(H_B). Its multipliers include every finite sum of bounded plaque operators tensor bounded H_B operators. P, R and [D_0,P] are such multipliers. If j is in J then j(D_0-z) inverse is scalar compact: for a separated tensor this is a compact plaque factor times a compact normal/tree resolvent factor, and norm density handles all j. The resolvent identity gives

\[
 j(D_{\rm diag}-z)^{-1}
 =j(D_0-z)^{-1}+j(D_0-z)^{-1}R(D_{\rm diag}-z)^{-1},
                                                        \tag{PT.22}
\]

so the localized diagonal resolvents are compact on either side. This is the finite-factor argument; tensoring a compact normal resolvent with the identity on an infinite return space would not prove it.

On the dense algebraic core a=P a_0 P, with a_0 a finite sum of smooth plaque rank-one kernels tensor smooth finite crossed coefficients, the actual convolution action preserves P Dom D_0 and

\[
 [D_G,\Pi_A(a)]=P[D_0,a]P,
 \qquad [D_0,a]\in J\text{ bounded}.             \tag{PT.23}
\]

Expand [D_0,Pa_0P]. The outer P commutators are bounded multipliers of J, while a_0 and [D_0,a_0] lie in J by (PT.14). This gives both the domain assertion and the J extension. For F_G=D_G(1+D_G^2)^(-1/2), the localized square defect a(1+D_G^2) inverse is compact by (PT.22). Apply the two first-order resolvent argument of (PT.15) with C_a having its J extension. Both products in its integrand are compact by (PT.22), and the same integrable 5||C_a||/[4(1+t^2)] bound proves [F_G,a] compact. Norm density extends both defects to every element of the whole reduced graph algebra. The self-adjointness defect is zero.

For each smooth finite-chart module vector eta the rectangular graph error K_eta=D_G T_eta-T_eta D_B is bounded by the same finite transition calculation. Its mixed first-order resolvent error has norm at most ||K_eta||/|Im z| squared and is compact because the source D_B resolvent is compact. Averaging and integrating gives the compact full bounded-phase connection error and its adjoint. Dense module vectors extend these errors by bounded creation norms; positivity for the zero Morita operator is zero. Hence

\[
 z_G=[{}_A E_B,0]\otimes_B[\Pi_B,F_B]
       \in KK^q(A,\mathbb C).                   \tag{PT.24}
\]

Choose a unit-norm smooth scalar bump in a single base chart and let xi be its section times 1_B. Then p=theta_xi,xi and i(b)=theta_xi b,xi identify B with pAp. The exact correspondence unitary is

\[
 E^*\longrightarrow pA,\qquad
 \overline\eta\longmapsto\theta_{\xi,\eta}.      \tag{PT.25}
\]

Its inner product is \(\theta_{\eta,\zeta}\) because \(\langle\xi,\xi\rangle_B=1\); its right A action and left B action agree by the rank-one identities; every pa equals \(\theta_{\xi,a^*\xi}\), so it is onto. The normalized bump also identifies the actual transverse counting correspondence with this module on finite return columns; both the counting norm and plaque bump integral are exactly one. Thus \(i_*b_U\) is the original local transverse class, not a different conveniently chosen K-class. Equations (PT.18), (PT.24) and (PT.25) prove

\[
 i^*z_G=[\Pi_B,F_B],\qquad
 (i_*b_U)\otimes_Az_G=1.                        \tag{PT.26}
\]

This completes the reduced scalar graph-column construction for the stated proper-tree family, including every coefficient transition, domain, localized defect and normalization interface.

### 11D.7. A strict extension: a compact suspension with finite tree stabilizers

Take Gamma=F2 times C2, with C2={1,c}. Let T be the degree-four Cayley tree of F2; Gamma acts through its F2 factor. Every vertex and edge stabilizer is exactly C2. Use the already proved faithful free pair rho:F2 into SU(2) of SG.1, and let

\[
 N=S^1\times SU(2),\qquad
 (g,c^\epsilon)(t,n)=(t+\epsilon\pi,\rho(g)n).
                                                        \tag{PT.27}
\]

This is a faithful free smooth compact action: a nontrivial F2 translation has no fixed point in SU(2), and the remaining nonidentity C2 translation has no fixed point in S1. The product left-invariant frame gives a preserved orientation and metric on TN of rank four. Choose the coherent normal spin-c lift with central character chi(g,c^epsilon)=(-1)^epsilon. The spin frame part is trivial in this global frame. Its square determinant character is trivial, while its full inverse phase remains chi inverse=chi. Forgetting that phase would change the specified orientation transport even though its determinant character is one.

For q=4, k_q=6, so the full inverse is C4 with its usual grading and **negative** physical generators -L_tj. This is the exact PT.1 sign; the positive rank-three Pauli sign of the old SU(2) example is not copied into rank four. The tree is an even completion and leaves this degree and sign intact. The embedding of a vertex/edge orbit is explicitly

\[
 J\delta_g=(\delta_{(g,1)}+\delta_{(g,c)})/\sqrt2.
                                                        \tag{PT.28}
\]

The surface group of a closed genus-three surface surjects onto Gamma by sending its three a-generators to (a,1),(b,1),(1,c), and all b-generators to identity; every surface commutator therefore maps to identity. The associated connected regular cover X supplies (PT.19), and V is a compact connected six-dimensional foliated manifold with leaf dimension two and normal dimension four. PT.26 gives its source-oriented degree-four reduced scalar graph class with local Bott pairing +1. It uses proper finite stabilizers rather than three full regular tree orbits. Its discrete auxiliary and graph-column subspace are explicit.

This example uses the all-length free-pair proof of Lemma 11.9, the elementary surface-cover construction and the full normal inverse and calibration of Section 11A.

### 11D.8. Exact remaining scope

The two newly filled interfaces are: (1) an index-one reduced auxiliary for every locally finite tree action with finite vertex stabilizers, using literal coset-average embeddings rather than an assumption of K-amenability; and (2) normal Dirac return commutators and graph induction for arbitrary smooth compatible normal connections, retaining all bounded connection changes and inverse central phases.

![Finite-stabilizer embedding, whole inverse phase and original pairing](../figures/kt-proper-tree-normal-inverse.png)

**Figure 11D.1.** PT.1, PT.5–PT.18 and PT.25–PT.28. The tree is the exact radius-two window of the infinite degree-four F2 Cayley tree. Its vertices and edges have C2 stabilizer in the strict extension Gamma=F2 times C2. Each nonroot positive vertex maps to its orange negative parent edge; the unmatched root keeps the full normal class. The regular coset-average vector has precisely two orthogonal terms of coefficient 1/sqrt(2). The plotted curves are the exact product-block formula for a continuous normal eigenvalue parameter, not a computation of the S1 times SU(2) Dirac spectrum. Rank four has k=6 and negative physical inverse generators; its central phase is retained even though its determinant character is trivial. PT.25 is the actual normalized counting/plaque correspondence. [Reproducible source](../reproduction/countable-tree/draw_proper_tree.py) and [editable SVG](../figures/kt-proper-tree-normal-inverse.svg).

The general normal spin-c hypothesis does not imply the proper-tree hypothesis. For an arbitrary holonomy groupoid, an auxiliary controlling all regular return multiplicities with scalar index +1 has still not been constructed here. Nor has the discrete D_T been replaced by a local finite-order differential operator on the smooth graph. The existing full-graph circle and finite-potential obstructions remain exact constraints on that latter step. The conditional maximal-descent obstruction in Section 11A.10 still does not refute all normalized reduced classes. No universal K-amenability, reduced descent of the compact normal implementing representation, or full graph spin-c structure is assumed.

The historical question and original local transverse class are from Alain Connes, *A survey of foliations and operator algebras* ([author’s text](https://alainconnes.org/wp-content/uploads/foliationsfine.pdf)). The full normal inverse and local calibration are supplied by Section 11A, Theorem 7.17 and Lemma 7.20, with their stated hypotheses. The exposition, proofs, exercises and diagram expression use [CC0 1.0](https://creativecommons.org/publicdomain/zero/1.0/). Actual bundled font glyphs and external software retain their separate component terms.

### 11D.9. Proper weights remove local finiteness

The locally finite proof of Section 11D.2–11D.3 remains valid with its exact distance estimate. The following strengthening shows that the auxiliary tree need only be **countable**, connected and have finite vertex stabilizers. Local finiteness, a cocompact tree action, and finite generation of Gamma are unnecessary for this auxiliary. The normal and graph construction of Section 11D.1 and 11D.4–11D.6 then applies with proper weights.

#### 11D.9.1. A proper length on an arbitrary countable group

Let Gamma be countable. Enumerate a symmetric set of generators containing every group element, with identity omitted. If an inverse pair occurs twice, identify that pair. Give its i-th pair positive integer cost at least i. Define ell(g) to be the minimum total cost of a word representing g, and ell(1)=0. Every element has such a word, so its value is finite; a nonempty set of nonnegative integers has a minimum. Word concatenation and reversal give

\[
 \ell(gh)\le\ell(g)+\ell(h),\qquad
 \ell(g^{-1})=\ell(g),\qquad
 \ell(g)>0\quad(g\ne1).                          \tag{CT.1}
\]

For every R only finitely many generators have cost at most R and every word of cost at most R has at most R letters. Thus only finitely many words and group elements have ell at most R. The same construction for finite Gamma is finite, and for the trivial group use ell=0. This proves the needed proper group length directly; no amenability or word-metric theorem is used.

#### 11D.9.2. A proper weight on all tree vertices

Let T be a countable connected simplicial tree, not necessarily locally finite. Gamma acts by automorphisms with finite vertex stabilizers; inversions are again allowed. Select one vertex v_j in each orbit, j=1,2,..., with a finite list if there are finitely many orbits. Put

\[
 w_0(gv_j)=j+\min\{\ell(a):av_j=gv_j\}.
                                                        \tag{CT.2}
\]

This is independent of the representing g, since its minimizing set depends only on the vertex. The minimum exists by CT.1. Its sublevel set w_0 at most R is contained in

\[
 \bigcup_{j\le R}\{a v_j:\ell(a)\le R-j\},       \tag{CT.3}
\]

a finite union of finite sets. This proves properness. If a realizes the minimum for v, then ha sends the same orbit representative to hv and has cost at most ell(h)+ell(a). Applying the same argument to h inverse gives

\[
 |w_0(hv)-w_0(v)|\le\ell(h).                     \tag{CT.4}
\]

Fix the root o and set w(o)=0, w(v)=w_0(v) for v other than o. All nonroot weights are positive, their bounded sublevel sets are finite, and modifying one value changes no properness statement. Outside the root and its h inverse translate, (CT.4) remains valid for w. Those two vertices already belong to the finite parent-map exceptional set below.

#### 11D.9.3. Exact weighted operator and return bound

Keep the **whole** root-parent bijection Q of PT.5; it is valid on every connected tree, regardless of valence. Replace M by M_w delta_v=w(v)delta_v. Define D_w with lower QM_w and upper \(M_wQ^*\). The root is its only kernel, in positive grading. Its remaining blocks are [[0,w(v)],[w(v),0]]. Their square-summability domains give the self-adjoint direct sum with a finite-block core. CT.3 makes every bounded spectral window finite-dimensional. The tail of the square resolvent after the w at most R cutoff is bounded by 1/(1+R^2), so the resolvent is scalar compact. The even phase still has Index Q=+1.

For fixed h the parent labels agree unless hv is on the finite simplicial geodesic segment [o,ho]. Local finiteness is not used: between two vertices of a connected simplicial tree the unique geodesic has finitely many edges. Let d_h be that segment length and let

\[
 E_h=\{v:hv\in[o,ho]\},\qquad
 C_h=\max_{v\in E_h}\bigl(w(hv)+w(v)\bigr).       \tag{CT.5}
\]

The set has exactly d_h+1 vertices, so C_h is finite. Off it, neither input nor output is the root, the edge-output map is injective, and (CT.4) bounds the common edge coefficient by ell(h). On it the commutator column norm is at most C_h. Therefore the literal finite-column and weighted-partial-permutation estimate is

\[
 \|QM_w L_h^V-L_h^E QM_w\|
       \le\ell(h)+C_h\sqrt{d_h+1}.               \tag{CT.6}
\]

Adjointing the lower equation for h inverse gives the upper bound. The graph-core approximation in Section 11D.2 again proves invariance of the full closed domain. The unweighted phase commutator has only these finitely many exceptional columns, hence is finite rank. The normalized transform differs compactly from that phase because w(v)/sqrt(1+w(v)^2) tends to one off finite weight sublevel sets.

Every edge stabilizer is finite by the endpoint kernel argument. The explicit coset-average isometries of PT.10 therefore embed the entire vertex and edge representations into countably many regular copies. No local finiteness or weight enters this reduced-norm proof. Thus this auxiliary has all five required properties: an even graded reduced group representation, self-adjoint compact-resolvent generator, bounded return commutators, and scalar index +1. The proper weight modifies the generator, not its phase or normalization.

#### 11D.9.4. The full graph theorem and what it does not imply

Replace D_T in PT.11 by D_w. Its square with the full normal factor is D_w squared plus D_N squared. Compactness follows from their two finite bounded spectral windows. PT.14 uses (CT.6) for each fixed return label. The total scalar commutator proof, noninvariant connection terms, projection-domain argument and J-localized resolvent proof are unchanged. The fixed-tree-column product calculation PT.16 now uses w(v) instead of n(v), and PT.17 uses |D_w|. Its positive root and the complete creation/positivity checks therefore give the full restriction [D_N] and original plaque pairing +1, in the same right-C_q order and with the same inverse z^{-1} spin phases.

Consequently the complete PT.1–PT.26 suspension conclusion is valid whenever Gamma admits an action on a **countable connected tree with finite vertex stabilizers**, with no local finiteness requirement. The operator remains a transverse Dirac plus a discrete proper tree generator. This condition is not inferred from the normal spin-c structure, and the discrete term remains unconverted to a smooth finite-order graph differential expression. The historical arbitrary-groupoid target remains open.

#### 11D.9.5. An exact infinite-valence example and why distance fails

Start with the F2 Cayley tree, and at each base vertex g attach one leaf (g,j) for every integer j at least one. Call this larger tree T_infinity. It is countable, connected and has infinite valence at every base vertex. Gamma=F2 times C2 acts by left translation on g and fixes j, with C2 acting trivially on the tree. Every vertex and edge stabilizer is C2, so it can serve as the auxiliary for the compact normal suspension PT.27.

At root o=1 in the base tree choose the explicit weights

\[
 w(g)=|g|,\qquad w(g,j)=|g|+j.                   \tag{CT.7}
\]

All nonroot weights are positive and the weight at most R set has cardinality

\[
 B(R)+\sum_{j=1}^R B(R-j),\qquad
 B(0)=1,\quad B(r)=1+\sum_{n=1}^r4\cdot3^{n-1}.
                                                        \tag{CT.8}
\]

This is finite. Its return difference is at most |h| both for base vertices and for attached leaves. The exceptional geodesic [o,ho] lies in the base tree, so its weights are at most |h| and the original PT.8 bound with ell(h)=|h| applies exactly. Each attached leaf is paired with its own parent edge by Q; only the base root is unmatched. The full tree phase still has index +1.

In contrast, the **unweighted graph distance** gives distance one to every (1,j). Its compact-resolvent claim fails: infinitely many orthogonal leaf/edge blocks then have the same eigenvalues +1 and -1. Applying the resolvent to their normalized eigenvectors gives orthogonal outputs of the same nonzero norm. CT.7 replaces those depths by j, so every bounded spectral window is finite and its tail norm tends to zero. This is an exact analytic distinction between infinite valence and a proper auxiliary weight, not a finite numerical extrapolation.

![A proper weight resolves the infinite-valence compactness failure](../figures/kt-countable-tree-proper-weight.png)

**Figure 11D.2.** CT.5–CT.8. The attached labels j=1,2,3 are an explicitly finite schematic of all j at least one. Every attached leaf has its own parent edge. The plotted root-leaf block norm is exactly 1/(1+w^2), with distance w=1 or proper weight w=j. The exact finite-window counts include both base vertices and attached leaves in the entire infinite tree. The return mismatch remains confined to the finite base segment; the proper weight controls the added infinite multiplicity. These displayed formulas illustrate the all-label proof, not a numerical replacement for it. [Reproducible source](../reproduction/countable-tree/draw_countable_tree.py) and [editable SVG](../figures/kt-countable-tree-proper-weight.svg).

## 11E. Translation oscillators and a smooth source-column graph class

The translation oscillator has an even scalar index-one cycle. The normal-product and graph constructions below use the full inverse orientation of K-theory of the leaf space, Theorem 7.17, the connection and positivity criterion in K-theory of the leaf space, Section 0, and the compact-base graph correspondence in K-theory of the leaf space, Section 11D. The normal Dirac and local calibration assumptions are specified in Section 11E.9.

**Theorem 11E.1 (the translation auxiliary).** For every integer \(d\ge1\), lattice translations of \(\Gamma=\mathbb Z^d\) on \(L^2(\mathbb R^d,\Lambda^*\mathbb C^d)\) admit a self-adjoint, smooth first-order oscillator \(D\) that is odd on this graded Hilbert space. Its resulting scalar KK cycle has even degree. \(D\) has exact domain \(H^1\cap\{\psi:|x|\psi\in L^2\}\), compact resolvent, one positive Gaussian kernel and scalar index +1. Its lattice representation is a literal amplification of the left regular representation, with exact reduced norm. Every fixed translation preserves the domain; its commutator with \(D\) extends boundedly with exact norm \(|m|\), and its bounded-phase commutator is compact. Thus it is a scalar cycle of \(C_r^*(\mathbb Z^d)\), pairing +1 with its unit.

**Theorem 11E.2 (the full normal product and graph class).** Supply the full inverse normal module and compact normal Dirac specified in Section 11E.9. For a coherent isometric \(\mathbb Z^d\) action, this auxiliary gives a reduced scalar degree-\(q\) crossed-product cycle with full restriction \([D_N]\), without an invariant connection assumption. The compact-base graph correspondence then gives the normalized graph-column class. For the particular suspension \(V=(\mathbb R^d\times N)/\mathbb Z^d\), with germ-effective normal action, the induced cycle is represented, up to a proved bounded perturbation, on the **ordinary entire smooth holonomy graph** by a first-order operator differentiating in the source-leaf \(r\) and normal \(n\) directions, with oscillator potential \(r-t\). It does not differentiate in the range \(t\) direction, and is not full-tangent elliptic. Its exact anisotropic domain, smooth core, reduced representation, localized scalar defects and original local Bott pairing +1 are proved below. Arbitrary holonomy groupoids remain outside these theorems.

### 11E.1. Exterior algebra and the physical expression

Use Lebesgue measure \(dx\), the standard Hermitian exterior algebra \(E_d=\Lambda^*\mathbb C^d\), ordered wedge basis \(e_I\), and grading \(\gamma=(-1)^{\deg}\). Let \(\epsilon_j\) be left wedge by \(e_j\) and \(\iota_j\) its Hilbert-space adjoint, so on an ordered wedge \(\iota_j\) removes \(e_j\) with sign \((-1)^{\text{number of preceding positions}}\), or gives zero if absent. Directly on that basis,

\[
 \{\epsilon_j,\epsilon_k\}=\{\iota_j,\iota_k\}=0,
 \quad\{\epsilon_j,\iota_k\}=\delta_{jk},
 \quad C_j=\epsilon_j-\iota_j,
 \quad E_j=\epsilon_j+\iota_j.                 \tag{TO.1}
\]

Thus \(C_j^*=-C_j\), \(E_j^*=E_j\), \(\{C_j,C_k\}=-2\delta_{jk}\), \(\{E_j,E_k\}=2\delta_{jk}\) and \(\{C_j,E_k\}=0\). Every \(C_j\) and \(E_j\) is odd. The number operator \(N_E=\sum_j\epsilon_j\iota_j\) has eigenvalue \(|I|\) on \(e_I\). On Schwartz functions, or smooth compactly supported sections, put

\[
 a_j=\partial_j+x_j,\qquad a_j^*=-\partial_j+x_j,
 \quad D=\sum_j(\epsilon_j a_j+\iota_j a_j^*)
       =\sum_j(C_j\partial_j+E_jx_j).         \tag{TO.2}
\]

Integration by parts gives formal symmetry. The positive degree-zero to degree-one block in dimension one is \(a=\partial+x\); its adjoint is the upper block \(-\partial+x\). No scalar phase is inserted into either block.

### 11E.2. Exact square, signs and grading

The different scalar coordinates commute. In one coordinate, \(a_j^*a_j=-\partial_j^2+x_j^2-1\) and \(a_ja_j^*=-\partial_j^2+x_j^2+1\). Cross-coordinate terms cancel by (TO.1). The number projections \(\epsilon_j\iota_j\) and \(\iota_j\epsilon_j\) respectively select occupied and empty wedges. Consequently

\[
 D^2=-\Delta+|x|^2+2N_E-d,
 \qquad\gamma D=-D\gamma.                    \tag{TO.3}
\]

Equivalently the derivative-potential cross term in coordinate \(j\) is \(C_jE_j=2\epsilon_j\iota_j-1\). This independently fixes the negative constant on the empty wedge and the positive constant on the occupied wedge. The principal symbol \(i\sum_jC_j\xi_j\) is self-adjoint and squares to \(|\xi|^2\). The confining potential \(\sum_jE_jx_j\) is odd and self-adjoint. It is not a leafwise multiplication-only operator.

### 11E.3. Complete Hermite basis, without a spectral citation

Set \(\phi_0(x)=\pi^{-d/4}\exp(-|x|^2/2)\). For completeness its normalization is elementary: the square of the positive integral \(\int_{\mathbb R}e^{-u^2}\,du\) equals the plane integral of \(e^{-(u^2+v^2)}\); polar coordinates give \(2\pi\int_0^\infty re^{-r^2}\,dr=\pi\). Thus the one-coordinate integral is \(\sqrt{\pi}\), products give the \(d\)-coordinate integral, and \(\phi_0\) has \(L^2\) norm one. Direct differentiation gives \(a_j\phi_0=0\). Since \([a_j,a_k^*]=2\delta_{jk}\),

\[
 \phi_\nu=\frac{(a_1^*)^{\nu_1}\cdots(a_d^*)^{\nu_d}}
 {\sqrt{2^{|\nu|}\nu_1!\cdots\nu_d!}}\phi_0,
 \quad a_j\phi_\nu=\sqrt{2\nu_j}\phi_{\nu-e_j},
 \quad a_j^*\phi_\nu=\sqrt{2(\nu_j+1)}\phi_{\nu+e_j}.
                                                        \tag{TO.4}
\]

These identities follow by moving \(a_j\) through the creation powers; the term with \(a_j\phi_0\) vanishes. Repeated integration by parts gives orthogonality of unequal indices and squared norm \(2^{|\nu|}\nu!\) before normalization. In each coordinate creation on \(P(x)e^{-x^2/2}\) produces \((2xP-P')e^{-x^2/2}\), so these polynomials have nonzero leading coefficients and span all polynomials times the Gaussian.

Here is completeness. If \(f\) in \(L^2\) is orthogonal to every \(\phi_\nu\), all moments of \(h=fe^{-|x|^2/2}\) vanish. The function \(F(z)=\int h(x)e^{z\cdot x}\,dx\) is entire in \(z\in\mathbb C^d\): Cauchy-Schwarz with the Gaussian bounds every derivative uniformly on compact \(z\) sets. Its Taylor coefficients at zero vanish, hence \(F=0\). In particular the Fourier transform of the \(L^1\) function \(h\) vanishes. Fourier uniqueness here requires only the elementary Gaussian identity: for \(t>0\),

\(g_t(x)=(4\pi t)^{-d/2}e^{-|x|^2/(4t)}=(2\pi)^{-d}\int e^{-t|\xi|^2}e^{i\xi\cdot x}\,d\xi.\)

The one-dimensional identity follows by differentiating its Gaussian integral in \(x\), solving the resulting first-order equation and evaluating at \(x=0\); products give \(d\) dimensions. Fubini now gives \(h*g_t=0\). The positive mass-one Gaussian is an \(L^1\) approximate identity: translations of compact continuous functions are continuous in \(L^1\), such functions are \(L^1\)-dense, and its mass outside each fixed ball tends to zero. Thus \(h*g_t\) tends to \(h\) in \(L^1\). Therefore \(h=0\), and \(f=0\). This proves completeness, not just orthogonality.

For \(k=0,1,\ldots\) let \(H_k\) be the span of \(\phi_\nu e_I\) with \(|\nu|+|I|=k\). It is finite-dimensional. Equations (TO.1),(TO.4) show \(D\) preserves \(H_k\) and

\[
 D_k^*=D_k,\qquad D_k^2=2k\,1_{H_k},
 \quad H_0=\mathbb C\phi_0 e_\varnothing,
 \quad\sum_{k\ge0}\dim H_k z^k
      =\frac{(1+z)^d}{(1-z)^d}.              \tag{TO.5}
\]

In particular a bounded spectral window has finitely many basis vectors. These are entire spaces, not numerical truncations standing in for a completeness argument.

### 11E.4. Self-adjoint closure, exact domain and both cores

Define \(D\) as the orthogonal direct sum of these finite self-adjoint matrices, with

\[
 \operatorname{Dom}D=
 \left\{\psi=\sum_k\psi_k:
       \sum_k 2k\|\psi_k\|^2<\infty\right\}
 =\{\psi\in H^1(\mathbb R^d,E_d):|x|\psi\in L^2\}.
                                                        \tag{TO.6}
\]

The first domain gives self-adjointness directly: \((D_k-z)^{-1}\), \(\operatorname{Im}z\ne0\), have uniform norm at most \(1/|\operatorname{Im}z|\), and their direct sum has range exactly the displayed domain. The adjoint acts by the same matrices with the same square-summability condition. Finite Hermite sums form a graph core by weighted-tail convergence.

To prove the second equality, denote its right-hand side by \(Q\) with norm squared \(\|\psi\|^2+\|\nabla\psi\|^2+\|x\psi\|^2\). This is complete: weak derivatives and multiplication by \(x\) are closed in \(L^2\). Smooth compact support is \(Q\)-dense. Cut off at radius \(R\); the gradient error is bounded by \(O(R^{-1})\|\psi\|\) on the cutoff annulus and all other tails tend to zero. Then mollify the compactly supported result, in a fixed slightly larger compact set, in \(H^1\); multiplication by \(x\) is bounded there. This proves the asserted density.

For a smooth compactly supported scalar component \(u\), completeness of the Hermite basis and integration by parts give the coefficient of \(a_ju\) against \(\phi_\mu\) as \(\sqrt{2(\mu_j+1)}\) times its coefficient against \(\phi_{\mu+e_j}\). Parseval therefore gives \(\sum_j\|a_ju\|^2=\sum_\nu2|\nu||u_\nu|^2\). Also integration by parts gives \(\sum_j\|a_ju\|^2=\|\nabla u\|^2+\|xu\|^2-d\|u\|^2\). Componentwise, (TO.3) consequently gives

\[
 \|D\psi\|^2=\|\nabla\psi\|^2+\|x\psi\|^2
                  +\langle\psi,(2N_E-d)\psi\rangle
             =\sum_k2k\|\psi_k\|^2.         \tag{TO.7}
\]

\(Q\)-density extends the coefficient identity to \(Q\). Conversely finite Hermite sums whose weighted coefficients are square-summable are Cauchy in \(Q\) by that identity, proving the reverse inclusion. Thus the formal \(C_c^\infty\) expression closes to precisely the direct-sum operator, and \(C_c^\infty\) is a graph core as well. Schwartz functions are another core. This specifies the completed operator, not just a formal expression.

### 11E.5. Compact resolvent, Gaussian kernel and scalar index

On the complement of \(\bigoplus_{k\le K}H_k\), the norm of \((1+D^2)^{-1}\) is at most \(1/(1+2(K+1))\); the norm of \((D-i)^{-1}\) is at most \(1/\sqrt{1+2(K+1)}\). The cutoffs have finite rank, so both operators are scalar compact. The only kernel is \(\phi_0e_{\varnothing}\), which is even. Every \(k>0\) block is invertible and odd, identifying its equal positive and negative graded spaces.

\[
 F=D(1+D^2)^{-1/2},\quad S=D|D|^{-1}\text{ off the kernel},
 \quad S=0\text{ on the kernel},\quad
 \operatorname{Index}F^+=1.                  \tag{TO.8}
\]

Here \(F-S\) is compact, because on \(k>0\) its block coefficient differs from \(S\) by \(\sqrt{2k}/\sqrt{1+2k}-1\), tending to zero off finite windows. \(F^+\) has one-dimensional kernel, no cokernel and a lower bound \(\sqrt{2/3}\) on the kernel complement, so its Fredholm index is literally +1. For the scalar algebra, \(S\) splits into the positive one-dimensional identity cycle and a degenerate odd-unitary cycle on the complement. Thus its scalar KK class is the identity even before invoking an abstract index isomorphism.

### 11E.6. All lattice translations and their closed domains

For \(m\) in \(\mathbb Z^d\) set \((T_m\psi)(x)=\psi(x-m)\), and put \(c_+(m)=\sum_jE_jm_j\). These are even unitaries and \(T_mT_l=T_{m+l}\). Derivatives commute with \(T_m\) and \(x_jT_m-T_mx_j=m_jT_m\). Hence, with the signs fixed,

\[
 [D,T_m]=c_+(m)T_m,\qquad
 T_mDT_m^{-1}=D-c_+(m),\qquad
 \|[D,T_m]\|=|m|.                            \tag{TO.9}
\]

\(c_+(m)^2=|m|^2\) gives the exact norm, including zero for \(m=0\). \(T_m\) preserves \(H^1\) and \(\|xT_m\psi\|\le\|x\psi\|+|m|\|\psi\|\). Applying the reverse translation proves equality of full domains. The formula initially computed on smooth compact support therefore extends on \(\operatorname{Dom}D\). No uniform-in-\(m\) boundedness is asserted or needed.

### 11E.7. Compact phase defects, with an integrable resolvent proof

If a bounded \(V\) preserves \(\operatorname{Dom}D\) and \(C=[D,V]\) is bounded, let \(s=1+t^2\) and \(Q_+=(D-i\sqrt{s})^{-1}\), \(Q_-=(D+i\sqrt{s})^{-1}\). On the full domain, and then as bounded maps,

\[
 [Q_\pm,V]=-Q_\pm C Q_\pm,\qquad
 D(s+D^2)^{-1}=(Q_++Q_-)/2,
 \|[D(s+D^2)^{-1},V]\|\le\|C\|/(1+t^2).
                                                        \tag{TO.10}
\]

The first identity is multiplication of the closed resolvent equations; it needs no domain invariance of C. Each commutator is compact because the resolvents are compact. Spectral calculus gives the strong integral \(F=(2/\pi)\int_0^\infty D(1+t^2+D^2)^{-1}\,dt\): the scalar integral at eigenvalue \(\lambda\) is \(\lambda/\sqrt{1+\lambda^2}\). The commutator integrals converge in compact norm by (TO.10), and strong convergence of the uncommuted truncations identifies their limit with \([F,V]\). In particular \([F,T_m]\) is scalar compact for every fixed \(m\). Also \(F^*=F\) and \(F^2-1=-(1+D^2)^{-1}\) is compact. Density then gives all defects for the completed represented algebra, once its norm is verified next.

### 11E.8. Literal regular completion and exact reduced norm

Let \(C=[0,1)^d\) with Lebesgue measure and \(K_0=L^2(C,E_d)\). The disjoint measurable partition \(\mathbb R^d=\bigsqcup_{n\in\mathbb Z^d}(n+C)\) gives an onto isometry

\[
 J:L^2(\mathbb R^d,E_d)\longrightarrow
       \ell^2(\mathbb Z^d)\otimes K_0,
 \quad(J\psi)_n(y)=\psi(n+y),\quad
 JT_mJ^{-1}=\lambda_m\otimes1.              \tag{TO.11}
\]

The inverse glues these functions almost everywhere, and the lattice-boundary set has measure zero. For a finitely supported scalar coefficient \(a\), the Hilbert-basis tensor norm estimate gives

\[
 \left\|\sum_m a_mT_m\right\|
    =\left\|\sum_m a_m\lambda_m\right\|
    =\|a\|_{C_r^*(\mathbb Z^d)}.             \tag{TO.12}
\]

The upper inequality is Parseval in an orthonormal basis of \(K_0\); tensoring a nearly norm-attaining regular vector with any unit vector in \(K_0\) gives the reverse inequality. Thus the extension is faithful, nondegenerate and genuinely reduced, not a maximal integrated representation or a Hilbert-module descent assertion. Combining Sections 11E.5–11E.7 proves the auxiliary theorem and unit pairing +1. In the cube coordinates the domain is the glued image \(JQ\); the \(H^1\) matching across neighbouring cube faces is retained. No independent cube boundary conditions are invented.

### 11E.9. Supplied full inverse normal orientation

Let \(N\) be a compact connected smooth manifold without boundary, \(\dim N=q\), with a fixed metric and a specified **full** spin-c normal orientation. Retain the full inverse module of Theorem 7.17, its Riemannian density, its grading, and finite right \(C_q\) coefficient in the original order. In a prescribed original transition \(zs\), its local form is

\[
 \mathcal D_N\cong\Pi^{k_q}C_q,\quad k_q=q(q-1)/2,
 \quad c_{\rm inv}(e^j)=(-1)^{k_q+1}L_{t_j},
 \quad zs\longmapsto z^{-1}\rho_t(s).        \tag{TO.13}
\]

An original even line twist \(H\) becomes \(H^*\). The entire module is used, including two-torsion information and the last \(\text{right }C_1\) in odd rank. The oscillator has even KK degree and causes no Clifford block permutation or replacement by a chosen minimal normal spinor. Rank zero retains the supplied full graded orientation line, rather than an automatic trivialization.

Assume the normal data include a smooth compatible, formally symmetric Dirac \(D_N\) on \(H_N=L^2(N,\mathcal D_N)\), with self-adjoint domain \(H^1\), smooth core, scalar compact resolvent, full inverse principal symbol, and original positive local transverse Bott class \(b_U\) satisfying \(\langle b_U,[D_N]\rangle=1\). These normal Dirac and local calibration properties are hypotheses of the normal-product and graph statements; no general existence assertion is made.

Suppose \(\mathbb Z^d\) acts on \(N\) by isometries \(h_m\), with a coherent lift to that entire full structure. Its induced even unitaries \(U_m\) retain all inverse central phases and densities. A compatible connection need not be invariant. \(U_mD_NU_m^{-1}\) and \(D_N\) have the same principal symbol, so their difference is a smooth zeroth-order endomorphism, bounded on compact \(N\). Smooth diffeomorphism and bundle pullback preserve \(H^1\). Thus \(B_m=[D_N,U_m]\) is bounded and \(U_m\operatorname{Dom}D_N=\operatorname{Dom}D_N\). Each \(m\) has its own bound. This argument uses symbol cancellation, not averaging over the noncompact group.

### 11E.10. Completed normal product and reduced crossed representation

On \(H_B=H_D\widehat\otimes H_N\) put

\[
 D_B=D\otimes1+\gamma\otimes D_N,
 \quad\Pi(f)=1\otimes M_f,\quad
 \Pi(u_m)=T_m\otimes U_m,
 \qquad D_B^2=D^2\otimes1+1\otimes D_N^2.    \tag{TO.14}
\]

This is a scalar Hilbert-space construction with its finite right \(C_q\) coefficient retained. Finite oscillator energy blocks and finite normal spectral windows are invariant finite-dimensional spaces for \(D_B\). On each, the displayed symmetric operator squares to the sum of the two nonnegative energies. Their orthogonal direct sum is self-adjoint, with exact domain \(\operatorname{Dom}(D\otimes1)\cap\operatorname{Dom}(1\otimes D_N)\), equivalently square-summability of that sum of energies. Finite spectral sums form a core; approximating each normal component in its supplied \(H^1\) graph norm by smooth sections shows finite Hermite sums tensor smooth normal sections form a core. Bounded spectral windows are finite, so the resolvent is scalar compact.

Apply \(J\otimes1\) from (TO.11), then the unitary \(W\) with \((W\xi)_n=U_n^{-1}\xi_n\), including the untouched \(K_0\) factor. Direct substitution gives

\[
 W(\lambda_m\otimes U_m)W^{-1}=\lambda_m\otimes1,
 \qquad W(1\otimes M_f)W^{-1}|_n=M_{f\circ h_n}.
                                                        \tag{TO.15}
\]

This is the regular induced representation from the faithful multiplication representation of \(C(N)\) on \(H_N\), amplified by \(K_0\). To bind it to the pointwise reduced norm, a finite crossed coefficient acts, after measurable local trivializations, as the direct integral over \(n_{\rm normal}\in N\) of its defining point-regular representation, with finite spin coefficient multiplicity. Its norm is the essential supremum of those norms. That equals the supremum: at any chosen normal point, test a nearly norm-attaining finite-support regular vector. Its output for a finite crossed coefficient has finitely many coordinates, continuously depending on the point. The inequality persists on a neighbourhood of positive Riemannian measure, where a unit local spin section supplies the same test. The reverse bound is immediate. Thus

\[
 \|\Pi(a)\|=\sup_{y\in N}\|\pi_y^{\rm reg}(a)\|
       =\|a\|_{C(N)\rtimes_r\mathbb Z^d}.    \tag{TO.16}
\]

This proves faithful reduced factorization for \(B=C(N)\rtimes_r\mathbb Z^d\) without declaring its implementing compact-normal representation reduced. Scalar multiplication removes no central phase; the phase is carried in \(U_n\) and cancels only inside the explicit absorption unitary.

For smooth \(f\), \([D_B,\Pi(f)]=\gamma\otimes[D_N,M_f]\). For each fixed \(m\),

\[
 [D_B,\Pi(u_m)]
   =c_+(m)T_m\otimes U_m+\gamma T_m\otimes B_m,
 \qquad\|[D_B,\Pi(u_m)]\|\le|m|+\|B_m\|.  \tag{TO.17}
\]

These formulas and the two exact domains prove preservation on the smooth finite-group-support core. Section 11E.7 now proves the scalar compact phase defects; density gives every element of \(B\).

### 11E.11. Full normal restriction, creation equations and positivity

The auxiliary phase \(S\), externally extended to \(C(N)\), is the identity class: its positive Gaussian rank-one \(C(N)\) summand is exactly the identity module, and its complement is a degenerate odd-unitary cycle. This retains its sign rather than deducing a product from a finite normal index.

For each nonzero energy \(k\) choose graded paired bases of \(H_k\) so \(D_k=\begin{pmatrix}0&\sqrt{2k}\\\sqrt{2k}&0\end{pmatrix}\) and \(\gamma=\operatorname{diag}(1,-1)\). On the corresponding normal spectral parameter \(\nu\) the total transform is

\[
 (1+2k+\nu^2)^{-1/2}
       \begin{pmatrix}\nu&\sqrt{2k}\\\sqrt{2k}&-\nu\end{pmatrix}.
                                                        \tag{TO.18}
\]

For a homogeneous finite oscillator vector, its creation-map connection with \(F_N=D_N(1+D_N^2)^{-1/2}\) is compact: the off-diagonal entry tends to zero as \(|\nu|\) tends to infinity, and the difference of the diagonal entries from \(\pm\nu/\sqrt{1+\nu^2}\) does too. Compact normal resolvent proves compactness. The positive Gaussian column is exactly \(F_N\). Coefficient functions follow from the compact normal phase commutators. Finite sums, continuity of creation norms and density extend both the connection equation and its adjoint to every module vector, with the negative-column grading sign retained.

The lifted first phase has \(SD=DS=|D|\) and anticommutes with \(\gamma\). Since the two squares commute,

\[
 \{S\otimes1,F_B\}
  =\frac{2|D|\otimes1}{\sqrt{1+D^2\otimes1+1\otimes D_N^2}}
  \ge0,\qquad
 [\Pi,F_B]|_{C(N)}=[D_N],\quad
 \langle b_U,[\Pi,F_B]|_{C(N)}\rangle=1.      \tag{TO.19}
\]

The bounded Kasparov product criterion of K-theory of the leaf space, Definition 0.3 and Theorems 0.4–0.7 proves the restriction identity from the full connections and positivity. All right \(C_q\) order, grading and inverse transition phases remain unchanged.

### 11E.12. General compact-base suspension: graph-column completion

Let \(X\) be a connected regular \(\mathbb Z^d\)-cover of a closed connected smooth \(M\). For a germ-effective normal action set \(V=(X\times N)/\mathbb Z^d\), with horizontal foliation. Germ effectivity means no nonzero group element is the identity on an open normal chart; it identifies the actual holonomy groupoid with the suspension graph, rather than leaving an unremoved path-groupoid kernel.

Use the compact-base graph correspondence in K-theory of the leaf space, Section 11D.5, equations PT.20 and SG.6–SG.9b: the flat right-\(B\) module \(E=L^2(M,\mathcal L_B)\) has \(A=C_r^*(G)=\mathcal K_B(E)\), equals \(P(K\otimes B)\) in finitely many buffered flat charts, and is full. Its exact reduced norm is proved there by separated-kernel Schur approximation, source-fibre identification and a normalized plaque bump. That correspondence is independent of the auxiliary tree and applies with \(\Gamma=\mathbb Z^d\). This step uses the hypotheses of that correspondence.

On the free scalar completion put \(D_0=1_K\otimes D_B\). A partition with \(\sum_i\chi_i^2=1\) gives projection entries \(\chi_i\chi_j u_{m_{ij}}\), with actual coordinate/density changes. On each compact overlap only finitely many locally constant \(m_{ij}\) occur. Equations (TO.17) give \([D_0,P]\) bounded and \(P\operatorname{Dom}D_0\subset\operatorname{Dom}D_0\). Remove the bounded self-adjoint off-diagonal \(R=PD_0(1-P)+(1-P)D_0P\). Then

\[
 D_{\rm diag}=D_0-R,\qquad
 D_G=P D_0P|_{P\operatorname{Dom}D_0},\qquad
 H_G=E\otimes_BH_B=P(K\otimes H_B).          \tag{TO.20}
\]

For \(|\operatorname{Im}z|>\|R\|\) the Neumann resolvent of \(D_0-R\) proves self-adjointness on the unchanged domain; \(P\) reduces it. Its compressed domain is exactly \(P\operatorname{Dom}D_0\). Finite Hermite/smooth-normal vectors tensor smooth compact chart profiles form a free graph core, and projection is graph-norm continuous. Their projections therefore form the compressed core. This is a genuine completion with an explicit auxiliary \(L^2(\mathbb R^d)\) factor. For a general \(M\) it is not identified with a physical leaf coordinate.

### 11E.13. Graph-column localized compacts and Morita product

Let \(J_0=\mathcal K(K)\otimes\mathcal B(H_B)\). For \(j\) in \(J_0\), \(j(D_0-z)^{-1}\) is scalar compact: prove it for separated tensors using the compact \(D_B\) resolvent, then use norm density. \(P\), \(R\) and \([D_0,P]\) are bounded multipliers of \(J_0\). The resolvent identity gives \(j(D_{\rm diag}-z)^{-1}\) compact as well. A smooth finite chart/rank-one/finite crossed coefficient \(a=Pa_0P\) has \([D_0,a]\) in \(J_0\), by expansion and (TO.17), and preserves the compressed domain. The mixed version of Section 11E.7 has both \(QCQ\) terms compact by these localized resolvents and norm bounded by \(\|C\|/(1+t^2)\). It proves every localized scalar phase defect; density handles \(A\).

For a smooth finite-chart module vector \(\eta\), \(D_GT_\eta-T_\eta D_B\) is bounded by its finite transition commutators. The rectangular resolvent formula \(Q_GT_\eta-T_\eta Q_B=-Q_G(D_GT_\eta-T_\eta D_B)Q_B\) has compact right-hand side because \(Q_B\) is compact, and norm at most \(\|\mathrm{error}\|/s\). The phase integral gives the connection equation and its adjoint. Density extends them. Product positivity for the zero Morita operator is zero. Therefore

\[
 z_G=[{}_AE_B,0]\otimes_B[\Pi,F_B]\in KK^q(A,\mathbb C).
                                                        \tag{TO.21}
\]

The normalized plaque bump \(\xi\), with \(\langle\xi,\xi\rangle_B=1\), gives \(p=\theta_{\xi,\xi}\) and \(i(b)=\theta_{\xi b,\xi}\). Its inverse counting correspondence is \(E^*\) identified with \(pA\) by \(\eta\)-bar to \(\theta_{\xi,\eta}\), with the same inner product, action and onto rank-one identities. It is the original transverse class, not a chosen alternative. Hence the restriction \(i^*z_G=[\Pi,F_B]\), and the push-forward \(i_*b_U\) satisfies \((i_*b_U)\otimes_A z_G=1\), by (TO.19). This proves the complete generic graph-column adaptation at its stated suspension scope.

### 11E.14. A particular actual smooth graph, rather than renamed coordinates

Now specialize **the cover** to \(X=\mathbb R^d\) and \(M=\mathbb T^d=\mathbb R^d/\mathbb Z^d\), with the same germ-effective normal action. A lifted horizontal path has endpoints \((r,n),(t,n)\). Two such lifted paths give the same holonomy arrow precisely up to the common deck action: the only additional possible identification would be a return label with identity normal germ, excluded by germ effectivity. Thus

\[
 V=(\mathbb R^d\times N)/\mathbb Z^d,\quad
 G=(\mathbb R_t^d\times\mathbb R_r^d\times N)/\mathbb Z^d,
 \quad m(t,r,n)=(t+m,r+m,h_m n),
 \quad r_G=[(t,n)],\quad s_G=[(r,n)].        \tag{TO.22}
\]

The diagonal action is free and proper because it is so on \(t\). \(G\) is a smooth Hausdorff manifold. Its coefficient bundle is \(E_d\otimes\) the full inverse normal module at \(n\), with diagonal transition \(1\otimes\) the actual inverse lift. Along left convolution it is transported by the identity at the common lifted \(n\); these transports compose exactly. Ordinary entire-graph \(L^2\) uses the quotient of \(dt\,dr\,d\operatorname{vol}_N\), realized by \(t\in[0,1)^d\), \(r\in\mathbb R^d\), \(n\in N\), with the diagonal boundary identification. This coordinate is the **actual source-leaf \(r\) of (TO.22)**. It is not available for the arbitrary compact-base cover of Section 11E.12 just by relabelling the auxiliary.

Choose a smooth covariant family of compatible normal Diracs \(D_N(t)\) with \(D_N(t+m)=U_mD_N(t)U_m^{-1}\); an explicit such family is constructed in Section 11E.18. On equivariant smooth compact graph sections define

\[
 \mathscr D=\sum_j[C_j\partial_{r_j}+E_j(r_j-t_j)]\otimes1
                     +\gamma\otimes D_N(t).
                                                        \tag{TO.23}
\]

It is invariant under the diagonal deck action and formally symmetric for the stated density. It is first order, with finite-rank smooth coefficients, and has the full prescribed normal principal symbol. Its symbol squares to \(|\xi_r|^2+|\eta_n|^2\). There is no derivative in \(t\); consequently it is not elliptic on all \(TG\) and is outside the ordinary full-tangent ansatz of GD.1. The noncompact source coordinate allows a proper potential, a fact absent from the compact source-circle obstruction.

### 11E.15. Exact anisotropic domain and smooth compact graph core

Use \(y=r-t\) on each range chart. There are no derivatives in \(t\), so differentiation in \(r\) equals differentiation in \(y\). The oscillator and the normal expression anticommute, and

\[
 \mathscr D^2=-\Delta_y+|y|^2+2N_E-d+D_N(t)^2,
 \quad\operatorname{Dom}\mathscr D=
 \{\psi:\nabla_r\psi,\ (r-t)\psi,\ \nabla_n\psi\in L^2(G)\}.
                                                        \tag{TO.24}
\]

The last norm is defined in finitely many normal/range coefficient charts and any fixed compatible reference normal connection; equivalent choices differ by bounded terms on the compact range base. **No \(t\) derivative is required.** The covariance glues these local definitions. On each range chart the family differs from a fixed supplied normal \(D_N\) by a uniformly bounded smooth self-adjoint zeroth-order endomorphism \(K_t\). The two large imaginary Neumann inverses prove \(D_N+K_t\) self-adjoint on the unchanged \(H^1\) domain; its resolvent identity against \(D_N\) proves scalar compact resolvent. Its normal \(H^1\) graph estimates are uniformly equivalent on the compact base. At each \(t\) the completed sum is consequently self-adjoint by the finite-energy/normal spectral argument of Section 11E.10, translated to \(y\). Its measurable direct integral is self-adjoint: the fibre resolvents have \(\text{norm}\le1/|\operatorname{Im}z|\) and give inverse operators on the stated square-integrable graph domains. Integration of the fibre graph identity gives exactly (TO.24) and equivalent graph norms; no base compactness is used as a false global compact-resolvent assertion.

Smooth compact graph sections form a core. Cut off in invariant \(y\), with derivative error \(O(R^{-1})\) in \(r\); the weighted and derivative tails tend to zero. On the resulting compact graph region use finitely many coefficient charts and a smooth partition. Mollify in \(r\),\(n\) in their \(H^1\) norms and in \(t\) in \(L^2\). Smooth dependence of the normal coefficients and their bounded derivatives gives convergence in the normal first-order norm; one can first approximate by smooth finite \(t\) profiles times smooth \((r,n)\) sections on buffered patches, using density in \(L^2(t;H^1(r,n))\) and the bounded coefficient maps \(H^1\) to \(L^2\). Mollification does not impose a \(t\) derivative on the limit. The locally constant flat transitions and smooth cutoffs preserve the completed norm. This proves the full global graph core and the exact closed extension of (TO.23).

### 11E.16. Actual convolution, domain preservation and bounded commutators

For a smooth compactly supported graph coefficient \(a\), left convolution in lifted coordinates is

\((\Pi(a)\psi)(t,r,n)=\int_{\mathbb R^d}a(t,t^\prime,n)\psi(t^\prime,r,n)\,dt^\prime,\)

with the prescribed coefficient transport. Compact graph support bounds \(|t^\prime-t|\) uniformly and yields the two finite Schur integrals, uniformly over the compact normal/range base. Both input and output use the same lifted source \((r,n)\). Thus \(\partial_r\) commutes with the kernel; the oscillator potential commutator is \(\sum_jE_j(t^\prime_j-t_j)a\). The normal principal coefficients are the same at \((t,n)\) and \((t^\prime,n)\). The normal commutator kernel is its normal derivative Clifford term plus \(a\) times the smooth zeroth-order difference \(D_N(t)-D_N(t^\prime)\). On finitely many range charts and the finitely many return labels met by the support, all these coefficients and their Schur integrals are bounded. Therefore \([\mathscr D,\Pi(a)]\) has a bounded Schur-kernel extension. This proves principal cancellation rather than assuming it.

\(\Pi(a)\) maps smooth compact graph sections to smooth compact graph sections: the source variable stays fixed, and bounded \(|t^\prime-t|\) shifts the invariant \(y\) support only by a bounded amount. Approximate any completed-domain vector by that core. The identity \(\mathscr D\Pi(a)\psi=\Pi(a)\mathscr D\psi+C_a\psi\) and closedness prove domain preservation on the **whole** domain. This also proves covariance and convolution compatibility for the completed expression.

Its norm is genuinely reduced independently of the oscillator calculus. Decomposing ordinary \(L^2\)(\(G\)) over source units gives the direct integral of the source-fibre regular representations. Its norm is at most their defining supremum. For a smooth compact coefficient, a compactly supported nearly norm-attaining source-fibre test varies continuously in a local source chart; kernel continuity and the two Schur bounds retain its norm on a neighbourhood of positive source volume. Thus essential supremum equals supremum. Density gives \(\|\Pi(a)\|=\|a\|_r\) on the entire reduced graph algebra. This agrees with Sections 11E.12–11E.13's correspondence representation.

### 11E.17. Localized scalar compactness despite the undifferentiated range

Global compact resolvent is **not** asserted: \(t\) is an undifferentiated compact range variable. On finitely many buffered range patches trivialize the normal bundle and set \(y=r-t\). The fibre operator \(\mathscr D_t\) is an oscillator-plus-normal Dirac with compact resolvent \(Q_t(z)\), by Section 11E.10. It depends norm-continuously on \(t\): differences of the normal operators are bounded zeroth-order terms whose norms tend to zero, and the resolvent identity applies. Compactness is uniform in the needed sense because a continuous image of the compact range-patch closure in the compact-operator norm is totally bounded.

The translation sign can be fixed before taking operator norms. Write \(\phi_{t^\prime}(y,n)=\psi(t^\prime,y+t^\prime,n)\). Holding \(r=y+t\) fixed gives

\[
 y'=r-t'=y+t-t',\qquad
 \psi(t',r,n)=\phi_{t'}(y+t-t',n)
             =(T_{t'-t}\phi_{t'})(y,n),
 \qquad (T_\delta f)(y)=f(y-\delta).
\]

Thus the coordinate substitution is by \(t-t^\prime\), while the operator factor under our fixed translation convention is \(T_{t^\prime-t}\). Its potential commutator is still \(t^\prime-t\). In these coordinates \(\Pi(a)Q(z)\) has integral kernel in the two compact range coordinates of the form \(M_{a(t,t^\prime,n)}\) times this bounded \(y\)-translation, a fixed return lift \(U_m\), and \(Q_{t^\prime}(z)\). In buffered chart representatives the translation includes the actual return label in \(t^\prime\); the same formula applies and that label's full \(U_m\) remains. Only finitely many \(m\) occur. Every such fibre operator is scalar compact. It is norm-continuous: translations are strongly continuous, hence norm-continuous after composition with a compact operator (first check finite rank, then norm approximation); the other factors are norm-continuous as described. The kernel is supported in buffered range charts. A continuous compact-operator-valued kernel on compact coordinate sets defines a scalar compact integral operator: approximate it uniformly by finite sums of separated continuous range functions times fixed finite-rank fibre operators. The integral-norm error is bounded by the two range Schur bounds. These approximants are finite-rank on the total Hilbert space. Smooth partitions and their finite sum give

\[
 \Pi(a)(\mathscr D-z)^{-1},\quad
 (\mathscr D-z)^{-1}\Pi(a)\quad\hbox{scalar compact}.
                                                        \tag{TO.25}
\]

The second assertion follows by adjointing the first for \(a^*\text{ and }\overline z\). These arguments provide compactness in the undifferentiated \(t\) direction via the actual kernel; compact normal/oscillator fibres alone would not do so.

In TO.10 with \(C_a\) the Schur commutator, a compact kernel on the range base remains of this type after multiplying by bounded local connection/potential coefficients. Consequently \(QC_aQ\) is compact: apply the same compact-kernel proof to \(C_aQ\), then multiply by \(Q\). The bound \(\|C_a\|/(1+\tau^2)\) is integrable. The phase integral therefore gives \([F_{\mathscr D},\Pi(a)]\) compact. The square defect \(\Pi(a)(1+\mathscr D^2)^{-1}\) is compact by (TO.25). Self-adjointness defect is zero. Density extends these defects to **every element of \(C_r^*(G)\)**. This is a scalar Kasparov cycle, with the finite right \(C_q\) orientation coefficient reduced only in the supplied order.

### 11E.18. Exact identification with the induced oscillator expression

Take buffered coordinate charts \(U_i\) on \(\mathbb T^d\) with smooth lifts \(z_i:U_i\to\mathbb R^d\) and real \(\chi_i\) supported there, \(\sum_i\chi_i^2=1\). For a global lifted \(t\) over \(s=t\bmod\mathbb Z^d\), define \(m_i(t)=t-z_i(s)\in\mathbb Z^d\) wherever \(\chi_i(s)\) is nonzero. It is locally constant. Put

\(b(t)=\sum_i\chi_i(s)^2m_i(t),\)

and \(D_N^{\rm ind}(t)=\sum_i\chi_i(s)^2U_{m_i(t)}D_NU_{m_i(t)}^{-1}.\)

Each weighted term extends smoothly across its chart boundary; \(b\) is smooth, \(b(t+m)=b(t)+m\) and \(b(t)-t\) is periodic and bounded. The normal family is compatible, smooth and covariant; its connections are the affine weighted sum of compatible transported connections. Thus it supplies the existence claimed in Section 11E.14 without presuming an invariant connection.

Identify \(H_G=E\otimes_BH_B\) with equivariant \(H_B\)-valued functions \(\xi(t)\) satisfying \(\xi(t+m)=(T_m\otimes U_m)\xi(t)\). The unitary to ordinary graph sections is simply \(\xi(t)(r,n)=\psi(t,r,n)\) in (TO.22); their equivariance, measure, full normal transition and convolution actions agree literally. Its onto inverse uses that same measurable fundamental domain. There is no additional auxiliary cube left after identifying **this actual cover** \(X=\mathbb R^d\).

The partition isometry sends \(\xi\) to its local components \(\chi_i(s)(T_{-m_i(t)}\otimes U_{-m_i(t)})\xi(t)\). Its range projection is the actual \(P\) of Section 11E.12. Since these weights have no \(r\),\(n\) derivative, conjugating \(PD_0P\) back by this isometry gives exactly

\[
 D_{\rm ind}(t)=\sum_j[C_j\partial_{r_j}+E_j(r_j-b_j(t))]
                       +\gamma D_N^{\rm ind}(t),
 \quad \mathscr D-D_{\rm ind}
       =\sum_jE_j(b_j(t)-t_j)
          +\gamma(D_N(t)-D_N^{\rm ind}(t)).   \tag{TO.26}
\]

For \(D_N(t)=D_N^{\rm ind}(t)\), only the first periodic bounded term remains. More generally any smooth covariant compatible normal family differs from it by a bounded zeroth-order endomorphism on the compact base. Thus the actual graph expression is a **proved** bounded perturbation of the Morita-induced auxiliary, with the same exact anisotropic domain. This is not an assumed coordinate substitution.

### 11E.19. Bounded perturbation, equality of classes and original pairing

Write \(K=\mathscr D-D_{\rm ind}\), a bounded self-adjoint covariant zeroth-order endomorphism from (TO.26). Its commutator with a compact smooth convolution kernel is bounded by the same two Schur bounds. For \(D_s=D_{\rm ind}+sK,\ 0\le s\le1\), the bounded-perturbation Neumann resolvents give self-adjointness on the unchanged domain. Localized compact resolvents persist: \(\Pi(a)Q_s=\Pi(a)Q_0-\Pi(a)Q_0sKQ_s\). The square and commutator phase arguments of Section 11E.17 apply throughout (the commutator remains a smooth bounded Schur kernel). Moreover \(Q_s-Q_u=-(s-u)Q_sKQ_u\). Averaging at \(\pm i\sqrt{1+\tau^2}\) and integrating bounds \(\|F_s-F_u\|\) by \(|s-u|\|K\|\). This is a norm-continuous cycle homotopy. Localized phase differences are compact by that same resolvent identity. Hence the ordinary smooth graph cycle represents exactly \(z_G\) of (TO.21), and its original transverse local Bott pairing is +1, with the full normal class in the point-transversal corner. A scalar auxiliary index alone is not used to infer this normalization.

### 11E.20. Original families and the finite-stabilizer tree distinction

For each \(d\) take \(N=\mathbb T^d\) with its flat metric and full framing spin-c structure. Let \(h_m(n)=n+\sqrt2m\bmod\mathbb Z^d\). This is a coherent isometric normal action and is germ-effective: a translation germ is the identity only for an integer displacement; \(\sqrt2m\) integer forces \(m=0\) coordinatewise. Optional central character \(e^{i\beta\cdot m}\) gives an actual coherent full lift; its inverse phase \(e^{-i\beta\cdot m}\) is retained in (TO.13)-(TO.26). \(V\) has dimension \(2d\) and \(G\) dimension \(3d\). The trivial compatible normal connection is invariant; adding any smooth compatible skew-Hermitian line connection on \(N\) gives the noninvariant case covered by the proof. No normal spectrum or global normal index is inferred from a drawing.

For \(d=2\) the oscillator's energy multiplicity is \(\dim H_0=1\) and \(\dim H_k=4k\) for \(k\ge1\), by \((1+z)^2/(1-z)^2\). Each positive-energy block has equal even/odd dimension \(2k\). Its scalar index remains one on the entire infinite space, not by dropping the odd modes.

For \(d\ge2\), \(\mathbb Z^d\) cannot act on a connected tree with finite vertex stabilizers. Indeed its torsion-free property makes every finite vertex stabilizer trivial. The stabilizer of an unoriented edge has endpoint-permutation kernel in a vertex stabilizer and image of order at most two, so it is finite and hence also trivial. There are no inversions. The quotient graph is therefore covered by the tree with bijective vertex stars. Choose a spanning tree of the connected quotient and lift its unique paths from a root. Each oriented edge outside the spanning tree then gives a deck element by comparing its lifted endpoint with the selected lift. Lifting paths shows these elements generate the deck group. A reduced nonempty word in these edges cannot give the identity: the corresponding lifted closed edge path, after deletion of immediate reversals and the unique spanning-tree segments, would be a nontrivial reduced closed path in a tree. No such path exists, since the unique path between any two vertices cannot contain a closed reduced subpath. Thus the deck group is free on those edges. An abelian free group has at most one generator: two distinct free generators have a nonempty reduced commutator. It is therefore trivial or \(\mathbb Z\). This excludes \(\mathbb Z^d\) for \(d\ge2\). The oscillator theorem thus adds a family outside the finite-stabilizer-tree condition, rather than merely changing the weight on that family.

### 11E.21. Scope retained

The auxiliary theorem holds for every \(d\ge1\). The normal crossed-product theorem needs the supplied compact full metric spin-c normal data and coherent isometric action. Generic compact-base covers have a normalized graph-column cycle but retain a separately identified auxiliary coordinate. A smooth actual-graph finite-order realization is proved here only for the specific \(\mathbb R^d\)-over-\(\mathbb T^d\) suspension with germ-effective normal action and the smooth covariant normal family constructed above. It is degenerate in the range directions, not an ordinary full graph tangent Dirac. These distinctions preserve Section 11C's exact full-tangent obstruction and source-column alternative (GD.1 and GD.7).

This does not narrow or close the historical general holonomy-groupoid graph-Dirac question. It does not prove that its transverse spin-c premise supplies the present isometric action or lattice-suspension geometry, nor assert a universal groupoid index-one auxiliary. This section leaves the general problem unfinished.

Further reading: Heath Emerson, *An Introduction to C*-Algebras and Noncommutative Geometry* (2024), ISBN 978-3-031-59850-0, Sections 11.4–11.5, pp. 517–532. Bundled fonts and software retain the terms stated in the linked component notices.

### 11E.22. Reproducible illustration

![Translation oscillator, regular completion and actual source-column mechanism](../figures/kt-translation-oscillator-graph.png)

**Figure 11E.1.** Equations TO.1–TO.26 and Sections 11E.3–11E.8, 11E.14–11E.19. Panel 1 shows the exact two-dimensional ground-state squared amplitude \(\pi^{-1}e^{-|x|^2}\), with level circles \(|x|^2=1,4\); its empty-wedge line is even. Panel 2 displays the exact \(d=2\) multiplicities only for \(k=0,\ldots,5\), while the completeness and tail arguments cover every level. Panel 3 is the literal cube partition defining \(J\), with the selected translation \(m=(2,1)\); the completed \(H^1\) face matching remains in its domain. Panel 4 uses the exact one-dimensional Gaussian and its translate by +1, with the operator commutator sign and norm. Panel 5 uses the exact one-coordinate squared Gaussian profile, \(\pi^{-1/2}e^{-(r-t)^2}\), with the orange centre line \(r=t\). It suppresses the other leaf coordinates and the compact normal variable: at fixed actual source \(r=3/2\), convolution takes range \(t^\prime=3/4\) to \(t=1/4\), so \(y^\prime=3/4\) becomes \(y=5/4\) and the potential change is +1/2. Its horizontal derivative arrow is \(\partial_r\); no range derivative is introduced. Panel 6 separates the completed domains, compact fibre resolvents and convolution-produced scalar compactness in the undifferentiated range variable. No global graph compact resolvent or normal index is computed by these plots. The original local Bott normalization follows from the full module/connection/positivity and exact induced-class argument. Further reading is Emerson's Sections 11.4–11.5, cited above. [Editable SVG](../figures/kt-translation-oscillator-graph.svg) · [Reproducible generator](../reproduction/translation-oscillator-graph/draw_translation_oscillator.py) · Component terms.


Portable reproduction files and complete component notices:

- [kt-translation-oscillator-graph.svg](../figures/kt-translation-oscillator-graph.svg)
- COMPONENT-TERMS.md
- [FIGURE-CHECKS.json](../reproduction/translation-oscillator-graph/FIGURE-CHECKS.json)
- FONT-NOTICE.txt
- [FORMULA-CHECKS.json](../reproduction/translation-oscillator-graph/FORMULA-CHECKS.json)
- MATPLOTLIB-LICENSE.txt
- NUMPY-LICENSE.txt
- PILLOW-LICENSE.txt
- PYTHON-LICENSE.txt
- [README.md](../reproduction/translation-oscillator-graph/README.md)
- [check_translation_oscillator_formulas.py](../reproduction/translation-oscillator-graph/check_translation_oscillator_formulas.py)
- [draw_translation_oscillator.py](../reproduction/translation-oscillator-graph/draw_translation_oscillator.py)
- DejaVuSans-Bold.ttf
- DejaVuSans-Oblique.ttf
- DejaVuSans.ttf
- DejaVuSansDisplay.ttf
- DejaVuSansMono.ttf
- LICENSE_DEJAVU.txt
- LICENSE_STIX.txt
- STIXGeneral.ttf
- STIXGeneralItalic.ttf
- STIXNonUniIta.ttf
- STIXSizFiveSymReg.ttf
- STIXSizFourSymReg.ttf
- STIXSizOneSymReg.ttf
- STIXSizThreeSymReg.ttf
- STIXSizTwoSymReg.ttf
- [requirements.txt](../reproduction/translation-oscillator-graph/requirements.txt)
- LICENSE.txt
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- [LICENSE.md](../reproduction/translation-oscillator-graph/software-notices/numpy/numpy/random/src/splitmix64/LICENSE.md)
- [verify_translation_oscillator_reproduction.py](../reproduction/translation-oscillator-graph/verify_translation_oscillator_reproduction.py)


## 11F. Local differential operators and the finite tree coefficient

The tree cycle of Sections 11A and 11D has the required reduced representation and transverse normalization. Its auxiliary operator moves return labels. The translation oscillator of Section 11E instead admits an actual smooth source-column differential expression. We now prove two restrictions on converting the finite tree coefficient into such an expression. The first detects movement between disjoint plaques; the second detects the normal symbol at a single point. Neither assumes a spin-c structure on the leaf tangent bundle.

We use the full oriented normal spinor coefficient of Section 11A, including its specified spin-c line and the odd degree convention. All symbols below act on the physical odd scalar model. Adding a different Clifford factor or replacing this physical module is a different construction.

### 11F.1. Locality means preservation of support

Let \(X\to\Sigma\) be the connected regular cover in Section 11A, with deck group \(\Gamma=F_r\), positive generators \(a_1,\ldots,a_r\), and \(r\geq1\). Choose a covering disc \(U\subset\Sigma\), a lift \(z:U\to X\), and a smaller disc \(W\) with compact closure in \(U\). The sets

\[
 W_g=g^{-1}z(W),\qquad g\in F_r,                         \tag{LD.1}
\]

are pairwise disjoint. This is the return-label convention of the natural source-column identification in Section 11A.4. The group acts by deck transformations on \(X\); its action on the normal manifold and full spinor coefficient supplies the accompanying unitary transport.

**Lemma 11F.1 (support of a differential expression).** A smooth differential operator of any finite order between finite-rank bundles over a manifold \(Y\) satisfies

\[
 \operatorname{supp}(Lu)\subseteq\operatorname{supp}u
       \quad(u\in C_c^\infty(Y)).                       \tag{LD.2}
\]

The inclusion survives a pointwise smooth bundle isomorphism and a common smooth change of base coordinates on input and output.

**Proof.** Outside the closed support of \(u\), the section is identically zero on a neighbourhood. All its derivatives vanish there. In any bundle frame a finite-order differential expression is a finite sum of smooth coefficient matrices times these derivatives, so its value vanishes there as well. A pointwise invertible bundle map preserves the support. A common diffeomorphism transports both supports by that same map. These observations prove every assertion. In particular a choice of closed operator domain cannot change (LD.2) on the smooth compact core.

The unoriented Cayley-tree edges have the unique form

\[
 (h,j)\longleftrightarrow\{h,ha_j\},\qquad
 \ell^2(\mathcal E)=\bigoplus_{j=1}^r\ell^2(F_r).       \tag{LD.3}
\]

For uniqueness, reversing an edge with label \(a_j\) would require a positive generator to equal \(a_j^{-1}\), which does not occur in a free group with the chosen positive basis. Different positive labels cannot describe the same oriented edge. Thus the vertex module has one regular orbit and the edge module has \(r\) regular orbits.

Let \(Q\delta_e=0\), and, for \(g\neq e\), let \(Q\delta_g\) be the edge joining \(g\) to its parent in the tree rooted at \(e\). The parent deletes the last letter of the reduced word. For \(M\delta_g=|g|\delta_g\), the lower block of the tree operator is \(QM\). In particular,

\[
 QM\delta_{a_j}=\delta_{(e,j)}.                         \tag{LD.4}
\]

**Proposition 11F.2 (the literal tree expression is nonlocal).** Under the regular-orbit identifications (LD.1)–(LD.3), the tree term of Section 11A cannot equal a finite-order smooth differential operator on the natural source-column manifold \(X\times N\), even on its smooth compact core. This conclusion also holds after pointwise bundle isomorphisms and a common diffeomorphism of \(X\times N\).

**Proof.** In the vertex copy choose a nonzero smooth section supported in \(W_{a_j}\times N\), with compact support in the first factor. In return coordinates it is \(\phi(x)\delta_{a_j}\otimes v(n)\), where \(\phi\in C_c^\infty(W)\) is nonzero and \(v\) is a nonzero smooth normal section. Formula (LD.4) sends it to the \(j\)-th edge copy supported in \(W_e\times N\). The normal transport is unitary and cannot annihilate the section. The two leaf supports are disjoint. Thus (LD.2) fails.

The normal differential part \(D_N\otimes\gamma_T\) preserves the original leaf support and acts in the vertex copy. It cannot cancel the nonzero edge output in the different plaque. To see the same fact for the partition-projected realization, choose the partition with one plaque function equal to one on a neighbourhood of \(\operatorname{supp}\phi\) and the others zero there. On that neighbourhood the projection is the identity on the chosen coordinate component. The argument therefore concerns the actual column expression, rather than an artifact of overlapping charts. Lemma 11F.1 gives the invariance under the stated changes of description.

This assertion concerns the unbounded differential expression. The bounded transform of a local differential operator usually does not preserve supports. No locality assertion about that transform is used here. A unitary that moves the edge copy to a different base point is also outside the pointwise and common-coordinate identifications in this proposition.

### 11F.2. An odd normal Clifford module cannot accept an unbalanced leaf symbol

Let \(\nu\) be an oriented Euclidean normal space of odd dimension \(q=2m+1\). Let \(S\) be a physical normal spinor space, with Hermitian Clifford matrices satisfying

\[
 c_j^*=c_j,\qquad c_jc_k+c_kc_j=2\delta_{jk},\qquad
 (-i)^m c_1\cdots c_q=1_S.                             \tag{LD.5}
\]

The last equation fixes the positive normal volume convention. The opposite irreducible convention merely exchanges the two signs in the argument below. For the three-dimensional model, \(S=\mathbb C^2\) and \(c_j\) are the positive Pauli matrices of Section 11A.

Take a finite auxiliary coefficient

\[
 E=E^+\oplus E^-,\qquad \gamma_E=1\oplus(-1),\qquad
 r_\pm=\dim E^\pm,\qquad
 a_j=c_j\otimes\gamma_E.                              \tag{LD.6}
\]

The two auxiliary summands have physical dimensions \(\dim(S)r_+\) and \(\dim(S)r_-\). They are not the grading of the ordinary odd scalar cycle. In a bundle version the ranks are evaluated at one base point, and the matrices are evaluated in an oriented normal frame.

**Theorem 11F.3 (normal volume balance).** If an endomorphism \(B\) of \(S\otimes E\) is invertible and anticommutes with every \(a_j\), then \(r_+=r_-\). Consequently a first-order Dirac-type source-column symbol for an orthogonal leaf/normal splitting, which restricts to (LD.6) on normal covectors, cannot exist on this physical module when the auxiliary ranks differ and the leaf dimension is positive.

**Proof.** Define the normalized normal volume

\[
 \Omega=(-i)^m a_1\cdots a_q=1_S\otimes\gamma_E.       \tag{LD.7}
\]

The equality follows from (LD.5) and \(\gamma_E^q=\gamma_E\). Moving \(B\) through the \(q\) factors gives \(B\Omega=-\Omega B\), because \(q\) is odd. Hence \(B\) maps the positive volume eigenspace into the negative one and conversely. Invertibility makes these restrictions inverse isomorphisms. Their dimensions are therefore equal, proving the rank equality.

For the stated Dirac-type symbol write its normal restriction as \(a(\xi)=\sum_j\xi_j a_j\), and choose a nonzero leaf covector \(\eta\). Put \(B=\sigma(\eta,0)\). The squared symbol identity

\[
 \sigma(\eta,\xi)^2=(|\eta|^2+|\xi|^2)1              \tag{LD.8}
\]

and linearity imply \(B^2=|\eta|^2 1\). Comparing the coefficient of \(t\) in the same identity for \((\eta,t e_j)\) gives \(Ba_j+a_jB=0\). Thus \(B\) is the invertible anticommuting endomorphism already excluded.

The pointwise proof is unchanged by arbitrary smooth zeroth-order potentials or changes of connection: neither changes the principal symbol. A full spin-c transition conjugates the Clifford matrices by the actual spinor transport and preserves the oriented volume. A common line tensor factor multiplies both volume eigenspace ranks by the same nonzero rank. Adding the same finite number of positive and negative normal copies leaves \(r_+-r_-\) unchanged. None of these modifications removes the obstruction.

The argument makes no claim that an invariant normal spin-c structure supplies a leaf Clifford structure. It instead tests an additional candidate leaf symbol, if one is proposed.

### 11F.3. In normal dimension three, even a general Hermitian elliptic extension is impossible

For \(q=3\) the balance condition does not require the mixed Clifford relation (LD.8). Write

\[
 c(n)=\begin{pmatrix}
 n_3&n_1-i n_2\\n_1+i n_2&-n_3
 \end{pmatrix},\qquad n\in S^2.                       \tag{LD.9}
\]

The positive eigenspace is a complex line. On the northern and southern hemispheres use the unit vectors

\[
 v_N(n)=\frac{(1+n_3,n_1+i n_2)}{\sqrt{2(1+n_3)}},
 \qquad
 v_S(n)=\frac{(n_1-i n_2,1-n_3)}{\sqrt{2(1-n_3)}}.     \tag{LD.10}
\]

They are defined on neighbourhoods of their respective closed hemispheres. Direct multiplication by (LD.9) gives eigenvalue \(+1\). At \(n=(\cos\theta,\sin\theta,0)\),

\[
 v_S=e^{-i\theta}v_N.                                 \tag{LD.11}
\]

The negative eigenline has transition \(e^{i\theta}\). Indeed if \(v=(v_1,v_2)\), the vector \((-\overline{v_2},\overline{v_1})\) is its unit orthogonal complement, has eigenvalue \(-1\), and its phase transforms by the complex conjugate.

**Theorem 11F.4 (Hermitian elliptic extension test).** Let \(p\geq1\), and let

\[
 \sigma:\mathbb R^p\oplus\mathbb R^3
       \longrightarrow\operatorname{Herm}(\mathbb C^2\otimes E)
                                                               \tag{LD.12}
\]

be continuous away from zero, with \(\sigma(\eta,\xi)\) invertible for every \((\eta,\xi)\neq0\). Suppose its restriction to unit normal covectors is exactly \(c(\xi)\otimes\gamma_E\). Then \(r_+=r_-\). In particular no formally self-adjoint first-order elliptic differential operator in all source leaf and normal directions can have this prescribed normal principal symbol with unequal auxiliary ranks. No Clifford identity in the leaf direction is required.

**Proof.** Fix one unit leaf direction and restrict (LD.12) to the sphere

\[
 S^3=\{(t,\xi):t^2+|\xi|^2=1\},
 \qquad (t,\xi)\longmapsto\sigma(t\eta_0,\xi).        \tag{LD.13}
\]

Its matrices are Hermitian and invertible. Put \(A=\sigma(t\eta_0,\xi)\) on this sphere. Continuity of the inverse follows from its finite-matrix adjugate formula, so compactness gives constants \(M<\infty\) and \(g>0\) with \(\|A\|\leq M\) and \(\|A^{-1}\|\leq g^{-1}\). Thus every eigenvalue has absolute value at least \(g\). The positive spectral projection is continuous, as can be seen directly from

\[
 P=\frac{1+\operatorname{sgn}A}{2},\qquad
 \operatorname{sgn}A=\frac{2}{\pi}\int_0^\infty
            A(A^2+s^2 1)^{-1}\,ds.
\]

Indeed, a finite Hermitian matrix has an orthonormal eigenbasis by the following elementary induction. Maximize its real Rayleigh quotient on the unit sphere. Differentiating at a maximizing vector in each orthogonal direction and its multiple by \(i\) makes both real and imaginary components of the corresponding matrix coefficient vanish. That vector is therefore an eigenvector; its orthogonal complement is invariant by Hermitian symmetry, and induction applies to that complement. In this basis the elementary arctangent integral gives \((2/\pi)\int_0^\infty\lambda/(\lambda^2+s^2)\,ds=\operatorname{sgn}\lambda\) for every nonzero real \(\lambda\). For \(0\leq s\leq1\), the matrix integrand has norm at most \(M/g^2\); for \(s\geq1\), it has norm at most \(M/s^2\). Each integrand is continuous in the base point, by matrix inversion. These uniform bounds make the integral a uniform norm limit of continuous finite-interval integrals. Hence \(P\) is continuous. Its trace is its finite integer rank, and is constant on the connected sphere. Locally its range has continuous frames: if \(\|P(x)-P(x_0)\|<1\), projection of a basis of \(\operatorname{ran}P(x_0)\) into \(\operatorname{ran}P(x)\) is injective and therefore a basis of that equal-rank space. This proves that the ranges form a vector bundle.

We next construct a frame on the closed upper hemisphere, without an extension theorem or a smooth functional-calculus assumption. Identify that hemisphere with the closed unit ball in \(\mathbb R^3\) by the explicit smooth coordinates

\[
 b\longmapsto
 \left(\frac{1-|b|^2}{1+|b|^2},\frac{2b}{1+|b|^2}\right),
 \qquad |b|\leq1;
 \qquad b=\frac{\xi}{1+t}.
\]

Write \(P(b)\) for the projection in these coordinates. Extend it to all of \(\mathbb R^3\) by the direct radial formula \(P_{\mathrm{ext}}(b)=P(b/\max(1,|b|))\). This is a continuous bounded Hermitian projection everywhere, and agrees with \(P\) on the closed unit ball. Let \(\varphi\geq0\) be a real smooth function supported in the unit ball with integral one; for example normalize \(\exp(-1/(1-|h|^2))\) inside that ball and set it to zero outside. All its boundary derivatives vanish, because every power of \((1-|h|^2)^{-1}\) times that exponential tends to zero at the boundary. Define the Hermitian smooth matrix

\[
 A_\varepsilon(b)=\int\varphi(h)P_{\mathrm{ext}}(b-\varepsilon h)\,dh.
\]

Smoothness follows by rewriting this as convolution with \(\varphi_\varepsilon(b-y)\) and differentiating that smooth compactly supported kernel. Each derivative is a finite integral bounded by the corresponding kernel derivative's \(L^1\) norm, since \(\|P_{\mathrm{ext}}\|\leq1\). Uniform continuity of \(P_{\mathrm{ext}}\) on the ball of radius two gives \(\|A_\varepsilon(b)-P_{\mathrm{ext}}(b)\|<\delta=1/16\) for all \(|b|\leq3/2\), by taking \(\varepsilon<1/2\) sufficiently small. This estimate follows by bounding each integrand difference by the modulus of continuity at distance \(\varepsilon\); no smooth extension of \(P\) is needed.

Let \(\mathcal C\) be the counterclockwise circle \(|z-1|=1/2\), and put

\[
 P'(b)=\frac{1}{2\pi i}\int_{\mathcal C}
                   (z1-A_\varepsilon(b))^{-1}\,dz.
\]

Here all smoothness and approximation claims follow from finite-matrix estimates. For any projection \(R\), \((z1-R)^{-1}=z^{-1}(1-R)+(z-1)^{-1}R\), so its norm on \(\mathcal C\) is at most two. The Neumann series applied with \(R=P_{\mathrm{ext}}(b)\) shows that \((z1-A_\varepsilon(b))^{-1}\) exists on \(\mathcal C\), uniformly for \(|b|\leq3/2\), with norm at most \(2/(1-2\delta)\). More generally the same Neumann argument at real numbers outside the \(\delta\)-neighbourhood of \(\{0,1\}\) shows that the real spectrum of \(A_\varepsilon\) lies in the two intervals \([-\delta,\delta]\) and \([1-\delta,1+\delta]\). Hermitian diagonalization therefore makes the contour integral the orthogonal projection onto the second interval. To evaluate its scalar integral, expand \((z-\lambda)^{-1}\) in a geometric series on \(\mathcal C\): the integral is one for \(|\lambda-1|<1/2\) and zero for \(|\lambda-1|>1/2\), since integration selects its \((z-1)^{-1}\) coefficient.

The inverse matrix is smooth in \(b\), with first derivative \(R_z(\partial A_\varepsilon)R_z\), where \(R_z=(z1-A_\varepsilon)^{-1}\). Every higher derivative is a finite sum of products of such inverses and derivatives of \(A_\varepsilon\), by repeated differentiation of the inverse identity. On each compact subset of \(|b|<3/2\) these products are uniformly bounded in \(z\in\mathcal C\). Differentiation under the fixed finite-length contour is thus valid to every order. This proves that \(P'\) is smooth on a neighbourhood of the closed unit ball.

The resolvent identity also gives an explicit approximation estimate. On the unit ball compare the inverses for \(A_\varepsilon\) and \(P\). Their difference has norm at most \(4\delta/(1-2\delta)\). Since the contour length is \(\pi\), integration yields \(\|P'-P\|\leq2\delta/(1-2\delta)=1/7<1\). Projection of either range into the other is therefore injective: if, for example, \(P'v=0\) and \(Pv=v\), then \(\|v\|\leq\|P'-P\|\|v\|\), forcing \(v=0\). The two injections give equal finite ranks.

To trivialize the smooth projection \(P'\), use the paths \(P_s(b)=P'(sb)\), \(0\leq s\leq1\). In the following equation \(P\) denotes this smooth path:

\[
 \dot U=[\dot P,P]U,\qquad U(0)=1.                    \tag{LD.14}
\]

For completeness this equation has a unique bounded solution continuously depending on \(b\). Put \(K(s,b)=[\partial_sP_s(b),P_s(b)]\). This continuous matrix family is bounded, say \(\|K(s,b)\|\leq L\), on \([0,1]\) times the closed ball. The successive integral series

\[
 U(s,b)=1+\sum_{n\geq1}
 \int_{0\leq s_n\leq\cdots\leq s_1\leq s}
   K(s_1,b)\cdots K(s_n,b)\,ds_n\cdots ds_1
\]

converges uniformly: its \(n\)-th term has norm at most \(L^n s^n/n!\), and \(\|U(s,b)\|\leq e^L\). Each term is continuous in \((s,b)\), as the substitution \(s_j=s t_j\) makes its integration simplex fixed. Substitution into \(U=1+\int_0^sK(v,b)U(v,b)\,dv\), justified by that bound, shows that the sum is continuously differentiable in \(s\) and solves (LD.14). If two solutions existed, their difference would satisfy the homogeneous integral equation. For each fixed \(b\), iterating it bounds that difference by \(L^n/n!\) times its supremum over \(s\in[0,1]\) for every \(n\), hence makes it zero. Thus no ODE existence or uniqueness result is being assumed.

The coefficient \(K\) is skew-Hermitian, so differentiating \(U^*U\) gives zero and \(U\) is unitary. Differentiation of \(P_s^2=P_s\) gives \(P_s\dot P_sP_s=0\) and \([[\dot P_s,P_s],P_s]=\dot P_s\). Therefore \(\partial_s(U^*P_sU)=U^*(\dot P_s-[K,P_s])U=0\), which proves \(P_s=UP'_0U^*\). Transporting a fixed orthonormal basis of \(\operatorname{ran}P'(0)\) by \(U(1,b)\) gives a continuous global frame of \(\operatorname{ran}P'(b)\), including the boundary. Applying \(P(b)\) to that frame gives a continuous basis of \(\operatorname{ran}P(b)\), by the injectivity and equal ranks proved above. This supplies the required frame on the entire closed hemisphere.

On the equatorial normal sphere the positive eigenbundle is

\[
 (L_+\otimes E^+)\oplus(L_-\otimes E^-).              \tag{LD.15}
\]

Here \(E^\pm\) are fixed vector spaces at the one base point where the principal symbol is being tested. Taking its determinant, (LD.11) gives equatorial transition

\[
 g(\theta)=e^{-i(r_+-r_-)\theta}.                     \tag{LD.16}
\]

A trivial determinant line on \(S^2\) must have a transition of winding zero in these two hemisphere frames. We give the phase and homotopy argument explicitly. For a continuous nowhere-zero loop \(f:[0,2\pi]\to\mathbb C\), normalize \(u=f/|f|\). Uniform continuity permits a finite partition \(0=\theta_0<\cdots<\theta_k=2\pi\) such that \(\operatorname{Re}(u(\theta)/u(\theta_j))>0\) for every \(\theta\in[\theta_j,\theta_{j+1}]\). On the open right half-circle define the continuous angle \(\alpha(z)=\arctan(\operatorname{Im}z/\operatorname{Re}z)\in(-\pi/2,\pi/2)\). Choose any real \(\phi(0)\) with \(e^{i\phi(0)}=u(0)\), and recursively set

\[
 \phi(\theta)=\phi(\theta_j)
       +\alpha\bigl(u(\theta)/u(\theta_j)\bigr)
       \quad(\theta_j\leq\theta\leq\theta_{j+1}).
\]

This defines a continuous phase lift with \(e^{i\phi}=u\). Two lifts differ by a continuous function with values in \(2\pi\mathbb Z\), hence by a constant. Since the endpoints of the loop agree, \(\phi(2\pi)-\phi(0)\) is an integer multiple of \(2\pi\). Their quotient by \(2\pi\) is the winding, independent of the initial lift.

For a continuous homotopy of nowhere-zero loops, normalize it on the compact rectangle \([0,1]\times[0,2\pi]\). Uniform continuity supplies one partition in \(\theta\) with the same right-half-circle condition for every homotopy parameter. The initial-value path \(u(s,0)\) itself has a continuous real lift on \([0,1]\), constructed by the same finite-interval procedure. With those initial values, the displayed recursion now gives a phase \(\phi(s,\theta)\) continuous in both variables. Its endpoint difference divided by \(2\pi\) is a continuous integer-valued function of \(s\); the intermediate value property makes it constant. This proves homotopy invariance. Lifts also prove additivity for a quotient: the difference of two phase lifts is a lift of the normalized quotient loop, so \(\operatorname{wind}(f/h)=\operatorname{wind}(f)-\operatorname{wind}(h)\).

If a nowhere-zero function \(f\) is defined on the closed disc, the loops \(f((1-s)e^{i\theta})\), \(0\leq s\leq1\), give a continuous nowhere-zero homotopy from its boundary values to its constant centre value. Thus its boundary winding is zero. Now compare the two hemisphere determinant frames with a hypothetical global nonzero determinant section. Writing that section as each frame times a nowhere-zero coefficient \(a_N\) or \(a_S\), both coefficients extend over their respective closed discs. The frame relation gives \(g=a_N/a_S\) along their common boundary. The quotient rule and the disc contraction prove \(\operatorname{wind}(g)=0\).

The bundle on the upper hemisphere of (LD.13) is trivial, so its restriction to the equator, and its determinant, are trivial. Formula (LD.16) has winding \(-(r_+-r_-)\). Hence \(r_+=r_-\). This proves the theorem. The matrix-valued principal symbol of a formally self-adjoint operator is Hermitian; ellipticity supplies the invertibility in (LD.12), so the differential-operator conclusion follows at every base point.

The assertion also excludes arbitrary continuous Hermitian symbol extensions, not only linear ones. It does not exclude a nonelliptic or anisotropic expression, an operator whose normal symbol changes, or a different smooth physical coefficient.

### 11F.4. Application to the actual SU(2)/F2 coefficient

For \(F_2\), (LD.3) gives

\[
 E^+=\mathbb C,\qquad E^-=\mathbb C^2,
 \qquad r_+-r_-=-1.                                  \tag{LD.17}
\]

The physical normal module has rank two. Its normal-volume eigenspaces therefore have dimensions two and four. Theorem 11F.3 rules out adding a longitudinal Clifford symbol on this unchanged module. More strongly, Theorem 11F.4 rules out any Hermitian full-source elliptic principal symbol extending its three-dimensional normal Pauli symbol. The obstruction occurs at one point, so it is unaffected by the choice of leaf metric, compatible connection, smooth potential, or global topology of the coefficient line.

For the Cayley tree of \(F_r\), the difference is \(1-r\). The literal support obstruction of Proposition 11F.2 holds for every \(r\geq1\). The normal-symbol obstruction holds for \(r\neq1\). A free cocompact action on another tree with \(v\) vertex orbits and \(e\) edge orbits gives the same necessary balance \(v=e\), if its chosen physical column model really consists of those finite regular-orbit copies. No such finite-copy identification is asserted for infinite quotient graphs or nontrivial stabilizers.

These results explain a precise limitation of the tree route. Its already proved scalar index \(+1\) is compatible with an unbalanced pointwise normal coefficient because the tree operator is nonlocal. It cannot be replaced on that same module by a full-source elliptic local operator. They do not show that every local finite-order source-column completion has zero index or fails to exist.

### 11F.5. A balanced local coefficient can have index one

The rank obstruction has a genuine escape already present in Section 11E. In one translation dimension the auxiliary physical space has one positive and one negative copy. With \(\gamma_E=\sigma_3\), take

\[
 D_{
 \mathrm{osc}}=-i\sigma_2\partial_x+x\sigma_1
  =\begin{pmatrix}0&-\partial_x+x\\\partial_x+x&0\end{pmatrix}.
                                                               \tag{LD.18}
\]

The source leaf principal matrix \(\sigma_2\) anticommutes with \(c_j\otimes\sigma_3\) after the normal tensor factor is restored. At the auxiliary level the two volume eigenspaces have equal dimensions; in normal rank three their physical dimensions are both two. Thus both balance tests pass. Section 11E.3–5 proves the complete self-adjoint domain, Hermite completeness and compact resolvent of (LD.18), and Section 11E.14–19 proves its actual smooth graph realization, convolution-domain estimates and localized scalar compacts, including the undifferentiated range variable.

The index sign can also be read directly from its two equations. The positive-to-negative operator \(\partial_x+x\) kills \(\pi^{-1/4}e^{-x^2/2}\). Every distributional solution is a scalar multiple of that Gaussian, by multiplying the equation by \(e^{x^2/2}\). The adjoint equation \(-\partial_x+x=0\) instead gives a multiple of \(e^{x^2/2}\), which is not square integrable unless its coefficient is zero. The compact-resolvent result of Section 11E supplies Fredholmness, so the scalar index is \(+1\), despite the equal pointwise ranks.

Thus balanced ranks are necessary for the tested longitudinal extensions and are compatible with normalization one. They are not sufficient: the global representation, domain, convolution estimates, localized compacts and pairing still need their own proofs. The general graph construction under a holonomy-invariant transverse spin-c structure remains unproved beyond the geometries for which these interfaces have been established.

![Plaque movement and the normal-volume balance test](../figures/kt-local-tree-differential-obstruction.png)

**Figure 11F.1.** Proposition 11F.2 and Theorems 11F.3–4, equations (LD.1)–(LD.18). The left panel is a schematic of two disjoint lifted plaques in the source cover; rectangles are not a metric drawing of the covering surface. The literal tree block sends the return label \(a\) to the edge orbit based at \(e\), and hence moves the displayed support. The middle panel shows the exact two-versus-four physical normal-volume dimensions for the SU(2)/F2 coefficient. The right panel shows the balanced two-versus-two dimensions and the normalized one-dimensional Gaussian of the translation oscillator. The dimension comparison is proved, and the Gaussian is its exact normalized kernel; no finite plot or word sample replaces the proof. [Editable SVG](../figures/kt-local-tree-differential-obstruction.svg) · [Reproducible figure source](../reproduction/local-tree-differential-obstruction/draw_local_tree.py) · [Component terms](../reproduction/local-tree-differential-obstruction/COMPONENT-TERMS.md).

Further reading: Heath Emerson, *An Introduction to C*-Algebras and Noncommutative Geometry*, Sections 7.8, 9.1 and 11.5, [DOI](https://doi.org/10.1007/978-3-031-59850-0). These sections discuss complex Clifford modules, local differential symbols and equivariant translation Bott cycles. The complete arguments above establish the support and symbol restrictions used here; the crossed-product discussion does not turn an arbitrary holonomy groupoid into a translation action.

## 11G. Affine source oscillators and hyperbolic return holonomy

An orthogonal linear part does not prevent an oscillator from providing the missing even factor in a graph class. We first extend the translation construction to free cocompact affine Euclidean actions. A nonabelian example has a Klein-bottle base, a compact oriented normal torus, the full normal spin-c transitions, and an actual local first-order graph expression with transverse Bott pairing one. We then prove a different restriction: a hyperbolic return prevents a specified finite-rank smooth graph expression from having bounded commutators with all smooth convolution kernels. The latter restriction needs careful interpretation of the invariant normal premise.

The historical question is Alain Connes's graph-Dirac problem in *A survey of foliations and operator algebras*, Section 8(2), [p. 29](https://alainconnes.org/wp-content/uploads/foliationsfine.pdf#page=29). Sections 11A–11F supply the earlier constructions and the precise distinctions between a discrete completion, a normal symbol, a full-tangent symbol and bounded unbounded-operator commutators.

### 11G.1. The affine auxiliary and its actual regular representation

Let a countable discrete group \(\Gamma\) act freely, properly and cocompactly on \(X=\mathbb R^d\), \(d\geq1\), by

\[
 g x=O_gx+b_g,\qquad O_g\in O(d),\qquad
 b_{gh}=b_g+O_gb_h.                                      \tag{NH.1}
\]

No orientation of \(X/\Gamma\) is required. Put \(E_d=\Lambda^*\mathbb C^d\), graded by degree parity \(\gamma\). In the standard orthonormal covector frame put \(C_j=\varepsilon_j-\iota_j\) and \(E_j=\varepsilon_j+\iota_j\). The symbol conventions and complete closure of

\[
 D_{\mathrm o}=\sum_j(C_j\partial_{x_j}+E_jx_j)
 \quad\hbox{on }H_{\mathrm o}=L^2(\mathbb R^d,E_d)          \tag{NH.2}
\]

are those of Section 11E.1–5. In particular its exact domain is \(\{u\mid \nabla u,|x|u\in L^2\}\), smooth compact sections form a core, its resolvent is scalar compact, and its positive kernel is the one-dimensional normalized Gaussian \(\pi^{-d/4}e^{-|x|^2/2}\); its negative kernel is zero. The complete Hermite proof there establishes index \(+1\), rather than an index inferred from a finite sample.

Let \(R_g=\Lambda^*(O_g)\) be the unitary covector transport, and set

\[
 (L_gu)(x)=R_g u(g^{-1}x),\qquad
 [D_{\mathrm o},L_g]=E(b_g)L_g,
 \quad E(v)=\sum_jv_jE_j.                                \tag{NH.3}
\]

These are even unitaries with \(L_gL_h=L_{gh}\). Orthogonality preserves Lebesgue measure and intertwines the exterior and contraction matrices. Thus \(d+d^*=\sum C_j\partial_j\) commutes with \(L_g\). Also \(L_g E(x)L_g^{-1}=E(x-b_g)\). This proves the displayed commutator and its bound \(\|[D_{\mathrm o},L_g]\|=|b_g|\). The equality uses \(E(v)^2=|v|^2\). Pullback preserves the derivative norm, while \(|gx|\leq |x|+|b_g|\) and the analogous inverse inequality preserve the weighted domain. The identity therefore holds on the full closed domain.

Here the reduced representation is proved literally. A measurable fundamental set \(Q\subset X\) exists: take finitely many evenly covered base charts, partition their images into Borel sets, and select one lift of each. Properness gives covering charts; freeness gives uniqueness of the selected translates. Chart boundaries can be selected measurably and have measure zero, or assigned once to a neighbouring piece. The sets \(gQ\) partition \(X\) modulo null sets. Define

\[
 (Ju)_g(x)=R_g^{-1}u(gx),\quad x\in Q,\qquad
 J:H_{\mathrm o}\longrightarrow
       \ell^2(\Gamma)\otimes L^2(Q,E_d).                  \tag{NH.4}
\]

Summing the integrals over \(gQ\) proves that \(J\) is an isometry. Its inverse assigns \(R_g\xi_g(x)\) to \(gx\), so it is onto. Substitution gives \((JL_hu)_g=(Ju)_{h^{-1}g}\). Thus the actual auxiliary representation is \(\lambda_\Gamma\otimes1\). Its infinite regular multiplicity is retained; compactness of the oscillator resolvent is a property of the operator on the whole \(X\), not a tensor compactness assertion about \(1\otimes D\).

### 11G.2. The complete normal product

Use the supplied compact normal data of Section 11E.9: a compact connected \(N\), its fixed metric, entire inverse spin-c module \(\mathcal D_N\), original grading and right \(C_q\) order, a compatible self-adjoint \(D_N\) with exact \(H^1\) domain, smooth core, scalar compact resolvent, and the original source-calibrated local class \(b_U\) with \(\langle b_U,[D_N]\rangle=1\). The transitions remain \(z^{-1}\rho_t(s)\), with the parity \(q(q-1)/2\), dual orientation line and last right \(C_1\) in odd rank; no minimal spinor replacement is made. Suppose \(\Gamma\) acts on \(N\) by isometries \(h_g\), with a coherent lift to this full structure, and let \(U_g\) be its even unitaries. Suppose the action is germ-effective and free. The example below verifies these action hypotheses.

**Theorem 11G.1 (affine graph oscillator).** Under (NH.1) and the complete normal hypotheses just stated, the actual compact suspension (NH.9) has the smooth finite-rank local first-order graph expression (NH.11). Its closure is self-adjoint on exactly (NH.12), its compact smooth convolution commutators are bounded and preserve that domain, and its bounded transform is a degree-\(q\) scalar reduced cycle with localized scalar compact defects. Its original local transverse Bott product is \(+1\), with the full normal inverse coefficient and determinant line retained. The operator differentiates the source and normal variables and leaves the range undifferentiated.

**Proof.** A compatible normal connection need not be invariant. The same-symbol comparison in Section 11E.9 proves that \(B_g=[D_N,U_g]\) is bounded and that \(U_gH^1=H^1\). On \(H_B=H_{\mathrm o}\widehat\otimes H_N\) define

\[
 D_B=D_{\mathrm o}\otimes1+\gamma\otimes D_N,
 \quad \Pi(f)=1\otimes M_f,
 \quad \Pi(u_g)=L_g\otimes U_g.                           \tag{NH.5}
\]

The two unbounded summands anticommute. The finite Hermite/normal spectral block proof of Section 11E.10 gives self-adjointness on the intersection of their domains, compact resolvent, and the complete smooth product core. It uses only these two operators, and therefore applies without a lattice assumption. Equation (NH.3) gives

\[
 [D_B,\Pi(u_g)]=E(b_g)L_g\otimes U_g
                    +\gamma L_g\otimes B_g.              \tag{NH.6}
\]

For smooth \(f\), the commutator is \(\gamma\otimes[D_N,M_f]\). The product domain is preserved by these coefficients and hence by every smooth finite crossed coefficient.

After \(J\otimes1\), apply \((W\xi)_g=U_g^{-1}\xi_g\), leaving the \(L^2(Q,E_d)\) factor untouched. Then

\[
 W(\lambda_h\otimes U_h)W^{-1}=\lambda_h\otimes1,
 \qquad W(1\otimes M_f)W^{-1}|_g=M_{f\circ h_g}.            \tag{NH.7}
\]

This is the regular induced representation of \(C(N)\rtimes_r\Gamma\). The point-regular reduced norm equals its direct-integral norm: a nearly norm-attaining finite-support vector at any normal point, in a local spin frame, has the same lower bound on a neighbourhood of positive normal volume, by continuity of the finitely many output coefficients. Thus essential supremum equals supremum. The reverse inequality is the defining regular bound. Amplification by \(L^2(Q,E_d)\) changes neither norm. Consequently \(\|\Pi(a)\|=\|a\|_r\); no K-amenability assertion or descent of \(U\) alone is used.

Compact resolvent and (NH.6) give the scalar phase defects by the integrable first-order resolvent argument of Section 11E.7. The restriction to \(C(N)\) is the full normal class. Indeed the positive Gaussian supplies its identical normal block; paired positive-energy Hermite blocks give exactly (TO.18), with compact creation errors, and the lifted oscillator phase has anticommutator \(2|D_{\mathrm o}|/\sqrt{1+D_{\mathrm o}^2+D_N^2}\geq0\). The complete connection and positivity proof of Section 11E.11 applies unchanged, including its right Clifford order. We have proved

\[
 [\Pi,F_B]|_{C(N)}=[D_N],\qquad
 \langle b_U,[\Pi,F_B]|_{C(N)}\rangle=1.                  \tag{NH.8}
\]

### 11G.3. A local operator on the actual affine suspension graph

Put \(M=X/\Gamma\), \(V=(X\times N)/\Gamma\), with diagonal action \((x,n)\mapsto(gx,h_gn)\), and horizontal foliation. These are compact smooth manifolds without boundary. Since \(X\) is simply connected, horizontal paths lift to \(X\). Their holonomy depends only on their return label. Germ effectivity removes exactly the possible kernel of this label. Hence the actual holonomy groupoid, not an auxiliary renamed space, is

\[
 G=(X_t\times X_r\times N)/\Gamma,\qquad
 g(t,r,n)=(gt,gr,h_gn),\quad
 r_G=[t,n],\quad s_G=[r,n].                             \tag{NH.9}
\]

The diagonal action is proper and free because its action on \(t\) is. Arrows multiply by \((t,t',n)(t',r,n)=(t,r,n)\), after making the common lift agree. The bundle on \(G\) is the quotient of \(E_d\otimes\mathcal D_N\) with the actual even transition \(R_g\otimes U_g\). Along left convolution, transport at common \((r,n)\) is the identity in lifted coordinates, and composes exactly. The ordinary graph density is the quotient of \(dt\,dr\,d\operatorname{vol}_N\), invariant under all these isometries.

Choose finitely many buffered charts of \(M\), smooth lifts \(z_i\), and real \(\chi_i\) with \(\sum\chi_i^2=1\). On the support of \(\chi_i\), the unique \(g_i(t)\) satisfying \(t=g_i(t)z_i([t])\) is locally constant. Put

\[
 b(t)=\sum_i\chi_i([t])^2g_i(t)0,
 \quad D_N(t)=\sum_i\chi_i([t])^2
                         U_{g_i(t)}D_NU_{g_i(t)}^{-1}.      \tag{NH.10}
\]

The compactly supported weighted terms extend smoothly across chart boundaries. The affine identity and the sum of the weights give \(b(gt)=gb(t)\), whereas \(D_N(gt)=U_gD_N(t)U_g^{-1}\). Thus \(|b(t)-t|\) is bounded on the whole cover: its norm descends to compact \(M\). The weighted connections in (NH.10) are compatible; transported Diracs have the same normal principal symbol. Their differences are smooth zeroth-order terms. On each compact range chart the family therefore differs from a fixed supplied normal operator by bounded terms, uniformly with its smooth parameters.

On equivariant smooth compact graph sections set

\[
 \mathscr D=\sum_j(C_j\partial_{r_j}+E_j(r_j-t_j))\otimes1
                         +\gamma\otimes D_N(t).           \tag{NH.11}
\]

Orthogonal covariance of the exterior matrices, covariance of \(D_N(t)\), and \(gr-gt=O_g(r-t)\) show that this is a global smooth finite-rank expression. It is odd and formally symmetric, with the full supplied normal Clifford symbol and the actual source-leaf Clifford symbol. There is no range derivative in \(t\). It is consequently an anisotropic graph operator, outside Section 11C's full-tangent ansatz. Explicitly, in the convention (NH.19), its principal symbol and square are

\[
 \sigma_{\mathscr D}(\xi_r,\eta_n)
   =iC(\xi_r)\otimes1+\gamma\otimes\sigma_{D_N}(\eta_n),
 \qquad \sigma_{\mathscr D}^2
   =\bigl(|\xi_r|^2+|\eta_n|_N^2\bigr)1.
\]

Here \(iC_j\) is Hermitian because \(C_j^*=-C_j\); its anticommutation with \(\gamma\) cancels the mixed term. The supplied normal symbol is unchanged, and the \(t\) covector has zero symbol.

Here are its complete analytic interfaces. With \(y=r-t\), the oscillator and normal summands anticommute, so the fibre identity is the one in (TO.24). Its exact closed domain is

\[
 \operatorname{Dom}\mathscr D=
 \{\psi:\nabla_r\psi,\ |r-t|\psi,\ \nabla_n\psi\in L^2(G)\}.
                                                               \tag{NH.12}
\]

Normal derivatives use local compatible connections; on the compact range charts all choices differ by bounded terms. Orthogonal and normal unitary transitions make the stated norms equivalent across overlaps. Each range fibre has the complete oscillator-plus-normal self-adjoint domain and compact resolvent. Its normal bounded perturbations have self-adjointness by the two large imaginary Neumann inverses. The fibre resolvents have norm at most \(1/|\operatorname{Im}z|\); their measurable direct integral proves self-adjointness on exactly (NH.12). Integrating the fibre square/graph estimate proves both inclusions in that domain, without requiring a \(t\) derivative.

Smooth compact graph sections are a core. Cut off in invariant \(|r-t|\); the derivative error in \(r\) is \(O(1/R)\), and all completed tails vanish. On the resulting compact region, use finitely many buffered charts. Finite smooth range profiles times smooth source/normal sections are dense in \(L^2(t;H^1(r,n))\), by simple-function approximation followed by ordinary convolution on the finite charts. Smooth coefficient multiplication maps \(H^1(r,n)\) continuously to \(L^2\), uniformly on those charts. Mollification in \(r,n\) and in \(t\) therefore converges in the full graph norm, including the bounded weight on that region. This is exactly the core argument of Section 11E.15; the additional fixed orthogonal transition matrices preserve every norm used.

For \(a\in C_c^\infty(G)\), convolution is

\[
 (\Pi_G(a)\psi)(t,r,n)=\int_Xa(t,t',n)\psi(t',r,n)\,dt'.    \tag{NH.13}
\]

In lifted coordinates the source \((r,n)\) is unchanged. The source derivative commutes with the kernel; the potential commutator is \(E(t'-t)a\). The normal derivative gives its derivative of \(a\), plus the zeroth-order difference \(a(D_N(t)-D_N(t'))\); the normal principal matrices cancel. Compact graph support bounds \(|t'-t|\). On finitely many compact range charts it meets only finitely many return labels: properness of the covering action makes the set of translates intersecting two chosen compact lifts finite. Smooth coefficients there have finite uniform Schur integrals. Thus the full commutator has a bounded Schur-kernel extension. Convolution maps the smooth compact core to itself, and closedness extends domain preservation and the commutator identity to (NH.12).

The representation is reduced: decomposition by source units is the direct integral of the actual source-fibre regular representations. A nearly norm-attaining compact source-fibre test continues on a source neighbourhood of positive volume, using kernel continuity and Schur estimates, so its essential supremum equals the defining supremum. This proves \(\|\Pi_G(a)\|=\|a\|_r\).

Localized compactness retains the undifferentiated range. On a buffered range chart use \(y=r-t\). The fibre resolvent \(Q_t(z)\) is scalar compact and norm-continuous: after that translation the oscillator is fixed and the normal differences are bounded zeroth-order terms, so the resolvent identity applies. Its compact range parameter set has a totally bounded compact-operator image. Holding \(r\) fixed under convolution gives

\[
 y'=r-t'=y+t-t',\qquad (T_\delta u)(y)=u(y-\delta),
 \quad\hbox{the operator factor is }T_{t'-t}.               \tag{NH.14}
\]

On overlaps, finitely many fixed affine/exterior/normal unitaries are inserted. Strong continuity of translations becomes norm continuity after composition with a compact operator, by finite-rank approximation. Multiplication coefficients are norm-continuous. Thus \(\Pi_G(a)Q(z)\) has a continuous compact-operator-valued kernel in two compact range coordinates. Uniform approximation of this kernel by separated continuous range functions times fixed finite-rank fibre operators gives finite-rank total operators; the two Schur bounds control the error. A finite partition proves both localized resolvents compact, the other by adjointness. The commutator kernel has the same form, so \(Q[D,\Pi_G(a)]Q\) is compact. The integrable phase bound \(\|[D,\Pi_G(a)]\|/(1+\tau^2)\) proves the compact phase commutator; the square defect follows from the localized resolvent. Density proves every scalar Kasparov defect on \(C_r^*(G)\).

### 11G.4. The class and the original transverse normalization

The affine changes do not leave an unidentified auxiliary factor. Equivariant \(H_B\)-valued functions on \(X_t\), with transition \(L_g\otimes U_g\), are exactly the ordinary graph sections \(\psi(t)(r,n)=\psi(t,r,n)\). Their equivariance and their quotient measures agree, including \(R_g\). This is an onto unitary with inverse defined on the same fundamental set.

The compact-base graph correspondence of Section 11D.5 is independent of its tree auxiliary. It identifies \(A=C_r^*(G)\) with \(\mathcal K_B(E)\), \(B=C(N)\rtimes_r\Gamma\), and \(E=P(K\otimes B)\), using finitely many flat charts and the normalized plaque bump. Its norm and counting correspondence are the ones proved there, with this same germ-effective suspension graph. Projection entries are the finite flat return unitaries times the partition weights. Equation (NH.6) makes \([1\otimes D_B,P]\) bounded.

For clarity the analytic induction still has its scalar compactness proof. Remove the bounded self-adjoint off-diagonal \(R=PD_0(1-P)+(1-P)D_0P\), \(D_0=1_K\otimes D_B\). Neumann resolvents make \(D_0-R\) self-adjoint on its unchanged domain and reduce \(P\). For \(J_0=\mathcal K(K)\otimes\mathcal B(H_B)\), separated tensors give \(j(D_0-z)^{-1}\) scalar compact; norm density handles all \(j\in J_0\). The projection and off-diagonal terms are multipliers of \(J_0\). Resolvent identities preserve those localized compacts. Smooth finite chart coefficients have bounded commutators in \(J_0\), giving the phase defects by the same integrable bound. A smooth finite-chart creation map satisfies \(D_GT_\eta-T_\eta D_B\) bounded. Its rectangular resolvent difference is compact, because the right resolvent of \(D_B\) is compact, with bound \(\|\mathrm{error}\|/(1+\tau^2)\). Integration gives both creation connection equations. Density extends them, and positivity for the zero Morita operator is zero. Therefore this induced cycle is \([{}_AE_B,0]\otimes_B[\Pi,F_B]\).

Conjugating its compression by the partition isometry gives, since the weights have no source or normal derivative,

\[
 D_{\mathrm{ind}}(t)=\sum_j(C_j\partial_{r_j}+E_j(r_j-b_j(t)))
                         +\gamma D_N(t),\qquad
 \mathscr D-D_{\mathrm{ind}}=E(b(t)-t).                    \tag{NH.15}
\]

This calculation uses \(L_gD_{\mathrm o}L_g^{-1}=\sum C_j\partial_j+E(r-g0)\), proved in (NH.3); the orthogonal part is carried by the exterior transition. The difference is a bounded self-adjoint covariant endomorphism. The Neumann/resolvent proof and localized compactness on its straight-line path give the same-domain norm-continuous phase homotopy, as in Section 11E.19. Hence the actual local graph expression represents exactly the induced class.

Finally the normalized plaque bump \(\xi\), \(\langle\xi,\xi\rangle_B=1\), gives \(p=\theta_{\xi,\xi}\) and \(i(b)=\theta_{\xi b,\xi}\). The map \(\overline\eta\mapsto\theta_{\xi,\eta}\) identifies the actual inverse counting correspondence with \(pA\), preserving both actions, inner product and onto rank-one span, as in (SG.18a). It is the original transversal correspondence. Thus (NH.8) proves

\[
 z_G=[\Pi_G,F_{\mathscr D}]\in KK^q(C_r^*(G),\mathbb C),
 \qquad (i_*b_U)\otimes_A z_G=1.                          \tag{NH.16}
\]

This completes the theorem. It covers the stated affine suspension data. It does not supply a proper Euclidean action for an arbitrary holonomy groupoid.

### 11G.5. A nonabelian example outside the tree condition

Consider the transformations of \(\mathbb R^2\)

\[
 a(x,y)=(x+1,y),\qquad b(x,y)=(-x,y+1).
 \quad ba b^{-1}=a^{-1}.                                 \tag{NH.17}
\]

Every word has form \(a^mb^n\), by the relation, and its action is \((x,y)\mapsto(m+(-1)^nx,y+n)\). Distinct pairs give distinct maps, so this is an exact presentation of the acting group. Nontrivial elements have no fixed point: \(n\neq0\) changes \(y\), while \(n=0,m\neq0\) translates \(x\). Only finitely many pairs can move one compact set to meet another, proving properness. The half-open square \([0,1)^2\), with boundary assignments of measure zero, is a fundamental set. Its horizontal edges are identified with opposite orientations and its vertical edges by translation; the quotient is the closed Klein bottle.

Take the flat oriented \(N=\mathbb T^3\), with its full framed spin-c structure, and set \(\alpha=\sqrt2\), \(\beta=\sqrt3\), read modulo one. Define

\[
 h_a(\theta)=(\theta_1+\alpha,\theta_2,\theta_3),\qquad
 h_b(\theta)=(-\theta_1,\theta_2+\beta,-\theta_3).            \tag{NH.18}
\]

The same relation holds. For \(a^mb^n\), the second coordinate changes by \(n\beta\); if \(n\neq0\) this is nonzero modulo one. If \(n=0,m\neq0\), the first changes by \(m\alpha\neq0\). Irrationality of these two square roots follows from unique prime factorization in a purported squared rational equality. Thus the action is free, faithful and germ-effective. Both maps are orientation-preserving normal isometries; the derivative of \(b\) is \(\operatorname{diag}(-1,1,-1)\).

In the original real Clifford convention \(e_j^2=-1\), let \(s_b=e_1e_3\in\operatorname{Spin}(3)\), projecting to this half-turn, and let \(s_a=1\). Its square is \(-1\). The assignments \(a^mb^n\mapsto s_b^n\) satisfy the relation and give a coherent lift. Take the nontrivial central character \(z_g=e^{in\pi/5}\) for \(g=a^mb^n\). The Spin-c lift is the class of \((s_b^n,z_g)\), and its determinant character is \(z_g^2=e^{2in\pi/5}\), since the determinant map on Spin-c is \([(s,z)]\mapsto z^2\). This is nontrivial already on \(b\). The full inverse normal module retains \(e^{-in\pi/5}\rho_t(s_b^n)\), and the inverse determinant line retains \(e^{-2in\pi/5}\); neither is replaced by a trivial line. At \(b^2\) the original spin factor is \(-1\), with original central phase \(e^{2i\pi/5}\); the inverse coefficient factor is \(e^{-2i\pi/5}\rho_t(-1)\). The identity derivative of \(b^2\) does not erase this factor. All are constant unitary transports on the flat normal bundle, so they preserve its compatible connection. They descend on the suspension to an actual holonomy-invariant metric Spin-c structure with the declared determinant line.

Use the flat source-calibrated full normal inverse from Theorem 7.17 and Section 11E.9. Its Fourier matrices square to the flat frequency norm times the identity; the Fourier decomposition gives its exact \(H^1\) self-adjoint domain and compact resolvent. The full lifted flat connection is invariant under (NH.18), so the induced normal operators agree. Its original local Bott sign is the supplied \(+1\), in the unchanged inverse coefficient order. All inputs of (NH.16) now hold, with \(d=2,q=3\). We obtain a compact five-dimensional foliated suspension, a seven-dimensional actual graph, and a finite-rank local first-order graph expression of degree three with original transverse pairing \(+1\).

This group is nonabelian: \(a\neq a^{-1}\) as translations, but conjugation by \(b\) sends one to the other. It contains the subgroup \(\langle a,b^2\rangle\cong\mathbb Z^2\), as the formula for \((m,2n)\) proves. It cannot act on a connected tree with finite vertex stabilizers: restriction would give such an action of this \(\mathbb Z^2\), ruled out by the complete spanning-tree/free-group argument in Section 11E.20. It is therefore outside Section 11D's finite-stabilizer-tree scope and outside Section 11E's acting group \(\mathbb Z^d\). It is virtually abelian and belongs to the explicitly wider affine-isometry family; no assertion about a non-virtually-abelian group follows.

### 11G.6. Oscillatory extraction of the source symbol

We now give the precise bounded-commutator consequence used for return holonomy. Let \(Y\) be either the entire smooth graph of a suspension, or its restriction to a smooth complete source transversal. Use the regular graph Hilbert completion with longitudinal Haar density and a smooth positive source density. Let \(\mathcal F\to Y\) be a smooth positive finite-rank Hermitian coefficient, with smooth unitary convolution transports composing exactly. Let \(D\) be a first-order differential expression on \(C_c^\infty(Y,\mathcal F)\), with symbol convention

\[
 D(e^{iL\Phi}\psi)=e^{iL\Phi}(D\psi+L\sigma_D(d\Phi)\psi).
                                                               \tag{NH.19}
\]

Assume convolution preserves this core and every \([D,\pi(k)]\), \(k\in C_c^\infty(G)\), has a bounded extension. No closure, compactness or Fredholm conclusion is assumed.

**Lemma 11G.2 (source-symbol covariance).** For a real smooth source function \(\phi\), multiplication by \(\sigma_D(d(\phi\circ s))\) commutes on compact sections with every such convolution. In local lifted coordinates \((u,v,z)\), where left convolution changes \(u\) and fixes \((v,z)\), this gives

\[
 B(u,v,z)W(u,u';v,z)=W(u,u';v,z)B(u',v,z),
 \quad B=\sigma_D(d(\phi\circ s)).                        \tag{NH.20}
\]

**Proof.** The unitary multiplication \(M_L=e^{iL\phi\circ s}\) commutes with \(\pi(k)\), because source is unchanged under left convolution. For a fixed compact smooth \(\psi\), both it and \(\pi(k)\psi\) lie in the core. Equation (NH.19) therefore gives the exact identity

\[
 [[D,\pi(k)],M_L]\psi
   =L M_L[\sigma_D(d(\phi\circ s)),\pi(k)]\psi.            \tag{NH.21}
\]

Its left side has norm at most \(2\|[D,\pi(k)]\|\,\|\psi\|\), uniformly for all positive integers \(L\). Division by \(L\) proves that the right commutator kills every compact smooth \(\psi\). This is a compactly supported oscillatory test with fixed input norm, rather than a formal comparison of orders.

The integral kernel of that zero operator is \(k(u,u',z)(B(u,v,z)W-WB(u',v,z))\). A nonzero continuous matrix entry at one triple would remain, after multiplying by a fixed scalar phase and testing a fixed pair of vectors in local frames, with positive real part on smaller product neighbourhoods. Nonnegative input and kernel bumps supported there would give a nonzero output on an open set of positive density. This contradicts the zero operator. Smooth kernels can be chosen nonzero at any prescribed composable pair. Their kernel entries therefore vanish pointwise, proving (NH.20). Smooth source functions whose differentials realize any local covector give the same assertion for every pulled-back source covector. \(\square\)

For a Hermitian Clifford symbol of a positive graph metric, squaring (NH.20) makes its source-covector norm independent of \(u\), for a fixed \((v,z)\). This is source-symbol covariance. By itself it is not equality with a norm at the range point or invariance of a metric already chosen on \(V\).

### 11G.7. A complete hyperbolic-return obstruction at that interface

Take \(S^1=\mathbb R/\mathbb Z\) and the smooth increasing lift

\[
 F(x)=x+\frac{1}{4\pi}\sin(2\pi x),\quad
 f=F\bmod1,\quad p=0,\quad f'(p)=\lambda=\frac32.           \tag{NH.22}
\]

Its derivative lies in \([1/2,3/2]\), so its strictly increasing inverse is smooth, and \(F(x+1)=F(x)+1\). On \((0,1/2)\) it moves every point forward, and on \((1/2,1)\) backward, retaining the two fixed endpoints. Hence no nonzero power is the identity on any open set. At the two fixed points its derivative is respectively \((3/2)^n\) and \((1/2)^n\), also nonidentity for \(n\neq0\). The action is germ-effective.

Define the compact suspension by \((t+1,x)\sim(t,f(x))\). Its graph is

\[
 G=(\mathbb R_u\times\mathbb R_v\times S^1_z)/\mathbb Z,
 \quad (u,v,z)\mapsto(u+1,v+1,f^{-1}z),\quad
 r=[u,z],\quad s=[v,z].                                  \tag{NH.23}
\]

Use the longitudinal Haar density \(du\), and any smooth positive density on the compact source suspension. A source chart \(v\in J\) near zero gives the graph norm \(\int|\psi(u,v,z)|^2\,du\,d\nu(v,z)\). For the source-transversal restriction set \(v=0\) and use any smooth positive density in \(z\). The same proof below covers both spaces.

**Theorem 11G.3 (hyperbolic-return obstruction).** In this regular physical graph completion, no expression satisfying the hypotheses of Lemma 11G.2 has a Hermitian symbol that is invertible on the source transverse covector \(dz\) near \((u,v,z)=(0,0,p)\). In particular this excludes every positive full-metric Clifford first-order symbol, with any positive finite coefficient rank and composing smooth unitary transports. Smooth zeroth-order potentials, arbitrary connections and leaf-derivative terms cannot repair that interface.

**Proof.** In a smaller source neighbourhood, choose a global smooth real source function equal to the coordinate \(z\), independent of \(v\). Lemma 11G.2 transports \(B(u,v,z)=\sigma_D(dz)\) unitarily along every fixed source column. Its least singular value equals that of \(B(0,v,z)\). Invertibility and continuity give a constant \(c>0\), uniform on a smaller compact source neighbourhood, such that

\[
 \|B(u,v,z)e\|\geq c\|e\|
 \quad\hbox{for every }u\hbox{ and those }(v,z).             \tag{NH.24}
\]

No assertion that the graph metric is a pulled-back normal metric was needed.

Choose a small range flow box at \([0,p]\) with coordinates \((t,x)\). Let \(\alpha\) be a real smooth longitudinal bump of \(L^2(dt)\) norm one and supported in that box, and let \(\chi(x)=1\) on a smaller normal interval. The one fixed compact graph kernel for the identity plaque arrows is \(k(t,t',x)=\alpha(t)\alpha(t')\chi(x)\). It is zero outside that buffered flow-box component. On the lifts with \(u=n+t,u'=n+t'\) it is exactly \(\alpha(u-n)\alpha(u'-n)\chi(f^nz)\). This is one compact kernel on the quotient; it is not a sequence of differently supported algebra elements.

Take a smooth real function \(h\) on \(V\), equal to \(x\) near the smaller box and independent of \(t\) throughout the support of \(\alpha\). Its range multiplication \(M_{h\circ r}\) is bounded. Since \(hk\) is again a compact smooth graph kernel, both commutators on the left of

\[
 [D,\pi(hk)]-M_{h\circ r}[D,\pi(k)]
       =[D,M_{h\circ r}]\pi(k)
       =-i\sigma_D(d(h\circ r))\pi(k)                     \tag{NH.25}
\]

are bounded. The harmless scalar \(-i\) here follows from the real-multiplication form of (NH.19); its modulus is one. Thus the last operator must be bounded. This identity removes every range derivative of the kernel and every zeroth-order or connection term simultaneously.

For each positive \(n\), the derivative of \(f^n\) at \(p\) is \(\lambda^n\). Choose a nonempty interval \(I_n\) about \(p\), within the fixed source chart, on which \((f^n)'\geq\lambda^n/2\), \(f^nI_n\) lies in the region \(\chi=1\), and \(h_x=1\). These follow from continuity for each fixed \(n\); no uniform interval is asserted. Choose a smooth unit coefficient vector in a source reference frame, and a smooth source amplitude \(\eta_n(v,z)\) supported in \(J\times I_n\) with \(L^2(d\nu)\) norm one. In the transversal case omit \(v\). Use the composing unitary transport to define the compact smooth input

\[
 \psi_n(u',v,z)=\alpha(u'-n)\eta_n(v,z)
                  W(u',n;v,z)e(v,z).
\]

Its norm is one. On its support the transported rank-one kernel gives exactly
\(\pi(k)\psi_n=\alpha(u-n)\eta_n(v,z)W(u,n;v,z)e(v,z)\), also of norm one. On this output, \(h\circ r=h(f^nz)=f^nz\) in the chosen local coordinate, so
\(\sigma_D(d(h\circ r))=(f^n)'(z)B(u,v,z)\). Equations (NH.24)–(NH.25) therefore give an output norm at least \(c\lambda^n/2\). This diverges on unit inputs, contradicting boundedness of the single fixed operator in (NH.25). The proof uses actual compact smooth tests; their supports move down the graph, and their transverse widths may shrink.

For a different positive fixed derivative \(0<\lambda<1\), repeat the argument with negative return labels, so \((f^{-n})'(p)=\lambda^{-n}\). The same conclusion holds whenever \(\lambda\neq1\). \(\square\)

The theorem does not assert that the source metric from (NH.24) equals a return-invariant metric. Exact covariance alone fixes its norm along a source column. The additional bounded range-multiplication identity and the unbounded return derivatives supply the contradiction. This distinction prevents an incorrect deduction of invariance from source covariance alone.

### 11G.8. The invariant spin-c premise and the remaining general problem

The hyperbolic suspension has an oriented topologically trivial normal line. Its positive frame holonomy cocycle even has an explicit coherent lift to \(\mathbb R_{>0}\times U(1)\): send a positive derivative \(a\) to \((a,1)\). Multiplicativity follows from the chain rule. Equivalently take the positive frame bundle times \(U(1)\), with holonomy acting by its actual derivative on the first factor. This describes a topological oriented-frame lifting, not a metric-preserving unitary Clifford transport.

The distinction is decisive. If a fixed positive normal metric and Hermitian Clifford module have unitaries \(U_h\) satisfying \(U_hc(\xi)U_h^{-1}=c((dh^{-1})^*\xi)\), squaring gives equality of the two inverse-metric norms. At a one-dimensional fixed point with derivative \(\lambda>0\), the transported covector is \(\lambda^{-1}\xi\); equality forces \(\lambda^{-2}|\xi|^2=|\xi|^2\), hence \(\lambda=1\). Thus (NH.22) admits no such invariant positive metric/full unitary Clifford lift. An orientation or a trivial determinant line cannot supply it.

The survey's invariant spin-c premise must therefore not be replaced silently by mere triviality of the normal line. Under the metric/unitary reading, the hyperbolic example is excluded and Theorem 11G.3 clarifies an attempted extension's boundary. Under a topological oriented-frame reading, the displayed lift exists, but the theorem still only excludes the specified regular finite-rank local expression with bounded unbounded-operator convolution commutators and a nondegenerate source transverse symbol. It does not exclude a different graph completion, an infinite-rank module, a degenerate source symbol, a nonlocal operator, or an expression whose unbounded commutators are unbounded but whose bounded phase has the required Kasparov defects. Sections 11C.7 and 11F preserve precisely these distinctions.

The affine theorem supplies a genuine additional eligible isometric family and a nonabelian actual local graph construction. It uses a free proper cocompact Euclidean cover and supplied complete normal inverse data. A general holonomy-invariant transverse spin-c groupoid need not supply that cover, its oscillator, its literal regular auxiliary or the same global source geometry. The general local graph-Dirac construction and original pairing under only the historical invariant normal premise remain unfinished.

![Affine oscillator and the hyperbolic-return boundary](../figures/affine-oscillator-and-return-holonomy.png)

**Figure 11G.1.** Equations (NH.1)–(NH.25). The left square is an exact fundamental region for (NH.17), with translation on vertical edges and reversed horizontal-edge identification. The middle panel shows the exact two-dimensional auxiliary Gaussian \(\pi^{-1/2}e^{-|y|^2/2}\), \(y=r-t\), and the normal action (NH.18); the torus coordinates are a formula, not a metric drawing of \(\mathbb T^3\). The right panel samples the exact derivative \((f^n)'(p)=(3/2)^n\). The fixed kernel and normalized tests of Theorem 11G.3 give the proved bound \(c(3/2)^n/2\). The tests have shrinking transverse supports; the panel plots their analytic norm bound, not those supports. That suspension fails the invariant metric/unitary Clifford premise and is not a counterexample to that version of the historical problem. [Editable SVG](../figures/affine-oscillator-and-return-holonomy.svg) · [Figure source](../reproduction/normal-holonomy/draw_normal_holonomy.py) · [Component terms](../reproduction/normal-holonomy/COMPONENT-TERMS.md).

## 11H. A normal Dirac on one actual source column over each compact leaf

A source column can remove the second leaf coordinate without removing the prescribed normal inverse. The following result makes this precise for compact smooth bundles with a global transverse section. It includes twisted bundles and varying fibre densities. Its operator is a first-order differential expression on the actual source-column manifold. It is not an operator obtained solely by naming an abstract Morita product.

### 11H.1. The geometric data and the claim

Let \(p:V\to B\) be a smooth locally trivial bundle of connected compact manifolds without boundary, with \(V\) and \(B\) compact and without boundary, and let \(\sigma:B\to V\) be a smooth section. Fibres and their tangent bundles need not be orientable or spin-c. Give the fibres a smooth positive density \(\mu_b\), and \(B\) a Riemannian metric with density \(\nu\). The foliation is \(F=\ker dp\). Its normal bundle is \(p^*TB\).

On \(B\), supply exactly the full inverse normal module \(\mathcal D_B\), normal Dirac \(D_B\), grading and final right \(C_q\) coefficient of Section 11E.9, with \(q=\dim B\). In particular the original transition \(zs\) acts on this entire inverse module as \(z^{-1}\rho_t(s)\); the original determinant line becomes its dual, and an original even line twist becomes its dual. Retain the parity \(\Pi^{q(q-1)/2}\), the sign of the full inverse principal symbol, and the order of every right Clifford factor. Assume the supplied \(D_B\) is odd, compatible and formally symmetric, self-adjoint on \(H^1(B,\mathcal D_B)\) with smooth core and scalar compact resolvent. Assume its original positive local transverse Bott class \(b_U\) has pairing one. These are the supplied normal inverse and calibration hypotheses, rather than conclusions inferred from a chosen minimal spinor.

The transverse metric and this entire unitary spin-c structure on \(p^*TB\) are holonomy invariant: a leaf path leaves its base point fixed, and its normal germ and its pullback coefficient transport are the identity. Thus this family satisfies the positive-metric invariant premise. A chosen connection on \(B\) need not have any additional invariance along unrelated base loops.

**Theorem 11H.1.** For these data and any smooth horizontal complement to \(F\), there is an odd, formally symmetric first-order differential operator on the finite-rank full inverse normal bundle \(D_H\) on the actual smooth source-column manifold \(G|_{s\,\sigma(B)}\). Its closure has the maximal distributional domain, is self-adjoint, and has smooth core. Actual smooth convolution has bounded commutators with \(D_H\). The ordinary scalar localized resolvents and all scalar Kasparov defects are compact. Its reduced class is exactly the holonomy Morita transport of the supplied full normal inverse, and its original local transverse Bott pairing is \(+1\).

Changing the horizontal complement or fibre density does not require a leaf spin-c structure. The proof identifies the class of each resulting operator by the same specified correspondence. It does not claim a bounded difference between all such differential expressions.

### 11H.2. The actual graph and its reduced completion

In a bundle chart, the transverse coordinate of a path in one fibre is constant. All such paths induce the identity transverse germ. Two paths with the same endpoints are therefore identified by holonomy, including paths that differ by a nontrivial loop in a fibre. Connectedness of the fibre guarantees that any two fibre points can be joined. Consequently

\[
 G=V\times_B V,\qquad
 r(v,w)=v,\quad s(v,w)=w,\quad
 (v,w)(w,z)=(v,z).                                      \tag{SC.1}
\]

This proves that \(G\) is the actual holonomy groupoid, rather than a fundamental-path groupoid with an unremoved isotropy kernel. The section \(T=\sigma(B)\) is a compact embedded complete transversal: \(dp\,d\sigma=1\), it meets every leaf once, and \(G|_T\) is the unit groupoid \(B\). The map

\[
 \Phi:V\longrightarrow Y=G|_{sT},\qquad
 \Phi(v)=(v,\sigma(pv))                                  \tag{SC.2}
\]

is a smooth bijection with smooth inverse \(r\). Hence \(Y\) is genuinely this source-column manifold. There is no second independently integrated source endpoint in (SC.2).

Put \(E=L^2(V/B,\mu)\), the Hilbert \(C(B)\)-module completion of smooth scalar functions with
\(\langle\eta,\zeta\rangle(b)=\int_{V_b}\overline\eta\,\zeta\,d\mu_b\).
Put \(H_B=L^2(B,\mathcal D_B,\nu)\). The unitary from the balanced Hilbert-space tensor product to the physical column space is

\[
 E\otimes_{C(B)}H_B\longrightarrow
 H=L^2(V,p^*\mathcal D_B,\mu_b\nu(b)),\qquad
 \eta\otimes u\longmapsto \eta(v)u(pv).                  \tag{SC.3}
\]

Its inner products agree by the fibre integral. Its range is dense: on a finite bundle-and-coefficient trivializing cover, smooth sections are approximated by finite sums of separated smooth fibre profiles and smooth base-coefficient profiles. To see this without a separability assertion standing in for a proof, take a fine coordinate grid on each compact chart support, a product partition of unity, and freeze the coefficient of a smooth section on the fibre part of each cell. Uniform approximation follows from uniform continuity. The resulting finite sums have the displayed form after the base partition is included in one factor. Smooth sections are dense in \(L^2\), by finite-chart convolution. Thus the isometry is onto.

For \(k\in C^\infty(V\times_B V)\), actual convolution on \(H\) is

\[
 (K_k u)(b,x)=\int_{V_b}k(b,x,y)u(b,y)\,d\mu_b(y),
 \qquad \|K_k\|\le
 \sqrt{\sup_{b,x}\int|k|\,d\mu_b(y)\,
             \sup_{b,y}\int|k|\,d\mu_b(x)}.              \tag{SC.4}
\]

The coefficient fibres at \(x,y\) are the same full \((\mathcal D_B)_b\), so the coefficient transport is the identity and composes exactly. No determinant phase is omitted; it is in the common base coefficient, and both slots transform by the same base transition.

Here the reduced graph algebra is
\(A=C_r^*(G)=\mathcal K_{C(B)}(E)\).
Indeed each source-fibre regular representation is the scalar integral-kernel operator on \(L^2(V_b,\mu_b)\), independent of the selected source point in that fibre. Its defining reduced norm is \(\sup_b\|K_{k,b}\|\). In a bundle chart, multiplying by the square root of the density ratio identifies all these fibre spaces with one fixed \(L^2\) space. A smooth kernel then varies continuously in the two Schur norms, so its operator norm is continuous in \(b\). Hence the essential supremum for (SC.3) equals that supremum: a strict lower bound at a base point persists on a neighbourhood of positive \(\nu\)-measure. Tensoring by the nonzero finite normal coefficient does not change the norm. This proves literal reduced descent and faithfulness.

Every smooth rank-one kernel \(\eta(b,x)\overline{\zeta(b,y)}\) is convolution. Conversely, product partitions in compact fibre charts approximate any smooth \(k\) uniformly by finite sums of such kernels. One first partitions in \(b,x,y\), freezes the smooth coefficient on the fibre-coordinate grid, and puts the base partition into either profile. Finite chart volume makes uniform error small in both Schur norms. Smooth profiles approximate continuous \(E\)-profiles in the fibre supremum norm, by local smoothing and partitions. The two norm closures are thus equal. In particular the equality with \(\mathcal K(E)\) is not a maximal-to-reduced inference or an amenability assumption.

### 11H.3. A unitary fibre connection and the physical first-order expression

A smooth complement to \(\ker dp\) exists: choose a smooth metric on compact \(V\) and take its orthogonal complement. In a bundle coordinate chart \((u,x)\), its lift of \(\partial_{u^a}\) is
\(X_a=\partial_{u^a}+\sum_j A_a^j(u,x)\partial_{x^j}\).
Define the intrinsic density derivative \(\beta_a\) by the Lie derivative of the vertical density under this lift. In the chart it is

\[
 \beta_a=\partial_a\log\mu+
                   \operatorname{div}_{\mu_b} A_a,
 \qquad
 \nabla^E_a=X_a+\tfrac12\beta_a.                         \tag{SC.5}
\]

The expression \(\partial_a\log\mu\) uses the density coefficient in the chart. The sum is the coordinate-independent infinitesimal change of the vertical density under the horizontal flow. This also follows by changing variables in the defining fibre integrals, so (SC.5) patches under every fibre-coordinate transition.

Integration by parts on a compact boundaryless fibre gives

\[
 \partial_a\langle\eta,\zeta\rangle
   =\langle\nabla^E_a\eta,\zeta\rangle+
                         \langle\eta,\nabla^E_a\zeta\rangle.
                                                               \tag{SC.6}
\]

Explicitly the integral of \(A_a(\overline\eta\zeta)\) is
\(-\int(\operatorname{div}_{\mu_b} A_a)\overline\eta\zeta\);
the two halves of \(\beta_a\) then leave exactly the density derivative \(\partial_a\mu\). This proves unitarity of the fibre connection, including varying density. Compact fibres also give existence of the horizontal flow for each bounded base-curve interval: in finitely many bundle charts its smooth vertical vector fields have bounded coordinate coefficients; successive local ODE solutions extend until the interval ends because their images stay in compact \(V\).

Write the supplied normal expression in a coefficient frame as
\(D_B=-i\sum_a M^a(u)\partial_a+W(u)\), with the entire original normal connection, determinant connection and density terms in \(W\). The \(M^a\) are the prescribed Hermitian inverse Clifford matrices. Define

\[
 D_H=-i\sum_a M^a(u)
           \bigl(X_a+\tfrac12\beta_a\bigr)+W(u)
 \quad\hbox{on }C^\infty(V,p^*\mathcal D_B).             \tag{SC.7}
\]

Invariantly this is the same Clifford contraction of the tensor connection
\(\nabla^E\otimes1+1\otimes\nabla^{\mathcal D_B}\) used for \(D_B\).
The connection product rule proves (SC.7) on separated sections and shows it patches under both bundle coordinates and full coefficient transitions. The even fibre profile introduces no extra Clifford factor or block permutation. Equation (SC.6) and the supplied symmetry of \(D_B\) prove formal symmetry. Its grading and right \(C_q\) action are exactly those of the supplied normal inverse.

For a covector \(\zeta\in T_v^*V\), its symbol is the supplied normal Clifford action on the restriction of \(\zeta\) to the chosen horizontal lift. Its square is that horizontal covector's squared normal norm. In particular, on the normal covectors \(p^*T_b^*B\) it is the entire prescribed positive-metric normal symbol. It vanishes on covectors annihilating the horizontal complement. There is no claim of full ellipticity on \(V\), and no assumption of a spin-c structure on \(TV\).

### 11H.4. Self-adjoint closure with the exact anisotropic domain

The domain is

\[
 \operatorname{Dom}D_H=
 \{u\in H:\ D_Hu\text{ as a distribution is in }H\}.
                                                               \tag{SC.8}
\]

Here is a complete core and self-adjointness argument that does not replace this domain by \(H^1(V)\). Reduce a local coefficient frame and density to Euclidean \(L^2\) by their smooth positive square-root factors. For a compactly supported local first-order expression
\(L=\sum_j a_j(z)\partial_j+a_0(z)\), mollification \(J_\epsilon u=\rho_\epsilon*u\) satisfies a uniformly \(L^2\)-bounded commutator. Integration by parts in the convolution variable expresses its derivative part as kernels

\[
 (a_j(z)-a_j(w))\,\partial_j\rho_\epsilon(z-w)
       +(\partial_j a_j)(w)\rho_\epsilon(z-w).           \tag{SC.9}
\]

An immaterial overall sign depends on the chosen commutator order. Both Schur integrals of the first term are bounded independently of \(\epsilon\): the smooth coefficient is Lipschitz on the buffered chart and
\(\int|h|\,|\partial_j\rho_\epsilon(h)|\,dh\) is constant under scaling. The second term and the zeroth-order commutator have the same uniform bounds. The inequality (SC.4), applied in Euclidean coordinates, proves the \(L^2\) bound. For smooth functions the commutator tends to zero because \(LJ_\epsilon u\) and \(J_\epsilon Lu\) both tend to \(Lu\). Density and the uniform bound prove strong convergence to zero for every \(L^2\) input.

Take a finite partition \(\chi_j\) subordinate to buffered frames, with auxiliary cutoffs equal to one on their supports. Smooth each \(\chi_j u\) inside the larger chart and sum. For \(u\) in the maximal domain, \(L(\chi_j u)=\chi_j Lu+[L,\chi_j]u\) is in \(L^2\). Formula (SC.9) shows convergence of each smoothed term and its image to \(\chi_j u\) and \(L(\chi_j u)\). The auxiliary cutoffs have no additional limiting error because they are one on the original support and the mollifier supports shrink. Summing yields \(u_n\to u\) and \(D_Hu_n\to D_Hu\); the partition commutators sum to zero. Thus smooth sections are a core for the maximal domain.

For a formally symmetric smooth expression on a compact boundaryless manifold the adjoint of its smooth-core operator has exactly the maximal domain: the defining adjoint identity against every smooth test section is precisely the distributional identity \(D_Hu\in L^2\). The preceding graph approximation identifies maximal and minimal domains. Hence the closure equals its adjoint and is self-adjoint on (SC.8). This argument also proves closedness, without a prescribed boundary condition or a full elliptic estimate for \(D_H\).

### 11H.5. Principal cancellation in actual convolution

Let \(K_k\) be (SC.4). Differentiation of the fibre integral, followed by integration by parts in the input variable \(y\), gives

\[
 [\nabla^E_a,K_k]=K_{\delta_a k},\qquad
 \delta_a k=\partial_a k+
       A_a(x)\cdot\partial_x k+A_a(y)\cdot\partial_y k
                  +\tfrac12(\beta_a(x)+\beta_a(y))k.
                                                               \tag{SC.10}
\]

For clarity the input derivative initially contributes
\(-\int k A_a(y)\partial_yu\,d\mu_b(y)\).
Integration by parts replaces it by
\(\int(A_a(y)\partial_yk+\operatorname{div}_{\mu_b}A_a(y)\,k)u\,d\mu_b(y)\).
The differentiated density supplies \(\partial_a\log\mu\,k\), and subtraction of the input half-density connection leaves precisely \(\beta_a(y)k/2\). The output half is \(\beta_a(x)k/2\). This proves the signs in (SC.10).

At common \(b\), \(W\) and every \(M^a\) commute with the scalar kernel and with its fibre integration. Thus

\[
 [D_H,K_k]=-i\sum_a M^a K_{\delta_a k}=:C_k.             \tag{SC.11}
\]

These are smooth matrix kernels on the compact fibre product, with bounded Schur integrals. The commutator is bounded on \(H\). Convolution sends smooth sections to smooth sections. Approximating a maximal-domain section by the core in Section 11H.4 and using (SC.11) proves that it preserves (SC.8) and that the same commutator identity holds there. Coefficient-frame changes conjugate the entire matrix kernel, so the full determinant connection is retained in this argument, rather than set equal to zero.

### 11H.6. Scalar compactness after convolution

Although \(D_H\) need not have compact resolvent, smooth fibre convolution removes the undifferentiated fibre multiplicity. Here are the required global estimates.

Choose finitely many smooth vertical vector fields \(Z_j\) spanning \(\ker dp\) at every point, by local frames and a finite subordinate partition. There is a constant \(C\) such that

\[
 \|w\|_{H^1(V)}
 \le C\left(\|D_Hw\|+\sum_j\|Z_jw\|+\|w\|\right)
 \quad (w\in C^\infty(V,p^*\mathcal D_B)).               \tag{SC.12}
\]

To prove it, the squared norm of the joint principal symbol on \(w\) is the horizontal Clifford norm squared plus \(\sum_j|\zeta(Z_j)|^2\), times \(\|w\|^2\). If this vanishes, \(\zeta\) annihilates both horizontal and vertical tangent spaces and hence is zero. On the unit cotangent sphere, compactness gives a uniform positive lower bound. For frozen constant coefficients, Fourier transformation and Plancherel therefore bound the gradient norm by the norm of the joint first-order expression. In a sufficiently small chart the variation of each principal coefficient is as small as required to absorb its gradient error on the left. All zeroth-order terms are bounded there. A finite chart partition globalizes the estimate; commutators with its cutoffs are bounded zeroth-order terms. This proves (SC.12), including its coefficient-bundle version.

The estimate holds whenever the displayed derivatives are distributions in \(L^2\). Apply the same local mollification as in (SC.9) to the finite augmented list \(D_H,Z_j\). Their commutators are uniformly bounded and strongly tend to zero. The smoothed approximations converge in the joint graph norm and are uniformly bounded in \(H^1\) by (SC.12). A weakly convergent subsequence in \(H^1\) has the prescribed \(L^2\) limit, establishing the estimate for that limit. This step does not assert that \(D_H\) alone controls every derivative.

Put \(R=(D_H-i)^{-1}\). For \(w=K_kRu\), (SC.11) gives
\(D_Hw=K_kD_HRu+C_kRu\), bounded in \(L^2\) by a constant times \(\|u\|\). Each \(Z_jK_k\) is also a bounded smooth matrix fibre kernel, obtained by differentiating the output variable and its frame. Therefore (SC.12) gives

\[
 K_k(D_H-i)^{-1}:H\longrightarrow H^1(V,p^*\mathcal D_B)
       \quad\hbox{bounded, hence scalar compact into }H. \tag{SC.13}
\]

The last implication has an elementary local proof. After a finite chart partition, extend the compactly supported sections by zero to a fixed cube and periodically to a torus. A bounded \(H^1\) family has Fourier tail squared at frequencies \(|n|>R\) at most \(C/(1+R^2)\). Its finitely many retained Fourier coefficients lie in a bounded finite-dimensional set. These finite-rank approximations prove compactness into \(L^2\), and the finite partition proves the assertion on \(V\). Adjointness yields the other localized resolvent. The same argument works at every nonreal parameter and for every smooth matrix fibre kernel, including \(C_k\).

For \(F_H=D_H(1+D_H^2)^{-1/2}\), the square defect is a localized resolvent product. The phase commutator is a norm-convergent integral of compact operators: with \(a=\sqrt{1+t^2}\),
\([ (D_H\mp ia)^{-1},K_k]=-(D_H\mp ia)^{-1}C_k(D_H\mp ia)^{-1}\).
Its norm is at most \(\|C_k\|/(1+t^2)\), and it is compact by (SC.13). The strong formula
\(F_H=(2/\pi)\int_0^\infty D_H(D_H^2+1+t^2)^{-1}\,dt\)
then gives the actual compact commutator by integrating the differences in norm. No norm convergence of the uncommuted integral is asserted. Density of smooth kernels and boundedness of \(F_H\) give every defect for every \(a\in A\). The self-adjointness defect is zero. These are ordinary scalar compacts on \(H\), not just fibrewise or Hilbert-module compacts.

### 11H.7. Exact Morita class, full coefficient and the positive local pairing

Construct a smooth real nonnegative fibre profile \(\xi\) with fibre norm one. In each of finitely many section neighbourhoods choose a nonnegative vertical bump that is positive at \(\sigma(b)\); multiply by a base partition and sum. Its fibre squared norm is smooth and strictly positive. Dividing by its positive square root gives

\[
 \langle\xi,\xi\rangle=1,\quad
 p_\xi=\theta_{\xi,\xi}\in A,\quad
 i:C(B)\longrightarrow p_\xi A p_\xi,\quad
 i(f)=\theta_{\xi f,\xi}.                              \tag{SC.14}
\]

These are smooth kernels. The creation map \(T_\eta:H_B\to H\) for a smooth profile is \(T_\eta u=\eta u\), bounded in general and isometric for \(\xi\). The tensor product rule proves

\[
 D_HT_\eta-T_\eta D_B
       =-i\sum_a M^aT_{\nabla^E_a\eta}=:E_\eta,
 \qquad \|E_\eta\|<\infty.                             \tag{SC.15}
\]

The formula is first proved on the smooth normal core. It extends to the normal domain by closedness. The fibre norm of each differentiated smooth profile is bounded on compact \(B\).

The rectangular resolvent identity
\[
 (D_H-z)^{-1}T_\eta-T_\eta(D_B-z)^{-1}
       =-(D_H-z)^{-1}E_\eta(D_B-z)^{-1}                 \tag{SC.16}
\]
is compact, because the right resolvent is scalar compact on \(H_B\). At \(z=\pm i\sqrt{1+t^2}\) its norm is at most \(\|E_\eta\|/(1+t^2)\). Integrating the averaged identities gives
\(F_HT_\eta-T_\eta F_B\) compact; its adjoint is the other creation connection equation. Smooth profiles are dense in \(E\), and creation is continuous in the module norm, so the connection equations hold for every profile. The positivity condition for the Morita cycle with operator zero is identically zero. Hence, in the original coefficient order,

\[
 [H,K,F_H]=[{}_AE_{C(B)},0]\otimes_{C(B)}[H_B,M,F_B]
       \in KK^q(A,\mathbb C).                          \tag{SC.17}
\]

There is also an exact compressed unbounded expression. Since \(\xi\) is real and \(\nabla^E\) is real, \(\langle\xi,\nabla^E_a\xi\rangle\) is real. Differentiating its norm in (SC.6) makes twice this number zero. Thus

\[
 T_\xi^*D_HT_\xi=D_B,\qquad
 \{u:T_\xi u\in\operatorname{Dom}D_H\}
                     =H^1(B,\mathcal D_B).             \tag{SC.18}
\]

For the forward inclusion of domains use (SC.15). For the reverse, its distributional identity, followed by \(T_\xi^*\), gives \(D_Bu\in L^2\); the stipulated maximal normal domain is \(H^1\). The bounded commutator with \(p_\xi\) also allows removal of its off-diagonal blocks by a bounded self-adjoint perturbation, on the unchanged domain, and proves that the compressed operator is self-adjoint. The two sufficiently large imaginary Neumann inverses establish that perturbation statement. No equality of bounded transforms of compressions is presumed; the connection calculation above supplies their compact comparison.

The inverse holonomy correspondence is exactly \(E^*\), since the transversal has one intersection per leaf and no return isotropy. Its unitary onto the essential correspondence of \(i\) is
\(\overline\eta\mapsto\theta_{\xi,\eta}\in p_\xi A\).
The right inner product is preserved because
\((\theta_{\xi,\eta})^*\theta_{\xi,\zeta}=\theta_{\eta,\zeta}\);
the right action follows by adjointing the second slot; and the left \(f\) action is \(i(f)\). It is onto because \(p_\xi a=\theta_{\xi,a^*\xi}\). The actual transverse counting norm has one term in each fibre, while the normalized plaque integral is one. These are the same normalization, so this is the original counting correspondence, not a separately chosen \(K\)-class. Equations (SC.17)–(SC.18) give

\[
 i^*[H,K,F_H]=[D_B],\qquad
 (i_*b_U)\otimes_A[H,K,F_H]=1.                         \tag{SC.19}
\]

The full inverse determinant, even line twists, grading and final right \(C_q\) have been retained at every step. No second even/odd longitudinal index factor multiplies this one. This proves Theorem 11H.1.

### 11H.8. Open transversal chart ends and the exact gluing domain

Even in this compact family, taking an unrestricted sum of open transverse charts and closing each local operator separately is unsafe. On \((0,1)\), \(-i\,d/dt\) with compact smooth core has minimal domain \(H^1_0\), adjoint domain \(H^1\), and deficiency dimensions \((1,1)\): its two deficiency equations have solutions \(e^{-t}\) and \(e^t\), both in \(L^2\). Here are the endpoint details. The derivative graph norm is the \(H^1\) norm, and the integral representation bounds endpoint traces continuously by it, so limits of compact smooth functions have zero traces. Conversely, if \(u(0)=0\), then \(|u(t)|^2\le t\int_0^t|u'|^2\). A cutoff changing on \((\epsilon,2\epsilon)\) with derivative bounded by \(C/\epsilon\) has derivative-error squared at most \(C'\int_0^{2\epsilon}|u'|^2\), tending to zero. The same proof applies at the other zero endpoint; the ordinary tails also vanish. Mollifying the resulting interior functions proves density. Finally the adjoint identity is the distributional derivative in \(L^2\), which is exactly \(H^1\), with no trace restriction. Thus an incomplete chart does not acquire a self-adjoint domain merely from a normal symbol.

Here is the exact gluing in the present family. Choose buffered base charts \(U_j\), real smooth \(\chi_j\) compactly supported in them, with \(\sum_j\chi_j^2=1\). On the actual disjoint source-chart columns put
\[
 H_{\rm ch}=\bigoplus_j L^2(p^{-1}U_j,p^*\mathcal D_B),
 \quad Ju=(\chi_j u)_j,\quad P=JJ^*,\quad
 P_{jk}=\chi_j\chi_k.                                  \tag{SC.20}
\]
The entries identify the same \(v\) on an overlap, with its actual coefficient transition and density; multiplication outside the overlap is zero. \(J\) is an onto unitary from \(H\) to \(PH_{\rm ch}\), and \(P^2=P=P^*\) follows from the sum of squares. It commutes with chart convolution, since the \(\chi_j\) depend only on \(b\). The local differential expressions \(D_{\rm loc}\) are defined on compact smooth chart cores. The vector \(Ju\) lies in those cores for smooth global \(u\), even near chart ends, because each \(\chi_j\) has buffered support. Direct calculation gives
\[
 J^*D_{\rm loc}Ju
  =D_Hu-i\sum_a M^a
                      \Bigl(\sum_j\chi_j\partial_a\chi_j\Bigr)u
  =D_Hu.
\]
Therefore \(PD_{\rm loc}P\) on the specified core \(J C^\infty(V,p^*\mathcal D_B)\) closes to \(JD_HJ^*\), self-adjoint on exactly \(J\operatorname{Dom}D_H\). This proves its domain and compactness by the already proved onto unitary. It does not choose unrelated self-adjoint extensions of the separate charts. The projected chart completion is explicitly identified with the global physical source column; the unrestricted direct sum is not silently equated with it.

### 11H.9. A twisted compact-fibre example with a nontrivial determinant

Take
\[
 V=(\mathbb R\times S^2)/((t+1,x)\sim(t,-x)),
 \qquad B=\mathbb R/\mathbb Z,\quad p[t,x]=[t].
\]
The antipodal map reverses the orientation of \(S^2\). Thus the vertical tangent orientation is not preserved around the base; the construction genuinely needs no vertical spin-c orientation. A smooth section exists: choose a smooth path \(\gamma\) from the north pole to the south pole, constant near each endpoint, and extend by \(\gamma(t+1)=-\gamma(t)\). The flat complement \(X=\partial_t\) and round fibre density give \(\beta=0\).

Give the original rank-one oriented normal spin-c structure the central phase \(e^{i\kappa}\) around the base, with \(\kappa=\pi/5\). Its original determinant has phase \(e^{2i\kappa}\); the full inverse module has phase \(e^{-i\kappa}\), dual determinant phase \(e^{-2i\kappa}\), and retains the entire graded right \(C_1\). In the constant frame let \(J_N\) be its supplied Hermitian inverse normal symbol, \(J_N^2=1\), with its original grading and right action. Choose the flat compatible normal connection, represented on the interval by this seam condition rather than deleting its phase. Then
\[
 D_H=-iJ_N\partial_t,\qquad
 u(1,x)=e^{-i\kappa}u(0,-x).
                                                               \tag{SC.21}
\]
Its exact domain is \(H^1([0,1];L^2(S^2)\otimes\mathcal D_1)\) with the displayed trace condition. Indeed the distributional \(t\) derivative in \(L^2\) gives an absolutely continuous Hilbert-valued representative by integrating that derivative; the difference has zero weak derivative and is a constant. Its norm and trace are bounded by its derivative graph norm. Integration by parts across the seam gives exactly the stated condition for the maximal global domain. This identifies (SC.8) explicitly here.

The antipodal pullback \(A u(x)=u(-x)\) is a self-adjoint unitary. Its positive and negative eigenspaces are both infinite-dimensional: smooth disjoint pairs of small caps, with equal or opposite values on antipodes, give arbitrarily many independent functions. On these two subspaces the permitted frequencies are respectively \(2\pi n-\kappa\) and \(2\pi(n+\tfrac12)-\kappa\). The eigenvalues of (SC.21) are those frequencies times the finite eigenvalues of \(J_N\), with infinite fibre multiplicity. Completeness follows from the orthogonal \(A\)-splitting and the ordinary Fourier basis after multiplication by the corresponding scalar phase. The resolvent is therefore not scalar compact before localization. Nevertheless every smooth graph kernel localized resolvent is scalar compact by (SC.13), and (SC.19) is the original positive local pairing one.

![The actual compact-fibre source column, chart projection, antipodal seam and localized compactness](../figures/source-column-normal.png)

**Figure 11H.1.** Equations (SC.1)–(SC.21) and Exercises 128–130. Panel A is a leaf schematic: fixing the actual source \(\sigma(b)\) leaves the one manifold \(Y\cong V\). Panel B plots the exact smooth normalized bump weights with support radius \(0.36\) in base-circle charts of radius \(0.38\), and the overlap entry \(P_{01}=\chi_0\chi_1\); their projection implements the proven global domain. Panel C shows meridian projections of the actual \(S^2\) seam and both determinant phases. Panel D plots frequencies divided by \(\pi\) for the two antipodal sectors, each with infinite fibre multiplicity; smooth convolution nonetheless maps its localized resolvent to \(H^1\), compactly into \(L^2\). The supplied full normal inverse, rather than another leaf factor, has original local pairing \(+1\). [Editable SVG](../figures/source-column-normal.svg) · [Reproduction source](../reproduction/source-column-normal/draw_source_column.py) · [Reproduction instructions](../reproduction/source-column-normal/README.md) · [Font and software terms](../reproduction/source-column-normal/COMPONENT-TERMS.md).

### 11H.10. What remains for general holonomy

This result uses a compact smooth leaf space \(B\), compact fibres, no return isotropy, a global section, and a supplied normal Dirac on that entire closed base. These hypotheses produce the actual \(Y\cong V\), compact Schur bounds, global symmetric differential domain, and normalized one-term counting correspondence. They cannot be inferred from an invariant transverse metric or a unitary normal spin-c lift alone.

For a general complete transversal, its open components may be incomplete and its return groupoid may be infinite. Morita induction gives an abstract class only after a reduced scalar transverse cycle has been supplied; it does not prove that a bare normal Dirac with infinitely many regular return copies has localized scalar compacts. Section 11A's reduced tree completion and the local-potential obstruction retain their exact scopes. In that suspension the normalized plaque corner has infinitely many normal copies; local multiplication cannot replace its nonlocal return completion with positive local pairing. Our proof avoids that obstruction because its actual transversal has one point per leaf and no return labels, not because every acting group is K-amenable or because a compact-transversal implementing representation automatically descends.

The remaining general interface is a compatible global self-adjoint realization on actual complete-transversal columns, with ordinary scalar compactness after the *whole* reduced convolution and the original Morita counting normalization, in the presence of arbitrary eligible return holonomy. A proposed construction must prove those interfaces and its actual graph realization with normal Dirac geometry. A local finite-order differential expression is one sufficient realization; an induced or pseudodifferential construction must prove its own geometric comparison. Theorem 11H.1 provides them on the stated compact-bundle family. It does not close the historical general graph-Dirac problem.

## 11I. Transverse fundamental classes and the reduced norm

The transverse fundamental-class question has a distinction between the algebra on which a class is defined and the operator used to represent it. Hilsum and Skandalis construct the general K-oriented projection class on the **maximal** foliation algebra. Their notation explicitly differs from Connes's reduced foliation algebra; see their original paper, pp. 326–327 and Section IV, pp. 350–355. Thus the general maximal construction is established. This statement does not give a reduced scalar graph representative with the prescribed local transverse Bott pairing.

Connes and Moscovici's metric-bundle construction gives a hypoelliptic spectral triple for a geometric crossed-product representation. Their 1995 paper, p. 12, distinguishes that metric-bundle algebra from the original transverse algebra; Theorem I.1 on p. 16 states bounded commutators and localized compact resolvents. A reduced-norm estimate is an additional requirement. The following proof shows that this requirement can fail even for invariant transverse metric and spin-c data.

### 11I.1. A general necessary condition for the geometric representation

Let a discrete group \(\Gamma\) act by homeomorphisms on a compact Hausdorff space \(N\), preserving a Borel probability measure \(\mu\). On \(H=L^2(N,\mu)\), put

\[
 (U_g\xi)(x)=\xi(g^{-1}x),\qquad
 \pi(fu_g)=M_fU_g,\qquad
 \alpha_g(f)(x)=f(g^{-1}x).
 \tag{RN.1}
\]

Here \(u_gfu_g^{-1}=\alpha_g(f)\). This is the geometric covariant representation of the algebraic crossed product. The reduced norm is the supremum of the regular point representations on \(\ell^2(\Gamma)\):

\[
 (\pi_x(f)\eta)(h)=f(hx)\eta(h),\qquad
 (\lambda_g\eta)(h)=\eta(g^{-1}h).
 \tag{RN.2}
\]

They have the covariance in (RN.1). Taking the supremum over all \(x\in N\) is faithful on \(C(N)\); equivalently one can use their direct sum. In particular the norm of \(\sum c_gu_g\) is exactly the norm of \(\sum c_g\lambda_g\) in \(C_r^*(\Gamma)\).

**Theorem 11I.1.** If the representation in (RN.1) extends to \(C(N)\rtimes_r\Gamma\), then \(\Gamma\) has a left-invariant mean on \(\ell^\infty(\Gamma)\). Consequently a nonamenable group with an invariant probability measure does not have this reduced geometric representation. This assertion concerns the displayed representation, not every scalar K-homology class.

**Proof.** The constant function \(\mathbf1\) is a unit vector fixed by all \(U_g\). Contractivity of the extended representation and (RN.2) imply

\[
 \left|\sum_g c_g\right|
 \leq \left\|\sum_g c_gU_g\right\|
 \leq \left\|\sum_g c_g\lambda_g\right\|.
 \tag{RN.3}
\]

The algebraic trivial character therefore extends continuously to a unital multiplicative functional \(\varepsilon:C_r^*(\Gamma)\to\mathbb C\). The extended map remains multiplicative and preserves the involution by approximating both factors by finite sums. It is positive because \(\varepsilon(v^*v)=|\varepsilon(v)|^2\); positive elements are squares by continuous functional calculus. Its norm is one.

Use the norm-preserving Hahn–Banach extension to extend \(\varepsilon\) to a functional \(\psi\) on \(B(\ell^2\Gamma)\) with \(\|\psi\|=\psi(1)=1\). Such an extension is positive. For completeness, if \(z=z^*\), the norm-convergent exponential satisfies
\(\psi(e^{itz})=1+it\psi(z)+O(t^2)\). Its modulus is at most one, for both signs of \(t\), so \(\operatorname{Im}\psi(z)=0\). If \(0\leq z\leq1\), then \(\|1-z\|\leq1\), giving \(|1-\psi(z)|\leq1\) and hence \(\psi(z)\geq0\). Scaling proves positivity on the whole cone. The required calculus and positive-functional inequalities are proved in Order, local units and quotients of C*-algebras, Sections 5, 8 and Lemma 13.1; the extension theorem is [Hahn–Banach, Baire and the basic theorems on Banach spaces](https://kokunoyumeto.github.io/open-math-courses-public/courses/foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.html#oa-fnd-hb-02).

For \(u=\lambda_g\) we have \(\psi(u)=\psi(u^*)=1\), and therefore

\[
 \psi((u-1)^*(u-1))
 =\psi((u-1)(u-1)^*)=0.
 \tag{RN.4}
\]

The positive form \((a,b)\mapsto\psi(b^*a)\) satisfies Cauchy–Schwarz: expand \(\psi((a+tb)^*(a+tb))\geq0\) and minimize in the complex scalar \(t\); if a diagonal value is zero, first add a positive number and then let it decrease to zero. Applying this inequality with \(u-1\) and \(u^*-1\) shows, for every bounded \(T\),

\[
 \psi((u-1)T)=0,\qquad \psi(T(u^*-1))=0,
 \qquad \psi(uTu^*)=\psi(T).
 \tag{RN.5}
\]

For \(f\in\ell^\infty(\Gamma)\), let \(M_f\) be diagonal multiplication on \(\ell^2(\Gamma)\), and define \(m(f)=\psi(M_f)\). This is a positive linear functional with \(m(1)=1\). Since
\(\lambda_gM_f\lambda_g^*=M_{f(g^{-1}\,\cdot)}\), (RN.5) gives

\[
 m(f(g^{-1}\,\cdot))=m(f).
 \tag{RN.6}
\]

This is precisely a left-invariant mean, which is the amenability asserted in the theorem. No faithfulness of \(\mu\), freeness, ergodicity, elliptic estimate or Fredholm operator was used. \(\square\)

The theorem still applies if \(H\) is replaced by a geometric Hilbert space containing the trivial representation as an invariant subspace. It also applies after a central unitary character is removed from the group coefficients: the coefficient change \(u_g\mapsto\chi(g)u_g\) preserves both the full and reduced norms. The next calculation proves that claim and gives an exact numerical obstruction.

### 11I.2. An exact reduced-norm gap in eligible transverse geometry

Use the explicit free pair in [Section 11A](#section-11a):
\(\rho(a)=(1+2i)/\sqrt5\), \(\rho(b)=(1+2j)/\sqrt5\) in \(\operatorname{SU}(2)\). Lemma 11.9 supplies its all-word injectivity proof. Let
\(N=\operatorname{SU}(2)\times S^1\), with \(\Gamma=F_2=\langle a,b\rangle\) acting by \(\rho(g)\) on the first factor and by the identity on the second. The action is smooth, free and germ-effective: a nonidentity left translation has no fixed point. The product oriented left-invariant orthonormal frame is fixed, so the full rank-four metric spin structure is invariant.

A supplied character \(\chi:\Gamma\to U(1)\) gives the coherent spin-c lift \([1,\chi(g)]\) in this frame. Its determinant character is \(\chi(g)^2\). The inverse normal coefficient carries \(\chi(g)^{-1}\), rather than being determined just by that square. In the conventions of Theorem 7.17 its complete rank-four inverse is \(C_4\), with physical generators \(-L_{t_j}\), since \(4(4-1)/2=6\). None of those coefficient signs or phases is changed in the following norm calculation.

There is an actual compact foliation with this transversal: use the genus-two surface cover \(\Sigma_0\) and surjection onto \(F_2\) from (SG.1), and form
\(V=(\Sigma_0\times N)/F_2\). Properness and freeness follow from the deck action; compactness follows from the compact base surface and compact fibre. Kernel loops have identity holonomy, and the remaining germs are exactly the displayed translations. Thus the actual complete-transversal holonomy groupoid is \(N\rtimes F_2\). This example satisfies the invariant metric/spin-c premise; a hyperbolic transverse return is not being used.

Let \(\mathcal P\) be the manifold of positive-definite symmetric \(4\times4\) matrices. The bundle of transverse metrics is
\(P=N\times\mathcal P\) in the fixed frame. The lifted action is

\[
 g(n,Q)=(gn,Q).
 \tag{RN.7}
\]

Equip \(P\) with product measure \(d\mu_N\,d\nu\), where \(\mu_N\) is normalized Haar measure times normalized circle measure and \(\nu\) is any positive smooth locally finite measure on \(\mathcal P\). The natural metric-bundle density also has this form in the invariant frame. In the exterior coefficient used for the triangular signature construction, the degree-zero component is a trivial invariant line. Its geometric representation therefore contains \(L^2(P)\); adding finite exterior or spin coefficients cannot remove the norm of this subrepresentation. Tensoring this component with the inverse character line gives

\[
 (U_g^\chi\xi)(n,Q)
 =\chi(g)^{-1}\xi(g^{-1}n,Q).
 \tag{RN.8}
\]

Choose \(q\in C_c^\infty(\mathcal P)\), \(0\leq q\leq1\), equal to one on a nonempty relatively compact open ball, and extend it constantly on \(N\). Such a bump is supplied by Local tools for bundles and transport, Section 3. Choose a nonzero smooth \(\varphi\) supported in that ball and normalize its \(L^2(\nu)\)-norm to one. Define the actual compactly supported algebra element

\[
 v_\chi=\frac14\sum_{s\in\{a,a^{-1},b,b^{-1}\}}\chi(s)u_s,
 \qquad z=qv_\chi.
 \tag{RN.9}
\]

Since \(q\) is invariant and \(\chi(s^{-1})=\overline{\chi(s)}\), \(z=z^*\). Its geometric norm is one: the unit vector
\(\mathbf1_N\otimes\varphi\) is fixed by every \(\chi(s)U_s^\chi\), and \(q\varphi=\varphi\), so \(\pi(z)\) fixes it. The reverse bound follows from the triangle inequality for four unitaries and \(\|q\|_\infty=1\).

In a regular point representation at \((n,Q)\), \(q\) is the scalar \(q(Q)\). The diagonal unitary
\((W\eta)(h)=\chi(h)\eta(h)\) satisfies
\(W\lambda_sW^*=\chi(s)\lambda_s\). Taking the defining supremum over \((n,Q)\), rather than using a geometric norm, yields

\[
 \|z\|_r=\left\|\frac{\lambda_a+\lambda_{a^{-1}}
                         +\lambda_b+\lambda_{b^{-1}}}{4}\right\|
          =\frac{\sqrt3}{2}<1=\|\pi(z)\|.
 \tag{RN.10}
\]

Here is a full proof of the middle equality. Let \(A\) be the unnormalized adjacency operator on the Cayley tree and set \(h(g)=3^{-|g|/2}\). At a nonroot vertex there is one neighbour of length \(|g|-1\) and three of length \(|g|+1\), so \(Ah=2\sqrt3\,h\). At the root \(Ah=4/\sqrt3\leq2\sqrt3\,h\). For every finitely supported vector \(\eta\), the inequality

\[
 2|\eta(x)\eta(y)|
 \leq |\eta(x)|^2\frac{h(y)}{h(x)}
       +|\eta(y)|^2\frac{h(x)}{h(y)}
 \tag{RN.11}
\]

on each unoriented edge proves
\(|\langle A\eta,\eta\rangle|\leq2\sqrt3\|\eta\|^2\).
The operator \(A\) is bounded and self-adjoint (a sum of four unitaries). There is also a direct operator-norm bound, avoiding a spectral-radius shortcut. Cauchy–Schwarz at each vertex gives
\[
 |A\eta(x)|^2
 \leq\left(\sum_{y\sim x}\frac{h(y)}{h(x)}\right)
       \left(\sum_{y\sim x}\frac{h(x)}{h(y)}|\eta(y)|^2\right).
\]
The first factor is at most \(C=2\sqrt3\). Summing the second factor over \(x\) gives \(\sum_y|\eta(y)|^2(Ah)(y)/h(y)\leq C\|\eta\|^2\). Hence \(\|A\eta\|^2\leq C^2\|\eta\|^2\) on the dense finite-support space, proving \(\|A\|\leq2\sqrt3\).

Let \(h_R=h\,\mathbf1_{\{|g|\leq R\}}\). There are \(4\cdot3^{j-1}\) vertices at level \(j\geq1\). There are that many edges from level \(j-1\) to level \(j\), and each contributes \(3^{-j+1/2}\) to the product of endpoint weights. Hence, for every integer \(R\geq1\),

\[
 \|h_R\|^2=1+\frac{4R}{3},\qquad
 \langle Ah_R,h_R\rangle=\frac{8R}{\sqrt3},\qquad
 \frac{\langle (A/4)h_R,h_R\rangle}{\|h_R\|^2}
  =\frac{2R/\sqrt3}{1+4R/3}
  \longrightarrow\frac{\sqrt3}{2}.
 \tag{RN.12}
\]

This proves the matching lower bound, so (RN.10) is exact. In particular, reduced continuity would force the false inequality \(1\leq\sqrt3/2\). The full norm of \(z\) is exactly one: every covariant representation has norm at most one by (RN.9), and the displayed geometric representation attains it.

This obstruction persists before any Dirac operator is chosen. It therefore cannot be repaired by changing a compatible connection, a self-adjoint domain, a normalizing function, or a localized-resolvent proof while retaining this same geometric representation. It does not exclude the regular tree completion of Section 11A, which changes the representation and already has the prescribed local pairing. Nor does it identify the triangular signature coefficient with the calibrated inverse spin-c coefficient. The character-twisted calculation checks the full phase explicitly; the norm obstruction itself is independent of that calibration.

### 11I.3. A scalar index does not identify an equivariant unit

**Proposition 11I.2.** Forgetting an equivariant action can take two different equivariant Fredholm indices to the same scalar index \(+1\).

**Proof.** Let \(C_2=\{1,c\}\). Take the graded finite-dimensional Hilbert space \(H_+=\mathbb C\), \(H_-=0\), the scalar representation of \(\mathbb C\), and the odd operator \(F=0\). All Kasparov defects are compact because the space is finite-dimensional, and \(F\) is exactly equivariant. With \(c\) acting as \(+1\), the equivariant index is the trivial line. With \(c\) acting as \(-1\), it is the sign line. Both underlying ordinary Fredholm indices are one.

They are distinct equivariant indices. For any equivariant Fredholm operator, split source and target by the commuting projections
\[
 P_\pm=\frac{1\pm U_c}{2}.
 \tag{RN.13}
\]
The restricted operators are Fredholm: an equivariant parametrix is obtained by averaging an arbitrary parametrix over \(C_2\), and then restricting it to each summand.

The pair also distinguishes the full equivariant scalar KK classes, including Hilbert-module homotopies. First compress to the essential scalar summand and symmetrize \(F\), as in [Theorem 0.14](#scalar-classification-through-the-stable-corona). Equivariance makes \(U_cFU_c-F\) compact. Replace \(F\) by \((F+U_cFU_c)/2\); this is an odd self-adjoint compact perturbation and its straight-line path preserves every cycle defect. It commutes exactly with \(P_\pm\). Restriction gives two ordinary scalar KK cycles. On an equivariant homotopy module the same averaging gives two homotopy modules over \(C([0,1])\). Theorem 0.14 identifies their endpoint integer indices, so the pair is invariant under arbitrary such homotopies, as well as stabilization, unitary equivalence and addition of degenerate cycles. For the trivial line the pair is \((1,0)\); for the sign line it is \((0,1)\). Equivalently their characters at \(c\) are \(+1\) and \(-1\). Ordinary forgetting sums the two integers, giving one in each case. \(\square\)

This is a statement about equivariant indices, not a holonomy counterexample made by adding ineffective isotropy. Emerson's Theorem 11.5.10 uses more than a scalar oscillator index: for \(G=\mathbb R\), or a countable subgroup of \(\mathbb R\), it deforms the translation action by \(g:x\mapsto x+tg\) to the trivial action. That actual equivariant homotopy identifies the oscillator class with the unit before descent. Its hypotheses and homotopy cannot be inferred for an arbitrary holonomy groupoid from a single local Bott pairing.

### 11I.4. The exact remaining construction

The maximal projection class and the metric-bundle spectral construction are established results with the algebraic scopes stated above. A universal claim that the transverse fundamental-class construction is still historically open is therefore too broad.

The remaining construction here is specific: a suitable representative in
\(KK^q(C_r^*(G),\mathbb C)\) for every holonomy groupoid with the supplied full invariant transverse spin-c data, with the entire inverse coefficient, a proved closed domain and core, the required commutator interface, **ordinary scalar** localized compacts, and the original oriented local transverse Bott pairing \(+1\). Morita or source-column transport must use the actual normalized counting/plaque correspondence. A full graph tangent Dirac and a normal Dirac on source columns have different domains and multiplicities; neither name supplies these estimates.

The norm calculation above does not rule out such a class: it excludes one natural geometric representation, which the already constructed tree representative replaces successfully. Equally, a maximal class, a signature spectral triple on \(P\), or an ordinary scalar index one does not supply that general reduced graph representative. No universal nonexistence result or universal normalized reduced construction follows from these distinctions.

![Exact geometric and reduced norms, and the equivariant index distinction](../figures/transverse-reduced-norm.png)

**Figure 11I.1.** Equations (RN.7)–(RN.13). The first panel is a coordinate schematic of the metric bundle: returns change \(n\) and fix the positive metric matrix \(Q\); the compact cutoff is in \(Q\). The second panel compares the proved norms \(1\) and \(\sqrt3/2\). The third plots the exact finite-vector Rayleigh quotients (RN.12), with their proved limiting line. The last panel shows two different \(C_2\)-equivariant indices whose ordinary indices are both \(+1\). No finite plot proves an infinite norm or an equivariant unit identity. [Reproducible figure source](../reproduction/transverse-reduced-norm/draw_norm.py) · [Editable SVG](../figures/transverse-reduced-norm.svg) · [Component terms](../reproduction/transverse-reduced-norm/COMPONENT-TERMS.md).

Further reading: [Hilsum–Skandalis, *Morphismes K-orientés d'espaces de feuilles et fonctorialité en théorie de Kasparov*](https://www.numdam.org/article/ASENS_1987_4_20_3_325_0.pdf), pp. 326–327 and 350–355; [Connes–Moscovici, *The local index formula in noncommutative geometry*](https://repo-archives.ihes.fr/FONDS_IHES/I_Prepublications/CONNES/1994-1998/M_95_19/M_95_19.pdf), preprint pp. 12–16; [Emerson, *An Introduction to C*-Algebras and Noncommutative Geometry*](https://doi.org/10.1007/978-3-031-59850-0), Section 11.5, pp. 524–529, especially Lemma 11.5.8 and Theorem 11.5.10.

## 11J. A scalar index-one class from a proper factorization

Original exposition and proofs in this reading are dedicated to the public domain under CC0. The group is discrete and countable, and all coefficient algebras are separable and graded. Hilbert-module inner products are linear in the second variable.

We use the construction, naturality and product law for full and reduced descent proved in Descent and the K-theory of crossed products, Sections 1–3, especially Theorems 2.2 and 3.2 and equation (3.9). The ordinary and equivariant products and their associativity are proved in Equivariant KK-theory and the Green–Julg theorem. The norm conventions are those of Reduced crossed products and Fell's absorption principle. These results are used with their coefficient algebras retained.

### 11J.1. A proper coefficient has only one crossed-product norm

A proper \(\Gamma\)-algebra means an algebra \(P\) with an equivariant nondegenerate central homomorphism
\[
 C_0(X)\longrightarrow ZM(P)
 \tag{AU.1}
\]
for a second-countable locally compact Hausdorff proper \(\Gamma\)-space \(X\). Properness means that \((g,x)\mapsto(gx,x)\) is proper. Finite stabilizers are allowed.

**Lemma 11J.1 (a square-normalized cutoff).** There is a nonnegative continuous function \(c:X\to[0,1]\) such that
\[
 \sum_{g\in\Gamma}c(g^{-1}x)^2=1.
 \tag{AU.2}
\]
For each compact \(K\subset X\), only finitely many \(g\) have \(c(g^{-1}x)\ne0\) at some \(x\in K\).

**Proof.** The quotient \(Q=X/\Gamma\) is locally compact Hausdorff and second countable. The closed orbit relation and openness of the quotient prove this, as in Lemma 9.1 of Proper actions, free actions and the orbit space. In particular \(Q\) is paracompact. For each orbit choose a nonnegative compactly supported bump \(\phi_i\) on \(X\), positive at a point of that orbit. The open sets
\(V_i=q(\{\phi_i>0\})\) cover \(Q\). Choose a locally finite partition of unity \(\rho_i\) subordinate to a locally finite refinement of these sets, relabeling the corresponding bumps. We may arrange \(\operatorname{supp}\rho_i\subset V_i\); the partition is obtained by a locally finite relatively compact shrinking and normalizing its nonnegative bumps.

Set
\[
 b(x)=\sum_i \rho_i(q(x))\phi_i(x),\qquad
 S(x)=\sum_g b(g^{-1}x)^2.
 \tag{AU.3}
\]
The first sum is locally finite. On a compact \(K\subset X\), only finitely many indices \(i\) can occur, because the partition is locally finite on the compact \(q(K)\). For each such \(i\), properness makes the transporter from \(\operatorname{supp}\phi_i\) to \(K\) finite. Thus the second sum is locally finite near every compact set. Every orbit has some point where \(b>0\), since an index with \(\rho_i(q(x))>0\) has \(q(x)\in V_i\). Consequently \(S\) is continuous and strictly positive. It is invariant by reindexing. The function \(c=b/S^{1/2}\) satisfies (AU.2), lies in \([0,1]\), and has the asserted local support property. No compactness of \(Q\) is assumed. \(\square\)

**Theorem 11J.2 (proper coefficient quotient).** The canonical surjection
\[
 q_P:P\rtimes_{\max}\Gamma\longrightarrow P\rtimes_r\Gamma
 \tag{AU.4}
\]
is an actual *-isomorphism.

**Proof.** Let \((\pi,U)\) be any nondegenerate covariant representation on \(H\). Central nondegeneracy in (AU.1) extends \(\pi\) to the bounded continuous central multipliers used below. Put \(c_g(x)=c(g^{-1}x)\). The finite partial sums of \(\sum c_g^2\) converge strictly to \(1\): they are bounded by \(1\), and are eventually exactly \(1\) on each compact set. Multiplication by a compactly supported \(C_0(X)\)-function therefore converges in norm; its products with \(P\) are dense by (AU.1). This proves strict convergence on \(P\). A nondegenerate representation carries bounded strict convergence to strong convergence. Hence
\[
 J:H\longrightarrow H\otimes\ell^2\Gamma,\qquad
 J\xi=\sum_g\pi(c_g)\xi\otimes\delta_g
 \tag{AU.5}
\]
is an isometry. There is no convergence claim for the unweighted sum of the multipliers themselves.

Centrality gives \(J\pi(a)=(\pi(a)\otimes1)J\). Our convention is \(\alpha_h(f)(x)=f(h^{-1}x)\). Thus
\(\alpha_h(c_{h^{-1}g})=c_g\), and covariance gives
\[
 JU_h=(U_h\otimes\lambda_h)J,\qquad
 \lambda_h\delta_g=\delta_{hg}.
 \tag{AU.6}
\]
Indeed the output coordinate \(g\) on the right is
\(U_h\pi(c_{h^{-1}g})\xi=\pi(c_g)U_h\xi\).
Every integrated polynomial in \((\pi,U)\) is therefore a compression of its diagonal amplification.

The unitary
\[
 W(\xi\otimes\delta_g)=U_g^{-1}\xi\otimes\delta_g
 \tag{AU.7}
\]
sends that diagonal amplification to the regular covariant representation:
the coefficient at coordinate \(g\) is \(\pi(\alpha_{g^{-1}}(a))\), and a group element acts by \(1\otimes\lambda_h\). The regular norm bounds the norm of this representation even when \(\pi\) is not faithful: this is the representation-independence assertion in the reduced-crossed-product reading. Thus every full covariant norm is at most the reduced norm. The reverse inequality holds because regular covariant representations are among the full ones. Equality on the dense finite convolution algebra proves injectivity of (AU.4), and its defining quotient is onto. Grading is preserved throughout. \(\square\)

### 11J.2. The forgetful unit is sufficient

Write \(M=C_{\max}^*\Gamma\), \(R=C_r^*\Gamma\), \(q:M\to R\). Write \(u_m:\mathbb C\to M\) and \(u_r:\mathbb C\to R\) for the unit inclusions. The trivial character \(\epsilon:M\to\mathbb C\) exists on the full algebra, with \(\epsilon(u_g)=1\).

**Theorem 11J.3 (the proper-factorization construction).** Suppose \(P\) is a proper algebra as above and
\[
 \eta\in KK^\Gamma(\mathbb C,P),\qquad
 d\in KK^\Gamma(P,\mathbb C),\qquad
 \operatorname{For}(\eta\otimes_Pd)=1_{\mathbb C}.
 \tag{AU.8}
\]
The classes here have even degree; \(P\) may have a nontrivial grading. There is an even class
\[
 \begin{aligned}
 \kappa={}&j_r(\eta)\otimes_{P\rtimes_r\Gamma}[q_P]^{-1}\\
 &\otimes_{P\rtimes_{\max}\Gamma}j_{\max}(d)
       \otimes_M[\epsilon]\ \in KK(R,\mathbb C)
 \end{aligned}
 \tag{AU.9}
\]
such that
\[
 [u_r]\otimes_R\kappa=1_{\mathbb C}.
 \tag{AU.10}
\]
Here \([q_P]^{-1}\) is the class of the inverse *-isomorphism from Theorem 11J.2. It is not an assumed inverse for \(q\).

**Proof.** The coefficient square (3.9) of the descent reading gives, with all intermediate algebras displayed,
\[
 [q]\otimes_R j_r(\eta)
   =j_{\max}(\eta)\otimes_{P\rtimes_{\max}\Gamma}[q_P].
 \tag{AU.11}
\]
After multiplying by the remaining factors in (AU.9), cancellation of the actual coefficient isomorphism and the product law for full descent give
\[
 [q]\otimes_R\kappa
   =j_{\max}(\gamma)\otimes_M[\epsilon],
 \qquad \gamma=\eta\otimes_Pd.
 \tag{AU.12}
\]
This is an identity from \(M\) to \(\mathbb C\).

We verify explicitly the scalar evaluation of its right side. Represent \(\gamma\) by an equivariant graded Hilbert-space cycle \((H,U,F)\), on its unital essential part, with \(F\) self-adjoint, odd and contractive. Full scalar descent has core \(C_c(\Gamma,H)\), right coefficient \(M\), and inner product
\[
 \langle\xi,\zeta\rangle(h)
     =\sum_g\langle\xi(g),\zeta(gh)\rangle_H.
 \tag{AU.13}
\]
After tensoring the right coefficient with \(\epsilon\), the map
\[
 \xi\otimes z\longmapsto z\sum_g\xi(g)
 \tag{AU.14}
\]
is isometric: applying \(\epsilon\) to (AU.13) sums over \(g,h\), giving the Hilbert-space inner product of the two finite vector sums. This map respects right balancing, since augmenting a convolution multiplies its vector sum by \(\epsilon\) of the coefficient. Vectors supported at the identity prove that it has dense range and hence is onto. The descended operator is pointwise \(F\), so (AU.14) carries it to \(F\). Left multiplication by a group element \(h\) sends \(\xi(g)\) to \(U_h\xi(h^{-1}g)\); its vector sum is \(U_h\sum_g\xi(g)\). Thus the balanced cycle retains the representation \(U\) of \(M\). Pulling its source back by \(u_m\) removes that representation and leaves precisely the ordinary graded cycle \((H,F)\). The same argument handles degenerate summands and homotopies. Consequently
\[
 [u_m]\otimes_Mj_{\max}(\gamma)\otimes_M[\epsilon]
       =\operatorname{For}(\gamma).
 \tag{AU.15}
\]
Finally \(q u_m=u_r\), and (AU.12), (AU.15) and (AU.8) prove (AU.10). No reduced trivial character was used. \(\square\)

**Corollary 11J.4 (an integral unit splitting).** The class \([1_R]\) generates a direct \(\mathbb Z\)-summand of \(K_0(R)\).

**Proof.** The index pairing \(\ell(x)=x\otimes_R\kappa\) is a homomorphism \(K_0(R)\to K_0(\mathbb C)=\mathbb Z\), and (AU.10) says \(\ell([1_R])=1\). For every \(x\),
\[
 x=\ell(x)[1_R]+\bigl(x-\ell(x)[1_R]\bigr),
 \qquad K_0(R)=\mathbb Z[1_R]\oplus\ker\ell.
 \tag{AU.16}
\]
The second summand lies in \(\ker\ell\), and a multiple of the unit in that kernel has coefficient zero. This is an integral splitting, without a universal-coefficient theorem. \(\square\)

An infinite-order unit, or a unit that is not a nontrivial integer multiple, is a weaker algebraic assertion than the splitting in (AU.16). A real trace with value \(1\) on the unit is not a substitute for the integral homomorphism \(\ell\), and an arbitrary homomorphism on \(K_0\) has not thereby been lifted to a KK-class.

### 11J.3. Ordinary identity and equivariant identity

The hypothesis (AU.8) concerns the ordinary forgetful class. It does not require \(\gamma=1\) in \(KK^\Gamma(\mathbb C,\mathbb C)\). If that stronger identity holds, (AU.12) becomes \([q]\otimes_R\kappa=[\epsilon]\). Together with the dual Green–Julg correspondence for discrete groups, that is a weakly regular representative of the equivariant unit and implies K-amenability. Neither this stronger identity nor invertibility of the group quotient follows from (AU.10).

For precision, suppose \(\Gamma\) is infinite and has property \(T\). Its full Kazhdan projection \(p\) satisfies \(q(p)=0\) and \(\epsilon(p)=1\). The full proof is Proposition 8.4 of Descent and the K-theory of crossed products: a finite Kazhdan gap gives a continuous spectral projection, and the regular representation has no invariant vector. For every class \(\kappa\in KK(R,\mathbb C)\), \(p\otimes_M[q]\otimes_R\kappa=0\). Therefore \([q]\otimes_R\kappa\ne[\epsilon]\). This obstruction addresses the equivariant identity, not the ordinary unit index. It does not contradict Theorem 11J.3 with only the forgetful hypothesis.


### 11J.4. The exact existence premise and diagram

Theorem 11J.3 proves the scalar consequence of a supplied proper factorization with forgetful product one. It does not construct that factorization for an arbitrary countable group or holonomy groupoid. The latter existence theorem is an additional prerequisite. For related geometric constructions see Guentner, Higson and Weinberger, [*The Novikov Conjecture for Linear Groups*](https://math.hawaii.edu/~erik/papers/linear.pdf), Theorem 4.1 and Section 6, and Tu, [*La conjecture de Baum–Connes pour les feuilletages moyennables*](https://citeseerx.ist.psu.edu/document?doi=486238991671dba031f9cb078eeba25443307a0a&repid=rep1&type=pdf), Sections 6–9. These reading references do not replace a proof of the required existence premise.

![The cutoff isometry and the four typed factors of the scalar unit-index construction](../figures/proper-auxiliary.png)

**Figure 11J.1.** Lemma 11J.1, Theorems 11J.2–11J.3 and Corollary 11J.4, equations (AU.1)–(AU.16). Panel A shows the exact square normalization, compression isometry and absorption unitary; all finite stabilizers are retained. Panel B shows the sources and targets of all four KK factors and the exact forgetful evaluation. The inverse is the proper coefficient quotient's actual inverse; no inverse of the group quotient is assumed. This is an algebraic diagram, not a claim that all such factorizations exist. Original mathematical expression is CC0 1.0; bundled glyphs retain their component terms.

[Editable SVG](../figures/proper-auxiliary.svg) · [Reproducible source](../reproduction/proper-regular-auxiliary/draw_proper.py).

## 11K. Regularization by a scalar index-one auxiliary

The regular representation supplies the reduced norm. It does not, by itself, supply the auxiliary Fredholm index or a Dirac operator on a holonomy graph. The following result separates these assertions. It applies to every countable discrete action satisfying its stated auxiliary premise, rather than to a selected family of groups.

The KK products, scalar classification and homotopy rules used below are proved in K-theory of the leaf space, Section 0, especially Scalar classification through the stable corona. The prescribed inverse coefficient is the one in Theorem 7.17, under The actual coefficient transfer of a Thom inverse. These are proof prerequisites, not replacements by literature citations.

### 11K.1. An exact norm-preserving map

Let \(\Gamma\) be a countable discrete group acting by automorphisms \(\alpha\) on a separable unital C*-algebra \(A\). Write

\[
 B_m=A\rtimes_{\max}\Gamma,\qquad B_r=A\rtimes_r\Gamma,
 \qquad R=C_r^*(\Gamma),\qquad q:B_m\longrightarrow B_r.
 \tag{RA.1}
\]

The convention is \(u_ga u_g^{-1}=\alpha_g(a)\), and \(\lambda_g\delta_h=\delta_{gh}\). Denote the coefficient inclusions by \(i_m:A\to B_m\) and \(i_r:A\to B_r\). All tensor products of algebras in this subsection are minimal spatial tensor products.

**Theorem 11K.1 (regular diagonal map).** The formula

\[
 \Delta_r\!\left(\sum_g a_gu_g\right)
       =\sum_g i_m(a_g)u_g\otimes\lambda_g
 \tag{RA.2}
\]

extends to an isometric *-homomorphism \(B_r\to B_m\otimes R\). In particular

\[
 \Delta_r i_r(a)=i_m(a)\otimes1_R.
 \tag{RA.3}
\]

No amenability, exactness, nuclearity, freeness or invariant measure is assumed.

**Proof.** Choose a faithful unital representation \(\rho\) of \(B_m\) on a Hilbert space \(H\). Set \(\pi=\rho i_m\) and \(U_g=\rho(u_g)\). The coefficient inclusion is faithful: a faithful representation of \(A\) has its regular covariant extension, whose restriction to the identity coordinate is that representation. Its norm is among the norms defining the maximal completion. Therefore \(\pi\) is faithful on \(A\).

On \(H\otimes\ell^2(\Gamma)\), define the onto unitary

\[
 W(\xi\otimes\delta_h)=U_h^{-1}\xi\otimes\delta_h,
 \qquad W^*(\xi\otimes\delta_h)=U_h\xi\otimes\delta_h.
 \tag{RA.4}
\]

Coordinate substitution gives

\[
 \begin{aligned}
  \bigl(W(\pi(a)\otimes1)W^*\eta\bigr)_h
       &=\pi(\alpha_{h^{-1}}(a))\eta_h,\\
  W(U_g\otimes\lambda_g)W^*&=1_H\otimes\lambda_g.
 \end{aligned}
 \tag{RA.5}
\]

For the second equation, the output at \(h\) before applying \(W\) is \(U_gU_{g^{-1}h}\eta_{g^{-1}h}=U_h\eta_{g^{-1}h}\). The factor \(U_h^{-1}\) cancels it. Thus the represented right-hand side of (RA.2) is unitarily the regular covariant representation associated to the faithful \(\pi\). Its norm is the defining reduced norm of the polynomial.

The representation \(\rho\otimes\lambda\) is faithful on the minimal tensor product, by its spatial-norm construction from faithful factor representations. Consequently (RA.2) has exactly that reduced norm in \(B_m\otimes R\). Covariance verifies multiplication and involution on polynomials: the term for \((a_gu_g)(b_hu_h)\) is \(a_g\alpha_g(b_h)u_{gh}\otimes\lambda_{gh}\). It agrees with multiplying their two images. Completion therefore gives an isometric *-homomorphism. Formula (RA.3) is its identity-coordinate term. \(\square\)

This proof is an explicit absorption argument. It does not assert that the original geometric representation \((\pi,U)\) is reduced. The representation being regularized acts on the larger tensor Hilbert space.

### 11K.2. Exactly which auxiliary index is sufficient

Let \(u:\mathbb C\to R\) be the unit homomorphism. An **ordinary scalar index-one auxiliary** means an even class \(\kappa\in KK(R,\mathbb C)\) satisfying

\[
 [u]\otimes_R\kappa=1_{\mathbb C}
       \quad\hbox{in }KK(\mathbb C,\mathbb C).
 \tag{RA.6}
\]

By scalar classification this is equivalent to \(\langle[1_R],\kappa\rangle=+1\). A representative is a graded scalar Fredholm module with a representation of \(R\); its group representation is thus weakly contained in the regular representation. Equation (RA.6) is a KK left inverse for the unit inclusion. Theorem 11J.3 supplies it from a proper factorization with forgetful product one. It is not a consequence of countability or the existence of \(\lambda\).

**Theorem 11K.2 (reduced replacement preserving the ordinary normal class).** Let \(d_m\in KK^q(B_m,\mathbb C)\), with its complete coefficient and grading, and suppose (RA.6). Then

\[
 d_r=[\Delta_r]\otimes_{B_m\otimes R}(d_m\boxtimes\kappa)
       \in KK^q(B_r,\mathbb C)
 \tag{RA.7}
\]

satisfies the entire ordinary restriction identity

\[
 [i_r]\otimes_{B_r}d_r=[i_m]\otimes_{B_m}d_m
       \quad\hbox{in }KK^q(A,\mathbb C).
 \tag{RA.8}
\]

If \(b\in KK^q(\mathbb C,A)\) has prescribed normal Bott product \(b\otimes_A[i_m]\otimes d_m=+1\), then

\[
 (b\otimes_A[i_r])\otimes_{B_r}d_r=+1.
 \tag{RA.9}
\]

**Proof.** Let \(j:B_m\to B_m\otimes R\), \(j(x)=x\otimes1\). Equation (RA.3) gives \(\Delta_r i_r=j i_m\). Naturality and associativity of the proved KK product imply

\[
 \begin{aligned}
 [i_r]\otimes d_r
   &=[i_m]\otimes[j]\otimes(d_m\boxtimes\kappa)\\
   &=[i_m]\otimes\bigl(d_m\boxtimes([u]\otimes_R\kappa)\bigr)\\
   &=[i_m]\otimes d_m.
 \end{aligned}
 \tag{RA.10}
\]

The second equality is the tensor functoriality on the second, scalar auxiliary factor. That factor is even and equals the scalar unit by (RA.6). No two odd factors were interchanged, and no orientation sign has been inserted. Compose this full equality with the original \(b\), in its original coordinate order, to obtain (RA.9). \(\square\)

The theorem preserves much more than a single integer: it preserves the ordinary \(A\)-K-homology restriction as a class. It does **not** assert \([q]\otimes d_r=d_m\), nor an equivariant restriction identity. The stronger claims require their own hypotheses.

For a compact transversal \(N\), take \(A=C(N)\). If the given restriction is the full inverse normal class, its right Clifford block and its spin-c line are retained in the first factor of (RA.7). With \(k_q=q(q-1)/2\), its local finite module remains

\[
 \Pi^{k_q}C_q,\qquad
 c_{\rm phys}(e_j)=(-1)^{k_q+1}L_{t_j},\qquad
 zs\longmapsto z^{-1}\rho_t(s).
 \tag{RA.11}
\]

These are Theorem 7.17's actual inverse, not a newly chosen positive spinor convention. The determinant of the inverse line has phase \(z^{-2}\); it cannot recover the full phase \(z^{-1}\). The even auxiliary changes neither the transverse degree nor that inverse order. Under a diagonal return its additional scalar representation acts as a separate tensor factor; it does not erase the normal factor's coherent transport. This observation does not make its equivariant KK class the identity.

### 11K.3. An analytic realization when the additional operator data are supplied

The abstract class (RA.7) is not a claimed finite-order graph differential operator. Here is a complete unbounded realization at the discrete-action level under precise extra premises.

Assume a full degree-\(q\) normal cycle \(d_m\) is represented in the usual graded Clifford picture by \((H_N,D_N,\pi_N,U)\). Retain the finite right \(C_q\) block until its prescribed parity encoding. Assume:

- \(D_N\) is odd and self-adjoint, with compact resolvent, and has a supplied core \(\mathcal C_N\). Its coefficient representation and grading are fixed.
- On a dense \(\Gamma\)-invariant smooth *-algebra \(A_0\subset A\), \(\pi_N(a)\) preserves its domain and \([D_N,\pi_N(a)]\) is bounded. Each \(U_g\) is even, preserves the domain, and \([D_N,U_g]\) is bounded.
- \(\kappa\) has a graded representative \((H_K,\gamma_K,D_K,V)\), where \(D_K\) is odd, self-adjoint with compact resolvent and core \(\mathcal C_K\), \(V\) factors through \(R\), each \(V_g\) is even, preserves the domain, and \([D_K,V_g]\) is bounded. Its scalar index is \(+1\).

For a compact metric spin-c manifold, the normal commutator hypotheses admit a noninvariant connection. The given coherent metric spin-c lift transports the first-order Clifford symbol. The difference of the transported Dirac and the original Dirac therefore has zero principal symbol, so is a smooth zeroth-order endomorphism. Compactness bounds it. This does not assume an invariant connection, and does not provide a uniform bound as \(g\) varies; a finite group polynomial needs only its finitely many bounds.

Theorem 11L.1 and Corollary 11L.2 supply the stated auxiliary operator, domain and core from any given even scalar class \(\kappa\). They preserve its exact unit pairing and reduced representation. Thus these auxiliary operator data do not impose a further existence premise once the class is supplied. The specified normal cycle and its graph transport still require their own construction.

Place the even auxiliary first and the old normal Clifford block second. Define

\[
 \begin{aligned}
  H&=H_K\widehat\otimes H_N,\\
  D&=D_K\widehat\otimes1+\gamma_K\widehat\otimes D_N,\\
  \Pi\!\left(\sum_g a_gu_g\right)
      &=\sum_g V_g\widehat\otimes\pi_N(a_g)U_g,\\
  \operatorname{Dom}D
      &=\operatorname{Dom}(D_K\otimes1)
           \cap\operatorname{Dom}(1\otimes D_N).
 \end{aligned}
 \tag{RA.12}
\]

All operators here use the displayed ordinary Hilbert-space action of the graded sum, including \(\gamma_K\) exactly once. The finite right Clifford action is unchanged.

**Proposition 11K.3.** The closure on \(\mathcal C_K\odot\mathcal C_N\) is self-adjoint on exactly (RA.12), has compact resolvent, and its bounded transform is a degree-\(q\) reduced scalar Kasparov cycle. It represents (RA.7) for the supplied \(d_m,\kappa\).

**Proof: closure, domain and compactness.** The two odd summands anticommute. On the algebraic core,

\[
 D^2=D_K^2\otimes1+1\otimes D_N^2,\qquad
 \|D\xi\|^2=\|(D_K\otimes1)\xi\|^2
                   +\|(1\otimes D_N)\xi\|^2.
 \tag{RA.13}
\]

Here are the compact spectral facts needed for the closure. For a nonzero positive compact operator \(T\), let \(\lambda=\|T\|\) and choose unit vectors \(v_n\) with \(\langle Tv_n,v_n\rangle\to\lambda\). Positivity and \(T^2\leq\lambda T\) give \(\|(T-\lambda)v_n\|^2\leq\lambda(\lambda-\langle Tv_n,v_n\rangle)\to0\). Compactness gives a convergent subsequence of \(Tv_n\); the displayed estimate then gives a unit limit of \(v_n\), which is a \(\lambda\)-eigenvector. Remove its whole eigenspace and repeat on the orthogonal complement. Every positive eigenspace is finite-dimensional: otherwise its orthonormal vectors have images a fixed distance apart. The successive positive eigenvalues tend to zero by the same compactness test. The residual orthogonal space is the kernel, because its restricted norm is bounded by all successive eigenvalues; if the procedure stops, a positive residual norm would yield another eigenvector by the first argument. This proves the needed compact positive spectral theorem.

Apply it to \(T=(1+D_K^2)^{-1}\) and \((1+D_N^2)^{-1}\). Their kernels are zero. A positive \(T\)-eigenvector belongs to \(\operatorname{Ran}T\subset\operatorname{Dom}D^2\) and satisfies \(D^2v=(\lambda^{-1}-1)v\). Thus each squared spectral window is finite-dimensional and invariant under \(D\) and its grading; finite-dimensional self-adjoint diagonalization supplies its \(D\)-eigenvectors. The scalar functional calculus and its product/domain rules are proved in Spectral products, transport and inverse domains, through its linked earlier ordinary calculus proofs.

On each joint window, the expression in (RA.12) is a finite-dimensional self-adjoint matrix and its square is (RA.13). The orthogonal direct sum of these matrices is self-adjoint: its two resolvents are the direct sums of the inverses of the blocks, each bounded by one at \(\pm i\), and their ranges give its full closed domain.

The square-norm identity says that this domain is exactly the intersection in (RA.12). Joint spectral truncations converge in its graph norm. The stated factor cores approximate each finite elementary tensor with both factor graph norms, so their algebraic tensor is a core for that closed operator. This establishes the claimed closure, rather than merely a formal square.

There are only finitely many joint eigenvectors satisfying \(\nu_K^2+\nu_N^2\leq L\), for each finite \(L\). Spectral truncation of \((1+D^2)^{-1/2}\) therefore gives finite-rank approximations with error at most \((1+L)^{-1/2}\). It is an ordinary scalar compact operator. Keeping a finite-dimensional Clifford coefficient does not turn this argument into compactness only relative to an infinite coefficient algebra. Consequently the resolvents and \(F^2-1=-(1+D^2)^{-1}\), for \(F=D(1+D^2)^{-1/2}\), are scalar compact.

**Proof: norm and domain-preserving commutators.** The representation is the tensor representation of (RA.2), with the two factors flipped to put the even auxiliary first. Theorem 11K.1 makes it contractive for the exact reduced crossed-product norm; faithfulness of this particular tensor representation is unnecessary. It remains a *-representation of every \(B_r\). For each coefficient term, direct substitution gives

\[
 \begin{aligned}
 [D,V_g\otimes\pi_N(a)U_g]
   ={}&[D_K,V_g]\otimes\pi_N(a)U_g\\
     &+\gamma_KV_g\otimes
       \bigl([D_N,\pi_N(a)]U_g+\pi_N(a)[D_N,U_g]\bigr).
 \end{aligned}
 \tag{RA.14}
\]

These are bounded operators. The factor domain assumptions preserve the intersection domain; (RA.13) extends the same assertion from elementary tensors to its graph closure. Thus every smooth finite polynomial preserves the actual domain and has a bounded commutator.

Put \(s=(1+t^2)^{1/2}\). The bounded transform is the strong limit of
\((2/\pi)\int_0^T D(D^2+s^2)^{-1}\,dt\). The resolvent identity gives a commutator integrand equal to

\[
 -\frac1\pi\left((D-is)^{-1}[D,a](D-is)^{-1}
                  +(D+is)^{-1}[D,a](D+is)^{-1}\right).
 \tag{RA.15}
\]

Its norm is at most \(2\|[D,a]\|/(\pi(1+t^2))\); it is compact, because either resolvent is scalar compact. Hence its integral converges in compact-operator norm, and the strong limit identifies it with \([F,a]\). Self-adjointness is exact. Polynomial density, boundedness of \(F\), and contractivity of \(\Pi\) extend all compact defects to every \(a\in B_r\). In particular every localized defect is an ordinary scalar compact, not merely a module or measured-trace defect.

**Proof: the represented product.** Before restricting along \(\Delta_r\), use the tensor representation of \(R\otimes B_m\). A homogeneous vector \(\xi\) in a bounded squared spectral window of \(D_K\) defines the graded creation map \(T_\xi\eta=\xi\otimes\eta\). In the normal spectral variable \(\nu\), the off-diagonal creation term from \(D_K\xi\) is divided by \((1+\nu_K^2+\nu^2)^{1/2}\), and tends to zero as \(|\nu|\to\infty\). The diagonal term differs from the signed normal phase by
\(\nu/(1+\nu_K^2+\nu^2)^{1/2}-\nu/(1+\nu^2)^{1/2}\), which also tends to zero. Each finite auxiliary window has only finitely many such columns. Normal compact resolvent therefore makes the creation difference and its adjoint compact. Density of these homogeneous windows, with \(\|T_\xi\|=\|\xi\|\), proves the connection equations for every creation vector.

For \(F_K=D_K(1+D_K^2)^{-1/2}\), the product positivity is the exact positive operator

\[
 \{F_K\otimes1,F\}
  =\frac{2D_K^2\otimes1}
       {(1+D_K^2\otimes1)^{1/2}
        (1+D_K^2\otimes1+1\otimes D_N^2)^{1/2}}\geq0.
 \tag{RA.16}
\]

The cross term cancels by oddness of \(D_K\). The two positive denominators commute by their joint spectral construction. For a creation vector \(\xi\otimes b\) in the first module \(H_K\otimes B_m\), its creation map is \(T_\xi\pi_N(b)\). The already proved connection error for \(T_\xi\), multiplied by \(\pi_N(b)\), is compact; the additional error is \(T_\xi[F_N,\pi_N(b)]\), up to its prescribed homogeneous sign, and is compact because the supplied \(d_m\) is a Kasparov cycle. Finite sums of these vectors are dense in that entire module. Thus the connection is for the full first module, not only its scalar columns. The connection and positivity criterion of Section 0 proves that this is the actual external product \(\kappa\boxtimes d_m\). Flipping the even auxiliary past the degree-\(q\) factor incurs no Koszul sign. Pullback along the flipped \(\Delta_r\) gives exactly (RA.7). This proves the proposition, including its degree, inverse coefficient and original local pairing. \(\square\)

For a compact-base suspension with its specified flat return charts, the complete graph-column induction proved in A general sufficient interface, and its precise limit applies to (RA.12). Its normalized counting and plaque-bump correspondence is still required. An arbitrary holonomy groupoid has not been replaced here by a globally acting discrete group, nor have arbitrary graph overlaps or a wholly local finite-order graph expression been constructed.

### 11K.4. Two automatic regular constructions that do not give index one

**Proposition 11K.4.** On an infinite group, neither the bare regular factor nor its elementary paired confining length operator supplies (RA.6).

**Proof.** If \(K\neq0\) is a compact operator, then \(K\otimes1_{\ell^2\Gamma}\) is not compact. Choose a unit vector \(v\) with \(Kv\neq0\) and distinct \(g_n\). The images \(Kv\otimes\delta_{g_n}\) are orthogonal with fixed positive norm, so have no convergent subsequence. Thus \(D_N\otimes1\) leaves infinite regular multiplicity in its localized normal resolvents whenever the localized normal factor is nonzero. Regular norm continuity alone does not prove scalar compactness.

For a finitely generated \(\Gamma\), take word length \(\ell\) and \(w(g)=1+\ell(g)\). On \(H_K^+=H_K^-=\ell^2(\Gamma)\), represent the group by \(\lambda\oplus\lambda\), grade by \(\operatorname{diag}(1,-1)\), and put

\[
 D_w=\begin{pmatrix}0&M_w\\M_w&0\end{pmatrix},\qquad
 \operatorname{Dom}D_w=
 \{(\xi,\eta):w\xi,w\eta\in\ell^2(\Gamma)\}.
 \tag{RA.17}
\]

It is the orthogonal sum of self-adjoint two-by-two matrices, with eigenvalues \(\pm w(g)\). Finite word balls prove compact resolvent. Its finite-support core is a graph core. Since \(|\ell(hg)-\ell(g)|\leq\ell(h)\), each group commutator is bounded and preserves this domain. It is therefore an actual reduced even scalar cycle by the same resolvent integral. But \(M_w\) is onto and has bounded inverse \(M_{1/w}\): its kernel and cokernel are zero. Its scalar index is zero, so (RA.6) fails. For a countable group, enumerate its nonidentity elements and their inverses with positive integer weights tending to infinity, assigning the same weight to an element and its inverse. Define \(\ell(g)\) to be the minimum total weight among finite words representing \(g\), with \(\ell(e)=0\). Every element has such a word, and the minimum exists among nonnegative integers. At a fixed total weight, only finitely many letters and finitely many words can occur. Thus the sublevels are finite. Concatenation and reversal prove \(\ell(gh)\leq\ell(g)+\ell(h)\) and \(\ell(g^{-1})=\ell(g)\); hence \( |\ell(hg)-\ell(g)|\leq\ell(h)\). This supplies the same example without finite generation.

Adding an unmatched positive **trivial** representation would change the scalar index to one, but also adds that representation to the group action. If it were weakly regular, the algebraic trivial character would extend to \(R\). The fully proved invariant-mean argument in A general necessary condition for the geometric representation would make \(\Gamma\) amenable. Thus this repair fails for a nonamenable group. A different unmatched construction inside a genuinely reduced Fredholm module requires a proof of its compact group commutators and unit index; an arbitrary enumeration and unilateral shift do not supply those assertions. \(\square\)

### 11K.5. Ordinary index one and the equivariant unit

An even Fredholm module for \(R\) also determines an equivariant Fredholm class \(e_\kappa\in KK^\Gamma(\mathbb C,\mathbb C)\): the group acts by its represented unitaries and its commutators with the phase are compact. Condition (RA.6) says only

\[
 \operatorname{Res}^{\Gamma}_{\{e\}}e_\kappa=1_{\mathbb C}.
 \tag{RA.18}
\]

Representing the equivariant unit \(e_\kappa=1_\Gamma\) by weakly regular representations is the stronger K-amenability premise. The scalar classification theorem proves (RA.18); it does not promote it to this equivariant identity. If a target prescribes an equivariant normal class \(d_N^\Gamma\), the diagonal external product has class \(d_N^\Gamma\otimes e_\kappa\). Equality with \(d_N^\Gamma\) requires this exact product identity. The full spin-c coefficient in its first factor and its inverse phase can be retained even when that global equivariant identity is absent.

Here is a complete conditional test that separates the two assertions without importing an arithmetic lattice theorem.

**Proposition 11K.5 (Kazhdan projection and a specified descent).** Suppose \(\Gamma\) is infinite, and a finite set \(S\subset\Gamma\) and \(c>0\) satisfy the following uniform spectral-gap hypothesis: in every unitary representation \(U\),

\[
 \sum_{s\in S}\|(1-U_s)\xi\|^2\geq c\|\xi\|^2
 \quad\text{for }\xi\perp H^\Gamma,
 \qquad \ker\sum_{s\in S}(1-U_s)^*(1-U_s)=H^\Gamma.
 \tag{RA.19}
\]

Then there is a projection \(p\in C^*_{\max}(\Gamma)\) whose image in every representation is the orthogonal projection onto its invariant vectors, and whose reduced image is zero. No weakly regular equivariant Fredholm class can equal \(1_\Gamma\). Nevertheless this argument does not exclude (RA.6), which pairs with the full unit rather than \(p\).

If \(d_m\in KK^0(B_m,\mathbb C)\) satisfies a nonzero pairing with the image of \(p\) in \(B_m\), then that specified \(d_m\) has no reduced descent. It can still have a reduced replacement of Theorem 11K.2 if its auxiliary premise holds.

**Proof.** Put \(L=\sum_{s\in S}(1-u_s)^*(1-u_s)\) in the full group algebra. In a faithful universal representation, (RA.19) splits it into zero on the invariant space and an operator at least \(c\) on its orthogonal complement. Its spectrum is contained in \(\{0\}\cup[c,4|S|]\). Choose a real continuous function \(h\) which is one at zero and zero on that interval. Functional calculus gives the projection \(p=h(L)\). Functional calculus commutes with every representation. Thus its represented projection is precisely onto invariants; the trivial representation sends it to one.

There is no nonzero invariant vector in \(\ell^2\Gamma\): an invariant vector is constant on \(\Gamma\), and a nonzero constant is not square-summable on an infinite group. The regular image of \(p\) is consequently zero. The same is true in every representation factoring through \(R\).

To see that this tests equivariant **classes**, integrate an equivariant scalar Fredholm cycle into a Fredholm cycle of \(C^*_{\max}(\Gamma)\). Compact group commutators imply compact commutators for finite polynomials; density extends them to the full algebra. An equivariant KK homotopy gives the same integration on its entire interval module, so the resulting ordinary maximal class is homotopy invariant. The equivariant unit integrates to the trivial character, whose \(p\)-pairing is one. A weakly regular cycle integrates to a class pulled back from \(R\), whose \(p\)-pairing is zero. They cannot be equal.

For a general quotient \(q:B_m\to B_r\), associativity gives, for every \(x\in K_*(B_m)\) and putative descent \(d_m=[q]\otimes d_r\),

\[
 x\otimes_{B_m}d_m=q_*x\otimes_{B_r}d_r.
 \tag{RA.20}
\]

The image of \(p\) is killed by this quotient, so a nonzero pairing contradicts (RA.20). If an invariant compact-resolvent even geometric Dirac represents \(d_m\), the pairing is the Fredholm index on its invariant subspace: its represented \(p\) is that subspace's projection and commutes with the Dirac and grading. This index must actually be calculated for the specified full inverse normal coefficient; a global index of some differently twisted Dirac is not a substitute.

By contrast, an ordinary unit-index auxiliary asks for \(\langle[1_R],\kappa\rangle=1\). Its maximal pullback pairs \(p\) as zero and the unit as one. There is no contradiction between these two integers. Theorem 11K.2 does not promise to reproduce the obstructed maximal class, so the obstruction to **that descent** does not invalidate its distinct reduced replacement. \(\square\)

In particular, a proposed arithmetic suspension needs complete providers for infinitude, the uniform gap, the faithful compact action and the exact invariant Dirac index. Even after these are supplied, the argument refutes only the specified maximal descent unless the target itself forces equality with that maximal class. The local Bott pairing in Connes's question does not state such an equality.

### 11K.6. A proper intermediate algebra and the exact local identity

There is a second useful formulation, independent of a globally acting group. Let \(q_A:A_m\to A_r\), \(q_P:P_m\to P_r\) be quotient maps, and suppose \([q_P]\) has a proved KK inverse \(v\in KK(P_r,P_m)\). Let

\[
 \eta_m\in KK^k(A_m,P_m),\quad
 \eta_r\in KK^k(A_r,P_r),\quad
 D_m\in KK^k(P_m,A_m),
 \qquad [q_A]\otimes\eta_r=\eta_m\otimes[q_P].
 \tag{RA.21}
\]

For \(d_m\in KK^q(A_m,\mathbb C)\), put

\[
 \gamma_m=\eta_m\otimes D_m,\qquad
 d_r=\eta_r\otimes v\otimes D_m\otimes d_m.
 \tag{RA.22}
\]

All tensor products are in this displayed typed order. Associativity, naturality and \([q_P]\otimes v=1_{P_m}\) prove

\[
 [q_A]\otimes d_r=\gamma_m\otimes d_m.
 \tag{RA.23}
\]

Consequently a specified local class \(b_m\in KK^q(\mathbb C,A_m)\) has reduced pairing one if

\[
 b_m\otimes\gamma_m=b_m,
 \qquad b_m\otimes d_m=+1.
 \tag{RA.24}
\]

Indeed its reduced image is \(b_m\otimes[q_A]\), and its product with (RA.22) is \(b_m\otimes\gamma_m\otimes d_m=+1\). These equalities require neither \(\gamma_m=1_{A_m}\) nor equality of the new maximal pullback with \(d_m\). They do require a proved intermediate quotient inverse and the **actual local** identity (RA.24), together with any extra full-class restriction required by the problem.

Calling an intermediate algebra proper does not prove these typed identities or construct an unbounded graph representative. Kasparov–Skandalis's primary paper constructs a proper intermediate algebra and Dirac/dual-Dirac elements under its proper bolic-action hypotheses; its Theorem 5.2 concerns reduced assembly injectivity, and Section 8 constructs its Dirac element. Those hypotheses are additional geometric inputs, not consequences stated for every holonomy-invariant transverse spin-c groupoid. This is a literature comparison, not an invocation of its unproved prerequisites as a proof of the present missing existence assertion. [Primary paper, pp. 165–167, 179–183 and 200–201](https://arxiv.org/pdf/math/0402374).

### 11K.7. The metric-bundle properness test

**Proposition 11K.6 (compact invariant metric section).** Let an infinite discrete \(\Gamma\) act smoothly on a nonempty compact manifold \(N\), preserving a Riemannian metric \(g\). Let \(P\to N\) be the bundle of positive-definite metrics with the transported action. Then the action on \(P\) is not proper, even if the action on \(N\) is free.

**Proof.** The invariant metric gives an equivariant section \(s(n)=g_n\). It is an embedding: the bundle projection composed with it is the identity. Its image is compact and closed because \(N\) is compact and \(P\) is Hausdorff. For the action map \(m(\gamma,p)=(\gamma p,p)\), the compact subset \(K=s(N)\times s(N)\subset P\times P\) has exact inverse image

\[
 m^{-1}(K)=\Gamma\times s(N).
 \tag{RA.25}
\]

Indeed the second coordinate forces \(p\in s(N)\), and equivariance puts \(\gamma p\) in the same section; the converse is immediate. This inverse image has the infinite discrete \(\Gamma\) as a continuous surjective image, so it cannot be compact. Properness would require that inverse image to be compact. No ergodicity, stabilizer assumption or freeness was used. \(\square\)

The eligible free \(F_2\) action on \(SU(2)\times S^1\) in Section 11I is an actual compact spin-c example of this lemma. The transverse linear cocycle has compact image, while the displayed metric-bundle action is nonproper. The different regular tree class in Section 11A remains valid.

Hilsum–Skandalis's Proposition 4.7, printed p. 355, compares its **maximal** constructions in the almost-isometric case. Its amenable group is the transverse vector bundle's structural subgroup of a linear group. The gamma class of that structural group pulls back to the identity. It is not an assertion that the holonomy groupoid is amenable, that its full-to-reduced quotient is a KK equivalence, or that its metric-bundle action is proper. The full-algebra convention is stated on pp. 326–327. Proposition 11K.6 explains the properness distinction by a complete geometric proof.

### 11K.8. Exact disposition for arbitrary eligible holonomy

The norm part of the regular-auxiliary proposal is proved by Theorem 11K.1. For a discrete action, an even scalar unit-index class suffices to preserve the entire ordinary normal restriction and its original Bott \(+1\), by Theorem 11K.2. Proposition 11K.3 supplies the exact self-adjoint domain, core, commutators and ordinary scalar compactness. Section 11L supplies its auxiliary operator data from the given scalar class; the specified normal data and subsequent graph realization remain necessary. Proposition 11K.4 proves why the most immediate regular/confining substitutes fail to supply its index. Proposition 11K.5 separates the obstruction to a prescribed maximal descent from a different normalized reduced class and from its equivariant unit requirement.

What remains for the arbitrary eligible holonomy question is a construction of the necessary auxiliary for that whole scope, with its exact normal restriction requirement; coherent groupoid field, overlap and domain data; the actual normalized counting/plaque correspondence; and an identification with a suitable operator on the actual graph. A compact-generated pseudogroup is not silently a global discrete action, and a regular Hilbert module over its coefficient algebra is not silently a scalar Fredholm cycle with ordinary compact defects. A measured or module index is not the scalar index in (RA.6).

The metric bundle is not an automatic proper substitute, by Proposition 11K.6. Hilsum–Skandalis's almost-isometric structural group is a subgroup of the transverse linear group. Its amenability does not identify it with the discrete holonomy group or furnish the missing proper action. Their Proposition 4.7 is a maximal construction and comparison.

Thus no full-scope reduced graph construction or counterexample to Connes's entire stated problem is proved here. The general maximal construction is established in Hilsum–Skandalis; an assertion that the whole historical transverse fundamental-class problem remains open in the current literature is not justified by this remaining analytic interface.

Original proof expression: CC0 1.0. Human reading references: [Connes, original question and local class, pp. 28–29](https://alainconnes.org/wp-content/uploads/foliationsfine.pdf); [Hilsum–Skandalis, full-algebra convention pp. 326–327, metric-bundle identities pp. 339–340 and Proposition 4.7 p. 355](https://www.numdam.org/article/ASENS_1987_4_20_3_325_0.pdf); [Kasparov–Skandalis, proper bolic-action construction](https://arxiv.org/pdf/math/0402374). Heath Emerson, *An Introduction to C*-Algebras and Noncommutative Geometry* (2024), Section 11.5, pp. 524–529, gives the translation oscillator's equivariant unit identity under its stated group-in-the-line hypothesis; that group has not been changed to an arbitrary return group here.

### 11K.9. Reproducible illustration

![The exact regular diagonal map and its distinct scalar and equivariant requirements](../figures/regular-auxiliary.png)

**Figure 11K.1.** Theorem 11K.1 and equations (RA.2)–(RA.11), Proposition 11K.4 and the invariant metric-section lemma. Panel A gives the actual absorption unitary and its exact norm equality. Panel B records equality of the whole ordinary normal restriction, the full inverse phase and grading, and the separate equivariant product premise. Panel C is a schematic of the compact metric section; the three arrows represent selected returns of an infinite group, rather than the whole positive-metric fibre or a finite group. Its displayed inverse image is exact. Panel D displays four members of an infinite orthogonal sequence with fixed positive norm, proving failure of bare tensor compactness, and the actual paired-length spectrum and zero index. No existence of an arbitrary unit-index auxiliary is inferred. The structural-group distinction is in Hilsum–Skandalis, Proposition 4.7, p. 355; the original local normalization is in Connes, pp. 28–29. Original diagram and generator, CC0 1.0; bundled glyphs retain their complete component notice.

[Editable SVG](../figures/regular-auxiliary.svg) · [Reproducible source](../reproduction/proper-regular-auxiliary/draw_auxiliary.py).

## 11L. Compact-resolvent representatives with prescribed group commutators

The existence of a scalar KK class and the choice of an unbounded operator representing it are separate questions. Once the class is given, a countable discrete group creates no additional obstruction to choosing compact resolvent and bounded commutators with every group element. The following construction also gives a unital reduced representation and an explicit invariant graph core.

The controlled inverse and bounded-transform arguments are proved in *Unbounded Kasparov modules and spectral triples*, Section 3, Lemma 3.1 and Theorem 3.2. Their inputs are the full convex and nested-cutoff proofs in *Kasparov's technical theorem*, Section 1, the compact-operator and countability proof in *Graded C*-algebras, Clifford algebras and graded Hilbert modules*, Lemma 2.0, and the parity-preserving normalization in *Kasparov modules and the groups KK(A,B)*, Theorem 3.2. We spell out the prescribed-family choice and the removal of the zero-representation summand. That removal is needed because an exact odd unitary obtained by doubling need not have a unital source representation.

### 11L.1. The complete auxiliary provider

Give a unital C*-algebra the trivial grading. A representation below is complex-linear and *-preserving. Compactness means ordinary Hilbert-space compactness.

**Theorem 11L.1 (a prescribed countable family).** Let \(A\) be a separable unital C*-algebra, let \(S\subset A\) be countable, and let \(\kappa\in KK(A,\mathbb C)\) be even. There are a separable graded Hilbert space \(H_K\), an even unital representation \(\pi_K:A\to\mathcal B(H_K)\), and a closed densely defined odd self-adjoint operator \(D_K\) such that

\[
 (D_K\pm i)^{-1}\in\mathcal K(H_K),\qquad
 [H_K,\pi_K,D_K(1+D_K^2)^{-1/2}]=\kappa.
 \tag{UP.1}
\]

Every member of the unital *-algebra generated by \(S\) preserves \(\operatorname{Dom}D_K\) and has a bounded commutator with \(D_K\). The family may be enlarged by any prescribed countable dense subset of \(A\). Consequently the operator is an unbounded Kasparov representative on a dense unital *-algebra. No finite summability, prescribed eigenvalue growth, differential order or uniform commutator bound over \(S\) is asserted. A finite-dimensional or zero representative is allowed.

**Proof: the prescribed countable tests.** Start with a countably generated graded scalar Kasparov cycle representing \(\kappa\). Such a scalar module is a separable Hilbert space. The normalization theorem just linked replaces it, without changing its class, by a cycle \((\widetilde H,\phi,F)\) with

\[
 F=F^*=F^{-1},\quad F\text{ odd},\quad
 [F,\phi(a)]\in\mathcal K(\widetilde H)\quad(a\in A).
 \tag{UP.2}
\]

The normalization can add an oppositely graded summand with zero representation. Thus \(P=\phi(1)\) is an even orthogonal projection, and need not be the identity.

Choose a countable family \((a_j)\) containing \(S\), its adjoints, \(1\), and a countable dense subset and its adjoints. In the controlled inverse proof, this is the family to be used from the outset. Nothing there requires choosing the family independently of \(S\). With an even positive injective compact \(h\) of dense range commuting with \(F\), its nested quasicentral cutoffs \(u_n\in C^*(h)\) can satisfy

\[
 \begin{split}
 u_{n+1}u_n&=u_n,\qquad u_n\longrightarrow1\text{ strongly},\\
 \|[u_n,\phi(a_j)]\|&\leq2^{-n},\\
 \|[F,\phi(a_j)](1-u_n)\|&\leq2^{-n}
                      \quad(j\leq n).
 \end{split}
 \tag{UP.3}
\]

Here the commutator tests derive the compact ideal, and the second test is a compact approximate-identity test. Both are covered by the linked nested-cutoff proof. Each cutoff vanishes on a neighborhood of zero in \(\sigma(h)\). The continuous inverse defined on the nonzero spectrum is

\[
 r=1+\sum_{n\geq1}(1-u_n),\qquad l=r^{-1},\qquad
 D=Fl^{-1},\qquad\operatorname{Dom}D=\operatorname{Ran}l.
 \tag{UP.4}
\]

The sum is locally finite away from zero. Nesting makes every later cutoff equal to one wherever a previous one is positive. Each finite group of cutoffs vanishes sufficiently near zero, so \(r\to\infty\) there. Accordingly \(l\) extends by zero at zero and is an even injective compact operator of dense range commuting with \(F\).

For clarity, the domain assertion follows from a convergent inverse equation, rather than a bounded-sequence test. Put \(R_N=1+\sum_{n=1}^N(1-u_n)\). Then \(lR_N\to1\) strongly and \(R_N\xi\to l^{-1}\xi\) for \(\xi\in\operatorname{Ran}l\). For fixed \(j\), (UP.3) makes \([R_N,\phi(a_j)]\) converge in norm. Thus

\[
 R_N\phi(a_j)\xi\longrightarrow
       \phi(a_j)l^{-1}\xi+C_j\xi.
 \tag{UP.5}
\]

Multiply this equation by \(l\). Its left side tends to \(\phi(a_j)\xi\), proving that this vector lies in \(\operatorname{Ran}l\), with the displayed inverse image. Similarly \([F,\phi(a_j)]R_N\) converges in norm. Therefore

\[
 [D,\phi(a_j)]
   =F[l^{-1},\phi(a_j)]+[F,\phi(a_j)]l^{-1}
 \tag{UP.6}
\]

extends boundedly. Products satisfy the domain Leibniz rule. Adjoints were included among the tests, so the whole generated *-algebra has the asserted domain and commutator properties.

The inverse resolvents and bounded transform are exactly

\[
 (D\pm i)^{-1}=l(F\mp il)(1+l^2)^{-1},\qquad
 F_D=F(1+l^2)^{-1/2}.
 \tag{UP.7}
\]

The two resolvent formulas prove self-adjointness on \(\operatorname{Ran}l\), as in the linked theorem; they are compact. The difference \(F_D-F\) is compact because its scalar coefficient \((1+l^2)^{-1/2}-1\) vanishes at zero. Hence \((\widetilde H,\phi,F_D)\) still represents \(\kappa\).

**Proof: bounded off-diagonal removal.** Since \(1\) was a controlled test, \(P\) preserves \(\operatorname{Dom}D\) and \(C=[D,P]\) extends boundedly. Put \(Q=1-P\). The domain decomposes as \(P\operatorname{Dom}D\oplus Q\operatorname{Dom}D\). On it define the off-diagonal part

\[
 B=PDQ+QDP=C(2P-1),\qquad D_0=D-B.
 \tag{UP.8}
\]

The right-hand formula gives an everywhere bounded extension of \(B\), of norm at most \(\|C\|\). Symmetry of \(D\) gives \(C^*=-C\), first by testing domain vectors and then by density. Its diagonal blocks vanish, so \(C(2P-1)=-(2P-1)C\). Consequently \(B^*=B\). Both \(C\) and \(B\) are odd because \(P\) is even. Moreover \([B,P]=C\), so \(D_0\) commutes with \(P\) on the full domain.

We include the perturbation and class checks. The operator \(D_0\), on \(\operatorname{Dom}D\), is symmetric and closed, since its graph norm is equivalent to that of \(D\). For \(\rho>\|B\|\),

\[
 (D_0\pm i\rho)^{-1}
  =(D\pm i\rho)^{-1}
       \bigl(1-B(D\pm i\rho)^{-1}\bigr)^{-1}.
 \tag{UP.9}
\]

The final inverse exists by its Neumann series. Surjectivity of both operators and the symmetric adjoint-domain argument prove self-adjointness of \(D_0\). Formula (UP.9) gives compact resolvents at these two points. The resolvent identity then gives compactness at every nonreal point. Every controlled \(a\) preserves the same domain, and

\[
 [D_0,\phi(a)]=[D,\phi(a)]-[B,\phi(a)]
 \tag{UP.10}
\]

is bounded.

For a self-adjoint operator \(T\), write
\(Q_\rho(T)=T(T^2+\rho^2)^{-1}
=\tfrac12((T-i\rho)^{-1}+(T+i\rho)^{-1})\).
The bounded-transform integral in the linked Theorem 2.2 is a strong integral. Its difference here converges in norm, since the resolvent identity gives

\[
 \|Q_\rho(D_0)-Q_\rho(D)\|\leq\|B\|\rho^{-2},
 \quad
 F_{D_0}-F_D=\frac2\pi\int_0^\infty
    \bigl(Q_{\sqrt{1+t^2}}(D_0)
          -Q_{\sqrt{1+t^2}}(D)\bigr)\,dt.
 \tag{UP.11}
\]

Each integrand is compact, and the bound is integrable. Thus the difference is compact. The compact perturbation proof in *Kasparov modules and the groups KK(A,B)*, Proposition 3.1 shows that \((\widetilde H,\phi,F_{D_0})\) represents the same class.

**Proof: the essential compression.** The domain commutation with \(P\) and the uniqueness of resolvent solutions show that both resolvents of \(D_0\) commute with \(P\). Therefore the two summands reduce \(D_0\). Define

\[
 \begin{split}
 H_K&=P\widetilde H,\qquad
 \pi_K(a)=\phi(a)|_{H_K},\\
 D_K&=D_0|_{H_K},\qquad
 \operatorname{Dom}D_K=P\operatorname{Dom}D.
 \end{split}
 \tag{UP.12}
\]

This operator is odd and self-adjoint, with compact resolvents obtained by restricting those of \(D_0\). Its representation is even and unital. Source operators commute with \(P\), so (UP.10) restricts to the required bounded commutators on \(H_K\), including domain preservation. Functional calculus is block diagonal. The other block has identically zero source representation, hence is a degenerate Kasparov cycle; Proposition 2.3 of the same lesson proves its class is zero. The remaining block therefore represents exactly \(\kappa\), proving (UP.1). \(\square\)

### 11L.2. All elements of a countable discrete group

**Corollary 11L.2 (the reduced scalar auxiliary).** Let \(\Gamma\) be any countable discrete group and \(R=C_r^*(\Gamma)\). For every even \(\kappa\in KK(R,\mathbb C)\) there is a representative as in (UP.1), with an even unitary representation

\[
 V_g=\pi_K(\lambda_g),\qquad
 V_g\operatorname{Dom}D_K=\operatorname{Dom}D_K,\qquad
 C_g=[D_K,V_g]\in\mathcal B(H_K)\quad(g\in\Gamma).
 \tag{UP.13}
\]

It factors through the reduced algebra. The dense algebra can be taken to contain the complete complex group algebra \(\mathbb C\Gamma\). There is a countably generated algebraic graded graph core invariant under every \(V_g\).

**Proof.** The linear span of the countable family \(\{\lambda_g:g\in\Gamma\}\) is norm dense in \(R\), so \(R\) is separable and unital. Apply Theorem 11L.1 with this entire family, rather than only a selected generating set. Unitality gives \(V_g^*V_g=V_gV_g^*=1\), and multiplication gives \(V_gV_h=V_{gh}\). Apply domain preservation to both \(g\) and \(g^{-1}\) to get the equality in (UP.13). For a finite group-algebra element,

\[
 [D_K,\pi_K(\textstyle\sum_g c_g\lambda_g)]
       =\sum_g c_g C_g,
 \qquad
 C_{gh}=C_gV_h+V_gC_h.
 \tag{UP.14}
\]

Both equations hold on the full domain and extend boundedly. Each fixed group element has a finite bound; these formulas impose no common bound as \(g\) varies. Reduced continuity is retained because \(\pi_K\) is a representation of \(R\) itself.

For the core, choose a countable dense set \((x_j)\) in \(H_K\), and put \(\xi_j=(D_K-i)^{-1}x_j\). The map \(D_K-i\) is an isometry from its domain with norm \((\|\xi\|^2+\|D_K\xi\|^2)^{1/2}\) onto \(H_K\). Hence the span of the \(\xi_j\) is graph dense. Let \(\gamma_K\) be the grading, and \(P_\pm=(1\pm\gamma_K)/2\). These two projections preserve the domain and are bounded for its graph norm because \(D_K\gamma_K=-\gamma_KD_K\). Now set

\[
 \mathcal C_K=\operatorname{span}_{\mathbb C}
       \{V_gP_\pm\xi_j:g\in\Gamma,\ j\geq1\}.
 \tag{UP.15}
\]

All these vectors lie in the domain. The core is graded, contains the graph-dense original span, and is invariant under all group elements. Countability makes its displayed generating family countable. Explicitly,

\[
 \|V_g\xi\|_{\operatorname{Dom}D_K}
       \leq(1+\|C_g\|)\|\xi\|_{\operatorname{Dom}D_K}.
 \tag{UP.16}
\]

This follows by applying the triangle inequality to the two-component vector \((V_gD_K\xi+C_g\xi,V_g\xi)\). Thus each group operator also acts continuously on the complete graph domain. The core is not asserted to be invariant under \(D_K\) itself or under higher powers of \(D_K\). Neither assertion is needed for a graph core. \(\square\)

### 11L.3. The scalar sign and the product premise

The construction preserves the whole class \(\kappa\), rather than merely its unit pairing. In particular, if the supplied class satisfies

\[
 \langle[1_R],\kappa\rangle=+1,
 \tag{UP.17}
\]

then the compressed representative has the same sign. One can also see the scalar Fredholm operator directly. Its bounded transform is odd, so write \(F_K^+:H_K^+\to H_K^-\) for the positive-to-negative block. It is Fredholm by the proved scalar classification in *K-theory of the leaf space*, Scalar classification through the stable corona. The even operator \((1+D_K^2)^{-1/2}\) maps \(H_K\) bijectively onto \(\operatorname{Dom}D_K\): it is \((D_K-i)^{-1}\) times the everywhere unitary continuous-calculus function \((D_K-i)(1+D_K^2)^{-1/2}\). Consequently

\[
 F_K^+=D_K^+(1+D_K^2)^{-1/2}|_{H_K^+},
 \qquad \operatorname{ind}D_K^+=\operatorname{ind}F_K^+=+1.
 \tag{UP.18}
\]

Here the domain of \(D_K^+\) carries its graph norm. The equality of indices follows from equality of ranges and the bijective domain change; its kernel and cokernel are the same finite-dimensional spaces. If (UP.17) is not assumed, the identical argument gives the actual integer \(\langle[1_R],\kappa\rangle\).

This supplies precisely the auxiliary-operator premise of the regular scalar product: compact resolvent, an odd self-adjoint operator, an even reduced group representation, full domain preservation, bounded commutators and a graph core. In the auxiliary-first product

\[
 D_K\widehat\otimes1+\gamma_K\widehat\otimes D_N,
 \tag{UP.19}
\]

the supplied even scalar class contributes no extra Clifford degree or exchange sign. The normal factor retains its actual inverse phase \(z^{-1}\), determinant coefficient, grading and local Bott value \(+1\). Existence of a class satisfying (UP.17), the full normal/holonomy field construction, and any prescribed equivariant restriction identity are separate premises. This theorem does not manufacture those premises or a finite-order differential operator on a holonomy graph.

### 11L.4. A finite model of the compression

The compression is not cosmetic. Consider \(\widetilde H^+=\mathbb C^2\), \(\widetilde H^-=\mathbb C^2\), with ordered coordinates \((e_1^+,e_2^+,e_1^-,e_2^-)\), and put

\[
 T=\begin{pmatrix}2&1\\1&3\end{pmatrix},\quad
 F=\begin{pmatrix}0&I_2\\I_2&0\end{pmatrix},\quad
 l=\begin{pmatrix}T^{-1}&0\\0&T^{-1}\end{pmatrix},\quad
 P=\operatorname{diag}(1,1,1,0),\quad \phi(a)=aP.
 \tag{UP.20}
\]

The positive matrix \(T\) has eigenvalues \((5\pm\sqrt5)/2\), so \(l\) is positive and invertible. Its scalar compactness is automatic in finite dimension. It commutes with \(F\), and \(D=Fl^{-1}\) has lower-left block \(T\). Equation (UP.8) removes only the couplings to \(e_2^-\). On \(P\widetilde H\) the compressed positive-to-negative block is

\[
 D_K^+=\begin{pmatrix}2&1\end{pmatrix}:\mathbb C^2\to\mathbb C.
 \tag{UP.21}
\]

Its kernel is the line spanned by \((1,-2)\), its range is all of \(\mathbb C\), and its index is \(+1\). Although \(F\) before compression is an exact odd unitary, its representation is not unital. After compression the representation is unital and the index-one kernel remains. This is an exact finite-dimensional illustration of the construction, not a model for the growth or summability of the general operator.

### 11L.5. The exact finite compression diagram

![The full off-diagonal removal and the index-one essential compression](../figures/auxiliary-essential-compression.png)

**Figure 11L.1.** Theorem 11L.1, equations (UP.8)–(UP.12), and the exact finite model (UP.20)–(UP.21). The panels show every entry of the doubled operator, its removed off-diagonal part and its three-dimensional essential compression, in the stated coordinate order. Orange outlines mark the fourth coordinate, whose source representation is zero. The represented positive and negative dimensions are two and one; the row (2,1) has the displayed kernel and no cokernel. The norm √10 is the exact norm of the removed coupling vector (1,3). The finite diagram illustrates the general class-preserving proof; it asserts no eigenvalue growth or summability. Original mathematical expression is CC0 1.0; bundled glyphs retain their component terms.

[Editable SVG](../figures/auxiliary-essential-compression.svg) · [Reproducible source](../reproduction/proper-regular-auxiliary/draw_proper.py).

## 11M. Regular fields, unit-space balancing and the graph Dirac interface

Sections 11J–11L prove the proper-factorization unit calculation and the scalar regular-auxiliary realization for a discrete action, with a single group algebra and an ordinary external KK product. An arbitrary holonomy groupoid has range
and source units. These cannot be replaced by one group without changing
the construction. We give the correct regular-field norm argument and two
precise limitations of a proposed groupoid transfer. They concern the tensor
and the operator, respectively; they do not exclude every normalized reduced
graph class.

### 11M.1. The original normalization and the stronger auxiliary premise

In Connes's survey, page 28 defines the transverse class using an
orientation-preserving disk and its positive Bott generator. Page 29 asks
for a graph Dirac K-homology class whose pairing with that class is one,
under the holonomy-invariant normal spin-c hypothesis. Thus the required
integer is the **original oriented local transverse Bott pairing**. The
normal inverse coefficient, its grading, determinant phase and final right
Clifford factor must be the prescribed ones of Section 11E.9.

Neither source passage separately requires an auxiliary to represent the
equivariant unit, or a fixed maximal class to descend, or a full restriction
to every function on a chosen closed transversal to equal one supplied
normal cycle. These can be sufficient comparison premises for a particular
route. They are stronger data than the original pairing requirement.
Proposition 11I.2 and Section 11K.5 distinguish ordinary scalar index from equivariant
index; it does not turn an equivariant-unit equality into a necessary
condition for every solution of the survey problem.

### 11M.2. Why ordinary balancing is the wrong algebra

**Proposition 11M.1 (the diagonal needs different unit-space typing).**
Let the pair groupoid on two units have algebra \(A=M_2(\mathbb C)\), and
let \(D=\mathbb C^2\subset A\) be its diagonal unit algebra. Quotienting
\(A\otimes A\) by the two-sided ideal generated by
\(d\otimes1-1\otimes d\), \(d\in D\), gives the zero algebra.
Nevertheless

\[
 \Delta(e_{ij})=e_{ij}\otimes e_{ij},\qquad
 P=e_{11}\otimes e_{11}+e_{22}\otimes e_{22}
 \tag{GR.1}
\]

defines an injective homomorphism into the corner
\(P(A\otimes A)P\), carrying the unit to \(P\).
This corner is not that balanced quotient.

There is also a continuous obstruction to treating the corner as an
ordinary scalar product-space diagonal. Let \(M\) be a nonempty compact
positive-dimensional smooth manifold with a smooth positive volume measure,
and put \(H=L^2(M)\). Any operator \(T\in B(H\otimes H)\) satisfying

\[
 (M_h\otimes1)T=(1\otimes M_h)T,
 \qquad h\in C(M),
 \tag{GR.2}
\]

is zero. In particular no nonzero homomorphism from the pair-groupoid
algebra \(K(H)\) into this ordinary product representation can have every
image satisfy (GR.2).

**Proof.** Identify \(A\otimes A\) with the matrices on the basis
\(v_{11},v_{12},v_{21},v_{22}\). For \(p=e_{11}\), the balancing generator
\(p\otimes1-1\otimes p\) has diagonal entries \(0,1,-1,0\).
Multiplying it on the left by \(E_{a,12}\) and on the right by
\(E_{12,b}\) gives every matrix unit \(E_{a,b}\). Its two-sided ideal is
therefore the entire algebra. This also explains the defect in regarding
\(A\) as a central \(D\)-algebra: the unit functions are not central
when arrows change the unit.

On the other hand,
\((e_{ij}\otimes e_{ij})(e_{kl}\otimes e_{kl})
=\delta_{jk}e_{il}\otimes e_{il}\), and involution reverses the two
indices. The four displayed matrices act as the matrix units on
\(\operatorname{span}\{v_{11},v_{22}\}\); hence their map is injective,
their unit is \(P\), and this is precisely the corner. The compressed
balancing generators vanish, while the uncompressed quotient is zero.

For the continuous assertion identify \(H\otimes H=L^2(M\times M)\)
using the product volume. Choose a countable collection \(h_j\in C(M)\)
separating points; for example a countable dense collection in \(C(M)\)
does so. For every vector \(v\), equation (GR.2) says
\((h_j(x)-h_j(y))(Tv)(x,y)=0\) almost everywhere for every \(j\).
Discard the countable union of these null sets. Outside it, \(Tv\) is
supported where all \(h_j(x)=h_j(y)\), namely on the diagonal.
That diagonal has product measure zero: every singleton has zero smooth
volume, so the nonnegative iterated integral of its indicator is zero.
Thus \(Tv=0\), for every \(v\), proving the claim. No theorem about all
homomorphisms \(K(H)\to B(H\otimes H)\) is asserted: the specific unit
compatibility (GR.2) is essential. \(\square\)

The pair groupoid of a compact manifold is already a holonomy groupoid
of the one-leaf foliation, with zero normal bundle and its invariant
spin-c structure. Its scalar K-homology is not obstructed by this
proposition. Passing to a point transversal gives its familiar rank-one
Morita corner. What fails is the proposed **ordinary** balanced tensor
or product-measure diagonal, not that Morita construction. For a
positive-dimensional transversal, a fibrewise Hilbert tensor uses a
measure over one unit variable, rather than product measure over two
independent copies. The next theorem uses that correct space explicitly.

### 11M.3. A genuine groupoid regular-field absorption

Let \(\mathcal G\rightrightarrows X\) be a second-countable Hausdorff
étale groupoid. Write
\(\mathcal G^x=r^{-1}(x)\),
\(\mathcal G_x=s^{-1}(x)\). Let \(\mu\) be a sigma-finite positive
Borel measure on \(X\) invariant under every partial bisection, and let
\(V_x\) be a measurable finite-dimensional Hermitian field with a
measurable unitary groupoid action

\[
 U_\gamma:V_{s\gamma}\longrightarrow V_{r\gamma},
 \qquad U_{\gamma\eta}=U_\gamma U_\eta.
 \tag{GR.3}
\]

This can be the **whole** normal inverse field, with its right Clifford
action and grading; it is not just its determinant or a minimal spinor.
The regular auxiliary is the field
\(\ell^2(\mathcal G^x)\), with
\(L_\gamma\delta_\eta=\delta_{\gamma\eta}\) for
\(r\eta=s\gamma\). Each \(L_\gamma\) maps between its specified
unit fibres. It is not a single representation of one global group.

Define the two counting measures on the arrow space by

\[
 \int b\,d\nu_r=\int_X\sum_{r g=x}b(g)\,d\mu(x),
 \qquad
 \int b\,d\nu_s=\int_X\sum_{s g=x}b(g)\,d\mu(x).
 \tag{GR.4}
\]

All sums retain the arrow labels, including distinct isotropy arrows.

**Theorem 11M.2 (typed regular-field norm).** On

\[
 \mathscr H_r=\int_X^\oplus
       V_x\otimes\ell^2(\mathcal G^x)\,d\mu(x)
       =L^2(\mathcal G,\nu_r;r^*V),
 \tag{GR.5}
\]

the convolution expression

\[
 (\Pi_U(f)\xi)(g)
    =\sum_{r\gamma=r g}f(\gamma)U_\gamma\xi(\gamma^{-1}g),
       \qquad f\in C_c(\mathcal G),
 \tag{GR.6}
\]

has the actual reduced norm bound
\(\|\Pi_U(f)\|\le\|f\|_r\). It extends to a representation of
\(C_r^*(\mathcal G)\). The onto unitary proving it is

\[
 (J\xi)(g)=U_g^{-1}\xi(g),\qquad
 J:\mathscr H_r\longrightarrow
    \int_X^\oplus\bigl(\ell^2(\mathcal G_x)\otimes V_x\bigr)\,d\mu(x).
 \tag{GR.7}
\]

This is a fibrewise absorption statement. It does not provide a
homomorphism into an ordinary spatial tensor of two groupoid algebras,
an auxiliary Fredholm operator, or a scalar Kasparov cycle.

**Proof.** A countable open bisection cover exists by second countability
and étaleness. Subtract preceding members to partition the arrows into
countably many Borel partial bisections, without duplicating a label.
On one such piece, the source-to-range map preserves \(\mu\) by
invariance. Integrating a nonnegative function over its range or over
its source therefore gives the same number. Summing the pieces by
monotone convergence proves \(\nu_r=\nu_s\), including infinite
integrals. This is the only measure interchange in (GR.7).

Each \(U_g\) is unitary and measurable. Hence (GR.7) preserves the
norm by (GR.4), carries the coefficient at \(r g\) to the coefficient
at \(s g\), and has the measurable onto inverse
\((J^{-1}\eta)(g)=U_g\eta(g)\). If the normal inverse transitions
are \(z^{-1}\rho_t(s)\), these entire transitions enter \(U_g\);
applying their inverse in \(J\) identifies the two pullbacks. It does
not trivialize the original line, replace its determinant phase, or
alter the grading or final right Clifford action.

The cocycle identity gives, with all maps typed,
\[
 U_g^{-1}U_\gamma U_{\gamma^{-1}g}=1_{V_{s g}}.
 \tag{GR.8}
\]
Consequently \(J\Pi_U(f)J^{-1}\) is
\[
 (\lambda(f)\eta)(g)
       =\sum_{r\gamma=r g}f(\gamma)\eta(\gamma^{-1}g).
 \tag{GR.9}
\]
The source of \(\gamma^{-1}g\) is \(s g\), so this decomposes over
the source unit \(x\) as \(\lambda_x(f)\otimes1_{V_x}\), the regular
source-column representation. On each fibre its norm is at most
\(\|\lambda_x(f)\|\): expand in an orthonormal basis of the finite
coefficient space and sum the squared component norms. The direct
integral norm is at most the supremum over \(x\), which is the
defining reduced norm. Compact support has a finite bisection cover,
so its convolution sums first define bounded operators on the dense
finite-section core; equivalently the regular bound just proved
extends that formula. Matrix multiplication of the sums proves the
convolution law. Taking the adjoint and using \(\nu_r=\nu_s\) gives
the groupoid involution, or follows from the regular form (GR.9).
Norm completion proves the representation assertion. \(\square\)

**Lemma 11M.5 (the eligible metric separates holonomy arrows).**
For a smooth foliation on a second-countable Hausdorff manifold with a holonomy-invariant
positive transverse metric, its holonomy groupoid is Hausdorff.
The holonomy groupoid of any countable complete transversal is also
second-countable, Hausdorff and étale. Its local Riemannian volume has the invariance
required in (GR.4).

**Proof.** A transversal obtains its metric from its tangent
identification with the normal bundle. Metric invariance says
that each transverse holonomy map is a local Riemannian isometry.
A local isometry's germ is determined by its value and first
derivative. Here is the needed local argument. The Koszul formula
determines the Levi-Civita connection from the metric, Lie brackets
and derivatives of the metric; a local metric isometry preserves
each of its terms and therefore the connection. It carries a
geodesic with initial velocity \(v\) to the geodesic with initial
velocity its derivative applied to \(v\).

For completeness these initial data give unique short geodesics.
In a relatively compact coordinate box the smooth Christoffel
coefficients and their first derivatives are bounded. The integral
equation for position and velocity on a sufficiently short interval
is a contraction on a closed uniform ball of continuous paths;
its coefficient Lipschitz constant times the interval length is
less than one. Successive substitution converges uniformly,
gives a solution, and gives uniqueness by the same contraction
estimate. For sufficiently small initial \(v\), the solution
exists up to time one and stays in the box: the velocity equation
has a bound \(C|\dot x|^2\), so integration keeps
\(|\dot x|\le2|v|\) and \(|x(t)-x(0)|\le2|v|\) after reducing
the velocity ball. Differencing the integral equation first bounds
the change of its solution by a constant times the change of \(v\).
Its derivative is the unique solution of the linear variational
integral equation with coefficient the derivative of the smooth
position/velocity vector field along the solution. On each short
interval this linear integral equation is a contraction too.
Subtracting that candidate linear change from the actual change
leaves a remainder bounded by the contraction constant times
itself, plus \(o(|\Delta v|)\), by the uniform first-order
remainder for the vector field on the compact box. Absorbing
the contraction factor proves differentiability. The same
estimate and continuity of the vector-field derivative prove
continuous dependence of this derivative on \(v\); concatenate
the finitely many short intervals up to time one. At the constant
zero-velocity solution the variational equation is explicit:
the linearized position is \(tv\), and the quadratic velocity
term has zero linearization. Thus the time-one position map has
derivative the identity at \(v=0\). On a smaller ball its remainder
in the time-one map \(v\mapsto x+v+R(v)\) thus has Lipschitz
constant at most \(1/2\). For \(w\) in a sufficiently small
ball, \(v=w-R(v)\) is again a contraction of the velocity
ball into itself. Hence these geodesic endpoints fill a
neighbourhood of \(x\).

Two local isometries with the same value and derivative at \(x\)
therefore agree on all of these endpoints, by uniqueness of the
transported geodesic. They have the same germ. A holonomy arrow
is determined by its source, range and transverse holonomy germ.
Consequently the continuous map sending an arrow to
its source, range and derivative between their normal fibres
is injective. Continuity follows in the defining smooth graph
charts: the transverse holonomy map and its derivative vary
smoothly there. The bundle of normal linear isometries over the
product of the unit manifold with itself is Hausdorff. Distinct
arrow images therefore have disjoint open neighbourhoods,
whose inverse images separate the arrows. This proves
Hausdorffness; an embedding or properness of that map is not
needed.

For an immersed disjoint union of transversal charts \(T\),
the restricted groupoid is the fibre product
\(T\times_r G\times_s T\). It is a subspace of a Hausdorff
product. Its local holonomy charts are bisections, making it
étale. A countable small foliation atlas generates its holonomy
charts by finite words of overlap maps. There are countably
many such words; restricting their domains to a countable
coordinate basis supplies a countable bisection basis. Thus
the transversal groupoid is second-countable, as required
in Theorem 11M.2. Its bisection maps are the local isometries already
proved, so change of variables preserves Riemannian volume.
A countable collection of relatively compact charts gives a
sigma-finite unit measure. This proves the lemma. \(\square\)

Thus Theorem 11M.2 supplies its regular-field norm calculation
for complete transversals throughout this eligible metric scope,
without an additional unproved Hausdorffness premise. The
full spin-c phases still enter the unitary \(U_g\); the lemma
only separates the underlying holonomy arrows by their metric
jets and does not erase central phases in the coefficient.
The smooth full holonomy graph has positive-dimensional source
fibres and is not being silently declared étale. Its comparison uses
the actual holonomy Morita correspondence, not a global-action
replacement.

### 11M.4. The canonical regular unit vector does not normalize an auxiliary

**Proposition 11M.4 (the canonical doubled family has index zero).**
In the continuous regular field \(E_x=\ell^2(\mathcal G^x)\) of a
Hausdorff étale groupoid, let \(e_x=\delta_{1_x}\) and

\[
 P_x=|e_x\rangle\langle e_x|,
 \qquad L_\gamma e_{s\gamma}=\delta_\gamma.
 \tag{GR.19}
\]

The unit vector is not invariant at a nonunit arrow. Its covariance
defect is nevertheless finite rank. On the identically acted-on
graded double \(E\oplus E\), the canonical self-adjoint odd family

\[
 T=\begin{pmatrix}0&2P-1\\2P-1&0\end{pmatrix}
 \tag{GR.20}
\]

has fibrewise chiral index zero and is homotopic to a degenerate
family through locally compact square and covariance defects.
It does not supply an ordinary scalar index-one auxiliary.

**Proof.** Left translation sends \(1_{s\gamma}\) to \(\gamma\).
Hence \(L_\gamma P_{s\gamma}L_\gamma^{-1}\) is the projection
onto \(\delta_\gamma\), whose difference from \(P_{r\gamma}\)
has rank at most two. The field \(e\) is continuous: the units are
open and closed in a Hausdorff étale groupoid, so multiplying the
unit indicator by any compactly supported continuous unit function
gives a continuous compactly supported regular section. Rank-one
operators made from these sections show that \(aP\) is compact
for \(a\in C_0(X)\). On a bisection the same description applies
to \(\delta_\gamma\), proving continuity and local compactness of
the translated defect. Noninvariance itself therefore does not
prohibit a permissible compact covariance defect.

The off-diagonal block \(2P-1\) is an invertible involution. Its
kernel and cokernel are both zero. More explicitly set

\[
 B_u=-1+2uP,
 \quad T_u=\begin{pmatrix}0&B_u\\B_u&0\end{pmatrix},
 \quad B_u^2-1=4u(u-1)P,
 \qquad 0\le u\le1.
 \tag{GR.21}
\]

This is norm continuous. Its square defect after unit localization
is compact, and its translated defect is \(2u\) times the
rank-at-most-two defect above. Unit multiplication commutes with
every \(B_u\). At \(u=0\) the off-diagonal block is \(-1\),
with zero square, adjoint, commutator and covariance defects. At
\(u=1/2\) its kernel and cokernel are each exactly the same unit
line, so its index remains zero there as well. This proves the
claimed homotopy and index. It does not exclude a different
index-one family or require that such a family be an equivariant
unit. \(\square\)

### 11M.5. A regular representation and bounded commutators still need compacts

**Theorem 11M.3 (the unconfined normal graph operator fails the scalar test).**
There is a compact connected suspension with holonomy-invariant positive
normal metric and a coherent full unitary spin-c lift for which the
unconfined normal Dirac on the entire smooth graph has:

- the actual reduced regular convolution representation;
- a self-adjoint maximal normal-derivative domain with a smooth compact core;
- bounded unbounded-operator commutators with every smooth compact graph kernel;
- but a fixed smooth compact convolution kernel for which
  \(\Pi(k)(1+D^2)^{-1}\) is not an ordinary scalar compact operator.

Therefore regular absorption and the normal symbol alone do not give
this graph Kasparov cycle or its prescribed Bott pairing.

**Proof: geometry and full coefficient.** Let
\(N=\mathbb R/2\pi\mathbb Z\), choose
\(\alpha=2\pi(\sqrt2-1)\), \(\beta=\pi/5\), and let
\(j\in\mathbb Z\) act by

\[
 (t,n)\longmapsto(t+j,n+j\alpha),\qquad
 V=(\mathbb R\times N)/\mathbb Z.
 \tag{GR.10}
\]

The leaf tangent is the \(t\) direction. The quotient is compact
because \([0,1]\times N\) meets every orbit, and it is a smooth
manifold because integer translations act freely and properly on
the first factor. The normal metric is \(dn^2\), the orientation
is increasing \(n\), and every return is an isometry with derivative
one. The lift \([1,e^{ij\beta}]\) is coherent; its determinant is
\(e^{2ij\beta}\). For this rank-one normal bundle, in its complete
inverse module use the full \(C_1\) coefficient and final right \(C_1\) action of
Section 11E.9, with inverse transition \(e^{-ij\beta}\rho_t(1)\)
and inverse determinant \(e^{-2ij\beta}\). We retain the prescribed
physical inverse Clifford generator, call it \(A\), with
\(A=A^*\), \(A^2=1\), and the existing grading/sign calibration.
No minimal-spinor substitution or determinant-only comparison is made.

The actual holonomy graph is
\[
 G=(\mathbb R_t\times\mathbb R_r\times N)/\mathbb Z,
 \qquad j(t,r,n)=(t+j,r+j,n+j\alpha).
 \tag{GR.11}
\]
The arrow represented by \((t,r,n)\) goes from \([r,n]\) to
\([t,n]\). There are no nonidentity leaf loops: irrationality makes
every leaf an injectively immersed copy of \(\mathbb R\). Distinct
returns have distinct transverse germs, so (GR.11) retains exactly
the holonomy arrows. The transversal at \(t=0\) has groupoid
\(N\rtimes\mathbb Z\); this is the actual suspension example,
not a model for every holonomy groupoid.

On the graph, coefficient functions obey the inverse transition
\[
 \psi(t+j,r+j,n+j\alpha)=e^{-ij\beta}\psi(t,r,n).
 \tag{GR.12}
\]
The invariant density is \(dt\,dr\,dn\), with the fixed finite
Clifford coefficient. A fundamental range strip identifies the scalar
Hilbert space with
\(L^2([0,1]_t\times\mathbb R_r\times N;C_1)\).
The graph action is
\[
 (\Pi(k)\psi)(t,r,n)=\int_{\mathbb R}
             k(t,t',n)\psi(t',r,n)\,dt'.
 \tag{GR.13}
\]
Here \(k(t,t',n)\) is the deck-invariant scalar kernel of the
arrow from \([t',n]\) to \([t,n]\), with the unitary inverse
coefficient transport in the chosen cover frame. In that frame the
transport along this lifted leaf is identity; the deck phases in
(GR.12) remain in the global coefficient.

**Proof: reduced norm and closed domain.** The same quotient volume
can instead be integrated in a fundamental **source** strip
\(0\le r<1\). This is verified by partitioning into integer strips
and applying the measure-preserving deck maps; nonnegative sums
justify the interchange. Fixing \([r,n]\), (GR.13) is precisely the
regular integral-kernel representation on its full source fibre
\(\mathbb R_t\), tensored with the fixed finite normal coefficient.
Thus
\(\|\Pi(k)\|\le\sup_{v\in V}\|\lambda_v(k)\|=\|k\|_r\).
No geometric representation on the compact transversal is substituted
for this regular representation.

In the range strip put
\[
 D=A(-i\partial_n),\qquad
 \operatorname{Dom}D=
 \{\psi\in L^2:G\text{-distributional }\partial_n\psi\in L^2\}.
 \tag{GR.14}
\]
The symbol is the prescribed inverse normal generator, with no
\(t\) or \(r\) derivative. The constant deck phase and rotation
in (GR.12) make this a globally defined physical first-order
normal expression. Fourier series on the closed normal circle
identify its domain with
\(\sum_{m\in\mathbb Z}(1+m^2)\|\psi_m\|_{L^2(t,r;C_1)}^2<\infty\).
The operator on coefficient \(m\) is the self-adjoint matrix
\(mA\). The bounded inverse matrices \((mA\pm i)^{-1}\) have
norm at most one and produce a domain vector for every \(L^2\)
right-hand side, because
\(\|mA u\|^2+\|u\|^2=\|(mA\pm i)u\|^2\).
Hence both ranges \(D\pm i\) are all of \(L^2\), proving
self-adjointness on exactly (GR.14).

For the core, first truncate Fourier modes and \(r\) tails.
In the range strip cut off a small neighbourhood of the endpoints
\(t=0,1\). This costs no derivative term, since \(D\) has no
\(t\) derivative; absolute continuity of the normal graph-norm
integral makes the removed tails small. Mollify \(t,r\) and use
finite Fourier normal sections. The resulting compact smooth
sections supported in the interior of the strip extend by
(GR.12) to globally smooth compact sections of the quotient.
They converge in the graph norm. This proves the core and does
not select arbitrary extensions at open transverse chart ends:
the normal variable here is a closed circle. A general open
transversal is not given this closed normal domain by the norm
absorption theorem.

On that core differentiation under (GR.13) gives
\[
 [D,\Pi(k)]=-iA\Pi(\partial_n k).
 \tag{GR.15}
\]
The phases are constant and \(A\) commutes with them. The differentiated
kernel is again smooth with compact quotient support. Its two
regular Schur bounds are finite, so (GR.15) extends to a bounded
scalar operator. The core identity and closure show that \(\Pi(k)\)
preserves (GR.14) and that the commutator identity holds there.

**Proof: one fixed localized defect.** Choose a real
\(\xi\in C_c^\infty(I)\), \(I=(1/4,3/4)\),
with \(\int\xi(t)^2dt=1\). Define
\[
 k(t,t',n)=\sum_{j\in\mathbb Z}
                \xi(t-j)\xi(t'-j).
 \tag{GR.16}
\]
The sum is locally finite, deck invariant, and independent of \(n\).
It descends to a smooth compact graph kernel: its quotient support
has representatives in \(\overline I\times\overline I\times N\).
It is self-adjoint and idempotent under convolution. To check
idempotence, the disjoint integer supports remove all mixed indices
and each surviving integral of \(\xi^2\) equals one. In the
fundamental range strip its action is exactly the rank-one plaque
projection in \(t\), tensored with identities on \(r,n\) and the
unchanged normal coefficient.

Take a constant normal section \(u\) of norm one in
\(L^2(N;C_1)\), and a real
\(\zeta\in C_c^\infty((-1/8,1/8))\) of norm one. For distinct
integers \(j\), prescribe in the fundamental range strip
\[
 \psi_j(t,r,n)=\xi(t)\zeta(r-j)u(n).
 \tag{GR.17}
\]
Extending by (GR.12) gives smooth compact graph sections. Their
supports in \(r\) are disjoint in this strip, so the \(\psi_j\)
are orthonormal. They satisfy
\(D\psi_j=0\) and \(\Pi(k)\psi_j=\psi_j\). Therefore
\[
 \Pi(k)(1+D^2)^{-1}\psi_j=\psi_j,
 \qquad \|\psi_j-\psi_l\|=\sqrt2\quad(j\ne l).
 \tag{GR.18}
\]
The image of this unit sequence has no convergent subsequence.
The fixed localized square defect is not compact. For the bounded
phase \(F=D(1+D^2)^{-1/2}\),
\(\Pi(k)(1-F^2)=\Pi(k)(1+D^2)^{-1}\), so a necessary scalar
Kasparov condition fails. No Bott pairing of this failed cycle
can be assigned. This proves every assertion. \(\square\)

![Unit-space tensor typing and the undifferentiated source multiplicity](../figures/groupoid-regular-interface.png)

**Figure 11M.1.** Proposition 11M.1, Theorems 11M.2–11M.3 and
(GR.1)–(GR.21). The finite corner and zero balanced quotient are
different objects. The typed triangle retains the source of
\(\gamma^{-1}g\) and the full coefficient transport of (GR.8).
The graph panel uses the actual range strip \(0\le t<1\), displays
only finitely many supports of (GR.17) in the entire source line,
and suppresses the closed normal circle and finite Clifford factor.
The fixed kernel averages the range coordinate; it does not average
the source coordinate. Every displayed unit vector and image has
norm one, and every distinct pair of images has distance \(\sqrt2\).
These finite panels illustrate the fully proved infinite sequence;
they do not replace its proof. [Editable SVG](../figures/groupoid-regular-interface.svg)
· [Generator](../reproduction/groupoid-regular-interface/draw_interface.py)
· [Component terms](../reproduction/groupoid-regular-interface/COMPONENT-TERMS.md).

### 11M.6. What the regular auxiliary would still have to supply

Theorem 11M.2 settles the scalar reduced-norm calculation for its
correctly typed regular field. It supplies no odd auxiliary generator
with an even local index-one comparison, and no connection between
such a generator and the normal differential operator. Proposition
GR.4 proves that merely doubling the canonical regular unit line
does not supply that integer. Theorem 11M.3
proves that the zero auxiliary fails even on an eligible entire
smooth graph, despite its exact normal symbol, self-adjoint domain
and bounded convolution commutators. The translation oscillator
of Section 11E successfully replaces it for this particular
suspension; hence this example is not a counterexample to the
historical graph problem.

A proposed general transfer must specify the fibrewise/correspondence
tensor and its representation, not the zero quotient in Proposition
GR.1. It must construct a globally compatible auxiliary and a closed
physical graph operator, or prove an exact comparison to one. Open
transversal ends, determinant connections and all domain/core
conditions survive that requirement. The actual normalized
counting/plaque Morita correspondence then has to carry the
prescribed transverse Bott class to a cycle whose pairing is one.
Neither a measurable absorption unitary, nor a fibrewise compact
resolvent, nor an ordinary scalar auxiliary index establishes these
global operator and class comparisons by itself. Conversely, the
source does not authorize declaring a stronger equivariant-unit
identity necessary for every possible solution. The arbitrary
eligible reduced graph construction remains outside the results
proved here.

Human-source context: Connes, *A survey of foliations and operator
algebras*, [pages 28–29 in the author's text](https://alainconnes.org/wp-content/uploads/foliationsfine.pdf),
for the original transverse class, its pairing and graph problem.
The regular-field, tensor, metric-jet and localized-defect proofs
above are independently written. Existing exact comparison routes
are Checked Morita product, plaque restriction and local normalization,
the actual oscillator graph construction of Sections 11E.14–11E.19,
and Exact Morita class, full coefficient and the positive local pairing,
with their stated domains and hypotheses.

## 11N. Coarse embeddings of finitely generated compact-Lie subgroups

Groups below have their abstract discrete topology. We construct a coarse Hilbert embedding for every finitely generated subgroup of a compact Lie group, including dense subgroups. The argument uses valuations of a rational function field; no extension theorem for valuations in an algebraic extension is required.

We use the complete earlier results Matrix coefficients and the Peter–Weyl theorem, Theorem 5.2, for a faithful finite-dimensional representation of a compact Lie group, and Amenability and the equality of full and reduced crossed products, the abelian and closed-normal-extension propositions and the compact-uniform Reiter implication. The latter supplies compactly supported almost invariant vectors for a finite-dimensional solvable Lie group.

### 11N.1. Enough scalar valuations

A multiplicative absolute value \(d\) is called discrete here if its nonzero range is \(e^{a\mathbb Z}\), \(a>0\), and \(d(x+y)\leq\max(d(x),d(y))\). An archimedean value is the usual absolute value through an embedding into \(\mathbb C\).

Call a countable family \((d_j)\) **finite on bounded intersections for \(R\)** if, for every family of finite positive bounds \(B_j\),
\[
 \{r\in R:d_j(r)\leq B_j\text{ for all }j\}
       \text{ is finite}.
 \tag{UE.1}
\]
There is no requirement that only finitely many \(B_j\) are imposed.

**Lemma 11N.1.** For every finitely generated subring \(R\) of
\(K_r=\mathbb Q(t_1,\ldots,t_r)\), with algebraically independent \(t_i\), there is such a family consisting of discrete and archimedean values.

**Proof.** For \(r=0\), every finitely generated subring of \(\mathbb Q\) lies in \(\mathbb Z[1/D]\) for some positive integer \(D\). Use the ordinary absolute value and all \(p\)-adic values. A bound for each of the finitely many primes dividing \(D\) bounds the exponent of that prime in a reduced denominator. The ordinary bound then bounds the integer numerator. This gives finitely many rationals.

Suppose the result holds for a countable field \(K=K_{r-1}\), and let \(S\subset K(t)\) be finitely generated. Factor the finitely many denominators of its generators in the Euclidean ring \(K[t]\). After adjoining their leading coefficients and inverses and the coefficients of their finitely many monic irreducible factors to a finitely generated ring \(R\subset K\), we have
\[
 S\subset R[t][p_1^{-1},\ldots,p_l^{-1}],
 \tag{UE.2}
\]
where \(p_i\) are distinct monic irreducibles with coefficients in \(R\). Let \((d_j)\) be a family for \(R\) from the induction hypothesis.

Extend every discrete \(d_j\) by the Gauss rule
\[
 d_j\!\left(\sum_i a_it^i\right)=\max_i d_j(a_i).
 \tag{UE.3}
\]
This rule is multiplicative: divide each of two polynomials by a coefficient of maximum value; their coefficients are then in the valuation ring and each reduction modulo its maximal ideal is a nonzero polynomial. Their product is nonzero over the residue field. The product consequently has maximum coefficient value \(1\). Undoing the two scalar divisions gives multiplicativity. Extend to rational functions by division. The nonarchimedean inequality and discreteness are retained.

For an archimedean embedding \(\sigma_j:K\to\mathbb C\), choose countably many distinct complex numbers \(z_{j,n}\) transcendental over the countable field \(\sigma_j(K)\). The assignments \(t\mapsto z_{j,n}\) extend \(\sigma_j\) to injective maps \(K(t)\to\mathbb C\), giving countably many archimedean values. Such numbers exist because the algebraic closure of a countable field inside \(\mathbb C\) is countable.

Add the degree value and the finitely many irreducible-factor values
\[
 d_\infty(a/b)=2^{\deg a-\deg b},\qquad
 d_{p_i}(a/b)=2^{-\operatorname{ord}_{p_i}(a/b)}.
 \tag{UE.4}
\]
Polynomial factorization proves their multiplicativity and the nonarchimedean inequality; their nonzero ranges are discrete.

An element of (UE.2) can be written as
\(s=a/\prod_i p_i^{k_i}\), \(a\in R[t]\), with no \(p_i\) dividing \(a\) when \(k_i>0\). Cancellation leaves the numerator in \(R[t]\): division by a monic polynomial with coefficients in \(R\) uses only additions and multiplications in \(R\). Bounds for \(d_{p_i}(s)=2^{k_i}\) bound every \(k_i\). The bound for \(d_\infty(s)\) then bounds \(\deg a\), say by \(m\).

For a discrete \(d_j\), the rational-function bound and the finite possibilities for the denominator exponents bound \(d_j(a)\). Equation (UE.3) bounds every coefficient of \(a\). For an archimedean \(\sigma_j\), take any \(m+1\) of the distinct \(z_{j,n}\). Their bounds on \(s(z_{j,n})\), together with the bounded denominator exponents, bound the \(m+1\) values \(a(z_{j,n})\). Inverting their fixed Vandermonde matrix bounds every coefficient \(\sigma_j(a_i)\). Thus every coefficient of \(a\) is bounded under every original \(d_j\). By (UE.1) for \(R\), each coefficient belongs to a fixed finite set. With its degree bounded, \(a\) has only finitely many possibilities. The denominator also has finitely many possibilities. This proves the induction. \(\square\)

### 11N.2. A Hilbert embedding for one value

For a discrete value on a field \(K\), put
\[
 \ell_d(g)=\log\max_{i,j}\{d(g_{ij}),d((g^{-1})_{ij})\},
       \qquad g\in GL(n,K).
 \tag{UE.5}
\]
For an archimedean value, use its image in \(GL(n,\mathbb C)\) and put
\[
 \ell_d(g)=\log\max\{\|g\|,\|g^{-1}\|\}.
 \tag{UE.6}
\]
These are nonnegative symmetric length functions. For (UE.5), matrix multiplication and the nonarchimedean inequality give subadditivity without a factor \(n\); \(gg^{-1}=1\) gives nonnegativity. For (UE.6), use the operator-norm product inequality.

**Lemma 11N.2 (positive kernels give a pseudo-metric embedding).** Suppose a group with length \(\ell\) has normalized real positive kernels \(\phi_j(x,y)\in[0,1]\) such that
\[
 \ell(x^{-1}y)\leq j\ \Longrightarrow\
       1-\phi_j(x,y)\leq2^{-j},
 \qquad
 \ell(x^{-1}y)>B_j\ \Longrightarrow\ \phi_j(x,y)=0.
 \tag{UE.7}
\]
It has a map \(F\) to a real Hilbert space whose difference is uniformly bounded on each \(\ell\)-ball, and whose bounded difference implies a bound for \(\ell(x^{-1}y)\).

**Proof.** A positive kernel is the Gram matrix of vectors \(v_x^j\): give the finite formal span of \(\delta_x\) the form \(\sum c_xc_y\phi_j(x,y)\), quotient its null space and complete. Cauchy–Schwarz justifies the quotient. The vectors have norm one. Define
\[
 F(x)=\bigoplus_j(v_x^j-v_e^j).
 \tag{UE.8}
\]
For fixed \(x\), all sufficiently large \(j\) satisfy \(\ell(x)\leq j\), and their squared summands are
\(2(1-\phi_j(x,e))\leq2^{1-j}\). Thus (UE.8) exists. If \(\ell(x^{-1}y)\leq A\), its difference has squared norm at most \(2\lceil A\rceil+\sum_{j>A}2^{1-j}\), a finite bound depending only on \(A\). If \(\ell(x^{-1}y)>\max_{j\leq m}B_j\), its first \(m\) squared summands each equal \(2\), so its squared difference is at least \(2m\). Choosing \(m\) above half any proposed squared-difference bound proves the converse. \(\square\)

**Lemma 11N.3.** The group \(GL(n,K)\) has an embedding of the kind in Lemma 11N.2 for either value in (UE.5)–(UE.6).

**Proof for an archimedean value.** Every complex invertible matrix has a decomposition \(g=ph\), with \(h\) unitary and \(p\) upper triangular with positive real diagonal. One obtains it by orthonormalizing its rows from the last row upwards, so multiplication on the right by \(h^{-1}\) is triangular and its diagonal can be made positive. The length (UE.6) is unchanged by multiplication on either side by a unitary matrix. Hence any chosen triangular representatives satisfy
\(\ell(g_1^{-1}g_2)=\ell(p_1^{-1}p_2)\).

The triangular group \(P\) is solvable and locally compact. Its strictly upper triangular unipotent part has the finite filtration by vanishing superdiagonals; its successive quotients are additive complex vector groups. Its quotient by that part is \((\mathbb R_{>0})^n\), an abelian group. Closed-normal-extension and abelian amenability in the cited lesson therefore make \(P\) amenable. The compact-uniform Reiter implication supplies nonnegative compactly supported unit \(L^2(P)\)-vectors \(\xi_j\) almost fixed on each prescribed compact set. Their coefficients
\(\varphi_j(p)=\langle\xi_j,\lambda_p\xi_j\rangle\)
are real, nonnegative, positive definite, and vanish off a compact set. The length balls of \(P\) have compact closure: both the matrix and its inverse are bounded, the positive diagonal entries are bounded above and away from zero, and a closed bounded matrix set is compact. Thus the coefficients may be chosen to satisfy (UE.7), and their compact supports have bounded length. Pull their Gram kernels back through \(g\mapsto p\) and apply Lemma 11N.2.

**Proof for a discrete value.** Choose \(\pi\) with \(d(\pi)=e^{-a}\), \(a>0\), and let \(\mathcal O=\{z:d(z)\leq1\}\). Put
\[
 H=GL(n,\mathcal O),\quad
 A=\{\operatorname{diag}(\pi^{k_1},\ldots,\pi^{k_n}):k_i\in\mathbb Z\},
 \quad P=AN,
 \tag{UE.9}
\]
where \(N\) is the upper unipotent group. Here \(GL(n,\mathcal O)\) means that both the matrix and its inverse have entries in \(\mathcal O\).
We have \(GL(n,K)=PH\). To prove this, move an entry of largest value in the last row into the last column. Every other last-row entry divided by it belongs to \(\mathcal O\). Integral elementary column operations clear that row. Repeat on the upper block. Finally split each nonzero diagonal entry into a power of \(\pi\) times an \(\mathcal O\)-unit. All the column operations and diagonal units belong to \(H\).

Every \(h\in H\) has length zero. Subadditivity on both sides consequently proves bi-\(H\) invariance of the length and, as before, equality of the lengths of differences of chosen triangular representatives.

Let \(N_M\) be generated by the elements of \(N\) of length at most \(M\). Every such generator has
\(d(n_{ij})\leq e^M\) for \(i<j\). All products and inverses lie in the subgroup described by
\(d(n_{ij})\leq e^{(j-i)M}\).
Matrix multiplication proves closure: each term in its \(ij\) entry has exponent bound \((k-i)M+(j-k)M=(j-i)M\), and the nonarchimedean sum retains the bound. The finite geometric series for the inverse of a unipotent matrix proves the same bound for inverses. Thus
\[
 \ell(n_0)\leq(n-1)M\qquad(n_0\in N_M).
 \tag{UE.10}
\]

In the permutation representation of \(P\) on \(\ell^2(P/N_M)\), take the unit vector which is constant on the cosets represented by the finite box
\[
 A_M=\{\operatorname{diag}(\pi^{k_i}):a|k_i|\leq M/4\},
 \tag{UE.11}
\]
and zero elsewhere. The cosets are distinct since \(A\cap N_M=\{e\}\). Its coefficient \(\varphi_M\) is normalized, nonnegative and positive definite. If \(\ell(n)\leq2C\) and \(M\geq4C\), then \(n\) fixes every coset in (UE.11): for its representative \(a_1\),
\[
 \ell(a_1^{-1}na_1)\leq M/2+2C\leq M,
\]
so that conjugate belongs to \(N_M\).

For \(g=an\) with \(\ell(g)\leq C\), triangular diagonals give
\(\ell(a)\leq C\), and then \(\ell(n)\leq2C\). For \(M\geq4C\),
\(\varphi_M(an)=\varphi_M(a)\).
Writing \(L=\lfloor M/(4a)\rfloor\), the overlap fraction for a diagonal exponent shift \(r\in\mathbb Z^n\) is
\[
 \prod_{i=1}^n
       \max\left(0,1-\frac{|r_i|}{2L+1}\right).
 \tag{UE.12}
\]
The bound \(a|r_i|\leq C\) makes this tend uniformly to \(1\) as \(M\to\infty\).
If the coefficient is nonzero, then
\(g=a_2n_0a_1^{-1}\) for \(a_1,a_2\in A_M\) and \(n_0\in N_M\).
Equations (UE.10)–(UE.11) bound its length by
\((n-\tfrac12)M\). Thus each coefficient has bounded length support. Choose \(M\) successively to meet (UE.7), pull the kernels back from \(P\), and apply Lemma 11N.2. \(\square\)

### 11N.3. Assembly for all eligible subgroups

**Theorem 11N.4.** Every finitely generated subgroup of a compact Lie group, with any finite word metric, admits a coarse embedding into a separable real Hilbert space.

**Proof.** The faithful unitary representation from the cited Peter–Weyl lesson puts the compact Lie group in \(GL(n,\mathbb C)\). Let \(K\) be the subfield generated over \(\mathbb Q\) by the finitely many matrix entries of a symmetric generating set of \(\Gamma\). It is a finitely generated field of characteristic zero. Choose a maximal algebraically independent subset \(t_1,\ldots,t_r\) of a finite field-generating list. Every remaining generator is algebraic over \(K_0=\mathbb Q(t_1,\ldots,t_r)\); adjoining finitely many algebraic elements gives a finite extension \(K/K_0\), of some degree \(b\).

Regard \(K^n\) as a vector space of dimension \(nb\) over \(K_0\), and choose a \(K_0\)-basis. The faithful action of \(GL(n,K)\) on this vector space gives an injective homomorphism to \(GL(nb,K_0)\). This regular-representation step avoids extending any valuation across \(K/K_0\).

Let \(R\subset K_0\) be generated by the entries of the finitely many generator matrices and their inverses. It is finitely generated and contains every entry of every element of \(\Gamma\). Take the countable values \(d_j\) from Lemma 11N.1. For each, Lemma 11N.3 gives a Hilbert map \(F_j\), translated so \(F_j(e)=0\), whose differences are bounded on each \(\ell_j\)-ball and control \(\ell_j\) in the other direction. The finite set of generators has finite \(\ell_j\)-lengths. Hence there is a finite \(A_j\geq1\) such that
\[
 \|F_j(x)-F_j(xs)\|\leq A_j
       \quad(x\in\Gamma,\ s\text{ a generator}).
 \tag{UE.13}
\]
For a word of length \(l\), the triangle inequality gives
\(\|F_j(x)-F_j(xg)\|\leq l A_j\).
Put \(\varepsilon_j=2^{-j}/A_j\) and define
\[
 F(g)=\bigoplus_j\varepsilon_jF_j(g).
 \tag{UE.14}
\]
The preceding word bound proves convergence at every \(g\) and the upper estimate
\[
 \|F(x)-F(xg)\|\leq |g|
             \left(\sum_j4^{-j}\right)^{1/2}.
 \tag{UE.15}
\]

If that difference is at most \(C\), every \(j\)-difference is at most \(C/\varepsilon_j\). The converse bound of Lemma 11N.2 bounds \(\ell_j(g)\) independently of \(x\). For a discrete value, (UE.5) directly bounds every entry of \(g\). For an archimedean value, (UE.6) bounds its operator norm and hence each entry. Thus (UE.1) bounds every entry to a finite set. Only finitely many matrices, hence group elements \(g\), are possible.

This proves the uniform properness of differences. Define the lower control at radius \(r\) as the infimum of all differences with \(|g|\geq r\), capped above by \(r\); include \(0\) at \(r=0\). It is finite and nondecreasing, and tends to infinity because a bounded difference allows only a finite set of \(g\). The upper control is the finite linear bound (UE.15). These are the coarse controls in (CI.1). Only the countable span of the images of \(\Gamma\) in each summand is needed, so the receiving Hilbert space is separable. The proof did not use that the inclusion of the abstract group in the compact Lie group is closed or discrete in the Lie topology. \(\square\)

**Corollary 11N.5 (characteristic-zero matrix groups).** Every finitely generated subgroup of \(GL(n,F)\), with \(F\) any characteristic-zero field and with a finite word metric, has a coarse embedding in a separable real Hilbert space.

**Proof.** Let \(K\) be the field generated over \(\mathbb Q\) by the entries of a symmetric finite matrix generating set. Choose a transcendence basis from that finite list; the remaining entries give a finite algebraic extension of \(K_0=\mathbb Q(t_1,\ldots,t_r)\). Restriction of scalars gives an injective matrix representation over \(K_0\). The finite ring of generator and inverse entries then satisfies Lemma 11N.1. Lemmas 11N.2–11N.3 and (UE.13)–(UE.15) give the embedding and both controls exactly as in Theorem 11N.4. These steps use only the matrix inclusion over \(K_0\), rather than a compact ambient group. The countable-span argument makes the Hilbert space separable. \(\square\)

### 11N. Reading

Guentner, Higson and Weinberger, [The Novikov Conjecture for Linear Groups](https://math.hawaii.edu/~erik/papers/linear.pdf), author draft of 15 May 2003, Sections 2–4. The proof here uses only characteristic zero, factors the finite algebraic extension into a matrix representation, constructs its positive-kernel Hilbert maps directly, and retains the abstract group metric.


## 11O. From a coarse embedding to a proper Hilbert-field cone

The argument constructs the compact coefficient base and the proper locally compact space, including varying and infinite-dimensional fibres. It does not assert the existence of a Dirac inverse for the resulting field.

Let \(\Gamma\) be finitely generated, with a finite symmetric generating set and word length \(|g|\). Suppose \(f:\Gamma\to H\) is a map to a real Hilbert space such that
\[
 \rho_-(|x^{-1}y|)\leq\|f(x)-f(y)\|
      \leq\rho_+(|x^{-1}y|),
 \qquad \lim_{r\to\infty}\rho_-(r)=\infty,
 \tag{CI.1}
\]
where the two functions are finite, nonnegative and nondecreasing. The domain metric is the abstract discrete word metric. A bounded orbit in a compact Lie group is not itself a map satisfying (CI.1).

### 11O.1. A compact coefficient base

For \(g\in\Gamma\) set
\[
 a_g(x)=\|f(x)-f(xg)\|^2.
 \tag{CI.2}
\]
These are bounded real functions, since \(|x^{-1}xg|=|g|\). Let \(B\) be the unital C*-subalgebra of \(\ell^\infty(\Gamma)\) generated by all right translates of all \(a_g\). This is separable: its generating set is countable. Its spectrum \(Y\) is compact metrizable, and the evaluations at the points of \(\Gamma\) have dense image. To see density directly, a continuous function vanishing at every evaluation would be the zero bounded function under the faithful inclusion \(B\subset\ell^\infty(\Gamma)\).

Right translation extends to a right action \(y\mapsto yg\) on \(Y\). Write \(k(y,g)\) for the continuous extension of \(a_g\). For all \(y\),
\[
 \rho_-(|g|)^2\leq k(y,g)\leq\rho_+(|g|)^2,\qquad
 k(y,e)=0,\qquad k(yg,g^{-1})=k(y,g).
 \tag{CI.3}
\]
All these assertions hold on the dense evaluation image and then by continuity. Moreover, for real numbers \(c_1,\ldots,c_n\) with sum zero,
\[
 \sum_{i,j}c_ic_j k(yg_i,g_i^{-1}g_j)\leq0.
 \tag{CI.4}
\]
At an evaluation \(y=x\), the expression is
\[
 \sum_{i,j}c_ic_j\|f(xg_i)-f(xg_j)\|^2
       =-2\left\|\sum_i c_i f(xg_i)\right\|^2.
\]
The zero-sum condition cancels both single-index squared-norm terms. Taking continuous limits proves (CI.4).

Let \(X=\operatorname{Prob}(Y)\), with its weak probability topology, and extend the right action by pushforward. This is a compact metrizable convex space. Define
\[
 K(\mu,g)=\int_Y k(y,g)\,d\mu(y),\qquad
 q_\mu(h,l)=K(\mu h,h^{-1}l).
 \tag{CI.5}
\]
Each function of \(\mu\) is continuous. Integration preserves (CI.3) and (CI.4); thus \(q_\mu\) is a symmetric conditionally negative kernel with zero diagonal, and
\[
 \rho_-(|h^{-1}l|)^2\leq q_\mu(h,l)
       \leq\rho_+(|h^{-1}l|)^2.
 \tag{CI.6}
\]
The parameter \(\mu\) is a probability, not an invariant probability under \(\Gamma\).

For every finite subgroup \(F\), choose \(y_0\in Y\) and let
\(\mu_F=|F|^{-1}\sum_{h\in F}\delta_{y_0h}\).
This is \(F\)-fixed. The affine path
\[
 (\mu,t)\longmapsto(1-t)\mu+t\mu_F
 \tag{CI.7}
\]
is an \(F\)-equivariant contraction of the entire \(X\). It does not produce a fixed point for an infinite nonamenable group.

### 11O.2. The Hilbert field and its exact affine action

For each \(\mu\), take the real vector space of finite formal combinations
\(\sum_h c_h\delta_h\) with \(\sum_hc_h=0\), and put
\[
 \left\langle\sum_hc_h\delta_h,\sum_ld_l\delta_l\right\rangle_\mu
       =-\frac12\sum_{h,l}c_hd_lq_\mu(h,l).
 \tag{CI.8}
\]
Conditional negativity makes this positive semidefinite. Quotient its null space and complete to \(H_\mu\). The Cauchy–Schwarz inequality for the semidefinite form follows by requiring nonnegativity of the quadratic polynomial for \(v+tw\); a null vector is consequently orthogonal to every vector. The quotient is therefore well defined.

Write \(e_h^\mu=[\delta_h-\delta_e]\). Then
\[
 \begin{aligned}
 \langle e_h^\mu,e_l^\mu\rangle
   &=\frac12\bigl(q_\mu(e,h)+q_\mu(e,l)-q_\mu(h,l)\bigr),\\
 \|e_h^\mu-e_l^\mu\|^2&=q_\mu(h,l).
 \end{aligned}
 \tag{CI.9}
\]
Their finite spans are dense, and their Gram entries depend continuously on \(\mu\). This defines a separable continuous real Hilbert field: start with finite sums of these sections with continuous scalar coefficients, take local uniform limits in fibre norm, and use (CI.9) to see that all norm functions are continuous. The evaluation spans remain dense in each fibre. No constant-rank or locally trivial Hilbert-bundle assertion is needed.

Use arrows \((\mu,g)\) with source \(\mu g\), range \(\mu\), and product
\((\mu,g)(\mu g,h)=(\mu,gh)\).
The formula
\[
 L_{\mu,g}e_h^{\mu g}=e_{gh}^{\mu}-e_g^\mu
 \tag{CI.10}
\]
preserves all Gram products, because
\(q_{\mu g}(h,l)=q_\mu(gh,gl)\).
It is onto: each \(e_l^\mu\) is the image of
\(e_{g^{-1}l}^{\mu g}-e_{g^{-1}}^{\mu g}\).
Thus it extends to an orthogonal isomorphism \(H_{\mu g}\to H_\mu\).
The affine maps
\[
 A_{\mu,g}v=L_{\mu,g}v+e_g^\mu
 \tag{CI.11}
\]
compose exactly, since \(L_{\mu,g}e_h^{\mu g}+e_g^\mu=e_{gh}^\mu\).
Their displacement at the origins has squared norm
\[
 \|e_g^\mu\|^2=K(\mu,g)\geq\rho_-(|g|)^2.
 \tag{CI.12}
\]
This is a metrically proper affine action of the transformation groupoid over \(X\). It is not a proper affine isometric action of the abstract group on one Hilbert space with a fixed base point.

### 11O.3. A locally compact proper space without finite fibre dimension

Consider the set
\[
 Z=\{(\mu,v,t):\mu\in X,\ v\in H_\mu,\ 0\leq\|v\|^2\leq t\}.
 \tag{CI.13}
\]
Give it the topology determined by \(\mu\), \(t\), and the real coordinates
\(\langle v,e_h^\mu\rangle\), for \(h\in\Gamma\).
The extra coordinate \(t\) is squared total height. In particular this is a weak-fibre cone topology; it is not the norm topology of an infinite-dimensional Hilbert ball.

**Theorem 11O.1.** The space \(Z\) is second countable, locally compact Hausdorff and sigma-compact. The affine action extends continuously to \(Z\) by
\[
 (\mu g,v,t)\longmapsto
 \left(\mu,A_{\mu,g}v,\
 t+2\langle L_{\mu,g}v,e_g^\mu\rangle+\|e_g^\mu\|^2\right).
 \tag{CI.14}
\]
Its action-relation map is proper.

**Proof.** For \(R\geq0\), let \(Z_R=\{t\leq R\}\). Each coordinate has
\(|\langle v,e_h^\mu\rangle|\leq\sqrt R\,\rho_+(|h|)\).
View \(Z_R\) in the compact countable product
\[
 X\times[0,R]\times
     \prod_{h\in\Gamma}
       [-\sqrt R\,\rho_+(|h|),\sqrt R\,\rho_+(|h|)].
 \tag{CI.15}
\]
Its image is exactly the set whose coordinates \(z_h\) satisfy, for every finite rational coefficient family,
\[
 \left|\sum_h c_h z_h\right|^2
       \leq t\sum_{h,l}c_hc_l
                       \langle e_h^\mu,e_l^\mu\rangle .
 \tag{CI.16}
\]
One implication is Cauchy–Schwarz. Conversely the inequalities, extended to real coefficients by continuity, make
\(\sum c_he_h^\mu\mapsto\sum c_hz_h\)
a well-defined functional of norm at most \(\sqrt t\) on the dense span. Extension and the Hilbert-space representation of a bounded functional give a unique \(v\) of norm at most \(\sqrt t\) with these coordinates. The uniqueness follows from density. Thus (CI.16) also proves that the image is closed; each inequality has continuous Gram entries. Hence \(Z_R\) is compact metrizable. The coordinates separate points. Neighborhoods \(t<R\) have closure in \(Z_R\), proving local compactness. Countably many \(Z_n\) exhaust \(Z\), and the countable coordinate topology proves second countability.

The final coordinate in (CI.14) is nonnegative and at least the squared norm of the new vector: their difference is exactly \(t-\|v\|^2\). All other coordinates are continuous. More explicitly, for an input vector \(v\in H_{\mu g}\) with coordinates \(z_h\), they are
\[
 \begin{aligned}
 \langle A_{\mu,g}v,e_l^\mu\rangle
   &=z_{g^{-1}l}-z_{g^{-1}}
                    +\langle e_g^\mu,e_l^\mu\rangle,\\
 t_{\rm out}&=t-2z_{g^{-1}}+K(\mu,g).
 \end{aligned}
 \tag{CI.17}
\]
These formulas follow by applying the inverse of (CI.10); in particular
\(L_{\mu,g}^{-1}e_g^\mu=-e_{g^{-1}}^{\mu g}\).
They prove continuity and the composition law, including the sign of the height correction.

If both an input and its output belong to \(Z_R\), their vector norms are at most \(\sqrt R\). The triangle inequality in (CI.11) therefore gives
\[
 \rho_-(|g|)\leq\|e_g^\mu\|
          \leq \|A_{\mu,g}v\|+\|v\|
          \leq2\sqrt R.
 \tag{CI.18}
\]
Only finitely many \(g\) satisfy this, because the word metric is proper and \(\rho_-\) tends to infinity. Every compact subset of \(Z\) has bounded continuous height, so the transporter of two compact sets is finite. The inverse image of their product under the action-relation map is a closed subset of a finite union of copies of the compact input set, and is compact. This proves properness. \(\square\)

This theorem supplies a genuine proper locally compact central-space candidate. It does not supply the graded Bott algebra over that space, an equivariant Dirac class, or a Kasparov product inverse. Infinite-dimensional fibre compactness in (CI.15) relies on the weak cone topology and cannot be substituted for compact resolvent of an oscillator.

### 11O. Reading

For the broader source route see Guentner–Higson–Weinberger, [The Novikov Conjecture for Linear Groups](https://math.hawaii.edu/~erik/papers/linear.pdf), and Jean-Louis Tu, [La conjecture de Baum–Connes pour les feuilletages moyennables](https://citeseerx.ist.psu.edu/document?doi=486238991671dba031f9cb078eeba25443307a0a&repid=rep1&type=pdf), Section 6. The proof above constructs its compactification from the countable bounded kernels and proves compactness directly with Gram inequalities, retaining the varying-field and right-action conventions throughout.


## 11P. A confining oscillator over a varying Hilbert field

Inner products on complex modules are linear in the second variable. The operator is odd; its Kasparov cycle has even degree.

Let \(X\) be compact metrizable, let \(E\) be the continuous-section module of a countably generated real Hilbert field over \(X\), and let a countable discrete group \(\Gamma\) act on that field by continuous affine isometries. Write the action, with range \(x\) and source \(xg\), as
\[
 v\longmapsto L_{x,g}v+b_g(x),\qquad
 b_{gh}(x)=b_g(x)+L_{x,g}b_h(xg).
 \tag{WF.1}
\]
Each \(b_g\) is a continuous section. The action need not be proper for this theorem. In the coarse-embedding application the field and cocycle are those constructed in Section 11O.

We use the Hilbert-module completion and compact-operator proofs in Graded C*-algebras, Clifford algebras and graded Hilbert modules, Lemma 2.0 and Lemma 6.0a, and the completed regular-operator calculus in Unbounded Kasparov modules and spectral triples, Lemma 0.0. The convexity argument below proves the extra automorphism version of approximate-identity selection; it does not assume that a semilinear action is a module-linear multiplier.

### 11P.1. A weight adapted to the entire countable action

**Lemma 11P.1.** There is a positive regular operator \(\Theta\geq1\) on \(E\), commuting with real conjugation after complexification, such that
\[
 (1+\Theta)^{-1}\in\mathcal K(E),\quad
 L_g\Theta L_g^{-1}-\Theta\in\mathcal K(E),\quad
 b_g\in\operatorname{Dom}\Theta
 \tag{WF.2}
\]
for every \(g\). The differences and the sections \(\Theta b_g\) are continuous in the base, in module norm. Domains transform covariantly.

**Proof.** Choose real generating sections \(\xi_j\), rescaled to norm at most one, and let
\(h=\sum_j2^{-j}\theta_{\xi_j,\xi_j}\).
This is a positive compact operator. Its action has dense range: if \(r_\epsilon=\epsilon(h+\epsilon)^{-1}\), positive order gives
\[
 \|r_\epsilon\xi_j\|^2
 =\|r_\epsilon\theta_{\xi_j,\xi_j}r_\epsilon\|
 \leq2^j\|r_\epsilon h r_\epsilon\|
 \leq2^j\epsilon/4.
 \tag{WF.3}
\]
Thus \(r_\epsilon\to0\) on the generators and, by their density and the uniform bound, on all sections. Continuous scalar functions of \(h\) which vanish near zero and equal one on successively larger portions of its nonzero spectrum give a commuting compact approximate identity \(e_n\), with
\(e_ne_{n+1}=e_n\).
For example choose plateau cutoffs with the support of the \(n\)-th cutoff inside the one-region of the next.

Conjugation by the semilinear \(L_g\) induces an automorphism \(\alpha_g\) of \(\mathcal K(E)\). Both \(e_n\) and \(\alpha_g(e_n)\) are approximate identities. Their difference tends weakly to zero. Here is the needed functional argument. Every positive functional has a GNS vector representation; an approximate identity tends strongly to the identity in that representation on its nondegenerate subspace. Hence its functional value tends to the norm of the functional. Write a bounded complex functional as a linear combination of four positive functionals, or apply the same argument in its vector-functional representation, to obtain weak convergence for every functional. This elementary functional decomposition is proved in the convex-quasicentrality proof of Kasparov's technical theorem, Theorem 1.1.

For finitely many \(g\), the tuple \((\alpha_g(e_n)-e_n)_g\) therefore tends weakly to zero. Hahn–Banach separation says that zero lies in the norm-closed convex hull of every tail: a separating real functional would contradict that weak limit. Simultaneously impose \((1-e_n)b_g\to0\) and the approximate-identity tests on the first finitely many generators; these components already converge in norm. Their joint tuple also has zero in every tail's closed convex hull.

Enumerate \(\Gamma\). Inductively choose a finite convex combination \(v_n\) from a tail beyond every index used in \(v_{n-1}\), so that, for the first \(n\) group elements,
\[
 \|\alpha_g(v_n)-v_n\|\leq2^{-n},
 \qquad \|(1-v_n)b_g\|\leq2^{-n}.
 \tag{WF.4}
\]
Let the least index tend to infinity. The \(v_n\) form a compact approximate identity, commute, and satisfy \(v_nv_{n+1}=v_n\). Their real scalar coefficients retain conjugation symmetry.

Define \(\Theta=1+\sum_{n\geq1}(1-v_n)\) by functional calculus of \(h\). At each nonzero spectral value only finitely many terms survive; near zero the sum tends to infinity. Its reciprocal is a continuous function of \(h\) vanishing at zero, so is compact. Multiplication by this reciprocal has dense range by (WF.3); it defines a positive regular operator with that inverse and with \(\Theta\geq1\). Finite cutoff ranges are a core.

For a fixed \(g\), the series
\[
 \alpha_g(\Theta)-\Theta
       =\sum_n(v_n-\alpha_g(v_n)),\qquad
 \Theta b_g=b_g+\sum_n(1-v_n)b_g
 \tag{WF.5}
\]
converge in norm after their finitely many initial terms, by (WF.4). The first sum is compact. For the domain assertion use the bounded partial sums and their resolvents: both positive sums converge in strong resolvent sense by the scalar cutoff functional calculus, while their differences converge in norm. The resolvent identity consequently identifies the conjugated limit with the bounded self-adjoint perturbation of the original limit. Its domain is the original domain and its difference is exactly the first series in (WF.5). The second sum puts \(b_g\) in the closed operator's domain. All sums are in the section module or compact module algebra, so their stated base continuity is included. \(\square\)

### 11P.2. The completed Fock operator

Complexify \(E\) and form
\[
 \mathcal F=\bigoplus_{p,q\geq0}
       \operatorname{Sym}^p(E_{\mathbb C})
          \otimes_{C(X)}\Lambda^q(E_{\mathbb C}),
 \qquad \text{parity}=q\bmod2.
 \tag{WF.6}
\]
Symmetric and antisymmetric powers are the ranges of the finite permutation averaging projections on the completed interior tensor powers. Thus (WF.6) is a countably generated Hilbert \(C(X)\)-module. Its vacuum is a rank-one even copy of \(C(X)\).

In a fibre diagonalize \(\Theta\), with eigenvalues \(\lambda_i\geq1\). Compactness in (WF.2) makes each bounded-eigenvalue subspace finite dimensional and \(\lambda_i\to\infty\) in an infinite fibre. Let \(a_i,a_i^*\) be bosonic annihilation and creation, and \(\varepsilon_i,\varepsilon_i^*\) exterior creation and annihilation. Set
\[
 Q_i=\sqrt2(a_i\varepsilon_i+a_i^*\varepsilon_i^*),
 \quad
 D=\sum_iQ_i,\quad B(A)=\sum_i\lambda_i(A)Q_i
 \tag{WF.7}
\]
when \(A\) is a diagonal positive weight. The definition for an arbitrary self-adjoint \(A\) is obtained by orthogonal covariance of these sums; no eigenbasis is chosen continuously in \(X\).

The bosonic and exterior relations give, on finite occupation vectors,
\[
 Q_i^2=2(N_i^{\mathrm b}+N_i^{\mathrm f}),\quad
 Q_iQ_j+Q_jQ_i=0\ (i\ne j),\quad
 D^2=2N,\quad B(\Theta)^2=2\sum_i\lambda_i^2N_i.
 \tag{WF.8}
\]
Here \(N_i=N_i^{\mathrm b}+N_i^{\mathrm f}\) and \(N=\sum_iN_i\).
For clarity, on exterior degree zero in the \(i\)-th mode, \(Q_i^2=2a_i^*a_i\); on degree one it is \(2a_ia_i^*=2(N_i^{\mathrm b}+1)\). This fixes the sign and the even vacuum.

For \(0<s\leq1\), define the odd operator
\[
 C_s=sD^3+B(1+s\Theta).
 \tag{WF.9}
\]
Use the algebraic finite-particle sections made from vectors in bounded spectral cutoff ranges of \(\Theta\) as a core. On a sector of total occupation \(N=n\), all its summands preserve that sector, and
\[
 C_s=\sum_i(1+2sn+s\lambda_i)Q_i,\qquad
 C_s^2=2\sum_i(1+2sn+s\lambda_i)^2N_i.
 \tag{WF.10}
\]
The operator \(D^2\) commutes with every term. Since all the scalar coefficients are positive,
\[
 C_s^2\geq s^2D^6+B(1+s\Theta)^2\geq D^2.
 \tag{WF.11}
\]

**Lemma 11P.2.** Formula (WF.9) closes to a regular self-adjoint operator, has compact resolvent for \(s>0\), and has exactly the vacuum as its kernel. The stated finite-particle cutoff sections are a core.

**Proof.** On any finite-particle, bounded-\(\Theta\) cutoff the sums define bounded operators and (WF.10) holds by the relations. Increasing both cutoffs exhausts a dense submodule. The operators and their squares agree on overlaps. On each occupation sector the resolvents defined by
\[
 (C_s\pm i)^{-1}=(C_s\mp i)(1+C_s^2)^{-1}
 \tag{WF.12}
\]
are bounded by one; the formulas on cutoffs extend to bounded adjointable operators on their dense union and hence everywhere, with adjoints obtained by changing the sign. Their ranges are dense, since the cutoff union is contained in those ranges. These mutually adjoint resolvents define the closed self-adjoint regular operator. A section in its graph domain is approximated in graph norm first by total-occupation cutoffs, then by the single-particle spectral cutoffs, using (WF.10) and dominated convergence. This proves the core assertion and avoids a fibrewise-only domain claim.

For compactness, a continuous spectral cutoff of \(C_s^2\) supported below \(R\) permits only \(N\leq R/2\). By (WF.10), every occupied mode then has \(\lambda_i\leq\sqrt{R/2}/s\). Choose a compact operator \(k=\chi(\Theta)\), with a continuous finite-support cutoff equal to one on that interval. On each permitted particle sector the energy cutoff equals its product with the corresponding tensor power of \(k\); that tensor power is compact. Indeed tensor products of rank-one maps are rank-one maps, their norm limits are compact, and compression by the symmetric/exterior projections retains compactness. There are only finitely many sectors. The spectral cutoff is therefore compact. Approximate \((1+C_s^2)^{-1}\) uniformly by these cutoffs to prove compact resolvent.

Equation (WF.10) vanishes precisely when every \(N_i=0\). This is the even vacuum. The first nonzero sector has \(C_s^2\geq2\), giving the uniform gap \(\sqrt2\) on its orthogonal complement. \(\square\)

The construction is a section-module construction: tensor operations and continuous functional calculus preserve sections, and all bounded cutoff computations use finitely many tensor factors. It remains valid when fibre ranks change or are infinite. Compactness here is compactness in \(\mathcal K(\mathcal F)\); a weakly compact Hilbert ball is not being treated as a compact-resolvent proof.

### 11P.3. Affine covariance and compact phase defects

The linear maps \(L_g\) act naturally on both Fock factors. A real translation \(b\) acts on the bosonic factor by the Weyl unitary \(T_b\). One can construct it without a Lebesgue measure in infinite dimensions: on exponential vectors \(e(z)=\bigoplus z^{\otimes n}/\sqrt{n!}\), use their inner product \(\langle e(z),e(w)\rangle=\exp\langle z,w\rangle\) and the formula
\[
 T_be(z)=\exp(-\|b\|^2/4-\langle b,z\rangle/\sqrt2)\,
                    e(z+b/\sqrt2).
 \tag{WF.13}
\]
Substitution in the exponential-vector inner product proves isometry; \(T_{-b}\) is its inverse. The real translations multiply exactly, \(T_bT_c=T_{b+c}\). Density of exponential vectors proves strong continuity. These identities also hold for continuous sections and give adjointable semilinear field maps.
Define \(U_g=T_{b_g}L_g\). Equation (WF.1) proves the exact group law with no central phase.

For a constant real vector \(b\), (WF.13) gives
\[
 T_bDT_b^{-1}=D-c_b,\quad
 T_bB(A)T_b^{-1}=B(A)-c_{Ab},\quad
 \|c_b\|=\|b\|,
 \qquad c_b=\varepsilon_b+\varepsilon_b^*.
 \tag{WF.14}
\]
The equality \(c_b^2=\|b\|^2\) proves the norm. In a finite-dimensional fibre the same normalization is the usual translation \(f(x)\mapsto f(x-b)\) of the exterior oscillator on Lebesgue \(L^2\); its even ground state is proportional to \(\exp(-\|x\|^2/2)\).

For any bounded self-adjoint real \(S\), the relations prove
\[
 B(S)^2=2\,d\Gamma(S^2)\leq\|S\|^2D^2.
 \tag{WF.15}
\]
Diagonalization proves this for finite rank and approximation on finite-particle vectors proves it for bounded \(S\). The comparison is \(\leq\), not \(\geq\).

Put \(C=C_1\). Linear conjugation changes it by \(B(L_g\Theta L_g^{-1}-\Theta)\). Translation changes it by
\[
 (D-c_b)^3-D^3-c_{(1+\Theta)b}.
 \tag{WF.16}
\]
The final term has norm \(\|(1+\Theta)b\|\), bounded by \(\|b\|+\|\Theta b\|\); the identity summand must be retained.

Every \(c_b\) changes total particle number by at most one and has norm \(\|b\|\). Consequently
\(\|D^kc_b\xi\|\leq\|b\|\|(D^2+2)^{k/2}\xi\|\)
for \(k=1,2\). Expanding the cube in (WF.16), with the factors kept in their original order, proves
\[
 \|((D-c_b)^3-D^3)\xi\|
 \leq C_b\|(1+D^2)\xi\|,
 \tag{WF.17}
\]
where one may take a finite polynomial bound in \(\|b\|\), for example
\(C_b=10(\|b\|+\|b\|^2+\|b\|^3)\).
This follows term by term from
\(-D^2c_b-Dc_bD-c_bD^2+Dc_b^2+c_bDc_b+c_b^2D-c_b^3\);
the number-shift estimate bounds each of the first three by \(2\|b\|(1+D^2)\), and each remaining term by its stated polynomial times \(1+D^2\). Bounds are uniform over the compact base for a fixed \(g\).

We use the following full phase estimate. If self-adjoint regular \(A,B\) have compact resolvents and their difference \(E=B-A\), initially on a common core, satisfies
\(\|E(1+A^2)^{-p/2}\|\leq M\) for some \(p<1\), then
\[
 B(1+B^2)^{-1/2}-A(1+A^2)^{-1/2}
                  \in\mathcal K.
 \tag{WF.18}
\]
To prove it, set \(a=(1+t^2)^{1/2}\). The resolvent identity and spectral calculus give
\(\|(B\pm ia)^{-1}E(A\pm ia)^{-1}\|
 \leq C_pMa^{p-2}\).
The integral representation
\[
 A(1+A^2)^{-1/2}
 =\frac1\pi\int_0^\infty
       \big((A-ia)^{-1}+(A+ia)^{-1}\big)\,dt
\]
holds strongly, by evaluating the scalar integral; its difference for \(A,B\) converges in norm because \(p<1\). Each resolvent difference is compact, since one resolvent is compact and the other factors are bounded. Norm integration proves (WF.18). The same estimates give norm continuity when the bound for \(E\) tends to zero.

By (WF.11), and because \(D^2\) commutes with \(C\),
\[
 D(1+C^2)^{-1/6},\qquad
 (1+D^2)(1+C^2)^{-1/3}
                  \quad\text{are bounded}.
 \tag{WF.19}
\]
Use \(p=1/3\) for (WF.15) and \(p=2/3\) for (WF.17). The completed-domain comparison needed for the resolvent identity can also be checked here. If \(E(1+C^2)^{-p/2}\) is bounded with \(p<1\), then \(\|E(C\pm i\lambda)^{-1}\|=O(\lambda^{p-1})\). For large \(\lambda\), the Neumann series for \(1+E(C\pm i\lambda)^{-1}\) gives both inverse resolvents of \(C+E\) on \(\operatorname{Dom}C\). The symmetry identity makes their adjoints agree; it proves that this operator is closed, self-adjoint and regular. Its graph norm is equivalent to that of \(C\), since the relative bound can be made less than one. The original cutoff core consequently remains a core. Linear conjugation contains this core in its domain by (WF.2). For translation, the Weyl image of a finite-particle vector is a polynomial times an exponential vector. Its number moments of every order are finite; its single-particle weighted moment is finite because \(b\in\operatorname{Dom}\Theta\). Expanding the exponential series gives \(\|\Theta b\|^2\) times its convergent factorial series, with only finitely many additional polynomial factors. Hence the original core is also in the translated domain and the translation formulas hold there. The unitarily conjugated self-adjoint operator extends the just constructed self-adjoint closure, so the two agree and their domains agree. Apply (WF.18) separately to these linear and translation comparisons. This proves every fixed-\(g\) compact phase defect. The domains of the conjugated operators are exactly the unitary images of the completed domains. No assertion that an arbitrary unbounded first-order perturbation has a compact phase defect is used.

It follows that \((\mathcal F,C(1+C^2)^{-1/2})\) is an even \(\Gamma\)-equivariant Kasparov \(C(X),C(X)\)-cycle. Scalar base functions commute exactly; \(1-F^2\) is compact by Lemma 11P.2.

### 11P.4. Its precise class

**Theorem 11P.3.** This cycle has class \(1_{C(X)}\) in \(KK^\Gamma(C(X),C(X))\).

**Proof.** First scale \(b_g\) to \(u b_g\), \(0\leq u\leq1\), keeping the operator \(C\) fixed. This is still an affine action by (WF.1). The Weyl maps are strongly continuous in \(u\). Estimates (WF.16)–(WF.19), applied to the difference of two scaled translations, show that the compact phase defects are norm continuous in \(u\). Hence the section module over \(X\times[0,1]\) is a Kasparov homotopy to the linear action.

For the linear action remove the vacuum and replace the bounded phase by its exact sign, using the gap \(\sqrt2\). The difference is compact by functional calculus, so the class is unchanged. On this complement consider \(\operatorname{sign}(C_s)\) from (WF.9), \(0<s\leq1\), and at \(s=0\) use \(\operatorname{sign}(D)\).

These signs are strongly continuous down to zero: on finite-particle, bounded-\(\Theta\) sections the operators converge in norm on their invariant sectors and retain the gap. Density and the common bound one prove strong and adjoint-strong continuity on all sections. For \(s_0>0\) they are norm continuous. Indeed (WF.10) reduces the sign on each joint occupation space to a normalized Clifford vector with coefficients \(1+2sn+s\lambda_i\). Their relative coefficient change is at most \(|s-s_0|/s_0\); the Clifford norm identity and normalization bound the sign change by \(2|s-s_0|/s_0\).

The compact linear equivariance defects also tend to zero in norm at \(s=0\). To check the endpoint rather than merely assume it, write \(S=L_g\Theta L_g^{-1}-\Theta\). The conjugated \(C_s\) differs by \(sB(S)\). On the complement, \(|C_s|\geq\sqrt2\), \(|C_s|\geq|D|\), and \(|C_s|\geq s|D|^3\). In the sign-resolvent integral with parameter \(a\geq0\), the part \(0\leq a\leq1\) is bounded by a constant times \(s\|S\|\). For \(a\geq1\),
\[
 \|D(C_s\pm ia)^{-1}\|
 \leq c\,s^{-1/3}a^{-2/3}.
 \tag{WF.20}
\]
This is the scalar maximum of \(z/(s^2z^6+a^2)^{1/2}\), using commutation with \(D^2\). Multiplying by the other resolvent, of norm at most \(a^{-1}\), and by \(s\|S\|\), gives the integrable bound
\(c\,s^{2/3}\|S\|a^{-5/3}\).
The entire defect is therefore \(O(\|S\|(s+s^{2/3}))\). For every \(s>0\) it is compact by the same argument as (WF.18). Together with norm continuity away from zero, this proves a continuous compact-valued defect on the closed parameter interval.

At \(s=0\), \(D\) is exactly invariant under the linear action. Its sign on the vacuum complement squares to one, is self-adjoint and commutes with \(C(X)\). The complement cycle is degenerate. The remaining vacuum line is even, with its natural action on \(C(X)\), and is the identity cycle. This proves the theorem. \(\square\)

This identity has coefficients \(C(X)\) on both sides. It supplies neither a reduced scalar character nor a map \(KK^\Gamma(C(X),C(X))\to KK^\Gamma(\mathbb C,\mathbb C)\) by evaluation at an arbitrary point of \(X\). Such an evaluation is equivariant only at a fixed point. In particular the theorem by itself does not factor the identity through a proper Bott algebra.

### 11P. Reading

Jean-Louis Tu, [La conjecture de Baum–Connes pour les feuilletages moyennables](https://citeseerx.ist.psu.edu/document?doi=486238991671dba031f9cb078eeba25443307a0a&repid=rep1&type=pdf), K-Theory 17 (1999), 215–264, Sections 8.4 and 9. The proof above uses a section-module weight, states complete domains and resolvent bounds, retains the identity part of the translation potential, and proves the compact-defect homotopy endpoint. Its conclusion is the coefficient identity just stated, not the later proper-factorization theorem.

### 11P. Figure

![Coarse geometry, proper cone and the coefficient oscillator identity](../figures/coarse-field-unit.png)

**Figure 11P.1.** The first two panels show the faithful characteristic-zero linearization and the kernel/probability construction, with their exact scope. The third is the one-dimensional sample \(g=3\), \(v=-1.5\), \(t=3\) of the squared-height action; its excess remains \(0.75\). The fourth gives the exact energies from (WF.10) at \(s=1\), \(\lambda=(1,3)\), including the unique even vacuum. The fifth plots the integrable bound shape at \(p=2/3\), rather than a measured operator error. The final panel distinguishes Theorem 11P.3’s coefficient identity from the separate proper scalar factorization, now constructed in Sections 11V–11Y. The complete accompanying proofs state all domains, supports and quantifiers. [Vector figure](../figures/coarse-field-unit.svg); [reproducible generator](../reproduction/general-auxiliary-providers/reproduce.py); [component terms](../reproduction/general-auxiliary-providers/COMPONENT-TERMS.md).


## 11Q. A locally compact universal proper base and its probability map

Let \(\Gamma\) be countable discrete. Its topology here is independent of any compact Lie group in which it embeds.

### 11Q.1. The mass threshold

Let \(Q\) be the space of positive functions \(\mu:\Gamma\to[0,\infty)\) with total mass at most one, in the topology of pointwise convergence. Set
\[
 M=\{\mu\in Q:1/2<\|\mu\|_1\leq1\},\qquad
 (h\mu)(g)=\mu(h^{-1}g).
 \tag{PM.1}
\]
The topology is the weak-* topology from \(c_0(\Gamma)^*=\ell^1(\Gamma)\); it is not the norm topology. The total mass need not be continuous.

**Theorem 11Q.1.** The space \(M\) is second countable, locally compact Hausdorff and sigma-compact; its \(\Gamma\)-action is proper. It is convex and, for every finite subgroup \(F\), is \(F\)-equivariantly contractible. Every second-countable locally compact proper \(\Gamma\)-space has an equivariant map to \(M\), unique up to equivariant homotopy.

**Proof.** The set \(Q\) is a closed subset of the compact countable product \([0,1]^\Gamma\): positivity is explicit, and the conditions \(\sum_{g\in E}\mu(g)\leq1\) for all finite \(E\) are closed and equivalent to the total bound. On \(Q\), pointwise convergence agrees with weak-* convergence, since a \(c_0\) function is uniformly approximable by finite-support functions and all masses are bounded by one. The function \(\|\mu\|_1=\sup_{E\ {\rm finite}}\sum_E\mu(g)\) is lower semicontinuous. Thus \(M\) is open in the compact metrizable \(Q\), so is locally compact, second countable and sigma-compact. All translations are homeomorphisms.

For a compact \(K\subset M\), there is one finite \(E\subset\Gamma\) with
\[
 \mu(E)>1/2\qquad(\mu\in K).
 \tag{PM.2}
\]
For each \(\mu\), choose such a finite set; its strict inequality defines an open coordinate neighborhood. A finite subcover of \(K\) gives a finite union \(E\), which has the required bound for all of \(K\). If \(\mu\in K\) and \(h\mu\in K\), then
\(\mu(E)>1/2\) and \(\mu(h^{-1}E)>1/2\).
Disjointness would give total mass greater than one. Therefore
\[
 E\cap h^{-1}E\ne\varnothing,\qquad h\in EE^{-1}.
 \tag{PM.3}
\]
The transporter is finite. For two compact sets apply the argument to their union. The inverse image of their product under the action-relation map is closed inside finitely many copies of a compact input set, and hence compact. This is properness, including finite nontrivial stabilizers.

Convex combinations of two positive measures in \(M\) have mass strictly greater than \(1/2\) and at most one. The combinations are continuous in the stated coordinate topology. For finite \(F\), let \(\nu_F=|F|^{-1}\sum_{f\in F}\delta_f\). It is fixed by \(F\), and
\((\mu,t)\mapsto(1-t)\mu+t\nu_F\)
is the required equivariant contraction.

For a proper space \(Z\), take a continuous square cutoff \(c\) with
\(\sum_g c(g^{-1}z)^2=1\) and translates locally finite over compact subsets. Such cutoffs are constructed in Lemma 11J.1 in Section 11J, the proper-factorization construction. Define
\[
 \mu_z(g)=c(g^{-1}z)^2.
 \tag{PM.4}
\]
This is a probability in \(M\), its coordinates are continuous, and
\(\mu_{hz}(g)=\mu_z(h^{-1}g)\).
Thus it is an equivariant map. Any two equivariant maps \(Z\to M\) are joined by their pointwise convex combination, which is continuous and remains in \(M\). This proves the stated universal property without a finite-dimensional, free-action or closed-subgroup hypothesis. \(\square\)

### 11Q.2. Mapping to the coefficient probability base

Let \(Y\) be a nonempty compact metrizable \(\Gamma\)-space and \(X=\operatorname{Prob}(Y)\). Use left actions in this section; a right action from the coarse-field construction is converted by \(h\cdot y=yh^{-1}\).

**Proposition 11Q.2.** Every second-countable locally compact proper \(\Gamma\)-space \(Z\), including \(M\), has a continuous equivariant map \(\Phi:Z\to X\).

**Proof.** Fix \(y_0\in Y\) and a square cutoff \(c\) on \(Z\). Set
\[
 \Phi(z)=\sum_g c(g^{-1}z)^2\delta_{g\cdot y_0}.
 \tag{PM.5}
\]
The coefficients are nonnegative and sum to one. On any compact subset of \(Z\) only finitely many of them occur. For each continuous function on \(Y\), its integral against (PM.5) is consequently a locally finite continuous sum; this is exactly weak probability continuity. Substituting \(g=hk\) in (PM.5) proves \(\Phi(hz)=h\Phi(z)\). No invariant probability for the whole group is used. \(\square\)

The superficially simpler formula
\(\|\mu\|_1^{-1}\sum_g\mu(g)\delta_{g\cdot y_0}\)
is not a justified substitute for (PM.5): its denominator can fail to be continuous in \(M\). For example
\(\mu_n=\tfrac34\delta_e+\tfrac14\delta_{g_n}\),
with \(g_n\) escaping every finite set, tends to \(\tfrac34\delta_e\) in \(M\), while its masses are one and the limit's mass is \(3/4\). The square-cutoff formula avoids that defect.

### 11Q.3. What this interface does and does not supply

Theorem 11Q.1 and Proposition 11Q.2 supply the actual proper central space and the continuous equivariant probability-base map required in the compact-base transfer route. Pullback of \(C(X)\) along \(\Phi\) acts in \(C_b(Z)=M(C_0(Z))\), and is nondegenerate: on a compact support its continuous image in \(X\) is compact, so a function equal to one on that image acts as the identity there. For this unital \(C(X)\) the constant one already proves the assertion.

They do not construct a proper coefficient algebra with a Dirac class. In particular an equivariant map \(M\to X\) gives a homomorphism \(C(X)\to M(C_0(M))\), not a scalar equivariant evaluation \(C(X)\to\mathbb C\). Two stronger forms of the separate factorization interfaces in the source route are:

1. a proper algebra \(A\), a class \(D\in KK^\Gamma(A,\mathbb C)\), and a family \(\theta\in RKK^\Gamma(M;\mathbb C,A)\) with \(\theta\otimes_A D=1\);
2. a proper algebra \(B\) for the affine Hilbert field over \(X\), and classes \(\eta_X\in KK^\Gamma_X(C(X),B)\), \(d_X\in KK^\Gamma_X(B,C(X))\) with the supplied coefficient product identity.

The existing coefficient oscillator identity proves the homotopy calibration needed for the second construction's eventual product. It is not either of these two factorization statements. Sections 11V–11Y construct both proper algebras and the actual equivariant factors. Their transfer uses only the ordinary coefficient identity and each ordinary universal-family calibration; neither stronger equivariant product identity in the preceding list is required. This completes the discrete scalar-unit construction under the coarse-field hypotheses. It does not replace the original anchored holonomy factorization of Section 11R.

### 11Q. Reading

Jean-Louis Tu, *The Gamma Element for Groups which Admit a Uniform Embedding into Hilbert Space* (2004), pp.276 and 282–284, [complete freely readable original scan](https://www.yumpu.com/en/document/view/50923940/the-gamma-element-for-groups-which-admit-a-uniform-embedding-). The proof here establishes the proper measure base, its topology and the cutoff probability map directly; the Dirac and Bott factorization interfaces remain explicitly separate.

### 11Q. Figure

![The half-mass properness test and the cutoff probability map](../figures/proper-probability-base.png)

**Figure 11Q.1.** The left panel is the coordinate projection \(a=\mu(E)\), \(b=\mu(h^{-1}E)\) when the two sets are disjoint. The total bound gives \(a+b\leq1\), which has no point with both coordinates strictly greater than \(1/2\). Thus two compact-set half-mass tests force overlap and the finite transporter bound. The right panel records the normalized square-cutoff coefficients and their locally finite, equivariant map to \(\operatorname{Prob}(Y)\). Neither panel treats total mass as continuous or supplies a scalar evaluation. [Vector figure](../figures/proper-probability-base.svg); [generator](../reproduction/general-auxiliary-providers/reproduce.py); [component terms](../reproduction/general-auxiliary-providers/COMPONENT-TERMS.md).


## 11R. Coherent proper factorization and the local transverse restriction

A scalar class for each countable return group is not yet a class for
the holonomy groupoid. We give the exact groupoid transfer formula,
prove its unit-space restriction on the actual descended modules,
and identify an obstruction to turning a field oscillator's compact
operator algebra into the missing proper coefficient. We also
construct the proper weak cone from genuinely coherent groupoid
kernel data. Sections 11S–11T construct one actual coherent proper extension throughout the eligible foliation scope. A compact auxiliary kernel route, original-unit-normalized Bott/Dirac factors and the physical graph operator remain distinct requirements.

Throughout, \(\mathcal G\rightrightarrows T\) is second-countable,
locally compact, Hausdorff and étale. A complete transversal of a
compact foliation with invariant positive transverse metric has
these properties, by Lemma 11M.5, the eligible metric separates holonomy arrows. No global discrete action replaces
\(\mathcal G\). Write

\[
 C=C_0(T),\quad A_m=C^*(\mathcal G),\quad
 A_r=C_r^*(\mathcal G),\quad q_A:A_m\longrightarrow A_r,
 \quad i_m:C\longrightarrow A_m,\quad i_r:C\longrightarrow A_r.
 \tag{CA.1}
\]

The unit inclusions are actual homomorphisms: compactly supported
unit functions extend by zero on the open-and-closed unit space.
Their full and reduced norms are the supremum norm. They are
nondegenerate, by the unit approximate identity on compact
convolution supports. Full and reduced groupoid descent below
retain their coefficient algebras. Their existence, product law
and natural quotient square are stated explicitly as inputs to
the transfer theorem, rather than inferred from the corresponding
discrete-group theorem.

### 11R.1. The unit restriction of a descended field

Let \(E\) be a countably generated graded Hilbert \(C\)-module with
the canonical nondegenerate left \(C\)-action, and a continuous
even unitary groupoid action \(U_g:E_{s g}\to E_{r g}\).
Let \((E,F)\) be a base-linear equivariant Kasparov cycle.
Thus \(F\) is odd, adjointable and \(C\)-linear; its localized
square and adjoint defects are compact, and its localized
groupoid covariance defects have the required compactness and
continuity. Denote its ordinary forgetful class by
\(\gamma_0\in KK(C,C)\). In particular forgetting means keeping
the whole \(C\)-module, rather than selecting one fibre.

**Theorem 11R.1 (unit restriction is extension of scalars).**
For the full descent of this cycle,

\[
 [i_m]\otimes_{A_m}j_m(E,F)
       =\gamma_0\otimes_C[i_m]
       \quad\hbox{in }KK(C,A_m).
 \tag{CA.2}
\]

The equality has an explicit unitary on the completed modules.
No invariant measure, global scalar character or equivariant
evaluation of \(T\) at a point is used.

**Proof.** Use the descent core of continuous compactly supported
sections \(\xi(g)\in E_{r g}\). Its right scalar convolution and
inner product are

\[
 \begin{split}
 (\xi*k)(g)&=\sum_{s h=s g}\xi(gh^{-1})k(h),\\
 \langle\xi,\zeta\rangle(h)
       &=\sum_{s g=r h}\langle\xi(g),\zeta(gh)\rangle_{E_{r g}}.
 \end{split}
 \tag{CA.3}
\]

All products are composable, and all arrow labels remain in the
sums. In particular distinct isotropy arrows have not been
identified. Define on elementary tensors

\[
 \Phi:E\otimes_{C,i_m}A_m\longrightarrow E\rtimes_m\mathcal G,
 \qquad \Phi(e\otimes k)(g)=e(r g)k(g).
 \tag{CA.4}
\]

Balancing is correct: multiplication of \(k\) on the left by
\(i_m(a)\) is \(a(r g)k(g)\), the same as replacing \(e\)
by \(ea\). The inner product of two such sections at \(h\) is

\[
 \sum_{s g=r h}\overline{k(g)}
       \langle e(r g),f(r g)\rangle\,l(gh)
       =\bigl(k^*i_m(\langle e,f\rangle)l\bigr)(h).
 \tag{CA.5}
\]

This is exactly the interior-tensor inner product. It proves
positivity on this dense core and that \(\Phi\) extends
isometrically.

It is onto. A compact arrow support has a finite bisection
cover. Locally a continuous pullback section is approximated
in norm by a unit section evaluated at the range, times a
continuous arrow function; a finite partition of unity on
the support gives finite sums of this form. The continuous
Hilbert-field definition supplies the required unit sections,
without a local trivialization or a fixed fibre rank. Uniform
error \(\epsilon\) on a support covered by \(N\) bisections
has module norm at most \(N\epsilon\): both absolute fibre
sums for its inner product are at most \(N^2\epsilon^2\).
Thus those elementary sections are dense in the completed
descent module.

The descended left action is

\[
 (\Lambda_E(f)\xi)(g)
       =\sum_{r h=r g}f(h)U_h\xi(h^{-1}g).
 \tag{CA.6}
\]

For a unit function \(a\), this is exactly
\(a(r g)\xi(g)\). Hence pulling back by \(i_m\) gives the
ordinary left \(C\)-action on the first factor of (CA.4).
The descended operator is pointwise
\((\widetilde F\xi)(g)=F_{r g}\xi(g)\); under \(\Phi\) it
is \(F\otimes1\). Extension of scalars along the homomorphism
\(i_m\) represents the ordinary Kasparov product on the
right of (CA.2). To see that there is no loss of local
compactness, localize a rank-one operator of \(E\) by a
unit function: its tensor extension factors through
left multiplication by \(i_m(a)\in A_m=\mathcal K(A_m)\),
and is a compact operator on \(E\otimes_C A_m\).
Approximation by these localized rank-one operators gives
the same assertion for every compact defect. Thus this
is equality of actual cycles, not merely equality of
their fibre indices. This proves (CA.2). \(\square\)

The same core identification works for reduced descent when
that descent is supplied. The full statement is the one
needed to evaluate the transfer formula.

### 11R.2. The exact coherent transfer certificate

Let \(P\) be a separable graded \(\mathcal G\)-algebra over
\(T\), and set \(P_m=P\rtimes_m\mathcal G\),
\(P_r=P\rtimes_r\mathcal G\). A proper coefficient structure
means a nondegenerate equivariant central homomorphism
\(C_0(Z)\to ZM(P)\), compatible with the anchor to \(T\),
where \(Z\) is a second-countable locally compact Hausdorff
proper \(\mathcal G\)-space.

**Theorem 11R.2 (general groupoid transfer with its exact local premise).**
Suppose all the following data are supplied.

1. Classes \(\eta\in KK_{\mathcal G}^k(C,P)\) and
   \(d\in KK_{\mathcal G}^k(P,C)\), for the same
   \(k\in\mathbb Z/2\), with product
   \(\gamma=\eta\otimes_P d\) represented as in Theorem 11R.1.
2. Their full and reduced descents, the product law for
   full descent, and the exact naturality identity
   \([q_A]\otimes j_r(\eta)=j_m(\eta)\otimes[q_P]\).
3. An actually proved inverse
   \(v\in KK(P_r,P_m)\) to the coefficient quotient
   \([q_P]\), in particular
   \([q_P]\otimes_{P_r}v=1_{P_m}\).
4. A supplied maximal normal class
   \(d_m\in KK^q(A_m,\mathbb C)\), retaining the entire
   inverse normal coefficient, and an original local
   oriented disk class \(b_U\in KK^q(\mathbb C,C)\).

Put \(d_N=[i_m]\otimes_{A_m}d_m\), and let
\(\gamma_0=\operatorname{For}(\gamma)\). Define

\[
 d_r=j_r(\eta)\otimes_{P_r}v
       \otimes_{P_m}j_m(d)\otimes_{A_m}d_m
       \in KK^q(A_r,\mathbb C).
 \tag{CA.7}
\]

Then

\[
 [i_r]\otimes_{A_r}d_r
       =\gamma_0\otimes_C d_N
       \quad\hbox{in }KK^q(C,\mathbb C).
 \tag{CA.8}
\]

If \(\gamma_0=1_C\), the full ordinary normal restriction is
preserved. For the original local Bott requirement the weaker
precise premises

\[
 b_U\otimes_C\gamma_0=b_U,\qquad
 b_U\otimes_C d_N=+1
 \tag{CA.9}
\]

already suffice:
\((b_U\otimes_C[i_r])\otimes_{A_r}d_r=+1\).
Neither an equivariant unit nor invertibility of \(q_A\) is
required.

**Proof.** Track the factors of (CA.7) in their displayed
order: they run from \(A_r\) to \(P_r\), then \(P_m\),
then \(A_m\), then \(\mathbb C\), with degrees \(k,0,k,q\).
No factors are interchanged and the total degree is \(q\).
Naturality, cancellation of the supplied coefficient inverse
and the full product law give

\[
 [q_A]\otimes_{A_r}d_r
    =j_m(\eta)\otimes_{P_m}j_m(d)\otimes_{A_m}d_m
    =j_m(\gamma)\otimes_{A_m}d_m.
 \tag{CA.10}
\]

Since \(q_A i_m=i_r\), restriction of this equality to \(C\)
and Theorem 11R.1 give
\([i_r]\otimes d_r=\gamma_0\otimes[i_m]\otimes d_m\),
which is (CA.8). Multiplication by \(b_U\), followed by the
two exact equalities (CA.9), gives the claimed integer.
The even \(\gamma\) changes no transverse degree or Clifford
order. Nothing evaluates \(T\) at a point equivariantly.
The full inverse line and finite right Clifford factor are
still in the supplied \(d_m\); (CA.8) is a class equality
and does not replace them by a determinant-only module.
\(\square\)

Properness of \(P\), by itself, is not a proof of every
input just listed. In particular the arbitrary-groupoid
descent and proper coefficient quotient provider must be
supplied at that scope. The already proved discrete-group
cutoff isomorphism is not silently its proof for this
groupoid. A normalized holonomy Morita correspondence
transfers the resulting transversal class and its Bott
pairing to the smooth graph algebra when that exact
correspondence is supplied. This abstract transfer does
not establish a finite-order physical graph operator.

The condition \(\gamma_0=1_C\) is optional for the historical
local integer. For example a finite graded index bundle
representative \(W^+-W^-\) is trivial over a sufficiently
small disk \(U\). If its rank difference is one there,
its action on \(b_U\) is
\((\operatorname{rank}W^+-\operatorname{rank}W^-)b_U=b_U\):
choose frames on that disk, tensor the Bott module by
each frame, and obtain the indicated finite direct sum
with its original sign. Such an index bundle can have
nontrivial global transitions. Fibrewise rank one alone
must not therefore be asserted to give the stronger
whole-\(C\) identity.

### 11R.3. An elementary compact algebra is proper exactly when its base is

This result applies to general groupoid fields, not just
to group actions or constant-rank bundles.

**Theorem 11R.3 (the compact-operator coefficient criterion).**
Let \(Y\) be a second-countable locally compact Hausdorff
\(\mathcal G\)-space with anchor \(p:Y\to T\). Let
\(\mathcal E\) be a full countably generated continuous
Hermitian Hilbert field over \(Y\), with nonzero fibres
and a continuous unitary \(\mathcal G\)-action. Put
\(Q=\mathcal K(\mathcal E)\). Then \(Q\) has a proper
\(\mathcal G\)-algebra structure as defined above if
and only if the action on \(Y\) itself is proper.

Thus an oscillator's elementary compact algebra over a
nonproper compact probability base cannot be the missing
proper coefficient, even when its oscillator cycle is
the identity with those coefficients.

**Proof: the centre.** A multiplier of
\(\mathcal K(\mathcal E)\) is an adjointable field
operator. Here is the needed identification. For a
compact-operator approximate identity \(u_n\), the
operators \(Mu_n\) have norm at most \(\|M\|\).
On vectors \(a\xi\), \(a\in\mathcal K(\mathcal E)\),
they converge to \((Ma)\xi\), since
\(u_n a\to a\) in norm. Such vectors have dense span,
because \(u_n\to1\) on the section module. Uniform
boundedness therefore defines the limit operator
on every section. Applying the same construction
to \(M^*\) proves that the two limits are adjoints.
Conversely each adjointable operator multiplies
the rank-one operators on both sides, so gives
a multiplier. This proves the identification.
If it is central, its value at each \(y\) commutes with
every compact operator on \(\mathcal E_y\). This value
is scalar. Indeed commutation with each rank-one
projection preserves every one-dimensional subspace;
applying this to two vectors and their sum makes their
scalars equal. The same proof covers finite and infinite
dimension. Write that scalar \(c(y)\).

Near any \(y_0\) choose a continuous section \(\xi\)
nonzero there. Its norm stays positive nearby, and

\[
 c(y)=\frac{\langle\xi(y),M\xi(y)\rangle}
              {\|\xi(y)\|^2}
 \tag{CA.11}
\]

is continuous. Its supremum is bounded by \(\|M\|\).
Conversely multiplication by any \(c\in C_b(Y)\) is a
central adjointable operator. Thus
\(ZM(Q)=C_b(Y)\), including when the field rank varies.

Suppose a nondegenerate central homomorphism
\(a:C_0(Z)\to ZM(Q)\) gives a proper coefficient
structure. For every \(y\), its scalar value
\(a(f)(y)\) is a nonzero character of \(C_0(Z)\).
It cannot be zero: nondegeneracy on \(Q\) passes to
the nonzero compact fibre \(Q_y\), so the central
approximate identity acts there as one. A nonzero
character of \(C_0(Z)\) is evaluation at a unique
point \(z_y\). For instance extend it to the
one-point compactification; its kernel is a
codimension-one maximal ideal, whose common zero
is a single point by the compact-space finite
intersection argument and point separation. It
cannot be the added point because the original
character is nonzero.

All \(f(z_y)=a(f)(y)\) are continuous in \(y\).
Compactly supported functions supported in an
arbitrary neighbourhood of \(z_y\) show that
\(y\mapsto z_y\) is continuous. Compatibility
with \(C_0(T)\) makes it an anchored map
\(p_Z(z_y)=p(y)\); equivariance gives

\[
 z_{g y}=g z_y.
 \tag{CA.12}
\]

Consequently every proposed proper central structure
gives an actual continuous equivariant map
\(z:Y\to Z\).

**Proof: properness pulls back through that map.**
For compact \(K,L\subset Y\), let

\[
 S_{K,L}=\{(g,y):y\in K,\ gy\in L\}.
 \tag{CA.13}
\]

The proper action on \(Z\) has compact transporter
between the compact sets \(z(K)\) and \(z(L)\).
Its projection on the arrow space is a compact
set \(B\subset\mathcal G\). By (CA.12), every
element of \(S_{K,L}\) lies in \(B\times K\).
The source-anchor composability condition is
closed, since \(T\) is Hausdorff. The condition
\(gy\in L\) is closed by action continuity and
compactness of \(L\). Thus \(S_{K,L}\) is a
closed subset of \(B\times K\), hence compact.
Any compact subset of \(Y\times Y\) is closed
and lies in the product of its two compact
projections. Applying the same argument shows
that the action-relation map is proper on all
such compact sets. The action on \(Y\) is
therefore proper.

Conversely if \(Y\) is a proper \(\mathcal G\)-space,
use \(Z=Y\) and its canonical central multiplication
on \(Q\). A \(C_0(Y)\) approximate identity converges
on the section module and hence on its compact
rank-one operators, so this multiplication is
nondegenerate. It is anchored and equivariant.
This is a proper coefficient structure. Both
directions are proved. \(\square\)

No obstruction to a different, non-elementary
proper Bott algebra is asserted. That is precisely
where the additional geometry of a proper
Hilbert-field cone has to enter the coefficient
algebra and its Kasparov factors.

### 11R.4. A coherent kernel supplies a proper groupoid cone

We formulate the input on an actual groupoid
\(\mathcal H\rightrightarrows Y\), which may be
the transformation groupoid
\(\mathcal G\ltimes Y\) of an auxiliary space.
It is second-countable, locally compact,
Hausdorff and étale. Suppose
\(\psi:\mathcal H\to[0,\infty)\) is continuous,
is zero on units, obeys
\(\psi(h^{-1})=\psi(h)\), and satisfies:

\[
 \sum_{i,j}c_i c_j\psi(h_i^{-1}h_j)\le0
 \quad(rh_i=y,\ \sum_i c_i=0).
 \tag{CA.14}
\]

Also suppose it is proper relative to its two
unit maps: for compact \(K,L\subset Y\) and
finite \(R\), the set

\[
 \{h:s h\in K,\ r h\in L,\ \psi(h)\le R\}
 \quad\hbox{is compact}.
 \tag{CA.15}
\]

These are coherent continuous kernel assumptions,
not consequences of separate coarse embeddings
chosen on each orbit or isotropy group.

**Theorem 11R.4 (the groupoid weak cone).**
These data construct a continuous separable real
Hilbert field \(H_y\), an affine unitary action
of \(\mathcal H\), and a second-countable
locally compact Hausdorff proper
\(\mathcal H\)-space

\[
 Z=\{(y,v,t):v\in H_y,\ \|v\|^2\le t,\ t\ge0\}.
 \tag{CA.16}
\]

The topology is the weak-field coordinate
topology described below, not the norm
topology of an infinite-dimensional ball.
If \(\mathcal H=\mathcal G\ltimes Y\), the same
space is a proper \(\mathcal G\)-space over
the composite anchor to \(T\).

**Proof: field and action.** For each \(y\),
take finite zero-sum combinations of
\(\delta_h\), \(h\in\mathcal H^y\), with
semidefinite inner product

\[
 \left\langle\sum_h a_h\delta_h,\sum_k b_k\delta_k\right\rangle_y
       =-\frac12\sum_{h,k}a_h b_k\psi(h^{-1}k).
 \tag{CA.17}
\]

Positivity follows from (CA.14). Cauchy–Schwarz
follows by applying positivity to the quadratic
polynomial of \(u+\lambda v\). A null vector is
orthogonal to all vectors, so quotienting by the
null space and completing gives a real Hilbert
space \(H_y\). Let
\(e_h=[\delta_h-\delta_{1_y}]\). Direct expansion
gives

\[
 \langle e_h,e_k\rangle_y
       =\tfrac12(\psi(h)+\psi(k)-\psi(h^{-1}k)),
 \qquad \|e_h\|^2=\psi(h).
 \tag{CA.18}
\]

A countable bisection basis and countable
compactly supported bump functions in its
range domains produce countably many global
sections \(\sigma_j\): on such a domain
they are a bump times \(e_{h(y)}\), extended
by zero. Gram entries are continuous by
(CA.18). Their evaluation spans are dense
in every fibre: a bisection through any
arrow and a bump nonzero at its range
recover its \(e_h\) up to a nonzero scalar.
Finite linear combinations with continuous
unit coefficients, followed by local uniform
norm completion, define the continuous
Hilbert field. No constant dimension or
globally indexed arrow enumeration is assumed.

For \(g:y\to z\), define

\[
 L_g e_h=e_{gh}-e_g,\qquad
 A_g v=L_gv+e_g,\qquad
 \|e_g\|^2=\psi(g).
 \tag{CA.19}
\]

The Gram products are preserved because
\((gh)^{-1}(gk)=h^{-1}k\). The map is onto:
\(e_l=L_g(e_{g^{-1}l}-e_{g^{-1}})\).
It extends to an orthogonal isomorphism
\(H_y\to H_z\). The identities
\(L_gL_h=L_{gh}\) and
\(L_g e_h+e_g=e_{gh}\) prove exact affine
composition. On bisection generators these
maps are continuous by the displayed Gram
formula; approximation by those generators
proves continuous field action.

**Proof: the topology and local compactness.**
Give \(Z\) its initial topology from the
unit \(y\), the height \(t\), and the
coordinates \(\langle v,\sigma_j(y)\rangle\).
For compact \(K\subset Y\) and \(R<\infty\),
the set \(Z_{K,R}\) is a closed subspace of
the compact countable product

\[
 K\times[0,R]\times
   \prod_j[-\sqrt R\,M_j,\sqrt R\,M_j],
 \qquad M_j=\sup_{y\in K}\|\sigma_j(y)\|.
 \tag{CA.20}
\]

Indeed its coordinate image consists exactly
of the inequalities, for every finite
rational coefficient list,

\[
 \left|\sum_j a_j z_j\right|^2
    \le t\sum_{j,k}a_j a_k
                   \langle\sigma_j(y),\sigma_k(y)\rangle.
 \tag{CA.21}
\]

Cauchy–Schwarz gives one implication.
Conversely these inequalities, extended to
real coefficients by continuity, define a
well-defined bounded functional on the
dense span of the \(\sigma_j(y)\), of norm
at most \(\sqrt t\). Here one can construct its
representing vector explicitly. Gram--Schmidt on
those generators, omitting zero remainders,
gives a finite or countable orthonormal dense
family \(u_n\). If the functional is \(\ell\),
put \(a_n=\ell(u_n)\). Applying the bound to
\(\sum_{n\le N}a_nu_n\) gives
\(\sum_{n\le N}a_n^2\le t\). Consequently
\(v=\sum_na_nu_n\) converges, has squared
norm at most \(t\), and represents the
functional on the dense span and then on
the entire fibre. Uniqueness follows from
density. Thus (CA.21)
characterizes the image and makes it
closed. This proves compactness of
\(Z_{K,R}\) and also separation of points
by its coordinates.

Around a point of \(Z\), choose a unit
neighbourhood with compact closure and
an upper height bound larger than its
height. The resulting open neighbourhood
has closure in one of these compact
sets. This proves local compactness.
The unit space and countable real
coordinate product are second-countable;
the subspace coordinate topology makes
\(Z\) second-countable and Hausdorff.

**Proof: action and properness.** Extend
the affine action by

\[
 g\cdot(y,v,t)=
 \bigl(z,L_gv+e_g,\,
       t+2\langle L_gv,e_g\rangle+\psi(g)\bigr).
 \tag{CA.22}
\]

The new height minus the new squared
norm is exactly \(t-\|v\|^2\), so the
point remains in \(Z\). This identity
and (CA.19) prove the composition law,
including the cross-term sign.

The action is continuous in the weak
coordinate topology. For a target
section \(\sigma_j\), its output
coordinate is
\(\langle v,L_g^{-1}\sigma_j(rg)\rangle+
 \langle e_g,\sigma_j(rg)\rangle\).
The transported section is continuous
on each arrow neighbourhood. Approximate
it locally in norm by finite sums of
the source sections \(\sigma_k(sg)\)
with continuous arrow coefficients.
On a convergent input, \(t\) is bounded,
so the error of its pairing with \(v\)
is at most \(\sqrt R\) times the norm
error. Finite coordinate pairings are
continuous, and approximation proves
continuity of the output coordinate.
The same argument with \(L_g^{-1}e_g\)
proves continuity of the height cross
term. This establishes continuity of
every coordinate in (CA.22).

If an input and its output lie in
compact subsets with heights at most
\(R_1,R_2\), then

\[
 \sqrt{\psi(g)}
    =\|e_g\|
    \le\|L_gv+e_g\|+\|v\|
    \le\sqrt{R_2}+\sqrt{R_1}.
 \tag{CA.23}
\]

Their unit projections are compact.
By (CA.15) all such arrows belong
to a compact set. The action
transporter is a closed subset of
that compact arrow set times the
compact input set, hence compact.
This proves properness, without
finite fibre dimension or freeness.

For a transformation groupoid
\(\mathcal G\ltimes Y\), each such
compact arrow set also has compact
projection on \(\mathcal G\).
The same closed-subset argument
therefore proves properness of
the underlying \(\mathcal G\)-action
on \(Z\). All labels, anchors and
isotropy are retained. \(\square\)

![Coherent transfer and the exact proper-cone height](../figures/groupoid-compatible-auxiliary.png)

**Figure 11R.1.** Theorem 11R.2's coefficient-retaining
transfer and Theorem 11R.4's proper cone. The
right panel is an exact one-dimensional fibre
slice, not a drawing of an infinite-dimensional
norm ball. Its affine translation has \(L=1\),
\(b=1\), \((v,t)=(-1/2,3/4)\) and output
\((1/2,3/4)\). Both have excess height \(1/2\);
the cross term in (CA.22) is indispensable.
The geometric illustration supplies no missing
kernel or Kasparov factor. [Editable SVG](../figures/groupoid-compatible-auxiliary.svg)
· [Generator](../reproduction/groupoid-compatible-auxiliary/draw_compatible.py)
· [Component terms](../reproduction/groupoid-compatible-auxiliary/COMPONENT-TERMS.md).

### 11R.5. Exact conclusion for the general holonomy problem

The groupoid restriction identity in Theorem 11R.1 and
the transfer certificate in Theorem 11R.2 are general;
no chosen global acting group appears.
Theorem 11R.4 constructs a proper central-space
candidate from coherent kernel data,
including variable infinite-dimensional
fibres. Theorem 11R.3 proves that the elementary
compact algebra of a field oscillator
over a nonproper base cannot substitute
for the required proper coefficient.

Sections 11S–11T supply a single actual coherent proper extension on the full holonomy groupoid, with all arrow labels and isotropy retained. Its field source is \(C_0(M_{\mathcal G})\), rather than the original \(C_0(T)\). Section 11U supplies the proper-coefficient quotient isomorphism. Sections 11Z–11AB prove the original-anchor proper algebra, continuous family, ordinary Dirac calibration, equivariant Dirac lift and descent identities. Sections 11AC–11AD construct the original-unit dual and local comparison from the explicit labelled-kernel input. Complete Riemannian holonomy and the original-unit kernel supplies its general eligible-holonomy instance. A suitable operator on the actual graph or graph-column completion, with normal Dirac geometry, a closed domain, ordinary localized compact defects and the original local pairing, still requires a proof. A local finite-order expression is one sufficient route; the source does not impose full-tangent ellipticity or that particular expression. The original local disk Bott integer is the requirement; a stronger equivariant-unit identity has not been added to it.

Human-source context: Connes, [*A survey of foliations and operator algebras*, pages 28–29](https://alainconnes.org/wp-content/uploads/foliationsfine.pdf),
for the original local class, positive
pairing and graph question. The
complete arguments here are independently
expressed. The discrete and field
providers retain their own hypotheses;
no proof of the full historical
target is claimed here.

## 11S. A universal proper base on the actual holonomy groupoid

It generalizes the
range-fibre measure construction, rather than replacing a pseudogroup
by a global acting group. The groupoid is
\(\mathcal G\rightrightarrows T\), second countable, locally compact,
Hausdorff and étale. Its arrow labels, anchors and isotropy are retained.
Every eligible metric holonomy transversal satisfies these hypotheses
by the existing Lemma 11M.5. The construction in this section does not
need that metric hypothesis once these topological properties hold.

We use Haar measure on locally compact groups,
Theorem 2.2 and the positive-norm paragraph preceding Theorem 2.4,
for the positive-functional/Radon-measure bridge on arbitrary locally
compact Hausdorff spaces. Its proof does not require a group structure.
The weak-* compact
unit ball used below can also be obtained directly: choose a countable
dense set in \(C_0(\mathcal G)\); bounded functional values lie in a
compact countable product, and the closed linearity, positivity and
norm inequalities characterize its positive unit ball. Uniform
approximation reconstructs its functional from those coordinates.
Thus separability gives both compactness and a metrizable topology.

### 11S.1. The range-fibre subprobability ball

Let \(Q\) consist of pairs \((y,\mu)\), with \(y\in T\) and a
positive Radon measure \(\mu\) on the arrow space of total mass at
most one, supported on \(\mathcal G^y=r^{-1}(y)\). Give it the
topology of its inclusion in \(T\) times the positive weak-* unit
ball of \(C_0(\mathcal G)^*\). The anchor is
\(p_Q(y,\mu)=y\). Define

\[
 M_{\mathcal G}=\{(y,\mu)\in Q: \tfrac12<\mu(\mathcal G)\le1\}.
 \tag{GP.1}
\]

The measure has countably many possible atomic locations in each
range fibre, because an étale second-countable fibre is countable
and discrete. Its topology is nevertheless defined on the whole
arrow space. No simultaneous continuous enumeration of all fibres
is required. In particular different isotropy arrows give different
locations in the measure.

**Theorem 11S.1 (topology and the groupoid action).** The space
\(Q\) is second countable, locally compact and Hausdorff; its
anchor to \(T\) is proper. The space \(M_{\mathcal G}\) is open in
\(Q\), hence second countable, locally compact, Hausdorff and
sigma-compact. Left multiplication of arrows defines a continuous
\(\mathcal G\)-action on both spaces:

\[
 g\cdot(y,\mu)=(r g,(L_g)_*\mu),\qquad
 L_g(h)=gh,\quad y=s g.
 \tag{GP.2}
\]

**Proof of the topology.** Support on the range fibre is equivalent
to the identities

\[
 \mu((a\circ r)f)=a(y)\mu(f),
 \qquad a\in C_c(T),\quad f\in C_c(\mathcal G).
 \tag{GP.3}
\]

One direction follows from the support condition. For the other,
if an arrow has range different from \(y\), choose a unit bump
\(a\) equal to one near its range and zero near \(y\). Equation
(GP.3), tested on nonnegative arrow bumps in that neighbourhood,
forces its measure to be zero. A countable arrow cover proves that
the complement of \(r^{-1}(y)\) has measure zero. All terms in
(GP.3) are continuous in \((y,\mu)\). Consequently \(Q\) is
closed in the stated product. Over a compact \(K\subset T\), it
is closed in the compact product \(K\) times the positive weak-*
unit ball, so \(p_Q^{-1}(K)\) is compact. Compact unit
neighbourhoods then prove local compactness. The two factors have
countable bases; so does \(Q\).

The total mass equals

\[
 \mu(\mathcal G)=
 \sup\{\mu(f):f\in C_c(\mathcal G),\ 0\le f\le1\}.
 \tag{GP.4}
\]

This follows from Radon inner regularity and compact-set bump
functions. It is a supremum of continuous functions, hence lower
semicontinuous. The strict half-mass condition in (GP.1) is open.
Second-countable locally compact spaces have countable compact
exhaustions: cover by a countable family of relatively compact
open sets and take successive finite unions of their closures.
This also proves the stated sigma-compactness.

**Proof of action continuity.** For fixed \(g\), left
multiplication is a homeomorphism between the two closed discrete
range fibres, with inverse \(L_{g^{-1}}\). It preserves total
mass. The actions therefore stay in \(Q\) and \(M_{\mathcal G}\)
and satisfy the exact composition law.

Continuity with a varying arrow is the essential step. Choose an
open bisection \(B\) through \(g_0\), with section
\(b:s(B)\to B\), and a compactly supported bump
\(\chi\in C_c(s(B))\) equal to one on a neighbourhood of
\(s g_0\). For a test function \(f\in C_c(\mathcal G)\), set

\[
 \widetilde f(h)=
 \begin{cases}
 \chi(rh)f(b(rh)h),&rh\in s(B),\\
 0,&rh\notin s(B).
 \end{cases}
 \tag{GP.5}
\]

This is continuous. It has compact support: if its value is
nonzero, \(b(rh)\) lies in the compact set
\(b(\operatorname{supp}\chi)\), and \(b(rh)h\) lies in
\(\operatorname{supp}f\). Thus \(h\) belongs to the product
of the inverse of the former compact set and the latter compact
set. Composable pairs form a closed subset because \(T\) is
Hausdorff, and multiplication maps that compact subset to a
compact set. Hence \(\widetilde f\in C_c(\mathcal G)\).

On the neighbourhood of \(g_0\) where \(\chi(sg)=1\), for a
measure anchored at \(s g\), we have the exact equality

\[
 ((L_g)_*\mu)(f)=\mu(\widetilde f).
 \tag{GP.6}
\]

The right side is one fixed weak-* coordinate, independent of
the varying arrow within this bisection neighbourhood. The output
anchor is \(r g\), also continuous. Testing all \(C_c\) functions,
then using uniform approximation and the mass bound one for
\(C_0\) functions, proves action continuity. \(\square\)

The anchor of \(M_{\mathcal G}\) need not be proper. Its fibres
can contain measures with mass escaping to infinity. The
continuous nonequivariant section
\(y\mapsto(y,\tfrac34\delta_{1_y})\) proves that every unit
fibre is nonempty; it is not used as an equivariant section.

### 11S.2. Half mass proves properness without freeness

**Theorem 11S.2 (proper action).** The action-relation map

\[
 \mathcal G* M_{\mathcal G}\longrightarrow
 M_{\mathcal G}\times M_{\mathcal G},
 \qquad (g,m)\longmapsto(gm,m)
 \tag{GP.7}
\]

is proper.

**Proof.** Let \(K,L\subset M_{\mathcal G}\) be compact. For
each of their points, (GP.4) supplies a compactly supported
function with values in \([0,1]\) whose integral exceeds
\(1/2\). This inequality persists on an open neighbourhood.
A finite subcover supplies \(f_1,\ldots,f_n\). The one
compactly supported function
\(f=1-\prod_{j=1}^n(1-f_j)\) dominates them all. Put
\(E=\operatorname{supp}f\), a compact subset of the arrow
space. Then every measure in \(K\cup L\) satisfies
\(\mu(E)>1/2\).

If \(m=(s g,\mu)\in K\) and \(gm\in L\), both
\(\mu(E)>1/2\) and \(\mu(L_g^{-1}E)>1/2\) hold. They
cannot have disjoint measured supports, since total mass is at
most one. Choose \(h\in E\cap L_g^{-1}E\) in the relevant
range fibre, and let \(k=gh\in E\). Consequently

\[
 g=kh^{-1}\in EE^{-1}:=
 \{kh^{-1}:k,h\in E,\ s k=s h\}.
 \tag{GP.8}
\]

The set on the right is compact: the composability constraint is
closed in \(E\times E\), and the displayed product is
continuous. The transporter is a closed subset of
\(EE^{-1}\times K\): source-anchor composability is closed,
and the condition \(gm\in L\) is closed by GP.1. It is
therefore compact. Every compact subset of
\(M_{\mathcal G}^2\) is closed and lies in the product of its
compact projections, so the same argument proves properness on
all compact subsets. No isotropy arrows were quotiented out and
no freeness premise was used. \(\square\)

At a unit \(y\), every finite subgroup \(F\subset\mathcal G_y^y\)
has the fixed point
\(|F|^{-1}\sum_{f\in F}\delta_f\). Convex combination with it
contracts the entire fibre \((M_{\mathcal G})_y\)
\(F\)-equivariantly. Both the strict half-mass bound and the
upper mass bound persist under convex combination.

### 11S.3. Universal maps from every proper groupoid space

**Theorem 11S.3 (universal property).** Every second-countable
locally compact Hausdorff proper \(\mathcal G\)-space \(Z\)
has a continuous anchored equivariant map
\(Z\to M_{\mathcal G}\). Any two such maps are joined by a
continuous anchored equivariant homotopy.

**Proof: the square cutoff with its exact support condition.**
The proper action quotient \(Z/\mathcal G\) is Hausdorff and
locally compact. For Hausdorffness, the proper action relation is
closed and the quotient map is open: saturation by an étale
bisection is a partial homeomorphism. Disjoint neighbourhoods in
the product outside the relation have disjoint saturated images.
Images of relatively compact open neighbourhoods give compact
neighbourhoods in the quotient. The open quotient map also
preserves second countability. Hence the quotient admits a
countable locally finite subordinate partition of unity.

Choose countably many nonnegative \(\xi_j\in C_c(Z)\) such
that the open sets \(q(\{\xi_j>0\})\) cover the quotient, and
choose a locally finite partition \(\tau_j\) with closed
supports inside those open sets. The orbit sum

\[
 S_j(z)=\sum_{r h=p_Z(z)}\xi_j(h^{-1}z)^2
 \tag{GP.9}
\]

is finite on every orbit evaluation and continuous. Locally over
a compact neighbourhood in \(Z\), properness bounds all arrows
which carry \(\operatorname{supp}\xi_j\) into that
neighbourhood by a compact set. A finite bisection cover, with
arrow bumps forming a partition, writes the sum locally as a
finite sum of continuous bisection branches. Compact intersection
with any fixed range fibre is finite, because that fibre is
closed and discrete. This proves the two assertions.
Reindexing left multiplication gives
\(S_j(gz)=S_j(z)\). The sum is positive precisely on the
saturation of \(\{\xi_j>0\}\).

Define a term with \(S_j=0\) to be zero, and put

\[
 c(z)^2=\sum_j
       \frac{\tau_j(q(z))\xi_j(z)^2}{S_j(z)}.
 \tag{GP.10}
\]

The closed support of \(\tau_j\) lies where \(S_j>0\), so
each term is continuous, including where it is extended by
zero. Local finiteness makes \(c\) continuous. For a compact
\(D\subset Z\), only finitely many indices occur over
\(q(D)\). If \(c(h^{-1}z)\ne0\) for \(z\in D\), then
\(h^{-1}z\) lies in the union of the compact supports of
their \(\xi_j\). Properness of the action therefore bounds
all pairs \((h,z)\) of this form in one compact subset of
the inverse-composability space
\(\{(h,z):r h=p_Z(z)\}\). Explicitly start with the
compact proper-action transporter of pairs \((h,w)\)
with \(s h=p_Z(w)\), \(w\) in that compact union and
\(hw\in D\), and apply \((h,w)\mapsto(h,hw)\).
This is the required support condition;
it is stronger than merely having a finite sum at each point.
Reindexing (GP.9) now gives exactly

\[
 \sum_{r h=p_Z(z)}c(h^{-1}z)^2=\sum_j\tau_j(q(z))=1.
 \tag{GP.11}
\]

**Proof: map and uniqueness.** Put

\[
 \Phi(z)=\left(p_Z(z),
   \sum_{r h=p_Z(z)}c(h^{-1}z)^2\delta_h\right).
 \tag{GP.12}
\]

This is a probability measure on the actual range fibre, hence
belongs to \(M_{\mathcal G}\). Its test against an arrow
function is a locally finite continuous sum by the compact
support condition and the bisection argument just proved.
Thus it is continuous in exactly Theorem 11S.1's topology. For
\(g:p_Z(z)\to y\), replace \(h\) by \(gk\) to obtain

\[
 \Phi(gz)=g\Phi(z).
 \tag{GP.13}
\]

All products are composable and the coefficient
\(c(k^{-1}z)^2\) is unchanged. Any two anchored maps take
their values in the same range-fibre measure space. Their
pointwise convex combination is continuous in every test
coordinate, equivariant, and has mass in \((1/2,1]\).
It is the asserted homotopy. \(\square\)

### 11S.4. Exact contribution to the general graph interface

Theorems 11S.1–11S.3 construct one genuine universal proper
\(\mathcal G\)-space for the entire eligible holonomy scope.
They supply the central proper space itself and all its
topological, overlap and isotropy data. In particular
\(C_0(M_{\mathcal G})\) is a separable proper groupoid algebra
with its own canonical central structure. Theorem 11R.3 also
makes the compact algebra of any full nonzero equivariant
Hilbert field on this base proper.

These statements do not supply classes
\(\eta\in KK_{\mathcal G}(C_0(T),P)\) and
\(d\in KK_{\mathcal G}(P,C_0(T))\) with the local product
identity of Theorem 11R.2. The anchor only gives
\(C_0(T)\to M(C_0(M_{\mathcal G}))\); it is not a reverse
Dirac class. Contractibility of each convex fibre supplies
no equivariant continuous choice of its contraction point.

This proper base can be used in the universal proper Dirac
construction. A compact auxiliary kernel base, if used to
construct a different proper Bott coefficient and the required
factors, must still have one continuous compatible kernel on the
whole transformation groupoid. The compact-Lie discrete
embedding provider does not by itself supply that continuous
family. Thus the completed all-holonomy proper-space provider
and the unfinished compact-kernel/factorization provider have
different, explicit types.


![The groupoid half-mass test and its typed overlap arrows](../figures/groupoid-proper-base.png)

**Figure 11S.1.** Theorem 11S.2's left panel is the exact
two-coordinate set-mass projection under the stated
disjointness premise. The strict quadrant has no point
in the unit-mass triangle. The right panel retains the
three unit anchors and the actual composable arrows:
\(h:x\to y\), \(g:y\to z\), \(k=gh:x\to z\).
Thus \(g=kh^{-1}\), with both \(h,k\in E\).
The arrows are a schematic of the composability identity,
not an embedding of the groupoid in a plane. The bottom
panel gives GP.3's actual cutoff map and the exact
remaining Dirac-factor type.
[Editable SVG](../figures/groupoid-proper-base.svg) ·
[Generator](../reproduction/groupoid-proper-base/draw_proper_base.py) ·
[Component terms](../reproduction/groupoid-proper-base/GP-COMPONENT-TERMS.md).

Human-source context: Jean-Louis Tu,
[*The Gamma Element for Groups which Admit a Uniform Embedding into Hilbert Space*](https://www.yumpu.com/en/document/view/50923940/the-gamma-element-for-groups-which-admit-a-uniform-embedding-)
(2004), pp. 282–284, for the discrete measure-base route;
Jean-Louis Tu,
[*La conjecture de Baum–Connes pour les feuilletages moyennables*](https://citeseerx.ist.psu.edu/document?doi=486238991671dba031f9cb078eeba25443307a0a&repid=rep1&type=pdf),
K-Theory 17 (1999), 215–264,
Section 1, for proper groupoid spaces and cutoffs. The
proofs above retain the actual varying range fibres and
prove the displayed continuity and compact-transporter
steps directly. These sources do not assert this packet
closes the historical graph question.

## 11T. A coherent affine field on the universal proper extension

The Theorems 11S.1–11S.3 constructs \(M=M_{\mathcal G}\) for
every second-countable locally compact Hausdorff étale groupoid
\(\mathcal G\rightrightarrows T\). It is a locally compact
proper \(\mathcal G\)-space. We now supply one actual coherent
continuous conditionally negative kernel on the entire
transformation groupoid \(\mathcal H=\mathcal G\ltimes M\).
The base \(M\) is retained throughout; this is not a kernel
on \(\mathcal G\) with its original unit space \(T\), or on a
compact auxiliary extension. All arguments use the explicitly proved universal-base provider in Section 11S.

### 11T.1. One proper continuous height on the entire base

**Lemma 11T.1.** There is a continuous proper function
\(a:M\to[0,\infty)\).

**Proof.** A second-countable locally compact Hausdorff space
has a compact exhaustion \(K_n\) with
\(K_n\subset\operatorname{int}K_{n+1}\). To obtain it,
start with a countable relatively compact open cover; at
each stage cover the previous compact set and the next
finitely many cover members by finitely many relatively
compact open neighbourhoods, and take the union of their
closures. Choose \(u_n\in C_c(M)\), with
\(0\le u_n\le1\), equal to one on \(K_n\), and supported
inside \(\operatorname{int}K_{n+1}\). Set

\[
 a(m)=\sum_{n\ge1}(1-u_n(m)).
 \tag{UPL.1}
\]

Near any point, all sufficiently large \(u_n\) are one;
therefore the sum is locally finite and continuous. If
\(m\notin K_{N+1}\), its first \(N\) terms equal one,
so \(a(m)\ge N\). For \(N>R\), the closed sublevel
\(\{a\le R\}\) lies in the compact \(K_{N+1}\).
It is compact. Thus \(a\) is proper. \(\square\)

No equivariant height or fixed point is assumed. Such a
height is chosen once on the whole \(M\), rather than
independently on unit fibres.

### 11T.2. The coherent kernel and literal affine line

For an arrow \(h:m\to n\) of \(\mathcal H\), define

\[
 b_h=a(n)-a(m),\qquad \psi(h)=b_h^2.
 \tag{UPL.2}
\]

**Theorem 11T.2.** The function \(\psi\) is continuous,
symmetric, zero on units, conditionally negative in the
exact (CA.14) convention, and proper relative to the two
unit maps in the exact (CA.15) convention. The trivial
real line field \(H_m=\mathbb R\), with linear action
\(L_h=1\), has the continuous affine action

\[
 A_h(v)=v+b_h,
 \qquad b_{hk}=b_h+b_k
 \quad(s h=r k).
 \tag{UPL.3}
\]

In particular this supplies coherent groupoid field data
on a single actual extension for the whole eligible
holonomy scope.

**Proof.** The continuity, unit and inverse assertions
follow directly from the two continuous unit maps and
the one function \(a\). For arrows \(h_i:m_i\to n\)
with one common range and real \(c_i\) with sum zero,

\[
 \begin{aligned}
 \sum_{i,j}c_i c_j\psi(h_i^{-1}h_j)
 &=\sum_{i,j}c_i c_j(a(m_i)-a(m_j))^2\\
 &=-2\left(\sum_i c_i a(m_i)\right)^2\le0.
 \end{aligned}
 \tag{UPL.4}
\]

The two single-index square terms vanish exactly because
\(\sum_i c_i=0\). No orbitwise choices are made.
The transformation groupoid \(\mathcal H\) is proper
by Theorem 11S.2. For compact \(K,L\subset M\), the entire
transporter \(\{h:s h\in K,r h\in L\}\) is compact.
Its intersection with the closed set \(\{\psi\le R\}\)
is compact. This is precisely the relative properness
needed in CA.15. It does not assert absolute properness
of \(\psi\) on an arrow space with noncompact units.

Composition telescopes the differences of \(a\), giving
(UPL.3). Each affine map is an isometry of its specified
real fibres, and its dependence on the arrow and vector
is continuous. This proves the entire assertion.
\(\square\)

### 11T.3. The exact proper cone and its change of coordinates

**Theorem 11T.3.** The space

\[
 Z=\{(m,v,t):m\in M,\ v\in\mathbb R,\ t\ge v^2\}
 \tag{UPL.5}
\]

is second countable, locally compact and Hausdorff. It
has the continuous proper \(\mathcal G\)-action

\[
 h\cdot(m,v,t)=
 (n,v+b_h,t+2vb_h+b_h^2),\qquad h:m\to n.
 \tag{UPL.6}
\]

**Proof.** The cone is a closed subset of
\(M\times\mathbb R\times[0,\infty)\), so has the
stated topological properties. The excess height is
preserved:

\[
 (t+2vb_h+b_h^2)-(v+b_h)^2=t-v^2.
 \tag{UPL.7}
\]

The action stays inside the cone, is continuous, and
composes by (UPL.3). There is an exact global
homeomorphism

\[
 (m,v,t)\longmapsto(m,w,e)
       =(m,v-a(m),t-v^2),\qquad
 Z\cong M\times\mathbb R\times[0,\infty).
 \tag{UPL.8}
\]

Its inverse is
\((m,w,e)\mapsto(m,w+a(m),e+(w+a(m))^2)\).
Under it the action sends \((m,w,e)\) to
\((hm,w,e)\): both added coordinates are invariant.
For compact input and output subsets, their projections
on \(M\) are compact. Theorem 11S.2 bounds all transporting
arrows by a compact set. The cone transporter is then
a closed subset of that compact arrow set times the
compact input set. This proves properness of the
original \(\mathcal G\)-action, with the composite
anchor to \(T\). No arrow labels or isotropy are
removed by the coordinate change. \(\square\)

The finite-dimensional cone has its ordinary product
topology. The general weak-field cone of Theorem 11R.4 remains
available for infinite-dimensional kernels. Here the
displayed literal line field supplies (UPL.5) directly,
so no fibre-rank inference from a GNS construction is
needed.

### 11T.4. Exact coefficient type and remaining local index

This completes an actual coherent proper-kernel and
affine-field construction on
\(\mathcal G\ltimes M_{\mathcal G}\) for every eligible
holonomy groupoid. Both \(C_0(M)\) and \(C_0(Z)\)
are separable proper coefficients, and the latter has
the explicit central space \(Z\). The elementary
compact algebra of a full nonzero equivariant Hilbert
field over \(Z\) is proper by Theorem 11R.3.

The source of a field Bott or oscillator identity on
this extension is \(C_0(M)\). The original transfer
requires source \(C_0(T)\). Pullback along the anchor
\(M\to T\) is a multiplier homomorphism; it is not
the class from \(C_0(T)\) through a proper algebra and
back required in Theorem 11R.2. The affine action (UPL.3) is a
coboundary on this proper base. It does not supply a
compact-base coarse kernel for the original groupoid,
a universal Dirac inverse, or the positive local Bott
integer of the original graph. The deep universal
Dirac construction therefore remains the distinct
required provider rather than being asserted from
the convexity of \(M\) or from this field alone.


**Illustration for Theorem 11T.3.** The right panel of [Figure 11R.1](#11r-4-a-coherent-kernel-supplies-a-proper-groupoid-cone) is
also the exact fibre slice of Theorem 11T.3 when \(b_h=1\).
Its input \((-1/2,3/4)\), output \((1/2,3/4)\)
and excess height \(1/2\) exhibit (UPL.6)–(UPL.7).
For this new construction the affine displacement
is the actual difference of the single global
height function; the base is the universal proper
\(M\). The existing figure's conditional-transfer
panel keeps the separate original-unit Dirac
requirement visible. Its generator, exact font
notices and component terms are linked in Section 11R.

## 11U. The full and reduced norms of every proper groupoid coefficient
The argument works on the actual groupoid and completed Hilbert modules. It uses no groupoid disintegration theorem, invariant unit measure or discrete-group norm theorem.

Let \(\mathcal G\rightrightarrows T\) be second countable,
locally compact, Hausdorff and étale. Let \(P\) be a separable
graded \(\mathcal G\)-algebra, with its nondegenerate
\(C_0(T)\)-structure and continuous even fibre isomorphisms
\(\alpha_g:P_{s g}\to P_{r g}\). Suppose it is proper in the
precise central sense: a second-countable locally compact
Hausdorff proper \(\mathcal G\)-space \(Z\), anchored by
\(p:Z\to T\), and a nondegenerate even equivariant central
homomorphism

\[
 \phi:C_0(Z)\longrightarrow ZM(P)
 \tag{PN.1}
\]

compatible with the anchor are supplied. Faithfulness of
\(\phi\), freeness, compactness of \(Z\), and constant
fibre dimension are not assumed. Multipliers associated to
functions on \(Z\) below always mean their images under
the unique nondegenerate multiplier extension of \(\phi\).

Write \(\mathcal A=C_c(\mathcal G,r^*P)\). Its convolution
and involution are

\[
 \begin{aligned}
 (f*a)(l)&=\sum_{r h=r l}f(h)\alpha_h(a(h^{-1}l)),\\
 f^*(l)&=\alpha_l(f(l^{-1})^*).
 \end{aligned}
 \tag{PN.2}
\]

The full crossed product \(B_m\) is the universal completion
for the \(I\)-bounded Hilbert-space star representations
of this convolution algebra, where

\[
 \|f\|_I=\max\left\{
 \sup_x\sum_{r g=x}\|f(g)\|,
 \sup_x\sum_{s g=x}\|f(g)\|\right\}.
 \tag{PN.3}
\]

We construct the reduced completion on a Hilbert \(P\)-module
explicitly. This also fixes the reduced norm without a
measure or representation-disintegration convention.

### 11U.1. The source-column regular Hilbert module

Let \(E_c=C_c(\mathcal G,s^*P)\). Define

\[
 (\xi p)(g)=\xi(g)p(sg),\qquad
 \langle\xi,\zeta\rangle(x)
       =\sum_{s g=x}\xi(g)^*\zeta(g).
 \tag{PN.4}
\]

The sum is a continuous compactly supported unit section:
a compact arrow support has a finite bisection cover,
and a partition subordinate to that cover reduces the
sum to finitely many continuous unit branches. It is
positive in every coefficient fibre and hence in \(P\).
Quotient by its zero norm and complete to the Hilbert
\(P\)-module \(E\). Its fibre at \(x\) is the source
column \(\ell^2(\mathcal G_x)\otimes P_x\). The module
is countably generated: a countable relatively compact
bisection basis and a countable dense set of unit
sections of the separable \(P\) give a countable
generating family by source pullback and arrow bumps.

For \(f\in\mathcal A\), put

\[
 (\Lambda(f)\xi)(g)=
   \sum_{r h=r g}\alpha_g^{-1}(f(h))\xi(h^{-1}g).
 \tag{PN.5}
\]

Every summand lies in \(P_{s g}\), because
\(s(h^{-1}g)=s g\). Products of the two compact arrow
supports are compact, so the expression maps \(E_c\)
to \(E_c\).

**Lemma 11U.1 (regular norm and coefficient inclusion).**
The operator in (PN.5) is adjointable, with adjoint
\(\Lambda(f^*)\), is multiplicative, and satisfies
\(\|\Lambda(f)\|\le\|f\|_I\). Let
\(B_r=\overline{\Lambda(\mathcal A)}\subset\mathcal L(E)\).
Then \(\|f\|_r=\|\Lambda(f)\|\le\|f\|_m\), and the
unit coefficient inclusion \(i_P:P\to M(B_m)\) is
faithful and nondegenerate.

**Proof.** At a fixed source \(x\), the matrix indexed
by \(g,l\in\mathcal G_x\) has entries

\[
 A_{g,l}=\alpha_g^{-1}(f(gl^{-1}))\in P_x.
 \tag{PN.6}
\]

Its row sums of entry norms are bounded by the first
quantity in (PN.3), say \(R\); its column sums are
bounded by the second quantity, say \(S\). For a finite
column \((\xi_l)\), the C*-module Cauchy–Schwarz
inequality gives, row by row,

\[
 \left(\sum_l A_{g,l}\xi_l\right)^*
 \left(\sum_l A_{g,l}\xi_l\right)
 \le R\sum_l\|A_{g,l}\|\xi_l^*\xi_l.
 \tag{PN.7}
\]

To verify it directly, form the row operator with
entries \(A_{g,l}/\sqrt{\|A_{g,l}\|}\), omitting zero
entries, and the column with entries
\(\sqrt{\|A_{g,l}\|}\xi_l\). The row operator times
its adjoint is at most \(R1\), which proves (PN.7)
by positive operator order. Summing the rows gives
\(\langle\Lambda(f)\xi,\Lambda(f)\xi\rangle
\le RS\langle\xi,\xi\rangle\) in every unit fibre,
therefore in \(P\). Thus the norm is at most
\(\sqrt{RS}\le\|f\|_I\). Finite source columns are
dense; the same bound extends the operator.

The adjoint matrix entry is
\(\alpha_l^{-1}(f(lg^{-1})^*)\), exactly the
\((g,l)\)-entry of \(\Lambda(f^*)\). Reindexing finite
convolution sums, using \(\alpha_{gh}=\alpha_g\alpha_h\),
proves multiplication. These identities hold on the
dense core and extend to \(E\).

For the full-norm comparison take a faithful ordinary
Hilbert-space representation \(\pi\) of \(P\), obtained
as the direct sum of GNS representations of positive
states separating its positive elements. On
\(E\otimes_{P,\pi}H_\pi\), the operators
\(\Lambda(f)\otimes1\) are \(I\)-bounded star
representations. The induced representation of
\(\mathcal L(E)\) is faithful: if \(T\xi\ne0\), the
nonzero positive element \(\langle T\xi,T\xi\rangle\)
has a nonzero value on some vector under faithful
\(\pi\); the tensor \(T\xi\otimes v\) then has
positive norm. Hence it is isometric. The universal
full norm bounds \(\|\Lambda(f)\|\). This argument
uses GNS for a single C*-algebra, not a decomposition
of a groupoid representation.

On unit-supported coefficient functions the \(I\)-norm
is \(\sup_x\|p(x)\|=\|p\|\). A unit source-column
vector at \(1_x\) shows the reverse bound for their
regular operators. Unit functions extend by zero on
the open-and-closed unit space. Completion therefore
gives a faithful coefficient inclusion in \(B_m\)
(and likewise in \(B_r\)). An approximate identity
of \(P\) acts uniformly on the finite local unit
sections covering any compact convolution support,
so it converges on both sides of \(\mathcal A\).
The coefficient inclusion is nondegenerate.
\(\square\)

### 11U.2. The correctly typed cutoff and a dense localized core

Theorem 11S.3 supplies one continuous square cutoff
\(c:Z\to[0,1]\), with

\[
 \sum_{r g=p(z)}c(g^{-1}z)^2=1.
 \tag{PN.8}
\]

For each compact \(D\subset Z\), the pairs
\((g,z)\) with \(z\in D\), \(r g=p(z)\) and
\(c(g^{-1}z)\ne0\) lie in one compact subset of
the inverse-composability space
\(\{(g,z):r g=p(z)\}\). This is Theorem 11S.3's support
condition; it is not a condition in the differently
typed source action space. The bound \(c\le1\)
also follows directly from the identity-arrow term
in (PN.8).

The central homomorphism extends to bounded continuous
functions by a concrete strict multiplier extension.
For a contractive approximate identity \(u_i\) in
\(C_0(Z)\), the operators \(\phi(cu_i)\) have norm
at most \(\|c\|_\infty\). On the dense span of
\(\phi(f)p\) they converge to the rule
\(\phi(f)p\mapsto\phi(cf)p\), since
\(u_i f\to f\) in norm. Uniform boundedness extends
this limit to every coefficient. Applying the same
argument to \(\overline c\) gives its adjoint.
Thus the limit is a central multiplier. The same construction applies
fibrewise and commutes with transport. Thus (PN.8)
becomes the strict multiplier identity

\[
 \sum_{r g=y}\alpha_g(c_{s g}^2)=1_{M(P_y)}.
 \tag{PN.9}
\]

Here \(c_x\) is the central multiplier on \(P_x\)
induced by \(c|_{Z_x}\). To check strict convergence,
first localize a coefficient by a function supported
on a compact subset of \(Z_y\). The cutoff support
condition bounds all contributing arrows in a compact
set, whose intersection with the closed discrete
range fibre is finite. The sum is then exactly one
on that localization. Those localizations are dense
by nondegeneracy of (PN.1). The partial sums are
positive contractions, so convergence extends
strictly to every coefficient and its adjoint.

Let \(\mathcal A_Z\subset\mathcal A\) consist of
sections having compact central \(Z\)-support:
\(a=\rho a\) for some \(\rho\in C_c(Z)\), where
\((\rho a)(l)=\phi_{r l}(\rho|_{Z_{r l}})a(l)\).
This core is \(I\)-norm dense. Indeed a central
\(C_c(Z)\) approximate identity converges on every
unit coefficient. On a compact arrow support,
finitely many bisections and their local unit
sections make the convergence uniform. A support
covered by \(N\) bisections has \(I\)-norm at most
\(N\) times its uniform coefficient bound. Multiplying
by these central functions therefore approximates
in \(I\)-norm, and a bump equal to one on their
compact supports places each approximation in
\(\mathcal A_Z\).

This is a two-sided star ideal of \(\mathcal A\).
For a left convolution by \(f\), the new central
support lies in the compact image of the composable
pairs in \(\operatorname{supp}f\times\operatorname{supp}\rho\)
under the action. A right convolution retains the
first factor's central support. Taking the adjoint
transports the compact central support by the
compact inverse arrow support. All three compactness
claims concern continuous images of closed subsets
of compact products. A bump equal to one on each
resulting compact set supplies the required central
localizer.

### 11U.3. The completed double-arrow module and cutoff isometry

Form the interior tensor product

\[
 \mathscr R=E\otimes_{P,i_P}B_m.
 \tag{PN.10}
\]

The pair coordinates of an elementary tensor are

\[
 (\xi\otimes a)(g,k)=\xi(g)a(k)\in P_{s g},
 \qquad s g=r k.
 \tag{PN.11}
\]

This is balanced over \(P\): its left coefficient
action on \(a\) is multiplication by \(p(rk)\),
the same coefficient as the right action on \(\xi\).
Every continuous compactly supported section of
this pair space is in the completed span of these
elementary sections. For a detailed local argument,
choose bisections through \(g\) and \(k\). Their
composable pair is parametrized by the middle unit
\(x=s g=r k\). A supported section on this chart
is a local section of \(P_x\); unit sections and
scalar bumps approximate it in norm. Products of
two unit sections span a dense subset of each
\(P\), by its approximate identity. The two factors
and the scalar bumps give exactly (PN.11). A finite
partition on the compact pair support gives finite
elementary approximations. No fibre trivialization
or constant rank is involved.

Their interior-tensor inner product, written in
pair coordinates, is

\[
 \langle K,L\rangle(h)=
 \sum_{s k=r h}\alpha_k^{-1}\left(
       \sum_{s g=r k}K(g,k)^*L(g,kh)\right).
 \tag{PN.12}
\]

For elementary tensors this follows by expanding
\(a^*i_P(\langle\xi,\zeta\rangle)b\) with (PN.2).
It also proves that uniform local approximations
converge in the completed module. More explicitly,
if a pair section has uniform norm at most
\(\epsilon\), and its first and second arrow
supports have covers by \(N_g\) and \(N_k\)
bisections, respectively, its squared inner
product has coefficient norm at most
\(N_gN_k\epsilon^2\) at each arrow. Its support
lies in the second-support product \(K_k^{-1}K_k\),
covered by \(N_k^2\) bisections. Thus its module
norm is at most
\(\sqrt{N_gN_k^3}\,\epsilon\), by (PN.3).
Consequently (PN.12) extends to the claimed pair
sections without a new positivity assumption.

For \(a\in\mathcal A_Z\), define

\[
 (Wa)(g,k)=c_{s g}\alpha_g^{-1}(a(gk)),
 \qquad s g=r k.
 \tag{PN.13}
\]

It is a continuous section on this exact pair
space. It has compact support. To see this, let
\(D\) be a compact central support for \(a\) and
\(K_a\) its compact arrow support. A nonzero
central product in (PN.13) requires a point
\(z\in D\) with \(c(g^{-1}z)\ne0\) and
\(r g=p(z)\). Theorem 11S.3 therefore bounds the first
arrow \(g\) in one compact set \(B_D\).
Since \(gk\in K_a\), the second arrow lies
in the compact product \(B_D^{-1}K_a\).
The support of the section is closed inside
the resulting compact pair set. Hence (PN.13)
is an actual vector of \(\mathscr R\), by the
completed-core construction above.

**Lemma 11U.2 (isometry and covariance).** The map
\(W\) extends to an even isometric right
\(B_m\)-module map
\(B_m\to\mathscr R\), and for every
\(f\in\mathcal A\) it intertwines left
convolution with the regular tensor action:

\[
 W(f*a)=(\Lambda(f)\otimes1)Wa.
 \tag{PN.14}
\]

**Proof: the norm.** Substitute (PN.13) into
(PN.12) and put \(l=gk\). All products are
composable, with \(s l=r h\), and one obtains

\[
 \begin{aligned}
 \langle Wa,Wb\rangle(h)
 &=\sum_{s l=r h}\alpha_l^{-1}\left[
   a(l)^*\left(\sum_{r g=r l}
      \alpha_g(c_{s g}^2)\right)b(lh)\right]\\
 &=\sum_{s l=r h}\alpha_l^{-1}(a(l)^*b(lh))
   =(a^**b)(h).
 \end{aligned}
 \tag{PN.15}
\]

The inner sum is (PN.9). On the compact central
supports occurring here its localization has
only finitely many contributing arrows at each
fixed unit, and the strict identity justifies
the same expression before localization. Thus
the equality holds in \(B_m\), not only in
individual fibres. It gives
\(\|Wa\|^2=\|a^*a\|=\|a\|_m^2\).
The dense core \(\mathcal A_Z\) completes to
the isometric map on all of \(B_m\).

The right action in pair coordinates is
convolution in the second arrow. On the core,
\[
 \begin{aligned}
 ((Wa)*b)(g,k)
 &=\sum_{r l=s g}
 c_{s g}\alpha_g^{-1}(a(gl))
                 \alpha_l(b(l^{-1}k))\\
 &=c_{s g}\alpha_g^{-1}((a*b)(gk))
   =W(a*b)(g,k).
 \end{aligned}
 \tag{PN.16}
\]

This proves right module linearity and extends
by continuity.

**Proof: regular intertwining.** The regular
tensor action in pair coordinates is
\[
 ((\Lambda(f)\otimes1)K)(g,k)=
  \sum_{r h=r g}\alpha_g^{-1}(f(h))K(h^{-1}g,k).
 \tag{PN.17}
\]

The two source units in this sum are both
\(s g\). Its cutoff multiplier is therefore
the same \(c_{s g}\) for every summand. It is
central. Also
\(\alpha_{h^{-1}g}^{-1}=\alpha_g^{-1}\alpha_h\).
Consequently the expression at \(K=Wa\)
is exactly

\[
 c_{s g}\alpha_g^{-1}
    \left(\sum_{r h=r g}f(h)
                    \alpha_h(a(h^{-1}gk))\right),
 \tag{PN.18}
\]

which is \(W(f*a)(g,k)\). The ideal property
ensures that this vector is defined on the
localized core. This proves (PN.14), then
extends it to the completed modules. No
adjoint of the isometry is needed for the
norm comparison. All cutoff multipliers
are even; the coefficient grading and
Clifford order are unchanged. \(\square\)

### 11U.4. The actual quotient is an isomorphism

**Theorem 11U.3 (full equals reduced for proper coefficients).**
For every \(P\) satisfying (PN.1),
\[
 \|f\|_m=\|f\|_r\quad(f\in\mathcal A),
 \qquad q_P:P\rtimes_m\mathcal G
         \xrightarrow{\cong}P\rtimes_r\mathcal G.
 \tag{PN.19}
\]

The isomorphism is graded. Its inverse gives
an even class \(v=[q_P^{-1}]\in KK(B_r,B_m)\)
with both inverse product identities.

**Proof.** For \(a\in B_m\), the isometry and
intertwining imply
\[
 \|f*a\|_m=\|W(f*a)\|
     \le\|\Lambda(f)\otimes1\|\,\|Wa\|
     \le\|\Lambda(f)\|\,\|a\|_m.
 \tag{PN.20}
\]

Left multiplication on a C*-algebra has
the norm of its multiplier: the reverse
bound follows by applying a contractive
approximate identity. Thus (PN.20) gives
\(\|f\|_m\le\|\Lambda(f)\|=\|f\|_r\).
Lemma 11U.1 gives the other inequality.
Equality on the dense convolution core
makes its quotient map an isometric onto
star homomorphism after completion.

The inverse homomorphism is even because
the quotient is even and bijective.
The Kasparov products of homomorphism
classes are their compositions, so
\([q_P]\otimes_{B_r}[q_P^{-1}]=1_{B_m}\)
and
\([q_P^{-1}]\otimes_{B_m}[q_P]=1_{B_r}\).
\(\square\)

This supplies exactly the proper-coefficient
quotient input in Theorem 11R.2 at the actual
groupoid scope. It does not assert that
the coefficient-free quotient
\(C^*(\mathcal G)\to C_r^*(\mathcal G)\)
is invertible. Neither it nor the proper
space alone provides the original-unit
Dirac factors, the local Bott integer,
or a physical graph operator. The full
normal inverse and its grading remain
in the other supplied factors of Theorem 11R.2.

![The cutoff isometry and its exact middle-unit coefficient](../figures/groupoid-proper-coefficient-norm.png)

**Figure 11U.1.** Lemma 11U.2's square commutes on the
completed \(B_m\)-modules. The right panel retains
\(k:x\to y\), \(g:y\to z\) and \(l=gk:x\to z\).
Both \(c_y\) and \(\alpha_g^{-1}(a(gk))\) lie
in the multiplier/coefficient algebra of the middle
unit \(P_y\). In the convolution sum, an arrow
\(h:w\to z\) changes \(g\) to
\(h^{-1}g:y\to w\), retaining that same cutoff.
The bottom panel displays the strict identity
in \(M(P_y)\) and the actual norm comparison of
Theorem 11U.3. This is a typed schematic; it does
not replace \(\mathcal G\) by a planar graph or
assert the existence of the remaining Dirac factors.
[Editable SVG](../figures/groupoid-proper-coefficient-norm.svg) ·
[Generator](../reproduction/groupoid-proper-coefficient-norm/draw_proper_norm.py) ·
Component terms.

Human-source context: Jean-Louis Tu,
[*La conjecture de Baum–Connes pour les feuilletages moyennables*](https://citeseerx.ist.psu.edu/document?doi=486238991671dba031f9cb078eeba25443307a0a&repid=rep1&type=pdf),
K-Theory 17 (1999), 215–264, Section 1,
states the proper-groupoid norm equality
and cutoff framework. The argument here
constructs its source-column module,
double-arrow completion and cutoff
intertwiner explicitly. The complete
Theorem 11S.3 cutoff proof is its exact geometric
provider, and Theorem 11R.2 retains the other
factorization inputs separately.

## 11V. Proper source localization for a contractible-kernel quotient

All groups below are countable and discrete, all algebras are separable and graded, and all homomorphisms preserve grading. A proper algebra has a nondegenerate equivariant central action of \(C_0(W)\), where \(W\) is a second-countable locally compact Hausdorff proper group space. This text supplies the localization step needed to lift a Dirac class. It does not by itself construct a Bott deformation.

### 11V.1. A supported equivariant completely positive section

**Lemma 11V.1.** A square cutoff \(c\) on \(W\) can be chosen together with a continuous \(d:W\to[0,1]\) such that \(dc=c\), and the translates of \(\operatorname{supp}d\) meet each compact set only finitely often.

**Proof.** Use the locally finite quotient partition and compact bumps in the square-cutoff proof in the already supplied Section 11J, Lemma 11J.1. Denote their quotient partition functions by \(\rho_i\) and their compact bumps by \(\phi_i\). Enlarge each compact support of \(\phi_i\) to a relatively compact open \(O_i\). Enlarge the quotient supports of \(\rho_i\) to a locally finite open family \(V_i\). The closed locally finite set
\[
 L=\bigcup_i\bigl(q^{-1}(\operatorname{supp}\rho_i)\cap
                              \operatorname{supp}\phi_i\bigr)
 \tag{PX.1}
\]
contains \(\operatorname{supp}c\). It is contained in the open set
\(O=\bigcup_i(q^{-1}V_i\cap O_i)\).
Only finitely many \(V_i\) meet the image of a given compact \(K\). For each of these indices, the proper transporter of \(\overline O_i\) to \(K\) is finite. Thus the translates of \(\overline O\) have the asserted local finiteness; the same argument applies if the quotient enlargements are first shrunk to have closures in a locally finite family. A metrizable locally compact space is normal. Shrink \(O\) about its closed subset \(L\) and use the continuous separating function to choose \(d=1\) on \(L\), with support in that shrink. The transporter assertion follows from the enlargement just constructed. This proves \(dc=c\) and the stronger support property. \(\square\)

**Theorem 11V.2.** Suppose
\[
 0\longrightarrow I\longrightarrow E\xrightarrow q Q\longrightarrow0
 \tag{PX.2}
\]
is an equivariant extension of proper \(C_0(W)\)-algebras, with the same central structure map and with \(q\) respecting it. If \(q\) has an ordinary completely positive contractive section \(s\), it has an equivariant such section.

**Proof.** Make \(s\) even by averaging its two grading conjugates. Put
\(c_g=\alpha_g(c)\), \(d_g=\alpha_g(d)\). For a finite set \(F\subset\Gamma\) set
\[
 S_F(a)=\sum_{g\in F}c_g\,
   \alpha_g^E\!\left(s\bigl(\alpha_{g^{-1}}^Q(d_g a d_g)\bigr)\right)c_g.
 \tag{PX.3}
\]
Every term is completely positive. At every matrix level a positive input of norm at most one gives a positive output at most
\(\sum_{g\in F}c_g^2\leq1\)
in the corresponding multiplier algebra: the input compression is contractive, \(s\) is contractive, and the two output cutoffs are central. Hence every \(S_F\) is a completely positive contraction.

If \(a\) is compactly localized over \(W\), meaning \(a=f a\) for some compactly supported central \(f\), only finitely many input compressions \(d_ga d_g\) are nonzero, by Lemma 11V.1. Therefore (PX.3) stabilizes for this dense subspace of \(Q\). Uniform contractivity shows that the net stabilizing there converges in norm on every \(a\in Q\); approximate \(a\) by a compactly localized input and use the two contraction bounds to control any tail. Its limit \(S:Q\to E\) is a completely positive contraction. This argument proves norm convergence, rather than presuming strict convergence gives an element of \(E\).

The quotient of each summand is
\(c_gd_ga d_gc_g=c_g^2a\), since \(d_gc_g=c_g\).
Thus
\[
 qS(a)=\sum_g c_g^2a=a.
 \tag{PX.4}
\]
The last equality is the square-cutoff strict approximate-identity identity, applied in the quotient algebra. Replacing \(g\) by \(hg\) in (PX.3) proves
\(\alpha_h^E S(a)=S(\alpha_h^Q a)\)
first for compactly localized inputs and then by continuity. The map is even. \(\square\)

Both input and output localizations in (PX.3) are essential: output localization alone need not give norm convergence, and using the same cutoff twice in both positions would give \(\sum_g c_g^4\), rather than (PX.4).

### 11V.2. Exactness for the source extensions used here

We use the following already proved ordinary foundations with their precise scope: the completely positive lifting theorem in Ext groups, absorption and Brown–Douglas–Fillmore theory, Lemmas 3.0a--3.0e; the Stinespring extension cycle, cone excision and both exact sequences in Exact sequences in KK and the universal coefficient theorem, Sections 3--4; its countable additivity and telescope construction in Sections 5 and 9; and the equivariant technical theorem and product in Equivariant KK-theory and the Green–Julg theorem, Lemma 2.4 and Theorems 2.5--2.6.

Here is the equivariant adaptation at the exact interfaces consumed below. For an equivariant completely positive section the Stinespring completion has action
\[
 U_g(a\otimes j)=\alpha_g(a)\otimes\alpha_g(j).
 \tag{PX.5}
\]
Complete positivity and covariance preserve its inner product. The cyclic isometry \(Vj=1\otimes j\) is equivariant, so \(VV^*\) and the extension involution are invariant. Their commutator defects are the ordinary multiplicativity defects, which remain compact. Convex interpolation of equivariant sections is equivariant and proves independence and naturality with the same two-by-two rotation of cyclic isometries; it requires no complemented cyclic submodule.

In cone excision, every cone homomorphism, scalar taper, reflection and rotation is equivariant. If the algebras have a common proper central space, the suspended-ideal section is made equivariant by Theorem 11V.2; all its cone algebras still have that space. The auxiliary cone-to-cone section is equivariant once the original section is. For a finite group one may instead average each section over that finite group. The locally compact technical theorem supplies approximate identities with both commutator and fixed-group-element defects tending to zero, so the operator normalization and Puppe-cycle constructions have the required equivariance defects. The extension involutions themselves use (PX.5) and are exactly invariant. Consequently the two cone inverse identities and the boundary-product exact sequences hold for these extensions in \(KK^\Gamma\), with the same ordered Clifford signs as in the ordinary proof.

More explicitly, write \(e:I\to C_q\) for the ideal-to-cone map and \(u\) for the negative of the suspended extension class followed by positive inverse suspension. Pullback to \(e\) is the ordinary coefficient cone, equivariantly contracted by its scalar parameter, so
\[
 [e]u=1_I.
 \tag{PX.6}
\]
The auxiliary extension from the cone of \(C_q\) onto the cone of \(Q\) has both middle and quotient equivariantly contractible. The same left-inverse assertion makes its kernel cone have zero equivariant KK groups, by Puppe exactness. Puppe exactness for \(e\) then produces a right inverse \(w\), and associativity gives \(w=w([e]u)=u\). Thus
\[
 u[e]=1_{C_q}.
 \tag{PX.7}
\]
Replacing the mapping cone by the ideal gives both cyclic sequences, and their arrows are product with the invariant extension class from (PX.5). The odd interchange cancels the negative left-endpoint sign exactly as in the written ordinary proof. This explains the equivariant adaptations rather than treating an ordinary theorem as an equivariant one without checking it.

### 11V.3. Discrete open-subgroup reciprocity in the source variable

Let \(F\leq\Gamma\) be finite and \(D\) an \(F\)-algebra. The algebra \(\operatorname{Ind}_F^\Gamma D\) consists of \(D\)-valued functions on \(\Gamma\), covariant on right \(F\)-cosets and vanishing at infinity on \(\Gamma/F\).

**Lemma 11V.3.** For every \(\Gamma\)-algebra \(B\), there is a natural isomorphism
\[
 KK^\Gamma(\operatorname{Ind}_F^\Gamma D,B)
        \simeq KK^F(D,\operatorname{Res}_F B).
 \tag{PX.8}
\]
It applies in both degrees and commutes with coefficient maps.

**Proof.** In the forward direction take the essential part of a cycle. The central \(c_0(\Gamma/F)\)-multipliers give mutually orthogonal adjointable projections \(P_r\), whose ranges sum to the essential module. Restrict to the identity-coset range and to \(F\). This gives an \(F\)-cycle with source \(D\).

Conversely average the operator of an \(F\)-cycle over the finite group, using the localized compact perturbation formula proved in the equivariant lesson. For representatives \(r\in\Gamma/F\), take a copy of its module with right multiplication twisted by \(\alpha_{r^{-1}}\) and inner product twisted by \(\alpha_r\). Their Hilbert-module direct sum is a \(B\)-module. A group element sends the summand \(r\) to the summand \(hr\), using the \(F\)-transition in the chosen representatives. Those transitions satisfy the exact cocycle identity, so this is an action. Put the averaged operator on each summand and let the induced source act at its coset value. The operator is invariant. Each localized defect is a diagonal compact field with a \(c_0\) norm tail; finite truncation proves that it is compact on the sum. This produces a countably generated \(\Gamma\)-cycle.

For the other composite, replace an original operator \(T\) by its coset compression \(T^{\rm diag}=\bigoplus_rP_rTP_r\). For a homogeneous source element \(a\) supported on coset \(r\), exact identities give
\((T-T^{\rm diag})\pi(a)=(1-P_r)[T,\pi(a)]_{\rm gr}\) and
\(\pi(a)(T-T^{\rm diag})=[\pi(a),T]_{\rm gr}(1-P_r)\).
The omitted opposite products vanish because \(\pi(a)=P_r\pi(a)=\pi(a)P_r\), so both displayed operators are compact by the cycle commutator condition. Finite support and then the \(c_0\) source tail give two-sided local compactness for every source element. The difference between each diagonal block and the translate of the identity block is locally compact by the original equivariance defect. These differences have common norm bound two after contraction normalization. Multiplication by a \(c_0\) source element again makes their sum compact. After the finite \(F\)-average, the induced operator therefore differs from the original by a two-sided locally compact operator. Straight interpolation is an equivariant cycle homotopy. The identity-coset composite is visibly the original finite-group cycle up to its averaging homotopy. All constructions are unchanged on interval modules and under coefficient extension. This proves (PX.8), including relations and naturality. \(\square\)

This is reciprocity for an open subgroup in a discrete group. It is distinct from the compact ambient-group reciprocity stated in the earlier lesson.

### 11V.4. The localization theorem

**Theorem 11V.4.** Let \(q:E\to Q\) be an equivariant quotient with an ordinary completely positive contractive section. Suppose its kernel \(J\) is equivariantly KK-contractible, including after restriction to finite subgroups. For every nuclear proper \(\Gamma\)-algebra \(A\), the map
\[
 q_*:KK^\Gamma_k(A,E)\longrightarrow KK^\Gamma_k(A,Q)
 \tag{PX.9}
\]
is an isomorphism for \(k=0,1\).

**Proof.** For a finite subgroup \(F\), average the given section to obtain an \(F\)-equivariant section. The exact sequences from Section 11V.2 and the zero groups for \(J\) make \(q_*\) an isomorphism for every \(F\)-source. Lemma 11V.3 implies the same statement for every induced source \(\operatorname{Ind}_F^\Gamma D\).

Choose a countable slice cover of \(W\). Its members are \(U_i=\Gamma V_i\), where \(V_i\) is invariant under a finite stabilizer \(F_i\), and \(gV_i\cap V_i=\varnothing\) for \(g\notin F_i\). To construct a slice at \(w\), first choose a precompact neighborhood. Its compact transporter is finite by properness. For each transporter element not fixing \(w\), shrink to separate the point from its translate; intersect the finitely many shrinks and then the finitely many stabilizer translates. This is the required \(F_i\)-invariant slice. Second countability gives a countable cover.

Put
\[
 A_n=C_0(U_1\cup\cdots\cup U_n)A.
 \tag{PX.10}
\]
The first algebra is induced from \(F_1\), by the disjoint slice decomposition. In the extension
\[
 0\longrightarrow A_{n-1}\longrightarrow A_n
                 \longrightarrow A_n/A_{n-1}\longrightarrow0,
 \tag{PX.11}
\]
the quotient is the image of \(C_0(U_n)A\). It is induced from \(F_n\): an invariant ideal of an induced algebra corresponds to its identity-slice ideal, and evaluation and disjoint translation identify the quotient with the induced identity-slice quotient. This follows directly on compactly supported coset sections, then by norm completion.

All algebras in (PX.11) are nuclear and retain the common proper central space \(W\). Ideals and quotients of a nuclear algebra are nuclear; equivalently use the finite-matrix approximation theorem, compress its approximating maps by an ideal approximate identity for ideals, and use the completely positive lifting and quotient approximation for quotients. The ordinary nuclear lifting theorem supplies a section of (PX.11), and Theorem 11V.2 makes it equivariant. Its two contravariant exact sequences with coefficients \(E\) and \(Q\) commute with \(q_*\). The two outer source types are already in the isomorphism class, by induction and the induced-source case. An exact-sequence chase, or the five lemma on five consecutive cyclic terms, gives (PX.9) for \(A_n\).

Finally \(A=\overline{\bigcup_n A_n}\). To check passage to this increasing union without assuming contravariant continuity, use the telescope of the earlier exact-sequence lesson, Section 9. Each eventual path family carries the same central \(C_0(W)\)-action. It is nondegenerate: a compactly localized approximation to the finitely many initial paths and their eventual endpoint uniformly approximates an eventual family; these families are dense. The telescope contraction is equivariant and \(C_0(W)\)-linear. Its quotient-at-infinity extension has nuclear quotient \(A\), so its ordinary section becomes equivariant by Theorem 11V.2. Its integer-evaluation extension has the already explicit equivariant convex-interpolation section. Thus its two exact sequences and source additivity give, in both degrees,
\[
 0\longrightarrow \varprojlim{}^1 KK^\Gamma_{k-1}(A_n,B)
 \longrightarrow KK^\Gamma_k(A,B)
 \longrightarrow \varprojlim KK^\Gamma_k(A_n,B)
 \longrightarrow0.
 \tag{PX.12}
\]
Here \(\varprojlim\) and \(\varprojlim{}^1\) are the kernel and cokernel of
\((z_n)\mapsto(z_n-i_n^*z_{n+1})\) on the product of the stage groups. The telescope sign is the positive suspension of identity minus shift; the proof is the two adjacent scalar intervals, which are equivariant and carry the unchanged action. Source additivity holds with the same direct-sum projections and localized compact estimates; those projections are invariant because each summand is invariant.

All stage maps for \(q_*\) are isomorphisms and commute with restriction. They conjugate the two product operators just displayed, so induce isomorphisms on both kernels and cokernels. The short-exact-sequence chase in (PX.12) proves (PX.9) for \(A\). This accounts explicitly for the possible \(\varprojlim{}^1\) term. \(\square\)

### 11V.5. The deformation specialization

Suppose the proposed deformation has the exact form
\[
 0\longrightarrow C_0((0,1])\widehat\otimes
          \mathcal K(\mathcal F)\widehat\otimes S
 \longrightarrow\mathcal E\xrightarrow{h_0}\mathcal B
 \longrightarrow0,
 \tag{PX.13}
\]
with an equivariant Hilbert \(C(X)\)-module field \(\mathcal F\), an exact affine-unitary action, and a nuclear proper Bott quotient \(\mathcal B\). The kernel is equivariantly Morita equivalent to \(C_0((0,1])\widehat\otimes C(X)\widehat\otimes S\). Its cone contraction is \(f(t)\mapsto f(ut)\), with zero at \(u=0\). The contraction is equivariant on the coefficient algebra, and hence gives zero KK groups for the kernel. The ordinary section follows from nuclearity of \(\mathcal B\). Theorem 11V.4 therefore gives a unique
\[
 z\in KK^\Gamma(\mathcal B,\mathcal E),\qquad z[h_0]=1_{\mathcal B}.
 \tag{PX.14}
\]
Evaluation at one, Fock Morita equivalence, and the even map \(S\to\mathbb C\), \(f\mapsto f(0)\), define the desired equivariant Dirac from this lift.

Equation (PX.14) is conditional on actually constructing (PX.13), proving its equivariance and identifying its kernel. This text closes the proper-source localization interface, not those analytic deformation assertions.


### 11V. Reading

Tu, *La conjecture de Baum--Connes pour les feuilletages moyennables*, original printed pp.232--234 and 251, motivates the proper-source exactness interface. The proof here uses discrete slices, supported completely positive averaging and an equivariant telescope, with the exact adaptations written above. The original source's nuclear equivariant exactness citation is not used as a missing-proof substitute. The companion construction supplies the infinite-field Bott algebra and its deformation.

## 11W. A proper Bott algebra and its equivariant deformation

**Theorem 11W.1.** For the compact-base affine field in Section 11O, the proper nuclear Bott algebra and actual equivariant deformation below give even classes with the ordinary coefficient product (BD.22). All positive-time domains, zero-endpoint norms and transported finite-frame estimates are part of the construction.


The hypotheses are those already proved in Section 11O: a compact metrizable \(\Gamma\)-space \(X\), a separable continuous real Hilbert field \(E\), and a continuous affine action with exact cocycle and uniform proper displacement. We require only an equivariant factorization whose product becomes the identity in ordinary KK. We do not claim that its equivariant product is the identity.

### 11W.1. Stabilize before defining the algebra

The real version of the actual operator-range stabilization proof in Graded algebras and Hilbert modules, Theorem 6.2, gives
\[
 E\oplus C(X,\ell^2_{\mathbb R})\simeq C(X,H),
 \qquad H=\ell^2_{\mathbb R}.
 \tag{BD.1}
\]
Here is the real check. Choose real generators, use the real maps \(R_j\), repeat them with the real factors \(2^{-j},4^{-j}\), and take the positive square root of \(T^*T\). Its continuous functional calculus commutes with conjugation. The dense-range polar rule \(U(|T|v)=Tv\) therefore commutes with conjugation and restricts to an orthogonal real module isomorphism. No continuous choice of eigenvectors or constant original field rank is involved.

Extend the original linear action by identity on the new summand and the cocycle by zero, then transport them through (BD.1). We obtain strongly continuous orthogonal maps \(L_g(x)\) on the fixed fibre and continuous vectors \(b_g(x)\), satisfying the exact affine identities. The original displacement bound is unchanged:
\[
 \inf_{x\in X}\|b_g(x)\|\longrightarrow\infty
                          \quad(g\longrightarrow\infty).
 \tag{BD.2}
\]

The compact-module weight proof in Section 11P can now start with the fixed diagonal compact operator \(h e_j=2^{-j}e_j\). Its nested cutoff approximate identities are diagonal and independent of \(x\). Finite convex combinations stay so. The quasicentrality test is performed in \(C(X,\mathcal K(H))\), simultaneously with all the cocycle sections. Consequently that proof produces a fixed diagonal positive weight \(\Theta e_j=\lambda_je_j\), with \(\lambda_j\geq1\), \(\lambda_j\to\infty\), and
\[
 S_g(x)=L_g(x)\Theta L_g(x)^{-1}-\Theta
        \in C(X,\mathcal K(H)),
 \qquad \Theta b_g\in C(X,H).
 \tag{BD.3}
\]
All domains in (BD.3) are the closed regular domains from that proof. In particular \(L_g(x)\operatorname{Dom}\Theta=\operatorname{Dom}\Theta\). This convenient diagonal choice is made after stabilization, not by diagonalizing a varying field.

### 11W.2. The phase-space Bott limit

Write \(S=C_0(\mathbb R)\), graded by reflection. For a finite affine plane \(V\subset H\), let
\[
 C(V)=C_0(V\times V_0,\operatorname{End}\Lambda^*V_{0,\mathbb C}).
 \tag{BD.4}
\]
The matrix grading is exterior degree. On the phase space of a vector plane \(W\), put
\(\beta_W(q,p)=\varepsilon(q+ip)+\varepsilon(q+ip)^*\).
The creation and contraction relations give \(\beta_W^2=\|q\|^2+\|p\|^2\), fixing the outward \(q+ip\) convention.

For \(V\subset V'=V\oplus W\), the connecting homomorphism is
\[
 a\widehat\otimes f\longmapsto
 a\widehat\otimes f(T+\beta_W).
 \tag{BD.5}
\]
The summands anticommute, so their square is \(T^2+\beta_W^2\). Functional calculus gives a graded homomorphism. It is injective: restriction to \((q,p)=(0,0)\) in the added plane is a left inverse. For two added orthogonal planes the odd sum is associative; computing the image of its resolvents, or of its affiliated coordinate \(T\), gives the same map in either order. Thus (BD.5) is an inductive system. Call its limit \(A(H)\).

The fixed coordinate planes \(V_n=\operatorname{span}(e_1,\ldots,e_n)\), with origin zero, have dense union in this limit. To check this for an arbitrary finite affine plane, approximate its origin and orthonormal basis by vectors in \(V_n\), and apply the finite Gram-matrix inverse square root to restore orthonormality. The resulting finite affine isometries approach the original one. Their effects on a compactly supported symbol and on the compactly supported functional-calculus factor in (BD.5) tend to zero uniformly. Indeed the square of that factor bounds the entire added phase radius on its support; on this bounded radius the finite coordinate and matrix formulas are uniformly continuous. The comparison can be made in an ambient plane of dimension at most twice the given dimension plus one. Its constants are therefore fixed, not growing with \(n\). Dense compactly supported symbols and functional-calculus functions prove the assertion for every element. This also proves norm continuity of the affine functor on finite generating planes when the finite basis vectors converge in norm.

Hence \(A(H)\) is separable. It is nuclear: its finite stages are finite matrix-valued commutative algebras with the reflection grading, and the graded matrix corner description preserves their nuclear finite-matrix models. Given a finite set in the limit, approximate it inside one stage, take that stage's completely positive finite-matrix model, extend the first map to the limit by the proved matrix-valued completely positive extension theorem, and include the second map into the limit. This gives finite-matrix approximation of the given set.

Set
\[
 \mathcal B=C(X)\widehat\otimes A(H).
 \tag{BD.6}
\]
The affine maps \(L_g(x)v+b_g(x)\) induce its equivariant action by the just defined functor. The norm-continuity check is on the finite generating planes: their base-dependent origins and basis vectors have compact images, and the preceding finite-dimensional uniform-continuity argument is uniform over the compact base. Thus it gives actual automorphisms of the section algebra, rather than only fibre automorphisms. The cocycle law gives the exact group law. Nuclearity of \(C(X)\) follows from finite partitions and evaluation maps; tensoring its models with those above gives nuclearity of \(\mathcal B\).

### 11W.3. Its proper central space

Use the phase cone
\[
 Z=\{(x,v,p,r):x\in X,\ v,p\in H,
                         \|v\|^2+\|p\|^2\leq r\}.
 \tag{BD.7}
\]
Its topology is base, weak coordinates in the two fibres, and the continuous total squared height \(r\). The compact-product Gram-inequality proof in the supplied coarse-interface text, Section 11W.3, applies to \(H\oplus H\): every bounded-height set is compact metrizable, and open bounded-height neighborhoods have compact closures. Thus \(Z\) is second countable and locally compact. The action is
\[
 (x,v,p,r)\longmapsto
 (gx,L_gv+b_g,L_gp,
                      r+2\langle L_gv,b_g\rangle+\|b_g\|^2).
 \tag{BD.8}
\]
It preserves the excess \(r-\|v\|^2-\|p\|^2\). Strong continuity of \(L_g(x)\), norm continuity of \(b_g(x)\), and bounded weak-coordinate tests prove continuity on bounded-height sets, hence everywhere. If a point and its translate have height at most \(R\), then \(\|b_g(x)\|\leq2\sqrt R\). Equation (BD.2) gives finitely many possible \(g\). The closed action relation inside their finite union of compact input sets proves properness.

There is a nondegenerate equivariant central map
\[
 C_0(Z)\longrightarrow ZM(\mathcal B).
 \tag{BD.9}
\]
To prove it, at a finite plane use the scalar functions of its two projected coordinates and of \(T^2\). Under (BD.5), the final variable becomes \(T^2+\|q_W\|^2+\|p_W\|^2\), which is exactly the residual squared height in the cone projection. These scalar functions remain central in every larger stage. Their cylinder functions separate the points of \(Z\); bounded-height compactness and a radial taper give their dense span in \(C_0(Z)\). The injective maps at the finite stages therefore complete to (BD.9). Radial total-height cutoffs are an approximate identity on every compactly supported finite generator, and hence on \(\mathcal B\). This proves nondegeneracy. The affine coordinate and height formulas give equivariance. The construction retains the momentum coordinate: the proper central space of the phase-space Bott algebra is not just the position cone.

### 11W.4. The finite quantum provider and a uniform weight comparison

For a finite vector plane \(V\), let \(\mathcal Q(V)\) be the Euclidean semiclassical algebra with zero fibre \(C(V)\) and positive fibre \(\mathcal K(L^2(V,\Lambda^*V))\). Its explicit Gaussian completely positive section, exact cone kernel, exterior oscillator and resolvents are proved in Deformations and the analytic index, Lemmas 22.7–22.8, with the actual Schur/Fourier kernel estimates in Dirac classes and the cotangent Dolbeault element, Lemma CI.10. Those finite proofs, including the \(q+ip\) sign, are the finite input used here.

In particular, on \(L^2(V,\Lambda^*V)\),
\[
 D_{V,t}=\sum_j\big((q_j+t\partial_j)\varepsilon_j
                         +(q_j-t\partial_j)\varepsilon_j^*\big),
 \qquad D_{V,t}^2=-t^2\Delta+\|q\|^2+t(2N-\dim V).
 \tag{BD.10}
\]
Its zero symbol is \(\beta_V\). The exact heat-kernel and resolvent proof in that finite provider makes \(f(T+D_{V,t})\) a section for every \(f\in S\). Consequently finite orthogonal inclusion defines the exact homomorphism of finite deformation algebras using this odd sum. This is not an assertion based solely on the oscillator index.

Let \(\mathcal F_t\) be the completed Fock space obtained by Gaussian inclusions of these finite \(L^2\) spaces, with Gaussian variance \(t\). Dilation identifies it with the completed Fock space from the supplied weighted-oscillator text, and \(D_t=\sqrt t D\), \(B_t(A)=\sqrt t B(A)\). The completed-domain proofs there apply after this dilation.

We need the uniform estimate
\[
 \|f_0(T+B_t(A_1))-f_0(T+B_t(A_2))\|
                                  \leq3\|A_1-A_2\|,
 \quad A_i\geq1,
 \tag{BD.11}
\]
where \(f_0(u)=u(1+u^2)^{-1/2}\) and the difference of the weights is bounded. Here is the order check and estimate. The creation/contraction identity is
\[
 B_t(S)^2\leq\|S\|^2D_t^2.
 \tag{BD.12}
\]
It is the corrected \(\leq\) identity from the supplied oscillator proof. All these operators commute with \(D_t^2\). On its spectral subspace of value \(\mu\), the perturbation has norm at most \(\|S\|\sqrt\mu\), and \((T+B_t(A_i))^2\geq T^2+\mu\). Use the three-term phase-resolvent identity proved with bounded perturbations in the supplied source reading. For its first and third terms the norm integral is bounded by
\[
 \|S\|\sup_{\mu\geq0}\frac1\pi\int_0^\infty
 \frac{\lambda^{-1/2}\sqrt\mu}{1+\lambda+T^2+\mu}\,d\lambda
 \leq\|S\|.
 \tag{BD.13}
\]
For its middle term place the two phase numerators against their adjacent square-root resolvents, of norm at most one, and obtain the same integral. Add the three bounds. Initially this is on finite occupation cores; their completed resolvents and the uniform estimates extend it to the closed operators. For \(f\in S\), approximate \(f\circ f_0^{-1}\) on \([-1,1]\) by polynomials, with its endpoint values zero. Equation (BD.11) then proves continuity for the general \(f\)-calculus, uniformly in \(t\). No inequality in the wrong direction is used.

### 11W.5. The infinite deformation algebra

For a finite affine \(V\subset\operatorname{Dom}\Theta\), define, at \(t>0\),
\[
 \Phi_{V,t}(a\widehat\otimes f)
 =a_t\widehat\otimes
              f(T+B_t(1+t\Theta_{V^\perp})),
 \tag{BD.14}
\]
and at zero use its Bott image in \(A(H)\). Here \(a\) is a section of \(\mathcal Q(V)\), the tail weight is the compression of \(\Theta\), and the Gaussian factorization is \(\mathcal F_t=\mathcal F_{V,t}\widehat\otimes\mathcal F_{V^\perp,t}\). It is a homomorphism on that finite deformation algebra, because the two factors graded commute.

Positive-time continuity needs a separate check, since \((t-s)\Theta\) is unbounded. Use the Gaussian dilation to identify all positive Fock fibres with \(\mathcal F_1\). In a joint occupation block the tail Clifford coefficients are \(\sqrt{2n_i}\,w_i(t)\), where \(w_i(t)=\sqrt t(1+t\lambda_i)\). On \([a,b]\subset(0,1]\),
\[
 \left|\frac{d}{dt}\log w_i(t)\right|
 =\frac1{2t}+\frac{\lambda_i}{1+t\lambda_i}
 \leq\frac3{2a}.
 \tag{BD.23}
\]
Consequently the coefficient ratios tend uniformly to one as \(t-s\to0\), independently of \(i\) and of the occupations. Include the unchanged scalar \(T\) coefficient. If \(|w-v|\leq\varepsilon|v|\), with \(\varepsilon\leq1/2\), the triangle inequality and \(\big|\sqrt{1+|w|^2}-\sqrt{1+|v|^2}\big|\leq|w-v|\) give
\(\|c(w)/\sqrt{1+|w|^2}-c(v)/\sqrt{1+|v|^2}\|\leq4\varepsilon\).
Taking the supremum over the occupation blocks and \(T\) proves norm continuity of the phase. Polynomial approximation of \(f\circ f_0^{-1}\), with its zero endpoint values, proves norm continuity for every \(f\in S\). Compactness of each value follows from the weighted compact-resolvent proof. This proves positive-time norm continuity of (BD.14); it does not apply the bounded-difference estimate to an unbounded difference.

For the fixed coordinate planes, include all these sections and the ideal
\[
 J=C_0((0,1],\mathcal K(\mathcal F_t)\widehat\otimes S).
 \tag{BD.15}
\]
If \(V\subset V'\), split the tail into the added finite plane and the remaining tail. Replacing the original tail weight by zero on the added plane changes the single-particle weight by a bounded finite-rank operator. After multiplying by \(t\), (BD.11) makes the functional-calculus error tend to zero as \(t\downarrow0\). The exact finite quantum inclusion from Section 11W.4 therefore identifies the two lifts modulo \(J\). The same argument treats the off-diagonal blocks for a non-coordinate plane: \(V\subset\operatorname{Dom}\Theta\) makes both finite off-diagonal blocks bounded, by the adjoint of the finite-column map \(\Theta|_V\). Thus the finite lift images form an increasing system modulo \(J\). Their union plus \(J\) is a star algebra; its closure is \(\mathcal E_\Theta\).

The norm at zero is the correct quotient norm. The tail map \(f\mapsto f(T+B_t(1+t\Theta_{V^\perp}))\) is injective, since restriction to the even tail vacuum is \(f(T)\). Minimal tensor injectivity therefore gives
\(\|\Phi_{V,t}(\beta)\|=\|\beta_t\|\).
The finite Euclidean continuous-field proof gives the limit \(\|\beta_0\|\). All finite words reduce modulo \(J\) to a common finite stage, so their zero norms have the same limit. Uniform closure proves this for every element. Zero evaluation is onto the dense finite Bott stages and has closed range. An element with zero value has norm tending to zero and is a positive compact field, hence belongs to \(J\). This proves the exact sequence
\[
 0\longrightarrow J\longrightarrow\mathcal E_\Theta
             \xrightarrow{h_0}A(H)\longrightarrow0.
 \tag{BD.16}
\]
Separability follows from the coordinate stages and the countable compact-field ideal. Positive evaluation is onto the compact fibre by (BD.15).

For clarity, the construction is independent of using only coordinate planes. If \(V\subset\operatorname{Dom}\Theta\), approximate its origin and orthonormal basis in the graph norm of \(\Theta\) by their spectral-coordinate truncations, followed by the finite Gram inverse square root. The correcting finite-rank orthogonal rotation \(R_n\) satisfies
\(\|R_n-1\|\to0\) and \(\|\Theta(R_n-1)\|+\|(R_n-1)\Theta\|\to0\).
For an explicit rotation, put \(P=P_V\) and \(P'=P_{V_n}\), and take the orthogonal polar part of \(P'P+(1-P')(1-P)\). Its denominator is \((1-(P'-P)^2)^{1/2}\). When \(\|P'-P\|<1/2\), the inverse square-root series converges uniformly. The finite frame formulas and their graph-norm differences therefore give the two displayed graph bounds. In particular \(R_n\Theta R_n^{-1}-\Theta\to0\) in operator norm.

Here is the uniform geometric estimate needed for the finite deformation factors. Work in the common finite ambient plane, of dimension at most \(2\dim V+1\), including the origin displacement. A dense section algebra consists of smooth compactly supported tangent-coordinate kernels and their products with the added oscillator Gaussian kernels in the finite inclusion formula (22.21). Differentiate those formulas once in the finite translation and orthogonal-frame parameters. The differentiated Gaussian kernels have a common bound
\(C(1+|q|+|v|)^N\exp[-c(|q|^2+|v|^2)]\), with \(c>0\) independent of \(0\leq t\leq1\); the compactly supported kernels have a common compact majorant. Near identity, a finite frame change and translation act by \((q,v)\mapsto(R^{-1}(q-a),R^{-1}v)\), together with conjugation by the finite exterior matrix. The mean-value theorem adds at most one polynomial factor. Its row and column integrals are uniformly finite, so
\[
 \sup_{0\leq t\leq1}\|k^{R,a}_t-k_t\|_I
       \leq C_k(\|R-1\|+\|a\|),
 \tag{BD.24}
\]
where \(\|\cdot\|_I\) is the maximum of the two absolute kernel-integral bounds. The Schur estimate gives the same operator-norm control at positive time, and the zero-time Fourier estimate gives it at zero. Density and contractivity extend uniform frame continuity to every finite deformation section. This is an estimate in the frame parameter, beyond continuity in \(t\) alone.

Combine (BD.24) with the preceding graph rotation and the bounded tail-weight difference estimate (BD.11). The lifts for the approximating planes converge in the full section norm to (BD.14). This proves their membership in \(\mathcal E_\Theta\), including the zero endpoint.

The same argument is uniform for compact families of graph-continuous finite frames: their graphs have compact image, their spectral truncations converge uniformly, and the Gram inverse square roots retain a uniform lower bound. Scalar partitions on \(X\) then give sections for every such family.

### 11W.6. The action on the deformation

At positive time let
\[
 U_{g,t}(x)=T_{b_g(x)/\sqrt t}L_g(x),
 \tag{BD.17}
\]
where the real Weyl maps are the exact unitary maps of the supplied Fock proof. They satisfy the group law without a phase. On compact positive parameter intervals they are strongly continuous in base and parameter; conjugation is norm continuous on compact operators.

Conjugating (BD.14) transports its finite plane to the affine plane \(gV\) and its tail weight to the corresponding compression of \(g\Theta\). The actual transported plane is graph-continuous, by (BD.3). Section 11W.5 proves that its geometric lift belongs to the algebra. The bounded difference of the two tail weights is the compression of \(S_g\); its functional-calculus error tends uniformly to zero by (BD.11), since the deformation uses \(1+t\Theta\). Thus it lies in \(C(X)\widehat\otimes J\). This proves invariance of the entire section algebra under (BD.17), with its stated affine action at zero. Applying the inverse element proves equality. The automorphisms have the exact group law on every fibre, so jointly faithful fibre evaluation proves it in the algebra.

Set \(\mathcal E=C(X)\widehat\otimes\mathcal E_\Theta\), with this base-dependent action. We have constructed the actual equivariant extension
\[
 0\longrightarrow C(X)\widehat\otimes J
 \longrightarrow\mathcal E\xrightarrow{h_0}\mathcal B
 \longrightarrow0.
 \tag{BD.18}
\]
The kernel is equivariantly Morita equivalent, using the module with action (BD.17), to \(C_0((0,1])\widehat\otimes C(X)\widehat\otimes S\). Its scalar cone contraction is equivariant. The ordinary section exists by the nuclear lifting theorem for \(A(H)\), tensored with identity on \(C(X)\). The proper-source theorem in the companion text therefore produces a unique \(z\in KK^\Gamma(\mathcal B,\mathcal E)\) with \(z[h_0]=1\). Define
\[
 d_X=z[h_1]m_{\mathcal F_1}[\operatorname{ev}_0]
                      \in KK^\Gamma(\mathcal B,C(X)).
 \tag{BD.19}
\]
The equivariant Fock Morita module in this formula has the original exact affine action at time one. The evaluation map on \(S\) is even. No equivariant section of (BD.18) or equivariant inverse of \(h_0\) in the whole KK category has been asserted.

### 11W.7. The dual class and the ordinary product

The origin-point Bott coordinate defines an odd regular multiplier \(T_0\) of \(A(H)\), with \((1+T_0^2)^{-1}\) in the algebra. Changing the origin by \(b\) changes it by bounded Clifford multiplication of norm \(\|b\|\). The bounded-perturbation resolvent identity and its norm-integrable phase difference therefore show that \((\mathcal B,f_0(T_0))\) is an equivariant \(C(X),\mathcal B\)-cycle. Write its class as \(\eta_X\).

After forgetting equivariance, (BD.18) is the tensor product of (BD.16) by \(C(X)\). Its cone ideal has zero KK groups, so its ordinary zero evaluation is a KK-equivalence. Uniqueness implies that the ordinary image of \(z\) is its inverse. It suffices to lift the origin Bott cycle in ordinary KK.

Use at positive time the odd multiplier
\[
 \widetilde T_t=T+tD_t^3+B_t(1+t\Theta),
 \qquad \widetilde T_0=T_0.
 \tag{BD.20}
\]
Here no equivariance of this lifting cycle is required. To verify the ordinary lift, put \(A_t=T+B_t(1+t\Theta)\). Its compactly supported functional calculus is the origin-plane lift (BD.14). Its even radial cutoffs form an approximate identity of \(\mathcal E_\Theta\): they do so on every fibre, including zero, and the norm of each cutoff error is a decreasing continuous function of the compact parameter interval. Finite-stage norm continuity and uniform closure give that continuity; the elementary finite-subcover argument for monotone continuous functions makes it uniform.

The cubic phase is also norm continuous on positive intervals. On a block of total occupation \(n\), its Clifford coefficients are \(\sqrt{2n_i}\sqrt t(1+2t^2n+t\lambda_i)\), in the fixed dilation. For \(t\in[a,b]\),
\[
 \frac{d}{dt}\log\!\big(\sqrt t(1+2t^2n+t\lambda_i)\big)
 =\frac1{2t}+\frac{4tn+\lambda_i}{1+2t^2n+t\lambda_i}
 \leq\frac5{2a}.
 \tag{BD.25}
\]
The relative-vector phase bound after (BD.23) is therefore uniform in \(n,i,T\). It proves the positive-time phase continuity, and polynomial approximation proves the positive-time compact resolvent continuity used below.

On a cutoff of \(A_t^2\) below \(R\), \(D_t^2\leq R\), and the perturbation \(tD_t^3\) has norm at most \(tR^{3/2}\). All these cutoffs commute with the involved operators, because the occupation-square operators commute. The bounded phase estimate gives a localized phase error at most \(3tR^{3/2}\). It therefore is a compact field vanishing at zero after localization. This proves that the phase of (BD.20) is a multiplier of the section algebra with zero phase \(f_0(T_0)\).

The exact completed Fock square gives
\[
 \widetilde T_t^2\geq T^2+B_t(1+t\Theta)^2=A_t^2.
 \tag{BD.21}
\]
On the cutoff just used the inverse-resolvent difference tends to zero. On its complement both inverse resolvents have norm at most \((1+R)^{-1}\). Let \(R\) tend to infinity after the small-time limit. Their difference is consequently a compact field vanishing at zero. Thus \((1+\widetilde T^2)^{-1}\in\mathcal E_\Theta\). The two adjoint resolvents define the completed self-adjoint regular multiplier: on the cutoff core they have the usual inverse identities, their ranges contain the dense cutoff union, and those identities extend by the uniform resolvent bound. These checks give the ordinary lifting Kasparov cycle, rather than only a fibrewise operator family.

At time one followed by \(\operatorname{ev}_0\), its operator is precisely \(D^3+B(1+\Theta)\), the completed oscillator already proved. Its only kernel is its even vacuum, with a uniform gap on the complement. Replacing its phase on the complement by the exact sign changes it by a compact operator and leaves a degenerate ordinary cycle. The vacuum contributes \(+1\). Tensoring this argument with \(C(X)\) gives
\[
 \operatorname{For}_\Gamma(\eta_X\otimes_{\mathcal B}d_X)
                              =1_{C(X)}\quad\hbox{in }KK(C(X),C(X)).
 \tag{BD.22}
\]
This equality is deliberately the ordinary one. It needs no unproved scalar equivariant evaluation on \(X\), and no claim that the equivariant product equals its identity.


### 11W. Reading

The construction follows the infinite phase-space Bott/deformation mechanism in Tu's 1999 original, Sections 7--9, while using real stabilization to remove the varying-rank field issue, the independently completed weight and oscillator estimates, and the separately proved proper-source localization. The finite quantization input is the complete Euclidean provider already in the course. The corrected bounded-weight comparison and the identity part of translation are written explicitly. The conclusion used later is (BD.22), not the stronger equivariant gamma identity.

## 11X. A universal proper algebra with an ordinary calibrated Dirac family

**Theorem 11X.1.** Every countable discrete group has the proper nuclear algebra, equivariant Dirac and continuous universal family below, with the ordinary positive calibration (UD.16). The simplex Morita grading is shifted throughout.


Let \(\Gamma\) be countable discrete, with a proper left-invariant metric. The construction is independent of uniform embedding. Its conclusion is a proper nuclear algebra \(A\), an equivariant \(D\in KK^\Gamma(A,\mathbb C)\), and a family \(\theta\) over a universal proper base, such that every ordinary parameter evaluation satisfies \(\theta_mD=+1\). That is the weaker universal interface needed for the scalar transfer; an equivariant RKK product identity is not used.

For finite \(\Gamma\), take \(A=\mathbb C\), \(D=1\), and the constant identity family. Give \(A\) its central \(C_0(M)\)-action by evaluation at the fixed uniform probability \(m_\Gamma=|\Gamma|^{-1}\sum_g\delta_g\) in the half-mass space. This action is nondegenerate and equivariant. The one-point action is proper, and the ordinary product is \(+1\). The construction below treats infinite \(\Gamma\), so \(\ell^2(\Gamma)\) is the infinite real Hilbert space required by the deformation input. Together these cases give the stated countable scope, including the central map needed for transfer.

### 11X.1. Finite probabilities with a continuous radius coordinate

Put
\[
 Z=\{(\mu,t):\mu\text{ is a probability on }\Gamma,
                     \operatorname{diam}(\operatorname{supp}\mu)\leq t\},
 \tag{UD.1}
\]
with coordinatewise probability convergence and \(t\in[0,\infty)\). Every support here is finite by metric properness. Near a point choose \(g\) with \(\mu(g)>a>0\), and bound \(t<T\). All probabilities in this neighborhood have support in the finite ball \(B(g,T)\). Its closed smaller neighborhood is a compact finite-dimensional probability simplex times a compact interval. Thus \(Z\) is locally compact and second countable. Its action is proper by the strict half-mass compact transporter argument in the supplied proper probability-base proof: a compact set has one finite coordinate set carrying mass greater than one half, so its transporter lies in the finite set \(EE^{-1}\).

Every proper second-countable \(\Gamma\)-space \(W\) maps equivariantly to \(Z\). The square-cutoff probability \(\mu_w(g)=c(g^{-1}w)^2\) has finite support locally uniformly on \(W\). Its support diameter is locally bounded, invariant, and hence has a continuous invariant upper bound: use a locally finite quotient cover on whose inverse images the diameter is bounded by constants \(T_i\), and take \(1+\sum_i\rho_iT_i\). Every positive partition term bounds the diameter at that point, so the weighted average does as well. Pair this function with \(\mu_w\). In particular there is a map from the proper half-mass space \(M\) already constructed to \(Z\).

### 11X.2. The nearest-face algebra

Let \(H=\ell^2_{\mathbb R}(\Gamma)\), with its left permutation action. For each finite nonempty \(\sigma\subset\Gamma\), write \(H_\sigma\) for the affine span of its basis vertices and \(q_\sigma:H\to|\sigma|\) for nearest-point projection onto their probability simplex. Projection depends only on the finite affine coordinates, so is continuous in the weak topology.

For two finite subsets define the open cylinder
\[
 Y_{\sigma\tau}=\operatorname{int}\{v:q_\sigma(v)=q_\tau(v)\}.
 \tag{UD.2}
\]
Equality of the two projections is equivalent to
\(q_{\sigma\cup\tau}(v)\in|\sigma\cap\tau|\).
Indeed a nearest point satisfies \(\langle v-y,z-y\rangle\leq0\) for every point in the simplex. If it is nearest in both simplices, this inequality holds on their convex hull. Conversely a point nearest in the union and lying in the intersection is nearest in each. This also proves
\[
 Y_{\sigma\rho}\cap Y_{\rho\tau}\subset Y_{\sigma\tau}.
 \tag{UD.3}
\]
The interiors can be taken in the weak or norm topology, since the conditions involve only the finite union of the coordinates.

Use the phase-space Bott algebra \(A(H)\) from Section 11W. It has central finite-position-coordinate multipliers. Put \(\Omega_{\sigma\tau}=Y_{\sigma\tau}\cap H_{\sigma\cup\tau}\), an open subset of the finite-dimensional affine plane. The cylinder \(Y_{\sigma\tau}\) is its inverse image under affine orthogonal projection. Let
\(I_{\sigma\tau}=C_0(\Omega_{\sigma\tau})A(H)\),
with the functions acting through those finite coordinates. This notation does not put an undefined \(C_0\)-algebra on the full weak Hilbert cylinder. Define the simplex Hilbert space and ambient algebra by
\[
 \mathcal H=(\Lambda^*H_{\mathbb C}\ominus\Lambda^0H_{\mathbb C})[1],
 \qquad A_1=A(H)\widehat\otimes\mathcal K(\mathcal H),
 \tag{UD.4}
\]
and
\[
 A=\{a\in A_1:a_{\sigma\tau}\in I_{\sigma\tau}
                                      \text{ for all }\sigma,\tau\}.
 \tag{UD.5}
\]
The grading on the simplex line \(L_\sigma\) is \(|\sigma|-1\), not \(|\sigma|\). The bracket in (UD.4) records this shift. Reversing the module grading leaves the grading automorphism of its compact algebra unchanged, but changes its Morita correspondence; that correspondence will retain (UD.4).

Equation (UD.3) gives \(I_{\sigma\rho}I_{\rho\tau}\subset I_{\sigma\tau}\), and symmetry gives the adjoint condition. Finite matrix corners therefore form a star algebra. Truncating the compact matrix factor approximates every element of (UD.5), so it is exactly their closed union. This proves it is a separable C-star algebra. Its diagonal entries are all of \(A(H)\), and their approximate identities show that its faithful inclusion in \(A_1\) is nondegenerate.

Here is the nuclearity check, which cannot follow merely from inclusion in a nuclear algebra. A finite matrix corner is a central algebra over the phase cone of \(H\). At a point, the nonzero ideals \(I_{\sigma\tau}\) specify an equivalence relation by (UD.3); its fibre is the direct sum of the corresponding finite full matrix blocks over the fibre of \(A(H)\). Those coefficient fibres are nuclear quotients of \(A(H)\), so every finite-corner fibre is nuclear.

We recall the elementary central patching proof used here. For a finite set in a central \(C_0(T)\)-algebra, first localize it over a compact subset of \(T\). A nuclear fibre gives a completely positive matrix factorization of its images. Lift the second finite-matrix map by the proved finite Choi-matrix lifting lemma; compose the first map with fibre evaluation. At that point the factorization error is small. The norm of a fibre of any fixed algebra element is upper semicontinuous, so the same error remains small on a neighborhood. Cover the compact set finitely and take a scalar partition with sum at most one and equal to one on the localization. The direct sum of the first maps is contractive; the partition-weighted sum of the lifted second maps is completely positive and contractive. On the covered points its error is the weighted sum of the small errors; the localization controls the outside tail. This is finite-matrix approximation. Thus the finite corner is nuclear. Increasing-corner approximation and matrix-valued extension give nuclearity of \(A\), as in the Bott-limit proof.

The proper half-mass space \(M\) acts centrally on \(A\) by
\[
 (m_fa)_{\sigma\tau}(v)=f(q_\sigma(v))a_{\sigma\tau}(v),
                         \qquad f\in C_0(M).
 \tag{UD.6}
\]
On \(Y_{\sigma\tau}\), \(q_\sigma=q_\tau\), so this is central and preserves multiplication. It is equivariant by permutation of the vertices. On each finite matrix corner a compactly supported \(f\) equal to one on the finitely many compact simplices \(|\sigma|\) acts as identity. Hence the map is nondegenerate. The supplied properness proof for \(M\) proves that \(A\) is a proper \(\Gamma\)-algebra.

### 11X.3. An explicit nearest-face estimate

For a finite simplex, the projection of a vector \(v\) has coordinates
\[
 q_\sigma(v)_i=(v_i-a)_+,
                    \qquad\sum_{i\in\sigma}(v_i-a)_+=1.
 \tag{UD.7}
\]
When \(v\) lies in its affine span the coordinate sum of \(v\) is one, so the uniquely determined threshold is nonnegative. The threshold exists and is unique because the displayed sum is continuous, strictly decreasing until it reaches zero, and crosses one. To verify the projection formula, put \(y_i=(v_i-a)_+\). Then \((v_i-y_i)y_i=ay_i\), so \(\langle v-y,y\rangle=a\) and \(\langle v-y,e_i-y\rangle=v_i-y_i-a\leq0\). This is the nearest-point inequality.

Suppose \(F\subset\sigma\), \(v_i\leq0\) for \(i\notin F\), and some \(k\in F\) has \(v_k>1+\sqrt2\|v-w\|\). Apply (UD.7) to \(w\). For \(i\notin F\),
\[
 w_i-a\leq 1+|w_i-v_i|+|w_k-v_k|-v_k
             \leq1+\sqrt2\|v-w\|-v_k<0.
 \tag{UD.8}
\]
Thus the projection has no coordinate outside \(F\). The strict bound persists in a neighborhood, so \(w\in Y_{\sigma F}\). The constant \(\sqrt2\) comes from precisely the two tested coordinates.

### 11X.4. The family cycle

For \((\mu,t)\in Z\), put \(\xi_\mu=\sqrt\mu\in H\), and let \(Q\) remove the exterior vacuum. Define
\[
 F_\mu=Q(\varepsilon(\xi_\mu)+\varepsilon(\xi_\mu)^*)Q.
 \tag{UD.9}
\]
It is odd and self-adjoint, and
\(F_\mu^2=1-P_{\xi_\mu}\).
This follows from the Clifford square before compression; the only lost vacuum-to-one-form transition is the line \(\xi_\mu\). That line is even in (UD.4). Thus its ordinary index is \(+1\).

Set
\[
 \begin{split}
 D(g,\mu)&=\sum_h\mu(h)d(g,h),\quad
 c_g=\min(1,(D(g,\mu)-t)_+),\quad c_\sigma=\max_{g\in\sigma}c_g,\\
 w_{\sigma,g}&=6\min(2c_\sigma,1)c_g,\\
 a_\sigma&=(1-\sum_{g\in\sigma}w_{\sigma,g})\mu
                                  +\sum_{g\in\sigma}w_{\sigma,g}e_g,\\
 \chi_\sigma&=(2c_\sigma-1)_+.
 \end{split}
 \tag{UD.10}
\]
In particular \(c_g=0\) on \(\operatorname{supp}\mu\). If \(\sigma\triangle\tau\) is in that support, then \(c_\sigma=c_\tau\), \(a_\sigma=a_\tau\), and \(\chi_\sigma=\chi_\tau\). Only such pairs occur in (UD.9).

Let \(f_c(u)=\max(-1,\min(u,1))\). It is an odd bounded multiplier of \(S\); it is not an element of \(S\). Its defect \(1-f_c^2\) is in \(S\). Let \(\beta\) be the diagonal multiplier with entries \(i_{a_\sigma}(f_c)\), the origin-point Bott multiplier. Define
\[
 P_{\mu,t}=\beta+\chi(1-\beta^2)^{1/2}(1\widehat\otimes F_\mu).
 \tag{UD.11}
\]
The finite-support observations after (UD.10) give the graded anticommutation of \(\beta\) and \(F_\mu\), and the ordinary commutation of \(\chi\) with both. Hence this is an odd self-adjoint contraction.

It is a multiplier of \(A\). The only off-diagonal coefficients have a factor \((1-\beta_\sigma^2)^{1/2}\), so their position support lies in the radius-one ball about \(a_\sigma\). If \(\chi_\sigma\ne0\), a maximum coordinate outside \(\operatorname{supp}\mu\) in (UD.10) is greater than three, and all coordinates outside \(\sigma-\operatorname{supp}\mu\) are nonpositive. Equation (UD.8), with radius one and \(3>1+\sqrt2\), puts that entire support strictly in
\(Y_{\sigma\cup\operatorname{supp}\mu,\sigma-\operatorname{supp}\mu}\).
This is contained in \(Y_{\sigma\tau}\) for every coefficient connected by \(F_\mu\). Thus the required support ideal is satisfied. On a compact parameter set only finitely many \(\mu\)-coordinates occur, as in Section 11X.1, so multiplication of a fixed finite matrix by (UD.11) involves only finitely many matrix entries. Their support condition and (UD.3) put the product in \(A\). Uniform norm bounds and finite-matrix density extend this multiplier action to all of \(A\), on both sides.

The localized compact defect is checked in the ambient algebra and then in \(A\). Put
\(P'=\beta+(1-\beta^2)^{1/2}(1\widehat\otimes F_\mu)\).
The difference \(P-P'\) has only finitely many matrix entries locally in parameters: \(\chi_\sigma<1\) implies \(\sigma\subset B(\operatorname{supp}\mu,t+1)\), a finite set. These entries are in \(A(H)\), because the square-root Bott defect is compact in that algebra. Also
\(1-(P')^2=(1-\beta^2)(1\widehat\otimes P_{\xi_\mu})\)
has finite matrix support. Thus \(1-P^2\in A_1\). Since \(P\) preserves \(A\), the defect does too. Multiply it by a column-diagonal approximate identity of \(A(H)\). Its \(\sigma,\tau\) coefficient belongs to \(I_{\sigma\tau}\), and converges in norm to the defect's coefficient because that coefficient is already in \(A(H)\). Closedness gives the required coefficient ideal; finite truncation then puts the defect in \(A\).

All entries of these localized defects are norm continuous in parameters. The finite supported input/output test also gives adjoint-strong continuity of \(P\) as a module multiplier. The formulas are equivariant under vertex permutations and left metric translations. Therefore (UD.11) defines
\[
 \theta_Z\in RKK^\Gamma(Z;\mathbb C,A).
 \tag{UD.12}
\]
Composing its parameter with the map \(M\to Z\) from Section 11X.1 gives \(\theta_M\). Pullback here means the explicit family module over \(M\); compact subsets have compact image, so the localized defect proof just given applies even if the parameter map is not proper.

### 11X.5. The Dirac and its ordinary calibration

Apply the deformation construction of Section 11W to the linear Hilbert action on \(H\), using \(b=0\). Its positive kernel is an equivariant cone up to Fock Morita equivalence. Tensor the extension with \(\mathcal K(\mathcal H)\). Nuclearity and properness of the source \(A\), together with the proved proper-source theorem, give a unique lift
\[
 z_A\in KK^\Gamma(A,\mathcal E_\Theta\widehat\otimes\mathcal K(\mathcal H)),
 \qquad z_A[h_0\widehat\otimes1]=[i:A\to A_1].
 \tag{UD.13}
\]
Define
\[
 D=z_A[h_1\widehat\otimes1]
           m_{\mathcal F_1\widehat\otimes\mathcal H}
                  [\operatorname{ev}_0]\in KK^\Gamma(A,\mathbb C).
 \tag{UD.14}
\]
The Morita correspondence retains the shifted grading (UD.4).

Fix an ordinary parameter \((\mu,t)\). In the ambient algebra, remove \(\chi\) using the compact difference above. Move each origin \(a_\sigma\) linearly to \(\mu\). The anticommutation with \(F_\mu\) persists, since the origins remain equal on its connected entries. Their finite-entry multiplier tests give a strict continuous homotopy; the compact defect has the fixed finite matrix support \(P_{\xi_\mu}\). At the endpoint the operator is
\[
 i_\mu(f_c)+(1-i_\mu(f_c)^2)^{1/2}(1\widehat\otimes F_\mu).
 \tag{UD.15}
\]
On the orthogonal complement of \(\xi_\mu\), its square is exactly one, so that summand is degenerate for the scalar source. On the even line \(\xi_\mu\) it is the origin Bott cycle. Changing the clipped phase to the smooth phase changes it by an element of \(A(H)\); straight interpolation of contractions is an allowed compact perturbation homotopy. Thus after ordinary Morita equivalence, \(\theta_{\mu,t}[i]\) is exactly the origin Bott class \(\eta_\mu\), with positive sign.

The ordinary image of (UD.13) is the unique lift through the ordinary cone-kernel KK-equivalence. It consequently is the restriction of the full ordinary Hilbert deformation Dirac. The origin \(\mu\) has finite coordinate support and lies in the weight domain. Its Bott class is homotopic to the zero-origin class by the bounded origin perturbation. The completed cubic oscillator calibration in Section 11W gives
\[
 \theta_m\otimes_A\operatorname{For}_\Gamma(D)=+1
                                      \quad(m\in M).
 \tag{UD.16}
\]
This is an ordinary evaluation of a family, not evaluation at an equivariant scalar point of \(M\). No such point is assumed.


### 11X. Reading

Tu's 2004 original, printed pp. 275--281, supplies the nearest-face and exterior-family mechanism. The algebra, nuclearity, supports, finite parameter topology and shifted sign are proved above. The universal construction uses the earlier complete finite Euclidean quantization input and the newly supplied proper-source localization. It does not cite Tu's gamma-existence theorem as a proof of the required factorization.

## 11Y. The ordinary product is enough for the full compact-base transfer

**Theorem 11Y.1.** Every countable discrete group with the compact-base proper affine field constructed in Section 11O has the even proper factorization (ST.4) and ordinary product (ST.8). Thus the reduced scalar unit pairs as +1. Section 11N supplies the field for every finitely generated abstract compact-Lie subgroup and every finitely generated characteristic-zero linear group.


Use the compact convex probability base \(X\), the proper Bott algebra \(\mathcal B\), and the classes \(\eta_X,d_X\) from the Bott deformation proof. Its actual conclusion is
\[
 \operatorname{For}_\Gamma(\eta_Xd_X)=1_{C(X)}.
 \tag{ST.1}
\]
Use also the proper universal algebra \(A\), its central half-mass space \(M\), the family \(\theta_M\), and the equivariant \(D:A\to\mathbb C\). Every ordinary parameter of that family satisfies \(\theta_mD=+1\). Neither stronger equivariant product identity is a premise below.

Let \(W\) be the proper central phase cone of \(\mathcal B\). The supplied universal proper-space construction gives an equivariant map \(\psi:W\to M\). Pull back the explicit family \(\theta_M\) to \(W\), then extend its coefficients along the central action on \(\mathcal B\). This gives
\[
 T_{\mathcal B}\in KK^\Gamma(\mathcal B,\mathcal B\widehat\otimes A).
 \tag{ST.2}
\]
The module is the coefficient extension of the family module; for a compactly localized element of \(\mathcal B\), its image parameters lie in a compact subset of \(M\), so the family's compact defects extend to compact module defects. Approximation by such elements proves all cycle conditions. The central functions commute with the left source, and the exact group actions give equivariance. This explains the balanced extension rather than substituting an untyped pullback notation.

The square-cutoff probability map from the already supplied proper probability-base proof gives an equivariant \(\Phi:M\to X\). Its pullback functions act centrally on \(A\) through (UD.6). Therefore
\[
 e_A:C(X)\widehat\otimes A\longrightarrow A,
               \qquad f\widehat\otimes a\longmapsto\Phi^*(f)a
 \tag{ST.3}
\]
is an equivariant homomorphism. It is nondegenerate, since the constant one of \(C(X)\) acts as identity. Write \(u:\mathbb C\to C(X)\) for the constant homomorphism, and define
\[
 \eta=[u]\eta_X\,T_{\mathcal B}\,(d_X\widehat\otimes1_A)[e_A]
                         \in KK^\Gamma(\mathbb C,A),
 \qquad d=D\in KK^\Gamma(A,\mathbb C).
 \tag{ST.4}
\]
All factors have even degree. Their successive sources and coefficients are
\[
 \mathbb C\to C(X)\to\mathcal B
 \to\mathcal B\widehat\otimes A
 \to C(X)\widehat\otimes A\to A\to\mathbb C.
 \tag{ST.5}
\]
This supplies one separable graded proper algebra, with genuinely equivariant classes of the required types.

To compute the ordinary product, fix \(m_0\in M\) and \(x_0\in X\). After forgetting \(\Gamma\), the map \(\psi:W\to M\) contracts to the constant \(m_0\), by the continuous convex interpolation in the actual weak half-mass topology. Pulling the explicit family along that homotopy and extending coefficients as above gives
\[
 \operatorname{For}(T_{\mathcal B})
                      =1_{\mathcal B}\widehat\otimes\theta_{m_0}.
 \tag{ST.6}
\]
This homotopy is valid without properness of the map: on a compact source support, its image times the compact homotopy interval is compact, exactly the condition used for the localized defects.

Similarly \(X\) is convex in its weak probability topology, so \(\Phi:M\to X\) contracts ordinarily to \(x_0\). The homomorphisms (ST.3) consequently give
\[
 \operatorname{For}([e_A])=[\operatorname{ev}_{x_0}]\widehat\otimes1_A.
 \tag{ST.7}
\]
The even exterior-product interchange and (ST.1), (ST.6), (ST.7) now compute
\[
 \begin{split}
 \operatorname{For}(\eta d)
 &= [u]\operatorname{For}(\eta_Xd_X)
                         [\operatorname{ev}_{x_0}]
                           (\theta_{m_0}\otimes_A D)\\
 &= [u]1_{C(X)}[\operatorname{ev}_{x_0}]\cdot1_{\mathbb C}
   =1_{\mathbb C}.
 \end{split}
 \tag{ST.8}
\]
The two scalar evaluations in this computation occur only after forgetting equivariance. Before that step every factor in (ST.4) is equivariant, and neither a \(\Gamma\)-fixed probability nor an equivariant scalar evaluation has been invented.

For each finitely generated abstract subgroup of a compact Lie group, the already completed uniform-embedding proof supplies the Hilbert-field hypotheses and (BD.2). The present construction then supplies (ST.4)--(ST.8) for the full stated scope, including dense subgroups and nontrivial finite stabilizers. It makes no topological closed-subgroup assumption and no equivariant \(\gamma=1\) assertion.

The same completed uniform-embedding proof applies to every finitely generated linear group over a field of characteristic zero. Start with its given faithful finite matrix representation. Its finitely many generator coefficients lie in a finitely generated characteristic-zero field \(K\); the regular representation over a finite extension of a purely transcendental subfield is the same faithful characteristic-zero reduction used there. The coarse construction, probability base, Hilbert field and the present factorization then apply unchanged. No positive-characteristic linear-group conclusion is inferred.

Finally apply the already complete scalar proper-factorization theorem. For \(R=C_r^*\Gamma\), \(M_\Gamma=C^*_{\max}\Gamma\), and the actual proper coefficient quotient \(q_A\), define
\[
 \kappa=j_r(\eta)[q_A]^{-1}j_{\max}(D)[\epsilon]
                             \in KK(R,\mathbb C).
 \tag{ST.9}
\]
Its exact descent square and ordinary full-character evaluation give
\([u_R]\kappa=+1\).
The character is on the full group algebra; there is no reduced trivial character in the construction. Thus the unit generates a direct integral summand of \(K_0(C_r^*\Gamma)\), with the splitting induced by this KK class.


### 11Y. Reading

The compact-base transfer mechanism is due to Tu, *The Gamma Element for Groups which Admit a Uniform Embedding into Hilbert Space*, original pp.282--284. The proof above specializes it to the weaker ordinary normalization actually required by the reduced scalar-unit construction, and supplies all types and ordinary contractions explicitly. The result remains a discrete-group theorem. A coherent étale-groupoid factorization requires additional anchored kernel and coefficient data and is not asserted here.

![The exact proper factorization, ordinary contractions and even simplex kernel](../figures/proper-factorization.png)

**Figure 11Y.1.** The upper row gives the exact even equivariant sources and coefficients of (ST.4)–(ST.5). The lower left shows the ordinary contractions (ST.6)–(ST.8). The lower right is the exact sample \(\mu=(1/4,3/4)\) of (UD.9), with the shifted grading (UD.4): its kernel is even and its complement is paired by the displayed Clifford matrix. The positive sign and full coefficient chain are proved in Sections 11V–11Y. [Vector figure](../figures/proper-factorization.svg) · [Generator](../reproduction/proper-factorization/reproduce.py) · [Component terms](../reproduction/proper-factorization/COMPONENT-TERMS.md).


## 11Z. An anchored proper algebra and a continuous calibrated family

The construction keeps the original unit space, the actual arrow action and every isotropy label. It gives a separable graded continuous proper algebra, its genuinely continuous family and a whole-base ordinary product equal to the identity. The ordinary Dirac and its unit-space dual are actual classes. Section 11AA proves the separately displayed equivariant lift.

### 11Z.1. Supported finite probabilities on the actual range fibres

Throughout
\(\mathcal G\rightrightarrows T\) is second countable,
locally compact, Hausdorff and étale. The unit space is
open and closed. The Hausdorff property makes the unit
space closed; the étale property makes it open. All
arrow labels, including distinct isotropy arrows, stay
in the construction.

#### 11Z.1.1. A continuous length and its compact sublevels

**Lemma AG.1.** There is a continuous symmetric length
\(\ell:\mathcal G\to[0,\infty)\), zero exactly on units,
subadditive on composable pairs, and at least one on
nonunits. For every finite \(R\), its nonunit sublevel
\(\{g\notin T:\ell(g)\le R\}\) is compact.

**Proof.** The clopen nonunit space is second countable
and locally compact. A compact exhaustion and supported
bumps give a continuous proper nonnegative function
\(u\) there, by the complete sum construction in CL.1.
Put \(w(1_y)=0\) and
\(w(g)=1+\max(u(g),u(g^{-1}))\) on nonunits. This is
continuous, symmetric, and its nonunit sublevels are
compact. Define

\[
 \ell(g)=\inf\left\{
      \sum_{j=1}^n w(g_j):g=g_1\cdots g_n\right\}.
 \tag{AG.1}
\]

Unit factors can be omitted. A factorization of a
nonunit of cost at most \(R\) has at most
\(\lfloor R\rfloor\) nonunit factors, all in the
compact \(\{w\le R\}\). Composable tuples form closed
subsets of finite products because \(T\) is Hausdorff.
The infimum for a nonunit is consequently attained:
use factorizations of cost at most \(w(g)+1\), pass
to a fixed number of factors, and take a convergent
subsequence. Symmetry follows by reversing and
inverting a factorization; concatenation proves
subadditivity. Each nonunit has \(\ell(g)\ge1\).

For lower semicontinuity, a convergent sequence of
nonunits with bounded lengths has minimizing
factorizations in the same finite compact tuple
spaces. A convergent subsequence of those tuples
factors the limiting arrow, with cost the limit of
the costs. Thus \(\ell\) at that arrow is at most
the lower limit. Units have open neighbourhoods
consisting of units, so their continuity is immediate.

For upper semicontinuity at a nonunit, choose bisections
through the finitely many arrows in one minimizing
factorization. Their composable product is an open
bisection through \(g\). The individual factors of
nearby arrows in that product depend continuously
on the source unit. Their summed costs converge to
\(\ell(g)\) and bound \(\ell\) from above.
This proves continuity. Finally the nonunit length
sublevel is closed and is contained in the finite
union of those compact product images, with at most
\(\lfloor R\rfloor\) factors. It is compact.
\(\square\)

The full length sublevel can contain the noncompact
unit space. What is used is its compact nonunit part
and compactness after either unit map is restricted
to a compact subset of \(T\).

#### 11Z.1.2. The actual vertex and Hilbert fields

Let
\(V=\mathcal G\times\mathbb N_0\), anchored by the
range of its arrow coordinate, with left action
\(k\cdot(g,n)=(kg,n)\). For \(v=(g,n),v'=(h,m)\)
in the same range fibre define

\[
 d_y(v,v')=\ell(g^{-1}h)+|n-m|.
 \tag{AG.2}
\]

This is a metric, continuous on the fibre-pair space
and exactly left invariant. The triangle inequality
follows from the composable product
\(g^{-1}h=(g^{-1}a)(a^{-1}h)\). It separates
distinct isotropy labels. A bounded ball in a fixed
vertex fibre is finite: the length part is contained
in a compact arrow set intersected with a closed
discrete range fibre, and the integer labels have
a finite interval bound.

Let \(H_y=\ell^2_{\mathbb R}(V_y)\). Compactly
supported bisection sections, with one integer
label at a time, generate its continuous real
Hilbert \(C_0(T)\)-module \(H\). The inner product
is the range-fibre counting sum. Finite bisection
covers make these sums continuous on compact
supports, and define the completion. It is
countably generated and has the exact continuous
unitaries

\[
 L_k e_{(g,n)}=e_{(kg,n)}:
       H_{s k}\longrightarrow H_{r k}.
 \tag{AG.3}
\]

The unit vertices \((1_y,n)\) give an actual
orthonormal standard submodule \(C_0(T,\ell^2_\mathbb R)\).
Its orthogonal complement is the nonunit-vertex
module. The real operator-range stabilization
proof, applied to that countably generated complement,
therefore gives an ordinary real \(C_0(T)\)-unitary
\(H\simeq C_0(T,\ell^2_\mathbb R)\). This
trivialization is used only after forgetting the
groupoid action. The vertex labels and the action
(AG.3) remain the data of the equivariant construction.
No ordinary basis chosen in this trivialization
is asserted to be an equivariant basis.

#### 11Z.1.3. The supported finite-probability parameter

Define

\[
 Y=\{(y,\mu,t):\mu\text{ is a probability on }V_y,
            \operatorname{diam}_{d_y}\operatorname{supp}\mu\le t\},
 \qquad t\ge0.
 \tag{AG.4}
\]

Its topology is anchor, vague probability tests on
\(V\), and the radius coordinate \(t\). Every
support is finite by the finite-ball property.

**Theorem AG.2.** The space \(Y\) is second countable,
locally compact and Hausdorff. Near every parameter
its probabilities use only finitely many moving
vertex sheets, so \(\xi_\mu=\sqrt\mu\in H_y\)
is norm continuous there. The \(\mathcal G\)-action
\(k(y,\mu,t)=(rk,k\mu,t)\) is continuous and proper.

**Proof.** Fix a positive atom at a vertex
\(v_0=(g_0,n_0)\) of the chosen parameter. Choose
a relatively compact vertex bisection around it
and a nonnegative bump on that bisection whose
integral is positive. Retain a smaller neighbourhood
where this integral is bounded below and \(t<R\).
Every support vertex is within radius \(R\) of
some vertex in that fixed compact bisection support.
The difference arrow in (AG.2) lies either in the
compact nonunit length sublevel or is a unit at
the compact source image of that bisection. Products
therefore bound the arrow coordinates in one
compact set; integer coordinates lie in the finite
interval \([\max(0,n_0-R),n_0+R]\), enlarged by
one if necessary to contain integer endpoints.

Intersect that compact vertex set with the range
fibre of the base unit of the parameter. It is
finite. Choose pairwise disjoint bisections through
these finitely many vertices. After reducing the
unit neighbourhood, there are no other points of
the compact set above it: otherwise a convergent
subsequence would give a missed vertex above the
original unit. Thus all nearby probabilities have
support on these finitely many moving sheets.
The topology there is the topology of their finite
probability coordinates, base and \(t\).

Take a smaller unit neighbourhood with compact
closure inside that common sheet domain, a closed
radius interval, and a closed positive-atom test
bound. The diameter condition is closed in this
finite model: whenever both coordinates are positive,
the distance of the two continuous vertex sections
must be at most \(t\); the condition is closed even
when one of the coordinates vanishes. Consequently
this smaller closed neighbourhood is compact.
This proves local compactness. The positive-
functional countable-coordinate topology and the
anchor separate points and give a countable base,
as in GP.1. Finite square-root coordinates prove
norm continuity of \(\xi_\mu\).

On a bisection of \(\mathcal G\), left translation
sends the finitely many support sheets to finitely
many continuous sheets and preserves (AG.2).
This proves action continuity. Forgetting integer
labels and summing the finitely many local coordinates
gives a continuous anchored equivariant probability
map \(Y\to M_{\mathcal G}\). Its mass is one.
GP.2 makes the action on that target proper. For
compact input/output subsets of \(Y\), their images
bound the transporting arrows in one compact set;
the transporter on \(Y\) is closed inside its
product with the compact input set. This proves
properness. \(\square\)

#### 11Z.1.4. A map from the supported universal proper base

Apply GP.3 to its actual proper space
\(M=M_{\mathcal G}\). Let \(c\) be its properly
supported square cutoff. Define the probability
on \(V_{p(m)}\)

\[
 \mu_m(g,n)=
 \begin{cases}c(g^{-1}m)^2,&n=0,\\0,&n>0.\end{cases}
 \tag{AG.5}
\]

The normalization, continuity and exact covariance
are the GP.3 proof. The cutoff support condition
makes the vertex support locally uniformly compact;
finite bisection covers give the same local finite
sheet control as above. Its diameter is therefore
locally bounded and is invariant. Choose a locally
finite quotient cover of \(M/\mathcal G\) on whose
inverse images this diameter is bounded by finite
constants \(R_i\), and a subordinate quotient
partition \(\tau_i\). The continuous invariant
function

\[
 t(m)=1+\sum_i\tau_i([m])R_i
 \tag{AG.6}
\]

bounds the diameter, since every positive summand's
constant bounds it at that point. Consequently
\(\Psi(m)=(p(m),\mu_m,t(m))\) is a continuous
anchored equivariant map \(M\to Y\).
Compact subsets have compact image, even when
this map is not proper.

### 11Z.2. The anchored nearest-face algebra and its continuous family

The standing hypothesis is a Hausdorff,
second-countable, locally compact, étale groupoid
\(G\rightrightarrows T\). Write \(G^y=r^{-1}(y)\).
The completed geometry AG.1–AG.6 in Section 11Z.1, supported finite probabilities supplies
\(V_y=G^y\times\mathbb N_0\), its actual left-invariant metric,
the continuous real Hilbert module \(H_y=\ell^2_{\mathbb R}(V_y)\),
the supported parameter \(Y\), and the anchored equivariant map
\(\Psi:M_G\to Y\). Distinct isotropy arrows are distinct vertices.
All balanced tensor products below use the original anchor to \(T\).

The conclusions proved here are an actual separable nuclear graded
proper \(G\)-algebra \(P\) and an actual continuous family
\(\theta_M\). The equivariant Dirac uses the lifting condition in Section 11Z.5, proved in Section 11AA. The ordinary calibrated Dirac is constructed
in Section 11Z.3, the whole-base ordinary calibration; the precise equivariant
lifting assertion, now proved in Section 11AA, is in Section 11Z.5, the precise equivariant lifting condition.

#### 11Z.2.1. The Bott and simplex fields, including their signs

Let \(\mathcal B(H)\) be the phase-space Bott \(C_0(T)\)-algebra.
On a finite affine plane \(W\subset H_y\) its stage is
\(S\widehat\otimes C_0(W\times W^0,
\operatorname{End}\Lambda^*W^0_{\mathbb C})\); the odd finite
Bott multiplier is
\(\varepsilon(q+ip)+\varepsilon(q+ip)^*\).
Orthogonal affine inclusions use the finite Bott homomorphisms,
including the odd scalar \(T_S\) from \(S=C_0(\mathbb R)\).
These are the finite stages and connecting maps proved in the
ordinary analytic provider A proper Bott algebra and its equivariant deformation, BD.4–BD.6.
The position coordinate, the momentum coordinate and \(T_S\)
are separate coordinates.

There is a concrete section-algebra definition here. Use local
finite orthonormal vertex frames from bisections, the finite-stage
sections, and compactly supported unit-space scalar multiples;
complete their union in the norm \(\sup_y\|a_y\|\). Two frames
are compared after passing to their finite-dimensional common
span. The finite Gram-rotation estimate in the analytic provider
gives norm continuity of this comparison, so the definition is
independent of the frames. Equivalently, the ordinary real
trivialization in AG.3 identifies this algebra, as an ordinary
\(C_0(T)\)-algebra, with \(C_0(T)\otimes A(H_0)\), where
\(H_0=\ell^2_{\mathbb R}\). This equivalence does not trivialize
the action: it transports \(L_k:H_{sk}\to H_{rk}\) into a
continuous field of orthogonal maps. Finite-frame continuity
makes its induced Bott action point-norm continuous. Countably
many local frames and scalar bumps show separability. The finite
Bott stages, their matrix approximation and increasing-stage
approximation prove nuclearity exactly as in that ordinary provider.

Define the continuous graded simplex Hilbert module

\[
 \mathscr H=(\Lambda^*H_{\mathbb C}\ominus\Lambda^0H_{\mathbb C})[1],
 \qquad \mathcal A_1=\mathcal B(H)
                    \widehat\otimes_{C_0(T)}\mathcal K(\mathscr H).
 \tag{AF.1}
\]

For a finite nonempty \(\sigma\subset V_y\), let
\(L_\sigma\) be its exterior line. Its degree is \(|\sigma|-1\).
Local ordered vertex sheets choose a wedge basis of this line;
changing that order multiplies the basis by the permutation sign.
There is no global ordering of the vertices. Matrix coefficients
are intrinsically in
\(\mathcal B(H_y)\widehat\otimes\operatorname{Hom}(L_\tau,L_\sigma)\).
The action on these lines is the actual exterior action of \(L_k\),
so its signs and isotropy action are retained. The reversal in
(AF.1) leaves the grading automorphism of the compact algebra
unchanged but changes its Morita module. Every Morita module used
below retains precisely (AF.1).

#### 11Z.2.2. The closed matrix algebra

Let \(q_\sigma:H_y\to|\sigma|\) be metric nearest projection
onto the probability simplex of its basis vertices. It depends
only on the finite position coordinates in \(\sigma\). Put

\[
 Y_{\sigma\tau}=\operatorname{int}
       \{v:q_\sigma(v)=q_\tau(v)\}.
 \tag{AF.2}
\]

The equality holds exactly when
\(q_{\sigma\cup\tau}(v)\in|\sigma\cap\tau|\).
Indeed a nearest point \(a\) satisfies
\(\langle v-a,z-a\rangle\le0\) on its simplex. If the same
point is nearest in both simplices, the inequality holds on
their convex hull. Conversely a nearest point in that hull
which belongs to both simplices is nearest in both. Consequently

\[
 Y_{\sigma\rho}\cap Y_{\rho\tau}\subset Y_{\sigma\tau},
 \qquad Y_{\sigma\tau}=Y_{\tau\sigma}.
 \tag{AF.3}
\]

These cylinders depend on finitely many coordinates, so norm
and weak interiors give the same finite cylinder. Write
\(\Omega_{\sigma\tau}=Y_{\sigma\tau}\cap
\operatorname{Aff}(\sigma\cup\tau)\) and
\(I_{\sigma\tau}=C_0(\Omega_{\sigma\tau})\mathcal B(H_y)\),
with this function algebra acting through affine orthogonal
projection onto that finite plane. This is a well-defined closed
ideal of the Bott algebra. It is not a \(C_0\)-algebra of a
full weak Hilbert space.

**Theorem AF.1.** There is an actual continuous
\(C_0(T)\)-algebra with fibres

\[
 P_y=\{a\in\mathcal A_{1,y}:a_{\sigma\tau}
                    \in I_{\sigma\tau}\otimes
                       \operatorname{Hom}(L_\tau,L_\sigma)\}.
 \tag{AF.4}
\]

It is separable, nuclear, graded and invariant under the actual
\(G\)-action. Its faithful inclusion \(i:P\to\mathcal A_1\)
is nondegenerate.

**Proof.** First take finitely many distinct moving vertex sheets
over a relatively compact open unit set and finitely many of
their simplex lines. The ideals in (AF.4) then are fixed finite
coordinate cylinder ideals in the local Bott field; their
continuous compactly supported local sections form the required
finite corner. Equation (AF.3) gives
\(I_{\sigma\rho}I_{\rho\tau}\subset I_{\sigma\tau}\), so
matrix multiplication closes these local sections. Symmetry gives
closure under adjoint; the condition also respects the grading.
Products of two different local corners are compared on the
intersection of their unit domains, using their finite common
vertex set. The closed algebra generated by these local corners
defines \(P\) inside \(\mathcal A_1\).

Its evaluation at a unit is exactly (AF.4). Finite truncation of
the compact exterior matrix factor approximates any element of
(AF.4). All its finitely many vertices continue as distinct local
sheets through that unit, and the coefficient ideals there have
the constant finite-coordinate description above. Each
coefficient can therefore be approximated by the local sections
used to generate \(P\). Evaluation has closed range and is onto.
If a section vanishes at a unit, approximate it by finitely many
local sections and multiply by unit-space bumps vanishing there;
norm continuity makes the error arbitrarily small. The evaluation
kernel is consequently \(C_0(T\setminus\{y\})P\), not an
unidentified larger kernel. Its fibre norm is continuous because
it is the ambient Bott-section norm. This proves the asserted
continuous \(C_0(T)\)-field.

A countable bisection cover of \(V\), its finite subsets, and
countable dense coefficient stages give separability. For
nuclearity, fix a local finite corner using a finite vertex set
\(F\). Regard its coefficient algebra as a central algebra over
the local unit set times \(\operatorname{Aff}(F)\). The central
action is nondegenerate: embed a coefficient stage in a plane
containing \(F\) and use its position decay to approximate by
finite-coordinate cutoffs. At each point of this finite base,
the open ideals \(I_{\sigma\tau}\) specify an equivalence
relation on the active simplex indices by (AF.3); the fibre is
a direct sum of the corresponding full finite matrix blocks of
the nuclear coefficient fibre. It is nuclear.

Here is the central patching step, so nuclearity is not inferred
from being a subalgebra of a nuclear algebra. Localize a finite
set over a compact subset of that finite base. A nuclear fibre
has a cpc finite-matrix approximation. Lift its matrix-to-fibre
map by positive Choi-matrix lifting and compose its other map
with fibre evaluation. For each fixed algebra element the fibre
norm is upper semicontinuous, so the resulting error remains
small on a neighbourhood. Cover the compact set finitely and
use a scalar partition with sum at most one and equal to one
on the localization. The direct sum of the first maps and the
partition-weighted sum of the lifted second maps are cpc and
give the same small error. The localized outside tail is small.
Thus every finite corner is nuclear. Increasing finite-corner
approximation and cpc extension into matrix algebras give
nuclearity of \(P\).

Left translation is an isometry of the actual vertex field;
it sends \(q_\sigma\) to \(q_{k\sigma}\), hence sends its
ideals and exterior lines to the corresponding ones. Its action
is continuous on local finite corners, then on their completion.
Finally \(I_{\sigma\sigma}=\mathcal B(H_y)\). Diagonal
approximate identities in these coefficient algebras, on
increasing finite exterior corners and localized unit patches,
are a strict approximate identity of \(\mathcal A_1\) inside
\(P\). This proves nondegeneracy. \(\square\)

**Lemma AF.1a (local anchored finite-matrix model).** Each finite
moving-vertex corner on a sufficiently small unit patch is
\(C_0(U)\)-isomorphic to \(C_0(U)\otimes P_F^0\), where
\(P_F^0\) is the fixed finite-corner nearest-face algebra on
\(H_0\). In particular \(P\) itself has local, then global,
\(C_0(T)\)-linear cpc finite-matrix approximations through
finite sums of \(C_0(T)\otimes M_n\). This conclusion concerns
\(P\), not its arbitrary anchored source quotients.

**Proof.** In the ordinary trivialization, a finite moving
orthonormal vertex frame is norm continuous. Near a fixed unit
it is close to its frame there. The orthogonal polar part of
the map between the two frame projections, corrected by the
finite frame isometry, gives a norm-continuous real orthogonal
operator equal to identity off the span of those two frames.
The inverse square-root denominator converges by its series
when the projections differ by less than one. It transports
every vertex of the finite set to its fixed vertex and hence
every simplex line, with its exterior orientation, to the
fixed one. The induced Bott isomorphism transports its finite
nearest-face cylinder ideals exactly. This proves the local
constant-corner assertion.

The fixed corner is nuclear by AF.1. Tensor its cpc finite-matrix
approximations with \(C_0(U)\). To test an arbitrary finite
set in \(P\), first approximate it by finitely many such
corners with compact unit support. Compress to those corners
by the diagonal even finite-line projections in \(M(P|_U)\).
Choose local scalar input bumps equal to one on the supports
of a finite subordinate partition. They make the first maps
vanish at the patch boundary, so these extend by zero to
\(C_0(T)\otimes M_n\). Multiply the second maps by the
partition members and extend their outputs by zero into \(P\).
The direct sum first map is cpc; the sum second map is cpc
because the nonnegative scalar partition has sum at most one.
Their error is the partition-weighted small corner error,
plus the already small localization and truncation errors.
All maps are \(C_0(T)\)-linear. If graded approximations are
needed, replace each matrix factor by \(M_n\oplus M_n\)
with flip grading, send \(a\) to
\((\phi(a),\phi(\gamma_Pa))\), and send \((b,c)\) to
\((\psi(b)+\gamma_P\psi(c))/2\); approximate the finite
set together with its grading image. This gives even cpc
factorizations with the same error. It does not supply an
anchored section for a quotient such as the one in LB.1's
boundary record. \(\square\)

#### 11Z.2.3. The actual proper central action

Forget only the integer label in a probability on \(V_y\):
\(\nu(g)=\sum_{n\ge0}\mu(g,n)\). On a finite simplex this
is a finite sum and has mass one in \(G^y\). Define

\[
 \nu_\sigma(v)(g)=\sum_{n\ge0}q_\sigma(v)(g,n),\qquad
 (m_fa)_{\sigma\tau}(v)
      =f(p(v),\nu_\sigma(v))a_{\sigma\tau}(v),
                      \quad f\in C_0(M_G).
 \tag{AF.5}
\]

Here \(p(v)=y\), and the argument on the right is the actual
point \((y,\nu_\sigma(v))\in M_G\); it has mass one.
No isotropy arrow is forgotten or combined with a different
arrow. Only copies bearing different integer labels of the
same arrow are added.

**Theorem AF.2.** Formula (AF.5) is an even, nondegenerate,
anchored equivariant homomorphism
\(C_0(M_G)\to ZM(P)\). Thus \(P\) is proper at the
whole groupoid scope of the standing hypothesis.

**Proof.** On every permitted off-diagonal coefficient cylinder
the two nearest probabilities agree, so \(\nu_\sigma=\nu_\tau\).
The multiplier therefore commutes with every coefficient, respects
products and adjoints, and preserves the ideals. On the overlap
of unit patches it is the same intrinsic probability, so these
local definitions agree. The norm is at most \(\|f\|\).
Boundedness and local-corner density extend it to a central
multiplier of \(P\). The anchor in (AF.5) gives the original
\(C_0(T)\)-action by restriction of its multiplier action.
Left translation commutes with the integer-label sum and the
nearest projection, proving equivariance with the exact arrows.

For a compactly supported finite local corner, all its unit
anchors lie in a fixed compact subset of \(T\); each of its
finitely many moving simplices has compact probability image
over that subset. Their union is a compact subset of \(M_G\).
A function in \(C_c(M_G)\) equal to one on this union acts
as identity on that corner. Corner approximation proves
nondegeneracy. GP.2 proves that \(M_G\) itself is a proper
\(G\)-space, so this is the required actual proper central
structure. PN.1–PN.3 now apply to \(P\) and give its actual
graded full-to-reduced quotient isomorphism. \(\square\)

#### 11Z.2.4. The exact nearest-face bound

For any finite simplex, even when the vector is not in its
affine span, its projection is

\[
 q_\sigma(v)_j=(v_j-a)_+,\qquad
                 \sum_{j\in\sigma}(v_j-a)_+=1.
 \tag{AF.6}
\]

The threshold is uniquely determined. The function of \(a\)
is continuous, strictly decreasing while positive, and passes
through one. For the proposed point \(z\),
\((v_j-z_j)z_j=az_j\) and
\(v_j-z_j-a\le0\); summing gives the nearest-point
half-space inequalities. If \(\sum_{j\in\sigma}v_j=1\)
then \(a\ge0\), but this extra hypothesis is not used next.

If \(F\subset\sigma\), \(v_j\le0\) outside \(F\), and
some \(k\in F\) satisfies
\(v_k>1+\sqrt2\|v-w\|\), the threshold for \(w\) obeys
\(w_k-a\le1\). Hence, for \(j\notin F\),

\[
 w_j-a\le1+|w_j-v_j|+|w_k-v_k|-v_k
         \le1+\sqrt2\|w-v\|-v_k<0.
 \tag{AF.7}
\]

The strict condition persists in a neighbourhood of \(w\).
Thus \(w\in Y_{\sigma F}\). This is an exact two-coordinate
bound with the stated constant, rather than a projection diagram
standing in for the proof.

#### 11Z.2.5. A locally finite, strictly continuous cycle

For \((y,\mu,t)\in Y\), let
\(\xi_\mu=\sum_{v\in V_y}\sqrt{\mu(v)}e_v\). Its norm
is one; its support is finite. With \(Q\) removing the
exterior vacuum, define on \(\mathscr H_y\)

\[
 F_\mu=Q(\varepsilon(\xi_\mu)+\varepsilon(\xi_\mu)^*)Q,
 \qquad F_\mu^2=1-P_{\xi_\mu}.
 \tag{AF.8}
\]

The full Clifford square is one. Compression loses exactly the
vacuum-to-\(\xi_\mu\) transition, which proves the second
identity. The operator is odd and self-adjoint. Its deficient
line is even, because its degree is \(1-1=0\) in (AF.1).
All wedge permutation signs in the off-diagonal entries come
from this actual creation and contraction operator.

For \(v\in V_y\), and nonempty finite \(\sigma\subset V_y\),
put

\[
\begin{split}
 D(v,\mu)&=\sum_w\mu(w)d_y(v,w),\quad
 c_v=\min(1,(D(v,\mu)-t)_+),\quad c_\sigma=\max_{v\in\sigma}c_v,\\
 w_{\sigma,v}&=6\min(2c_\sigma,1)c_v,\\
 a_\sigma&=(1-\sum_{v\in\sigma}w_{\sigma,v})\mu
                         +\sum_{v\in\sigma}w_{\sigma,v}e_v,\\
 \chi_\sigma&=(2c_\sigma-1)_+.
\end{split}
 \tag{AF.9}
\]

The origins \(a_\sigma\) are real finite affine vectors, not
asserted to be probabilities. On \(\operatorname{supp}\mu\),
\(D(v,\mu)\le\operatorname{diam}\operatorname{supp}\mu\le t\),
so \(c_v=0\). If \(F_\mu\) connects two simplex lines, their
symmetric difference is in that support, and therefore their
\(c_\sigma,a_\sigma,\chi_\sigma\) agree.

Let \(f_c(u)=\max(-1,\min(u,1))\), an odd bounded multiplier
of \(S\), with \(1-f_c^2\in S\). Let
\(\beta_\sigma=i_{a_\sigma}(f_c)\) be the Bott multiplier
at that origin, and \(\beta\) their diagonal matrix. Set

\[
 \Pi_{\mu,t}=\beta+\chi(1-\beta^2)^{1/2}
                               (1\widehat\otimes F_\mu).
 \tag{AF.10}
\]

**Theorem AF.3.** These operators are actual odd self-adjoint
contractive multipliers of \(P_y\). They define a continuous
equivariant family cycle

\[
 \theta_Y\in KK^{G\ltimes Y}
       (C_0(Y),P_Y),\qquad
 P_Y=C_0(Y)\otimes_{C_0(T)}P.
 \tag{AF.11}
\]

Explicit substitution by \(\Psi\) gives

\[
 \theta_M\in KK^{G\ltimes M_G}(C_0(M_G),P_M),\qquad
 P_M=C_0(M_G)\otimes_{C_0(T)}P.
 \tag{AF.12}
\]

In relative notation this is
\(RKK^G(M_G;C_0(T),P)\): both source and coefficient have
been pulled back over the same original unit space.

**Proof.** Equality of the origins on the connected entries
gives graded anticommutation of \(\beta\) with
\(1\widehat\otimes F_\mu\). The diagonal \(\chi\) commutes
with both. Their square consequently has norm at most one,
so (AF.10) is the stated self-adjoint contraction.

For an off-diagonal entry with \(\chi_\sigma>0\), choose a
vertex at which \(c_\sigma>1/2\) is attained. That vertex is
outside \(\operatorname{supp}\mu\), its coordinate in
\(a_\sigma\) is greater than three, and every coordinate
outside \(F=\sigma\setminus\operatorname{supp}\mu\) is
nonpositive: the coefficient of \(\mu\) is negative because
the sum of the weights is already greater than three. The
square-root Bott defect in this entry has position support in
the radius-one ball about \(a_\sigma\). Equation (AF.7) and
\(3>1+\sqrt2\) put that support strictly in
\(Y_{\sigma\cup\operatorname{supp}\mu,F}\). All simplex
indices connected by this entry lie between \(F\) and that
union, so their projections agree there. Thus the entry lies
in the required \(I_{\sigma\tau}\). A row of \(F_\mu\)
toggles only the finitely many vertices in \(\operatorname{supp}\mu\).
Multiplication of any finite corner on either side therefore
has only finitely many new entries, and (AF.3) preserves their
ideals. The uniform norm bound and corner density extend both
actions to the entire algebra. This proves the multiplier claim.

For the compact defect first work in the ambient algebra and
replace \(\chi\) by one, obtaining \(\Pi'\). If
\(\chi_\sigma<1\), all its vertices satisfy
\(D(v,\mu)<t+1\), hence lie within \(t+1\) of some support
vertex. AG.2 gives finitely many such moving vertices on a
small parameter neighbourhood. The difference \(\Pi-\Pi'\)
therefore has finite matrix support there, with coefficient in
\(\mathcal B(H_y)\). Also

\[
 1-(\Pi')^2=(1-\beta^2)
                            (1\widehat\otimes P_{\xi_\mu})
 \tag{AF.13}
\]

has finite matrix support. Thus \(1-\Pi^2\in\mathcal A_{1,y}\).
It multiplies \(P_y\). Multiplication by a column-diagonal
coefficient approximate identity of \(P_y\) puts each of its
entries in \(I_{\sigma\tau}\); those entries converge in
norm because they already belong to \(\mathcal B(H_y)\).
The ideals are closed, and finite truncation then puts the
defect in \(P_y\) itself.

On a small parameter neighbourhood AG.2 supplies finitely many
moving support sheets and a uniform metric-ball bound. The
finite defect entries are norm continuous by (AF.9), finite
frame Bott continuity and square-root continuity. Testing
\(\Pi\) or its adjoint on any fixed finite local corner uses
only finitely many further entries, each continuous as a
multiplier. Density and the uniform contraction bound prove
strict continuity on the whole module field. The locally
compact defect multiplied by any \(f\in C_c(Y)\) is therefore
an element of \(P_Y=\mathcal K(P_Y)\); approximation extends
this to \(f\in C_0(Y)\). The source multiplication is central,
so its commutator is zero. These are the actual Kasparov cycle
conditions on the completed Hilbert \(P_Y\)-module.

The metric, nearest projections, square roots, affine origins
and exterior Clifford operator all transform by the actual
unitary \(L_k\); hence (AF.10) is exactly covariant. This
proves (AF.11). Finally \(\Psi\) is continuous and equivariant,
and sends compact subsets of \(M_G\) to compact subsets of
\(Y\). Substituting its parameter in the preceding module and
defect proof proves (AF.12) directly, including compactness
after multiplication by \(C_0(M_G)\). Properness of \(\Psi\)
is not required. \(\square\)

#### 11Z.2.6. Reading and proof inputs

The finite nearest-face, clipped Bott and shifted exterior
mechanism is the one proved in the discrete construction in A universal proper algebra with an ordinary calibrated Dirac family, UD.2–UD.15, following
Jean-Louis Tu, [The Gamma Element for Groups which Admit a Uniform
Embedding into Hilbert Space](https://link.springer.com/chapter/10.1007/3-7643-7314-8_18)
(2004), printed pp. 275–281. The admitted original-page scan
and its independent completed proof are bound in the provider
record; no primary-source expression is selected here. The present proof supplies
the actual groupoid vertex field, moving-sheet topology,
orientation-line action, anchored section algebra and completed
family; it does not apply the discrete theorem to an invented
global group. The finite Bott and oscillator foundations are
the already checked ordinary providers BD.4–BD.6, BD.10,
BD.11–BD.16 and BD.20–BD.25. The equivariant Dirac lifting condition is stated in Section 11Z.5, the precise equivariant lifting condition.

### 11Z.3. Whole-base ordinary calibration with the original anchor

Use the standing hypothesis and
the actual \(G\)-algebra \(P\), ambient algebra \(\mathcal A_1\),
and family of Section 11Z.2, the anchored algebra and family. Forgetting \(G\) here retains
the original \(C_0(T)\)-linearity. Write \(KK_T\) for this
ordinary anchored category and \(KK_Y\) for its balanced
base change to \(Y\). All classes below are even.

#### 11Z.3.1. An actual ordinary anchored Dirac

Fix the real ordinary trivialization
\(J:H\simeq C_0(T,H_0)\) proved in AG.3. It induces the
ordinary anchored Bott isomorphism
\(J_B:\mathcal B(H)\to C_0(T)\otimes A(H_0)\).
Let
\(d_\infty\in KK(A(H_0),\mathbb C)\) be the ordinary
infinite Hilbert Bott Dirac constructed by the completed
deformation and cubic oscillator in BD.16 and BD.20–BD.25.
Its actual zero-evaluation inverse exists because that ordinary
extension is semisplit with a contractible cone kernel. The
proved even oscillator vacuum identity is
\(\eta_0d_\infty=+1\). This ordinary class does not use the
discrete equivariant proper-source theorem.

The full shifted simplex module in (AF.1) gives the even Morita
class
\(m_{\mathscr H}\in KK_T(\mathcal A_1,\mathcal B(H))\).
It is full because every fibre contains its actual unit-vertex
one-form line. Its source compact algebra has the grading from
that shifted module. Define the actual class

\[
 D_0=[i]\otimes_{\mathcal A_1}m_{\mathscr H}
      \otimes_{\mathcal B(H)}[J_B]
      \otimes_{C_0(T)\otimes A(H_0)}
                          (1_{C_0(T)}\otimes d_\infty)
                      \in KK_T(P,C_0(T)).
 \tag{OC.1}
\]

Every domain and codomain in (OC.1) is over the same original
unit space. There is no evaluation of \(C_0(M_G)\) to a scalar
or equivariant choice of a point in \(M_G\).

#### 11Z.3.2. The continuous family identity

Balanced pullback along a continuous anchor \(p:W\to T\)
is used on source and coefficient simultaneously. Its class
construction does not assert that
\(C_0(T)\to M(C_0(W))\) defines a \(KK(C_0(T),C_0(W))\)
class for a nonproper \(p\). For completeness, a pulled-back
cycle is localized by the new source \(C_0(W)\): on the compact
support of \(h\in C_c(W)\), take \(f_T\in C_c(T)\) equal
to one on the compact anchor image. The old compact defect
localized by \(f_T\) pulls back to finite-rank approximants;
multiplying their vectors by suitable square roots of scalar
bumps supported near that compact set gives compact operators
on the new module. This verifies the new localized defect.
Partitions and approximation prove the same for all source
elements. Tensor-product connections commute with this
construction, so balanced pullback preserves the products
used below.

**Theorem OC.1.** For the actual family in (AF.11) one has

\[
 \operatorname{For}_G(\theta_Y)
                \otimes_{P_Y}p_Y^*D_0=1_{C_0(Y)}
                                      \quad\text{in }KK_Y.
 \tag{OC.2}
\]

The same whole-base equality holds for the actual family on
\(M_G\):

\[
 \operatorname{For}_G(\theta_M)
                \otimes_{P_M}p_M^*D_0=1_{C_0(M_G)}
                                      \quad\text{in }KK_{M_G}.
 \tag{OC.3}
\]

**Proof.** Apply the ambient inclusion to (AF.10). Its compact
difference from \(\Pi'\), proved locally in AF.3, is compact
after every \(C_0(Y)\) localization; linear interpolation
removes \(\chi\) on the entire family. All intermediate
operators are self-adjoint contractions, their difference from
\(\Pi'\) is locally compact, and their cycle defects are
therefore locally compact.

Next move every origin by
\(a_{\sigma,s}=(1-s)a_\sigma+s\mu\), \(0\le s\le1\).
On every pair connected by \(F_\mu\) the origins stay equal,
so graded anticommutation persists. Testing on any fixed finite
local corner uses finitely many origin functions continuous in
\((y,\mu,t,s)\). It gives a strict continuous family over
the full homotopy base. Its compact defect is (AF.13), with
the locally fixed finite support of \(P_{\xi_\mu}\).
Thus this is an actual ordinary \(C_0(Y)\)-linear homotopy,
not a collection of independent fibre homotopies.

At its endpoint the ambient operator is

\[
 i_\mu(f_c)+(1-i_\mu(f_c)^2)^{1/2}
                                      (1\widehat\otimes F_\mu).
 \tag{OC.4}
\]

The line \(\mathbb C\xi_\mu\) is a continuous global even
line field: (AG.4)'s locally finite sheets give norm continuity
and \(\xi_\mu\) is a globally defined unit section. Its
orthogonal complement has \(F_\mu^2=1\); on that complement
(OC.4) squares exactly to one and commutes exactly with the
source. That summand is an exactly degenerate cycle. On the
even line it is the origin Bott cycle with positive sign.
Applying the shifted Morita class in (OC.1) therefore gives
the origin Bott class \(\eta_\mu\) over the whole base \(Y\).

Move that origin by \((1-s)\mu\) to zero in the Hilbert
field. Each \(\mu\) has finite support and \(\|\mu\|_2\le1\);
locally its finite coordinates vary continuously. The finite
Bott bounded-translation estimate gives a strict continuous
homotopy with locally compact defects. Under \(J_B\) its
endpoint is the balanced base change of
\(1_{C_0(T)}\otimes\eta_0\). Changing the clipped phase
to the smooth phase is a locally compact perturbation, by the
finite Bott defect calculus. The actual completed ordinary
oscillator identity \(\eta_0d_\infty=+1\), tensored with
\(C_0(Y)\), now proves (OC.2). Explicit substitution by
\(\Psi:M_G\to Y\), or the balanced localization argument
above, proves (OC.3). \(\square\)

#### 11Z.3.3. Original-unit and local consequences

There is an ordinary anchored section
\(s_0:T\to Y\),
\(s_0(y)=(y,\delta_{(1_y,0)},0)\). The unit vertex is a
continuous sheet and its diameter is zero. This section need
not be equivariant. Pulling the actual family back on source
and coefficient gives an actual even ordinary anchored class

\[
 \eta_0^T=s_0^*\operatorname{For}_G(\theta_Y)
                \in KK_T(C_0(T),P),\qquad
 \eta_0^T\otimes_PD_0=1_{C_0(T)}.
 \tag{OC.5}
\]

After forgetting the anchor as well, every original local
transverse class \(b_U\in KK(\mathbb C,C_0(T))\) consequently
satisfies

\[
 b_U\otimes_{C_0(T)}(\eta_0^T\otimes_PD_0)=b_U.
 \tag{OC.6}
\]

These are whole-class identities, stronger than a list of
positive point indices. They prove the precise ordinary and
local normalization for the pair actually constructed here.
They do not assert that \(\eta_0^T\) is equivariant. If a
later actual equivariant Dirac \(D\) has
\(\operatorname{For}_G(D)=D_0\), (OC.2)–(OC.6) remain valid
with its ordinary image. To use Theorem 11R.2 with an actual
equivariant dual class \(\eta\), its ordinary product or its
specified local \(b_U\)-product must still be shown to equal
the one above. Neither a section of a proper base nor a fibre
scalar \(\kappa\) supplies that equivariant class or comparison.

### 11Z.4. The actual anchored groupoid deformation interface

Its standing hypotheses and
notation are those of AF.1–AF.13. It supplies the actual
equivariant extension with an anchored ordinary cpc section needed by
the localization condition in Section 11Z.5. Section 11AA supplies that bridge. The analytic finite quantization,
completed Fock operators, resolvent bounds and ordinary Bott
calibration are the already checked providers BD.10–BD.25.
The proof below checks the groupoid and noncompact-unit
prerequisites for using those analytic constructions.

#### 11Z.4.1. A weight at the actual continuous arrow scope

Use the real ordinary trivialization \(J:H\simeq C_0(T,H_0)\)
from AG.3. Set
\(U_k=J_{rk}L_kJ_{sk}^{-1}\). It is a strongly continuous
field of real orthogonal operators on \(H_0\), with the exact
composition rule on composable pairs. It is not asserted to
be an action of a global discrete group. Let \(K_j\) be an
increasing compact exhaustion whose interiors cover the arrow space. Choose
countably many compactly supported local vertex sections
\(\zeta_j\) generating \(H\). Their images under \(J\)
are continuous functions with compact Hilbert-space image.

**Lemma DI.1.** There is a positive diagonal self-adjoint
\(\Theta\ge1\) on \(H_0\), with compact inverse, such that

\[
 U_k\Theta U_k^{-1}-\Theta=S_k\in\mathcal K(H_0),
 \tag{DI.1}
\]

where \(S_k\) is norm continuous on the arrow space and
bounded on each compact subset. Each \(J\zeta_j\) is a
continuous section of \(\operatorname{Dom}\Theta\) in the
graph norm. The domain is transported exactly by \(U_k\).

**Proof.** Let \(P_n\) be the increasing real finite-coordinate
projections on \(H_0\). For any compact arrow set \(K\),
both fields \(P_n\) and \(U_kP_nU_k^{-1}\) are bounded
approximate identities of \(C(K,\mathcal K(H_0))\). For the
second statement conjugate a continuous compact-operator field
back by \(U_k\); it is again norm continuous. Its compact
image is uniformly approximated by the first coordinate
projections, proving strict convergence on the whole compact
set. Norm continuity of the conjugation on compact operators
follows by finite-rank approximation and strong continuity
of \(U_k\) and \(U_k^{-1}\).

The difference of these two approximate identities tends weakly
to zero in that Banach space. Indeed a positive bounded
functional takes either approximate identity to its norm;
this follows from the strong convergence in its GNS
representation. A bounded functional is a linear combination
of positive functionals, so their difference has limit zero
against every functional. The elementary Hahn–Banach separation
proof of Mazur's lemma now puts zero in the norm closure of
the convex hull of every tail of these differences.

Inductively choose finite convex combinations \(v_n\) of
coordinate projections, with all projection indices beyond
the last used at step \(n-1\), so that

\[
 \sup_{k\in K_n}\|U_kv_nU_k^{-1}-v_n\|<2^{-n},\qquad
 \sup_y\|(1-v_n)J\zeta_j(y)\|<2^{-n}\quad(j\le n).
 \tag{DI.2}
\]

The second bound is achieved by first going far enough out
the projection tail; compactness of each section's Hilbert
image makes it uniform, and every convex combination in that
tail retains it. Mazur's lemma then achieves the first bound.
The combinations are real diagonal positive contractions;
\(v_n\ge v_{n-1}\), and each fixed coordinate is exactly one
eventually. Define

\[
 \Theta=1+\sum_{n\ge1}(1-v_n).
 \tag{DI.3}
\]

Each diagonal coordinate has a finite sum. Outside the union
of the first \(N\) finite supports its eigenvalue is at least
\(N+1\), so its inverse is compact. Equation (DI.2) makes
the series for \(\Theta J\zeta_j\) converge uniformly for
each fixed \(j\); each finite sum is continuous. This proves
graph-norm continuity. The series
\(S_k=\sum_n(v_n-U_kv_nU_k^{-1})\) converges in operator
norm uniformly on each compact arrow set, because all
sufficiently large \(n\) satisfy (DI.2) there. Every term
is a continuous compact-operator field, proving the claimed
compactness and norm continuity.

To verify the unbounded identity, take the increasing bounded
partial sums in (DI.3). Their conjugates differ from them by
partial sums converging in norm to \(S_k\). The closed
positive quadratic forms consequently have the same form
domain up to this bounded perturbation, and their associated
self-adjoint operators differ by the same bounded operator.
The associated operator domains agree. Thus
\(U_k\operatorname{Dom}\Theta=\operatorname{Dom}\Theta\)
and (DI.1) is an identity of completed operators. Dividing a
generator section by its positive scalar bump on a smaller
patch gives the same graph-continuity assertion for every
local finite vertex frame used in AF.1. \(\square\)

#### 11Z.4.2. The actual section algebra and its action

Let \(\mathcal E_\Theta\) be the ordinary infinite deformation
algebra on \(H_0\), generated by the finite-plane lifts and
the positive compact-field ideal, as explicitly constructed
in BD.14–BD.16. Write

\[
 0\to J_\Theta=C_0((0,1],\mathcal K(\mathcal F_t)
                                      \widehat\otimes S)
   \to\mathcal E_\Theta\xrightarrow{h_0}A(H_0)\to0.
 \tag{DI.4}
\]

The ordinary analytic proof gives an exact quotient, not merely
correct fibre indices: finite tail-vacuum restriction proves
the zero quotient norm, the finite-stage lift norm tends to
its zero norm, and an element with zero value is precisely
a positive compact field vanishing at zero. In particular the
kernel in (DI.4) is the stated completed cone ideal.

Set \(\mathcal E_H=C_0(T)\otimes\mathcal E_\Theta\), identifying
its zero quotient with \(\mathcal B(H)\) through \(J_B\).
At \(t>0\) use the actual real Fock unitary induced by \(U_k\)
between source and range fibres; at zero use its Bott functor
action. The orthogonal action has no Weyl translation or phase.

**Theorem DI.2.** These are a continuous exact groupoid action
on \(\mathcal E_H\), and its zero evaluation is an actual
anchored equivariant extension

\[
 0\to C_0(T)\otimes J_\Theta\to\mathcal E_H
              \xrightarrow{h_{0,T}}\mathcal B(H)\to0.
 \tag{DI.5}
\]

**Proof.** A finite-plane lift has positive-time tail weight
\(1+t\Theta\). Conjugating by \(U_k\) transports its finite
frame and replaces its tail weight by the corresponding
compression of \(1+tU_k\Theta U_k^{-1}\). Its difference
from the geometric lift using \(1+t\Theta\) is bounded by
the compression of \(tS_k\). The actual completed phase
estimate BD.11 is
\(\|f_0(T_S+B_t(A_1))-f_0(T_S+B_t(A_2))\|
\le3\|A_1-A_2\|\), following from the corrected bound
\(B_t(S)^2\le\|S\|^2D_t^2\). Hence this tail error tends
to zero uniformly for arrows in a compact set. Polynomial
approximation in the smooth phase transfers the same statement
to every \(f\in S\). At positive time it is a continuous
compact-operator field. It therefore belongs to the positive
ideal after unit localization.

The transported finite frame is graph continuous by DI.1.
The graph-truncation and finite Gram-rotation estimates BD.24
show that its geometric lift belongs to the same ordinary
section algebra, uniformly on compact families. Their proof
uses the Gaussian kernel Schur bounds in the finite common
ambient plane and applies equally to a compact family of
arrows, with source and range explicitly retained. Compact
unit support reduces every section test to such compact
families. Finite local partitions therefore give membership
of the transported sections in \(\mathcal E_H\).

The same finite-frame estimate and the norm-continuous compact
field \(S_k\) prove continuity as the arrow varies, in the
full section norm over \(0\le t\le1\). Positive-time
continuity in \(t\) itself is the separate relative-weight
estimate BD.23; no bounded-difference estimate is applied to
the unbounded difference \((t-s)\Theta\). Inversion gives
invariance onto the whole algebra. Exact Fock composition at
positive time and exact Bott composition at zero give the
groupoid composition law on all fibres. Fibre evaluation is
jointly faithful, so that law holds on the completed algebra.
The anchor acts by the same scalar on every fibre and is
preserved. The exactness of the ordinary tensor extension
and the kernel action just proved establish (DI.5).
\(\square\)

#### 11Z.4.3. The supplied section and kernel for the bridge

The ordinary nuclear Bott algebra \(A(H_0)\) has a cpc
section \(s_0\) of (DI.4) by the proved ordinary nuclear
lifting provider. Choose that section once. Averaging it
with the two grading automorphisms replaces it by the actual
graded cpc section
\((s_0+\gamma_Es_0\gamma_B)/2\), which still splits \(h_0\).
Use this graded choice in the formulas below. Unlike a source
filtration quotient, this anchored coefficient extension has
a section for a specific reason: it is the ordinary constant
tensor extension under \(J\).

Tensor (DI.5) with \(\mathcal K(\mathscr H)\) over \(C_0(T)\).
Let

\[
 E_1=\mathcal E_H\widehat\otimes_{C_0(T)}\mathcal K(\mathscr H),
 \quad Q=\mathcal A_1,\quad
 q=h_{0,T}\widehat\otimes1:E_1\longrightarrow Q.
 \tag{DI.6}
\]

The exact quotient is unchanged by this nuclear compact-module
tensor factor. Under the ordinary real trivialization and its
shifted exterior trivialization it is exactly
\(1_{C_0(T)}\otimes h_0\otimes1_{\mathcal K(\mathscr H_0)}\).
Consequently

\[
 s_q=1_{C_0(T)}\otimes s_0\otimes
                          1_{\mathcal K(\mathscr H_0)}
 \tag{DI.7}
\]

transported back by those ordinary trivializations is an actual
graded \(C_0(T)\)-linear cpc section of \(q\). It is not
asserted to be equivariant.

The completed kernel \(I_1=\ker q\) has the exact groupoid
Fock and shifted exterior action. The full module
\(\mathcal F_t\widehat\otimes\mathscr H\), with its actual
unitary action, gives the equivariant graded Morita equivalence

\[
 I_1\sim_M^G C_0((0,1])\widehat\otimes C_0(T)
                                      \widehat\otimes S.
 \tag{DI.8}
\]

The groupoid acts only on the original \(T\) anchor on the
right; the interval and \(S\) actions are trivial. There is
an explicit anchored equivariant cone contraction
\(H_u(f)(t)=f(ut)\), \(0\le u\le1\), interpreting
\(f(0)=0\). It starts at zero and ends at identity, is
point-norm continuous, and commutes with the groupoid and
the grading. Thus the kernel is genuinely equivariantly
contractible up to this Morita module.

At positive endpoint, evaluation and the same shifted module
give the actual equivariant classes

\[
 [h_1]:E_1\to\mathcal K_{C_0(T)}(\mathcal F_1)
                 \widehat\otimes S
                 \widehat\otimes_{C_0(T)}\mathcal K(\mathscr H),
 \quad m_{\mathcal F_1\widehat\otimes\mathscr H},\quad
 [\operatorname{ev}_0]:C_0(T)\otimes S\to C_0(T).
 \tag{DI.9}
\]

These interfaces, together with the actual proper continuous
source \(P\) and its inclusion \(i:P\to Q\), are precisely
what the localization condition in Section 11Z.5 requires. The section (DI.7) is supplied. No extra section
for arbitrary source-filtration quotients is supplied or
assumed.

### 11Z.5. Exact equivariant lifting boundary

The actual objects and interfaces proved in Sections 11Z.1–11Z.4 are:

- AG.1–AG.6: genuine range-fibre vertex metric and Hilbert
  field, supported parameter \(Y\), and equivariant
  \(\Psi:M_G\to Y\).
- AF.1–AF.13: genuine continuous nuclear graded proper
  \(G\)-algebra \(P\), its nondegenerate inclusion
  \(i:P\to Q=\mathcal B(H)\otimes_T\mathcal K(\mathscr H)\),
  and the actual continuous family \(\theta_M\).
- OC.1–OC.6: actual ordinary anchored
  \(D_0\in KK_T(P,C_0(T))\), the whole-base family identity,
  and an ordinary anchored dual with product
  \(1_{C_0(T)}\).
- DI.1–DI.9: actual groupoid deformation
  \(q:E_1\to Q\), its actual graded anchored cpc section,
  equivariant cone-kernel Morita module, and endpoint classes.

Section 11AA.8 proves the exact lifting assertion

\[
 q_*:KK^G(P,E_1)\longrightarrow KK^G(P,Q)
             \text{ is bijective.}
 \tag{LB.1}
\]

It must be proved in the original anchored graded groupoid
category. The supplied ordinary section of \(q\) is
\(C_0(T)\)-linear; that premise is not missing. What fails
in copying the discrete filtration proof is a newly introduced
source-quotient premise. Ordinary nuclearity does not imply
that source quotients have anchored cpc sections. For example
take the unit groupoid \(G=T=[0,1]\), a proper groupoid, and
the nuclear source quotient
\(C([0,1])\to\mathbb C_{\{0\}}\). If an anchored section
sent \(1\) to \(a\), then every function vanishing at zero
would annihilate \(a\). The continuous function \(a\) would
be supported on the singleton \(\{0\}\), whose interior is
empty; hence \(a=0\), contradicting evaluation at zero.
This invalidates that filtration argument, not (DI.7).

Restriction of
(DI.7) to an actual open finite-action subgroupoid
\(F\ltimes U\), followed by the finite group average, gives
a local anchored equivariant semisplitting. Its mapping cone
is locally KK-contractible. Sections 11AA.4–11AA.8 supply the actual anchored reciprocity, open-cover exactness and source telescope with its full Milnor term, and thereby prove (LB.1) for the full proper source.

By (LB.1), now proved in Section 11AA.8, there is a unique actual

\[
 z_P\in KK^G(P,E_1),\qquad z_P\otimes_{E_1}[q]=[i],
 \tag{LB.2}
\]

and the consequent actual original-unit Dirac is

\[
 D=z_P\otimes[h_1]\otimes
      m_{\mathcal F_1\widehat\otimes\mathscr H}
            \otimes[\operatorname{ev}_0]
                   \in KK^G(P,C_0(T)).
 \tag{LB.3}
\]

Equations (LB.2) and (LB.3) define the actual classes proved in Section 11AA.8. After
forgetting \(G\), the anchored quotient in DI.6 is the
constant tensor of the ordinary cone-kernel KK-equivalence.
Its inverse is the tensor of that ordinary inverse. The
actual lift in (LB.2) has ordinary image fixed by this
inverse, and its ordinary endpoint class is precisely OC.1's
\(D_0\). The proved whole-base OC.2–OC.6 identities
therefore give the correct positive ordinary/local products
for that actual lift.

Even after this lift is supplied, Sections 11Z.1–11Z.4 do not
construct the compact anchored coarse-kernel/Bott route that
produces the equivariant dual class needed for the general
original-unit transfer. In particular it does not produce a
scalar equivariant evaluation from the proper base, identify
the actual descended local class with a chosen scalar
\(\kappa\), or close the entire FOL055 graph-Dirac theorem.

![Actual arrow transport, the exact nearest-face bound and the whole-base ordinary calibrated family](../figures/anchored-nearest-face-family.png)

**Figure 11Z.1.** Left translation retains the actual source/range and isotropy labels in (AG.2)–(AG.3). The finite-plane panel is an orthogonal isometric view of the exact test in (AF.7), with squared distance 0.0608 and a radius-one disk. The lower row gives (OC.3), with both algebras balanced over the original T and the even deficient line from (AF.8). Sections 11Z.1–11Z.5 give the full proofs and the precise equivariant lifting condition, proved in Section 11AA. [Vector figure](../figures/anchored-nearest-face-family.svg) · [Generator](../reproduction/anchored-universal-family/draw_anchored_universal.py) · [Finite checks](../reproduction/anchored-universal-family/self_check.py) · [Reproduction instructions](../reproduction/anchored-universal-family/REPRODUCTION.md) · [Component terms](../reproduction/anchored-universal-family/COMPONENT-TERMS.md).


## 11AA. Original-anchor localization and the equivariant Dirac lift

The actual nearest-face source now admits the required lift through its original deformation quotient. The proof retains C0(T)-linearity, ineffective isotropy, both Clifford degrees and the Milnor term in the countable passage. The endpoint defines an actual equivariant Dirac class whose whole-base ordinary image is D0.

### 11AA.1. Anchored étale technical theorem, products and the exactness actually used

Put R=C0(T), where T is the unit space of
the Hausdorff second-countable locally compact étale groupoid G.
Algebras are separable graded R-algebras, modules are countably generated,
and every representation and map is R-linear and even. The groupoid
action on a module is on its actual coefficient fibres, not on an
unanchored module substituted for it. An arrow defect means a continuous
compact-operator section on G after multiplication by a source section;
its restriction to a compact arrow set has the section sup norm.

The ordinary proofs being extended are the existing complete local
proofs in Kasparov’s technical theorem §§1–2 (technical theorem), Connections and the existence of the Kasparov product §§1–5
(connections, products, essential replacement), Homotopy, associativity, the index pairing and KK-equivalence (homotopies
and associativity), and Exact sequences in KK and the universal coefficient theorem §§1–5 (Puppe, excision, additivity).
Below we give the additional anchored and arrow arguments at every
interface. Their scalar operator equations and Clifford conventions
are retained. No theorem about a nuclear source quotient is used.

#### 11AA.1. KG.1. Local transport of operator data

If B is an open bisection, transport by B is a unitary identification
between the restrictions of a G-Hilbert module to s(B) and r(B), with
the coefficients transported by B. For X in its adjointable algebra
and χ in Cc(r(B)), define χ X_B on r(B) by this identification, and
extend it by zero. It is a bounded adjointable operator on the original
module: χ makes both the operator and its adjoint vanish at the boundary,
and its norm is at most ||χ|| ||X||. It commutes with R. For compact X
it is compact, first by transporting rank-one operators and then by
norm approximation. These assertions also apply between two modules.

Take a countable basis of bisections closed under inverse and product,
and countable dense sets of compact unit bumps in their range domains.
Closing a separable R-subalgebra of adjointable operators under these
local transports gives a separable R-algebra with an actual G-action
whenever the transported operators satisfy the same defining algebra
relations. Here R acts nondegenerately: use compact unit bumps when
adjoining an operator that is only a multiplier. On a bisection, the
action is exactly the just-defined isomorphism of the two restricted
R-algebras. It therefore sends continuous fibre sections to continuous
fibre sections; these local identifications agree on overlaps and their
products agree by the module action identity. This proves the bundle
continuity and cocycle law. It does not presume that the norm of a
strictly continuous operator field is continuous or identify its
R-algebra fibre with an operator fibre when that identification fails.
The map to operators on the actual module is used only after the
R-algebra and its local action have been constructed.

#### 11AA.1. KG.2. Convex approximate invariance on compact arrow sets

Let J be a sigma-unital G-algebra. Choose a nested even functional-
calculus approximate identity e_j as in Kasparov’s technical theorem Lemma 1.2, with
e_j e_i=e_i for sufficiently later j. For compact K⊂G, the sections

    g ↦ e_j(r(g)),     g ↦ α_g(e_j(s(g)))

are positive contractive approximate identities of the C*-algebra of
continuous sections of r*J over K. To check this assertion uniformly,
cover K by finitely many bisections and unit cutoffs. A compact section
on such a chart, pulled back to units or transported back to source
units, is locally a J-section. Extend it with the unit cutoff and use
the approximate-identity property in J. The finite cover proves both
left and right uniform convergence on K. The same argument works for
upper semicontinuous C*-bundles; it uses sections and finite charts,
not a common fibre trivialization.

In a representation of this section algebra both approximate identities
converge strongly to the same support projection. The bounded-functional
vector-coefficient argument of Kasparov’s technical theorem Theorem 1.1 consequently makes
their difference weakly zero in its full Banach dual. Combine this
linear difference test with each finite ordinary derivating-multiplier
commutator tuple. Hahn–Banach separation in the finite direct sum gives
a *single* convex combination of any sufficiently late common tail
such that

    sup(g∈K)||α_g(v(s(g)))−v(r(g))|| < ε,                 (KG.1)

and all the ordinary commutator and approximate-identity tests hold.
Compact test families reduce to finite nets. Choosing later indices
beyond the preceding finite functional-calculus support retains the
nested product equation. Averaging the grading conjugates retains all
these bounds and yields even v. This proves the anchored étale version
of the convex approximate-invariance lemma, rather than substituting
the group argument C(K,J) for the pulled-back bundle algebra.

#### 11AA.1. KG.3. The invariant separation theorem

Let J and A1,A2⊂M(J) be sigma-unital graded R-algebras with genuine
local G-actions as in KG.1, with A1 A2⊂J. Let the separable tested
multiplier space Δ derive A1. Include its adjoints and the localized
translates needed below. They commute with R. Apply the complete
strict-series construction of Kasparov’s technical theorem Theorem 2.1:

    b_n=(v_n−v_(n−1))^(1/2),
    N=Σ_(n≥1) b_n u_n b_n  (bounded strict sum),   M=1−N. (KG.2)

Here u_n∈A1 and v_n∈J retain exactly its ordinary tests, including
||u_n x−x||<2^−n, its prospective commutator tolerances and its
annihilation tests on u_j X2 for j≤n+1. KG.2 permits the extra tests,
on a compact exhaustion K_n of G,

    sup_Kn||α_g(u_n)−u_n||<2^−n,
    sup_Kn||α_g(b_n)−b_n||<2^−n.                       (KG.3)

The second follows by uniform polynomial approximation to square root:
choose the invariance tolerance of v_n prospectively small enough for
the square roots at both steps n and n+1. This is compatible with the
identical prospective commutator selection in the ordinary proof.
Its inequalities (2.7)–(2.12) prove strict convergence, MA1⊂J,
NA2⊂J and [M,Δ]⊂J with no change, since the identities take place in
the original anchored multiplier algebra.

On K_n the arrow difference of b_n u_n b_n has norm ≤3·2^−n. On each
fixed compact K_m the difference series is therefore uniformly norm
convergent after finitely many terms. Each initial term is an actual
continuous r*J-section by KG.1. Thus

    α_g(M(s(g)))−M(r(g)) ∈ J_(r(g))                   (KG.4)

is a continuous compact-arrow section; bounded strict continuity
identifies the difference series with the multiplier difference.
All multipliers remain R-linear. For separable data the construction
already has countably many tests. If a sigma-unital separation algebra
is larger, the ordinary separable reduction is closed additionally
under the countable localized transports in KG.1, retaining its
strictly positive elements. Its nondegenerate multiplier inclusion
and the annihilation conclusions extend exactly as in Kasparov’s technical theorem; (KG.4)
remains in the original J. This proves the used separation theorem.

#### 11AA.1. KG.4. Connections and the anchored groupoid product

Use the *relative* unitization D_R^+=D⊕R, with
(d,f)(d',f')=(dd'+fd'+f'd,ff'). Its extension to a representation
sends (d,f) to ψ(d)+f·1. Fibrewise this is the forced unitization.
It is a genuine R-algebra with the transported base action, and it
retains the original module and tensor balancing. Ordinary graded
stabilization of E1 as a D_R^+-module commutes with the central R-action;
its projection does also. The diagonal operator and its Grassmann
compression in Connections and the existence of the Kasparov product (1.6)–(1.7) therefore produce an odd self-adjoint
R-linear F2-connection G0. Compact creation errors are proved by its
finite-coordinate calculation, unchanged; the relative scalar
coordinates have zero commutator because F2 is R-linear.

A transported G0 is again a connection for the transported F2. The
arrow defect of F2 is two-sided locally compact, so the creation
compactness identity

    (T_x R)^*(T_x R)=R* ψ(<x,x>) R

makes the transported G0 a connection for F2 as well. Consequently
each localized bisection difference (G0)_B−G0 is a zero-connection.
Its creation errors are *continuous* compact sections: the connection
error depends boundedly in norm on x, local transport of compact
operators is continuous, and the F2 arrow error is continuous by the
cycle condition. Finite bisection charts verify this on every compact
arrow set. The R*R compactness test is applied on the actual pullback
Hilbert modules localized by χ on such a bisection: its right side
is an element of that compact-operator section algebra by the original
cycle condition. The test therefore proves compactness in that
section algebra, rather than only in each fibre. No norm-continuous global orbit of G0, and no unanchored
group convolution, is needed.

Put E=E1⊗_D E2, J=K(E), S=F1⊗1, φ=φ1⊗1, and
A1=K(E1)⊗1+J. In A2 put J and the **compact-unit-localized** ordinary
defect generators f(G0²−1), f[S,G0]_gr, f[G0,φ(a)]_gr, f∈Cc(T),
together with all the compact-unit-localized bisection differences
of G0. A multiplier-only defect is not adjoined without that bump:
for noncompact T it could make the R-action degenerate. Saturate this algebra and
the derivating tests by KG.1. The zero-connection calculus in
Connections and the existence of the Kasparov product Lemmas 1.1–1.2 proves A1 A2⊂J. This persists on every
localized translate: the first algebra is invariant, and translating
the annihilation identities or replacing the translated second
operator by its locally compact difference preserves them. The
operators S,G0,φ(A) and their local translates derive A1 by the same
connection calculation. Countable generation and compact-unit
localization give sigma-unitality by the positive generator sum used
in Connections and the existence of the Kasparov product Theorem 3.1. Thus KG.3 applies to these actual data.

The usual operator

    F0=M^(1/2) S + N^(1/2) G0                        (KG.5)

and its self-adjoint sandwich have the connection, defect and localized
positivity equations proved in Connections and the existence of the Kasparov product (3.5)–(3.7), now tested after
source sections and creation vectors. To recover a test with an
unlocalized defect Z from its generators fZ, use f a→a for a in
the source, and f x→x for a creation vector x. R is nondegenerate
on both source and modules and commutes with every tested operator.
Thus the corresponding operator errors have norm bounds by a fixed
operator constant times ||a−fa|| or ||x−fx||; their compact localized
approximants converge in norm. Only these source and creation tests,
not N Z∈J for every unlocalized Z, are required for the product axioms.
Positivity is recovered by the same source approximation and the
closedness of the positive quotient cone. For an arrow,
expand its new defect into the four terms

    (α M^(1/2)−M^(1/2)) α S + M^(1/2)(α S−S)
      +(α N^(1/2)−N^(1/2)) α G0 + N^(1/2)(α G0−G0). (KG.6)

The first and third are compact-arrow sections by (KG.4) and quotient
functional calculus. The second, after source multiplication, is
compact because the first-input arrow defect lies in A1 and M
annihilates A1 modulo J. The last is compact on every bisection
after unit localization, since those zero-connection differences were
actual A2 generators. Each resulting compact is a continuous section:
the first/third are norm-uniform sums as proved above, and the other
two are the actual localized compact conclusions in the corresponding
chart. A finite chart cover checks any compact arrow test. This proves
the groupoid cycle condition for (KG.5); the sandwich has the same
localized compact change and class.

For uniqueness and homotopy comparison repeat the exact separation
and positive-anticommutator homotopies of Connections and the existence of the Kasparov product Theorem 4.1 and
Homotopy, associativity, the index pairing and KK-equivalence, adjoining localized arrow differences of both compared
operators. They are zero-connections by the preceding calculation,
and so satisfy the same annihilation hypothesis. Equation (KG.6)
then proves the arrow condition for the comparison operator and its
functional-calculus homotopies. Performing these tests on interval
coefficient modules proves homotopy independence and bilinearity.
The simultaneous connections in the associativity proof have the
same invariant first algebra and zero-connection defects, so the
same enlargement and estimate apply at both separation steps. The
ordinary three-tensor unitaries are R-balanced and intertwine the
diagonal G-action. This proves existence, uniqueness, associativity,
homomorphism naturality and the signed external flips in anchored KK_G.
The proof is an explicit étale adaptation of the already written
ordinary operator proof, with KG.1–KG.6 supplying its new prerequisites.

#### 11AA.1. KG.5. Essential replacement without an orthogonal complement

For a cycle (E,ψ,F) let E0=closure(ψ(D)E). In the relative unitization
put L={f∈C([0,1],D_R^+):f(1)∈D}. The actual tensor module

    L⊗_(ψ_R^+) C([0,1],E)
      ≅ {ξ∈C([0,1],E):ξ(1)∈E0}                    (KG.7)

is proved by the inner-product and endpoint-approximation argument of
Connections and the existence of the Kasparov product Proposition 5.1. It is anchored, countably generated and
has the pointwise G-action. Left multiplication by constant D is
compact on L. Take the ordinary anchored Grassmann connection G0
on (KG.7), as in KG.4's connection construction. **F is not promoted
to a cycle for D_R^+:** its square and arrow defects may fail after
the new scalar unit. Only its compact commutators with D_R^+ are
used in constructing this connection; the new scalar commutators
are zero by R-linearity.

Localize its square defect to constant d∈D, approximated by sums
d1 d2*. On the tensor module its representation is approximated
by T_(d1) T_(d2)*, and the two connection equations move G0²−1
through these creations to (F²−1) localized by d2 and d1. This is
compact by the *original* D-cycle condition and the R*R creation
test. Its source commutators are compact because D acts compactly
on L, by the same connection calculation. For an arrow, transport
the Grassmann connection and the creation vector back to the range
unit. On a constant creation vector d∈D, the difference of the two
connection equations leaves only the compact errors and
T_d(α_g F−F). This latter operator is compact by its R*R test and
the original arrow defect after ψ(d*d). Both creation sides therefore
give compact differences; multiplying by the finite sums
T_(d1)T_(d2)* proves (α_g G0−G0)ρ(d) and its opposite product compact.
They are continuous compact-arrow fields on every bisection: the
connection errors are bounded norm-continuous functions of their
creation vectors, transported compacts are continuous, and the
original localized arrow defect is continuous. Finite chart tests
and the norm approximation by the d1 d2* sums finish the proof.
Thus G0 itself is an anchored equivariant cycle for source D on
(KG.7), with no product using the scalar unitization as a cycle.

At 0 it is E with operator differing from F two-sided locally
compactly for ψ(D). The actual L-vector
ℓ_f(t)=(1−t)(0,f), f∈Cc(T), evaluates to the scalar f·1 at 0;
its creation connection identifies the initial operator with F
after multiplication by f, modulo compacts. Then d≈fd gives the
original source tests. The preceding arrow R*R test, applied on
χ-localized bisection pullback modules, takes place in their compact
section algebra, so the homotopy's defect is not just fibrewise compact.
At 1 it has essential
representation on E0. The compact initial perturbation and the actual
interval cycle give the desired equivariant homotopy. Multiplication
maps A_d:E→E0 have the compact connection errors in (5.1a), with
adjoints defined by multiplication followed by inclusion. No projection
onto E0 has been assumed.

#### 11AA.1. KG.6. Puppe and the semisplit extensions used in localization

For an anchored equivariant homomorphism f:A→B use its actual cone
C_f={(a,g):g(0)=f(a),g(1)=0}. Every substitution, moving junction and
scalar cone homotopy in Exact sequences in KK and the universal coefficient theorem Lemma 1.1 preserves R, G and grading.
The gluing module in its (2.1) has the diagonal pointwise G-action:
the matching coefficient map is anchored equivariant, so this action
preserves its inner product and matching condition. Countability and
compact gluing have the same rank-one approximation proof. Compact
arrow defects glue by that proof, uniformly on compact arrow charts.
The null-homotopy gluing and source half-interval argument therefore
give both Puppe sequences; the essential replacement used there is
KG.5. Positive suspension/inverse suspension are the earlier explicit
scalar cycles, externally tensored with the original R-algebra, with
trivial action on their scalar parameter. Their inverse identities and
Clifford signs hold in KK_G by the same invariant cycles/homotopies
and KG.4. This verifies the simultaneous suspension used in source
Puppe, rather than inferring it from fibre groups.

Suppose 0→J→A→B→0 has an even anchored *equivariant* cpc section σ.
Use the relative unitization B_R^+ in the Stinespring construction:
H is the completion of B_R^+⊙_R J with inner product
x* L(b*c)y, L(b,f)=mσ(b)+f·1. Fibrewise contractivity of σ makes this
relative unital extension completely positive. The formula

    U_g(b⊗x)=α_g(b)⊗α_g(x),    Vx=1_fibre⊗x           (KG.8)

defines the actual continuous action and the even cyclic isometry.
For precision V is obtained using an approximate identity of R;
the formula is fibre notation for its anchored adjointable limit.
The section covariance preserves the inner product; bisection
elementary tensors prove continuity. V* (b⊗x)=L(b)x, so P=VV* is
exactly invariant. The multiplicativity defect in J makes
(1−P)ρ(b)V compact by the R*R compactness test, hence [P,ρ(b)]
compact. Thus (2P−1)⊗ε gives the odd extension cycle. The full dilation
rotation of Exact sequences in KK and the universal coefficient theorem (3.6) is anchored and invariant and proves
pullback naturality without a complemented cyclic submodule.

The one additional quotient in the excision proof is
CA→C_q, h↦(h(0),qh), with ideal SJ. Its ordinary anchored section
is the explicitly written formula (3.8), with an even quasicentral
approximate identity u_t of J. It is R-linear because each multiplier
commutes with R. In the present proof this extension has either a
common proper central space (MV and telescope) or an action of the
finite group F over U (the restriction of the deformation q). In the
former case the supported averaging lemma below makes the section
exactly G-equivariant. In the latter case its finite F-average does.
Asymptotic invariance of u_t is *not* mistaken for exact covariance
of the section. The auxiliary cone-to-cone section (4.4) is already
equivariant once σ is. All its cones retain the same proper centre,
or the same finite action, as required for this extra averaging.

The left inverse of J→C_q is the negative suspended extension class
of CA→C_q followed by positive inverse suspension. Pullback is CJ,
so the scalar cone sign gives [e]u=1_J. The auxiliary extension with
contractible middle and quotient and both Puppe sequences give zero
KK groups for C_e and therefore a right inverse w of e. Associativity
gives w=w([e]u)=(w[e])u=u. Thus e is a genuine anchored equivariant
KK-equivalence and both semisplit cyclic sequences follow. This is
the actual two-inverse proof, not a general excision assertion for
arbitrary merely ordinary semisplit groupoid extensions.

#### 11AA.1. KG.7. Countable source additivity

For A=⊕A_n with each coordinate invariant, a sequence of normalized
G-cycles gives its direct-sum cycle. Finite source support gives finite
sums of compact defects and compact-arrow sections. The uniform
contraction bounds and ||a_n||→0 give the compact and continuous norm
limits for every source element, also on compact arrow sets. This
proves surjectivity of KK_G^k(A,B)→∏KK_G^k(A_n,B).

For injectivity use KG.5 to make the source essential. The actual
orthogonal coordinate submodules E_n=closure(π(A_n)E) then have dense
sum, hence adjointable coordinate projections P_n; they are G-invariant.
Diagonal compression has two-sided locally compact difference, by
Exact sequences in KK and the universal coefficient theorem (5.2), and its arrow defects are the corresponding diagonal
compact-arrow fields. Finite-support approximation gives the required
uniform tail estimates. It is connected to the original cycle by the
straight locally compact operator homotopy. Countably many normalized
coordinate null homotopies have a direct sum with the same finite-support
and uniform-tail proof. Hence all zero coordinate classes imply zero
global class. Clifford coefficients give both degrees. This proves
the additivity used by the anchored telescope, without assuming
contravariant continuity and without deleting a Milnor lim1 term.

### 11AA.2. Anchored finite-matrix approximation for the actual nearest-face source

This original prerequisite proof is CC0. Let \(G\rightrightarrows T\)
be Hausdorff, second countable, locally compact and étale. The source
\(P\) is the actual graded nearest-face algebra of AF.1--AF.13 in
`groupoid-compatible-auxiliary-construction/[The actual anchored proper algebra and continuous family](k-theory-of-the-leaf-space.html#section-11z)`.
The proof uses its explicit local moving-vertex corner models. It does
not use a section of a source quotient.

Write \(R=C_0(T)\). A finite-dimensional graded algebra may be a finite
direct sum of matrix algebras. We prove the following precise anchored
approximation statement: for every finite \(\mathcal F\subset P\) and
\(\varepsilon>0\), there are even \(R\)-linear cpc maps
\[
 P\xrightarrow{\alpha}R\otimes F
       \xrightarrow{\beta}P,
 \qquad \|\beta\alpha(a)-a\|<\varepsilon\quad(a\in\mathcal F),
 \tag{AN.1}
\]
with \(F\) finite dimensional and graded. Neither map is claimed
equivariant. This is anchored nuclearity of the source itself; it
does not imply anchored semisplitting of its closed-supported quotients.

#### 11AA.2. 1. The local corner really has a constant anchored model

Choose finitely many distinct moving vertex sheets over an open
\(U\subset T\). Include all the nonempty simplex lines of their
vertex set \(F_U\), and let \(p_U\) be that finite exterior projection.
Compression
\[
 P|_U\longrightarrow p_U(P|_U)p_U,
                     \quad a\longmapsto p_Uap_U
 \tag{AN.2}
\]
is even, cpc and \(C_0(U)\)-linear. It preserves \(P\): the compressed
coefficients are precisely a subset of its prescribed cylinder ideals.
It need not be a homomorphism.

Shrink \(U\) so the vertex sheets remain distinct and ordered. Their
orthonormal real frame identifies their finite span with a fixed
Euclidean space. The orthogonal tail is also ordinarily trivial over
\(U\). To see this with the actual field, reserve the infinitely many
unit vertices \((1_y,n)\) with integer labels larger than every label
in \(F_U\). They give an actual standard real summand in that tail.
Real operator-range stabilization of its countably generated complement
then gives a trivialization of the tail while keeping the selected
finite frame fixed. This is an ordinary anchored trivialization, and
does not remove the groupoid action.

In these coordinates all the nearest-face cylinder ideals and all the
finite exterior matrix lines are fixed. Cofinal finite Bott stages
containing the selected finite span split off its position/momentum
factor; their remaining connecting maps are the tail Bott maps.
Consequently the corner in (AN.2) is
\[
 p_U(P|_U)p_U\simeq C_0(U)\otimes D_U,
 \tag{AN.3}
\]
where \(D_U\) is the fixed finite matrix cylinder algebra over the
infinite tail Bott algebra. The grading on its finitely many exterior
lines is fixed as well. AF.1's ordinary nuclearity proof applies to
\(D_U\): its finite-coordinate central fibres are finite sums of full
matrix blocks of nuclear Bott fibres; positive Choi lifting and scalar
patching prove finite-matrix approximation, and increasing coefficient
stages retain it. This uses the actual finite-coordinate ideals, not
nuclear-subalgebra inheritance.

#### 11AA.2. 2. Localized finite-corner approximation

First replace every \(a\in\mathcal F\) by \(b=f_Ta\), with one
\(0\le f_T\le1\) in \(C_c(T)\), to within an arbitrarily small
error. Such localization follows from the nondegenerate \(R\)-action.
Put \(K=\operatorname{supp}f_T\), a compact unit set.

For each \(y\in K\), finite exterior truncation of the finitely many
\(b_y\) gives a finite vertex set for which the compression error is
small. Continue these vertices as distinct local sheets. Ambient field
norm continuity of \(P\) shows that the same compression error stays
small on a neighborhood of \(y\). Choose finitely many such
\(U_i\) covering \(K\), with finite projections \(p_i\) and models
\(C_0(U_i)\otimes D_i\).

Choose \(\phi_i\in C_c(U_i)\), \(\phi_i\ge0\), such that
\(\sum_i\phi_i\le1\) on \(T\) and \(\sum_i\phi_i=1\) on
\(K\). Take \(0\le\chi_i\le1\) in \(C_c(U_i)\), equal to
one on \(\operatorname{supp}\phi_i\). The local sections
\(\chi_ip_i b p_i\), written in (AN.3), have compact image in
\(D_i\). Nuclearity of \(D_i\) therefore gives cpc maps
\[
 D_i\xrightarrow{\alpha_i^0}F_i
       \xrightarrow{\beta_i^0}D_i
 \tag{AN.4}
\]
approximating these entire compact images uniformly. Indeed take a
finite norm net of each image; the two contraction bounds extend the
finite-net approximation to the whole compact image.

For grading preservation enlarge the approximated sets by their
grading conjugates. If \(\alpha^0,\beta^0\) are initially ordinary,
replace their finite factor by \(F_i\oplus F_i\) with the flip
grading, and set
\[
 \alpha^{\rm gr}(a)=(\alpha^0(a),\alpha^0(\gamma a)),\qquad
 \beta^{\rm gr}(x,y)=\tfrac12\bigl(\beta^0(x)+\gamma\beta^0(y)\bigr).
 \tag{AN.5}
\]
These are cpc and even. Their composition is the grading average of
the original approximation, so it has the same error on the enlarged
sets. Use this choice in (AN.4).

#### 11AA.2. 3. Anchor-preserving patching of the maps

Define \(\alpha_i:P\to R\otimes F_i\) by compression,
multiplication by \(\chi_i\), the model (AN.3), and
\(1_{C_0(U_i)}\otimes\alpha_i^0\), followed by extension by zero.
The compactly supported input cutoff makes this extension by zero an
actual continuous section. Every operation is \(R\)-linear and cpc.
Their direct sum \(\alpha=(\alpha_i)_i\) is cpc, since the norm of a
finite direct sum is the maximum of its component norms.

For \((x_i)_i\in\bigoplus_i(R\otimes F_i)\), define
\[
 \beta((x_i)_i)=\sum_i\phi_i\,
       j_i(1\otimes\beta_i^0)(x_i|_{U_i}),
 \tag{AN.6}
\]
where \(j_i\) is the corner inclusion after its fixed coordinate
identification. Multiplication by \(\phi_i\) ensures vanishing at
the boundary of \(U_i\), so each summand extends by zero to \(P\).
The maps are anchored and even. At every matrix level a positive input
of norm at most one has output bounded by
\(\sum_i\phi_i p_i\le\sum_i\phi_i1\le1\) in the corresponding
multiplier algebra. Thus \(\beta\) is cpc.

On \(\operatorname{supp}\phi_i\), the input cutoff equals one.
The error of \(\beta\alpha(b)\) relative to
\(\sum_i\phi_ip_i b p_i\) is therefore bounded by the maximum of
the uniform finite-model errors. The error of this latter sum relative
to \(b\) is bounded by the maximum local compression error, because
the scalar partition sums to one wherever \(b\) can be nonzero.
Outside \(K\) all \(b_y=0\), and \(R\)-linearity makes both
approximations zero there. The original localization error contributes
at most twice \(\|a-b\|\). Choose all three errors small enough to
obtain (AN.1).

In particular, if \(\pi:P\to\mathcal L_B(\mathcal E)\) is an
anchored representation on a Hilbert module, then
\(\pi\beta\alpha\to\pi\) point-norm through anchored cpc
finite factors. All compact-vector compressions of \(\pi\) inherit
these anchored finite-factor approximations. This verifies the nuclear
representation prerequisite at its actual source; it does not by
itself prove lifting of degenerate equivariant cycles or operator
homotopies.

### 11AA.3. The proper source identity on an anchored regular module

This original prerequisite proof is CC0. Let \(P\) be a separable
graded \(G\)-algebra with a nondegenerate even central anchored map
\(C_0(Z)\to ZM(P)\), where \(Z\to T\) is a proper LCH
Hausdorff \(G\)-space. Use a nonnegative continuous square cutoff
\(c\) with
\[
 \sum_{g\in G^{p(z)}}c(g^{-1}z)^2=1,
 \tag{RC.1}
\]
and proper support: for each compact \(K\subset Z\), the arrows
tested by \(z\in K\) and \(c(g^{-1}z)\ne0\) lie in a compact
arrow set. The supplied GP cutoff proof constructs precisely this
support property. No invariant or equivariant point of \(Z\) is
chosen.

#### 11AA.3. 1. The actual regular Hilbert module and source representation

Let \(\mathcal R_y=\ell^2(G^y)\widehat\otimes P_y\). The
continuous Hilbert \(P\)-module is the completion of compactly
supported local bisection sections \(\zeta(g)\in P_{rg}\), with
\[
 \langle\zeta,\eta\rangle_y=\sum_{g\in G^y}\zeta(g)^*\eta(g),
 \quad (U_k\zeta)(g)=\alpha_k\zeta(k^{-1}g),
 \quad (\pi_{\mathcal R}(a)\zeta)(g)=a_{rg}\zeta(g).
 \tag{RC.2}
\]
Finite bisection covers make the compactly supported sums continuous;
completion gives a countably generated module. The right action is
pointwise multiplication by \(P_{rg}\). Formula (RC.2) preserves
the inner product with its exact coefficient transport and has the
composition law. Its action and representation are continuous on the
dense local sections, hence on the completion. Every operation is
over the original unit anchor, and \(\pi_{\mathcal R}\) is an
even equivariant representation.

For \(g\in G^y\), write \(c_g(z)=c(g^{-1}z)\) on \(Z_y\),
interpreted as a central multiplier of \(P_y\). Define
\[
 (V\xi)(g)=c_g\xi_y,
 \qquad V^*\zeta=\sum_{g\in G^y}c_g\zeta(g).
 \tag{RC.3}
\]
For \(\xi\) localized centrally over a compact subset of \(Z\),
proper cutoff support places the first formula in a compact arrow
set, covered by finitely many bisections. It consequently defines an
actual regular-module section. These inputs are dense in \(P\).
Equation (RC.1) gives \(\langle V\xi,V\eta\rangle=\xi^*\eta\).
The displayed adjoint formula on compactly supported local sections
satisfies \(\langle V\xi,\zeta\rangle=\langle\xi,V^*\zeta\rangle\).
The Cauchy--Schwarz bound from (RC.1) gives \(\|V^*\zeta\|\le\|\zeta\|\);
both maps therefore extend to the completions and are adjoints.
Thus \(V^*V=1_P\).

Centrality gives \(V(a\xi)=\pi_{\mathcal R}(a)V\xi\).
For \(k\) and \(g\in G^{rk}\),
\(\alpha_k(c_{k^{-1}g})(z)=c(g^{-1}z)\), by the exact action
on the anchored space. It follows that \(U_kV=V\alpha_k\).
The projection \(p=VV^*\) is therefore even, equivariant and
commutes with the source representation. Moreover
\[
 p\pi_{\mathcal R}(a)=V M_aV^*\in\mathcal K(\mathcal R),
 \tag{RC.4}
\]
since multiplication by \(a\) is compact on the standard
Hilbert \(P\)-module \(P\). This is a compact-operator identity
on completed modules, not a fibre index calculation.

#### 11AA.3. 2. An explicit regular representative of the identity

Put \(\mathcal H=\bigoplus_{n\ge0}\mathcal R\). On each copy
the projection \(p\) gives the genuine orthogonal decomposition
\(\mathcal R=VP\oplus C\), where \(C=(1-p)\mathcal R\).
The source representation and the groupoid action preserve it.
Define an even equivariant unitary
\[
 W:\mathcal H\longrightarrow\mathcal H\ominus(VP)_0
 \tag{RC.5}
\]
by sending \((VP)_n\) to \((VP)_{n+1}\), using the identical
copy of \(V\), and sending \(C_n\) identically to \(C_n\).
This countable direct-sum permutation is an actual adjointable
unitary, intertwines the source, and is exactly equivariant. It uses
no source additivity theorem or Milnor assertion.

Use the graded module \(\mathcal H\widehat\oplus\mathcal H[1]\),
with source representation \(\operatorname{diag}(\pi(a),\pi(\gamma_P(a)))\)
and odd operator
\[
 F=\begin{pmatrix}0&W\\W^*&0\end{pmatrix},\qquad
 F^2=\begin{pmatrix}1-p_0&0\\0&1\end{pmatrix}.
 \tag{RC.6}
\]
Here the first occurrence of \(W\) includes its range in the first
even copy; \(p_0\) projects onto \((VP)_0\). The operator is
odd, self-adjoint, contractive and invariant. For homogeneous \(a\),
\(W\pi(\gamma_P(a))=(-1)^{|a|}\pi(a)W\), so its graded source
commutator is zero. The grading conjugation in the second diagonal
is essential for a genuinely graded source. Its square defect
localized by \(a\in P\) is compact
by (RC.4). Hence this is an actual anchored \(G\)-Kasparov
\((P,P)\)-cycle.

Its \((VP)_0\) summand is the even cycle \((P,\mathrm{id},0)\)
via \(V\). On the orthogonal complement \(F\) is an exact odd
involution commuting with the source and action, so that summand
is degenerate. Consequently
\[
 [\mathcal H\widehat\oplus\mathcal H[1],\pi,F]
                    =1_P\quad\text{in }KK_G(P,P).
 \tag{RC.7}
\]
The underlying module is a countable graded regular module with
original coefficients \(P\). It has not been replaced by
\(C_0(Z)\).

For the actual nearest-face source, the representation in this
identity cycle also has the anchored finite-factor approximations
of AN.1, simply by composing their output maps with
\(\pi_{\mathcal H}\). This supplies the regular-identity and
anchored nuclearity prerequisites of the direct source argument.
It does not yet supply the lifting of degenerate cycles and operator
homotopies needed to conclude coefficient half-exactness.

### 11AA.4. Supported averaging of an anchored section on a common proper centre

Let G be the specified Hausdorff second-
countable LCH étale groupoid, with unit space T and R=C0(T). Suppose
0→J→E→A→0 is an anchored graded G-extension whose three algebras
have the same even equivariant nondegenerate central action of C0(Z),
for a proper G-space Z→T, and the maps respect that action. Suppose
the quotient has an even ordinary *R-linear* cpc section s. No
C0(Z)-linearity or equivariance of s is assumed.

#### 11AA.4. SA.1. The support enlargement actually required

Write π:Z→Z/G. Use the square-cutoff construction GP.9–GP.11 in
A universal proper groupoid base with supported cutoffs: a locally finite partition τ_j on Z/G,
compact bumps ξ_j on Z, and the strictly positive invariant orbit
sums on their partition supports give

    c(z)^2=Σ_j τ_j(πz) ξ_j(z)^2 / S_j(z),
    Σ_(g∈G^y) c(g^−1 z)^2=1  (z∈Z_y).               (SA.1)

The quotient is Hausdorff, second countable and locally compact;
the proper action relation is closed and the quotient map is open.
Hence the quotient is paracompact and permits the indicated locally
finite partition and shrinkings. Enlarge its partition supports to
a locally finite family V_j, and enlarge each compact bump support
to a relatively compact open O_j. The closed locally finite set

    L=⋃_j (π^−1(supp τ_j) ∩ supp ξ_j)

contains supp c and is contained in O=⋃_j(π^−1(V_j)∩O_j).
Choose the enlargements with closures subordinate to a locally finite
family. Normality of the metrizable LCH space Z gives a continuous
d:Z→[0,1], equal to one on L, with support inside a closed shrinking
of O. For every compact K⊂Z/G, supp d∩π^−1(K) is a closed subset
of finitely many compact sets closure(O_j), hence compact. Thus the
restriction of π to supp d is proper, and dc=c.

In particular, for compact C⊂Z the set

    {g: some z∈C has g^−1 z∈supp d}                 (SA.2)

is compact. Indeed g^−1 z belongs to the compact set
supp d∩π^−1(πC), and properness of (g,w)↦(gw,w) makes the corresponding
arrow-point pairs compact. Their arrow projection is (SA.2). This
is the support assertion used below, stronger than strict convergence
of an orbit sum.

#### 11AA.4. SA.2. The actual anchored fibre formula

R-linearity gives cpc fibre maps s_y:A_y→E_y: s sends the ideal
C0(T\{y})A into C0(T\{y})E, by its linearity, continuity and an
approximate identity of that ideal. For a∈A and g∈G^y write
c_g(z)=c(g^−1z), d_g(z)=d(g^−1z), z∈Z_y. Set

    S(a)_y=Σ_(g∈G^y) c_g α_g^E
             [s_(s(g))(α_(g^−1)^A(d_g a_y d_g))] c_g. (SA.3)

All expressions use the *original* fibres over s(g) and r(g).
They are well-defined although s is not C0(Z)-linear. Each summand
is completely positive. At every matrix level, a positive norm-one
input gives a positive output bounded by c_g^2, since s_y is cpc
and 0≤d_g≤1. Every partial positive sum is thus bounded by
Σ_g c_g²≤1; the normalization acts as a strict multiplier identity.

First take a=f a with f∈Cc(Z); such inputs form a dense linear
subspace. The input compression in (SA.3) vanishes outside the compact
arrow set (SA.2) for C=supp f. Cover that set by finitely many
bisections and a nonnegative compact arrow partition whose sum is
one there. On one such chart the selected arrow is a continuous
function of its range unit. Transport of d_g a_y d_g to source units
is a continuous local A-section. A unit cutoff extends it to an
actual A-section before applying the global R-linear map s.
Transport back and multiplication by c_g yield a continuous local
E-section. The compact arrow partition weight makes it vanish at
the boundary of the chart, so its zero extension is an actual
E-element. Its norm is bounded by ||a|| times that weight. The finite
sum of these sections is precisely (SA.3). Over each fixed range
unit only finitely many arrows in the compact set occur; étaleness
and the finite bisection cover give the finite sum, rather than a
globally labelled discrete group sum.

The same matrix bound applies to the finite section sum. Uniform
contractivity and density consequently extend it to a cpc map
S:A→E in the norm topology of the *actual algebra*. In particular
strict multiplier convergence has not been substituted for this
norm-convergence and section-membership argument.

#### 11AA.4. SA.3. Section, covariance, anchor and grading

Applying the quotient to (SA.3) gives

    q S(a)_y=Σ_g c_g d_g a_y d_g c_g
             =Σ_g c_g² a_y=a_y,                     (SA.4)

because d_g c_g=c_g and the central square normalization is one.
This holds first for localized inputs, then for all a by continuity.
For k:y→y', replacing g by kg in the normalized counting sum gives
α_k S_y(a_y)=S_(y')(α_k a_y). The enlarged input cutoff transports
in exactly the same way as the output cutoff. This proves *exact*
G-equivariance. The bisection construction in SA.2 supplies the
continuity over arrows required for that equality.

For h∈R, its fibre value at r(g) passes through both transports and
s_(s(g)) as the same scalar. Thus S(ha)=hS(a). Central even cutoffs
and the even s make S even. Therefore q has an actual even anchored
equivariant cpc section. The theorem applies to the telescope and
the suspended-ideal cone quotients because those retain the same
proper central Z; it makes no assertion about an extension with
unrelated centres or without an anchored ordinary section.

### 11AA.5. Anchored source Mayer--Vietoris without a closed-quotient section

Its source \(A\) has an even invariant
central action by a proper \(G\)-space \(Z\to T\). Let
\(U,V\subset Z\) be invariant open sets covering the central support
of \(A\), and put \(I=C_0(U)A\), \(J=C_0(V)A\). Work with the
original \(C_0(T)\)-action and the actual grading. Thus \(I+J=A\).

Choose an invariant scalar partition \(\phi_I+\phi_J=1\) subordinate
to \(U,V\), with \(0\le\phi_I,\phi_J\le1\). It is obtained from
the two open sets on the Hausdorff paracompact quotient of the proper
central support. Both are bounded central multipliers of \(A\).
In particular \(\phi_I=0\) outside \(U\) and \(\phi_I=1\)
outside \(V\). No section of \(A\to A/I\) or \(A\to A/J\)
is assumed.

#### 11AA.5. 1. An explicit homotopy pullback

Define the anchored graded equivariant algebra
\[
 D=\{f\in C([0,1],A):f(0)\in I,\ f(1)\in J\}.
 \tag{MV.1}
\]
Variable-parameter functional calculus in its central multiplier
algebra defines a homomorphism
\[
 r:D\longrightarrow I\cap J,
                  \qquad r(f)=f(\phi_I).
 \tag{MV.2}
\]
For precision, a polynomial section \(f(t)=\sum t^na_n\) is sent
to \(\sum\phi_I^na_n\); uniform approximation defines it on all
continuous sections. Centrality gives multiplicativity and adjoint
preservation and the norm bound is one. On the quotient \(A/I\),
\(\phi_I=0\) and \(f(0)=0\), so its image is zero. On \(A/J\),
\(\phi_I=1\) and \(f(1)=0\). Thus the range is in both ideals.

Let \(i:I\cap J\to D\) be the constant section homomorphism.
Then \(ri=1\). Define
\[
 H_u(f)(t)=f\bigl((1-u)t+u\phi_I\bigr),
                         \qquad 0\le u\le1.
 \tag{MV.3}
\]
The same central functional calculus proves that each \(H_u\) is a
homomorphism. At \(t=0\), its image modulo \(I\) is \(f(0)=0\).
At \(t=1\), its image modulo \(J\) is \(f(1)=0\). Hence it
preserves \(D\). It is point-norm continuous in \(u\), first on
polynomials, then on their uniform approximations. It starts at
identity and ends at \(ir\). Central evenness, original anchor
linearity and invariance of \(\phi_I\) make the homotopy graded,
anchored and equivariant. Consequently
\[
 D\simeq I\cap J
 \quad\hbox{by actual anchored equivariant homotopy equivalence.}
 \tag{MV.4}
\]

#### 11AA.5. 2. The extension used is explicitly anchored semisplit

Endpoint evaluation gives
\[
 0\longrightarrow SA\longrightarrow D
       \xrightarrow{(e_0,e_1)}I\oplus J\longrightarrow0,
 \tag{MV.5}
\]
with the actual even equivariant \(C_0(T)\)-linear cpc section
\[
 s(a,b)(t)=(1-2t)_+a+(2t-1)_+b.
 \tag{MV.6}
\]
The two nonnegative scalar coefficients have disjoint interior
supports and sum at most one. Positivity at every matrix level and
contractivity follow directly. The endpoints are \(a,b\), so the
map splits evaluation. Its kernel consists precisely of sections
vanishing at both endpoints, namely \(SA=C_0((0,1),A)\).

The anchored equivariantly semisplit coefficient and source exactness
construction applies to (MV.5). Combining it with (MV.4) gives the
source Mayer--Vietoris exact sequence. For the particular inference
used in localization, if
\[
 KK_G^k(I,N)=KK_G^k(J,N)=KK_G^k(I\cap J,N)=0
                                     \quad(k=0,1),
 \tag{MV.7}
\]
the exact sequence of (MV.5) makes \(KK_G^k(SA,N)=0\), hence
\(KK_G^k(A,N)=0\) in both degrees. This step uses an explicit
equivariant anchored section, not ordinary nuclear lifting.

This argument is compatible with restriction of the invariant
central support to \(U\cup V\); a source need not have central
support on all of \(Z\). Countable covers still require a separate
anchored source telescope argument. Equation (MV.7) alone does not
assert contravariant continuity or discard a Milnor term.

### 11AA.6. Proper étale slices and anchored induction reciprocity

G is Hausdorff second-countable LCH étale,
T=G^(0), and R=C0(T). All algebras and modules have their original
anchors and gradings. We use the anchored product, homomorphism unit
and homotopy constructions in Anchored étale technical theorem, products and the exactness actually used KG.1–KG.5.
The source's proper central G-space is Z→T. The argument supplies
the precise finite-action induction needed for localization.

#### 11AA.6. SR.1. An actual finite-action subgroupoid near a proper orbit point

Fix z∈Z, x=p(z). Properness of
(g,w)↦(gw,w) makes F={g∈G_x^x:gz=z} compact; étaleness makes it
discrete, so F is finite. Choose disjoint bisections B_f about its
arrows, with B_e in the units. Their partial homeomorphisms have
the multiplication and inverse relations of F as germs at x.
These are actual relations on small neighborhoods: products of
bisections with the same arrow agree on an open neighborhood of
that arrow, since source is a local homeomorphism. There are only
finitely many relations. Shrink a common relation domain O about x.
Choose V about x with every image ρ_f(V) contained in O and all
compositions there defined. Then

    U=⋂_(f∈F) ρ_f(V)

is open, contains x, and is invariant under every ρ_h: on these
sets ρ_h ρ_f=ρ_(hf), so the intersection is permuted. Restricted
bisections define an actual action of the finite *abstract* group
F on U. Their disjoint union H is an open subgroupoid of G|U,
isomorphic to F⋉U. It retains the isotropy arrows themselves even
if some act trivially on unit points. H is also closed in G|U:
it is a finite union of the continuous source sections U→G|U,
and a convergent net with sources tending to a point of U has its
limit on one of these sections, by Hausdorffness.

The inverse image of (z,z) under the proper action map consists
exactly of F. The open set of arrow-point pairs with arrow in H
contains it. A proper map is closed; the image of its complement
therefore misses an open neighborhood of (z,z). Choose S0 about z,
anchored inside U, so S0×S0 is in this neighborhood. Every arrow
returning S0 to S0 belongs to H. Put S=⋃_f fS0. It is open and
H-invariant. If g returns S to S, then f2^−1 g f1 returns S0 to
S0 for two H-arrows f1,f2; thus g∈H as well. Consequently

    G×_H S → G·S,        [g,w]↦gw                    (SR.1)

is bijective. The action map is open on bisection charts, and the
finite free quotient defining its domain is open. Hence (SR.1) is
a homeomorphism. This proves the full finite-action slice assertion,
including the actual base U and return-arrow test.

This repairs a possible ambiguity in a germ-shrinking argument:
restricting all bisections to an arbitrary common open neighborhood
does not alone prove that their partial maps preserve that neighborhood.
The finite intersection U above is the invariant neighborhood.

#### 11AA.6. SR.2. The continuous coset bundle and induced coefficient algebra

For such H=F⋉U put G_U=s^−1(U) and C=G_U/H. Right H-translation
is a free finite group action on G_U: equality gh=g forces h to
be the unit arrow, including at points with isotropy. Its quotient
is Hausdorff LCH and the finite quotient map is proper. A small
bisection disjoint from its nonidentity right translates gives a
chart in C on which its range anchor r:C→T is a homeomorphism.
Thus C is an étale countable-fibre bundle. Compact subsets of C
have compact sets of arrow representatives. The identity-coset
image over U is clopen in C|U, since its inverse image is the
open and closed H⊂G|U established above.

For an H-algebra D define Ind_H^G D by continuous covariant sections
f(g)∈D_(s(g)), with

    f(gh)=α_(h^−1)f(g),

and whose norm vanishes at infinity on C. Products and grading are
pointwise. The range anchor gives the *original* C0(T)-structure;
G acts by left translation. This is the section algebra of the
associated bundle G_U×_H D→C, and its direct-sum range fibres.
Finite coset charts, compact unit bumps and a countable bisection
basis prove separability. No replacement of its coefficient fibres
by scalars is made.

For S in SR.1 let D=C0(S)P, regarded as an H-algebra over U.
The homeomorphism (SR.1) induces an actual anchored graded G-isomorphism

    C0(G·S)P ≅ Ind_H^G D.                            (SR.2)

Here is its section-algebra check. Distinct cosets at one range
unit give disjoint open subsets gS in Z_y: an intersection would
make h^−1g return S and therefore belong to H. They cover (G·S)_y.
On the component gS restrict the P-section and transport it back
by g to D_(s(g)). Changing g by h uses exactly the H-covariance.
Compact central support has compact image under G·S≅G×_H S→C;
it therefore gives the required c0 coset norm tail. On a coset
bisection chart, transporting a local D-section and extending it
on its open component gives the inverse local P-section. A compact
coset partition gives finite sums of these; such sections are dense
on both sides. The two constructions are isometric, multiplicative,
even and inverse, proving (SR.2). They keep p(z)=r(g), not s(g),
as the endpoint anchor. An invariant open subset of G·S corresponds
to an H-invariant open subset of S and has the same induced form.

#### 11AA.6. SR.3. Induction of classes and its tensor unitary

An H-cycle over U has an operator that can be averaged over the
finite group F. Its average is exactly H-invariant and R-linear.
The difference is two-sided locally compact for the source by its
finite equivariance defects, so straight interpolation preserves
its class. Assume this averaged representative in what follows.

Induce its module as the associated bundle of its Hilbert modules
over C, with right coefficient Ind B. Inner products, source action
and operator are pointwise at the coset value. Covariance uses the
H-action and the invariant operator; the left G-action and grading
are exact. Compact creation and cycle defects follow on a finite
coset chart from the original compact defects. Approximation of a
source element by compact coset support removes the remainder in
norm, with uniform bounded-operator estimates. Compact unit bumps
make the local defects actual compact module operators. These charts
also prove countable generation and continuity. This defines Ind
on cycles and interval homotopies, hence on KK_H classes.

For two modules its natural even anchored G-unitary is

    Ind E1 ⊗_(Ind B) Ind E2
          → Ind(E1⊗_B E2),                           (SR.3)

given on matching coset charts by [g,ξ]⊗[g,η]↦[g,ξ⊗η]. The central
C0(C)-action of the intermediate coefficient forces different
coset components to be orthogonal. Thus the formula is balanced.
Inner products agree exactly; compact coset chart partitions and
the ordinary tensor density prove surjectivity. A finite H-average
of a product operator remains a product operator once the input
operators are invariant: connection errors are compact and localized
positivity is preserved by the finite convex average. Its induced
operator satisfies the product conditions under (SR.3). Creation
errors reduce to finitely many charts, then to the norm tail of
the creation vector. Positivity modulo global compacts is checked
by the compact corrections on those finitely many charts and a
nonnegative chart partition; the c0 tail and closedness of the
positive quotient cone give the limit. Uniqueness from KG.4 proves
that Ind preserves composition, including Clifford degrees.

#### 11AA.6. SR.4. Unit and counit are actual coefficient correspondences

For an H-algebra D the homomorphism

    η_D:D→Res_H Ind_H^G D                            (SR.4)

inserts its value on the clopen identity coset and zero elsewhere.
It is even, anchored over U and H-equivariant. For a G-algebra B
let K_B be the completion of compactly supported coset sections
ξ(c)∈B_(r(c)), with counting inner product

    <ξ,ζ>_y=Σ_(c∈C_y) ξ(c)*ζ(c).

Thus K_B=ℓ²(C)⊗_R B on the original coefficient fibres. G acts
by left coset translation and α on B. An element of Ind Res B
acts diagonally after transporting its value f(g) to α_g f(g)
in B_(r(g)); this is independent of the representative. Its
entries are actual B-elements with c0 coset norm tail. Its action
is compact: approximate by compact coset support, use finitely
many coset charts and compact unit bumps, and approximate each
B-valued diagonal by rank ones of the standard B-module (K(B)=B).
The norm tail removes the complement. Therefore

    ε_B=[K_B, diagonal action, 0]
            ∈KK_G(Ind Res B,B)                      (SR.5)

is an actual anchored even correspondence.

For an H-class x first use KG.5's actual anchored equivariant
interval replacement to make its source representation essential,
and then take the finite H-average. This is a homotopy replacement,
not projection onto an uncomplemented submodule. The unit naturality

    [η_D]⊗Res Ind x = x⊗[η_(D')]

is the identity-coset tensor identification. The projection acts
from the induced *right coefficient* C0(C)-action, which is
nondegenerate on its Hilbert module. Its complement has zero source
action. This makes no assertion about a complemented source-essential
submodule of an arbitrary cycle.

For a G-class x:E, B→B', first make its source B-representation
essential by the actual KG.5 homotopy. Restrictions to U remain
essential, by compact unit cutoffs and the dense source action.
Now B⊗_ψ E→E is the surjective multiplication unitary. This is
required: without replacement its image would only be closure(ψ(B)E),
which need not be complemented. On this justified representative,
the tensor module of ε_B followed by x
is ℓ²(C)⊗_R E. Its product operator is the diagonal range-fibre
operator F: for a creation vector its connection errors are the
original commutators with the vector's B-valued entries; finite
chart approximation and the vector norm tail make them compact.
The first operator is zero, so positivity is vacuous. The other
order, Ind Res x followed by ε_(B'), is identified with that same
module by transporting its source-fibre vectors using U_g^E.
The transported inner products and coefficient actions agree.
Since the counit has compact intermediate left action, the tensor
of the induced operator with one is a product operator: its
creation maps are compact by the R*R test, and positivity is 2S²≥0.

The difference of the two resulting operators, multiplied on either
side by an Ind Res B source element, is compact. On compact coset
support choose compact arrow representatives and finitely many
bisections. The differences there are the original continuous
localized G-equivariance defects of F, plus its finite H-average
defects; hence they are continuous compact fields. The c0 source
tail gives norm convergence of their compact sum, uniformly on
compact arrow tests. The difference is bounded by two after
normalization. Straight interpolation gives

    Ind Res x ⊗ ε_(B') = ε_B ⊗ x.                   (SR.6)

This proves counit naturality on classes, including homotopies and
coefficient homomorphisms, with explicit tensor operators.

#### 11AA.6. SR.5. Both triangles and the Hom adjunction

In Res ε_B after η_(Res B), the only nonzero source action is on
the clopen identity-coset summand, canonically B|U. Its complement
has zero source and zero operator and is degenerate. Thus

    [η_(Res B)]⊗Res ε_B=1_(Res B).                  (SR.7)

In Ind η_D followed by ε_(Ind D), the standard module has an
outer coset and an inner coefficient coset. The source insertion
selects their diagonal. The diagonal in C×_T C is clopen: it is
closed by Hausdorffness and open because C→T is étale. Its actual
adjointable projection selects a module canonically Ind D, whose
inner product has exactly one outer-coordinate term at each inner
coset. The complement again has zero source. This proves

    Ind[η_D]⊗ε_(Ind D)=1_(Ind D).                  (SR.8)

The two Hom maps are now defined on every class without compressing
its source representation:

    R(z)=[η_D]⊗Res z,
    L(x)=Ind x⊗ε_B.

Unit naturality and (SR.7) give RL(x)=x; counit naturality and
(SR.8) give LR(z)=z. Hence the actual anchored natural isomorphism is

    KK_G^k(Ind_H^G D,B) ≅ KK_H^k(D,Res_H B), k=0,1.  (SR.9)

All sources, targets and restrictions in this formula retain their
original coefficient anchors. It is finite-action open-subgroupoid
reciprocity; it is not a claim about arbitrary closed subgroupoids.

#### 11AA.6. Sources and proof scope

The actual admitted Bönicke–Proietti paper 2202.08067v3, printed
pp.26–27, Proposition 3.2 and Remark 3.3, was read for the local
proper-action comparison; its §§2.1–2.2 describe a crossed-product
induction model. The definitions and proofs above instead give the
associated-coset section model, its tensor unitaries and both
correspondence triangles explicitly. No unproved Morita comparison
with that paper's induction model is used. Every new groupoid
prerequisite here has a local proof, with KG.4 supplying composition.

### 11AA.7. The anchored source telescope, including its Milnor term

G,R and gradings are as in Anchored étale technical theorem, products and the exactness actually used. P is the actual
nearest-face source, with the anchored finite-dimensional approximation
in The actual anchored proper algebra and continuous family AF.1a, equivalently Anchored finite-matrix approximation for the actual nearest-face source
AN.1–AN.6. In particular its output factors have *P-valued* finite
Choi entries, due to their compact unit output bumps. Let P_n be
increasing invariant open central ideals with dense union P; all
retain the same proper central G-space Z. No section of P/P_n is used.

#### 11AA.7. AT.1. The two actual telescope extensions

Put P_0=0 and let

    Tel={f∈C([0,∞],P): f(0)=0,
             f(t)∈P_n when n−1<t≤n}.

The infinite endpoint has the norm-continuous one-point compactification
topology. Conditions at an integer match the adjacent stages because
P_n⊂P_(n+1). This is a closed anchored graded G-algebra. Its R and
C0(Z)-actions are pointwise and nondegenerate: the image of a continuous
path is a norm-compact subset of P, and either central approximate
identity converges uniformly on that set. Infinite evaluation κ is
onto. For a∈P choose a_n∈P_n with a_n→a, a_0=0, and interpolate
linearly between a_n and a_(n+1) on [n,n+1]. Let Tel0=ker κ and
Q_src=⊕_(n≥1)P_n. Then

    0→Tel0→Tel→P→0,
    0→S Q_src→Tel0→Q_src→0                          (AT.1)

are actual anchored graded G-extensions. The last map is evaluation
at the positive integers; its kernel has the P_n-suspension on
(n−1,n). It is onto by the interpolation section described below.
Integer evaluation is on Tel0, not on Tel whose tail need not vanish.

#### 11AA.7. AT.2. Contractibility of Tel

For x(t)=t/(1+t), x(∞)=1, define h_u by x(h_u(t))=u x(t):

    h_u(t)=ut/(1+(1−u)t),
    h_u(∞)=u/(1−u) for u<1,    h_1(∞)=∞.

Since h_u(t)≤t and the stage ideals are increasing, f∘h_u still
belongs to Tel. Uniform continuity on the compact x-interval proves
point-norm continuity in u; this is an actual homotopy of homomorphisms
from zero to identity. It changes only the scalar parameter and so
preserves R, C0(Z), grading and G.

#### 11AA.7. AT.3. The anchored ordinary section of κ

For a prescribed finite set the actual AF.1a factors are

    P —α→ R⊗F —β→ P,     β(f⊗b)=f k(b),             (AT.2)

where F is a finite direct sum of matrices (with the indicated even
grading) and k:F→P is cpc with actual P-valued entries. Indeed each
local output has the form φ_i β_i^0(b) with φ_i compactly supported
in its unit chart; thus it extends to an element of P, and
k(1)≤Σ_i φ_i≤1. This element assertion, rather than multiplier-valued
Choi entries, is essential to what follows.

The finite Choi construction lifts k through κ to a cpc map k~:F→Tel.
For completeness lift the entries of the square root of its positive
Choi matrix, form their X*X, and normalize the image of the finite
unit by functional calculus min(1,t^−1/2). This is the ordinary finite
matrix lifting in Ext groups, absorption and Brown–Douglas–Fillmore theory Lemma 3.0a; normalization does not change
the quotient because k(1)≤1. Include the block-diagonal expectation
if F has several matrix summands and average the grading conjugates.
Now β~(f⊗b)=f k~(b) is actual Tel-valued, R-linear and even. Its finite
Choi matrix is positive fibrewise; ||k~||≤1 proves contractivity.
Thus β~α lifts βα through actual anchored cpc maps.

Here is the summable gluing of these approximate lifts. Choose dense
unit-ball elements a_j and anchored cpc approximations φ_n=β_n α_n
with ||φ_n(a_j)−a_j||<2^−n−3 for j≤n. Given a lift L_n of φ_n and
an anchored lift T_(n+1) of φ_(n+1), choose an even positive contraction
e∈Tel0 quasicentral on their finitely tested values and sufficiently
late to compress T_(n+1)(a_j)−L_n(a_j) within its quotient norm plus
the chosen tolerance. The quotient-norm compression formula and
quasicentral selection are proved in Ext groups, absorption and Brown–Douglas–Fillmore theory Lemmas 3.0b–3.0c.
Set v=e^(1/2), w=(1−e)^(1/2), and

    L_(n+1)(a)=v L_n(a)v+w T_(n+1)(a)w.              (AT.3)

This is cpc, even and R-linear since the ideal cutoff commutes with R.
It lifts φ_(n+1): κ(v)=0 and κ(w)=1. The exact estimate comes from

    L_(n+1)(a)−L_n(a)
      =w(T_(n+1)(a)−L_n(a))w
         +v[L_n(a),v]+w[L_n(a),w].

The quotient term has norm at most the sum of the two approximation
errors; choose the compression and commutator tolerances to make the
whole norm <2^−n for j≤n. These are finitely many tests at each step.
Summable tails and uniform contraction give a point-norm limit
s:P→Tel, even and R-linear cpc, with κs=id. This proves an actual
anchored section from the special output factors (AT.2); ordinary
nuclearity alone has not been used to infer an anchored lift.

#### 11AA.7. AT.4. Both extensions are equivariantly anchored semisplit

Tel,Tel0,P and κ share the proper centre Z established in AT.1.
Supported averaging of an anchored section on a common proper centre SA.1–SA.3 therefore makes s exactly
G-equivariant. Its enlarged input cutoff gives compact arrow support
on compact central inputs, so the averaged output is an actual Tel
element. This is the first section in (AT.1).

For the second, put a_0=0 for a norm-null sequence a_n∈P_n, and
interpolate between a_n and a_(n+1) on [n,n+1]. The two scalar
weights are nonnegative and sum to one. It is cpc at every matrix
level, takes values in the right stage and tends to zero at ∞.
It is exactly G-equivariant, even and anchored because the stage
inclusions are so and the scalar parameter has trivial action.
This supplies the second section directly.

#### 11AA.7. AT.5. The exact boundary and the Milnor sequence

Both source six-term sequences apply by KG.6. Tel is contractible,
so the first identifies KK_G^k(P,N) with KK_G^(k−1)(Tel0,N).
For the second extension, the two adjacent interval endpoints give
the positive-suspension class of 1−F_src, where F_src:Q_src→Q_src
moves coordinate n to n+1 by the stage inclusion. To verify the sign,
the interval (n−1,n) is increasing; its upper endpoint contributes
the identity on P_n and its lower endpoint contributes the negative
inclusion from P_(n−1). Their scalar extension cycles are exactly
the two Stinespring cones in Exact sequences in KK and the universal coefficient theorem Lemma 9.1, with the same
ordered Clifford factors. The scalar taper, endpoint rotations and
finite Choi lifts here are anchored and invariant, so that actual
two-endpoint proof gives this same boundary in KK_G, not a fibre
index inference.

KG.7's source additivity identifies KK_G^j(Q_src,N) with
∏_n KK_G^j(P_n,N). F_src* sends (z_n) to (ι_n* z_(n+1)). Thus
the unspliced exact sequence is

    0→lim1 KK_G^(k−1)(P_n,N)→KK_G^k(P,N)
       →lim KK_G^k(P_n,N)→0.                         (AT.4)

Here lim is the kernel of 1−F_src* on the product, and lim1 is its
cokernel. This *proves* the required Milnor statement from the two
actual sections and contractibility; it assumes neither general
contravariant continuity nor vanishing of lim1. If every stage group
is zero in both degrees, both the kernel and cokernel are zero, and
KK_G^k(P,N)=0. That is the only limit inference used in localization.

### 11AA.8. The actual anchored proper-source lift for the nearest-face algebra

G is Hausdorff second-countable LCH étale,
T=G^(0), R=C0(T). P,Q,E1 and q are the actual algebras and
map in The actual anchored proper algebra and continuous family and
The actual original-anchor deformation. We preserve their original anchors and
gradings.
The inputs consumed are precisely:

- P is separable, graded and proper over its actual central Z=M_G;
  its AF.1a anchored finite-matrix approximation has P-valued output
  Choi entries, as also proved in Anchored finite-matrix approximation for the actual nearest-face source.
- q:E1→Q is an actual even anchored equivariant quotient with the
  DI.7 supplied even *ordinary R-linear* cpc section. It is not
  assumed equivariant.
- Its kernel I1 is DI.8's actual equivariant Fock/shifted-exterior
  Morita partner of C0((0,1])⊗R⊗S. The interval and S actions are
  trivial, and f(t)↦f(ut), extended by zero at t=0, is an exact
  anchored equivariant homotopy from zero to identity.

The source-localization prerequisites are proved in Section 11AA:
KG.1–KG.7, SA.1–SA.3, MV.1–MV.7, SR.1–SR.9 and AT.1–AT.5. No
proper KK/E comparison, closed-supported source quotient section,
coefficient substitution by C0(M_G), or fibre-index whole-base
argument is used.

#### 11AA.8. PL.1. Every finite-action restriction of q is a KK-equivalence

Let H=F⋉U⊂G be any finite-action slice subgroupoid in SR.1. Restrict
all *original* coefficient fields to U and H. The supplied ordinary
R-linear section restricts to an ordinary C0(U)-linear cpc section:
its linearity sends the open-unit ideal to that ideal. Its finite
F-average is exactly H-equivariant and still anchored, even and cpc.
The kernel's actual G-Morita equivalence and cone contraction restrict
to H. Thus 1_(Res I1)=0 in KK_H, including both Clifford degrees.

KG.6 applies to this finite-action semisplit extension. Its two
cyclic sequences imply that

    (Res q)_*:KK_H^k(D,Res E1)→KK_H^k(D,Res Q)

is bijective for every separable anchored H-source D and k=0,1.
For D=Res Q lift its identity to an inverse; for D=Res E1 injectivity
proves the other inverse identity. Hence Res q is an actual KK_H-
equivalence. The homomorphism cone N=C_q restricts to C_(Res q),
because open-unit restriction commutes with these path algebras and
matching conditions. KG.6's Puppe sequence therefore gives

    KK_H^k(D,Res N)=0 for all D and k=0,1.             (PL.1)

This argument proves a whole open-base groupoid statement, rather
than deriving it from individual fibre groups.

#### 11AA.8. PL.2. Local proper-source groups vanish on every slice open

The proper central space Z has a countable cover by invariant opens
V_j=G·S_j of SR.1. Countability follows by selecting a countable
subcover from second countability of Z. For an invariant open W⊂V_j,
SR.2 gives an actual source isomorphism

    C0(W)P ≅ Ind_(H_j)^G D_W,

where D_W=C0(S_j∩W)P has its original H_j anchor over U_j. By
SR.9 and (PL.1),

    KK_G^k(C0(W)P,N)
      ≅KK_(H_j)^k(D_W,Res_(H_j) N)=0.                (PL.2)

The arbitrary intersection with other invariant slice opens is
allowed here; it is simply an H_j-invariant open restriction of
the same S_j, with the same return-arrow condition.

#### 11AA.8. PL.3. Finite unions vanish by the explicitly semisplit open MV

Put W_n=V_1∪⋯∪V_n and P_n=C0(W_n)P. We prove by induction that
KK_G^k(P_n,N)=0 in both degrees. The case n=1 is (PL.2). For the
induction step let I=P_n and J=C0(V_(n+1))P as ideals in P_(n+1).
Their intersection is C0(W_n∩V_(n+1))P, which vanishes by (PL.2)
inside the one slice V_(n+1). The source MV.1–MV.6 extension has
the *explicit anchored equivariant* taper section, and its path
algebra is actually anchored equivariantly homotopy equivalent to
I∩J. MV.7 therefore gives

    KK_G^k(P_n,N)=0 for every n and k=0,1.             (PL.3)

At no stage is a section of P_(n+1)/P_n assumed. In particular the
ordinary-nuclearity/closed-quotient counterexample has not entered
this proof.

#### 11AA.8. PL.4. The countable passage retains lim1 and gives the lift

The union of the P_n is dense in P. Indeed compactly central-
localized source elements are dense, and their compact central
supports have a finite subcover by the V_j. AT.1–AT.5 now applies.
Its first section comes from the *actual* AF.1a output factors,
summable anchored Choi gluing, and SA supported averaging. Its
second section is exact interval interpolation. Thus the full
anchored Milnor sequence, with its lim1 term, has all zero stage
groups by (PL.3), and gives

    KK_G^k(P,C_q)=0, k=0,1.                           (PL.4)

Both Puppe degrees for the actual homomorphism q then yield the
required isomorphism

    q_*:KK_G(P,E1)→KK_G(P,Q).                         (PL.5)

The same assertion holds with one Clifford degree. The proof used
the original E1,Q and their C0(T)-actions throughout; their central
supports were never replaced by those of P.

#### 11AA.8. PL.5. The actual equivariant Dirac class and its ordinary image

Let i:P→Q be the actual inclusion in AF. Then (PL.5) defines a
unique actual class

    z_P=q_*^−1[i] ∈KK_G(P,E1),    z_P⊗[q]=[i].       (PL.6)

Compose it with DI's actual t=1 endpoint, its Fock Morita class,
and the positive scalar S-evaluation in the stated Clifford order.
This is the actual anchored equivariant class D:P→R in LB.2; the
definition no longer contains an unproved lift premise.

After forgetting G, DI.6's ordinary trivialization makes q the
constant R-tensor of the ordinary Bott-deformation quotient. Its
anchored ordinary section and contractible kernel give an inverse
by KG.6 with the unit groupoid (which needs no equivariant averaging).
Thus the ordinary preimage of [i] is unique *in KK_T*, not just
fibrewise. Define that ordinary lift explicitly as [i] followed by
the anchored inverse of q. Its constant-tensor endpoint composition
is the definition of the ordinary infinite Dirac in OC.1, with the
same shifted-simplex Morita module and ordered scalar evaluation.
Consequently the endpoint composition is exactly

    For_G(D)=D0 ∈KK_T(P,R).                           (PL.7)

OC.2–OC.3's whole-base products with the continuous θ-family apply
to this equality. In particular the ordinary anchored calibration
needed by the ensuing factorization is inherited without any claim
that individual fibre indices determine a whole-base class. This proves the lift and the Dirac endpoint interface. Section 11AB supplies the full and reduced descent and coefficient-quotient argument.

![Explicit open-cover homotopy, the clopen coefficient diagonal, the continuous telescope and the original-anchor lift](../figures/anchored-localization.png)

**Figure 11AA.1.** The first and third panels give exact labelled scalar and c0(N) samples of the homotopy and telescope formulas. The second panel shows the clopen diagonal on the actual coefficient module; its complement has zero source. The last panel records the whole-base proof with the original E1 and Q. Proof locators: MV.1–MV.7, SR.4–SR.9, AT.1–AT.4 and PL.1–PL.7. [Vector figure](../figures/anchored-localization.svg) · [Generator](../reproduction/anchored-localization/render_localization.py) · [Exact data](../reproduction/anchored-localization/figure-data.json) · [Reproduction instructions](../reproduction/anchored-localization/REPRODUCTION.md) · [Component terms](../reproduction/anchored-localization/COMPONENT-TERMS.md).

## 11AB. Full and reduced descent over the original groupoid anchor

Descent is proved on the actual crossed modules for every separable graded anchored coefficient of a Hausdorff second-countable LCH étale groupoid. The completed tensor unitary preserves products, while coefficient base change gives the exact full-to-reduced quotient square. The original product existence is supplied by Section 11AA.1.

### 11AB.1. Full and reduced descent on the actual anchored étale modules

Let \(G\rightrightarrows T\) be Hausdorff,
second countable, locally compact and étale. All coefficient algebras
are separable graded \(C_0(T)\)-algebras with nondegenerate central
anchor, and all actions are continuous and even. Modules are countably
generated. Inner products are linear in the second variable. A bundle
may be upper semicontinuous; constant rank and a global enumeration of
the arrow fibres are not hypotheses.

For a coefficient \(B\), write \(B_\tau=B\rtimes_\tau G\), where
\(\tau=m,r\). Convolution, involution, full universal \(I\)-norm and
source-column reduced norm are the actual constructions (PN.2)–(PN.7)
of *The full and reduced norms of every proper groupoid coefficient*.
The proof of its Lemma PN.1 precedes properness and applies to every
coefficient here. In particular
\(i_{B,\tau}:B\to B_\tau\) is faithful and nondegenerate, and its
image is in the crossed algebra itself. Properness of \(B\) is not used
in this section. This last étale fact will make all ordinary module
compacts descend to compacts.

Let \(E\) be an equivariant Hilbert \(B\)-module, with actual arrow
unitaries \(U_g:E_{sg}\to E_{rg}\) covering \(\beta_g\).
Let \(\pi:A\to\mathcal L_B(E)\) be an even anchored equivariant
homomorphism. It may be degenerate.

#### 11AB.1. DS.1. The range-fibre core and its positive completion

Put \(\mathcal X_E=C_c(G,r^*E)\). For
\(b\in C_c(G,r^*B)\) define

\[
\begin{aligned}
(\xi b)(g)&=\sum_{rh=rg}\xi(h)\beta_h(b(h^{-1}g)),\\
\langle\xi,\eta\rangle(g)
 &=\sum_{rh=rg}\beta_h\bigl(
   \langle\xi(h^{-1}),\eta(h^{-1}g)\rangle_{B_{sh}}\bigr).
\end{aligned}
\tag{DS.1}
\]

Both vectors in the displayed inner product lie in \(E_{sh}\).
In the first line \(b(h^{-1}g)\in B_{sh}\), so transport by
\(\beta_h\) makes its product with \(\xi(h)\in E_{rg}\) legal.
Finite bisection covers make every sum finite locally. Products of
compact arrow supports are compact because composability is a closed
condition and multiplication is continuous. The sums are continuous
sections with compact support, by local bisection partitions.

**Lemma DS.1.** The null quotient and completion of (DS.1) in either
crossed norm is the actual interior tensor module

\[
 X_{E,\tau}=E\widehat\otimes_{B,i_{B,\tau}}B_\tau,
 \qquad
 e\widehat\otimes b\longmapsto
       [g\mapsto e(rg)b(g)].
\tag{DS.2}
\]

**Proof.** The displayed map is balanced: the two values of
\((ed)\otimes b=e\otimes i_B(d)b\) are both
\(e(rg)d(rg)b(g)\). For elementary sections, expansion of (DS.1)
gives exactly \(b^*i_B(\langle e,f\rangle)c\). Thus all algebraic
Gram positivity and the null relation follow from the interior tensor
product, rather than from an unproved linking crossed-product theorem.

Here is a useful completion estimate. A section supported on one
bisection has inner product with itself supported on units: in (DS.1),
the two arrows \(h^{-1}\) and \(h^{-1}g\) have the same range, and
range injectivity on the bisection forces them equal. Its norm is
therefore at most its uniform vector norm. If a common compact support
has an \(N\)-bisection partition, each partition member has uniform
norm at most \(\varepsilon\), so the section norm is at most
\(N\varepsilon\). Initially this estimate holds for elementary
sections; it makes their local uniform approximations Cauchy.

On a bisection, the range coordinate identifies a pullback section with
a local section of the Hilbert bundle of \(E\). Such sections are locally
uniformly approximated by global module sections, with compact unit
bumps. The dense equality \(EB=E\) permits each of those to be
approximated by finite sums \(e_jd_j\). Multiplying the scalar arrow
partition into \(d_j(rg)\) gives precisely (DS.2). A finite partition
on a compact support and the preceding estimate prove density in both
the core completion and the interior tensor module. Both directions of
the identification are consequently isometric and onto. A countable
dense family of \(E\), coefficients and relatively compact bisection
bumps makes this module countably generated. No original fibre is
replaced by a fixed-rank model. \(\square\)

For any \(T\in\mathcal L_B(E)\), the pointwise rule
\((\widetilde T\xi)(g)=T_{rg}\xi(g)\) is, under (DS.2),
\(T\otimes1\). It is a contractive adjoint-preserving homomorphism
on adjointable operators. Such operators commute with the anchor:
\(B\)-linearity first gives commutation with coefficient products and
then with their strict scalar multiplier limits.

**Lemma DS.2 (the étale compact extension).** For two such modules,
\(K\mapsto K\otimes1\) sends
\(\mathcal K_B(E_2,E_1)\) into
\(\mathcal K_{B_\tau}(X_{E_2,\tau},X_{E_1,\tau})\).

**Proof.** Approximate both vectors of a rank-one operator by finite
sums \(e_jb_j\) and \(f_kc_k\). On elementary tensors

\[
 \theta_{eb,fc}\otimes1
  =\theta_{e\otimes i_B(b),\,f\otimes i_B(c)}.
\tag{DS.3}
\]

The equality follows by inserting
\(\langle fc,z\rangle=c^*\langle f,z\rangle\).
Its two vectors are actual module vectors because \(i_B(b),i_B(c)\)
belong to \(B_\tau\), not just to its multiplier algebra. Rank-one
approximation and contractivity prove the assertion. It includes the
rectangular and graded versions. \(\square\)

#### 11AB.1. DS.2. The full left representation, including upper semicontinuous norms

On the core define

\[
 (\Lambda_\pi(a)\xi)(g)
  =\sum_{rh=rg}\pi_{rg}(a(h))U_h\xi(h^{-1}g).
\tag{DS.4}
\]

The transport sends \(E_{sh}\) to \(E_{rg}\). This formula commutes
with the right action in (DS.1). Directly reindexing the finite sums
gives multiplication and
\(\Lambda_\pi(a)^*=\Lambda_\pi(a^*)\): use
\(U_h\pi_{sh}(d)U_h^{-1}=\pi_{rh}(\alpha_h(d))\),
\(U_hU_k=U_{hk}\), and
\(\langle U_hx,U_hy\rangle=\beta_h(\langle x,y\rangle)\).
For example the product sum has indices \((h,k)\) with
\(rh=rg,rk=sh\); substitution \(l=hk\) gives precisely
\(\alpha_h(a'(h^{-1}l))\) in the coefficient convolution.
The adjoint substitution changes \((h,g)\) to
\((h^{-1},h^{-1}g)\); applying the inner-product covariance gives
the stated involution, with no Haar or modular factor in counting
measure.

If \(a\) is supported on one bisection,
\(\Lambda(a)^*\Lambda(a)=\widetilde\pi(a^**a)\) has unit
support. It is bounded by \(\|a\|_\infty^2\). This proves
boundedness on one bisection before using a universal crossed norm;
finite partitions prove boundedness on every compact support.
For a scalar \(c\) on one bisection the same calculation gives an
adjointable operator \(V_c\) with norm at most \(\|c\|_\infty\).
It uses the module action alone; its adjoint is \(V_{c^*}\).

**Lemma DS.3 (weighted bisection Schur bound).** Let
\(R=\sup_y\sum_{rh=y}\|a(h)\|\) and
\(S=\sup_x\sum_{sh=x}\|a(h)\|\). On both completed modules,

\[
 \|\Lambda_\pi(a)\|\le\sqrt{RS}\le\|a\|_I.
\tag{DS.5}
\]

**Proof.** Choose finitely many \(0\le\rho_i\le1\), compactly
supported inside bisections, with \(\sum_i\rho_i=1\) on
\(\operatorname{supp}a\) and \(\sum_i\rho_i\le1\) everywhere.
First suppose \(p(h)=\|a(h)\|\) is continuous. On the range of the
\(i\)-th bisection put
\(c_i(rh)=\rho_i(h)p(h)\), extending by zero, and set
\(d_i(h)=\rho_i(h)a(h)/\sqrt{c_i(rh)}\), with value zero at zeros.
The estimate \(\|d_i(h)\|\le\sqrt{\rho_i(h)p(h)}\) proves
continuity at those zeros. Compact supports strictly inside the
bisection prove continuity of all zero extensions. Then
\(a=\sum_i\sqrt{c_i}\,d_i\).

The row of scalar unit multipliers \((\sqrt{c_i})_i\) has norm at
most \(\sqrt R\). The column \((\Lambda(d_i))_i\) has squared
norm at most \(S\), since
\(\sum_i\Lambda(d_i)^*\Lambda(d_i)\) is the unit coefficient
representation of
\(\sum_{sh=x,i}\alpha_h^{-1}(d_i(h)^*d_i(h))\).
Each positive coefficient is at most
\(\rho_i(h)p(h)1\) in its multiplier algebra. This proves the
column bound without a nondegeneracy assumption on \(\pi\).
Their row-column product is \(\Lambda(a)\).

For an upper semicontinuous bundle, the function \(p\) is upper
semicontinuous and zero off the compact support. There are continuous
compactly supported majorants \(p_n\downarrow p\) on one fixed
relatively compact neighbourhood of that support. Here is a construction.
Choose a compact metrizable neighbourhood \(K'\) and a scalar bump
\(\chi\) equal to one on the support, with support in its interior.
On \(K'\), the functions
\(h_n(x)=\sup_{z\in K'}(p(z)-n d(x,z))_+\) are continuous,
decrease, and converge to \(p(x)\). Upper semicontinuity proves the
last assertion by taking maximizers and a convergent subsequence.
Use \(p_n=\chi h_n\), extended by zero. A compatible metric exists
because the groupoid is second countable LCH Hausdorff. These functions
majorize \(p\), and their pointwise limit is exactly \(p\).

Repeat the factorization with \(p_n\). Its row and column bounds are
\(R_n=\sup_y\sum_{rh=y,i}\rho_i(h)p_n(h)\) and
\(S_n=\sup_x\sum_{sh=x,i}\rho_i(h)p_n(h)\). The finite counting
sums are continuous, decrease, and have fixed compact unit supports.
Their suprema decrease to \(R,S\). Otherwise some decreasing compact
upper-level sets would all be nonempty and have a common point, in
contradiction with the pointwise limit. Thus
\(\|\Lambda(a)\|\le\inf_n\sqrt{R_nS_n}=\sqrt{RS}\).
All identities were on the completed tensor modules of DS.1.
\(\square\)

Compose this \(I\)-bounded star representation with a faithful
ordinary representation of \(\mathcal L(X_{E,m})\), obtained by
tensoring with a faithful GNS sum of \(B_m\). Its faithfulness follows
by testing the positive element \(\langle T\xi,T\xi\rangle\),
as in PN.1. The full universal norm therefore bounds
\(\|\Lambda_\pi(a)\|\). Consequently (DS.4) extends to an actual
representation of \(A_m\) on \(X_{E,m}\). This proof uses no
disintegration of a groupoid representation.

#### 11AB.1. DS.3. Reduced norm through actual source columns

Let \(\mathcal R_B\) be the PN.4 source-column Hilbert \(B\)-module.
The representation \(\lambda_B:B_r\to\mathcal L_B(\mathcal R_B)\)
is faithful by the definition of the reduced completion. Tensoring a
Hilbert \(B_r\)-module with this correspondence is faithful on its
adjointable operators: for \(T\ne0\), a nonzero positive inner product
\(d\) has \(\lambda_B(d)\ne0\), and some column \(v\) has
\(\langle v,\lambda_B(d)v\rangle\ne0\). Thus its tensor test
detects \(T\).

Let \(\mathcal R_E\) be the completion of
\(C_c(G,s^*E)\) with right pointwise \(B\)-action and inner product
\(\sum_{sg=x}\langle v(g),w(g)\rangle_{B_x}\).
There is an even unitary

\[
\begin{aligned}
X_{E,r}\widehat\otimes_{B_r,\lambda_B}\mathcal R_B
 &\longrightarrow \mathcal R_E,\\
 e\otimes b\otimes\zeta&\longmapsto
 [g\mapsto U_g^{-1}e(rg)(\lambda_B(b)\zeta)(g)].
\end{aligned}
\tag{DS.6}
\]

**Proof of this unitary.** Associativity reduces the left side to
\(E\otimes_{B,i_B}\mathcal R_B\). Unit multiplication on the
regular column is \(\beta_g^{-1}(d(rg))\), so the rule
\(e\otimes\zeta\mapsto U_g^{-1}e(rg)\zeta(g)\) is balanced.
Its inner product at \(x\) is
\(\sum_{sg=x}\zeta(g)^*\beta_g^{-1}(\langle e,f\rangle(rg))
\eta(g)\), exactly the interior-tensor inner product. Bisection
partitions, transport by \(U_g\), and the density \(EB=E\) give
dense range, with the same finite-support estimate as DS.1. This proves
isometry and surjectivity. On general core tensors the rule is
\(\sum_{rh=rg}U_g^{-1}\xi(h)\zeta(h^{-1}g)\); every coefficient
now belongs to \(B_{sg}\), so no mistyped transport of a source
coefficient is present.

Under (DS.6), the induced left operator is

\[
 (L_\pi(a)v)(g)=
 \sum_{rh=rg}\pi_{sg}(\alpha_g^{-1}(a(h)))v(h^{-1}g).
\tag{DS.7}
\]

Indeed \(U_g^{-1}\pi_{rg}(a(h))U_h
=\pi_{sg}(\alpha_g^{-1}a(h))U_{h^{-1}g}^{-1}\).
On the source fibre at \(x\) its matrix, indexed by \(g,l\in G_x\),
is \(\pi_x(\alpha_g^{-1}(a(gl^{-1})))\). The PN.6 regular
matrix for \(A\) has exactly those entries before applying \(\pi_x\).
Every finite rectangular compression is a matrix over \(A_x\), and
the matrix amplification of the possibly degenerate star homomorphism
\(\pi_x\) is contractive. Hence its norm is at most the corresponding
regular matrix norm, and at most \(\|a\|_{A_r}\). A finite input
column has finite output for a compact bisection-covered support, so
these finite compressions give the bound on all finite columns and then
on their completion. Module norms are suprema of the fibre norms; thus
the same bound holds globally on \(\mathcal R_E\).

Faithfulness of the preceding tensor test now proves

\[
 \|\Lambda_\pi(a)\|_{\mathcal L(X_{E,r})}
                 \le\|a\|_{A_r}.
\tag{DS.8}
\]

This is the actual reduced left representation. It uses neither reduced
exactness nor a complemented essential submodule. In particular it does
not assume \(\overline{\pi(A)E}\) has an orthogonal complement.

#### 11AB.1. DS.4. Compact convolution kernels

**Lemma DS.4.** If
\(k\in C_c(G,r^*\mathcal K_B(E_2,E_1))\), then

\[
 (K_k\eta)(g)=\sum_{rh=rg}k(h)U_h^2\eta(h^{-1}g)
\tag{DS.9}
\]

is compact between the two crossed modules for both completions.
Its norm is at most \(\|k\|_I\), with the coefficient operator norm
in the two counting sums.

**Proof.** Apply DS.3's Schur calculation to the equivariant coefficient
\(\mathcal K(E_1\oplus E_2)\), represented on \(E_1\oplus E_2\),
and take the rectangular corner. This proves the bound without asserting
a full crossed-linking-algebra isomorphism. On each bisection, approximate
the compact-operator section locally by finite sums
\(\theta_{e(rh),f(rh)}c(h)\). Upper semicontinuity of the norm of
the error gives uniform approximation on a finite compact-support
partition. The resulting operator is
\((\theta_{e,f}\otimes1)V_c\). It is compact by DS.2. An error
\(\varepsilon\) supported on \(N\) bisections has \(I\)-norm at
most \(N\varepsilon\); hence the operators converge in norm. Closedness
of the rectangular compact space proves the claim. These are genuine
continuous compact sections; pointwise compactness without that local
continuity would not suffice. \(\square\)

The module and norm proofs retain the original unit coefficient at
every stage. The group-only linking and regular-representation arguments
in *Descent and the K-theory of crossed products*, Sections 1–3, are
orientation for this construction; they are not substituted for the
typed groupoid proofs above.

### 11AB.2. Cycles, tensor products and the coefficient quotient under anchored descent

The standing hypothesis, algebras, modules,
crossed norms and inner-product convention are those of the preceding
module construction (DS.1–DS.9). Every source and coefficient starts over the same
original \(C_0(T)\). The target of descent is ordinary KK of the actual
crossed algebras; it does not evaluate the original anchor at a point.

#### 11AB.2. DS.5. The actual cycle convention and its descent

An anchored equivariant cycle \((E,\pi,F)\) has odd adjointable
\(F\), ordinary compact commutator, square and adjoint defects, and
the following localized arrow condition. For each compactly supported
continuous section \(a(g)\in A_{rg}\),

\[
 g\longmapsto\pi_{rg}(a(g))
       (F_{rg}-U_gF_{sg}U_g^{-1})
       \quad\text{is in }C_c(G,r^*\mathcal K(E)).
\tag{DS.10}
\]

The two-sided version follows by taking adjoints after self-adjoint
normalization. Continuity means continuity as a compact-operator bundle
section, with uniform norm control on a fixed compact arrow support.
It does not assert norm continuity of the unlocalized adjointable
operator orbit. The ordinary compact conditions are conditions on
\(\mathcal K_B(E)\), not merely on its individual fibres.

The ordinary self-adjoint normalization and clipping proof retains
the anchor because all adjointable operations commute with \(C_0(T)\).
It retains (DS.10) as follows. Polynomial differences of transported
and original operators are finite sums containing an arrow defect.
Move the source representation through the other factors, using its
ordinary compact commutator. The localized defect then is compact.
Multiplying a continuous compact section by a bounded strictly
continuous adjointable field is norm continuous: check rank-one
sections on their vectors and adjoints, then approximate. Uniform
polynomial approximation on the bounded spectra gives the same
conclusion for continuous functional calculus. The standard compact
perturbation homotopies have precisely these localized equations.
We may therefore use self-adjoint contractions below.

**Theorem DS.5 (full and reduced descent of cycles).** On the modules
constructed in DS.1–DS.3,

\[
 j_\tau[E,\pi,F]
   =[X_{E,\tau},\Lambda_\pi,\widetilde F]
      \in KK(A_\tau,B_\tau),\qquad \tau=m,r.
\tag{DS.11}
\]

These assignments preserve sums, unitary equivalence, degeneracy and
homotopy, and give group homomorphisms on the homotopy groups of actual
anchored equivariant cycles.

**Proof of the cycle conditions.** For homogeneous
\(a\in C_c(G,r^*A)\), the descended graded commutator is (DS.9)
with coefficient

\[
 [F_{rh},\pi_{rh}(a(h))]_{\rm gr}
 +(-1)^{|a|}\pi_{rh}(a(h))
       (F_{rh}-U_hF_{sh}U_h^{-1}).
\tag{DS.12}
\]

The second term is a continuous compact section by (DS.10). The first
is one because the bounded compact-valued map
\(a\mapsto[F,\pi(a)]_{\rm gr}\) pulls back locally to the arrow
bundle: approximate the varying section on finitely many bisections
by global unit sections and scalar arrow bumps. The same argument
applies to the square coefficient
\((F_{rh}^2-1)\pi_{rh}(a(h))\), and before normalization to
\((F_{rh}-F_{rh}^*)\pi_{rh}(a(h))\). Lemma DS.4 makes all three
descended operators compact. Density of the convolution core and
closedness of the compact ideal give the whole-source conditions.
All representations are bounded in their actual source norm by
DS.3 and (DS.8).

Even unitary equivalences and direct sums intertwine the formulas.
When every original defect is zero, the displayed kernels are zero,
so a degenerate cycle descends to a degenerate cycle. Interval
homotopies are checked in DS.8 below; thus these assignments are
well-defined group homomorphisms. \(\square\)

#### 11AB.2. DS.6. The exact crossed tensor unitary

Let \(E_1:A\to B\) and \(E_2:B\to C\) be actual equivariant
correspondences, with representations \(\pi_1,\pi_2\). Put
\(E=E_1\widehat\otimes_BE_2\), with its diagonal arrow action.
The formula

\[
\begin{aligned}
 W_\tau:X_{E_1,\tau}\widehat\otimes_{B_\tau}X_{E_2,\tau}
                &\longrightarrow X_{E,\tau},\\
 W(\xi\widehat\otimes\eta)(g)
 &=\sum_{rh=rg}\xi(h)\widehat\otimes
                           U_h^2\eta(h^{-1}g)
\end{aligned}
\tag{DS.13}
\]

is an even unitary for each \(\tau\), intertwining both right
coefficient convolution and the left source representation.

**Proof: balancing.** Expanding \(W((\xi b)\otimes\eta)\), write
the first convolution arrow as \(l\), and put \(h=lk\). Its term is
\(\xi(l)\beta_l(b(k))\otimes U_{lk}^2\eta(k^{-1}l^{-1}g)\).
The tensor balancing and covariance of \(\pi_2\) change this to
\(\xi(l)\otimes U_l^2\pi_2(b(k))U_k^2
\eta(k^{-1}l^{-1}g)\). Summing is exactly
\(W(\xi\otimes\Lambda_{\pi_2}(b)\eta)\).
The same finite reindexing proves right \(C_\tau\)-linearity.

**Proof: the precise inner product.** Expand the right-side inner
product by (DS.1), first with arrow \(l\), and then the two tensor
sums with arrows \(h,k\), where \(rh=rk=sl\). Its term is

\[
 \gamma_l\left\langle
 U_h^2\eta(h^{-1}l^{-1}),\,
 \pi_2(\langle\xi(h),\xi'(k)\rangle_B)
 U_k^2\eta'(k^{-1}l^{-1}g)
 \right\rangle.
\tag{DS.14}
\]

Here the two vectors and represented coefficient lie at the unit
\(sl\). Use covariance of \(U_h^2\), and substitute
\(u=lh\), \(v=h^{-1}k\), so that
\(h^{-1}l^{-1}=u^{-1}\) and
\(k^{-1}l^{-1}g=v^{-1}u^{-1}g\). For fixed \(u,v\) the coefficient
sum over \(sh=su=rv\) becomes

\(\sum_{sh=rv}\beta_h^{-1}
 (\langle\xi(h),\xi'(hv)\rangle_B)
 =\langle\xi,\xi'\rangle(v)\).

The resulting expression is exactly
\(\langle\eta,\Lambda_{\pi_2}
 (\langle\xi,\xi'\rangle)\eta'\rangle(g)\), the interior-tensor
inner product on the left. All substitutions are composable-arrow
bijections of finite compact-support sums. There is no permutation of
homogeneous tensor factors, so no hidden Clifford sign occurs in this
calculation. Equality of these algebraic \(C\)-crossed inner products
proves isometry in each completion.

**Proof: dense range and left intertwining.** If \(\xi\) is supported
on the actual unit bisection, with \(\xi(1_y)=e(y)\), then
\(W(\xi\otimes\eta)(g)=e(rg)\otimes\eta(g)\).
Take a compact unit bump equal to one on the range of a compact target
support, so that this unit section has compact support. Local tensor
density and finite bisection partitions approximate every compact
section of \(r^*E\) by such sections. The \(N\varepsilon\) estimate
in DS.1 converts this approximation to module norm approximation.
Thus \(W\) is onto, including when \(\pi_2\) is degenerate.
Substitute (DS.4) and put \(h=lk\) in the left-action sum; the diagonal
unitary is \(U_l^1\otimes U_l^2\), so this gives the same left action
on both sides of (DS.13). \(\square\)

#### 11AB.2. DS.7. Connections and positivity for an actual equivariant product

Suppose \((E_1,\pi_1,F_1)\) and \((E_2,\pi_2,F_2)\) are normalized
cycles and \(F\) is an actual anchored equivariant product operator on
\(E\). Precisely, it is an equivariant cycle, an \(F_2\)-connection,
and, with \(S=F_1\otimes1\) and \(\pi=\pi_1\otimes1\),
\(\pi(a)(SF+FS)\pi(a)^*\ge0\) modulo \(\mathcal K(E)\) for
every \(a\in A\).

**Theorem DS.7 (descent preserves actual products).**

\[
 j_\tau(x\widehat\otimes_B y)
       =j_\tau(x)\widehat\otimes_{B_\tau}j_\tau(y),
                         \qquad \tau=m,r.
\tag{DS.15}
\]

**Proof: the connection.** For a homogeneous creation section
\(\xi\in\mathcal X_{E_1}\), its creation operator after (DS.13)
is the convolution sum with \(T_{\xi(h)}U_h^2\). The error against
\(\widetilde F_2\) has coefficient

\[
\begin{aligned}
 d_\xi(h)={}&F_{rh}T_{\xi(h)}
       -(-1)^{|\xi|}T_{\xi(h)}F_{2,rh}\\
 &+(-1)^{|\xi|}T_{\xi(h)}
            (F_{2,rh}-U_h^2F_{2,sh}(U_h^2)^{-1}).
\end{aligned}
\tag{DS.16}
\]

The first line is compact by the actual connection condition. Its
dependence on the vector is bounded and norm continuous, so local
unit-section approximation proves continuity as a rectangular compact
bundle section. For the second line, approximate \(\xi(h)\) uniformly
on a finite compact bisection cover by vectors
\(e_j(rh)b_j(h)\), with \(b_j\in C_c(G,r^*B)\). Then
\(T_{e_jb_j}=T_{e_j}\pi_2(b_j)\). Condition (DS.10) on the second
cycle makes the localized defect compact and continuous. The uniform
error is bounded by \(2\|F_2\|\) times the vector error, so the limit
has the same property. This establishes genuine compact-section
continuity without a global norm-continuous orbit of \(F_2\).
Lemma DS.4 makes the descended creation error compact. Adjointing
gives the adjoint creation error, with the same graded sign. Core
vector density and boundedness extend the connection condition to
every vector of the first crossed module.

**Proof: positivity by unit approximate identities.** Under (DS.13)
the first descended operator tensored with one is exactly
\(\widetilde S\); putting \(F_1\xi(h)\) into the tensor sum proves
this without commuting \(F_1\) with \(U_h^1\). Set \(T=SF+FS\),
an even self-adjoint operator. Lemma DS.2 shows that
\(L(E)\to L(X_{E,\tau})\), \(R\mapsto\widetilde R\), sends
all original compacts into crossed compacts. It therefore induces
a star homomorphism of the two Calkin algebras, preserving positivity.
The original product criterion yields
\(\widetilde{\pi(a)T\pi(a)^*}\ge0\) in the crossed Calkin algebra.

Choose an even positive contractive approximate identity \(a_n\) of
\(A\). For every \(z\in A_\tau\),
\(z i_A(a_n)\to z\) and \(i_A(a_n)z\to z\) in crossed norm.
For a core element this follows by uniform approximation on finitely
many compact bisection supports, and the \(I\)-bound; density proves
it for all \(z\). This is the nondegenerate unit-inclusion assertion
of PN.1, applied before any representation. In particular it remains
valid when \(\pi\) is degenerate. Hence

\[
\begin{aligned}
 \Lambda(z)\widetilde T\Lambda(z)^*
  &=\lim_n \Lambda(z)\widetilde{\pi(a_n)}
        \widetilde T\widetilde{\pi(a_n)}\Lambda(z)^*\\
  &\ge0\pmod{\mathcal K(X_{E,\tau})}.
\end{aligned}
\tag{DS.17}
\]

Each approximating compression is positive in the quotient, and its
positive cone is norm closed. No reduced exactness and no algebra
generated by globally norm-continuous translates of \(T\) is used.
The connection and positivity criteria now identify \(\widetilde F\)
with the ordinary Kasparov product of the two descents, proving
(DS.15). \(\square\)

**Exact product-existence input.** The existence, uniqueness and
associativity of the *original anchored equivariant* product are a
separate foundation. The exact input is KG.1–KG.4 of
*Anchored étale technical theorem, products and the exactness actually
used*: local bisection transport,
convex approximate invariance in the actual pulled-back compact
operator bundle, the strict-series technical partition with continuous
compact arrow defects, and relative-anchor connection/product
construction. Its ordinary operator proof inputs are Kasparov’s technical theorem, Connections and the existence of the Kasparov product and Homotopy, associativity, the index pairing and KK-equivalence;
KT-KK-17's group product theorem is not a groupoid substitute. The
proof of DS.7 above applies to every actual product representative
independently of how that representative was obtained. The categorical
existence/associativity assertion uses this exact separately proved
foundation, rather than an inferred consequence of the descent
formulas. The KG.1–KG.4 provider is a complete proof at this scope.
In particular its
multiplier-only separation defects are compact-unit-localized before
forming the genuine anchored algebras, and their original source and
creation tests are recovered by nondegenerate unit bump approximation.
No unresolved product-existence premise remains for this supplied
standing hypothesis. This proof consumes that technical foundation;
proper-source localization remains a separate theorem.

#### 11AB.2. DS.8. Interval homotopies and coefficient naturality

For \(I=[0,1]\), with trivial action on its coordinate, there are
actual graded isomorphisms

\[
 (C(I)\otimes B)\rtimes_\tau G
                  \cong C(I)\otimes B_\tau.
\tag{DS.18}
\]

**Proof.** A finite scalar partition of the interval uniformly
approximates a compact-arrow section by finite sums \(u_j(t)b_j(g)\)
on a common arrow support. The \(I\)-norm estimate controls the
approximation. In the full completion, constant-interval coefficient
sections give a copy of \(B_m\), isometric because endpoint evaluation
is a contractive reverse map. The unit multiplier inclusion gives a
commuting central copy of \(C(I)\). For
\(z(t)=\sum_j u_j(t)d_j\), approximate by
\(\sum_l\tau_l(t)z(t_l)\), with \(\tau_l\ge0\) and
\(\sum_l\tau_l=1\). In any representation of the commuting copies
the row \((\sqrt{\tau_l})\) and diagonal \((z(t_l))\) give norm
at most \(\max_l\|z(t_l)\|\). Passage to the limit gives
\(\|z\|\le\sup_t\|z(t)\|\); evaluation gives the reverse bound.
This proves the full norm equality and surjectivity using the finite
core approximation. It is a direct proof of the interval tensor norm,
not an assertion of groupoid disintegration or full exactness.

For the reduced completion the source-column module is
\(C(I)\otimes\mathcal R_B\), by the same local partition and inner
product proof. Its regular operator is the continuous interval field
of the PN.5 operators. Its norm is the supremum of their norms: the
upper bound holds pointwise, and localized interval bump columns give
the reverse bound at every interval value. The finite core approximation
again gives onto. This proves the reduced assertion without a reduced
quotient-exactness assumption. \(\square\)

For a homotopy module over \(C(I)\otimes B\), evaluating a compact
arrow section at \(t\) gives an even unitary
\(X_{E,\tau}\otimes_{\operatorname{ev}_t}B_\tau
\cong X_{E_t,\tau}\). The inner products agree by (DS.1) and
evaluation; local vector lifting and finite bisection partitions prove
dense range, hence onto. The descended operator and representation
evaluate to the original endpoint descents. Thus DS.5 really preserves
interval homotopies on completed modules.

For an anchored equivariant homomorphism \(v:B\to C\), convolution
of its coefficient sections defines \(v_\tau:B_\tau\to C_\tau\).
The full map is contractive by composing every \(I\)-bounded
representation with the core map. For the reduced map apply (DS.8)
to the standard module \(C\), with representation \(v\). Its core
image lies in \(C_\tau\), and the reduced estimate gives the desired
homomorphism. No injectivity, nondegeneracy or reduced exactness is
assumed. The tensor unitary (DS.13), with \(E_2=C\), identifies
coefficient change before descent with tensoring after descent.
Pullback on the source commutes immediately with (DS.4). These are
both coefficient naturalities of DS.11.

For the actual homomorphism cycle \((B,v,0)\), the core is the
coefficient convolution algebra, and the descended representation is
the just constructed crossed homomorphism. Therefore

\[
 j_\tau[v]=[v_\tau],\qquad j_\tau(1_A)=1_{A_\tau}.
\tag{DS.19}
\]

#### 11AB.2. DS.9. The exact full-to-reduced quotient square

Let \(q_B:B_m\to B_r\) be the canonical quotient. Associativity
in (DS.2) gives an actual even unitary

\[
 X_{E,m}\widehat\otimes_{B_m,q_B}B_r
                  \cong X_{E,r}.
\tag{DS.20}
\]

On the convolution core it sends \(\xi\otimes c\) to \(\xi_r c\).
Its inner product is the image under \(q_B\) of (DS.1), and the
right coefficient completion has dense range. The left representation
is the reduced (DS.4) and the pointwise operator is the same
\(\widetilde F\). Consequently, for every actual class,

\[
 [q_A]\widehat\otimes_{A_r}j_r(x)
       =j_m(x)\widehat\otimes_{B_m}[q_B]
                      \quad\text{in }KK(A_m,B_r).
\tag{DS.21}
\]

This is a module identification, not a conclusion from reduced
ideal exactness. If \(B\) is proper, PN.3 separately makes \(q_B\)
an isomorphism. Equation (DS.21) supplies exactly the quotient square
required by CA.7–CA.10. It does not make the coefficient-free quotient
\(C^*(G)\to C_r^*(G)\) invertible, produce original-unit Dirac factors,
or turn an ordinary forgetful product into an equivariant identity.

For a cycle from \(C_0(T)\) to itself with the canonical central source
action, restriction of (DS.11) along \(i_{T,\tau}\) is precisely
\(E\otimes_{C_0(T)}C_0(T)_\tau\) with the ordinary forgetful operator.
Hence

\[
 [i_{T,\tau}]\widehat\otimes j_\tau(x)
   =\operatorname{For}_G(x)\widehat\otimes_{C_0(T)}[i_{T,\tau}].
\tag{DS.22}
\]

This verifies the completed-module interface of CA.2 directly. It
retains the whole original unit algebra and evaluates no individual
unit fibre.

#### 11AB.2. DS.10. Clifford degrees and the full categorical conclusion

Every core grading is coefficient grading, and every arrow unitary
is even. Finite auxiliary Clifford algebras have trivial groupoid
action. There are natural even identifications

\[
 (B\widehat\otimes\mathrm{Cl}_p)\rtimes_\tau G
       \cong B_\tau\widehat\otimes\mathrm{Cl}_p.
\tag{DS.23}
\]

Here the core map sends \(b(g)\otimes c\) to the same coefficient.
Multiplying homogeneous elementary terms gives exactly
\((-1)^{|c||b'|}(b*b')(g)\otimes cc'\), and the adjoint gives
\((-1)^{|b||c|}b^*(g)\otimes c^*\). Thus the map preserves the
graded star algebra. Here are both norm arguments, including the
one-generator Clifford algebra whose grading interchanges its two
ungraded scalar summands.

The ordinary graded tensor product with a finite Clifford algebra has
one C*-norm. Its elementary matrix verification starts with
\(B\widehat\otimes\mathrm{Cl}_1\): the generator \(\varepsilon\)
satisfies \(\varepsilon b\varepsilon=\gamma_B(b)\), so the algebra
is the two-element grading crossed algebra. Its regular matrix is
\(b\mapsto\operatorname{diag}(b,\gamma_B b)\) and
\(\varepsilon\mapsto\left(\begin{smallmatrix}0&1\\1&0\end{smallmatrix}\right)\).
The full and regular norm equality for this finite action follows
directly from PN.3, with its proper one-point central space and constant
square cutoff \(1/\sqrt2\). Thus this matrix norm is the universal
one as well. Iterating the Clifford generators gives the finite
graded tensor norm; the usual fixed ordered Clifford matrix
identifications give the same norm at every step. This is a finite
ordinary coefficient fact, proved before its use here.

For full descent, the copy of \(B_m\) and the constant Clifford
multiplier copy inside \((B\widehat\otimes\mathrm{Cl}_p)_m\)
graded-commute: check the displayed signs on the core, and use
nondegeneracy of the unit coefficient inclusion to extend the constant
Clifford multipliers. The preceding universal finite graded tensor
norm gives a homomorphism from
\(B_m\widehat\otimes\mathrm{Cl}_p\) into this crossed algebra.
The reverse core map is \(I\)-bounded. Indeed a one-bisection
coefficient has its star-square in the unit algebra
\(B\widehat\otimes\mathrm{Cl}_p\), whose representation into
\(B_m\widehat\otimes\mathrm{Cl}_p\) is contractive. The same
row-column factorization as DS.3 gives the \(I\)-bound on a general
compact support. Compose with a faithful ordinary representation and
use the full universal norm. The two maps are inverse on the dense
core, proving the full isomorphism.

For reduced descent, take the standard Clifford coefficient module
\(E=B\widehat\otimes\mathrm{Cl}_p\), viewed as a Hilbert module
over itself, and form the finite matrix model of its regular source
columns. The arrow action on the Clifford coordinates is trivial.
Consequently its source-column matrix is exactly the preceding finite
graded matrix model applied entry by entry to the PN.6 regular B
matrix; conversely every such finite model is present. The faithful
finite regular matrix model therefore gives equality of the two
reduced norms. Finite Clifford coordinate expansion and the common
compact-support approximation prove surjectivity. This proves (DS.23)
without treating an odd Clifford generator as an ungraded scalar.

The modules and \(W\) are even. Their graded inner products and
balancing move no Clifford factor past another. When combining degrees,
use the same fixed ordered identification
\(\mathrm{Cl}_p\widehat\otimes\mathrm{Cl}_q\cong
\mathrm{Cl}_{p+q}\) before and after descent. An actual interchange
introduces precisely the standard signed graded flip, on both sides.
Thus the cycle maps, tensor/product identity and quotient square hold
in both parities and preserve every original right Clifford factor.

With the separately supplied completed KG.1–KG.4 existence/uniqueness
foundation, (DS.11), (DS.15), (DS.18)–(DS.23) give the full and reduced
graded descent functors at this actual anchored étale scope. Every
analytic descent requirement is proved above. The KG foundation is
the only product-existence input; it is not replaced by a
group theorem or a citation to an external groupoid theorem.

Human-source orientation: Alistair Miller,
[*Functors between Kasparov categories from étale groupoid correspondences*](https://arxiv.org/html/2303.02089v2),
Section 2.4, gives the standard equivariant cycle and product scope.
The range-fibre completion, reduced matrix proof, exact tensor sums,
localized compact kernels, étale positivity proof, interval norm and
quotient square here are supplied in original expression, with their
complete proofs above and the stated completed technical foundation.

![Typed groupoid descent modules, reduced source columns, compact tensor extension and the quotient square](../figures/anchored-etale-descent.png)

**Figure 11AB.1.** The arrow diagram keeps every source and range in the range-fibre formula. The source-column matrix has actual E_x entries. Étale unit coefficients make ordinary rectangular compacts descend to compacts, and the lower right gives the completed quotient module and the two equal KK products. This is a labelled schematic, with no claimed metric scale. Proof locators: DS.1–DS.9, DS.13–DS.17 and DS.20–DS.22. [Vector figure](../figures/anchored-etale-descent.svg) · [Generator](../reproduction/anchored-etale-descent/draw_descent.py) · [Finite checks](../reproduction/anchored-etale-descent/finite_checks.py) · [Exact data](../reproduction/anchored-etale-descent/figures/anchored-etale-descent-data.json) · [Reproduction instructions](../reproduction/anchored-etale-descent/REPRODUCTION.md) · [Component terms](../reproduction/anchored-etale-descent/COMPONENT-TERMS.md).


## 11AC. Labelled geometric kernels over the original unit space

This text has two different conclusions. GK.1–GK.3 establish the actual
metric/frame geometry and the failures of two proposed shortcuts. GK.4–GK.7
prove the compact-fibre kernel construction from its exact coarse input,
including the whole original unit space and an ordinary section. GK.8 gives
that input for genuine discrete action groupoids. GK.9 points to the complete
general eligible-holonomy proof in Section 11AF; GK.10 proves equivalence
of that exact input with the requested interface.

Let G ⇒ T be a Hausdorff second-countable locally compact étale groupoid. For
the holonomy applications, T is a smooth complete transversal to a regular
foliation on a compact manifold. Its supplied normal data include a positive
holonomy-invariant metric and the actual unitary Spin-c transport. Multiplication
is gh when s(g)=r(h). G_x=s^{-1}(x), G^x=r^{-1}(x). Distinct arrows, including
isotropy arrows, are distinct elements throughout.

The corrected compactness requirement is relative to T: an auxiliary anchor
p:Y→T is proper and surjective, with nonempty compact fibres. It is not necessary
that Y itself be compact when T is noncompact. A continuous section y0:T→Y
is an ordinary section; it need not be equivariant. The original full normal
inverse module is pulled back, not replaced by a numerical determinant.

### 11AC.1. GK.1. What the invariant positive metric proves

A holonomy germ h:(T,x)→(T,z) is a local isometry. The metric determines its
Levi-Civita connection: the Koszul identity determines each inner product
⟨∇_XY,Z⟩ from the metric, its derivatives, and brackets. Pullback by an isometry
preserves all these expressions, and hence the connection. Therefore h sends
geodesics to geodesics with the same parameter. The coordinate geodesic ODE
has unique local solutions: on a compact coordinate box its first-order right
side is locally Lipschitz, and integration on a sufficiently short time interval
is a contraction. Its solution depends smoothly on initial data by successive
integration and differentiation. At zero the derivative of exp_x is the identity;
the local inverse follows by the same contraction argument applied to
v ↦ y−(exp_x(v)−v). Thus

\[
h(\exp_x v)=\exp_z(dh_xv)
\tag{GK.1}
\]

for sufficiently small v. Value and derivative determine the germ. The map

\[
j:G\longrightarrow\operatorname{Iso}(TT),\qquad
j(g)=(s(g),r(g),dh_{s(g)})
\tag{GK.2}
\]

is consequently injective and continuous. Continuity is checked in an actual
holonomy bisection, where source, target and derivative are smooth. The target
is a Hausdorff locally compact bundle with O(q) as its local matrix fibre. None
of this proves that j is a topological embedding or a proper map.

The orthonormal frame bundle P→T is Hausdorff, second countable and locally
trivial with compact fibre O(q). Its anchor is proper. Indeed, cover a compact
K⊂T by finitely many relatively compact trivializing neighborhoods. Choose
smaller neighborhoods whose closures lie in those charts and still cover K.
The inverse image of each such closure is a closed subset of a product of two
compact spaces. The inverse image of K is a closed subset of their finite
union. The action g·u=dh(u) is continuous and composes by the chain rule.
An isotropy arrow fixing a frame has identity derivative and value, and hence
is the identity germ by (GK.1). This is freeness on frames; it is not properness
of the G-action. Nor does a general frame bundle have an ordinary global
section. If T is a disjoint union of coordinate balls, orthonormalization of
each coordinate frame does supply a section on that particular T.

The actual Spin-c lift is independent additional data. If its transport is
[(s_g,z_g)], the inverse normal factor keeps z_g^{-1}ρ_t(s_g), the right Clifford
action, and the grading. The derivative does not determine z_g. An orientation
or topological K-orientation alone does not supply the invariant positive
metric: squaring unitary Clifford covariance forces preservation of its
inverse-metric covector norm. At a one-dimensional fixed return of derivative
λ>0 it forces λ^{-2}|ξ|²=|ξ|² and hence λ=1.

### 11AC.2. GK.2. Compact jet geometry loses coarse labels

Put α=(√5−1)/2 and let Z act on the unit circle T=R/Z by n·x=x+nα. This
action is germ effective: nα∈Z implies n=0. Its holonomy germ groupoid is the
actual action groupoid Z⋉T, with arrow (n,x):x→x+nα. The invariant metric is
dx² and every derivative is 1. The suspension over the base circle is a regular
Riemannian foliation on a compact torus, so this is eligible metric holonomy.
The coherent Spin-c central character z_n=exp(inπ/5) may be supplied on its
trivial normal line. Its determinant character is exp(2inπ/5), and its inverse
normal factor is exp(−inπ/5)ρ_t(1). These are not erased by the derivative 1.

With F_0=0,F_1=1,F_{m+1}=F_m+F_{m−1}, direct induction using α²=1−α gives

\[
F_m\alpha-F_{m-1}=(-1)^{m-1}\alpha^m.
\tag{GK.3}
\]

For fixed x, j(F_m,x) tends to the identity jet at x, while (F_m,x) escapes
every compact subset of Z×T: a compact subset has a finite projection to
discrete Z. Indeed j^{-1}(T×T×{1}) is the whole noncompact groupoid. The
bounded squared chord distance

\[
\psi_{\rm jet}(n,x)=4\sin^2(\pi n\alpha)\le4
\tag{GK.4}
\]

is CND (it is the squared displacement of a vector under rotations of R²),
but is not proper relative to the two unit maps. Its value on F_m tends to
zero. Passing to the frame bundle does not repair this: on an oriented
one-frame the derivative action is still trivial, and an entire labelled
return sequence remains. In contrast

\[
\psi_{\rm label}(n,x)=n^2
\tag{GK.5}
\]

is a continuous CND function with compact sublevels over compact unit sets.
For a real zero-sum family c_i, Σ c_ic_j(n_j−n_i)²=−2(Σ c_in_i)². Its
sublevel restricts n to finitely many integers. Thus this example obstructs
the jet shortcut, not the desired labelled-kernel theorem.

More generally a continuous function factoring through a compact jet
closure is bounded. A uniformly proper kernel cannot be obtained solely by
pulling back that finite-dimensional compact geometry in this example.


### 11AC.3. GK.3. A local isometry need not be a global isometry

This example isolates another insufficient inference; it is not a claimed
counterexample for a complete holonomy pseudogroup. On R² put

\[
u(v)=\begin{cases}
\varepsilon\exp[-1/(1-|v|^2)],&|v|<1,\\
0,&|v|\ge1,
\end{cases}\quad \varepsilon>0,
\qquad ds^2=e^{2u(v)}(dv_1^2+dv_2^2).
\tag{GK.6}
\]

All derivatives of the inner expression tend to zero at |v|=1 because an
exponential beats every reciprocal polynomial; this is a smooth metric. It
is complete, since 1≤e^u≤exp(ε/e), and its distance is bounded above and
below by these constant multiples of Euclidean distance. Its Gaussian
curvature is −e^{-2u}Δu. Inside the disk Δu is analytic and is nonzero at
the centre, so it vanishes on no nonempty open subset of the disk. The set
of points having a flat neighborhood is exactly the connected exterior
E=R²\overline{B(0,1)}.

A nonzero small translation v↦v+a on a small ball sufficiently far from
the disk is a local isometry. If it extended to a global isometry H, H would
map E onto E, because the flat-neighborhood locus is intrinsic. A local
Euclidean isometry on E has constant orthogonal derivative: preservation
of the zero Levi-Civita connection gives ∂_i∂_jH=0, and connectedness makes
H(v)=Qv+b on E. Agreement on the original ball forces Q=1,b=a. But a
nonzero translation does not map the complement of a centred unit disk
onto itself, a contradiction.

Consequently local isometry, first-jet uniqueness and completeness of the
underlying Riemannian manifold do not imply that the given local maps lie
in one global isometry group. A separate complete-pseudogroup globalization
argument is required. Smooth coefficient functions also cannot be inserted
without proof into the discrete valuation construction: their ring has
zero divisors (disjoint nonzero bump functions), and a p-adic norm is not
continuous in the real topology. For example 1+p^{-n}→1 in R, while its
p-adic norm is p^n. Neither the field hypothesis nor compact-arrow bounded
continuous norms follows from smooth matrix entries.

### 11AC.4. GK.4. The exact same-source coarse input

Let k:G×_sG→[0,∞) be continuous, symmetric, zero on the diagonal, and CND
on each actual source fibre. Thus for h_i∈G_x and real c_i with Σc_i=0,

\[
\sum_{i,j}c_ic_jk(h_i,h_j)\le0.
\tag{GK.7}
\]

Its required controls are these:

1. For every compact C⊂G there is U_C<∞ such that
   k(gh,h)≤U_C for all g∈C and h∈G^{s(g)}.
2. For compact K⊂T and R<∞ there is a compact C(K,R)⊂G containing
   every g with s(g),r(g)∈K for which k(gh,h)≤R for some h∈G^{s(g)}.

The second condition retains g, not its jet and not its unlabelled endpoint
pair. No G-invariance of k is required. A continuously varying family of
Hilbert maps f_x:G_x→H_x with squared distance k and the same two controls
is sufficient. Conversely a CND k has such fibrewise Gram realizations;
GK.7 below proves the continuity after compactification. These controls are
the additional geometric input, not a consequence claimed from GK.1.

### 11AC.5. GK.5. A proper compact-fibre spectrum with an actual section

Assume GK.4. There exists a separable self-adjoint C*-algebra A of bounded
continuous functions on G, containing r*C0(T), on which that central algebra
acts nondegenerately. Its spectrum Y has the required proper surjective
anchor p:Y→T and continuous section y0(x)=ev_{1_x}. We now construct A and
the action, including the local rather than globally bounded case.

Choose a countable basis of relatively compact open bisections B whose
closures lie in larger bisections. For each such larger bisection choose
countably many compactly supported unit cutoffs f in s(B), sufficient that
each x∈s(B) has some f identically 1 near x. Add an ordinary countable
dense self-adjoint generating set for C0(T) chosen in Cc(T). Write b(x) for the arrow in
B with source x. Include the functions

\[
a_{B,f}(h)=\begin{cases}
f(rh)k(b(rh)h,h),&rh\in s(B),\\
0,&\text{otherwise}.
\end{cases}
\tag{GK.8}
\]

They are continuous, including the cutoff boundary, and bounded: the arrows
b(supp f) form a compact subset of the larger bisection, so the first
control bounds k there. Include cutoff translations of all generators:
if d is a bounded continuous function and χ∈C_c(r(B)), include

\[
(\alpha_{B,\chi}d)(h)
 =\chi(rh)d(b^{-1}(rh)h)
\quad(rh\in r(B)),
\tag{GK.9}
\]

extended by zero. Use countable dense cutoff sets and repeat saturation
countably often. All functions remain bounded and continuous. Each generator
is supported over a compact unit set. Consequently multiplication by a unit
approximate identity converges in norm on every generator, then on their
*-algebra and its closure A. This proves the asserted nondegeneracy. Countable
generation proves separability.

For an arbitrary a∈A, its translation localized by any χ∈C_c(r(B)) belongs
to A. To check this, first approximate χ by the chosen dense cutoffs and a
by polynomials in the generators. Translation is a local *-homomorphism;
in a product insert an extra cutoff identically 1 on supp χ. Approximate
that extra cutoff from its countable set. This proves membership on
polynomials and then by the norm estimate ||α_{B,χ}a||≤||χ||||a||.
The chosen bisection basis suffices for any bisection by covering the support
of χ by finitely many smaller members and a unit partition of unity. This
also proves invariance on the local ideals.

Every evaluation at h∈G is a nonzero character, because some r*f is nonzero
there. These characters are dense in Y. Otherwise local compactness and
regularity of Y would give a nonzero a∈C0(Y)=A supported outside their
closure. Its function on G would vanish at every h, contradicting the
faithful concrete inclusion A⊂C_b(G). This argument allows extra points
over a limiting unit; no claim of fibrewise evaluation density is needed.

The nondegenerate inclusion r*C0(T) defines p. It is faithful since there
are unit evaluations at every x, and p(ev_h)=rh. For a compact K⊂T choose
f∈C_c(T) equal to 1 on K. On p^{-1}(K) its Gelfand transform is 1. The
set {|f̂|≥1} is compact because f̂∈C0(Y). Hence p^{-1}(K), a closed subset,
is compact. This proves properness. The unit-evaluation section is continuous:
every a∈A restricts continuously to h=1_x; Gelfand transforms determine the
topology. Its existence proves surjectivity and nonempty fibres. A separable
commutative C*-algebra has a second-countable locally compact Hausdorff
spectrum (use a countable generating family for its compact neighborhood
topologies); these properties are actual properties of Y.

The local ideal translation in (GK.9) gives a homeomorphism
p^{-1}(sB)→p^{-1}(rB). At evaluations it is ev_h↦ev_{b(rh)h}. Different
bisections defining the same germ agree on an open unit neighborhood, so
their translations agree after every central cutoff there. They therefore
agree at all spectral points over that neighborhood. Multiplication and
inversion agree on the dense evaluation set and then everywhere. Thus these
homeomorphisms define an actual G-action on Y. On each bisection its
continuity is the continuity of this homeomorphism; the bisections cover G,
which proves continuity of the full action. No arrow is identified with a
different arrow by this construction.

### 11AC.6. GK.6. The continuous proper CND function on G⋉Y

For g∈G and p(y)=s(g), use a bisection B containing g and a cutoff f=1
near s(g). Set

\[
\psi(g,y)=\widehat{a_{B,f}}(y).
\tag{GK.10}
\]

This is independent of the choices: on their common neighborhood the two
functions agree at every h with range in that neighborhood, so their
localized difference is the zero element of A. In a smaller bisection
neighborhood (GK.10) is evaluation of one fixed element of A, and is
continuous. At evaluations it is exactly

\[
\psi(g,ev_h)=k(gh,h),\qquad rh=s(g).
\tag{GK.11}
\]

Nonnegativity and ψ(unit,y)=0 follow by localized positive functional
calculus. Symmetry follows from k(gh,h)=k(h,gh), with the reverse arrow
(g^{-1},g·y), first on evaluations and then on the dense set over each open
unit neighborhood.

To prove CND, let (g_i,y_i) have the same range z in G⋉Y, so
r(g_i)=p(z) and y_i=g_i^{-1}z. Choose bisections of all g_i and restrict their
ranges to a common open U. They give locally varying arrows g_i(x) for
x∈U. The finite matrix

\[
\psi\bigl(g_i(x)^{-1}g_j(x),g_j(x)^{-1}z\bigr)
\tag{GK.12}
\]

is continuous for z∈p^{-1}(U). At z=ev_h, rh=x, it is
k(g_i(x)^{-1}h,g_j(x)^{-1}h). All arguments have the same actual source
s(h), and (GK.7) gives its CND inequality. The evaluations with rh∈U are
dense in the open set p^{-1}(U), by total density from GK.5. The finite
inequality passes to all z. This is the groupoid CND condition, including
distinct isotropy arrows.

The relative properness proof must also allow spectral points arising at
limiting units. Let L_s,L_r⊂Y be compact and R<∞. Choose a compact unit
neighborhood K containing p(L_s)∪p(L_r) in its interior. Take the compact
C=C(K,R+1) supplied by GK.4. If s(g),r(g) are in that interior and g∉C,
shrink its bisection neighborhood to avoid C and to keep both endpoints
in K. For every arrow in this neighborhood, (GK.8) without a cutoff has
value >R+1 on all regular evaluations. Localized positivity makes its
spectral value ≥R+1. Therefore ψ(g,y)≤R with y∈L_s,g·y∈L_r implies
g∈C. The sublevel

\[
\{(g,y):y\in L_s,\ g\!\cdot\!y\in L_r,\ \psi(g,y)\le R\}
\tag{GK.13}
\]

is closed in the compact fibre product C×_{s,p}L_s. It is compact. This is
properness relative to the two Y-unit maps. It does not assume that G is
proper, that the action on Y is proper, or that T is compact.

### 11AC.7. GK.7. The coherent continuous affine Hilbert field

Write H=G⋉Y. For z∈Y, start with the real finite zero-sum combinations of
δ_a, a∈H^z. Put e_a=δ_a−δ_{1_z} and

\[
\langle e_a,e_b\rangle_z
 =\tfrac12\{\psi(a)+\psi(b)-\psi(a^{-1}b)\}.
\tag{GK.14}
\]

The CND inequality is precisely nonnegativity of this Gram form. Quotient
its null vectors and complete, obtaining E_z. The norm of e_a is √ψ(a).
For a:z'→z define on the dense span

\[
L_ae_b=e_{ab}-e_a,\qquad A_av=L_av+e_a.
\tag{GK.15}
\]

Polarization of (GK.14), using (ab)^{-1}(ac)=b^{-1}c, proves that L_a is an
isometry; L_{a^{-1}} is its inverse. Direct calculation gives
L_aL_b=L_{ab} and e_{ab}=e_a+L_ae_b. Hence A_aA_b=A_{ab} exactly,
with ||A_a0−0||²=ψ(a). No choices of individual fibre identifications occur.

A local bisection of H gives a local section e_{b(z)} over its range.
Finite linear combinations with continuous real coefficients have continuous
Gram entries and therefore continuous norms. Their fibrewise spans are
dense by construction. Declare continuous sections to be their local uniform
limits; compatibility of the norm and the local approximation axiom follows
directly, producing a continuous real Hilbert field. Countable bisection bases
give a countable generating family. Formula (GK.15) on those local generators,
and its constant norm estimate, prove continuity of the linear and affine
actions. Properness of its displacement is exactly GK.6.

The projection H→G retains every original arrow. Pulling back the whole
normal inverse module supplies the same transport U_g on (g,y); in particular
z_g^{-1}ρ_t(s_g), the right Clifford action and the grading are retained.
This geometric result supplies no eta or Bott KK-class by itself. Their
whole-base product and anchored probability comparison require separate proofs.

### 11AC.8. GK.8. An actual action-groupoid provider

Let a countable discrete Γ act continuously on T. Suppose a Hilbert map
f:Γ→E has proper right-invariant coarse controls:

\[
\rho_-(|\gamma\lambda^{-1}|)^2
 \le\|f(\gamma)-f(\lambda)\|^2
 \le\rho_+(|\gamma\lambda^{-1}|)^2,
\qquad\rho_-(t)\longrightarrow\infty.
\tag{GK.16}
\]

Here |·| is a proper length; in particular its bounded sets are finite. A
left-invariant embedding is converted to this convention by γ↦f_0(γ^{-1}).
On the actual source fibre of Γ⋉T, identify (γ,x) by its Γ-label and put

\[
k((\gamma,x),(\lambda,x))
 =\|f(\gamma)-f(\lambda)\|^2.
\tag{GK.17}
\]

This is continuous: source-fibre-pair charts have fixed discrete labels. It
is CND by the squared-distance identity −2||Σc_if(γ_i)||². If
g=(δ,γx) and h=(γ,x), k(gh,h) is between ρ_-(|δ|)² and ρ_+(|δ|)²,
independent of γ and x. Compact arrow sets have finitely many δ-labels,
proving the upper control. A sublevel gives finitely many δ-labels; with
source in compact K, it lies in a finite union {δ}×K. Imposing the compact
range condition gives a closed subset. This proves GK.4 at the original
whole unit-space scope, and GK.5–GK.7 give Y,y0,ψ and the coherent field.

In this case one may equivalently use Y=T×Z, where Z is the compact
metrizable closure of Γ evaluations in the countable algebra generated by
all left translates of γ↦||f(δγ)−f(γ)||². Left translation acts continuously
on Z, y0(x)=(x,ev_e), and ψ((δ,x),z) is the corresponding generator's
evaluation. The same-source proof above proves all its groupoid claims;
the discrete theorem is used only to supply the map f.

The discrete compact-Lie theorem in Section 11N supplies f for every
finitely generated subgroup of a compact Lie group, with its proper word
length. Therefore GK.8 applies to genuine isometric suspensions having that
actual action-groupoid transversal and their supplied invariant normal
Spin-c transport. For a connected normal manifold and an effective global
isometry action, (GK.1) proves germ effectiveness, so no action labels are
lost when passing to its germ groupoid. A groupoid merely locally modelled
on such actions does not automatically satisfy (GK.17).

### 11AC.9. GK.9. The eligible-holonomy kernel theorem

Complete frame geometry and local matrix-monodromy labels and Label-preserving positive-kernel patching and frame transfer prove the exact GK.4 input for the supplied invariant positive normal metric on the original compact regular foliation, with its full invariant normal Spin-c data and any complete transversal. They prove uniform germ extension, the closed-local-group lemma, actual local Lie development, finite-label compactness, continuous CND patching and compact-frame transfer. The extension to a supplied complete bundle-like metric is proved in EG.9.

The theorem retains the whole original unit algebra and every actual arrow and inverse phase. The matrix proof in Corollary 11N.5 and real Ado in Low-degree cohomology, Whitehead’s lemmas and the Levi decomposition are the ordinary inputs; neither an injective jet map nor an arbitrary local-isometry globalization is used as a substitute for these geometric proofs. GK.5–GK.7 consequently give the required proper compact-fibre Y/psi interface at this eligible scope. The physical graph operator is a separate analytic construction.

### 11AC.10. GK.10. The coarse input is equivalent to the requested interface

The same-source condition is not an unnecessarily stronger replacement
target. Conversely, suppose the requested Y,p,y0,ψ exist with precisely
the properties stated in GK.5–GK.6. For h,l∈G_x define

\[
k(h,l)=\psi\bigl(hl^{-1},\ l\!\cdot\!y_0(x)\bigr).
\tag{GK.18}
\]

This is continuous on G×_sG by continuity of the section, action, inverse,
multiplication and ψ. It is symmetric and zero on the diagonal by the
groupoid symmetry and normalization. The arrows (h,y0(x)) have a common
source; their inverses have common range y0(x). Applying CND to those
inverses gives exactly (GK.7), so k is a same-source CND kernel.

Its controls follow from the actual proper anchor and relative properness.
For compact C⊂G, C×_{s,p}Y is compact: it is closed in
C×p^{-1}(s(C)), and the latter factor is compact. The continuous ψ has a
finite maximum on it. Since

\[
k(gh,h)=\psi\bigl(g,\ h\!\cdot\!y_0(s(h))\bigr),
\tag{GK.19}
\]

this maximum is U_C. For compact K⊂T the two Y-unit sets p^{-1}(K) are
compact. The ψ-sublevel over those two sets is compact by its assumed
relative properness. Its projection to G is compact and contains every
g occurring in the second control of GK.4 by (GK.19). This gives C(K,R).

Consequently GK.4 and the requested Y,p,y0,ψ interface are equivalent at
the original whole-base scope. Section 11AF proves these conditions
for the original eligible holonomy scope. The equivalence retains that
precise geometric input. Probability convexification
and KK transfer require no openness or local triviality of p; those
constructions are outside the present geometric result.

![Retained irrational-rotation labels and the compact-fibre kernel construction](../figures/labelled-geometric-kernel.png)

**Figure 11AC.1.** The Fibonacci identity is exact; circle and chord positions are numerical samples. The label kernel is n², while the bounded jet kernel loses the escaping labels and independent inverse central phase. The lower panel is a typed schematic of GK.4–GK.8. The general eligible-holonomy input GK.9 is proved in Section 11AF. [Vector figure](../figures/labelled-geometric-kernel.svg) · [Generator](../reproduction/labelled-geometric-kernel/draw_kernel.py) · [Finite checks](../reproduction/labelled-geometric-kernel/finite_checks.py) · [Sample data](../reproduction/labelled-geometric-kernel/figures/labelled-geometric-kernel-data.json) · [Reproduction](../reproduction/labelled-geometric-kernel/REPRODUCTION.md) · [Component terms](../reproduction/labelled-geometric-kernel/COMPONENT-TERMS.md).

## 11AD. The original-unit dual from a labelled geometric kernel

Assume the precise labelled-kernel input of Section 11AC.4. The following constructions then give an actual even eta over C0(T), with For_G(eta D)=1_C0(T) and the original local Bott normalization. The hypothesis is proved for the actual action cases in Section 11AC.8 and for the general eligible-holonomy scope in Section 11AF. The physical graph operator remains unproved.

### 11AD.1. Anchored probability compactification and proper kernel integration

G is Hausdorff second-countable LCH étale,
with unit space T and R=C0(T). This theorem specifies and proves the
probability part of the original-unit dual construction. Its geometric
input is explicit; a transverse metric or frame bundle by itself is not
asserted to supply it.

Assume a Hausdorff second-countable LCH G-space Y with a **proper**
surjective anchor p:Y→T, a continuous ordinary section y0:T→Y, and
a continuous nonnegative function k on the actual transformation
groupoid H_Y=G⋉Y. Write its arrows as (g,y), with range y, source
g^−1y and p(y)=r(g). Require

\[
\begin{gathered} k(1,y)=0,\quad k(g,y)=k(g^{-1},g^{-1}y),\\ \sum_{i,j}c_ic_j k(g_i^{-1}g_j,g_i^{-1}y)\le0\quad(r(g_i)=p(y),\ \sum_i c_i=0).\end{gathered}
\tag{PK.1}
\]

and require, for compact K,L⊂Y and finite A,

\[
\{(g,y):y\in K,\ g^{-1}y\in L,\ k(g,y)\le A\}\text{ is compact}.
\tag{PK.2}
\]

Neither an equivariant section y0 nor a proper jet map is required.
Section 11AF and GK.5–GK.7 supply Y,y0,k for general eligible holonomy. The constructions below are actual for
every input satisfying (PK.1)–(PK.2); they are not an assertion that
that geometric premise has already been supplied.

#### 11AD.1. PK.1. The locally compact probability space over the original units

Let X_t=Prob(Y_t), with probabilities defined, equivalently for the
construction, as positive unital linear functionals on C(Y_t).
The fibre Y_t is compact and nonempty by properness and surjectivity.
Put X=⊔_t X_t and topologize it by the anchor t and the functions

\[
\mu\longmapsto\mu(f|_{Y_t}),\qquad f\in C_0(Y).
\tag{PK.3}
\]

This is a Hausdorff second-countable topology. Restrictions of C0(Y)
separate fibre probabilities; a countable norm-dense set in C0(Y)
suffices for its topology since all functionals have norm one.
Countable base tests in T give a countable topology, and distinct
unit anchors or a separating fibre function distinguish any two points.

The anchor p_X:X→T is proper. To see this for compact K⊂T, Y_K is
compact. Probabilities on Y_K form a compact metrizable space: choose
a countable norm-dense scalar subalgebra of C(Y_K); all positive
unital functionals are a closed subset of the product of its bounded
coordinate intervals, using their linearity, positivity and unit
equations. A limit extends uniquely by the norm-one bound to all
of C(Y_K). Thus the countable compact-product argument proves
compactness without using a measure-selection theorem.

Inside K×Prob(Y_K), the conditions of support over the one unit t
are closed. They can be written

\[
\mu(((a\circ p)-a(t))f)=0,\qquad a\in C_0(T),\ f\in C(Y_K).
\tag{PK.4}
\]

They say exactly that the functional factors through C(Y_t): the
ideal of functions vanishing on Y_t is generated by these central
unit differences. For precision, on the compact support of a function
away from Y_t choose finitely many unit functions separating its
anchor points from t, form their positive sum of squared differences,
and divide on that support. Cut off an arbitrary function vanishing
on Y_t away from its small-norm neighbourhood; approximation gives
the whole ideal. Thus (PK.4) describes X_K as a closed subset of
that compact product. The coordinate topology is the one in (PK.3),
by uniform approximation of fibre functions by restrictions of C0(Y).
This proves p_X proper. A relatively compact unit neighbourhood now
gives a neighbourhood in X with compact closure, proving LCH.

Fibrewise convex combinations are continuous by (PK.3). The actual
ordinary section

\[
x_0(t)=\delta_{y_0(t)}.
\tag{PK.5}
\]

is continuous. Every anchored section into a Hausdorff space is proper:
the inverse image of a compact set is closed and contained in its
compact unit-anchor image. In particular x0 is proper. We obtain
actual even anchored homomorphisms

\[
\begin{gathered}u:R\to C_0(X),\quad f\mapsto f\circ p_X,\\v_0:C_0(X)\to R,\quad a\mapsto a\circ x_0,\qquad v_0u=\mathrm{id}_R.\end{gathered}
\tag{PK.6}
\]

The first exists as an algebra-valued homomorphism precisely because
p_X is proper. Its fibre scalar map is G-equivariant. The second
is ordinary anchored; no equivariance of x0 is inferred.

#### 11AD.1. PK.2. The actual action and the integrated kernel

G acts on X by pushforward: (gμ)(f)=μ(f∘g) on Y_(s(g)). On a bisection
and a compact unit neighbourhood the function f∘g is a continuous
function on the corresponding part of Y. Multiply by a compact unit
bump, extend by zero, and use properness of p to obtain a C0(Y)
test. Equation (PK.3) proves action continuity. The exact action
identity is inherited from the action on Y, with no collapse of
distinct isotropy arrows.

Define on H_X=G⋉X

\[
K(g,\mu)=\mu(y\mapsto k(g,y)),\qquad \mu\in X_{r(g)}.
\tag{PK.7}
\]

The fibre test is continuous and bounded because Y_(r(g)) is compact.
The function K is continuous over the whole arrow space. Locally
g is the selected arrow g_t on a bisection. Then y↦k(g_(p(y)),y)
is continuous on the corresponding open part of Y; a compact unit
bump extends it to C0(Y), by properness of p. Near the selected
arrow (PK.7) is exactly the test (PK.3) of that extension. This proves
continuity without assuming a global enumeration of arrow fibres.

Symmetry, zero on units and conditional negativity follow by direct
substitution. For arrows (g_i,μ) with common range μ,

\[
\sum_{i,j}c_ic_j K(g_i^{-1}g_j,g_i^{-1}\mu)=\mu\left(y\mapsto\sum_{i,j}c_ic_j k(g_i^{-1}g_j,g_i^{-1}y)\right)\le0.
\tag{PK.8}
\]

The equality uses pushforward and actual composable arrows; positivity
of the functional gives the inequality. Thus integration supplies
one coherent CND kernel on H_X, not separate orbit kernels.

#### 11AD.1. PK.3. Properness survives integration for this actual anchor

Let C,D⊂X be compact and A finite. Their unit anchors C_T,D_T are
compact. Put C_Y=p^−1(C_T), D_Y=p^−1(D_T); these are compact by the
input's proper anchor. If μ∈C, g^−1μ∈D and K(g,μ)≤A, some y∈Y_(r(g))
satisfies k(g,y)≤A. Otherwise the continuous function on the compact
fibre would have minimum >A and its probability value would be >A.
For that y, y∈C_Y and g^−1y∈D_Y. Consequently g belongs to the
compact projection of the set in (PK.2) for C_Y,D_Y,A.

The desired bounded-kernel arrows (g,μ) are now a closed subset of
the product of this compact arrow set with C: composability is closed,
g^−1μ∈D is closed by the continuous action, and K≤A is closed by
continuity. They are therefore compact. We have proved

\[
\{(g,\mu):\mu\in C,\ g^{-1}\mu\in D,\ K(g,\mu)\le A\}\text{ is compact}.
\tag{PK.9}
\]

The argument needs the proper anchor of Y. Conditional negativity
alone, or a proper kernel tested only on arbitrarily selected compact
subsets of a nonproper Y, would not justify this step.

#### 11AD.1. PK.4. An equivariant probability map from a proper space

Let Z→T be any second-countable LCH proper G-space. The completed
proper-space provider A universal proper groupoid base with supported cutoffs GP.9–GP.11 gives an
actual square cutoff c, with its compact-arrow support test. Define

\[
\Phi(z)=\sum_{g\in G^{p(z)}}c(g^{-1}z)^2\,g x_0(s(g))\in X_{p(z)}.
\tag{PK.10}
\]

This is a probability in its compact fibre: every summand is a positive
functional and the scalar weights sum to one. For a compact subset
of Z, all active arrow-point pairs belong to one compact set by
GP.11. A finite bisection cover therefore gives a locally finite
continuous counting sum for each test (PK.3). Both the section x0
and the action on it are continuous on these charts. This proves
Φ continuous in exactly the actual topology of X. Replacing g by hg
proves Φ(hz)=hΦ(z). All unit anchors and isotropy labels remain.

After forgetting G, the actual anchored homotopy

\[
\Phi_s(z)=(1-s)\Phi(z)+s x_0(p(z)),\qquad 0\le s\le1.
\tag{PK.11}
\]

stays in the same compact probability fibre and is continuous by
(PK.3). Compact source supports times [0,1] have compact image in X.
Properness of Φ itself is not needed for the multiplier/class use.

#### 11AD.1. PK.5. The actual coefficient collapse needed for the dual

Apply PK.4 to the actual proper central M_G of the earlier P. Its
central structure and Φ yield an even anchored equivariant homomorphism

\[
e_P:C_0(X)\otimes_R P\to P,\qquad f\otimes a\mapsto\Phi^*(f)a.
\tag{PK.12}
\]

The anchor makes it balanced; centrality gives multiplicativity and
the graded adjoint identity. Contractivity on the minimal tensor
product follows from the commuting representation of the commutative
nuclear C0(X). It has algebra-valued output since a∈P and Φ*(f) is
a bounded central multiplier. It is nondegenerate: on a compact
central support choose f∈Cc(X) equal to one on its compact Φ-image;
then extend by source approximation. Thus its standard P-module
defines its homomorphism class. This is an actual coefficient collapse,
not an equivariant scalar evaluation.

The homotopy (PK.11) gives ordinary anchored homomorphisms e_(P,s).
For an algebraic f⊗a with compact central support, uniform continuity
on its compact homotopy image proves point-norm continuity. Density
and contraction extend this to the whole balanced tensor product.
At s=1 it is v0⊗_R id_P. Consequently

\[
\operatorname{For}_G[e_P]=[v_0]\boxtimes_R1_P\quad\text{in }KK_T(C_0(X)\otimes_RP,P).
\tag{PK.13}
\]

### 11AD.2. Actual anchored Bott factors from a coherent probability-kernel input

The earlier inputs are Original-anchor localization and the equivariant Dirac lift, including Anchored KK foundations, Supported anchored averaging, Central Mayer–Vietoris, Proper slices and reciprocity, and The anchored telescope. These give precisely the product and special-source exactness used below.

Use the actual X,K constructed in
Anchored probability compactification and proper kernel integration from its explicit geometric input. Set H=G⋉X
and R_X=C0(X). All algebras retain this original X-anchor until the
explicit forgetting step BA.10. Every Kasparov class below has even
degree; its displayed operator is odd. No equivariant gamma identity
is required or inferred.

The fixed-Hilbert proof in The actual weighted Bott deformation and its proper-source lift, BD.4–BD.25, supplies the actual finite
quantization, completed weighted operators, resolvent estimates and
ordinary cubic-vacuum homotopy have complete proofs there. The arguments below establish the groupoid, noncompact-base, source-lift and whole-base interfaces.

#### 11AD.2. BA.1. A genuine continuous Hilbert field over X

For each x∈X take zero-sum finite combinations of the actual arrows
h∈H^x, with inner product

\[
\left\langle\sum_h a_h\delta_h,\sum_k b_k\delta_k\right\rangle_x=-\frac12\sum_{h,k}a_hb_kK(h^{-1}k).
\tag{BA.1}
\]

Conditional negativity gives positivity. Applying it to v+su proves
Cauchy–Schwarz, hence null vectors are orthogonal to the entire space.
Quotient and complete to E_x. The sections e_h=[δ_h−δ_(1_x)] have

\[
\langle e_h,e_k\rangle=\frac{K(h)+K(k)-K(h^{-1}k)}2,\qquad\|e_h\|^2=K(h).
\tag{BA.2}
\]

A countable bisection basis of H and compact unit bumps supply
countably many global sections, a bump times the selected e_h.
Their Gram entries are continuous and their evaluation spans are
dense. Local uniform completion therefore defines a continuous real
Hilbert field and a countably generated real R_X-module. No global
arrow enumeration or locally constant fibre rank has been assumed.

For h:x→x', set

\[
L_he_k=e_{hk}-e_h,\qquad b_h=e_h,\qquad A_hv=L_hv+b_h.
\tag{BA.3}
\]

The Gram identity proves isometry and
e_l=L_h(e_(h^−1l)−e_(h^−1)) proves surjectivity. The affine cocycle
identity follows on generators, hence on completions. Bisection
generators and (BA.2) prove strong continuity of L and norm continuity
of b over H. In particular ||b_h||²=K(h), with PK.9's relative properness.

#### 11AD.2. BA.2. Ordinary real stabilization preserves the anchor

Real graded module stabilization gives

\[
E\oplus C_0(X,\ell^2_{\mathbb R})\cong C_0(X,H_0),\qquad H_0=\ell^2_{\mathbb R}.
\tag{BA.4}
\]

Its full operator-range proof is the stabilization theorem in Kasparov’s stabilization theorem: choose real generators, make the dense-range operator
and its positive square root with real scalar coefficients, and use
the real polar unitary. Every operator is R_X-linear; the proof works
for a sigma-unital nonunital C0(X) just as for a compact base.
The new summand has identity linear arrow maps and zero cocycle.
Transport by this *ordinary linear* trivialization preserves the
zero section and (BA.3), including the displacement norm. Thus L_h
is a strongly continuous orthogonal H0-valued field, not a globally
equivariant trivialization. The ordinary algebra will be constant;
its equivariant action still contains the actual L_h,b_h.

#### 11AD.2. BA.3. A single diagonal weight with compact-arrow estimates

Take a fixed diagonal compact h0 e_j=2^−j e_j on H0 and its finite
diagonal nested plateau approximate identities e_n. For compact
A⊂H, e_n and L_h e_n L_h^−1 are approximate identities of
C(A,K(H0)): uniformity follows from compactness of the image of
each norm-continuous compact-operator test and strong continuity
of L_h. Their represented differences converge strongly to zero,
hence weakly in the full Banach dual, by the bounded-functional
support-projection argument proved in earlier KG.2.

Choose a compact exhaustion A_n of H, and use the common-tail
convex separation in that argument, also testing the norm-compact
vector set {b_h:h∈A_n}. It produces *fixed diagonal finite-rank*
positive contractions v_n, independent of x, with v_n v_(n+1)=v_n,
least coordinate-cutoff index tending to infinity, and

\[
\sup_{h\in A_n}\|L_hv_nL_h^{-1}-v_n\|\le2^{-n},\qquad\sup_{h\in A_n}\|(1-v_n)b_h\|\le2^{-n}.
\tag{BA.5}
\]

Define by the common scalar spectral calculus

\[
\Theta=1+\sum_n(1-v_n),\qquad\Theta e_j=\lambda_j e_j.
\tag{BA.6}
\]

Each λ_j is finite, λ_j≥1 and λ_j→∞. Its inverse is compact on H0
and has dense range, so Θ is positive self-adjoint and regular on
the constant field, with finite-coordinate core. On a noncompact X
this reciprocal is a multiplier-valued compact field: **only its
compact-unit-localized versions lie in K_(R_X)(C0(X,H0))**. Global
module compactness is not inferred from fibre compactness.

For each compact arrow set the tails of

\[
\begin{gathered}S_h=L_h\Theta L_h^{-1}-\Theta=\sum_n(v_n-L_hv_nL_h^{-1}),\\\Theta b_h=b_h+\sum_n(1-v_n)b_h.\end{gathered}
\tag{BA.7}
\]

are uniformly norm summable, by (BA.5). Their initial terms are
continuous. Thus S_h is compact and norm continuous over H and
Θb_h is norm continuous. The bounded partial sums have strong
resolvent limits; their norm-convergent differences and the resolvent
identity identify the conjugated limit as Θ+S_h. Consequently
L_h Dom Θ=Dom Θ. In particular
Θ L_h e_j=L_h Θe_j−S_h L_h e_j, so moving finite vertex frames and
the translated origins L_hv+b_h are graph continuous on compact
arrow sets. These are the actual domain tests consumed by the
deformation, not a continuous fibre diagonalization.

#### 11AD.2. BA.4. The anchored Bott algebra and its actual proper centre

Define B=R_X⊗A(H0), using the phase-space Bott limit in earlier
BD.4–BD.6. A finite vector plane has the exterior matrix symbol

\[
\beta(q,p)=\varepsilon(q+ip)+\varepsilon(q+ip)^*,\qquad\beta^2=\|q\|^2+\|p\|^2.
\]

Its connecting map uses the graded odd sum T_S+β of the added
orthogonal plane, with the original reflection-graded scalar S.
Affine finite-frame norm continuity, proved there by a common
finite ambient plane, makes (BA.3) an actual continuous action on B:
for a compactly supported finite symbol and compact arrow test the
transported origins and frame vectors have compact norm image, so
the finite norm estimates are uniform. Compact unit support and
finite-stage approximation handle every B-section. Composition is
the exact affine cocycle identity. B is separable and nuclear by
the finite commutative matrix models and increasing-stage approximation.

Its central space is the **phase** cone

\[
W=\{(x,v,p,r):\|v\|^2+\|p\|^2\le r\}.
\tag{BA.8}
\]

with base, weak coordinates and continuous total height r. Over
compact C⊂X and bounded height it is a closed subspace of a compact
countable product; the finite Gram inequalities describe it exactly.
This proves second countability, LCH and compact bounded-height
pieces. The action is

\[
(x,v,p,r)\longmapsto(hx,L_hv+b_h,L_hp,r+2\langle L_hv,b_h\rangle+\|b_h\|^2).
\tag{BA.9}
\]

The excess height is preserved. Strong/norm arrow continuity and
bounded weak-coordinate tests prove continuity, including the cross
term. If input and output heights are ≤R, ||b_h||≤2√R. Their X-units
lie in fixed compact sets, so PK.9 bounds all such H-arrows in a
compact set. The closed action relation then proves W is proper.
Projecting transformation arrows to G proves the same properness
for the G-action over the composite original anchor to T.

At finite Bott stages the scalar projected coordinates and T_S²
give the central cone functions. The connecting homomorphism sends
residual squared height to T_S²+||q_added||²+||p_added||². Thus the
cylinder functions are compatible, even and central. They separate
points of W; compact height/base localization and scalar approximation
give C0(W). These are exactly the complete BD.9 central-map tests,
now with compact unit bumps in X. Radial height cutoffs combined
with a C0(X) approximate identity act as identity on each compactly
supported finite generator. Hence

\[
C_0(W)\longrightarrow ZM(B).
\tag{BA.10}
\]

is actual, anchored, equivariant and nondegenerate. The momentum
coordinate has not been removed from the proper central space.

#### 11AD.2. BA.5. The actual groupoid quantum deformation

Ordinarily let E_B=R_X⊗E_Θ, where E_Θ is the completed fixed-Hilbert
deformation in earlier BD.10–BD.16. Its zero quotient and ideal are

\[
\begin{gathered}0\to I_B\to E_B\xrightarrow{q_B}B\to0,\\I_B=C_0((0,1])\otimes R_X\otimes\mathcal K(\mathcal F_t)\otimes S.\end{gathered}
\tag{BA.11}
\]

The finite graph-frame and positive-time ratio tests BD.14–BD.24
are uniform on compact arrow sets by (BA.7). Explicitly positive
occupation coefficients are √t(1+tλ_j); on [a,b]⊂(0,1] their log
derivatives are at most 3/(2a), uniformly in j and occupation.
At zero a finite-rank weight change has functional-calculus error
≤3t||weight change||. This is the corrected estimate
B_t(S)²≤||S||²D_t²; its direction is ≤. Transported finite origins
and frames are graph continuous, so the finite semiclassical
Schur/Fourier estimates also apply uniformly on those compact sets.
These are norm estimates in the actual section algebra, not just
separate positive-time fibre estimates.

On positive Fock fibres use the real Weyl and exterior action

\[
U_{h,t}=T_{b_h/\sqrt t}\Gamma(L_h).
\]

Its exponential-vector formula is earlier WF.13. Real translations
have exactly T_b T_c=T_(b+c), with no central phase; therefore the
affine cocycle proves U_(h,t)U_(k,t)=U_(hk,t). Dense exponential
vectors give strong continuity for t>0. The identity part of the
translated weighted potential is retained:

\[
U_{h,t}B_t(1+t\Theta)U_{h,t}^{-1}=B_t(1+t\Theta+tS_h)-c_{(1+t\Theta+tS_h)b_h}.
\tag{BA.12}
\]

Changing the linear weight and the origin is precisely the finite
affine generator transport used in BD.17–BD.18; (BA.7) and the
norm-uniform finite tests above make it preserve E_B and extend
continuously to zero with its actual Bott action. Checking on the
dense finite generators, then on their products and closure, gives
a genuine anchored H-action and the exact groupoid composition law.
This explicitly checks the new groupoid interface instead of assuming
that a countable-group action proof already has it.

The ordinary real trivialization makes q_B=1_(R_X)⊗h0. The ordinary
fixed-Hilbert nuclear section s0 is chosen even; thus
s_B=1_(R_X)⊗s0 is an actual even **R_X-linear** cpc section of q_B.
It is not assumed equivariant. The full Fock module gives the actual
H-equivariant graded Morita equivalence

\[
I_B\sim_M^H C_0((0,1])\otimes R_X\otimes S.
\tag{BA.13}
\]

The interval and S actions on its right are trivial. The scalar
contraction f(t)↦f(ut), f(0)=0, is anchored and exactly equivariant.

#### 11AD.2. BA.6. The stronger anchored finite-output premise for this source

B, as an ordinary R_X-algebra, is constant with nuclear coefficient
A(H0). It has actual even cpc finite factors

\[
B\xrightarrow{\alpha}R_X\otimes F\xrightarrow{\beta}B,\qquad\beta(f\otimes m)=f k(m).
\tag{BA.14}
\]

approximating any prescribed finite set, where k:F→B has B-valued
Choi entries. Here is the construction. First localize the finite
set by a compact unit bump in X; its finitely many section images
on that compact set are norm compact. A nuclear finite coefficient
model approximates them uniformly. Choose a compactly supported
output partition equal to one there and an input bump equal to
one on its support. Tensor the coefficient maps by R_X, multiply
the output by that compact bump and extend by zero. Output Choi
entries now belong to B itself; the positive output-unit sum is
at most one, hence the maps are cpc. Localizing the grading image
and doubling F with flip grading gives even maps, with the same
error. The outside tail was already arbitrarily small. This proves
(BA.14), not just ordinary nuclearity, and supplies exactly the
telescope-lifting premise.

#### 11AD.2. BA.7. The actual proper Bott source lift

For clarity, apply the earlier KG/SR/MV/AT arguments with groupoid H,
unit algebra R_X, source B and centre W. All their hypotheses are
checked above. The relevant generic inference is proved as follows.
Restrict (BA.11) to a finite-action proper slice F⋉U of W. Finite
averaging makes its supplied anchored section equivariant; its
actual Morita-cone kernel is zero in restricted KK. Semisplit
excision therefore makes its mapping cone locally KK-zero for
every restricted source. The proved anchored induction adjunction
transfers this vanishing to each induced open ideal of B, including
its invariant intersections. The explicit central open-cover
homotopy and endpoint taper extension propagate it to every finite
union of slice opens.

Choose a countable cover of W and the increasing ideals B_n of
finite unions. Their union is dense. The two actual anchored
telescope extensions have the first ordinary section obtained from
the B-valued output factors (BA.14), finite Choi lifting and summable
anchor-preserving gluing. Its common proper centre W makes that
section exactly equivariant by the earlier supported averaging proof.
The integer endpoint section is the exact scalar interpolation.
The full source Milnor sequence has both lim and lim1 zero because
all stage groups vanish in both degrees. Both Puppe degrees then give

\[
(q_B)_*:KK_H^j(B,E_B)\cong KK_H^j(B,B),\qquad j=0,1.
\tag{BA.15}
\]

This is the complete instantiated localization chain. It does not
infer anchored source-quotient sections from ordinary nuclearity;
it uses the special actual factors (BA.14). It does not change the
earlier nearest-face source theorem or its files.

Let z_B be the unique preimage of 1_B. The actual positive endpoint,
its full Fock Morita class and the graded scalar S evaluation give

\[
d_X=z_B\otimes[h_1]\otimes m_{\mathrm{Fock}}\otimes[\mathrm{ev}_0]\in KK_H(B,R_X).
\tag{BA.16}
\]

Every factor has even degree with the same outward q+ip and ordered
Clifford convention as the completed fixed-Hilbert provider.

#### 11AD.2. BA.8. The actual dual and its compact arrow defects

The zero-origin odd regular Bott multiplier T0 has compactly supported
functional calculus in A(H0). On B its phase F0=T0(1+T0²)^−1/2 gives
the standard graded B-module cycle with source R_X. Every source-
localized resolvent is in B; globally on a noncompact X it is only
a multiplier. Linear conjugation preserves the origin and affine
translation changes T0 by bounded Clifford multiplication of norm
||b_h||. The bounded-perturbation phase-resolvent integral expresses
the resulting difference as a norm-convergent integral of B-compact
terms after compact unit localization. Its three terms have the
same finite integral bounds as BD.11–BD.13, now for a bounded origin
change. On compact arrow sets these bounds and (BA.7) are uniform,
and finite-frame approximation gives continuous compact sections.
This proves the actual H-cycle, not only its pointwise indices:

\[
\eta_X=[B,\text{source multiplication},F_0]\in KK_H(R_X,B).
\tag{BA.17}
\]

#### 11AD.2. BA.9. The ordinary product is a whole-base identity

After forgetting H while retaining the X-anchor, q_B is the actual
constant tensor of the fixed deformation. Its anchored section and
cone kernel make it a KK_X-equivalence for the unit groupoid. Thus
the ordinary image of z_B is 1_(R_X)⊗[h0]^−1. The ordinary origin
cycle is 1_(R_X)⊗η0 under the *linear* real stabilization (BA.4).
The fixed-Hilbert lifting cycle is the completed regular operator

\[
\widetilde T_t=T_S+tD_t^3+B_t(1+t\Theta).
\]

Its cutoffs, adjoint resolvents and zero norm limit are the actual
BD.20–BD.25 proof. At t=1 followed by ev0 its operator is
D³+B(1+Θ), with unique even vacuum and uniform gap √2 on its
complement. Its exact-sign change on the complement is compact and
makes that complement degenerate. Tensoring this actual homotopy
and finite-rank vacuum decomposition with R_X gives the actual
standard R_X-module identity over *all X*. Compactness is correctly
localized by f∈C0(X), not asserted for bare constant resolvents.
Consequently

\[
\operatorname{For}_H(\eta_X\otimes_Bd_X)=1_{R_X}\quad\text{in }KK_X(R_X,R_X).
\tag{BA.18}
\]

The proof tensors one completed homotopy; it does not infer a
whole-base equality from equal fibre indices. It is an ordinary
identity. No equivariant gamma=1 was used.

#### 11AD.2. BA.10. Forgetting the extra X anchor preserves actual G-cycles

An H=G⋉X algebra with nondegenerate central X-action is a G-algebra
over T by retaining the same section algebra and transporting its
X-fibres over each G-bisection. The same construction on an H-Hilbert
module preserves its inner product and right coefficient action:
on a G-bisection it is the continuous H-action on the corresponding
open X-part, sending C0-sections to C0-sections by a homeomorphism.
Its local transports agree and satisfy the exact G-cocycle law.
Countable generation, grading and R-linearity are unchanged.

For the arrow defect test, approximate a source element by f_X a,
f_X∈Cc(X). On a compact G-arrow chart, the corresponding composable
arrow/point pairs with x in supp f_X form a compact subset of H.
The actual H-compact defect is therefore a continuous compact field
there. Pullback on that bisection and a compact unit bump identify
it with a section of the same compact algebra K_B(E), now over T.
Finite charts and the source norm tail give the actual G-defect
condition. This is section-algebra compactness, not merely compactness
on each x-fibre. It proves the forgetting operation on cycles and
interval homotopies. The interior tensor module and all creation/
positivity equations are unchanged; a product operator for H is a
product operator for G by the same argument. Uniqueness in earlier
KG.4 proves preservation of products and homomorphisms.

Thus (BA.16)–(BA.17) give actual even G-classes
η_X:C0(X)→B and d_X:B→C0(X), over the original T. Their ordinary
product identity is still (BA.18), now regarded in KK_T. These,
PK.6 and PK.12, are exactly the typed factors consumed by the
original-unit transfer.

### 11AD.3. Ordinary transfer with the original unit algebra

Let \(G\rightrightarrows T\) be the
specified Hausdorff second-countable LCH étale groupoid and put
\(R=C_0(T)\). All tensor products below are graded and balanced over
the displayed original anchor. All transfer factors are even.

The correctly anchored conclusion of the transfer is

\[
 \operatorname{For}_G(\eta D)=\varepsilon\quad\text{in }KK_T(R,R),
 \qquad
 \varepsilon=[u]\operatorname{For}_G(\eta_Xd_X)[e_0].
 \tag{AS.1}
\]

Here \(e_0\) evaluates along an **ordinary section** of a compact
probability bundle over \(T\), rather than at a global scalar point.
Whole-base ordinary normalization of the compact-base factor gives
\(\varepsilon=1_R\). The exact weaker local requirement is
\(b_U\varepsilon=b_U\). The proof below constructs all transfer
factors and proves (AS.1); it does not invent the geometric kernel, which remains an explicit hypothesis.

#### 11AD.3. AS.2. The precise supplied interfaces

The following inputs suffice.

1. **Actual nearest-face data.** Section 11Z supplies
   its separable graded proper \(G\)-algebra \(P\), with the
   nondegenerate even central action of \(C_0(M_G)\), the actual
   continuous exactly covariant family
   \(\theta_M\in KK^{G\ltimes M_G}(C_0(M_G),P_M)\),
   \(P_M=C_0(M_G)\otimes_RP\), and the localized lift supplies
   \(D\in KK_G(P,R)\). Its ordinary image is \(D_0\), and
   the whole-family identity is (OC.3):
   \(\operatorname{For}_G(\theta_M)p_M^*D_0=1_{C_0(M_G)}\)
   in the ordinary relative category. This is an actual family
   identity, not a list of fibre indices.
2. **Compact probability bundle.** \(p_X:X\to T\) is an actual
   proper anchor of a Hausdorff second-countable locally compact
   \(G\)-space with nonempty compact convex probability fibres.
   The topology makes fibrewise convex combinations continuous,
   including zero weights. It has a supplied continuous ordinary
   section \(x_0:T\to X\). For example these requirements hold
   for a fibre-probability space on a supplied proper compact-fibre
   space \(K\to T\), with its weak topology, if the section is
   actually supplied. The existence of this \(K\), its kernel,
   or its section is not inferred here from a derivative group or
   from \(M_G\).
3. **Actual Bott factorization over that bundle.** Put \(A=C_0(X)\).
   Section 11AD.2 supplies a separable graded \(G\)-algebra
   \(B\), with its original \(R\)-anchor and its \(C_0(X)\)-action,
   and even classes
   \(\eta_X\in KK_{G\ltimes X}(A,B)\),
   \(d_X\in KK_{G\ltimes X}(B,A)\). Forgetting the extra base gives
   classes in \(KK_G\) used below. The algebra \(B\) has a supplied
   even nondegenerate central action of a second-countable LCH
   **proper** \(G\)-space \(W\to T\), compatible with its anchor.
4. **Ordinary compact-base calibration.** For a whole-base result,
   Section 11AD.2 proves
   \(\operatorname{For}_G(\eta_Xd_X)=1_A\) in \(KK_T(A,A)\).
   An identity first proved in \(KK_X\) implies this by forgetting
   the extra base. For the requested local conclusion it suffices
   instead to supply the weaker whole-class identity
   \(b_U[u]\operatorname{For}_G(\eta_Xd_X)[e_0]=b_U\).
   Separate fibre vacuum indices do not supply either assertion.

The proper-space square-cutoff construction in GP.9–GP.11 gives the
continuous equivariant \(\psi:W\to M_G\). The same cutoff construction
on \(M_G\), combined with input 2, gives the equivariant map
\(\Phi:M_G\to X\) in AS.4 below. Thus these maps require no further
equivariant section or fixed probability.

A literal globally compact \(X\) cannot have a surjective anchor to
noncompact \(T\). More significantly, compact fibres without a
**proper anchor** do not suffice for the constant homomorphism used
below. This is a necessary typing hypothesis of this transfer model.

#### 11AD.3. AS.3. Two actual anchored homomorphisms

Properness of \(p_X\) gives the homomorphism

\[
 u:R\longrightarrow A,\qquad u(f)(x)=f(p_X(x)).
 \tag{AS.2}
\]

Indeed the inverse image of the compact support of a unit bump is
compact in \(X\); approximation gives \(u(f)\in C_0(X)\) for
every \(f\in R\). It is nondegenerate, even, \(R\)-linear and
\(G\)-equivariant. Its zero-operator homomorphism cycle on the
standard \(A\)-module has compact left action because \(u(f)\in A\).
If the anchor were merely continuous, \(u\) would in general take
values only in \(M(A)\), and this cycle would not be justified.

The ordinary section gives

\[
 e_0:A\longrightarrow R,\qquad e_0(f)(y)=f(x_0(y)).
 \tag{AS.3}
\]

A continuous section of a continuous Hausdorff anchor is proper:
for compact \(L\subset X\), its inverse image is closed and lies
in the compact set \(p_X(L)\). Thus \(e_0(f)\in C_0(T)\).
The homomorphism is even and \(R\)-linear, and \(e_0u=1_R\).
It is used only after forgetting \(G\); no equivariance is claimed.

#### 11AD.3. AS.4. The anchored probability map and central collapse

Let \(c\) be a square cutoff on the actual proper space \(M_G\),
chosen with support proper over its orbit space as in GP.9–GP.11.
For \(m\in(M_G)_y\) set

\[
 \Phi(m)=\sum_{g\in G^y}c(g^{-1}m)^2\,
                          g\cdot x_0(sg)\in X_y.
 \tag{AS.4}
\]

The weights are nonnegative and sum to one. Over a compact central
set \(L\subset M_G\), the relevant arrows lie in a compact set:
\(g^{-1}m\) belongs to the compact part of \(\operatorname{supp}c\)
over the compact orbit image of \(L\), and properness of the action
relation then bounds \((g,g^{-1}m)\). A fixed range fibre meets that
compact arrow set in finitely many points. A finite bisection cover
gives locally finitely many continuous branches of (AS.4).

For a probability bundle, continuity can be checked without selecting
points at a zero weight: against a bounded weak-coordinate test each
branch contributes its continuous weight times its probability
integral, with a zero extension at the chart boundary. A compact arrow
partition separates overlapping charts and leaves the same sum.
Equivalently the supplied continuous barycentre operation, with its
zero-weight extension, gives this argument directly. Properness of
\(p_X\) keeps all points over a compact unit image in one compact
set. These finite chart checks prove continuity of (AS.4) in the actual
probability topology. Substitution \(g=hk\) proves
\(\Phi(hm)=h\Phi(m)\) exactly. This uses the ordinary section
inside an equivariant cutoff average; it does not assert that the
section itself is equivariant.

Extend the nondegenerate central representation of \(C_0(M_G)\)
on \(P\) to bounded continuous multipliers. Then

\[
 e_P:A\otimes_RP\longrightarrow P,\qquad
             f\otimes p\longmapsto\Phi^*(f)p
 \tag{AS.5}
\]

is an actual even anchored equivariant homomorphism. The two
representations commute, and nuclearity of the commutative factor
gives the minimal tensor homomorphism. Its \(R\)-balancing follows
from \(p_X\Phi=p_M\). Each image is in \(P\), since a central
multiplier times \(p\) is an element. It is nondegenerate, indeed
surjective: \(e_P(u(f)\otimes p)=fp\), and a unit approximate
identity of \(R\) converges on \(P\). No evaluation of
\(C_0(M_G)\) to \(R\) or to a scalar occurs in (AS.5).

#### 11AD.3. AS.5. The actual graded coefficient-extension cycle

Pull the explicit family along \(\psi\), on its source and its
coefficient simultaneously. Write its cycle as
\((E_{\theta,W},F_{\theta,W})\) over
\(P_W=C_0(W)\otimes_RP\). In the actual nearest-face construction
this is the standard \(P_W\)-module with its strict continuous
family multiplier. The argument also applies to the displayed
countably generated family module.

This pullback does not require a proper \(\psi\). For
\(h\in C_c(W)_+\), its compact support has compact image in \(M_G\).
Choose a function in \(C_c(M_G)\) equal to one on that image.
The old defect localized by this function is compact. Pull its
finite-rank approximants back and multiply their vectors by compact
bumps on the new base, including the square root of the new source
cutoff. Their inner products are actual \(P_W\)-elements and they
prove the new compact defect after \(h\). The strict continuity is
checked on the pulled-back local coefficient generators. Uniform
operator bounds and approximation complete this argument on the
whole module. Covariance uses the equivariance of \(\psi\).

Let \(C=B\otimes_RP\), and use the genuine nondegenerate
multiplier homomorphism

\[
 \rho:P_W\longrightarrow M(C),\qquad
       h\otimes p\longmapsto h^B\otimes p.
 \tag{AS.6}
\]

Its nondegeneracy follows by the central approximate identity on
\(B\) and the coefficient approximate identity on \(P\), tested
on finite elementary tensors. Define

\[
 E_B=E_{\theta,W}\otimes_\rho C,\quad
 F_B=F_{\theta,W}\otimes1,
\]
\[
 \pi_B(b)(\xi\otimes c')
       =(-1)^{|b||\xi|}\xi\otimes(b\otimes1)c'.
 \tag{AS.7}
\]

The sign is essential. Since \(\rho(P_W)\) graded-commutes with
\(B\otimes1\), this formula respects the tensor relation and
defines the bounded graded adjointable left representation. It has
the stated adjoint and multiplication identities. The operator
\(F_B\) graded-commutes with \(\pi_B(B)\) exactly. Anchors and
gradings remain the original ones. Elementary tensor actions of
\(G\) preserve the inner product by covariance of \(\rho\); local
bisection generators prove their continuity. The actual family is
exactly covariant, so \(F_B\) is exactly covariant.

Here the multiplier homomorphism (AS.6) need **not** have compact
left action, and arbitrary family compacts are **not** asserted to
remain compact on tensor extension. The correct source-localized
argument is as follows. Take homogeneous \(b,\xi\); linearity and
parity decomposition give the general case. Let \(T_\xi:C\to E_B\) be the creation
map and let \(L_b\) multiply by \(b\otimes1\) on \(C\). Then

\[
 (T_\xi L_b)^*(T_\xi L_b)
       = (b^*\otimes1)\rho(\langle\xi,\xi\rangle)
                                      (b\otimes1)\in C.
 \tag{AS.8}
\]

To check element membership, approximate the actual
\(P_W\)-valued inner product by finite sums \(h_i\otimes p_i\).
The corresponding expression is a sum of
\(b^*h_i^Bb\otimes p_i\), actual elements of \(B\otimes_RP\).
The norm limit belongs to \(C=\mathcal K(C)\).

The compact-root criterion now gives
\(T_\xi L_b\in\mathcal K(C,E_B)\). Explicitly, approximate it
by \(T_\xi L_b f_\delta((T_\xi L_b)^*(T_\xi L_b))\), where
\(f_\delta\) vanishes near zero and increases to one away from
zero. The second factor is compact and the norm error is bounded
by \(\sup_{t\ge0}\sqrt t\,|1-f_\delta(t)|\to0\).
For a homogeneous family rank one,

\[
 (\theta_{\xi,\zeta}\otimes1)\pi_B(b)
       =(-1)^{|b||\zeta|}T_\xi L_bT_\zeta^*,
 \tag{AS.9}
\]

which is compact. Finite ranks and norm approximation give the
same statement for every family compact, and adjoints give the
opposite source product. Factorization of a source into sums of
products gives the equivalent two-sided creation proof.

Finally take a central approximate identity \(h_j\in C_c(W)\).
The original family has
\(h_j(F_{\theta,W}^2-1)\in\mathcal K(E_{\theta,W})\).
For \(b_j=h_j^Bb\), (AS.9) therefore proves
\((F_B^2-1)\pi_B(b_j)\) compact. The bound by
\(\|F_B^2-1\|\,\|b-b_j\|\) removes the cutoff. These
are actual compact operators on the completed module, not merely
fibrewise compact operators. The same finite-rank proof applies
on interval and localized bisection pullback modules. Countable
generation follows from countable generators of the family and
of the standard \(C\)-module. Thus

\[
 T_B=[E_B,\pi_B,F_B]\in KK_G(B,B\otimes_RP)
 \tag{AS.10}
\]

is the correctly anchored coefficient-extension class.

#### 11AD.3. AS.6. Ordinary contractions to sections

Only in this section forget the \(G\)-action, retaining \(R\).
Take the actual ordinary half-mass section

\[
 m_0(y)=(y,\delta_{1_y})\in M_G.
 \tag{AS.11}
\]

The clopen unit sheet makes it continuous. It has mass one and is
not asserted to be equivariant. The alternative
\(\tfrac34\delta_{1_y}\) gives the same ordinary comparison,
by their fibrewise convex homotopy. Define the ordinary anchored
family pullback \(\theta_0=m_0^*\operatorname{For}_G(\theta_M)\)
with **both** source and coefficient changed to the new base
\(T\). Its actual type and normalization are

\[
 \theta_0\in KK_T(R,P),\qquad \theta_0D_0=1_R.
 \tag{AS.12}
\]

For precision its multiplier is the actual family multiplier at
\(m_0(y)\); a compact unit support has compact image under \(m_0\).
Family compact finite-rank approximants there give the new source
compactness. The same pullback of product connections proves
preservation of the products used in (AS.12). Equivalently this is
the explicit balanced base-change proof in OC.2 applied to its
whole-family identity OC.3. No inappropriate homomorphism class for
the nonproper anchor \(M_G\to T\) is used.

The half-mass fibres are convex. The anchored ordinary homotopy

\[
 H_s(w)=(1-s)\psi(w)+s\,m_0(p_W(w)),\qquad 0\le s\le1,
 \tag{AS.13}
\]

stays in the same fibre \((M_G)_{p_W(w)}\), with mass strictly
greater than \(1/2\) and at most one. Its continuity is directly
the continuity of the weak arrow tests and of the anchor. The image
of a compact subset of \(W\) times the interval is compact in
\(M_G\). Substitution in the actual family therefore gives a
completed interval family with continuous localized compact defects.
Applying AS.5 on that interval proves a homotopy of actual anchored
\(B\)-cycles. At its endpoint the parameter depends only on the
unit anchor; the canonical tensor identification then gives

\[
 \operatorname{For}_G(T_B)=1_B\boxtimes_R\theta_0
                          \quad\text{in }KK_T(B,B\otimes_RP).
 \tag{AS.14}
\]

This does not require properness of \(\psi\) or of the contraction
map: it requires compact parameter images on each compact central
source support, which the displayed continuous homotopy provides.
At the endpoint the two descriptions identify elementary family
vectors with the ordinary exterior-tensor vectors; their inner
products, left \(B\)-actions and operators agree. Central approximate
identities supply density and hence the unitary identification.
For the actual standard nearest-face module both tensor completions
are simply the standard \(B\otimes_RP\)-module: the dense map
\(\xi\otimes c'\mapsto\rho(\xi)c'\) gives this identification,
and the endpoint operator is the same unit-dependent multiplier
\(1_B\otimes\Theta(m_0(y))\) in both descriptions.

Similarly the fibre-probability homotopy

\[
 K_s(m)=(1-s)\Phi(m)+s\,x_0(p_M(m))\in X_{p_M(m)}
 \tag{AS.15}
\]

contracts to the **section over the same unit**, not to a global
point. It gives actual homomorphisms
\(e_s(f\otimes p)=K_s^*(f)p\). They form an anchored point-norm
homotopy into \(C([0,1],P)\): first approximate \(p\) by a compact
central localization, then use continuity of \(fK_s\) uniformly
on that compact set times the interval. Uniform boundedness removes
the localization. The endpoint is

\[
 \operatorname{For}_G([e_P])=[e_0]\boxtimes_R1_P
                          \quad\text{in }KK_T(A\otimes_RP,P).
 \tag{AS.16}
\]

#### 11AD.3. AS.7. Construction and the exact class comparison

Use the actual equivariant factors

\[
 \eta=[u]\eta_XT_B(d_X\boxtimes_R1_P)[e_P]
                                      \in KK_G(R,P),
 \qquad d=D\in KK_G(P,R).
 \tag{AS.17}
\]

Their successive coefficient algebras are

\[
 R\longrightarrow A\longrightarrow B
 \longrightarrow B\otimes_RP
 \longrightarrow A\otimes_RP\longrightarrow P\longrightarrow R.
 \tag{AS.18}
\]

There is one final proper algebra \(P\). Its proper centre is the
original \(M_G\); none of the tensor products replaces its
\(C_0(T)\)-anchor. Let
\(\gamma_X=\operatorname{For}_G(\eta_Xd_X)\in KK_T(A,A)\)
and put \(\varepsilon=[u]\gamma_X[e_0]\in KK_T(R,R)\).

All degrees are even. Balanced exterior-product interchange,
associativity and the homomorphism units give, using (AS.14)–(AS.16),

\[
 \begin{aligned}
 \operatorname{For}_G(\eta)
 &=[u]\operatorname{For}_G(\eta_X)
       (1_B\boxtimes_R\theta_0)
       (\operatorname{For}_G(d_X)\boxtimes_R1_P)
       ([e_0]\boxtimes_R1_P)\\
 &=[u]\operatorname{For}_G(\eta_X)
       (\operatorname{For}_G(d_X)[e_0]\boxtimes_R\theta_0)\\
 &=\bigl([u]\gamma_X[e_0]\bigr)\otimes_R\theta_0
   =\varepsilon\otimes_R\theta_0.
 \end{aligned}
 \tag{AS.19}
\]

Every factor in this calculation has a displayed source and target
over \(T\). In particular the balanced unit is \(R\), not
\(\mathbb C\). Applying the proved \(\operatorname{For}_G(D)=D_0\)
and (AS.12) yields (AS.1). If the supplied ordinary compact-base
calibration is \(\gamma_X=1_A\), then

\[
 \varepsilon=[u][e_0]=1_R,\qquad
 \operatorname{For}_G(\eta)=\theta_0,\qquad
 \operatorname{For}_G(\eta D)=1_R.
 \tag{AS.20}
\]

The comparison proves the needed normalization without an
equivariant identity for \(\eta_Xd_X\), an equivariant contraction
to \(m_0\) or \(x_0\), or an equivariant evaluation of the proper
centre. Before forgetting \(G\), (AS.17) uses only the actual
equivariant family extension and the central homomorphism (AS.5).

#### 11AD.3. AS.8. The precise local transverse consequence

The original transverse class has type
\(b_U\in KK^\ell(\mathbb C,R)\). It is not being assigned an
invented \(R\)-linear scalar source. Let \(F_T\) forget the anchor
of an ordinary anchored cycle, without changing either coefficient
algebra. Then the exact assertion is

\[
 b_U\otimes_R F_T\operatorname{For}_G(\eta D)
       =b_U\otimes_RF_T(\varepsilon).
 \tag{AS.21}
\]

The right side equals \(b_U\) under the weaker local input in
AS.2(4), and in particular under (AS.20). This is the correctly
typed meaning of \(b_U\operatorname{For}_G(\eta D)=b_U\).
Forgetting the anchor at this last composition introduces no
evaluation or substitution of the algebra \(R\).

The probability and Bott constructions in Sections 11AD.1–11AD.2 supply the stated transfer inputs from the explicit kernel hypothesis in Section 11AC.4. This does not assert that every eligible holonomy groupoid satisfies that hypothesis.

### 11AD.4. Kernel-to-original-unit dual with the exact transverse normalization

Section 11AA supplies
the actual even D∈KK_G(P,R), R=C0(T), and For_G(D)=D0 in KK_T.
Section 11Z gives a continuous family with the whole-base ordinary product
For_G(θ_M) p_M*D0=1 in the relative category. Neither assertion is
inferred from individual fibre indices.

The theorem below closes the remaining KK/family mechanism from the
**explicit** geometric input in Anchored probability compactification and proper kernel integration. Section 11AF supplies that input for the original eligible holonomy scope,
so the theorem applies there. Its ordinary unit comparison does not itself
supply the physical graph operator.

#### 11AD.4. OU.1. Construction from an actual coherent compact-kernel input

Assume PK.1–PK.2: an actual proper surjective Y→T with ordinary
section y0 and its continuous relatively proper CND kernel on G⋉Y.
PK.1–PK.9 construct X=Prob_T(Y), its **proper** original anchor,
the ordinary section x0, and the continuous relatively proper kernel
K on the actual G⋉X. BA.1–BA.18 construct the actual proper Bott
algebra B and actual even classes

\[
\begin{gathered}\eta_X\in KK_{G\ltimes X}(C_0(X),B),\quad d_X\in KK_{G\ltimes X}(B,C_0(X)),\\\operatorname{For}_{G\ltimes X}(\eta_Xd_X)=1_{C_0(X)}\quad\text{in }KK_X.\end{gathered}
\tag{OU.1}
\]

Forgetting the extra X anchor gives their actual G-classes over T,
by the bisection and compact-source proof in BA.10. The whole-base
ordinary identity remains valid after forgetting to KK_T. The proper
phase centre W of B has its actual universal cutoff map ψ:W→M_G.

PK.6 gives the actual equivariant u:R→C0(X). PK.10–PK.12 give the
actual equivariant central collapse e_P:C0(X)⊗_R P→P. The complete
graded coefficient-extension proof in Ordinary transfer with the original unit algebra
AS.5 gives the actual even family class T_B:B→B⊗_R P. In particular
its multiplier-valued coefficient representation is not mistakenly
assumed to preserve compacts: the R*R identity AS.8 proves compactness
after the actual B-source action.

Define the actual even class

\[
\eta=[u]\otimes\eta_X\otimes T_B\otimes(d_X\boxtimes_R1_P)\otimes[e_P]\in KK_G(R,P).
\tag{OU.2}
\]

Its factors have successive original-anchor types

\[
R\longrightarrow C_0(X)\longrightarrow B\longrightarrow B\otimes_RP\longrightarrow C_0(X)\otimes_RP\longrightarrow P.
\tag{OU.3}
\]

Every homomorphism and class in (OU.2) is actually equivariant.
There is no equivariant evaluation of C0(M_G) or of X to a scalar,
and no replacement of the original endpoint R by C0(M_G).

#### 11AD.4. OU.2. Whole-base ordinary calibration

Let m0(t)=(t,δ_(1_t)) be the ordinary half-mass section and
θ0=m0*For_G(θ_M)∈KK_T(R,P). Balanced base change of the actual
whole-family identity gives θ0 D0=1_R. The explicit ordinary
section-valued homotopies in AS.6 give

\[
\begin{gathered}\operatorname{For}_G(T_B)=1_B\boxtimes_R\theta_0,\quad\operatorname{For}_G[e_P]=[v_0]\boxtimes_R1_P,\\v_0:C_0(X)\to R,\qquad v_0u=\mathrm{id}_R.\end{gathered}
\tag{OU.4}
\]

The homotopies contract each parameter inside the fibre over its
**same unit**. Compact source supports times the homotopy interval
have compact parameter image, so all compact defects are actual
completed-module compact operators. The mass on M_G need not be
continuous; the homotopy uses its weak arrow tests, not a mass
normalization. The second homotopy uses the continuous fibrewise
probability barycentre established by PK.3.

All factors are even. Balanced exterior-product interchange and
associativity, with the displayed R unit, give the actual equality

\[
\operatorname{For}_G(\eta)=[u]\operatorname{For}_G(\eta_Xd_X)[v_0]\otimes_R\theta_0=\theta_0\quad\text{in }KK_T(R,P).
\tag{OU.5}
\]

because (OU.1) is the whole-base identity and v0u=id_R. Multiplying
by the actual For_G(D)=D0 proves

\[
\operatorname{For}_G(\eta\otimes_PD)=1_R\quad\text{in }KK_T(R,R).
\tag{OU.6}
\]

This is an ordinary identity proved by completed homotopies and
the tensor-vacuum identity in BA.9. It makes no equivariant gamma=1
assertion. In fact the transfer comparison remains valid with any
ordinary compact-base product γ_X, yielding
For_G(ηD)=ε=[u]γ_X[v0]; the exact weaker local input is b_U ε=b_U,
as proved in AS.19–AS.21.

#### 11AD.4. OU.3. The original transverse class is preserved with its full degree

The original disk class has type b_U∈KK^q(C,R), including its supplied
orientation and full right Clifford factor. It is not an R-linear
scalar-source class. Let F_T:KK_T(R,R)→KK(R,R) forget just the
ordinary anchor, keeping both original algebras. Equation (OU.6)
gives the exactly typed assertion

\[
b_U\otimes_R F_T\operatorname{For}_G(\eta\otimes_PD)=b_U.
\tag{OU.7}
\]

No odd factor is moved in this multiplication: η,D and their
comparison are even. Thus neither the transverse degree q nor its
Clifford order/sign changes. Equation (OU.7) is precisely the
normalization required by the original-unit transfer certificate.
The descent and maximal normal class use
it in CA.9. This construction does not supply the physical graph
operator or replace its entire inverse coefficient by its determinant.

#### 11AD.4. OU.4. Exact present application boundary

The original-unit KK mechanism is proved from PK's geometric input.
GK.4–GK.7 prove the implication from a
continuous CND source-pair kernel on actual G×_s G, with uniform
bounded compact-generator increments and the specified relatively
compact low-value increments, to exactly the Y,y0,k required here.
Their source-point notation and our range-point notation agree by
the explicit groupoid homeomorphism

\[
k(g,y)=\psi(g,g^{-1}y),\qquad p(y)=r(g).
\]

It sends the common-range CND matrix in (PK.1) to the matrix
ψ(g_i^−1g_j,g_j^−1y) in (GK.12), and sends the compact sublevel
in (PK.2) to (GK.13). Thus no opposite-action or unit convention
is silently substituted at this interface.
No openness or local triviality of Y→T is required by our theorem.
The metric's injective first-jet map and compact-fibre frame bundle
do not themselves supply that source-pair kernel. The general eligible-holonomy construction in Section 11AF supplies the
geometric application prerequisite. Thus the actual eta and its whole-base
ordinary unit comparison are constructed there; a suitable physical
graph operator still requires its own normal connection, domain and comparison.

![Section-valued ordinary contractions and the original coefficient chain](../figures/anchored-transfer.png)

**Figure 11AD.1.** The upper panels are exact two-unit samples of the contractions to sections; their endpoints remain in different original unit fibres. The lower panel records the actual coefficient chain and ordinary comparison. Proof locators: AS.11–AS.21 and OU.1–OU.7. The geometric kernel hypothesis is retained. [Vector figure](../figures/anchored-transfer.svg) · [Generator](../reproduction/anchored-transfer/draw_anchored_transfer.py) · [Exact data](../reproduction/anchored-transfer/figure-data.json) · [Reproduction](../reproduction/anchored-transfer/REPRODUCTION.md) · [Component terms](../reproduction/anchored-transfer/COMPONENT-TERMS.md).


## 11AE. Smooth labelled kernels and normal analytic control

Use Labelled geometric kernels over the original unit space for the GK hypotheses and Actual anchored Bott factors from a coherent probability-kernel input for the continuous Gram field. The real form of Kasparov’s stabilization theorem supplies the ordinary stabilization below.

### 11AE.1. Smooth approximation with both coarse controls

Let G be the actual Hausdorff second-countable
smooth étale holonomy groupoid over the smooth transversal T. Assume exactly
GK.4's continuous same-source CND kernel k on G times_s G, with its bounded
compact-arrow increments and relatively compact low-increment sets. This
argument does not construct that input; Section 11AF supplies it. It proves that this
input can be made smooth on the actual labelled pair manifold, with explicit
preserved bounds. It makes no claim of uniform transverse derivatives at
spectral compactification points.

#### SK.1. A continuous single Hilbert-valued arrow map

For x in T use the real zero-sum Gram realization of k on G_x. On the
generator e_h=delta_h-delta_(1_x), its inner product is

\[
\langle e_h,e_l\rangle=\frac{k(h,1_x)+k(l,1_x)-k(h,l)}2.
\]

Conditional negativity proves positivity and Cauchy–Schwarz; quotient null
vectors and complete. Bisection sections multiplied by compact source bumps
give a countable dense generating family with continuous Gram entries.
Their local uniform completion is a continuous real Hilbert field E over T.
In particular h maps to e_h continuously as a section of s*E on G. This
is an actual field statement, including arrows with identical endpoints
and distinct isotropy labels; no label is identified with another.

Apply the proved real C0(T)-linear stabilization to
E plus C0(T,H0), with H0 a fixed countably infinite real Hilbert space.
The ordinary linear unitary identifies the result with C0(T,H0).
The arrow map becomes a norm-continuous map f:G to H0, with

\[
f(1_x)=0,\qquad \|f(h)-f(l)\|^2=k(h,l)\quad(s(h)=s(l)).
\tag{SK.1}
\]

This ordinary trivialization is not asserted to intertwine a G-action.
The same-source kernel needs no such assertion. The field construction
uses precisely the countable bisection/Gram proof used in BA.1, with the
source convention of GK.4.

#### SK.2. Actual smooth approximation on the labelled arrow manifold

For any epsilon>0 there is a smooth H0-valued map F on G such that

\[
F(1_x)=0,\qquad \|F(h)-f(h)\|<\varepsilon\quad(h\in G).
\tag{SK.2}
\]

Here is the global approximation proof. The unit sheet is open in a smooth
étale groupoid and closed because G is Hausdorff. It can be treated separately,
with F=0 there. On its complementary smooth manifold, continuity gives an
open cover U_i with sample points h_i and
||f(h)-f(h_i)||<epsilon for h in U_i. Construct a countable locally finite
relatively compact refinement and smooth subordinate partition chi_i.
For completeness, take a compact exhaustion with nested interiors, cover
each successive closed annulus by finitely many coordinate balls whose
closures are in its slightly enlarged annulus, and shrink the balls while
retaining a cover. Each point meets only finitely many enlarged annuli.
Smooth coordinate bumps supported in the unshrunk balls and positive on
the shrunk balls have a locally finite positive sum. Dividing them by that
sum gives the desired partition. Refining each ball into one of the U_i
keeps the approximation test.

Put F(h)=sum_i chi_i(h) f(h_i). The sum is locally finite. Thus it is smooth
in the norm topology of H0, with every derivative given by a finite local
sum. Convexity and sum_i chi_i=1 give (SK.2). Extending by zero across the
unit sheet is smooth because the sheet and its complement are clopen.
Neither a global enumeration of an arrow fibre nor a global group action
has been used.

#### SK.3. Smooth CND and explicit compact-control preservation

Define on the actual smooth fibre product

\[
k_{\mathrm{sm}}(h,l)=\|F(h)-F(l)\|^2\quad(s(h)=s(l)).
\tag{SK.3}
\]

It is smooth, symmetric, nonnegative and zero on the diagonal. For each
finite real zero-sum family c_i in the same source fibre,

\[
\sum_{i,j}c_ic_jk_{\mathrm{sm}}(h_i,h_j)=-2\left\|\sum_i c_iF(h_i)\right\|^2\le0\quad\left(\sum_i c_i=0\right).
\]

This proves conditional negativity. From (SK.1)–(SK.2),

\[
\left|\sqrt{k_{\mathrm{sm}}(h,l)}-\sqrt{k(h,l)}\right|\le2\varepsilon.
\tag{SK.4}
\]

Consequently a compact-arrow upper bound U_C in GK.4 becomes

\[
U_{C,\mathrm{sm}}=(\sqrt{U_C}+2\varepsilon)^2.
\tag{SK.5}
\]

If s(g),r(g) are in a compact K and k_sm(gh,h)<=R, then
k(gh,h)<=(sqrt(R)+2 epsilon)^2. Thus the original compact set
C(K,(sqrt(R)+2 epsilon)^2) contains every such actual g. This
is the required lower/proper control for k_sm. Equivalently any original
distance lower control rho_- becomes max(0,rho_- -2 epsilon).
All source units, endpoints and arrow labels are unchanged.

#### SK.4. Exact analytic boundary

GK.5–GK.7 can therefore use a smooth same-source input from
the continuous kernel proved in Section 11AF. This does not prove that its compact-fibre spectrum
has a smooth locally trivial bundle structure. It does not bound the
transverse derivatives of k_sm(gh,h) uniformly in the unrestricted arrow
h. Local finite sums are smooth at each actual labelled arrow, but the
partition's derivatives can grow along escaping arrows. In particular
one may not conclude that differentiating the spectral-field action gives
a bounded normal-Dirac connection defect. The closed domain and compact
phase comparison for a physical graph operator remain separate obligations.

The full inverse normal coefficient, grading and right Clifford order are
not modified by this scalar kernel approximation. No K-homology class is
relabelled as a geometric graph operator by this theorem.

### 11AE.2. A smooth kernel with a normal flow discontinuous in norm

Smoothness of the actual labelled kernel supplies neither a norm-continuous
normal flow on its compactification algebra nor the compact bounded-phase
commutator of the untwisted normal derivative on the regular columns. The
following example proves both failures on eligible metric holonomy. It does
not refute existence of a different suitable geometric graph operator.

#### CK.1. Actual holonomy and a smooth labelled Hilbert map

Let T=R/Z with metric dx² and let the discrete group Z act by irrational
rotation x↦x+nα, α=(√5−1)/2. Its actual germ groupoid is G=Z⋉T:
irrationality makes the action germ effective. The mapping-torus suspension
is a regular Riemannian foliation of a compact torus. This is the eligible
example in GK.2. Multiplication is (δ,x+nα)(n,x)=(δ+n,x).
Keep the supplied inverse normal character exp(−inπ/5) and its entire
right-Clifford module; the scalar construction below does not alter it.

Set ε=1/8 and define the single smooth real-valued arrow map

\[
 F(n,x)=n+\varepsilon\sin(2\pi n^2x),\qquad F(0,x)=0,
 \qquad \varepsilon=\frac18.
 \tag{CK.1}
\]

Each integer n selects one smooth circle component of the arrow manifold.
For two arrows with the same actual source put

\[
 k((n,x),(m,x))=|F(n,x)-F(m,x)|^2.
 \tag{CK.2}
\]

This is smooth on G×_sG. It is symmetric, zero on the diagonal and CND:
for real c_i with Σc_i=0 the quadratic sum equals
−2|Σc_iF(n_i,x)|². Distinct labels remain distinct; the Hilbert map is
not substituted for the arrow space.

#### CK.2. Both coarse controls hold uniformly

For composable g=(δ,x+nα), h=(n,x), the two sine values differ by at
most two. The triangle inequality and its reverse therefore give

\[
 \max(0,|\delta|-\tfrac14)
 \le \sqrt{k(gh,h)}
 \le |\delta|+\tfrac14.
 \tag{CK.3}
\]

A compact subset of G has only finitely many δ-labels. The upper bound
in GK.4 is consequently max_{δ∈labels(C)}(|δ|+1/4)², with zero for an
empty C. If k(gh,h)≤R, (CK.3) gives |δ|≤√R+1/4. With both original
units in a compact K⊂T, the relevant g lie in the finite union of
{δ}×K for those labels. This is compact; the actual endpoint condition
is closed. Thus the complete lower control of GK.4 holds. GK.5–GK.7
apply and supply the proper compact-fibre spectrum and proper CND
transformation kernel. The example satisfies those hypotheses.

#### CK.3. One bounded generator has unbounded normal derivatives

For the global bisection δ=1 take the unit cutoff identically one, which
belongs to C_c(T) since T is compact. Its generator (GK.8), in the actual
coordinates h=(n,x), is

\[
 a(n,x)=\bigl[1+\varepsilon
       \{\sin(2\pi(n+1)^2x)-\sin(2\pi n^2x)\}\bigr]^2.
 \tag{CK.4}
\]

It is bounded and smooth on G, and belongs to the concrete compactification
algebra A⊂C_b(G). Yet

\[
 a(n,0)=1,\qquad
 \partial_xa(n,0)=4\pi\varepsilon(2n+1)
                 =\frac\pi2(2n+1).
 \tag{CK.5}
\]

There is no uniform derivative bound in the unrestricted arrow h.
Continuity on every fixed smooth bisection does not supply such a bound.

#### CK.4. Even norm continuity of the normal unit flow fails

The maps R_t(n,x)=(n,x+t) are genuine smooth groupoid automorphisms:
circle translation commutes with all the irrational rotations. On bounded
functions put σ_t a=a∘R_t. These maps preserve the supremum norm of C_b(G).
For n=4m≥4 set t_n=1/(4n+2). The sine difference in (CK.4) at x=t_n is

\[
 \begin{aligned}
 \Delta_n
 &=2\sin(\pi(2n+1)t_n)
        \cos(\pi(2n^2+2n+1)t_n)\\
 &=2\cos\left(\pi\left(\frac14+\frac1{8n+4}\right)\right)
 \ge1.
 \end{aligned}
 \tag{CK.6}
\]

Here n/2=2m removes an even integral multiple of π, and
1/4+1/(8n+4)≤5/18<1/3. The cosine lower bound is therefore 1/2.
It follows exactly that

\[
 t_n\longrightarrow0,\qquad
 \|\sigma_{t_n}a-a\|_\infty
 \ge a(n,t_n)-a(n,0)
 \ge(1+\tfrac18)^2-1=\frac{17}{64}.
 \tag{CK.7}
\]

Thus this normal flow is not norm continuous on the generator. If a
concrete C*-algebra containing a and its translates admits this flow as
automorphisms, it cannot be a strongly continuous C*-dynamical action.
The conclusion also applies to any proposed spectral lift intertwining
those concrete functions and asserted to induce norm continuity.
It does not assume that the original A is invariant under every R_t;
failure of that invariance would itself prevent the proposed lift.

#### CK.5. The normal derivative on regular columns has no bounded commutator

Consider the specified scalar column completion
H=L²(T,dx)⊗ℓ²(Z), on which A acts by multiplication a(n,x).
Let D=−i∂_x⊗1. Its Fourier realization multiplies the coefficient at
(j,n) by 2πj. The closed domain is the set of coefficients satisfying
Σ_{j,n}(1+4π²j²)|u_{j,n}|²<∞; finite Fourier/label sums are a core.
This gives a self-adjoint operator directly by the diagonal resolvents.
On smooth finite-label vectors the commutator with (CK.4) is
−i multiplication by ∂_xa. For every n, continuity near x=0 and (CK.5)
give an arc where its magnitude is at least half its value there.
Choose a unit smooth vector supported in that arc in column n. Its
commutator norm is at least π(2n+1)/4. These bounds diverge, so the
commutator has no bounded extension. No assertion about the domain of
multiplication on all of Dom D is needed for this core contradiction.

#### CK.6. The bounded-phase commutator is also noncompact

Use the actual bounded transform F_D=D(1+D²)^−1/2. Its value on Fourier
mode j is f(j)=2πj/√(1+4π²j²). For n≥4 put u_n=e^{2πix}⊗δ_n.
These unit vectors are orthogonal. In the Fourier series of a(n,x), the
coefficient at −(n+1)² is exactly iε. The linear sine term gives iε;
the squared sine difference has only the frequencies
0, ±2n², ±2(n+1)², ±(2n+1), ±(2n²+2n+1).
None equals −(n+1)² for integer n≥4. For the possibly closest equality
(n+1)²=2n², n²−2n−1=0 has no integer root.

Consequently the coefficient of [F_D,M_a]u_n at Fourier mode
1−(n+1)² in the same column is iε{f(1−(n+1)²)−f(1)}. Since that
first mode is negative and f(1)>1/2, its magnitude is at least ε/2.
Thus

\[
 \|[F_D,M_a]u_n\|\ge\frac1{16},\qquad n\ge4.
 \tag{CK.8}
\]

An orthonormal sequence converges weakly to zero: its scalar inner
products with any fixed vector tend to zero by the sum-of-squares bound.
A compact operator sends it to a norm-null sequence, first for finite
ranks and then by operator-norm approximation. Equation (CK.8)
therefore proves noncompactness. The calculation takes place on the
specified completed columns and uses a bounded commutator; it is not
an inference from unbounded commutators alone.

Tensoring this column calculation with a fixed finite inverse-spinor/right-
Clifford block preserves its lower bounds. For any unit Clifford involution c,

\[
 [F_D\otimes c,M_a\otimes1]=[F_D,M_a]\otimes c,
 \qquad \|c\|=1.
 \tag{CK.9}
\]

The inverse character exp(−inπ/5) remains in arrow transport and does
not change these scalar multiplication tests. Neither its determinant
nor a scalar index replaces the complete inverse normal coefficient.

#### CK.7. What this obstruction establishes

The smooth-kernel approximation SK.1–SK.5 preserves coarse controls.
The example above shows that those controls and smoothness still do not
justify norm differentiation of the compactification algebra or a naive
normal Dirac product on these columns. A carefully chosen kernel, a
different completed geometric connection, a confining auxiliary, or
another operator with a proved class comparison could repair the problem.
Indeed the same irrational rotation also has the kernel n² independent
of x. The result is a failure of the proposed implication, not a theorem
that an eligible normal graph-Dirac construction cannot exist.

![Smooth labelled kernel, exact coarse controls and failure of normal norm continuity](../figures/smooth-kernel-normal-flow.png)

**Figure 11AE.1.** Every irrational-rotation arrow label and the full inverse normal phase is retained. The labelled map at x=1/4, derivative coefficients, Fourier coefficient i/8 and margins 17/64 and 1/16 are exact; the upper-right curves are numerical samples. The lower-right panel displays the orthonormal-column test that proves noncompactness. Proof locators: CK.1–CK.9; compare SK.1–SK.5 for the preserved coarse controls. A different kernel or geometric connection may repair this failure. [Vector figure](../figures/smooth-kernel-normal-flow.svg) · [Generator](../reproduction/smooth-kernel-normal-flow/draw_normal_flow.py) · [Exact finite checks](../reproduction/smooth-kernel-normal-flow/finite_checks.py) · [Data](../reproduction/smooth-kernel-normal-flow/figure-data.json) · [Reproduction](../reproduction/smooth-kernel-normal-flow/README.md) · [Component terms](../reproduction/smooth-kernel-normal-flow/COMPONENT-TERMS.md).

## 11AF. Complete Riemannian holonomy and the original-unit kernel

The following proofs supply the general geometric input of GK.4 on the actual eligible holonomy groupoid. Every original arrow, isotropy label, unit fibre and inverse normal coefficient is retained. The stronger normal-connection premises and the physical graph operator remain separate.

### 11AF.1. Complete frame geometry and local matrix-monodromy labels

Let (M,F) be a regular smooth foliation on a compact smooth manifold without
boundary, with its supplied positive holonomy-invariant normal metric and
full invariant normal Spin-c structure. This is the original eligible scope.
Let T be any Hausdorff second-countable complete transversal, and G⇒T its
actual holonomy germ groupoid. No presentation as a discrete action is assumed.

We prove the local hypotheses of EK.1 on H=G⋉P, where P→T is the oriented
normal orthonormal frame bundle. These are local label models, not a claim
that all local isometries globalize to one global action. The only Lie-algebra
input is the real Ado theorem, Low-degree cohomology, Whitehead’s lemmas and the Levi decomposition, Theorem 7.2. Its
complete proof uses finite weighted PBW modules and field descent; it supplies
a faithful Lie-algebra matrix map, not a groupoid theorem. The discrete matrix
embedding input is UE.1–UE.15 at its characteristic-zero matrix scope.

#### EG.1. The actual compact frame foliation and its complete transverse fields

Let N→M be the oriented normal frame bundle. Its fibre SO(q) is compact, so
N is compact. The positive metric descends to each local transversal and
the local holonomy maps are local isometries. The Koszul formula makes their
transverse Levi-Civita connections agree. Consequently the solder form σ
and this connection form ω on N are globally defined and invariant under
lifted holonomy. In foliation coordinates they contain no leaf-coordinate
differentials. The simultaneous map

\[
(\sigma,\omega):TN\longrightarrow\mathbb R^q\oplus\mathfrak{so}(q)
\tag{EG.1}
\]

is surjective; its kernel is the tangent space of the lifted foliation F_N,
whose leaves are the actual frame-parallel lifts of leaf paths. Thus it gives
a global transverse parallelism of dimension d=q+q(q−1)/2.

Choose a smooth complement to TF_N and let X_1,…,X_d be representatives
of its dual transverse frame. They are foliate: in each foliation chart their
normal coefficients depend only on the transverse coordinates, so
[X_i,Y]∈TF_N for every leaf-tangent field Y. Their flows preserve F_N; this
follows by differentiating the transported tangent distribution and using
the bracket identity, or directly by the projectable coordinate ODE.
They are complete because N is compact. Every lifted holonomy map commutes
with their induced local transverse flows and preserves the parallelism.

The groupoid on P obtained from this frame foliation is precisely H=G⋉P.
A leaf path sends u to dh(u). Conversely any such arrow is its actual lifted
path. If two lifted arrows have the same source and target frame, their
underlying germs have the same value and derivative; (GK.1) makes them
equal. H is therefore principal. This observation retains all original
g-arrows: distinct isotropy elements of G send a given frame to distinct
frames. No supplied Spin-c phase has been quotiented or reconstructed from
the derivative; it remains the original transport on each g.

We will use new small frame transversals as well as P. Their union gives
an actual Hausdorff second-countable étale holonomy enlargement containing
H as a full subgroupoid. Hausdorffness follows from the same first-jet
separation; countably many coordinate transversals and the countable local
holonomy-chart construction give second countability. Its restriction back
to P is the original H, not a different original anchor.

#### EG.2. Uniform extension from complete transverse flows

Here is a direct completeness proof for the parallelized frame pseudogroup.
In a transversal near a point z use ordered flow coordinates

\[
\Phi_z(t)=\exp(t_d\xi_d)\cdots\exp(t_1\xi_1)(z),
\tag{EG.2}
\]

where ξ_i are the local transverse fields. At t=0 their derivatives form
a basis, so these are local diffeomorphisms with uniform parameter boxes
when z ranges in a sufficiently small relatively compact neighborhood.
This uniform statement follows from the inverse-function contraction with
a uniform bound on the derivative and its modulus of continuity.

Given a leaf path c from z to v, apply the globally defined ordered flows
of X_i to c. Each resulting path stays in one leaf, because every flow is
foliate. At its endpoints append the small plaque paths to the chosen
transversals. For all parameters in one box depending only on the endpoint
neighborhoods, the endpoints are Φ_z(t),Φ_v(t). These are actual leaf paths
throughout the box. The smooth endpoint map
Φ_v∘Φ_z^{-1} is an actual holonomy map at each of its points: around any
fixed parameter the path has a finite foliation-chart chain, and its nearby
paths give exactly that chain's holonomy. The endpoint boxes do not depend
on the length or chart count of c, since the complete global flows exist
on the whole path for all these parameters.

For z near x, the flow-coordinate image contains one common neighborhood
U of x; for v near y, all target endpoints lie in a fixed transverse chart.
Thus every holonomy germ with source in U and range near y extends over U.
This is the actual completeness-pair condition. It proves uniform extension,
not global extension of a germ to an isometry of a transverse manifold.

The same argument shows that every holonomy map preserving the parallelism
is determined by its value at one point: it has the formula Φ_v∘Φ_z^{-1}.
The formula depends smoothly on z,v on its common endpoint domains.

#### EG.3. Closure equivalence and a closed local group

In a transversal define x R y when y lies in the closure of the actual
holonomy orbit of x. A limiting source can be fixed: if x_n→x and y_n→y
with actual arrows x_n→y_n, use the small ordered flow taking x_n to x.
Apply the same global foliate flows to their paths. The new target tends
to y, and the new source is exactly x. Hence closure of the full endpoint
relation is the same as closure in each fixed-source orbit.

The common extension formula in EG.2 then gives, for x R y, a local
parallelism-preserving diffeomorphism H_{x,y}=Φ_y∘Φ_x^{-1}: it is the smooth
limit of the extended actual holonomy maps with fixed source x. Its inverse
gives y R x by evaluating nearby inverse maps at y. Actual holonomy maps
preserve orbit closures by continuity. Therefore y R x and y R z imply
x R z: the whole orbit of y, and then its closure, is contained in the orbit
closure of x. R is an equivalence relation. It is closed, by the source-fixing
flow argument; it is invariant under every global foliate flow X_i and under
the limiting local maps H_{x,y}.

Fix x, identify it with 0 in a flow chart, and put C=R_x in a small chart.
For y near 0, the smooth candidate H_y=Φ_y∘Φ_0^{-1} is defined on a common
neighborhood, whether or not y∈C. Set μ(y,z)=H_y(z). For y∈C it is an
actual limiting parallelism map. Hence C is closed under the local product
μ and the local inverse; associativity on C follows because a composition
of parallelism maps is determined by its value at 0. In fact for y∈C its
left translation and inverse compose to the identity on the whole common
coordinate neighborhood. The ambient μ is smooth and obeys
μ(y,0)=y, μ(0,z)=z. We prove the required local Lie conclusion explicitly.

#### EG.4. The finite-dimensional closed-local-group lemma

Suppose C is a closed subset near 0⊂R^d with the preceding smooth ambient
product, inverse and actual local left translations. Put

\[
Y_v(z)=D_2\mu(z,0)v,
\qquad E(tv)=\operatorname{Flow}_{Y_v}^{t}(0).
\tag{EG.3}
\]

Let V be the vectors v for which E(tv)∈C for all sufficiently small positive
and negative t. If y_n∈C\{0} tends to 0 with y_n/||y_n||→v, its repeated
products with step ||y_n|| converge to E(tv), uniformly on a small fixed time
interval. Indeed the recurrence is
z_{k+1}=z_k+||y_n||Y_v(z_k)+o(||y_n||), uniformly on a compact box, by
Taylor expansion of μ. The error estimate for the Euler scheme follows from
the local Lipschitz constant L: the errors satisfy
e_{k+1}≤(1+Lδ)e_k+δ o(1)+O(δ²), whence e_k≤const·(o(1)+δ) for kδ bounded.
The iterates stay in C and within the box for that interval. Closedness gives
E(tv)∈C. Negative times use the inverse, whose first derivative is −1.

V is a closed linear subspace. Scalar multiplication follows from linearity
of Y_v in v. For v,w∈V, the product E(δv)E(δw) is
δ(v+w)+O(δ²); its repeated products give E(t(v+w)) by the same Euler estimate.
Thus v+w∈V. Continuous dependence of the ODE and closedness of C prove that
limits of vectors in V remain in V. Here the time interval can be fixed for
bounded v: an initially short curve E(tv) in C extends by repeated products
as long as the ODE remains in the fixed coordinate box. Left translations
by C preserve Y_v, since they preserve the original parallelism and
D_2μ(z,0)v is its constant-v frame combination. Thus these products are
exact flow concatenations. A bound on ||v|| gives a uniform small time box,
which justifies the stated passage to limits. Conversely each nonzero v∈V is a tangent
direction of C because E(tv)=tv+O(t²). Hence all normalized limiting tangent
directions of C are exactly the unit directions in V.

Choose a basis v_1,…,v_k of V and let S(t) be the ordered product of E(t_iv_i).
It lies in C and has derivative of rank k at zero. For a complement V' of V,
the map (t,w)↦μ(S(t),w), w∈V', has invertible derivative at zero. If its
value p lies in C, its inverse left translation gives w=μ(S(t)^{-1},p)∈C.
There is a neighborhood with C∩V'={0}; otherwise normalized nonzero points
there tending to zero would have a unit tangent direction in V'∩V, impossible.
Thus, in that neighborhood, C is exactly S(t). It is an embedded manifold,
and its inherited smooth associative local operations make it a local Lie
group. This proves the local Lie conclusion without invoking a Lie-pseudogroup
or local Hilbert-fifth theorem.

The same construction applies at every point. The ordered foliate flows
carry R_x to R_{Φ_t x}; hence their dimensions are locally constant and
their tangent planes depend smoothly on x. Jointly near (x,x), the relation
is parameterized smoothly by a source point and its transported S(t).
Its local classes are the leaves of a smooth regular distribution D.

The quotient by R is Hausdorff: for unrelated x,y, closedness of R gives
neighborhoods U,V with (U×V)∩R empty; their saturations are disjoint and
open, since the local maps H_{x,y} make saturation of an open set open.
Local foliation coordinates for D give its smooth quotient charts. No
extra return relation identifies two transverse coordinates in a small
such chart: the joint parameterization above represents every nearby R-pair
by a sufficiently small S(t), whose path stays in a larger foliation chart.
The transverse coordinate is constant along that path. Shrinking the original
chart yields precisely one local quotient coordinate for each class.
These open quotient charts have smooth changes of coordinates because their
projections are submersions. Countable transverse charts give second countability.

Adding the original leaf coordinates gives a smooth submersion

\[
Q:N\longrightarrow W=N/\overline{F_N}
\tag{EG.4}
\]

to a Hausdorff second-countable manifold. Its fibres are the actual leaf
closures. They are compact connected submanifolds, since N is compact and
the closure of a connected leaf is connected. The map q:P→W is its actual
restriction. No assertion that the original unframed closure dimensions are
constant is needed.

#### EG.5. Foliated product neighborhoods, with an actual globalization proof

Every complete flow of X_i sends a leaf closure to a leaf closure, by EG.3.
It therefore descends to a smooth flow on W; smoothness follows from the
submersion charts in EG.4. Choose at w a constant linear combination of the
X_i for each basis vector of the transverse quotient T_wW. Their descended
ordered flows give a coordinate map t↦w(t) on some neighborhood V of w.
Let Ψ_t be the corresponding actual ordered global flows on N. If F_w=Q^{-1}w,

\[
F_w\times V\longrightarrow Q^{-1}V,
\qquad (z,w(t))\longmapsto\Psi_t(z)
\tag{EG.5}
\]

is a diffeomorphism. Its inverse is y↦(Ψ_t^{-1}y,w(t)), with t the smooth
coordinate of Q(y). Indeed Q(Ψ_t z)=w(t), independently of z∈F_w, and Ψ_t
is a global diffeomorphism with inverse on the entire fibre. It preserves
the original frame foliation, since each X_i is foliate. Thus the restricted
foliation is exactly F_w's foliation times V. This is the needed actual
local globalization and compatible product statement. It is not a discrete
action theorem imported for the original groupoid.

#### EG.6. The flat Lie coframe on each actual closure

In the transverse parallelism coordinates, the tangent plane of a closure
has a fixed subspace E_w⊂R^d along that closure. Actual and limiting holonomy
maps preserve the parallelism, and their orbits are dense in the closure,
so those coordinate planes agree throughout it. Pick a basis of E_w and
let η_1,…,η_ℓ be the corresponding constant linear combinations of X_i,
restricted to F_w. They are tangent to F_w, foliate and give its full
transverse frame relative to the original leaf foliation.

The normal bracket coefficients [X_i,X_j] modulo TF_N are smooth basic
functions. They are constant along each original leaf, and therefore on
its closure F_w. Brackets of the η_a stay tangent to F_w, so their normal
coefficients are constant numbers in E_w. They define a real Lie algebra
𝔩_w; Jacobi follows by taking normal brackets of the actual foliate fields.
Let θ be the dual 𝔩_w-valued form on F_w, zero on its leaf tangent space.
The actual formula for exterior differentiation gives

\[
d\theta+\tfrac12[\theta,\theta]=0.
\tag{EG.6}
\]

On η_a,η_b it is exactly the bracket definition; on a mixed leaf/η pair it
is zero because the transverse frame is basic. These pairs span TF_w.

Use Low-degree cohomology, Whitehead’s lemmas and the Levi decomposition, Theorem 7.2 over R to choose a faithful finite-dimensional
matrix map ι:𝔩_w→gl_m(R). Set A=ιθ. Its rank is ℓ and
dA+A∧A=0. The rank-zero case is a point transversal with trivial labels.

#### EG.7. Matrix development and countable, discrete monodromy labels

For a path c in F_w solve P'=P A(c'), P(0)=1. This matrix linear ODE has
an invertible solution on each compact time interval, by successive integration;
its inverse solves the inverse equation. Concatenation multiplies transports.
Flatness makes transport invariant under endpoint-fixed homotopy. To check it,
for a two-parameter path put B=P^{-1}∂_sP−A(∂_sc). Differentiating and using
flatness gives ∂_tB=[B,A(∂_tc)], with B(0)=0. Thus B=0; at a fixed final
endpoint ∂_sP=0. Subdivision handles piecewise smooth paths and homotopies.
Coordinate-ball approximation gives the same fundamental group as continuous
paths: finite subdivisions and straight homotopies in those balls perform
the approximation while fixing endpoints.

It follows that the universal path-cover has a smooth developing map D,
with D^{-1}dD=A, and deck monodromy in a countable subgroup
Λ_w⊂GL_m(R). Countability can be proved without finite triangulation:
a countable coordinate-ball cover has only countably many path components
of its overlaps, since those overlaps are open in a second-countable manifold.
Choose one connecting path for each such component and each pair of balls.
Every loop is homotopic to a finite word in this countable graph, using finite
subdivision and contraction inside its balls. Countability is sufficient
for EK.1; no finite-generation assertion is needed.

Near a chosen frame-transversal point, a local primitive D is constant along
leaves and has faithful differential on the transverse η-space. It is a
diffeomorphism from a small ℓ-dimensional transversal O onto a local integral
manifold of M↦Mι(𝔩_w) in GL_m(R). This matrix distribution is involutive
because [ιv,ιu]=ι[v,u]. Its local integral chart can also be constructed by
ordered matrix flows; their tangent spans remain invariant by the bracket
ODE, and the inverse-function theorem gives the rank-ℓ chart. Thus no global
Lie-group integration theorem is hidden in this coordinate step.

For an actual holonomy arrow h with source and range in O,

\[
D(rh)=\lambda(h)D(sh),\qquad \lambda(h)\in\Lambda_w.
\tag{EG.7}
\]

To see membership, close its leaf path by the chosen local primitive paths
from a fixed base point; leaf paths have A=0, and the resulting loop transport
is λ(h) or its inverse, according to the closing orientation. On a connected
holonomy chart, preservation of θ gives h* A=A. Differentiation of
D(hz)D(z)^{-1} is then zero, so λ is locally constant into discrete Λ_w.
Composition and inverse in (EG.7) prove that λ is a groupoid homomorphism.
For a fixed source and label, the range and germ are unique, since D is
injective on O and a parallelism map is determined by its value. This keeps
the actual germ labels; it does not measure their size in the matrix Lie topology.

#### EG.8. The compactness of finitely many labels

Choose O much smaller than a completeness source neighborhood O_0 and a
local primitive target chart O_1 at the same point. The domains can be chosen
so that every germ with both endpoints in O extends over the connected
O_0 and has image in O_1: completeness gives the extension, and its isometry
property bounds the image of short source paths by their length, uniformly
in the germ. Take O_0 a small geodesic ball and O_1 a larger ball; small
endpoint balls and the finite local distance bound ensure this inclusion.
Completeness here is EG.2 applied to F_w and its complete fields
η_a, and the positive transverse metric declares that frame orthonormal.
The developing coordinates are injective on O_1. A germ with label δ then
has on all O_0 the formula

\[
\tau_\delta(z)=D^{-1}\bigl(\delta D(z)\bigr),
\tag{EG.8}
\]

because its derivative satisfies h*A=A and O_0 is connected. Thus any
δ admitted at one point of O defines that same actual map on the whole
O_0. Its arrows with endpoints in O are exactly the restriction of this
map to O∩τ_δ^{-1}O. For compact K,L⊂O, their portion is the closed set
{z∈K:τ_δ(z)∈L} in one genuine bisection. It is compact. Finitely many
labels give a finite compact union. This proves the proper-label condition
of (EK.1), not merely an injectivity assertion for jets.

Use the foliated product (EG.5) to transport O,O_0,O_1 and every actual
map τ_δ across V. The same discrete group Λ_w and labels work on O×V,
and its finite-label compactness holds on arbitrary compact endpoints in
that product. This is an actual holonomy isomorphism induced by the
foliated diffeomorphism (EG.5), so all transition and inverse phases remain
the ones of the original arrows. Different product neighborhoods need no
common identification of their label groups: EK.4 uses invariant outer
weights on W and proves compatible kernel patching directly.

Every original frame leaf is dense in its closure. It meets a foliation box
whose full local transversal is O; the short plaque path in that box then
meets O itself. Therefore its orbit meets O×{v} in each product fibre, so
the enlarged open transversal O×V meets
every orbit over V. Étaleness gives the bisections β_i required in EK.2.
A locally finite cover of W by these V, subordinate cutoffs and the actual
holonomy enlargements consequently satisfies all EK.1 hypotheses for H.
EK.4 and compact-frame averaging EK.5 now supply the continuous CND kernel
on the actual G×_sG with both GK.4 controls.

This argument constructs local matrix-monodromy labels and patches their
kernels; it does not assert that the whole original holonomy is one discrete
action, that its jet map is proper, or that an arbitrary K-orientation supplies
the positive invariant metric. The supplied full inverse Spin-c transport
is unchanged on every original arrow.

#### EG.9. Exact extension to complete bundle-like manifolds

The preceding EG.1–EG.8 theorem assumes compact M and is unconditional at that scope. The same geometry proof also applies to a second-countable
M with a supplied complete bundle-like metric extending the invariant normal
metric. Here is the replacement for every use of compactness.

On its oriented normal frame bundle put the connection metric
π* g_M+⟨ω,ω⟩, using a fixed Ad-invariant positive inner product on so(q).
The projection decreases path length and its compact-fibre anchor is proper.
A Cauchy sequence projects to a convergent sequence in complete M; eventually
its projection lies in a compact coordinate neighborhood. Its frame preimage
is compact, so it has a convergent subsequence in N. The original sequence
then converges, proving metric completeness of N. The horizontal normal and
vertical representatives of the frame dual to (σ,ω) have fixed norm 1.
They remain foliate as in EG.1. A bounded vector field on a complete metric
manifold has a complete flow: a putative finite terminal trajectory is
Cauchy by its speed bound, converges, and the local ODE extends it past that
time. Thus every X_i is complete. All arguments EG.2–EG.5 apply unchanged.

Closure fibres are now closed connected submanifolds, with no compactness
claim. Each η_a in EG.6 is a constant linear combination of the unit X_i;
it is bounded on N and tangent to its closed closure fibre, so its complete
global flow stays in that fibre. This supplies the completeness used in EG.8.
Their Lie coframe and development are global on that fibre. Its monodromy
group is countable by EG.7; EK.1's explicit countable-linear construction
requires no finite generation. Local product neighborhoods EG.5 use the
globally complete X_i rather than a proper-fibration theorem, so they also
require no compact closure fibre. Locally finite partitions in EK.2–EK.4
replace finite partitions and prove the same whole-base controls.

The global derivative refinement in EK.6 specifically retains its compact
manifold and finite good-transversal hypotheses. This extension supplies
the continuous, locally smooth source-pair kernel, not that stronger uniform
derivative bound. An abstract complete Riemannian pseudogroup without a
supplied complete foliation realization is not silently identified with this
extension. An arbitrary incomplete noncompact eligible foliation is outside this particular geometric provider; completeness is not added to the frozen graph-class problem as a universal hypothesis.

### 11AF.2. Label-preserving positive-kernel patching and frame transfer

This is the kernel construction needed after the local geometry. It is an
actual source-pair construction; no equivariant scalar evaluation or groupoid
replacement by a discrete action is used. The local label hypotheses below
must be proved for the eligible holonomy geometry before EK.5 is applied.

Let H⇒Z be Hausdorff second-countable LCH étale. Suppose q:Z→W is a continuous
invariant map to a second-countable manifold. Choose a locally finite smooth
partition χ_a on W with supp χ_a contained in an open set V_a. Put Z_a=q^{-1}V_a.
For each a assume an étale enlargement E_a containing H|Z_a as a full subgroupoid,
an open unit set U_a meeting every orbit of Z_a, and a continuous homomorphism

\[
\lambda_a:E_a|U_a\longrightarrow\Lambda_a
\tag{EK.1}
\]

to a countable characteristic-zero linear group with its discrete topology.
Its required proper-label condition is: for compact K,L⊂U_a and finite
S⊂Λ_a, the set of arrows with source in K, range in L and label in S is
compact. No properness of the map into the Lie topology is assumed.

#### EK.1. Countable linear labels have the required positive kernels

The completed scalar and matrix lemmas UE.1–UE.15 supply coarse Hilbert
embeddings for finitely generated characteristic-zero linear groups. To
apply them, take the field generated by the finitely many generator matrix
entries, regard its finite algebraic extension as matrices over a rational
function field, and apply the values, triangular kernels and weighted sum
UE.13–UE.15. Every step after the faithful matrix inclusion uses just these
matrix data; no compactness of the matrix subgroup occurs. This proves
the stated linear-group consequence of that input. It also supplies
the following kernels for any countable subgroup Λ; here is the direct-limit
argument, so no permanence theorem is imported.

Write Λ=∪Λ_m with Λ_m increasing finitely generated subgroups. If F_m is
the input map controlled by a^{-1}b, put f_m(a)=F_m(a^{-1}); its difference
is controlled by a b^{-1}. Thus f_m is the required right-coarse Hilbert
map, with uniform bounds for every finite right-difference set. For t>0 the kernel

\[
u(a,b)=\exp[-t\|f_m(a)-f_m(b)\|^2]
\tag{EK.2}
\]

is normalized, real, nonnegative and positive definite. Indeed expand
exp(2t⟨v,w⟩) as the sum of its tensor-power Gram kernels and multiply by
exp(−t||v||²)exp(−t||w||²); the norm-convergent series proves positivity
on every finite matrix. It is a Gram kernel of unit vectors.

Extend it to Λ by using one orthogonal Hilbert summand for each right coset
Λ_m c, writing a=h_a c there and using f_m(h_a). Distinct cosets have inner
product zero. In the same coset a b^{-1}=h_a h_b^{-1}, so right-coarse control
is retained. For any prescribed finite D⊂Λ and ε>0 take m with D⊂Λ_m and
t small enough. The extended kernel satisfies

\[
1-u(a,b)\le\varepsilon\quad(a b^{-1}\in D),
\qquad
u(a,b)\le\eta\quad(a b^{-1}\notin S_\eta),
\tag{EK.3}
\]

where for each η>0 the exceptional S_η⊂Λ is finite. The second assertion
uses the proper lower control in Λ_m; outside its cosets the kernel is zero.
These statements are uniform in a,b. Only these positive-kernel statements,
not an invariant CND function on Λ, will be used.

#### EK.2. Transfer across actual local bisections

Choose countably many continuous source sections β_i in E_a with source
domains covering Z_a and ranges in U_a. Each is locally a bisection; global
injectivity of its range is unnecessary. Existence follows from orbit fullness
and étaleness, initially using bisections if convenient. Choose
a locally finite partition φ_i on Z_a subordinate to their source domains,
with each support closed inside that domain. Here is a concrete locally
finite construction. Exhaust the unit space by compact sets contained in
the interiors of their successors; cover each compact annulus by finitely
many precompact charts subordinate to the source domains. Shrink inside
those charts and choose nonnegative bumps on their enlargements. The
resulting supports are locally finite, and their positive sets cover.
Divide every bump by their positive sum. This gives the required φ_i with
compact individual supports. In the smooth case divide smooth bumps by
the square root of their squared sum instead; their squares are φ_i.
The same construction on W gives the χ_a and its square version.
For x=s(h), define the actual label

\[
c_{ij}(h)=\lambda_a\bigl(\beta_i(rh)h\beta_j(x)^{-1}\bigr).
\tag{EK.4}
\]

It is used only where φ_i(rh)φ_j(x) is nonzero. These are actual composable
arrows in the enlargement. Its restriction to the original H does not change
an original arrow or its isotropy label.

For a positive kernel u as in EK.1 choose unit Gram vectors v(c). Put

\[
V_{j,h}=\sum_i\phi_i(rh)v(c_{ij}(h)),\qquad
W_{j,h}=V_{j,h}/\|V_{j,h}\|,
\]
\[
Q_{a,u}(h,l)=\sum_j\phi_j(x)\langle W_{j,h},W_{j,l}\rangle,
\qquad s(h)=s(l)=x.
\tag{EK.5}
\]

All sums are locally finite. The denominator is nonzero, since u≥0 and
u(c,c)=1 imply ||V_{j,h}||²≥Σ_iφ_i(rh)²>0. Local norms are bounded away
from zero. The resulting kernel is continuous: labels are locally constant
on actual arrow charts, the weights are continuous, and only finitely many
terms occur locally. On each source fibre it is normalized positive definite,
as a direct-sum Gram kernel. It takes values in [0,1].

The key cancellation retains unrestricted h and j. For a left increment g,

\[
c_{ij}(gh)c_{kj}(h)^{-1}
=\lambda_a\bigl(\beta_i(rg)g\beta_k(sg)^{-1}\bigr).
\tag{EK.6}
\]

The right side is independent of h and of the source choice j. This is why
ordinary patching of unbounded embeddings is not used: a change of their
range weights would multiply unbounded vectors. The vectors in (EK.5) are
unit vectors, and the common-source choices cancel in their differences.

For a compact arrow set C and its compact unit endpoints K, finitely many
i,k have cutoff supports meeting K. Include C and all unit arrows at K
when collecting their increment labels. The β-images of their closed support portions
in K are compact. Continuity of λ into a discrete group therefore gives a
finite set D(C) of all the labels in (EK.6), including g=1_z for z∈K. If u
has defect at most ε≤1/2 on D(C), all the constituent unit vectors for h
and gh, with a fixed j, are within √(2ε) of one another. Convex averaging
gives ||V_{j,gh}−V_{j,h}||≤√(2ε) and each norm is at least √(1−ε).
Normalization gives

\[
1-Q_{a,u}(gh,h)\le8\varepsilon
\quad(g\in C),
\tag{EK.7}
\]

uniformly over all h and all active j. The factor 8 follows from
||v/||v||−w/||w||||≤2||v−w||/min(||v||,||w||) and the half squared-distance
formula for unit vectors. No compactness assumption on s(h) is present.

#### EK.3. Lower control survives normalization and patching

Fix compact K⊂Z_a. Let M be an upper bound for the number of active range
weights at every point of K; local finiteness and compactness make M finite.
Thus ||V_{j,h}||≥1/√M when rh∈K, uniformly in j and s(h). Take η=1/(2M)
in (EK.3). Outside a compact exceptional set of left increments g with
s(g),r(g)∈K, every label in (EK.6) is outside S_η. Indeed, for each of the
finitely many active i,k, a label in S_η makes
δ=β_i(rg)gβ_k(sg)^{-1} belong to the compact labelled-arrow set supplied
by (EK.1), with endpoints in the compact images
β_i(K∩supp φ_i),β_k(K∩supp φ_k). Multiplication
g=β_i(rg)^{-1}δβ_k(sg) projects a closed fibre product of compact sets to
a compact set. The finite union over i,k is compact.

Outside this union every cross inner product in V_{j,gh},V_{j,h} is at
most η. Their convex averages have inner product at most η, and division
by their norms increases it by at most M. Consequently

\[
Q_{a,u}(gh,h)\le\tfrac12
\tag{EK.8}
\]

uniformly in h. Finite-label properness, rather than the topology of the
linear matrix image, is the compactness input in this calculation.

#### EK.4. The whole-base continuous CND sum

Choose increasing compact arrow and unit exhaustions C_n,K_n, with interiors
covering H,Z, C_n containing the units of K_n and the endpoints of C_n in
K_n. For a with supp χ_a meeting q(K_n), restrict to the compact portions
over supp χ_a, which lie inside V_a. There are finitely many such a.
Choose u_{n,a} by EK.1 to have defect ≤2^{-n}/16 on all the finite labels
D_a(C_n). For the other a choose any such positive kernel. Define

\[
Q_n(h,l)=\sum_a\chi_a(q(x))Q_{a,u_{n,a}}(h,l),\qquad
k_H(h,l)=\sum_{n=1}^{\infty}(1-Q_n(h,l)),
\quad x=s(h)=s(l).
\tag{EK.9}
\]

The outer weights are invariant, so they are unchanged under left increments.
Every Q_n is continuous, normalized, real nonnegative and positive definite
on the actual source fibres. For g in a fixed compact C, eventually g∈C_n
and (EK.7) bounds 1−Q_n(gh,h) by 2^{-n}. The finite initial terms are at
most 1. This proves uniform upper control on every compact left-increment
strip. A compact subset of H×_sH has compact image h l^{-1}; the same bound
gives locally uniform convergence. Hence k_H is continuous, symmetric,
nonnegative and zero on the diagonal. For a real zero-sum family,
Σc_ic_j(1−Q_n(h_i,h_j))=−Σc_ic_jQ_n(h_i,h_j)≤0. Passage through the finite
convergent sum proves CND on each source fibre.

For compact unit endpoints K and a bound R, choose N with N/2>R. Only
finitely many a are active over q(K). EK.3, applied to the compact unit
sets K_a=K∩q^{-1}(supp χ_a)⊂Z_a and to all n≤N, gives one compact
exceptional set D(K,N) of g-arrows. Outside it
Q_n(gh,h)≤1/2 for n≤N and every h, so k_H(gh,h)≥N/2>R. Thus

\[
\{g:s(g),r(g)\in K,\ \exists h\ k_H(gh,h)\le R\}
\subseteq D(K,N).
\tag{EK.10}
\]

This is precisely both GK.4 controls at the whole original H-unit scope.
The construction tolerates different label groups and Hilbert spaces on the
overlapping local models; invariant outer weights preserve CND and the lower
bound. It requires no identifications between those Hilbert spaces.

#### EK.5. Return to the original holonomy anchor by compact-frame averaging

Let G⇒T be actual eligible holonomy and P→T its oriented normal orthonormal
frame bundle. The compact fibre is SO(q), with its probability Haar measure
m_x; q=0,1 have their evident trivial fibre conventions. Local trivializations
and bi-invariance of Haar give a continuous probability field, independent
of frame choices, invariant under every derivative transport.

Suppose the preceding construction has supplied k_H for H=G⋉P. For h,l∈G_x
lift both arrows from the same u∈P_x and set

\[
k_G(h,l)=\int_{P_x}k_H\bigl((h,u),(l,u)\bigr)\,dm_x(u).
\tag{EK.11}
\]

The compact-fibre integration is continuous in source-pair charts by local
trivialization and uniform continuity on their compact frame factors. The
same-source CND inequality holds inside every integrand and passes through
the probability integral. Symmetry and the zero diagonal also pass.

For compact C⊂G, the set of its lifted left increments (g,v), v∈P_{s(g)},
is compact, since P→T is proper. Upper control for k_H therefore bounds
k_G(gh,h) uniformly in h. For compact K⊂T, P_K is compact. If
k_G(gh,h)≤R, some u has its continuous integrand ≤R. The lifted left
increment is (g,h·u), with both endpoints in P_K. EK.10 places it in one
compact subset of H. Projection to G gives the required compact exceptional
set, uniformly in h. This proves GK.4 for the original G, without a frame
section and without changing its unit algebra.

No original arrows are quotient-identified. Distinct isotropy arrows remain
distinct g in (EK.11), even when their derivative moves frames. The normal
inverse module is still its supplied module with transport
z_g^{-1}ρ_t(s_g), right Clifford action and grading; the scalar kernel does
not replace or trivialize any phase. Metric frame averaging does not turn a
weaker K-orientation into invariant positive metric data.

#### EK.6. Smooth source-pair models and the exact derivative scope

All local geometry in EG.1–EG.8 is smooth, labels are locally constant, and
the unit partitions may be chosen as squares: χ_a=b_a²,φ_i=a_i² with smooth
b_a,a_i and Σb_a²=Σa_i²=1. Hence (EK.5) has an explicit unit-vector model
in one fixed countable Hilbert direct sum:

\[
T_n(h)=\bigoplus_{a,j}b_a(q(sh))a_j(sh)W_{a,n,j,h},
\qquad
F_H(h)=2^{-1/2}\bigoplus_n(T_n(h)-T_n(1_{sh})).
\tag{EK.12}
\]

Each T_n is locally a smooth finite sum of fixed label vectors. Its norm is
1 and its Gram kernel is Q_n. Thus ||F_H(h)−F_H(l)||²=k_H(h,l) for the same
source, with exactly the sign and normalization in (EK.9).

One may choose the defects ε_n faster than before to obtain smoothness on
all actual arrow charts. Exhaust compact arrow/source-pair charts; arrange
C_n also to contain the compact increment images hl^{-1} of the first n
such charts. At step n, bound all cutoff derivatives through order n on
those compact sets and all their closed cutoff-support portions.
For a fixed j, all constituent unit vectors of h and its increment are within
√(2ε_n), by (EK.6), including every index in a cutoff support. Subtract one
of these locally constant reference vectors. Derivatives of the convex weights
sum to zero, so derivatives of V minus that reference are bounded by a cutoff
jet constant times √ε_n. Its norm differs from 1 by O(ε_n); differentiation
of its positive inverse square root gives the same estimate for the normalized
W. Product differentiation of the source and outer square weights gives

\[
\|\partial^r(T_n(gh)-T_n(h))\|
 \le C_{r,L}\sqrt{\varepsilon_n}
\tag{EK.13}
\]

on each prescribed compact source-pair chart L, for r≤n. Constants depend
only on its finite cutoff jets, not on the label vectors or the dimension of
their Hilbert space. Choose ε_n≤2^{-n}/(1+max_{r≤n,L≤n}C_{r,L}²).
The direct sum in (EK.12) and all its derivatives then converge locally in
Hilbert norm. This proves a smooth fixed-Hilbert source-pair model, and a
smooth scalar k_H. Compact-frame averaging is smooth by differentiation
under the fixed compact Haar integral, giving a smooth k_G on G×_sG.

An explicit model on the original units is the ordinary Hilbert source bundle
\(\mathscr H_x^G=L^2(P_x,m_x;\mathscr H)\), where \(\mathscr H\) is the fixed
receiving space in (EK.12), with \(F_G(h)(u)=F_H((h,u))\). A local frame
trivialization identifies this with \(L^2(SO(q))\otimes\mathscr H\). The
same-source distance squared is exactly k_G by (EK.11). Smooth joint frame
derivatives on compact source-pair charts imply Hilbert-norm smoothness in
these trivializations: the compact Haar factor bounds the derivative norms
and justifies difference quotients and differentiation under the integral.

There is also a useful uniform derivative refinement for the usual finite
complete coordinate transversal of the original compact manifold. Choose its
balls with compact closures inside larger transverse coordinate balls. In
EG.5, each compact F_w has a finite foliation-chart cover and finite local
paths into O. Their source sections extend across the full product V. A finite
square partition on F_w, pulled through that product, supplies the a_i. A
finite cover of the compact W supplies b_a. On each closed outer-cutoff
support the product parameters stay in a compact subset of V. The restrictions
to the chosen transversal extend over its compact coordinate closures, so
all jets of the used cutoffs and the products b_a a_i are uniformly bounded.
Global range injectivity is why source sections,
rather than unnecessarily global bisections, were allowed in EK.2.

Holonomy preserves the frame parallelism, so differentiation of a range cutoff
along a source-pair chart is the same ordered transverse derivative at that
range point; no long-arrow derivative growth occurs. Now the constants in
(EK.13) can be taken as C_{r,C}, uniformly in every unrestricted h, when g
ranges in a compact arrow chart C. Source cutoff derivatives are bounded
on every used outer support by the preceding finite construction; range derivatives are
bounded on the compact endpoints of C. The label cancellation is independent
of h and j. Fast defect choices therefore give finite bounds for every
transverse derivative of compact-arrow increments of F_H, uniformly in h.
For the finitely many initial n not satisfying the small-defect estimates,
the finite range partitions still give ||V||≥1/√M globally; their unit
feature vectors and bounded cutoff jets give uniform bounds for all their
normalized derivatives. Thus the initial terms cause no missing bound.
For F_G, all frame lifts of a compact G-increment chart form a compact
H-increment set. Normal coordinate differentiation in a fixed frame
trivialization is a combination of the frame-parallelism derivatives with
bounded coefficients and jets: the chosen normal charts have compact
closures and SO(q) is compact. The proved H-bounds consequently give the
same uniform unrestricted-h compact-increment derivative bounds for F_G
in its original-unit L² frame trivializations. Matrix-label multiplication
is not asserted to act unitarily on the receiving feature space.

This refinement concerns the actual source-pair model at that finite good
transversal, which is still an open and generally noncompact whole unit space.
For an arbitrary complete transversal the locally smooth model and GK.4
controls hold, but such a global derivative bound is not inferred without
bounded cutoff jets. Neither statement supplies a smooth connection on the
final kernel-spectrum/Gram field or a globally norm-continuous equivariant
Fredholm operator. Those analytic constructions need their own proofs.

### 11AF.3. The general kernel and its original-unit consequences

Let (M,F) be a regular smooth foliation on a compact manifold without
boundary. Its normal bundle has a specified positive holonomy-invariant
metric and a specified full invariant normal Spin-c structure. Let T be
any Hausdorff second-countable smooth complete transversal and let G⇒T
be its actual holonomy germ groupoid. Multiplication is gh when s(g)=r(h).
Write G_x=s^{-1}(x). The invariant metric separates germs by their first
jets, so G is Hausdorff; that fact supplies no properness assertion.

**Theorem (the general eligible-holonomy kernel).** There is a continuous
real kernel k:G×_sG→0,∞), symmetric, zero on the diagonal and conditionally
negative definite on each actual source fibre G_x. It has these two controls:

1. For every compact C⊂G there is U_C<∞ such that k(gh,h)≤U_C for all
   g∈C and all h with r(h)=s(g).
2. For every compact K⊂T and every R<∞, the set
   {g:s(g),r(g)∈K and ∃h with r(h)=s(g), k(gh,h)≤R} is contained in
   one compact subset of G.

Both estimates include every unrestricted h and every original isotropy
arrow. The original source and unit algebra remain G and C0(T).

The proof has three parts. [The frame geometry, EG.1–EG.5
constructs complete global transverse flows on the actual compact normal
frame foliation. Their path action gives uniform germ extension. Its closed
local return groups are proved to be Lie groups by the Euler/product argument
in EG.4. Thus closure fibres form a smooth quotient W and have actual
foliated product neighborhoods. EG.6–EG.8
constructs a flat Lie coframe, faithful matrix development and countable
discrete monodromy labels. Uniform germ extension proves that finitely many
labels with compact endpoints give a compact arrow set.

The kernel construction, EK.1–EK.4
uses positive unit-vector kernels on those countable linear label groups.
It transfers them through actual source sections. The exact identity

\[
\lambda(\beta_i(rg)gh\beta_j(sh)^{-1})
\lambda(\beta_k(sg)h\beta_j(sh)^{-1})^{-1}
=\lambda(\beta_i(rg)g\beta_k(sg)^{-1})
\tag{EH.1}
\]

removes h and the common source choice j from every tested left-increment
label. Convex normalization gives unit-vector kernels and uniform upper
defects; the proved finite-label compactness gives their uniform lower
defects. Invariant partitions on W patch different local label groups while
preserving both estimates and positive definiteness. The summable series
in (EK.9) is the continuous CND kernel on H=G⋉P, where P→T is the actual
oriented normal frame bundle. Compact-frame averaging, EK.5
then gives

\[
k(h,l)=\int_{P_{sh}}k_H((h,u),(l,u))\,dm_{sh}(u),
\qquad s(h)=s(l),
\tag{EH.2}
\]

where m_x is the invariant probability Haar field on the SO(q)-torsor P_x.
A bounded average has an integrand at most that bound; the lifted increment
then lies in the proved compact exceptional set. Projection yields the
second control on the whole original G. This finishes the theorem.

The only non-elementary ordinary inputs are the characteristic-zero
matrix embedding proof UE.1–UE.15 in K-theory of the leaf space, Section 11N
and real Ado, Low-degree cohomology, Whitehead’s lemmas and the Levi decomposition, Theorem 7.2.
The matrix proof applies after any supplied faithful characteristic-zero
matrix inclusion; compactness of a matrix subgroup is not used. Ado supplies
only a faithful Lie-algebra matrix map. All needed actual groupoid extension,
local Lie structure, development and gluing are proved here. The local-Lie
route is situated in [Álvarez López–Masa, Sections 9 and 12](https://arxiv.org/html/1311.3511)
and [Haefliger–Porto Silva](https://arxiv.org/abs/1706.01551); no theorem from
those sources replaces a step of this proof.

The same theorem holds when compact M is replaced by a second-countable
manifold carrying a supplied complete bundle-like metric extending its
invariant normal metric. EG.9
proves every needed replacement: the frame connection metric is complete,
its transverse fields are bounded and complete, closure fibres are closed,
and countable monodromy and locally finite patching suffice. No compactness
of those closure fibres is claimed.

EK.6
also supplies a locally smooth source-pair Hilbert model on H and an explicit
ordinary L²(P_x) source-bundle model on the original G. On a finite good
coordinate transversal for compact M it proves bounds for every transverse
derivative of compact-arrow increments, uniformly in unrestricted h.
For an arbitrary complete transversal it asserts local smoothness and the
two stated controls. It supplies no connection on the compactified spectrum
or final Gram field; those remain separate analytic constructions.

The theorem feeds exactly the labelled-kernel input GK.4. The proved GK.5–GK.7
construction produces a Hausdorff second-countable G-space p:Y→T with proper
surjective anchor, compact nonempty fibres, a continuous ordinary section,
and a continuous CND function proper relative to both Y-unit maps. Those
are the precise inputs of the original-unit probability/Bott construction
PK.1–PK.13. Thus its general eligible geometric premise is supplied by the
theorem. No physical graph operator is inferred from a scalar kernel.

The specified normal inverse transport is still z_g^{-1}ρ_t(s_g), with its
original right Clifford action and grading. The scalar kernel neither
quotients arrows nor identifies central phases. In particular a weaker
K-orientation alone does not supply the positive invariant metric premise.

![Actual frame geometry, labelled cancellation, convex features and original-anchor compact control](../figures/eligible-kernel-patching.png)

**Figure 11AF.1.** EG.5–EG.8 and EK.2–EK.5. The arrows show the actual source/range order and exact cancellation of h and the common source choice. The feature sample has v0=(1,0), v1=(3/5,4/5), V=(4/5,2/5), squared norm 4/5 and W=(2/√5,1/√5); plotted positions are numerical renderings of those exact coordinates. The frame quotient and Haar-transfer panels are typed schematics of the proved constructions. Related human readings are [Álvarez López–Masa](https://arxiv.org/html/1311.3511) and [Haefliger–Porto Silva](https://arxiv.org/abs/1706.01551). [Vector figure](../figures/eligible-kernel-patching.svg) · [Generator](../reproduction/eligible-kernel-patching/reproduce.py) · [Finite checks](../reproduction/eligible-kernel-patching/finite-checks.json) · [Data](../reproduction/eligible-kernel-patching/eligible-kernel-patching-data.json) · [Reproduction](../reproduction/eligible-kernel-patching/REPRODUCTION.md) · Component terms.

## 11AG. Geometric normal connections and compact-frame descent

Use the exact frame and label geometry in Complete Riemannian holonomy and the original-unit kernel. Fix a finite good coordinate complete transversal T for the compact foliation in that theorem, its oriented normal frame bundle P, and H=G⋉P over Z=P. All actual arrows and full inverse normal Spin-c data are retained. The fields below use new, explicitly constructed compact convex bases; no connection is assigned to an arbitrary KK representative or to the earlier patched-kernel spectrum.

The discrete compact affine field and the proper probability input are proved in Labelled geometric kernels over the original unit space, Anchored probability compactification and proper kernel integration, and Actual anchored Bott factors from a coherent probability-kernel input. Separability, finite-product approximation and density of the observable algebras use The Stone–Weierstrass theorem for functions vanishing at infinity, Proposition 15.1. The geometric operator still requires a confining averaged weight, differentiable inverse-module passage and its closed-domain class comparison.

### 11AG.1. Local affine bundles and normal connections

#### LF.1. Exact geometric input and convention

Let H restricted to Z_a be the actual frame groupoid over an invariant
open set Z_a=q^{-1}(V_a). Use EG.8's actual enlargement 𝓔_a, open full
transversal U_a, and discrete matrix-monodromy functor

\[
\lambda:\mathcal E_a|_{U_a}\longrightarrow\Lambda.
\tag{LF.1}
\]

Here Lambda is the actual countable abstract matrix subgroup. For compact
K,L in U_a, arrows with endpoints in K,L and labels in a finite subset
of Lambda form a compact set. Choose actual smooth source sections
beta_i on open sets O_i covering Z_a, with range in U_a, and a smooth
locally finite partition phi_i subordinate to those domains. Each
support is closed inside its domain. These are the source sections and
partitions proved in EK.2. No global range injectivity is required.

For x in O_i intersect O_j and an actual arrow g:y->x define

\[
\begin{gathered}t_{ij}(x)=\lambda(\beta_i(x)\beta_j(x)^{-1}),\\c_{ij}(g)=\lambda(\beta_i(rg)g\beta_j(sg)^{-1}).\end{gathered}
\tag{LF.2}
\]

These functions are locally constant into the discrete Lambda on their
actual domains. Composition gives t_ij t_jk=t_ik and
c_ij(g)c_jk(h)=c_ik(gh). In particular c_ij(1_x)=t_ij(x),
which need not be the identity if two different charts are used.
This last fact is why an origin correction is indispensable below.

All sections and labels used here are actual ones in 𝓔_a. We do not
assert that every Lambda-label occurs as an arrow above every x.

#### LF.2. A compact coefficient and affine field for countable Lambda

Here is the precise input needed from the existing discrete construction.
There is a compact metrizable Lambda-space Y, Q=Prob(Y), a continuous
real Hilbert field H_Q, and affine onto isometries

\[
A_\delta(q)v=L_\delta(q)v+b_\delta(q):H_q\longrightarrow H_{\delta q}.
\tag{LF.3}
\]

with the exact group cocycle law. For every R, only finitely many delta
have ||b_delta(q)||<=R for some q in Q. Each fixed displacement is a
continuous section and is bounded on compact Q. The field is separable
and countably generated. The action on Q is linear on probabilities.

For completeness, this input applies to the countable group here, not
just to finitely generated groups. EK.1 produces normalized nonnegative
positive kernels u_n on Lambda, arbitrarily close to 1 for any prescribed
finite right-difference set and at most 1/2 off a finite right-difference
set. Choose an increasing finite exhaustion D_n and defects epsilon_n
with sum epsilon_n finite. Write u_n as Gram kernels of unit vectors v_n.
Then

\[
f(a)=2^{-1/2}\bigoplus_n(v_n(a)-v_n(1)).
\]

is defined in one countable real Hilbert space. Indeed a fixed a belongs
to D_n eventually, so the squared tail norm is at most sum epsilon_n.
Its same-source squared-distance kernel is

\[
k(a,b)=\sum_n(1-u_n(a,b)).
\tag{LF.4}
\]

Every finite right-increment strip has uniformly bounded sum: its tail
is bounded by sum epsilon_n, and its finite initial part is bounded by
its number of terms. For given R choose N>2R. Outside the finite union
of the first N exceptional difference sets, the first N terms each
are at least 1/2. Thus k(delta h,h)>R there, uniformly in h.
This proves both GK.4 controls for the actual discrete groupoid Lambda.

Apply the proved concrete spectrum construction GK.5–GK.7 to (LF.4).
Its unit base is a point, so Y is compact. Its CND action function is
proper uniformly on Y by the just-proved finite exceptional-set bound.
PK.3–PK.9's probability averaging retains this bound, and BA.1's Gram
construction gives (LF.3). Those proofs construct the continuous Gram
field and exact affine action; no invariant CND function of Lambda
or invariant probability on Y is asserted. This establishes the whole
input in (LF.3) from the countable matrix-label theorem. The labels in
this auxiliary space are elements of the actual monodromy group; they
are not invented original holonomy arrows.

#### LF.3. The actual compact bundle and groupoid action

Glue O_i times Q by

\[
(x,q_j)_j\sim(x,t_{ij}(x)q_j)_i.
\tag{LF.5}
\]

The cocycle and inverse identities give a Hausdorff locally trivial
bundle X_a->Z_a with compact metrizable fibre Q. More explicitly, on
an O_j chart its quotient map identifies every other representative
uniquely with (x,t_ji(x)q_i). This continuous inverse proves its local
product topology. Distinct base points are separated by unit charts;
distinct points over one base point are separated inside one product
chart. A countable chart cover and countable product bases prove second
countability. A finite cover of a compact base set by relatively compact
subcharts places its inverse image in a finite union of compact products;
it is closed there. Thus the anchor is proper and onto.

For g:y->x, take source chart j and range chart i and send q_j to
c_ij(g)q_j. Changing charts uses exactly the identities from (LF.2),
so this defines one action of the actual H restricted to Z_a on X_a.
It is continuous in each arrow chart since c_ij is locally constant,
and the actual normal map is smooth. Composition and inverse are exact.
No original arrow, including an isotropy arrow, has been collapsed.

An ordinary smooth-normal probability section exists. In chart j set

\[
x_{0,j}(x)=\sum_i\phi_i(x)t_{ji}(x)q_0.
\]

where q0 is any fixed probability on Y. Each term with nonzero phi_i
is defined, local finiteness makes the sum continuous, and convexity
makes it a probability. The transition identity makes these sections
agree by (LF.5). In a local product chart it is a smooth finite convex
combination of constant probabilities. Contraction to x0 within each
convex fibre is the actual whole-base ordinary contraction. No
equivariance of x0 or of this contraction is claimed.

#### LF.4. The affine bundle, its linear field and a genuine origin

Glue H_Q by the affine maps A_(t_ij). Their linear parts glue an actual
continuous real Hilbert field E_a over X_a. Positivity and continuity
of its local Gram fields follow from H_Q; countably many charts and
local scalar bumps give countable generation. Affine gluing gives a
torsor F_a with this linear field. The actual H action is A_(c_ij(g))
on F_a and L_(c_ij(g)) on E_a.

In a j chart the i-chart zero is the vector
b_(t_ji)(q_i), where q_i=t_ij q_j. Define the affine origin

\[
\omega_j(x,q_j)=\sum_i\phi_i(x)b_{t_{ji}(x)}(q_i),\qquad q_i=t_{ij}(x)q_j.
\tag{LF.6}
\]

This is an affine weighted average of the local zero sections. An
affine map preserves such an average precisely because sum phi_i=1;
therefore omega_i=A_(t_ij)omega_j. It is a genuine section of the
affine torsor, not of the incorrectly trivialized linear bundle.
It is continuous and smooth in the normal variable in every flat
chart: the finitely many active labels are locally constant and only
the smooth weights depend on that variable. Its norm in any selected
chart is uniformly bounded above a compact unit set, for all q, by
compactness of Q and finite chart/label covers.

Translate F_a by omega to the linear field E_a. Its actual affine
action then has displacement

\[
B_g=A_g\omega_{\mathrm{source}}-\omega_{\mathrm{range}}.
\tag{LF.7}
\]

In i,j coordinates this means

\[
\begin{gathered}B_g(q_j)=b_{c_{ij}(g)}(q_j)+L_{c_{ij}(g)}(q_j)\omega_j(y,q_j)\\-\omega_i(x,c_{ij}(g)q_j).\end{gathered}
\]

The formula is independent of coordinates, satisfies the exact affine
cocycle identity, and gives B_(1_x)=0. In particular a nonidentity
transition t_ij(1_x) does not create spurious identity displacement.

#### LF.5. Properness retains the actual long-arrow labels

Let K be compact in Z_a. Use finitely many chart supports meeting K.
All their closed support portions lie compactly in the beta domains;
their beta-images are compact in U_a. Enlarge K finitely if necessary
to the compact chart neighborhoods used in this test. In these
finitely many chart trivializations (LF.6) has a common norm bound W_K.
Consequently, if source and range of g lie in K and ||B_g||<=R,

\[
\|b_{c_{ij}(g)}(q_j)\|\le R+2W_K.
\tag{LF.8}
\]

LF.2 restricts c_ij(g) to a finite subset of Lambda. The original
proper-label condition in LF.1 places the conjugated arrow
beta_i(rg)g beta_j(sg)^{-1} in a compact set. Multiplying by the
compact beta support portions and their inverses gives a compact
set of original g. Finite i,j choices give a finite compact union.
This argument retains g itself, not its frame jet or endpoint pair.

The function ||B_g||^2 is continuous and CND on the transformation
groupoid: for common-source affine translates v_l, a zero-sum family
has squared-distance sum -2||sum_l a_l v_l||^2. Equation (LF.8)
proves relative properness. A compact original g-test and compact
X_a fibres also bound ||B_g|| above. Thus X_a with its field and
action is an actual local PK/BA input, with a stronger normal
connection. It is not inferred by evaluating C0(M_H) to a scalar.

#### LF.6. A metric, holonomy-covariant normal connection

In each O_i times Q chart take the dense finite smooth-normal
coefficient sections of H_Q and differentiate only the O_i coordinate.
The transition L_(t_ij)(q_j) and the Q transition t_ij q_j are
independent of that coordinate on each overlap component. Therefore
the local operators agree as a connection on E_a. They satisfy the
Leibniz and metric identities by direct differentiation of the Gram
inner products. They are defined on one dense sheaf, not chosen
separately on unit fibres. The affine connection on F_a is defined
the same way; its translations A_(t_ij) are also constant-normal.

Actual holonomy transport in a g-chart consists of the actual smooth
normal map and the constant-normal map L_(c_ij). The chain rule proves
exact covariance of this connection with the pushed normal vector
field. This does not suppose a label acts unitarily on the original
receiving feature space in EK.12: it uses the separately constructed
Gram field's actual linear isometries (LF.3).

In these charts b_(c_ij) is constant-normal. Differentiating (LF.7)
therefore differentiates only omega and its pullback by the actual
normal map. On compact arrow charts these derivatives of every fixed
order are uniformly bounded in q, by finite labels, compact Q and
the finite smooth coefficient bounds. They are norm-continuous
Hilbert-module derivatives. Choose the ordinary real stabilization supplied by Kasparov’s stabilization theorem over Q independently
of the normal coordinate. It preserves all these statements.
This supplies FJ's analytic flat-field interface locally without its
unnecessary all-arrow-label product-domain assumption.

#### LF.7. A genuine smooth graph weight on this bundle

After that constant-normal stabilization, apply BA.3 to the countable
Lambda action over compact Q. Choose one fixed diagonal positive
self-adjoint Theta>=1 with compact inverse. Its conjugation defects
S_delta=L_delta Theta L_delta^{-1}-Theta are compact and norm-continuous
in q, and Theta b_delta is norm-continuous for every delta. The real
stabilizing summand carries the identity linear action. All these
sections are independent of the normal variable in flat coordinates.

Each chart has this weight. On an overlap the transported chart weight
differs from the other by the bounded compact S_(t_ij), so their
domains coincide and their graph norms are equivalent locally.
Set Theta_a=sum_i phi_i Theta_i on the common domain. In a j chart it
is Theta_j plus a locally finite sum of bounded compact fields. It
is self-adjoint on Dom Theta_j: the Neumann-resolvent argument for a
bounded self-adjoint perturbation at imaginary modulus greater than
its norm proves this directly. Positivity follows on that domain by
the convex sum of weights >=1, hence Theta_a>=1. Its inverse is compact
locally in X_a: the bounded perturbation resolvent identity expresses
it through a compact resolvent of Theta_j. Unit-localized module
compactness, not global module compactness, is the asserted property.

The same local resolvent identity proves continuous field resolvents.
All normal derivatives of Theta_a-Theta_j are bounded compact fields,
since they differentiate only phi_i. Equations (LF.6)–(LF.7), the
common domains, and Theta_j b_delta's norm continuity prove that
omega and B_g are smooth in the graph norm, with locally uniform
bounds of every normal derivative on compact arrow charts. Linear
transports preserve the common domain; their weight differences are
bounded compact fields, smooth-normal by the same finite formula.
Thus the actual affine finite frames/origins satisfy the deformation's
graph-domain tests with a geometric normal connection.

#### LF.8. Exact global and physical boundary

This proves an actual flat compact-base affine-field provider over
each EG.5–EG.8 invariant closure-product neighborhood Z_a of the
frame groupoid. It handles partial label domains by functor pullback,
not by declaring those domains to be full. The underlying auxiliary
compact base is different from the globally patched kernel spectrum.
LF.6–LF.7 consequently do not prove a connection on that old spectrum.

Different V_a have different actual monodromy groups. Global patching through vanishing weights
proves the first global field and connection.
Compact-frame connection descent
returns a particular covariant connection to the original units.
Passing connections through the proper-source inverse classes and the
actual graph-column modules still requires proof. A local KK class or a
local weight is not substituted for the global graph operator.
Its self-adjoint completed domain, normal principal Clifford action,
bounded-phase defects and exact original Bott pairing remain required.
No equivariant gamma=1 is inferred. The full inverse coefficient,
auxiliary-first grading and final right Clifford block are unchanged.

### 11AG.2. Global patching through vanishing weights

#### GJ.1. Flat cutoffs and the collapsed compact product

Use the invariant smooth closure map q:Z->W from EG.4. For compact
original M, W is compact and the EG.5 neighborhoods V_a have a finite
subcover. The same local construction below works with a locally finite
cover. Choose smooth functions 0<=rho_a<=1 with support closed inside
V_a and whose positive sets cover W. Set

\[
\begin{gathered}\beta_a=\begin{cases}e^{-1/\rho_a^2},&\rho_a>0,\\0,&\rho_a=0,\end{cases}\\B=(\sum_a\beta_a^2)^{1/2},\qquad b_a=\beta_a/B,\qquad w_a=b_a^2.\end{gathered}
\tag{GJ.1}
\]

B is positive. All derivatives of beta_a and b_a vanish where rho_a=0:
on a compact coordinate set they are finite sums of an exponential
times reciprocal powers of rho_a and bounded smooth derivatives,
and exp(-1/rho_a^2) dominates every such power. This also proves
their difference-quotient extension at the zero set. Sum w_a=1.
All these functions are invariant under H because they depend on q.

LF gives compact convex bundles X_a->Z_a=q^{-1}V_a. At a unit x,
retain a point z_a in (X_a)_x if w_a(qx)>0 and collapse that factor
to one vertex if w_a(qx)=0. Denote the resulting product by X_x.
Its topology is the following explicit local cone topology. On an
open U meeting a support of w_a, shrink U inside Z_a and trivialize
X_a there. Its a-factor is the image of U times Q_a under

\[
(x,z_a)\longmapsto(x,w_a(qx),w_a(qx)z_a).
\tag{GJ.2}
\]

in the cone on the compact convex probability space Q_a. At zero
weight all points are identified. Convex coordinate functionals of
Q_a determine the cone topology. Locally finitely many supports can
meet U; the other factors are fixed vertices. These descriptions
agree under the LF transition homeomorphisms, which are affine and
preserve weight. They therefore define one Hausdorff second-countable
space X and a proper surjective anchor p:X->Z. To verify properness,
cover a compact unit set by finitely many compact subcharts of (GJ.2).
Their preimages are closed images of compact products of finitely
many Q_a, and hence compact. Each X_x is a compact convex product
in the weighted cone coordinates; this includes collapsed factors.

The actual H action acts separately on its active X_a factors.
Invariant q preserves each weight. Continuity at a collapsed factor
follows directly from (GJ.2): its radius tends to zero regardless of
its Q_a-coordinate, and the latter lies in a compact space. Away from
zero this is LF's continuous action. These facts also give joint
continuity on every compact arrow chart. Composition and inverses
are the actual ones. No frame arrow has been relabelled or removed.

The sections x0_a of LF.3 give a continuous section x0 of X by their
weighted coordinates; continuity at zero again follows from (GJ.2).
This is an ordinary continuous section; no differentiability of its
composition with an arbitrary continuous function on Q_a is asserted.
The fibrewise affine homotopy to x0 is a genuine whole-base contraction.
Its maps are proper: an output lying above a compact unit set has
input above that same set, whose inverse image under p is compact.

#### GJ.2. A continuous Hilbert field and one exact affine action

At z in X_x put

\[
E_z=H_{\mathrm{std}}\oplus\bigoplus_{a:w_a(qx)>0}(E_a)_{z_a}.
\tag{GJ.3}
\]

where H_std is a fixed infinite real Hilbert space with identity H
action. The standard summand is an optional linear stabilization;
its affine displacement is zero. Continuous generating sections are
the standard axes and b_a times LF's local generating sections.
Their Gram entries extend continuously through a collapsed factor
as zero. Finite sums with continuous scalar coefficients, followed
by norm completion, define a continuous countably generated Hilbert
field: positivity holds in the explicit fibres, its displayed
generator norms are continuous, and their spans are dense in every
active fibre. This is not a constant-rank bundle assertion.

The linear H action is the direct sum of LF's linear actions. It is
continuous on those generators; density proves continuity on the
field. Put

\[
C_g=\bigoplus_a b_a(q(sg))B_g^a.
\tag{GJ.4}
\]

The summands vanish to infinite order when they collapse. There are
locally finitely many summands, so (GJ.4) is an actual continuous
section in the target field of each transformation arrow. The exact
affine cocycle identity follows from LF.4 and invariance of b_a.
It thus gives one actual global affine action, rather than a sum of
length functions lacking a linear action.

Its squared displacement is CND and relatively proper. Indeed

\[
\|C_g\|^2=\sum_a w_a(q(sg))\|B_g^a\|^2.
\tag{GJ.5}
\]

For compact unit K finitely many supports occur, say at most N.
For every source of g one weight is at least 1/N. If ||C_g||<=R,
that active a satisfies ||B_g^a||<=sqrt(N)R. Its source and range
belong to the compact portion K intersect q^{-1}(supp rho_a), which
lies inside Z_a. LF.5 places g in a compact original-arrow set.
A finite union proves properness. On compact arrow tests all active
LF displacements are bounded uniformly in their compact Q_a fibres,
which proves the upper bound. The standard summand changes neither
bound. The argument retains the actual long arrow, not its derivative.

#### GJ.3. The dense normal algebra and metric connection

Here is an explicit connection through the zero-weight strata; no
local product of X itself is inferred at a collapsed factor. In a
chart U as in (GJ.2), its full potential product U times product Q_a
surjects onto X|U. Use the dense algebra generated by smooth unit
functions and functions

\[
d(x)f(z_a).
\tag{GJ.6}
\]

where f is continuous on Q_a and d is smooth and flat on {rho_a=0}.
For membership in the connection domain require all normal
derivatives of d to vanish there. Include local scalar cutoffs and
all derivatives of these functions. In particular b_a and all its
derivatives qualify. The algebra is dense in C0(X): compact unit
localization reduces to a finite potential product, ordinary
continuous functions are approximated by functions away from each
zero-weight locus plus unit-only functions, and any such supported
function can be multiplied by a flat cutoff identically one on its
support. Product-coordinate functions separate the remaining compact
fibres; the elementary finite-product approximation gives density.

Differentiate the lifted functions (GJ.6) in x with their Q_a
coordinates fixed in the LF flat charts. Every derivative descends
through collapsed factors because it vanishes there. Relations are
respected: a relation vanishing on X vanishes on the whole potential
product for every x, so its ordinary derivatives vanish there as
well. Overlap changes are constant-normal on every LF factor;
the chain rule therefore makes these derivatives consistent.
This is a dense normal derivation algebra with the actual smooth
normal vector fields and coordinate changes.

The module connection domain consists of local finite sums of these
scalar coefficients times standard axes and flat-weighted LF
generating sections. All their normal derivatives are continuous
sections. For example a flat coefficient d times a local E_a
section can be approximated by cutting off rho_a>=epsilon; its
norm tends uniformly to zero near collapse, so it lies in the module
generated by b_a sections even when d/b_a is unbounded. Derivatives
have the same property. The lifted finite-product metric connection
descends by the same relation argument as (GJ.6). It is dense,
satisfies the metric and Leibniz identities, and transforms by the
actual normal chain rule. LF covariance and invariance of b_a give
its exact covariance under H. Formula (GJ.4) is smooth in this
connection with all compact-arrow normal jets controlled by finitely
many LF tests and flat coefficient derivatives.

This is a connection on the actual field over X. It asserts neither
that its derivations generate complete normal flows on X nor that
an arbitrary ordinary stabilization is differentiable.

#### GJ.4. A weight whose compact inverse survives collapse

Take LF.7's local weights Theta_a>=1. They have common graph domains
in flat charts, compact inverse locally, and bounded smooth-normal
perturbation derivatives. Fix K0>=1 and, when rho_a>0, set

\[
\tau_a=1+K_0\rho_a^{-4},\qquad T_a=\tau_a I+b_a\Theta_a.
\tag{GJ.7}
\]

On H_std choose a fixed positive diagonal compact-inverse weight.
Let T be the direct sum of these weights on (GJ.3). At each active
factor the domain is Dom Theta_a; the scalar b_a is strictly positive
there. T_a>=tau_a and

\[
\|(T_a+i)^{-1}\|\le1/\tau_a\le\rho_a^4/K_0.
\tag{GJ.8}
\]

Hence the resolvent disappears in norm at a collapsed factor. Off
that factor the ordinary bounded-perturbation resolvent identities
give continuous adjointable resolvents. Equation (GJ.8) extends them
by zero through collapse. The direct-sum resolvent identities define
a densely defined regular self-adjoint T on the whole Hilbert
module; its domain is the range of (T+i)^{-1}. Density follows on
the stated generating sections by active-factor cutoffs, finite
Theta cores and the standard axes. The inverse and adjoint-resolvent
identities then prove self-adjointness and regularity directly.

Every compact-unit localization of (T+i)^{-1} is a module compact.
For a factor a, cut off rho_a<epsilon; its norm error is at most
epsilon^4/K0. On the remaining compact set b_a has a positive lower
bound. LF compactness of its resolvent there supplies finite-rank
approximations, whose sections may be expressed by the b_a
generators. Finitely many factors suffice on compact units. The
standard diagonal resolvent is compact there too. Letting epsilon
tend to zero proves the assertion on the completed module.

The scalar addition tau_a is necessary here: using Theta_a alone
would leave nonzero inverse norm at a disappearing factor. The
factor b_a in front of Theta_a has a second purpose. Actual arrow
conjugation gives

\[
U_gT_{\mathrm{source}}U_g^{-1}-T_{\mathrm{range}}=\bigoplus_a b_aS_g^a.
\tag{GJ.9}
\]

because both tau_a and b_a are invariant. The right side is a
genuine unit/arrow-localized compact field: S_g^a is LF's compact
field, and b_a forces its norm to vanish at collapse. The same is
true of all normal derivatives. With a scalar multiple of Theta_a
that instead tends to infinity, this compactness conclusion would
generally fail. Formula (GJ.9) proves preservation of the common
weight domain and equivalence of its transported graph norms.

#### GJ.5. Graph jets and an explicit relative derivative estimate

The generating sections b_a xi and C_g have smooth T graph norms.
The terms in T(b_a xi) are tau_a b_a xi and b_a^2 Theta_a xi;
every derivative of the first coefficient remains flat, and the
second uses LF's graph-smooth cores. The same argument applies to
(GJ.4). Their span is a local graph core: first cut off a graph-domain
section away from each collapsing factor, then use LF's finite
coordinate graph cores there. The cutoffs converge both on the
section and on T times it, by continuity and compact localization.

There is also a useful quantitative normal estimate. In a flat
chart suppose a unit normal vector v satisfies

\[
|v\rho_a|\le L,\qquad|v\log B|\le J,\qquad\|v\Theta_a\|\le H.
\tag{GJ.10}
\]

These constants are finite on the closed finite good chart tests;
v Theta_a means the bounded derivative of its perturbation of the
constant local weight. Since 0<rho_a<=1,

\[
\frac{|vb_a|}{b_a}\le2L\rho_a^{-3}+J,\qquad\frac{|v\tau_a|}{\tau_a^2}\le\frac{4L\rho_a^3}{K_0}.
\]

For lambda>=1, lambda/(tau+b lambda)^2<=1/(4tau b).
Spectral calculus therefore gives

\[
\|(vT_a)T_a^{-2}\|\le\frac{4L\rho_a^3+(2L\rho_a+J\rho_a^4)/4+H\rho_a^4}{K_0}.
\tag{GJ.11}
\]

For the last term use b_a<=1 and T_a>=tau_a. The expression tends
to zero at collapse. Each of its operator terms is compact off
collapse by its decaying scalar spectral function or LF compact
perturbation. The same cutoff proof as GJ.4 makes this a genuine
localized module compact. Choosing K0 large makes its norm as small
as required on a prescribed finite compact normal atlas. This is an
actual relative derivative estimate, not the false assertion that
v T_a is bounded. It is one usable form estimate for a later normal
Dirac/auxiliary sum, whose operator domain still must be proved.

#### GJ.6. Exact provider and next boundary

GJ.1–GJ.5 patch all local monodromy groups on the actual frame
groupoid. They provide one proper compact convex anchor with an
ordinary section/contraction, one proper affine field with exact
action, a dense covariant metric normal connection, and an actual
regular graph weight with localized compact inverse, compact
conjugation defects and controlled graph jets. Variable fibre rank
at zero cutoffs is handled explicitly; no derivative is assigned
to a nonexistent coordinate at a collapsed point.

Compact-frame connection descent
proves an original-unit affine field and connection with the full
inverse Spin-c coefficient retained. A differentiable
Bott/deformation/localization construction on this graph-domain field
has not yet been carried through the inverse KK modules, and a closed
physical graph Dirac sum has not been proved.
In particular (GJ.11) alone proves neither essential self-adjointness
nor the Fredholm compactness of that sum. It supplies an explicit
estimate for that remaining work. The already proved original-unit
normalization belongs to an actual KK class; it is not yet a proof
that a new geometric sum represents that class. The final right
Clifford order, grading and whole inverse coefficient remain required.

### 11AG.3. Compact-frame connection descent

#### FD.1. The pinned probability base over the original units

Let P->T be the oriented normal frame bundle, with compact fibre SO(q)
and normalized invariant probability m_x on P_x. The supplied positive
normal metric makes the actual G frame transports preserve m_x.
Write X_H->P for GJ's proper compact convex auxiliary anchor. Put

\[
X_{G,x}=\{\mu\in\operatorname{Prob}(X_H|_{P_x}):(\mathrm{frame\ anchor})_*\mu=m_x\}.
\tag{FD.1}
\]

These are nonempty compact convex spaces. Nonemptiness follows by
pushing m_x through the actual continuous section x0 of GJ.1.
Give their union X_G the unit coordinate and the weak integrals of
C0(X_H). For compact K in T, (X_H)|P_K is compact, since both anchors
are proper. Probabilities on that compact metrizable space form a
compact metrizable space. The support-over-one-unit and the Haar
marginal constraints are closed: test them against a countable dense
family of continuous base/frame functions. The marginal m_x varies
continuously in x in ordinary local frame trivializations. Therefore
the inverse image of K in X_G is compact. These tests separate points,
so X_G is Hausdorff; countably many relatively compact unit charts
and countably many C0(X_H) tests prove second countability and local
compactness. The anchor to T is proper and onto.

The action of an original arrow g pushes mu through the actual
map (u,z)->(g u,(g,u)z) of X_H. Frame Haar invariance preserves
(FD.1). On a compact arrow chart all relevant points lie in compact
anchor preimages, and ordinary uniform continuity of each test
function proves joint continuity of this pushforward. Composition
and inversion are exact original-arrow identities. No frame section
or equivariant scalar evaluation is used.

The section

\[
\mu_0(x)=(x_0|_{P_x})_*m_x.
\tag{FD.2}
\]

is continuous by compact Haar integration in local frame charts.
Fibrewise convex contraction to it is an ordinary whole-base
contraction. All its maps are proper by the compact anchor argument.
Neither the section nor this contraction is asserted equivariant.

#### FD.2. Averaged Hilbert fields and the exact affine action

Let E_H over X_H, U_h and C_h be GJ's field, linear action and
affine cocycle. For continuous compactly supported field sections
xi,eta define

\[
\langle\xi^\#,\eta^\#\rangle_\mu=\int\langle\xi(z),\eta(z)\rangle\,d\mu(z).
\tag{FD.3}
\]

Finite Gram matrices are positive by integration, their entries
are continuous by the weak topology, and a countable generating
set of sections suffices. Quotient the null vectors and complete,
obtaining a countably generated continuous real Hilbert field E_G
over X_G. Its fibre is the corresponding L2 completion; this is
an assertion about continuous generating sections and the completed
module, not only separately defined fibre Hilbert spaces.

The original G action transports these sections by the actual
linear maps U_(g,u), together with its pushforward of mu. It is
an onto isometry of the Gram spaces by change of variables. It
respects their null spaces, so extends to the completed fields.
Compact bisection tests, compact anchor preimages and density of
the generating sections prove its continuity. Define its target
displacement at g mu by the section

\[
z′=(g,u)z\longmapsto C_{(g,u)}(z).
\tag{FD.4}
\]

Its squared norm is

\[
\Psi(g,\mu)=\int\|C_{(g,u)}(z)\|^2\,d\mu(u,z).
\tag{FD.5}
\]

This section belongs to the field: locally it is an actual
continuous section on compact anchor sets, approximated by finite
global generating sections using scalar partitions and density.
The same change-of-variable calculation and the exact H affine
cocycle law prove the original G affine cocycle law. Its norms and
Gram entries are continuous by those compact tests. Consequently
Psi is a continuous CND transformation-groupoid function.

#### FD.3. Properness still measures the original arrow

Fix compact original unit K and R. All frame endpoints above K
are in the compact P_K. If Psi(g,mu)<=R, at least one point (u,z)
in the compact support fibre has ||C_(g,u)(z)||^2<=R: otherwise
the minimum of the continuous integrand would exceed R and so
would its probability integral. GJ.2's uniform lower control
places the actual lifted frame arrow (g,u) in one compact set of
H. Projection to G places the original g in a compact set. This
is uniform in mu and preserves every original isotropy/return label.

For a compact original g-test, all its frame lifts are compact
because P->T is proper. GJ.2's upper bound then bounds (FD.5)
uniformly in mu. Thus (FD.1)–(FD.5) give an actual original-unit
compact convex proper affine-field input. The required ordinary
section and whole-base contraction are FD.1; fibrewise indices
have not been substituted for these statements.

#### FD.4. Horizontal lifts and a faithful differentiable algebra

The invariant normal Levi-Civita connection gives a horizontal
lift v_hat to P of each smooth normal vector field v on T.
An actual holonomy isometry preserves that connection: the
Koszul identity determines it from the metric and brackets, so
the transported connection has the same identity and agrees.
Thus g_*v_hat=(g_*v)_hat. Horizontal parallel transport on P
acts by an orthogonal frame rotation, hence preserves m_x.

Use GJ.3's dense flat observable algebra A_H and dense field
connection core, with local frame cutoffs; include all their
horizontal derivatives. For f in A_H put f^#(mu)=integral f dmu.
Define on the algebra generated by these functions and smooth
unit functions

\[
\delta_v(f^\#)=(\delta_{\widehat v}f)^\#.
\tag{FD.6}
\]

and extend by the Leibniz rule. This algebra is dense in C0(X_G).
On compact anchors, the linear tests from the dense A_H separate
probabilities; products and constants approximate continuous
functions on their compact image. Unit cutoffs then prove global
density. The same argument provides a countable dense system.

The following path test proves that (FD.6) respects every relation;
it does not presume a globally defined normal flow on X_H. Fix
mu above x. Cover compact P_x by finitely many compact subcharts
inside GJ.1's potential-product charts and choose a smooth
partition chi_j(u) subordinate to them. Lift the measure chi_j mu
to each full potential product over that compact frame subchart.
Such a lift exists for every continuous surjection of compact
metrizable spaces: pullback embeds C(target) in C(source), extend
its positive probability functional with the same norm and unit
value by the real Hahn–Banach theorem, Corollary 2.3, and apply
Haar measure on locally compact groups, Theorem 2.2 to represent the positive functional. Positivity of the extension follows directly:
for 0<=a<=1, norm(1-a)<=1 and its unit value is 1, so its value
on a is nonnegative. Scaling proves positivity for every positive
a. The extension agrees on frame functions, so retains the
marginal chi_j(u)m_x. Zero-mass pieces are omitted.

Move the frame u along the horizontal lift of a short integral
curve of v, keeping each potential Q_a coordinate fixed in the
LF flat charts. The compact subcharts stay in larger charts for
one common small time interval. Push each lifted measure through
the potential-product quotient at that time and sum. This gives
an actual path mu_t in X_G: its frame marginal is m_(x(t)), since
all pieces use the same Haar-preserving horizontal transport and
sum chi_j=1. At t=0 it is mu. For each f in A_H the lifted
function is smooth, with all derivatives descending at collapsed
coordinates by GJ.3. Differentiation under the fixed lifted
measures gives

\[
\frac{d}{dt}f^\#(\mu_t)=(\delta_{\widehat v}f)^\#(\mu_t).
\tag{FD.7}
\]

Its value depends only on mu_t, not on the particular lifted
measures: the derivative test functions already descend to X_H.
If a polynomial relation among the observables vanishes on X_G,
it vanishes on all these paths; differentiation at zero proves
that its assigned derivative vanishes at every mu. This proves
well-definedness of (FD.6), and also all iterated normal
derivatives. It proves no completeness or uniqueness of the
chosen paths on X_G.

#### FD.5. An actual dense closable metric normal connection

On the generating field sections in FD.2 define

\[
\nabla_v(\xi^\#)=(\nabla_{\widehat v}\xi)^\#.
\tag{FD.8}
\]

with scalar Leibniz rule for FD.4's algebra. These are actual
continuous averaged sections. To verify module/null relations,
use the same finite lifted measures as FD.4. In each potential
chart the LF stabilizations are independent of the normal
coordinate, so the flat-weighted vector sections are norm-smooth
maps into one fixed Hilbert space at its Q parameters. They and
their derivatives vanish on any collapsed coordinate. They
therefore give differentiable vectors in the fixed L2 spaces of
the lifted measures. A module relation zero on X_G is the zero
L2 vector along every path mu_t; its derivative at zero is also
zero. This proves that (FD.8) is independent of representatives.
The local constructions agree under the actual LF coordinate
changes and sum independently of the auxiliary partition used
for the path test.

Metricity follows by differentiating (FD.3) under those fixed
lifts and applying GJ.3's metric identity. The same calculation
gives the scalar Leibniz identity with (FD.6). The domain is
dense by the countable continuous Gram generators and the
dense scalar observable algebra. Original holonomy covariance
follows from the horizontal-lift identity in FD.4, GJ's exact
H covariance and measure pushforward. Thus (FD.8) is a genuine
covariant normal connection on this particular original-unit
affine field.

The local derivations and connection operators are closable in
the relevant supremum module norms. Indeed, if f_n->0 and
delta_v f_n->h uniformly, integrate (FD.7) on any of the short
paths. Its left side tends to zero and the right side to
integral h(mu_t)dt. Continuity then forces h(mu)=0 at every
starting point. For field sections use their fixed-L2 lifts
and (FD.8): uniform convergence of sections and connection
derivatives makes the vector integral zero, then continuity
gives the derivative vector zero at the starting mu. This
proves closability, rather than merely a formal Leibniz rule.

The averaged displacement (FD.4) has all these local normal
connection jets. Its H source-pair section and all its jets
are continuous on compact lifted arrow/anchor tests, and Haar
compactness plus GJ.3's bounds justify each fixed-L2 derivative.
The conclusion is norm-smoothness in this dense connection
structure; it is not a claim of a normal product trivialization
of the entire probability space.

#### FD.6. Exact original coefficient scope

The field in FD.2 is even real auxiliary data. The physical
normal-spinor factor is still the supplied inverse factor with
transport z_g^{-1}rho_t(s_g), its stated grading and generators,
and its final right Clifford block. Haar averaging uses the
metric frame bundle to construct auxiliary data only. It does
not replace the original coefficient algebra by a frame algebra,
erase a Spin-c phase or evaluate C0(M_G) equivariantly to a scalar.

FD.1–FD.5 supply an actual original-unit proper compact convex
affine field and a covariant normal connection, derived from the
eligible local geometry. They may be used as new input for a
differentiable Bott/Dirac construction. They are not a proof that
the connection has passed to the chosen proper-source inverse KK module,
or that a transported bounded graph-column operator is normal-smooth.

#### FD.7. Compact inverse is not preserved by this averaging

Pointwise averaging of GJ's weight is not declared a compact
graph weight on E_G. A concrete special case proves the issue.
Let a compact frame fibre be S1 with Haar measure, let its
Hilbert field be a constant H0, and let Theta0 on H0 have compact
inverse with a unit eigenvector e1. On

\[
L^2(S^1)\otimes H_0.
\]

For the Fourier notation, complexify the real auxiliary field. If a real operator is compact, its finite-rank approximants complexify with the same norm: for a real operator A, the squared norm on x+iy is ||Ax||²+||Ay||², so the complexified operator has exactly the original norm. Thus compactness would be preserved. The same obstruction can alternatively be written with the normalized real sine and cosine modes.

The pointwise operator is 1 tensor Theta0. Its inverse sends
the orthonormal vectors exp(inu) tensor e1 to the same nonzero
multiple of themselves. Those vectors are weakly zero, so its
inverse is not compact. The same issue can occur in additional
probability multiplicities. Compactness on each integrated
point is insufficient for compactness of the integrated operator.

A weight selected jointly for the averaged field and its normal
connection jets, or another proved confining construction, is
therefore still required. Actual anchored Bott factors from a coherent probability-kernel input, BA.3, supplies an abstract weight for its
continuous affine action, but no differentiability of such a
selection is asserted here. The differentiable Bott/deformation
and proper-source inverse passage, self-adjoint physical graph
sum with completed domains, localized scalar compactness and
its original Bott class comparison also remain necessary. A
generic countable unboundedification does not discharge them.

![Actual affine origins, flat collapse, graph-weight estimates and compact-frame connection descent](../figures/normal-connection-frame-descent.png)

**Figure 11AG.1.** LF.4–LF.6, GJ.1–GJ.5 and FD.1–FD.7. The affine example has transition 3, weights 2/3 and 1/3, origins −2 and 1, and exactly zero identity displacement. The cone and Haar-probability panels are typed schematics. The graph samples render the conditional inequalities with L=J=H=1 and K0=100, with exact endpoint 23/400; they do not select a foliation metric with those illustrative constants. The Fourier example uses the complexified auxiliary field, fixed auxiliary eigenvalue 2 and inverse norm 1/2 on every integer frame mode; compactness of the real operator would imply compactness after complexification. Human context: [Connes, A survey of foliations and operator algebras](https://alainconnes.org/wp-content/uploads/foliationsfine.pdf) and [Kaad–van Suijlekom, Riemannian submersions and factorization of Dirac operators](https://arxiv.org/abs/1610.02873v1); no external operator theorem supplies the unfinished physical sum. [Vector figure](../figures/normal-connection-frame-descent.svg) · [Generator](../reproduction/normal-connection-frame-descent/reproduce.py) · [Finite checks](../reproduction/normal-connection-frame-descent/finite-checks.json) · [Data](../reproduction/normal-connection-frame-descent/figure-data.json) · [Reproduction](../reproduction/normal-connection-frame-descent/REPRODUCTION.md) · [Component terms](../reproduction/normal-connection-frame-descent/COMPONENT-TERMS.md).

## 11AH. Normalized proper fields and actual graph columns

Let R=C0(T). Use the actual probability base X and A=C0(X), the anchored Bott algebra B, the nearest-face proper algebra P with centre M_G, and its equivariant Dirac class D from Actual anchored Bott factors from a coherent probability-kernel input and Kernel-to-original-unit dual with the exact transverse normalization. Complete Riemannian holonomy and the original-unit kernel supplies their geometric premise at the original eligible scope. The product gamma=eta D has ordinary whole-base class 1_R; no equivariant gamma=1 is asserted.

The proper coefficient quotient theorem is proved in The full and reduced norms of every proper groupoid coefficient. The essential-replacement and anchored product proofs are in Original-anchor localization and the equivariant Dirac lift. The completed full/reduced counting-plaque correspondence is proved in K-theory of the leaf space, Theorem 11A.1, with its common section core and quotient-tensor unitary (A.G.4). Its inverse normal coefficient is the specified one in Section 11E.9. Regular fields, unit-space balancing and the graph Dirac interface proves the normal-field absorption and scalar regular norm used below. The square cutoff and its compact support control GP.11 are proved in A universal proper base on the actual holonomy groupoid. These constructions keep the original unit algebra and actual graph coordinates throughout.

### 11AH.1. A proper-covariant normalized field and its reduced norm

#### PF.1. Keep the actual standard family modules

Choose an actual representative (K_X,F_X) of d_X in KK_G(B,A).
It is a countably generated graded Hilbert A-module with its actual
continuous G-action and nondegenerate B-source representation, after
the proved essential-replacement homotopy if needed. Its A-anchor is
its right scalar action. Choose an actual representative (E_D,F_D)
of D in KK_G(P,R), with nondegenerate source representation pi_D.
Essential replacement here is the proved KG.5 interval construction;
it preserves the class and G-action, and does not assume a complemented
source submodule.

The dual eta_X uses the standard B-module. The actual nearest-face
family theta uses the standard P_M-module. Its coefficient extension
T_B therefore uses the standard B tensor_R P module, by AS.14's
dense multiplication unitary. In that unitary the signed left B-action
AS.7 becomes left multiplication by B tensor 1: commuting b past a
homogeneous P-family vector cancels exactly the displayed tensor sign.

Consequently the **underlying module**, before choosing the proved
product operator for eta, is canonically

\[
E_\eta=K_X\otimes_{A,\Phi^*}P.
\tag{PF.1}
\]

Here Phi*:A to ZM(P) is the actual equivariant central map in PK.12.
This is not an arbitrary Hilbert-space identification. The first two
standard-module products identify b tensor c with (b tensor 1)c.
The third identifies c tensor z with pi_(d_X exterior 1_P)(c)z.
The last multiplication map is

\[
(\xi\otimes p)\otimes q\longmapsto\xi\otimes pq.
\]

Its balancing relation is exactly
a xi tensor p = xi tensor Phi*(a)p. Its inner-product equality is

\[
\langle\xi\otimes p,\zeta\otimes q\rangle=p^*\Phi^*(\langle\xi,\zeta\rangle_A)q.
\]

with the standard graded convention (A is even). Its range is dense
by the P approximate identity. Thus it extends to an onto even unitary.
All elementary maps intertwine the actual G-actions, original R-anchor
and right coefficients. They identify completed modules, not just fibres.
The transferred operator supplied by the proved product theorem is
transported by these particular unitaries; none of its cycle conditions
is inferred from an abstract Hilbert-space isomorphism.

#### PF.2. An actual nondegenerate proper coefficient representation

There is a genuine left representation of P on (PF.1),

\[
\lambda_\eta(p)(\xi\otimes q)=(-1)^{|p||\xi|}\xi\otimes pq.
\tag{PF.2}
\]

The central even Phi*(A) makes it respect the balancing relation. The
finite-column inner-product matrix is positive and commutes with P;
the inequality p*p<=||p||^2 proves its norm bound. The adjoint and
multiplication identities then prove a graded *-representation.
The P approximate identity converges on its dense elementary tensors,
so it is nondegenerate. It is G-covariant by covariance of Phi* and
of the module actions. No commutator of this representation with the
eta product operator is asserted here or needed in the norm argument.

The underlying module of the actual equivariant product gamma=eta D
is correspondingly

\[
E_\gamma=K_X\otimes_{A,\pi_D\Phi^*}E_D.
\tag{PF.3}
\]

The unitary is (xi tensor p) tensor e -> xi tensor pi_D(p)e.
The same inner-product calculation and nondegeneracy of pi_D prove
it is onto and even. The extension of pi_D to M(P) exists by that
nondegeneracy. It sends Phi*(A) to operators commuting with pi_D(P).
Hence there is a nondegenerate graded covariant representation

\[
\lambda_\gamma(p)(\xi\otimes e)=(-1)^{|p||\xi|}\xi\otimes\pi_D(p)e.
\tag{PF.4}
\]

The same finite-column estimate proves boundedness on the balanced
completion. Nondegeneracy follows first on elementary tensors from
the P approximate identity and then by uniform boundedness. The G
action is the actual diagonal action on the balanced tensor: on each
bisection, transport both factors and the A-balancing relation. Inner
products are preserved; elementary-section continuity and density
prove continuity on the completed module. Thus this is an actual
continuous original-anchor field, not a collection of independent
representations on its unit fibres.

Use the proved anchored product construction to obtain its actual
odd adjointable product operator F_gamma. The result is

\[
\begin{gathered}[E_\gamma,F_\gamma]=\eta D\in KK_G(R,R),\\\operatorname{For}_G[E_\gamma,F_\gamma]=1_R\in KK_T(R,R).\end{gathered}
\tag{PF.5}
\]

The latter is the proved OU.6 whole-base homotopy statement. It is
not an assertion that F_gamma commutes with G, that gamma=1 in KK_G,
or that a fieldwise scalar kernel count proves the identity.

#### PF.3. Proper covariance gives the reduced norm before any Dirac choice

We prove the scalar groupoid multiplier assertion used here, including
its reduced direction. There are actual nondegenerate homomorphisms

\[
\iota_m:C^*(G)\longrightarrow M(P\rtimes_mG),\qquad\iota_r:C_r^*(G)\longrightarrow M(P\rtimes_rG).
\tag{PF.6}
\]

given on compact convolution tests by coefficient identity multipliers.
On the dense crossed-P core they act by the actual groupoid convolution.
The I-norm estimate constructs the full multiplier and its adjoint;
the unit R approximate identity proves nondegeneracy.

For the reduced bound, take a source unit x and a representation of P_x.
The corresponding regular crossed-P representation is on
ell2(G_x) tensor H_x. In source-trivialized coordinates the transported
coefficient identity is the identity on H_x, so the multiplier of f is
exactly lambda_x(f) tensor 1. Its norm is at most ||f||_r. Taking the
supremum of the actual regular coefficient representations proves the
bounded multiplier on P crossed_r G; the dense convolution and adjoint
identities extend to C_r*(G). This proves iota_r without assuming it
from the full norm.

The quotient square q_P iota_m(f)=iota_r(q_G f)q_P holds on the dense
core, hence for multipliers. Since P has the actual proper central
space M_G, the proved proper-coefficient theorem makes q_P an
isomorphism. Therefore iota_m itself factors through C_r*(G).

Now let H_N=L2(T,mu;S_inv) be the actual normal-spinor Hilbert space,
with its full final right C_q coefficient, supplied unitary inverse
transport and invariant smooth normal volume. Invariance is the
Jacobian-one property of the given transverse isometries. Partial
bisections act by their actual unitary transports, between their
open source and range Hilbert-space restrictions. Form the genuine
balanced Hilbert-space completion

\[
H_{\mathrm{trans}}=E_\gamma\otimes_R H_N.
\tag{PF.7}
\]

It has the nondegenerate representation lambda_gamma(P) tensor 1.
Its partial groupoid transports are the diagonal transports of both
factors. They preserve the Hilbert-space inner product by invariance
of mu; local bisection elementary sections prove continuity. The
covariance and composition laws hold on those sections and then on
the completion. Integrated on compact bisection partitions, this is
a covariant representation of P crossed_m G. Its extension to the
multiplier iota_m gives the exact diagonal groupoid representation
Pi on (PF.7). The factorization just proved gives

\[
\|\Pi(f)\|\le\|f\|_r\qquad(f\in C_c(G)).
\tag{PF.8}
\]

Thus the normal coefficient's nonregular geometric action has been
coupled to a concrete proper-covariant normalized field. The norm
assertion is independent of choosing an unbounded operator. It does
not replace P by C0(M_G), or treat that space as a scalar coefficient.

#### PF.4. Actual graph completion and preserved coefficient order

Let G_full be the smooth physical holonomy graph on V and let Z be
its actual source-transversal graph-column correspondence to G on T.
Use the already proved reduced imprimitivity module E_Z, built from
the counting and normalized plaque half-density cores. Induction is

\[
H_{\mathrm{graph}}=E_Z\otimes_{C_r^*(G)}H_{\mathrm{trans}}.
\tag{PF.9}
\]

Its dense vectors are the actual compact graph-column half-density
sections tensor elementary vectors from (PF.7), modulo the displayed
convolution relation and inner-product null vectors. Left physical
graph convolution acts on the first factor and is bounded at its
actual reduced norm by the proved imprimitivity construction. This
specifies a graph-correspondence completion geometrically. It is not
the claim that an arbitrary Hilbert space becomes a graph Hilbert
space when assigned a new name.

All E_gamma factors are even KK factors. Keep their tensor grading
first, S_inv second and the right C_q block last. The inverse transition
on the latter is the supplied z_g^(-1)rho_t(s_g); its degree and
physical generators are unchanged. The plaque half-density module
is even and introduces no additional Clifford block. The normalized
counting/plaque comparison sends the local positively oriented Bott
class to the original physical graph class. Once an actual operator
on (PF.9) is proved to represent the transferred product, OU.7 and
that correspondence give its original pairing +1 without a sign
permutation or loss of the inverse phase.

#### PF.5. Exact operator obligation left by this construction

(PF.1)–(PF.9) settle actual field/module, original-anchor, reduced-norm
and coefficient interfaces from the explicit kernel input. They do not
by themselves construct a smooth normal connection on E_gamma, an
unbounded auxiliary with controlled connection commutators, or a
closed physical graph Dirac domain. The source-linear product operator
F_gamma is an actual KK-cycle operator, but it is not declared a
geometric differential expression. Tensoring it alone with 1 on H_N
does not provide ordinary scalar compactness. Fibrewise compactness
in the R-module also does not provide scalar compactness after (PF.7).

A suitable graph construction still needs a concrete geometric
connection/operator on this specified completion, or another specified
graph completion, with a proved self-adjoint domain/core, bounded-phase
defects and the exact class comparison. A local finite-order expression
is sufficient if supplied; no full-tangent finite-order interpretation
has been added to the source. Smooth labelled kernels and normal analytic control does not
silently supply bounded derivatives of the spectral field. A generic
countable unboundedification of F_gamma does not close these geometric
interfaces.

### 11AH.2. Equivariant cutoff maps and literal regular graph coordinates

Let R=C0(T), let M=M_G be the actual proper central space, and let
E be a countably generated continuous G-Hilbert R-module carrying a
nondegenerate equivariant even representation of C0(M). A graded E
can be treated in each grading. Denote the multiplier action of a
square cutoff c on M by C; 0<=c<=1 since its identity-arrow term is
one term in a sum equal to one. This action is well defined by the
nondegenerate extension to bounded continuous multipliers.
For PF.3's actual E_gamma this central representation is nondegenerate:
if phi_j is a C0(M) approximate identity, phi_j p -> p in P, and
nondegeneracy of lambda_gamma(P) proves phi_j e -> e on its dense
P-generated submodule, then on all E_gamma.

#### RG.1. The actual range-regular module with source coefficients

Complete C_c(G,s*E) in the R-valued counting inner product

\[
\langle\xi,\eta\rangle(x)=\sum_{rg=x}\langle\xi(g),\eta(g)\rangle_{sg}.
\tag{RG.1}
\]

The right unit action is f(r g), not f(s g). Denote the completion
by R_G(E). It is countably generated by compact bisection sections
and a countable generating family of E. The actual arrow transport
is

\[
(L_h\xi)(g)=\xi(h^{-1}g)\qquad(rg=rh).
\tag{RG.2}
\]

Its coefficient fibre E_(s g) stays the same. Compact bisections and
elementary sections give continuous module maps; the cocycle and
inner-product identities are exact.

If S:E -> E' is any ordinary even R-linear unitary, then

\[
(\mathcal R_G(S)\xi)(g)=S_{sg}\xi(g).
\tag{RG.3}
\]

is an actual **G-equivariant** unitary of these regular modules.
Indeed s(h^(-1)g)=s(g), so the same S_(s g) occurs on the two sides
of (RG.2). No equivariance of S on its original coefficient field
has been assumed. This is source-coefficient absorption, with the
source/range unit types retained.

#### RG.2. The geometric cutoff isometry and its adjoint

Define on actual arrow fibres

\[
(Ve)(g)=C_{sg}U_g^{-1}e_{rg}.
\tag{RG.4}
\]

This defines an adjointable R-linear isometry E -> R_G(E).
The following support proof is needed for the actual completion.
First take e localized by phi in C_c(M). By GP.11, the pairs
(g,m), m in supp(phi), c(g^(-1)m) nonzero, lie in a compact set.
Thus (RG.4) applied to phi e has compact arrow support. In each
bisection its coefficient is an actual continuous section: after
the compact central localization the multiplier product is in
C0(M), and its transported action on e has the proved continuous
section topology. Finite bisection partitions give membership in
the compact core of (RG.1). General e are norm limits of such
central localizations, by nondegeneracy of C0(M).

For a localized e the cutoff identity gives the exact norm equality

\[
\langle Ve,Ve\rangle(x)=\left\langle e_x,\left[\sum_{rg=x}U_gC_{sg}^2U_g^{-1}\right]e_x\right\rangle=\langle e_x,e_x\rangle.
\tag{RG.5}
\]

The sum is the multiplier action of sum_g c(g^(-1)m)^2=1.
It is strict on the localized core, with compact-arrow support as
above; approximation extends it to E. This proves (RG.4) on the
completed modules, not only on its unit fibres.

The adjoint on the compact core is

\[
(V^*\xi)_x=\sum_{rg=x}U_gC_{sg}\xi(g).
\tag{RG.6}
\]

Finite-row Cauchy–Schwarz and sum_g U_g C^2 U_g^(-1)<=1 show
||V*xi||<=||xi||. The compact-arrow core gives continuous sections
and the inner-product adjoint identity. Therefore (RG.6) extends
boundedly and is the adjoint on the full completion. In particular
V*V=1. Direct substitution proves

\[
VU_h=L_hV.
\tag{RG.7}
\]

Thus the projection VV* and its complement are actual G-invariant
adjointable projections. This is a proper-space cutoff embedding;
its support, continuity and onto-range assertions were all checked.

#### RG.3. The standard regular field also has the proper centre

Write H_G=R_G(R) and H_G^infinity for countably many copies. Its
even nondegenerate C0(M) action is diagonal at the actual measures

\[
m_g=\delta_g\in M_{rg},\qquad(\rho_M(\phi)\xi)(g)=\phi(\delta_g)\xi(g).
\tag{RG.8}
\]

The map g -> delta_g is continuous in the weak compact-arrow tests.
Its image of a compact arrow support is compact. A central compact
bump equal to one there proves nondegeneracy on the dense core.
The equality h delta_g=delta_(hg) proves covariance with (RG.2).
All labels, including isotropy arrows, remain distinct.

Ordinarily H_G contains R as its unit-arrow summand, because G's
unit space is clopen. Accordingly H_G^infinity contains the standard
R^infinity as an adjointable ordinary summand. Its complementary
module is countably generated. Kasparov’s stabilization theorem then gives

\[
E\oplus H_G^\infty\cong R^\infty\quad\text{ordinarily over }R.
\tag{RG.9}
\]

This unitary is R-linear on the whole module. It is not an arbitrary
identification of its integrated Hilbert space.

#### RG.4. Equivariant stabilization onto regular arrow coordinates

Put E_1=E plus H_G^infinity, with the direct proper central action.
Use (RG.4) for E_1, then use its ordinary unitary (RG.9) on **source
coefficients** through (RG.3). This gives an actual equivariant
adjointable isometric embedding

\[
E_1\hookrightarrow\mathcal R_G(R^\infty)=H_G^\infty\quad G\text{-equivariantly}.
\tag{RG.10}
\]

Its orthogonal complement Q is a continuous G-Hilbert R-module.
Thus H_G^infinity=E_1 plus Q, with an actual equivariant module
unitary. Taking countably many copies and shifting the E_1 copies
gives the explicit absorption

\[
E_1\oplus H_G^\infty\cong H_G^\infty\quad G\text{-equivariantly over }R.
\tag{RG.11}
\]

For detail, identify the second summand with
(E_1 plus Q)^infinity, regroup as E_1^infinity plus Q^infinity,
and send the extra first E_1 to the first E_1 coordinate, shifting
the old E_1 coordinates by one. Regroup back. Every map is even,
R-linear and G-equivariant, and is onto by its inverse coordinate
shift. The identification of countably many copies of H_G^infinity
with H_G^infinity is a fixed bijection of the copy indices. It does
not permute an arrow label with an unrelated graph point.

Apply this argument to each grading of the actual E_gamma in PF.3.
First add the degenerate equivariant cycle
(H_G^infinity plus Pi H_G^infinity, swap), and then a second such
cycle. Its square is one, and it commutes with the R-source and
with G exactly; its class is zero. Formula (RG.11) in both gradings
gives an actual equivariant regular-coordinate representative

\[
\begin{gathered}(H_G^\infty\oplus\Pi H_G^\infty,F_{\mathrm{reg}})\text{ represents }\gamma=\eta D\in KK_G(R,R),\\\operatorname{For}_G[H_G^\infty\oplus\Pi H_G^\infty,F_{\mathrm{reg}}]=1_R.\end{gathered}
\tag{RG.12}
\]

The operator F_reg is the actual product operator plus the two
degenerate swaps, transported by the explicitly constructed module
maps. All equivariant compact **sections** and ordinary module
compactness are preserved by these adjointable unitaries. This is
not the bare doubled regular field with zero or invertible length
operator, which Sections 11K/11M correctly show has zero index.

#### RG.5. Literal normal-spinor graph Hilbert space

Tensor (RG.12) with the supplied finite full normal inverse Hilbert
space H_N=L2(T,mu;S_inv), keeping its right C_q block last. The
actual range-counting Hilbert-space completion is

\[
L^2(G,\nu_r;r^*S_{\mathrm{inv}})\otimes\ell^2(\mathbb N)\quad\text{in each auxiliary grading}.
\tag{RG.13}
\]

To verify this particular unitary, send a finite arrow section xi
tensor a normal section v to the section g -> xi(g)v(r g).
Its integrated norm is exactly (RG.1)'s inner product integrated
against |v|^2. Finite bisection sections and normal unit bumps have
dense range. Thus it is onto by elementary-section density. Groupoid
convolution is the actual counting convolution with the prescribed
normal transport. Section 11M's absorption, or the proper norm proof
PF.3, gives its exact reduced bound. No arbitrary Hilbert-space
unitary has been used to rename coordinates.

The full inverse phase z_g^(-1)rho_t(s_g), its grading and physical
Clifford generators remain in S_inv. The auxiliary copy index is
even; it changes neither the transverse degree nor the final right
Clifford order. Physical full-graph induction is then the actual
normalized correspondence PF.9, with its specified plaque half
densities and convolution relation.

#### RG.6. Exact boundary: a phase field is still not the graph Dirac

(RG.12) proves that the normalized auxiliary can be placed on the
literal regular arrow field by equivariant, anchor-preserving,
cutoff/source-coordinate maps. This closes an actual representation
and module-interface gap. It does not prove that F_reg is smooth in
normal graph coordinates. Choosing its generic countable unbounded
representative would still not prove a geometric Dirac-type operator,
nor the domain of a sum with the physical normal differential.

The graph operator still requires a geometric normal connection,
controlled auxiliary generator and closed completed domain, or a
proved geometric bounded-phase alternative, with the class comparison.
Smooth labelled kernels and normal analytic control
proves why individually smooth kernel values do not establish the
bounded-phase commutator condition. The connection in
Geometric normal connections and compact-frame descent
concerns its explicitly constructed field; its transfer to F_reg still
requires a differentiable product construction.

### 11AH.3. The probability step preserves the geometric normal connection

#### PJ.1. Actual endpoints and the proper convex probability anchor

Write Y=X_G from FD.1, p:Y->T its proper anchor, E_Y its FD.2 field,
and Psi the proper CND affine displacement from FD.5. Put

\[
Q_x=\operatorname{Prob}(Y_x),\qquad Q=\bigsqcup_xQ_x.
\tag{PJ.1}
\]

Use the original unit coordinate and the weak C0(Y) tests. For compact
K in T all measures are supported in the compact p^{-1}K. Their
support-over-one-unit constraint is closed, as tested against unit
functions. Thus Q_K is compact metrizable. This proves that Q->T is
proper, onto, second countable and LCH by the same compact unit-chart
argument as FD.1. Its fibres are compact convex. Its actual G action
is pushforward. The ordinary continuous section is

\[
q_0(x)=\delta_{\mu_0(x)}.
\tag{PJ.2}
\]

where mu0 is FD.2. Straight fibrewise contraction to q0 is a genuine
ordinary whole-base proper homotopy: its input and output unit are
the same and every compact unit inverse image is compact. No
equivariance of q0 or of this contraction is asserted.

#### PJ.2. The continuous averaged field and proper affine action

For sections xi,eta of E_Y define generators by

\[
\langle\xi^\flat,\eta^\flat\rangle_\nu=\int_{Y_x}\langle\xi,\eta\rangle\,d\nu.
\tag{PJ.3}
\]

Positivity, null-space quotient and completion give a countably
generated continuous real Hilbert field E_Q. Its field topology is
the continuous Gram topology of these generating sections. Actual
G pushforward and E_Y transport give onto linear isometries; compact
anchor tests and density prove their continuity. Average the actual
displacement b_g of E_Y in the target field. Its squared norm is

\[
\Psi_Q(g,\nu)=\int\Psi(g,\mu)\,d\nu(\mu).
\tag{PJ.4}
\]

The exact affine cocycle identity survives by change of variables
in (PJ.3). Thus this is an actual affine action and (PJ.4) is CND,
not just a positive scalar assignment. Compact original-arrow upper
bounds from FD.3 survive. If (PJ.4)<=R over compact original units,
the minimum of Psi on its compact support is at most R, so some mu
has Psi(g,mu)<=R. FD.3 then bounds the original g in a compact set.
This proves the exact proper affine input of PK/BA at its original
unit/anchor, without any equivariant scalar evaluation.

#### PJ.3. A compact local resolution of the connection paths

We spell out the measure-lift interface from FD.4. At x choose finite
compact frame subcharts, a smooth partition chi_j, and the finite
potential products P_j of their compact frame portions with their
active compact Q_a parameters. Their quotient maps onto the
corresponding portions of Y's inner frame probability construction
are continuous. Let C_x be the product of the compact spaces of
positive lifted measures on P_j whose frame marginals are
chi_j m_x. Summation after the potential-product quotient gives a
continuous map

\[
r_0:C_x\longrightarrow Y_x.
\tag{PJ.5}
\]

It is surjective: decompose an arbitrary mu as chi_j mu and apply
FD.4's positive norm-preserving extension to each pullback algebra.
The source spaces here are compact; all masses and marginal
constraints are fixed and closed. Zero-mass factors are removed.

For a short integral curve of a chosen smooth normal v, transport
the frames horizontally and leave potential Q_a coordinates fixed.
This gives continuous maps r_t:C_x->Y_(x(t)). For every function
f in FD.4's dense normal algebra, r_t^*f is norm-smooth on C_x;
its derivatives are exactly r_t^*(delta_v^j f). Indeed each
piece is an integral of a norm-smooth flat potential-product
function against a fixed lifted measure. Compact products give
uniform derivative bounds, and the derivatives already descend
at zero-weight strata. Finite sums and polynomial observable
products preserve these conclusions.

There is a fixed continuous Hilbert field over C_x containing the
lifts of all the E_Y connection-core sections along r_t. Construct
it by integrating the LF constant-normal local Hilbert fields over
the lifted potential measures, taking their finite direct sum.
Flat coefficients make an inactive component the zero vector.
For each core section xi its lifted family is norm-smooth in that
one section module; its derivative is the lift of nabla_v xi.
Its norm and Gram entries are exactly those of xi at r_t(s).
This is the fixed-field null-relation test used below. It does not
assert that the whole Y or E_Y has a normal product trivialization.

#### PJ.4. The connection on the outer probability field

For f in FD.4's algebra let f^flat(nu)=integral f dnu. Define

\[
\delta_v^Q f^\flat=(\delta_vf)^\flat,\qquad\nabla_v^Q\xi^\flat=(\nabla_v\xi)^\flat.
\tag{PJ.6}
\]

Extend the first by the scalar Leibniz rule on the algebra of
polynomial observables and smooth unit functions, and the second
by the corresponding module Leibniz rule on its finite generating
sections. These are dense domains: the inner dense scalar tests
separate outer probabilities on every compact fibre, and the
continuous Gram generators span the field. Unit localization
gives density on the noncompact whole base.

To prove every scalar and module relation, lift an arbitrary nu
in Q_x through the compact surjection (PJ.5) to a probability
nu_tilde on C_x. Existence is exactly the positive extension
argument: a unital faithful pullback algebra, norm-preserving
real Hahn–Banach extension, Corollary 2.3, and
Haar measure on locally compact groups, Theorem 2.2 for the compact measure representation.
Push this fixed measure through r_t, obtaining an actual path
nu_t in Q. Differentiation under nu_tilde, using PJ.3's uniform
norm jets, gives precisely (PJ.6). It is independent of the lift
because the derivative functions/sections already descend to Y.
A scalar relation vanishing on Q vanishes on all paths nu_t, so
its differentiated relation vanishes. A module relation is the
zero vector in the fixed integrated Hilbert space of nu_tilde
along this path; its derivative is also zero. This proves that
both operators in (PJ.6) are well defined on completed fibres
and their stated dense sheaves.

Differentiating (PJ.3) gives metricity and the scalar Leibniz
identity. Actual G covariance follows from FD's covariant
horizontal connection and its pushforward action. These local
operators are closable: if sections tend uniformly to zero
while their derivatives tend uniformly, integrate along the
fixed lifted paths just used. The limiting vector integral
is zero on each short interval, and continuity forces the
limiting derivative zero at its starting point. The same
argument works for scalar observables. This verifies closability,
not merely formal differentiation of probabilities.

All normal connection jets of the affine displacement survive
in E_Q: apply the same fixed lifted-measure norm derivatives to
its FD connection-core approximations and compact arrow tests.
The resulting bounds are at most the corresponding uniform
FD bounds before integration. No derivative of an arbitrary
continuous function on Prob(Y_x) is silently asserted; the
domain is the displayed dense polynomial observable algebra.

#### PJ.5. Exact consequence and remaining analytic work

The actual Y from FD is a valid proper compact-fibre CND input,
and its genuine PK probability space Q has the covariant dense
closable normal connection just proved. The original-unit
contraction is (PJ.2), hence a whole-base contraction with the
original C0(T) endpoint. Kernel-to-original-unit dual with the exact transverse normalization
may consume this new input with its exact coefficient types.

Neither averaging step preserves compact inverse by itself;
FD.7's exact Fourier example still applies. A joint confining
weight compatible with this connection, differentiable
Bott/deformation and inverse-module passage, the actual closed
physical graph sum and its original +1 class comparison remain
necessary. This lemma preserves the even auxiliary placement,
whole inverse normal phase and final right Clifford order.

![Exact proper cutoff and actual regular graph coefficient order](../figures/regular-graph-field.png)

**Figure 11AH.1.** RG.4–RG.13 and PF.7–PF.9. The exact C2 sample uses the displayed arrow/source-coefficient order, squared cutoff and four-entry embedding: V*V=1, LV=VU, and the invariant range projection has diagonal (1,0,1,0). The lower panel records actual module maps and the preserved inverse normal phase, grading and final right Clifford block. The general normalized class follows from eta D and exactly degenerate additions, rather than this finite sample. A normal connection on the final regularized product, geometric generator and closed graph domain remain required. Human context: [Connes, A survey of foliations and operator algebras](https://alainconnes.org/wp-content/uploads/foliationsfine.pdf). [Vector figure](../figures/regular-graph-field.svg) · [Generator](../reproduction/regular-graph-field/draw_regular_graph.py) · [Exact matrix checks and data](../reproduction/regular-graph-field/FIGURE-CHECKS.json) · [Reproduction](../reproduction/regular-graph-field/REPRODUCTION.md) · [Component terms](../reproduction/regular-graph-field/COMPONENT-TERMS.md). The connection-preserving probability step uses the measure-lift and Gram mechanism illustrated in Figure 11AG.1.

## 11AI. Intrinsic normal jets and the compact-frame quotient

Work at a finite good complete transversal of the original compact regular foliation with its holonomy-invariant positive normal metric and full invariant Spin-c data. Complete frame geometry and local matrix-monodromy labels proves the invariant frame parallelism; Label-preserving positive-kernel patching and frame transfer, EK.6, proves the actual fixed-Hilbert source-pair vector model and uniform compact-increment normal jets used here.

The scalar compactification and affine action are proved in Labelled geometric kernels over the original unit space, GK.4–GK.6. The compact approximate-unit selection and its localized weak convergence are proved in Original-anchor localization and the equivariant Dirac lift, KG.2; the graph weight and domain argument are proved in A confining oscillator over a varying Hilbert field, WF.1, and Actual anchored Bott factors from a coherent probability-kernel input, BA.3. The additional probability step, with its precise compact lifted-path premise, is proved in Normalized proper fields and actual graph columns, PJ.1–PJ.5. Here the vector jets define a genuine connection on an enlarged spectrum. They do not give a connection to the earlier scalar spectrum or estimate normal derivatives of the chosen confining weight.

Write K=SO(q), P->T for the actual oriented frame bundle, and H=G acting
on P for the actual frame étale groupoid. Write h.a for the usual right frame action and use the left K convention
k.u=u k^{-1}, so k.h=h.k^{-1}. K acts freely on P and on H, commutes with original G
transport, and H/K=G. These are auxiliary frame actions; the full
inverse Spin-c coefficient is not quotiented or changed.

### 11AI.1. An exactly K-covariant receiving feature space

Let F_H:H->H0 be EK.12's actual fixed-Hilbert feature map. On common
source arrows it has squared distance k_H; F_H(1_u)=0. Put

\[
\overline F(h)(a)=F_H(h.a),\qquad a\in K,\qquad H_1=L^2(K,H_0).
\tag{IJ.1}
\]

The Haar integral is normalized. It is a genuine fixed receiving
space. Its same-source squared distance is the compact-frame
average of k_H, and both GK.4 controls survive exactly as in EK.5:
compact lifted increments give upper bounds, and a small average
has one integrand below the same bound, retaining the original
lifted arrow in a compact set. No new CND theorem is imported.

For R_k v(a)=v(k^{-1}a), one has

\[
\overline F(k.h)=R_k\overline F(h),\qquad (R_kv)(a)=v(k^{-1}a).
\tag{IJ.2}
\]

R is an orthogonal representation by Haar change of variables.
The local and uniform vector jets of (IJ.1) are those of EK.6
integrated over compact K. All compact-increment derivatives in
the invariant frame-parallelism directions are bounded uniformly
in the unrestricted return arrow. Derivatives in a of the feature
increments have the same property, because they are the vertical
frame-parallelism derivatives. This verifies the norm derivative
tests used for R below, rather than applying strong continuity to
an arbitrary bounded set of H1 vectors.

### 11AI.2. Bounded increment vectors on the actual arrow manifold

The solder and Levi-Civita forms give the actual frame parallelism
xi_1,...,xi_d on P, invariant under every H bisection, by EG.1.
Lift xi_i to the arrow manifold through r; since r and s are local
diffeomorphisms and holonomy preserves the parallelism, its s
projection is also xi_i. Denote the resulting actual vector-field
derivative on H by partial_i. It is defined locally even though
the whole open transversal need not admit a complete flow.

Choose a countable basis of relatively compact original G
bisections B and lift them over all their frame fibres. These are
K-saturated H bisections. If b(x) is their range-parameterized
arrow, and chi is a smooth compact unit cutoff there, set

\[
v_{B,\chi}(h)=\chi(rh)\bigl[\overline F(b(rh)^{-1}h)-\overline F(h)\bigr].
\tag{IJ.3}
\]

Extend by zero outside the cutoff. The two arguments in the
brackets have the same source. EK.6 therefore proves that (IJ.3)
and every word in the partial_i derivatives are bounded H1-valued
functions, with continuous normal jets. Cutoff extension is smooth
because its support is closed inside the larger bisection domain.
The bounds are uniform on compact range-cutoff and increment tests.

Include all these derivative vectors, their cutoff left translates,
and their K transforms. Countably dense smooth cutoff families and
countably dense K parameters suffice: the just-proved norm jets
give all parameters by norm limits. Let V be the real algebraic
span of this countable system, then its module products with the
scalar algebra below. The extra derivative directions are actual
vectors, not merely differentiated squared-distance functions.

### 11AI.3. One proper enlarged spectrum and continuous Gram module

Let A_infty be the self-adjoint scalar algebra generated by
r*C_c^infty(P), all <v,w> for the above vector generators, all
their normal derivatives, and cutoff left/K translates. Repeat
this countable saturation. Leibniz differentiation closes the
algebra under each partial_i; its elements and all their specified
normal jets are bounded and continuous on H. Unit cutoffs give
compact range-unit support to each nonconstant generator. Take
its supremum-norm closure A in C_b(H).

The central r*C0(P) action is nondegenerate: a unit approximate
identity converges uniformly on each compact-range generator,
then on its products and norm closure. The spectrum Y_H is thus
properly anchored to P. Indeed, above compact C in P a unit bump
equal to one makes the preimage a closed subset of its compact
Gelfand level set. The unit evaluations give a continuous
section u->ev_(1_u) and prove surjectivity. Separability gives
second countability. Evaluations at actual h are globally dense
by the faithful concrete inclusion A subset C_b(H). No fibrewise
evaluation-density assertion is required.

The original GK scalar generators are included: the squared norm
of (IJ.3), where chi=1 on a smaller test, is exactly the localized
averaged kernel generator (choose the scalar cutoffs as smooth squares).
Thus this is an actual enlargement,
with a proper restriction map to that compactification. The usual
cutoff-ideal left translations define the actual H action as in
GK.5; the same norm estimates and finite bisection partitions
prove their extension from the countable saturation. K translations
act on A continuously in norm. The latter follows from the bounded
vertical jets of the scalar generators and compact K, then from
polynomial/norm approximation. The actions commute in the actual
transformation-groupoid sense.

On V A_infty set the A-valued inner product to its pointwise H1
inner product. Every finite Gram matrix is positive on actual
arrows and then on the whole spectrum by faithfulness. Quotient
the null space and complete. This gives a countably generated
continuous real Hilbert field E_H over Y_H, with

\[
\left\|\sum_jv_ja_j\right\|=\sup_h\left\|\sum_jv_j(h)a_j(h)\right\|_{H_1}.
\tag{IJ.4}
\]

The continuous Gram generators have dense spans in each defined
fibre, which proves the claimed field topology. The fibre rank
may change; adding all jet directions before completion supplies
one ambient field, without asserting constant rank.

### 11AI.4. The actual affine action, not a unitary label fiction

Linear H transport is cutoff precomposition by left translation:

\[
(L_gv)(h)=v(g^{-1}h),\qquad r(h)=r(g).
\tag{IJ.5}
\]

The scalar inner product is transported by that same actual local
algebra action. Saturation gives onto isometries on the completed
fields, with continuity proved on compact bisection generators
and then by density. This does not assert that a matrix label
acts orthogonally on H0.

At a target spectral point above r(g), let b_g be the section
represented near g by Fbar(g^{-1}h)-Fbar(h). Its cutoff versions
are (IJ.3), and different cutoffs/bisections agree on a unit
neighborhood of the same actual arrow. Direct subtraction gives

\[
L_gb_l+b_g=b_{gl}.
\tag{IJ.6}
\]

Indeed the intermediate Fbar(g^{-1}h) cancels. Its squared norm
is the pullback of the extended averaged CND kernel along that
restriction map, so GK.6's properness proof applies also at
limiting-unit spectral points. Thus the exact proper lower
and compact-increment upper controls of IJ.1 hold. This proves
one actual proper affine action, with the original arrow labels
and original anchor. In PK's source convention its function is
Psi(g,y_source)=||b_g(g y_source)||^2; b_g above is a target
section. The section is zero at a unit arrow.

K acts on the module by

\[
(W_kv)(h)=R_kv(k^{-1}.h).
\tag{IJ.7}
\]

Equations (IJ.2) and (IJ.3) give W_k b_g=b_(k.g).
Scalar products are K translated; the inverse map is W_(k^-1).
The representation and its continuity follow from the K norm
jets verified in IJ.1–IJ.2, not just pointwise strong continuity.
It intertwines (IJ.5) with the actual transformed arrow k.g.

### 11AI.5. The intrinsic dense closable metric normal connection

Define on the dense displayed scalar algebra and vector core

\[
\delta_i a=\partial_i a,\qquad\nabla_i v=\partial_i v.
\tag{IJ.8}
\]

Use the scalar/module Leibniz rules. If two finite expressions
define the same module element, their difference has zero
concrete vector function on every actual h by (IJ.4). Its
derivative is also zero there. Faithfulness again makes the
derivative the zero module element. This proves well-definedness
of the connection, including all algebraic null relations.
Differentiating the concrete inner products proves metricity.
It is C_c^infty(P)-linear in the normal vector-field argument
and satisfies the ordinary anchor Leibniz rule.

It is closable. Suppose v_n->0 and nabla_i v_n->w in module norm.
For each actual h choose a short integral curve of its partial_i
vector field. Uniform convergence from (IJ.4) permits integration
of the ordinary H1-valued derivatives along this curve. It gives
integral w(h_t)dt=0; continuity gives w(h)=0. This holds at every
actual arrow, so faithfulness in (IJ.4) implies w=0. The same
argument proves scalar closability. A uniform complete flow on
all return labels has not been assumed.

Every H bisection preserves the frame parallelism. Consequently
the normal chain rule in (IJ.5) intertwines this connection with
the pushed vector field exactly. Under K the parallelism changes
by the actual constant orthogonal/adjoint matrices of k, and R_k
is independent of the unit coordinate. The ordinary chain rule
in (IJ.7) proves K covariance too. All normal connection jets
of b_g are the actual vectors adjoined at IJ.2, with the proved
EK uniform bounds. Thus the connection is an actual field
connection derived from the source-pair vector estimate.

### 11AI.6. Compact-frame quotient preserves the field and connection

The K action on Y_H is free, since its anchor to the principal
frame space P is equivariant. Put Y=Y_H/K. It is Hausdorff and
second countable, by compact proper K action. Its anchor to T
is proper: above compact original K0 subset T its prequotient
lies above compact P_K0 and is compact by IJ.3. The unit
evaluation section is K-equivariant, so descends to one actual
continuous ordinary section T->Y.

E_H/K is a continuous real Hilbert field E over Y. This assertion
has an explicit local proof: choose a smooth frame section u(x)
on a normal chart. Each K orbit above that chart has a unique
representative above u(x). The restriction of E_H to those
representatives gives the local quotient field. Changes of frame
are its actual onto W_k maps, with the same continuous Gram
topology. The original G action and affine displacement descend
by IJ.6–IJ.7. The proper lower bound holds on actual original g:
pick any representative frame above its compact endpoints,
apply IJ.4's proper H bound, and project its compact transporter.
This uses compact P_K0, not a scalar evaluation.

Restrict the K-covariant connection to Levi-Civita horizontal
normal vectors. It descends to E: a horizontal lift of v is
K-equivariant, and the K chain rule in IJ.5 makes its derivative
independent of the chosen representative frame. Its dense core
is obtained by averaging IJ.5 core sections over compact K;
continuity and the uniform K jets permit their norm integrals.
These averages approximate every K-equivariant continuous
section by averaging dense core approximants, proving density.
Metricity, closability and original G covariance descend from
IJ.5, with the invariant Levi-Civita horizontal lift. Hence this
is another actual original-unit proper affine field with a
covariant normal connection, directly from the enlarged jet
spectrum. No L2(K) probability fibre has been added at this step.

### 11AI.7. A compact invariant weight exists, with a precise limitation

A continuous compact-resolvent graph weight can be made exactly
K invariant on E_H. Choose a real strictly positive compact h
on its countably generated module, by the weighted rank-one
generator sum of WF.1. Average alpha_k(h) over K, obtaining h_K.
It is compact and real. It remains strictly positive: a positive
functional vanishing on h_K vanishes on the continuous
nonnegative integrand alpha_k(h), hence on h at the identity;
strict positivity of h forces that functional to be zero. The
support-projection approximate-identity argument then proves
that the scalar plateau cutoffs of h_K form a compact approximate
identity. They commute and are exactly K invariant.

Apply WF.1/BA.3's joint-tail convex argument to these cutoffs,
with compact H-arrow and vector tests. For local bisections use
their compact unit cutoffs: the translated approximate identities
have the same local support projection, so their localized
differences tend weakly to zero; this is exactly KG.2's bounded
functional argument. Hahn–Banach convex separation gives the
summable localized action defects and displacement tails. Their
convex combinations remain K invariant and have nested plateaus.
The usual sum Theta=1+sum(1-v_n) is a positive regular operator,
has locally compact inverse and compact continuous transport
defects, puts b_g in its completed domain, and preserves that
domain under actual transports, by the resolvent argument WF.1.
This is a section-module construction with compact unit supports.

Theta descends through the quotient, and its localized compact
inverse stays compact. In the smooth frame trivializations of
IJ.6, the quotient fibre is literally a representative fibre,
so its resolvent is its ordinary restriction. Finite-rank
approximations restrict with the same norms, and compact unit
partitions prove localized module compactness. Thus quotient
does not create FD.7's Haar Fourier multiplicity.

This selection proves compact inverse and action-domain control,
but not bounded or relative-bounded normal derivatives of Theta.
The convex cutoffs were not selected with normal derivative
estimates. No differentiability of this invariant weight is
silently inferred from that of h or of the generating vectors.
The joint weight/connection interface is still a distinct proof
obligation. A later outer probability step also requires its own
joint selection; pointwise averaging would again be insufficient.

### 11AI.8. Comparison of the coefficient fields

IJ.1–IJ.6 use the actual EK vector jets to build a new enlarged
kernel spectrum with a genuine metric connection, preserving
the original unit anchor and full inverse coefficient. This construction
uses the actual faithful arrow functions, without assuming an all-label
product domain or adding the Haar-pinned probability field of FD. The enlargement's
restriction map to the scalar spectrum is proper, but no
connection is transferred backward through that map.

To use the existing dual proof one can take Y as its input and
pass to Prob(Y_x), with ordinary unit section and contraction.
The repeated Gram path argument PJ proves the connection step
when its required compact lifted-path resolution is supplied;
it is not automatically imported here merely from closability.
IJ.7 proves a continuous invariant compact weight before that
probability step, not the final physical confining generator.

Joint normal estimates for a confining weight, differentiable
Bott/deformation and inverse-module passage, and the actual
closed physical graph operator with original +1 class comparison
remain unproved. No equivariant gamma=1, fibre-index-to-whole-base
inference or generic unboundedification is used. The auxiliary
field remains even and the entire z_g^-1 rho_t(s_g) normal factor
and final right Clifford order stay unchanged.

![Compact-frame quotient and the different Haar Fourier field](../figures/compact-frame-quotient.png)

**Figure 11AI.1.** IJ.1–IJ.7 and FD.7. In the exact complexified principal-circle sample, P=S1 over one point and E_H=P×C has scalar weight 2 with trivial fibre rotation. The quotient E=C has rank-one inverse 1/2. The different Haar field L2(S1)×C has inverse 1/2 on every orthonormal Fourier mode, giving the full noncompact sequence of FD.7. The finite displayed modes illustrate that sequence; they do not replace its infinite proof. For the real sequence use sqrt(2)cos(nu), n>=1. Circle positions are 513 numerical samples of the stated exact parametrization with marker pi/3. The lower panel distinguishes the proved metric connection and continuous invariant compact weight from its unproved joint normal estimates. Human context: [Connes, A survey of foliations and operator algebras](https://alainconnes.org/wp-content/uploads/foliationsfine.pdf#page=29), the graph-operator problem. [Vector figure](../figures/compact-frame-quotient.svg) · [Generator](../reproduction/compact-frame-quotient/draw_compact_quotient.py) · [Toy checks](../reproduction/compact-frame-quotient/QUOTIENT-FIGURE-CHECKS.json) · [Reproduction](../reproduction/compact-frame-quotient/REPRODUCTION.md) · [Component terms](../reproduction/compact-frame-quotient/COMPONENT-TERMS.md).

## 11AJ. Smooth compact inverses and completed normal forms

Intrinsic normal jets and the compact-frame quotient proves the actual field, faithful norm, dense covariant normal connection and free compact-frame quotient used here. The compact-support and uniform vector-jet premises are the finite good compact-manifold scope of Complete Riemannian holonomy and the original-unit kernel, EK.6. The strict-positive compact and functional separation arguments are proved in A confining oscillator over a varying Hilbert field, WF.3, and Kasparov's technical theorem. The completed resolvent criterion is proved in Unbounded Kasparov modules and spectral triples.

The normal derivative below is a specified form on the completed weight domain. A one-sided operator bound and transport-domain preservation are different conclusions; the real block example distinguishes them explicitly. This result concerns the auxiliary coefficient field over its properly anchored spectrum. It does not replace the original unit algebra or supply the physical graph operator.

### 11AJ.1. Compact-operator differentiation on the actual jet field

Use the actual real field E_H over the enlarged proper spectrum Y_H of
Section 11AI, at its finite good complete transversal of the original
compact eligible manifold. Its dense normal core E_infty and the scalar
algebra A_infty are closed under every ordered frame-parallelism derivative.
Write A=C0(Y_H) after complexification; every operator preserves the
underlying real field. This is a coefficient module anchored to the
original frame units, rather than a replacement of the original unit algebra.
They have the faithful concrete arrow-function norm of (IJ.4). Write
delta_i for the scalar derivative and nabla_i for the corresponding metric
connection. Outside the core, use the graph closures of delta_i and
nabla_i supplied by IJ.5. Every chosen C0 section generator has bounded
derivatives of all orders. Compact range-unit cutoffs supply such generators globally;
proper anchoring turns their compact unit supports into compact spectral
supports. All completions below retain the full original unit space.

For real core vectors xi,eta, the finite-rank operator differentiates by

\[
\partial_i\theta_{\xi,\eta}=\theta_{\nabla_i\xi,\eta}+\theta_{\xi,\nabla_i\eta}.
\tag{CW.1}
\]

Here partial_i T means the core commutator nabla_i T-T nabla_i, not an
unbounded derivative whose domain is left implicit. Formula (CW.1) follows
by the scalar Leibniz rule and metricity; products satisfy the ordinary
operator Leibniz rule and adjoints differentiate to the corresponding
adjoints. The finite-rank algebra preserves the core.

This operator derivation is closable in compact-operator norm. If T_n tends
to zero and partial_i T_n tends to S, then for every core xi,
T_n xi tends to zero and nabla_i(T_n xi) tends to S xi, by the commutator
formula. Closability of nabla_i in IJ.5 gives S xi=0. Density of the core
then gives S=0. We use this closure, and its ordered iterates, throughout.
If T_n and all its derivative words converge in norm, closedness applied
successively shows that the limit lies in the domains of those derivative
words and has exactly their limits. This supplies the operator-domain
statement needed for the infinite sums below.

### 11AJ.2. One smooth invariant compact inverse

Let 0<epsilon<=1/2. Choose countably many real C0 core generators xi_j,
j>=1. The compact group K=SO(q) acts by the actual W_k of IJ.7. Under K
the solder/connection parallelism transforms by constant orthogonal
matrices, after using the invariant Euclidean and trace inner products
on R^q and so(q). Its core-vector jet norms on compact K orbits are finite.

Choose finite numbers D_(m,j), nondecreasing in m, which bound the norms of
all ordered derivatives through order m of W_k xi_j, uniformly in k.
Include the norms of xi_j themselves. At m=1 also include every derivative
in a frame direction of length at most one, rather than just the separate
basis derivatives. This only enlarges the bound by a finite frame constant.
The K chain rule in IJ.5 supplies these bounds from the actual bounded jet
generators, so no norm-continuous action on an arbitrary bounded vector
set is assumed. Set

\[
\begin{gathered}c_j=\dfrac{\varepsilon^2 2^{-j}}{1+\max_{0\le m\le j+1}D_{m,j}^2},\qquad h_0=\sum_{j\ge1}c_j\theta_{\xi_j,\xi_j},\\h=\int_KW_kh_0W_k^{-1}\,dk.\end{gathered}
\tag{CW.2}
\]

The Haar integral is normalized. The sum converges in compact-operator
norm, is real and positive, and has norm at most epsilon^2. Its generating
vectors have positive coefficients, so it is strictly positive in K(E_H):
the proof with epsilon(h_0+epsilon)^(-1) in WF.3 works with c_j in place
of 2^(-j). It first proves convergence on each xi_j and then on all E_H;
the finite-rank approximation shows the resulting cutoffs approximate
the identity on K(E_H).

The averaged h is still strictly positive. A positive functional vanishing
on h vanishes on the continuous nonnegative integrand at every k, hence
on h_0 at k=1. Since h_0 is strictly positive, that functional is zero.
If the support of h in the bidual were smaller than the unit, a nonzero
positive normal functional supported on its complement would restrict to
such a functional on K(E_H), a contradiction. Thus the support is full.
Scalar plateau functions of h therefore approximate the identity, first
in every nondegenerate representation and then in the usual strict
approximate-identity sense; equivalently the norm-closed hereditary
subalgebra generated by h is all K(E_H). To see the norm assertion directly, separation of that
closed hereditary subalgebra would give a nonzero positive functional
vanishing on h. Consequently h has dense range on E_H, by nondegeneracy
of the compact action and its cutoffs. No inference from zero orthogonal
complement alone is used for the Hilbert module.

Every fixed derivative word of h_0 converges in norm. The derivative of
order m of a rank-one term is the finite Leibniz sum of its two vector
jets, bounded by 2^m D_(m,j)^2. For j>=m, multiplying by c_j bounds it by
2^m epsilon^2 2^(-j); the remaining finitely many j are finite. K averaging
preserves these bounds. Continuous K dependence of the jet vectors gives
Riemann sums converging in every fixed derivative norm, and CW.1's closed
derivations identify the integral derivatives. Thus h belongs to every
specified operator derivative domain. It is exactly K invariant.

For the first derivative there is a more informative factorization. Let
L be the standard real Hilbert A-module with coefficient space
L2(K) tensor ell2(N), and define its densely specified column map and
derivative column by

\[
\begin{gathered}Sf=\sum_j\sqrt{c_j}\int_K(W_k\xi_j)f_j(k)\,dk,\\S_i^{\prime}f=\sum_j\sqrt{c_j}\int_K\nabla_i(W_k\xi_j)f_j(k)\,dk,\\h=SS^*,\qquad\partial_i h=S_i^{\prime}S^*+S(S_i^{\prime})^*.\end{gathered}
\tag{CW.3}
\]

The formula is first defined on finite simple A-valued functions.
Hilbert-module Cauchy-Schwarz and the summable squared vector bounds
give bounded adjointable extensions. Explicitly S* eta has coordinates
sqrt(c_j)<W_k xi_j,eta>; the same formula with the derivative vectors
gives (S_i')*. Finite-sum and simple-function approximation justifies the
displayed identities and Haar integrals. Their norm estimates are

\[
\|S\|\le\varepsilon,\qquad\|S_v^{\prime}\|\le\varepsilon,\qquad\|\partial_vh\|\le2\varepsilon^2.
\tag{CW.4}
\]

for every compactly supported normal/frame vector field v whose parallelism
coefficient vector has pointwise length at most one. For variable
coefficients this is the C0(P)-linear connection rule; no derivatives of
v enter a single connection derivative. The estimates hold on the whole
module, not only on one fibre.

### 11AJ.3. The reciprocal and smooth compact resolvents

Define Theta=h^(-1) with domain Ran h and Theta(h xi)=xi. Full support and
dense range make this a densely defined positive self-adjoint regular
operator. One can verify regularity directly: its mutually adjoint
resolvents are

\[
R_+=(\Theta+i)^{-1}=h(1+ih)^{-1},\qquad R_-=(\Theta-i)^{-1}=h(1-ih)^{-1}.
\tag{CW.5}
\]

They are bounded adjointable, have dense ranges, satisfy the resolvent
identity, and are adjoints of each other. The inverse graph of the bounded
positive injective h is closed; these two resolvents prove self-adjointness
and regularity by the completed resolvent criterion. Moreover Theta>=
epsilon^(-2), Theta^(-1)=h is compact, and both R_+ and R_- are compact.
All statements are module statements. K invariance holds on the completed
domain because W_k h=h W_k.

The resolvents belong to every operator derivative domain. Since ||h||<=1/4,
the Neumann series for (1+-ih)^(-1) and each of its fixed derivative words
converges in operator norm: at word length m the product-rule factors
grow at most polynomially in the series index, while powers of ||h||
decay geometrically. Closedness from CW.1 identifies the derivatives.
For the first derivative, direct cancellation gives

\[
\partial_vR_\pm=(1\pm ih)^{-1}(\partial_vh)(1\pm ih)^{-1}.
\tag{CW.6}
\]

The surrounding multipliers have norm at most one by positivity of h.
Thus these compact derivatives have norm at most 2 epsilon^2 for unit v.
This proves smooth resolvent control; it does not claim that partial_v
Theta is a bounded operator.

### 11AJ.4. The actual completed normal form and its bound

The range h E_infty is a graph core for Theta: approximate any xi in E_H
by core xi_n, so h xi_n tends to h xi and Theta h xi_n tends to xi.
It is also in Dom nabla_v. Indeed the finite sums and K-integral
approximations have the commutator formula on E_infty; convergence of h
and partial_v h in norm and closedness of nabla_v give
nabla_v(h xi)=h nabla_v xi+(partial_v h)xi.

On u=h xi,w=h eta with xi,eta in E_infty, define the weak metric derivative
of Theta by the fully specified expression

\[
\begin{aligned}q_v(u,w)&=\delta_v\langle u,\Theta w\rangle-\langle\nabla_vu,\Theta w\rangle-\langle\Theta u,\nabla_vw\rangle\\&=-\langle\Theta u,(\partial_vh)\Theta w\rangle.\end{aligned}
\tag{CW.7}
\]

The last identity follows by substituting the preceding commutator rule
and metricity. In particular it never applies Theta to nabla_v w, which
need not lie in Dom Theta. The last expression extends continuously to
Dom Theta with its graph norm, because partial_v h is bounded. It is
symmetric for a real v and real coefficients (and Hermitian after
complexification). This is the exact form interpretation used below.

Put C_v=||S_v'||<=epsilon. For u in Dom Theta, write z=Theta u and use
CW.3. The two cross terms of <z,partial_v h z> give, in A-positive order,
for every t>0,

\[
\pm q_v(u,u)\le t\langle u,\Theta u\rangle+\dfrac{C_v^2}{t}\langle\Theta u,\Theta u\rangle\qquad(t>0).
\tag{CW.8}
\]

This is the elementary inequality +-(<a,b>+<b,a>)<=
t<a,a>+t^(-1)<b,b>, with a=S* z and b=(S_v')* z.
Here <S* z,S* z>=<u,Theta u>, while
<(S_v')* z,(S_v')* z><=C_v^2<z,z>. The signs account for the minus
in (CW.7). This proof uses no polar decomposition of S; such a polar
decomposition need not be adjointable on a Hilbert module.

For every a>0 this implies the strict infinitesimal form bound

\[
\pm q_v(u,u)\le a\langle\Theta u,\Theta u\rangle+\dfrac{2C_v^4}{a^3}\langle u,u\rangle\qquad(a>0).
\tag{CW.9}
\]

If C_v=0 then q_v=0. Otherwise set t=2 C_v^2/a in (CW.8), and use the
scalar functional-calculus inequality t lambda<=(a/2)lambda^2+
t^2/(2a), valid for every lambda>=0. The two a/2 terms add to a and
the remaining constant is exactly 2 C_v^4/a^3. This proves the order
bound on the completed form domain, including its constants.

### 11AJ.5. Return to the original units

The actual compact-frame quotient of IJ.6 sends this invariant h to a
compact operator h_bar on the quotient field E over Y=Y_H/K. Smooth local
frame sections identify the quotient fibre with one representative
fibre. Thus h_bar, its resolvents and its normal form are their ordinary
restrictions, not operators on additional L2(K) modes. The horizontal
K-covariant connection makes (CW.7) independent of the frame choice,
and the same inequalities hold for unit original normal vectors.

Here global compactness is justified, not inferred from fibre compactness.
For a compact K-invariant operator on E_H, choose a compact unit cutoff
making its discarded tail arbitrarily small; the proper anchor and
compact K preserve that norm estimate downstairs. Cover the remaining
compact unit support by finitely many smooth frame charts. Finite-rank
approximants upstairs restrict on each representative field with the
same error. Multiply their vector sections by smaller unit cutoffs to
extend them downstairs, and use a finite unit partition to sum the local
rank-one approximants. Their total error is arbitrarily small. This
proves h_bar in K(E), and the same argument applies to its compact
resolvents and their compact first derivatives. Full support and dense
range descend by the identical local representative and approximate-unit
argument. Hence Theta_bar=h_bar^(-1) is positive self-adjoint regular,
with compact inverse, the completed core and (CW.9), on the whole
original-unit anchored C0(Y)-module. Only the auxiliary field is quotiented; the full
inverse normal Spin-c factor and the final right Clifford action stay
unchanged.

### 11AJ.6. Why this form estimate is not an operator estimate

Let H=ell2({n>=2}) tensor R2, with its constant real field over the circle
and flat connection d/dx. Set a_n=2^(-n), b_n=2^(-n^3),
J=[[0,-1],[1,0]], U(x)=exp(xJ), and

\[
\begin{gathered}h(x)=\bigoplus_{n\ge2}U(x)\operatorname{diag}(a_n,b_n)U(x)^*,\\\Theta(x)=h(x)^{-1},\qquad h^{\prime}(x)=[J,h(x)].\end{gathered}
\tag{CW.10}
\]

This is a real positive strictly positive compact operator field. Finite
block cutoffs converge uniformly on the compact image of every continuous
section; their finite positive eigenvalues give the dense-range assertion
on the section module. Its
derivatives of every order converge in norm, since on block n their
norms are bounded by 2^m a_n at order m. Its reciprocal is a positive
self-adjoint regular field with compact inverse and smooth resolvents.
The normal form has all the properties above: take
S(x)=direct_sum_n U(x)diag(sqrt(a_n),sqrt(b_n)), so h=SS* and S'=JS
is bounded. This example uses trivial auxiliary K. It is a Hilbert-field
example, not a claimed counterexample to the original foliation problem.

At x=0 write A_n=2^n,B_n=2^(n^3). Direct matrix multiplication gives

\[
\begin{gathered}\Theta_n^{\prime}=(A_n-B_n)\begin{pmatrix}0&1\\1&0\end{pmatrix},\\\|\Theta_n^{-1}\Theta_n^{\prime}\Theta_n^{-1}\|=a_n-b_n\longrightarrow0,\\\|\Theta_n^{\prime}\Theta_n^{-2}\|=\dfrac{B_n}{A_n^2}-\dfrac1{A_n}=2^{n^3-2n}-2^{-n}\longrightarrow\infty.\end{gathered}
\tag{CW.11}
\]

Thus the quadratic form is controlled, even with a compact two-sided
normalization, while the one-sided operator normalization is unbounded.
The latter expression is first defined on finite block vectors; its
displayed norms show that it has no bounded extension. One must not
replace (CW.9) by such an operator assertion.

Completed action domains are a separate condition as well. On the same
field let the finite group C2 act on the coefficient by the constant
orthogonal swap P=[[0,1],[1,0]], over the identity of the circle, with zero
affine displacement. This is a smooth covariant connection and a proper
finite group action. At x=0 choose the vector

\[
\begin{gathered}u_n=(2^{-n}/n,0),\qquad\|\Theta(0)u\|^2=\sum_{n\ge2}\dfrac1{n^2}<\infty,\\\|(\Theta(0)Pu)_n\|=2^{n^3-n}/n\not\longrightarrow0.\end{gathered}
\tag{CW.12}
\]

The vector lies in H and Dom Theta(0), whereas Pu is not in that domain.
Indeed sum 1/n^2<=sum 1/(n(n-1))=1 for n>=2, by telescoping, and
sum 2^(-2n)/n^2<=sum 4^(-n)=1/12. In contrast n^3-n>=2n and n<=2^n,
so 2^(n^3-n)/n>=2^n and cannot tend to zero.
This also gives a completed module-domain failure, not just a fibre test:
the continuous section u(x)=U(x)(u_n)_n equals
h(x)U(x)(1/n,0)_n, so it is in Ran h with the continuous image
Theta(x)u(x)=U(x)(1/n,0)_n. If Pu were in Ran h as a section, evaluation
at x=0 would put Pu(0) in Ran h(0), contradicting (CW.12).
Therefore the current construction does not supply the compact-arrow
transport-domain and displacement bounds of WF.2/BA.7. This finite-group
coefficient example illustrates that logical distinction; it does not
identify this auxiliary action with an eligible foliation's holonomy.

### 11AJ.7. The exact interface supplied

CW.1–CW.5 provide an actual original-unit real weight with a smooth compact
inverse, completed regular domain, compact norm-smooth resolvents and a
specified infinitesimal normal form estimate. CW.6 proves why neither a
one-sided operator estimate nor the action-domain condition follows.
A joint selection which retains the actual G compact-arrow defects and
displacement domains is still required, as are the subsequent probability
and differentiable inverse-module interfaces and the physical graph sum
with its original +1 comparison. The form estimate must be used as a form
estimate in any attempted sum-domain proof. It is not a completed graph
operator or a generic unbounded representative of the normalized class.
One-sided operator control is one possible sufficient route, rather than
a premise of the original Dirac problem. A complete form-method proof
could instead supply the actual sum-domain, localized compact-phase and
original-class comparison conclusions.

![Smooth compact inverse and the exact normal-form and action-domain distinction](../figures/normal-form-weight.png)

**Figure 11AJ.1.** CW.1–CW.12. The top panel shows the actual K-averaged jet columns, compact inverse, completed reciprocal domain and normal-form bound. The two matrix normalizations are separate operations on the same derivative. For n=2 their norms are exactly 63/256 and 63/4. With C=1/2 and a=1/4, the remainder is 8 and both positive-order matrices have determinant 133200. The bottom panel gives the continuous section and its completed domain, followed by the genuine domain failure under the auxiliary C2 swap. The infinite sequence and module proof accompany the figure; the five finite rational cases only check the displayed matrices. This auxiliary example is not a foliation counterexample. Human context: [Connes, A survey of foliations and operator algebras](https://alainconnes.org/wp-content/uploads/foliationsfine.pdf#page=29), end of Section 8. [Vector figure](../figures/normal-form-weight.svg) · [Generator](../reproduction/normal-form-weight/draw_normal_form.py) · [Exact finite checks](../reproduction/normal-form-weight/NORMAL-FORM-CHECKS.json) · [Reproduction](../reproduction/normal-form-weight/REPRODUCTION.md) · [Component terms](../reproduction/normal-form-weight/COMPONENT-TERMS.md).

## 11AK. Smooth orbit compacts and the completed oscillator cross form

Intrinsic normal jets and the compact-frame quotient proves the actual real Gram field, normal connection and original-unit quotient. Smooth compact inverses and completed normal forms proves the closed compact derivation and column construction. A confining oscillator over a varying Hilbert field supplies the completed bosonic/exterior Fock operator and its correctly oriented bound B(S)^2≤||S||^2D^2. The full inverse Spin-c coefficient, original G/T/\(C_0(T)\) endpoint and final right Clifford order are retained.

### 11AK.1. The precise input and an invariant initial compact

Let A=\(C_0(Y_H)\), with the underlying real coefficient algebra used before
complexification. E is IJ's real continuous Gram field. Let \(E_\infty\)
be the bounded-jet, compact-range-unit-supported core in IJ.2–IJ.5.
Let \(\partial_i\) be CW.1's closed compact-operator derivations and \(\nabla_i\)
the closed metric connection. Actual H bisections preserve the frame
parallelism, so every ordered derivative intertwines linear transport.
K=SO(q) has the actual W action of IJ.7, and its change of parallelism
uses constant orthogonal matrices. All compact-operator and column
limits below are limits on this whole A-module.

We shall need a little more than strict positivity of an initial
compact. For every chosen compact-cutoff displacement section \(\xi_j\),
we arrange


\[
\theta_{\xi_j,\xi_j}\le C_jh_0.
\tag{JD.1}
\]

with a finite constant C_j. The same list contains a countable real
generating system for E. This can be achieved while retaining every
normal jet and exact K invariance, as follows.

For a real core section \(\xi\), the map k -> W_k \(\xi\) and every normal jet
are norm smooth. This follows directly from IJ.1–IJ.2's vertical
feature estimates: differentiate R_k \(\xi\)(k^-1.h), using the vertical
frame derivatives and the fixed receiving-space translation. Its
derivative vectors are limits of difference quotients of existing
module elements, hence belong to E. Repeated normal derivatives have
the same property. Adjoin these K derivative vectors to the generator
list used to choose \(h_0\), without changing the completed field.

Here is the needed elementary evaluation estimate on compact K. Choose
a finite coordinate atlas, a smooth partition, and an integer
m>dim(K)/2. On a coordinate cube extend each compactly supported
coordinate function to a torus. Fourier Cauchy-Schwarz gives


\[
|f(0)|^2\le\left[\sum_{\ell\in\mathbb Z^d}(1+|\ell|^2)^{-m}\right]\sum_\ell(1+|\ell|^2)^m|\widehat f(\ell)|^2.
\]

The sum is finite: dyadic annuli contribute at most a constant times 2^(j(d-2m)), a summable geometric series. The smooth scalar functions here have rapidly decreasing Fourier coefficients by repeated integration by parts. Their torus Fourier polynomials converge in L2: the circle Fejer approximate-identity argument proved in Transverse measures of foliations, Theorem5.19, extends to the finite-dimensional torus by successive convolution in each coordinate. Thus the Fourier characters form the complete system to which Parseval is applied.  Express the coordinate derivatives through the global
left-invariant Lie-algebra frame; their finitely many coefficient
functions and derivatives are bounded on the compact coordinate
supports. Parseval from Hilbert spaces and compact operators, Section 4, and the finite coordinate partition therefore
give a finite constant C_K and the finite collection D_a of all
left-invariant derivative words through order m, for which


\[
|f(1)|^2\le C_K\sum_a\int_K|D_af(k)|^2\,dk.
\tag{JD.2}
\]

This also holds for vector-valued functions: apply the scalar argument
to every coordinate in an orthonormal basis and sum. Apply its scalar
version in a fibre to f(k)=<W_k \(\xi\),e>. Its derivatives differentiate
the displayed vector section. Thus, in A-positive operator order,


\[
\theta_{\xi,\xi}\le C_K\sum_a\int_K\theta_{D_a(W_k\xi),D_a(W_k\xi)}\,dk.
\]

For this invariant choice, D_a(W_k \(\xi\))=W_k \(\eta_a\) with a fixed core
vector \(\eta_a\) obtained by differentiating the norm-smooth W orbit of
\(\xi\) at the identity. Thus the integrals are exactly K-averaged
rank-one generators, as required for invariance of \(h_0\).
This inequality can equally be checked against a finite family of
test sections and the corresponding positive Gram matrices; hence it
is an order statement on the module, not just a numerical norm bound.
Choose the positive coefficients of this finite collection small
enough to obey CW's diagonal all-jet bounds. Their minimum is positive
for each fixed \(\xi\). CW's rank-one sum and K averaging therefore produce
a real positive strictly positive \(h_0\) in \(\mathcal K(E)\), with all normal jets,
which satisfies (JD.1) for every listed \(\xi_j\). Its norm can be made
arbitrarily small. K invariance follows by Haar change of variables.

This initial compact has an adjointable column \(S_0\) with \(h_0\)=\(S_0\) \(S_0\)*.
Its columns and every ordered normal derivative are square summable
in the following stronger, intrinsic sense. For a standard coefficient
module L, a column R:L->E has Hilbert-Schmidt norm


\[
\|R\|_{2,A}^2=\left\|\sum_j\langle Re_j,Re_j\rangle\right\|_A.
\]

The sum converges in A norm. For the K-integral columns, approximate
their continuous kernels by finite simple functions and use Parseval
on L2(K); the same definition and limit apply. Equivalently these
columns form the continuous Hilbert A-module E tensor_A conjugate(L).
The CW diagonal choice makes every derivative column converge in this
norm: the finite exceptional terms at any fixed derivative order are
finite, and the remaining squared jet norms are bounded by a geometric
tail. In particular \(S_0\) and its derivative columns are compact. No
adjointable polar decomposition of \(S_0\) is used.

### 11AK.2. Localized genuine-arrow completely positive operators

Choose a countable list of relatively compact original G bisections
B_i and smooth cutoffs 0<=\(\chi_i\)<=1 on their range domains, with support
closed inside those domains. Choose enough smaller charts that every
actual arrow has a neighborhood on which one of the \(\chi_i\) is one.
Include inverse charts in the list. Lift B_i over all frame fibres;
these are the actual K-saturated H bisections of IJ.2.

For T in \(\mathcal K(E)\), define


\[
\Phi_i(T)(y)=\chi_i(p(y))^2L_{g_i(y)}T(g_i(y)^{-1}y)L_{g_i(y)}^{-1}.
\tag{JD.3}
\]

on the lifted range domain, and zero outside. The notation uses the
actual frame arrow whose original arrow is B_i's range section.
\(\Phi_i\) is real, positive, completely positive and contractive. To
justify its target \(\mathcal K(E)\), first apply it to a rank-one operator. Local
transport gives a rank-one expression with its two vectors multiplied
by \(\chi_i\), extended continuously by zero. First multiply each source vector by the source pullback of the cutoff. This is a bounded continuous scalar multiplier whose support is closed inside the source unit domain. Its product remains a section vanishing at infinity in the actual spectral space, and belongs to the corresponding open-source ideal. The actual bisection action is a homeomorphism between the open source and range spectral spaces, so it transports that vanishing-at-infinity section to the open-range ideal. Inclusion of that ideal gives its continuous zero extension. Equivalently, compactly truncated source sections give this statement first, and norm approximation gives the general case. No properness of the spectral anchor map is inferred from properness of the groupoid action. Finite rank and norm approximation give the assertion
for all compact operators. Positivity at every matrix level is
preserved by conjugation and multiplication by a central square.

The maps commute with K, because the original bisection transport
commutes with the change of frame and \(\chi_i\) is an original-unit
function. Their normal derivatives obey the ordinary product rule.
Every derivative of the transported operator is its transported
derivative: this is the exact chain rule in IJ.5, not a unitary action
of a fictitious matrix label on the feature space. All additional
derivatives fall on \(\chi_i\).

Let J_(m,i) be a finite number bounding all ordered normal derivatives
through order m of \(\chi_i\), including the zero-order norm. Such numbers
exist on its compact supported frame-unit test. Choose positive a_i
with


\[
Q=\sum_i a_i<1,\qquad\sum_i a_iJ_{m,i}^2<\infty\quad(m\ge0).
\tag{JD.4}
\]

For example take


\[
a_i=\frac{2^{-i}}{8(1+\max_{0\le m\le i}J_{m,i}^2)}.
\]

The finitely many i<m are harmless. Set \(\Phi\)=sum_i a_i \(\Phi_i\), a bounded
real completely positive operator of norm at most Q.

### 11AK.3. One smooth orbit compact and its genuine order controls

Define the norm-convergent compact sum


\[
h=\sum_{n\ge0}\Phi^n(h_0)=h_0+\sum_i a_i\Phi_i(h).
\tag{JD.5}
\]

It is real, positive, K invariant, and strictly positive, since h>=\(h_0\)
and \(h_0\)'s cutoffs act nondegenerately on E. Its norm is at most
||\(h_0\)||/(1-Q), so scale \(h_0\) to ensure ||h||<=1/4.

Here every ordered normal derivative also converges. An n-fold word
in \(\Phi_i\) has a column obtained from \(S_0\) by actual composed transport
and the product of n scalar chi cutoffs, multiplied by the square root
of its word coefficient. For a derivative word of length m, expand
Leibniz through this product. There are at most (n+1)^m assignments of
the derivatives to the n cutoffs and the final \(S_0\). Cauchy-Schwarz on
this finite sum bounds its squared Hilbert-Schmidt norm by the number
of assignments times the sum of their squared norms. At most m
cutoff factors are differentiated. Upon summing word coefficients,
the undifferentiated factors contribute Q each, and the differentiated
ones contribute one of the finite sums in (JD.4). The final \(S_0\) factor
has one of its finitely many derivative bounds through order m.
Consequently the sum at word length n is bounded by


\[
C_m(n+1)^{2m}Q^{\max(n-m,0)}.
\tag{JD.6}
\]

for a finite constant C_m independent of n. This is summable. The
cutoff derivatives after intermediate transports have precisely the
same bounds, since the actual parallelism is preserved. There is no
uncontrolled derivative of a long return label in this calculation.

The resulting full word column S and every normal derivative S'_i
converge in Hilbert-Schmidt module norm. Thus


\[
h=SS^*,\qquad\partial_i h=S_i^{\prime}S^*+S(S_i^{\prime})^*.
\tag{JD.7}
\]

and all higher normal jets converge in compact norm by the product
rule and CW.1's closedness. This proves the simultaneous smoothness
claim rather than assuming that a general norm-convergent orbit sum
is differentiable.

The order controls are exact. The second equation in (JD.5) implies
\(\Phi_i\)(h)<=a_i^-1 h. On a neighborhood where \(\chi_i\)=1, write
b=L_g h_source L_g^-1 and a=h_target. Then


\[
b\le a_i^{-1}a.
\]

An inverse chart gives a<=a_j^-1 b. Every compact actual-arrow test
is covered by finitely many such neighborhoods, so both constants
can be chosen uniformly on that test. These are original G transport
controls after compact-frame quotient, with actual arrow labels
retained. They are not bounded additive defects of h^-1.

### 11AK.4. Completed module domains for 0<alpha<1/2

We prove the precise range theorem used here; ordinary fibrewise
Douglas factorization is insufficient at the endpoint.

Let a,b be bounded positive adjointable operators with dense range
on a Hilbert C0(Z)-module F, and suppose b<=C a. Fix 0<\(\alpha\)<1/2.
The operators need not be compact on a restricted open-unit ideal.
Then


\[
D=a^{-\alpha}b^\alpha.
\]

exists as a bounded adjointable operator on F, has norm at most
C^\(\alpha\), and satisfies a^\(\alpha\) D=b^\(\alpha\). If also a<=C' b, the reverse
operator exists and the two are inverses. Hence


\[
\operatorname{Ran}a^\alpha=\operatorname{Ran}b^\alpha.
\tag{JD.8}
\]

as completed module ranges, with their actual inverse-power graph
norms equivalent.

First prove the needed operator-monotone inequality. For 0<p<1,


\[
x^p=c_p\int_0^\infty t^{p-1}x(x+t)^{-1}\,dt.
\]

Here \(c_p\) is the reciprocal of the positive integral of u^(p-1)/(1+u) over (0,infinity). It is finite because the powers at zero and infinity are p-1>-1 and p-2<-1. Substitution t=xu gives the scalar identity; the zero case is immediate. The substitution u=1/v shows \(c_p\)=\(c_{1-p}\), which supplies the inverse-power integral below. Uniformly truncated integrals converge on bounded spectra by bounding x/(x+t) by one near zero and ||x||/t near infinity. Continuous functional calculus therefore gives the operator identity. If 0<=b<=C a, inverse order gives
b(b+t)^-1<=C a(C a+t)^-1. Integrating gives
b^p<=C^p a^p. Use p=2 \(\alpha\). Thus


\[
D_t=(a+t)^{-\alpha}b^\alpha,\qquad\|D_t\|\le C^\alpha\quad(t>0).
\tag{JD.9}
\]

Next the norm integral


\[
N=c_\alpha\int_0^\infty\lambda^{-\alpha}(a+\lambda)^{-1}b\,d\lambda.
\]

converges. For \(\lambda\)<=1, order b<=C a gives


\[
\|(a+\lambda)^{-1}b\|\le\frac{\sqrt C}{2\sqrt\lambda}\|b^{1/2}\|.
\]

Indeed square the first factor (a+\(\lambda\))^-1 b^1/2 and bound its
positive product by C a/(a+\(\lambda\))^2. For \(\lambda\)>=1 the norm is at
most ||b||/\(\lambda\). The integrability at zero is exactly \(\alpha\)<1/2.
Dominated convergence in the same formula with a+t shows
(a+t)^(-\(\alpha\))b -> N in operator norm as t decreases to zero.
It follows that D_t converges on the dense range of b^(1-\(\alpha\)),
because D_t b^(1-\(\alpha\))=(a+t)^(-\(\alpha\))b. The uniform bound (JD.9)
extends this strong convergence to every element of F.

The adjoints also converge strongly. On the dense range of a^(1-\(\alpha\)),


\[
D_t^*a^{1-\alpha}=b^\alpha(a+t)^{-\alpha}a^{1-\alpha}\longrightarrow b^\alpha a^{1-2\alpha}.
\]

in operator norm, since 1-2 \(\alpha\)>0 makes the scalar convergence
uniform down to zero. Uniform boundedness extends it to all F.
Strong limits of operators and their adjoints are adjointable and
mutually adjoint: pass to the limit in their A-valued inner products.
Call the limit D. The identity a^\(\alpha\) D=b^\(\alpha\) follows from strict
convergence of (a/(a+t))^\(\alpha\) and strong convergence of D_t. The
closed inverse graph of a^\(\alpha\) then identifies D with the actual
inverse-power expression, not merely a fibrewise bounded formula.
Apply the same argument in reverse; injectivity of a^\(\alpha\) and
b^\(\alpha\) proves the two limits are inverse. This proves (JD.8).

Apply this to (JD.3)'s actual transported h. Put


\[
\Theta_\alpha=h^{-\alpha},\qquad\operatorname{Dom}\Theta_\alpha=\operatorname{Ran}h^\alpha.
\tag{JD.10}
\]

On the whole original coefficient module, the inverse of this
positive injective dense-range compact has the
regular self-adjoint resolvents h^\(\alpha\)(1+-i h^\(\alpha\))^-1, exactly as
in CW.3. It has compact inverse and compact resolvents. On every
compact arrow test actual L_g preserves the completed domain, with
graph-norm transport bounds at most the appropriate C^\(\alpha\).
These graph transports are continuous. To check the potentially
singular limit, work on one compact bisection ideal. The convergent
integral used above has a common bound from its fixed comparison
constant. Thus it converges uniformly on that test on sections in
the dense range of the appropriate h^(1-\(\alpha\)). Uniform graph bounds
and strict cutoff approximation then give continuity on every
section. Composition is the actual L action on the domain and
therefore satisfies the groupoid identities; no new label action is
constructed.

### 11AK.5. Actual affine displacement graph domains below the endpoint

The initial inequality (JD.1) and h>=\(h_0\) give theta_(\(\xi\),\(\xi\))<=C h for
each chosen compact-cutoff displacement section. Consequently


\[
\|(h+\lambda)^{-1}\xi\|\le\frac{\sqrt C}{2\sqrt\lambda}.
\]

for \(\lambda\)<=1, and at most ||\(\xi\)||/\(\lambda\) for \(\lambda\)>=1. The norm
Bochner integral


\[
\Theta_\alpha\xi=c_\alpha\int_0^\infty\lambda^{-\alpha}(h+\lambda)^{-1}\xi\,d\lambda.
\tag{JD.11}
\]

converges for \(\alpha\)<1/2 and is a genuine E section. Multiplying by
h^\(\alpha\) gives \(\xi\), by the scalar identity and strict functional
calculus. In particular it proves membership in Ran h^\(\alpha\) rather
than merely bounded fibrewise graph norms. An explicit bound is


\[
\|\Theta_\alpha\xi\|\le c_\alpha\left[\frac{\sqrt C}{2(1/2-\alpha)}+\frac{\|\xi\|}{\alpha}\right].
\tag{JD.12}
\]

Each actual b_g agrees, on a smaller bisection, with one of the
chosen displacement sections where its cutoff equals one. Compact
arrow tests have finite such covers. Their graph sections are
continuous by (JD.11)'s uniform integrable bounds and continuity of
the ordinary compact resolvents. They have bounded graph norms by
(JD.12). All identities retain the original target-section convention
of IJ.4 and its source conversion in PK; no displacement identity
term is removed.

Thus for every \(\alpha\) in (0,1/2) this one h gives genuine actual G
domain covariance and displacement graph continuity. It does not
yet give the compact additive defects needed by the particular
cubic oscillator phase argument in WF.2/BA.7.

### 11AK.6. Compact first normal derivative of the square root

The Hilbert-Schmidt column hypothesis in (JD.7) gives an additional
normal result without a module polar decomposition:


\[
k=h^{1/2}\in\operatorname{Dom}\partial_i,\qquad\partial_i k\in\mathcal K(E).
\tag{JD.13}
\]

Only the first derivative is asserted. Arbitrary higher derivatives
of a square root at a changing-rank boundary do not follow from this
argument.

For t>0 put k_t=(h+t)^1/2-sqrt(t) I, a compact operator. Ordinary
holomorphic functional calculus on a neighborhood of the positive
spectrum and the closed derivation product rule show it lies in
Dom \(\partial_i\). One can prove this calculus rule by polynomial
approximation on a contour at positive distance from -t and the
resolvent identity; both the functions and their divided differences
converge uniformly. Its derivative X_t is the unique solution of


\[
(h+t)^{1/2}X_t+X_t(h+t)^{1/2}=S_i^{\prime}S^*+S(S_i^{\prime})^*.
\tag{JD.14}
\]

It is an element of the Hilbert-Schmidt A-module of endomorphisms
E tensor_A conjugate(E). Left and right multiplication by h are
commuting positive adjointable operators there. The solution is
their joint continuous functional calculus, or the norm integral
of the two positive exponential semigroups. For fixed t this has
bounded denominator at least 2 sqrt(t).

The crucial estimate is uniform in t:


\[
\|X_t\|_{2,A}\le\sqrt2\|S_i^{\prime}\|_{2,A}.
\tag{JD.15}
\]

To prove it, at each fibre diagonalize the compact h. This is a
calculation of a bound, not a choice of continuous eigenvectors.
In that fibre write S=h^1/2 V with V a coisometry on its supported
space and put B=S'_i V*. This fibrewise polar expression is used
only to estimate scalar sums; it is not asserted to be an
adjointable module polar decomposition. If the eigenvalues of h
are lambda_j, then


\[
(X_t)_{ij}=\frac{\sqrt{\lambda_j}B_{ij}+\sqrt{\lambda_i}\overline{B_{ji}}}{\sqrt{\lambda_i+t}+\sqrt{\lambda_j+t}}.
\]

Weighted Cauchy-Schwarz bounds the squared entry by


\[
\frac{\sqrt{\lambda_j}}{d_{ij}}|B_{ij}|^2+\frac{\sqrt{\lambda_i}}{d_{ij}}|B_{ji}|^2.
\]

because the sum of these two weights is at most one. On summing
i,j the two sums are equal and each coefficient is at most one.
This gives 2 sum_ij |B_(ij)|^2<=2||S'_i||_2^2. Zero-dimensional
fibres cause no exception. Supremum over the spectrum proves
(JD.15) in the Hilbert A-module norm.

The X_t are Cauchy in that norm. Since h has dense range on E,
left multiplication by h has dense range on the Hilbert-Schmidt
column module: approximate finitely many columns, then their
square-summable tail. Hence approximate S'_i there by h R.
For this approximating column, the right side of (JD.14) is
h R S*+S R* h. Its solution is obtained by the joint functions


\[
F_t(\lambda,\mu)=\frac{\lambda}{\sqrt{\lambda+t}+\sqrt{\mu+t}}.
\]

and the reversed function. F_t converges uniformly on the compact
square of spectra to F_0, extended by zero at (0,0). Indeed the
functions are bounded by sqrt(\(\lambda\)), and outside a small square
at (0,0) the denominators are uniformly separated from zero.
Thus the corresponding solutions converge in Hilbert-Schmidt
module norm. The error from replacing S'_i by h R is uniformly at
most sqrt(2)||S'_i-h R||_(2,A), by (JD.15). This proves the Cauchy
claim for the original columns. Their limit X is a continuous
Hilbert-Schmidt section, hence a compact operator. Since k_t->k
in compact norm and \(\partial_i\) k_t=X_t->X, CW.1's closedness gives
(JD.13).

Put Theta_(1/2)=k^-1 on Ran k. On k \(E_\infty\) define its weak metric
normal form by the CW.7 expression. The inverse commutator rule
now gives


\[
q_i(u,w)=-\langle\Theta_{1/2}u,(\partial_i k)\Theta_{1/2}w\rangle.
\tag{JD.16}
\]

The core is in Dom \(\nabla_i\) by the compact first-derivative rule,
and is a graph core of this reciprocal. The expression extends
to the whole completed inverse graph domain.

It has an infinitesimal A-positive-order bound relative to
Theta_(1/2)^2. Let e_n=f_n(k) be positive scalar plateaus equal
to zero near zero and strictly approaching one. Compactness of
M=\(\partial_i\) k implies ||M-e_n M e_n||->0. Given a>0, choose n
making this norm at most a. Then, for every u in Ran k,


\[
\pm q_i(u,u)\le a\langle\Theta_{1/2}u,\Theta_{1/2}u\rangle+\|M\|\|\Theta_{1/2}e_n\|^2\langle u,u\rangle.
\tag{JD.17}
\]

This follows by bounding the small remainder in positive order;
the commuting plateau moves past \(\Theta\) in the finite part, where
\(\Theta\) e_n is bounded. It is a completed form estimate and never
applies \(\Theta\) to \(\nabla_i\) u. The constant in the second term is
finite because e_n vanishes near zero.

All operators and estimates in JD.1–JD.6 are K covariant. Apply
CW.5's one-representative finite-frame-chart compactness argument
to h, k, their compact resolvents and compact derivatives. The
original-unit quotient preserves the stated domains and normal
forms. It introduces no Haar-mode multiplicity. The whole inverse
normal Spin-c coefficient and final right Clifford action remain
the separate physical factors, with their original order.

### 11AK.7. Why the endpoint is not a completed module conclusion

Here is a smooth real finite-group coefficient example satisfying
the order controls and the column hypotheses, yet failing completed
action covariance at \(\alpha\)=1/2. It is an auxiliary Hilbert-module
example, not a counterexample to Connes's foliation problem.

Let Z be the circle with coordinate x near 0, F=C(Z,ell2_R(N0)),
and let e0,e1,... be its standard basis. Put


\[
\varepsilon_n=2^{-n^3},\quad x_n=2^{-n},\quad r_n=2^{-n-4},\quad a=\operatorname{diag}(1,\varepsilon_1^2,\varepsilon_2^2,\ldots).
\]

Choose a fixed smooth bump 0<=phi<=1, supported in (-1,1), with
phi(0)=1. On the disjoint intervals (x_n-r_n,x_n+r_n), put
t_n(x)=epsilon_n phi((x-x_n)/r_n). Define an orthogonal involution
P(x) to equal diag(1,-1,-1,...) outside those intervals, and, on
the plane span(e0,en), to have block


\[
\begin{pmatrix}\sqrt{1-t_n(x)^2}&t_n(x)\\t_n(x)&-\sqrt{1-t_n(x)^2}\end{pmatrix}.
\tag{JD.18}
\]

All other tail vectors have eigenvalue -1. The intervals are
disjoint, since the distance of two successive centers is
2^(-n-1) while their radii sum to 3*2^(-n-5). Every derivative
of the n-th varying block is bounded by a constant depending
on its order times epsilon_n r_n^(-m); this tends to zero faster
than any power of x_n. Therefore P is norm smooth, flat to its
constant value at 0. Its involution gives an actual smooth C2
action over the identity of Z.

It is covariant for a smooth metric connection. Let U(x) be the
rotation through half arcsin(t_n(x)) on the varying plane and
identity elsewhere. Then P=U P0 U*, and
\(\nabla\)=d/dx-U' U* makes the U-frame parallel and P covariant.
Both U and its connection coefficient are norm smooth and flat
at 0 by the same estimates.

Use the actual localized-orbit compact


\[
h=a+\tfrac12PaP.
\tag{JD.19}
\]

It is real positive strictly positive compact and norm smooth
with every connection derivative. Strict positivity follows
from h>=a and the dense finite-block range of a. It has a
square-summable smooth column


\[
S=[a^{1/2},2^{-1/2}Pa^{1/2}].
\]

All derivative columns are Hilbert-Schmidt: the varying block
amplitudes are flat epsilon_n, while the fixed initial column
is Hilbert-Schmidt; its connection derivatives have the same
rapidly summable bounds. The geometric CP series for \(\Phi\)(T)=
(1/2) P T P equals (4/3)h. This harmless scalar factor has no
effect on domain preservation. In particular this is precisely
the type of orbit construction in (JD.5), not an arbitrary
unrelated fibre order example.

The order comparisons are uniform and exact:


\[
\tfrac12h\le PhP\le2h.
\tag{JD.20}
\]

Subtracting the left side from the middle gives (3/4) P a P;
subtracting the middle from the right gives (3/2) a. Thus
all fibrewise normalized transports have uniformly bounded
inverse-half graph norms. Nevertheless these fibres do not
assemble to the asserted module-domain action.

Indeed u=h e0 is a continuous section in Ran h and hence in
Ran h^1/2, with continuous preimage h^1/2 e0. At x=x_n set
\(\varepsilon\)=epsilon_n and r=sqrt(1-\(\varepsilon\)^2). The relevant block
of h is [[A,B],[B,C]], where


\[
A=\tfrac32-\tfrac{\varepsilon^2}2+\tfrac{\varepsilon^4}2,\quad B=\tfrac{r^3\varepsilon}2,\quad C=2\varepsilon^2-\tfrac{\varepsilon^4}2.
\tag{JD.21}
\]

Let \(\delta\)=sqrt(AC-B^2). The positive 2 by 2 square-root formula
is


\[
h^{1/2}=\frac{h+\delta I}{\sqrt{A+C+2\delta}}.
\]

It follows either by squaring and using h^2-(A+C)h+\(\delta\)^2 I=0,
or by direct multiplication. Hence the en coefficient of the
unique fibre preimage h^(-1/2) P h e0 is


\[
\frac{\varepsilon(A^2+\delta A-B^2)-rB(2A+\delta)}{\delta\sqrt{A+C+2\delta}}.
\tag{JD.22}
\]

Now A->3/2, B/\(\varepsilon\)->1/2, C/\(\varepsilon\)^2->2 and
\(\delta\)/\(\varepsilon\)->sqrt(11)/2. The ratio in (JD.22) tends to


\[
\frac{3/4}{\sqrt{33/8}}=\sqrt{3/22}>0.
\tag{JD.23}
\]

At x=0 the same fibre preimage is sqrt(3/2)e0 and has no tail
component. Since n tends to infinity, a continuous section
approaching that vector must have its en component tend to
zero. The nonzero limit (JD.23) is a contradiction. Thus


\[
Pu\notin\operatorname{Ran}h^{1/2}\quad\text{as a section of }F.
\]

Every individual fibre is in the corresponding range and obeys
the graph bound. The failure is precisely continuity of the
module preimage. This is why JD.4 proves strict strong limits
and restricts \(\alpha\)<1/2 instead of invoking fibre indices or
unqualified Hilbert-space range factorization.

### 11AK.8. Exact supplied scope and remaining physical interface

The actual IJ field has a real K-invariant smooth compact h
whose localized genuine-arrow transports are uniformly order
comparable. Its inverse powers for every 0<\(\alpha\)<1/2 have
completed actual G domains and displacement graph continuity.
The compact k=h^1/2 has a compact first normal derivative;
its reciprocal has the completed infinitesimal form (JD.17).
These statements descend to the actual original-unit field.

The proof has not established all of these properties for one
and the same oscillator weight. JD.7 shows that the usual
order comparison cannot supply the missing half-power module
domain. Also absent are a compact-additive oscillator transport
defect or a proved alternative localized phase estimate, a
closed physical normal/auxiliary sum with its grading and
maximal domain, passage through the actual probability and
inverse-module products, and the original Bott +1 comparison.
The existing whole-base ordinary KK identity remains an input;
it does not prove any of those analytic conclusions.

The route remains open to a sufficient form-domain graph sum.
Neither a one-sided operator estimate nor local finite order
is declared universally necessary. A complete replacement
must prove the actual sum-domain, local compactness, coefficient
commutator and class comparison for its specified operator.




### 11AK.9. The completed weak derivative and oscillator conventions

Let E be the actual real IJ field after the compact-frame quotient,
or its frame pullback when doing the invariant-coordinate calculation.
Let \(\Theta\) be a positive self-adjoint regular real weight with compact
inverse k=\(\Theta\)^-1. Suppose k has a compact first normal derivative
partial_v k for the actual closable metric connection. On k \(E_\infty\)
the completed weak derivative of \(\Theta\) is


\[
q_v(u,w)=-\langle\Theta u,(\partial_vk)\Theta w\rangle.
\]

Put \(M_v\)=-partial_v k. It is real self-adjoint compact. The form may
be denoted S_v=\(\Theta\) \(M_v\) \(\Theta\), but this notation defines a form;
it does not assert an operator \(\Theta\) partial_v \(\Theta\)^-2 or a domain
for \(\Theta\) nabla_v u. CW.1–CW.5 supply exactly this input with
\(\Theta\)=h^-1 and k=h. JD.6 supplies it separately with \(\Theta\)=h^-1/2.

Use the actual complexified Fock module


\[
\mathscr F(E)=\bigoplus_{p,q\ge0}\operatorname{Sym}^p(E_\mathbb C)\otimes\Lambda^q(E_\mathbb C),\qquad\text{parity}=(-1)^q.
\]

Write a_j for bosonic annihilation and epsilon_j for exterior
creation, in a finite orthonormal test frame. The real convention is


\[
\begin{gathered}Q_j=\sqrt2(a_j\varepsilon_j+a_j^*\varepsilon_j^*),\qquad D=\sum_jQ_j,\quad D^2=2N,\\B(A)=\sqrt2\sum_{i,j}(A_{ij}a_j\varepsilon_i+\overline{A_{ij}}a_j^*\varepsilon_i^*).\end{gathered}
\]

These formulas are finite-core specifications, continued by WF's
completed field domains. The identity D^2=2N uses total occupation
N=p+q. The confining odd operator at s=1 is


\[
C=D^3+B(1+\Theta).
\tag{FQ.1}
\]

Its completed regular self-adjoint construction is WF. On each
occupation block the exact square is


\[
\begin{gathered}C^2=2\sum_j(2N+1+\lambda_j)^2N_j\ge2d\Gamma_b(\Theta^2)+2d\Gamma_f(\Theta^2),\\C^2\ge D^2.\end{gathered}
\tag{FQ.2}
\]

Finite spectral tests of \(\Theta\) prove these positive-order bounds;
WF's common graph core and regular functional calculus pass them
to the completed domains. There is no assumption of continuous
eigenvector choices on the field.

### 11AK.10. Weighted contractions define the derivative on Dom C

For \(\psi\) in the finite weighted Fock core, let A_Theta \(\psi\) be the
array of bosonic contractions weighted by \(\Theta\), and let E_Theta
\(\psi\) be the array of exterior annihilations weighted by \(\Theta\).
Their squared A-valued norms are respectively


\[
\langle\psi,d\Gamma_b(\Theta^2)\psi\rangle,\qquad\langle\psi,d\Gamma_f(\Theta^2)\psi\rangle.
\]

For any bounded real self-adjoint M, define the quadratic form


\[
b_M(\psi,\psi)=2\sqrt2\operatorname{Re}\langle E_\Theta\psi,MA_\Theta\psi\rangle.
\tag{FQ.3}
\]

In a finite orthonormal frame this is precisely the weak
B(\(\Theta\) M \(\Theta\)) expression. Intrinsically the arrays live in
the appropriate E tensor Fock modules; M acts on their E factor.
Changing an orthonormal frame preserves the inner product, so
the definition is independent of its coordinates. The real
field, grading and adjoints are those in FQ.1.

Hilbert-module Cauchy-Schwarz in positive order gives


\[
\begin{aligned}\pm b_M(\psi,\psi)&\le\sqrt2\|M\|\bigl(\langle A_\Theta\psi,A_\Theta\psi\rangle+\langle E_\Theta\psi,E_\Theta\psi\rangle\bigr)\\&\le\frac{\|M\|}{\sqrt2}\langle C\psi,C\psi\rangle.\end{aligned}
\tag{FQ.4}
\]

One can prove the first inequality directly by positivity of
the 2 by 2 operator matrix [[||M|| I,M],[M*,||M|| I]], then
the ordinary 2ab<=a^2+b^2 inequality. Thus it is an A-positive
order statement, not merely a norm estimate of scalar fibres.
Polarization gives a bounded Hermitian form on the graph module
Dom C. Since the two contraction arrays are controlled by
(FQ.2), they extend continuously from WF's graph core to this
completed domain. This supplies its exact domain.

For M=\(M_v\), this form is the actual weak normal derivative of C.
To check that assertion on a dense core, use finite products of
vectors k \(\xi\), \(\xi\) in \(E_\infty\), and the finite occupation subspace.
They lie in Dom nabla_v and in the weighted oscillator core.
The induced Fock connection commutes with D and N: its metric
Leibniz rule differentiates tensor factors, while the canonical
real contraction/creation pairing is invariant. Differentiate
<\(\psi\),C phi> by that metric rule. The D^3 part cancels, and the
B(1+\(\Theta\)) part is exactly (FQ.3) with the weak derivative from
FQ.1. In this verification C phi and nabla_v phi are separately
defined; C is not applied to nabla_v phi. Its continuous
extension is therefore the completed derivative form just
constructed.

### 11AK.11. Compact normalization gives an infinitesimal cross form

If M is compact, then for every a>0 a finite K_a exists such that


\[
\pm b_M(\psi,\psi)\le a\langle C\psi,C\psi\rangle+K_a\langle\psi,\psi\rangle\quad(\psi\in\operatorname{Dom}C).
\tag{FQ.5}
\]

Choose a positive plateau e=f(k), vanishing near zero, so
||M-e M e||<=a/sqrt(2). Compactness and the strict approximation
of the identity by these plateaus justify this choice on the
whole module. The remainder in (FQ.4) is at most (a/2)||C \(\psi\)||^2.

For the finite part, \(S_e\)=\(\Theta\) e M e \(\Theta\) is a bounded compact
real self-adjoint operator, since \(\Theta\) e is bounded and compact.
Its form is exactly the ordinary B(\(S_e\)) form on the core. The
corrected WF inequality gives


\[
\langle B(S_e)\psi,B(S_e)\psi\rangle\le\|S_e\|^2\langle D\psi,D\psi\rangle.
\]

in the A-positive-order form sense. Consequently Cauchy-Schwarz,
D^2<=C^2, and t xy<= (a/2)x^2+t^2/(2a)y^2 give


\[
\pm\langle\psi,B(S_e)\psi\rangle\le\frac a2\langle C\psi,C\psi\rangle+\frac{\|S_e\|^2}{2a}\langle\psi,\psi\rangle.
\]

Adding the remainder proves (FQ.5), with the explicit permissible
constant K_a=||\(\Theta\) e M e \(\Theta\)||^2/(2a). The chosen plateau
depends on a; no uniform one-sided derivative bound is claimed.

The same estimate holds if the form is paired with a bounded
normal Clifford unitary and the auxiliary parity. Such unitaries
preserve both contraction energy norms and commute with C^2.
Apply the same two-array inequality to the transformed vector;
the finite part is bounded by ||B(\(S_e\))\(\psi\)||||\(\psi\)|| as before.

### 11AK.12. The exact form of a normal/auxiliary Dirac square

Suppose an actual Hilbert-space representation of these coefficient
modules is given, with a self-adjoint normal Dirac \(D_N\) from the
specified metric connection, correct density correction, and full
inverse normal Spin-c coefficient. Its normal Clifford operators
c_j are self-adjoint with c_j^2=1, and the normal principal part
is -i sum_j c_j nabla_j. The original parity shift and final right
Clifford order are fixed inputs from Section 11A; this argument
does not replace the inverse lift z_g^-1 rho_t(s_g) by its
determinant or alter that Clifford order.

On a common finite weighted smooth core consider the specific
geometric differential expression


\[
S_0=C\otimes1+\Gamma_{\rm aux}\otimes D_N.
\tag{FQ.6}
\]

All tensor signs are those of the displayed order. Since C is odd,
the cross term in its formal square is


\[
S_0^2=C^2+D_N^2+\Gamma_{\rm aux}[D_N,C]=C^2+D_N^2-i\sum_j\Gamma_{\rm aux}c_jB(S_j).
\tag{FQ.7}
\]

where S_j is the weak derivative form \(\Theta\) \(M_j\) \(\Theta\).
The spin/density lower-order connection terms commute with the
auxiliary C; the auxiliary part of the connection is exactly
the one used to define its derivative above. The sign in (FQ.7)
comes from C \(\Gamma_{\rm aux}\)=-\(\Gamma_{\rm aux}\) C. The product -i \(\Gamma_{\rm aux}\)
B(S_j) is Hermitian because \(\Gamma_{\rm aux}\) anticommutes with B(S_j).
Thus every cross summand is a Hermitian form of the type FQ.3.

FQ.3 proves a completed infinitesimal bound for their finite sum.
In particular, on


\[
V=\operatorname{Dom}D_N\cap\operatorname{Dom}C.
\]

the quadratic form


\[
Q(\psi)=\|D_N\psi\|^2+\|C\psi\|^2-i\sum_j\langle\psi,\Gamma_{\rm aux}c_jB(S_j)\psi\rangle.
\tag{FQ.8}
\]

is well defined. Assuming V is dense, it is lower semibounded
and closed. Here are the exact prerequisites and proof. The
intersection graph norm ||\(\psi\)||^2+||\(D_N\) \(\psi\)||^2+||C \(\psi\)||^2 is
complete, because \(D_N\) and C are closed. For any 0<a<1 the
finite summed cross form lies between +-a||C \(\psi\)||^2 plus
a finite multiple of ||\(\psi\)||^2, by splitting a among the
finitely many j in (FQ.5). Add a sufficiently large scalar
multiple of ||\(\psi\)||^2 to Q. Its norm is then equivalent to
the intersection graph norm, giving completeness and positivity.
This proves closedness directly, without a one-sided operator
anticommutator estimate.

For completeness, the associated positive self-adjoint energy
operator can be constructed from this closed form in Hilbert
space. With the shifted positive form inner product on V,
The Riesz–Frechet representation theorem proved in Hilbert spaces and compact operators, Theorem2.3, gives for every f in the physical Hilbert
space a unique Rf in V satisfying Q_shift(Rf,v)=<f,v>.
The inclusion V->H is bounded, so R is bounded; the same
identity gives R*=R>=0 and injectivity, since V is dense.
Its range is dense because its orthogonal complement equals
ker R. The inverse of R, minus the scalar shift, is the
self-adjoint operator represented by Q. Equivalently its
resolvents are obtained by the same positive form argument.
The inverse is self-adjoint on its range: its resolvents are R(1+-iR)^-1, with the required adjoint relation and dense ranges, as proved for bounded positive reciprocals in Smooth compact inverses and completed normal forms. This supplies the energy-operator statement with its actual domain.

It does not by itself prove self-adjointness of the first-order
expression (FQ.6). Nor does it prove that the square of a closed
Dirac extension equals this energy operator. A jointly dense
graph core, the range or deficiency-space argument for S_0,
localized compactness and the coefficient commutators are
additional conclusions. The positive square root of the energy
operator has no supplied odd grading or Bott comparison and
cannot be silently declared to be the requested Dirac operator.



The onto-resolvent proof of self-adjointness for the specified odd sum, under its exact differentiable physical realization premise, is given in A completed form-domain Dirac sum.


![Exact completed-domain and operator illustration](../figures/joint-domains-cross-form.png)

**Figure 11AK.1.** JD.1–JD.23 and FQ.1–FQ.8. The two top outputs follow separately from the same orbit compact. The exact half-power counterexample has escaped preimage amplitude tending to sqrt(3/22); the five graph points are numerical samples of its exact matrix formula, not a proof of the infinite limit. The contraction arrays and compact cutoff split give the completed oscillator cross form. Its energy-form consequence requires the stated physical normal Dirac. The coefficient example is not a foliation counterexample. Human context: [Connes, A survey of foliations and operator algebras](https://alainconnes.org/wp-content/uploads/foliationsfine.pdf#page=29), end of Section 8. [Vector figure](../figures/joint-domains-cross-form.svg) · [Generator](../reproduction/joint-domains-cross-form/draw_joint_domains.py) · [Exact finite checks](../reproduction/joint-domains-cross-form/JOINT-DOMAIN-CHECKS.json) · [Reproduction](../reproduction/joint-domains-cross-form/REPRODUCTION.md) · [Component terms](../reproduction/joint-domains-cross-form/COMPONENT-TERMS.md).

## 11AL. A sufficient compact-derivative bridge below the half-power

**Theorem (a sufficient common-power estimate).** The actual adjointable column factorization in FB.1, with 1/2-\(\alpha\)<\(\beta\)<1/2, makes h^\(\alpha\) compactly normal differentiable at the same inverse power used by the transport-domain theorem. The factorization is a premise; it has not been proved for the constructed orbit column.

Let E be the actual anchored real coefficient module, L the column
module, and \(\delta\) a closed compact-operator derivation coming from its
metric normal connection, as in CW.1. Suppose h=SS* is positive, compact
and strictly positive, with its actual derivative


\[
\delta h=S^{\prime}S^*+S(S^{\prime})^*.
\]

Fix 0<\(\alpha\)<1/2. Assume, for one \(\beta\) with


\[
1/2-\alpha<\beta<1/2.
\]

that there is an actual bounded adjointable column B:L->E such that


\[
S^{\prime}=h^\beta B.
\tag{FB.1}
\]

This is a completed module factorization, not a bounded-fibre assertion.
Then k=h^\(\alpha\) belongs to Dom \(\delta\), and \(\delta\) k is compact. In
particular the normal reciprocal form and FQ completed oscillator cross
form apply to the same weight \(\Theta\)=h^(-\(\alpha\)) as JD's transport and
affine-displacement domain construction, provided (FB.1) is established
for that construction and every normal direction used by the sum.

### 11AL.1. The norm-integrable compact derivative

For 0<\(\alpha\)<1 set


\[
c_\alpha=\left[\int_0^\infty\frac{u^{\alpha-1}}{1+u}\,du\right]^{-1}.
\]

The integral is finite and positive: its integrand is bounded by
u^(\(\alpha\)-1) near zero and u^(\(\alpha\)-2) near infinity. Substitution t=xu
gives, for every x>=0,


\[
x^\alpha=c_\alpha\int_0^\infty t^{\alpha-1}\frac{x}{x+t}\,dt.
\tag{FB.2}
\]

The zero case is immediate. Uniformly truncated integrals converge on
each bounded spectrum; near zero use x/(x+t)<=1, and near infinity use
x/(x+t)<=||h||/t. Thus (FB.2) is an identity of continuous functional
calculus for h. No \(\beta\)/reflection identity is required to fix the
normalizing constant.

Write R_t=(h+t)^-1. Functional calculus gives


\[
\|R_th^\beta\|\le t^{\beta-1},\qquad\|S^*R_t\|^2=\|R_thR_t\|\le\frac1{4t}.
\tag{FB.3}
\]

For the first inequality maximize u^\(\beta\)/(1+u); its maximum is
\(\beta\)^\(\beta\)(1-\(\beta\))^(1-\(\beta\))<=1. The second is the scalar maximum of
x/(x+t)^2. Use the actual factorization (FB.1) and its adjoint. For
0<t<=1,


\[
\|R_t(\delta h)R_t\|\le\|B\|t^{\beta-3/2}\quad(0<t\le1).
\tag{FB.4}
\]

Indeed each of its two terms is at most
||B|| t^(\(\beta\)-1)/(2 sqrt(t)). For t>=1 the simpler bound is
||\(\delta\) h||/t^2. Therefore the compact-operator Bochner integral


\[
X=c_\alpha\int_0^\infty t^\alpha R_t(\delta h)R_t\,dt.
\tag{FB.5}
\]

converges in norm: at zero its majorant is
||B|| t^(\(\alpha\)+\(\beta\)-3/2), integrable because \(\alpha\)+\(\beta\)>1/2; at
infinity it is ||\(\delta\) h|| t^(\(\alpha\)-2), integrable because \(\alpha\)<1.
Each integrand is compact, and \(\mathcal K(E)\) is norm closed. In particular


\[
\|X\|\le c_\alpha\left[\frac{\|B\|}{\alpha+\beta-1/2}+\frac{\|\delta h\|}{1-\alpha}\right].
\tag{FB.6}
\]

To prove that X is the actual closed derivative, rather than a formal
divided difference, take \(\tau\)>0 and put


\[
k_\tau=(h+\tau)^\alpha-\tau^\alpha I.
\]

This is compact, since its scalar function vanishes at zero. The
closed derivation and positive-distance resolvent calculus used in
CW.1 put k_tau in Dom \(\delta\). Differentiating (FB.2) on the spectrum
shifted away from zero gives


\[
\delta k_\tau=c_\alpha\int_0^\infty t^\alpha(h+\tau+t)^{-1}(\delta h)(h+\tau+t)^{-1}\,dt.
\tag{FB.7}
\]

One can first differentiate its truncated integrals using
\(\delta\) (h+\(\tau\)+t)^-1=-(h+\(\tau\)+t)^-1(\(\delta\) h)(h+\(\tau\)+t)^-1;
the \(\tau\)>0 bounds and the same tail estimate justify the norm limit.
The term \(\tau\)^\(\alpha\) I has derivative zero in the unitized derivation.

The scalar inequality gives ||k_tau-h^\(\alpha\)||<=\(\tau\)^\(\alpha\): the function
(x+\(\tau\))^\(\alpha\)-x^\(\alpha\) decreases on x>0, since its derivative is
\(\alpha\)[(x+\(\tau\))^(\(\alpha\)-1)-x^(\(\alpha\)-1)]<=0, and its value at zero is
\(\tau\)^\(\alpha\). Its values are nonnegative. Thus the displayed norm bound
follows directly from continuous functional calculus.
For each t>0 the integrand in (FB.7) converges in compact norm to
the integrand of (FB.5). Its norm is dominated on (0,1] by the same
majorant in (FB.4), because \(\beta\)-3/2<0 and \(\tau\)+t>=t; on [1,infinity)
the same large-t bound applies. Norm dominated convergence therefore
gives \(\delta\) k_tau->X. Closedness of \(\delta\) proves h^\(\alpha\) in Dom \(\delta\)
and \(\delta\) h^\(\alpha\)=X in \(\mathcal K(E)\).

All steps use the actual column factor, positive functional calculus,
closed derivative and completed module. They do not glue fibrewise
polar operators, select eigenvectors, assume a one-sided reciprocal
derivative, or apply \(\Theta\) to an unproved connection-domain vector.

### 11AL.2. The exact common-weight premise

JD.1–JD.5 already give the genuine-arrow and displacement domains for
\(\Theta\)=h^(-\(\alpha\)), 0<\(\alpha\)<1/2. If (FB.1) is proved for the same h and
its derivative columns, (FB.5) gives a compact normal derivative of
its reciprocal compact k=h^\(\alpha\). FQ.1–FQ.3 then give the completed
infinitesimal oscillator cross form on that same weight domain. This
still does not prove a compact transport phase defect, the actual
physical representation and closed odd Dirac sum, the differentiable
inverse-module passage, or the original Bott +1 comparison.

The needed factorization is not inferred from JD's smooth
Hilbert-Schmidt column, its fibre order comparisons, or the different
half-power argument. JD.7's smooth finite-group example shows why the
completed endpoint cannot be assumed. Other sufficient constructions
remain permitted; (FB.1) is one concrete bridge to investigate.



![Exact completed-domain and operator illustration](../figures/fractional-power-bridge.png)

**Figure 11AL.1.** FB.1–FB.7. The strict region 0<\(\alpha\)<1/2 and 1/2-\(\alpha\)<\(\beta\)<1/2 makes the small-resolvent majorant integrable. At \(\alpha\)=1/4 and \(\beta\)=3/8 the exact integral constants are 8 and 4/3; the diagonal boundary produces dt/t and is excluded. The lower panel identifies the compact derivative by closedness on the completed module. This sufficient result still requires the actual bounded adjointable factor B; the three rational samples do not prove that factorization. Human context: [Connes, A survey of foliations and operator algebras](https://alainconnes.org/wp-content/uploads/foliationsfine.pdf#page=29), end of Section 8. [Vector figure](../figures/fractional-power-bridge.svg) · [Generator](../reproduction/fractional-power-bridge/draw_fractional_bridge.py) · [Exact finite checks](../reproduction/fractional-power-bridge/FRACTIONAL-POWER-CHECKS.json) · [Reproduction](../reproduction/fractional-power-bridge/REPRODUCTION.md) · [Component terms](../reproduction/fractional-power-bridge/COMPONENT-TERMS.md).

## 11AM. A completed form-domain Dirac sum

**Theorem (the specified odd sum).** Under the exact differentiable physical realization in Section11AM.3, the actual oscillator C and maximal self-adjoint normal Dirac \(D_N\) have the self-adjoint odd sum C+\(\Gamma\) \(D_N\) on exactly Dom C intersect Dom \(D_N\). Its two onto resolvents, joint graph core and square-energy identity are proved below. The physical realization, groupoid phase compatibility, localized compactness and original Bott +1 are distinct obligations.

### 11AM.1. A bounded-commutator symmetric-product lemma

Let A be self-adjoint on a Hilbert space, B bounded self-adjoint,
and assume B Dom A is contained in Dom A and M=[A,B] is bounded.
Then the symmetric operator


\[
X=(BA+AB)/2=BA+M/2,\qquad\operatorname{Dom}X=\operatorname{Dom}A.
\tag{SD.1}
\]

is essentially self-adjoint. This lemma is abstract and needs no
extra local finite-order condition.

Here is a full domain proof. Put J_n=n^2(A^2+n^2)^-1. It is bounded
self-adjoint, converges strongly to one and maps the Hilbert space
into Dom A^2. On Dom A,


\[
[X,J_n]=[B,J_n]A+[M,J_n]/2.
\tag{SD.2}
\]

Both terms have bounded extensions, uniformly in n. Write
R_+=(A+i n)^-1 and R_-=(A-i n)^-1. Then
J_n=n(R_--R_+)/(2i) and [B,R_+-]=R_+- M R_+-.
Since ||R_+-||<=1/n and ||R_+- A||<=1,


\[
\|[B,J_n]A\|\le\|M\|,\qquad\|[X,J_n]\|\le2\|M\|.
\tag{SD.3}
\]

The extension in (SD.2) tends strongly to zero. On Dom A this
follows directly from strong convergence J_n->1 in [B,J_n]A
and [M,J_n]; uniform boundedness extends it to every vector.

For u in Dom X*, the distributional adjoint identity gives


\[
XJ_nu=J_nX^*u+[X,J_n]u.
\tag{SD.4}
\]

Indeed pair against every v in Dom X, commute X past J_n using
(SD.2), and use symmetry of J_n. Since J_n u lies in Dom A,
the left side is already defined. The right side tends to X*u,
and J_n u tends to u. Thus u belongs to Dom closure(X) and
closure(X)u=X*u. The reverse inclusion follows from symmetry.
Consequently closure(X)=X* is self-adjoint. This proves (SD.1),
including the core statement, rather than assuming maximality of
an unbounded product with a bounded coefficient.

### 11AM.2. Exact oscillator-resolvent derivative from the CW form

Use the actual CW field and \(\Theta\)=k^-1 with k=h real compact and
norm smooth in the closed normal derivations. Let


\[
C=D^3+B(1+\Theta).
\]

be the completed WF oscillator. FQ.1–FQ.3 prove that its weak
normal derivative form b_j has completed domain Dom C and, for
each a>0,


\[
|b_j(\psi,\psi)|\le a\|C\psi\|^2+K_{a,j}\|\psi\|^2.
\tag{SD.5}
\]

The A-positive-order version holds before the actual Hilbert-space
representation. Its constants are uniform on the whole coefficient
module. The desired consequence is an actual closed operator
derivative of \(R_\lambda\)=(C+i \(\lambda\))^-1, \(\lambda\) real nonzero:


\[
\partial_jR_\lambda=-R_\lambda b_jR_\lambda,\qquad\|\partial_jR_\lambda\|\le a+K_{a,j}/\lambda^2.
\tag{SD.6}
\]

The sandwich is defined through the form: its pairing at \(\xi\),\(\eta\)
is -b_j(R_-\(\lambda\) \(\xi\),\(R_\lambda\) \(\eta\)). It is bounded by (SD.5),
because ||C \(R_\lambda\)||<=1 and ||\(R_\lambda\)||<=1/|\(\lambda\)|.
At each Hilbert-space fibre, the Hermitian form order bound represents the form by a self-adjoint contraction in the graph inner product, using the Riesz theorem just cited. Each resolvent graph weight has norm at most sqrt(a+K_(a,j)/\(\lambda\)^2), so their mixed sandwich has norm at most a+K_(a,j)/\(\lambda\)^2. For the genuine regularized adjointable derivatives below, taking the supremum of these faithful fibre bounds gives the same module operator estimate. Their norm-convergent limit constructs the actual adjointable derivative; no general representation theorem for bounded Hilbert-module forms is assumed. It does not apply C to a normal
derivative of a vector.

We prove that this is the derivative in the closed compact-operator
derivation, not merely an asserted weak formula. For t>0 put


\[
\Theta_t=(k+t)^{-1},\qquad C_t=D^3+B(1+\Theta_t).
\tag{SD.7}
\]

Each \(\Theta_t\) is bounded with all normal jets, and the occupation
blocks of C_t are differentiable bounded operators. Truncating
occupation and using the cubic number bound proves its resolvent
derivative formula; the derivative is the normal weak B form with


\[
\partial_j\Theta_t=-\Theta_t(\partial_jk)\Theta_t.
\]

The induced Fock connection is closable: on the actual arrow
functions, its finite tensor rule is the derivative in the fixed
receiving Fock space. Uniform norm convergence and a short actual
arrow integral curve prove closability exactly as in IJ.5. Its
concrete Gram norm is faithful after tensor completion. The CW.1
commutator closability proof therefore also applies to bounded
adjointable Fock operators, without requiring compactness: if
T_n->0 and \(\partial_j\) T_n->U in operator norm, apply the closed
connection to T_n \(\xi\) for each core \(\xi\) to obtain U \(\xi\)=0. We use
this closed bounded-operator derivation for the resolvents below.

FQ's proof works uniformly in t: use the same compact plateau e
of k to split \(M_j\)=-\(\partial_j\) k. The weighted finite part is
\(\Theta_t\) e \(M_j\) e \(\Theta_t\), whose norm is bounded by
||\(\Theta\) e \(M_j\) e \(\Theta\)||. The remainder estimate only uses its
norm. Thus (SD.5) and (SD.6) hold for C_t with the same constants.

The resolvents R_(\(\lambda\),t) converge to \(R_\lambda\) in operator norm.
Here are estimates supplying this sometimes missing step. Use the joint finite-spectral Fock tests of \(\Theta\) at each faithful fibre; the resulting uniform estimates pass to the actual module norm. For Lambda
large, the subspace having any occupied one-particle mode of
weight above Lambda is invariant under both C and C_t, since
Q_j preserves total occupation in each mode. On that subspace
C^2>=2 Lambda^2 and C_t^2>=2[Lambda/(1+t Lambda)]^2. Their
resolvent norms are therefore at most constants depending on
\(\lambda\) times (Lambda^-1+t). On its complement,


\[
\|\Theta-\Theta_t\|\le t\Lambda^2,\qquad B(\Theta-\Theta_t)^2\le t^2\Lambda^4D^2.
\]

The ordinary resolvent identity there and D^2<=C^2,C_t^2 bound
the resolvent difference by t Lambda^2/|\(\lambda\)|. The vacuum
resolvents are identical. Choosing Lambda=t^(-1/3) proves norm
convergence with a bound O_lambda(t^(1/3)+t). The finite-spectral
core establishes these estimates, and WF's completed occupation
domains pass them to the whole field; no continuous diagonal
basis is chosen.

Their first derivatives also converge in operator norm. To see
this, first use the compact finite part e \(M_j\) e. Its coefficient
S_(e,t)=\(\Theta_t\) e \(M_j\) e \(\Theta_t\) converges in operator norm to
\(S_e\)=\(\Theta\) e \(M_j\) e \(\Theta\); the plateau vanishes near zero. All
these S commute with a fixed larger spectral cutoff of \(\Theta\)
which is one on the support of e. On the Fock subspace with
every occupied mode below a fixed larger cutoff, D R_(\(\lambda\),t)
converges in norm: truncate N, using the uniform estimate
||D R_(\(\lambda\),t)|| on N>L bounded by O(1/L), which follows from
C_t^2>=8N^3; then use the bounded-coefficient finite-N resolvent
identity. On the complementary high-mode subspace, B(S_(e,t))
acts only in the cutoff modes, while the high-mode occupation
lower bound makes the sandwich norm as small as desired,
uniformly as t->0. One convenient bound follows from


\[
\|B(S_{e,t})R_{\lambda,t}\|\le\|S_{e,t}\|\|DR_{\lambda,t}\|.
\]

and the second resolvent factor on that high-mode subspace,
whose norm is O(Lambda^-1+t). Thus the finite-part derivative
sandwich converges. The remainder \(M_j\)-e \(M_j\) e can be made
arbitrarily small; FQ.2 bounds its derivative sandwich uniformly
in t by its norm divided by sqrt(2). This proves the Cauchy
assertion for the entire derivative.

Closedness of this bounded-operator normal derivation now gives
(SD.6). The approximating C_t resolvents need not be compact;
no compactness is assumed for this closure argument. The
derivative has no vacuum scalar part. The compact first-jet
input and the displayed uniform bounds give the global bounded
derivative and its norm estimate, including a noncompact
coefficient base. In particular


\[
\|\partial_jR_\lambda\|\longrightarrow0\quad(|\lambda|\longrightarrow\infty).
\tag{SD.8}
\]

To prove the last limit, fix a in (SD.6), let |\(\lambda\)| tend to
infinity, and then let a decrease to zero. No rate for K_a is
required.

### 11AM.3. The exact differentiable physical realization premise

Let the actual inverse-module graph realization be a Hilbert
space with a self-adjoint normal Dirac A=\(D_N\) whose domain is
its maximal distributional domain. It must retain the original
coefficient representation, full inverse Spin-c transport,
grading and final right Clifford order. The auxiliary oscillator
C is self-adjoint on that same represented field. Let \(\Gamma\) be
its parity, so \(\Gamma\) C=-C \(\Gamma\) and \(\Gamma\) A=A \(\Gamma\).

The realization is required to carry the actual normal derivation
on the coefficient resolvents: its bounded coefficient operator
\(R_\lambda\) has


\[
[A,R_\lambda]=-i\sum_jc_j\partial_jR_\lambda.
\tag{SD.9}
\]

with the specified Clifford order, and the spin/density terms
commute with the auxiliary resolvent. The normal frame is finite
on the compact original manifold, or a finite buffered normal
atlas gives the equivalent bounded expression. This identity is
part of the differentiable coefficient passage, not a conclusion
from ordinary KK products or fibre indices.

Under that premise, (SD.6) gives a bounded commutator K_lambda
with norm tending to zero. It also gives


\[
R_\lambda\operatorname{Dom}A\subseteq\operatorname{Dom}A.
\tag{SD.10}
\]

The domain statement follows from the maximal distributional
domain: for u in Dom A, the metric weak product rule gives
A(\(R_\lambda\) u)=\(R_\lambda\) Au+K_lambda u in the Hilbert space.
This places \(R_\lambda\) u in that maximal domain. Density and
closedness extend the identity from its smooth local core.
Thus no domain is silently dropped in writing (SD.9).

The physical inverse-module passage supplying this premise is
not completed by the present proof. If it is supplied, SD.4
below proves the full self-adjoint graph sum without requiring
a one-sided derivative of \(\Theta\).

### 11AM.4. Onto resolvents of the specific odd graph sum

Define


\[
S=C+\Gamma A,\qquad\operatorname{Dom}S=\operatorname{Dom}C\cap\operatorname{Dom}A.
\tag{SD.11}
\]

This is symmetric and densely defined. Density follows, for
example, because \(R_\lambda\) maps Dom A into both domains and
has dense range; approximate any preimage by Dom A.

Set R=\(R_\lambda\) and K=[A,R]. The bounded operator


\[
B=-i\Gamma R.
\]

is self-adjoint. Indeed \(\Gamma\) R=-R* \(\Gamma\), using \(\Gamma\) C=-C
\(\Gamma\). Its commutator with A is -i \(\Gamma\) K, bounded. SD.1
therefore makes


\[
X=(BA+AB)/2\quad\text{on }\operatorname{Dom}A.
\]

essentially self-adjoint. Write its self-adjoint closure as
\(\overline X\). On Dom A the exact identity is


\[
\Gamma AR=iX+\tfrac12\Gamma K.
\tag{SD.12}
\]

The signs follow by expanding X=BA+[A,B]/2 and
[A,B]=-i \(\Gamma\) K. Choose |\(\lambda\)| sufficiently large that
||K||/2<1. Then


\[
L_0=1+i\overline X+\tfrac12\Gamma K.
\]

is bijective with bounded inverse: 1+i \(\overline X\) is bijective
with inverse norm at most one, and a norm-convergent Neumann
series treats the remaining bounded term. Its domain is Dom
\(\overline X\). This is the explicit range argument, not an inference
from a closed quadratic form.

The natural operator T=\(\Gamma\) A R is closed on


\[
\operatorname{Dom}T=\{\xi:R\xi\in\operatorname{Dom}A\}.
\]

For if xi_n->\(\xi\) and T xi_n->\(\eta\), then R xi_n->R \(\xi\) and
closedness of A identifies the limit. It extends (SD.12),
and hence extends i \(\overline X\)+(1/2)\(\Gamma\) K. In fact they are
equal. The natural operator 1+T is injective: if it kills
\(\xi\), put \(\psi\)=R \(\xi\). This lies in Dom A intersection Dom C,
and


\[
(S+i\lambda)\psi=(1+T)\xi=0.
\]

Symmetry of S gives ||(S+i \(\lambda\))\(\psi\)||>=|\(\lambda\)|||\(\psi\)||;
therefore \(\psi\)=0 and \(\xi\)=0. Any closed extension of the already
onto L0, whose extension is injective, equals L0: for any
element of the extension domain, solve its output using L0
and subtract; injectivity identifies the two preimages.
Thus 1+T=L0, and


\[
(S+i\lambda)R=1+T.
\tag{SD.13}
\]

is onto. R sends its preimages into Dom S. Repeat the same
argument with -\(\lambda\). Both S+i \(\lambda\) and S-i \(\lambda\) are
onto, so the densely defined symmetric S is self-adjoint
on exactly the displayed intersection domain. To verify
this familiar last criterion, if u belongs to Dom S*,
solve (S-i \(\lambda\))v=(S*-i \(\lambda\))u. The difference lies in
ker(S*-i \(\lambda\)), which is the orthogonal complement of
Ran(S+i \(\lambda\)), hence zero. Therefore u=v in Dom S.
This also proves closedness of S.

The original total grading preserves both domains and anticommutes
with each of the two displayed summands, so this self-adjoint sum
is odd on that actual grading. Its normal right Clifford action
has the original graded commutation convention, by the same
domain invariance and the unchanged tensor order. No Clifford
representation is moved across the inverse normal factor.

### 11AM.5. The common graph core and the represented energy form

The proof also identifies a completed graph core. Let \(\mathcal K_A\)
be any core of A. Then


\[
R_\lambda\mathcal K_A.
\]

is a core for the intersection graph norm of Dom S. To
show it, write \(\psi\)=R \(\xi\) with \(\xi\) in Dom T=Dom \(\overline X\).
Approximate \(\xi\) in the \(\overline X\) graph norm by vectors in Dom A,
using SD.1. For those vectors, approximate in the A graph
norm by \(\mathcal K_A\). The identities (SD.12)–(SD.13) imply convergence
of psi_n and S psi_n, and


\[
C\psi_n=(1-i\lambda R)\xi_n.
\]

converges to C \(\psi\). Thus A psi_n=\(\Gamma\)(S-C)psi_n converges
to A \(\psi\). This proves the stronger joint graph-core claim.

On this core the metric weak-resolvent rule (SD.6), inserted
in (SD.13), gives the same cross term as FQ.4's formal
calculation. This can be checked by pairing two core
vectors R \(\xi\),R \(\eta\): the mixed derivative sandwich is
exactly the form -R b_j R in (SD.6); all remaining terms
are the separately defined C and A actions. The equality
extends by joint graph continuity and FQ.3's form bound.
Consequently the represented energy form is precisely


\[
\|S\psi\|^2=\|C\psi\|^2+\|A\psi\|^2-i\sum_j\langle\psi,\Gamma c_jB(S_j)\psi\rangle.
\tag{SD.14}
\]

on Dom S, with the same tensor signs. Its associated
positive energy operator is S^2; it is no longer merely
a separately chosen Friedrichs square root. The result
supplies a full sufficient form-domain sum, conditional
on SD.3's exact physical realization premise.

### 11AM.6. The exact represented scope

SD.1 is unconditional Hilbert-space analysis. SD.2 is the
actual-CW oscillator-resolvent differentiability provider.
SD.4–SD.5 prove self-adjointness, the exact completed domain,
onto resolvents and the energy identity for the specific
odd geometric sum whenever SD.3 is supplied.

The original geometric Dirac result still needs the actual differentiable
inverse-module passage in SD.3, an action/phase-compatible
joint weight or adequate substitute, localized compactness
and coefficient commutators for the represented sum, and
its original Bott +1 comparison. The present theorem
does not claim those. It does not evaluate C0(M_G)
equivariantly to a scalar or assume equivariant gamma=1.




![Exact completed-domain and operator illustration](../figures/form-sum-resolvents.png)

**Figure 11AM.1.** SD.1–SD.14. The physical coefficient realization is an explicit premise. The genuine oscillator resolvent derivative produces a small bounded commutator; the self-adjoint symmetric product and Neumann series give onto resolvents of the exact sum and its completed intersection domain. In the finite Pauli example c=1, \(\lambda\)=2 and p=3, the exact norm squares are 1/5, 5/14 and 1/14. This finite example checks signs and constants; the complete infinite-dimensional proof and its physical premises accompany it. Human context: [Connes, A survey of foliations and operator algebras](https://alainconnes.org/wp-content/uploads/foliationsfine.pdf#page=29), end of Section 8. [Vector figure](../figures/form-sum-resolvents.svg) · [Generator](../reproduction/form-sum-resolvents/draw_form_sum.py) · [Exact finite checks](../reproduction/form-sum-resolvents/FORM-SUM-CHECKS.json) · [Reproduction](../reproduction/form-sum-resolvents/REPRODUCTION.md) · [Component terms](../reproduction/form-sum-resolvents/COMPONENT-TERMS.md).
## 11AN. A Dirac sum when the normal operator has a boundary

The normal operator need not be self-adjoint before the confining coefficient is added. The following theorem replaces that premise by a precise regularization of its maximal domain. It supplies an alternative to A completed form-domain Dirac sum. It remains a theorem about an actual represented operator and its specified domains; it does not construct the missing differentiable physical representation of the general holonomy groupoid.

All Hilbert spaces below are complex. A minimal normal operator means the closure of the specified densely defined symmetric normal core. Its adjoint is its maximal distributional operator whenever that identification has been proved. We distinguish the two domains throughout.

### 11AN.1. A symmetric product regularized by its coefficient

**Lemma.** Let \(A\) be a closed densely defined symmetric operator on a Hilbert space \(\mathcal H\). Let \(B\) be bounded, injective and self-adjoint. Suppose \(B\) preserves both \(\operatorname{Dom}A\) and \(\operatorname{Dom}A^*\), the commutator \(M=[A,B]\) extends boundedly, the identity \(A^*B=BA^*+M\) holds on \(\operatorname{Dom}A^*\), and

\[
B\operatorname{Dom}A^*\subset\operatorname{Dom}A.
\]

Then the following symmetric product is essentially self-adjoint, even if \(A\) is not self-adjoint:

\[
\begin{gathered}
X=\tfrac12(BA+AB)=BA+\tfrac12M,\qquad \operatorname{Dom}X=\operatorname{Dom}A,\\
\operatorname{Dom}X^*=\{u:Bu\in\operatorname{Dom}A^*\},\qquad
X^*u=A^*Bu-\tfrac12Mu .
\end{gathered}
\tag{NR.1}
\]

The commutator is skew-adjoint: test \(M\) against two vectors in \(\operatorname{Dom}A\), use symmetry of \(A,B\), and then use density and boundedness. Thus the displayed adjoint formula follows directly from the definition of the adjoint of \(BA\), with its full domain. The product is symmetric on that domain.

For \(\epsilon>0\) define bounded functions of \(B\) by

\[
Q_\epsilon=(B^2+\epsilon)^{-1},\qquad
J_\epsilon=B^2Q_\epsilon=1-\epsilon Q_\epsilon.
\tag{NR.2}
\]

These operators preserve the domains of both \(A\) and \(A^*\). Here is a domain justification. The commutator with \(B^2\) is \(BM+MB\), bounded on each domain. Choose \(L>\|B\|^2+\epsilon\) and expand
\((B^2+\epsilon)^{-1}=L^{-1}\sum_{n\ge0}[1-(B^2+\epsilon)/L]^n\).
The powers have norm at most \(r^n\) for \(r<1\). Their commutators have norm at most a constant times \(nr^{n-1}\), by the product rule. Hence both series converge in operator and commutator norm. Closedness of \(A\), respectively \(A^*\), proves domain preservation and the inverse commutator identity.

On \(\operatorname{Dom}A\), direct multiplication gives

\[
\begin{gathered}
[X,B]=\tfrac12(BM+MB),\\
[X,J_\epsilon]
=\epsilon Q_\epsilon\left[BMB+\tfrac12(B^2M+MB^2)\right]Q_\epsilon=:N_\epsilon,\\
\|N_\epsilon\|\le\tfrac54\|M\|.
\end{gathered}
\tag{NR.3}
\]

To check the constant, functional calculus gives
\(\|B^2Q_\epsilon\|\le1\), \(\|\epsilon Q_\epsilon\|\le1\), and
\(\|BQ_\epsilon\|\le(2\sqrt\epsilon)^{-1}\).
The two outside terms contribute at most \(\|M\|/2\) each, and the middle term contributes at most \(\|M\|/4\). This is a uniform bound, with no inverse of \(B\).

Moreover \(N_\epsilon\to0\) strongly. Since \(B\) is injective and self-adjoint, \(\operatorname{Ran}B^2\) is dense. First take \(u=B^2v\). In the term \(\epsilon Q_\epsilon B^2M Q_\epsilon u\), the right factor \(Q_\epsilon u\) stays bounded and the left factor has norm at most \(\epsilon\). In the term \(\epsilon Q_\epsilon M B^2Q_\epsilon u\), the vector following \(\epsilon Q_\epsilon\) converges, while \(\epsilon Q_\epsilon\to0\) strongly. The latter follows from its scalar functions and the zero spectral projection of \(B\). In the middle term, \(BQ_\epsilon u\) stays bounded and \(\|\epsilon Q_\epsilon B\|\le\sqrt\epsilon/2\). All three terms tend to zero. The uniform bound extends the convergence to \(\mathcal H\). Likewise \(J_\epsilon\to1\) strongly.

For \(u\in\operatorname{Dom}X^*\), the adjoint formula gives \(Bu\in\operatorname{Dom}A^*\). Domain preservation gives \(BQ_\epsilon u=Q_\epsilon Bu\in\operatorname{Dom}A^*\). The regularization premise therefore puts
\(J_\epsilon u=B(BQ_\epsilon u)\) in \(\operatorname{Dom}A\).
Testing NR.3 against \(\operatorname{Dom}X\) gives the adjoint-domain identity

\[
XJ_\epsilon u=J_\epsilon X^*u+N_\epsilon u
\longrightarrow X^*u,\qquad J_\epsilon u\longrightarrow u.
\tag{NR.4}
\]

For detail, NR.3 first says \(XJ_\epsilon v=J_\epsilon Xv+N_\epsilon v\) on \(\operatorname{Dom}X\). Both \(X\) and \(J_\epsilon\) are symmetric there, so the bounded extension \(N_\epsilon\) is skew-adjoint. Moving these operators in the inner product proves the same identity with \(X^*\) on its domain. Since \(J_\epsilon u\) is already in \(\operatorname{Dom}X\), it is exactly NR.4. Thus every vector in the adjoint graph is a limit of vectors in the original graph. Symmetry gives the reverse graph inclusion, proving \(\overline X=X^*\) and essential self-adjointness.

If \(\mathcal A_0\) is an \(A\)-graph core, it is also a core for \(\overline X\): \(X=BA+M/2\) is bounded for the \(A\)-graph norm. No self-adjoint extension of \(A\) was selected.

### 11AN.2. The completed sum on the minimal normal domain

**Theorem.** Let \(C\) be self-adjoint, and let \(\Gamma\) be a self-adjoint unitary with \(\Gamma C=-C\Gamma\). Let \(A\) be closed, densely defined and symmetric, with \(\Gamma\) preserving its domain and commuting with \(A\). Suppose that for both signs of one nonzero real \(\lambda\),

\[
\begin{gathered}
R_\lambda=(C+i\lambda)^{-1},\qquad
R_\lambda\operatorname{Dom}A\subset\operatorname{Dom}A,\qquad
R_\lambda\operatorname{Dom}A^*\subset\operatorname{Dom}A,\\
[A,R_\lambda]=K_\lambda,\qquad
A^*R_\lambda u=R_\lambda A^*u+K_\lambda u
\quad(u\in\operatorname{Dom}A^*),\qquad
\|K_\lambda\|<2 .
\end{gathered}
\tag{NR.5}
\]

The commutators in NR.5 are actual bounded operators. Then
\(S=C+\Gamma A\) is self-adjoint on exactly
\(\operatorname{Dom}C\cap\operatorname{Dom}A\).
The normal domain here is the minimal one specified by \(A\), rather than an assumed self-adjoint maximal normal domain. If a total grading is supplied for which both summands are odd, \(S\) retains that grading.

The hypotheses imply that \(\Gamma\) also preserves \(\operatorname{Dom}A^*\) and commutes there with \(A^*\), by taking adjoints of the domain identity for \(A\). Put \(B=-i\Gamma R_\lambda\). Anticommutation with \(C\) gives \(\Gamma R_\lambda=-R_\lambda^*\Gamma\), so \(B=B^*\). It is bounded and injective. All domain and regularization hypotheses of the lemma follow from NR.5 and the corresponding \(\Gamma\) identities. Its commutator is \(M=-i\Gamma K_\lambda\).

Let \(\overline X\) be the self-adjoint closure from the lemma. On \(\operatorname{Dom}A\),

\[
T=\Gamma AR_\lambda
=iX+\tfrac12\Gamma K_\lambda,\qquad
\operatorname{Dom}T=\{\xi:R_\lambda\xi\in\operatorname{Dom}A\}.
\tag{NR.6}
\]

The domain on the right is the natural domain of the left-hand product, not just its initial core. That natural \(T\) is closed, because \(A\) is closed and \(R_\lambda\) is bounded. It extends
\(\overline T_0=i\overline X+\Gamma K_\lambda/2\).
Indeed closure of the displayed identity on \(\operatorname{Dom}A\) gives this inclusion, including domains.

Self-adjointness of \(\overline X\) gives an onto inverse \((1+i\overline X)^{-1}\) of norm at most one, by spectral calculus. Therefore

\[
1+\overline T_0
=\left[1+\tfrac12\Gamma K_\lambda(1+i\overline X)^{-1}\right]
(1+i\overline X)
\tag{NR.7}
\]

is onto and invertible. The first factor has its norm-convergent geometric inverse because \(\|K_\lambda\|/2<1\). In particular its inverse has norm at most \((1-\|K_\lambda\|/2)^{-1}\).

The natural \(1+T\) is injective. If \((1+T)\xi=0\), then
\(\psi=R_\lambda\xi\) lies in \(\operatorname{Dom}C\cap\operatorname{Dom}A\), and \((S+i\lambda)\psi=0\). The sum is symmetric on this intersection: its \(C\) part is symmetric, while \(\Gamma A\) is symmetric because the bounded self-adjoint \(\Gamma\) commutes with \(A\) on its domain. Its imaginary inner product therefore gives \(\lambda\|\psi\|^2=0\), so \(\psi=0\) and then \(\xi=0\).

An injective extension of an onto operator has the same domain. For any \(\xi\in\operatorname{Dom}T\), choose \(\eta\in\operatorname{Dom}\overline T_0\) with
\((1+\overline T_0)\eta=(1+T)\xi\). Subtracting and using injectivity gives \(\eta=\xi\). Consequently \(T=\overline T_0\), and

\[
(S+i\lambda)^{-1}
=R_\lambda(1+i\overline X+\tfrac12\Gamma K_\lambda)^{-1},\qquad
\operatorname{Dom}S=\operatorname{Dom}C\cap\operatorname{Dom}A.
\tag{NR.8}
\]

Repeating for \(-\lambda\) gives both onto resolvents. The intersection is dense: it contains \(R_\lambda\operatorname{Dom}A\), whose closure contains the dense range of \(R_\lambda\). A densely defined symmetric operator with these two onto ranges is self-adjoint. To see the domain assertion directly, for \(u\in\operatorname{Dom}S^*\) choose \(v\in\operatorname{Dom}S\) with
\((S-i\lambda)v=(S^*-i\lambda)u\). Then \(u-v\) lies in \(\ker(S^*-i\lambda)\), which is the orthogonal complement of the onto range of \(S+i\lambda\), and hence is zero. Thus \(S^*=S\) on the displayed domain.

The theorem also supplies an actual joint graph core. If \(\mathcal A_0\) is an \(A\)-graph core, take \(R_\lambda\mathcal A_0\). For \(\psi\in\operatorname{Dom}S\), set \(\xi=(C+i\lambda)\psi\in\operatorname{Dom}\overline X\), using the proved equality of natural domains in NR.6. Approximate \(\xi\) in the \(\overline X\)-graph by \(\xi_n\in\mathcal A_0\). With \(\psi_n=R_\lambda\xi_n\),

\[
C\psi_n=(1-i\lambda R_\lambda)\xi_n,\qquad
(S+i\lambda)\psi_n=(1+i\overline X+\tfrac12\Gamma K_\lambda)\xi_n,
\qquad A\psi_n=\Gamma(S\psi_n-C\psi_n).
\tag{NR.9}
\]

Every right-hand side converges to its value at \(\psi\). Thus the approximation is in the \(C\)-, \(A\)- and \(S\)-graphs simultaneously. It does not require the separately non-self-adjoint normal operator to be replaced by a self-adjoint extension.

### 11AN.3. A complete interval example

Let \(I=(0,1)\), \(\mathcal H=L^2(I,\mathbb C^2)\), and use the usual Pauli matrices \(\sigma_1,\sigma_2,\sigma_3\), with \(\Gamma=\sigma_3\). Define

\[
h(x)=x^2(1-x)^2,\qquad \theta(x)=h(x)^{-1},\qquad
C=\sigma_1\theta,\qquad
A=-i\,d/dx,\quad\operatorname{Dom}A=H^1_0(I,\mathbb C^2).
\tag{NR.10}
\]

Here \(H^1\) consists of \(L^2\) functions with weak derivative in \(L^2\), and \(H^1_0\) is the closure of compactly supported smooth functions in that graph norm. The multiplication operator \(C\) has domain \(\{u:\theta u\in L^2\}\) and is self-adjoint. Its two resolvents are bounded pointwise matrix inverses, have dense ranges, and are adjoints of each other; or diagonalize the constant \(\sigma_1\) and use the real multiplication functions \(\theta,-\theta\). These descriptions prove the full multiplication domain.

For clarity, all normal boundary facts used here have elementary proofs. A function with weak derivative \(v\in L^2\) is almost everywhere \(c+\int_{1/2}^xv(t)\,dt\): subtract this primitive, test against derivatives of compactly supported test functions, and note that every test function of integral zero is the derivative of such a test function. The remaining distribution is constant. Thus \(H^1\) functions have absolutely continuous representatives and endpoint values, with
\(\|u\|_\infty\le\|u\|_2+\|u'\|_2\); choose a point at which \(|u|\le\|u\|_2\) and integrate the derivative. Integration by parts then identifies \(A^*=-i\,d/dx\) on all of \(H^1(I,\mathbb C^2)\).

The minimal domain is exactly the \(H^1\) functions vanishing at both endpoints. One direction follows from the continuous trace bound. For the converse cut off a zero-trace function within distance \(2\delta\) of the endpoints, using cutoffs with derivative at most a fixed constant divided by \(\delta\). Near zero,
\(|u(x)|^2\le x\int_0^x|u'(t)|^2dt\);
hence the squared norm of the cutoff-derivative term is bounded by a fixed constant times \(\int_0^{2\delta}|u'|^2\), and tends to zero. The analogous estimate holds near one. The other derivative and function tails tend to zero by integrability. Convolution inside the remaining compact interval approximates in \(H^1\) by compactly supported smooth functions. The convolution convergence follows from translation continuity in \(L^2\), proved first for compactly supported continuous functions and then by their density. This proves the stated minimal domain and its closedness.

This \(A\) is not self-adjoint. For every constant vector \(e\), \(e^x e\) and \(e^{-x}e\) are maximal-domain solutions of \(A^*u=-iu\) and \(A^*u=iu\), respectively, with nonzero boundary values. Neither belongs to the minimal domain.

The coefficient resolvents, however, regularize that domain:

\[
\begin{gathered}
R_\lambda=h\sigma_1(1+i\lambda h\sigma_1)^{-1},\qquad
\|R_\lambda(x)\|=\frac{h(x)}{\sqrt{1+\lambda^2h(x)^2}},\\
R_\lambda'=h'\sigma_1(1+i\lambda h\sigma_1)^{-2},\qquad
\|R_\lambda'(x)\|=\frac{|h'(x)|}{1+\lambda^2h(x)^2},\\
K_\lambda=-iR_\lambda',\qquad
\|K_\lambda\|\le\frac{\sqrt3}{9}<2,\qquad
\|K_\lambda\|\le 2|\lambda|^{-1/2}.
\end{gathered}
\tag{NR.11}
\]

Differentiate the bounded rational expression to obtain its actual derivative. Its derivative is bounded, its endpoint values are zero, and its derivative endpoint values are zero as well. If \(u\in H^1\), the product has weak derivative \(R_\lambda'u+R_\lambda u'\in L^2\) and zero traces. It therefore belongs to \(H^1_0\). This proves both domain conditions and the maximal commutator identity in NR.5, without an appeal to fibre indices.

The first norm constant is exact. Writing \(y=1-2x\) gives
\(|h'|=|y|(1-y^2)/2\), whose maximum on \([-1,1]\) is \(\sqrt3/9\), attained at \(|y|=1/\sqrt3\). For the second bound, \(|h'|\le2\sqrt h\); put \(t=|\lambda|h\) and use \(\sqrt t/(1+t^2)\le1\). Thus the commutators also decay in norm as \(|\lambda|\to\infty\). Finally \(B^2=h^2/(1+\lambda^2h^2)\) is positive and injective, as required in the lemma.

The theorem now proves that the specified first-order expression
\(S=\sigma_1\theta-i\sigma_3\,d/dx\) is self-adjoint on
\(\{u\in H^1_0:\theta u\in L^2\}\).
It also has compact resolvent, as the following direct argument shows.

On compactly supported smooth vectors its exact energy is

\[
\begin{gathered}
\|Su\|^2=\|u'\|^2+\|\theta u\|^2+
\int_I\theta'(x)\langle u(x),\sigma_2u(x)\rangle\,dx,\\
|\theta'|=|h'|\theta^2,\qquad \rho=\sqrt3/9<1,\\
\|u'\|^2+(1-\rho)\|\theta u\|^2
\le\|Su\|^2
\le\|u'\|^2+(1+\rho)\|\theta u\|^2 .
\end{gathered}
\tag{NR.12}
\]

The sign follows from \(\sigma_3\sigma_1=i\sigma_2\) and
\([A,C]=-i\sigma_1\theta'\). The cross term is a form, bounded in the displayed intersection norm. Compactly supported smooth vectors are dense in that norm. To prove this stronger assertion, use the preceding endpoint cutoffs. The potential and derivative tails tend to zero. For the new cutoff-derivative term, \(h(x)\le x^2\) near zero gives
\(\delta^{-2}\int_0^{2\delta}|u|^2\le16\delta^2\int_0^{2\delta}|\theta u|^2\to0\),
up to the fixed cutoff constant; the other endpoint is identical. On the remaining compact interval \(\theta\) is bounded, so ordinary \(H^1\) convolution suffices. Thus NR.12 extends to the entire completed intersection domain.

In particular the \(S\)-graph norm controls the \(H^1\) norm. The inclusion \(H^1(I,\mathbb C^2)\to L^2(I,\mathbb C^2)\) is compact by an explicit finite-rank approximation: partition into \(n\) equal intervals and replace a vector by its average on each. The fundamental theorem of calculus and Cauchy–Schwarz give \(\|u-P_nu\|_2\le n^{-1}\|u'\|_2\). Each \(P_n\) has rank \(2n\); their norm errors on the \(H^1\) unit ball tend to zero. By NR.8 the resolvent maps the Hilbert-space unit ball into a bounded \(S\)-graph ball, so this inclusion proves its compactness. No assertion about the graph class of a foliation is inferred from this interval example.

### 11AN.4. Application to the confining graph construction

Smooth orbit compacts and the completed oscillator cross form gives the completed oscillator cross forms. A completed form-domain Dirac sum, SD.5–SD.8, proves bounded actual oscillator-resolvent normal derivatives with norm tending to zero. In a differentiable physical realization these supply small commutators \(K_\lambda\), provided the stated maximal distributional identity is verified. The present theorem then offers a second normal-domain route: prove \(R_\lambda\operatorname{Dom}A^*\subset\operatorname{Dom}A\) for the specified minimal core. One need not first prove \(A=A^*\).

The interval calculation shows why this distinction matters: its normal operator has nonzero deficiency spaces, yet the confining coefficient produces the self-adjoint sum on an explicit completed domain. It does not prove the domain-regularization premise for the intrinsic varying field or its differentiable inverse-module representation. These are actual operator statements to establish in that realization. Nor does it prove a common groupoid phase-compatible weight, localized graph compactness or the original Bott \(+1\) comparison. The original full inverse Spin-c factor and final right Clifford action must be retained in any use of this route.

![Minimal normal domains and coefficient graph regularization](../figures/normal-boundary-regularization.png)

**Figure 11AN.1.** NR.1–NR.12. The resolvent-domain arrow is an explicit premise in the general theorem and is proved in the interval example. The coefficient cutoffs approximate the full adjoint graph with commutator bound 5/4. At the interval midpoint and lambda=2, B^2=1/260 and the cutoff with epsilon=1/260 is exactly 1/2. The graph curves are numerical samples of the displayed formulas; the domain, extremum and compactness statements have full proofs above. The example supplies no groupoid class comparison. Human context: [Connes, A survey of foliations and operator algebras](https://alainconnes.org/wp-content/uploads/foliationsfine.pdf#page=29), end of Section 8. [Vector figure](../figures/normal-boundary-regularization.svg) · [Generator](../reproduction/normal-boundary-regularization/draw_normal_boundary.py) · [Exact finite checks](../reproduction/normal-boundary-regularization/NORMAL-BOUNDARY-CHECKS.json) · [Reproduction](../reproduction/normal-boundary-regularization/REPRODUCTION.md) · [Component terms](../reproduction/normal-boundary-regularization/COMPONENT-TERMS.md).

## 11AO. Normal domains and scalar compactness on the actual regular graph

The regular coordinates of Normalized proper fields and actual graph columns, RG.1–RG.13, admit an actual normal differential. Finite-rank operators with compact arrow support regularize its maximal domain into its minimal domain. This yields a concrete sufficient test for A Dirac sum when the normal operator has a boundary. A separate argument below turns local module compactness of coefficient resolvents into scalar localized compactness of a completed sum.

These assertions concern the literal regular module and its geometric normal differential. They do not assert that the unitary absorption in RG.9–RG.12 preserves normal derivatives, or that the transferred normalized phase has the resolvents required below. In particular they do not identify a newly chosen auxiliary weight with the original normalized class.

### 11AO.1. The physical normal differential in regular coordinates

Let \(T\) be the finite good complete transversal of the original foliation, with its positive normal metric and full supplied inverse Spin-c bundle \(S_{\mathrm{inv}}\). Write \(G_T\) for its actual étale holonomy groupoid and \(R=C_0(T)\). The actual range-regular module and its integrated Hilbert space are

\[
\begin{gathered}
\mathcal E=\mathcal H_{G_T}^{\infty},\qquad
\langle\xi,\eta\rangle(x)=\sum_{rg=x}\sum_n\overline{\xi(g,n)}\eta(g,n),\\
\mathcal H=\mathcal E\otimes_R L^2(T,\mu;S_{\mathrm{inv}})
=L^2(G_T,\nu_r;r^*S_{\mathrm{inv}})\otimes\ell^2(\mathbb N).
\end{gathered}
\tag{RGN.1}
\]

Here \(\mu\) is the positive normal volume and \(\nu_r\) is its range-counting integral. The tensor identification is the elementary-section unitary proved in RG.13. Extra auxiliary gradings are treated in direct sum, without moving the final right Clifford block.

In this holonomy-invariant metric setting \(G_T\) is Hausdorff, by Regular fields, unit-space balancing and the graph Dirac interface, Section 11M.3. Its complete proof uses first-jet determination of local holonomy isometries and separates the actual arrow germs by their frame images. Thus Hausdorffness here is a proved property of the metric input, rather than an extra assumption on arbitrary holonomy groupoids.

Use the smooth compact bisection sections with finitely many copy coordinates as the auxiliary core \(\mathcal E_c\). The support of each such section is closed inside its chosen larger bisection. Finite sums are allowed. This is a dense module core: compact bisection sections generate the counting module, smooth unit cutoffs approximate continuous compactly supported coefficients, and finite copy truncations are dense. Tensoring them with smooth unit spinors gives a dense Hilbert-space core \(\mathcal H_c\).

On a bisection \(B\), use \(r:B\to r(B)\) to differentiate in normal coordinates. Choose a metric Clifford connection on the full finite inverse Spin-c bundle over \(T\); its pullback acts on every row with the same range unit. In a local orthonormal normal frame \(e_j\), put

\[
A_0=-i\sum_{j=1}^{q}\mathsf c_j\nabla_{e_j}
-\frac i2\sum_j\left[\nabla_{e_j}\mathsf c_j+
(\operatorname{div}_{\mu}e_j)\mathsf c_j\right],\qquad
A_{\min}=\overline{A_0|_{\mathcal H_c}},\qquad
A_{\max}=A_{\min}^* .
\tag{RGN.2}
\]

The physical \(\mathsf c_j\), grading shift, complete inverse lift and final right Clifford order are those fixed in Sections 11A and 11E. Their principal symbol is the Hermitian Clifford symbol for the supplied normal metric. No determinant line replaces \(S_{\mathrm{inv}}\).

The displayed correction is explicit: if \(D=-i\sum_j\mathsf c_j\nabla_{e_j}\), metric integration by parts gives \(D^\dagger=D-i\sum_j[\nabla_{e_j}\mathsf c_j+(\operatorname{div}_{\mu}e_j)\mathsf c_j]\), and \(A_0=(D+D^\dagger)/2\). Formula RGN.2 is independent of the bisection description of a graph point: both descriptions pull back the same normal connection by the actual map \(r\). Smooth changes of normal frame give the usual connection rule, and symmetrization uses the same intrinsic normal volume. In a density-normalized local spinor trivialization it is a first-order system \(D_T\) with Hermitian elliptic principal symbol. Integration by parts on compact bisection supports, followed by the range-counting sum, proves symmetry. Thus \(A_0\) is closable: if \(u_n\to0\) and \(A_0u_n\to v\), symmetry tested against the dense core gives \(v=0\).

The adjoint has the full local distributional characterization. A vector \(u\in\mathcal H\) belongs to \(\operatorname{Dom}A_{\max}\) exactly when the distribution obtained by RGN.2 in its bisection rows is a single vector of \(\mathcal H\). Testing compact bisection spinors proves necessity and identifies that vector. Conversely those distributional identities give \(\langle A_0v,u\rangle=\langle v,Du\rangle\) for every core \(v\), hence the adjoint-domain condition. This description supplies no equality between the minimal and maximal domains.

### 11AO.2. The finite-dimensional normal estimate

We need one local analytic fact, including its maximal-domain use. If \(D_T\) is the preceding smooth finite-dimensional Dirac system and \(w,D_Tw\) are locally in \(L^2\), then \(w\in H^1_{\mathrm{loc}}\). On a compact buffered coordinate test \(K\Subset\Omega\),

\[
\|w\|_{H^1(K)}
\le C_K\bigl(\|D_Tw\|_{L^2(\Omega)}+\|w\|_{L^2(\Omega)}\bigr),
\tag{RGN.3}
\]

with a slightly smaller buffered neighborhood if necessary.

Here is the complete local proof. Normalize the density by its smooth positive square root, adding only smooth bounded zeroth-order coefficients on a compact chart. Freeze the principal Clifford coefficients at its center. For the constant system \(D_f\), integration by parts and the Clifford anticommutation relations give
\(\|D_fv\|_2^2=\int g_f^{jk}\langle\partial_jv,\partial_kv\rangle\ge c^2\|\nabla v\|_2^2\)
for compactly supported smooth \(v\), with \(c>0\). The antisymmetric Clifford cross terms cancel because the coordinate derivatives commute. On a small enough ball, the difference of principal coefficients contributes at most \(c\|\nabla v\|_2/2\); the lower-order coefficients contribute at most \(C\|v\|_2\). Absorb the first term to get the smooth-vector estimate. A finite smooth partition proves its buffered version, since \([D_T,\chi]\) is bounded multiplication.

To justify the estimate for distributions, localize first by a smooth cutoff with support closed inside the chart, and extend the localized vector by zero. Mollify there. For a principal coefficient \(b_j\), the commutator of \(b_j\partial_j\) with convolution by \(\rho_\epsilon\) has the two kernels

\[
[b_j(x)-b_j(y)]\,\partial_{x_j}\rho_\epsilon(x-y),
\qquad
\rho_\epsilon(x-y)(\partial_jb_j)(y).
\]

The first has both absolute Schur integrals bounded uniformly: use the Lipschitz bound on \(b_j\) and
\(\int |z|\,|\partial_j\rho_\epsilon(z)|\,dz=\int |z|\,|\partial_j\rho(z)|\,dz\).
The second is bounded by \(\|\partial_jb_j\|_\infty\); a zeroth-order commutator is bounded by twice its coefficient norm. Thus the mollified Dirac images stay bounded in \(L^2\). The smooth estimate bounds all mollified coordinate derivatives. Their distributional limits define bounded functionals on \(L^2\) tests, so the Riesz–Frechet theorem, Theorem 2.3, represents those derivatives by \(L^2\) functions. This proves \(H^1\) regularity and RGN.3. Translation continuity gives convergence of the original mollifications in \(L^2\), first for continuous compact functions and then by density. No scalar self-adjoint boundary realization has entered this proof.

### 11AO.3. Finite graph coefficients regularize the maximal domain

For \(\eta\in\mathcal E_c\) and \(u\in\operatorname{Dom}A_{\max}\), the contraction
\(w_\eta=\langle\eta,u\rangle_R\) is a spinor over \(T\). It is a finite range-counting contraction on the support of \(\eta\), rather than an inner product in a renamed integrated Hilbert space. Test its distributional derivative against a compact unit spinor. The adjoint identity of RGN.2 and the auxiliary scalar Leibniz rule give

\[
D_Tw_\eta
=\langle\eta,A_{\max}u\rangle_R
-i\sum_j\mathsf c_j\langle\partial_{e_j}\eta,u\rangle_R .
\tag{RGN.4}
\]

The connection and density terms of \(D_T\) act on the unit spinor and therefore already occur in its left-hand side. The auxiliary \(\eta\) is spin-free. Counting Cauchy–Schwarz proves that every right-hand term is in \(L^2\), with bounds given by the module norms of \(\eta\) and its first derivatives. These norms are finite on its compact bisection support. RGN.3 therefore gives a bounded map from the \(A_{\max}\)-graph norm to \(H^1\) on any compact unit test containing that support.

Let \(T_0=\sum_{\ell=1}^m\theta_{\xi_\ell,\eta_\ell}\), with \(\xi_\ell,\eta_\ell\in\mathcal E_c\), act trivially on the full unit spinor. Its actual action and normal commutator are

\[
\begin{gathered}
T_0u=\sum_\ell\xi_\ell w_{\eta_\ell},\qquad
\partial_jT_0
=\sum_\ell\left(\theta_{\partial_j\xi_\ell,\eta_\ell}
+\theta_{\xi_\ell,\partial_j\eta_\ell}\right),\\
T_0\operatorname{Dom}A_{\max}\subset\operatorname{Dom}A_{\min},\qquad
A_{\min}T_0u=T_0A_{\max}u-i\sum_j\mathsf c_j(\partial_jT_0)u .
\end{gathered}
\tag{RGN.5}
\]

Indeed each \(w_{\eta_\ell}\) is \(H^1\) locally. Cover the compact unit support of \(\xi_\ell\) by finitely many buffered spinor charts, multiply by an inner partition and approximate the resulting \(H^1\) coefficients by compactly supported smooth coefficients in those charts. Their products with \(\xi_\ell\) belong to \(\mathcal H_c\). The scalar \(H^1\) convergence and the bounded first derivatives of \(\xi_\ell\) give convergence in the \(A_0\)-graph. This proves minimal-domain membership. The product rule, first on smooth coefficients and then by that graph convergence, proves the stated commutator formula.

The compact arrow support is essential in this argument. A finite fibre-rank assertion, or an operator with only compact range-unit support and arbitrary return-arrow tails, is not this hypothesis.

Define \(\mathcal K^1_c(\mathcal E)\) to be the completion of this finite core algebra in
\(\|T\|+\sum_j\|\partial_jT\|\), with the graph closures of its actual normal derivatives. The transverse components of the finite good transversal have fixed smooth frames; equivalently one may use their finitely many localized frame seminorms. Products and adjoints remain in the algebra by the metric Leibniz rule. For any \(T\) in this completion, RGN.5 passes to the limit:

\[
T\operatorname{Dom}A_{\max}\subset\operatorname{Dom}A_{\min},
\qquad
A_{\min}Tu=TA_{\max}u+K_Tu,\qquad
K_T=-i\sum_j\mathsf c_j\partial_jT .
\tag{RGN.6}
\]

To see the domain step, choose simultaneous graph-norm approximants \(T_n\) in the finite core algebra. For fixed \(u\in\operatorname{Dom}A_{\max}\), \(T_nu\to Tu\) and
\(A_{\min}T_nu\to TA_{\max}u+K_Tu\).
Closedness of \(A_{\min}\) proves both conclusions. The module operators integrate boundedly in RGN.1, so the convergence is also genuine Hilbert-space convergence. If \(T\) preserves an auxiliary grading, so do its limits. No completed fibre preimage, global normal flow or eigenvector selection was assumed.

### 11AO.4. The graph ideal contains actual smooth regularizers

Choose a countable generating family \(\xi_j\) in \(\mathcal E_c\), including its derivative words, and finite bounds \(d_{m,j}\) on their normal jets through order \(m\). Put

\[
c_j=\frac{\epsilon^2 2^{-j}}
{1+\max_{m\le j+1}d_{m,j}^2},
\qquad
h=\sum_{j\ge1}c_j\theta_{\xi_j,\xi_j},\qquad 0<\epsilon\le\tfrac12 .
\tag{RGN.7}
\]

Use bounds nondecreasing in \(m\), including the zero-order norm. Every fixed derivative word of the sum converges in compact norm: its Leibniz bound is at most \(2^m d_{m,j}^2\), and for \(j\ge m\) the weighted bound is at most \(2^m\epsilon^2 2^{-j}\). There are only finitely many exceptional terms. Finite core sums therefore put \(h\) in the simultaneous normal graph ideal, with all stated derivative words. The rank-one generator sum is strictly positive by the approximate-unit argument proved in A confining oscillator over a varying Hilbert field, WF.1. It has norm at most \(\epsilon^2\).

Consequently \(\Theta=h^{-1}\) is positive self-adjoint regular on its completed module range, and

\[
(\Theta+i\lambda)^{-1}
=h(1+i\lambda h)^{-1}\in\mathcal K^1_c(\mathcal E)
\qquad(\lambda\ne0).
\tag{RGN.8}
\]

Regularity and the range domain are proved for this reciprocal in Smooth compact inverses and completed normal forms, CW.5. Here graph-ideal membership can be checked directly. For fixed \(\lambda\), expand the inverse around a sufficiently large positive scalar multiple of \(1+\lambda^2h^2\), whose positive spectrum has a gap at zero. The geometric series and each first commutator series converge in norm, as in NR.2. Approximate \(h\) in its simultaneous graph norm by finite core sums. Multiplication, the inverse identity and the uniform spectral gap identify their graph limits. Polynomial approximants with zero constant term are finite core operators. Thus the displayed compact resolvent is in the graph ideal, with its actual normal derivatives.

RGN.6 proves that these resolvents map the maximal normal domain into the minimal one. This is a concrete construction on the actual arrow-and-copy module. It does not assert that this chosen reciprocal is the auxiliary generator representing RG.12's normalized phase.

### 11AO.5. The actual normal convolution commutators

On RGN.1 the scalar graph representation uses the complete inverse transport:

\[
(\pi(f)u)(g)
=\sum_{rh=rg}f(h)\,U_h^{\mathrm{inv}}u(h^{-1}g),
\qquad
U_h^{\mathrm{inv}}=z_h^{-1}\rho_t(s_h).
\tag{RGN.9}
\]

This is the exact range-regular convolution of RG.13, with its reduced norm and prescribed Clifford coefficient. For a smooth compact bisection kernel \(f\), differentiating in a range-normal direction gives cancellation of the two principal normal derivatives. The actual holonomy map is an isometry, and its entire inverse Spin-c transport intertwines the corresponding physical Clifford symbols. The remaining coefficients are the first derivatives of \(f\), the derivatives of \(U_h^{\mathrm{inv}}\), and the smooth difference of the two pulled-back connection/density terms. They depend on \(h\), not on an unrestricted return label \(g\).

Their matrix kernel \(k_f(h)\) is smooth and compactly supported. Covering its support by finitely many bisections bounds both counting sums. Counting Cauchy–Schwarz, followed by invariance of normal volume under the holonomy isometry, gives the actual operator estimate

\[
\|[A_0,\pi(f)]\|
\le
\left[
\sup_x\sum_{rh=x}\|k_f(h)\|\,
\sup_y\sum_{sh=y}\|k_f(h)\|
\right]^{1/2}<\infty .
\tag{RGN.10}
\]

For completeness, apply \(\|\sum_h k_hv_h\|^2\le(\sum_h\|k_h\|)(\sum_h\|k_h\|\|v_h\|^2)\) in each range row and integrate. In the second sum change variables from \(g\) to \(h^{-1}g\); counting and the preserved normal volume leave the source sum in RGN.10. Finite sums of bisection kernels follow by addition.

Convolution maps \(\mathcal H_c\) to itself: products of its compact arrow supports are compact, and locally they have smooth bisection coefficients and finite copy support. RGN.10 and closure prove preservation of \(\operatorname{Dom}A_{\min}\), with its stated bounded commutator. Apply the same argument to \(f^*\) and use adjoint-domain testing to get preservation of \(\operatorname{Dom}A_{\max}\) and its identical maximal commutator. These are ordinary Hilbert-space assertions about the physical normal differential. They do not estimate the commutator of an unproved transferred auxiliary generator.

### 11AO.6. Local module compacts give scalar localized compacts

Suppose an actual self-adjoint coefficient generator \(C\) on \(\mathcal E\), integrated in RGN.1, has its stated bounded adjointable resolvents \(R_\lambda=(C+i\lambda)^{-1}\). Suppose a closed normal realization \(A\), lying between \(A_{\min}\) and \(A_{\max}\), and the actual auxiliary parity \(\Gamma\) give the completed sum \(S=C+\Gamma A\), with the onto-resolvent proof of Section 11AM or 11AN. In particular its exact domain is \(\operatorname{Dom}C\cap\operatorname{Dom}A\). Assume local module compactness:

\[
\chi R_\lambda\in\mathcal K(\mathcal E)
\quad\text{for every }\chi\in C_c^\infty(T).
\tag{RGN.11}
\]

Then \(\chi(S+i\lambda)^{-1}\) is compact on the scalar Hilbert space \(\mathcal H\). This conclusion requires only the local condition RGN.11, not a global compact vacuum projection over a noncompact transversal.

Here is the proof. The constructed inverse has the form
\((S+i\lambda)^{-1}=R_\lambda L_\lambda^{-1}\), with \(L_\lambda^{-1}\) bounded. Its image has uniformly bounded \(A\)-graph norm on the Hilbert-space unit ball. Indeed
\(C(S+i\lambda)^{-1}=(1-i\lambda R_\lambda)L_\lambda^{-1}\) is bounded, while
\(A(S+i\lambda)^{-1}=\Gamma[1-i\lambda(S+i\lambda)^{-1}-C(S+i\lambda)^{-1}]\).
Multiplication by a fixed smooth unit cutoff has bounded normal commutator on the maximal domain. Thus the localized image has bounded \(A_{\max}\)-graph norm. No preservation of an arbitrarily chosen intermediate boundary realization is needed.

Choose a bounded compact approximate identity and approximate each of its operators in norm by a finite core operator \(e_n=\sum_\ell\theta_{\xi_{\ell,n},\eta_{\ell,n}}\). This gives a uniformly bounded finite core sequence with
\((1-e_n)\chi R_\lambda\to0\) in module operator norm. Such finite core approximants exist because \(\mathcal E_c\) is dense in \(\mathcal E\), hence its rank-one operators are norm dense in \(\mathcal K(\mathcal E)\). Approximate a contractive compact approximate identity to error \(1/n\); its strict action on the fixed compact \(\chi R_\lambda\) gives the claimed convergence.

For fixed \(n\), apply RGN.4 and RGN.3 to the contractions
\(w_{\eta_{\ell,n}}=\langle\eta_{\ell,n},\chi(S+i\lambda)^{-1}f\rangle_R\).
For \(\|f\|\le1\), these coefficients are bounded in \(H^1\) on each of the finitely many compact unit charts needed for the output vectors \(\xi_{\ell,n}\). The coefficient sets are precompact in \(L^2\) there. One elementary proof of this local Rellich statement is to extend a buffered compactly supported chart vector to a fixed cube, partition into \(n^q\) cubes of side \(\ell\), and use coordinate averages \(P_n\). Each coordinate average is an orthogonal projection; the commuting projections give an orthogonal telescoping decomposition of \(1-P_n\). The one-dimensional fundamental theorem and Cauchy–Schwarz bound each summand by \(\ell\|\partial_jv\|_2\). Therefore

\[
\operatorname{rank}P_n=r\,n^q,\qquad
\|v-P_nv\|_2\le\ell\|\nabla v\|_2 .
\tag{RGN.12}
\]

Here \(r\) is the finite spinor rank, and the fixed cube side is \(n\ell\). The norm errors tend to zero on the \(H^1\) unit ball. Finite partitions and smooth density/spinor trivializations preserve this conclusion on the compact unit test.

Multiplication by each fixed \(\xi_{\ell,n}\) is a bounded map from these unit \(L^2\) coefficients to \(\mathcal H\). Finite sums of their compact coefficient images prove that
\(e_n\chi(S+i\lambda)^{-1}\) is scalar compact. Finally

\[
\|\chi(S+i\lambda)^{-1}-e_n\chi(S+i\lambda)^{-1}\|
\le\|(1-e_n)\chi R_\lambda\|\,\|L_\lambda^{-1}\|
\longrightarrow0 .
\tag{RGN.13}
\]

Thus the localized inverse is scalar compact. In particular \(\pi(f)(S+i\lambda)^{-1}\) is compact for every compact graph kernel \(f\): choose \(\chi=1\) on its compact source-unit support, so \(\pi(f)=\pi(f)\chi\), and use boundedness of convolution. Self-adjointness then gives the usual localized square-resolvent compactness by bounded composition.

### 11AO.7. Exact use for the normalized graph class

RGN.1–RGN.6 construct the physical normal differential, identify its maximal distributional domain, and prove the finite-core graph-ideal regularization on RG.13's actual arrow coordinates. RGN.9–RGN.10 prove its ordinary convolution commutators with the full inverse Spin-c coefficient. RGN.11–RGN.13 prove scalar localized compactness for an actual completed sum with locally module-compact coefficient resolvents.

The unresolved transfer must now be checked against concrete operator ideals and domains. Module compactness alone is preserved by RG's adjointable unitaries. The simultaneous first-normal-derivative graph ideal is not automatically preserved by an ordinary module unitary. The intrinsic CW/JD field and its compact derivatives do not establish that the resolvents of a generator carrying the transferred normalized phase lie in this regular graph ideal, or satisfy another sufficient sum-domain route.

Consequently the differentiable inverse-module/phase transfer, a common groupoid phase-compatible generator and the original Bott \(+1\) comparison remain to be constructed. No all-label product-domain or one-sided reciprocal derivative is imposed. The graph-ideal criterion is one sufficient route, with local compactness treated at its actual local scope. Choosing the smooth positive \(h\) of RGN.7 by itself determines no normalized auxiliary KK class.

![Actual regular normal domains and scalar localized compactness](../figures/regular-normal-domains.png)

**Figure 11AO.1.** RGN.1–RGN.13. The top panels retain actual arrow/copy coordinates and the finite-core maximal-to-minimal domain test. The finite rank-one sample has contraction 17/5, contraction derivative 1 and operator derivative on u equal to (-2,3,0). The two-component unit-square sample v=(x,2y) shows exact cell means: column labels give x and left row labels give 2y; colours encode the two mean components. At n=4 the rank is 32 and squared error 5/192. This finite coefficient example does not change the full physical inverse Spin-c rank. The lower-right proof uses local module compactness and the already constructed self-adjoint sum, rather than fibre compactness alone. The normalized auxiliary generator and its differentiable transfer and original Bott +1 remain unproved. Human context: [Connes, A survey of foliations and operator algebras](https://alainconnes.org/wp-content/uploads/foliationsfine.pdf#page=29), end of Section 8. [Vector figure](../figures/regular-normal-domains.svg) · [Generator](../reproduction/regular-normal-domains/draw_regular_normal.py) · [Exact finite checks](../reproduction/regular-normal-domains/REGULAR-NORMAL-CHECKS.json) · [Reproduction](../reproduction/regular-normal-domains/REPRODUCTION.md) · [Component terms](../reproduction/regular-normal-domains/COMPONENT-TERMS.md).

## 11AP. One normally differentiable weight with completed groupoid domains

Smooth orbit compacts and the completed oscillator cross form supplies a positive compact \(h=SS^*\) on the actual intrinsic field, its genuine-arrow order comparisons, and the compact first normal derivatives of \(h^{1/2}\). A fixed power below \(1/2\) gives completed arrow domains, but its normal derivative was not established by that argument. We now construct one weight having both properties. Its defining function is a positive resolvent sum, rather than a fixed power.

The construction stays on that actual field. It does not change the original unit algebra, inverse Spin-c coefficient or final right Clifford order. In particular it supplies no differentiable absorption unitary or normalized auxiliary phase by itself.

### 11AP.1. A positive resolvent sum with compact normal derivatives

Work first before the compact-frame quotient, with the actual finite frame parallelism and the closed compact-operator derivatives \(\delta_i\). The input, proved in JD.1–JD.6, is

\[
h\in\mathcal K(E),\qquad h\ge0,\qquad\overline{hE}=E,
\qquad k=h^{1/2}\in\bigcap_i\operatorname{Dom}\delta_i,
\qquad\delta_i k\in\mathcal K(E).
\tag{OP.1}
\]

The compact-frame action fixes \(h\), and acts on the finite list of normal derivatives by its actual orthogonal changes of parallelism. Fix any \(\eta>0\). Since \(kE\) is dense, \(A_t=t(k+t)^{-1}\) tends strongly to zero as \(t\downarrow0\). For a rank-one compact, \(A_t\theta_{u,v}A_t=\theta_{A_tu,A_tv}\) tends to zero in norm. Finite sums and norm approximation therefore give \(\|A_t(\delta_i k)A_t\|\to0\). Choose positive \(t_n\), simultaneously for the finite derivative list, with

\[
t_n\le2^{-n}/4,\qquad
\max_i\|t_n^2(k+t_n)^{-1}(\delta_i k)(k+t_n)^{-1}\|
\le\eta2^{-n}\qquad(n\ge1).
\tag{OP.2}
\]

Each choice is possible by the just-proved norm limit. Put

\[
\psi(s)=\sum_{n\ge1}\frac{t_ns}{s+t_n},\qquad
\phi(x)=\psi(\sqrt{x}),\qquad
\kappa=\phi(h)=\psi(k).
\tag{OP.3}
\]

The series converges uniformly on \(s\ge0\), because its \(n\)-th term is at most \(t_n\). Thus \(\kappa\) is compact, positive, has norm at most \(1/4\), and has dense range: \(\psi(0)=0\) and \(\psi(s)>0\) for every \(s>0\). To verify the dense-range assertion directly, approximate a vector in \(kE\) by applying the bounded cutoff functions \(\psi(k)(\psi(k)+\epsilon)^{-1}\); on \(kE\) the convergence follows uniformly from the scalar functions, and density and contractivity extend it to all \(E\).

For a fixed \(t>0\), the inverse product rule of Smooth compact inverses and completed normal forms, CW.1, gives the actual closed derivative of \(tk(k+t)^{-1}\). Equations OP.2 then prove convergence in the simultaneous derivative graph norms:

\[
\delta_i\kappa
=\sum_{n\ge1}t_n^2(k+t_n)^{-1}(\delta_i k)(k+t_n)^{-1}
\in\mathcal K(E),\qquad
\|\delta_i\kappa\|\le\eta.
\tag{OP.4}
\]

This is first normal differentiability; arbitrary higher derivatives are not asserted. Every term is a function of the exactly compact-frame-invariant \(h\), so \(\kappa\) is exactly invariant too. The existing compact-frame quotient argument therefore descends it and its horizontal first derivatives to the original anchored intrinsic field.

There is an important additional scalar property. For \(s>0\),

\[
\frac{\psi(s)}s=\sum_{n\ge1}\frac{t_n}{s+t_n}\longrightarrow\infty
\quad(s\downarrow0),\qquad
g(x)=\frac{\sqrt{x}}{\phi(x)}\quad(x>0),\qquad g(0)=0.
\tag{OP.5}
\]

To prove divergence, retain any first \(N\) terms: each tends to one. Since \(N\) is arbitrary, the sum diverges. Consequently \(g\) is continuous and bounded on every compact interval \([0,L]\). This vanishing at zero supplies the completed module domain argument below. A constant multiple of \(\sqrt{x}\) would not have it.

### 11AP.2. The square has an explicit positive resolvent integral

For \(s,t>0\), define \(F_{s,t}(x)=x/[(\sqrt{x}+s)(\sqrt{x}+t)]\). Its exact integral is

\[
F_{s,t}(x)=\frac{s+t}{\pi}\int_0^\infty
\frac{x}{x+r}\frac{\sqrt r}{(r+s^2)(r+t^2)}\,dr
\qquad(x\ge0).
\tag{OP.6}
\]

Here is a complete elementary verification. For positive \(a,b\), partial fractions and the antiderivative of \(1/(u^2+a^2)\) give

\[
\int_0^\infty\frac{u^2}{(u^2+a^2)(u^2+b^2)}\,du
=\frac{\pi}{2(a+b)}.
\]

Taking the difference of this formula with \(b\) and with \(c\), then dividing by \(c^2-b^2\), gives

\[
\int_0^\infty\frac{u^2}{(u^2+a^2)(u^2+b^2)(u^2+c^2)}\,du
=\frac{\pi}{2(a+b)(a+c)(b+c)}.
\]

Equal parameters follow by dominated convergence. Substitute \(r=u^2\), \(a=\sqrt{x},b=s,c=t\) to obtain OP.6 for \(x>0\). The zero case is immediate. The integral converges at both endpoints; its density is \(O(\sqrt r)\) at zero and \(O(r^{-3/2})\) at infinity. Thus it also converges uniformly on bounded positive operator spectra and is a genuine norm functional-calculus identity.

For positive operators \(a\le b\), inverse order gives \(a(a+r)^{-1}\le b(b+r)^{-1}\). One may verify inverse order directly by conjugating by \((a+r)^{-1/2}\), whose conjugate of \(b+r\) is at least the identity, and then using its positive inverse. Formula OP.6, with its positive density, proves that \(F_{s,t}\) preserves operator order. Moreover for \(C\ge1\), \(Cx/(Cx+r)\le Cx/(x+r)\), so \(F_{s,t}(Ca)\le C F_{s,t}(a)\).

Squaring OP.3 gives the uniformly convergent positive double sum

\[
\phi(x)^2=\sum_{n,m\ge1}t_nt_mF_{t_n,t_m}(x).
\]

Each \(F_{s,t}\le1\), and \(\sum t_n<\infty\), which justifies the convergence and the interchange with bounded functional calculus. Therefore

\[
0\le b\le Ca,\ C\ge1
\quad\Longrightarrow\quad
\phi(b)^2\le C\phi(a)^2.
\tag{OP.7}
\]

No assertion that squaring an arbitrary operator-monotone function preserves operator order is used. OP.6 is the explicit positive integral for this particular square.

### 11AP.3. Completed inverse domains, with actual adjoints

Let \(a,b\) be positive bounded adjointable operators of dense range on a Hilbert module, with \(b\le Ca\), \(C\ge1\). They need not commute. For \(\epsilon>0\), put \(U_\epsilon=(\phi(a)+\epsilon)^{-1}b^{1/2}\). The order comparison gives

\[
\|U_\epsilon-U_\delta\|
\le\sqrt C\sup_{0\le x\le\|a\|}
\sqrt{x}\left|\frac1{\phi(x)+\epsilon}-\frac1{\phi(x)+\delta}\right|.
\tag{OP.8}
\]

This follows by multiplying \(b\le Ca\) on both sides by the self-adjoint difference of the two inverse functions and taking the norm of the resulting positive operator. The functions \(\sqrt{x}/(\phi(x)+\epsilon)\) converge uniformly to \(g(x)\): near zero use \(0\le g\\) and \(g(0)=0\); away from zero use the positive minimum of \(\phi\). Hence \(U_\epsilon\) converges in operator norm to an adjointable \(U\), with \(\|U\|\le\sqrt C\|g\|_\infty\) and \(\phi(a)U=b^{1/2}\).

Write \(\phi_N(x)=\sum_{n=1}^N t_n\sqrt{x}/(\sqrt{x}+t_n)\), and define the finite adjointable sum

\[
D_N=\sum_{n=1}^N t_n U(b^{1/2}+t_n)^{-1},
\qquad\phi(a)D_N=\phi_N(b).
\tag{OP.9}
\]

Since \(\phi_N(b)^2\le\phi(b)^2\le C\phi(a)^2\), the adjoint identity \(D_N^*\phi(a)=\phi_N(b)\) shows \(\|D_N^*\phi(a)v\|\le\sqrt C\|\phi(a)v\|\). Dense range of \(\phi(a)\) then gives \(\|D_N\|\le\sqrt C\). Both the operators and their adjoints converge strongly on the completed modules. Indeed

\[
D_N b^{1/2}=U\phi_N(b)\longrightarrow U\phi(b),\qquad
D_N^* a^{1/2}=\phi_N(b)g(a)\longrightarrow\phi(b)g(a)
\tag{OP.10}
\]

in operator norm. The two square-root ranges are dense; the uniform bounds extend convergence from these ranges to every vector. The limits satisfy the adjoint identity by taking inner-product limits, so the strong limit \(D\) is adjointable. Equation OP.9 gives \(\phi(a)D=\phi(b)\), and therefore

\[
D=\phi(a)^{-1}\phi(b)\in\mathcal L(E),\qquad\|D\|\le\sqrt C.
\quad
a\le C'b\ \Longrightarrow\
\operatorname{Ran}\phi(a)=\operatorname{Ran}\phi(b).
\tag{OP.11}
\]

For the last assertion apply the construction in the reverse direction. The two operators are inverses: multiplying their products by the injective \(\phi(a)\) or \(\phi(b)\) gives the corresponding identity. This proves equality of actual completed ranges with equivalent inverse graph norms. A fibrewise bounded inverse without the adjoint and section limits would not prove OP.11.

For an actual JD arrow take \(a=h_{r g}\) and \(b=L_g h_{s g}L_g^{-1}\). Its two original order comparisons now give covariance of the single domain \(\operatorname{Ran}\kappa\) and bounds \(\sqrt C,\sqrt{C'}\). On every compact arrow test the constants are uniform. The \(U_\epsilon\) limit in OP.8 is uniform there, because its scalar bound uses that same constant. For \(D_N\), OP.10 converges uniformly on dense square-root-generated sections; density and uniform boundedness give the actual continuous completed graph transports and their adjoints. Composition is precisely the original \(L\) action on these domains, so it satisfies the original groupoid identities. No replacement arrow action is introduced.

### 11AP.4. Displacements and normal forms use the same weight

For each chosen JD displacement section, \(\theta_{\xi,\xi}\le C_\xi h\). Apply this order inequality to the difference of the regularized inverse functions in OP.8. It proves a norm Cauchy sequence \((\kappa+\epsilon)^{-1}\xi\), whose limit is an actual module section \(u\), with \(\kappa u=\xi\) and

\[
\xi\in\operatorname{Ran}\kappa,\qquad
\|\kappa^{-1}\xi\|\le\sqrt{C_\xi}\|g\|_\infty.
\tag{OP.12}
\]

Compact arrow tests have finite covers by those original cutoff sections, so their graph sections are continuous with uniform local bounds. The same argument applies to every additional actual jet section included in JD.1's countable rank-one domination list; adding that list is allowed by its original diagonal compact-column construction, rather than a new assumption on the foliation.

Let \(\Theta=\kappa^{-1}\), the regular reciprocal proved by CW.5. OP.4 gives its completed weak normal form on \(\kappa E\):

\[
\mathfrak s_i(\kappa u,\kappa v)=-\langle u,(\delta_i\kappa)v\rangle,
\qquad
\|\mathfrak s_i(w,w)\|\le\epsilon\|\Theta w\|^2+C_\epsilon\|w\|^2
\quad(w\in\operatorname{Dom}\Theta).
\tag{OP.13}
\]

To prove the second statement, approximate the compact \(M=\delta_i\kappa\) by \(eMe\), where \(e=f(\kappa)\), \(0\le e\le1\), and \(f\) vanishes near zero. Such spectral cutoffs form an approximate identity on compact operators because \(\kappa\) has dense range. Choose \(e\) with \(\|M-eMe\|\le\epsilon\). The error has form norm at most \(\epsilon\|\Theta w\|^2\), and the retained part is the bounded operator \(\Theta eMe\Theta\), of norm at most \(\|\Theta e\|^2\|M\|\). This proves the stated inequality on the entire completed reciprocal domain. It gives the exact input for FQ's compact oscillator cross-form argument on this same weight.

Thus the normally controlled reciprocal, actual groupoid domains and actual affine-displacement domains are now available simultaneously. This route does not require FB.1's additional derivative-column factorization. It also does not establish compact additive transport defects, the normalized auxiliary phase, its differentiable inverse-module realization on RG.13, or the original Bott \(+1\) comparison. Those remain separate work.

### 11AP.5. A noncommuting two-row test

For \(0<\varepsilon<1\), let \(a=\operatorname{diag}(1,\varepsilon^2)\), \(P_\varepsilon=\left(\begin{smallmatrix}\sqrt{1-\varepsilon^2}&\varepsilon\\\varepsilon&-\sqrt{1-\varepsilon^2}\end{smallmatrix}\right)\), and \(b=P_\varepsilon aP_\varepsilon\). This is a genuine real orthogonal involution. The matrix \(a^{-1/2}P_\varepsilon a^{1/2}=\left(\begin{smallmatrix}\sqrt{1-\varepsilon^2}&\varepsilon^2\\1&-\sqrt{1-\varepsilon^2}\end{smallmatrix}\right)\) has squared Hilbert-Schmidt norm \(3-2\varepsilon^2+\varepsilon^4\le3\). Consequently \(b\le3a\), and conjugating by \(P_\varepsilon\) gives \(a\le3b\).

For the explicit resolvent sequence \(t_n=4^{-n}\), put \(A=\psi(1)=\sum_{n\ge1}1/(4^n+1)\le1/3\). The exact conjugated transport is

\[
\phi(a)^{-1}P_\varepsilon\phi(a)
=\begin{pmatrix}
\sqrt{1-\varepsilon^2}&\varepsilon\psi(\varepsilon)/A\\
\varepsilon A/\psi(\varepsilon)&-\sqrt{1-\varepsilon^2}
\end{pmatrix}.
\]

Both off-diagonal entries tend to zero as \(\varepsilon\downarrow0\), by OP.5 and boundedness of \(\psi\). By contrast the square-root weight alone has lower-left entry one. More quantitatively, when \(\varepsilon=4^{-m}\),

\[
\frac{\psi(\varepsilon)}\varepsilon
=m-\frac12+\sum_{n=m}^\infty\frac1{4^n+1}\ge\frac m2,
\qquad
0\le\frac{\varepsilon A}{\psi(\varepsilon)}\le\frac2{3m}.
\tag{OP.14}
\]

For the identity split the defining sum at \(n=m\), and pair \(1/(1+4^{-j})\) with \(1/(1+4^j)\), whose sum is one. The lower bound also follows by retaining the first \(m\) terms, each at least \(1/2\). These are noncommuting matrix tests for the domain mechanism, not a replacement of the actual auxiliary module by two rows. The explicit sequence illustrates the range theorem; the general normal derivative construction selects its sequence by OP.2.

![One normal reciprocal with completed genuine-arrow domains](../figures/common-normal-domain-weight.png)

**Figure 11AP.1.** OP.1–OP.14. The two upper panels show the actual compact normal derivative and completed adjointable range arguments on the intrinsic JD field. The lower panel is the explicit noncommuting two-row test with \(t_n=4^{-n}\): the square-root transport has lower-left coefficient one, while the enlarged weight has coefficient \(L_m=\varepsilon\psi(1)/\psi(\varepsilon)\), \(\varepsilon=4^{-m}\), bounded by \(2/(3m)\). Exact rational intervals enclose the 32 sampled coefficients; the interval tails are bounded by \(1/(3\cdot4^{24})\). These are scalar matrix tests for the proved domain mechanism. The actual normally controlled sequence is selected by OP.2. The normalized phase, differentiable inverse-module passage and original Bott \(+1\) are not inferred. Human context: [Connes, A survey of foliations and operator algebras](https://alainconnes.org/wp-content/uploads/foliationsfine.pdf#page=29), end of Section8. [Vector figure](../figures/common-normal-domain-weight.svg) · [Generator](../reproduction/common-normal-domain-weight/draw_common_weight.py) · [Exact finite checks](../reproduction/common-normal-domain-weight/COMMON-WEIGHT-CHECKS.json) · [Reproduction](../reproduction/common-normal-domain-weight/REPRODUCTION.md) · [Component terms](../reproduction/common-normal-domain-weight/COMPONENT-TERMS.md).

## 11AQ. Compact transport defects with a controlled normal core

The common reciprocal of One normally differentiable weight with completed groupoid domains gives completed arrow domains. We now slow its inverse sufficiently to obtain compact additive arrow defects, while retaining a precise normal-core estimate. The resulting oscillator has an equivariant coefficient phase. Its coefficient algebra remains the actual anchored algebra \(A=C_0(Y)\); no evaluation of this cycle at an arbitrary point of \(Y\) is made.

### 11AQ.1. Keep the actual field and refine the normal choice

Use the actual real JD field before compact-frame quotient and then its quotient. The square-root proof in Smooth orbit compacts and the completed oscillator cross form, JD.6, proves convergence in the Hilbert-Schmidt module, not just in operator norm. Thus the OP sequence may also be selected with

\[
\|t_n^2(k+t_n)^{-1}(\delta_i k)(k+t_n)^{-1}\|_{2,A}
\le\eta2^{-n},\qquad
\kappa=\sum_{n\ge1}\frac{t_nk}{k+t_n},
\qquad M_i=\delta_i\kappa\in\mathcal L^2_A(E).
\tag{SL.1}
\]

Here \(\mathcal L^2_A(E)\) denotes the completed Hilbert-Schmidt module \(E\otimes_A\overline E\), as defined in JD.1; its norm dominates operator norm and its elements are compact operators. This refinement is possible because \(t(k+t)^{-1}\to0\) strongly, and multiplying a Hilbert-Schmidt column on either side tends to zero in Hilbert-Schmidt norm. Prove the latter first for finite columns, then approximate their square-summable tail. All OP conclusions hold for this refined sequence. In particular \(\kappa\) is real, compact-frame invariant, has dense range, and \(\Theta_{\mathrm f}=\kappa^{-1}\ge4\) is regular self-adjoint.

On a compact arrow test the original order comparisons give, with constants \(C,C'\ge1\),

\[
\kappa_b^2\le C\kappa_a^2,\qquad
\kappa_a^2\le C'\kappa_b^2,\qquad
\operatorname{Dom}\Theta_{\mathrm f,a}=\operatorname{Dom}\Theta_{\mathrm f,b}.
\tag{SL.2}
\]

For an arrow \(g\), the symbols \(a,b\) mean the receiving operator and the transported source operator on that same receiving field. Thus \(\kappa_b=L_g\kappa_{sg}L_g^{-1}\). No unitary action on fictitious labels is substituted.

### 11AQ.2. The first logarithm has a bounded completed difference

Put \(H=\log(1+\Theta_{\mathrm f})\), and regularize with

\[
\Theta_{\mathrm f,\epsilon}=(\kappa^2+\epsilon)^{-1/2},
\qquad H_\epsilon=\log(1+\Theta_{\mathrm f,\epsilon}).
\tag{SL.3}
\]

The two comparisons in SL.2 persist after adding \(\epsilon I\). Inverse order followed by the square-root order theorem gives \(\Theta_{\mathrm f,a,\epsilon}\le\sqrt C\,\Theta_{\mathrm f,b,\epsilon}\), and the reverse inequality with \(C'\). The needed bounded square-root order theorem follows from its positive integral
\[
x^{1/2}=\frac1\pi\int_0^\infty t^{-1/2}\frac{x}{x+t}\,dt.
\]
Substitution \(t=xu^2\) and the arctangent integral verify the scalar identity; uniformly truncated positive integrals prove the operator identity and order assertion. Logarithmic order follows likewise from
\[
\log(1+x)=\int_0^\infty
\left(\frac1{1+t}-\frac1{1+t+x}\right)\,dt.
\]
The truncations converge uniformly on bounded spectra. Hence

\[
-\tfrac12\log C'\,I
\le H_{a,\epsilon}-H_{b,\epsilon}
\le\tfrac12\log C\,I,\qquad
L=\tfrac12\max(\log C,\log C').
\tag{SL.4}
\]

These bounded self-adjoint differences converge strongly, with their adjoints, on the actual module. On \(\operatorname{Dom}\Theta_{\mathrm f}\), convergence follows from
\(\kappa H_\epsilon\to\kappa H\) in operator norm: the scalar function \(x\log(1+x^{-1})\) extends by zero at \(x=0\), and the same is true of the regularized products. SL.2 gives a common dense domain. The uniform bound \(L\) extends convergence to every vector; taking inner-product limits proves that the limit \(B\) is adjointable and self-adjoint.

The common domain \(\operatorname{Dom}\Theta_{\mathrm f}\) is a graph core for either \(H\): spectral cutoffs of \(\Theta_{\mathrm f}\) approximate both a vector in \(\operatorname{Dom}H\) and its \(H\)-image, and their ranges lie in \(\operatorname{Dom}\Theta_{\mathrm f}\). On this core \(H_a=H_b+B\). A bounded summand changes the graph norm by at most its norm times the vector norm. Closing both graphs therefore proves the actual identity

\[
H_a=H_b+B,\qquad
\operatorname{Dom}H_a=\operatorname{Dom}H_b,\qquad
B\in\mathcal L(E),\quad \|B\|\le L.
\tag{SL.5}
\]

On compact arrow tests this bounded field \(B_g\) is strongly continuous with its adjoint. Indeed the regularized fields are continuous. Convergence is uniform on source sections generated by \(\kappa_{sg}\), using OP's uniformly bounded completed transport and the scalar norm limit \(\kappa H_\epsilon\to\kappa H\). Density and the bound \(L\) give the asserted strong continuity. Norm continuity of \(B_g\), or compactness of \(B_g\), is not asserted.

### 11AQ.3. Slow down again, with normal derivative tests

For \(s>0\), \(J_s(H)=H(H+s)^{-1}\to0\) strongly as \(s\to\infty\). Check this on \(\operatorname{Dom}H\) using the bound \(\|J_s(H)v\|\le\|Hv\|/s\), then use density and contractivity. On the Hilbert-Schmidt module it gives \(\|J_s(H)M_i\|_{2,A}\to0\).

Enumerate the actual cutoff displacements and their first normal derivatives on a countable compact-arrow exhaustion, including each local normal frame. Regard them as a countable list \(\xi_j\) of sections over the corresponding compact tests, with the supremum module norm on each test. Their intrinsic-core approximations and the contractive strong limit above give uniform convergence: approximate each continuous section by core sections on a finite cover of its compact test, and then bound the remainder by contractivity. This tests entire displacement sections, rather than merely a dense list of their point values. Choose increasing positive \(s_n\ge2^n\), testing the finite derivative list and the first \(n\) such sections, so that

\[
\max_i\|J_{s_n}(H)M_i\|_{2,A}\le\eta2^{-n},
\qquad
\|J_{s_n}(H)\xi_j\|\le2^{-n}\quad(j\le n).
\tag{SL.6}
\]

Every finite choice is possible by the strong limits just proved. This is a single sequence, chosen before any arrow is fixed. Set

\[
f(t)=\sum_{n\ge1}\frac{t}{t+s_n},\qquad
\Omega=1+f(H).
\tag{SL.7}
\]

The scalar series converges uniformly on bounded \(t\)-intervals because \(\sum s_n^{-1}\le1\). It is positive, increasing and concave, \(f(0)=0\), \(f(t)\le t\), and \(f(t)\to\infty\): any first \(N\) summands tend to one. Functional calculus defines the regular positive \(\Omega\ge1\). Its reciprocal is compact, since \(x\mapsto[1+f(\log(1+x^{-1}))]^{-1}\) extends by zero at \(x=0\). SL.6 puts \(f(H)M_i\) in the Hilbert-Schmidt module and puts \(f(H)\xi_j\) in the actual section module by norm-convergent sums. Thus \(\Omega M_i\) is a compact Hilbert-Schmidt operator; no boundedness of \(\Omega\) is inferred.

### 11AQ.4. Compact additive defects and completed arrow domains

For \(H_a=H_b+B\), the resolvent identity gives

\[
S=\sum_{n\ge1}s_n(H_a+s_n)^{-1}B(H_b+s_n)^{-1},
\qquad
\|S\|\le L\sum_{n\ge1}s_n^{-1}.
\tag{SL.8}
\]

Each term is compact, because the \(H\)-resolvents are compact. The norm bound follows from positivity of \(H_a,H_b\), and proves norm summability. On the common domain \(\operatorname{Dom}H\), summing the finite resolvent identities gives \(\Omega_a-\Omega_b=S\). That common domain is a graph core for either \(\Omega\), since \(f(t)\le t\) and spectral cutoffs of \(H\) approximate the \(\Omega\)-graph. Closing the graphs as in SL.5 proves

\[
\Omega_a=\Omega_b+S,\qquad
\operatorname{Dom}\Omega_a=\operatorname{Dom}\Omega_b,\qquad
S\in\mathcal K(E).
\tag{SL.9}
\]

For actual arrows, \(S_g=L_g\Omega_{sg}L_g^{-1}-\Omega_{rg}=-S\), because SL.8 defines \(S\) as the receiving-minus-transported difference. The field is norm continuous on compact arrow tests: the transported compact resolvents are norm continuous, \(B_g\) is strongly continuous with its adjoint, and multiplication by a compact operator turns such uniformly bounded strong continuity into norm continuity. Prove this last assertion on rank-one compacts, then use finite rank approximation. The uniformly summable tails in SL.8 finish the proof. The original action gives the actual domain transports and their composition law.

Since \(1+f(\log(1+t))\le1+t\), \(\operatorname{Dom}\Theta_{\mathrm f}\subset\operatorname{Dom}\Omega\), and \(\Omega\kappa\) is a bounded compact function of \(\kappa\). OP's completed sections \(\Theta_{\mathrm f} b_g\) therefore give norm-continuous sections \(\Omega b_g=(\Omega\kappa)\Theta_{\mathrm f} b_g\) with local uniform bounds. These are the actual affine displacements.

### 11AQ.5. The normal core is controlled without differentiating the slow reciprocal

Write

\[
q(x)=1+f(\log(1+x^{-1}))\quad(x>0),
\qquad \Omega=q(\kappa).
\tag{SL.10}
\]

Then \(q\) is decreasing, while \(xq(x)\) is increasing and extends continuously by zero at zero. To prove the last two statements, use \(f'(t)t\le f(t)\), a consequence of concavity and \(f(0)=0\), and
\[
\log(1+x^{-1})\ge(1+x)^{-1},\qquad
x\,\frac{d}{dx}\log(1+x^{-1})=-(1+x)^{-1}.
\]
They give \((xq(x))'\ge0\). Also \(f(t)\le t\) proves the zero extension. For the divided difference \(q^{[1]}\), including its diagonal derivative, these two monotonicities imply

\[
|q^{[1]}(u,v)|\,v\le q(u)\qquad(u,v>0).
\tag{SL.11}
\]

For \(u<v\), rearrange \(vq(v)\ge uq(u)\); for \(u>v\), rearrange the same inequality with the variables reversed. Diagonal values follow by taking the limit. This estimate is for this particular scalar function, not a general assertion about divided differences.

Regularize \(q_\epsilon(x)=q(x+\epsilon)\). The closed derivative functional calculus applies to the bounded smooth function \(q_\epsilon\) on the compact spectrum of \(\kappa\). On each fibre its Hilbert-Schmidt derivative has matrix entries \(q_\epsilon^{[1]}(u,v)(M_i)_{uv}\); multiplication on the right by \(\kappa\) gives an extra factor \(v\). This is the joint functional calculus of the commuting left and right \(\kappa\) actions on the Hilbert-Schmidt module. Fibre eigenvectors are used only to calculate its norm bound, not selected as continuous sections.

The coefficient is bounded by \(q(u)\), because SL.11 applied to \(u+\epsilon,v+\epsilon\) gives \(v|q_\epsilon^{[1]}(u,v)|\le q(u+\epsilon)\le q(u)\). Thus the weighted derivatives converge in Hilbert-Schmidt norm:

\[
\delta_i q_\epsilon(\kappa)\,\kappa\longrightarrow X_i,
\qquad
\|X_i\|_{2,A}\le\|\Omega M_i\|_{2,A}.
\tag{SL.12}
\]

Here is the needed convergence proof. Cut \(M_i\) on both sides by a continuous plateau \(e(\kappa)\) vanishing near zero. Since \(\Omega M_i\) is an actual Hilbert-Schmidt element, \(\Omega(M_i-eM_i e)\) can be made arbitrarily small in that norm. On the retained spectral square the divided differences converge uniformly. SL.11 bounds the two omitted errors uniformly by their weighted Hilbert-Schmidt norms. This proves the Cauchy claim and identifies its limit without a Hilbert-module Riesz representation theorem.

Also \(q_\epsilon(\kappa)\kappa\to\Omega\kappa\) in norm and \(q_\epsilon(\kappa)M_i\to\Omega M_i\) in Hilbert-Schmidt norm. The first follows from the continuous zero extension of \(xq(x)\); for the second cut off \(\Omega M_i\) near zero as above. Closedness of the derivative now gives the actual graph identity

\[
\Omega\kappa\in\operatorname{Dom}\delta_i,\qquad
\delta_i(\Omega\kappa)=X_i+\Omega M_i\in\mathcal K(E).
\tag{SL.13}
\]

On the completed fast domain the notation \((\delta_i\Omega)\kappa=X_i\) means this weighted derivative limit. It does not assert a bounded derivative of \(\Omega\), or a normal derivative of \(\Omega^{-1}\).

For an actual cutoff displacement \(b\), its derivative \(\nabla_i b\) is among the SL.6 tests, while \(\Theta_{\mathrm f}b\) is supplied by OP. Regularization gives \(\delta_i q_\epsilon(\kappa)b\to X_i \Theta_{\mathrm f}b\), and \(q_\epsilon(\kappa)\nabla_i b\to\Omega\nabla_i b\). Both limits are in module norm. Closing the connection therefore proves

\[
\Omega b\in\operatorname{Dom}\nabla_i,\qquad
\nabla_i(\Omega b)=X_i \Theta_{\mathrm f}b+\Omega\nabla_i b.
\tag{SL.14}
\]

These statements hold on compact arrow tests with the actual cutoffs and local uniform graph bounds. Exact compact-frame invariance of \(\kappa\) makes all its scalar functions invariant; the finite changes of parallelism preserve the construction and its horizontal quotient.

### 11AQ.6. The equivariant coefficient phase and its precise algebra

Apply A confining oscillator over a varying Hilbert field, WF.9–WF.20, with the actual weight \(\Omega\). On the completed real-field Fock module over \(A\), complexified with exterior parity, the regular odd operator and phase are

\[
C_\Omega=D^3+B(1+\Omega),\qquad
F_\Omega=C_\Omega(1+C_\Omega^2)^{-1/2}.
\tag{SL.15}
\]

SL.9 gives the compact norm-continuous linear weight defects and completed domains required by that proof; SL.14 gives the actual displacement core, with SL.9's graph domain bounds. The original affine cocycle acts by its actual linear transport and Weyl translation. Linear conjugation changes \(C_\Omega\) by \(B(S_g)\), controlled by \(|D|\); translation changes it by the fully retained noncommuting cubic expression of WF.16, controlled by \(1+D^2\). The exact square of the cubic oscillator controls \(|D|^3\), so these are relative powers \(1/3\) and \(2/3\). WF.18's integrable two-resolvent difference proves the compact phase defects and their norm continuity on compact arrow tests. On the noncompact coefficient base these statements are localized by \(a\in A\), as in the whole-base proof of BA.7; the vacuum projection is compact only after such localization.

For completeness, its coefficient class is \(1_A\). Scale the affine displacement by \(u\in[0,1]\); WF.16–WF.19 give the continuous compact defects of this equivariant homotopy. At zero the action is linear. Remove the even vacuum and replace the phase by the sign, whose difference is locally compact by the spectral gap. On the complement use \(\operatorname{sign}(D+s[D^3+B(\Omega)])\), \(s>0\), and \(\operatorname{sign}(D)\) at zero. Finite-occupation bounded-\(\Omega\) cores give strong continuity. The compact linear defect tends to zero in norm with the bound \(O(\|S_g\|(s+s^{2/3}))\), proved by the scalar maximum in WF.20 and its integrable resolvent formula. At zero the complement is exactly degenerate. The remaining even vacuum is the genuine identity cycle on \(A\); its localized square defect is multiplication by \(-a\), a compact operator on \(A\).

This is the equivariant coefficient identity at \(A=C_0(Y)\). It is not an assertion that the original \(\gamma\) at \(R=C_0(T)\) equals the equivariant identity, nor a scalar reduced character. The actual balanced proper factorization and its inverse-module realization must retain their own algebras and full inverse Spin-c coefficient. A closed physical sum on the literal graph and the original Bott \(+1\) comparison remain unproved. The slow phase and the fast normal reciprocal are two coordinated operators; SL.13–SL.14 do not identify their derivative domains with one another.

### 11AQ.7. An exact pair-swap model for the two logarithmic steps

On \(\bigoplus_{m\ge1}\mathbb C^2\), put \(\Theta_{\mathrm f,m}=\operatorname{diag}(4^m,2\cdot4^m)\), and let the real involution \(P\) swap the two coordinates in every pair. The reciprocal is compact and the order constants in SL.2 are both \(4\). The original difference \(P\Theta_{\mathrm f}P-\Theta_{\mathrm f}\) has entries \(4^m,-4^m\), and is unbounded. The first logarithmic difference has entries \(B_m,-B_m\), where
\[
H_m=\log(1+4^m),\qquad
B_m=\log\frac{1+2\cdot4^m}{1+4^m}\longrightarrow\log2.
\]
It is bounded and is not compact.

There is no normal variation in this example, so \(M_i=0\), and \(s_n=2^n\) is an admissible explicit slow sequence. The second difference has entries \(d_m,-d_m\), with
\[
d_m=\sum_{n\ge1}
\frac{2^n B_m}{(H_m+2^n)(H_m+B_m+2^n)}
\longrightarrow0.
\]
The first \(N\) terms have omitted tail at most \((\log2)2^{-N}\). For the limit choose \(N=\lfloor\log_2 H_m\rfloor\); the retained sum is at most \((\log2)(2^{N+1}-2)/H_m^2\), and the tail at most \((\log2)2^{-N}\). Hence
\[
0\le d_m\le\min\{\log2,\;4\log2/H_m\}.
\]
Thus the second difference is an actual compact diagonal operator, unlike the first. The plot below uses a finite sum with its proved tail bound; the compactness follows from the displayed estimate.

![Slow compact transport and the controlled normal core](../figures/slow-transport-weight.png)

**Figure 11AQ.1.** SL.1–SL.15. The upper panels retain the actual intrinsic JD field and distinguish the fast normal reciprocal \(\Theta_{\mathrm f}\) from the slow transport weight \(\Omega\). The pair-swap model has \(\Theta_{\mathrm f,m}=\operatorname{diag}(4^m,2\cdot4^m)\) and exact reciprocal-square order constants four. The original weight difference is unbounded; the orange first-log difference tends to \(\log2\) and is not compact. The blue slow difference uses 24 positive terms, with omitted tail at most \((\log2)2^{-24}\); the green bound \(\min(\log2,4\log2/H_m)\) proves its compact limit. These are numerical samples with a proved analytic tail bound. The coefficient identity is at \(A=C_0(Y)\), with its actual even vacuum. No original-unit evaluation or physical graph realization is inferred. Human context: [Connes, A survey of foliations and operator algebras](https://alainconnes.org/wp-content/uploads/foliationsfine.pdf#page=29), Section8. [Vector figure](../figures/slow-transport-weight.svg) · [Generator](../reproduction/slow-transport-weight/draw_slow_transport.py) · [Finite checks](../reproduction/slow-transport-weight/SLOW-TRANSPORT-CHECKS.json) · [Reproduction](../reproduction/slow-transport-weight/REPRODUCTION.md) · [Component terms](../reproduction/slow-transport-weight/COMPONENT-TERMS.md).

## 11AR. Weighted normal derivatives of the coefficient oscillator phase

Compact transport defects with a controlled normal core constructs the slow coefficient oscillator and its actual affine transport. We now prove its normal derivative estimate on the completed fast-oscillator core. The coefficient algebra is the same actual anchored algebra \(A=C_0(Y)\). The fast inverse below is used as a graph weight; no normal derivative of that inverse is required.

### 11AR.1. Two oscillators on the actual completed field

Keep SL's real intrinsic field, compact-frame quotient, closed metric connection and actual Fock completion. Denote its even vacuum projection by \(P_0\). On the positive-occupation complement put

\[
C_{\mathrm f}=D^3+B(1+\Theta_{\mathrm f}),\qquad
C_{\mathrm s}=D^3+B(1+\Omega),\qquad
R_{\mathrm f}=|C_{\mathrm f}|^{-1}(1-P_0),\qquad
W=\operatorname{sign}(C_{\mathrm s})(1-P_0).
\tag{PW.1}
\]

The inverse and sign are defined to be zero on the vacuum. Regularity, self-adjointness and the completed cutoff cores follow from A confining oscillator over a varying Hilbert field, WF.6–WF.12. Both oscillators have the even vacuum as their exact kernel and have the uniform complement gap \(\sqrt2\).

On total occupation \(N=n\), WF.10 gives, in a fibre calculation,

\[
C_{\mathrm f}^2=2\sum_j(1+2n+\theta_j)^2N_j,
\qquad
C_{\mathrm s}^2=2\sum_j(1+2n+\omega_j)^2N_j.
\tag{PW.2}
\]

Here \(\theta_j\) and \(\omega_j\) are scalar functions of the same positive compact \(\kappa\). The individual occupation operators commute with the corresponding mode Clifford generators. Thus \(C_{\mathrm f}^2\), and hence \(R_{\mathrm f}\), commute with \(C_{\mathrm s}\) and its resolvents. This is a strong completed commutation assertion: first verify it on finite occupation and bounded \(\kappa^{-1}\) cutoffs, then extend the bounded resolvents by their common dense cutoff core. No fibre eigenbasis is selected continuously.

Both \(R_{\mathrm f}\) and \(DR_{\mathrm f}\) are compact on this actual Fock module. Indeed a fast energy cutoff in the positive-occupation complement permits finitely many occupation sectors and only one-particle modes in a bounded \(\Theta_{\mathrm f}\) cutoff. Each sector has at least one particle; its required tensor power of a compact one-particle cutoff is compact by the rank-one tensor argument of WF.12. Summing the finitely many sectors gives a compact energy cutoff. Moreover

\[
|D|\le |C_{\mathrm f}|^{1/3},\qquad
\|DR_{\mathrm f}\,1_{\{|C_{\mathrm f}|>r\}}\|\le r^{-2/3}.
\tag{PW.3}
\]

The indicator in this estimate is fibre notation for the spectral norm bound; continuous energy plateaus give the module approximations. The vacuum has been removed from these compact operators. On a noncompact coefficient base \(P_0\) itself is compact only after multiplication by \(a\in A\).

### 11AR.2. The weighted one-particle derivative controls the Fock derivative

Use exactly SL.10–SL.13's bounded regularizations \(q_\epsilon(x)=q(x+\epsilon)\). Write

\[
A_{i,\epsilon}=\delta_iq_\epsilon(\kappa),\qquad
X_{i,\epsilon}=A_{i,\epsilon}\kappa\longrightarrow X_i,
\qquad
\|X_i\|_{2,A}\le\|\Omega M_i\|_{2,A}.
\tag{PW.4}
\]

The convergence is in Hilbert-Schmidt norm and therefore in operator norm. Each \(A_{i,\epsilon}\) is a bounded real self-adjoint operator. On the completed fast one-particle domain it equals \(X_{i,\epsilon}\Theta_{\mathrm f}\). Consequently its positive quadratic form satisfies
\(A_{i,\epsilon}^2\le\|X_{i,\epsilon}\|^2\Theta_{\mathrm f}^2\).
Apply the bosonic/exterior relations to this form inequality: second quantization sums it over each occupied factor. Since
\(B(A)^2=2d\Gamma(A^2)\) and \(C_{\mathrm f}^2\ge2d\Gamma(\Theta_{\mathrm f}^2)\), it gives

\[
Y_{i,\epsilon}=B(A_{i,\epsilon})R_{\mathrm f},\qquad
\|Y_{i,\epsilon}\|\le\|X_{i,\epsilon}\|,
\qquad
\|Y_{i,\epsilon}-Y_{i,\epsilon'}\|
\le\|X_{i,\epsilon}-X_{i,\epsilon'}\|.
\tag{PW.5}
\]

For the difference use the same form argument for the bounded real self-adjoint difference of the two \(A\)'s; its product with \(\kappa\) is the difference of the two \(X\)'s. These estimates extend the initially defined products from the cutoff core to actual bounded adjointable maps. Their adjoints are the corresponding completed adjoint products, obtained from the finite-core inner-product identity. Bounded module maps are not assigned adjoints by a Riesz representation claim.

Each \(Y_{i,\epsilon}\) is compact. WF.15 makes \(B(A_{i,\epsilon})D^{-1}(1-P_0)\) bounded, and PW.3 makes \(DR_{\mathrm f}\) compact. Their product is \(Y_{i,\epsilon}\). PW.5 therefore gives a norm limit \(Y_i\in\mathcal K(\mathcal F)\), with \(\|Y_i\|\le\|X_i\|\).

### 11AR.3. The sign derivative converges after fast weighting

Let \(C_\epsilon=D^3+B(1+q_\epsilon(\kappa))\), and let \(W_\epsilon\) be its sign, zero on the vacuum. The Fock connection is the tensor metric connection. Total occupation and the vacuum are parallel, and \(D\) is parallel: differentiating its bosonic/exterior contraction formula leaves the same contraction, by metricity. Thus the normal derivative of \(C_\epsilon\) on its core is \(B(A_{i,\epsilon})\).

There is no unbounded-domain assumption hidden in the next calculation. On a fixed occupation sector \(C_\epsilon\) is bounded, since \(q_\epsilon(\kappa)\) is bounded. Polynomial calculus, the Leibniz rule and the closed tensor connection prove the bounded inverse derivative there. Summing sectors proves the resolvent formula on the full finite-occupation connection core. At large resolvent parameter the inequality \(|C_\epsilon|\ge|D|^3\) gives the integrable bound \(O(\|A_{i,\epsilon}\|a^{-5/3})\) for the unweighted derivative of the sign integral. At bounded parameter the uniform complement gap applies. Hence \(W_\epsilon\) has a bounded normal derivative on the section core, with the completed product rule.

On the complement use the strong sign integral
\[
\operatorname{sign}(C_\epsilon)
=\frac1\pi\int_0^\infty
\big((C_\epsilon-ia)^{-1}+(C_\epsilon+ia)^{-1}\big)\,da.
\]
The scalar arctangent integral verifies its normalization. Differentiate the inverse, retaining its minus sign. Since \(R_{\mathrm f}\) commutes with these resolvents, the weighted derivative is

\[
Z_{i,\epsilon}=(\delta_iW_\epsilon)R_{\mathrm f}
=-\frac1\pi\int_0^\infty
\left(R_{\epsilon,-}(a)Y_{i,\epsilon}R_{\epsilon,-}(a)
+R_{\epsilon,+}(a)Y_{i,\epsilon}R_{\epsilon,+}(a)\right)\,da,
\quad R_{\epsilon,\pm}(a)=(C_\epsilon\pm ia)^{-1}(1-P_0).
\tag{PW.6}
\]

The derivatives and products are zero on the parallel vacuum. This integral converges in operator norm. Its integrand is compact by PW.5 and has norm at most \(2\|Y_{i,\epsilon}\|/(2+a^2)\). Therefore

\[
Z_{i,\epsilon}\in\mathcal K(\mathcal F),\qquad
\|Z_{i,\epsilon}\|\le\frac{\|Y_{i,\epsilon}\|}{\sqrt2}.
\tag{PW.7}
\]

The regularized \(C_\epsilon\) converge to \(C_{\mathrm s}\) in strong resolvent sense, with adjoints. To prove this on the actual section module, use finite occupation and a continuous \(\kappa\) plateau vanishing near zero. The scalar \(q_\epsilon\) converge uniformly there, so the bounded operators on that retained core converge in norm. Their inverses preserve the same core and converge there by the resolvent identity. Uniform resolvent bounds extend the limits to every section. The signs likewise converge strongly with their adjoints, using the common gap on the complement and density of the same cutoff core.

For every \(a\), multiplying these uniformly bounded strong resolvent limits on both sides of the compact \(Y_i\) gives a norm limit. Prove this assertion first on a rank-one map, using strong convergence on its two defining vectors and the adjoint limits; then approximate a compact by finite rank maps. PW.5 and the common integrable bound from PW.7 consequently prove

\[
Z_{i,\epsilon}\longrightarrow Z_i\in\mathcal K(\mathcal F)
\quad\hbox{in norm},\qquad
\|Z_i\|\le\frac{\|X_i\|}{\sqrt2}
\le\frac{\|\Omega M_i\|_{2,A}}{\sqrt2}.
\tag{PW.8}
\]

This proves a weighted normal derivative of the actual slow phase. It does not assume a derivative of \(R_{\mathrm f}\).

### 11AR.4. Close the normal derivative on the fast graph core

Use finite particle sections generated by \(\kappa\xi\), with \(\xi\) in IJ's dense normal core, and include the normal scalar-core sections of the vacuum. The metric tensor connection on these sections is closable by the fixed lifted-field argument of Intrinsic normal jets and the compact-frame quotient, IJ.2–IJ.5, applied to each finite tensor factor and then to finite direct sums.

These sections are a graph core for \(C_{\mathrm f}\). On the one-particle domain the graph map is \(\xi\mapsto(\kappa\xi,\xi)\), and density of the normal core proves graph density. On any fixed finite occupation sector, PW.2 bounds the graph norm by the sum of the one-particle fast graph norms. Approximate each factor on a bounded \(\kappa^{-1}\) cutoff and then remove the occupation cutoff as in WF.12. The product sections belong to the closed normal connection domain because \(\nabla_i(\kappa\xi)=M_i\xi+\kappa\nabla_i\xi\). Denote this joint core by \(\mathcal C\).

For \(v\in\mathcal C\), the regularized product rule is
\(\nabla_i(W_\epsilon v)=(\delta_iW_\epsilon)v+W_\epsilon\nabla_i v\).
On its positive-occupation component substitute
\(v=R_{\mathrm f}|C_{\mathrm f}|v\) only in the first term. The vacuum derivative contributes zero. PW.8 and strong convergence of the signs give module-norm limits of the value and derivative. Closedness of the connection proves

\[
Wv\in\operatorname{Dom}\nabla_i,\qquad
\nabla_i(Wv)=Z_i|C_{\mathrm f}|v+W\nabla_i v,
\qquad v\in\mathcal C.
\tag{PW.9}
\]

In particular the completed weighted commutator is compact:
\((\delta_iW)R_{\mathrm f}=Z_i\). This notation refers to PW.8 and the actual graph identity PW.9; it makes no assertion that \(\delta_iW\) itself is bounded. It also does not differentiate \(|C_{\mathrm f}|v\).

In a local orthonormal normal frame let \(D_0=\sum_i c(e_i)\nabla_i\) include the pulled-back full inverse normal-spinor connection, with the auxiliary factor first and the final right Clifford block last. Put \(A_N=\Gamma_{\mathcal F}D_0\). On the joint normal core, the odd auxiliary phase anticommutes with \(\Gamma_{\mathcal F}\), so the actual weak first-order identity is

\[
\{W,A_N\}v
=\Gamma_{\mathcal F}\sum_i c(e_i)Z_i|C_{\mathrm f}|v.
\tag{PW.10}
\]

Each Clifford factor is on the specified inverse normal block. Its norm is one in an orthonormal frame, giving a bound by \(\sum_i\|Z_i\|\,\||C_{\mathrm f}|v\|\). This is a coefficient normal-core identity; it does not declare a completed scalar physical sum or an original-unit inverse-module transfer.

### 11AR.5. The phase class and its exact remaining interface

Replacing SL.15's bounded phase by \(W\) changes it by a locally compact operator: their scalar difference is zero at the vacuum and tends to zero at infinite nonzero energy, and the positive-energy cutoffs of the slow oscillator are compact by WF.12. Hence the actual equivariant coefficient class remains

\[
[\mathcal F,W]=1_A\quad\hbox{in }KK_G(A,A),\qquad A=C_0(Y).
\tag{PW.11}
\]

Compact-frame invariance of both weights and of the tensor connection gives the same estimates after the actual horizontal quotient. The genuine compact-arrow section tests used by SL give their local uniform bounds. The original affine action, rather than the fast oscillator's phase, supplies the compact transport defects of \(W\).

The normal phase estimate on this coefficient field is now proved. The balanced realization at the original \(R=C_0(T)\), differentiable inverse-module passage, completed physical graph sum and original Bott \(+1\) comparison remain to be constructed. A coefficient identity at \(A\) does not identify those original-unit classes.

### 11AR.6. An exact two-mode Clifford check

For a finite check use the real two-mode field over the circle, with
\(\kappa(t)=\operatorname{diag}((4+\sin t)^{-2},1/25)\) and the scalar weight
\(\Omega(t)=1+\kappa(t)^{-1/2}=\operatorname{diag}(5+\sin t,6)\).
This commuting weight supplies the same weighted-derivative input of PW.4–PW.5. At \(t=0\), its derivative is \(A=\operatorname{diag}(1,0)\), while
\(X=A\kappa=\operatorname{diag}(1/16,0)\).

On occupation \(N_1=N_2=1\), total \(n=2\), let
\(Q_1=\sigma_x\otimes I\) and \(Q_2=\sigma_z\otimes\sigma_x\).
These real self-adjoint matrices square to one and anticommute. They are the mode operators of WF.7 divided by \(\sqrt2\). Thus at this point

\[
C_{\mathrm s}=\sqrt2(10Q_1+11Q_2),\qquad
C_{\mathrm f}=\sqrt2(21Q_1+30Q_2),\qquad
C_{\mathrm f}^2=2682I,\qquad
\dot W=\frac{121Q_1-110Q_2}{221\sqrt{221}}.
\tag{PW.12}
\]

Indeed \(W=(10Q_1+11Q_2)/\sqrt{221}\); differentiate its normalized coefficient vector. The Clifford norm identity gives
\(\|\dot W\|=11/221\),
\(\|B(A)R_{\mathrm f}\|^2=1/1341\le1/256=\|X\|^2\), and
\(\|\dot W R_{\mathrm f}\|^2=121/(221^2\cdot2682)\le1/2682\).
The last right-hand side is exactly \(\|B(A)R_{\mathrm f}\|^2/2\), the bound in PW.7. These are exact rational checks of the squared norms, with the occupation shift \(1+2n=5\) retained.

![Weighted normal derivative of the coefficient oscillator phase](../figures/weighted-normal-phase.png)

**Figure 11AR.1.** PW.1–PW.12. The upper panels show the actual weighted derivative limit and compact sign-resolvent sandwich. The centre records the completed normal graph identity on the fast core. In the two-mode circle example the blue curve is the normalized Clifford coefficient vector and the green arrow is its derivative at zero. The exact square and weighted norm calculations retain total occupation two and the shift five. The coefficient class remains at \(A=C_0(Y)\); the balanced original-unit realization remains unproved. Human context: [Connes, A survey of foliations and operator algebras](https://alainconnes.org/wp-content/uploads/foliationsfine.pdf#page=29), Section8. [Vector figure](../figures/weighted-normal-phase.svg) · [Generator](../reproduction/weighted-normal-phase/draw_weighted_phase.py) · [Exact finite checks](../reproduction/weighted-normal-phase/WEIGHTED-PHASE-CHECKS.json) · [Reproduction](../reproduction/weighted-normal-phase/REPRODUCTION.md) · [Component terms](../reproduction/weighted-normal-phase/COMPONENT-TERMS.md).

## 11AS. A coefficient auxiliary with normally controlled resolvents

Weighted normal derivatives of the coefficient oscillator phase gives the actual slow phase \(W\) and its compact weighted normal derivative on the fast oscillator core. We now combine that phase with a stronger magnitude. The result is an actual regular self-adjoint coefficient auxiliary whose resolvents preserve the closed normal connection domain. Its coefficient algebra is still \(A=C_0(Y)\).

### 11AS.1. Keep the phase and change the magnitude

Use the two completed oscillators of PW.1 and the parallel even vacuum \(P_0\). Put

\[
H=C_{\mathrm f}^2,\qquad L=WH,\qquad
\operatorname{Dom}L=\operatorname{Dom}H.
\tag{MR.1}
\]

PW.2 proves strong commutation of \(W\) with \(H\); on the positive-occupation complement \(W\) is a self-adjoint unitary. Both operators vanish on the vacuum. Thus the actual resolvents are

\[
\mathcal R_\pm(\lambda)=(L\pm i\lambda)^{-1}
=(WH\mp i\lambda)(H^2+\lambda^2)^{-1},\qquad
\|\mathcal R_\pm(\lambda)\|\le\lambda^{-1}.
\tag{MR.2}
\]

The adjoint identities follow from strong commutation. Their ranges contain the finite-occupation, bounded-fast-weight cutoff core, and also the entire vacuum module. They therefore have dense range. The completed resolvent calculus of Unbounded Kasparov modules and spectral triples, Lemma0.0, proves that \(L\) is regular self-adjoint. On the complement \(|L|=H\); its graph norm is exactly the \(H\)-graph norm. The same cutoff sections are a graph core. This is a completed module operator, rather than a collection of independent fibre operators.

For \(a\in A\), \(a\mathcal R_\pm\) is compact. The vacuum term is \(\mp i\lambda^{-1}aP_0\), compact because multiplication by \(a\) on its copy of \(A\) is compact. The complementary term is compact by the positive-energy cutoff argument of PW.3. The unlocalized vacuum projection is not assigned global compactness.

### 11AS.2. A uniform cutoff estimate for the fast inverse

For \(0<\epsilon\le1/4\) define
\(\Theta_\epsilon=(\kappa+\epsilon)^{-1}\),
\(C_{\mathrm f,\epsilon}=D^3+B(1+\Theta_\epsilon)\),
\(H_\epsilon=C_{\mathrm f,\epsilon}^2\), and
\(r_\epsilon=H_\epsilon^{-1/2}(1-P_0)\).
Set \(r=R_{\mathrm f}\). The inverses are zero on the vacuum. Since \(\|\kappa\|\le1/4\), \(\Theta_\epsilon\ge2\), and the uniform complement gap used by PW remains valid.

On occupation \(n\ge1\), retain \(a_n=1+2n\). If \(\theta_j\) is an occupied fast mode, set \(c_j=a_n+\theta_j\) and \(c_{j,\epsilon}=a_n+\theta_j/(1+\epsilon\theta_j)\). Then
\(c_j/(1+\epsilon c_j)\le c_{j,\epsilon}\le c_j\).
Write \(v=(\sum_jc_j^2N_j)^{1/2}\). Every occupied \(c_j\le v\), because its integer occupation is at least one. Hence the regularized norm is between \(v/(1+\epsilon v)\) and \(v\). Taking its reciprocal gives the uniform completed operator estimate

\[
0\le r_\epsilon-r\le\frac{\epsilon}{\sqrt2}(1-P_0),\qquad
\|r_\epsilon-r\|\le\epsilon/\sqrt2.
\tag{MR.3}
\]

The mode calculation is positive operator order on the joint occupation/\(\kappa\) cutoff core. Its bounded functional-calculus extensions prove the estimate on the actual module without a continuous eigenbasis.

Use the same slow regularization \(W_\epsilon\) as PW.6, and put \(L_\epsilon=W_\epsilon H_\epsilon\), with resolvents \(\mathcal R_{\epsilon,\pm}\). The same completed construction as MR.2 applies; the phase commutes strongly with this regularized magnitude. On the complement the two scalar functions in its resolvent are

\[
\frac{H_\epsilon}{H_\epsilon^2+\lambda^2}
=\frac{r_\epsilon^2}{1+\lambda^2r_\epsilon^4},\qquad
\frac1{H_\epsilon^2+\lambda^2}(1-P_0)
=\frac{r_\epsilon^4}{1+\lambda^2r_\epsilon^4}.
\tag{MR.4}
\]

They converge in norm by MR.3. Their limiting functions of the compact \(r\) are compact. PW's strong sign limits with adjoints consequently give \(\mathcal R_{\epsilon,\pm}\to\mathcal R_\pm\) in norm: multiply the strong phase limit by the compact limiting function and retain the unchanged vacuum term. No norm convergence of the unweighted signs is required.

### 11AS.3. Match the regularized phase to the regularized fast weight

Retain \(M_i=\delta_i\kappa\), \(A_{i,\epsilon}=\delta_iq_\epsilon(\kappa)\), and \(X_i\) from PW.4. SL.11's divided-difference estimate also gives

\[
\epsilon A_{i,\epsilon}\longrightarrow0\quad\hbox{in HS norm},\qquad
\widetilde X_{i,\epsilon}=A_{i,\epsilon}(\kappa+\epsilon)
\longrightarrow X_i,\qquad
\|\widetilde X_{i,\epsilon}\|\le\|\Omega M_i\|_{2,A}.
\tag{MR.5}
\]

Indeed the coefficient of \(\epsilon A_{i,\epsilon}\) on the left/right \(\kappa\) actions is bounded by \(q(u)\epsilon/(v+\epsilon)\). Cut \(\Omega M_i\) on both sides by a plateau vanishing near zero, exactly as in SL.12. On its retained spectral square that coefficient tends uniformly to zero; the omitted weighted HS tails are uniformly small. This proves the first limit. Adding PW.4 gives the second. For the bound apply SL.11 to \(u+\epsilon,v+\epsilon\): \((v+\epsilon)|q_\epsilon^{[1]}(u,v)|\le q(u)\).

The proof of PW.5–PW.7, now using \(\Theta_\epsilon\) and \(\widetilde X_{i,\epsilon}\), gives

\[
Z_{i,\epsilon}^{\mathrm{reg}}=(\delta_iW_\epsilon)r_\epsilon,
\qquad
\|Z_{i,\epsilon}^{\mathrm{reg}}\|
\le\|\Omega M_i\|_{2,A}/\sqrt2,
\qquad
Z_{i,\epsilon}^{\mathrm{reg}}\longrightarrow Z_i\quad\hbox{in norm}.
\tag{MR.6}
\]

Here is the required final convergence argument. The unweighted derivative exists by PW.6. Its sign integral has, for \(0\le a\le1\), paired derivative norm at most \(\|A_{i,\epsilon}\|/\sqrt2\), since \(\|D(C_\epsilon\pm ia)^{-1}\|\le1/2\). For \(a\ge1\), that norm is at most \(2\|A_{i,\epsilon}\|a^{-5/3}\), using \(|C_\epsilon|\ge|D|^3\). Integration gives

\[
\|\delta_iW_\epsilon\|\le2\|A_{i,\epsilon}\|,\qquad
\|(\delta_iW_\epsilon)(r_\epsilon-r)\|
\le\sqrt2\,\epsilon\|A_{i,\epsilon}\|\longrightarrow0.
\tag{MR.7}
\]

MR.5 proves the last limit. Combining it with PW.8 proves MR.6. The regularized products need not individually be compact; their actual norm limit \(Z_i\) is compact.

### 11AS.4. Two bounded annihilation rows control the magnitude derivative

On total occupation \(n\), let \(\mathcal A_n\) be the direct sum of the bosonic and exterior annihilation contractions. It maps into the one-particle module tensor the corresponding \((n-1)\)-particle remainders. Its basis-free construction is the symmetric or antisymmetric tensor contraction, multiplied by \(\sqrt p\) or \(\sqrt q\). Finite tensor algebra gives

\[
\mathcal A_n^*(T\otimes1)\mathcal A_n=d\Gamma_n(T),\qquad
\mathcal A_n^*\mathcal A_n=nI.
\tag{MR.8}
\]

Verify the first identity for a finite rank one-particle operator by the creation/annihilation relations. For a general bounded adjointable \(T\), compress by a compact approximate identity and approximate each compact compression in norm by finite rank maps. These maps converge on every one-particle vector, with their adjoints and a common norm bound. Passing the limits on algebraic tensors, and then using the bound \(\|d\Gamma_n(T)\|\le n\|T\|\), proves the identity on the completed finite sector. The direct sum of the target row modules is an actual Hilbert \(A\)-module; no continuous fibre basis is used.

The bounded inverse derivative gives \(\delta_i\Theta_\epsilon=-\Theta_\epsilon M_i\Theta_\epsilon\). On a fixed occupation sector, where the coefficients are bounded,

\[
\delta_iH_\epsilon
=-2\mathcal A_n^* \left(
\Theta_\epsilon(a_n+\Theta_\epsilon)M_i\Theta_\epsilon
+\Theta_\epsilon M_i\Theta_\epsilon(a_n+\Theta_\epsilon)
\right)\mathcal A_n.
\tag{MR.9}
\]

All tensor identities on the row remainder are implicit. The shift is the **source** occupation shift \(a_n=1+2n\), not the remainder's shift.

Fix one resolvent sign and abbreviate it by \(\mathcal R_\epsilon\). Define two rows, block by block,

\[
P_\epsilon=[\Theta_\epsilon(a_n+\Theta_\epsilon)\otimes1]
\mathcal A_n\mathcal R_\epsilon,
\qquad
Q_\epsilon=(\Theta_\epsilon\otimes1)\mathcal A_n\mathcal R_\epsilon,
\qquad
\|P_\epsilon\|\le\tfrac12,\quad
\|Q_\epsilon\|\le\frac1{2\sqrt\lambda}.
\tag{MR.10}
\]

To prove the first bound, on joint occupied modes use
\(d\Gamma[\Theta_\epsilon^2(a_n+\Theta_\epsilon)^2]
\le[d\Gamma(a_n+\Theta_\epsilon)^2]^2=H_\epsilon^2/4\).
The scalar inequality is \(\sum_jN_jc_j^4\le(\sum_jN_jc_j^2)^2\), with \(\theta_j^2c_j^2\le c_j^4\) and integer occupations. The second follows from \(d\Gamma(\Theta_\epsilon^2)\le H_\epsilon/2\) and the scalar maximum
\(\sup_{h\ge0}\sqrt h/(h^2+\lambda^2)^{1/2}=1/\sqrt{2\lambda}\).
These positive form estimates give bounded adjointable rows, by their finite-sector adjoint formulas and the uniform direct-sum bounds.

Also define \(\widehat P_\epsilon,\widehat Q_\epsilon\) by replacing the last \(\mathcal R_\epsilon\) in these rows with \(W_\epsilon\mathcal R_\epsilon^*\). They have the same bounds: the phase commutes with the positive mode energies controlling both row norms.

These rows have actual strong limits with adjoints. Check this first on finite-occupation, bounded-\(\kappa^{-1}\) source and row-target cutoffs. All weighted contractions and their adjoints then converge in norm on each test. Uniform row bounds extend both limits to the full completed source and row modules; their inner-product limits give the actual adjoint identities.

In fact the \(Q\)-rows converge in norm to compact operators. Their positive squares are bounded by

\[
Q_\epsilon^*Q_\epsilon\le K_\epsilon
=\frac{r_\epsilon^2}{2(1+\lambda^2r_\epsilon^4)}
\longrightarrow K\in\mathcal K(\mathcal F)\quad\hbox{in norm}.
\tag{MR.11}
\]

The same inequality holds for the hatted row. To justify the norm upgrade, choose a compact approximate-identity contraction \(e\) making \((1-e)K(1-e)\) small. MR.11 makes \(Q_\epsilon(1-e)\) uniformly small for sufficiently small \(\epsilon\). On \(e\), strong convergence gives norm convergence by rank-one approximation. The limit has the same small tail, and its product with \(e\) is compact. This proves both norm convergence and compactness. Only the \(P\)-rows need strong convergence with adjoints.

### 11AS.5. Pass the inverse derivative to the actual completed operator

On each fixed occupation sector every regularized coefficient is normally differentiable and bounded. The inverse derivative, with its exact order, is

\[
\delta_i\mathcal R_\epsilon
=-\mathcal R_\epsilon(\delta_iW_\epsilon)H_\epsilon\mathcal R_\epsilon
+2\left(\widehat P_\epsilon^*\mathfrak M_iQ_\epsilon
+\widehat Q_\epsilon^*\mathfrak M_iP_\epsilon\right),
\qquad \mathfrak M_i=M_i\otimes1.
\tag{MR.12}
\]

MR.9 explains the positive sign of the last two terms: its magnitude derivative has a minus sign, and differentiating the inverse contributes a second minus sign.

For the first term write \(J_\epsilon=H_\epsilon^{1/2}\mathcal R_\epsilon\) and \(T_\epsilon=H_\epsilon\mathcal R_\epsilon\). The self-adjoint derivative of \(W_\epsilon\) and MR.6 give its exact form \(-J_\epsilon(Z_{i,\epsilon}^{\mathrm{reg}})^*T_\epsilon\). The scalar bound is \(\|J_\epsilon\|\le1/\sqrt{2\lambda}\), while \(\|T_\epsilon\|\le1\).
Moreover \(J_\epsilon\) converges in norm to a compact: its complementary scalar functions of \(r_\epsilon\) are \(r_\epsilon/(1+\lambda^2r_\epsilon^4)\) and \(\lambda r_\epsilon^3/(1+\lambda^2r_\epsilon^4)\). The limiting functions vanish at zero. The \(T_\epsilon\) converge strongly with adjoints and are uniformly bounded. MR.6 and compact multiplication therefore give a compact operator-norm limit of this first term.

MR.10–MR.11 give compact norm limits of both remaining terms: a strongly convergent \(P\)-row or its adjoint is multiplied by a norm-convergent compact \(Q\)-row. The bounded \(\mathfrak M_i\) need not be compact on the entire row remainder. Their combined norm is at most \(\|M_i\|/\sqrt\lambda\). Thus MR.12 has an actual compact norm limit \(B_{i,\pm}(\lambda)\), with

\[
B_{i,\pm}(\lambda)\in\mathcal K(\mathcal F),\qquad
\|B_{i,\pm}(\lambda)\|
\le\frac{\|M_i\|+\tfrac12\|\Omega M_i\|_{2,A}}{\sqrt\lambda}.
\tag{MR.13}
\]

No unbounded product \(\mathcal R(\delta_iL)\mathcal R\) is assigned a domain by notation; this is the limit of the proved regularized expressions.

For fixed \(\epsilon\), the normal product rule for \(\mathcal R_\epsilon\) holds first on the finite-occupation normal core. Total-occupation cutoffs are parallel. The uniform full-module bound MR.13 and closedness of the tensor connection extend it to every vector in its normal graph domain. Now MR.4 and the operator-norm derivative limit give, again by closedness,

\[
\mathcal R_\pm(\lambda)\operatorname{Dom}\nabla_i
\subset\operatorname{Dom}\nabla_i,\qquad
\nabla_i(\mathcal R_\pm(\lambda)v)
=B_{i,\pm}(\lambda)v+\mathcal R_\pm(\lambda)\nabla_i v.
\tag{MR.14}
\]

This proves actual unweighted normal resolvent control on the completed coefficient module. Compact-frame invariance and the actual finite normal frame changes preserve the construction. On compact normal/arrow tests the constants are the corresponding uniform SL–PW constants. Coupling the full inverse normal-spinor block gives the local coefficient commutator bound by the sum of MR.13 over its orthonormal normal frame. The auxiliary grading stays first and the final right Clifford action stays last.

### 11AS.6. Its coefficient class and the original graph obligation

The bounded phase of \(L\) differs from \(W\) by a locally compact operator: on positive energy their scalar difference tends to zero at infinity, and it vanishes on the vacuum. The genuine affine action consequently has the same compact bounded-phase defects, with their actual compact-arrow continuity. PW.11 gives

\[
[\mathcal F,L(1+L^2)^{-1/2}]=1_A\quad\hbox{in }KK_G(A,A).
\tag{MR.15}
\]

This coefficient auxiliary is now regular self-adjoint, has locally compact resolvents, and has compact normal resolvent derivatives with the stated decay. Its original-unit balanced realization at \(R=C_0(T)\), differentiable inverse-module passage, completed scalar physical graph sum and original Bott \(+1\) comparison remain to be constructed. The coefficient graph-domain rule MR.14 alone does not identify a maximal normal graph domain with a minimal physical graph core. Any such identification needed by the chosen physical sum must be proved on that actual realization. An equivariant coefficient identity at \(A\) gives no equivariant original-unit evaluation.

### 11AS.7. Exact resolvent derivatives in the two-mode circle model

Use the exact circle example of PW.12. On \(N_1=N_2=1\), \(n=2\), at \(t=0\) its magnitude and derivative are \(H=2682I\) and \(\dot H=672I\): the derivative of \((4+\sin t)^2\) is eight, and \(4\cdot21\cdot8=672\). Its phase derivative has norm \(11/221\). Since \(\{W,\dot W\}=0\), differentiating MR.2 and using the real Clifford relations gives

\[
\|\dot{\mathcal R}_\pm(\lambda)\|^2
=\frac{672^2+2682^2\,121/221^2}{(2682^2+\lambda^2)^2}
\le\frac{49}{4096\lambda}.
\tag{MR.16}
\]

Here \(\|M_i\|=1/32\) and \(\|\Omega M_i\|_{2,A}=5/32\), so MR.13's constant is \(7/64\), exactly the squared right-hand side. To verify the left identity directly, the derivative inverse has the form \(\alpha W+\beta V+i\gamma I\), where \(V=\dot W/\|\dot W\|\) is a self-adjoint unitary anticommuting with \(W\). Its adjoint product is \((\alpha^2+\beta^2+\gamma^2)I\). The two magnitude terms combine to \(672^2/(2682^2+\lambda^2)^2\), and the tangent term contributes the other numerator. The inequality follows from the proved MR.13 estimate; every displayed integer-parameter sample can also be checked as an exact rational inequality.

![A coefficient auxiliary with normally controlled resolvents](../figures/fast-magnitude-resolvents.png)

**Figure 11AS.1.** MR.1–MR.16. The upper panels combine the actual slow phase with the fast squared magnitude and record the compact normal derivative bound. The central panel shows the completed limit mechanism. In the circle example, the blue curve is the resolvent derivative norm multiplied by the square root of its parameter; both axes are logarithmic. The green bound is exactly 7/64. Ten integer-parameter inverse identities and bounds are checked as exact fractions; the plotted curve uses numerical samples of the displayed formula. The coefficient class is at C0(Y); the original-unit graph realization remains unproved. Human context: [Connes, A survey of foliations and operator algebras](https://alainconnes.org/wp-content/uploads/foliationsfine.pdf#page=29), Section8. [Vector figure](../figures/fast-magnitude-resolvents.svg) · [Generator](../reproduction/fast-magnitude-resolvents/draw_fast_magnitude.py) · [Exact finite checks](../reproduction/fast-magnitude-resolvents/FAST-MAGNITUDE-CHECKS.json) · [Reproduction](../reproduction/fast-magnitude-resolvents/REPRODUCTION.md) · [Component terms](../reproduction/fast-magnitude-resolvents/COMPONENT-TERMS.md).

## 11AT. Normally controlled stabilization on the probability coefficients

Geometric normal connections and compact-frame descent, FD.1–FD.6, and The probability step preserves the geometric normal connection, PJ.1–PJ.5, construct a proper probability anchor and averaged real field with a dense covariant closed normal connection. We use that field here. Its measure-lift premise is proved before selecting a weight; no lift on the different spectrum of Section 11AI is presumed.

The result is a regular coefficient auxiliary and an isometric embedding whose compressed resolvents have compact normal derivatives. Its algebra is the probability algebra \(A=C_0(X)\). A compressed resolvent vanishes on the orthogonal complement of the embedding range, so it is not a resolvent on that whole ambient module. The balanced original-unit inverse module and physical graph operator remain separate constructions.

### 11AT.1. Fix the field and its completed column

Write \(Y_{\rm FD}=X_G\) for FD.1's Haar-pinned base and put \(X=\operatorname{Prob}_T(Y_{\rm FD})\), as in PJ.1. Its anchor to the original transversal \(T\) is proper; its ordinary section and whole-base contraction are PJ.2. Let \(E_X\) be PJ.3's averaged real field, with its actual affine \(G\)-action and PJ.4's closed metric normal connection. The dense scalar algebra consists of polynomial moment observables and smooth unit functions. The field core consists of averaged connection-core sections and their products with that algebra. FD.4 and PJ.3's actual compact lifted-measure resolutions prove the relations, closability and ordered normal bounds.

Pull this field back to \(P\times_T X\), where \(P\to T\) is the original oriented normal frame bundle. The free \(K=SO(q)\)-action changes the frame; its quotient is exactly \(X\). Horizontal derivatives are the lifted PJ connection; vertical derivatives act on frame coefficients. Compact frame charts and the lifted-measure paths give bounded ordered jets after compact unit localization. Holonomy covariance follows from PJ.4's chain rule and preservation of the parallelism. The \(K\)-orbits of core sections are norm smooth: in these charts they change smooth frame coefficients and a compact frame variable. Differentiation under the compact Haar integral gives their uniform orbit-jet bounds.

We check the application of the smooth orbit-column construction, JD.1–JD.6, to this particular field. Choose countably many real compact-unit core generators, including displacement jets on a compact-arrow exhaustion. Proper anchoring makes them compact coefficient sections. CW.2's square-summable all-jet choice makes a compact column \(S_0\); JD.2's compact-group evaluation estimate includes every prescribed rank-one vector below a finite multiple of \(S_0S_0^*\). For a cutoff bisection, transport its columns by the actual linear action and multiply them by the receiving cutoff. They remain compact core columns. Their derivatives are the transported derivatives plus the cutoff derivatives, because the actual parallelism and connection are preserved.

Choose positive word coefficients with sum \(Q<1\) and JD.4's simultaneous cutoff-jet sums. At word length \(n\), the squared derivative-column norm at order \(m\) is bounded by
\[
C_m(n+1)^{2m}Q^{\max(n-m,0)}.
\tag{PC.1}
\]
Leibniz assigns each derivative to one of the \(n\) cutoffs or the final column, giving at most \((n+1)^m\) terms. Cauchy–Schwarz contributes a second such factor. At most \(m\) cutoff factors are differentiated; their coefficient sums are the fixed finite jet sums. Each other factor contributes \(Q\). The final column has one of its fixed derivative bounds through order \(m\). This proves PC.1, which is summable. The full column and every ordered derivative therefore converge in Hilbert–Schmidt norm, hence in compact norm, on the completed module. No fibrewise polar map is selected.

Rescale the initial column and descend through the compact-frame quotient. Write
\[
T=S^*:E_X\longrightarrow\mathcal L,\qquad
h=T^*T,\qquad 0\le h\le\tfrac14,\qquad
\overline{hE_X}=E_X,\qquad T,\delta_iT\in\mathcal K.
\tag{PC.2}
\]
The target \(\mathcal L\) is the quotient of the actual standard column module. Its compact-frame integral factor is the associated \(L^2(P_x)\) field, with its Haar-preserving horizontal connection; word and generator indices are fixed countable sums. Its dense core consists of smooth compact-unit, smooth-frame, finite-index functions. Horizontal frame transport identifies their values along a short normal path in one Haar Hilbert space. Integration along that path proves closability; finite-index completion gives the direct-sum connection. The same argument applies to its finite tensor and Fock cores below. Thus PC.2 is a compact derivative between specified metric connection modules.

A local smooth frame section identifies each quotient fibre with its representative fibre and restricts finite-rank approximations. Compact unit partitions patch these approximations. This does not give a constant scalar weight to infinitely many Haar modes. On the finite good transversal, a disjoint union of coordinate balls, fixed frames identify the underlying \(\mathcal L\) with a countable standard coefficient module. Its geometric connection is retained.

Select \(\kappa=\phi(h)=\psi(\sqrt h)\) by OP.1–OP.12, with SL.1's Hilbert–Schmidt refinement, and set
\[
G_\phi=\sup_{0<x\le\|h\|}\frac{\sqrt x}{\phi(x)}<\infty,\qquad
C_\phi=16+4G_\phi^2.
\tag{PC.3}
\]
OP.5 proves continuous extension of this quotient by zero at zero. If the positive-occupation field is zero, the following construction is vacuous there and only the parallel vacuum remains. The completed arrow and displacement domains of OP are retained. The slow phase \(W\), fast operator \(C_{\rm f}\), and compact weighted phase derivative \(Z_i\) are those of SL/PW on this particular field. Their constructions use the closed connection, columns, compact-arrow controls and functional calculus just checked.

### 11AT.2. A smooth compact Fock map

Let \(\mathscr F_+(E_X)\) be the positive-total-occupation part of the completed bosonic/exterior field. Occupation \(n\ge1\) is the direct sum of the symmetric/exterior sectors of total degree \(n\). Define
\[
\mathcal T=\Gamma(T)|_{\mathscr F_+},\qquad
q=\mathcal T^*\mathcal T=\Gamma(h)|_{\mathscr F_+}.
\tag{PC.4}
\]
The target \(\mathscr H=\mathscr F_+(\mathcal L)\) has its tensor connection. Both maps are even and real. On occupation \(n\),
\[
\|\Gamma_n(T)\|\le\|T\|^n,\qquad
\|\delta_i\Gamma_n(T)\|\le n\|T\|^{n-1}\|\delta_iT\|.
\tag{PC.5}
\]
The derivative sum differentiates each of the \(n\) tensor factors. Symmetric and exterior projections have norm one and commute with the tensor connection. Tensor products of rank-one coefficient maps are rank one: their inner-product factors multiply to the tensor inner product. Norm approximation proves compactness of each tensor map and its derivative. PC.5's tails tend to zero. Finite occupation cutoff and finite-rank approximation consequently prove compactness of \(\mathcal T\) and its actual closed derivative on the completed whole module. In particular
\[
q,\ M_i:=\delta_iq\in\mathcal K(\mathscr F_+),\qquad
0\le q\le\tfrac14,\qquad \overline{q\mathscr F_+}=\mathscr F_+.
\tag{PC.6}
\]
Density follows first from tensor products of vectors in the dense range of \(h\), then from symmetric/exterior projection and finite occupation sums. It also proves injectivity of \(q\). The unlocalized zero-occupation identity is not assigned compactness.

The operators \(q,W,C_{\rm f}\) strongly commute. On finite joint cutoffs of \(h\), bosonic/exterior conversion preserves each total mode occupation and hence the product of its \(h\)-values. Both oscillator weights are functions of \(h\). The equality follows on the cutoff core and then through PW.2's completed resolvents and phase limits. Fibre eigenvectors can calculate this identity; no continuous eigenvector selection is used.

### 11AT.3. Dominate the fast phase weight

On such a cutoff write \(\mu_j\) for occupied \(h\)-values and \(N_j\) for the nonnegative integer total mode occupations. They obey \(\sum N_j=n\) and \(0<\mu_j\le1/4\). Put \(a_n=1+2n\) and \(\theta_j=\phi(\mu_j)^{-1}\). Then
\[
C_{\rm f}^2=2\sum_j(a_n+\theta_j)^2N_j,\qquad
q=\prod_j\mu_j^{N_j}.
\tag{PC.7}
\]
For the smallest occupied value \(\mu_*\), \(q^{-1}\ge4^{n-1}\mu_*^{-1}\) and \(q^{-1}\ge4^n\). Also \(\theta_j^2\le G_\phi^2\mu_j^{-1}\). The scalar bounds
\[
n\le4^{n-1},\qquad n(1+2n)^2\le4\cdot4^n
\]
give
\[
C_{\rm f}^2\le C_\phi q^{-1},\qquad
C_{\rm f}q^{1/2}\in\mathcal L,\qquad
\|C_{\rm f}q^{1/2}\|\le\sqrt{C_\phi}.
\tag{PC.8}
\]
For the second scalar bound check \(n=1,2\); for \(n\ge2\), the successive ratio is at most four, since
\(4n(2n+1)^2-(n+1)(2n+3)^2=12n^3-17n-9\ge0\).
The first bound follows by induction. Apply \((a+\theta)^2\le2a^2+2\theta^2\) in PC.7. The \(a_n\)-part is at most \(16q^{-1}\); the reciprocal part is at most \(4G_\phi^2q^{-1}\). Uniform cutoff bounds and closedness of \(C_{\rm f}\) prove the completed graph map in PC.8.

Moreover
\[
C_{{\rm f},\epsilon}q^{1/2}\longrightarrow C_{\rm f}q^{1/2}
\quad\hbox{in norm},\qquad C_{\rm f}q^{1/2}\in\mathcal K.
\tag{PC.9}
\]
At fixed occupation its reciprocal coefficient times \(\sqrt\mu\) is
\(\sqrt\mu/(\phi(\mu)+\epsilon)\), converging uniformly to OP.5's continuous zero extension. Cutting off occupied \(\mu\)-values near zero gives compact approximation. For the occupation tail, the \(a_n\)-part is bounded by a constant times \(n^{3/2}2^{-n}\); the reciprocal part is bounded by a constant times \(\sqrt{n/4^{n-1}}\). Both tend to zero. The same bounds hold for the cutoffs because their reciprocal coefficients decrease. First discard the occupation tail and then apply the fixed-occupation argument. This proves PC.9 on actual tensor finite-rank cutoffs.

### 11AT.4. A magnitude that fits the embedding

On positive occupation set
\[
L=Wq^{-2},\qquad \operatorname{Dom}L=\operatorname{Ran}q^2,\qquad
R_\pm(\lambda)=q^2(W\pm i\lambda q^2)^{-1}.
\tag{PC.10}
\]
The inverse in parentheses is
\((W\mp i\lambda q^2)(1+\lambda^2q^4)^{-1}\), of norm at most one. The two \(R_\pm\) are mutually adjoint, have norm at most \(\lambda^{-1}\), and range exactly \(\operatorname{Ran}q^2\). Their inverse and resolvent identities prove the completed regular self-adjoint closure, with \(|L|=q^{-2}\). Each resolvent is compact. Thus two actual factors are available:
\[
R_\pm=q^2B_\pm=B_\pm q^2,\qquad
B_\pm=(W\pm i\lambda q^2)^{-1},\qquad \|B_\pm\|\le1.
\tag{PC.11}
\]
On the parallel even vacuum keep \(L=0\) and its resolvent \(\mp i\lambda^{-1}\). Multiplication by \(a\in A\) makes it compact. The whole auxiliary is regular self-adjoint on the direct sum of that vacuum and PC.10's domain.

The bounded transform differs compactly from \(W\) on positive occupation by
\(W\{(1+q^4)^{-1/2}-1\}\), whose scalar function vanishes at zero. The genuine affine action and coefficient cycle class therefore remain those of the slow phase, with coefficient identity \(1_A\). This stronger magnitude does not by itself imply invariance of its unbounded domain under every arrow.

### 11AT.5. Compact normal resolvent derivatives

Use the bounded slow phases \(W_\epsilon\) of PW/MR and let \(D_{i,\epsilon}=\delta_iW_\epsilon\). Their weighted derivatives obey
\[
D_{i,\epsilon}q^{1/2}
=Z_{i,\epsilon}^{\rm reg}C_{{\rm f},\epsilon}q^{1/2}
\longrightarrow Y_i:=Z_iC_{\rm f}q^{1/2}\in\mathcal K,\qquad
\|Y_i\|\le\sqrt{C_\phi}\|Z_i\|.
\tag{PC.12}
\]
MR.7 and PC.9 give the norm convergence. The equality first holds on finite occupation cutoffs and extends by PC.8.

Choose scalars \(\rho_\epsilon\to0\) with
\(\sqrt{\rho_\epsilon}\max_i\|D_{i,\epsilon}\|\le\epsilon\), for the finite frame derivative list; take
\(\rho_\epsilon=\epsilon^2/(1+\max_i\|D_{i,\epsilon}\|^2)\).
Put \(u=q+\rho_\epsilon\), \(m=u^{-2}\), \(L_\epsilon=W_\epsilon m\), and
\(\widetilde R_\pm=(L_\epsilon\pm i\lambda)^{-1}\).
These are bounded differentiable operators at each cutoff. Since
\(\|u^{1/2}-q^{1/2}\|\le\sqrt{\rho_\epsilon}\), the operators
\(Y_{i,\epsilon}=D_{i,\epsilon}u^{1/2}\) converge in norm to PC.12's compact \(Y_i\). The phase cutoff error is controlled, without assuming an unweighted derivative of \(W\).

Differentiate the bounded inverse in its actual order:
\[
\begin{split}
\delta_i\widetilde R_\pm={}&
-\widetilde R_\pm u^{-1/2}Y_{i,\epsilon}^*m\widetilde R_\pm\\
&+\widetilde R_\pm W_\epsilon u^{-1}M_i u^{-2}\widetilde R_\pm
+\widetilde R_\pm W_\epsilon u^{-2}M_i u^{-1}\widetilde R_\pm .
\end{split}
\tag{PC.13}
\]
Indeed \(\delta m=-u^{-1}M_i u^{-2}-u^{-2}M_i u^{-1}\). The negative inverse derivative gives the two positive terms. In the phase term use the adjoint identity \(D_{i,\epsilon}=u^{-1/2}Y_{i,\epsilon}^*\). Neither middle factor has been commuted.

The scalar bounds are
\[
\|u^{-1}\widetilde R_\pm\|\le(2\lambda)^{-1/2},\quad
\|u^{-2}\widetilde R_\pm\|\le1,\quad
\|u^{-1/2}\widetilde R_\pm\|\le\sigma\lambda^{-3/4},\qquad
\sigma=(1/4)^{1/8}(3/4)^{3/8}.
\tag{PC.14}
\]
The first follows by maximizing \(v/(v^4+\lambda^2)^{1/2}\). For the last, put \(z=u^{-2}/\lambda\) and maximize \(z^{1/4}/(1+z^2)^{1/2}\), at \(z^2=1/3\).

Every term in PC.13 has a compact norm limit. The factors \(u^{-1}\widetilde R_\pm\) and \(u^{-1/2}\widetilde R_\pm\) converge in norm to compacts: their scalar functions of \(u\) vanish at zero, and strong-adjoint convergence of \(W_\epsilon\) becomes norm convergence after compact multiplication, by rank-one approximation. The factors \(u^{-2}\widetilde R_\pm\) are uniformly bounded and converge strongly with their adjoints. Their functions have a nonzero value at zero, so compact norm convergence is not assigned to them. The other middle factors are the compact \(M_i\) and the norm limit \(Y_i\). Compact-times-strong-adjoint convergence therefore proves the whole-product norm limits, giving
\[
\delta_iR_\pm(\lambda)\in\mathcal K,\qquad
\|\delta_iR_\pm(\lambda)\|
\le\sqrt2\|M_i\|\lambda^{-1/2}
+\sigma\sqrt{C_\phi}\|Z_i\|\lambda^{-3/4}.
\tag{PC.15}
\]
The regularized resolvents converge in norm to PC.10 by the same compact scalar functions and phase argument. Closedness of the normal connection proves
\[
R_\pm\operatorname{Dom}\nabla_i\subset\operatorname{Dom}\nabla_i,\qquad
\nabla_i(R_\pm v)=(\delta_iR_\pm)v+R_\pm\nabla_i v.
\tag{PC.16}
\]
At each cutoff this is the bounded inverse product rule on the core and its closure. Passing to both norm limits on each domain vector gives PC.16. The vacuum is parallel. These are completed domain assertions.

### 11AT.6. The weighted stabilization and its compact transfer

Let \(b=(1-q)^{1/2}\) and define
\[
J_Nv=(\mathcal T b^nv)_{0\le n\le N},\qquad
Jv=(\mathcal T b^nv)_{n\ge0}:
\mathscr F_+(E_X)\longrightarrow\mathscr H^\infty .
\tag{PC.17}
\]
The geometric sum gives \(J_N^*J_N=1-b^{2N+2}\). Dense range of \(q\) makes the last term strongly zero: check first on \(\operatorname{Ran}q\), where \(\|b^nq\|\to0\), then use density and contractivity. Thus \(J\) is an isometry. Its adjoint on finite columns is \(\sum b^n\mathcal T^*v_n\); the finite bounds extend it to the whole target. Both \(J_N\) and their adjoints converge strongly. For compact \(K\), \((J-J_N)K\) and \(K(J^*-J_N^*)\) converge in norm, first on rank-one maps and then by approximation.

The binomial series gives compact \(\delta_i b\), equivalently
\[
\delta_i b=-\int_0^\infty e^{-sb}M_i e^{-sb}\,ds,\qquad
\|\delta_i b\|\le\frac{\|M_i\|}{2\sqrt{1-\|q\|}}.
\]
Its norm integral converges by the lower bound on \(b\). Differentiating in \(s\) gives the Sylvester identity \(b\delta_i b+(\delta_i b)b=-M_i\); the same lower bound proves uniqueness and identifies the derivative.

Let \(\mathfrak M=\sum_{n\ge0}\|b^nq^2\|\) and
\(\mathfrak L=(\sum_{n\ge0}\|b^nq^2\|^2)^{1/2}\).
They are finite: for \(n\ge1\), the summands are at most \(16/(e^2n^2)\), by \((1-x)^{n/2}\le e^{-nx/2}\).
The differentiated \(n\)-th column, multiplied by \(q^2\), is
\[
(\delta_i\mathcal T)b^nq^2+
\sum_{k=0}^{n-1}\mathcal T b^k(\delta_i b)b^{n-1-k}q^2.
\]
The first column series is square summable. In the second put \(m=n-1-k\). For each fixed \(m\), the whole column is a coordinate shift of \(J(\delta_i b)b^mq^2\), with norm at most \(\|\delta_i b\|\|b^mq^2\|\). The \(m\)-sum converges in norm. Each fixed-\(m\) finite-column tail converges in norm because its right factor is compact. First discard a summably small \(m\)-tail and then the finitely many compact tails. This proves
\[
(\delta_iJ_N)q^2\longrightarrow Q_i\in\mathcal K,\qquad
\|Q_i\|\le\|\delta_i\mathcal T\|\mathfrak L+\|\delta_i b\|\mathfrak M.
\tag{PC.18}
\]
It is a weighted derivative, not a bounded derivative of \(J\) or \(JJ^*\).

Apply both exact factors PC.11. The compacts \(J_NR_\pm J_N^*\) and their derivatives converge in norm, giving
\[
\delta_i(JR_\pm J^*)=
Q_iB_\pm J^*+J(\delta_iR_\pm)J^*+JB_\pm Q_i^*\in\mathcal K.
\tag{PC.19}
\]
In the first term use \((\delta_iJ_N)R_\pm=(\delta_iJ_N)q^2B_\pm\), PC.18 and compact multiplication with \(J_N^*\). In the last use \(R_\pm=B_\pm q^2\) and the adjoint of PC.18. The middle uses PC.15 and compact multiplication on both sides. Closedness identifies the derivative and proves its full normal-domain product rule on the ambient module. Its norm is at most \(2\|Q_i\|+\|\delta_iR_\pm\|\). Add the parallel vacuum into a fixed even coefficient coordinate; its compressed resolvent has zero operator derivative. This keeps the positive kernel without demanding that a positive-occupation inverse weight dominate a zero-energy vacuum.

The range has its transported field and grading. PC.19 is nevertheless zero on the complementary module. Its whole-module operator and equivariant coefficient absorption are constructed in Completing the coefficient stabilization with a normally controlled complement. The differentiable \(B\)-source inverse module remains a separate construction. In PF.1–PF.3, \(K_X\) represents \(d_X\in KK_G(B,A)\). Equality of its right probability algebra with this \(A\) does not identify that module or its \(B\)-representation with this coefficient Fock cycle. No physical sum, equivariant scalar evaluation or original Bott \(+1\) has been inferred.

### 11AT.7. Exact scalar test

Take \(A=C_0((0,1/2])\), \(E=\mathcal L=A\), \(T(t)=t^2\). On occupation \(n\ge1\), \(\mathcal T=t^{2n}I\), \(q=t^{4n}I\). The single-mode slow phase is
\(Q=\left(\begin{smallmatrix}0&1\\1&0\end{smallmatrix}\right)\)
on the bosonic/exterior pair, independently of its positive oscillator coefficient. Thus
\[
L_n=Qt^{-8n},\quad
\|R_{\pm,n}\|=\frac{t^{8n}}{\sqrt{1+\lambda^2t^{16n}}},\quad
\|R_{\pm,n}'\|=\frac{8nt^{8n-1}}{1+\lambda^2t^{16n}},\quad
\|\delta q_n\|=4nt^{4n-1}.
\tag{PC.20}
\]
The inverse derivative is \(8nt^{-8n-1}R_{\pm,n}QR_{\pm,n}\), since \(L_n'=-8nt^{-8n-1}Q\). Its squared norm and PC.15's magnitude bound compare:
\[
\frac{64n^2t^{16n-2}}{(1+\lambda^2t^{16n})^2}
\le\frac{32n^2t^{8n-2}}{\lambda}.
\]
After canceling factors this is \(2a\le(1+a^2)^2\), \(a=\lambda t^{8n}\). Their difference is \((1-a)^2+a^2+a^4\ge0\). Also \(\sup_{n,t}\|\delta q_n\|=1/2\): \(4n2^{1-4n}\) decreases with \(n\), and equality occurs at \(n=1,t=1/2\).

The literal infinite stabilization columns give
\[
\|J_n'\|^2=\frac{4n^2}{t^2(1-t^{4n})},\quad
\|J_n'q_n^2\|^2=\frac{4n^2t^{16n-2}}{1-t^{4n}},\quad
\langle J_n,J_n'\rangle=0.
\tag{PC.21}
\]
Sum the geometric series and its first two derivatives for
\(t^{2n}(1-t^{4n})^{k/2}\), \(k\ge0\), to obtain these identities.
The unweighted derivative diverges at zero while its weighted version vanishes. At \(n=1,t=1/2,\lambda=256\), the exact resolvent derivative square is \(1/1024\), its bound square is \(1/512\), the weighted isometry derivative square is \(1/3840\), and \(\|B_\pm\|^2=1/2\).

There is an exact check of the compressed derivative without truncating the infinite column. In either \(Q\)-eigenspace, with \(r=(\pm t^{-8n}+i\lambda)^{-1}\) and \(a=\|J_n'\|\), it is
\(\left(\begin{smallmatrix}r'&ra\\ra&0\end{smallmatrix}\right)\)
on the orthonormal span of \(J_n,J_n'/a\), and zero outside. Its largest singular value squared is
\[
\frac{|r'|^2+2|r|^2a^2+
\sqrt{|r'|^4+4|r'|^2|r|^2a^2}}2.
\tag{PC.22}
\]
At that sample the value is \((19+\sqrt{345})/30720\). It is less than \(19/15360\), because \(\sqrt{345}<19\), whereas the square of the transfer bound \(1/\sqrt{1920}+1/32\) exceeds \(23/15360\). This two-dimensional matrix is a proved reduction of the infinite-column derivative.

![Normally controlled stabilization on the probability coefficients](../figures/probability-coercive-stabilization.png)

**Figure 11AT.1.** PC.1–PC.22. The upper panels use the actual FD/PJ probability field, smooth column and positive-occupation Fock map. The middle panels show the exact weighted resolvent factors and the completed compact derivative transfer. In the scalar example n=1 and lambda=256; both axes are logarithmic. The red unweighted column derivative diverges at zero, while the blue weighted column derivative and green compressed resolvent derivative vanish. The curves are201 numerical evaluations of the proved infinite-column formulas, not finite column norms. At t=1/2 the compressed derivative square is exactly (19+sqrt(345))/30720. Thirty-six exact matrix inverse, derivative and bound checks accompany the diagram. TF denotes the Fock map. The original inverse module and physical graph sum remain to be constructed. Human context: [Connes, A survey of foliations and operator algebras](https://alainconnes.org/wp-content/uploads/foliationsfine.pdf#page=29), Section8. [Vector figure](../figures/probability-coercive-stabilization.svg) · [Generator](../reproduction/probability-coercive-stabilization/draw_probability_stabilization.py) · [Exact checks](../reproduction/probability-coercive-stabilization/PROBABILITY-STABILIZATION-CHECKS.json) · [Reproduction](../reproduction/probability-coercive-stabilization/REPRODUCTION.md) · [Component terms](../reproduction/probability-coercive-stabilization/COMPONENT-TERMS.md).

## 11AU. Completing the coefficient stabilization with a normally controlled complement

The probability-coefficient construction, PC.1–PC.19, gives a regular auxiliary and compact normal derivatives of its compressed resolvents. Here we construct its orthogonal complement, place a confining operator on that complement, and obtain resolvents on the entire ambient coefficient module.

The coefficient algebra remains \(A=C_0(X)\), with the actual FD/PJ probability field. The result represents the equivariant coefficient identity \(1_A\). It does not identify the \(B\)-source inverse \(d_X\) in the proper factorization, PF.1–PF.3. That identification, the original \(C_0(T)\)-balanced graph sum, and its oriented Bott comparison require their own arguments.

### 11AU.1. A Hilbert–Schmidt choice of the Fock column

Use PC.1's word column, with its real metric connections, actual compact-frame quotient and dense-range \(h=T^*T\). The column and its first derivatives are in the completed Hilbert–Schmidt module. Multiplying the initial column by one positive constant preserves density, all word-jet estimates and every transport order comparison. Choose that constant so that
\[
\sup_{x\in X}\operatorname{Tr}h_x=\|T\|_{2,A}^2\le\rho\le\tfrac14.
\tag{CSB.1}
\]
Indeed each undifferentiated transported word has the same trace bound as its initial column, multiplied by its cutoff squares and word coefficient; the sum is bounded by the initial trace bound divided by \(1-Q\). Equivalently, rescale the already convergent Hilbert–Schmidt column. Make this choice before selecting PC.3's weight and phase. No restriction on the foliation or new measure-lift premise is imposed.

The square-root argument in Smooth orbit compacts and the completed oscillator cross form, JD.13–JD.15, proves convergence in Hilbert–Schmidt norm, so
\(k=h^{1/2}\) and \(\delta_i k\) belong to that module. Set
\[
E=\mathscr F_+(E_X),\qquad H=\mathscr F_+(\mathcal L),\qquad
V=\Gamma(T)|_+,\qquad q=V^*V,\qquad k_q=q^{1/2}=\Gamma(k)|_+.
\]
These are the same positive-occupation constructions as PC.4. On total occupation \(n\), there are at most \(n+1\) symmetric/exterior sectors. Projection onto one such sector contracts the Hilbert–Schmidt norm. On the full tensor product that squared norm is the product of the squared factor norms. Differentiating the \(n\) factors and applying the triangle inequality gives
\[
\|\Gamma_n(T)\|_{2,A}^2\le(n+1)\rho^n,\qquad
\|\delta_i\Gamma_n(T)\|_{2,A}^2
 \le(n+1)n^2\|\delta_iT\|_{2,A}^2\rho^{n-1}.
\tag{CSB.2}
\]
The identical estimate applies to \(k\) and \(\delta_i k\), since
\(\|k\|_{2,A}^2=\sup_x\operatorname{Tr}h_x\). Both series converge. Finite tensor cores, finite occupation truncation and the closed connection therefore prove
\[
V,\ \delta_iV,\ k_q,\ \delta_i k_q\in\mathcal K_2,\qquad
0\le q\le\tfrac14,\qquad \overline{qE}=E.
\tag{CSB.3}
\]
Here \(\mathcal K_2\) denotes the completed Hilbert–Schmidt coefficient module, whose norm dominates compact-operator norm. The estimates are fibre trace calculations used to bound actual continuous sections. Their finite-column approximations and uniform tails establish module membership; they do not select continuous eigenvectors or a module polar decomposition.

### 11AU.2. An explicit unitary complement

Write \(b=(1-q)^{1/2}\), \(d=(1-VV^*)^{1/2}\), and let \(S\) be the parallel unilateral coordinate shift on \(H^\infty=\bigoplus_{n\ge0}H\). The isometry from PC.17 is
\(Je=(Vb^ne)_{n\ge0}\). Define
\[
K_0y=(dy,-VV^*y,-VbV^*y,-Vb^2V^*y,\ldots)
     =(dy,-JV^*y),\qquad K_n=S^nK_0.
\tag{CSB.4}
\]
All maps are even, real and adjointable. The functional-calculus intertwining
\(dV=Vb\) follows first for polynomials from
\((VV^*)V=Vq\), and then by uniform approximation. Thus
\[
K_0^*K_0=d^2+VJ^*JV^*=1,\qquad
J^*K_0=V^*d-bV^*=0.
\]
For \(m\ge1\), the first overlapping term of \(K_0^*S^mK_0\) is
\(-Vb^mV^*\). Its remaining terms sum to
\(Vb^m(\sum_{\ell\ge0}qb^{2\ell})V^*=Vb^mV^*\).
The geometric series converges strongly to one by PC.17; multiplication by the compact \(V,V^*\) permits the norm limits in this computation. Consequently
\[
K_n^*K_m=\delta_{nm}1_H,\qquad J^*K_n=0.
\tag{CSB.5}
\]
To see the last assertion for every \(n\), observe
\(J^*S^n=b^nJ^*\). Finite column sums define a contractive map
\(K:H^\infty\to H^\infty\) with columns \(K_n\). The sum is an isometry by CSB.5. On a target vector supported in coordinates \(0,\ldots,N\), \(K_n^*\) vanishes for \(n>N\). This proves strong convergence of the finite adjoint columns on a dense core, hence on the entire target.

There is no missing range. At each coordinate use the unitary cell
\[
(e_n,y_n)\longmapsto
(z_n,e_{n+1})=(Ve_n+dy_n,\ be_n-V^*y_n):
E\oplus H\longrightarrow H\oplus E.
\tag{CSB.6}
\]
Its two row and column products are identities by \(dV=Vb\), \(b^2=1-q\) and \(d^2=1-VV^*\). Given a finitely supported output \(z_0,\ldots,z_N\), put \(e_{N+1}=0\) and apply the inverse cells backwards:
\(e_n=V^*z_n+be_{n+1}\), \(y_n=dz_n-Ve_{n+1}\).
Then \(e_0=J^*z\), \(y_n=K_n^*z\), and the forward cascade produces \(z\) with zero subsequent carrier. Thus the closed isometric range contains every finite output. We have the actual module unitary
\[
U=(J,K):E\oplus H^\infty\xrightarrow{\ \cong\ }H^\infty,\qquad
JJ^*+KK^*=1.
\tag{CSB.7}
\]
This is a specified infinite-column construction with specified adjoints, rather than an arbitrary Hilbert-space identification.

### 11AU.3. The endpoint derivative needed by the complement

The derivative of \(J\) itself need not be bounded. We instead prove that
\((\delta_iJ_N)V^*\) converges in Hilbert–Schmidt norm. Fix \(i\), put
\(X=\delta_i k_q\), and differentiate the finite columns:
\[
[(\delta_iJ_N)V^*]_n
=(\delta_iV)b^nV^*
+\sum_{k+m=n-1}Vb^k(\delta_i b)b^mV^*,\qquad 0\le n\le N.
\tag{CSB.8}
\]
For the first column sum, fibre trace cyclicity and
\(\sum_{n\ge0}qb^{2n}=1\) give squared Hilbert–Schmidt norm
\(\|\delta_iV\|_{2,A}^2\). Its tails tend to zero in that norm: the remaining trace is the squared Hilbert–Schmidt norm of
\((\delta_iV)b^{N+1}\); strong-adjoint convergence of \(b^{N+1}\) gives uniform norm convergence after Hilbert–Schmidt multiplication. This follows first on finitely many rank-one columns and then by a small Hilbert–Schmidt tail.

For the triangular sum work in a single fibre, using the eigenvalues
\(\mu,\nu\) of \(q\) only to calculate its bound. Put
\(a=\sqrt{1-\mu}\), \(c=\sqrt{1-\nu}\). The identity
\(b\,\delta_i b+(\delta_i b)b=-\delta_iq\), together with
\(\delta_iq=k_qX+Xk_q\), gives
\[
(\delta_i b)_{\mu\nu}
=-\frac{\sqrt\mu+\sqrt\nu}{a+c}X_{\mu\nu}.
\]
The two outer \(V\)'s contribute \(\sqrt{\mu\nu}\) in the squared Hilbert–Schmidt calculation. Their fibre polar isometries are used only for this scalar estimate. For \(0\le a,c<1\),
\[
\sum_{n\ge1}\left(\sum_{j=0}^{n-1}a^jc^{n-1-j}\right)^2
=\frac{1+ac}{(1-a^2)(1-c^2)(1-ac)}.
\tag{CSB.9}
\]
For \(a\ne c\), substitute \((a^n-c^n)/(a-c)\) and sum the three geometric series. Simplification gives the displayed expression. For \(a=c\), use
\(\sum_{n\ge1}n^2(a^2)^{n-1}=(1+a^2)/(1-a^2)^3\), obtained by differentiating the geometric series twice. This is also the continuous diagonal value.

It follows that the coefficient multiplying \(|X_{\mu\nu}|^2\) in the full triangular squared norm is
\[
F(\mu,\nu)=
\frac{(1+ac)(\sqrt\mu+\sqrt\nu)^2}
     {(1-ac)(a+c)^2}\le\frac83.
\tag{CSB.10}
\]
Indeed \(a+c\ge\sqrt3\), \(1+ac\le2\),
\(1-ac\ge(\mu+\nu)/2\), and
\((\sqrt\mu+\sqrt\nu)^2\le2(\mu+\nu)\).
The third inequality follows from
\(\sqrt{(1-\mu)(1-\nu)}\le1-(\mu+\nu)/2\).
These estimates also bound every finite triangular truncation.

They prove convergence on the completed module, as follows. Approximate \(X\) in Hilbert–Schmidt norm by
\(e_\tau(q)Xe_\tau(q)\), where \(e_\tau\) is a scalar plateau vanishing near zero. Dense range of \(q\), finite rank-one approximation and then a small Hilbert–Schmidt tail prove this approximation on both sides. On its retained spectral rectangle \(\mu,\nu\ge\tau\), the geometric tails in CSB.9 converge uniformly. The corresponding finite triangular operators are compositions of actual \(V\)'s, plateau functions and left/right functions of \(q\); their limiting bound is therefore a module bound. On the discarded part CSB.10 gives uniform error at most \(\sqrt{8/3}\|X-e_\tau Xe_\tau\|_{2,A}\). First make this small and then take the finite-column tail. Hence
\[
(\delta_iJ_N)V^*\longrightarrow A_i\in\mathcal K_2,\qquad
\|A_i\|_{2,A}\le
\|\delta_iV\|_{2,A}+\sqrt{8/3}\|\delta_i k_q\|_{2,A}.
\tag{CSB.11}
\]
No extension of an unbounded derivative of \(J\) was used.

The finite maps \(J_NV^*\) converge in operator norm to \(JV^*\), by compact multiplication. Their derivative's remaining term
\(J_N(\delta_iV)^*\) converges in Hilbert–Schmidt norm to
\(J(\delta_iV)^*\). The gapped square root \(d\) is differentiable and solves
\(d\,\delta_i d+(\delta_i d)d=-\delta_i(VV^*)\); its exponential-integral solution gives
\[
\|\delta_i d\|_{2,A}\le
\frac{\|V\|}{\sqrt{1-\|q\|}}\|\delta_iV\|_{2,A}
\le\frac1{\sqrt3}\|\delta_iV\|_{2,A}.
\]
Closing the metric connection in CSB.4 now proves
\[
\delta_iK_0\in\mathcal K_2,\qquad
\|\delta_iK_n\|\le C_i,\qquad
C_i=(2+1/\sqrt3)\|\delta_iV\|_{2,A}
       +\sqrt{8/3}\|\delta_i k_q\|_{2,A}.
\tag{CSB.12}
\]
The shift is parallel, so \(\delta_iK_n=S^n\delta_iK_0\).
Each \(K_n\) has the closed normal-domain product rule. A bounded derivative of the entire \(K\), \(J\), \(U\) or \(JJ^*\) is not required.

### 11AU.4. A parallel confining operator on the complementary input

The field \(\mathcal L\) of PC.2 is associated to the actual normal-frame bundle, with fixed countable indices and Haar Hilbert space. In its fixed real model \(L_0\), choose a countable dense collection of smooth frame functions with finite index support. Positive rank-one coefficients can be chosen so rapidly decreasing that the sum and every prescribed frame derivative converge in norm. Average that positive compact over \(K=SO(q)\). It is \(K\)-invariant and real. It is strictly positive: for any nonzero vector its unaveraged quadratic form is positive at the identity, remains positive on an open set of \(K\), and has positive Haar integral. Its kernel is zero, hence its range is dense, since the orthogonal complement of the range of a bounded self-adjoint operator is its kernel. Rescale it to an operator
\(r_0\) of norm at most \(1/2\).

This constant \(K\)-invariant operator defines a parallel compact in each frame chart. Transition functions preserve it, and the horizontal connection differentiates only the frame representation, with which it commutes. The summable frame derivatives ensure its compatibility with the smooth-frame core. The original holonomy frame action preserves it. On a noncompact coefficient base the assertion is \(a r_0\in\mathcal K_A\) for \(a\in A\), rather than global compactness of a constant field.

On \(H=\mathscr F_+(\mathcal L)\), put \(r=\Gamma(r_0)|_+\). Finite tensor approximations and the tail \(\|r_0\|^n\) prove localized compactness; dense range follows from finite tensor density. It is parallel, real and even, with norm at most \(1/2\). Let \(Q_H=\operatorname{sign}(D_H)|_+\) be the unweighted bosonic/exterior oscillator sign. The oscillator formulas in A confining oscillator over a varying Hilbert field give \(D_H^2=2N\). Thus \(Q_H\) is an odd real self-adjoint unitary on positive occupation, is parallel, and is invariant under the linear frame action. It commutes strongly with \(r\): bosonic/exterior conversion preserves each total mode occupation, as in PC.4–PC.6.

Define
\[
C_0=Q_Hr^{-2},\quad\operatorname{Dom}C_0=\operatorname{Ran}r^2,
\qquad
C_{\mathrm c}=\bigoplus_{m\ge0}a_m C_0,\quad a_m=2^{m+1}.
\tag{CSB.13}
\]
The explicit \(C_0\) resolvents
\(r^2(Q_H\pm i\lambda r^2)^{-1}\) have dense range, the resolvent identities and mutually adjoint values. They prove regular self-adjointness exactly as in PC.10. Moreover \(|C_0|\ge4\). The direct sum has domain consisting of coordinate domain vectors for which
\(\sum_m\langle a_mC_0y_m,a_mC_0y_m\rangle\) converges in \(A\). Its diagonal resolvents \(R_{\mathrm c,\pm}=\bigoplus_mR_{\mathrm c,m,\pm}\) are mutually adjoint, have dense range by the finite-coordinate graph core, and satisfy the resolvent identity. This proves the regular self-adjoint direct sum with that completed domain. Each diagonal resolvent is parallel and
\[
\|R_{\mathrm c,m,\pm}(\lambda)\|
\le(16a_m^2+\lambda^2)^{-1/2},\qquad
\sum_{m\ge0}\|R_{\mathrm c,m,\pm}\|^2\le\frac1{8\lambda}.
\tag{CSB.14}
\]
The second estimate uses \(16a_m^2+\lambda^2\ge8a_m\lambda\) and
\(\sum_m a_m^{-1}=1\). Its tails go to zero in operator norm, while each finite block is locally compact. Hence \(aR_{\mathrm c,\pm}\) is compact for every \(a\in A\). The complementary bounded transform differs locally compactly from the exactly invariant degenerate phase \(\bigoplus_mQ_H\). Indeed
\(|a_mC_0|\ge4a_m\), so the difference's block norms are at most
\(1/(32a_m^2)\). The phase is odd, self-adjoint, squares to one and commutes with both coefficient multiplication and the linear frame action.

### 11AU.5. Actual full resolvents and their normal domains

For a finite number of complementary columns differentiate
\(K_{\le M}R_{\mathrm c,\pm}K_{\le M}^*\). The diagonal operator is parallel. Define the row from the complementary input to the ambient target by its coordinate maps
\[
\mathcal A_{i,\pm}y
=\sum_{m\ge0}(\delta_iK_m)R_{\mathrm c,m,\pm}y_m.
\]
Its tail has norm at most
\(C_i(\sum_{m\ge M}\|R_{\mathrm c,m,\pm}\|^2)^{1/2}\).
For finite sums this follows from the C*-module row estimate
\(\|\,[B_m]\,\|^2=\|\sum_m B_mB_m^*\|\le\sum_m\|B_m\|^2\);
pass to the norm completion. Every coordinate map is compact by CSB.12, so \(\mathcal A_{i,\pm}\) is compact. The finite compressions themselves converge in norm because the diagonal resolvent tail tends to zero. In the differentiated terms compact-times-strong-adjoint convergence of \(K_{\le M}\) identifies the limit. Closedness gives
\[
\delta_i(KR_{\mathrm c,\pm}K^*)
=\mathcal A_{i,\pm}K^*+K\mathcal A_{i,\mp}^*
\in\mathcal K,\qquad
\|\delta_i(KR_{\mathrm c,\pm}K^*)\|
\le\frac{C_i}{\sqrt{2\lambda}}.
\tag{CSB.15}
\]
The opposite sign in the adjoint is essential.

Use PC.10's \(L\) on \(E\). On \(H^\infty\) the operator
\[
\widehat L_+=U(L\oplus C_{\mathrm c})U^*,\qquad
\operatorname{Dom}\widehat L_+
=U(\operatorname{Dom}L\oplus\operatorname{Dom}C_{\mathrm c})
\]
is regular self-adjoint and odd. Its actual whole-module resolvents are
\[
\widehat R_\pm=JR_\pm J^*+KR_{\mathrm c,\pm}K^*.
\tag{CSB.16}
\]
CSB.7 proves their inverse and resolvent identities by conjugation; they no longer vanish on the complementary range. They are locally compact. PC.19 and CSB.15 give compact operator derivatives on the whole ambient module, and the closed normal connection gives, for every domain vector,
\[
\widehat R_\pm\operatorname{Dom}\nabla_i
 \subset\operatorname{Dom}\nabla_i,\qquad
\nabla_i(\widehat R_\pm z)
=(\delta_i\widehat R_\pm)z+\widehat R_\pm\nabla_i z.
\tag{CSB.17}
\]
This is a completed-domain conclusion, not a statement just on finite coordinates.

There is useful decay. PC.18's proof works with \(q^p\) for every \(p>1\): \(\sum_n\|b^nq^p\|<\infty\), using
\(\|b^nq^p\|\le(2p/(en))^p\), and the differentiated triangular sum is a summable collection of shifted isometries with compact right factors. Let
\(Q_{i,3/2}=\lim_N(\delta_iJ_N)q^{3/2}\in\mathcal K\).
In PC.19's first and last terms use the two separate factors
\(R_\pm=q^{3/2}q^{1/2}B_\pm=B_\pm q^{1/2}q^{3/2}\).
Maximizing \(z^{1/4}/\sqrt{1+z^2}\) at \(z^2=1/3\) gives
\(\|q^{1/2}B_\pm\|\le\sigma\lambda^{-1/4}\), with PC.14's
\(\sigma=(1/4)^{1/8}(3/4)^{3/8}\). Therefore
\[
\|\delta_i\widehat R_\pm\|
\le2\sigma\|Q_{i,3/2}\|\lambda^{-1/4}
 +\left(\sqrt2\|\delta_iq\|+C_i/\sqrt2\right)\lambda^{-1/2}
 +\sigma\sqrt{C_\phi}\|Z_i\|\lambda^{-3/4}.
\tag{CSB.18}
\]
No unweighted derivative of \(U\) occurs in this estimate.

### 11AU.6. The vacuum and the genuine equivariant coefficient cycle

Adjoin the parallel even vacuum \(A\) to both sides. The unitary
\(\widehat U=1_A\oplus U\) identifies
\((A\oplus E)\oplus H^\infty\) with \(A\oplus H^\infty\).
Put zero on the vacuum. Its resolvent is \(\mp i/\lambda\), compact only after coefficient localization, with zero operator derivative.

Use the genuine affine action on the entire source \(A\oplus E\), and the genuine linear frame action on the complementary input. Transport their direct sum by \(\widehat U\). This defines a continuous genuine groupoid action on the target: local continuous source sections are carried to target sections by an actual adjointable unitary, and composition is preserved by conjugation. The affine action can mix vacuum and positive occupation, so no affine action on \(E\) alone is asserted. On the finite good transversal, the fixed frames of PC.2 identify \(\mathcal L\) with the constant separable Haar-and-index Hilbert space. Its positive Fock space and countable sum are again fixed separable Hilbert spaces. Choosing a fixed orthonormal basis identifies the underlying target with a countable standard coefficient module; this retains its geometric connection and transported action.

The full source bounded transform is the coefficient cycle \(1_A\) of The equivariant coefficient phase and its precise algebra, by PC.10's localized compact perturbation. The complementary transform is a localized compact perturbation of the degenerate phase proved above. Conjugation by \(\widehat U\) thus gives \(1_A\) on the entire target with the transported action. CSB.16–CSB.18 and the parallel vacuum give its actual normal resolvent control for the specified geometric target connection. We do not infer that this connection is covariant for the transported affine action, that \(\delta_i\widehat U\) is bounded, or that every groupoid arrow preserves this stronger unbounded domain. These properties were not used.

The left action here is scalar \(A\)-multiplication. It is not the \(B\)-representation of the inverse \(d_X\). In particular the completed complementary resolvents and coefficient identity do not supply the \(B\)-source deformation module, the original graph's balanced tensor product, its unbounded physical sum, or its Bott \(+1\) pairing.

### 11AU.7. A scalar complement that can be checked exactly

Take \(A=C_0((0,1/2])\), \(E=H=A\otimes\mathbb C^2\) with exterior parity, and \(Q=\begin{pmatrix}0&1\\1&0\end{pmatrix}\).
Let \(V=t^2I\), \(q=t^4I\), \(b=\sqrt{1-t^4}\), \(L=Qt^{-8}\), and \(C_0=4Q\). The complement is precisely
\(K_0=(b,-t^4,-t^4b,-t^4b^2,\ldots)\otimes I\).
Differentiating the three geometric-series identities
\(\sum x^n=(1-x)^{-1}\),
\(\sum nx^n=x(1-x)^{-2}\),
\(\sum n^2x^n=x(1+x)(1-x)^{-3}\) gives
\[
\|J'\|^2=\frac4{t^2(1-t^4)},\qquad
\|K_0'\|^2=\frac{8t^2}{1-t^4}.
\tag{CSB.19}
\]
For the second formula first differentiate with respect to \(s=t^2\). The first coordinate contributes \(s^2/(1-s^2)\). The squared derivative sum of the tail \(-s^2b^n\) is
\(4-4+(2-s^2)/(1-s^2)\); their sum is \(2/(1-s^2)\). Multiplication by \((ds/dt)^2=4t^2\) proves the formula. For the first formula use the same sums on \(J_n=sb^n\); they give \(1/[s^2(1-s^2)]\) before this chain factor. Thus \(K_0'\) vanishes at zero while \(J'\) is unbounded.

Here \(C_{\mathrm c,m}=2^{m+3}Q\). If \(D_{\mathrm s}(t,\lambda)\) is the exact compressed source derivative norm in PC.22, the entire resolvent has the proved bound
\[
\|\widehat R_\pm'(t,\lambda)\|
\le D_{\mathrm s}(t,\lambda)
 +\frac{\sqrt8\,t}{\sqrt{1-t^4}\sqrt{2\lambda}}.
\tag{CSB.20}
\]
The second term comes from the same row calculation as CSB.15 with the exact \(\|K_0'\|\); it is an upper bound, not the norm of a finite complementary truncation. At \(t=1/2,\lambda=256\),
\[
\|K_0'\|^2=\frac{32}{15},\qquad
D_{\mathrm s}^2=\frac{19+\sqrt{345}}{30720},\qquad
\left(\frac{\|K_0'\|}{\sqrt{2\lambda}}\right)^2=\frac1{240}.
\tag{CSB.21}
\]
The full bound therefore vanishes as \(t\downarrow0\). Meanwhile a bounded derivative of \(U\) would bound the derivative of its fixed \(E\)-input column \(J\), contradicting CSB.19. Normal control of the actual full resolvents does not require such a derivative.

![Full coefficient stabilization with a normally controlled complement](../figures/full-coefficient-complement.png)

**Figure 11AU.1.** CSB.1–CSB.21. The defect-cell unitary identifies the actual source plus its specified complement with the entire ambient coefficient module. The endpoint Hilbert–Schmidt estimate controls each complementary column; a parallel confining direct sum gives genuine full resolvents with compact normal derivatives. The scalar family has V=t²I, L=Q t^-8, complementary operators2^(m+3)Q and lambda=256. Both axes are logarithmic. Red is the exact unweighted column derivative, blue the exact first complementary-column derivative, and green the proved infinite full-resolvent derivative upper bound in CSB.20. Green is neither an exact full norm nor a finite-truncation norm. The curves use201 evaluations of the analytic formulas. Sixteen exact geometric-series/kernel checks,36 exact finite defect-cell unitary matrices including their terminal carrier, and three exact infinite-column norm formulas accompany the diagram. At t=1/2, the complementary derivative-bound square is1/240. The affine action is transported only after the vacuum is adjoined. Human context: [Connes, A survey of foliations and operator algebras](https://alainconnes.org/wp-content/uploads/foliationsfine.pdf#page=29), Section8. [Vector figure](../figures/full-coefficient-complement.svg) · [Generator](../reproduction/full-coefficient-complement/draw_coefficient_complement.py) · [Exact checks](../reproduction/full-coefficient-complement/COEFFICIENT-COMPLEMENT-CHECKS.json) · [Reproduction](../reproduction/full-coefficient-complement/REPRODUCTION.md) · [Component terms](../reproduction/full-coefficient-complement/COMPONENT-TERMS.md).

## 11AV. Normal finite jets for the stabilized Bott source

The anchored Bott construction in The anchored affine field and its Bott deformation first stabilizes a real affine field and then chooses a fixed diagonal weight. A continuous stabilizing unitary alone does not prove that its transported finite vectors have normal derivatives. We make those choices explicitly on the actual FD/PJ probability field.

The result is an actual affine action on a standard one-particle module, normally differentiable finite columns and displacement, and a fixed parallel weight with the graph jets consumed by the positive Bott fibre. The left Bott algebra is retained. We do not identify its inverse lift with the scalar coefficient oscillator.

### 11AV.1. A constant-column connection for the standard target

Put \(A=C_0(X)\), with \(X=\operatorname{Prob}_T(Y_{\mathrm{FD}})\), and let \(E=E_X\) have the actual closed metric normal connection and affine groupoid action of FD/PJ. Write \(L_g:E_{s(g)}\to E_{r(g)}\) for its linear part and \(b_g\in E_{r(g)}\) for its target displacement. Use PC.1's one-particle column
\[
T:E\longrightarrow\mathcal L,\qquad
h=T^*T\le\tfrac14,\qquad \overline{hE}=E.
\tag{NBJ.1}
\]
This \(h\) is the one-particle compact, not PC.4's Fock compact.

On each component of the finite good transversal choose the fixed smooth frame used in PC.2. It identifies the underlying Haar-and-index module \(\mathcal L\) with a constant separable real Hilbert space \(H_0'\). Equip this standard module with the constant-column connection
\(\nabla_i^0(a_j)_j=(\delta_i a_j)_j\), on the finite smooth scalar core and its closure. It is metric by the scalar product rule; closability follows coordinatewise from the closed scalar normal derivative and then finite-coordinate approximation. Its basic groupoid action transports scalar coefficients and is the identity on \(H_0'\). The scalar FD/PJ chain rule makes this action preserve the connection, with the prescribed change of normal directions.

This is a specified new target connection. On the smooth frame core the previous Haar connection is
\(\nabla_i^{\rm Haar}=\nabla_i^0+A_i\), where \(A_i\) is a finite sum of the skew-adjoint regular \(K\)-generators with smooth frame-connection coefficients. The fixed frames extend over slightly larger coordinate balls; the coefficients are bounded on the chosen balls. PC.1's all-frame-jet construction puts each regular \(K\)-generator applied to \(T\) in the Hilbert–Schmidt column module. Thus
\[
\delta_i^0T=\delta_i^{\rm Haar}T-A_iT\in\mathcal K_2.
\tag{NBJ.2}
\]
If compact-unit localization is used, the same formula and finite coefficient bounds hold there. The original word-column coefficient selection includes those bounds. The connection on \(E\) has not changed. Since \(A_i^*=-A_i\), the two extra terms in the derivative of \(T^*T\) cancel. Hence \(\delta_i h\) is unchanged. JD.13–JD.15 consequently still give the actual Hilbert–Schmidt derivative of \(h^{1/2}\). All subsequent column derivatives use \(\nabla^0\).

### 11AV.2. An endpoint estimate for transported columns

Apply the algebraic and Hilbert–Schmidt proofs CSB.4–CSB.12 directly to the one-particle \(T\) of NBJ.1–NBJ.2. Those proofs concern an arbitrary compact map with dense \(T^*T\), and do not require it to be a Fock map. Put
\[
b=(1-h)^{1/2},\quad d=(1-TT^*)^{1/2},\quad
Je=(Tb^ne)_{n\ge0},\quad
K_0y=(dy,-JT^*y),\quad K_n=S^nK_0.
\tag{NBJ.3}
\]
They give the real even unitary
\[
U=(J,K):E\oplus\mathcal L^\infty
   \xrightarrow{\ \cong\ }\mathcal L^\infty,\qquad
\delta_iK_0\in\mathcal K_2.
\tag{NBJ.4}
\]
The shift is parallel for \(\nabla^0\). In particular
\(\delta_iK_n=S^n\delta_iK_0\). CSB.11 gives
\[
(\delta_iJ_N)T^*\longrightarrow A_i\in\mathcal K_2,\qquad
\|A_i\|_{2,A}\le
\|\delta_iT\|_{2,A}+\sqrt{8/3}\|\delta_i h^{1/2}\|_{2,A}.
\tag{NBJ.5}
\]
Here \(A_i\) denotes this endpoint operator, rather than the frame-connection coefficient in NBJ.2.

There is a useful extension with no endpoint module polar decomposition. Let \(V:F\to E\) be an actual compact map and suppose
\(VV^*\le C h\). For finite \(N,M\), put \(Z=\delta_iJ_N-\delta_iJ_M\). Positivity gives
\[
\|ZV\|^2=\|ZVV^*Z^*\|
 \le C\|ZhZ^*\|=C\|ZT^*\|^2.
\tag{NBJ.6}
\]
The right side tends to zero. Thus \((\delta_iJ_N)V\) converges in compact norm to \(Q_{i,V}\), with
\(\|Q_{i,V}\|\le\sqrt C\,\|A_i\|_{2,A}\).
If \(V\) also has actual compact derivative, compact multiplication and PC.17's strong-adjoint limits give
\[
\delta_i(JV)=Q_{i,V}+J\,\delta_iV\in\mathcal K.
\tag{NBJ.7}
\]
Indeed \(J_NV\to JV\) in norm, and
\((\delta_iJ_N)V+J_N\delta_iV\) converges in norm; close the connection. Both estimates are uniform on any parameter test with uniform \(C\) and uniform compact/derivative limits. For a vector \(v\in E\), use the compact column \(A\to E,\ a\mapsto va\), and the condition \(\theta_{v,v}\le Ch\). This proves the corresponding actual vector statement. A bounded derivative of \(J\) has not been asserted.

### 11AV.3. The original transported one-particle columns

For a compact actual-arrow test, JD.5–JD.7's order comparisons give
\[
L_g h_{s(g)}L_g^*\le C h_{r(g)}
\tag{NBJ.8}
\]
with one finite \(C\) for that test. Retain the actual arrow labels and the original anchor. Let \(W_g^0\) be the basic standard action on \(\mathcal L\); it preserves the constant-column connection. Then
\[
V_g=L_gT_{s(g)}^*(W_g^0)^{-1},\qquad
V_gV_g^*=L_gh_{s(g)}L_g^*\le Ch_{r(g)}.
\]
This is compact. Its normal derivative is the transport of the source derivatives of \(T^*\), with the original finite change-of-normal-direction matrix. This follows from covariance of the two specified connections, not a derivative of an arbitrary stabilizing unitary. Thus \(V_g\) and its derivative are continuous compact sections, uniformly on each compact actual-arrow test. NBJ.6–NBJ.7 prove
\[
J_{r(g)}L_gT_{s(g)}^*(W_g^0)^{-1}
 \in\operatorname{Dom}\delta_i,\qquad
\delta_i\!\left(J_{r(g)}L_gT_{s(g)}^*(W_g^0)^{-1}\right)\in\mathcal K.
\tag{NBJ.9}
\]
Uniform convergence follows by multiplying NBJ.5's tail bound by \(\sqrt C\). This estimate controls the actual transported column at the endpoint; it does not assume \(L_g\operatorname{Ran}h^p=\operatorname{Ran}h^p\) for \(p>1\).

The displacement sections were among JD.2's compact-unit generators. Its compact-group evaluation argument gives
\(\theta_{b_g,b_g}\le C h_{r(g)}\) on a compact-arrow test after the specified cutoff. FD/PJ supplies their actual first normal jets. Therefore NBJ.7's vector version gives
\[
\widetilde b_g=J_{r(g)}b_g\in\operatorname{Dom}\nabla_i^0,
\qquad
\nabla_i^0\widetilde b_g
 =Q_{i,b_g}+J_{r(g)}\nabla_i b_g.
\tag{NBJ.10}
\]
These are norm-continuous vector sections with norm-continuous first jets on compact arrow tests. The original target convention is preserved.

### 11AV.4. An exact affine standard-module action with finite normal jets

Extend the original linear action by the basic identity-fibre action on \(\mathcal L^\infty\), and the displacement by zero on that summand. Transport through the actual \(U\) of NBJ.4:
\[
\mathcal V_g
 =U_{r(g)}(L_g\oplus W_g^{0,\infty})U_{s(g)}^*
 =J_rL_gJ_s^*+K_rW_g^{0,\infty}K_s^*.
\tag{NBJ.11}
\]
It is an orthogonal module action with exact groupoid composition. Together with \(\widetilde b_g\), it satisfies the original affine cocycle identity, since
\(U_s^*J_s b_h=(b_h,0)\). The action is continuous: actual continuous sections are transported by adjointable module unitaries, and the dense finite-coordinate core determines strong continuity.

Let \(P_N\) project onto the first \(N+1\) \(\mathcal L\)-coordinates in the target. It is parallel. Then
\[
(\mathcal V_g-W_g^{0,\infty})P_N\in\mathcal K,\qquad
\delta_i(\mathcal V_gP_N)\in\mathcal K.
\tag{NBJ.12}
\]
To prove this, first
\(J_s^*P_N\) is the finite row with entries \(b_s^jT_s^*\).
The order of these factors matters. Functional calculus gives \(b_s^jT_s^*=T_s^*d_s^j\): first verify \(h_sT_s^*=T_s^*(T_sT_s^*)\), then approximate the square-root powers uniformly by polynomials. Thus the transported entry is \(J_rL_gT_s^*d_s^j\). NBJ.9 controls its first factor; \(d_s^j\) is bounded with an actual compact derivative by the gapped square-root rule and the finite product rule. Insert the parallel basic action to interpret both factors on the target standard fibre. This proves the first summand's normal derivative. Equivalently, apply NBJ.6 directly to \(L_gb_s^jT_s^*\), whose square is bounded by \(L_gh_sL_g^*\le Ch_r\). Next \(K_s^*P_N\) is supported in complementary coordinates \(0,\ldots,N\), because \(K_n\) starts at coordinate \(n\). Its finitely many components and derivatives are those of \(K_n^*P_N\). NBJ.4 supplies the compact derivatives for both factors of the second summand. The basic action is parallel. This proves the derivative assertion with uniform compact limits on the arrow test.

For the compact difference, \(K_0-I_0=(d-1,-JT^*)\) is compact, where \(I_0\) inserts the first coordinate. Each \(K_n-I_n\) is compact. Hence on \(P_N\) the finite \(K_rW^0K_s^*\) sum differs compactly from \(W^{0,\infty}P_N\); the \(J\) summand is already compact. This proves NBJ.12. The same statements for inverses follow by using the inverse-arrow test.

Choose a fixed real orthonormal basis \((e_j)\) of the fixed model
\(\mathcal L^\infty\simeq A\otimes H_0\). NBJ.12 implies that
\(\mathcal V_ge_j\) has an actual normal derivative, continuous in norm on compact arrow tests. Each basis vector is parallel for \(\nabla^0\). No normal derivative of the full \(\mathcal V_g\), of \(U\), or of \(JJ^*\) is required.

### 11AV.5. A fixed diagonal selector also accommodates these jets

We give the uniform quasicentral choice needed here, including the compact-parameter issue. Let \(p_m\) be the fixed nested finite-coordinate projections in \(H_0\). For a compact arrow test \(C\), identify the standard fibres by their basic action and fixed frames, and regard
\([p_m,\mathcal V]\) as a continuous compact-operator section over \(C\). Finite-column continuity and its adjoint prove this norm continuity; the full \(\mathcal V\) need only be strongly continuous.

These commutator sections tend weakly to zero in
\(C(C,\mathcal K(H_0))\). For a fixed \(l\), their compression by \(p_l\) on both sides is exactly zero once \(m\ge l\). Finite compressions are norm dense on the dual: for a positive functional, its GNS representation takes the approximate identity \(p_l\) strongly to one, and Cauchy–Schwarz bounds the error in compression by a constant times
\((\|\varphi\|-\varphi(p_l))^{1/2}\). A general bounded functional is a linear combination of four positive functionals; applying the same estimate proves its compression approximation. The commutator sections are uniformly bounded by two. They consequently tend to zero under every bounded functional. The separating-functional argument now implies that zero belongs to the norm-closed convex hull of each tail. Apply this in the finite direct sum for finitely many compact tests. Thus finite convex combinations of the \(p_m\) can have uniformly small commutators on all those tests.

Enumerate a compact-arrow exhaustion and the norm-continuous compact parameter families
\[
\widetilde b_g,\quad\nabla_i^0\widetilde b_g,\quad
\mathcal V_ge_j,\quad\nabla_i^0(\mathcal V_ge_j),
\tag{NBJ.13}
\]
with all finite direction and basis choices and their inverse-arrow versions. On the fixed standard fibre, their values in a compact test form norm-compact vector sets. Finite projections approximate those sets uniformly. Local frame charts and finitely many scalar cutoffs give the same statement for the actual coefficient sections.

Choose finite convex combinations \(v_n\) of projections beyond a sufficiently large index, so that \(v_n\ge v_{n-1}\), they fix the first \(n\) basis vectors, and on the first \(n\) tests
\[
\|[v_n,\mathcal V_g]\|\le2^{-n},\qquad
\|(1-v_n)\xi\|\le2^{-n}
\tag{NBJ.14}
\]
for the first \(n\) vector families. The monotone choice is possible by taking the new tail beyond the largest projection used previously; every projection in that tail dominates \(v_{n-1}\). The weak-convex argument gives the commutator bounds, while the uniform projection-tail bounds survive every convex combination.

Define
\[
\Theta=1+\sum_{n\ge1}(1-v_n),\qquad \kappa_0=\Theta^{-1}.
\tag{NBJ.15}
\]
This is a fixed real diagonal operator, with constant eigenvalues
\(\lambda_j\ge1\) tending to infinity. Each fixed \(e_j\) is fixed by all sufficiently late \(v_n\), so its sum is finite. Outside the supports of the first \(N\) cutoffs the eigenvalue is at least \(N+1\); hence the reciprocal is a compact on \(H_0\). Finite-coordinate resolvents prove regular self-adjointness and the finite-coordinate graph core on the coefficient module. On a noncompact \(X\), the correct assertion is \(a\kappa_0\in\mathcal K_A\) for \(a\in A\).

The weight and its reciprocal are parallel for the specified constant-column connection, since their coefficients are constant. NBJ.14's vector tails are norm summable, so every family in NBJ.13 belongs to its actual graph domain. Its graph sections are norm continuous, uniformly on their compact tests. Finite spectral cutoffs of \(\Theta\), which commute with \(\nabla^0\), and closedness show
\[
\nabla_i^0(\Theta\widetilde b_g)
 =\Theta\nabla_i^0\widetilde b_g,\qquad
\nabla_i^0(\Theta\mathcal V_ge_j)
 =\Theta\nabla_i^0(\mathcal V_ge_j).
\tag{NBJ.16}
\]
Both sides are actual module vectors; a fibrewise bounded graph norm was not substituted for membership in the graph domain.

Also
\[
S_g=\mathcal V_g\Theta_s\mathcal V_g^*-\Theta_r
 =\sum_{n\ge1}(v_{n,r}-\mathcal V_gv_{n,s}\mathcal V_g^*)
\tag{NBJ.17}
\]
converges in compact norm on each compact arrow test. Its finite terms are compact and continuous. NBJ.14 gives a summable tail after its finite initial part. Strong resolvent limits of the bounded weight sums and the norm difference identify the conjugated regular limit as \(\Theta+S_g\). Thus
\(\mathcal V_g\operatorname{Dom}\Theta_s=\operatorname{Dom}\Theta_r\).
This supplies exactly BA.7's fixed-Hilbert weight and domain data, now with the additional normal graph jets NBJ.16. A compact norm derivative of the whole \(S_g\) has not been inferred.

### 11AV.6. The proper Bott source and its positive fibre remain distinct

The underlying real module in NBJ.4 is exactly the original affine \(E_X\) plus a countable standard summand with zero cocycle and identity fibre action. NBJ.11 is its actual real stabilization. This is an explicit choice of the real stabilization used in BA.4, with the original cocycle and anchor retained. In particular
\[
\|\widetilde b_g\|=\|b_g\|.
\tag{NBJ.18}
\]
The proper displacement bounds therefore stay unchanged.

Make the fixed diagonal choice in the anchored Bott construction by NBJ.15. Its algebra is still
\(B=A\widehat\otimes A(H_0)\), with the actual transported affine action. The finite phase-space symbol is still
\(\varepsilon(q+ip)+\varepsilon(q+ip)^*\).
NBJ.17 and the actual graph-continuous finite origins and frames verify the finite transported-plane premises of BA.5. The proper phase cone retains both position and momentum; NBJ.18 gives exactly its proper-arrow bound. The ordinary nuclear section and anchored finite-output factor arguments in BA.5–BA.7 then apply to this specified input, giving the actual extension \(q_B:E_B\to B\), its unique proper-source lift \(z_B\), and BA.16's class \(d_X\). This instantiates those existence arguments; it does not replace their source by \(A\).

On positive Fock fibres the ordinary lifting operator is precisely
\[
\widetilde T_t=T_S+tD_t^3+B_t(1+t\Theta),\qquad 0<t\le1.
\tag{NBJ.19}
\]
Here \(D_t\) is the unweighted real oscillator, \(\Theta\) is the fixed diagonal one-particle weight just chosen, and the scalar \(T_S\) has its original reflection grading. All are parallel for the tensor constant-column connection. Their completed graph and resolvent statements are the actual BD.20–BD.25 proof; finite joint cutoffs are parallel and exhaust its domain. Closing the connection gives
\[
\delta_i(\widetilde T_t\pm i\lambda)^{-1}=0,\qquad
\nabla_i^0 R_{t,\pm}u=R_{t,\pm}\nabla_i^0u
\quad(u\in\operatorname{Dom}\nabla_i^0).
\tag{NBJ.20}
\]
The vacuum and noncompact-base compactness stay localized as in BA.9. At \(t=1\), evaluation of \(S\) at zero gives \(D^3+B(1+\Theta)\), the actual positive ordinary calibration operator with its even vacuum. This is a concrete normally controlled positive-fibre model.

There is also actual first normal graph control of the affine action on each finite occupation vector. The finite-basis part uses
\(\mathcal V_ge_j\) and its jets in NBJ.13–NBJ.16. The Weyl exponential formula uses
\(\widetilde b_g/\sqrt t\) and its jets, all in the \(\Theta\)-graph domain on \(t\in[a,1]\), \(a>0\). Differentiating that formula gives one creation or annihilation in the differentiated displacement, and the finite product rule differentiates the finitely many transported basis vectors. The weighted occupation estimates in WF.13–WF.18 bound these vectors and their graph values; all coefficients are locally uniformly bounded by NBJ.16. Finite occupation approximation and closure give the actual graph derivatives on this dense core. This does not assert a bounded derivative of the full affine unitary.

The actual \(B\)-source inverse representative is not yet proved with a normal module connection: \(z_B\), followed by positive evaluation and Morita equivalence, must retain its nondegenerate \(B\)-representation and required normal module passage. The scalar source action of NBJ.19 is not that representation. Nor does NBJ.20 prove a normal derivative uniformly through the zero endpoint for arbitrary transported sections. The prescribed PF.1–PF.3 unitaries in Normalized proper fields and actual graph columns, original \(C_0(T)\) balancing, physical graph sum, full inverse Spin-c coefficient and original Bott \(+1\) comparison therefore require their respective proofs.

### 11AV.7. An exact finite-column test

The following scalar test isolates the mechanism without claiming to model a proper holonomy action. Take \(A=C_0((0,1/2])\), \(E=A\), \(\mathcal L=A\), \(T(t)=t^2\), and the ordinary derivative. Put \(s=t^2\), \(b=\sqrt{1-t^4}\); then \(J(t)=(s b^n)_{n\ge0}\) is a real unit vector. Transport the involution \(-1\) on \(E\), with identity on its added complement and zero displacement, through NBJ.4. The resulting standard-module involution is
\[
\mathcal V=1-2JJ^*,\qquad \mathcal V^2=1,\qquad
\mathcal Ve_0=e_0-2sJ.
\tag{NBJ.21}
\]
Differentiating the norm identity gives \(\langle J,J'\rangle=0\). CSB.19's exact geometric summation gives \(\|J'\|^2=4/[t^2(1-t^4)]\). The derivative of \(JJ^*\) acts on the orthogonal plane spanned by \(J,J'/\|J'\|\) as the two off-diagonal entries \(\|J'\|\). Consequently
\[
\|\mathcal V'\|^2=\frac{16}{t^2(1-t^4)},\qquad
\|(\mathcal Ve_0)'\|^2
 =4\bigl((s')^2+s^2\|J'\|^2\bigr)
 =\frac{16t^2(2-t^4)}{1-t^4}.
\tag{NBJ.22}
\]
The whole derivative diverges as \(t\downarrow0\), while this actual column derivative tends to zero. Both claims are exact infinite-column norm formulas. The endpoint factor has
\[
\|J'T^*\|^2=\frac{4t^2}{1-t^4};\qquad
\left.\|(\mathcal Ve_0)'\|^2\right|_{t=1/2}=\frac{124}{15},\quad
\left.\|\mathcal V'\|^2\right|_{t=1/2}=\frac{1024}{15},\quad
\left.\|J'T^*\|^2\right|_{t=1/2}=\frac{16}{15}.
\tag{NBJ.23}
\]
These formulas show why NBJ.12 requires only finite input columns. They do not supply a bounded derivative of the full transported action or any \(B\)-source representation.

![Normal finite columns and the fixed Bott weight](../figures/normal-bott-stabilization.png)

**Figure 11AV.1.** NBJ.1–NBJ.23. Domination by the actual one-particle orbit compact gives completed normal jets of transported source columns and displacement. The explicit real stabilization retains the actual affine action and target cocycle. A fixed diagonal selector includes these jets in its graph domain and is parallel for the specified constant-column connection. The scalar reflection test has T=t², V=1−2JJ* and zero displacement. It illustrates the finite-column mechanism; it is not a proper holonomy model. Red, green and blue are the exact infinite-column norm formulas for V′, (Ve0)′ and J′T*, numerically evaluated at201 points with both axes logarithmic. They are not finite-truncation norms. At t=1/2 their squares are1024/15,124/15 and16/15. Thirty-six exact finite reflection matrices retain their terminal carrier; six rational samples independently check the geometric infinite norm identities and orthogonality. The positive lifting operator retains its scalar coefficient action; its B-source inverse representation has not been identified. Human context: [Connes, A survey of foliations and operator algebras](https://alainconnes.org/wp-content/uploads/foliationsfine.pdf#page=29), Section8. [Vector figure](../figures/normal-bott-stabilization.svg) · [Generator](../reproduction/normal-bott-stabilization/draw_normal_bott_stabilization.py) · [Exact checks](../reproduction/normal-bott-stabilization/NORMAL-BOTT-STABILIZATION-CHECKS.json) · [Reproduction](../reproduction/normal-bott-stabilization/REPRODUCTION.md) · [Component terms](../reproduction/normal-bott-stabilization/COMPONENT-TERMS.md).

## 11AW. Transporting the completed normal connection

A continuous module unitary can transport a closed connection even when its derivative is unbounded. The resulting connection has its transported graph domain. For the actual real stabilization of Normal finite jets for the stabilized Bott source, it is covariant for the linear groupoid action. It differs from the constant-column connection used to make the Bott weight parallel.

### 11AW.1. Transport is an equality of completed domains

Let \(M,N\) be Hilbert modules over \(A=C_0(X)\), with an adjointable even unitary \(U:M\to N\). Let \(\nabla_i\) be the actual closed metric normal connection on \(M\), with its scalar normal derivation \(\delta_i\). Define
\[
\operatorname{Dom}\widehat\nabla_i=U\operatorname{Dom}\nabla_i,
\qquad \widehat\nabla_i(Uv)=U\nabla_i v.
\tag{CTN.1}
\]
This is a densely defined closed operator: norm convergence of \(Uv_n\) and \(U\nabla_i v_n\) is equivalent to convergence of \(v_n\) and \(\nabla_i v_n\), and closedness on \(M\) applies. If \(\mathcal C\) is the specified graph core on \(M\), then \(U\mathcal C\) is its actual graph core on \(N\). No assertion that arbitrary fixed-coordinate truncations form this core is needed.

The unitary is \(A\)-linear, so on its right scalar domain
\[
\widehat\nabla_i(ua)
 =(\widehat\nabla_i u)a+u\delta_i a.
\tag{CTN.2}
\]
Indeed write \(u=Uv\) and use the original completed scalar product rule for \(va\). Preservation of the module inner product gives the metric identity as well. The connection is not defined by differentiating \(U\) on arbitrary vectors.

Suppose the genuine linear arrow maps \(L_g\) on \(M\) preserve its connection, with the original change-of-normal-direction coefficients \(m_{ij}(g)\). Thus the source and target connections satisfy the actual semilinear domain rule
\(\nabla_{i,r}L_gv=\sum_jm_{ij}(g)L_g\nabla_{j,s}v\).
For the transported action \(\widehat L_g=U_rL_gU_s^*\), cancellation of the middle unitaries gives
\[
\widehat\nabla_{i,r}\widehat L_g u
 =\sum_jm_{ij}(g)\widehat L_g\widehat\nabla_{j,s}u,
\quad u\in U_s\bigcap_j\operatorname{Dom}\nabla_{j,s}.
\tag{CTN.3}
\]
Both sides lie in their actual completed domains because CTN.1 defines those domains. This proves covariance with the actual arrow labels; it does not require a bounded commutator of \(U\) with the constant-column derivative.

### 11AW.2. Application to the actual one-particle stabilization

Use exactly NBJ.3–NBJ.4:
\(M=E_X\oplus\mathcal L^\infty\), \(N=\mathcal L^\infty\),
\(U=(J,K)\). On \(M\) take the direct-sum connection
\(\nabla^M=\nabla^{E_X}\oplus\nabla^0\), where the second connection is the basic constant-column connection on the added standard summand. Its closedness, core and metric product rule follow componentwise from the two specified connections. The actual linear action \(L_g\oplus W_g^{0,\infty}\) is covariant by FD/PJ and NBJ.1. CTN.1–CTN.3 therefore give a genuine covariant metric connection \(\widehat\nabla\) on the stabilized real field for NBJ.11's \(\mathcal V_g\).

The individual inclusions are parallel for this connection:
\[
\widehat\nabla_i(Je)=J\nabla_i^{E_X}e,\qquad
\widehat\nabla_i(Ky)=K\nabla_i^0y.
\tag{CTN.4}
\]
These statements include the domains \(e\in\operatorname{Dom}\nabla_i^{E_X}\), \(y\in\operatorname{Dom}\nabla_i^0\). They follow from \(Je=U(e,0)\) and \(Ky=U(0,y)\). In particular both orthogonal projections \(JJ^*\) and \(KK^*\) preserve the completed connection domain and have zero connection commutator there. The target displacement satisfies
\[
\widehat\nabla_i\widetilde b_g
 =J_r\nabla_i^{E_X}b_g.
\tag{CTN.5}
\]
This is the same genuine cocycle as NBJ.10, with its other specified connection. Covariance in CTN.3 concerns the linear action. For an affine section, differentiating \(\mathcal V_gv+\widetilde b_g\) also contributes CTN.5; that contribution is retained.

### 11AW.3. Finite columns compare the two connections

Let \(P_N\) be NBJ.12's first \(N+1\) standard coordinate projection. For a parallel constant vector in its range, the adjoint source components are
\[
J^*P_N=(b^jT^*)_{j=0}^{N},\qquad
K^*P_N=(K_n^*P_N)_{n=0}^{N}.
\tag{CTN.6}
\]
The later complementary entries are zero because \(K_n\) starts at coordinate \(n\). Every first entry has an actual compact derivative: the gapped function \(b=(1-h)^{1/2}\) has its compact derivative by the square-root rule, and \(T^*\) has the actual compact normal derivative proved in NBJ.2. The finite product rule applies with the displayed factor order. Every complementary entry has its actual compact derivative by NBJ.4. Thus the finite-column map \(U^*P_N\), from the basic target connection to the source direct-sum connection, has an actual compact covariant derivative
\[
D_{i,N}=\nabla_i^M U^*P_N-U^*P_N\nabla_i^0
 \in\mathcal K_A(P_NN,M).
\tag{CTN.7}
\]
The assertion means the completed product rule for this bounded map. To check completion, first use finite scalar core sections, then approximate the input in the graph norm of \(\nabla_i^0\); the finite bounded derivative and closedness of \(\nabla_i^M\) give the rule on the completed input domain. Its norms and compact limits are uniform on the specified compact tests, by the finite input estimates already proved.

Consequently, for \(u\in P_N\operatorname{Dom}\nabla_i^0\),
\[
\widehat\nabla_i u=\nabla_i^0u+A_{i,N}u,
\qquad A_{i,N}=U D_{i,N}\in\mathcal K_A(P_NN,N).
\tag{CTN.8}
\]
Compactness is preserved by multiplication with the actual adjointable \(U\). For \(M\ge N\), these maps agree on the smaller parallel coordinate core: both equal the difference of the same two connections. They define the connection difference \(A_i\) on the union of those finite input cores. Its metric identity is
\[
\langle A_i u,v\rangle+\langle u,A_i v\rangle=0.
\tag{CTN.9}
\]
It follows by subtracting the two metric connection identities on this common core. This proves skew-symmetry on that core, not boundedness or skew-adjointness of a full operator on all of \(N\).

Each fixed real orthonormal basis vector belongs locally to this comparison domain; with a compact scalar cutoff it is an actual module vector. The finite-column argument therefore supplies the actual first normal jets used in finite Bott planes for the covariant connection as well. The graph core for \(\widehat\nabla\) remains \(U\mathcal C\), as specified in CTN.1. No graph-core conclusion is inferred merely from Hilbert-module density of these fixed basis vectors.

### 11AW.4. Completed resolvents transport without extra derivative terms

Let \(D=D^*\) be a regular operator on \(M\), and suppose its genuine resolvents \(R_\pm=(D\pm i\lambda)^{-1}\) preserve the completed \(\nabla_i\)-domain with
\[
\nabla_i R_\pm v=R_\pm\nabla_i v+C_{i,\pm}v,
\qquad C_{i,\pm}\in\mathcal K_A(M).
\tag{CTN.10}
\]
The unitary conjugate \(\widehat D=UDU^*\), on \(U\operatorname{Dom}D\), is self-adjoint regular. Its adjoint resolvents are exactly \(\widehat R_\pm=UR_\pm U^*\), with dense ranges and the original inverse identities. For \(u=Uv\in\operatorname{Dom}\widehat\nabla_i\), CTN.1 and CTN.10 give
\[
\widehat\nabla_i\widehat R_\pm u
 =\widehat R_\pm\widehat\nabla_i u
   +\widehat C_{i,\pm}u,
\qquad \widehat C_{i,\pm}=UC_{i,\pm}U^*.
\tag{CTN.11}
\]
The domain assertion is part of this computation. Compactness and the norm are preserved exactly. There are no derivatives of \(J\), \(K\), or \(U\) in this transported connection formula.

Apply it to Completing the coefficient stabilization with a normally controlled complement, with its actual Fock-space unitary and source operator \(L\oplus C_c\). This is that graded coefficient unitary, not NBJ.4's one-particle unitary. The source connection is the original coefficient Fock connection plus the specified parallel complementary connection. PC.15–PC.16 give the derivative and domain rule of the source \(L\)-resolvent; the complementary input resolvent is parallel by CSB.13–CSB.15. Thus for the transported full connection,
\[
\|\widehat C_{i,\pm}\|
 \le\sqrt2\,\|\delta_i q\|\lambda^{-1/2}
  +\sigma\sqrt{C_\phi}\,\|Z_i\|\lambda^{-3/4}.
\tag{CTN.12}
\]
The vacuum, if adjoined, has the original parallel scalar connection and contributes zero derivative. Compactness on a noncompact base remains localized as before. This sharper bound belongs to the transported connection. CSB.18's additional stabilization-transfer terms belonged to the different constant-column target connection and remain its correct bound. The proof does not infer covariance under an affine Fock action with a varying displacement, a left \(B\)-representation, or a physical graph sum.

### 11AW.5. Parallel weight and covariant connection are separate data

For the fixed diagonal \(\Theta\) in NBJ.15, the proved statement is \(\delta_i^0\Theta=0\). On a common finite graph vector \(u\), the comparison rule reads
\[
\widehat\nabla_i(\Theta u)-\Theta\widehat\nabla_i u
 =A_i\Theta u-\Theta A_i u,
\tag{CTN.13}
\]
provided the vectors on both sides are in their indicated domains. Finite input control of \(A_i\) alone does not bound this commutator or prove that \(\Theta\) preserves the entire transported normal domain. The actual inverse-module construction must retain that distinction.

An exact two-dimensional test makes it visible. On \(A=C_0(\mathbb R)\), set
\[
U(t)=\begin{pmatrix}\cos t&-\sin t\\\sin t&\cos t\end{pmatrix},
\quad L=\operatorname{diag}(1,-1),\quad
\mathcal V=ULU^*,\quad
A_U=U(U^*)'=\begin{pmatrix}0&1\\-1&0\end{pmatrix}.
\tag{CTN.14}
\]
The connection is \(\widehat\nabla=\partial_t+A_U\), with its actual domain obtained from CTN.1; boundedness of this particular constant matrix also identifies it with the usual first derivative domain. Differentiating \(\mathcal V\) gives \(\mathcal V'+[A_U,\mathcal V]=0\), exactly CTN.3. But for the fixed \(\Theta=\operatorname{diag}(1,2)\),
\[
\delta^0\Theta=0,\qquad
[A_U,\Theta]=\begin{pmatrix}0&1\\1&0\end{pmatrix},
\qquad \|[A_U,\Theta]\|=1.
\tag{CTN.15}
\]
This test is an exact finite connection calculation, not a substitute for the infinite Bott or groupoid construction. The prescribed inverse class, its nondegenerate source action and the original balanced graph realization are not yet proved with these data.

### 11AW.6. An exact infinite-resolvent comparison

Use CSB.19's scalar graded model \(T=t^2I\), \(q=t^4\), source \(L=Qt^{-8}\), constant odd sign \(Q\), and parallel complementary operators \(2^{m+3}Q\). Let \(\lambda=256\), \(0<t\le1/2\). The source resolvent has derivative norm \(8t^7/(1+256^2t^{16})\), by its scalar spectral values \(\pm t^{-8}\). Its complementary derivative is zero. CTN.11 preserves that norm for the full transported connection, so
\[
\|\widehat C_\pm(t)\|
 =\frac{8t^7}{1+256^2t^{16}}
 \le\frac{\sqrt2}{4}t^3,\qquad
\|\widehat C_\pm(1/2)\|=\frac1{32}.
\tag{CTN.16}
\]
The upper bound is CTN.12 with \(\delta q=4t^3\), \(Z_i=0\). The squared ratio of the exact norm to this bound is \(2u/(1+u^2)^2\), where \(u=256t^8\). It is at most one because
\((1+u^2)^2-2u=(u-1)^2+u^2+u^4\ge0\). These are genuine infinite full-resolvent statements for the transported connection. CSB.20's different constant-column bound is still
\(\sqrt{(19+\sqrt{345})/30720}+1/\sqrt{240}\) at \(t=1/2\). Comparing the bounds does not identify the two normal connections.

![Completed normal connection and source module](../figures/covariant-bott-connections.png)

**Figure 11AW.1.** CTN.1–CTN.16. Unitary transport specifies the completed normal domain and actual graph core, preserving the metric rule and linear covariance. Finite columns compare this covariant connection with the constant connection. Full resolvents conjugate with the same compact derivative norm. The exact rotation test has a fixed weight with nonzero transported commutator of norm one. The scalar infinite-resolvent model has T=t²I, L=Qt⁻⁸, parallel complementary operators 2^(m+3)Q and λ=256. Red is the proved upper bound for the constant-column connection; green is the exact derivative norm 8t⁷/(1+256²t¹⁶) for the transported connection; blue is its upper bound √2 t³/4. Both axes are logarithmic, and the curves evaluate the infinite formulas at 201 points. Four exact rational rotations and six exact rational infinite-resolvent samples check the displayed identities. At t=1/2 the green norm is 1/32. These connections are distinct, and the full inverse cycle remains unproved. Human context: [Connes, A survey of foliations and operator algebras](https://alainconnes.org/wp-content/uploads/foliationsfine.pdf#page=29), Section 8. [Vector figure](../figures/covariant-bott-connections.svg) · [Generator](../reproduction/covariant-bott-connections/draw_covariant_connections.py) · [Exact checks](../reproduction/covariant-bott-connections/COVARIANT-BOTT-CONNECTIONS-CHECKS.json) · [Reproduction](../reproduction/covariant-bott-connections/REPRODUCTION.md) · [Component terms](../reproduction/covariant-bott-connections/COMPONENT-TERMS.md).

## 11AX. An ordinary Bott-source representation and its normal domain

A completely positive section gives an actual nondegenerate source representation on a completed Hilbert module. Its zero quotient is the identity module. That module identity alone does not give an inverse Kasparov cycle: a parameter taper below makes the positive module vanish.

### 11AX.1. Exact anchored input

The anchored affine field and its Bott deformation, BA.5 and BA.11, specify, ordinarily,
\[
A=C_0(X),\quad C=A(H_0),\quad D=E_\Theta,\quad
B=A\widehat\otimes C,\quad E_B=A\widehat\otimes D,
\]
and a zero quotient \(q_0:D\to C\) with an even completely positive contractive section \(s_0:C\to D\). The section on the actual anchored algebras is
\(s_B=1_A\widehat\otimes s_0\). It is explicitly not presumed equivariant. The fixed diagonal weight and constant-column normal connection in Normal finite jets for the stabilized Bott source, NBJ.15–NBJ.20, make these ordinary coefficient algebras constant over \(X\). Their normal derivation is the given scalar derivation on \(A\), tensored with identity. No normal derivative of an arbitrary nuclear lifting choice on a varying algebra is used.

### 11AX.2. An actual nondegenerate left representation

On the algebraic space \(C\odot D\) define a right \(D\)-action and inner product by
\[
\langle c\otimes d,c'\otimes d'\rangle
   =d^*s_0(c^*c')d'. \qquad\tag{BSM.1}
\]
For a finite sum the Gram matrix \((c_i^*c_j)_{ij}\) is positive. Complete positivity of \(s_0\) makes its image matrix positive, so BSM.1 is a positive \(D\)-valued form. Quotient the null space and complete in its norm; call the Hilbert module \(K_0\). The ordinary Cauchy–Schwarz inequality for this form follows by applying positivity to the two-by-two Gram matrix and then its positive scalar regularization; hence the right action is bounded and completion is legitimate.

Left multiplication defines
\[
\pi_0(a)(c\otimes d)=ac\otimes d,\qquad
\pi_0:C\longrightarrow\mathcal L_D(K_0). \qquad\tag{BSM.2}
\]
The matrix \((c_i^*(\|a\|^2-a^*a)c_j)_{ij}\) is positive, so the squared inner product after left multiplication is at most \(\|a\|^2\) times the original one in the positive order. This proves boundedness, preservation of the null space and \(\|\pi_0(a)\|\le\|a\|\). BSM.1 gives the adjoint \(\pi_0(a^*)\). Associativity gives an exact star homomorphism; the cpc section itself has not been treated as a homomorphism.

Let \((u_n)\) be a positive contractive approximate identity of \(C\). On an elementary vector,
\[
\|\pi_0(u_n)(c\otimes d)-c\otimes d\|
 \le \|u_nc-c\|\,\|d\|\longrightarrow0.
\]
Uniform boundedness and the dense algebraic span extend this to all \(K_0\). Thus \(\pi_0\) is nondegenerate on the actual completed module. There is no appeal to a unitization support projection. Countable dense subsets in separable \(C,D\), followed by a \(D\) approximate identity, give a countable generating set. The grading
\(c\otimes d\mapsto\gamma_C(c)\otimes\gamma_D(d)\) preserves BSM.1 because \(s_0\) is even; it defines the completed module grading and makes \(\pi_0\) even.

Tensor with the actual unit algebra:
\[
K=A\widehat\otimes K_0,\qquad
\pi=1_A\widehat\otimes\pi_0:B\to\mathcal L_{E_B}(K).
\qquad\tag{BSM.3}
\]
Finite elementary sections and scalar approximate identities prove nondegeneracy and countable generation over \(E_B\). The \(A\)-action is the original central anchor, not evaluation at a point of \(X\).

### 11AX.3. The identity zero quotient

The internal tensor product \(K_0\otimes_{D,q_0} C\) has a unitary
\[
Q:(c\otimes d)\otimes a\longmapsto c\,q_0(d)\,a.
\qquad\tag{BSM.4}
\]
To check it, compute the inner product of two such images:
\[
a^*q_0(d)^*c^*c'q_0(d')a'
 =a^*q_0\big(d^*s_0(c^*c')d'\big)a',
\]
using \(q_0s_0=1_C\). This is precisely the balanced module inner product. Thus the map respects the null space and is isometric on the algebraic balanced core. Its range is dense: \(q_0(D)=C\), and the span of triple products in \(C\) is dense by an approximate identity. It extends to an onto unitary. Left multiplication in BSM.2 becomes the identity representation of \(C\) on \(C_C\). Tensoring with \(A\) gives
\[
K\otimes_{E_B,q_B}B\cong B_B
\quad\text{with its actual nondegenerate left }B\text{ action}.
\qquad\tag{BSM.5}
\]
This is an exact module quotient identity. It is not yet the assertion \([K,F]q_B=1_B\) in KK: an operator \(F\) with its compactness, grading and cycle conditions has not been supplied.

### 11AX.4. A completed parallel normal connection

View \(K\) as the constant section module \(C_0(X,K_0)\), with its \(C_0(X,D)\)-valued inner product. On finite sections \(u=\sum_j a_j\otimes v_j\) with \(a_j\) in the original smooth scalar normal core and fixed \(v_j\in K_0\), put
\[
\nabla_i u=\sum_j\delta_i a_j\otimes v_j.
\qquad\tag{BSM.6}
\]
This is well-defined because each bounded complex linear functional on the fixed Banach space \(K_0\) reduces the identity to the well-defined scalar derivation. It is closable: if \(u_n\to0\) and \(\nabla_i u_n\to v\) in section norm, every such functional gives scalar sections tending to zero whose derivatives tend to its value on \(v\). Closedness of the original scalar normal derivative makes that value zero. Bounded functionals separate the Banach-space points at every \(x\), so \(v=0\). Take its minimal closed extension from this dense core.

The scalar product rule proves the metric identity on finite sections and the right Leibniz rule for coefficient sections in the scalar tensor core. Completed graph limits preserve both identities whenever the indicated factors and derivatives converge in their norms. In particular for a finite smooth source symbol \(b=\sum_l a_l\otimes c_l\),
\[
\nabla_i(\pi(b)u)=\pi(\delta_i b)u+\pi(b)\nabla_i u,
\qquad u\in\operatorname{Dom}\nabla_i,
\qquad\tag{BSM.7}
\]
and
\[
\|\nabla_i(\pi(b)u)\|
 \le\|\delta_i b\|\,\|u\|+\|b\|\,\|\nabla_i u\|.
\]
For a general source in the norm-and-derivative closure of those symbols, approximate \(b\) in both norms and then close \(\nabla_i\). This proves actual domain preservation and the displayed bound; a bounded fibre norm was not substituted for membership in the completed connection domain.

The coefficient maps \(q_0\), positive evaluation \(h_t\) for fixed \(t>0\), and the ordinary Fock Morita coefficient module are constant over \(X\). Tensor BSM.6 with their parallel coefficient cores. The same norming-functional closability proof and scalar product rule give closed normal connections on the completed tensor products. BSM.4–BSM.5 are parallel on their dense balanced core and hence on completed graph domains. Put \(M_t=A\widehat\otimes\mathcal F_t\widehat\otimes S\), with its ordinary left \(A\widehat\otimes\mathcal K(\mathcal F_t)\widehat\otimes S\)-action and right \(A\widehat\otimes S\)-action. Positive evaluation followed by this actual ordinary Fock Morita module and \(S\) evaluation gives a Hilbert \(A\)-module
\[
K_t^{A}=K\otimes_{E_B,h_t}M_t
   \otimes_{A\widehat\otimes S,\,1_A\otimes\mathrm{ev}_0}A,
\qquad\tag{BSM.8}
\]
with a nondegenerate left \(B\)-representation and the same scalar normal product rule. Nondegeneracy follows by tensoring the strong approximate-identity convergence already proved. This module may be zero for a permissible cpc section; its existence alone is not an inverse-cycle theorem.

### 11AX.5. Transport compares different section modules

For an actual arrow \(g\), denote its algebra maps by \(\alpha_g^B\), \(\alpha_g^E\), and define the conjugate section
\[
s_B^g=\alpha_g^E\,s_B\,(\alpha_g^B)^{-1}.
\qquad\tag{BSM.9}
\]
Transporting an algebraic vector by
\[
c\otimes e\longmapsto\alpha_g^B(c)\otimes\alpha_g^E(e)
\qquad\tag{BSM.10}
\]
preserves the semilinear inner product **in the Stinespring module for \(s_B^g\)**, because
\(s_B^g(\alpha_g^B(c^*c'))=\alpha_g^E(s_B(c^*c'))\).
The inverse algebra maps give its completed inverse. Consequently this is a genuine semilinear unitary between the two specified modules. It is not a unitary action on the fixed module for \(s_B\) unless those sections agree or coherent identifications of the different completed modules are supplied. BA.5 does not assume that agreement. The original equivariant proper-source lift \(z_B\) cannot be replaced by this ordinary module construction on that basis.

### 11AX.6. The positive section-taper test

The deformation parameter is central and invariant. Let \(\chi(t)=1-t\). Multiplication by \(\chi\) is a positive central contraction of \(D\) with \(\chi(0)=1\) and \(\chi(1)=0\). Therefore
\[
s'_0(c)=\chi s_0(c)
\qquad\tag{BSM.11}
\]
is another even cpc section of the same zero quotient. Complete positivity follows from multiplication by the positive central \(\chi^{1/2}\) on both sides; \(q_0s'_0=1_C\). It is still a constant ordinary section over \(X\), so all of BSM.1–BSM.7 apply. But \(h_1s'_0=0\). The inner product of every algebraic vector in its positive Stinespring tensor module is therefore zero, by BSM.1. Its completed positive module in BSM.8 at \(t=1\) is the zero module.

The zero quotient module is nevertheless exactly \(B_B\), by BSM.5. Hence correct nondegenerate source representation, parallel normal connection and the identity **module** at the zero quotient do not identify the positive inverse. The missing datum is the actual even Kasparov lift operator and genuine equivariant module action, with their comparison to the unique proper-source lift. This is a concrete counterexample to deriving that class merely from the section's Stinespring module.

### 11AX.7. An exact finite Stinespring example

The finite arithmetic model for BSM.1–BSM.11 uses \(C=\mathbb C^2\), \(D=\mathbb C^2\oplus M_2\), quotient onto the first component and
\[
s(c_0,c_1)=((c_0,c_1),c_0H_0+c_1H_1),\quad
H_0=\operatorname{diag}(9/25,16/25),\quad
H_1=\operatorname{diag}(16/25,9/25).
\]
The matrix-component Stinespring space is \(\mathbb C^2\oplus\mathbb C^2\), with the exact nondegenerate diagonal representation and isometry
\[
\pi(c_0,c_1)=\operatorname{diag}(c_0I_2,c_1I_2),\quad
V=\begin{pmatrix}3/5&0\\0&4/5\\4/5&0\\0&3/5\end{pmatrix}.
\qquad\tag{BSM.12}
\]
Both effect matrices are positive and sum to identity. Direct rational multiplication gives \(V^*V=1\), \(P=VV^*=P^*=P^2\), \(V^*\pi(c)V=c_0H_0+c_1H_1\), and \(H_0-H_0^2=(144/625)I_2\). The compression is a cpc map with a nonzero multiplicative defect, while \(\pi\) is an exact homomorphism. This is a finite algebra illustration of the distinction; it is not a Bott inverse or an equivariant proper-space example. The rational matrix identities verify the finite example directly.


![Completed normal connection and source module](../figures/bott-source-module.png)

**Figure 11AX.1.** BSM.1–BSM.12. The actual ordinary source representation is nondegenerate on the completed Stinespring module, with the original central anchor and parallel normal connection. Its zero quotient is the identity source module. A central taper χ(t)=1−t retains that quotient and makes the positive module at t=1 vanish, so the module identity alone cannot establish the inverse class. Genuine arrow transport lands in the module of the conjugate section; an action on the fixed module needs its own coherent identifications. The finite example displays the exact homomorphic diagonal representation, its isometric compression and the compression defect (144/625)I₂. Sixteen exact representation products and four compression samples verify this illustration. It does not represent a Bott inverse or a proper groupoid. Human context: [Connes, A survey of foliations and operator algebras](https://alainconnes.org/wp-content/uploads/foliationsfine.pdf#page=29), Section 8. [Vector figure](../figures/bott-source-module.svg) · [Generator](../reproduction/bott-source-module/draw_bott_source_module.py) · [Exact checks](../reproduction/bott-source-module/BOTT-SOURCE-MODULE-CHECKS.json) · [Reproduction](../reproduction/bott-source-module/REPRODUCTION.md) · [Component terms](../reproduction/bott-source-module/COMPONENT-TERMS.md).

## 11AY. A genuine regular action on the Bott source module

The ordinary section module in The completed Bott source representation has an actual left source action, but the arrow transport in BSM.10 compares different section modules. Regular induction gives a fixed completed module with a genuine arrow action. Its zero quotient is a regular module, rather than the identity module. A proper cutoff then identifies an equivariant copy of the identity inside that quotient. These are module statements; an odd operator representing the inverse lift is a further requirement.

### 11AY.1. The completed regular module

Retain the anchored algebras \(B,E_B\), their genuine actions \(\alpha^B,\alpha^E\), and the equivariant quotient \(q_B:E_B\to B\) of The actual anchored Bott factors, BA.5–BA.15. Put \(E=E_B\). The even, nondegenerate representation
\(\pi:B\to\mathcal L_E(K)\) and quotient unitary
\(Q:K\otimes_{E,q_B}B\to B_B\) are BSM.3–BSM.5. No equivariance of the chosen cpc section or of \(K\) is assumed.

Work on the second-countable locally compact Hausdorff étale groupoid \(H\) with unit space \(X\) used by the anchored construction. This includes its transformation groupoids. In the metric holonomy setting the germ groupoid on a complete transversal is Hausdorff: the value and derivative of a local isometry determine its germ, by its intertwining of exponential maps. Two unequal germs therefore have unequal values or derivatives at their source; disjoint jet neighbourhoods separate them. The usual bisection charts give a continuous jet map, so this separation is a separation in the germ topology. Taking a transformation groupoid with a Hausdorff \(X\) preserves this property.

The regular core consists of compactly supported sections \(\xi\) of the pullback \(s^*K\) on \(H\). In a bisection chart these are the usual pullbacks of module sections multiplied by compact scalar bumps. Finite sums of such sections suffice. Its right action and \(E\)-valued inner product are
\[
 (\xi e)(g)=\xi(g)\alpha^E_{g^{-1}}(e_{r(g)}),\qquad
 \langle\xi,\zeta\rangle(x)
 =\sum_{g\in H^x}\alpha^E_g
       \bigl(\langle\xi(g),\zeta(g)\rangle_{E_{s(g)}}\bigr).
 \tag{RSM.1}
\]
Here \(H^x=r^{-1}(x)\); repetitions of an arrow in different core descriptions are combined before taking the sum.

A compact core support is covered by finitely many bisections. Each such bisection meets \(H^x\) at most once, so the sums are finite, uniformly on the support. The pullback and algebra-action definitions make the inner product a section of \(E\), and its support lies in a compact unit set. Positivity follows term by term; the matrix of inner products of any finite collection of vectors is positive by the same calculation. The Hilbert-module Cauchy–Schwarz argument used for BSM.1 therefore applies. Quotient the null vectors and complete. Denote the resulting graded Hilbert module by
\[
 \mathcal R_E(K)=
  \overline{C_c(H,s^*K)}^{\,\|\langle\xi,\xi\rangle\|^{1/2}}.
 \tag{RSM.2}
\]
All gradings are inherited from \(K,E\); the arrow index is even.

This module is countably generated. Choose a countable bisection basis, relatively compact refinements of it, a countable collection of scalar core approximations on each refinement, and a countable generating family for \(K\). Their bisection pullbacks form a countable family of module vectors. On one bisection, coefficients in a source-module approximation can be transported to its range by \(\alpha^E\), multiplied by a bump, and extended by zero to coefficients of \(E\). Thus their right module span is dense on that bisection. A finite partition subordinate to bisections proves density on every compact core support, and completion proves the assertion. Equivalently, one may first use countable dense subsets of these core spaces and then a countable approximate identity of \(E\); separability gives the same generating family.

### 11AY.2. An actual nondegenerate source action

Define
\[
 (\Pi(b)\xi)(g)
   =\pi_{s(g)}\bigl(\alpha^B_{g^{-1}}(b_{r(g)})\bigr)\xi(g).
 \tag{RSM.3}
\]
The positive inequality
\[
 \langle\Pi(b)\xi,\Pi(b)\xi\rangle
 \leq \|b\|^2\langle\xi,\xi\rangle
\]
follows from the representation bound in every summand of RSM.1. This proves boundedness and passage to the null quotient. The same inner-product calculation gives
\(\Pi(b)^*=\Pi(b^*)\). Multiplication is exact, so RSM.3 defines an even star homomorphism \(B\to\mathcal L_E(\mathcal R_E(K))\).

It is nondegenerate. On a bisection the transported representation is nondegenerate, since \(\pi\) is; finite elementary sections and compact scalar bumps give its nondegenerate section-module representation. A finite bisection partition treats every core vector. Thus \(\Pi(B)\mathcal R_E(K)\) is dense. This argument does not require an invariant approximate identity or a choice of global arrow enumeration.

For \(h:x\to y\), left translation gives
\[
 (U_h\xi)(g)=\xi(h^{-1}g),\quad g\in H^y;\qquad
 U_h(\xi e)=U_h\xi\,\alpha^E_h(e),\quad
 \langle U_h\xi,U_h\zeta\rangle=\alpha^E_h\langle\xi,\zeta\rangle.
 \tag{RSM.4}
\]
The first formula takes values in the same source fibre of \(K\), because \(s(h^{-1}g)=s(g)\). It therefore uses no arrow identification of the section modules. Reindexing RSM.1 proves both other identities. Translation by \(h^{-1}\) is the inverse. The formulas extend to semilinear unitaries on the completed fibres and satisfy
\[
 U_hU_l=U_{hl},\qquad
 U_h\Pi_x(b)U_h^{-1}=\Pi_y(\alpha^B_h(b)).
 \tag{RSM.5}
\]
These equalities hold first on the compact core and hence on the completion. Multiplication of bisections and continuous algebra actions prove continuity there; unitarity and dense core approximation extend it to the whole module. Thus this is one genuine equivariant \(B\)-\(E\) module, although the original cpc section was merely ordinary.

### 11AY.3. Its exact zero quotient

Let \(\mathcal L_H(B)\) be the regular Hilbert \(B\)-module with vectors \(y(g)\in B_{r(g)}\), inner product
\(\langle y,z\rangle(x)=\sum_{g\in H^x}y(g)^*z(g)\), and right multiplication at \(r(g)\). The quotient of RSM.2 has a unitary
\[
 \mathcal Q:\mathcal R_E(K)\otimes_{E,q_B}B
        \longrightarrow \mathcal L_H(B),\qquad
 \mathcal Q(\xi\otimes b)(g)
   =\alpha^B_g\bigl(Q_{s(g)}(\xi(g)\otimes1)\bigr)b_{r(g)}.
 \tag{RSM.6}
\]
The notation \(Q(\xi\otimes1)\) means the quotient image of \(\xi\), not a tensor with a presumed unit of \(B\). The map \(K\to K/K\ker q_B\cong B\) is the quotient map.

For two core vectors the target inner product is
\[
 b^*\,q_B(\langle\xi,\zeta\rangle)\,b',
 \tag{RSM.7}
\]
by the isometry of \(Q\), equivariance of \(q_B\), and RSM.1. This proves balancing, preservation of null vectors and isometry. On each bisection, the quotient map of the coefficient module has dense range on the pulled-back compact section core: local lifts followed by a finite scalar partition give arbitrary uniform approximations on compact supports. Right multiplication by a \(B\) approximate identity treats the balanced vectors. Finite bisection partitions therefore make the range dense in \(\mathcal L_H(B)\). Its isometric completion is onto.

Under this unitary the actual actions are
\[
 (b\,y)(g)=b_{r(g)}y(g),\qquad
 (\mathcal U_hy)(g)=\alpha^B_h\bigl(y(h^{-1}g)\bigr).
 \tag{RSM.8}
\]
Indeed \(Q\) intertwines the left \(B\)-representations, and
\(\alpha^B_g\alpha^B_{g^{-1}}\) cancels in RSM.3. For translation, write \(g=hl\); then \(\alpha_g=\alpha_h\alpha_l\), which proves the second formula. RSM.6 is consequently equivariant, not just an ordinary quotient isometry. It identifies a **regular** quotient. It does not identify that whole quotient with \(B_B\).

### 11AY.4. The proper cutoff in the quotient

Let \(W\) be the proper central \(H\)-space of the Bott source \(B\), with the nondegenerate equivariant map \(C_0(W)\to ZM(B)\) supplied by BA.8–BA.10. Use a nonnegative continuous cutoff \(c\) with
\[
 \sum_{g:r(g)=a(w)}c(g^{-1}w)=1.
 \tag{RSM.9}
\]
Here \(a:W\to X\) is its anchor. The required support condition is that, for every compact \(C\subset W\), the set
\(\{(g,w):w\in C,\ g^{-1}w\in\operatorname{supp}c\}\) is compact in \(H\ltimes W\). Here is the groupoid version of the normalized cutoff argument in Proper coefficient norms and regular diagonal transport, Lemma 11J.1. The orbit map is open because bisection translates are open. Properness makes the orbit relation closed and separates distinct orbits; relatively compact unit neighbourhoods give locally compact quotient neighbourhoods. The orbit space is consequently locally compact Hausdorff and second countable, hence has locally finite partitions subordinate to open covers. Choose compactly supported nonnegative bumps \(\phi_i\) on \(W\) whose positive sets meet every orbit, and a locally finite quotient partition \(\rho_i\) with \(\operatorname{supp}\rho_i\) contained in the orbit image of \(\{\phi_i>0\}\). Set
\(b(w)=\sum_i\rho_i([w])\phi_i(w)\) and
\(S(w)=\sum_{g:r(g)=a(w)}b(g^{-1}w)\).
On a compact \(C\subset W\) only finitely many partition indices occur. Properness puts all corresponding pairs between \(C\) and these compact bump supports in a compact set. A finite bisection cover of its arrow projection makes the sum continuous. Every orbit has a positive bump term, so \(S>0\), and reindexing proves invariance. Thus \(c=b/S\) is continuous, nonnegative and satisfies RSM.9. For the closed support, the same finite-index argument includes partition-boundary points; properness makes the compact-pair set closed inside the compact transporter. This gives the stated support condition. No compact orbit space or freeness has been assumed.

Central functional calculus defines an adjointable isometry
\[
 V_c:B_B\longrightarrow\mathcal L_H(B),\qquad
 (V_cb)(g)(w)=c(g^{-1}w)^{1/2}b(w).
 \tag{RSM.10}
\]
The \(w\)-notation denotes multiplication by a central scalar function on the \(C_0(W)\)-algebra; it presumes no faithful field of point evaluations.

To verify that RSM.10 is an actual module vector, first localize \(b\) by a compactly supported central function. The support condition then puts its active arrow coordinates in a compact set covered by finitely many bisections. These give a regular core vector and RSM.9 gives its squared norm exactly \(b^*b\). Nondegeneracy of the central action makes such \(b\)'s dense; the norm identity extends \(V_c\) to all \(B\). On a finite regular core put
\[
 V_c^*y=\sum_g c(g^{-1}\,\cdot\,)^{1/2}y(g).
 \tag{RSM.11}
\]
Finite bisection descriptions interpret this sum as a coefficient section. The row Cauchy–Schwarz inequality and RSM.9 give norm at most \(\|y\|\). This extends it to the whole module, and the core inner-product calculation verifies its adjoint identity. Therefore
\[
 p_c=V_cV_c^*=p_c^*=p_c^2,\qquad
 p_c\mathcal L_H(B)\cong B_B.
 \tag{RSM.12}
\]
The source action commutes with \(p_c\), since all cutoff entries are central. The identity
\(c(g^{-1}hw)=c((h^{-1}g)^{-1}w)\) gives
\(\mathcal U_hV_c=V_c\alpha^B_h\); thus the identification is genuinely equivariant.

For every \(b\in B\), \(b p_c\) is a compact **module** operator. To see this, approximate \(b\) by products \(b_1b_2\) using a contractive approximate identity. For an exact product,
\[
 b_1b_2p_c=\theta_{V_cb_1,\,V_cb_2^*}.
 \tag{RSM.13}
\]
The centrality of the cutoff makes this identity exact; its action on \(y\) is \(V_cb_1b_2V_c^*y\). Norm limits prove compactness for general \(b\). Neither \(p_c\) itself nor the regular identity is claimed compact when \(B\) is nonunital.

Two cutoffs yield the explicit equivariant unitary
\(V_{c'}V_c^*:p_c\mathcal L_H(B)\to p_{c'}\mathcal L_H(B)\). Its inverse is \(V_cV_{c'}^*\), and it intertwines the source action. Thus the identity submodule is independent of the cutoff, with an actual completed comparison map.

### 11AY.5. Positive evaluation and the operator requirement

Regular induction commutes with an equivariant coefficient correspondence. Let \(M\) be a genuine equivariant \(E\)-\(A'\) correspondence, with action \(T_g\). There is a unitary
\[
 \mathcal R_E(K)\otimes_E M
   \longrightarrow\mathcal R_{A'}(K\otimes_E M),\qquad
 \xi\otimes m\longmapsto
   \bigl[g\mapsto\xi(g)\otimes T_{g^{-1}}m_{r(g)}\bigr].
 \tag{RSM.14}
\]
For the inner product, covariance of \(M\)'s left action rewrites the summand on the right as
\(\langle m,\alpha^E_g(\langle\xi(g),\zeta(g)\rangle)m'\rangle\).
Its sum is exactly the inner product on the left. This proves balancing and isometry. Elementary tensor sections on a bisection are locally of the displayed form after transporting their \(M\)-factor; compact partitions and coefficient approximate identities give density and hence surjectivity. Reindexing the same core proves equivariance and intertwining of the nondegenerate \(B\)-action.

In particular the genuine positive Fock Morita correspondence in BA.11 may be used, retaining its affine Weyl action and its grading. RSM.14 gives an actual equivariant positive \(B\)-source module. It does not justify transporting BSM.6's parallel connection through that affine action without proving its derivative and domain rules.

The central taper BSM.11 remains a decisive test even after this regular induction. It is invariant, and its Stinespring module still has zero positive evaluation at \(t=1\), while RSM.6–RSM.13 still identify an equivariant identity submodule of its zero quotient. RSM.14 takes its positive evaluation to zero. Thus a genuine regular action and the quotient cutoff do not by themselves supply an odd operator \(F\) with Kasparov defects or a representative of the unique lift \(z_B\).

A further distinction is visible in the cutoff: \(p_c\) is a projection in the quotient regular module. No projection lifting it to \(\mathcal R_E(K)\) has been constructed. In particular RSM.13 cannot be used as compactness of an unspecified lifted operator. The inverse-cycle construction must supply that operator and comparison, its completed normal domains, the original \(C_0(T)\) balance and the physical graph sum.

### 11AY.6. A non-equivariant section with exact regular covariance

Here is a finite illustration in which the ordinary section is genuinely non-equivariant. Let \(H=\mathbb Z/2\), with nontrivial element \(a\), let \(W=\{0,1\}\) with \(a\) interchanging the points, and take
\[
 B=\mathbb C^2,\quad E=\mathbb C^2\oplus M_2,\quad
 s(b_0,b_1)=((b_0,b_1),b_0H_0+b_1H_1),
\]
where \(H_0=\operatorname{diag}(9/25,16/25)\) and
\(H_1=\operatorname{diag}(16/25,9/25)\).
The action on \(B\) interchanges coordinates. The action on \(E\) interchanges the quotient coordinates and fixes its \(M_2\) component. Then \(q\) is equivariant, whereas \(s\) is not:
\(\alpha_a^E s(e_0)-s(\alpha_a^B e_0)\) has matrix component
\(\operatorname{diag}(-7/25,7/25)\).

On the positive Stinespring space use the exact representation
\(\pi(b)=\operatorname{diag}(b_0I_2,b_1I_2)\). Its regular induction is
\[
 \Pi(b)=\operatorname{diag}(b_0I_2,b_1I_2,b_1I_2,b_0I_2),\qquad
 U_a=\begin{pmatrix}0&I_4\\I_4&0\end{pmatrix}.
 \tag{RSM.15}
\]
Thus \(U_a^2=1\) and \(U_a\Pi(b)U_a=\Pi(\alpha_a^B b)\) exactly, despite the section's non-equivariance.

Choose \(c=(9/25,16/25)\). The quotient vectors and projections, indexed by the two arrows, are
\[
 v(0)=\binom{3/5}{4/5},\quad
 v(1)=\binom{4/5}{3/5},\qquad
 p(0)=\frac1{25}\begin{pmatrix}9&12\\12&16\end{pmatrix},\quad
 p(1)=\frac1{25}\begin{pmatrix}16&12\\12&9\end{pmatrix}.
 \tag{RSM.16}
\]
Both vectors have norm one, both matrices are self-adjoint projections, and \(p(1)=Jp(0)J\), where \(J\) interchanges the two arrow coordinates. In row-first order the quotient action is \(J\otimes J\), retaining both the arrow permutation and coefficient action. These identities verify the equivariant cutoff copy of \(B_B\).

This example also explains why a common equivariant proper centre cannot be presumed in \(E\). A nondegenerate central extension \(C(W)\to ZM(E)\) of its quotient identity would take \(e_0\) in the \(M_2\) component to a central projection, either \(0\) or \(I_2\), and \(e_1\) to its complement. Equivariance would make the two components equal, since \(\alpha_a^E\) fixes \(M_2\); equality to the complement is impossible for a projection. Hence such an equivariant central extension does not exist, although the source centre and the quotient cutoff are valid. A lift requires more than copying a central cutoff into the extension.

![Regular source action and quotient cutoff](../figures/regular-bott-source.png)

**Figure 11AY.1.** RSM.1–RSM.16. The completed regular induction has a genuine arrow action and nondegenerate Bott source representation even when the ordinary cpc section is not equivariant. Its zero quotient is the regular coefficient module; the proper cutoff gives an equivariant identity submodule, with compactness after source localization. The exact Z/2 example retains both quotient actions and has no nondegenerate equivariant central extension into its matrix component. Sixteen exact rational representation products, four covariance samples, two cutoff projections and the full quotient isometry/action identities are checked. The invariant taper still makes the positive module vanish. No quotient projection lift or inverse operator is asserted. Human context: [Connes, A survey of foliations and operator algebras](https://alainconnes.org/wp-content/uploads/foliationsfine.pdf#page=29), Section 8. [Vector figure](../figures/regular-bott-source.svg) · [Generator](../reproduction/regular-bott-source/draw_regular_bott_source.py) · [Exact checks](../reproduction/regular-bott-source/REGULAR-BOTT-SOURCE-CHECKS.json) · [Reproduction](../reproduction/regular-bott-source/REPRODUCTION.md) · [Component terms](../reproduction/regular-bott-source/COMPONENT-TERMS.md).

## 11AZ. A genuine shift operator for positive evaluation

Positive evaluation in the anchored Bott deformation has a concrete even representative with a genuine arrow action. A sufficiently slow sequence of positive parameters makes the shift commutators compact, including the localized arrow commutators. Its homotopy to the endpoint retains the entire unit base. The coefficient grading appears explicitly in the operator.

The source of this first cycle is the deformation algebra \(E_B\), not its quotient \(B\). Composition with the proved proper-source lift gives a bounded \(B\)-source inverse cycle. The differentiable realization needed for the physical graph sum is a separate step.

### 11AZ.1. The actual positive representations

Use The anchored Bott algebra and quantum deformation, BA.8–BA.16, with
\(A=C_0(X)\), \(H=G\ltimes X\), deformation \(E=E_B\) and equivariant quotient \(q:E\to B\). The ordinary trivialization of the positive Fock module identifies its graded fibres with one fixed separable graded Hilbert space \(\mathcal F\). Write
\(\mathcal H_A=C_0(X,\mathcal F)\) and \(\Gamma\) for its grading. Positive evaluation, the actual Fock Morita module and evaluation at zero in the scalar suspension give even star homomorphisms
\[
 \rho_t:E\longrightarrow\mathcal K_A(\mathcal H_A),\qquad 0<t\leq1.
 \tag{PES.1}
\]
This is the representation of \(h_t\) on the Morita module followed by \(\mathrm{ev}_0\); it is not the cpc section \(s_B\).

For fixed \(t\), the evaluation of the ideal onto its positive fibre is onto, so \(\rho_t\) is nondegenerate. Its action on a compactly supported unit section is an actual compact \(A\)-module operator. Finite tensor sections in \(A\otimes E_\Theta\), followed by norm approximation, show that \(t\mapsto\rho_t(e)\) is norm continuous on each compact subinterval of \((0,1]\), for every \(e\in E\). No continuity at the omitted endpoint in this fixed compact-operator algebra is presumed.

The genuine action is the positive real Weyl action of BA.12–BA.13:
\[
 U_{g,t}=T_{b_g/\sqrt t}\,\Gamma_{\!F}(L_g),\qquad
 U_{g,t}\rho_t(e)U_{g,t}^{-1}=\rho_t(\alpha_g^E(e)).
 \tag{PES.2}
\]
The second equation is a fibre equation with the source and range of \(g\) retained. These unitaries are even, satisfy the exact arrow law and are strongly continuous on compact positive-parameter intervals. Multiplication by a compact coefficient operator changes this strong continuity into norm continuity: approximate that operator by finitely many rank-one maps, use vector continuity for their images and adjoints, and bound the remaining tail by twice its norm. Thus both
\((U_{g,t}-U_{g,t'})\rho_{t'}(e_s)\) and
\(\rho_t(e_r)(U_{g,t}-U_{g,t'})\)
are norm-continuous compact-arrow sections on a positive compact parameter square. This argument uses the actual semilinear coefficient maps and ordinary Fock trivialization.

### 11AZ.2. One slow sequence for a whole-base homotopy

Choose a norm-dense sequence \((e_j)\) in \(E\), including adjoints and grading images, and an increasing compact arrow exhaustion \((C_j)\). Finite bisection cutoffs may be used to express the tests on these compact sets; their union tests every arrow. There is a decreasing sequence
\[
 t_0=1,\qquad t_n>0,\qquad t_n\longrightarrow0,
 \qquad t_n(v)=v+(1-v)t_n,\quad0\leq v\leq1,
 \tag{PES.3}
\]
such that, for every \(e\in E\),
\[
 \sup_{0\leq v\leq1}
 \|\rho_{t_{n+1}(v)}(e)-\rho_{t_n(v)}(e)\|
       \longrightarrow0.
 \tag{PES.4}
\]
For every fixed compact arrow test and source section, the following two bounds also tend to zero, uniformly in \(v\):
\[
 \begin{split}
 \sup_{g\in C}
 \|(U_{g,t_{n+1}(v)}-U_{g,t_n(v)})
                \rho_{t_n(v)}(e_s)\|&\longrightarrow0,\\
 \sup_{g\in C}
 \|\rho_{t_{n+1}(v)}(e_r)
             (U_{g,t_{n+1}(v)}-U_{g,t_n(v)})\|&\longrightarrow0.
 \end{split}
 \tag{PES.5}
\]
The tests include the corresponding transported sections
\(\alpha_{g^{-1}}^E(e_r)\), and the inverse compact arrow sets. This makes the same conclusions available when a source factor is moved through PES.2.

Here is the construction, including the homotopy uniformity. At stage \(k\), subdivide the interval \([2^{-k},2^{1-k}]\) into finitely many decreasing steps. For its first \(k\) coefficient tests and arrow sets, all functions in PES.4–PES.5 are uniformly continuous when both positive parameters are in \([2^{-k},1]\); the compact-arrow sections just proved give exactly this uniform continuity. Choose the subdivision fine enough that each adjacent pair has all these errors at most \(1/k\), **for every** \(v\). This is possible because
\(|t_{n+1}(v)-t_n(v)|\leq|t_{n+1}-t_n|\), and the interpolated parameters remain in that positive interval. Concatenate the finite subdivisions, identifying common endpoints. There are only finitely many steps before any stage, and the last parameter of stage \(k\) is \(2^{-k}\). Consequently the sequence tends to zero, and a fixed test has errors tending to zero.

For arbitrary \(e\), approximation by the dense sequence completes the proof: the coefficient difference has norm at most \(2\|e-e_j\|\), and the arrow tests have the same bound because the unitaries have norm one. For transported coefficient sections on a fixed compact arrow set, continuity and finite bisection cutoffs give compact sets of coefficient values in the ordinary fixed algebra. Finite nets followed by that same estimate give uniform approximation. Thus no uncountable arrow family is silently reduced to isolated fibre tests. PES.3–PES.5 supply one sequence for the actual continuous module homotopy, not just a different sequence at each \(v\).

### 11AZ.3. The shift and its grading

Set \(\mathcal L=\ell^2(\mathbb N)\otimes\mathcal H_A\), with the ordinary nonnegative row index starting at zero. Let \(S\) be the forward shift, \(S\xi_n=\xi_{n+1}\), and \(P_0\) its row-zero projection. On
\[
 \mathcal E_+=\mathcal L,\quad
 \mathcal E_-=\mathcal L^{\mathrm{op}},\quad
 \mathcal E=\mathcal E_+\oplus\mathcal E_-
 \tag{PES.6}
\]
use the total grading \(\operatorname{diag}(\Gamma,-\Gamma)\). The represented source and operator are
\[
 \Pi_v(e)=
 \begin{pmatrix}
   \operatorname{diag}_n\rho_{t_n(v)}(e)&0\\
   0&\operatorname{diag}_n\rho_{t_n(v)}(e)
 \end{pmatrix},
 \qquad
 F=
 \begin{pmatrix}0&\Gamma S\\ \Gamma S^*&0\end{pmatrix}.
 \tag{PES.7}
\]
The diagonal source is an exact even star homomorphism. It is nondegenerate: first test a finite row vector and use the nondegeneracy of each \(\rho_t\); finite rows are dense and the approximate-identity operators are contractions. The shift is adjointable on the whole completed module, \(F=F^*\), \(F\) is odd, and
\[
 1-F^2=\begin{pmatrix}P_0&0\\0&0\end{pmatrix}.
 \tag{PES.8}
\]
Thus its localized square defect is compact, since \(\rho_{t_0(v)}(e)=\rho_1(e)\) is compact on row zero.

The factor \(\Gamma\) in PES.7 is necessary for the graded source. For homogeneous \(e\) of degree \(p\), \(\Gamma\rho_t(e)=(-1)^p\rho_t(e)\Gamma\), so the lower-left graded commutator is
\[
 \Gamma\bigl(S^*\operatorname{diag}\rho_{t_n(v)}(e)
          -\operatorname{diag}\rho_{t_n(v)}(e)S^*\bigr).
 \tag{PES.9}
\]
Its nonzero entries are compact coefficient differences between adjacent parameters. The upper-right entry has the same conclusion, with \(S\). A finite row truncation of this commutator is a finite matrix of compact coefficient operators; the row projections themselves need not be compact. PES.4 makes the commutator tails tend to zero uniformly in \(v\). Hence the full graded source commutator is compact. Without \(\Gamma\), the odd-source commutator would contain sums rather than differences and this proof would fail.

### 11AZ.4. A genuine equivariant cycle and homotopy

On both copies of \(\mathcal L\), let
\[
 \mathcal U_{g,v}=\operatorname{diag}_n U_{g,t_n(v)}.
 \tag{PES.10}
\]
Each is an even semilinear unitary. The exact arrow law holds row by row. Strong continuity on finite row vectors, uniform norm one and finite row approximation give continuity on the whole module. PES.2 makes \(\Pi_v\) exactly equivariant.

The arrow error of \(F\), after multiplication by a represented source element, is a weighted shift. To check its indices explicitly, abbreviate \(U_n=U_{g,t_n(v)}\), put \(a=\alpha_{g^{-1}}^E(e_r)\), and suppress the fibre anchors only in the following calculation. Its lower-left block from positive row \(n+1\) to negative row \(n\), localized by the source on that input row, is
\[
 \Gamma(U_nU_{n+1}^*-1)\rho_{t_{n+1}(v)}(e_r)
 =\Gamma(U_n-U_{n+1})\rho_{t_{n+1}(v)}(a)U_{n+1}^*.
\]
PES.5 initially tests \(\rho_{t_n(v)}(a)\) on this side. Replacing it by the displayed adjacent coefficient changes the norm by at most
\(2\|\rho_{t_{n+1}(v)}(a)-\rho_{t_n(v)}(a)\|\), which tends uniformly to zero by PES.4 and the transported compact coefficient tests. The upper-right block from negative row \(n\) to positive row \(n+1\) is
\[
 \Gamma(U_{n+1}U_n^*-1)\rho_{t_n(v)}(e_r)
 =\Gamma(U_{n+1}-U_n)\rho_{t_n(v)}(a)U_n^*;
\]
this is directly a PES.5 test. Thus all these tails tend to zero uniformly on compact arrow sets and uniformly in \(v\). Each finite part is a continuous compact-arrow section, by PES.1–PES.2. This proves the full cycle condition
\[
 (\mathcal U_{g,v}F\mathcal U_{g,v}^{-1}-F)
                 \Pi_{v,r(g)}(e_r)
       \in\mathcal K_A(\mathcal E)_{r(g)},
 \tag{PES.11}
\]
with the required section continuity. The adjoint or inverse-arrow calculation gives the other localization. Pointwise compactness alone has not been used as a substitute for these uniform tails.

To see that this is a genuine homotopy, use
\(C([0,1])\otimes\mathcal E\) as the Hilbert \(C([0,1])\otimes A\)-module. Each fixed row of \(\Pi_v(e)\) and \(\mathcal U_{g,v}\) is continuous. Finite row approximation makes their bounded operator fields strictly continuous and adjointable on this module. Norm continuity of the entire diagonal operator in \(v\) is neither required nor claimed. PES.4–PES.5 give compactness on the parameter module because the finite compact rows are norm continuous and their tails vanish uniformly. PES.8 is already a continuous compact row. We have proved one even anchored equivariant cycle homotopy, retaining every \(x\in X\).

### 11AZ.5. Endpoint and sign

At \(v=1\) all parameters equal one, so \(\Pi_1(e)\) and \(\mathcal U_{g,1}\) are constant in the row index. The positive-to-negative block is \(\Gamma S^*\). Its kernel is the positive row zero, and its range is all of \(\mathcal E_-\). The positive complement consists of rows \(n\geq1\); on that complement \(\Gamma S^*\) is an onto unitary to the negative module, with inverse \(S\Gamma\). This submodule has \(F^2=1\), exact graded source commutation and exact arrow invariance, hence is degenerate. The kernel is the actual graded Fock Morita module with representation \(\rho_1\), action \(U_{g,1}\) and zero operator. Therefore
\[
 [\mathcal E,\Pi_0,F]
   =[h_1]\otimes m_{\mathrm{Fock}}\otimes[\mathrm{ev}_0]
    \quad\text{in }KK_H(E,A).
 \tag{PES.12}
\]
The row-zero contribution has coefficient-class multiplicity \(+1\). This is not a scalar Fredholm index of the infinite-dimensional Fock kernel. The Fock grading, its genuine action and the source \(E\) remain present. This is not a scalar evaluation of \(X\), an equivariant unit \(1_A\), or a conclusion from separate fibre indices.

A square truncation of \(S^*\) must not be used to certify that sign. On \(N\) rows it has a positive row-zero kernel **and** a negative terminal-row cokernel. They have the same coefficient dimension, so its ordinary finite index is zero. The infinite operator has no terminal row. Formula PES.12 follows from the actual completed decomposition, not from a finite-matrix approximation to its index.

### 11AZ.6. Compact ideal action does not change the source

If \(i\in I=\ker q\), then \(\|\rho_t(i)\|\to0\) as \(t\downarrow0\), since \(I\) is the actual positive coefficient cone in BA.11. At \(v=0\) this proves
\[
 \Pi_0(I)\subset\mathcal K_A(\mathcal E).
 \tag{PES.13}
\]
The diagonal tail vanishes in norm and its finite rows are compact. But the action is generally nonzero. For a nonzero compact unit bump \(f\), an even scalar-suspension bump equal to one at zero, and the even Fock vacuum projection, the ideal section with positive value \(t fP_{\mathrm{vac}}\) has
\[
 \rho_{t_n}(i)=t_n fP_{\mathrm{vac}}\ne0
 \quad\text{on the nonzero support of }f.
 \tag{PES.14}
\]
This is an actual element of the ideal: its compact-operator norm tends to zero at \(t=0\). Consequently PES.7 does not factor as a representation of \(B\), even though its Calkin representation annihilates \(I\). The condition for an exact factorization is \(\Pi_0(I)=0\), not compactness of that ideal action.

The cycle is constructed from positive star representations, independently of the cpc section. Thus the section taper in BSM.11 does not turn this cycle into the zero positive module. It still prevents identifying a section module with the inverse cycle merely from its zero quotient.

### 11AZ.7. Composing the actual proper-source lift

BA.15 proves the existence and uniqueness of
\(z_B\in KK_H(B,E)\) with \(z_B[q]=1_B\). Fix a normalized representative \((\mathcal Z,\phi_z,F_z)\) of that proved class. Its existence follows from the proved membership in \(KK_H(B,E)\); this makes no claim that \(\mathcal Z\) is the ordinary section module or the regular module of Section 11AY. Put
\[
 \mathcal D=\mathcal Z\widehat\otimes_{E,\Pi_0}\mathcal E,
 \qquad \Phi(b)=\phi_z(b)\widehat\otimes1,\qquad
 S_z=F_z\widehat\otimes1.
 \tag{PES.15}
\]
The balanced tensor product has its genuine diagonal action and the original \(X\)-anchor. It is countably generated.

The Anchored groupoid product, KG.3–KG.4, gives an actual self-adjoint odd \(F\)-connection \(G_F\) on this tensor module and a positive even separator \(M\), with \(N=1-M\). Explicitly, use the relative unitization \(E\oplus A\), the countable homogeneous stabilization and Grassmann compression of the diagonal graded \(F\); then use KG.2's bounded strict series for \(N\), with the compact-arrow tests included. The self-adjoint product operator is
\[
 F_d=M^{1/4}S_zM^{1/4}
          +N^{1/4}G_FN^{1/4}.
 \tag{PES.16}
\]
All terms are bounded adjointable operators on the actual module \(\mathcal D\). The connection errors are compact by the stabilized finite-column calculation. With
\(\mathcal J=\mathcal K_A(\mathcal D)\),
\(\mathcal A_1=C^*(\mathcal K_E(\mathcal Z)\otimes1,\mathcal J)\)
and the compact-unit-localized ordinary and arrow defects of \(G_F\) as \(\mathcal A_2\), the zero-connection identities give
\(\mathcal A_1\mathcal A_2\subset\mathcal J\). These are exactly the hypotheses of KG.3, not an assumed smooth equivariant section.

Modulo \(\mathcal J\), square-root separator factors commute with the tested source, connection and first operator. The localized square defect is the sum of the separator-annihilated first-module and connection defects; the mixed anticommutator term is annihilated as well. The graded source commutator and self-adjoint defect have the same compactness. KG.4's compact-arrow tests give continuous localized equivariance defects. The product positivity is
\(\Phi(b)[S_z,F_d]_{\mathrm{gr}}\Phi(b)^*
 \equiv2\Phi(b)M^{1/2}\Phi(b)^*\geq0\)
modulo compacts. Thus the proved product criterion and uniqueness identify
\[
 [\mathcal D,\Phi,F_d]
    =z_B\otimes_E[\mathcal E,\Pi_0,F]
    =d_X\quad\text{in }KK_H(B,A).
 \tag{PES.17}
\]
This is a bounded genuine \(B\)-source inverse representative constructed from the proved lift and the explicit positive cycle. It is not obtained by declaring the \(E\)-representation in PES.7 to be a \(B\)-representation. The full normal connection, the derivative of the separator and connection operators, and their completed common graph domains have not been specified by the bounded product construction. Identifying a representative with those analytic controls remains necessary before using it in the original physical graph sum and Bott comparison.

### 11AZ.8. A rational two-arrow illustration

For the sequence tests take \(t_n=1/(n+1)\). On a two-dimensional positive coefficient space use the exact involutions
\[
 V(t)=\frac1{1+t^2}
       \begin{pmatrix}1-t^2&2t\\2t&t^2-1\end{pmatrix},
 \qquad V(t)^*=V(t),\quad V(t)^2=1.
 \tag{PES.18}
\]
They give the genuine action of \(\mathbb Z/2\) on
\(E=C([0,1],M_2)\) by conjugation at parameter \(t\), with quotient evaluation at zero. The ideal section \(i(t)=tI_2\) acts by the nonzero compact diagonal \(t_nI_2\). The source section \(e(t)=\operatorname{diag}(1+t,2-t)\) has the exact adjacent difference
\[
 e(t_{n+1})-e(t_n)
 =\operatorname{diag}\left(-\frac1{(n+1)(n+2)},
                           \frac1{(n+1)(n+2)}\right).
 \tag{PES.19}
\]
Differentiating the two scalar entries of \(V(t)\) gives a real orthogonal Pauli combination with norm
\(\|V'(t)\|=2/(1+t^2)\leq2\). Integration therefore bounds the adjacent unitary difference by \(2/((n+1)(n+2))\). The source norm on \([0,1]\) is at most two, so a corresponding localized arrow difference has norm at most \(4/((n+1)(n+2))\). These are bounds for the actual infinite weighted shifts, not their full unlocalized unitary differences in the Fock model. Exact rational matrices check the first six adjacent involution pairs.

The finite \(N\)-row backward shift retains both its row-zero kernel and terminal-row cokernel. The displayed finite projections have equal ranks. The sign \(+1\) in PES.12 is supplied by the infinite decomposition in Section 11AZ.5; the finite diagram illustrates why a terminal carrier cannot be discarded.

![Positive-evaluation shift, grading and the bounded inverse product](../figures/positive-evaluation-shift.png)

**Figure 11AZ.1.** PES.1–PES.19. Positive star representations and one slow parameter sequence give a genuine graded E-source shift cycle and whole-base homotopy. Composition with a representative of the proved proper-source lift yields the bounded B-source inverse class d_X. The coefficient grading, actual source and full unit base are retained. Twenty-four exact rational arrow tests cover six adjacent pairs and four homotopy values; four graded commutator tests and four finite square shifts retain both boundary carriers. The infinite coefficient-class multiplicity +1 follows from the completed endpoint decomposition. No normal derivative or physical graph-domain estimate is inferred from the finite diagram. Human context: [Connes, A survey of foliations and operator algebras](https://alainconnes.org/wp-content/uploads/foliationsfine.pdf#page=29), Section 8. [Vector figure](../figures/positive-evaluation-shift.svg) · [Generator](../reproduction/positive-evaluation-shift/draw_positive_evaluation_shift.py) · [Exact checks](../reproduction/positive-evaluation-shift/POSITIVE-EVALUATION-SHIFT-CHECKS.json) · [Reproduction](../reproduction/positive-evaluation-shift/REPRODUCTION.md) · [Component terms](../reproduction/positive-evaluation-shift/COMPONENT-TERMS.md).

## 11BA. A completed operator for positive evaluation

The positive-evaluation factor has an unbounded representative on the actual graded shift module. The construction below retains its genuine arrow action, supplies a self-adjoint regular domain and proves continuous compact source-localized arrow defects. Its ordinary constant-column normal connection is parallel. Transport gives the corresponding completed connection, without identifying it with the original connection.

### 11BA.1. A proper anchor is not a proper action

Let \(p:X\to T\) be the proper surjective probability anchor of The anchored probability base. Then
\[
 G\ltimes X\text{ is proper}\quad\Longleftrightarrow\quad G\text{ is proper}.
 \tag{QPS.1}
\]
Here properness of a groupoid means properness of its range-source map. For the forward implication, take compact \(C\subset T\times T\). The set \((p\times p)^{-1}(C)\) is compact because \(p\) is proper. Properness of the transformation groupoid makes the arrows whose endpoint pair lies there compact. Their continuous projection to \(G\) is onto \((r,s)^{-1}(C)\): for each such arrow choose any point in its nonempty source fibre. The latter set is therefore compact. For the reverse implication, a compact endpoint set in \(X\times X\) projects to a compact endpoint set in \(T\times T\). The allowed \(G\)-arrows and source points lie in the product of two compact sets. The endpoint and action equations define a closed subset of that product, hence a compact set. These are the required proper preimages.

For example, an infinite discrete group acting on a point has a proper probability anchor to a point and a nonproper transformation groupoid. Thus properness of this anchor does not supply an averaging argument on \(X\). The proper central space of the Bott source remains a different space.

### 11BA.2. Dense positive sections with finite oscillator ranges

Use \(E=E_B=A\widehat\otimes E_\Theta\), \(A=C_0(X)\), the representations \(\rho_t\) and the single sequence \(t_n(v)\) from The positive-evaluation shift. Let \(Q\) be the constant weighted Fock operator constructed below, and let \(P_m\) be increasing finite spectral projections of \(Q^2\), tending strongly to one. They are even.

There is a dense graded star subalgebra \(E_{\mathrm{fin}}\subset E_\Theta\) such that for every \(e\in E_{\mathrm{fin}}\) and compact \(J\subset(0,1]\), the ranges of all its positive coefficient evaluations on \(J\) lie in one finite \(Q\)-spectral space, on both sides. In particular
\[
 t\longmapsto Q\rho_t(e),\quad
 t\longmapsto \rho_t(e)Q
 \quad\text{are norm-continuous bounded compact fields on }J.
 \tag{QPS.2}
\]
In this formula the coefficient evaluation includes scalar-suspension evaluation; the same finite-range statement can first be made before that evaluation.

Here is a construction that preserves the zero quotient. For an element \(e\in E_\Theta\), its positive field lies in \(\mathcal K(\mathcal F)\widehat\otimes S\). On a positive compact parameter interval this is a norm-compact family. Therefore \(P_m e(t)P_m\to e(t)\) uniformly there, including the \(S\)-variable. Take a locally finite partition of unity on overlapping dyadic positive intervals and, on each interval, choose a sufficiently large \(m\). Make the compression error at most \(\min(\varepsilon,t)\). The compressed field minus \(e(t)\) is continuous, takes values in \(\mathcal K(\mathcal F)\widehat\otimes S\), and tends in norm to zero at \(t=0\). It is thus an actual element of the cone ideal. Adding it to \(e\) gives another element of \(E_\Theta\), with the same zero quotient and distance at most \(\varepsilon\).

On each positive compact interval only finitely many partition terms occur; a single larger \(P_m\) contains their ranges. Compress homogeneous elements, close under adjoints, and repeat the construction with arbitrarily small \(\varepsilon\) on a countable dense set. The rational star algebra they generate is countable and dense; its complex linear span is the required dense complex star subalgebra. Products and adjoints retain the finite-range property. Since \(QP_m\) is a bounded finite matrix, the original positive-field norm continuity proves QPS.2. No assertion that arbitrary compact operators preserve \(\operatorname{Dom}Q\) is used.

Tensor this algebra with the original scalar normal core in \(A\), as in The completed parallel connection. Its finite tensor span is a dense source core; the scalar derivative acts on the \(A\)-factor.

### 11BA.3. The fixed weighted Fock operator

Write \(\Theta e_j=\lambda_j e_j\), where \(\lambda_j\geq1\) and \(\lambda_j\to\infty\), with the graph and arrow identities BA.6–BA.7. On finite bosonic occupation vectors and finite exterior vectors put
\[
 Q=\sum_j\lambda_j(a_j\varepsilon_j+a_j^*\iota_j),
 \qquad a_j=(q_j+\partial_{q_j})/\sqrt2.
 \tag{QPS.3}
\]
The Hilbert space is the bosonic vacuum completion tensored with the exterior completion; it is the actual positive Fock space \(\mathcal F\). The exterior grading is \(\Gamma\). This is the positive scaling by \(1/\sqrt2\) of the outward \(q+ip\) convention. Creation and contraction identities give
\[
 Q^2=\sum_j\lambda_j^2(N_j^{\mathrm{bos}}+N_j^{\mathrm{ext}}),
 \qquad \Gamma Q=-Q\Gamma.
 \tag{QPS.4}
\]
Every fixed energy sublevel has only finitely many basis vectors: only coordinates with \(\lambda_j^2\) below that energy can occur, and each occupation is bounded. On each finite energy space \(Q\) is a symmetric finite matrix whose square is the displayed scalar energy. The orthogonal sum of these matrices is self-adjoint; its domain consists exactly of vectors with finite summed energy norm. Finite energy truncations are a graph core and its resolvent is compact. Its kernel is the single even vacuum. Thus the \(P_m\) used above exist and commute with \(Q\) and \(\Gamma\).

On the constant \(A\)-module these are constant operators. Fibre compactness alone does not assert global \(A\)-module compactness on noncompact \(X\); multiplication by an element of \(A\) does.

### 11BA.4. Slow row weights and small oscillator scales

Enumerate homogeneous elements \(e_j\) in the dense rational source core, including adjoints and products, and compact arrow tests as in PES.4–PES.5. Let \(d_n(j)\) denote the adjacent source error, and the corresponding transported and inverse-arrow localized errors, between rows \(n-1\) and \(n\), all uniform in \(v\). For every fixed test these tend to zero. Choose \(N_k\) increasing so that all first \(k\) errors are at most \(k^{-3}\) for \(n\geq N_k\). Taking \(w_n=k\) on \(N_k\leq n<N_{k+1}\), with positive weights on the finite initial set, gives
\[
 w_n\geq1,\quad w_n\longrightarrow\infty,\qquad
 w_n d_n(j)\longrightarrow0\quad\text{for every fixed test}.
 \tag{QPS.5}
\]
This construction is made after the sequence in PES.3; that sequence need not be changed. To retain arbitrary smooth scalar tensor factors, enumerate the countable rational coefficient algebra together with compact scalar cutoffs equal to one on successive unit test sets. On a fixed compact arrow test both endpoint sets are compact, so one such cutoff is one there. The adjacent parameter errors for a factor \(f\in A\) are bounded by \(\|f\|\) times the coefficient errors: its scalar evaluations do not depend on the parameter. These weighted limits therefore hold on the whole finite complex scalar tensor span, and not merely the countable enumerated samples. Transported tests retain the scalar norm bound by the semilinear action.

For \(n\geq1\), QPS.2 makes the following number finite:
\[
 M_n=\max_{j\leq n}\ \sup_v
 \sum_{u\in\{t_n(v),t_{n-1}(v)\}}
 \big(\|Q\rho_u(e_j)\|+\|\rho_u(e_j)Q\|\big).
 \qquad
 0<a_n\leq\min\left(2^{-n}\sqrt{t_n},
                  \frac{2^{-n}}{1+M_n}\right).
 \tag{QPS.6}
\]
Take \(a_0=1\). The \(t_n\) in the upper bound is the un-interpolated positive parameter; hence the bound holds for every \(t_n(v)\geq t_n\). Both scales are constant over \(X\).

### 11BA.5. Actual operator and completed domain

Keep exactly the PES.6 module and its total grading \(\operatorname{diag}(\Gamma,-\Gamma)\). On finite row, finite energy vectors define \(D\) by its positive row-zero block \(a_0Q\) and, for every \(n\geq1\), the block on positive row \(n\) and negative row \(n-1\):
\[
 D_n=
 \begin{pmatrix}a_nQ&w_n\Gamma\\ w_n\Gamma&a_nQ\end{pmatrix}.
 \tag{QPS.7}
\]
This is the same row pairing as the backward shift in PES.7. The negative row \(n-1\) has scale \(a_n\), rather than \(a_{n-1}\). Consequently
\[
 D_n^2=\operatorname{diag}(a_n^2Q^2+w_n^2,\,
                           a_n^2Q^2+w_n^2).
 \tag{QPS.8}
\]
The mixed terms cancel by \(\Gamma Q=-Q\Gamma\). Each \(D_n\) is self-adjoint on \(\operatorname{Dom}Q\oplus\operatorname{Dom}Q\), either by its finite energy blocks or by bounded perturbation within that fixed pair. Take their orthogonal direct sum, together with row zero. Its coefficient-valued energy identity is
\[
 \begin{split}
 \langle D\xi,D\xi\rangle_A
 ={}&a_0^2\langle Q\xi_0^+,Q\xi_0^+\rangle_A\\
 &+\sum_{n\geq1}
   \left[a_n^2\big(\langle Q\xi_n^+,Q\xi_n^+\rangle_A
                  +\langle Q\xi_{n-1}^-,Q\xi_{n-1}^-\rangle_A\big)\right.\\
 &\hspace{45mm}\left.
        +w_n^2\big(\langle\xi_n^+,\xi_n^+\rangle_A
                  +\langle\xi_{n-1}^-,\xi_{n-1}^-\rangle_A\big)\right].
 \end{split}
 \tag{QPS.9}
\]
On the \(A\)-module the domain is the norm completion of this core in \(\|\xi\|+\|D\xi\|\). Equivalently each row belongs to the constant \(Q\)-graph domain and the displayed coefficient-valued energy sum converges in \(A\)-norm. The graph contribution \(\|D\xi\|\) is the square root of the norm of that sum. A sum of separate row norms, or separate pointwise summability, does not replace this completed module condition.

The direct sum resolvents \((D\pm i)^{-1}\) are bounded adjointable contractions, are adjoints of each other, and have dense range. The finite complete-pair and finite energy truncations prove these statements and are a graph core. They commute with \(D\). The resolvent equations give self-adjointness and regularity on the completed Hilbert module; this is not merely a fibrewise self-adjointness claim.

For \(f\in A\),
\[
 f(D\pm i)^{-1}\in\mathcal K_A(\mathcal E),\qquad
 \|\text{resolvent tail after pair }N\|
 \leq\sup_{n>N}(1+w_n^2)^{-1/2}\longrightarrow0.
 \tag{QPS.10}
\]
Each retained pair has a compact Fock resolvent because \(a_n>0\); its finite energy approximants give finite rank. Row zero has the same property. Multiply these finite matrices by \(f\) to obtain actual \(A\)-module compacts; the uniform row tail proves the assertion. The global unlocalized resolvent need not belong to \(\mathcal K_A\) on noncompact \(X\).

### 11BA.6. Source domains and the represented class

The source representation remains \(\Pi_v\) from PES.7, without changing either of its diagonal copies. On the finite core its row-shift commutator has entries
\(w_n\Gamma(\rho_{t_n(v)}(e)-\rho_{t_{n-1}(v)}(e))\), with the appropriate adjoint block. Its diagonal oscillator commutators are
\(a_n[Q,\rho_{t_n(v)}(e)]_{\mathrm{gr}}\) on the positive row, and
\(a_{n+1}[Q,\rho_{t_n(v)}(e)]_{\mathrm{gr}}\) on the negative row. QPS.5–QPS.6 therefore give
\[
 [D,\Pi_v(e)]_{\mathrm{gr}}\in\mathcal K_A(\mathcal E)
 \quad(e\text{ in the dense finite tensor source core}),
 \tag{QPS.11}
\]
uniformly and continuously in \(v\). Every finite block is compact and continuous; the indicated weighted tails vanish. Each such source maps the graph core into \(\operatorname{Dom}D\), by QPS.2. The bounded commutator and closedness then prove domain preservation on the full completed domain, with graph bound
\(\|D\Pi_v(e)\xi\|\leq\|e\|\|D\xi\|+\|[D,\Pi_v(e)]_{\mathrm{gr}}\|\|\xi\|\).
Source-localized resolvents are compact for all \(e\in E\): factor by scalar approximate identities in the nondegenerate \(A\)-anchor and use QPS.10 and norm approximation.

Let \(B_w\) be the operator with row-zero block zero and pair blocks
\(\left(\begin{smallmatrix}0&w_n\Gamma\\ w_n\Gamma&0\end{smallmatrix}\right)\).
Its phase differs from the PES shift by source-locally compact weighted tails, since \(w_n/\sqrt{1+w_n^2}\to1\). Put \(F_D=D(1+D^2)^{-1/2}\). For a source block \(R=\rho_u(e)\) in either row of pair \(n\), functional calculus in \(Q\) gives the bound
\[
 \|(F_D-F_{B_w})R\|
 \leq \frac{2a_n\|QR\|}{\sqrt{1+w_n^2}}.
 \tag{QPS.12}
\]
Indeed its diagonal term has scalar norm bound \(a_n|Q|/\sqrt{1+w_n^2}\). For the off-diagonal difference use
\(0\leq1-(1+x^2)^{-1/2}\leq|x|\), with
\(x=a_n Q/\sqrt{1+w_n^2}\), and \(w_n/\sqrt{1+w_n^2}\leq1\).
This argument applies on the finite spectral range of \(R\) and then extends to its bounded product. QPS.6 gives vanishing tails. Row zero is compact after multiplication by its represented source. The adjoint calculation supplies the other localization. Density and the bound \(2\|e\|\) extend the result to every source:
\[
 (F_D-F)\Pi_v(e),\quad \Pi_v(e)(F_D-F)
 \in\mathcal K_A(\mathcal E),\qquad
 [\mathcal E,\Pi_0,F_D]=[h_1]m_{\mathrm{Fock}}[\mathrm{ev}_0]
 \quad\text{in }KK_H(E,A).
 \tag{QPS.13}
\]
All compact tails here are uniform in \(v\). Source-local compact perturbation preserves the cycle and class: expand its three defects after source multiplication, and use exact source covariance for the arrow terms. Transport of a compact module operator is a continuous compact-arrow section; approximate transported coefficient families uniformly on compact arrow sets by finite section tests. Linear bounded-operator homotopy gives the class comparison. PES.12 then identifies the endpoint class. Thus the action is genuinely equivariant at the level of the bounded transform on the actual completed module.

### 11BA.7. Genuine arrows preserve the oscillator and row domains

BA.7 gives \(L_g\operatorname{Dom}\Theta=\operatorname{Dom}\Theta\), bounded \(S_g\), and \(\Theta b_g\) continuous on compact arrow sets. Second quantization on bosons and exterior vectors yields
\[
 \|Q\,\Gamma_F(L_g)\zeta\|
       \leq(1+\|S_g\|)\|Q\zeta\|,\qquad
 \|Q T_v\zeta\|\leq\|Q\zeta\|+\|\Theta v\|\|\zeta\|,
 \quad v\in\operatorname{Dom}\Theta.
 \tag{QPS.14}
\]
For the first inequality, \(\Theta\geq1\) and
\(\Theta L_g=L_g\Theta-S_gL_g\) bound the one-particle energy form by \((1+\|S_g\|)^2\) times its original form. Summing this positive-form inequality in each bosonic and exterior particle slot gives the inequality for QPS.4. Close the finite particle form core. For the second, finite-coordinate real translation changes \(a_j\) by the scalar displacement \(v_j/\sqrt2\). Hence its change of \(Q\) is bounded Clifford multiplication of norm \(\|\Theta v\|/\sqrt2\), which gives the displayed larger bound. Finite-coordinate approximation of \(v\) in its \(\Theta\)-graph norm, strong Weyl continuity and closedness of \(Q\) prove the assertion for every such \(v\). Apply the inverse transformations to obtain equality of domains. These arguments also prove graph continuity: finite particle vectors use the graph continuity in BA.7, and uniform compact-arrow bounds extend it to every graph vector.

For \(U_{g,t}=T_{b_g/\sqrt t}\Gamma_F(L_g)\) this implies, uniformly on compact arrow sets,
\[
 \|Q U_{g,t}\zeta\|
 \leq C_K\|Q\zeta\|+C_Kt^{-1/2}\|\zeta\|.
 \tag{QPS.15}
\]
For example \(C_K=2(1+\sup_K\|S_g\|+\sup_K\|\Theta b_g\|)\) is sufficient; include the inverse compact set when using the inverse bound.

The action on negative row \(n-1\) uses \(t_{n-1}(v)\), while its oscillator scale is \(a_n\). Because \(t_{n-1}(v)\geq t_n\), both that row and positive row \(n\) satisfy \(a_n/\sqrt t\leq2^{-n}\). Row weights commute with the rowwise action. Summing QPS.15 in the completed norm QPS.9, and then applying the inverse arrow, proves
\[
 \mathcal U_{g,v}\operatorname{Dom}D_{s(g)}
          =\operatorname{Dom}D_{r(g)},\qquad
 \|D\mathcal U_{g,v}\xi\|
       \leq C'_K(\|D\xi\|+\|\xi\|).
 \tag{QPS.16}
\]
The same finite-pair approximation and graph continuity prove continuity of these completed graph maps.

There is also a continuous compact extension of the source-localized unbounded arrow difference
\[
 (\mathcal U_{g,v}D\mathcal U_{g,v}^{-1}-D)\Pi_{v,r(g)}(e_r)
 \quad\text{for the finite source core}.
 \tag{QPS.17}
\]
For its shift part, use the two explicit row formulas following PES.10, now multiplied by \(w_n\); QPS.5 includes exactly their transported and inverse tests and the weighted source difference needed to switch adjacent input coefficients. For its oscillator part, QPS.15 applied to \(U^{-1}\rho_t(e)\) bounds each row by
\(C_K a_n\|Q\rho_t(e)\|+C_K(a_n/\sqrt t)\|\rho_t(e)\|\), with the same bound for the subtracted term. QPS.6 makes these tails tend to zero. Each retained row is compact: \(\rho_t(e)\) has finite \(Q\)-spectral range and the transported finite vectors are \(Q\)-graph continuous. This proves norm continuity and compactness in the actual arrow section algebra, not just in each fibre. An unlocalized bounded difference of \(D\) and its translate is not asserted.

### 11BA.8. Completed parallel normal connection

On finite normal-smooth scalar sections with fixed Fock vectors use the actual constant-column connection \(\nabla_i=\delta_i\otimes1\), with its minimal closed domain. The norming-functional argument in BSM.6 proves closability here too. Every constant complete-pair or energy truncation commutes with this connection. Constant resolvents of \(D\) preserve the completed connection domain by core approximation and closedness. Thus
\[
 \nabla_i(D\pm i)^{-1}\xi=(D\pm i)^{-1}\nabla_i\xi,\qquad
 [\nabla_i,\Pi_v(e)]\xi=\Pi_v(\delta_i e)\xi.
 \tag{QPS.18}
\]
The second formula holds on the finite scalar tensor core and its norm-and-derivative closure, on the completed domains, by the scalar product rule and closedness. Finite complete-pair and energy truncations followed by approximation in the original scalar normal graph norm give a core for the joint domain of \(D\) and these connections. For \(f\in A\) in this scalar norm-and-derivative domain, multiplication by \(f\) makes both a constant compact resolvent and its normal derivative compact; the latter is \((\delta_i f)(D\pm i)^{-1}\).

For an arrow, the connection \(\widehat\nabla_g=\mathcal U_g\nabla_s\mathcal U_g^{-1}\) has exactly the transported completed domain and core, by Transporting completed connections. Transporting \(D\) at the same time preserves QPS.18 with no extra term. QPS.16 proves preservation of the oscillator-row domain. It does not identify \(\widehat\nabla_g\) with the original \(\nabla_r\), or prove that an unweighted derivative of \(\mathcal U_g\) is bounded.

### 11BA.9. Class, calibration and an exact finite model

Composition of QPS.13 with the proved lift \(z_B\), using the actual anchored product, represents \(d_X\) as in PES.15–PES.17. The source of \(D\) itself remains \(E\). The lift module and product connection still need their own normal domains before this factor can be used in the original \(B\)-source physical graph sum.

There is a separate ordinary scalar calibration on this same completed module: scalar \(A\)-multiplication has compact localized resolvents, commutes with \(D\), and its class is \(1_A\) after forgetting arrows. The kernel is positive row zero tensored with the even Fock vacuum. Every pair is invertible, and the orthogonal complement in row zero is invertible, with spectral gap at least one after enlarging the positive initial \(w_n\) if necessary. The phase deforms to the sign there by a uniform scalar functional-calculus homotopy; the sign complement is degenerate for scalar multiplication. This proves the whole-base ordinary equality, with the single vacuum line \(A\), rather than separate fibre index counts. It is not an equivariant scalar unit statement or a replacement for the \(B\)-source inverse.

For an exact finite illustration take
\[
 \Gamma_{\mathrm{toy}}=\operatorname{diag}(1,1,-1),\quad
 Q_{\mathrm{toy}}=
 \begin{pmatrix}0&0&0\\0&0&2\\0&2&0\end{pmatrix},\quad
 w_n=n,\quad a_n=\frac1{n+1}.
 \tag{QPS.19}
\]
The first vector is the even vacuum. Direct multiplication proves oddness, self-adjointness, QPS.8 and positivity of each pair square. These numerical choices illustrate the block identities; they are not the scales selected for the general source tests in QPS.5–QPS.6.

A square \(N\)-row truncation has a positive row-zero vacuum and a negative terminal-row vacuum. Their grades are opposite and its finite index is zero. All intermediate pairs are invertible. The completed infinite operator has no terminal row; its ordinary calibration is the whole-base decomposition just proved. Complete-pair projections used for its graph core differ from square row truncations and are not a finite square-matrix index certificate.

![Completed positive operator, paired graph domains and compact tails](../figures/quantum-positive-shift.png)

**Figure 11BA.1.** QPS.1–QPS.19. Matching the oscillator scale on positive row n and negative row n−1 cancels the cross terms and gives a self-adjoint regular completed operator. Its source remains the deformation algebra E. The A-valued energy sum, compact localized resolvents, source-domain rules, genuine arrow graph maps and continuous localized arrow differences are proved. The constant-column normal connection is parallel; the transported connection is not identified with the original one. Four exact rational paired blocks, four square truncations and a mismatched-scale test retain both finite vacuum carriers. The infinite ordinary calibration is proved by its whole-base vacuum decomposition, not by the finite index. These toy scales do not select the general source-control scales. Human context: [Connes, A survey of foliations and operator algebras](https://alainconnes.org/wp-content/uploads/foliationsfine.pdf#page=29), Section 8. [Vector figure](../figures/quantum-positive-shift.svg) · [Generator](../reproduction/quantum-positive-shift/draw_quantum_positive_shift.py) · [Exact checks](../reproduction/quantum-positive-shift/QUANTUM-POSITIVE-SHIFT-CHECKS.json) · [Reproduction](../reproduction/quantum-positive-shift/REPRODUCTION.md) · [Component terms](../reproduction/quantum-positive-shift/COMPONENT-TERMS.md).

## 11BB. An unbounded representative with the actual Bott source

The positive operator in Section 11BA has source \(E_B\). This section constructs an unbounded representative of the actual inverse class \(d_X\), with source \(B\), on the genuine product module. Its completed graph domain is preserved by the arrows, with continuous compact source-localized unbounded defects. The construction uses the proved lift class, rather than a presumed projection lift from the regular quotient.

Its normalization adds a summand on which \(B\) acts as zero. That summand is retained throughout the unbounded construction. Removing it with normal and graph-domain controls is a separate requirement for the physical realization.

### 11BB.1. The actual product and its normalization

Put \(A=C_0(X)\) and \(H=G\ltimes X\). The proper-source lift, BA.15–BA.16, proves the class \(z_B\) and its inverse factor. Choose an essential representative of \(z_B\), using Essential replacement, KG.5, before taking the actual balanced product in PES.15–PES.17. The resulting module \(K\) is countably generated over \(A\), with a genuine even \(H\)-action, an even nondegenerate representation \(\phi_0:B\to\mathcal L_A(K)\), and a cycle of class \(d_X\). Nondegeneracy follows on finite tensors from the source approximate identity on the first factor, and then on the completion by uniform boundedness.

The operator-normalization steps of Odd self-adjoint normalization give an odd self-adjoint contraction \(F_0\) on this same essential module, representing the same class. In particular the actual self-adjoint sandwich can be clipped to \([-1,1]\) by odd continuous calculus. Its difference vanishes at the two points where \(F_0^2=1\), so the original source-local square defects make that difference source-locally compact; polynomial approximation gives the localized arrow comparison. The doubling is only performed in the following explicit step. Set
\[
 C=(1-F_0^2)^{1/2},\quad
 \widetilde K=K\oplus K^{\mathrm{op}},\quad
 \widetilde\Gamma=\operatorname{diag}(\Gamma,-\Gamma),\quad
 \phi(b)=\operatorname{diag}(\phi_0(b),0),\quad
 F=\begin{pmatrix}F_0&C\\ C&-F_0\end{pmatrix}.
 \tag{IUR.1}
\]
The action on both copies is the original action \(U_g\). Since \(C\) commutes with \(F_0\) and is even, \(F\) is odd, self-adjoint and satisfies \(F^2=1\).

The off-diagonal terms are two-sided \(B\)-locally compact. Indeed
\((C\phi_0(b))^*C\phi_0(b)=\phi_0(b)^*(1-F_0^2)\phi_0(b)\) is compact; the compact-operator square-root criterion gives compactness of \(C\phi_0(b)\), and adjoints give the other side. This criterion follows by functional calculus in the adjointable algebra: if \(T^*T\) is compact, then \(T\) is a norm limit of \(T f_n(T^*T)\), which are compact for continuous cutoff functions vanishing near zero.

For arrows, the off-diagonal error is \((\alpha_g C-C)\phi_0(b)\). The two terms are compact after source transport. They are continuous compact-arrow sections: approximate the square root uniformly on \([0,1]\) by polynomials vanishing at zero, and use the original compact source-localized square and arrow defects in the finite polynomial identities. Finite bisection tests and norm limits give the required continuity. The diagonal arrow error is the original one.

The path adding \(tC\) in the off-diagonal blocks is a cycle homotopy. Its square, graded source commutator and arrow errors after multiplication by \(B\) have exactly the compact terms just proved, uniformly in \(t\). At zero the second summand has identically zero representation, so its class vanishes. Consequently
\[
 [\widetilde K,\phi,F]=d_X\quad\text{in }KK_H(B,A).
 \tag{IUR.2}
\]
The full \(B\)-representation \(\phi\) is degenerate on its second summand. The relative unitization \(B_A^+=B\oplus A\) has the nondegenerate equivariant representation \((b,f)\mapsto\phi(b)+f\,1_{\widetilde K}\). After forgetting the arrow action, this is a bounded cycle because \(F^2=1\) and \(F\) is \(A\)-linear; its scalar \(A\)-restriction is then degenerate and has ordinary class zero. It is not asserted to be an equivariant cycle for the enlarged source: the extra arrow test \((\alpha_gF-F)f\,1_{\widetilde K}\) has not been proved compact. Nondegeneracy of a relative unitization representation supplies neither that additional test nor a scalar calibration \(1_A\).

### 11BB.2. One compact inverse with all compact-arrow tests

Write \(J=\mathcal K_A(\widetilde K)\). If this module is zero, the construction is the zero operator on the zero module. Otherwise \(J\) is sigma-unital: a countable generating family \((\xi_j)\), normalized and scaled, gives a norm-convergent positive sum of rank-one operators whose range is dense in \(\widetilde K\). Average such a positive compact over grading and conjugation by \(F\). These two conjugations commute, even though \(F\) is odd. The resulting positive \(h_0\in J\) has dense range and commutes with \(F\) and the grading.

Take nested compact functional-calculus cutoffs of \(h_0\). The Controlled inverse construction and Convex approximate invariance, KG.2, allow one common-tail convex choice \(u_n\in C^*(h_0)\). Include a countable dense homogeneous source algebra, its products and adjoints, compact-unit-localized transports along a countable bisection basis closed under inverse and product, and a compact arrow exhaustion. The source closure under those selected transports and cutoffs is countable, as in KG.1. For \(j\leq n\) and arrows in the \(n\)-th compact test, require
\[
 \begin{gathered}
 u_nu_{n+1}=u_n,\quad 0\leq u_n\leq1,\quad
 u_n\longrightarrow1\text{ strictly},\\
 \|[u_n,\phi(b_j)]\|\leq2^{-n},\qquad
 \|[F,\phi(b_j)]_{\mathrm{gr}}(1-u_n)\|\leq2^{-n},\\
 \sup_g\|\alpha_g(u_n)-u_n\|\leq2^{-n},\qquad
 \sup_g\|(\alpha_gF-F)\phi(b_j)(1-u_n)\|\leq2^{-n}.
 \end{gathered}
 \tag{IUR.3}
\]
Restrictions to source and range fibres are implicit in the last line. Include inverse arrow sets and the corresponding adjoint tests.

Here each ordinary compact-source error belongs to \(J\). Each arrow-source error is a continuous compact section on the actual pullback module. Its compact-test approximate-identity convergence is uniform: cover the compact arrow set by finitely many bisections, localize at units, pull back or transport the compact sections, and use the approximate-identity property of \(J\). This is the finite-chart argument of KG.2, with arbitrary compact sections included among its tests. The multiplier commutator tests derive \(J\). The weak-convex separation for the direct sum of these finite tests gives a single common-tail convex combination. Later supports may be chosen beyond the preceding cutoff plateau; that preserves nesting. Every combination still commutes with \(F\) and grading. Thus IUR.3 is one construction on the anchored module, not an inference from isolated fibres.

Define
\[
 r=1+\sum_{n\geq1}(1-u_n),\quad l=r^{-1},\quad
 R_N=1+\sum_{n=1}^{N}(1-u_n).
 \tag{IUR.4}
\]
On the nonzero spectrum of \(h_0\) the sum is locally finite: once an earlier cutoff is positive, every sufficiently later cutoff is one. Near zero the first \(N\) cutoffs all vanish, so \(r\geq N+1\). Hence \(l\) is a continuous function of \(h_0\), extends by zero at zero, belongs to \(J\), and commutes with \(F\) and grading. It has dense range: each cutoff vector \(u_m\xi\) equals \(l(r u_m)\xi\), where \(r u_m\) is bounded, and the cutoff ranges are dense. Moreover \(lR_N\to1\) strictly.

### 11BB.3. Completed operator and exact resolvents

On the actual completed module put
\[
 D=Fl^{-1},\qquad
 \operatorname{Dom}D=\operatorname{Ran}l,\qquad
 \|D\xi\|=\|l^{-1}\xi\|.
 \tag{IUR.5}
\]
The norm identity uses the genuine adjointable unitary \(F\), not a pointwise norm. The inverse \(l^{-1}\) is a closed regular positive operator by its commuting cutoff calculus. Its products here are defined on exactly \(\operatorname{Ran}l\).

The two inverses are
\[
 (D\pm i)^{-1}=l(F\mp il)(1+l^2)^{-1},\qquad
 F_D=D(1+D^2)^{-1/2}=F(1+l^2)^{-1/2}.
 \tag{IUR.6}
\]
Their ranges lie in \(\operatorname{Ran}l\), and direct multiplication gives both resolvent identities there. They are bounded adjointable adjoints of one another. The operator is symmetric because \(l,F\) commute; surjectivity at both signs proves self-adjointness and regularity. Equivalently the resolvent graph projection gives the completed regular graph. Each resolvent is an actual element of \(J\), globally on the \(A\)-module, since \(l\in J\). Also
\[
 F_D-F\in J,\qquad
 [\widetilde K,\phi,F_D]=d_X.
 \tag{IUR.7}
\]
The compact difference follows because the scalar function
\((1+s^2)^{-1/2}-1\) vanishes at \(s=0\). The straight compact perturbation homotopy preserves all cycle conditions, including continuous compact-arrow defects, by the same finite-chart compact transport argument.

### 11BB.4. Source domains and compact unbounded commutators

For a controlled homogeneous \(b_j\), the first two source estimates in IUR.3 give norm-convergent compact operators
\[
 C_j=\lim_N[R_N,\phi(b_j)],\qquad
 L_j=\lim_N[F,\phi(b_j)]_{\mathrm{gr}}R_N.
 \tag{IUR.8}
\]
Only finitely many initial terms precede the \(j\)-th test; the remaining norms are summable. Every term is compact, so both limits are compact.

For \(\xi\in\operatorname{Ran}l\), the convergent equation
\[
 R_N\phi(b_j)\xi
 \longrightarrow\phi(b_j)l^{-1}\xi+C_j\xi
 \tag{IUR.9}
\]
proves actual domain preservation. Multiply it by \(l\): its left side tends to \(\phi(b_j)\xi\), by strict convergence of \(lR_N\). The resulting identity puts that vector in \(\operatorname{Ran}l\), with the displayed inverse image. Thus
\[
 [D,\phi(b_j)]_{\mathrm{gr}}=F C_j+L_j\in J.
 \tag{IUR.10}
\]
Products satisfy the graded domain Leibniz rule. Adjoints and all homogeneous word tests were included, so the complex star algebra they generate is a dense source core. Its operators preserve the full completed graph domain and have compact, hence bounded, graded commutators. The graph bound is
\(\|D\phi(b)\xi\|\leq\|b\|\|D\xi\|+\|[D,\phi(b)]_{\mathrm{gr}}\|\|\xi\|\).
The resolvents are already compact for the entire source; no finite summability, first-order expression or uniform bound over this dense algebra is claimed.

### 11BB.5. Genuine arrows preserve the full graph domain

For a compact arrow test the last two estimates of IUR.3 give
\[
 A_g=\lim_N(\alpha_gR_N-R_N)
       =\sum_n(u_n-\alpha_gu_n)
 \tag{IUR.11}
\]
uniformly in compact-operator section norm. Each finite part is continuous; their tails are uniformly summable. Thus \(A_g\) is a continuous compact-arrow section, with locally uniform bounds.

Transporting the convergent inverse equation, or applying it to the two partial sums, gives
\[
 U_g\operatorname{Ran}l_{s(g)}
     =\operatorname{Ran}l_{r(g)},\qquad
 U_g r_{s(g)}U_g^{-1}=r_{r(g)}+A_g
 \quad\text{on that completed domain}.
 \tag{IUR.12}
\]
For example, for a source-domain vector \(\xi\),
\(R_N(r(g))U_g\xi
 =U_gR_N(s(g))\xi-(\alpha_gR_N-R_N)U_g\xi\)
has a limit. Multiplication by \(l_{r(g)}\) proves membership in its range. Apply the inverse arrow to obtain equality. Since \(F\) is unitary, IUR.5 gives the exact graph estimate
\[
 \|D_{r(g)}U_g\xi\|
 \leq\|D_{s(g)}\xi\|+\|A_g\|\|\xi\|.
 \tag{IUR.13}
\]
Graph continuity follows on the whole domain, not just the original finite core: transport of a Hilbert-module graph vector is continuous in its module components, and IUR.12 expresses its second graph component using the bounded strongly continuous \(F,U\) and continuous compact \(A_g\). Equivalently use the cutoff core and the locally uniform graph bound. The action retains its original exact cocycle, coefficient semilinearity and source covariance; the operator need not commute with that action.

Put \(K_{g,j}=(\alpha_gF-F)\phi(b_j)\). The last source-arrow estimate gives the norm-convergent compact section
\(K_{g,j}r=K_{g,j}+\sum_nK_{g,j}(1-u_n)\), uniformly on a compact arrow test. On the completed domain the source-localized unbounded error is
\[
 (\alpha_gD-D)\phi(b_j)
 =(\alpha_gF)A_g\phi(b_j)
       +K_{g,j}r+(\alpha_gF-F)C_j.
 \tag{IUR.14}
\]
All three right-hand terms are continuous compact-arrow sections. For the last term, multiplication of a bounded strictly continuous field by a compact field is norm continuous: approximate the compact by finite rank sections and use vector continuity. The same argument treats the first. Hence IUR.14 has a bounded compact extension on the actual pullback modules. Adjoint and inverse tests supply the opposite localization.

### 11BB.6. A genuine graph core and the remaining normal condition

The span of \(u_n\widetilde K\) is a graph core. On \(\operatorname{Ran}l\), the operators \(u_n\) commute with \(r,F\) and tend strictly to one; both the vector and its graph image converge. Each \(r u_n\) is bounded. Countable dense homogeneous sections inside these ranges therefore give a countably generated core. Adjoin their images under the countable controlled source algebra, compact-unit cutoffs and the countable bisection basis with compact scalar cutoffs. IUR.9–IUR.13 put these vectors in the completed domain. Their span contains the original graph-dense core, is stable under those tested operations, and supplies local graph sections on every arrow chart.

This proves an actual \(B\)-source unbounded inverse representative, with its class, coefficient anchor, grading, genuine action and completed graph domain. Its whole doubled module still has the zero \(B\)-source summand in IUR.1. Essential replacement by KG.5 is an actual bounded equivariant class homotopy. It does not by itself identify a reducing projection for this unbounded \(D\) or give a bounded commutator with the projection onto the first summand.

Likewise \(h_0,u_n,l\) were selected by compact, source and arrow tests. No normal derivative of them has been prescribed. The normal-compatible lift module, derivative rules, source-essential unbounded replacement and the original physical graph sum still require their own construction. The ordinary positive factor in Section 11BA remains available with its parallel connection, but cannot supply these derivative premises for the arbitrary lift witness and the new regulator.

### 11BB.7. An exact finite normalization

Take a three-dimensional original module with grading \(\operatorname{diag}(1,1,-1)\), source \(\phi_0(a)=aI_3\), and
\[
 F_0=\begin{pmatrix}0&0&0\\0&0&1\\0&1&0\end{pmatrix},
 \quad C=\operatorname{diag}(1,0,0),\quad
 P=\operatorname{diag}(1,1,1,0,0,0),\quad
 r=\operatorname{diag}(2,3,3,2,4,4).
 \tag{IUR.15}
\]
On the doubled module, \(F\) is IUR.1, the grading is
\(\operatorname{diag}(1,1,-1,-1,-1,1)\), and the source is \(aP\).
Direct multiplication gives \(F=F^*=F^{-1}\), oddness, \(rF=Fr\), and
\(D^2=r^2=\operatorname{diag}(4,9,9,4,16,16)\).
The whole \(D\) is invertible. The source acts only on the first three coordinates. Removing its off-diagonal coupling in this finite example leaves an even vacuum and one invertible odd pair on that essential subspace, whose index is \(2-1=+1\). Thus an invertible normalized whole operator can retain a nonzero source class; declaring the entire doubled source representation nondegenerate would lose this distinction. Finite-dimensional bounded removal does not prove bounded removal or normal control for the infinite module.

![Actual Bott-source inverse, compact regulator and completed graph](../figures/inverse-cycle-regularization.png)

**Figure 11BB.1.** IUR.1–IUR.15. The actual B-source inverse class is retained through a unitary normalization and one compact controlled inverse. The domain is the completed range of l, its resolvents are module compact, the dense source core has compact unbounded commutators, and the genuine arrows preserve its completed graph with continuous compact localized errors. The second normalized summand has zero B-source representation. Sixteen exact finite source products, four covariance tests and the operator, regulator, square, arrow-error and essential-compression identities are checked. The whole finite operator is invertible, its scalar-unit index is zero and its represented essential index is +1. This finite bounded removal supplies no infinite essential or normal-domain replacement. Human context: [Connes, A survey of foliations and operator algebras](https://alainconnes.org/wp-content/uploads/foliationsfine.pdf#page=29), Section 8. [Vector figure](../figures/inverse-cycle-regularization.svg) · [Generator](../reproduction/inverse-cycle-regularization/draw_inverse_cycle_regularization.py) · [Exact checks](../reproduction/inverse-cycle-regularization/INVERSE-CYCLE-REGULARIZATION-CHECKS.json) · [Reproduction](../reproduction/inverse-cycle-regularization/REPRODUCTION.md) · [Component terms](../reproduction/inverse-cycle-regularization/COMPONENT-TERMS.md).

## 11BC. A proper-source average and an essential unbounded inverse

The proper centre of the Bott source can be used on its actual representation module. Averaging the bounded operator there gives exact arrow invariance. A controlled regulator and closed compression then remove the zero-source normalization summand, retaining the nondegenerate \(B\)-representation and the inverse class \(d_X\). The full closed graph domain, not only a reference core, is preserved by the arrows.

### 11BC.1. The proper centre on the represented module

Use \(A=C_0(X)\), \(H=G\ltimes X\), and the essential \(B\)-source product module \((K,\phi_0,F_0)\) of The actual inverse representative, before doubling. Its source representation is nondegenerate. The proper central \(H\)-space \(W\) is the phase centre in BA.8–BA.10, in The anchored Bott algebra. Its nondegenerate map \(C_0(W)\to ZM(B)\) therefore gives a nondegenerate even map
\[
 \varpi:C_0(W)\longrightarrow\mathcal L_A(K).
 \tag{EAI.1}
\]
For a central multiplier \(f\), define its action first on finite source-generated vectors by
\(\varpi(f)\phi_0(b)\xi=\phi_0(fb)\xi\).
The positive matrix inequality
\((b_i^*f^*fb_j)\leq\|f\|^2(b_i^*b_j)\), followed by the represented inner product, gives its norm bound and independence of the chosen finite description. The formula for \(f^*\) gives its completed adjoint. Nondegeneracy of \(\phi_0\) extends it to the whole module. Compact central approximate identities satisfy \(f_n b\to b\), so their operators converge strongly to one on that dense span and hence on \(K\). Source covariance proves equivariance on the same span. Thus EAI.1 commutes with \(\phi_0(B)\), is nondegenerate and equivariant, and includes the bounded central multipliers used below. This is an action on the actual module; it is not an extension of the centre to \(E_B\).

Take the continuous nonnegative cutoff \(c\) of RSM.9, with
\(\sum_g c(g^{-1}w)=1\) and its compact-pair support condition. These are fully proved in The proper quotient cutoff. Only the cutoff function and its proper support are used here; no quotient projection is lifted.

Let \(\mathcal L_H(K)\) be the completed regular \(A\)-module with coordinates \(y(g)\in K_{r(g)}\), inner product
\(\langle y,z\rangle(x)=\sum_{g\in H^x}\langle y(g),z(g)\rangle_{A_x}\).
Compact bisection sections form its core. Its action is
\((\mathcal U_hy)(g)=U_hy(h^{-1}g)\).
Positivity, countable generation and completion have the finite-bisection proof in RSM.1–RSM.8, now with the given genuine action on \(K\).

Put \(C_g=\varpi(c\circ g^{-1})^{1/2}\) on the range fibre. There is an adjointable equivariant isometry
\[
 V:K\longrightarrow\mathcal L_H(K),\quad
 (V\xi)(g)=C_g\xi_{r(g)},\qquad
 V^*y=\sum_g C_gy(g).
 \tag{EAI.2}
\]
First take \(\xi=\varpi(f)\eta\) with \(f\in C_c(W)\). The compact-pair support condition puts its active arrows in a compact set covered by finitely many bisections. Thus it is a regular module vector. The normalized cutoff sum gives
\(\langle V\xi,V\xi\rangle=\langle\xi,\xi\rangle\).
Such vectors are dense by EAI.1, so \(V\) extends as an isometry. On finite regular columns the formula for \(V^*\) is a contraction: the positive operator sum \(\sum C_g^2\leq1\) makes the finite row operator have norm at most one. Core inner products prove the adjoint identity. This gives the full completed adjoint and \(V^*V=1\).

Source centrality gives \(V\phi_0(b)=\Lambda(b)V\), where \(\Lambda(b)\) is the diagonal range-fibre source representation. Reindexing the cutoff sum gives \(\mathcal U_hV=VU_h\), with the actual coefficient semilinearity. Neither a unit of \(B\) nor a compact orbit space is required.

### 11BC.2. An exactly invariant bounded representative

On the regular module define
\[
 (\mathcal T y)(g)=U_gF_{0,s(g)}U_g^{-1}y(g),\qquad
 \overline F=V^*\mathcal T V.
 \tag{EAI.3}
\]
The diagonal \(\mathcal T\) is a self-adjoint odd contraction. Its action on the compact bisection core is continuous; boundedness and the corresponding adjoint action extend it to an adjointable operator on the completed module. Its transported value at \(hg\) is the transport of its value at \(g\), so it is exactly invariant. EAI.2 consequently proves
\[
 \overline F=\overline F^*,\quad\|\overline F\|\leq1,\quad
 \alpha_h(\overline F)=\overline F.
 \tag{EAI.4}
\]

We prove the class comparison with the original \(F_0\), including compactness on the whole module. For a compactly \(W\)-supported source \(b\), choose \(f\in C_c(W)\) equal to one on its support, and write \(M_f=\varpi(f)\). Then
\[
 (\mathcal T V-VF_0)\phi_0(b)
 =\mathcal T V\phi_0(b)-V M_fF_0\phi_0(b)
      -V[F_0,M_f]\phi_0(b).
 \tag{EAI.5}
\]
The last term is compact. Indeed
\([F_0,M_f]\phi_0(b)=[F_0,\phi_0(fb)]_{\mathrm{gr}}
 -M_f[F_0,\phi_0(b)]_{\mathrm{gr}}\), since \(f\) is even. Both terms are compact by the original source cycle condition.

The first two columns have compact arrow support, by the support condition applied to \(b\) and \(f\). Each of their differences has compact entries. An entry is
\[
 (\alpha_gF_0-F_0)C_g\phi_0(b)
    +[F_0,C_g]\phi_0(b)
    +C_g(1-M_f)F_0\phi_0(b).
 \tag{EAI.6}
\]
The first is the original localized arrow defect for the continuous source field \(c_g^{1/2}b\). The second is compact by the same multiplier commutator identity just used for \(f\). The last is compact because
\((1-M_f)F_0\phi_0(b)=(1-M_f)[F_0,\phi_0(b)]_{\mathrm{gr}}\).
These are continuous compact sections on the actual arrow charts. The multiplier functions times \(b\) are norm continuous on compact charts: the cutoff is uniformly continuous on the compact source-support transporter, and central multiplication has its scalar norm bound.

A compactly supported continuous compact-valued column is a compact \(A\)-module map. To verify this, split its support into finitely many bisection pieces. Range coordinates on one such chart are ordinary unit coordinates. Uniform finite rank approximation of its compact section gives sums \(\theta_{\text{row }\xi,\eta}\); the row vector is an actual compact regular core vector, and each \(\eta\) is a unit-module vector extended with a cutoff. Their norm limits give compactness of that column. Thus EAI.5 is compact. Multiplication by \(V^*\) proves
\[
 (\overline F-F_0)\phi_0(b),\quad
 \phi_0(b)(\overline F-F_0)\in\mathcal K_A(K)
 \quad(b\in B).
 \tag{EAI.7}
\]
For general \(b\), use the nondegenerate central action to approximate it by compactly \(W\)-supported sources, and the bound \(2\|b\|\). Adjoints give the other side. The straight locally compact perturbation homotopy preserves all source and continuous compact-arrow defects, as proved in IUR.1–IUR.2. Hence \((K,\phi_0,\overline F)\) has the original class \(d_X\). No averaging of a merely ordinary cpc section has occurred.

### 11BC.3. A regulator that controls the essential projection

Apply the explicit IUR.1 normalization to \(\overline F\), and retain its notation \(F,\widetilde K,\phi\). Let \(P\) project onto its first summand, \(Q=1-P\). Both \(F\) and \(P\) are now exactly invariant. Their source representation satisfies \(P\phi(b)=\phi(b)\), \(Q\phi(b)=0\).

In the single compact-inverse construction IUR.3 include the additional multiplier test \(P\) and compact-source tests
\([F,P]\phi(b_j)\), with adjoints and source transports. These latter operators are compact by the square-root defect test in IUR.1. The same common-tail choice gives, for every fixed controlled source,
\[
 [r,P]=C_P\in\mathcal K_A(\widetilde K),\qquad
 [F,P]\phi(b_j)r\in\mathcal K_A(\widetilde K).
 \tag{EAI.8}
\]
The first is the norm-convergent sum of the cutoff commutators with \(P\). The second is the norm-convergent compact tail series, exactly as in IUR.8. The inverse equation IUR.9, now with \(P\), proves that \(P,Q\) preserve \(\operatorname{Dom}r\). This enlarges the regulator tests without changing the class.

The reference off-diagonal part is bounded, self-adjoint and even:
\[
 B_r=PrQ+QrP=C_P(2P-1),\qquad
 r_{\mathrm d}=r-B_r.
 \tag{EAI.9}
\]
The formula follows from its two blocks. Since \(C_P^*=-C_P\) and \(C_P\) is off diagonal, the product is self-adjoint. The bounded-perturbation resolvent proof in UP.8–UP.12, in Compact-resolvent representatives, works on the Hilbert module by the Neumann series and the adjoint-domain argument of Regular calculus. It proves that \(r_{\mathrm d}\) is self-adjoint regular on \(\operatorname{Dom}r\), with compact module resolvent.

For a domain vector, its quadratic form is the sum of the two diagonal \(r\)-forms, so \(r_{\mathrm d}\geq1\). It commutes with \(P,Q\) on its domain, hence its resolvents do. Its restrictions
\[
 r_i=P_i rP_i,\quad
 \operatorname{Dom}r_i=P_i\operatorname{Dom}r,\qquad
 P_0=P,\ P_1=Q
 \tag{EAI.10}
\]
are positive regular operators with compact resolvents on their actual completed submodules.

### 11BC.4. Closed essential compressions are self-adjoint regular

Define initially on \(\operatorname{Dom}r_i\)
\[
 S_i=P_iDP_i=F_i r_i+T_i,\qquad
 F_i=P_iFP_i,\quad T_i=P_iF(1-P_i)rP_i.
 \tag{EAI.11}
\]
The last factor is a bounded compact reference off-diagonal block, so \(T_i\) is bounded compact. The operator \(S_i\) is symmetric by symmetry of \(D\) and domain preservation by \(P_i\). Moreover
\[
 K_i=[r_i,F_i]
 =-P_i r(1-P_i)FP_i+P_iF(1-P_i)rP_i
 \tag{EAI.12}
\]
is bounded compact, since \([r,F]=0\). This identity also proves the full reference-domain rule for \(F_i\).

Here is a completed regularity proof that does not presume a bounded off-diagonal part of \(D\). Put
\[
 v_\varepsilon=(1+\varepsilon r_i)^{-1},\qquad
 [S_i,v_\varepsilon]
 =\varepsilon v_\varepsilon K_i v_\varepsilon r_i
      +[T_i,v_\varepsilon].
 \tag{EAI.13}
\]
The right side is bounded by \(\|K_i\|+2\|T_i\|\), uniformly in \(\varepsilon>0\). It tends strongly to zero. For the first term, test a vector in \(\operatorname{Dom}r_i\), where its norm is at most \(\varepsilon\|K_i\|\|r_i\xi\|\); use density and the uniform bound for all vectors. The second tends strongly to zero because \(v_\varepsilon\to1\) strongly and \(T_i\) is bounded. Each \(v_\varepsilon\) maps the whole module into \(\operatorname{Dom}r_i\), and \(S_i v_\varepsilon\) is bounded adjointable.

Localize by any state of \(A\), as in The symmetric local-global criterion, Lemma 0.1. All these bounded adjointable maps descend to its Hilbert space. The localized range of \(v_\varepsilon\) lies in the graph closure of the localized \(S_i\): approximate any Hilbert-space vector by module vectors and use boundedness of both \(v_\varepsilon\) and \(S_i v_\varepsilon\). Denote the symmetric closure by \(S_\omega\) and its commutator by \(E_\varepsilon\). Testing against the dense original domain, and using \(E_\varepsilon^*=-E_\varepsilon\), gives
\[
 S_\omega v_\varepsilon\eta
 =v_\varepsilon S_\omega^*\eta+E_\varepsilon\eta
 \quad(\eta\in\operatorname{Dom}S_\omega^*).
 \tag{EAI.14}
\]
If \(S_\omega^*\eta=\pm i\eta\), the norm of
\((S_\omega\mp i)v_\varepsilon\eta\) tends to zero, while symmetry bounds it below by \(\|v_\varepsilon\eta\|\). Thus \(\eta=0\). Both deficiency spaces vanish, so \(S_\omega\) is self-adjoint. Module symmetry gives closability; localization of its module graph closure has this same Hilbert-space graph closure, since graph approximations descend in norm. The fully proved local-global criterion now gives self-adjoint regular closures
\[
 \overline S_0\text{ on }P\widetilde K,\quad
 \overline S_1\text{ on }Q\widetilde K,\qquad
 D_{\mathrm d}=\overline S_0\oplus\overline S_1.
 \tag{EAI.15}
\]
These submodules retain their original gradings. Their reference domains in EAI.10 are graph cores by definition of these closures. Their full closed domains can be larger; no equality to the reference domain is asserted.

### 11BC.5. Source-local resolvent comparison preserves the exact class

On \(\operatorname{Dom}r\) put
\[
 B_{\mathrm o}=PDQ+QDP=[D,P](2P-1),\quad
 D-D_{\mathrm d}=B_{\mathrm o}.
 \tag{EAI.16}
\]
This operator need not be bounded on the whole module. It is \(r\)-relative bounded, and \(B_{\mathrm o}l\) is bounded adjointable: use
\([D,P]=FC_P+[F,P]r\) and
\(r(2P-1)l=(2P-1)+2C_Pl\).
For a controlled \(b\), \(B_{\mathrm o}\phi(b)\) has the compact extension
\[
 FC_P\phi(b)+[F,P]\phi(b)r+[F,P]C_b,
 \qquad C_b=[r,\phi(b)].
 \tag{EAI.17}
\]
All three terms are compact by EAI.8 and IUR.8. Adjoints supply the opposite product on the core.

Write \(R(z)=(D-z)^{-1}\), \(R_{\mathrm d}(z)=(D_{\mathrm d}-z)^{-1}\). The original resolvent maps into \(\operatorname{Dom}r\), so the core equality EAI.16 proves
\[
 R_{\mathrm d}(z)-R(z)=R_{\mathrm d}(z)B_{\mathrm o}R(z).
 \tag{EAI.18}
\]
The product \(B_{\mathrm o}R(\pm is)\) is bounded by
\(M=\|B_{\mathrm o}l\|\), since \(D=Fr\), \(F^2=1\), and
\(\|rR(\pm is)\|\leq1\). For homogeneous \(b\), put
\(\epsilon=(-1)^{|b|}\) and \(c_b=[D,\phi(b)]_{\mathrm{gr}}\).
The graded resolvent identity is
\(R(z)\phi(b)=\epsilon\phi(b)R(\epsilon z)
 -\epsilon R(z)c_bR(\epsilon z)\).
Together with EAI.17 it gives, for \(s\geq1\),
\[
 \|(R_{\mathrm d}(\pm is)-R(\pm is))\phi(b)\|
 \leq \frac{\|B_{\mathrm o}\phi(b)\|+M\|c_b\|}{s^2}.
 \tag{EAI.19}
\]
Each localized difference is compact: both terms use either the compact extension EAI.17 or the compact \(c_b\) from IUR.10.

The bounded-transform difference integral, with
\(Q_s(T)=\tfrac12((T-is)^{-1}+(T+is)^{-1})\), consequently converges in source-local compact norm:
\[
 (F_{D_{\mathrm d}}-F_D)\phi(b)
 =\frac2\pi\int_0^\infty
       (Q_{\sqrt{1+t^2}}(D_{\mathrm d})
           -Q_{\sqrt{1+t^2}}(D))\phi(b)\,dt.
 \tag{EAI.20}
\]
The scalar bounded-transform integrals converge strongly; the integrable EAI.19 estimate identifies this compact norm limit with their difference. Adjoints give the other side. Density and the bound \(2\|b\|\) extend the comparison to every \(b\in B\).

Thus the original compact resolvents give source-local compact resolvents for \(D_{\mathrm d}\), and its bounded transform has the same genuine equivariant class \(d_X\). Continuous arrow defects follow from the proved source-local perturbation argument, using the original exact source covariance and compact transport. The \(Q\)-summand has zero \(B\)-representation, hence is a degenerate cycle. On the remaining actual \(K= P\widetilde K\), the representation is exactly the original nondegenerate \(\phi_0\). The closed essential operator \(\overline S_0\) therefore represents \(d_X\) with its actual \(B\)-source. It is not a scalar auxiliary replacement.

### 11BC.6. Completed source and full arrow graph domains

The source core preserves each reference domain and
\([S_0,\phi_0(b)]_{\mathrm{gr}}=P[D,\phi(b)]_{\mathrm{gr}}P\)
there. This is bounded compact. Core graph approximation and closedness of \(\overline S_0\) prove preservation of its entire completed domain, with the same bounded commutator. No new support projection or representation is introduced.

Exact invariance of the averaged normalized \(F\) is crucial for the full arrows. IUR.11–IUR.12 give the continuous compact-arrow section
\(A_g=\alpha_g r-r\). Since both \(F,P\) are invariant, on the reference core
\[
 \alpha_gD-D=F A_g,\qquad
 (\overline S_{0,r(g)}U_g-U_g\overline S_{0,s(g)})\xi
       =-P F A_gP\,U_g\xi.
 \tag{EAI.21}
\]
The right side is a globally bounded continuous compact-arrow field; it is not only bounded after a source factor. The reference core is invariant under \(U_g\), by IUR.12 and invariance of \(P\). Approximate any completed graph vector by that core and use closedness on the range side. This proves the actual completed inclusion; inverse arrows give equality:
\[
 U_g\operatorname{Dom}\overline S_{0,s(g)}
      =\operatorname{Dom}\overline S_{0,r(g)},\qquad
 \|\overline S_{0,r(g)}U_g\xi\|
 \leq\|\overline S_{0,s(g)}\xi\|
          +\|PF A_gP\|\|\xi\|.
 \tag{EAI.22}
\]
The two graph components are continuous by this equation, the original module action and compact section continuity. A compact arrow test gives a uniform graph bound. This is a proof on the full closure domain; preservation of the smaller reference domain alone would not give it.

The operator is odd, self-adjoint regular, has source-local compact resolvents, and has compact unbounded commutators on the dense source core. Its source action is nondegenerate and genuinely equivariant, and its full graph domain has EAI.22. A noncompact \(X\), nonunital \(B\), noncompact orbit space and finite stabilizers are all retained. Exact phase averaging requires the proper centre of \(B\); it does not apply merely because the probability anchor is proper.

### 11BC.7. What remains for the physical graph

This closes the source-essential unbounded representative and its arrow-domain passage. It has not prescribed a normal derivative of the lift module's generators, the averaged phase, the cutoff or the compact regulator. The normal connection, its completed joint domain and the estimates needed to combine this operator with the original inverse normal Dirac still require their own construction. Neither the compact-arrow graph bounds nor source-local compact resolvents imply those derivative assertions.

The original \(R=C_0(T)\)-balancing, full inverse coefficient \(z_g^{-1}\rho_t(s_g)\), exterior grading and final right Clifford order must be retained in that passage. The physical sum and original Bott comparison have not been replaced by this abstract source-essential cycle.

### 11BC.8. An exact finite average and compression

On a three-dimensional module with grading \(\operatorname{diag}(1,1,-1)\) take
\[
 F_0=\begin{pmatrix}0&0&0\\0&0&1\\0&1&0\end{pmatrix},
 \quad
 U=\begin{pmatrix}0&1&0\\1&0&0\\0&0&1\end{pmatrix},
 \quad
 \overline F=\frac12(F_0+UF_0U)
   =\begin{pmatrix}0&0&1/2\\0&0&1/2\\1/2&1/2&0\end{pmatrix}.
 \tag{EAI.23}
\]
Here \(H=\mathbb Z/2\), \(B=M_3\) with its indicated grading and conjugation action, \(A=\mathbb C\), and its centre \(W\) is a point. The proper cutoff is \(c=1/2\). The average is exactly invariant and its even kernel is spanned by \((1,-1,0)\).

In the doubled normalization put
\(C=(1-\overline F^2)^{1/2}\), let \(P\) project onto the first copy, and choose
\[
 r=2I+P+FPF,\qquad rF=Fr,\quad r\geq2,\qquad
 D=Fr=2F+FP+PF.
 \tag{EAI.24}
\]
This reference need not commute with \(P\). The first essential compression is \(4\overline F\), whose positive-to-negative block is \((2,2)\). It is onto, with even kernel \((1,-1)\) and ordinary index \(+1\). The action on that kernel is multiplication by \(-1\), so its ordinary index has not identified an equivariant scalar unit. The removed reference block has norm \(1/2\); the removed \(D\)-block has norm \(3\). All normalization, averaging, square, source-covariance and compression identities can be checked exactly in the field \(\mathbb Q(\sqrt2)\). This is a finite illustration of the full compression proof, not its regularity argument or a normal-domain estimate.

![Proper-source averaging and completed essential inverse](../figures/essential-averaged-inverse.png)

**Figure 11BC.1.** EAI.1–EAI.24. The proper central Bott source acts on the actual essential representation module. Its cutoff column averages the phase to exact arrow invariance while preserving the B-source class. A controlled reference, state-local regularity proof and integrable source-local resolvent comparison remove the zero-source normalization summand. The remaining source is nondegenerate, its closed operator represents d_X and its full closed graph domain is preserved by all arrows. The finite Z/2 illustration retains the nontrivial action by -1 on its even kernel. Sixteen source products, four covariances and the averaging, normalization, regulator, square and compression identities are checked exactly in Q(sqrt(2)). Its ordinary index +1 is not an equivariant scalar unit. The finite model supplies neither the full regularity proof nor the missing normal derivatives. Human context: [Connes, A survey of foliations and operator algebras](https://alainconnes.org/wp-content/uploads/foliationsfine.pdf#page=29), Section 8. [Vector figure](../figures/essential-averaged-inverse.svg) · [Generator](../reproduction/essential-averaged-inverse/draw_essential_averaged_inverse.py) · [Exact checks](../reproduction/essential-averaged-inverse/ESSENTIAL-AVERAGED-INVERSE-CHECKS.json) · [Reproduction](../reproduction/essential-averaged-inverse/REPRODUCTION.md) · [Component terms](../reproduction/essential-averaged-inverse/COMPONENT-TERMS.md).

## 11BD. A bounded normal comparison and the completed graph sum

An unbounded inverse need not have a bounded normal derivative for its sum with a normal Dirac to exist. A bounded comparison with a separated operator supplies the completed sum domain and the product class. It does not imply that the inverse preserves the normal derivative domain. The infinite example below proves both distinctions on the same module.

### 11BD.1. Exact separated data

Let \(B_0,C\) be separable and \(A_0\) be sigma-unital. Let \(K\) be a countably generated graded Hilbert \(A_0\)-module, with grading \(\Gamma\), nondegenerate even representation \(\sigma:B_0\to\mathcal L_{A_0}(K)\), and odd self-adjoint regular operator \(S_*\). Suppose that a dense homogeneous source algebra preserves its full domain, has bounded graded commutators, and satisfies
\[
 \sigma(b)(S_*\pm i)^{-1}\in\mathcal K_{A_0}(K).
 \tag{BNC.1}
\]
These are source-local assumptions; no global compact resolvent or unit of either algebra is imposed.

Let \(N\) be an odd self-adjoint operator on a graded Hilbert space \(\mathcal H\), with an even representation \(M\) of an ungraded normal-coordinate algebra \(C\). Its dense smooth source algebra has bounded commutators and
\(M(f)(N\pm i)^{-1}\) is compact. Retain the actual tensor gradings. On
\(\mathscr H=K\widehat\otimes\mathcal H\), regarded as a Hilbert \(A_0\)-module, the separated expression and source are
\[
 D_*=S_*\otimes1+\Gamma\otimes N,\qquad
 \Psi(b\otimes f)=\sigma(b)\otimes M(f).
 \tag{BNC.2}
\]
The grading is \(\Gamma\otimes\Gamma_{\mathcal H}\). There is also an odd version: take \(N\) on an ungraded Hilbert space and an ungraded first source \(B_0\). Use the same displayed operator on the ungraded total module. Its joint-domain, compactness and product proofs below give the odd class in that version. This formula specifies its factor order; no original physical Clifford factor is reordered by this model.

The continuous regular calculus and its tensor extension are fully proved in Regular calculus, Lemma 0.0. The two tensor-factor calculi commute. Finite products of scalar compact spectral cutoffs therefore give a nondegenerate calculus of \(C_0(\mathbb R^2)\); its scalar product functions have dense span by uniform approximation on compact rectangles followed by a cutoff. Bounded continuous functions extend on the dense cutoff ranges, with their adjoints, exactly as in that lemma.

### 11BD.2. The full sum domain is the joint domain

On the finite tensor core with both factors in their operator domains, anticommutation of \(S_*\) with \(\Gamma\) cancels the mixed terms. It gives the coefficient-valued identity
\[
 \langle D_*u,D_*u\rangle
   =\langle(S_*\otimes1)u,(S_*\otimes1)u\rangle
       +\langle(1\otimes N)u,(1\otimes N)u\rangle.
 \tag{BNC.3}
\]
This identity precedes taking the \(A_0\)-norm. Positivity bounds each summand's norm by the sum's norm; it is not an assertion that the norm of a sum is the sum of its norms.

The intersection of the two closed domains is complete for the joint graph norm. Each tensor operator is closed by the regular calculus. Commuting compact spectral cutoffs in \(S_*^2,N^2\) approximate every joint-domain vector together with both operator images. For a fixed pair of cutoffs, both operators are bounded on its range. Approximation there by finite elementary tensors, followed by those cutoffs, gives vectors of the finite core converging in the joint graph norm. Thus that core is dense in the intersection. BNC.3 identifies its completed sum graph with precisely this intersection:
\[
 \operatorname{Dom}D_*=
  \operatorname{Dom}(S_*\otimes1)\cap
  \operatorname{Dom}(1\otimes N).
 \tag{BNC.4}
\]

Here is also an onto-resolvent proof on this completed domain. Put
\(Q=(1+S_*^2\otimes1+1\otimes N^2)^{-1}\) in the joint calculus. Then
\[
 (D_*\pm i)^{-1}=(D_*\mp i)Q.
 \tag{BNC.5}
\]
The terms with numerators \(s,t,1\) are bounded joint-calculus functions, with the displayed grading factors. Their squared norms and the anticommutation identity bound each resolvent by one. Products by either tensor operator are bounded as well: the functions with numerators \(s^2,st,t^2,s,t\), divided by \(1+s^2+t^2\), are bounded. Hence its range lies in BNC.4. On the spectral core multiplication gives both inverse identities; cutoff approximation gives them on the completed domain and whole module. The two resolvents are mutually adjoint. The onto-resolvent adjoint argument of Regular calculus proves self-adjointness, and their commuting product gives regularity. Thus no completeness assertion has been inferred merely from a formal square.

### 11BD.3. Source-local compactness and the product

For \(h\in C_0(\mathbb R)\), BNC.1 implies \(\sigma(b)h(S_*)\) is compact: products and adjoints of resolvents span a dense subalgebra of \(C_0(\mathbb R)\), and the compact ideal is norm closed. The same holds for \(M(f)h(N)\). For a product joint function the localized calculus is therefore a tensor product of two compact operators. Rank-one tensors have the corresponding rank-one formula on \(K\widehat\otimes\mathcal H\); finite-rank approximation proves module compactness. Uniform product-function approximation proves it for every \(h\in C_0(\mathbb R^2)\).

The three scalar functions \(s/(1+s^2+t^2)\), \(t/(1+s^2+t^2)\), and \(1/(1+s^2+t^2)\) vanish at joint infinity. Move a grading past a homogeneous source using its actual degree. BNC.5 consequently gives
\[
 \Psi(b\otimes f)(D_*\pm i)^{-1}
       \in\mathcal K_{A_0}(\mathscr H).
 \tag{BNC.6}
\]
The source preserves BNC.4, and its graded commutator is the bounded sum
\[
 [D_*,\Psi(b\otimes f)]_{\rm gr}
   =[S_*,\sigma(b)]_{\rm gr}\otimes M(f)
       +\Gamma\sigma(b)\otimes[N,M(f)].
 \tag{BNC.7}
\]
Indeed \(\Gamma\sigma(b)=(-1)^{|b|}\sigma(b)\Gamma\), which cancels the two terms containing an uncommuted \(N\).

The Bounded-transform theorem now gives a cycle. Its product identification also has the full domain proof required by The unbounded product criterion, Theorem 5.4. Apply it with first module \(K\widehat\otimes C\) over \(A_0\widehat\otimes C\), source \(B_0\widehat\otimes C\), and first operator \(S_*\otimes1\). The second module is \(A_0\widehat\otimes\mathcal H\), with operator \(1\otimes N\). For homogeneous \(\xi\in\operatorname{Dom}S_*\) of degree \(p\) and smooth \(f\), the creation operator \(T_{\xi,f}\) has graded connection error
\((D_*T_{\xi,f}-(-1)^pT_{\xi,f}N)(a\otimes\eta)
=S_*\xi a\otimes M(f)\eta+(-1)^p\xi a\otimes[N,M(f)]\eta\).
Both terms are bounded creation operators. Taking adjoints on the same core gives the bounded opposite error. These vectors have dense source-generated span by nondegeneracy. BNC.4 gives the product's domain inclusion. Finally its quadratic-form anticommutator with \(S_*\otimes1\) is \(2(S_*^2\otimes1)\geq0\), by BNC.3. All three hypotheses of the already proved criterion hold. Hence
\[
 [\mathscr H,\Psi,F_{D_*}]
    =[K,\sigma,F_{S_*}]\boxtimes[\mathcal H,M,F_N].
 \tag{BNC.8}
\]
This is the separated product with the stated source and coefficient algebras, not an identification with a foliation graph that has not been unitarily compared to it.

### 11BD.4. A bounded comparison requires no normal derivative

Let \(V=V^*\) be a bounded adjointable operator on this same completed module, odd in the graded version. In the ungraded odd version no operator grading is required. Its norm is uniformly bounded; pointwise bounded fields with unbounded suprema are excluded. Define
\[
 D_v=D_*+vV,\quad 0\leq v\leq1,\qquad
 \operatorname{Dom}D_v=\operatorname{Dom}D_*.
 \tag{BNC.9}
\]
Each operator is closed and symmetric on the stated domain. For \(s>\|V\|\), factor \(D_v\pm is\) through \(D_*\pm is\). Its remaining factor is identity plus an operator of norm less than one, so a Neumann series gives onto inverse resolvents. The mutually adjoint resolvent and adjoint-domain argument of Regular calculus proves self-adjoint regularity; rescaling the regular calculus gives all nonreal resolvents. Its full domain is BNC.4 and the finite tensor core is still a graph core, because bounded perturbation gives equivalent graph norms. No differentiation of \(V\) has entered.

For \(R_v(z)=(D_v-z)^{-1}\), the same completed domain gives
\[
 R_v(z)-R_w(z)=-(v-w)R_v(z)V R_w(z),\qquad
 \|R_v(\pm is)-R_w(\pm is)\|
        \leq |v-w|\|V\|s^{-2}.
 \tag{BNC.10}
\]
The resolvent identity \(R_v=R_0-R_0vV R_v\), localized on the left, proves BNC.6 for \(D_v\). Adjoints give the right localization. Source domain preservation is unchanged, and the additional graded commutator with \(vV\) is bounded.

The bounded-transform integral used in The essential inverse comparison applies to this completed-domain resolvent identity. Its difference is norm integrable by BNC.10 with \(s=\sqrt{1+t^2}\). Strong scalar transform integrals identify that norm limit, giving
\[
 \|F_{D_v}-F_{D_w}\|\leq |v-w|\|V\|,\qquad
 (F_{D_v}-F_{D_0})\Psi(a)\in\mathcal K_{A_0}(\mathscr H).
 \tag{BNC.11}
\]
For the second assertion use the compact right-localized \(R_0\) in BNC.10, then norm integration and dense source approximation. Adjoints give the other side. Thus this is a class-preserving bounded cycle homotopy, and \(D_1\) has the separated product class BNC.8. If this same module/source already has a genuine groupoid action and \(F_{D_0}\) is a genuine cycle, the source-local compact perturbation argument of EAI.7 retains its genuine equivariant class. This does not assert equivariance for an unspecified normal module.

Given a unitary identification with a geometric module, transport the operators and the full domain BNC.4, as proved in Completed connection transport. The identification must retain the source, \(C_0(T)\)-balance, inverse normal coefficient and Clifford order. A fibre index or an ordinary abstract module isomorphism does not establish those requirements.

### 11BD.5. A derivative is additional information

For a multiplication field \(V(x)\), boundedness gives BNC.9; preservation of a normal derivative domain is a separate assertion. On a normal circle suppose, additionally, that \(V(x)\) is norm \(C^1\) with bounded derivative. On finite differentiable sections the product rule gives
\[
 (-i\partial_x)(Vu)=V(-i\partial_x)u-iV'u.
 \tag{BNC.12}
\]
Its bounded derivative proves preservation of the completed \(H^1\) domain: approximate by smooth finite sections, use this bound and close the derivative. For a norm \(C^1\) source field the identical argument gives its source product rule. Without that hypothesis BNC.12 is not justified on the full domain. The next example has every compact regulator and arrow-domain control, but fails this assertion.

### 11BD.6. An infinite represented-source counterexample

Take the normal circle with coordinate \(x\in[-\pi,\pi]\), endpoints identified, and
\(A=B=C(S^1)\). Let the fibre Hilbert space have even basis \(e_0,e_1,e_2,\ldots\), odd basis \(f_1,f_2,\ldots\), and put
\[
 a(x)=\frac14+\frac18\sqrt{|\sin(x/2)|},\qquad
 S(x)e_0=0,\quad S(x)|_{\langle e_1,f_1\rangle}=a(x)J,\quad
 S(x)|_{\langle e_n,f_n\rangle}=nJ\ (n\geq2),\quad
 J=\begin{pmatrix}0&1\\1&0\end{pmatrix}.
 \tag{BNC.13}
\]
Here \(1/4\leq a\leq3/8\). Write \(\mathcal V\) for this fibre Hilbert space. The source representation is scalar multiplication on \(C(S^1,\mathcal V)\). It is nondegenerate.

The operator is self-adjoint regular with completed coefficient-energy domain
\[
 \operatorname{Dom}S
 =\left\{u:\sum_{n\geq2}n^2
   \bigl(\langle u_{e_n},u_{e_n}\rangle+
           \langle u_{f_n},u_{f_n}\rangle\bigr)
       \text{ converges in }C(S^1)\right\}.
 \tag{BNC.14}
\]
The first three coordinates impose no extra condition. Define the diagonal reference \(r\) to be one on \(e_0,e_1,f_1\), and \(n\) on the \(n\)-th pair for \(n\geq2\). Its completed domain is BNC.14, its inverse is module compact and has dense range, and it is constant in the normal coordinate. Set \(F_0=Sr^{-1}\). It is an odd self-adjoint contraction, equal to \(J\) on all pairs \(n\geq2\). Its square defect and \(C=(1-F_0^2)^{1/2}\) are supported on the first three coordinates. In the actual doubled normalization
\[
 F=\begin{pmatrix}F_0&C\\C&-F_0\end{pmatrix},\quad
 \widetilde r=r\oplus r,\quad
 F^2=1,\quad [F,\widetilde r]=0,\quad
 P(F\widetilde r)P=S.
 \tag{BNC.15}
\]
The second copy has the opposite grading. The \(B\)-action is the first-copy action and zero on the second, as in EAI.8. The reference commutator with \(P\) is zero. The weighted tails \([F,P]\phi(b)\widetilde r\) and unbounded source commutators are finite-rank module operators because \(Cr\) has only those first three coordinates. Finite-pair spectral projections \(u_k\), retaining \(e_0\) and pairs \(1,\ldots,k\), give \(\widetilde r=1+\sum_{k\geq1}(1-(u_k\oplus u_k))\) on its domain; their relevant source/arrow commutators are zero and their compact tails vanish. Thus the common-tail controls, rather than only abstract compactness of a resolvent, are present.

Let \(G=\mathbb Z/2\) fix the circle. Its nontrivial element acts by \(-1\) on \(e_0\) and \(+1\) on every balanced pair, on both normalized copies. This is a genuine even action. The proper central space is the circle with this finite action and cutoff \(1/2\). All phases and references are exactly invariant. The full completed arrow domains are equal, with zero arrow error. The compressed source action is nondegenerate. The resolvents are module compact: the \(n\)-th pair norm is at most \(1/n\), uniformly on the circle, and the first finitely many blocks are continuous matrices. Their finite-pair truncations converge in norm. The paired part of the bounded phase differs compactly from the constant invertible pair phase, so the even represented-source index is \(+1\), with the sign action on its surviving kernel.

Nevertheless take a smooth scalar bump \(\chi\), equal to one near zero, and \(u=\chi e_1\). It belongs to the full domain BNC.14 and to the constant normal \(H^1\) domain. On a punctured neighbourhood,
\[
 a'(x)=\frac{\operatorname{sgn}(\sin(x/2))\cos(x/2)}
                 {32\sqrt{|\sin(x/2)|}},\qquad
 |a'(x)|^2\sim\frac1{512|x|}.
 \tag{BNC.16}
\]
The weak derivative of \(Su=a\chi f_1\) therefore is not square integrable near zero; the divergent integral is \(\int_0^\epsilon dx/x\). Its distributional derivative agrees with this locally integrable derivative, by integration by parts away from zero and the vanishing boundary contribution of the continuous \(a\). Thus \(Su\notin H^1\). On the continuous-section normal graph it fails more strongly: the derivative is unbounded. Exact arrows, a constant compact-inverse regulator, nondegenerate source action and all the compact weighted source tails have not implied preservation of either completed normal derivative domain.

This is an infinite scalar-source counterexample to that inference. Its source is expressly \(C(S^1)\), not the original phase-space Bott algebra \(B\), and its index is not an identification of the actual \(d_X\).

### 11BD.7. The same rough field still has a completed normal sum

Let \(S_*\) be BNC.13 with \(a\) replaced by the constant \(1/4\). On the physical Hilbert space \(L^2(S^1)\otimes\mathcal V\), take the ungraded circle Dirac \(N=-i\partial_x\), and
\[
 \mathscr D_*=S_*+\Gamma N,\qquad
 \mathscr D=\mathscr D_*+(a(x)-1/4)J_1,\qquad
 \|(a-1/4)J_1\|\leq1/8.
 \tag{BNC.17}
\]
Here \(J_1\) is \(J\) on the first balanced pair and zero elsewhere. This is the odd ungraded-source version of the graph calculation BNC.3–BNC.11; no extra spinor is silently inserted.

For completeness the Fourier vectors \(e^{ikx}/\sqrt{2\pi}\), \(k\in\mathbb Z\), are a complete orthonormal basis. Orthogonality is their direct integral. The Fejér kernel is
\(K_M(x)=M^{-1}|\sum_{j=0}^{M-1}e^{ijx}|^2\).
It is nonnegative; integrating its finite expansion gives integral \(2\pi\). For \(\delta\leq|x|\leq\pi\), the geometric-sum formula bounds it by \(1/(M\sin^2(\delta/2))\). Splitting its normalized convolution integral at \(\delta\), uniform continuity makes the near part arbitrarily small and this bound makes the far part tend to zero. Thus convolution approximates a continuous periodic function uniformly by trigonometric polynomials. Continuous functions are dense in \(L^2\): first approximate by finite interval step functions, then replace their finitely many jumps by continuous ramps on intervals of total length tending to zero. This proves completeness. The closure of \(N\) has domain \(\sum k^2|u_k|^2<\infty\); testing its adjoint against every mode proves equality of the adjoint domain, hence self-adjointness.

On the vacuum, \(\mathscr D_*\) has eigenvalues \(k\). On pair \(n\geq1\) its exact Fourier matrix and spectrum are
\[
 \begin{pmatrix}k&m_n\\m_n&-k\end{pmatrix},
 \quad m_1=1/4,\ m_n=n\ (n\geq2),\qquad
 \lambda_{n,k}^{\pm}=\pm\sqrt{m_n^2+k^2}.
 \tag{BNC.18}
\]
There are only finitely many eigenvalues in every bounded interval: \(n\) and \(|k|\) are bounded there, and each block has two eigenvalues. The Fourier-block core proves self-adjointness and compact resolvent by the diagonal inverse and its vanishing tails. BNC.3 gives its exact full domain \(H^1\cap\operatorname{Dom}S_*\), now with the coefficient energy summed in the physical \(L^2\) norm. The bounded term in BNC.17 preserves that completed sum domain, self-adjointness and compact resolvent. This conclusion is compatible with the failure \(Su\notin H^1\): the sum does not require applying the normal derivative after \(S\).

For smooth scalar \(f\) the source-domain rule and commutator are
\([\mathscr D,M_f]=-i\Gamma M_{f'}\), bounded on the full sum domain. Thus its bounded transform is an odd circle cycle. The path replacing \(a\) by \((1-v)/4+v a\) is the norm-continuous bounded perturbation homotopy BNC.11, preserving its genuine finite-group class. No normal derivative of that rough potential is used.

### 11BD.8. The infinite comparison, surviving carrier and sign

The class of \(\mathscr D_*\) can be identified on the entire module. Separate its vacuum and paired submodules. The vacuum is \(N\) with the sign action. On the paired submodule replace every mass by \(m_n/(1-v)\) for \(v<1\), and at \(v=1\) take the constant pair operator \(J\) as the bounded phase. For \(v<1\) its phase is
\[
 F_v(n,k)=
 \frac{(m_n/(1-v))J+k\Gamma_{\rm pair}}
       {\sqrt{1+(m_n/(1-v))^2+k^2}}.
 \tag{BNC.19}
\]
Each finite block converges to \(J\), so finite-block approximation and norm one prove strict continuity with its adjoint at the endpoint. Its square defect is compact and has norm at most
\((1+(m_1/(1-v))^2)^{-1}\), tending to zero.

The source commutator is compact throughout this homotopy. For a smooth \(f\), the resolvent integral and \([\mathscr D_v,M_f]=-i\Gamma M_{f'}\), together with the spectral gap \(m_1/(1-v)\), bound its norm by
\(\|f'\|/\sqrt{1+(m_1/(1-v))^2}\), tending to zero. Uniform compactness also needs the infinite normal and pair tails, not only this bound. First take \(f=e^{iqx}\). Its commutator entries are differences of BNC.19 at \(k\) and \(k+q\). Differentiation in the real \(k\)-variable bounds the phase derivative by \(2/\sqrt{m_n^2+k^2}\), uniformly in \(v\). Integrate along the interval of length \(|q|\). The bound tends to zero uniformly as \(n\to\infty\), or as \(|k|\to\infty\) beyond \(|q|\). Finite Fourier polynomials have the same conclusion. Convolution by Fejér kernels approximates a \(C^1\) function and its derivative uniformly, giving the general compact commutator with uniform parameter tails. Finite block entries are continuous. These facts prove compactness on the actual parameter Hilbert module, not just in separate fibres. The square defects have the same uniform tails by their explicit formula.

At the endpoint \(J^2=1\), it commutes with the scalar source and is exactly invariant; that entire paired cycle is degenerate. The completed genuine homotopy leaves exactly
\[
 [\mathscr D]=\operatorname{sgn}\otimes[N]
   \quad\text{in }KK^1_{\mathbb Z/2}(C(S^1),\mathbb C).
 \tag{BNC.20}
\]
For the explicitly negative winding unitary \(u(x)=e^{-ix}\), compress by the nonnegative-mode projection of \(N\). The compression is the backward shift on modes \(k\geq0\): it is onto and its kernel is the constant mode. Its ordinary index is \(+1\); its equivariant index is the sign representation. The positive winding has index \(-1\), in agreement with The circle pairing. Neither computation is the original foliation Bott comparison.

### 11BD.9. The remaining actual comparison

The sufficient comparison BNC.9–BNC.11 is weaker than a bounded normal derivative of the inverse. To use it for the inverse of Section 11BC, one must exhibit the actual completed normal module and a source-preserving unitary to a separated representative plus a uniformly bounded error. Alternatively a different completed normal sum criterion must be proved on that actual module. The abstract lift and compact regulator tests alone have not exhibited either choice.

The original \(R=C_0(T)\)-balance, full inverse coefficient \(z_g^{-1}\rho_t(s_g)\), exterior grading and final right Clifford order remain required, along with the genuine graph action and the original Bott \(+1\) comparison. BNC.8 identifies the model's product, and BNC.20 identifies its entire infinite example. They do not replace those original algebras, arrows or geometry.

![Bounded normal comparison and the completed infinite graph sum](../figures/bounded-normal-comparison.png)

**Figure 11BD.1.** BNC.1–BNC.20. The separated sum has its full completed joint domain; a uniformly bounded potential preserves self-adjoint regularity, localized resolvents and the product class without a potential derivative. The infinite circle model has a constant compact-inverse regulator, nondegenerate source action and exact arrows, but its mass a(x)=1/4+sqrt(abs(sin(x/2)))/8 fails normal H1 preservation. The left log plot samples the actual squared derivative at181 positive coordinates and its proved asymptote1/(512x), whose integral diverges. The right plot samples positive energies sqrt(n²+k²), n=2–6, k=-6–6; the whole Fourier/pair compact-resolvent tails are proved in the text. Twenty exact rational Fourier blocks, five exact resolvent comparisons, three normalization identities and six exact reference-series values are checked. The full infinite mass homotopy leaves the vacuum circle Dirac with sign action; negative winding has ordinary index+1 and equivariant sign character. This model has scalar circle source and does not identify the original Bott inverse or its physical graph. Human context: [Connes, A survey of foliations and operator algebras](https://alainconnes.org/wp-content/uploads/foliationsfine.pdf#page=29), Section 8. [Vector figure](../figures/bounded-normal-comparison.svg) · [Generator](../reproduction/bounded-normal-comparison/draw_bounded_normal_comparison.py) · [Exact checks](../reproduction/bounded-normal-comparison/BOUNDED-NORMAL-COMPARISON-CHECKS.json) · [Reproduction](../reproduction/bounded-normal-comparison/REPRODUCTION.md) · [Component terms](../reproduction/bounded-normal-comparison/COMPONENT-TERMS.md).

## 11BE. A standard carrier for the actual source-essential inverse

The actual inverse class can be put on a standard ordinary module without adding a summand with zero source action. A genuine degenerate cycle supplies a nondegenerate source representation on the added module. The resulting carrier has a completed metric normal connection and retains the inverse's full unbounded arrow domains. Normal differentiability of its represented source, unbounded operator and arrow maps is further information.

### 11BE.1. The actual inverse and fixed ordinary source

Retain \(A=C_0(X)\), \(H=G\ltimes X\), and \(B=A\widehat\otimes C\), with \(C=A(H_0)\), as in The anchored Bott source, BA.4–BA.11. This is the ordinary constant coefficient description; its genuine affine arrow action on \(B\) is still the original one. The scalar algebra \(A\) is evenly graded.

Use the actual nondegenerate essential module \((K,\phi_0,\overline S_0)\) of The completed essential inverse. Its bounded transform has class \(d_X\in KK_H(B,A)\). Its source-local compact resolvents, dense homogeneous source-domain algebra, compact unbounded source commutators and full closed arrow graph domains are proved in EAI.15–EAI.22. None of those data is replaced by a scalar-source model.

Choose a separable faithful nondegenerate ordinary representation \(\rho_u\) of \(C\). Here is the needed countability construction. For a countable dense family of nonzero positive elements \(c_j\), choose states whose values approach each \(\|c_j\|\) arbitrarily closely, and take their countable direct-sum GNS representation. The cyclic GNS space is the completion of \(C\) modulo the null space of \(\omega(a^*a)\), with left multiplication bounded by the positive inequality \(a^*c^*ca\leq\|c\|^2a^*a\). Approximate identities make it nondegenerate. Each cyclic space is separable; the chosen positive norm tests and density make the direct sum faithful. The existence of a norming positive state follows from C*-algebras, continuous functional calculus and positive cones, Lemma 13.1(7), with its complete positive-extension proof. No universal nonseparable representation is used.

Grade it by taking
\[
 L=(L_u\oplus L_u)\otimes\ell^2(\mathbb N),\quad
 \rho(c)=\begin{pmatrix}\rho_u(c)&0\\0&\rho_u(\gamma_Cc)\end{pmatrix}\otimes1,\quad
 \Gamma_L=\begin{pmatrix}0&1\\1&0\end{pmatrix}\otimes1.
 \tag{ANC.1}
\]
Thus \(\Gamma_L\rho(c)\Gamma_L=\rho(\gamma_Cc)\). Both parity spaces are countably infinite dimensional, when \(C\ne0\). For the zero source the actual essential module is zero and the construction can be omitted. The ordinary source module \(M_0=A\otimes L\) has the nondegenerate representation \(\rho_0=1_A\otimes\rho\).

### 11BE.2. Genuine regular induction keeps that source nondegenerate

Apply the regular construction of The completed regular source module, RSM.1–RSM.5, with scalar right coefficient \(A\). Its core is \(C_c(H,s^*M_0)\), completed for
\[
 \langle\xi,\eta\rangle(x)=
    \sum_{g\in H^x}\langle\xi(g),\eta(g)\rangle_{\mathbb C},\qquad
 (\Pi(b)\xi)(g)=
    \rho_{0,s(g)}(\alpha^B_{g^{-1}}b_{r(g)})\xi(g).
 \tag{ANC.2}
\]
The right scalar acts at the range coordinate; its transport to the source fibre has the same scalar value. A compact core support has a finite bisection cover, so these inner products are actual continuous \(A\)-sections. Positivity, the representation inequality and the core adjoint identity give an even adjointable star representation on the completed module \(M\). Nondegeneracy on a bisection follows from that of \(\rho_0\); a finite bisection partition and core density prove it on all \(M\). The countable bisection basis and countable generators of \(M_0\) prove countable generation as in RSM.2.

Left translation and the inherited grading are
\[
 (V_h\xi)(g)=\xi(h^{-1}g),\qquad
 (\Gamma_M\xi)(g)=\Gamma_L\xi(g),\qquad
 V_h\Pi(b)V_h^{-1}=\Pi(\alpha_h^Bb).
 \tag{ANC.3}
\]
Reindexing ANC.2 proves its semilinear isometry, adjoint inverse and exact arrow law. Bisection translation proves continuity on the compact core; norm one and finite core approximation extend it to the completed action. It is a genuine action on this fixed module and commutes with \(\Gamma_M\). No equivariance of the original ordinary \(\rho_0\) was assumed.

### 11BE.3. A degenerate augmentation with nonzero action on both copies

On \(Q=M\oplus M^{\rm op}\), put
\[
 \psi(b)=\begin{pmatrix}\Pi(b)&0\\0&\Pi(b)\end{pmatrix},\quad
 \Gamma_Q=\begin{pmatrix}\Gamma_M&0\\0&-\Gamma_M\end{pmatrix},\quad
 \Delta=\begin{pmatrix}0&\Gamma_M\\ \Gamma_M&0\end{pmatrix}.
 \tag{ANC.4}
\]
Both copies have the nondegenerate representation \(\Pi\). Direct block multiplication gives
\[
 \Delta=\Delta^*,\quad\Delta^2=1,\quad
 \Gamma_Q\Delta=-\Delta\Gamma_Q,\quad
 [\Delta,\psi(b)]_{\rm gr}=0.
 \tag{ANC.5}
\]
For degree \(p\), the last equality uses
\(\Gamma_M\Pi(b)=(-1)^p\Pi(b)\Gamma_M\).
The action \(\operatorname{diag}(V_h,V_h)\) commutes exactly with \(\Delta\), so every source and arrow cycle defect is zero. This is a genuine degenerate cycle, of class zero in \(KK_H(B,A)\), with nondegenerate source on its entire module. An off-diagonal identity would fail the odd-source graded commutator; the \(\Gamma_M\) factors cannot be dropped.

Consequently
\[
 E=K\oplus Q,\qquad
 \Phi=\phi_0\oplus\psi,\qquad
 F_E=F_{\overline S_0}\oplus\Delta,\qquad
 [E,\Phi,F_E]=d_X.
 \tag{ANC.6}
\]
The source action is nondegenerate on every summand. Direct sum does not change the original class or its source to a relative unitization.

### 11BE.4. An unbounded augmentation and the actual completed arrows

Use the compact inverse construction IUR.3–IUR.7 of The actual source regulator, now on \(Q\) with the exactly invariant odd unitary \(\Delta\). Choose its compact positive dense-range reference averaged by \(\Gamma_Q,\Delta\), and the single common-tail nested cutoffs \(u_n\). Include a countable dense homogeneous source algebra and its adjoints, products and compact-bisection transports. The ordinary commutator and compact-arrow tests give
\[
 r_Q=1+\sum_{n\geq1}(1-u_n),\quad
 l_Q=r_Q^{-1}\in\mathcal K_A(Q),\quad
 [r_Q,\psi(b_j)]\in\mathcal K_A(Q),\quad
 A_h^Q=\alpha_h r_Q-r_Q\in\mathcal K_A(Q)_{r(h)}.
 \tag{ANC.7}
\]
The last is a continuous unlocalized compact-arrow section, uniformly bounded on each compact arrow test. The inverse equation and uniform finite-bisection approximation prove the actual completed range and graph rules, as in IUR.9 and IUR.12. No extra weighted phase-source tail is needed here: both the graded source and unlocalized arrow phase errors of \(\Delta\) are identically zero. Every cutoff still commutes with \(\Delta\).

Define
\[
 T_Q=\Delta l_Q^{-1},\quad
 \operatorname{Dom}T_Q=\operatorname{Ran}l_Q,\qquad
 (T_Q\pm i)^{-1}=l_Q(\Delta\mp il_Q)(1+l_Q^2)^{-1}.
 \tag{ANC.8}
\]
The explicit mutually adjoint onto inverses prove self-adjoint regularity on this full domain and module compact resolvents. Its phase differs compactly from \(\Delta\), so its bounded-transform class is still zero. For the dense source core,
\([T_Q,\psi(b_j)]_{\rm gr}=\Delta[r_Q,\psi(b_j)]\) is compact. Exact phase invariance gives the unlocalized arrow error \(\Delta A_h^Q\); the inverse range equation and inverse arrows prove equality of the full arrow graph domains and their continuity.

Take the finite direct sum
\[
 D_E=\overline S_0\oplus T_Q,\qquad
 \operatorname{Dom}D_E=
       \operatorname{Dom}\overline S_0\oplus\operatorname{Dom}T_Q.
 \tag{ANC.9}
\]
Its resolvents are the two corresponding diagonal inverses. Hence it is self-adjoint regular, with source-local compact resolvents, compact unbounded source commutators on the common dense algebra, and genuine full closed arrow-domain preservation. The sum of the two compact continuous arrow errors retains the unlocalized bound. To obtain that common algebra, include the entire fixed countable source-test family of \(\overline S_0\) in the auxiliary regulator tests from the outset. The dense star algebra generated by that original family is then a common source core for both summands. No intersection of unrelated dense algebras is presumed dense, and the original \(\overline S_0\) need not be changed. All these constructions retain their bounded classes, giving \([E,\Phi,F_{D_E}]=d_X\).

### 11BE.5. The ordinary standard carrier comes from a represented summand

The unit-index submodule of \(M\) is adjointably complemented and is \(A\otimes L\). Indeed the units of a Hausdorff étale groupoid form an open and closed subspace: a unit bisection supplies openness, and a net of units with limit \(g\) also converges to the unit at \(r(g)\), proving closedness. Restriction to its unit coordinates and extension by zero are adjoints under ANC.2. Their product is the unit-coordinate projection. This ordinary submodule need not be invariant under left translation.

Both parity spaces of \(L\) are countably infinite. Separate coordinate bijections give
\[
 A\otimes L\cong\widehat H_A
       =\ell^2(A)\oplus\ell^2(A)^{\rm op}.
 \tag{ANC.10}
\]
Choose one such unit-coordinate submodule inside \(Q\). Its orthogonal complement \(Q'\) is countably generated: project the original generating family by its adjointable complement. Thus
\(E\cong\widehat H_A\oplus(K\oplus Q')\).
The full Graded stabilization theorem, Theorem 6.2, now supplies an even ordinary \(A\)-linear unitary
\[
 W:E\longrightarrow\widehat H_A.
 \tag{ANC.11}
\]
Its proof uses the degree-preserving operator-range construction and has no finite-rank hypothesis. The standard summand used here already had nondegenerate \(B\)-source action. No new zero-source module has been attached to obtain ANC.11.

Conjugate the entire data, including the action:
\[
 \Phi'(b)=W\Phi(b)W^*,\quad
 D'=WD_EW^*,\quad\operatorname{Dom}D'=W\operatorname{Dom}D_E,\quad
 U_h'=W_{r(h)}(U_h^K\oplus V_h\oplus V_h)W_{s(h)}^*.
 \tag{ANC.12}
\]
These are the actual source, completed operator and genuine arrows, not the identity action on chosen coordinates. Module unitaries transport compact operators by their rank-one formula. They transport full graph closures and inverse resolvents by their inverse identity. Strict continuous conjugation of a continuous compact-arrow field is norm continuous on compact arrow tests, by finite-rank approximation. Thus all the properties of ANC.9 survive, the source remains nondegenerate, and its exact genuine class remains \(d_X\). Non-equivariance of \(W\) does not cause a class loss, since the action is transported with it.

### 11BE.6. A completed metric normal connection on this actual carrier

Use the original scalar normal derivations \(\delta_i\) on \(A\) and their specified dense scalar core, as in The completed scalar connection. On finite coordinate sections of \(\widehat H_A\) put
\[
 \nabla_i^0\left(\sum_j a_j e_j\right)=
          \sum_j(\delta_i a_j)e_j.
 \tag{ANC.13}
\]
This is closable in the section norm. A sequence tending to zero with its derivatives convergent has every coordinate derivative tending to the corresponding coordinate of the proposed limit. Closedness of the scalar derivative makes each coordinate zero. Coordinate functionals separate standard-module sections, hence the limit is zero. Its minimal closed extension is densely defined. Its actual graph core is the original finite scalar core; taking the closure defines that core, rather than asserting a new joint operator core.

Constant parity projections commute with the connection on the core and on its closure. The scalar product rule gives the right Leibniz and metric identities there. Norm-and-derivative approximation, finite coordinate truncation and closedness extend the scalar rule and its bound:
\[
 \nabla_i^0(ua)=(\nabla_i^0u)a+u\delta_i a,\qquad
 \|\nabla_i^0(ua)\|\leq
   \|a\|\|\nabla_i^0u\|+\|\delta_i a\|\|u\|.
 \tag{ANC.14}
\]
For the metric identity, approximate both vectors in their graph norms. Inner products and their two derivative terms converge in \(A\); closedness of \(\delta_i\) gives the identity on the full domain. Central left scalar multiplication has the same rule.

This is a completed connection on the actual standard carrier of \(d_X\), with its actual nondegenerate \(\Phi'\) and action \(U'\). On \(E\) its equivalent connection is
\[
 \operatorname{Dom}\nabla_i^E=W^*\operatorname{Dom}\nabla_i^0,\qquad
 \nabla_i^E=W^*\nabla_i^0W.
 \tag{ANC.15}
\]
The Completed connection transport, CTN.1–CTN.2, proves closedness, the exact transported graph core, metric rule and scalar Leibniz rule. A derivative of the arbitrary stabilization unitary was not assumed.

### 11BE.7. Exact scope of the connection assertion

The central \(A\)-action and its composed original \(R=C_0(T)\)-action are retained by \(W\). Extending \(W\) as identity on the separately ordered normal spinor and Clifford factors therefore retains their original factor order. This construction supplies no replacement for the full inverse coefficient \(z_g^{-1}\rho_t(s_g)\).

It has not proved that \(\Phi'(b)\) preserves \(\operatorname{Dom}\nabla_i^0\) for a dense normal-smooth Bott source, that \(D'\) or its resolvents preserve that domain, or that \(U_h'\) is parallel for the original connections. The operator-domain statement in ANC.12 and the connection-domain statement in ANC.13 are separate; a dense completed joint graph core has not followed merely from their individual density. The Infinite normal-domain example explains why arrow and compact regulator controls alone cannot give the missing inference.

For each specified arrow the transported connection \(U_h'\nabla_s^0(U_h')^{-1}\) has precisely its transported completed domain, by CTN.1. This does not identify it with \(\nabla_r^0\). The remaining analytic task is to prove a source-preserving bounded normal comparison of BNC.9–BNC.11, or a different actual completed normal-sum criterion with the necessary source, arrow and coefficient controls. The actual carrier now has its specified closed metric connection; the physical sum and original Bott comparison still require their proofs.

### 11BE.8. An exact graded augmentation example

Let \(C=M_2\), \(\Gamma_L=\operatorname{diag}(1,-1)\), and let the nontrivial element of \(\mathbb Z/2\) act on \(C\) by \(\alpha(b)=\Gamma_Lb\Gamma_L\). On the two-arrow regular module put
\[
 \Pi(b)=\operatorname{diag}(b,\Gamma_Lb\Gamma_L),\quad
 V=\begin{pmatrix}0&I_2\\I_2&0\end{pmatrix},\quad
 \Gamma_M=\operatorname{diag}(\Gamma_L,\Gamma_L).
 \tag{ANC.16}
\]
Then \(V^2=1\), \(V\Pi(b)V=\Pi(\alpha(b))\), and \(V\Gamma_M=\Gamma_MV\). Both source parities are retained. On its doubled module use exactly ANC.4:
\[
 \psi(b)=\operatorname{diag}(\Pi(b),\Pi(b)),\quad
 U=\operatorname{diag}(V,V),\quad
 \Delta=\begin{pmatrix}0&\Gamma_M\\ \Gamma_M&0\end{pmatrix}.
 \tag{ANC.17}
\]
The source identity is \(I_8\), so every represented coordinate is essential. Exact block multiplication verifies the sixteen matrix-unit source products, four covariance identities, both degree signs in ANC.5, and exact invariance of \(\Delta\). If it is replaced by
\(\Delta_{\rm plain}=\begin{pmatrix}0&I_4\\I_4&0\end{pmatrix}\),
then for \(b=e_{01}\) its graded source commutator is \(2\psi(b)\Delta_{\rm plain}\), of norm two. Its squared absolute value is four times the corresponding input projection. Thus ordinary off-diagonal commutation cannot replace the graded source test.

The correct finite cycle is exactly degenerate, of class zero. It illustrates the represented augmentation and signs; it neither proves the countably infinite standard-module absorption nor identifies the actual inverse class by a finite index.

![Actual represented standard inverse carrier and completed metric normal connection](../figures/actual-normal-carrier.png)

**Figure 11BE.1.** ANC.1–ANC.17. A nondegenerate graded ordinary source representation is regularly induced to a genuine arrow module. Its doubled augmentation has nondegenerate source on both copies and exactly zero Kasparov defects, with the necessary grading factors in its odd phase. Adding its controlled unbounded representative preserves the actual source-essential inverse class d_X and full arrow graph domains. Its represented unit-index summand supplies ordinary graded stabilization to a standard carrier; the source, action and full operator are transported together. The finite scalar normal core gives a completed metric connection with its actual graph domain. Sixteen exact rational source products, four covariances, four graded source tests, four unbounded source formulas and four transported covariances verify the finite illustration. Omitting the grading factor produces an odd-source error of norm 2. The finite cycle is degenerate; it does not certify infinite absorption or an inverse index. Full source, operator and arrow normal derivatives, the joint domain and original physical graph remain unproved. Human context: [Connes, A survey of foliations and operator algebras](https://alainconnes.org/wp-content/uploads/foliationsfine.pdf#page=29), Section 8. [Vector figure](../figures/actual-normal-carrier.svg) · [Generator](../reproduction/actual-normal-carrier/draw_actual_normal_carrier.py) · [Exact checks](../reproduction/actual-normal-carrier/ACTUAL-NORMAL-CARRIER-CHECKS.json) · [Reproduction](../reproduction/actual-normal-carrier/REPRODUCTION.md) · [Component terms](../reproduction/actual-normal-carrier/COMPONENT-TERMS.md).

## 11BF. An equivariant regular carrier and its closed normal graph

The ordinary absorption in Section 11BE can be strengthened by the proper centre of the actual inverse. The resulting carrier has the canonical regular arrow action and a parallel normal connection. This proves arrow invariance and completeness of the intersection of the two closed graph domains. It leaves the density of that intersection, normal control of the represented Bott source and the completed physical sum to be proved.

Throughout, \(A=C_0(X)\), \(H=G\ltimes X\) and \(B=A\widehat\otimes\mathcal A(H_0)\) are the actual coefficient algebra, transformation groupoid and Bott source of Sections 11AD–11BC. Write \((K,\phi_0,\bar S_0)\) for the source-essential unbounded inverse in Section 11BC. Its class is \(d_X\in KK_H(B,A)\). Write \((Q,\psi,T_Q)\) for the genuinely degenerate, everywhere nondegenerate-source augmentation in ANC.1–ANC.9. In particular \(Q=M\oplus M^{\mathrm{op}}\), with both parity multiplicities countably infinite, and its action is regular translation. Set

\[
 E=K\oplus Q,\qquad \Phi=\phi_0\oplus\psi,\qquad
 D_E=\bar S_0\oplus T_Q,\qquad [E,\Phi,F_{D_E}]=d_X.
 \tag{CVC.1}
\]

The \(B\)-source on both summands is nondegenerate. The fixed dense source-test algebra is the original EAI algebra included in the augmentation tests. No intersection of unrelated dense algebras is taken.

### 11BF.1. Proper-source embedding and genuine absorption

The proper phase centre \(C_0(W)\) of \(B\) acts nondegenerately on \(K\). Its geometric cutoff, proved in RSM.9 and used in EAI.2, gives even positive multiplier operators \(C_x\) with the strict identity

\[
 \sum_{r(g)=x}U_g C_{s(g)}^2 U_g^{-1}=1_{K_x}.
 \tag{CVC.2}
\]

Use the range-counting module \(\mathcal R_H(K)\) of RG.1, whose row at \(g\) is in \(K_{s(g)}\) and whose right \(A\)-action is evaluated at \(r(g)\). Define

\[
 (V_K\xi)(g)=C_{s(g)}U_g^{-1}\xi_{r(g)},\qquad
 (V_K^*y)_x=\sum_{r(g)=x}U_g C_{s(g)}y(g).
 \tag{CVC.3}
\]

These formulas define an adjointable isometry and its adjoint on the actual completions. Here is why the cutoff argument applies to this inverse. First localize \(\xi\) by an element of \(C_c(W)\). Properness makes the arrows meeting that phase support and the transported cutoff support a compact set. On finitely many bisections the terms in CVC.3 are continuous sections. Their counting norm is \(\|\xi\|\), by CVC.2. Nondegeneracy of the phase centre approximates every \(\xi\) by these localizations. For the adjoint, finite-row Cauchy–Schwarz and CVC.2 give \(\|V_K^*y\|\leq\|y\|\) on compact rows. The adjoint identity and completion then give \(V_K^*V_K=1\). This is exactly the support-and-adjoint proof RG.4–RG.6, now with the original proper \(B\)-source on \(K\). Substitution gives

\[
 V_KU_h=L_hV_K,\qquad (L_hy)(g)=y(h^{-1}g).
 \tag{CVC.4}
\]

The two coefficients in the latter identity have the same source unit. Thus no derivative or equivariance of a subsequent ordinary stabilization map is needed.

By the proved graded Kasparov stabilization theorem, choose an even ordinary adjointable isometry \(j:K\longrightarrow\widehat{\mathcal H}_A\), where the standard module has infinitely many coordinates in each parity. Regularize it **at the source**:

\[
 (\mathcal R_H(j)y)(g)=j_{s(g)}y(g),\qquad
 I=\mathcal R_H(j)V_K:K\longrightarrow Q.
 \tag{CVC.5}
\]

In CVC.5 the last identification is a fixed graded bijection of the infinite copy indices in \(\mathcal R_H(\widehat{\mathcal H}_A)\) and \(Q\). The adjoint of \(\mathcal R_H(j)\) applies \(j_{s(g)}^*\) coordinatewise. The counting inner product proves adjointability and isometry. Since \(s(h^{-1}g)=s(g)\), it intertwines regular translation exactly, as in RG.3. Consequently \(I\) is a genuine even equivariant isometry. Put \(P=II^*\) and \(N=(1-P)Q\). The projection is adjointable and invariant; projected generators show that \(N\) is countably generated. Therefore

\[
 Q=I(K)\oplus N\cong K\oplus N
 \quad\text{as genuine graded }H\text{-Hilbert }A\text{-modules}.
 \tag{CVC.6}
\]

Neither \(j\), \(I\) nor \(P\) has been claimed normal-smooth. Also \(I\) need not intertwine the previously represented \(B\)-source on \(Q\). The complete source representation will be transported.

There is a genuine even absorption unitary \(K\oplus Q\cong Q\). To prove surjectivity as well as isometry, first identify \(Q\) with \(Q^\infty\) by a fixed bijection of its infinite copy indices, without changing any arrow label. Apply CVC.6 in each copy. The following map and its inverse are literal coordinate maps:

\[
\begin{aligned}
 K\oplus(K\oplus N)^\infty&\longrightarrow(K\oplus N)^\infty,\\
 (\xi_*;(\xi_n,\eta_n)_{n\geq0})
 &\longmapsto((\xi_*,\eta_0),(\xi_0,\eta_1),(\xi_1,\eta_2),\ldots).
\end{aligned}
 \tag{CVC.7}
\]

The inverse extracts the first \(K\)-coordinate, shifts the later \(K\)-coordinates back by one and keeps every \(N\)-coordinate. Norms are preserved by the orthogonal direct-sum inner product. Every map is even, \(A\)-linear and equivariant, because it acts only on copies of the same modules. Regrouping and the inverse fixed copy bijection give an even \(H\)-equivariant unitary \(\mathsf W:E\longrightarrow Q\).

### 11BF.2. The entire inverse survives on the canonical arrow carrier

Transport the full source and operator, including its domain:

\[
 \Phi_H(b)=\mathsf W\Phi(b)\mathsf W^*,\qquad
 D_H=\mathsf W D_E\mathsf W^*,\qquad
 \operatorname{Dom}D_H=\mathsf W\operatorname{Dom}D_E.
 \tag{CVC.8}
\]

Unitary transport proves self-adjoint regularity and the resolvent formula
\((D_H\pm i\lambda)^{-1}=\mathsf W(D_E\pm i\lambda)^{-1}\mathsf W^*\).
It preserves the bounded transforms, nondegenerate \(B\)-source, dense unbounded source core, graded source commutator formulas and source-local compact resolvents. For example, conjugating a rank-one map gives
\(\theta_{\mathsf W\xi,\mathsf W\eta}=\mathsf W\theta_{\xi,\eta}\mathsf W^*\); norm closure proves compact transport on the full modules. Since \(\mathsf W\) is equivariant, the arrow action on the target is its already specified \(L_h\), and

\[
 [Q,\Phi_H,F_{D_H}]=d_X,\qquad
 L_h\operatorname{Dom}D_{H,s(h)}
     =\operatorname{Dom}D_{H,r(h)}.
 \tag{CVC.9}
\]

The equality is the full closed operator-domain equality from EAI.23 and ANC.9, conjugated by CVC.8. Its error field is

\[
 \mathcal E_h=D_{H,r(h)}L_h-L_hD_{H,s(h)}
     =\mathsf W_{r(h)}
       (D_{E,r(h)}U_h-U_hD_{E,s(h)})\mathsf W_{s(h)}^*.
 \tag{CVC.10}
\]

It is an unlocalized continuous compact-operator field, not just compact after multiplying by a source element. Strictly continuous unitary transport of rank-one fields and their local norm approximations proves this assertion. Over each compact arrow test, its norm is uniformly bounded. The class in CVC.9 is the actual \(B\)-source inverse, not the class of a scalar unit or a bare regular length operator.

### 11BF.3. A parallel completed normal connection

Use the original transverse bundle \(\tau\) and the actual scalar normal core on \(A\) constructed in PJ.6. Write \(\mathscr A_N\) for its polynomial observable sheaf with smooth unit localizations. Its derivation \(\delta_N\) is metric-compatible and closable, and scalar transport along the original normal-smooth bisections is covariant. All these assertions were proved using the fixed lifted probability paths; no derivative of an arbitrary continuous function on \(X\) is presumed.

Finite bisection sections with coefficients in this sheaf and constant finite copy vectors define a core \(\mathscr C_{\rm reg}\subset Q\). More explicitly, over a bisection \(\mathcal B\), write

\[
 y(g)=\sum_{n=1}^m f_n(r(g))e_n,\qquad
 (\nabla^{\rm reg}y)(g)=
       \sum_{n=1}^m(\delta_N f_n)(r(g))e_n
       \in Q\otimes_A a^*\tau^*.
 \tag{CVC.11}
\]

The supports of \(f_n\) lie compactly inside \(r(\mathcal B)\), and the sections are extended by zero. Here \(a:X\longrightarrow G^{(0)}\) is the original-unit anchor. The fixed \(e_n\) are source coefficient vectors, unchanged under arrow translation. Chart changes in CVC.11 are the original normal covector changes. On an overlap two bisection descriptions are related by scalar normal transport, so PJ.6 gives the same derivative.

This is a dense core. Compact arrow sections are covered by finitely many bisections and approximated by finite copy vectors. PJ.6 gives scalar normal-core approximation on each compact unit set. Normal-smooth localizers may be chosen inside any prescribed unit open set: take finitely many separating observable and unit tests on the compact set, apply smooth scalar cutoffs to them, and multiply by a smooth original-unit bump. The chain rule follows by differentiating along the same lifted paths. This also verifies extension by zero where the support is compactly inside the chart. Finite such localizations approximate the original compact bisection section.

The counting inner product, the scalar product rule and finite row sums give

\[
\begin{aligned}
 \nabla^{\rm reg}(ya)&=(\nabla^{\rm reg}y)a+y\otimes\delta_Na,\\
 \delta_N\langle y,z\rangle
   &=\langle\nabla^{\rm reg}y,z\rangle+
     \langle y,\nabla^{\rm reg}z\rangle
       \qquad(y,z\in\mathscr C_{\rm reg}).
\end{aligned}
 \tag{CVC.12}
\]

Local normal vector fields evaluate the covector identities. Coincident arrows on overlapping bisections are combined before taking their inner products; hence cross terms satisfy the same scalar rule. Distinct arrows have zero cross term. Constant parity projections commute with this connection.

To prove closability, suppose \(y_k\to0\) in \(Q\) and \(\nabla^{\rm reg}y_k\to z\) in \(Q\otimes_A a^*\tau^*\). Pair with any fixed \(\eta\in\mathscr C_{\rm reg}\) and any local smooth normal vector field. CVC.12 shows that the corresponding scalar functions \(\langle\eta,y_k\rangle\) tend to zero and their derivatives tend to the corresponding component of \(\langle\eta,z\rangle\). Closability of the scalar PJ.6 derivations makes that component zero. Compact normal localizations, a local finite frame of \(\tau\) and density of \(\mathscr C_{\rm reg}\) separate every component of \(z\); thus \(z=0\). Take the **single vector-valued graph closure**:

\[
 \nabla_{\rm reg}=\overline{\nabla^{\rm reg}},\qquad
 \mathcal D_N=\operatorname{Dom}\nabla_{\rm reg}
 \subset Q.
 \tag{CVC.13}
\]

This defines a densely defined closed normal connection and its actual completed domain. Its core is exactly the graph closure of \(\mathscr C_{\rm reg}\). We do not identify it with an intersection of separately closed component domains without proof. For \(a\in\mathscr A_N\) with bounded normal derivative, graph approximation in CVC.12 gives \(ya\in\mathcal D_N\) and the completed Leibniz identity. The metric identity follows by the same approximation and closedness of the scalar derivation. These estimates also hold after compact normal localization.

Let \(h\) vary along an original normal-smooth bisection. The normal holonomy matrix \(m(h)\) is the direction change of PJ.6, acting on \(\tau^*\) by the corresponding covector transport. On core sections, regular translation and scalar covariance give

\[
 \nabla^{\rm reg}_{r}L_h
       =(L_h\otimes m(h))\nabla^{\rm reg}_{s}.
 \tag{CVC.14}
\]

There is no derivative of \(U_h\) on a varying internal coefficient vector: the regular source vector at \(h^{-1}g\) is the very same fixed vector at \(g\). There is also no derivative of \(j\), \(P\), the cutoff or \(\mathsf W\) in CVC.14.

This identity extends to the **full completed local normal domains**. On compact bisection tests, both scalar pullback and covector transport are bounded, and they take the stated core to the corresponding core. Apply them to a sequence converging in the graph norm of CVC.13 and use closedness. This proves one inclusion of domains and the identity there. Apply the same argument to the inverse bisection for the reverse inclusion. The original holonomy-invariant normal metric makes covector transport isometric. Thus, with compact support and the natural source/range identifications,

\[
 L_h\mathcal D_{N,s}=\mathcal D_{N,r},\qquad
 \|\nabla_{\rm reg,r}L_hy\|
       =\|\nabla_{\rm reg,s}y\|.
 \tag{CVC.15}
\]

The notation in CVC.14–CVC.15 denotes pullback of **sections along a bisection**, or the corresponding localized graph sheaves. A normal derivative is not an \(A\)-linear fibre operator, so these equalities do not purport to define a domain of that derivative on an isolated unit fibre.

### 11BF.4. Completeness and arrow invariance of the joint graph

On the whole regular module define

\[
 \mathcal D_{\cap}=\operatorname{Dom}D_H\cap\mathcal D_N,\qquad
 \|y\|_{\cap}=\|y\|+\|D_Hy\|+\|\nabla_{\rm reg}y\|.
 \tag{CVC.16}
\]

This is a complete normed graph space. If \(y_k\) is Cauchy in this norm, its three components have limits \(y,u,v\) in the complete ambient modules. Closedness of \(D_H\) gives \(y\in\operatorname{Dom}D_H\) and \(D_Hy=u\); closedness of \(\nabla_{\rm reg}\) gives \(y\in\mathcal D_N\) and \(\nabla_{\rm reg}y=v\). Hence \(y_k\to y\) in the stated graph norm. The argument proves completeness; it makes no density claim.

Take the local joint graph domains by the same compact bisection localization as above. Equations CVC.9 and CVC.15 preserve both full domains, so their intersection is preserved in both directions. If \(C\) bounds \(\|\mathcal E_h\|\) on the chosen compact test, then

\[
 L_h\mathcal D_{\cap,s}=\mathcal D_{\cap,r},\qquad
 \|L_hy\|_{\cap}\leq(1+C)\|y\|_{\cap}.
 \tag{CVC.17}
\]

Indeed the ordinary norm and normal covector norm are preserved, while CVC.10 bounds the operator component by \(\|D_Hy\|+C\|y\|\). The inverse arrow gives the inverse graph bound. Compact-field continuity in CVC.10 and exact normal covariance in CVC.14 give continuity of these graph maps in the corresponding section topology.

This closes the mismatch between an arbitrary transported arrow action and the geometric normal connection. It does **not** show that \(\mathcal D_{\cap}\) is dense in \(Q\), that a specified source-generated family is a core for both operators, that \(\Phi_H(b)\) preserves it, or that \(D_H\) admits the mixed normal estimates needed for a self-adjoint regular sum. The rough infinite example in Section 11BD already shows why compact regulator and arrow estimates do not supply missing normal-domain control.

### 11BF.5. Exact reflection example and the original physical requirements

For an exact local normal example, let the two-arrow action be reflection \(x\mapsto-x\). On a two-row scalar section write \((L_a y)(x)=\operatorname{swap}(y(-x))\). The normal direction matrix is \(m(a)=-1\). With \(y(x)=(x,x^2)\), direct differentiation gives

\[
 (L_ay)(x)=(x^2,-x),\qquad
 \partial_x(L_ay)(x)=(2x,-1)
       =-L_a(\partial_xy)(x).
 \tag{CVC.18}
\]

The sign is the original normal covector transport, not an error term. Reflection squares to one, and its derivative covariance squares to the identity. Compactly supported even smooth localization gives the same identity on genuine compact normal sections. Finite rational samples check the displayed values and the sign. The infinite copy shift CVC.7 is independently proved by its explicit inverse; a finite truncation would not prove it onto.

All carrier maps above are \(A\)-linear, so the original \(R=C_0(T)\) balance remains the composed anchor balance. Tensoring these even maps with the identity on the separately ordered normal spinor factors preserves their grading and final right Clifford block. The physical comparison must still retain the full inverse coefficient \(z_g^{-1}\rho_t(s_g)\). No scalar auxiliary class, unitization, changed normal lift or equivariant \(\gamma=1\) assertion has been substituted. The source-preserving normal product, passage to the original physical graph Hilbert space and original Bott \(+1\) comparison remain required.

![Equivariant regular inverse carrier, normal reflection and complete invariant joint graph](../figures/covariant-inverse-carrier.png)

**Figure 11BF.1.** CVC.1–CVC.18. Proper cutoff embedding and regularization at source coefficients give an equivariant complemented copy of the actual inverse. An explicitly invertible infinite copy shift absorbs it into the canonical regular carrier, retaining the entire B-source class d_X. The completed vector normal connection is parallel for original holonomy. The joint graph intersection is complete and invariant; its density and a joint core remain unproved. Exact rational cutoff/isometry/equivariance tests, five reflection samples and six copy-index checks illustrate the formulas. The reflection sign is the normal covector transport. Finite checks do not establish infinite absorption or the original physical inverse. Human context: [Connes, A survey of foliations and operator algebras](https://alainconnes.org/wp-content/uploads/foliationsfine.pdf#page=29), Section 8. [Vector figure](../figures/covariant-inverse-carrier.svg) · [Generator](../reproduction/covariant-inverse-carrier/draw_covariant_inverse_carrier.py) · [Exact checks](../reproduction/covariant-inverse-carrier/COVARIANT-INVERSE-CARRIER-CHECKS.json) · [Reproduction](../reproduction/covariant-inverse-carrier/REPRODUCTION.md) · [Component terms](../reproduction/covariant-inverse-carrier/COMPONENT-TERMS.md).

## 11BG. A parallel invariant phase and a dense normal operator core

There is another represented operator in the actual inverse class \(d_X\) with a dense intersection of its operator and normal domains. It uses the exactly invariant normalized phase before the essential compression in Section 11BC. Its source representation retains the normalization's zero-source summand. The nondegenerate source-essential representative of Sections 11BC–11BF remains a separate available representative. We will prove the class, full new operator domain and a normal-compatible operator core; we will not infer unbounded source commutators, unbounded arrow-domain preservation or the physical Bott product.

### 11BG.1. The actual invariant phase on a regular carrier

Let \(A=C_0(X)\), \(H=G\ltimes X\) and \(B=A\widehat\otimes\mathcal A(H_0)\) be the original objects. EAI.3–EAI.7 give the actual source-essential bounded representative \((K,\phi_0,\overline F)\), with class \(d_X\), exactly invariant self-adjoint odd contraction \(\overline F\) and nondegenerate proper phase centre \(\varpi:C_0(W)\to\mathcal L_A(K)\). Apply IUR.1's odd unitary normalization:

\[
 \widetilde K=K\oplus K^{\mathrm{op}},\quad
 \widetilde\phi=\begin{pmatrix}\phi_0&0\\0&0\end{pmatrix},\quad
 \widehat F=
 \begin{pmatrix}
 \overline F&(1-\overline F^2)^{1/2}\\
 (1-\overline F^2)^{1/2}&-\overline F
 \end{pmatrix}.
 \tag{PNC.1}
\]

The copied grading is \(\operatorname{diag}(\Gamma_K,-\Gamma_K)\). Functional calculus makes the off-diagonal entries even on \(K\), so they are odd between the two opposite gradings. The two commuting functions of \(\overline F\) give \(\widehat F^*=\widehat F\) and \(\widehat F^2=1\). Exact invariance of \(\overline F\) gives exact invariance of its square root and \(\widehat F\). The complete class-preserving normalization is proved in The actual inverse representative, IUR.1–IUR.2. In particular the zero-source copy is part of this specific normalized cycle.

Use \(\operatorname{diag}(\varpi,\varpi)\) as a **separate** nondegenerate proper central-space action on all of \(\widetilde K\). It commutes with \(\widetilde\phi(B)\), respects the grading and is equivariant. On the zero-source copy it is the copied original \(\varpi\), not an action inferred from the zero representation of \(B\). It need not commute with \(\widehat F\).

Add ANC.1–ANC.5's represented regular degenerate cycle \((Q,\psi,\Delta)\), whose source is nondegenerate on both copies and whose phase is exactly invariant. Put

\[
 Z=\widetilde K\oplus Q,\qquad
 \Psi=\widetilde\phi\oplus\psi,\qquad
 F_Z=\widehat F\oplus\Delta,\qquad
 [Z,\Psi,F_Z]=d_X,\quad F_Z^2=1.
 \tag{PNC.2}
\]

Both grading parts have the nondegenerate proper \(W\)-action just specified. Also \(Q_+\) is an ungraded standard regular module with countably infinite copy multiplicity. The proper cutoff, source-coordinate embedding and onto copy-shift proof CVC.2–CVC.7, applied to \(\widetilde K_+\) and \(Q_+\), give a genuine equivariant \(A\)-linear unitary \(W_+:Z_+\to M\), where \(M\) has the canonical regular arrow action \(L\). The ungraded stabilization input is the same proved Kasparov stabilization theorem. No normal derivative of \(W_+\) is assumed.

Since \(F_Z\) is odd, self-adjoint and unitary, its off-diagonal part \(F_{+-}:Z_-\to Z_+\) is an onto equivariant unitary, inverse to \(F_{-+}\). Define

\[
 W_- =W_+F_{+-},\qquad
 \mathcal W=\operatorname{diag}(W_+,W_-):Z\longrightarrow M\oplus M,\qquad
 J=\begin{pmatrix}0&1\\1&0\end{pmatrix}.
 \tag{PNC.3}
\]

This is even, onto, \(A\)-linear and genuinely equivariant. On \(Z_+\), \(W_-F_{-+}=W_+\); on \(Z_-\), \(W_+F_{+-}=W_-\). These two identities prove

\[
 \mathcal WF_Z\mathcal W^*=J,\quad
 \Phi(b)=\mathcal W\Psi(b)\mathcal W^*,\quad
 [M\oplus M,\Phi,J]=d_X,\quad
 U_h=\operatorname{diag}(L_h,L_h).
 \tag{PNC.4}
\]

The grading is now \(\Gamma=\operatorname{diag}(1,-1)\). The full source representation and its graded compact commutators are transported. It is generally degenerate: unitary transport does not remove the annihilated submodule in PNC.1. In particular neither a unit of \(B\) nor nondegeneracy of \(\Phi\) is claimed.

### 11BG.2. A completed connection parallel to both phase and arrows

On each copy of \(M\), use the regular vector-valued normal connection constructed from the original scalar PJ.6 sheaf in CVC.11–CVC.15. Take their direct sum, with one common vector-valued graph closure and finite normal-core bisection/copy sections \(\mathscr C_N\). Its completed domain is \(\mathcal D_N\). Constant copy interchange gives

\[
 \nabla J= (J\otimes1)\nabla,\qquad
 J\mathcal D_N=\mathcal D_N,\qquad
 \nabla_rU_h=(U_h\otimes m(h))\nabla_s.
 \tag{PNC.5}
\]

The last equality concerns full local bisection **section** graph domains, with the original normal covector transport. The core equalities extend by graph approximation and the inverse copy interchange or inverse bisection. The connection is metric, respects \(\Gamma\) and has the completed scalar Leibniz identity. Pulling it back by \(\mathcal W^*\) gives an exactly phase-parallel connection on the original normalized augmented module; its domain is exactly \(\mathcal W^*\mathcal D_N\). This does not differentiate \(\mathcal W\).

Choose a countable normal-core generating family \(\eta_j\) in \(M\). It can be chosen with compact bisection support and all normal derivative words in the original sheaf: use the finite polynomial observables, smooth original-unit localizers and their derivative words. Let \(k_{m,j}\geq0\) bound the operator norms of \(\theta_{\eta_j,\eta_j}\) and all of its normal derivative words through order \(m\), in the finitely many original localized frames. Include the vector-valued first derivative in this bound. Every such bound is finite on the chosen support. Make them nondecreasing in \(m\), and, for \(0<\epsilon\leq1/2\), put

\[
 c_j=\frac{\epsilon^2\,2^{-j}}
 {1+\max_{m\leq j+1}k_{m,j}},\qquad
 h=\sum_{j\geq1}c_j\theta_{\eta_j,\eta_j},\qquad
 H_0=\operatorname{diag}(h,h).
 \tag{PNC.6}
\]

The scalar Leibniz and metric identities differentiate a rank-one operator by
\(\delta\theta_{\xi,\eta}=\theta_{\nabla\xi,\eta}+\theta_{\xi,\nabla\eta}\), with the second term interpreted componentwise and paired with the corresponding normal covector. Each fixed derivative-word series converges in compact operator norm: for \(j\geq m\) its term is at most \(\epsilon^2 2^{-j}\), with only finitely many exceptional terms. In particular \(h\) belongs to the simultaneous compact normal graph algebra, \(\|h\|\leq\epsilon^2\) and \(\|\delta h\|\leq\epsilon^2\).

The finite rank-one commutator identities and their norm limits prove that \(h\) preserves \(\mathcal D_N\) on its single copy and
\(\nabla h y=h\nabla y+(\delta h)y\). Thus the assertion is on the full closed domain, rather than only on the algebraic generators.

### 11BG.3. The inverse and its full normal-compatible operator core

The range of \(h\) is dense. This is stronger than merely having zero kernel. Set \(e_t=h(h+t)^{-1}\). Positivity gives \(\theta_{\eta_j,\eta_j}\leq c_j^{-1}h\), so

\[
 \|(1-e_t)\eta_j\|^2
 \leq c_j^{-1}\|(1-e_t)h(1-e_t)\|
 \leq \frac{t}{4c_j}\longrightarrow0.
 \tag{PNC.7}
\]

The last inequality is the scalar maximum of \(t^2s/(s+t)^2\) on \(s\geq0\). The generating span and \(\|1-e_t\|\leq1\) extend strong convergence to all of \(M\). Since \(e_t M\subset\operatorname{Ran}h\), this proves dense range. If \(hy=0\), the positive inner product sum gives \(\langle\eta_j,y\rangle=0\) for every \(j\), hence \(y=0\). Therefore

\[
 \Theta=h^{-1},\qquad
 \operatorname{Dom}\Theta=\operatorname{Ran}h
 \tag{PNC.8}
\]

is the completed positive self-adjoint regular inverse. For completeness, its two resolvents are \(h(1\pm i\lambda h)^{-1}\), bounded adjoints with dense range. They prove the full adjoint domain and regularity by the onto-resolvent criterion proved in CW.5; the graph is parametrized by \((h\xi,\xi)\). Nothing is inferred from fibrewise eigenvectors.

On \(M\oplus M\) define

\[
 T=JH_0^{-1},\qquad
 \operatorname{Dom}T=\operatorname{Ran}H_0,\qquad
 T(H_0\xi)=J\xi.
 \tag{PNC.9}
\]

The constant even copy diagonal \(H_0\) commutes with \(J\). Constant copy diagonalization reduces \(T\) to \(\Theta\oplus(-\Theta)\), proving self-adjoint regularity on precisely that domain. The grading \(\Gamma\) anticommutes with \(T\). Its full resolvents are

\[
 R_\lambda=(T+i\lambda)^{-1}
 =H_0(J+i\lambda H_0)^{-1}\in\mathcal K_A(M\oplus M),
 \quad \|R_\lambda\|\leq\min(\epsilon^2,|\lambda|^{-1}).
 \tag{PNC.10}
\]

The inverse \(B_\lambda^{-1}=(J+i\lambda H_0)^{-1}\) preserves \(\mathcal D_N\). To check the domain assertion, use
\(B_\lambda^{-1}=(J-i\lambda H_0)(1+\lambda^2H_0^2)^{-1}\).
The positive inverse is a norm-convergent geometric series about a sufficiently large scalar. Its differentiated series converges too: a product of \(n\) factors of norm at most \(r<1\) has derivative bound at most \(nr^{n-1}\) times the fixed factor derivative. Closedness of \(\nabla\) proves domain preservation and the inverse derivative formula. Differentiating PNC.10 on that full domain gives, in the indicated order,

\[
 \delta R_\lambda
 =J B_\lambda^{-1}(\delta H_0)B_\lambda^{-1},
 \qquad \|\delta R_\lambda\|\leq\epsilon^2.
 \tag{PNC.11}
\]

Indeed \(1-i\lambda H_0B_\lambda^{-1}=JB_\lambda^{-1}\). One must not commute \(\delta H_0\) through \(H_0\) or its inverse. The constant \(J\) commutes with their copy-diagonal derivative. The norm bound uses \(\|B_\lambda^{-1}\|\leq1\), which follows by diagonalizing \(J\) and applying scalar calculus to \(\pm1+i\lambda h\). Hence

\[
 R_\lambda\mathcal D_N\subset
 \operatorname{Dom}T\cap\mathcal D_N,\qquad
 \|\nabla R_\lambda y\|
 \leq\|R_\lambda\|\,\|\nabla y\|+\epsilon^2\|y\|.
 \tag{PNC.12}
\]

The uniform derivative estimate in \(\lambda\) alone does not give convergence in the joint graph norm as \(|\lambda|\to\infty\). The weighted compactness argument in Section 11BG.7 supplies that convergence.

Fix a nonzero \(\lambda\). Then

\[
 \mathscr C_\lambda=R_\lambda\mathscr C_N
 \subset\operatorname{Dom}T\cap\mathcal D_N
 \tag{PNC.13}
\]

is dense in \(M\oplus M\) and is a graph core for \(T\). To prove the full core assertion, take \(z\in\operatorname{Dom}T\), put \(\xi=(T+i\lambda)z\), and approximate \(\xi\) in module norm by \(\xi_n\in\mathscr C_N\). Then \(R_\lambda\xi_n\to z\) and
\(TR_\lambda\xi_n=(1-i\lambda R_\lambda)\xi_n\to Tz\).
This proves approximation of every vector in the full operator graph by vectors also in the completed normal domain. It also proves density, since the resolvent has dense range. The weighted argument in Section 11BG.7 proves that this same family is a core for the full intersection graph norm. A core for the normal connection alone and a completed normal sum require separate arguments.

### 11BG.4. The bounded transform is still the actual inverse

Functional calculus gives

\[
 F_T=T(1+T^2)^{-1/2}
   =J(1+H_0^2)^{-1/2},\qquad
 F_T-J\in\mathcal K_A(M\oplus M).
 \tag{PNC.14}
\]

The compactness follows because the continuous scalar function
\((1+s^2)^{-1/2}-1\) vanishes at \(s=0\), and \(H_0\) is compact. Its polynomial approximants with zero constant term are compact. Consequently the norm-continuous path \(J+u(F_T-J)\), \(0\leq u\leq1\), consists of bounded represented Kasparov cycles: every source commutator and square defect is the original compact defect plus products containing the compact difference. The same holds for the arrow defects, using exact invariance of \(J\). Thus

\[
 [M\oplus M,\Phi,F_T]
 =[M\oplus M,\Phi,J]=d_X
 \quad\text{in }KK_H(B,A).
 \tag{PNC.15}
\]

This is an equality of the entire actual source-typed equivariant class, not a finite model or a fibre index. Source-local compact resolvents hold, in fact all the resolvents in PNC.10 are module compact. The operator \(T\) has not thereby acquired a dense algebra of bounded **unbounded** source commutators. We call it a self-adjoint regular operator whose bounded transform represents \(d_X\); the additional unbounded Kasparov source requirements must be checked separately.

The bounded arrow error is explicitly compact:

\[
 \alpha_h(F_T)-F_T
 =\alpha_h(F_T-J)-(F_T-J).
 \tag{PNC.16}
\]

The action on compact rank-one fields and their local norm approximations proves continuity on actual arrow charts. These are full bounded-cycle errors. They do not imply \(U_h\operatorname{Dom}T_s=\operatorname{Dom}T_r\) or boundedness of the unbounded error \(T_rU_h-U_hT_s\).

### 11BG.5. Source compression and the physical comparison remain separate

Let \(P_0\) be the original invariant projection onto \(K\subset\widetilde K\) from EAI.8, extended by the identity on the represented augmentation \(Q\), and put \(P'=\mathcal W P_0\mathcal W^*\). It is an actual even invariant adjointable projection with \(\Phi(B)(M\oplus M)\subset P'(M\oplus M)\); the complementary original copy is annihilated by \(\Phi\). In general

\[
 P'\mathcal D_N\subset\mathcal D_N,\quad
 P'\operatorname{Dom}T\subset\operatorname{Dom}T,\quad
 [\nabla,P']\text{ bounded}
 \quad\text{are not established by PNC.1–PNC.16}.
 \tag{PNC.17}
\]

The construction proves a full joint operator/normal graph core **before** that essential compression, by Section 11BG.7. EAI.12–EAI.24 give another source-essential unbounded representative with full arrow graph domains; those properties are not silently transferred to the new \(T\). A complete solution must retain both sets of needed controls on one represented operator, or use a proved source-preserving comparison that avoids such a transfer.

All maps here are \(A\)-linear and therefore preserve the composed original \(R=C_0(T)\) anchor balance. Their even extensions by the identity retain the full inverse normal coefficient \(z_g^{-1}\rho_t(s_g)\), exterior grading and final right Clifford order. The separate original-\(R\) module and physical graph passage remain required. No non-equivariant probability evaluation is used to assert an original groupoid action on a physical Hilbert space.

### 11BG.6. Exact finite normalization and resolvent order

Take the graded source \(M_2(\mathbb C)\) with grading \(\gamma=\operatorname{diag}(1,-1)\), represented on \(K=\mathbb C^2\), and let \(f=(3/5)\sigma_1\). Then \((1-f^2)^{1/2}=(4/5)1\). PNC.1 has a four-dimensional exactly unitary phase, grading \(\operatorname{diag}(1,-1,-1,1)\) and source \(\operatorname{diag}(b,0)\). Its positive indices are \(0,3\) and negative indices \(1,2\). Choosing \(W_+\) to read coordinates \(0,3\) gives \(W_-=W_+F_{+-}\) exactly as in PNC.3. The transformed source identity has rank two, not rank four.

For an independent finite normal resolvent test use

\[
 h(x)=\frac1{32}
 \begin{pmatrix}2+x&x/4\\x/4&3-x\end{pmatrix},
 \qquad -1\leq x\leq1,\qquad \lambda\in\{1,2,4\}.
 \tag{PNC.18}
\]

The diagonal entries are positive and the determinant is
\((6+x-17x^2/16)/32^2\geq63/(16\cdot32^2)>0\), so it is positive invertible throughout this interval. Rational samples test the two full inverse identities and PNC.11 by realifying \(J+i\lambda H_0\). Since \(h(x)\) and \(h'(x)\) do not generally commute, this checks the order of the differentiated resolvent. Sixteen source products check the transported finite representation. These finite matrices illustrate normalization, the retained zero-source subspace and resolvent algebra; they do not certify the infinite original inverse or its physical Bott pairing.

### 11BG.7. A core for the full operator and normal graph

The compact vector derivative in PNC.6 supplies a weighted estimate stronger than the uniform bound in PNC.11. It proves a joint graph core for this actual normalized inverse, on its full domain, without any global choice of constant arrow coordinates.

Write \(\mathcal M=M\oplus M\), and let \(\mathcal V_N\) be the completed vector-valued normal target of \(\nabla\). An operator followed by \(\otimes1\) acts on that target. For real \(\lambda\geq1\) put

\[
 u_\lambda=i\lambda R_\lambda
       =1-JB_\lambda^{-1},\qquad
 \beta_\lambda=
 \|(B_\lambda^{-1}\otimes1)\delta H_0\|,
 \qquad \beta_\lambda\longrightarrow0 .
 \tag{PNC.19}
\]

Here \(B_\lambda^{-1}\) is a contraction. On the dense range of \(H_0\),
\(\|B_\lambda^{-1}H_0\eta\|\leq\lambda^{-1}\|\eta\|\): diagonalize the constant \(J\) and use \(s/|\pm1+i\lambda s|\leq\lambda^{-1}\). The contraction bound extends convergence \(B_\lambda^{-1}\xi\to0\) to every \(\xi\in\mathcal M\). On the vector normal target the same conclusion follows first for elementary tensors and then by density. In particular this is a statement about the completed section modules, independent of a global normal frame.

PNC.6 gives a compact operator \(\delta H_0:\mathcal M\to\mathcal V_N\). Multiplying it on the left by a uniformly bounded strongly convergent-to-zero family gives convergence in operator norm. To prove this assertion directly, for a rank-one map \(\theta_{a,b}\) the product is \(\theta_{(B_\lambda^{-1}\otimes1)a,b}\), with norm at most \(\|(B_\lambda^{-1}\otimes1)a\|\|b\|\). Finite sums therefore converge to zero, and uniform boundedness extends the conclusion to their compact norm limits. This proves PNC.19. It also gives \(\|\delta R_\lambda\|\leq\beta_\lambda\to0\) by PNC.11. Multiplication by \(\lambda\) prevents inferring raw derivative decay for \(u_\lambda\) from that bound; the weighted limit below addresses the needed full-domain convergence.

The ordered derivative in PNC.11 now yields

\[
\begin{aligned}
 (\delta u_\lambda)H_0
 &=i\lambda(JB_\lambda^{-1}\otimes1)
       (\delta H_0)B_\lambda^{-1}H_0,\\
 \|(\delta u_\lambda)H_0\|&\leq\beta_\lambda,\\
 \|(\delta u_\lambda)\xi\|
 &\leq\beta_\lambda\|T\xi\|,
       \qquad \xi\in\operatorname{Dom}T .
\end{aligned}
 \tag{PNC.20}
\]

The first bound uses \(\lambda\|B_\lambda^{-1}H_0\|\leq1\) at the right and the compact product defining \(\beta_\lambda\) at the left. It does not move \(\delta H_0\) through either inverse. For the second bound write the unique \(\xi=H_0\eta\), so \(\|\eta\|=\|T\xi\|\). This weighted bound does not require the raw operator norms \(\|\delta u_\lambda\|\) to converge or even to remain uniformly bounded.

PNC.12 gives preservation of \(\mathcal D_N\) by \(u_\lambda\). Functional calculus, or the expression \(1-JB_\lambda^{-1}\), gives strong convergence \(u_\lambda\to1\) and \(\|u_\lambda\|\leq1\). Also \(u_\lambda\) preserves the full \(T\)-domain and commutes with \(T\) there: if \(\xi=H_0\eta\), then \(u_\lambda\xi=H_0u_\lambda\eta\) and \(Tu_\lambda\xi=Ju_\lambda\eta=u_\lambda T\xi\). Consequently, for every \(\xi\) in the full intersection,

\[
\begin{aligned}
 &u_\lambda\xi\longrightarrow\xi,\qquad
 Tu_\lambda\xi\longrightarrow T\xi,\\
 &\nabla u_\lambda\xi-\nabla\xi
   =(u_\lambda\otimes1-1)\nabla\xi
      +(\delta u_\lambda)\xi\longrightarrow0,
 \qquad \xi\in\operatorname{Dom}T\cap\mathcal D_N .
\end{aligned}
 \tag{PNC.21}
\]

The last limit follows from strong convergence on the vector target and PNC.20. Thus the regularization converges in the complete joint graph norm
\(\|\xi\|+\|T\xi\|+\|\nabla\xi\|\). This intersection is complete because \(T\) and \(\nabla\) are both closed on their stated full domains.

For each fixed real \(\mu\ne0\), the family \(\mathscr C_\mu=R_\mu\mathscr C_N\) in PNC.13 is in fact a core for that entire joint graph. Let \(\xi\) be any vector in the intersection and set \(z_\lambda=u_\lambda\xi\). Its resolvent preimage is

\[
\begin{aligned}
 \eta_\lambda&=(T+i\mu)z_\lambda\\
 &=i\lambda\xi+i\lambda i(\mu-\lambda)R_\lambda\xi
       \in\mathcal D_N,\qquad
 z_\lambda=R_\mu\eta_\lambda,\\
 &\overline{\mathscr C_\mu}^{\,\|\cdot\|+\|T\cdot\|+\|\nabla\cdot\|}
       =\operatorname{Dom}T\cap\mathcal D_N .
\end{aligned}
 \tag{PNC.22}
\]

The displayed bounded formula proves \(\eta_\lambda\in\mathcal D_N\); it does not assume that \(T\xi\) lies in the normal domain. For each fixed \(\lambda\), approximate \(\eta_\lambda\) by \(\eta_{\lambda,n}\in\mathscr C_N\) in the full normal graph norm. PNC.12 proves normal graph convergence after applying \(R_\mu\). Ordinary norm convergence and \(TR_\mu=1-i\mu R_\mu\) give convergence of the vector and its \(T\)-image as well. Taking \(\lambda=1,2,\ldots\) and then choosing one \(n\) at each step so that this latter joint error is at most \(1/\lambda\) gives a sequence from the single fixed family \(\mathscr C_\mu\) converging to \(\xi\) in all three graphs, by PNC.21.

If a grading-stable core is desired, use
\(\mathscr C_\mu+\mathscr C_{-\mu}\). The normal core respects \(\Gamma\), and oddness of \(T\) gives \(\Gamma R_\mu=-R_{-\mu}\Gamma\). This sum is therefore grading-stable and is still a full joint core.

There is also an even positive regularization. For \(t>0\), put \(e_t=H_0(H_0+t)^{-1}\). The same inverse-derivative argument as in PNC.11 preserves the full normal domain and gives

\[
\begin{aligned}
 a_t&=\|(t(H_0+t)^{-1}\otimes1)\delta H_0\|
         \longrightarrow0\quad(t\downarrow0),\\
 \delta e_t&=t((H_0+t)^{-1}\otimes1)
                (\delta H_0)(H_0+t)^{-1},\\
 \|(\delta e_t)\xi\|&\leq a_t\|T\xi\|,
        \qquad \xi\in\operatorname{Dom}T .
\end{aligned}
 \tag{PNC.23}
\]

Indeed \(t(H_0+t)^{-1}\) is a contraction and tends strongly to zero on the dense range of \(H_0\), then everywhere. The compact rank-one argument proves \(a_t\to0\), and \(\|(H_0+t)^{-1}H_0\|\leq1\) gives the weighted bound. By PNC.7, \(e_t\to1\) strongly. These even contractions commute with \(T\) on its full domain, so \(e_t\xi\to\xi\), \(Te_t\xi\to T\xi\) and \(\nabla e_t\xi\to\nabla\xi\) for every vector in the full joint domain.

These are full joint-domain statements for the normalized operator PNC.9, whose bounded transform has the entire actual class \(d_X\). The zero-source copy is still present. They do not prove normal or operator domain preservation by the essential projection \(P'\), or transfer the result to the different source-essential operator \(S_N\) of Section 11BH. The source and unbounded arrow tests in Section 11BG.5 and the original physical normal sum remain required.

![Parallel actual normalized inverse phase and full joint graph core](../figures/parallel-normal-inverse.png)

**Figure 11BG.1.** PNC.1–PNC.23. The pre-compression normalized actual inverse has an exactly invariant odd unitary phase and a copied nondegenerate proper centre. Equivariant positive/negative carrier maps make its phase and arrows parallel for two equal completed regular normal connections. A compact normal-smooth inverse gives an operator in the same class d_X whose resolvents preserve the normal domain. A compact weighted derivative estimate makes the regularizations converge in all three graphs. For every fixed nonzero resolvent parameter, its image of the normal core is a core for the full joint operator/normal graph. Sixteen exact finite source products, source rank two and fifteen rational realified resolvent tests check normalization and derivative order. The zero-source copy is retained; source-essential compression, unbounded source/arrow controls and the original physical product remain separate. Human context: [Connes, A survey of foliations and operator algebras](https://alainconnes.org/wp-content/uploads/foliationsfine.pdf#page=29), Section 8. [Vector figure](../figures/parallel-normal-inverse.svg) · [Generator](../reproduction/parallel-normal-inverse/draw_parallel_normal_inverse.py) · [Exact checks](../reproduction/parallel-normal-inverse/PARALLEL-NORMAL-INVERSE-CHECKS.json) · [Reproduction](../reproduction/parallel-normal-inverse/REPRODUCTION.md) · [Component terms](../reproduction/parallel-normal-inverse/COMPONENT-TERMS.md).

## 11BH. A normal-compatible graph core for the source-essential inverse

The source-essential inverse can be constructed on the canonical regular carrier with a dense operator graph core lying in its completed normal domain. The construction retains its actual nondegenerate Bott source, its class and its full unbounded arrow domains. It uses normal-smooth diagonal cutoffs before averaging the regulator with the invariant phase. The phase itself need not be normal-smooth.

### 11BH.1. Start with the actual bounded inverse

Use the essential module \(K\), nondegenerate source \(\phi_0\), and exactly invariant self-adjoint odd contraction \(\overline F\) of EAI.1–EAI.7. Add the genuinely degenerate represented augmentation \((Q,\psi,\Delta)\) of ANC.4–ANC.5. The bounded data are

\[
 E=K\oplus Q,\qquad \Psi=\phi_0\oplus\psi,\qquad
 f_E=\overline F\oplus\Delta,\qquad
 [E,\Psi,f_E]=d_X\in KK_H(B,A).
 \tag{ENC.1}
\]

Both source actions are nondegenerate; both phases are exactly invariant. The proper \(C_0(W)\)-action of EAI.1 extends through the nondegenerate source on each summand. The genuine even absorption in CVC.2–CVC.7 applies to these same modules and actions independently of any unbounded operator. Transport the *entire* bounded cycle by that absorption onto its regular carrier \(M\). Denote the resulting source by \(\Phi\), phase by \(f\), and canonical arrows by \(L_h\). Then

\[
 \overline{\Phi(B)M}=M,\qquad
 f=f^*,\quad \|f\|\leq1,\quad
 \alpha_h f=f,\qquad [M,\Phi,f]=d_X.
 \tag{ENC.2}
\]

No derivative of the absorption unitary has been used. On \(M\), use exactly the closed vector normal connection \(\nabla_{\rm reg}\) and core \(\mathscr C_{\rm reg}\) of CVC.11–CVC.15. Its normal covariance is an equality of local section graph domains along bisections.

Normalize the phase on the opposite-graded double:

\[
 \widetilde M=M\oplus M^{\rm op},\quad
 \widetilde\Phi(b)=\operatorname{diag}(\Phi(b),0),\quad
 F=\begin{pmatrix}f&(1-f^2)^{1/2}\\(1-f^2)^{1/2}&-f\end{pmatrix},
 \quad P=\operatorname{diag}(1,0).
 \tag{ENC.3}
\]

IUR.1–IUR.2 prove that \(F\) is an odd self-adjoint unitary of the actual class \(d_X\). It is exactly invariant, as is \(P\). The two copies have equal regular normal connections, so \(P\) preserves their full normal domain and commutes with that connection. We have kept the original copied coordinates; no phase-flattening change of normal connection has occurred.

### 11BH.2. A diagonal compact seed with local positive floors

Choose countably many homogeneous vectors \(\xi_j\in\mathscr C_{\rm reg}\), each supported on one relatively compact bisection and one fixed copy coordinate. Choose their scalar normal-smooth bumps so that they are nonzero near every arrow/copy coordinate. Such a countable cover exists by second countability and the normal localizers proved in CVC.11. Their \(A\)-linear span is dense: subdivide a compact arrow support into finitely many bisections and bump neighborhoods, use a partition of unity, and approximate the resulting scalar coefficients by the dense scalar core.

For each \(j\), let \(M_j\) bound the operator norms of \(\theta_{\xi_j,\xi_j}\) and its finitely many tested normal derivative words through order \(j+1\). Enumerate compact normal charts and derivative words together, so every fixed test is included eventually. These bounds are finite by compact support and the scalar metric/product identities. Set

\[
 c_j=\frac{2^{-j}}{1+M_j},\qquad
 h_*=\sum_{j\geq1}c_j\theta_{\xi_j,\xi_j}
       \in\mathcal K_A(M),\qquad
 H_*=\operatorname{diag}(h_*,h_*).
 \tag{ENC.4}
\]

Each fixed derivative series converges in operator norm: after its finite initial part, each norm is at most \(2^{-j}\). In particular \(h_*\) preserves the full closed normal domain, with a bounded first normal commutator \(D_Nh_*\). To see this on the actual minimal domain, first apply the rank-one metric/product identity on \(\mathscr C_{\rm reg}\), pass to the operator and derivative norm limits, and then approximate any vector in the closed connection graph by that core. Closedness gives the claimed domain preservation and identity.

A single bisection has at most one arrow in each range fibre. A single-copy rank one therefore has no off-diagonal arrow or copy entry. Thus \(h_*\) is diagonal. Write its scalar entry at \((g,n)\) as \(h_*(g,n)\). The covering bumps give

\[
 h_*(g,n)=\sum_{\xi_j\ {\rm in\ coordinate}\ n}
                c_j|\xi_j(g)|^2>0.
 \tag{ENC.5}
\]

Uniform convergence makes these entries continuous. On every fixed compact arrow support in finitely many copy coordinates, they have a positive minimum. This is a *local* floor; there is no positive global lower bound.

The range is dense, not merely of zero kernel. If \(\eta\) has such finite compact support, a continuous cutoff \(e\), zero near zero and equal to one above its local floor, satisfies \(e(h_*)\eta=\eta\). Also \(e(h_*)=h_*a(h_*)\) for bounded continuous \(a\). Hence every such vector lies in \(\operatorname{Ran}h_*\), and these vectors are dense. The copied seed \(H_*\) commutes with \(P\) and the grading.

### 11BH.3. Normal-preserving cutoffs and one common-tail choice

Choose smooth scalar plateaus \(\chi_k\), with \(0\leq\chi_k\leq1\), zero on \([0,a_k]\) and one on \([b_k,\|H_*\|]\), where

\[
 a_k=2^{-2k-2},\qquad b_k=2^{-2k-1},\qquad
 b_{k+1}<a_k,\qquad e_k=\chi_k(H_*).
 \tag{ENC.6}
\]

Extend \(\chi_k\) smoothly with compact support beyond the spectrum. The \(e_k\) are compact, even, commute with \(P\), and form an approximate identity for \(\mathcal K_A(\widetilde M)\); density of \(\operatorname{Ran}H_*\) proves their strict convergence. They satisfy \(e_ke_l=e_k\) for \(l>k\).

Each \(e_k\) preserves the full normal domain. This uses smooth Fourier calculus, not a general operator-Lipschitz assertion for \(C^1\) functions. The power series first proves the completed product identity for \(e^{itH_*}\). Fourier inversion then gives

\[
 \|D_N\chi_k(H_*)\|
 \leq \frac1{2\pi}\int_{\mathbb R}|t|\,|\widehat\chi_k(t)|\,dt
                  \,\|D_NH_*\|<\infty.
 \tag{ENC.7}
\]

The ordered derivative of the exponential is
\((i\int_0^t e^{i(t-s)H_*}D_NH_*e^{isH_*}\,ds)\), with the natural amplification on normal covectors. It follows by differentiating powers and summing their absolutely convergent series. Integrate that graph identity using the displayed integrable bound and closedness of the connection. No uniform derivative bound in \(k\) is claimed.

Choose a countable dense homogeneous source-test algebra \((b_j)\), with products, adjoints and the compact-bisection transports used in EAI. Include the bounded multiplier tests \(F,\widetilde\Phi(b_j)\), and the compact tests
\(k_j=[F,\widetilde\Phi(b_j)]_{\rm gr}\) and
\(t_j=[F,P]\widetilde\Phi(b_j)\), with their adjoints. The latter are compact by IUR.1. Include compact arrow exhaustions, inverse arrows and their localized compact-field tests.

The common-tail convex separation proved in KG.2 and used in IUR.3 gives finite convex combinations \(v_n\) of the \(e_k\). Choose all indices for \(v_{n+1}\) beyond those for \(v_n\). For \(j\leq n\) and the \(n\)-th compact arrow test, require

\[
 \begin{gathered}
 v_nv_{n+1}=v_n,\quad 0\leq v_n\leq1,\quad v_n\to1\ {\rm strictly},\\
 \|[v_n,F]\|,\ \|[v_n,\widetilde\Phi(b_j)]\|\leq2^{-n},\\
 \|k_j(1-v_n)\|,\ \|t_j(1-v_n)\|\leq2^{-n},\qquad
 \sup_h\|\alpha_hv_n-v_n\|\leq2^{-n}.
 \end{gathered}
 \tag{ENC.8}
\]

This convex argument applies to the present approximate identity as well. A multiplier's cutoff commutators lie in the compact ideal and converge weakly to zero; simultaneous weak-convex separation controls finitely many of them. Compact tails converge in norm. On a compact arrow test the finite-bisection localization and compact-field approximation are precisely KG.2's uniform tests. One common tail treats their finite direct sum. Later indices preserve every preceding plateau. All \(v_n\) remain functions of \(H_*\), commute with \(P\) and grading, preserve the normal domain, and have bounded individual normal commutators.

Make this construction adaptive in one additional finite test. After \(v_m\) is selected, the operator \(q_m=[F,v_m]\) is compact. At stage \(n>m\), require
\(\|q_m(1-v_n)\|\leq2^{-n-m}\), together with its adjoint test, for every \(m<n\). There are only finitely many new known compact tails at that stage. Their approximate-identity convergence and the same common-tail convex argument supply them without changing any previous requirement. No future cutoff is used to choose a preceding one. This additional choice will control the cutoffs on the full essential graph in Section 11BH.10.


### 11BH.4. The reference domain has a normal-compatible graph core

Use the same locally finite reciprocal calculus as IUR.4:

\[
 r_*=1+\sum_{n\geq1}(1-v_n),\qquad
 l_*=r_*^{-1}\in\mathcal K_A(\widetilde M),\qquad
 \operatorname{Dom}r_*=\operatorname{Ran}l_*.
 \tag{ENC.9}
\]

The scalar sum is locally finite at every positive spectral value, is at least one, and tends to infinity at zero. Its reciprocal extends continuously by zero there. The cutoff ranges prove dense range of \(l_*\), and the adjoint resolvents of the reciprocal prove positive self-adjoint regularity on this exact range domain. The products \(r_*v_m\) are bounded: on the support of \(v_m\), all later plateaus are one, leaving only finitely many summands.

Let \(\widetilde{\mathscr C}_{\rm reg}\) be the copied finite normal core and put

\[
 \mathscr E=\operatorname{span}\{v_m\eta:
                       m\geq1,\ \eta\in\widetilde{\mathscr C}_{\rm reg}\}
       \subset\operatorname{Dom}r_*\cap\operatorname{Dom}\widetilde\nabla_{\rm reg}.
 \tag{ENC.10}
\]

For any \(\xi\in\operatorname{Dom}r_*\), strict convergence and commutation give \(v_m\xi\to\xi\) and \(r_*v_m\xi=v_mr_*\xi\to r_*\xi\). At each fixed \(m\), approximate \(\xi\) in ordinary module norm by \(\eta_j\in\widetilde{\mathscr C}_{\rm reg}\). Boundedness of both \(v_m\) and \(r_*v_m\) gives graph convergence to \(v_m\xi\). A diagonal choice proves that \(\mathscr E\) is a graph core for \(r_*\). Its normal-domain inclusion follows from ENC.7, without any uniform derivative bound or joint-graph approximation. The local floors in ENC.5 also show that each original finite-core vector is fixed by all sufficiently late \(v_m\).

### 11BH.5. Average the regulator, keeping its full domain

The multiplier tests in ENC.8 give compact bounded commutators
\(C_F=[r_*,F]\) and \(C_b=[r_*,\widetilde\Phi(b)]\), and compact extensions \(k_jr_*,t_jr_*\). Here a commutator is the norm-convergent series of cutoff commutators. The weighted product is
\(k_j+\sum k_j(1-v_n)\), with a finite initial part and summable tail; use the identical argument for \(t_j\).

These identities preserve the *full* reference domain. Indeed set \(R_N=1+\sum_{n\leq N}(1-v_n)\). For \(\xi=l_*\eta\), \(R_N\xi\to\eta\), and
\(R_NF\xi=FR_N\xi+[R_N,F]\xi\) converges. If \(R_Nz\to w\), then \(l_*R_Nz\to z\) by the bounded strict identity \(l_*R_N\to1\); hence \(z=l_*w\in\operatorname{Dom}r_*\). Apply this to \(z=F\xi\). The same argument works for the source multipliers and their adjoints.

Now define

\[
 r=\tfrac12(r_*+Fr_*F)=r_*+B_F,\qquad
 B_F=\tfrac12F C_F\in\mathcal K_A(\widetilde M),\qquad
 \operatorname{Dom}r=\operatorname{Dom}r_*.
 \tag{ENC.11}
\]

\(B_F\) is bounded self-adjoint and even: \(FC_F=Fr_*F-r_*\) on the dense common domain, and its bounded extension has that adjoint. The bounded-perturbation resolvent proof makes \(r\) self-adjoint regular on the displayed full domain. Both averaged forms are at least one, so \(r\geq1\). It commutes exactly with \(F\), because conjugation interchanges the two averaged terms. Its inverse is compact by
\(r^{-1}=l_*-l_*B_Fr^{-1}\).

The \(r\)- and \(r_*\)-graph norms are equivalent by the bounded difference. Thus \(\mathscr E\) is still an \(r\)-graph core. Source commutators and weighted compact tests survive:
\([r,\widetilde\Phi(b)]=C_b+[B_F,\widetilde\Phi(b)]\), and
\(k_jr=k_jr_*+k_jB_F\), with the same formula for \(t_j\). No normal derivative of \(F\), \(C_F\) or \(B_F\) is needed.

### 11BH.6. Essential compression and its actual operator core

Since \(P\) commutes with \(r_*\), it preserves \(\operatorname{Dom}r\), and

\[
 [r,P]=[B_F,P]\in\mathcal K_A(\widetilde M),\qquad
 r_0=PrP=r_*|_{P\widetilde M}+PB_FP,\quad
 \operatorname{Dom}r_0=P\operatorname{Dom}r_*.
 \tag{ENC.12}
\]

The diagonal-removal argument EAI.9–EAI.10 makes this a positive self-adjoint regular reference operator on \(M=P\widetilde M\). Set \(D=Fr\) on \(\operatorname{Dom}r\). Exact commutation with \(F\) gives self-adjoint regularity, compact resolvents and a compact bounded-transform difference from \(F\), by IUR.5–IUR.7.

Initially compress on the displayed reference domain:

\[
 S_0=PDP=f r_0+T_0,\qquad
 T_0=PF(1-P)rP\in\mathcal K_A(M),\qquad
 S_N=\overline{S_0}.
 \tag{ENC.13}
\]

The off-diagonal reference block of \(r\) is bounded compact by ENC.12. Exact \([r,F]=0\) gives the compact reference commutator EAI.12. Thus EAI.13–EAI.15's uniform resolvent-commutator and localized deficiency-space argument applies verbatim to this regulator, and proves that \(S_N\) is odd self-adjoint regular. Its full closed domain may be larger than \(\operatorname{Dom}r_0\).

\(P\mathscr E\) is an \(r_0\)-graph core. This follows from the copied seed, \(Pv_n=v_nP\), the core proof ENC.10 on the first copy, and the bounded perturbation in ENC.12. Since \(S_0=f r_0+T_0\), convergence in that reference graph implies convergence in the \(S_0\)-graph. Every full \(S_N\)-graph vector is already a limit of reference graph vectors by the definition of the closure. Approximate those in turn by \(P\mathscr E\). Therefore

\[
 P\mathscr E\subset\operatorname{Dom}S_N\cap\operatorname{Dom}\nabla_{\rm reg},
 \qquad
 \overline{P\mathscr E}^{\,\|\cdot\|+\|S_N\cdot\|}
          =\operatorname{Dom}S_N.
 \tag{ENC.14}
\]

This is a core for the full essential operator, contained in the actual completed normal domain. In particular their intersection is dense in \(M\).

### 11BH.7. Source, class and full arrow domains

The compact source commutators and weighted tests proved after ENC.11 give

\[
 [D,\widetilde\Phi(b)]_{\rm gr}
   =F[r,\widetilde\Phi(b)]+[F,\widetilde\Phi(b)]_{\rm gr}r
      \in\mathcal K_A(\widetilde M).
 \tag{ENC.15}
\]

The source preserves the reference domain. Compression and closed graph approximation then prove preservation of the entire \(\operatorname{Dom}S_N\), with compact graded commutator \(P[D,\widetilde\Phi(b)]_{\rm gr}P\), for the dense tested algebra.

All hypotheses of the source-local comparison EAI.16–EAI.20 now hold: \([r,P]\) is bounded compact, and \([F,P]\widetilde\Phi(b)r\) is compact. Its off-diagonal operator times \(r^{-1}\) is bounded, and times \(\widetilde\Phi(b)\) is compact. The resolvent identity and its \(s^{-2}\) localized bound therefore give source-local compact resolvents and a source-locally compact bounded-transform difference from \(D\). Remove the zero-source second summand *after* this comparison. It follows that

\[
 [M,\Phi,F_{S_N}]=d_X,\qquad
 \overline{\Phi(B)M}=M,\qquad
 \Phi(b)(S_N\pm i)^{-1}\in\mathcal K_A(M)\quad(b\in B).
 \tag{ENC.16}
\]

This uses the actual represented inverse and its full source throughout.

The arrow tests in ENC.8 give a continuous unlocalized compact field
\(A_h^{\rm ref}=\alpha_h r_*-r_*\), with full reference-domain preservation, by the partial-sum and inverse-range proof IUR.11–IUR.13. Exact invariance of \(F\) gives

\[
 A_h=\alpha_h r-r
     =\tfrac12(A_h^{\rm ref}+F A_h^{\rm ref}F),\qquad
 S_{N,r(h)}L_h-L_hS_{N,s(h)}
       =-PF A_hP\,L_h.
 \tag{ENC.17}
\]

The field is compact, continuous and uniformly bounded on every compact arrow test, before a source localization. Exact invariance of \(P\), reference-domain preservation and graph-core approximation prove the equality and inclusion on the full closed operator domain; the inverse arrow proves reverse inclusion. The sign is from the convention \(\alpha_h r=L_hr_{s(h)}L_h^{-1}\). Thus \(L_h\operatorname{Dom}S_{N,s(h)}=\operatorname{Dom}S_{N,r(h)}\) and the full graph norm is locally uniformly bounded. Compact-field continuity and that bound extend graph continuity from reference sections to all graph sections.

Intersect with CVC.15's full completed normal section domains. Closedness of both graphs, ENC.14 and ENC.17 prove

\[
 \begin{gathered}
 \mathcal D_{\cap}=\operatorname{Dom}S_N\cap\operatorname{Dom}\nabla_{\rm reg}
        \text{ is dense and complete for }
        \|\xi\|+\|S_N\xi\|+\|\nabla_{\rm reg}\xi\|,\\
 L_h\mathcal D_{\cap,s}=\mathcal D_{\cap,r},\qquad
 \|L_h\xi\|_{\cap}\leq(1+C)\|\xi\|_{\cap}
 \quad\text{on a compact test with }\|PF A_hP\|\leq C.
 \end{gathered}
 \tag{ENC.18}
\]

These joint normal domains are domains of sections along bisections, not domains of a normal derivative on an isolated unit fibre.

### 11BH.8. What this core does and does not prove

The same actual nondegenerate inverse now has unbounded source controls, full arrow graph controls and a dense operator graph core inside its completed covariant normal domain. Its original \(A\)- and composed \(R=C_0(T)\)-actions remain fixed. The precise remaining analytic obligations include

\[
 \begin{gathered}
 \text{normal-source domain preservation and a suitable dense joint graph core,}\\
 \text{mixed normal/operator estimates for the completed physical sum,}\\
 \text{the original }R\text{-balanced graph passage and Bott }+1
 \text{ pairing with }z_g^{-1}\rho_t(s_g).
 \end{gathered}
 \tag{ENC.19}
\]

ENC.14 is an operator graph core; it is not a core for the complete joint graph norm. ENC.7 gives individual cutoff normal derivatives, without bounds uniform along the regulator. ENC.11 involves the possibly normal-rough phase, so it does not prove normal preservation by \(r\), \(S_N\), or their resolvents. The normal-resolvent preservation of PNC.11 belongs to PNC's different representative with its retained zero-source copy; it has not been transferred to \(S_N\). No physical sum or original Bott comparison is inferred from density alone.

### 11BH.9. Exact finite symmetrization

An independent finite model uses two arrow rows with two parity coordinates each:

\[
 \begin{gathered}
 \gamma=\operatorname{diag}(1,-1,1,-1),\quad
 f=\tfrac35\operatorname{diag}(\sigma_x,\sigma_x),\quad C=\tfrac45I_4,\quad
 F=\begin{pmatrix}f&C\\C&-f\end{pmatrix},\\
 h_*(x)=\tfrac1{32}\operatorname{diag}(2+x,3-x,4+x,5-x),\quad
 -1\leq x\leq1,\quad r_*=\operatorname{diag}(h_*^{-1},h_*^{-1}),\\
 r=\tfrac12(r_*+Fr_*F),\qquad
 P=\operatorname{diag}(I_4,0),\qquad
 \Phi(b)=\operatorname{diag}(b,b).
 \end{gathered}
 \tag{ENC.20}
\]

Here \(\sigma_x\) is the two-by-two coordinate interchange. The arrow interchanges the two row blocks on each copy, and commutes with \(F\). Every entry of \(h_*(x)\) is at least \(1/32\). Rational samples verify exact phase/grading identities, the symmetrized commutator \([r,F]=0\), the compact-perturbation formulas for \(r,r_0\), the compressed identity \(PFrP=f r_0+T_0\), and the full arrow-error algebra ENC.17. Sixteen matrix-unit products check the nondegenerate essential source. The matrices illustrate these formulas; they do not certify the infinite inverse class or a physical normal sum.


### 11BH.10. Normal-preserving cutoffs on the full essential graph

Keep the adaptive choice just specified after ENC.8. For a fixed \(m\), \(q_m=[F,v_m]\) has norm at most \(2^{-m}\). The first \(m\) terms of its reference weighted series are bounded individually by this norm; every later term has the added bound \(2^{-n-m}\). Thus

\[
 \begin{gathered}
 q_mr_*=q_m+\sum_{n\geq1}q_m(1-v_n)\in\mathcal K_A(\widetilde M),
 \qquad
 \|q_mr_*\|\leq(m+1)2^{-m}+2^{-2m},\\
 E_m=[D,v_m]=F[B_F,v_m]+q_mr\in\mathcal K_A(\widetilde M),\\
 \|E_m\|\leq\|[B_F,v_m]\|
       +(m+1+\|B_F\|)2^{-m}+2^{-2m}\longrightarrow0 .
 \end{gathered}
 \tag{ENC.21}
\]

The weighted series is first an identity on \(\operatorname{Dom}r_*\), by its bounded partial sums and convergence; its norm-convergent compact extension is the one displayed. Since \(r=r_*+B_F\), it gives the second product as well. The reference cutoff commutes with \(r_*\) and preserves its full domain, so \([r,v_m]=[B_F,v_m]\) there. Insert this in \([Fr,v_m]\) to get the stated identity. Finally \(B_F\) is compact and \(v_m\to1\) strictly, so both \((1-v_m)B_F\) and \(B_F(1-v_m)\) tend to zero in norm. This proves the limit; no normal derivative of \(F\) has appeared.

Write \(w_m=v_m|_M\), using the constant first-copy projection. It preserves the reference domain and commutes with \(P\). Compression of ENC.21 gives \(e_m=PE_mP\). Approximate any \(\xi\in\operatorname{Dom}S_N\) by the reference graph core; boundedness of \(w_m,e_m\) and closedness of \(S_N\) then prove

\[
 \begin{gathered}
 w_m\operatorname{Dom}S_N\subset\operatorname{Dom}S_N,\qquad
 [S_N,w_m]=e_m\in\mathcal K_A(M),\quad \|e_m\|\to0,\\
 w_m\xi\to\xi,\qquad
 S_Nw_m\xi=w_mS_N\xi+e_m\xi\to S_N\xi
       \quad(\xi\in\operatorname{Dom}S_N).
 \end{gathered}
 \tag{ENC.22}
\]

In fact \(w_m\) maps the whole module into the reference domain, because \(r_*v_m\) is bounded. The images here therefore lie in that original reference domain even if the full essential domain is larger. ENC.7 simultaneously says that \(w_m\) preserves the full completed normal domain. Consequently these are normal-preserving approximations of *every full essential operator graph vector*. They preserve the joint intersection as well.

There is an exact test for convergence in the joint graph. Put \(B_m^N=D_Nw_m\), a bounded map \(M\to M\otimes_A a^*\tau^*\) for each fixed \(m\). On the full normal domain the completed product identity and its two norm limits give

\[
 \begin{gathered}
 \nabla_{\rm reg}w_m\xi-\nabla_{\rm reg}\xi
    =(w_m\otimes1-1)\nabla_{\rm reg}\xi+B_m^N\xi,\\
 w_m\xi\longrightarrow\xi\text{ in }
       \|\cdot\|+\|S_N\cdot\|+\|\nabla_{\rm reg}\cdot\|
 \quad\Longleftrightarrow\quad
       \|B_m^N\xi\|\longrightarrow0
       \qquad(\xi\in\mathcal D_{\cap}).
 \end{gathered}
 \tag{ENC.23}
\]

The first two graph components converge by ENC.22. The first term in the normal component converges to zero because the contractions \(w_m\otimes1\) tend strongly to one on elementary covector tensors and hence on their completion. This proves both implications in ENC.23. On every finite compact normal-core vector, the local floor of ENC.5 makes a sufficiently late cutoff identically one near its support, so \(B_m^N\xi=0\) eventually.

For example, an additional uniform relative estimate
\(\|B_m^N\xi\|\leq C(\|\xi\|+\|\nabla_{\rm reg}\xi\|)\) on the full normal domain would imply the limit for every such vector. Approximate it in the normal graph by finite-core vectors, use their eventual zero boundary term, and then use the displayed bound on the remainder. This estimate has **not** been proved for the present cutoff family. Individual bounded derivatives do not supply it. Thus ENC.22 proves the full operator graph approximation, and ENC.23 identifies the remaining simultaneous normal error; neither asserts a full joint graph core, normal-source stability or the physical sum.

### 11BH.11. A full joint core in global regular coordinates

Suppose the actual regular carrier in CVC.11 has a global description
\(M=A\widehat\otimes\ell^2(I)\), where \(A=C(X)\), \(X\) is compact, and
\(I\) is countable with the prescribed infinite parity multiplicities.
Require that the finite normal core is the finite-coordinate scalar core
and that its connection differentiates its scalar coefficients only.
These conditions hold for a global discrete action on a compact
transversal, with the corresponding compact probability base and the
constant regular arrow-and-copy coordinates. They are additional
conditions on the carrier; a general étale groupoid need not have these
global coordinates.

Enumerate its coordinates and choose \(0<\lambda_i\leq1\) tending to zero.
On both normalized copies choose the same diagonal seed:

\[
 H_*=\operatorname{diag}(h_0,h_0),\qquad
 h_0(e_i a)=\lambda_i e_i a,\qquad
 D_NH_*=0 .
 \tag{ENC.24}
\]

Finite coordinate truncations converge to \(h_0\) in operator norm, so it
is compact on the full \(A\)-module. Its range contains every finite
coordinate vector and is dense. Each cutoff \(e_k=\chi_k(H_*)\) has finite
coordinate support, since its lower threshold is positive. It has zero
normal derivative. The common-tail convex choices of ENC.8, including
the adaptive tests after each already selected cutoff, therefore retain
all source, phase, projection and compact-arrow requirements while
retaining zero normal derivative. The functional plateau and nesting
requirements are unchanged.

Write \(w_m=v_m|_M\) for these choices. Commutation on the finite scalar
core extends to the full closed vector connection by graph
approximation. More explicitly, if \(\eta_j\to\xi\) and
\(\nabla_{\rm reg}\eta_j\to\nabla_{\rm reg}\xi\), then
\(w_m\eta_j\to w_m\xi\) and their normal derivatives converge to
\((w_m\otimes1)\nabla_{\rm reg}\xi\). Closedness gives

\[
 \begin{gathered}
 w_m\mathcal D_N\subset\mathcal D_N,\qquad
 \nabla_{\rm reg}w_m\xi=(w_m\otimes1)\nabla_{\rm reg}\xi,\\
 w_m\xi\longrightarrow\xi\quad\hbox{in }
 \|\cdot\|+\|S_N\cdot\|+\|\nabla_{\rm reg}\cdot\|
 \quad(\xi\in\operatorname{Dom}S_N\cap\mathcal D_N).
 \end{gathered}
 \tag{ENC.25}
\]

The operator component is exactly ENC.22. The normal component follows
from the displayed identity and strong convergence of the contractions
\(w_m\otimes1\) on the completed covector module. In particular the
boundary term of ENC.23 is zero, even when the invariant phase and its
compact correction are normal-rough.

There is also an actual joint *core*, rather than just a joint
approximation by reference-domain vectors. The full finite normal core
\(\mathscr C_{\rm reg}\) is contained in the reference operator domain:
the reference weight of each fixed coordinate is finite, and the
remaining regulator and compression terms are bounded. For fixed \(m\),
\(S_Nw_m\) extends to a bounded adjointable operator on all of \(M\).
Indeed \(r_*w_m\) is bounded, and the compression formula ENC.12 gives
the assertion by multiplying its remaining bounded terms by \(w_m\).
Approximate any joint-domain \(\xi\) in the normal graph by
\(\eta_j\in\mathscr C_{\rm reg}\). For fixed \(m\),

\[
 \begin{gathered}
 w_m\eta_j\longrightarrow w_m\xi,\qquad
 S_Nw_m\eta_j\longrightarrow S_Nw_m\xi,\qquad
 \nabla_{\rm reg}w_m\eta_j\longrightarrow\nabla_{\rm reg}w_m\xi,\\
 w_m\eta_j\in\mathscr C_{\rm reg},\qquad
 \overline{\mathscr C_{\rm reg}}^{\,\|\cdot\|+\|S_N\cdot\|
          +\|\nabla_{\rm reg}\cdot\|}
       =\operatorname{Dom}S_N\cap\mathcal D_N .
 \end{gathered}
 \tag{ENC.26}
\]

The first and third limits use boundedness and ENC.25; the second uses
the just proved bounded operator \(S_Nw_m\). First choose \(m\) by
ENC.25 and then \(j\) by these three limits. This proves the last
equality on the *entire* essential domain, including any enlargement
relative to the reference domain. It does not differentiate \(F\) or
\(B_F\). The finite-core images remain finite-coordinate scalar-core
vectors because the cutoffs are constant diagonal matrices with finite
support.

Thus the actual source-essential inverse has a full joint
operator/normal core in this global compact-coordinate scope. Its
source and full arrow graph controls, class \(d_X\), and original
anchors are those of ENC.15–ENC.18. Normal preservation by the represented
source, mixed estimates for the physical sum, its actual original-unit
balance and its Bott comparison are still separate. ENC.24 has not
been constructed on every general holonomy carrier.

### 11BH.12. Smooth compact seeds can have a persistent boundary term

The seed choice matters even on a trivial groupoid with a compact
one-dimensional unit space. Let \(A=C(S^1)\), with its standard metric and parameter
\(t\in\mathbb R/(2\pi\mathbb Z)\). Use one even root coordinate and
countably many paired even/odd coordinates. Its normal connection is
the closure of componentwise differentiation on finite smooth
coordinates. Let \(f\) be zero on the root and the constant odd swap on
each pair. With central nondegenerate \(A\)-source it has compact square
defect at the root and represents \(1_A\).

Put \(\zeta(s)=e^{-1/s}\) for \(s>0\), and zero for \(s\leq0\), and set
\(\chi(u)=\zeta(u-1)/(\zeta(u-1)+\zeta(2-u))\).
The denominator is positive at every real \(u\). This is a smooth
nondecreasing cutoff, zero for \(u\leq1\), one for \(u\geq2\), with
\(\chi(3/2)=1/2\) and \(\chi'(3/2)=2\). Use the root seed value one and,
on both coordinates of pair \(n\geq1\), the value

\[
 h_n(t)=2^{-2n^2-2}\left(\frac32+\frac14\sin(nt)\right),
 \qquad \chi_k(s)=\chi(s/a_k),\quad a_k=2^{-2k-2}.
 \tag{ENC.27}
\]

This positive seed is compact, has dense range and has compact
norm-continuous normal derivatives of every order. For any derivative
order \(j\), its tail is bounded by a constant times
\(2^{-2n^2-2}n^j\), which tends to zero and is summable.
Differentiated diagonal truncations therefore converge in norm.

Choose \(N_m=2^{2^m}\), \(m\geq1\), and \(v_m=\chi_{N_m^2}(H_*)\).
On pair \(n<N_m\) it is one; on pair \(n>N_m\) it is zero. On pair
\(N_m\) it equals \(\chi(3/2+\sin(N_mt)/4)\). It is one on the root.
These observations follow respectively from the ratios being at least
\(5\), at most \(7/16\), and equal to the displayed middle value.
They prove finite support, strict convergence and
\(v_mv_{m+1}=v_m\).
All source, phase and projection commutators vanish; there are no
nonidentity arrow tests. The adaptive commutator tests vanish as well.
Each individual cutoff preserves the full normal domain.

The weight \(r_*=1+\sum_m(1-v_m)\) is a positive diagonal regular
operator. Each coordinate sum is finite. Its inverse is compact:
outside the first \(N_m\) pairs its weight is at least \(m+1\).
The full range domain and onto resolvents follow by coordinate
functional calculus. Since it commutes with the normalized constant
phase, symmetrization changes nothing. Essential compression is the
closed odd operator \(S=f r_*\), zero on the root and a weighted swap on
each pair. Its bounded transform differs compactly from \(f\), so its
class is still \(1_A\). In particular,

\[
 m\leq r_{N_m}(t)\leq m+1,\qquad
 r_{N_m}(0)=m+\frac12,\qquad
 \left.\frac{d}{dt}v_m\right|_{(N_m,\,t=0)}=\frac{N_m}{2}.
 \tag{ENC.28}
\]

Define a constant section \(\xi\) whose even coordinate at pair \(N_m\)
is \(1/N_m\), with all other coordinates zero. Its finite truncations
converge in the normal graph and its normal derivative is zero.
It belongs to the full \(S\)-domain because
\(\sum_m(m+1)^2/N_m^2<\infty\), uniformly bounding the squared tail of
\(S\xi\). Yet

\[
 \|\xi\|+\|\nabla_{\rm reg}\xi\|
     =\left(\sum_{m\geq1}N_m^{-2}\right)^{1/2}<\infty,\qquad
 \|(D_Nv_m)\xi\|\geq\frac12\quad(m\geq1).
 \tag{ENC.29}
\]

The second inequality is the single \(N_m\) coordinate at \(t=0\);
the module norm is the supremum of the fibre \(\ell^2\)-norm.
Consequently these cutoffs approximate the full operator graph but
fail to approximate this vector in the joint graph. Their normal
derivative norms are at least \(N_m/2\), although every derivative of
the compact seed is compact. The asserted uniform relative estimate
cannot follow from those seed properties: apply it to a constant unit
section in pair \(N_m\).

This example does not obstruct existence of a joint core. Constant
finite-coordinate projections supply one here, as in ENC.24–ENC.26.
It isolates a failed cutoff choice while preserving a positive
unit-index source cycle, its compact resolvents and all trivial
arrow/source controls. Neither that scalar unit-index example nor the
global-coordinate core proves the physical class for arbitrary
holonomy.


![Actual source-essential inverse with a dense normal-compatible full operator core](../figures/essential-normal-core.png)

**Figure 11BH.1.** ENC.1–ENC.29. The actual bounded inverse and its nondegenerate B-source are absorbed onto the canonical regular arrow carrier. Diagonal normal-smooth cutoffs produce a reference operator graph core inside the completed normal domain. Averaging the regulator with its invariant phase gives exact phase commutation through a bounded compact perturbation. Essential compression retains a full operator graph core inside the normal domain, the actual class d_X, source-local compact resolvents, unbounded source commutators and full arrow graph domains. The joint operator/normal intersection is dense, complete and invariant. Adaptive weighted self-commutator tests additionally make each normal-preserving cutoff preserve and approximate the entire essential operator graph; simultaneous normal convergence is equivalent to the explicit boundary term in ENC.23 tending to zero. Five rational symmetrization/compression/arrow samples and sixteen source products check the finite formulas. Constant normal seed coordinates supply a full joint core in the global compact-coordinate scope, ENC.24–ENC.26. The oscillatory circle seed in ENC.27–ENC.29 has compact derivatives of every order but a normal cutoff error at least 1/2 on one joint-domain vector. The full general joint core, normal-source and mixed estimates, physical sum and original Bott pairing remain. Human context: [Connes, A survey of foliations and operator algebras](https://alainconnes.org/wp-content/uploads/foliationsfine.pdf#page=29), Section 8. [Vector figure](../figures/essential-normal-core.svg) · [Generator](../reproduction/essential-normal-core/draw_essential_normal_core.py) · [Exact checks](../reproduction/essential-normal-core/ESSENTIAL-NORMAL-CORE-CHECKS.json) · [Reproduction](../reproduction/essential-normal-core/REPRODUCTION.md) · [Component terms](../reproduction/essential-normal-core/COMPONENT-TERMS.md).

## 11BI. The physical normal class on compact isometric matrix-group suspensions

The proper scalar factorization can be made regular before any normal variable
is introduced. This gives a completed physical operator for every suspension
in the following family, including actions with fixed points. It uses the
original inverse normal coefficient and the original transverse counting
correspondence.

### 11BI.1. A regular index-one auxiliary, with its full domain

Let \(\Gamma\) be a finitely generated characteristic-zero linear group, or
a finitely generated abstract subgroup of a compact Lie group. Give it the
discrete topology. Section 11Y constructs an even proper factorization
\(\gamma=\eta D\) with \(\operatorname{For}_\Gamma\gamma=1_{\mathbb C}\).

**Lemma 11BI.1.** There is a graded Hilbert space with exactly the regular
action
\[
 H_K^\pm=\ell^2(\Gamma)\otimes\ell^2(\mathbb N),\qquad
 V_g^\pm=\lambda_g\otimes1,
 \qquad \operatorname{ind}D_K^+=1,
 \tag{MGS.1}
\]
and an odd self-adjoint \(D_K\) with compact scalar resolvent. Every \(V_g\)
preserves its full domain, \([D_K,V_g]\) is bounded, and there is a graded,
countably generated, group-invariant graph core.

**Proof.** Use the actual product module of the proper factorization.
The standard-module multiplication and balanced inner-product calculations
PF.1–PF.4 apply to the discrete group with unit space a point: the product
module is \(K_X\otimes_{C(X),\,\pi_D\Phi^*}E_D\), and its second-factor
representation of the proper algebra is nondegenerate and covariant.
In particular its proper central \(C_0(M)\)-action is nondegenerate.
These assertions concern the underlying module; they impose no commutation
of the product operator with that additional centre.

Apply the square-cutoff isometry RG.4–RG.7 to this module. Its finite,
centrally localized rows have compact group support, its adjoint is the
displayed cutoff row sum, and the cutoff identity makes \(V^*V=1\).
Apply ordinary stabilization to the source coefficient of each regular row,
as in RG.3. This is genuinely group equivariant because left translation
does not change that coefficient. Add the two graded degenerate regular
swaps and use the onto copy shift RG.10–RG.12. The result is an actual cycle
\((H_K,V,F_K)\) on the space and action in (MGS.1), representing \(\gamma\).
Its ordinary class, and hence its scalar index, is \(+1\).

Integrating the displayed regular group action gives a unital representation
of \(C_r^*\Gamma\). The compact group defects of this scalar equivariant
cycle are compact commutators for finite group polynomials; polynomial
density extends them to all of \(C_r^*\Gamma\). Thus it is an actual
reduced scalar cycle, not just a representation with a reduced norm bound.

Now follow the construction in Theorem 11L.1 with this very representation
and all \(\lambda_g\) as the countable tests. Its normalization adds a
second summand with zero source action. The source projection is therefore
the projection onto the original \(H_K\). The bounded off-diagonal
removal UP.11 and essential compression UP.12 return to that same Hilbert
space and retain its exact \(V_g=\lambda_g\otimes1\). Compact resolvents,
the class and scalar index survive. Corollary 11L.2 gives the full domain
and invariant core:
\[
 V_g\operatorname{Dom}D_K=\operatorname{Dom}D_K,\qquad
 [D_K,V_g]\in\mathcal B(H_K),\qquad
 \mathcal C_K=\operatorname{span}\{V_gP_\pm(D_K-i)^{-1}x_j\}.
 \tag{MGS.2}
\]
Here \((x_j)\) is norm dense. The resolvent identifies the Hilbert norm
with the graph norm, and the two grading projections are graph bounded;
thus this is a core. It is invariant because it includes every \(g\).
The scalar index is preserved by the zero-source normalization and bounded
perturbation, as proved in UP.17–UP.18. No common bound over all \(g\),
invariant Fredholm phase, or equivariant identity \(\gamma=1_\Gamma\)
has been used. \(\square\)

### 11BI.2. The actual suspension and every holonomy label

Let \(L\) be a connected closed smooth manifold, \(\widehat L\to L\) a
connected regular cover with deck group \(\Gamma\) as above, and \(N\) a
connected closed Riemannian \(q\)-manifold. Suppose \(\Gamma\) acts
faithfully on \(N\) by smooth isometries, preserving a specified coherent
full metric spin-c structure. Put
\[
 V=(\widehat L\times N)/\Gamma,\qquad
 F=\text{the images of }T\widehat L,\qquad
 T=\{[z_0,n]:n\in N\},
 \tag{MGS.3}
\]
where \(z_0\) is one lift of a base point. The diagonal quotient is smooth
and compact: the deck action is free and proper, and evenly covered
base charts identify it with a bundle with compact fibre \(N\).
It carries the stated smooth foliation and compact complete transversal.

**Lemma 11BI.2.** Its actual holonomy groupoid and its source column are
\[
 G=(\widehat L\times\widehat L\times N)/\Gamma,\qquad
 G|_T=\Gamma\ltimes N,\qquad
 G|_{sT}\cong\widehat L\times N.
 \tag{MGS.4}
\]
These statements retain stabilizers of points of \(N\).

**Proof.** An isometry of connected \(N\) with identity germ at a point
is identity everywhere. Indeed it fixes that point and has identity
derivative; the identity
\(f(\exp_x v)=\exp_{f(x)}df_xv\) first proves identity in a normal ball.
The set of points where its value and derivative are identity is closed
by continuity and open by the same exponential argument. Connectedness
makes it all of \(N\). Faithfulness consequently makes different group
labels different germs, even when their values agree at one point.

A leaf path lifts to a path in \(\widehat L\) at constant normal
coordinate. Changing the chosen lift changes both endpoints and that
coordinate by the same deck transformation. The transverse germ is the
corresponding global isometry. All paths between two fixed lifted
endpoints have the same transverse germ; conversely a further endpoint
identification would force the intervening deck isometry to have identity
germ and therefore be the identity. This proves the first quotient in
(MGS.4), with range \([y,n]\) and source \([z,n]\).

The return labelled \(g\), from \(n\) to \(gn\), is
\([g^{-1}z_0,z_0,n]\). Every deck label occurs because the cover is
connected. The resulting label/germ identification is smooth on the
return sheets and identifies their étale topologies. Finally
\((y,n)\mapsto[y,z_0,n]\) is onto the source column: move the source
lift to \(z_0\) by its unique deck transformation. It is one-to-one
because the deck action fixes \(z_0\) only for the identity.
Its inverse in lift charts is smooth. This also proves the source-column
topology, rather than only a fibrewise bijection. \(\square\)

### 11BI.3. Construction on the full normal coefficient

**Theorem 11BI.3.** The suspension in (MGS.3) has a scalar degree-\(q\)
reduced class \(z_G\) represented by a self-adjoint operator on completed
actual holonomy columns with countable even auxiliary multiplicity.
Its transverse principal symbol is the specified full inverse normal
Clifford symbol. Smooth compact holonomy convolution preserves its full
domain and has bounded commutators. Both localized scalar resolvents and
all scalar Kasparov defects are compact. Its restriction through the
actual transversal correspondence is the full normal inverse, and its
original positive local transverse Bott pairing is \(+1\).

Choose compatible Hermitian connections on the prescribed inverse module
of Section 11E.9 and define its formally symmetric normal Dirac \(D_N\).
Partitions of unity give such connections without an invariance premise.
The smooth-core/maximal-domain argument SC.9, the elliptic estimate
SC.12 with no vertical directions, and the finite-chart Fourier-tail
compactness proof SC.13 give
\[
 H_N=L^2(N,\mathcal D_N),\qquad
 \operatorname{Dom}D_N=H^1(N,\mathcal D_N),\qquad
 (D_N\pm i)^{-1}\in\mathcal K(H_N).
 \tag{MGS.5}
\]
The smooth sections form a core. Here the elliptic estimate controls every
coordinate derivative because the full normal symbol squares to the
metric norm. Its maximal-domain extension follows from the same
bounded mollifier commutator SC.9. This proves the actual \(H^1\) domain;
no domain on an incomplete chart is closed separately.

Keep, from Theorem 7.17 and Section 11E.9, all of
\[
 \Pi^{q(q-1)/2}C_q,\qquad
 c_{\rm phys}(e_j)=(-1)^{q(q-1)/2+1}L_{t_j},\qquad
 [s_g,z_g]\longmapsto z_g^{-1}\rho_t(s_g).
 \tag{MGS.6}
\]
Thus the inverse determinant is the dual of the given determinant, and
the full phase is \(z_g^{-1}\), not just its square. The right \(C_q\)
factor is last. The supplied complete local duality calculation of
Section 11E.9 gives \(b_U[D_N]=1\) for the original positive local Bott
class. This uses that full inverse calculation, not a new choice of
minimal spinor.

Write \(U_g\) for the specified inverse action on \(H_N\). It preserves
smooth sections and \(H^1\). Both \(D_N\) and \(U_gD_NU_g^{-1}\)
have the same transported first-order symbol by isometry and the
coherent Clifford lift. Their difference is therefore a smooth
zeroth-order endomorphism; it is bounded on compact \(N\). Hence
\([D_N,U_g]\) is bounded for every fixed \(g\), even if the connection
is not invariant.

Use Lemma 11BI.1 and put the even auxiliary first:
\[
 \begin{split}
 H_B&=H_K\widehat\otimes H_N,\qquad B=C(N)\rtimes_r\Gamma,\\
 D_B&=D_K\otimes1+\gamma_K\otimes D_N,\\
 \Pi_B(fu_g)&=V_g\otimes M_fU_g,\\
 \operatorname{Dom}D_B
   &=\operatorname{Dom}(D_K\otimes1)
                   \cap\operatorname{Dom}(1\otimes D_N).
 \end{split}
 \tag{MGS.7}
\]
Proposition 11K.3 proves self-adjoint closure on
\(\mathcal C_K\odot C^\infty(N,\mathcal D_N)\), this exact intersection
domain, compact scalar resolvent and bounded domain-preserving
commutators with every smooth finite group polynomial. Its proof uses
the exact square sum, finite squared spectral windows, and the actual
connection and positivity equations RA.13–RA.16. In particular it
computes the whole ordinary restriction, not only its index:
\[
 [H_B,\Pi_B,F_B]|_{C(N)}=[D_N],\qquad b_U[H_B,\Pi_B,F_B]=1.
 \tag{MGS.8}
\]
The scalar factor has even index \(+1\), so it changes neither the degree
nor any full inverse transition.

Here \(\Pi_B\) is genuinely regular and faithful. Apply the absorption
unitary \(W(\delta_t\otimes u)=\delta_t\otimes U_t^{-1}u\).
It changes the diagonal group action to translation of \(t\); the
coefficient at \(t\) is \(U_t^{-1}M_fU_t=M_{f\circ t}\).
In source coordinates this is the defining regular representation at
each \(n\), with range normal point \(tn\). Integration against normal
volume has full support. For a finite group polynomial, the norm at
\(n\) is lower semicontinuous: apply it to a finite group-support vector,
whose image norm is continuous, and take the supremum of these vectors.
Every strict lower bound thus persists on an open set of positive volume.
Its essential supremum is its supremum. Copy and nonzero finite spin
multiplicities do not change it, so
\[
 \|\Pi_B(b)\|=\|b\|_r\quad(b\in B).
 \tag{MGS.9}
\]
Polynomial density proves the equality on all \(B\).

### 11BI.4. Physical graph induction and the exact closed domain

Choose finitely many buffered flat base charts \(U_i\), lifts \(z_i\),
and real smooth \(\chi_i\) with \(\sum_i\chi_i^2=1\).
The lift transition \(h_{ji}\), defined by \(z_j=h_{ji}z_i\), is locally
constant and has finite image on the compact support of
\(\chi_i\chi_j\). Form the flat right-\(B\) bundle with transitions
left multiplication by \(u_{h_{ji}}\), and let \(E\) be its leaf-\(L^2\)
module, with the actual base density. Set \(K=\bigoplus_iL^2(U_i)\).
The buffered chart map and its adjoint give
\[
 E=P(K\otimes B),\qquad
 A=C_r^*(G)=\mathcal K_B(E),\qquad
 H_G=E\otimes_B H_B=P(K\otimes H_B).
 \tag{MGS.10}
\]
This is the actual groupoid algebra and source column. Indeed a smooth
kernel \([y,z,n]\) becomes its finite chart \(B\)-valued kernel.
Convolution and adjoint agree by integration in the middle plaque
variable and the return covariance. Separated leaf kernels approximate
each compact chart kernel in both Schur norms; they are the finite-rank
module kernels. Tensoring \(E\) with the regular representation at \(n\)
gives exactly \(L^2(G_n)\) in lift coordinates. Right translation from
an arbitrary unit to \(T\) is an onto unitary of regular columns, so
the supremum over \(T\) is the full reduced norm. These observations
prove the onto isometric algebra equality in (MGS.10), as in SG.6–SG.9b;
point stabilizers were not removed in any step.

There is also an explicit physical coordinate identification. In chart
\(i\), an auxiliary regular label \(t\) and normal chart point \(n_i\)
correspond to
\[
 y=t^{-1}z_i(x),\qquad n=t^{-1}n_i .
 \tag{MGS.11}
\]
The inverse spin transport identifies the chart coefficient at \(n_i\)
with the full coefficient at \(n\). On overlap, the chart transition is
\((t,n_i)\mapsto(h_{ji}t,h_{ji}n_i)\), so both \(y\) and \(n\) in
(MGS.11) are unchanged. The leaf density and isometric normal volume
are preserved; summing the squared chart weights gives norm one.
Conversely partitioning an actual column and using (MGS.11) yields
those weighted chart components. Thus this is an onto unitary onto
the completed \(L^2(\widehat L\times N)\) columns, with the full
source-normal coefficient and the stated even copy multiplicities.
It is not a renaming of a merely weakly regular representation.

Put \(D_0=1_K\otimes D_B\). Each entry of \(P\) is a buffered overlap
coordinate/density identification, multiplied by
\(\chi_i\chi_j\) and the finite transition \(V_h\otimes U_h\).
The leaf weights have zero normal derivative. Equations (MGS.2) and
the proved bounded normal transport error therefore show that
\(P\operatorname{Dom}D_0\subset\operatorname{Dom}D_0\) and
\([D_0,P]\) is bounded. Let
\[
 R=P D_0(1-P)+(1-P)D_0P,\qquad
 D_G=(D_0-R)|_{H_G},\qquad
 \operatorname{Dom}D_G=P\operatorname{Dom}D_0 .
 \tag{MGS.12}
\]
The two off-diagonal terms extend boundedly, and \(R\) is self-adjoint.
For \(|\operatorname{Im}z|>\|R\|\), its Neumann resolvent proves that
\(D_0-R\) is self-adjoint on the unchanged full domain. \(P\) reduces
it, proving the stated exact domain and self-adjointness of \(D_G\).
Projection of the algebraic factor core with smooth compact leaf
profiles is graph dense: \(P\) is graph bounded, and that unprojected
core is graph dense by tensor spectral truncation and smooth leaf
approximation. Thus this also supplies a physical graph core.

The ordinary expression has a normal differential part and a
discrete auxiliary part. Its normal symbol, in the specified chart
and full coefficient order, is
\[
 \sigma_\perp(D_G)(\xi)=\gamma_K\otimes\sigma(D_N)(\xi),
 \qquad \sigma_\perp(D_G)(\xi)^2=|\xi|^2 .
 \tag{MGS.13}
\]
The overlap covariance of the Clifford symbol proves this formula
on the actual projected columns. The inverse scalar phase, determinant,
grading and right \(C_q\) in (MGS.6) are retained. There is no assertion
that the auxiliary, which acts on the displayed return/copy completion,
is a finite-order differential operator in every smooth graph coordinate.

For completeness, the scalar compactness and product do not use compactness
of the free global resolvent. Use
\(\mathcal J=\mathcal K(K)\otimes\mathcal B(H_B)\).
Every \(j(D_0-z)^{-1}\), \(j\in\mathcal J\), is compact: for an
elementary tensor it is a compact leaf factor tensored with the scalar
compact normal/auxiliary resolvent, and norm approximation proves the
assertion. \(P,R,[D_0,P]\) are finite sums of bounded leaf operators
and bounded normal/auxiliary operators, hence multiply \(\mathcal J\)
on both sides. The resolvent identity consequently gives compact
\(j(D_0-R-z)^{-1}\), on both sides by adjointness.

A smooth compact holonomy kernel has finitely many return labels in the
finite lift charts. Its normal derivative and its auxiliary commutator
are bounded Schur kernels, by (MGS.2), the full inverse normal product
rule, and the bounded errors \([D_N,U_g]\).
Both it and that commutator belong to \(\mathcal J\).
One can verify domain preservation first for separated kernels and pass
to the actual kernel in operator-plus-commutator norm: after buffered
partitions, smooth periodic Fourier approximation in the compact chart
variables converges with the first normal derivatives. The two Schur
bounds convert this \(C^1\) convergence to precisely that graph bound.
Closedness passes the commutator formula to the full domain in (MGS.12).
Compressing gives the same statements on \(H_G\).
The two imaginary resolvent identities and their integrable
\(O((1+t^2)^{-1})\) bound, as in SG.15 and RA.15, therefore prove compact
phase commutators. The square defect is a localized resolvent product.
Density proves every scalar Kasparov defect for all \(a\in A\).

For each smooth finite-chart module profile \(\eta\), its creation map
has bounded error \(D_GT_\eta-T_\eta D_B\): expand its finitely many
normal coefficients and return labels and use \([D_0,P]\).
The rectangular resolvent identity makes its bounded-transform creation
error compact, with integrable bound
\(\|D_GT_\eta-T_\eta D_B\|/(1+t^2)\).
Dense module profiles give the connection equations for every profile.
The Morita operator is zero, so positivity is zero. This proves the
actual class
\[
 z_G=[{}_AE_B,0]\otimes_B[H_B,\Pi_B,F_B],\qquad
 (i|_{C(N)})^*z_G=[D_N],\qquad (i_*b_U)z_G=1 .
 \tag{MGS.14}
\]
Here \(i(b)=\theta_{\eta b,\eta}\) for a real normalized plaque bump
\(\langle\eta,\eta\rangle=1_B\).
The map \(\overline\zeta\mapsto\theta_{\eta,\zeta}\) is the onto
unitary \(E^*\to p_\eta A\): its inner product is
\(\theta_{\zeta,\zeta'}\), its right action agrees by adjointing the
second slot, and every \(p_\eta a\) is
\(\theta_{\eta,a^*\eta}\). On finite return columns it is the
original transversal counting correspondence, since both its counting
norm and the plaque integral of \(|\eta|^2\) are one. Density extends
that very identification. Thus the last equality is the original
source-oriented transverse Bott pairing, not a substituted corner
class chosen only to have a numerical index. This proves the theorem.
\(\square\)

### 11BI.5. Fixed poles in an irrational sphere suspension

Take \(L=S^1\), \(\widehat L=\mathbb R\), \(\Gamma=\mathbb Z\), and
\(N=S^2\) with its round metric. Let the generator rotate by angle
\(2\pi\alpha\), \(\alpha\notin\mathbb Q\). The action is faithful.
Its two poles are fixed by every group element, but no nonzero power
has identity germ there: its tangent derivative rotates by
\(2\pi k\alpha\). Thus each pole has the full return isotropy
\(\mathbb Z\), while its holonomy cover is \(\mathbb R\).

Use the full spin-c structure induced by the oriented tangent complex
line. Its structural homomorphism is
\(e^{i\theta}\mapsto[e^{\theta e_1e_2/2},e^{i\theta/2}]\);
adding \(2\pi\) changes both entries by minus one, so this is well
defined. The rotations act on its tangent frames and therefore give
the coherent lift. Its determinant is the tangent complex line;
the inverse uses its dual, the original parity
\(\Pi^{2(2-1)/2}=\Pi\), and the full inverse phase in (MGS.6).
An invariant compatible round connection can be used in this example.

Here a completely explicit auxiliary avoids the abstract construction.
Use \(H_K^\pm=\ell^2(\mathbb Z)\) and \(V e_n^\pm=e_{n+1}^\pm\).
Define
\[
 D_K^+e_n^+=
 \begin{cases}
  -n e_n^- & n<0,\\
  0 & n=0,\\
  n e_{n-1}^- & n>0,
 \end{cases}
 \qquad D_K^-=(D_K^+)^* .
 \tag{MGS.15}
\]
The paired block weights are \(|n|\) on the negative half and \(n\)
on the strictly positive half; only \(e_0^+\) is unpaired.
Orthogonal block resolvents prove self-adjointness on the weighted
square-summability domain, finite-coordinate core and compact resolvent.
The inverse of \(D_K^+\) on its range sends \(e_m^-\) to
\((-m)^{-1}e_m^+\) for \(m<0\), and
\((m+1)^{-1}e_{m+1}^+\) for \(m\ge0\).
It is bounded into the graph domain; the range is all \(H_K^-\).
Thus the kernel is exactly the positive root and the index is \(+1\).

In \([D_K^+,V]\), inputs \(e_{-1}^+,e_0^+\) have outputs
\(-e_0^-,e_0^-\). All other inputs have mutually orthogonal
unit outputs. In \([D_K^-,V]\), the input \(e_{-1}^-\)
has output \(e_1^+-e_0^+\); all other outputs are orthogonal
unit vectors outside those two coordinates. Consequently
\[
 \|[D_K,V]\|=\sqrt2,\qquad
 \|[D_K,V^k]\|\le\sqrt2\,|k|,\qquad
 \ker D_K^+=\mathbb Ce_0^+,\quad \ker D_K^-=0.
 \tag{MGS.16}
\]
The weighted domain is invariant under each shift, and telescoping
proves the bound for positive and negative \(k\).
Theorem 11BI.3 now gives the physical degree-two scalar graph
cycle and original Bott pairing one for this compact three-manifold
foliation. Fixed poles have not become unit-groupoid points.

![The actual source column, fixed-point germs and regular index-one auxiliary](../figures/matrix-suspension-dirac.png)

**Figure 11BI.1.** MGS.1–MGS.16. The top row shows the actual holonomy
quotient and its source column. The sphere panel records the derivative
of a nonzero return at a fixed pole; the different labelled germs remain.
The auxiliary panel gives the exact pairing in (MGS.15), the unmatched
positive root, block weights and commutator norm. The bottom row retains
the entire original inverse coefficient and gives the checked normal
symbol, full domain and actual source-oriented Bott pairing.
The sphere is a schematic of the rotation action, not a spectrum plot.
[Vector figure](../figures/matrix-suspension-dirac.svg) ·
[Generator](../reproduction/matrix-suspension-dirac/draw_matrix_suspension_dirac.py) ·
[Exact checks](../reproduction/matrix-suspension-dirac/MATRIX-SUSPENSION-CHECKS.json) ·
[Reproduction](../reproduction/matrix-suspension-dirac/REPRODUCTION.md).
Historical graph-Dirac question: Connes, *A survey of foliations and operator
algebras*, Section 8, pp.28–29. The proof and diagram here are original CC0
expression.

This theorem completes the physical construction on the displayed family,
with no separate scalar-existence, regular-coordinate, normal-domain,
graph-induction or Bott-normalization premise. An arbitrary eligible
holonomy groupoid need not be one global compact suspension.
Its locally labelled models still require a compatible physical
construction over overlaps. The general target is retained.

## 11BJ. A boundary-controlled joint core after ordinary stabilization

A compact probability fibre does not require a compact transversal. We construct a source-essential inverse with a full joint operator/normal graph core over the actual proper probability base of a finite good transversal. The carrier is standardized ordinarily over \(A\), and its genuine groupoid action is transported with it. The new connection has a complete joint core. Its covariance under that transported action is a separate requirement, as the explicit example below shows.

### 11BJ.1. The actual source and the choice of connection

Take a finite good complete transversal \(T=\bigsqcup_{\ell=1}^L T_\ell\), with coordinate boxes \(T_\ell=(0,1)^q\) whose closures lie inside larger transverse charts. Its metric and coordinate norms are uniformly equivalent on those closures. Keep the actual proper compact-fibre probability anchor \(a:X\to T\), \(A=C_0(X)\), \(H=G\ltimes X\), source algebra \(B\) and inverse class \(d_X\) of PK.3–PK.6 and PJ.1–PJ.6. The normal derivative is the original polynomial-observable sheaf derivative, with the fixed lifted-probability paths of PJ.3–PJ.4. No product trivialization of \(X\) is assumed.

Start with the exactly invariant, source-essential bounded inverse on the canonical regular carrier from EAI and CVC. This graded carrier has infinitely many unit-arrow coordinates in each parity, a complemented ordinary copy of the graded standard \(A\)-module. Its remaining ordinary summand is countably generated. The proved Kasparov stabilization theorem therefore gives an even ordinary onto \(A\)-linear unitary to

\[
 \mathcal M=\mathcal H_A^+\oplus\mathcal H_A^-,
 \qquad \mathcal H_A^\pm=\ell^2(\mathbb N)\otimes A,
 \qquad
 U_h=V_{r(h)}L_hV_{s(h)}^* .
 \tag{OJC.1}
\]

Indeed the carrier is \(\mathcal H_A\oplus N\), and stabilization of \(N\) identifies this with \(\mathcal H_A\). Transport its entire source \(\Phi\), bounded invariant contraction \(f\), and grading by the same \(V\). This preserves the actual class \(d_X\), source nondegeneracy, original anchor and genuine continuous \(H\)-action. The cocycle in OJC.1 follows by cancellation of \(V_{s(h)}^*V_{s(h)}\). No equivariance or normal derivative of \(V\) is assumed.

On this carrier choose the finite-coordinate scalar normal core \(\mathscr C\), with coefficients in the original compactly localized polynomial-observable sheaf, and close its single vector-valued coordinate connection. Denote the result by \(\nabla_0\) with domain \(\mathcal D_0\). The scalar PJ.6 closability argument, coordinate inner-product identity, and finite-coordinate approximation prove density, closability, metricity and the completed Leibniz identity exactly as in CVC.11–CVC.13. Constant coordinate and parity projections commute with \(\nabla_0\). Pulling this connection back by \(V\) defines a connection on the original module, with precisely the pulled-back domain; this does not differentiate \(V\). It does not identify that connection with the canonical regular one.

### 11BJ.2. A vector estimate at the coordinate boundary

For \(x\in T_\ell\), let \(d(x)=\min_j\{x_j,1-x_j\}\). For \(\xi\in\mathcal D_0\) put

\[
 b_\xi(t)=\sup_{\mu\in X,\ d(a(\mu))\leq t}
                      \|\nabla_0\xi(\mu)\|,
 \qquad b_\xi(t)\longrightarrow0\quad(t\downarrow0).
 \tag{OJC.2}
\]

The limit follows because \(\nabla_0\xi\) is a section vanishing at infinity. The anchor image of a compact subset of \(X\) is compact inside the finite open boxes, hence has positive boundary distance. Outside a compact set the norm is uniformly small.

There is a constant \(C_T\), independent of \(\xi,\mu\), such that

\[
 \|\xi(\mu)\|\leq C_Td(a(\mu))\,b_\xi(d(a(\mu))),
                 \qquad \xi\in\mathcal D_0 .
 \tag{OJC.3}
\]

First let \(\xi\in\mathscr C\). Move its unit coordinate along the straight shortest segment to a closest face; this segment has length \(d(x)\) and stays in the same boundary strip. PJ.3–PJ.4 lift each initial probability to fixed compact potential/frame coordinates, carry a fixed lifted measure along this segment, and realize the coefficient functions in one fixed field. Finite standard coordinates give one fixed Hilbert space for the resulting vector-valued functions. Their derivative is the lifted \(\nabla_0\xi\) evaluated on that coordinate direction, whose norm is bounded by \(C_Tb_\xi(d(x))\). The coefficient section is zero sufficiently near the face, because its unit support is compact inside the box. Stop at such an interior point before reaching the face. The segment actually traversed is compact inside the box, and its probability preimage is compact by properness of the anchor. A finite collection of the PJ.3 local resolutions covers it; their horizontal frame transport and fixed potential coordinates provide the successive short lifts. At a subdivision endpoint choose a resolution of the probability already reached, so the lifted scalar coordinate vector remains continuous. Only finitely many short intervals are needed on this compact segment. Their vector derivatives obey the same estimate. The total length is at most the distance to the face, and integration up to the zero endpoint proves OJC.3. No extension of the probability space across the boundary is required. The norm and derivative norm of each lift are exactly the corresponding section norms, so the estimate is independent of the choice of lifted measure.

Approximate a full \(\mathcal D_0\)-vector by \(\mathscr C\) in the single vector graph. For each fixed \(\mu\) the vector norms converge, and the boundary-strip derivative suprema differ by at most the global derivative error. Passing to the limit proves OJC.3 on the full domain. The argument estimates the whole Hilbert-valued coordinate vector along one path; it does not sum separate pointwise bounds at different paths.

Choose a smooth \(\theta:\mathbb R\to[0,1]\), zero on \((-\infty,1]\) and one on \([2,\infty)\), and set \(\varepsilon_k=4^{-k}\),

\[
 \chi_k(x)=\prod_{j=1}^q
     \theta(x_j/\varepsilon_k)
     \theta((1-x_j)/\varepsilon_k),\qquad
 E_k=\chi_k(a(\mu))\,Q_k ,
 \qquad E_kE_l=E_k\quad(l>k),
 \tag{OJC.4}
\]

where \(Q_k\) projects onto the first \(k\) coordinates in each parity. Use the same choices on both normalized copies below. In normal rank zero take \(\chi_k=1\); the finite unit space has no boundary term. Properness of \(a\) makes \(\chi_k\circ a\in A\), so \(E_k\) is a compact even positive contraction. The product identity holds because a nonzero earlier factor has every face distance greater than \(\varepsilon_k\), whereas every later factor is already one there. These operators form a compact approximate identity: scalar unit cutoffs converge strictly in \(A\), and coordinate projections converge strongly on the standard module and its compact ideal.

The derivative of \(\chi_k\) is supported in \(d(x)\leq2\varepsilon_k\), has norm at most \(C_\theta/\varepsilon_k\), and vanishes when \(d(x)\leq\varepsilon_k\). The finite atlas bounds make \(C_\theta\) uniform. OJC.3 consequently gives

\[
 \|(\delta_NE_k)\xi\|
       \leq 2C_TC_\theta b_\xi(2\varepsilon_k)\longrightarrow0,
 \qquad
 \|(\delta_NE_k)\xi\|\leq C_0\|\nabla_0\xi\|,
 \quad C_0=2C_TC_\theta .
 \tag{OJC.5}
\]

Each \(E_k\) preserves the full normal domain by the completed product rule. The constant \(Q_k\) is a contraction, so the estimate uses the whole-vector bound OJC.3. Thus \(E_k\xi\to\xi\) in the normal graph for every \(\xi\in\mathcal D_0\). Their normal graph operator norms are bounded by \(1+C_0\). Every finite compact normal-core vector is fixed by sufficiently late \(E_k\), on a neighborhood of its support.

### 11BJ.3. Convex source and arrow tests preserve the boundary control

Normalize \(f\) exactly as in IUR.1, on
\(\widetilde{\mathcal M}=\mathcal M\oplus\mathcal M^{\rm op}\), with source \(\widetilde\Phi=\operatorname{diag}(\Phi,0)\), invariant odd unitary phase \(F\), and first-copy projection \(P\). The coordinate connection is copied and \(P\) is parallel. Apply the common-tail convex selection KG.2 to the copied approximate identity OJC.4, using precisely ENC.8's phase/source commutators, weighted compact tests, compact-arrow exhaustions and adaptive tests for each already chosen \(q_m=[F,v_m]\). It gives finite convex combinations \(v_n\) of increasingly late \(E_k\), with

\[
\begin{gathered}
 v_nv_{n+1}=v_n,\quad v_n\longrightarrow1\ {\rm strictly},\\
 \|[v_n,F]\|,\ \|[v_n,\widetilde\Phi(b_j)]\|\leq2^{-n}
                       \quad(j\leq n),\\
 \|q_m(1-v_n)\|\leq2^{-n-m}\quad(m<n),\\
 \|(\delta_Nv_n)\xi\|\longrightarrow0,\qquad
 \|(\delta_Nv_n)\xi\|\leq C_0\|\nabla_0\xi\|
                         \quad(\xi\in\mathcal D_0).
\end{gathered}
 \tag{OJC.6}
\]

All the additional compact source tails and the compact-arrow uniform defects have the bounds of ENC.8. KG.2's argument applies to the genuine transported action OJC.1 on \(\mathcal K_A(\widetilde{\mathcal M})\); it does not require a normal derivative of that action. Each selection uses only finitely many known tests. Taking all indices beyond the preceding finite support retains the product identity in OJC.4.

The last two statements survive every such selection. Their constants are uniform before selection, and convexity preserves the estimate. If all indices in a convex combination are at least \(k\), its derivative applied to \(\xi\) is bounded by \(C_0b_\xi(2\varepsilon_k)\). This tends to zero even though the individual operator norms \(\|\delta_Nv_n\|\) may grow.

Define

\[
 r_*=1+\sum_{n\geq1}(1-v_n),\qquad
 l_*=r_*^{-1}\in\mathcal K_A(\widetilde{\mathcal M}),\qquad
 r=\tfrac12(r_*+Fr_*F)=r_*+B_F,\quad
 B_F=\tfrac12F[r_*,F]\in\mathcal K_A .
 \tag{OJC.7}
\]

Here the scalar coordinate sum is locally finite on every fixed finite-coordinate compact unit set. Late plateaus are exactly one there. Its reciprocal is positive with dense range; finite compact coordinate vectors have bounded reference weight. Also
\(\|l_*(1-v_N)\|\leq1/N\): if \(v_N<1\), every preceding plateau \(v_n\), \(n<N\), is zero by their product identities. Since \(l_*v_N\) is compact, the bound proves compactness of \(l_*\). The full positive self-adjoint regular reference domain is \(\operatorname{Ran}l_*\), by the explicit reciprocal resolvents and graph parameterization of ENC.9. The norm-convergent cutoff commutator series preserves that entire domain by ENC.11's partial-sum argument. Thus \(B_F\) is bounded self-adjoint and \(r\) has the same full domain.

Essentially compress \(D=Fr\) as in ENC.12–ENC.16:

\[
 S=\overline{PDP},\qquad
 [\mathcal M,\Phi,F_S]=d_X,\qquad
 \overline{\Phi(B)\mathcal M}=\mathcal M,\qquad
 \Phi(b)(S\pm i)^{-1}\in\mathcal K_A(\mathcal M).
 \tag{OJC.8}
\]

All hypotheses of those proofs are present: \(P\) commutes with the reference cutoffs; \([r,P]\) is bounded compact; the source commutators and the source-local weighted phase/projection defects have summable tails. The localized resolvent/deficiency argument proves odd self-adjoint regularity on the actual closed essential domain, which may exceed the reference domain. The source-local bounded-transform comparison removes the zero-source copy and retains the entire represented inverse. The dense tested source algebra preserves the full \(S\)-domain and has compact unbounded graded commutators. The compact-arrow tests, exact invariance of \(F,P\), and reference graph approximation prove the full transported arrow graph-domain equality and continuous compact displacement errors, as in ENC.17. None of these arguments differentiates \(F\), \(V\) or \(U_h\).

### 11BJ.4. The full source-essential joint core

Put \(w_m=v_m|_{\mathcal M}\) and \(\mathscr K=\operatorname{span}\{w_m\eta:\eta\in\mathscr C,\ m\geq1\}\). The adaptive calculation ENC.21–ENC.22 applies to OJC.7: \([S,w_m]\) is compact with norm tending to zero, \(w_m\) preserves the full \(S\)-domain, and \(Sw_m\) extends boundedly to the entire module. The last assertion follows from boundedness of \(r_*v_m\) and the compressed formula on the reference domain. Normal graph preservation and OJC.6 simultaneously give

\[
\begin{gathered}
 w_m\xi\longrightarrow\xi,\quad
 Sw_m\xi\longrightarrow S\xi,\quad
 \nabla_0w_m\xi\longrightarrow\nabla_0\xi
          \quad(\xi\in\operatorname{Dom}S\cap\mathcal D_0),\\
 \overline{\mathscr K}^{\,\|\cdot\|+\|S\cdot\|+\|\nabla_0\cdot\|}
       =\operatorname{Dom}S\cap\mathcal D_0 .
\end{gathered}
 \tag{OJC.9}
\]

For the last equality, approximate \(\xi\) in the normal graph by \(\eta_j\in\mathscr C\). At each fixed \(m\), uniform normal graph boundedness makes \(w_m\eta_j\to w_m\xi\) in that graph. Boundedness of \(Sw_m\) makes the operator component converge from ordinary module norm convergence. Choose one \(j\) at each \(m\) so that the combined error is at most \(1/m\), and use the three limits. The intersection is complete because both operators are closed, and dense because \(\mathscr K\) is already an operator core, by ENC.14. This proves the full joint core on the source-essential represented inverse over the actual noncompact proper probability base.

### 11BJ.5. Ordinary standardization does not prove normal covariance

There is an explicit example on actual eligible holonomy. Let \(T=\mathbb R/2\pi\mathbb Z\), with irrational rotation by \(\alpha=2\pi(\sqrt5-1)/2\), and \(G=\mathbb Z\ltimes T\). On the regular standard carrier with arrow coordinate \(m\in\mathbb Z\) and copy coordinate \(n\geq1\), let \(L_k\) be regular translation. The ordinary \(C(T)\)-linear unitary
\(V(x)e_{m,n}=e^{in\sin x}e_{m,n}\) is adjointable: finite coordinates are smooth, and the bounded unitary tails extend continuity and the adjoint identity to every section. Transport gives

\[
 U_k(x)=V(x)L_kV(x-k\alpha)^*,\qquad
 U_1(x)e_{m,n}
    =e^{in\psi(x)}e_{m+1,n},\quad
 \psi(x)=\sin x-\sin(x-\alpha).
 \tag{OJC.10}
\]

The cancellations prove the genuine cocycle and all original arrow labels remain. The constant vector \(\xi=\sum_{n\geq1}n^{-1}e_{0,n}\) belongs to the full coordinate normal domain, by finite-coordinate graph approximation with derivative zero. But \(U_1\xi\) is not norm differentiable at zero. For every fixed \(N\), the squared norm of its difference quotient is at least the sum of the first \(N\) coordinate squares, whose limit is \(N|\psi'(0)|^2\). Since \(\psi'(0)=1-\cos\alpha>0\), these lower bounds are unbounded as \(N\to\infty\). A vector in the closed coordinate derivative domain is norm differentiable by the integrated core derivative identity, so \(U_1\xi\) is outside that domain.

Consequently genuine arrow continuity and ordinary \(A\)-linear standardization do not imply preservation of the chosen normal domain. The full inverse spin-c coefficient can be retained as a separate finite factor throughout this example. OJC.1–OJC.9 supply a full joint core for one actual source-essential inverse; they do not transfer it to CVC's covariant regular connection or prove normal-source stability. A normal-compatible carrier comparison, the mixed physical estimates, the original \(C_0(T)\)-balanced graph passage and original Bott \(+1\) comparison remain required. The completed physical suspension construction in Section 11BI remains a separate fully proved case.

![Boundary estimate, source-essential joint approximation and the actual transported-arrow domain test](../figures/ordinary-joint-core.png)

**Figure 11BJ.1.** OJC.1–OJC.10. The boundary panel shows one straight unit-coordinate path and the full vector estimate, with actual lifted probability coordinates kept fixed. The approximation panel retains the source-essential class \(d_X\), its full operator domain and all three graph limits. The last panel uses the genuine irrational-rotation arrow and its transported copy phase; finite samples of the first-coordinate sums illustrate the divergent difference-quotient bound, whose infinite proof is OJC.10. The connection covariance and physical Bott comparison are not inferred. [Vector figure](../figures/ordinary-joint-core.svg) · [Generator](../reproduction/ordinary-joint-core/draw_ordinary_joint_core.py) · [Checks](../reproduction/ordinary-joint-core/ORDINARY-JOINT-CORE-CHECKS.json) · [Reproduction](../reproduction/ordinary-joint-core/REPRODUCTION.md).

## 11BK. A small inverse commutator closes a normal sum

A normal sum can be closed using the derivative of a bounded inverse, rather than a derivative of its unbounded mass. We prove the full domain and the adjoint-domain argument. The canonical normalized carrier of Section 11BG supplies the small inverse and its weighted derivative limit. Application to the original physical graph still requires a self-adjoint normal operator on that graph, the source action and the original balanced identification. These are separate hypotheses; the conclusion below does not supply them.

### 11BK.1. Exact data and the two resolvent signs

Let \(\mathscr H\) be a complex Hilbert space. Let \(N=N^*\) be a self-adjoint normal operator and \(J=J^*=J^{-1}\) a bounded operator with

\[
 J\operatorname{Dom}N=\operatorname{Dom}N,\qquad NJ=-JN.
 \tag{SNC.1}
\]

In a graded coefficient construction this is the actual copy phase and the normal grading factor: the connection commutes with \(J\), whereas the physical operator \(N=\Gamma c\nabla\) anticommutes with it. Confusing these two identities gives the wrong resolvent sign.

Let \(H=H^*\geq0\) be bounded, injective and of dense range, commuting with \(J\). Suppose \(H\) preserves \(\operatorname{Dom}N\) and its ordinary commutator extends to a bounded operator

\[
 K=[N,H],\qquad a=\|H\|,\quad \kappa=\|K\|,
 \quad a+\kappa<\sqrt{1-\kappa}.
 \tag{SNC.2}
\]

In particular \(\kappa<1\). The adjoint identity on \(\operatorname{Dom}N\), followed by bounded extension, gives \(K^*=-K\); SNC.1 and \([J,H]=0\) give \(KJ=-JK\). Thus \(JK\) is self-adjoint.

Put \(A=H^{-1}\) with full domain \(\operatorname{Ran}H\), and \(T=JA\) on that domain. The spectral inverse of a bounded positive injective operator is self-adjoint on precisely its range; its resolvents, including the dense range assertion, follow by scalar calculus on \(H\). Since \(J\) commutes with \(H\), diagonalizing \(J\) proves self-adjointness of \(T\). For real \(\lambda\ne0\),

\[
 B_\lambda=J+i\lambda H,\quad R_\lambda=(T+i\lambda)^{-1}
       =H B_\lambda^{-1},\quad
 \|B_\lambda^{-1}\|\leq1,\quad
 \|B_\lambda^{-1}H\|\leq|\lambda|^{-1}.
 \tag{SNC.3}
\]

Assume a dense normal graph core \(\mathscr C\subset\operatorname{Dom}N\cap\operatorname{Dom}T\), whose vectors have \(N\xi\in\operatorname{Dom}T\) and \(T\xi\in\operatorname{Dom}N\). Require the same mixed-domain property for \(R_\mu\mathscr C\), for one fixed real \(\mu\ne0\). Finally require the norm limit

\[
 \beta_\lambda=\|B_{-\lambda}^{-1}K\|\longrightarrow0
           \quad(\lambda\longrightarrow+\infty).
 \tag{SNC.4}
\]

These are inspectable domain and norm hypotheses. If \(K\) is Hilbert-space compact, the limit follows directly from finite-rank approximation and strong convergence of \(B_{-\lambda}^{-1}\) to zero. Hilbert-module compactness after spatial passage is not automatically Hilbert-space compactness. Section 11BK.5 gives the appropriate coefficient-norm proof instead.

### 11BK.2. The full joint graph core

The inverses in SNC.3 preserve \(\operatorname{Dom}N\). Indeed,
\(B_\lambda^{-1}=(J-i\lambda H)(1+\lambda^2H^2)^{-1}\).
For the positive inverse, expand a geometric series about a scalar larger than \(1+\lambda^2a^2\). Its differentiated series converges: the derivative of a product of \(m\) contractions of norm at most \(r<1\) is bounded by \(mr^{m-1}\) times the one-factor commutator. Closedness of \(N\) gives preservation and the inverse commutator identity. The other factors preserve the domain by SNC.1–SNC.2.

On this full domain, direct multiplication gives

\[
 N R_\lambda=-R_{-\lambda}N+C_\lambda,\qquad
 C_\lambda=J B_{-\lambda}^{-1}K B_\lambda^{-1}.
 \tag{SNC.5}
\]

To check the sign, \(NB_\lambda=-B_{-\lambda}N+i\lambda K\), hence
\(NB_\lambda^{-1}=-B_{-\lambda}^{-1}N+i\lambda B_{-\lambda}^{-1}KB_\lambda^{-1}\).
Use \(1+i\lambda H B_{-\lambda}^{-1}=JB_{-\lambda}^{-1}\) when differentiating \(H B_\lambda^{-1}\). This proves SNC.5 in its stated order; neither \(K\) nor \(H\) is moved through the other inverse.

Let \(u_\lambda=i\lambda R_\lambda\). Both \(u_\lambda\) and \(u_{-\lambda}\) tend strongly to one by the spectral calculus of \(T\). For \(\xi\in\operatorname{Dom}T\), write \(\xi=H\eta\), with \(\|\eta\|=\|T\xi\|\). SNC.3–SNC.5 give

\[
 N u_\lambda\xi=u_{-\lambda}N\xi+i\lambda C_\lambda\xi,
 \qquad \|i\lambda C_\lambda H\|\leq\beta_\lambda,
 \qquad \|i\lambda C_\lambda\xi\|
       \leq\beta_\lambda\|T\xi\|.
 \tag{SNC.6}
\]

Consequently \(u_\lambda\xi\to\xi\), \(Tu_\lambda\xi\to T\xi\) and \(Nu_\lambda\xi\to N\xi\) for every vector of the full joint domain. This uses the weighted limit SNC.4. No convergence of the raw norms \(\lambda\|C_\lambda\|\) is inferred.

The family \(R_\mu\mathscr C\) is a core for the entire joint domain. For a joint vector, put
\(\eta_\lambda=(T+i\mu)u_\lambda\xi\).
The resolvent identity gives
\(\eta_\lambda=i\lambda\xi+i\lambda i(\mu-\lambda)R_\lambda\xi\in\operatorname{Dom}N\).
Approximate this vector in the normal graph by vectors of \(\mathscr C\). At fixed \(\mu\), SNC.5 bounds \(R_\mu\) on that graph, while \(TR_\mu=1-i\mu R_\mu\) is bounded. Apply \(R_\mu\), and then choose the approximation error after \(\lambda\). This two-stage sequence approximates all three graph components. Closedness of \(T\) and \(N\) makes their joint graph complete.

### 11BK.3. A quadratic mixed estimate

On the mixed-domain core, the inverse identity is
\([N,A]=-AKA\). Using SNC.1,

\[
 NT+TN=A J K A,\qquad
 \|(N+T)\xi\|^2
 =\|N\xi\|^2+\|T\xi\|^2+
       \langle A\xi,JK A\xi\rangle.
 \tag{SNC.7}
\]

Here the last term is real because \(JK\) is self-adjoint. Its absolute value is at most \(\kappa\|A\xi\|^2=\kappa\|T\xi\|^2\). The joint core proved above passes this identity and its inequalities to every full joint-domain vector. Thus

\[
 \|N\xi\|^2+(1-\kappa)\|T\xi\|^2
 \leq\|(N+T)\xi\|^2
 \leq\|N\xi\|^2+(1+\kappa)\|T\xi\|^2.
 \tag{SNC.8}
\]

Define \(D=N+T\) on precisely \(\operatorname{Dom}N\cap\operatorname{Dom}T\). SNC.8 makes its graph norm equivalent to the complete joint graph norm. It is therefore closed and symmetric, and \(R_\mu\mathscr C\) is a graph core for it. This argument proves closedness; self-adjointness still needs an adjoint-domain proof.

### 11BK.4. The adjoint-domain proof

The bounded inverse \(H\) preserves \(\operatorname{Dom}D\), and
\([D,H]=K\) there, because \(TH=HT=J\). If \(\xi\in\operatorname{Dom}D^*\) and \(y=D^*\xi\), the adjoint identity gives
\(H\xi\in\operatorname{Dom}D^*\) and
\(D^*H\xi=Hy+K\xi\).
Also \(H\xi\in\operatorname{Dom}T\) and \(TH\xi=J\xi\).
Testing against the normal graph core \(\mathscr C\subset\operatorname{Dom}D\), the functional for \(N\) is consequently represented by
\(Hy+K\xi-J\xi\).
Since \(\mathscr C\) is an \(N\)-core and \(N=N^*\), this proves \(H\xi\in\operatorname{Dom}N\). We have obtained the full-domain identity

\[
 H\operatorname{Dom}D^*\subset\operatorname{Dom}D,\qquad
 DH\xi=H D^*\xi+K\xi.
 \tag{SNC.9}
\]

Now take either deficiency equation \(D^*\xi=\pm i\xi\). Apply SNC.8 to \(H\xi\). Its lower bound is \(\sqrt{1-\kappa}\|\xi\|\), since \(\|TH\xi\|=\|\xi\|\). SNC.9 gives the upper bound \((a+\kappa)\|\xi\|\). SNC.2 forces \(\xi=0\). Both deficiency spaces vanish, hence the closed symmetric Hilbert-space operator \(D\) is self-adjoint. This last implication is the usual onto-resolvent criterion: \(D\pm i\) have closed range by symmetry, their orthogonal complements are the opposite deficiency spaces, so both ranges are the whole space.

In addition SNC.8 and \(\|T\xi\|\geq a^{-1}\|\xi\|\) give
\(\|D\xi\|\geq a^{-1}\sqrt{1-\kappa}\|\xi\|\).
Self-adjointness makes its range both closed and dense, so it is onto and

\[
 \|D^{-1}\|\leq\frac{a}{\sqrt{1-\kappa}},\quad
 \|TD^{-1}\|\leq(1-\kappa)^{-1/2},\quad
 \|ND^{-1}\|\leq1.
 \tag{SNC.10}
\]

These conclusions are Hilbert-space assertions. The absence of Hilbert-space deficiency vectors by itself is not a proof of regularity for an arbitrary Hilbert-module sum.

### 11BK.5. What the canonical normalized inverse supplies

Keep the actual \(B\)-source inverse, canonical regular carrier, phase \(J\), holonomy-parallel connection and annihilated source summand of PNC.1–PNC.23. Choose the generating vectors in PNC.6 as single-copy, compact-bisection normal-smooth bumps, as in ENC.4–ENC.5. The resulting seed is diagonal, has a positive floor on each compact finite-coordinate support, and retains the entire normalized inverse class. Functional calculus on each such floor supplies the mixed normal core above. This enlarges the scalar core by its smooth functional calculus without changing its closure or its connection; it does not use the ordinary trivialization of Section 11BJ.

Suppose that a verified physical realization carries this normal core to a core of a self-adjoint \(N=\Gamma c\nabla\), retains \(NJ=-JN\), and carries the coefficient product identity to \([N,H]=\Gamma c(\delta H)\). Clifford contraction has a fixed finite norm \(C_c\). Scale the seed so that
\(\|H\|,\|\delta H\|\leq[16(1+C_c)]^{-1}\).
Then \(a+\kappa\leq1/16\), \(\kappa\leq1/16\), and SNC.2 holds strictly. Positive scaling does not change the dense range or the inverse class established in PNC.18.

The coefficient derivative is module compact by PNC.6. The strong convergence of \(B_{-\lambda}^{-1}\) on the coefficient module, finite-rank approximation and uniform contraction bound prove
\(\|B_{-\lambda}^{-1}\Gamma c(\delta H)\|\to0\)
in that module's operator norm. Any compatible nondegenerate spatial realization is contractive on these bounded products, so the same bound gives SNC.4 physically. It does not claim that the spatial image of a module-compact operator is Hilbert-space compact. The mixed core and all required inverse identities are coefficient identities before realization.

Thus, once that actual self-adjoint normal realization and those core identifications are proved, SNC.1–SNC.10 give the full normal sum and its domain on the canonical normalized carrier. They bypass a derivative of \(H^{-1}\) as a bounded operator. They do not identify the source-essential compressed connection of Section 11BH with this one, or establish preservation by \(\Phi(b)\). Spatial source-local compactness, the actual holonomy cycle, the original \(C_0(T)\)-balance and the full inverse normal coefficient with its ordered Clifford factors still need their own proofs. FD.7's integrated-multiplicity obstruction remains relevant. No general physical graph class or original Bott \(+1\) pairing is certified by this analytic lemma alone.

### 11BK.6. A self-adjoint sum need not preserve the source domain

An infinite model exhibits the remaining source issue. On \(L^2(S^1)\otimes\ell^2(\mathbb N;\mathbb C^2)\), let \(J\) swap each pair, \(\Gamma=\operatorname{diag}(1,-1)\), \(N=\Gamma(-i\partial_x)\), and \(H\) be \((16n)^{-1}\) on pair \(n\geq1\). Then \(a=1/16\), \(K=0\), and finite Fourier/pair vectors are the required core. The completed sum has Fourier blocks
\(\left(\begin{smallmatrix}k&16n\\16n&-k\end{smallmatrix}\right)\)
and eigenvalues \(\pm\sqrt{k^2+256n^2}\). Its exact full domain is the square-summability of \((k^2+256n^2)\|u_{n,k}\|^2\). Only finitely many eigenvalues lie in each bounded interval, so the resolvent is compact.

Define \(\alpha(x)=\sqrt{|\sin(x/2)|}\), continuously and periodically on \([-\pi,\pi]\). Represent \(B=C(S^1)\) by \(b(e^{i\alpha(x)})\) on the first even coordinate only, with zero action on every other coordinate. This is an even star representation with the normalization's annihilated complement. On the coefficient module over \(C(S^1)\), its graded commutators with \(J\) are finite rank, and the phase of \(T=JH^{-1}\) differs compactly from \(J\). Its represented even source class has multiplicity \(+1\); the opposite copy has zero source. This is a model of a legitimate normalized source cycle, not an identification with the original \(d_X\).

For the smooth input function \(b(z)=z\) and the constant first even vector \(\xi\), \(\Phi(b)\xi=e^{i\alpha(x)}\xi\) is not in the sum domain. Indeed away from zero
\(|\alpha'(x)|^2=\cos^2(x/2)/(16|\sin(x/2)|)\sim1/(8|x|)\).
Its weak derivative is locally integrable but not square integrable. Continuity and integration by parts away from zero identify it as the distributional derivative. Thus \(e^{i\alpha}\notin H^1\), whereas \(\xi\) belongs to the full sum domain. Even compact resolvents, the correct normalized source multiplicity and the self-adjoint sum do not give the source-domain or product conclusion. The source in this example is explicitly the model circle algebra.

![Small inverse commutator, full joint domain and adjoint deficiency estimate](../figures/small-inverse-normal-sum.png)

**Figure 11BK.1.** SNC.1–SNC.10. The mixed term is controlled by \(\kappa\|T\xi\|^2\); the inverse sends every adjoint-domain vector into the full joint domain. The strict lower/upper deficiency bounds force both deficiency spaces to vanish. The circle Fourier blocks illustrate the exact infinite-domain proof and a remaining represented-source obstruction. Rational finite block and gap checks illustrate arithmetic only. The original graph realization, source action, spatial compactness and Bott comparison remain distinct obligations. Human context for that target: [Connes, A survey of foliations and operator algebras](https://alainconnes.org/wp-content/uploads/foliationsfine.pdf#page=29), Section 8. [Vector figure](../figures/small-inverse-normal-sum.svg) · [Generator](../reproduction/small-inverse-normal-sum/draw_small_inverse_normal_sum.py) · [Exact checks](../reproduction/small-inverse-normal-sum/SMALL-INVERSE-NORMAL-SUM-CHECKS.json) · [Reproduction](../reproduction/small-inverse-normal-sum/REPRODUCTION.md) · [Component terms](../reproduction/labelled-geometric-kernel/REPRODUCTION.md).

## 11BL. Flat-label operator families and the transverse disk comparison

A local original-unit operator can retain the compact arrow errors that an
infinite regular isotropy representation requires. Its ordinary family class
still depends on its transition functions. The following construction proves
the operator domains, computes that class for finite label groups, and proves
the complete comparison on a disk inside one flat chart. The disk comparison
does not require a global trivialization of the family.

### 11BL.1. A normally controlled operator on the actual unit charts

Use the precise input LF.1–LF.2 in Section 11AG.
Thus \(Z_a\) has smooth chart domains \(O_i\), locally constant labels
\(t_{ij}:O_i\cap O_j\to\Lambda\), and actual arrow labels \(c_{ij}(g)\).
They satisfy \(t_{ij}t_{jk}=t_{ik}\) and
\(c_{ij}(g)c_{jk}(h)=c_{ik}(gh)\).
The smooth locally finite partition \((\phi_i)\) has
\(\operatorname{supp}\phi_i\subset O_i\), with that support closed in \(Z_a\).

Let \(H=H^+\oplus H^-\) have the graded regular amplified action \(V\) of
\(\Lambda\). Let \(D_\Lambda\) be odd and self-adjoint, with compact scalar
resolvents, ordinary index \(+1\), and common full domain \(\mathcal D\), such that

\[
 V_\lambda\mathcal D=\mathcal D,\qquad
 [D_\Lambda,V_\lambda]\in\mathcal B(H)\quad(\lambda\in\Lambda).
 \tag{FBC.1}
\]

The regular unbounded construction MGS.1–MGS.2 in Section 11BI supplies these
inputs in its stated scalar scope. Only these operator inputs are used here;
no equivariant identity for its scalar class is presumed.

Glue the Hilbert bundle \(E\) by \(\xi_i=V_{t_{ij}}\xi_j\). The actual arrows
act by \(V_{c_{ij}(g)}\), together with their unit maps. Its normal connection
is the flat connection in these charts. Define, in chart \(j\),

\[
 \begin{split}
 K_j(x)&=\sum_i\phi_i(x)
       \bigl(V_{t_{ji}(x)}D_\Lambda V_{t_{ij}(x)}-D_\Lambda\bigr),\\
 D_j(x)&=D_\Lambda+K_j(x),\qquad \operatorname{Dom}D_j(x)=\mathcal D.
 \end{split}
 \tag{FBC.2}
\]

**Theorem 11BL.1.** Formula FBC.2 gives an odd self-adjoint regular operator
on \(\Gamma_0(Z_a,E)\), with nondegenerate central \(C_0(Z_a)\)-source,
source-local compact resolvents and smooth compact resolvents in the normal
directions. Every actual arrow preserves the full transported graph domain.
Its bounded transform has continuous compact arrow errors. Its unitwise
ordinary index is \(+1\).

**Proof.** Every summand in \(K_j\) is bounded, odd and self-adjoint by
FBC.1. A compact subset of \(O_j\) meets only finitely many active supports.
Each of its intersections with an active support is a compact subset of the
overlap where the label is defined, so only finitely many labels occur there.
All normal derivatives of \(K_j\) therefore exist in operator norm and are
bounded on that compact set: they differentiate only the smooth weights.
The support condition permits extension of an inactive term by zero.
On the common domain, FBC.2 can also be written
\(D_j=\sum_i\phi_iV_{t_{ji}}D_\Lambda V_{t_{ij}}\). Consequently

\[
 D_k=V_{t_{kj}}D_jV_{t_{jk}}.
 \tag{FBC.3}
\]

This is equality of operators and full domains, since every transition
preserves \(\mathcal D\). At fixed \(x\), bounded perturbation gives a closed
symmetric operator with the same graph domain. More explicitly, for
\(|b|>\|K_j(x)\|\),
\((D_j+ib)=(1+K_j(D_\Lambda+ib)^{-1})(D_\Lambda+ib)\)
has an onto inverse by the geometric series, for both signs. These onto
ranges prove self-adjointness. They also prove compactness of the resolvent;
the resolvent identity gives compactness at every nonreal parameter.
That identity gives, for a fixed normal vector field \(v\),

\[
 vR_{j,z}=-R_{j,z}(vK_j)R_{j,z},\qquad R_{j,z}=(D_j-z)^{-1}.
 \tag{FBC.4}
\]

Repeated differentiation gives norm-continuous compact values for every
fixed normal derivative word on a compact chart test.

The resolvents glue by FBC.3 and have uniform norm at most
\(|\operatorname{Im}z|^{-1}\). They are adjointable on the section module,
with adjoint the conjugate-parameter resolvent. Their range is precisely
the continuous sections \(\xi\) for which \(D\xi\) is continuous, with both
sections vanishing at infinity. Indeed the image satisfies
\(D R_z\xi=\xi+zR_z\xi\); conversely a section in that graph is
\(R_z(D-z)\xi\). The range is dense because compactly supported chart
sections with values in the dense domain \(\mathcal D\) are dense.
The two onto resolvents on that domain prove self-adjoint regularity.
A compactly supported continuous compact-operator field is a module
compact: approximate it uniformly on finitely many charts by constant
finite-rank operators, and use scalar supported partitions. This proves
source-local compactness of these resolvents and their derivatives.

For an arrow with source chart \(j\) and range chart \(i\), the error is
\([D_\Lambda,V_{c_{ij}(g)}]+K_i(rg)V_{c_{ij}(g)}-V_{c_{ij}(g)}K_j(sg)\).
It is bounded and continuous, uniformly on compact arrow tests. The
inverse arrow proves equality of the transported full domains. For two
self-adjoint operators with bounded difference \(B\), the difference of
their bounded transforms is the norm-convergent difference integral
obtained from
\(x(1+x^2)^{-1/2}=(2/\pi)\int_0^\infty x/(x^2+1+t^2)\,dt\).
Write the integrand as the half-sum of the two resolvents at
\(\pm i\sqrt{1+t^2}\). The resolvent identity bounds the difference
by \(\|B\|/(1+t^2)\). Compact resolvents make each difference compact;
the integrable bound proves compactness and local norm continuity of
the phase error. The same argument compares \(D_j\) with \(D_\Lambda\),
so their ordinary indices agree.

Finally, FBC.4 proves preservation of the full closed local normal graph.
For a smooth chart vector, \(R_z\xi\) is a smooth Hilbert-valued section,
and \(\nabla(R_z\xi)=R_z\nabla\xi+(\nabla R_z)\xi\). Such a section belongs
to the closed normal domain by finite-vector smooth approximation.
Apply the locally bounded derivative estimate and closedness to a
normal graph approximation of an arbitrary vector. Compact localization
gives the statement for the sheaf of normal domains. Smooth compact unit
functions times a dense \(D_\Lambda\)-graph core give a dense intersection
of the two domains. This proof establishes neither the domain equality
of separately ordered mixed products nor self-adjointness of a physical
sum with the transverse Dirac operator. \(\square\)

### 11BL.2. The whole-base index for finite labels

Let now \(\Lambda\) be finite and the base \(Y\) compact. Its transition
cocycle defines the principal finite covering \(P\to Y\). For a finite
dimensional representation \(W\), write \(P\times_\Lambda W\) for its
associated vector bundle, with the same transition convention as \(E\).

**Theorem 11BL.2.** Average the full operator domain by

\[
 D^{\mathrm{av}}=|\Lambda|^{-1}\sum_{\lambda\in\Lambda}
 V_\lambda D_\Lambda V_\lambda^{-1},\qquad
 W^\pm=\ker(D^{\mathrm{av}})\cap H^\pm.
 \tag{FBC.5}
\]

Then the ordinary central family class \(\epsilon\) of FBC.2 is

\[
 \epsilon=[P\times_\Lambda W^+]-[P\times_\Lambda W^-].
 \tag{FBC.6}
\]

Here a vector bundle denotes its central section-module class in
\(KK_Y(C(Y),C(Y))\). Its index, obtained by forgetting that anchor and
precomposing with the scalar unit \(\mathbb C\to C(Y)\), is the displayed
virtual vector bundle in \(K^0(Y)\). In particular an equivariant
index \([W^+]-[W^-]=[\mathbb C_{\mathrm{triv}}]\) suffices for the unit family.
The ordinary dimension difference \(1\) alone does not.

**Proof.** FBC.1 makes FBC.5 a bounded self-adjoint perturbation of
\(D_\Lambda\) on its exact domain. It is invariant on that domain by
reindexing the finite sum. It has compact resolvents by the previous proof.
Its two kernels are finite dimensional invariant subspaces. The operator
\(D^{\mathrm{av}}\) therefore defines a constant, compatible operator in
every chart. Interpolate \(D_j\) to it linearly. The interpolation has the
same full domains, glues by FBC.3, and has compact resolvents and continuous
phase defects by the preceding estimates, uniformly in the compact
homotopy parameter. It is a genuine section-module homotopy.

On the orthogonal complement of its kernel, the compact-resolvent spectrum
has a positive gap away from zero. Replace the bounded transform there by
its sign: their difference is compact since the scalar difference tends to
zero along the unbounded spectrum. Interpolation is a compact perturbation
homotopy. The sign on that complement is an invariant odd involution,
commuting with the central source, so that summand is degenerate. The
remaining finite kernel has zero operator and grading \(W^+\oplus W^-\).
Its associated bundles give exactly FBC.6. Taking dimensions proves the
rank statement. Associating vector bundles respects direct sums and
equivariant isomorphisms, so an equality in the representation group gives
the stated sufficient unit criterion, including stable cancellation.
\(\square\)

### 11BL.3. A regular index-one family with a nontrivial index line

The distinction in Theorem 11BL.2 already occurs for
\(\Lambda=\mathbb Z/2=\{1,s\}\). In each grade use
\(\ell^2(\Lambda)\otimes\ell^2(\mathbb N_0)\). Put
\(e_{\pm,n}=2^{-1/2}(\delta_{1,n}\pm\delta_{s,n})\), so the action of
\(s\) is respectively \(+1\) and \(-1\). Define

\[
 \begin{split}
 D^+e_{+,n}&=(n+1)e_{+,n}^-,\\
 D^+e_{-,0}&=0,\qquad D^+e_{-,n}=n e_{-,n-1}^-\quad(n\ge1),
 \end{split}
 \tag{FBC.7}
\]

and take \(D^-=(D^+)^*\), with \(D\) the odd two-by-two matrix.
The full domain consists of the vectors whose squared weighted
coefficients sum to a finite number, with weights \(n+1\) on both
trivial-character grades, \(n\) on the positive sign grade, and \(n+1\)
on the negative sign grade. There is no added constraint on the single
zero-weight coefficient.

**Proof of the operator assertions.** There are self-adjoint two-dimensional
blocks of eigenvalues \(\pm(n+1)\) on the trivial-character pairs and
\(\pm n\), \(n\ge1\), on the sign pairs, plus the single positive zero
line \(\mathbb C e_{-,0}\). Testing an adjoint vector against every finite
block gives exactly the weighted domain just stated, proving
self-adjointness with that full domain. Inverting the blocks at \(\pm i\)
gives onto ranges; their norms tend to zero along the blocks, proving
compactness. The action preserves each block and the whole weighted
domain, and commutes exactly with \(D\). Thus every hypothesis FBC.1 holds,
the source is nondegenerate in both regular amplified grades, and the
ordinary index is \(+1\). Its equivariant index is the sign representation,
since its only kernel is that positive sign line.

Use the double cover \(S^3\to\mathbb{RP}^3\), \(u\mapsto[u]\). Its
associated sign line is

\[
 L=(S^3\times\mathbb C)/((u,z)\sim(-u,-z)).
 \tag{FBC.8}
\]

The local constructor has \(K_j=0\), since FBC.7 is exactly invariant.
Theorem 11BL.2 gives its whole-base index \([L]\), and every point still
has index \(+1\).

**Proof that the whole-base class is different.** A nowhere-zero section of
\(L\), after unit normalization and pullback, would give a continuous
\(f:S^3\to S^1\) with \(f(-u)=-f(u)\). The sphere \(S^3\) is simply
connected: uniformly approximate a loop by a finite piecewise geodesic
loop; short geodesics give a homotopy to that approximation. The finitely
many arcs avoid some point of \(S^3\). Stereographic projection then puts
the loop in \(\mathbb R^3\), where it contracts. Hence \(f\) lifts to a
continuous real argument \(\psi\). The continuous function
\(\psi(-u)-\psi(u)\) belongs to \(\pi+2\pi\mathbb Z\), and is constant
because \(S^3\) is connected. Replacing \(u\) by \(-u\) negates that
constant, forcing zero, a contradiction. Thus \(L\) is nontrivial.

If \([L]=[1]\) in \(K^0(\mathbb{RP}^3)\), the defining group completion
would give a finite vector bundle \(B\) with
\(L\oplus B\cong1\oplus B\). Taking top exterior powers gives
\(L\otimes\det B\cong\det B\); tensoring with its inverse trivializes
\(L\), contradicting the preceding proof. Therefore \([L]\ne[1]\).
The tensor square \(L^{\otimes2}\) is trivial by multiplication of its
two sign coordinates. There is also the stronger explicit stable identity
\(L\oplus L\cong1^2\): for \(u=(z_1,z_2)\in S^3\subset\mathbb C^2\),
the matrix
\(U(u)=\left(\begin{smallmatrix}z_1&-\overline z_2\\z_2&\overline z_1\end{smallmatrix}\right)\)
is unitary and \(U(-u)=-U(u)\). The map \([u,w]\mapsto([u],U(u)w)\)
is a well-defined unitary trivialization of the associated double sign
bundle. Thus \([L]-[1]\) is a nonzero additive order-two class.

This example can be given the exact abstract LF input type. Let the
original groupoid be the unit groupoid of \(Y=\mathbb{RP}^3\). Enlarge its
unit space to \(Y\sqcup S^3\), with anchor the identity on \(Y\) and the
double cover on \(S^3\), and take the pair groupoid of equal anchor values.
It is a compact Hausdorff étale groupoid. The open full subspace \(S^3\)
has the free deck-action groupoid; label its arrows by their deck element.
Local cover lifts are exactly the sections \(\beta_i\), and LF.2 gives
the double-cover transition cocycle. The finite-label compactness property
holds since the whole groupoid is compact. Its restriction to the original
\(Y\) is the unit groupoid. This is a counterexample to deriving a whole-base
unit from the local constructor's stated inputs. It does not assert that
the specifically constructed class \(\eta D\) in OU.6 has this sign index,
nor refute OU.6, nor identify this freely chosen enlargement with EG.8.

### 11BL.4. A complete disk comparison, with the supplied Bott degree

**Theorem 11BL.3.** Let \(U\) be a transverse coordinate disk with
\(\overline U\subset O_j\), and let \(R=C_0(Z_a)\), \(R_U=C_0(U)\).
For the ordinary central class \(\epsilon\) of Theorem 11BL.1,
\(\epsilon|_U=1_{R_U}\) in the anchored ordinary category. If
\(j:R_U\to R\) is extension by zero and
\(\beta_U\in KK^d(\mathbb C,R_U)\) is the specified full transverse
Bott class for this disk, with \(d=\dim U\), then \(b_U=\beta_U[j]\) satisfies

\[
 b_U\otimes_R\operatorname{For}(\epsilon)=b_U.
 \tag{FBC.9}
\]

**Proof.** The one flat chart trivializes \(E|_U\), not merely its individual
fibres. In that trivialization \(D_j=D_\Lambda+K_j\), and \(K_j\) is bounded
on \(\overline U\). The full-domain interpolation
\(D_\Lambda+tK_j\), \(0\le t\le1\), is a regular homotopy over
\(C_0(U\times[0,1])\). The resolvent and phase estimates in Theorem 11BL.1
prove its completed continuity and its source-local compact defects,
uniformly in the homotopy parameter. The constant endpoint has class
\(1_{R_U}\): split its finite positive and negative kernels from the
complement, replace its bounded transform on that complement by its sign
by compact perturbation, and discard the resulting central degenerate
swap. The remaining constant graded finite bundle has dimension difference
\(+1\). This proves an identity of the entire family over \(U\).

Centrality gives the ideal base-change identity
\([j]\otimes_R\operatorname{For}(\epsilon)
=\operatorname{For}(\epsilon|_U)\otimes_{R_U}[j]\).
To check the modules, tensor a supported section from \(R_U\) with
\(\Gamma_0(Z_a,E)\). Multiplication identifies its completed image with
the sections over \(U\), extended by zero. Supported bumps approximate
every such section; the inner product is unchanged. The resolvent and
phase are the restrictions, because the source is central. This is the
displayed identity at the actual completed-cycle level.
Associativity now gives FBC.9. The inserted family is even, so no odd
factor is moved, no degree changes, and no Clifford factor is discarded.
\(\square\)

For an actual original transverse disk of dimension q, this comparison has d=q. A disk in the frame-unit space has its own dimension; identifying its comparison with the original transverse class still requires the compact-frame passage.

The example FBC.8 therefore has a nonunit whole-base class and still the
exact local disk comparison FBC.9. This is the same weaker type of
normalization identified in AS.19–AS.21 and OU.7; it is proved here for
the local original-unit family itself. Application to the original physical
graph still requires compatibility between the closure models, the actual
compact-frame quotient, the entire inverse normal Spin-c coefficient,
the self-adjoint physical normal sum, its source domains and spatial compact
defects. A chart on the frame-unit space is not automatically a descended
disk on the original transversal. These statements supply no such descent
or global identification with the particular OU.6 class.

![Flat-label family index and disk calibration](../figures/whole-base-flat-label-class.png)

**Figure 11BL.** The double cover is represented by two points \(u,-u\)
over the same \([u]\); the arrow is its covering map, not a projection of
the drawn shapes. The unmatched positive sign line is the kernel of
FBC.7. FBC.8 and the antipodal argument prove its nontriviality; the displayed
unitary proves its additive order two. Restriction to one flat disk uses
the full homotopy of Theorem 11BL.3. The drawing is a schematic of these
exact objects and comparisons. [Editable SVG](../figures/whole-base-flat-label-class.svg)
and [reproducible arithmetic](../reproduction/whole-base-flat-label-class/REPRODUCTION.md)
accompany the proof. The historical graph-Dirac problem is posed by Connes,
*A survey of foliations and operator algebras*, Section 8; the arguments in
this section are original.

## 11BM. A complete weighted normal cycle on the original transversal

For this construction, additionally assume a positive holonomy-invariant symbol metric and coherent smooth even unitary arrow transports on the entire inverse normal Spin-c coefficient. The invariant metric is an extra hypothesis: the full ML_q^c lift alone does not imply it, as the actual fixed-germ calculation in Section 11CM demonstrates. Choose a compatible Hermitian Clifford connection; its invariance is unnecessary. Let \(T\) be the labelled countable complete transversal, a union of original \(q\)-dimensional open coordinate disks. Here complete means that it meets every leaf, not that its metric is complete. All manifolds in this construction are without boundary. No compactness, metric completeness or finite generation is assumed.

**Theorem 11BM.1 (the weighted maximal normal cycle).** There is an actual weighted self-adjoint normal Dirac on \(L^2(T,S_{\rm inv})\), with scalar localized compact resolvents and compact phase commutators for every actual compact holonomy kernel. It defines

\[
d_m\in KK^q(C^*_{\max}(G_T),\mathbb C).
\]

Its restriction to the original positive \(q\)-disk pairs with the original Bott generator as \(+1\), with the full inverse line, grading and final right Clifford block retained. Its representation is maximal-continuous. A reduced physical graph cycle still requires the actual transfer and graph passage; this theorem does not claim that the maximal geometric representation is reduced-continuous.

**Proof.** The following domain, compactness, phase and local-comparison arguments supply every assertion.

### 11BM.1. The actual finite-coefficient normal expression

Let N be the formally symmetric odd Dirac expression on S for the chosen compatible connection and density dvol_g. It is elliptic in **all q directions of T**. Set

\[
 C(df)=[N,M_f],\qquad C(df)^*=-C(df),\qquad
 \|C(df)_x\|=|df_x|_g
\]

for real f, with the physical Clifford convention already fixed in S. A harmless fixed finite Clifford contraction constant can be included if that convention is normalized differently. The expression is right-C_q linear and preserves the specified degree.

The auxiliary specialization is H_aux^+=C, H_aux^-=0 and D_aux=0. It does not remove the normal right Clifford coefficient or choose a new minimal spinor. WFD's infinite-dimensional spectral truncations are unnecessary in this specialization.

For an actual holonomy bisection h:O_s→O_r, the transport on sections is

\[
 (U_h u)(x)=U_{h_{h^{-1}x}}u(h^{-1}x),\qquad x\in O_r.
\]

The local isometry preserves the g-volume and the full coefficient transport is unitary. Thus this is a unitary between the two open-set L² spaces, not a claim of a globally defined group action. The two transported principal symbols agree, hence

\[
 U_hNU_h^{-1}=N+B_h
\]

on O_r, where B_h is a smooth odd right-C_q-linear zeroth-order endomorphism. It is bounded on compact buffers. Since the two expressions are formally symmetric, B_h is pointwise self-adjoint. This uses the whole inverse transport, not merely its determinant.

### 11BM.2. Completeness weighting and the full domain

A smooth proper exhaustion r:T→[0,∞) exists: take a countable locally finite precompact chart cover and its smooth partitions, with coefficients increasing outside successive compact exhaustions. Such a construction also accommodates a countable disjoint union of components. Define

\[
 \alpha=(1+|dr|_g^2)^{-1/2}>0,
 \qquad A=\alpha^{1/2}N\alpha^{1/2}
       =\alpha N+\tfrac12 C(d\alpha).
 \tag{WN.1}
\]

The expression is formally symmetric. The separate term C(dα)/2 is skew-adjoint; it is the product-rule correction, not a separately self-adjoint potential.

The metric g_α=α^{-2}g is complete on each component. Indeed |dr|_{g_α}=α|dr|_g≤1. A g_α-Cauchy sequence has bounded r and is contained in a compact sublevel set. There the metrics are uniformly equivalent, so the sequence converges. Choose ζ smooth, compactly supported in [0,2), equal to one on [0,1], and set η_n=ζ(r/n). Then

\[
 [A,\eta_n]=\alpha C(d\eta_n),\qquad
 \|[A,\eta_n]\|\le\|\zeta'\|_\infty/n.
 \tag{WN.2}
\]

Initially use C_c^∞(T,S). Its adjoint domain is exactly the maximal distributional domain: testing the Hilbert-space adjoint equation against every compact smooth coefficient section is the distributional assertion Au∈L². Finite coefficient charts make this an ordinary finite-dimensional distribution statement.

On compact interior buffers α is bounded below and every coefficient is smooth and bounded. Local elliptic regularity gives

\[
 \|\chi u\|_{H^1}\le C_\chi
 (\|\widetilde\chi Au\|_{L^2}+\|\widetilde\chi u\|_{L^2}),
 \qquad \widetilde\chi=1\text{ near supp }\chi,
 \tag{WN.3}
\]

for maximal-domain u. One direct proof freezes the principal matrices in a sufficiently small chart. The constant symbol squares to the positive metric norm, so Fourier transformation bounds the gradient by the frozen expression. Small coefficient variation is absorbed; the bounded zeroth-order terms and cutoffs give the remaining terms. To pass to distributions, smooth on a larger buffer. For each first-order coefficient a_j, the mollifier commutator kernel is

\[
 (a_j(x)-a_j(y))\partial_j\rho_\epsilon(x-y)
   +(\partial_j a_j)(y)\rho_\epsilon(x-y).
\]

Its two Schur integrals are uniformly bounded by the local Lipschitz bound and ∫|v||∂ρ(v)|dv. The commutator tends strongly to zero, first on smooth inputs and then on all L² inputs by density and the uniform bound. The smoothed estimates yield a bounded H¹ family whose weak limit is χu. Conversely, compactly supported H¹ vectors are approximated in the expression graph norm by smoothing and finite chart partitions. Hence compactly localized minimal and maximal domain sheaves coincide with H¹_loc.

For u in the maximal global domain, η_nu is compactly supported H¹ and therefore belongs to the minimal closure. The completed product rule is

\[
 A(\eta_nu)=\eta_nAu+[A,\eta_n]u.
\]

Dominated convergence and WN.2 show convergence of η_nu to u and of their A-images to Au. Thus the minimal closure equals its adjoint and is self-adjoint. Its exact full domain is

\[
 \operatorname{Dom}A
 =\{u\in L²(T,S):u\in H^1_{loc},\ Au\in L²\text{ distributionally}\}.
 \tag{WN.4}
\]

No global unweighted H¹ domain, uniform bound at infinity, or artificial boundary condition has been asserted.

### 11BM.3. Ordinary scalar local compactness

Let z be nonreal. For u=(A-z)^{-1}v, the resolvent bound and Au=v+zu give a uniform bound on the right side of WN.3. Thus χ(A-z)^{-1} maps the L² unit ball into a bounded H¹ set with support in a fixed compact buffer. Finite coefficient Rellich compactness makes this an ordinary Hilbert-space compact operator. For completeness, flatten the finite charts and confine to a cube: Fourier truncation is finite rank and its H¹ tail is bounded by the reciprocal frequency cutoff. A finite partition and extension within buffers give the same compactness on T.

Adjoints give (A-z)^{-1}χ compact as well. Approximation by compact smooth cutoffs proves M_f(A-z)^{-1} compact for every f∈C_0(T). Consequently M_f g(A) is compact for every scalar g∈C_0(R), by resolvent functional calculus and norm approximation. This is scalar compactness, not a field/module-only statement. Right-C_q-linear scalar compacts are coefficient-module compacts as well, because C_q is finite-dimensional; its finitely many matrix-block Morita identifications verify this directly.

For f∈C_c^∞(T), [A,f]=αC(df) is bounded with compact support. The rectangular resolvent identity and the norm-integrable bound ||[A,f]||/(1+τ²) prove [F_A,f] compact, where

\[
 F_A=A(1+A²)^{-1/2}.
\]

Thus the restriction to C_0(T) is already a geometric scalar normal Dirac cycle.

### 11BM.4. Positive rescaling: derivative and compactness checked

Let f be real and compactly supported and put A_t=e^{tf/2}Ae^{tf/2}. These positive multipliers and their inverses preserve Dom A by the completed first-order product rule. More explicitly,

\[
 A_t=e^{tf}A+\tfrac t2e^{tf}[A,f].
\]

Hence Dom A_t=Dom A, all A_t are self-adjoint by the same cutoff/local argument, and their graph norms are uniformly equivalent on compact t-intervals. Define B_t=[A_t,f]=e^{tf}[A,f]. Then

\[
 \dot A_t=\tfrac12(fA_t+A_tf)=fA_t+\tfrac12B_t.
\]

Write R_z=(A_t-z)^{-1}. Since [R_z,f]=-R_zB_tR_z, differentiation on the common graph gives precisely

\[
 \dot R_z=-fR_z-zfR_z²
       +\tfrac12R_zB_tR_z+zR_zB_tR_z².
 \tag{WN.5}
\]

The following graph-domain step justifies differentiation of the bounded phase.

For s=(1+τ²)^{1/2}, z_±=±is, use the strong spectral formula

\[
 F_{A_t}=\frac1\pi\int_0^\infty(R_{z_+}+R_{z_-})\,dτ.
\]

The leading differentiated terms are -fA_tR_z². On any vector v in the common graph domain,

\[
 \|fA_tR_z²v\|\le\|f\|\,\|A_tv\|/(1+τ²),
\]

uniformly on compact t-intervals. This is the integrable bound needed to differentiate the strong integral on graph vectors; a raw operator-norm O(1/s) bound would be insufficient. Scalar spectral integration of these leading terms gives f A_t(1+A_t²)^{-3/2}. The result is

\[
 \dot F_{A_t}=fA_t(1+A_t²)^{-3/2}
 +\frac1\pi\int_0^\infty\sum_\pm
 \bigl(\tfrac12R_{z_\pm}B_tR_{z_\pm}
       +z_\pm R_{z_\pm}B_tR_{z_\pm}²\bigr)\,dτ.
 \tag{WN.6}
\]

The remainder has norm at most 3||B_t||/(1+τ²). Each integrand is compact: B_t is a compactly supported finite coefficient multiplier, so B_tR_z is compact by Section 4. The leading term is compact because x(1+x²)^{-3/2} belongs to C_0(R) and f is compactly supported.

A_t is norm-resolvent continuous: A_t-A_u is a norm-small map from the common graph domain to L², uniformly locally in t,u, and the resolvent identity gives the claim. B_t is norm continuous. These facts give norm continuity of WN.6, by C_0 functional calculus for its leading term and the uniform integrable remainder bound. The fundamental theorem on common graph vectors, followed by density, now gives the norm integral

\[
 F_{A_1}-F_A=\int_0^1\dot F_{A_t}\,dt\in\mathcal K.
 \tag{WN.7}
\]

This comparison does not require the completeness weight to be holonomy invariant or [A,π(k)] to be bounded.

Also, if V is a bounded self-adjoint compactly supported odd coefficient endomorphism, then A_1+V is self-adjoint on the same domain and

\[
 R_{A_1+V,z}-R_{A_1,z}=-R_{A_1+V,z}V R_{A_1,z}
\]

is compact with an integrable O(1/s²) bound in the same two-resolvent phase formula. Thus F_{A_1+V}-F_{A_1} is compact.

### 11BM.5. Real holonomy bisections: domains and phases

Let k∈C_c^∞(H) be supported in one bisection h:O_s→O_r, and let T_k=π(k)=M_{k_r}U_h. Choose compact source/range buffers around its support. On the range buffer put β=α∘h^{-1}. The transported expression is

\[
 U_h A U_h^{-1}=\beta^{1/2}(N+B_h)\beta^{1/2}.
\]

Choose f∈C_c^∞(O_r), extended by zero to T, equal to log(β/α) on a slightly larger buffer than supp k_r. Then A_1=e^{f/2}Ae^{f/2} equals β^{1/2}Nβ^{1/2} there. Extend β B_h from that buffer to a compactly supported smooth odd self-adjoint endomorphism V. Let A_2=A_1+V. It agrees with the transported expression near the range support and is globally self-adjoint. Sections 4–5 give F_{A_2}-F_A compact.

For every u∈Dom A, its restriction near the source support is H¹. Smooth compact multiplication and the actual smooth local transport carry it to compactly supported H¹ in the range buffer. Hence T_ku∈Dom A_2. On that full domain,

\[
 A_2T_k-T_kA=C_k=[A_2,M_{k_r}]U_h,
 \tag{WN.8}
\]

where C_k is bounded and has compact source and range support. The formula first holds for compact smooth sections and passes to the full domain by local H¹ approximation. This is the required domain proof; neither a global U_h nor prior invariance of Dom A under a partial uncut transport is assumed.

The completed rectangular resolvent identity is

\[
 (A_2-z)^{-1}T_k-T_k(A-z)^{-1}
 =-(A_2-z)^{-1}C_k(A-z)^{-1}.
\]

Its norm is at most ||C_k||/s² for z=±is. It is compact because C_k has compact source support and its right localized resolvent is compact. Integrating yields F_{A_2}T_k-T_kF_A compact. Since F_{A_2}-F_A is compact, [F_A,T_k] is compact.

Every compact smooth arrow kernel is a finite sum of kernels supported in actual bisections, using a smooth finite partition on its compact support. Therefore the statement holds for all C_c^∞(H). The potentially unbounded scaling-difference term in [A,π(k)] has not been dropped: it is bypassed by the proved positive-rescaling phase comparison.

### 11BM.6. The maximal integrated representation and all KK defects

On H=L²(T,S,dvol_g), define

\[
 (\pi(k)u)(x)=\sum_{h:r(h)=x}k(h)U_hu(s(h)).
 \tag{WN.9}
\]

Compact arrow supports have a finite bisection cover, so the sums define bounded operators. Coherence U_{gh}=U_gU_h gives convolution multiplication. Change of variables under the actual local isometries gives π(k*)=π(k)*. Write

\[
 I_r(k)=\sup_x\sum_{r(h)=x}|k(h)|,
 \qquad I_s(k)=\sup_x\sum_{s(h)=x}|k(h)|.
\]

Pointwise Cauchy–Schwarz, followed by change of variables on the finitely many bisections, gives

\[
 \|\pi(k)u\|²\le I_r(k)I_s(k)\|u\|².
 \tag{WN.10}
\]

Thus π is an I-norm-bounded star representation and extends to the actual universal maximal completion C_max^*(H). Its unit-space action is multiplication, nondegenerate by compact unit cutoffs. It need not be faithful; faithfulness is unnecessary for the cycle.

For a compact kernel k choose χ∈C_c^∞(T), equal to one on its compact source set. Then π(k)=π(k)M_χ, so π(k)(A-z)^{-1} is compact. Hence π(k)(1-F_A²)=π(k)(1+A²)^{-1} is compact. F_A is self-adjoint. Section 6 proves the commutator defect. Maximal-norm density and ||π(a)||≤||a||_max extend all three defects to every a∈C_max^*(H). Consequently

\[
 d_m=[L²(T,S),\pi,F_A]\in KK^q(C_{max}^*(H),C)
 \tag{WN.11}
\]

with its full right C_q realization. The underlying Hilbert space is separable under the stated second-countability and finite coefficient hypotheses, so the countable-generation condition is met. Invariant metric germ separation supplies Hausdorffness of H as in Lemma 11M.5; no compactness was used there.


Let \(T\) be the original \(q\)-dimensional transversal, \(A=\alpha^{1/2}N\alpha^{1/2}\) the weighted normal Dirac, and \(F_A=A(1+A^2)^{-1/2}\). The preceding construction proves its scalar analytic defects and exact self-adjoint domain. Let \(U\) be an original positively oriented coordinate \(q\)-disk. We prove that its restriction is the prescribed local full analytical inverse and hence that

\[
b_U\otimes_{C_0(T)}[i_m]\otimes_{A_m}[H_N,\Pi_m,F_A]=+1.
\tag{WB.1}
\]

Choose a smaller positive coordinate ball \(V\) with closure in a compact interior buffer of \(U\). Lemma 6.0e, formula (150), gives the exact small-disk extension identity \(b_U=b_V[j_{VU}]\). It therefore suffices to compare the phases after restriction to \(C_0(V)\), while preserving the original positive orientation of \(U\).

### 11BM.7. Completed bounded locality

Suppose two self-adjoint finite-coefficient Dirac realizations have the same expression on an interior buffer, identified by the actual coefficient and half-density unitary. For a smooth compact identification cutoff \(\chi\) there, put \(T_\chi=\chi\) followed by that local unitary and extension by zero. Local maximal-domain regularity puts \(T_\chi\) of the full source domain in the full target domain. Their rectangular difference is the bounded compactly supported coefficient

\[
A_1T_\chi-T_\chi A_2=C_\chi.
\tag{WB.2}
\]

The resolvent identity is therefore valid on the completed domains:

\[
(A_1-z)^{-1}T_\chi-T_\chi(A_2-z)^{-1}
=-(A_1-z)^{-1}C_\chi(A_2-z)^{-1}.
\tag{WB.3}
\]

For \(z=\pm i\sqrt{1+t^2}\), its norm is at most \(\|C_\chi\|/(1+t^2)\); scalar source-localized compactness makes every term compact. Integrating the two-resolvent phase formula gives

\[
F_{A_1}T_\chi-T_\chi F_{A_2}\in\mathcal K.
\tag{WB.4}
\]

This uses equality on a buffer and its cutoff derivative, not equality of global domains or a global unitary between the two realizations.

For completeness, restriction to \(C_0(V)\) may be made on the common essential space \(L^2(V,S_{\rm inv})\). If \(P_V\) is its Hilbert-space projection in a larger realization, compress the phase to \(P_VFP_V\). For \(a\) compactly supported in \(V\), choose \(\chi=1\) near its support and \(\operatorname{supp}\chi\Subset V\), within the expression-agreement buffer. Then

\[
aF(1-P_V)=a[\chi,F](1-P_V)
\tag{WB.5}
\]

is compact. Its adjoint counterpart is compact as well. Removing the cross blocks consequently preserves the localized square and commutator defects and gives the essential restricted cycle. Formula (WB.4), together with the two cutoff commutators, says that the two compressed phases differ by a \(C_0(V)\)-locally compact bounded operator. Their straight-line homotopy is a cycle: every new commutator and square cross term becomes compact after multiplication by \(a\), because both one-sided localized differences are compact. Norm density extends this to every \(a\in C_0(V)\). This proves equality of the two local KK classes on the smaller disk; it does not claim that agreement on one buffer makes the difference locally compact everywhere in \(U\).

### 11BM.8. A complete reference which agrees on the buffer

Choose the original positive coordinate frame and its full inverse coefficient trivialization on \(U\). Transfer the density to ordinary coordinate Lebesgue measure by the positive half-density unitary. On the chosen buffer the principal symbol of \(A\) is \(\alpha\,c_{\rm inv,g}(\xi)\). It is the inverse-Clifford symbol of the positive conformal metric \(g'=\alpha^{-2}g\), with the full module transferred by its positive metric isometry.

Extend this positive metric smoothly to all \(\mathbb R^q\), equal to the Euclidean metric outside a larger compact coordinate set. For example a smooth cutoff convex combination of its local positive matrix and the Euclidean matrix stays positive. Extend the compatible coefficient connection in the same way. The corresponding formally symmetric reference Dirac has the required symbol on the buffer. Its difference from the transported \(A\) there is a smooth self-adjoint zeroth-order endomorphism. A compactly supported extension of that endomorphism makes a reference expression \(A'\) agree exactly with \(A\) on the smaller buffer.

All coefficients and their first derivatives are bounded on this reference space; its symbol is uniformly elliptic and Euclidean outside a compact set. The same local estimates and Euclidean-speed exhaustion cutoffs prove that its self-adjoint domain is \(H^1(\mathbb R^q,S_{\rm inv})\). Thus (WB.2)–(WB.5) apply and identify its local class with the actual weighted phase. No extension of actual holonomy arrows is used in this reference comparison; it is a comparison after restriction to the original unit algebra.

### 11BM.9. Metric homotopy is an interval cycle, without a false norm-phase claim

Join the extended positive metric to the Euclidean one through positive metrics, fixed outside the larger compact set. Along this metric family choose the induced Levi-Civita Clifford connections; interpolate their scalar determinant-connection parts and the zeroth-order correction to the flat connection and zero correction. This constructs compatible connections along the family, rather than taking an unjustified convex combination for different Clifford structures. Transport the whole inverse module by the positive metric isometries. Under the fixed coefficient and half-density identifications this gives self-adjoint Dirac expressions \(A_s\), \(0\le s\le1\), on the common domain \(H^1\). Uniform ellipticity and bounded coefficients give uniformly equivalent graph norms. Moreover \(A_s-A_t:H^1\to L^2\) tends to zero in norm as \(s\to t\). The resolvent identity therefore proves norm-resolvent continuity.

Their bounded phases are uniformly bounded and strongly continuous. To see the latter directly, test on a compact smooth vector. Uniform bounds on \(\|A_sv\|\) control its large spectral tails, while norm-resolvent continuity controls compact spectral functions. Approximation of the bounded transform on a compact spectral interval then gives strong continuity; density treats every vector. Self-adjointness gives strong continuity of the adjoints too.

This is enough for an adjointable operator on \(C([0,1],L^2)\): for a continuous vector section, first compare with its value at one parameter and then use strong continuity and the uniform operator bound. The adjoint has the same property. A claim of operator-norm continuity of all the phases is unnecessary.

For \(a\in C_c^\infty\), the square defect is \(-a(1+A_s^2)^{-1}\), a norm-continuous compact field by scalar localized compactness and resolvent functional calculus. The coefficient \([A_s,a]\) is norm continuous, bounded, and compactly supported uniformly in \(s\). The rectangular phase-commutator integral has the uniform integrable bound \(\|[A_s,a]\|/(1+t^2)\). Its resolvents are norm continuous in \(s\) and its integrands are scalar compact, so \([F_{A_s},a]\) is a norm-continuous compact field. Norm density gives all \(a\in C_0(\mathbb R^q)\). Thus the family is an actual interval KK cycle and proves equality with the flat inverse class.

### 11BM.10. The full inverse and the positive Bott sign

The Euclidean endpoint is the actual full inverse of Theorem 7.17. Write the raw Thom pair as \(\beta_q\in KK(\mathbb C,C_0(\mathbb R^q)\widehat\otimes C_q)\), \(\alpha_q\in KK(C_0(\mathbb R^q)\widehat\otimes C_q,\mathbb C)\). Theorem 6.0c and Corollary 6.0d prove \(\beta_q\alpha_q=1\) through the complete oscillator comparison and its single positive Gaussian vacuum. Theorem 7.17 identifies the flat inverse with \(d_q=\mathcal T_q(\alpha_q)\). Its local full coefficient is \(\Pi^{k_q}C_q\), \(k_q=q(q-1)/2\), its physical generators are \((-1)^{k_q+1}L_{t_j}\), and its full transition is \(zs\mapsto z^{-1}\rho_t(s)\).

Apply the prescribed right reduction \(R_q\) of consecutive even generator pairs, retaining the last \(C_1\) when \(q\) is odd. Lemma 7.20 proves the exactly typed native product

\[
R_q(\beta_q)\,\cdot\,R_q\mathcal T_q(\alpha_q)=+1.
\tag{WB.6}
\]

In the odd–odd case the target order is the second-cycle coefficient followed by the externally retained first-cycle coefficient; no unrecorded graded flip is made. Lemma 7.11 supplies the analytical phase with this same full orientation, and Lemma 7.14 gives its restriction to the positive smooth ball \(V\). Thus (WB.6) is the required bridge from the raw inverse identity to the original parity product, with the full inverse line, grading and physical generator signs retained.

Sections 1–3 identify the actual weighted phase with precisely that endpoint inverse after restriction to \(V\). Multiplying by \(b_V\), followed by the exact positive disk extension \(b_U=b_V[j_{VU}]\), proves (WB.1) for the original \(U\). Rank zero is the discrete unit-space case: retain the full graded line in the original orientation and its dual in the inverse. Their even line evaluation is the unit. An odd stable inverse line is not paired with an unrelated bare even point class. This supplies the local normalization of the actual maximal normal cycle, without claiming its geometric representation descends to the reduced algebra.


![A positive weight completes an incomplete normal metric and retains its original phase](../figures/weighted-normal-cycle.png)

**Figure 11BM.** Theorem 11BM.1, WN.1–WN.11 and WB.1–WB.6. In the exact one-dimensional example \(T=(0,1)\), \(g=dx^2\), \(r=1/x+1/(1-x)\), and \(\alpha=(1+|r'|^2)^{-1/2}\), the original endpoints are a finite distance away but the positive rescaled metric is complete. The plotted values of \(\alpha\) are numerical samples of this exact formula; the divergent endpoint distance is proved by \(\alpha^{-1}\ge|r'|\). Positive scaling cancels from the normalized inverse-Clifford phase. The actual domain is local \(H^1\) with \(Au\in L^2\), rather than global unweighted \(H^1\). The diagram records the original disk comparison and makes no reduced-norm or physical graph claim. Historical context: [Connes], Section 8, fundamental class; full inverse conventions: Theorem 7.17.

Editable SVG · Reproduction source and exact formula.

## 11BN. Direct diagonal transfer on incomplete flat holonomy

**Theorem 11BN.1.** Let \(G\rightrightarrows T\) be a Hausdorff second-countable locally compact étale groupoid. Suppose it has a continuous cocycle to a countable discrete group \(\Gamma\), whose neutral algebra has injective full-to-reduced quotient. A cocycle with unit kernel satisfies this condition. If \(C_r^*(\Gamma)\) has an even scalar class pairing with its unit as \(+1\), there is an explicit transfer

\[
\Theta\in KK(C_r^*(G),C^*_{\max}(G)),\qquad
[i_r]\Theta=[i_m].
\]

For every transversely flat foliation with the full coherent invariant metric Spin-c normal data, a countable coordinate cover supplies such a transfer after its actual Čech enlargement. On returning to the original transversal, the transfer has the original positive-disk Bott restriction. The weighted normal class of Theorem 11BM.1 therefore gives a reduced class with the original Bott pairing \(+1\). No compactness or complete transverse metric is assumed. In particular this covers every codimension-one foliation with these data: its invariant one-dimensional metric has local arc-length coordinates.

The full inverse Spin-c line and the ordered final Clifford block remain in the normal factor. The transfer is even. Its actual convolution coefficients retain their original domains throughout.

The conditional-expectation method belongs to the theory of graded operator algebras. For historical context, see Ruy Exel, [*Amenability for Fell bundles*, Sections 2–3, especially Propositions 3.6–3.7](https://arxiv.org/pdf/funct-an/9604009). The actual-arrow diagonal, neutral-algebra and original-disk module comparisons are proved here.

**Proof.** We construct the diagonal map, prove its exact kernel and unit restriction, then construct the flat labels and return through the actual cover correspondence.

### 11BN.1. Exact direct diagonal theorem on actual arrows

Let `G => T` be Hausdorff, second countable, locally compact and etale.
Write

\[
 R=C_0(T),\qquad A_m=C^*_{\max}(G),\qquad A_r=C_r^*(G),
 \qquad q_G:A_m\longrightarrow A_r.
 \tag{IHT.1}
\]

The unit inclusions are `i_m` and `i_r`. Let

\[
 c:G\longrightarrow\Gamma,\qquad
 c(gh)=c(g)c(h),\qquad c^{-1}(e)=T,
 \tag{IHT.2}
\]

be continuous, with `Gamma` countable and discrete. No global action of
`Gamma` on `T` is assumed. In particular no unused label is promoted to
an original holonomy arrow.

For clarity IHT.2 implies only an actual partial action. If `g,h` have
the same source and the same label, `gh^{-1}` has label `e`, hence is a
unit, and `g=h`. The analogous same-range argument holds. Consequently
each open label component `G_delta=c^{-1}(delta)` is a bisection, whose
source and range are open subsets of `T`. Composition and inversion give
the partial-action identities on exactly the domains where the original
products exist. The following proof does not require globalization of
this partial action.

Put `B=C_r^*(Gamma)`, let `lambda_delta` be its regular unitary, and let
`tau` be its canonical faithful trace. Every compact core kernel has a
finite label decomposition `f=sum_delta f_delta`: a compact set has finite
image under a continuous map to a discrete space. Define on that core

\[
 \sigma(f)=\sum_\delta f_\delta\otimes\lambda_\delta
              \quad\hbox{in }A_m\otimes_{\min}B.
 \tag{IHT.3}
\]

The cocycle identity proves multiplication and involution in IHT.3,
using the actual convention `s(g)=r(h)` for `gh`.

#### 11BN.1.1. Full-norm extension, without global absorption coordinates

Choose a faithful separable representation `rho` of `A_m`. Its
disintegration, already proved in the companion KT-CP-14, "The full norm
and disintegration" through (14E.18), has a quasi-invariant measure on
`T`, a measurable Hilbert field `H_x`, and an actual measurable unitary
arrow action `U_g:H_{s(g)} -> H_{r(g)}`. Replace that field by

\[
 H_x\otimes\ell^2(\Gamma),\qquad
 U_g\otimes\lambda_{c(g)}.
 \tag{IHT.4}
\]

This is a genuine unitary groupoid representation: the first and second
factors both satisfy the same actual composition law. Its integrated
representation has the same scalar Radon--Nikodym correction as `rho`.
After the constant second factor is taken outside the direct integral,
the integrated operator on the finite-label core is
`sum_delta rho(f_delta) tensor lambda_delta`. The proved disintegration
theorem makes it I-contractive and hence bounded by the maximal norm.
The faithful spatial representation `rho tensor lambda` therefore proves
that IHT.3 extends to a contractive star homomorphism

\[
 \sigma:A_m\longrightarrow A_m\otimes_{\min}B.
 \tag{IHT.5}
\]

This step uses a theorem already proved in the course, with its exact
measurable representation hypotheses. It does not use the global-action
unitary `W` from RA.4 where the partial arrow domains would make that
formula meaningless.

#### 11BN.1.2. Exact reduced factorization and its kernel

The proved Hausdorff-etale expectation is

\[
 E_r:A_r\longrightarrow R,
 \qquad E_m=E_rq_G:A_m\longrightarrow R.
\]

Its reduced faithfulness can also be checked directly. On the source
fibre `G_x=s^{-1}(x)`, the regular representation satisfies

\[
 E_r(a^*a)(x)=\|\lambda_x(a)\delta_{1_x}\|^2.
\]

Right multiplication by an actual `h in G_x` is an onto unitary from
`ell^2(G_{r(h)})` to `ell^2(G_x)`, sending `delta_{1_{r(h)}}` to
`delta_h` and intertwining the two left regular representations. Thus
`E_r(a^*a)=0` annihilates every basis vector of every regular fibre, and
`a=0` in `A_r`. This is also KT-CP-14, Theorem 14.4.

IHT.2 makes the trace slice exactly the unit expectation:

\[
 (\mathrm{id}\otimes\tau)\sigma(a)=i_mE_m(a)
                       \qquad(a\in A_m).
 \tag{IHT.6}
\]

Indeed on the core the trace discards every nonidentity label, and the
identity component is exactly the unit space. Continuity gives IHT.6
on the completion.

The slice `id tensor tau` is faithful on positive elements of
`A_m tensor_min B`. Here is an elementary version of that fact. In the
faithful representation `rho tensor lambda`, a positive element with
zero slice has square root annihilating every `xi tensor delta_e`.
It commutes with all `1 tensor right-regular_gamma`, since the left and
right regular actions commute. These operators carry `delta_e` to every
coordinate vector. The square root therefore annihilates the dense span
of all `xi tensor delta_gamma`, so is zero.

For `a in A_m`, apply IHT.6 to `a^*a`. It follows in both directions that

\[
 \sigma(a)=0
 \quad\Longleftrightarrow\quad E_m(a^*a)=0
 \quad\Longleftrightarrow\quad q_G(a)=0.
 \tag{IHT.7}
\]

Thus `ker sigma=ker q_G`, not merely one inclusion. Taking the quotient
gives an isometric star homomorphism

\[
 \Delta_r:A_r\longrightarrow A_m\otimes_{\min}B,
 \qquad
 \Delta_r\Big(\sum_\delta f_\delta\Big)
       =\sum_\delta f_\delta\otimes\lambda_\delta,
 \qquad
 \Delta_ri_r(a)=i_m(a)\otimes1_B.
 \tag{IHT.8}
\]

All domains remain in the first factor `f_delta`. An arrow approaching a
missing continuation is not declared to have an endpoint arrow at that
continuation. Neither closedness nor properness of the label map is used.

#### 11BN.1.3. A concrete normalized transfer module

Suppose an even scalar class

\[
 \kappa\in KK(B,\mathbb C),\qquad
 [u_B]\otimes_B\kappa=1_{\mathbb C}
 \tag{IHT.9}
\]

is supplied, where `u_B` is the unit homomorphism. Define

\[
 \Theta_c=[\Delta_r]\otimes_{A_m\otimes B}
                     (1_{A_m}\boxtimes\kappa)
                         \in KK(A_r,A_m).
 \tag{IHT.10}
\]

The factor order is exactly the displayed order, and the auxiliary is even.
An actual representative of IHT.10 is the right `A_m` Hilbert module

\[
 \mathcal E_c=A_m\otimes H_\kappa,\qquad F_c=1\otimes F_\kappa,
 \qquad
 \varphi_c(f)=\sum_\delta L_{f_\delta}\otimes V_\delta,
 \tag{IHT.11}
\]

where `(H_kappa,V,F_kappa)` represents IHT.9 and `V` is a unital representation
of the reduced group algebra. Such a representative can always be chosen:
for any representative put `p=V(1_B)`. The cycle condition at `1_B` makes
`[p,F_kappa]` compact, so deleting the off-diagonal blocks is a compact
perturbation. The `(1-p)` block has zero left action and is degenerate;
discarding it leaves the unital representation on `pH_kappa`, with the
same class and unit pairing. The norm bound is IHT.8. Its compact
defects are finite sums of `a_delta tensor compact_scalar`, hence belong
to `K(E_c)=A_m tensor K(H_kappa)`. For example the commutator terms are
`L_f_delta tensor [F_kappa,V_delta]`; the square-defect terms are
`L_f_delta tensor V_delta(F_kappa^2-1)`. Maximal/reduced norm density
extends this check to every source coefficient. Compact unit cutoffs
make the left action nondegenerate. This is an explicit bounded cycle,
not only an existential KK expression.

By IHT.8, even external-product functoriality, and IHT.9,

\[
 [i_r]\otimes_{A_r}\Theta_c=[i_m]
                           \quad\hbox{in }KK(R,A_m).
 \tag{IHT.12}
\]

This proves a whole original-unit restriction identity in this scope.
It does not claim `q_G Theta_c=1_{A_m}`, K-amenability, an equivariant
gamma identity, or a proper groupoid action pulled back from `Gamma`.

#### 11BN.1.4. A weaker neutral-component interface also suffices

The unit-kernel assumption can be replaced by the following exact
condition. For any continuous discrete-group cocycle `c`, let
`N=c^{-1}(e)` and let `C` be the norm closure in `A_m` of its neutral
compact core. Require that `q_G` be injective on `C`. The same formulas
IHT.8--IHT.12 then hold. Here is the additional proof, with no assumption
that the neutral groupoid is the units.

First the reduced diagonal formula always exists for such a cocycle.
On `ell^2(G_x) tensor ell^2(Gamma)` define the actual source-fibre unitary

\[
 W_x(\delta_h\otimes\delta_t)
                  =\delta_h\otimes\delta_{c(h)^{-1}t}.
 \tag{IHT.12a}
\]

For a left multiplication by an arrow `g`, the output coordinate is
`h=gk` and `c(h)=c(g)c(k)`. Conjugating
`lambda_x(f_delta) tensor lambda_delta` by IHT.12a consequently gives
`lambda_x(f_delta) tensor 1`. Taking the direct sum of all source-fibre
representations, which is faithful on `A_r`, and the faithful regular
representation of `B`, proves an isometric homomorphism

\[
 \sigma_r:A_r\longrightarrow A_r\otimes B,
 \qquad (q_G\otimes1)\sigma=\sigma_rq_G.
 \tag{IHT.12b}
\]

This is absorption on actual arrow fibres. It does not require a globally
defined `Gamma` action, and no label coordinate changes the normal chart.

The slice `E_c=(id tensor tau)sigma` takes values in `C`: on the core
it selects exactly the neutral component. Its completed range remains
in that closed subalgebra. If `q_G(a)=0`, IHT.12b implies
`q_G(E_c(a^*a))=0`. Injectivity on `C` gives `E_c(a^*a)=0`, and the
faithful slice proof in Section 2.2 gives `sigma(a)=0`. Conversely, if
`sigma(a)=0`, the unit expectation of `E_c(a^*a)` is zero. That unit
expectation equals `E_m(a^*a)`, since every unit has neutral label.
Reduced expectation faithfulness gives `q_G(a)=0`. This proves IHT.7
under the stated weaker condition.

One concrete sufficient condition is amenability of the actual neutral
wide subgroupoid `N`. Its open inclusion gives a homomorphism
`C^*max(N)->A_m`: any full representation of `G` restricts to the
neutral core with its I-norm bound. Its reduced inclusion is faithful,
as is checked by splitting source fibres into right `N` cosets in the
regular representation: its invariant pieces are `N h`, and actual right
translation by `h` identifies each piece with the neutral source fibre
`N_{r(h)}`. The piece `N_x` gives the reverse norm bound. The diagram
with these two inclusions and
the full-to-reduced quotients commutes on the core. If `N` is amenable,
the proved amenable norm equality makes its own quotient an isomorphism.
Here amenability means the actual Haar/Reiter condition of KT-CP-14,
(14.32), whose full/reduced norm equality is Theorem 14.7.
The composition from `C^*max(N)` to `A_r(G)` is therefore injective;
it follows both that the full inclusion is injective and that `q_G`
is injective on its image `C`. This verifies the required interface in
that case. No amenability of the full groupoid or of `Gamma` is assumed.

This extension is a sufficient theorem, not a claim that every eligible
curved incomplete groupoid admits such a cocycle. A nontrivial neutral
component also shows why injectivity of isotropy labels is not logically
necessary for the direct transfer.

### 11BN.2. Supplying the scalar auxiliary for the exact label group

For a countable characteristic-zero linear group, EK.1 and LF.2 give
normalized positive kernels and the uniformly proper same-source
kernel LF.4, including the uniformity in every right multiplier. Their
finite-exhaustion proof is already countable, not just finitely generated.
The proved GK.5--GK.7 spectrum, PK probability averaging and BA Gram-field
construction yield the compact convex probability base and the proper
affine Hilbert field required by Theorem 11Y.1. That theorem's first
sentence applies to every countable group with this supplied field.
Its subsequent finitely generated examples are not a restriction of
that first sentence.

Consequently the actual even proper factorization `(eta,D)` in ST.4
has ordinary product one by ST.8. With its proper coefficient `P`,
the proper full-to-reduced quotient `q_P` is an isomorphism and

\[
 \kappa=j_r(\eta)[q_P^{-1}]j_{\max}(D)[\epsilon]
                \in KK(C_r^*\Gamma,\mathbb C)
 \tag{IHT.13}
\]

has the unit pairing IHT.9, by Theorem 11J.3 / ST.9. `epsilon` is the
trivial character of the full group algebra. No reduced trivial character,
amenability, or ordinary index substituted for a KK class occurs here.

This uses the existing complete countable-label proof rather than a new
relative Ado theorem or an unsupported extension of a finitely generated
statement.

### 11BN.3. Actual flat development labels on arbitrary incomplete domains

Suppose the fixed invariant transverse metric is locally Euclidean of
dimension `q`. Take a countable atlas of connected oriented normal
coordinate disks

\[
 u_i:O_i\longrightarrow\Omega_i\subset\mathbb R^q,
 \qquad \widetilde T=\coprod_iO_i.
\]

The actual flat germ groupoid is Hausdorff. On each chart-pair its arrow
maps continuously to its source point and affine expression. This map is
injective, since an affine expression and source point determine its target
and its germ. Distinct arrows in that chart-pair can therefore be separated
by the Hausdorff source/affine-coordinate space; chart indices separate
different pairs. Its countable germ atlas also gives second countability.

Let `p:tilde T -> T` be the open local-diffeomorphism cover and let
`tilde G=p^*G` be its actual Cech pullback. An arrow has the form
`(i,g,j):(s(g),j) -> (r(g),i)`, with `s(g) in O_j`, `r(g) in O_i`.
In those two charts its normal holonomy germ is an oriented Euclidean
local isometry and therefore has the unique affine expression

\[
 u_i h_g u_j^{-1}(v)=A_gv+b_g,
                  \qquad A_g\in SO(q),\quad b_g\in\mathbb R^q.
 \tag{IHT.14}
\]

The expression is constant along each connected local bisection:
an isometry between Euclidean open sets has constant orthogonal derivative
and is affine on each connected component. This can be proved by preservation
of the flat Levi-Civita connection, which gives zero second coordinate
derivatives. Hence the label below is locally constant. It is not an
extension claiming that the affine map is an actual arrow wherever that
affine expression can be written.

Assign

\[
 c(i,g,j)=(A_g,b_g,i-j)
       \quad\hbox{in }\operatorname{Isom}^+(\mathbb R^q)\times\mathbb Z.
 \tag{IHT.15}
\]

Composition uses

\[
 (A,b,m)(A',b',m')=(AA',b+Ab',m+m'),
\]

so the coordinate-change formula and the actual germ composition prove
that IHT.15 is a functor. At a unit the affine part is the identity and
`i-j=0`. Conversely if IHT.15 is the identity, `i=j` and the coordinate
holonomy germ is the identity on that same chart. It is the actual
holonomy unit. Thus `c^{-1}(e)=tilde T`. This is a statement about the
holonomy groupoid of germs; a separate monodromy groupoid retaining
holonomy-trivial path loops would require a different hypothesis.

There are only countably many labels. Indeed a locally constant map on a
second-countable space has countable image, by assigning a basis element
to each nonempty open fibre. Let `Gamma` be the group generated by that
image. It is a countable subgroup of the displayed group. The faithful
matrix map

\[
 (A,b,m)\longmapsto
 \operatorname{diag}\left(
       \begin{pmatrix}A&b\\0&1\end{pmatrix},
       \begin{pmatrix}1&m\\0&1\end{pmatrix}\right)
                     \in GL_{q+3}(\mathbb R)
 \tag{IHT.16}
\]

is an exact homomorphism and is injective. Thus Section 3 supplies IHT.9
for this exact group with its discrete topology. This construction permits
arbitrarily small holonomy domains and arbitrary degenerating overlaps.
The first factor of IHT.8 keeps every actual domain; no finite-label
compactness is used.

The full inverse Spin-c coefficient is not encoded by replacing it with
`(A,b,m)`. It remains in the independent original normal cycle `x_N`.
In particular a central lift phase `z^{-1}` stays in its coherent arrow
action, even when the geometric affine label has trivial derivative.

#### 11BN.3.1. Return to the original complete transversal and its original disk

The actual cover equivalence gives the proved full and reduced Morita
modules

\[
 X_r\in KK(A_r(G),A_r(\widetilde G)),\qquad
 X_m\in KK(A_m(G),A_m(\widetilde G)).
\]

Use the same compact-core correspondence for both, with their separately
proved full and reduced norms. Then

\[
 \Theta_G=X_r\otimes_{A_r(\widetilde G)}\Theta_c
                      \otimes_{A_m(\widetilde G)}X_m^*
                  \in KK(A_r(G),A_m(G)).
 \tag{IHT.17}
\]

For the precise local normalization, take the original disk `U` inside
one `O_j`, and write `j_U:C_0(U)->R` for zero extension. Restricting the
left source of the equivalence to `U` identifies its compact core with
the lifted-arrow core having range in the copy `(U,j)`. The map sends
an equivalence arrow `(g,(s(g),k))`, with `r(g) in U`, to
`((r(g),j),g,(s(g),k))`. It preserves the right action, convolution
inner product and counting convention term by term; completing it gives
an onto unitary on the essential source module. The same statement holds
for the full module. No adjointable characteristic projection of an open
disk is assumed. The essential-source replacement here is valid because
the operator is zero and the left action is compact: if `E_0` is the
closed source span inside `E`, the module
`{xi in C([0,1],E): xi(1) in E_0}` is a Kasparov homotopy from `E` to
`E_0`. Left compact operators are approximated by rank-one operators
whose two vectors belong to `E_0`, by applying a source approximate
identity on both sides. Those constant vectors lie in the homotopy
module, so its left action is compact. The modules are countably generated:
the original module is norm separable over a separable coefficient algebra,
and the displayed homotopy is a closed subspace of its norm-separable
continuous-section space. A countable norm-dense set of vectors generates
its right-module closure. This proves the replacement
without assuming a complemented essential submodule. In KK notation
the resulting identities are

\[
 [j_U][i_{r,G}]X_r=[j_{U,j}][i_{r,\widetilde G}],\qquad
 [j_{U,j}][i_{m,\widetilde G}]X_m^*=[j_U][i_{m,G}].
 \tag{IHT.18}
\]

IHT.12, IHT.17 and IHT.18 give the actual original-disk identity

\[
 [j_U][i_{r,G}]\Theta_G=[j_U][i_{m,G}].
 \tag{IHT.19}
\]

An arbitrary original positive calibration disk U need not lie in one
flat metric chart. Choose a smaller positive coordinate disk V with compact
closure in U and contained in one O_j. Apply IHT.19 to V. The exact positive
disk extension identity of Lemma 6.0e is
\[
 b_U=b_V[j_{VU}],\qquad
 b_U[j_U][i_{r,G}]\Theta_G=b_U[j_U][i_{m,G}].
 \tag{IHT.19a}
\]
Indeed \([j_{VU}][j_U]=[j_V]\), so the second equality is IHT.19 multiplied
by b_V. This retains the original U and its prescribed positive Bott
class without assuming that U itself is a flat metric chart. No whole-base
identity on a frame-unit enlargement is needed.

### 11BN.4. Section 11BM and the literal physical product

Keep the weighted maximal normal cycle of Section 11BM on the original
transversal, in its full Clifford picture,

\[
 x_N\in KK^q(A_m(G),\mathbb C),\qquad
 b_U[j_U][i_{m,G}]x_N=+1.
\]

IHT.19 now proves

\[
 x_r=\Theta_G\otimes_{A_m(G)}x_N\in KK^q(A_r(G),\mathbb C),
 \qquad b_U[j_U][i_{r,G}]x_r=+1.
 \tag{IHT.20}
\]

The auxiliary transfer is even. The original order and the full inverse
normal factor are retained: with `k_q=q(q-1)/2`, the local inverse finite
module remains `Pi^{k_q} C_q`, its physical generator is
`(-1)^{k_q+1} L_{t_j}`, and a full lift `zs` acts as
`z^{-1} rho_t(s)`. Its right Clifford block is contracted only by the
course's fixed contraction. There is no determinant-only reconstruction
of the inverse phase or an orientation sign from exchanging odd factors.

The bounded product is taken on the actual interior tensor product
`E_Theta tensor_{A_m} L^2(T,S_inv)`. Including the graph counting/plaque
correspondence gives the literal physical carrier

\[
 \mathcal E_Z\otimes_{A_r(G)}\mathcal E_\Theta
                         \otimes_{A_m(G)}L^2(T,S_{\rm inv}).
 \tag{IHT.21}
\]

The bounded product construction in Section 0 supplies the actual creation-map
connection to the normal phase on each compact physical coefficient
column. It does not identify IHT.21 by an arbitrary abstract Hilbert-space
unitary. No new completed unbounded normal-sum domain is claimed here:
the normal factor uses its closed completed domain from Section 11BM.
The connection retains that normal phase on columns, but by itself does
not prove that the separator-selected bounded product lies in a specified
normal pseudodifferential calculus. That operator-type issue remains
separate from the now proved transfer and Bott equality in the flat scope.

### 11BN.5. The exact continuation-radius shadowing countertest

Use the actual manifold and foliation

\[
 M=\mathbb R^2\setminus\{(0,0),(2,0)\},\qquad
 TF=\mathbb R\partial_x,\qquad g_N=dy^2,
 \tag{IHT.22}
\]

with the trivial coherent full rank-one normal Spin-c lift. Take the
leaf-meeting complete transversal `T=T_L coproduct T_M coproduct T_R`,
where the three vertical copies of `R` sit at `x=-1,1,3`, respectively.
At `y!=0` all three points at coordinate `y` are on one horizontal leaf.
At `y=0` they belong to three different leaves. Thus the actual etale
holonomy groupoid has three unit components `R`, six cross components
`R\{0}`, and no cross arrow at `y=0`. Each arrow between distinct copies
has identity normal germ where it actually exists.

For a nonunit arrow at height `y`, the maximal centered connected
source interval on which that actual cross bisection extends has radius
`|y|`. A larger centered interval meets zero, where this cross holonomy
does not exist. Unit germs extend arbitrarily far on their own copy.
Cap these radii by one and set

\[
 \rho(h)=\begin{cases}1&h\hbox{ unit},\\
                         \min(1,|y|)&h\hbox{ cross at }y,
          \end{cases}\qquad
 f(h)=-\log\rho(h),\qquad
 k_\rho(h,l)=|f(h)-f(l)|^2\quad(s(h)=s(l)).
 \tag{IHT.23}
\]

`f` is continuous on the actual arrow components. IHT.23 is a continuous
normalized symmetric CND kernel, as the squared distance of a real
Hilbert map. Its compact-increment upper estimate holds. On a nonunit
compact arrow test, `|y|` has a positive minimum; all possible radius
differences are at most `max(0,-log|y|)`. A unit increment has difference
zero. There are only finitely many component types.

Now let `y_n>0` tend to zero and use actual arrows

\[
 h_n:(-1,y_n)\longrightarrow(1,y_n),\qquad
 g_n:(1,y_n)\longrightarrow(3,y_n),\qquad
 g_nh_n:(-1,y_n)\longrightarrow(3,y_n).
 \tag{IHT.24}
\]

Both `h_n` and `g_nh_n` inherit the same tiny radius `y_n`, hence

\[
 k_\rho(g_nh_n,h_n)=0.
 \tag{IHT.25}
\]

The sources and ranges of `g_n` lie in the compact unit set consisting
of the three segments `|y|<=1`. But `g_n` escapes every compact subset
of its actual punctured cross-arrow component: its only possible
coordinate limit is the missing cross arrow at zero. IHT.25 therefore
violates GK.4's proper low-increment condition even at `R=0`. The
common-source test arrow is `h_n`, not the unit at `s(g_n)`.

For comparison the relative cost

\[
 k(h,l)=\begin{cases}0&h=l,\\1+y^{-2}&h\ne l\end{cases}
                        \qquad(s(h)=s(l))
 \tag{IHT.26}
\]

is defined on those same actual source fibres; distinct arrows occur
only at `y!=0`. It is continuous on their actual components. For any
real zero-sum family, combine coefficients belonging to the same arrow
into totals `a_j`; then

\[
 \sum_{i,j}a_ia_jk(h_i,h_j)
       =-(1+y^{-2})\sum_j a_j^2\le0.
\]

It is therefore CND. Cancellation shows `gh=h` exactly when `g` is a
unit. Every nonunit left increment consequently has cost `1+y^{-2}`,
independently of `h`. Compact cross-arrow tests are bounded away from
zero, giving GK.4's upper control. A bound `1+y^{-2}<=R` is impossible
for `R<=1`; for `R>1` it forces `|y|>=(R-1)^{-1/2}`. Compact unit
endpoints also bound `|y|` above. Taking the closed pieces of the six
cross components in that bounded interval, plus units over the compact
test set, gives an actual compact `C(K,R)`. This proves the lower
control, uniformly in all `h`.

This is a successful all-actual-arrow kernel for this example. It
retains all three arrow types per nonzero source fibre. It proves that
radius shadowing is a defect of the proposed scalar feature, not an
obstruction to a provider, and not evidence for an extra complete-metric
hypothesis. The direct flat transfer above also applies here, without
needing to construct IHT.26.



![Actual incomplete holonomy, relative costs and the normalized direct transfer](../figures/direct-flat-transfer.png)

**Figure 11BN.** Theorem 11BN.1 and IHT.8–IHT.26. The two-puncture horizontal foliation retains the three zero-height leaves and only the actual positive-height cross arrows. At \(y=1/2\), \(h\) and \(gh\) have the same continuation radius \(1/2\), so the radius-feature cost is zero, whereas the relative CND cost is exactly \(5\). At \(y=1/4\) that cost is \(17\); as \(y\to0\), it detects every nonunit increment uniformly in the common-source arrow. The transfer diagram records the actual kernel labels in the maximal first factor and the even auxiliary in the second; its two original-disk paths agree in KK. These are exact coordinate and type diagrams, not a globalization of the partial domains. Human context: Exel, *Amenability for Fell bundles*, Sections 2–3. Original diagram expression: CC0 1.0.

Editable SVG · Reproduction source and exact data.

## 11BO. Joint Dirac geometry for the discrete transfer

**Theorem 11BO.1.** Suppose the invariant metric and full coherent inverse Spin-c normal data of Theorem 11BM.1 are given, and the original transversal or an actual countable Čech cover has the discrete cocycle and normalized even auxiliary class of Theorem 11BN.1. The transfer-normal product has an explicit closed self-adjoint jointly weighted Dirac operator on its actual tensor carrier. Its completed domain is exactly local normal \(H^1\), local auxiliary graph domain and the distributional \(L^2\) operator condition. Its localized resolvents are ordinary scalar compacts. The phase has the actual product connections, positivity and holonomy covariance.

After Čech return, a bounded auxiliary correction glues both the operators and their complete domains on the original transversal. On the entire physical counting/plaque carrier, Parseval compression of this concrete normal phase gives an explicit bounded graph phase with scalar Kasparov defects and the inverse normal Clifford high-frequency limit on every compact physical column. It represents the literal reduced graph product, has degree \(q\) and pairs with the original positive disk as \(+1\). In particular it applies to the incomplete flat foliations of Theorem 11BN.1. Compactness and transverse completeness are not added.

The full inverse line, physical signs and final right Clifford factor are retained. The construction starts with the actual transfer module and proves its multiplication maps. The historical graph-normal question is due to Connes; all expressions and proofs below are independently written.

The joint differential expression on the actual covered normal carrier is

\[
 A=\alpha^{1/2}\bigl(D_\kappa\otimes1+
                  \Gamma_\kappa\otimes N_{\rm inv}\bigr)\alpha^{1/2}.
 \tag{DJ.1}
\]

**Proof.** We construct the controlled auxiliary, close the joint operator, verify the product, return through the actual cover, and then test its explicit physical graph phase.

### 11BO.1. Explicit scalar auxiliary construction

Let `B=C_r^*(Gamma)`, with Gamma countable, and let even
`kappa in KK(B,C)` satisfy `[u_B]kappa=+1`. Take a bounded graded
representative on its essential unital source space. Its odd
self-adjoint Fredholm operator can be replaced by its sign, zero on
its finite kernel, by a compact perturbation. Add a finite zero-source
space with opposite graded partners for all kernel vectors, and pair
these vectors. On the resulting space `tilde H` one has

\[
 F=F^*,\quad F^2=1,\quad F\text{ odd},\quad
 [F,\phi(a)]\in\mathcal K,\qquad P=\phi(1).
 \tag{DJ.2}
\]

The finite pairing changes no class. `P` need not be identity; an odd
unitary on a unital source of nonzero index is not being assumed.

Pair a basis `e_j` of `tilde H^+` with `Fe_j` in `tilde H^-`, giving
the pair the same positive eigenvalue tending to zero. This defines
an even injective compact `h_0` of dense range commuting with `F`.
Enumerate `1`, all group unitaries and their adjoints, and a countable
dense source family as `a_j`.

Finite spectral projections of `h_0` approximate the compact ideal.
Their compact commutators with each fixed bounded source operator tend
weakly to zero: trace-class tests and strong convergence prove this.
Hahn--Banach separation puts zero in the norm-closed convex hull of
each tail for any finite family of these commutators. Choose finite
convex combinations of late projections, also late enough for all
current compact tail tests. All those projections equal one on the
previous cutoff support. Including the first n basis pairs at stage n
gives positive finite-rank even cutoffs, commuting exactly with F, with

\[
 u_{n+1}u_n=u_n,\quad u_n\to1\text{ strongly},\quad
 \|[u_n,\phi(a_j)]\|\le2^{-n},\quad
 \|[F,\phi(a_j)](1-u_n)\|\le2^{-n}\quad(j\le n).
 \tag{DJ.3}
\]

Set `R_N=1+sum_{n<=N}(1-u_n)`, `r=1+sum_n(1-u_n)` and `l=r^{-1}`.
Nesting makes the sum finite at every fixed basis coordinate. The
weights tend to infinity off finite sets since all initial cutoff
supports are finite. Thus l is even injective compact with dense range
and commutes with F. On `Ran l`, `D=Fl^{-1}` is self-adjoint, with
compact everywhere resolvents

\[
 (D\pm i)^{-1}=l(F\mp il)(1+l^2)^{-1}.
 \tag{DJ.4}
\]

For `xi in Ran l`, `R_N xi -> l^{-1}xi`. DJ.3 makes
`[R_N,phi(a_j)]` norm convergent. The equation for
`R_N phi(a_j)xi`, multiplied by l, proves that `phi(a_j)xi` belongs
to `Ran l` with the resulting inverse image. This proves preservation
of the full completed domain. The compact tail tests also make
`[F,phi(a_j)]R_N` norm convergent. Therefore

\[
 [D,\phi(a_j)]
     =F[l^{-1},\phi(a_j)]+[F,\phi(a_j)]l^{-1}
 \tag{DJ.5}
\]

extends boundedly. Products obey the completed Leibniz rule and the
adjoints were controlled from the outset.

Since 1 was included, `C=[D,P]` is bounded. The off-diagonal operator

\[
 B_0=PD(1-P)+(1-P)DP=C(2P-1)
 \tag{DJ.6}
\]

is bounded, self-adjoint and odd. `D-B_0` is self-adjoint on the same
domain by the Neumann resolvent formula at sufficiently large imaginary
parameters; its resolvents remain compact. It commutes with P on that
domain. Hence

\[
 H_\kappa=P\widetilde H,\quad
 D_\kappa=(D-B_0)|_{P\operatorname{Dom}D},\quad
 V_\delta=\phi(\lambda_\delta)|_{H_\kappa}
 \tag{DJ.7}
\]

are an odd self-adjoint compact-resolvent operator and an even unital
reduced representation. Every V and its inverse preserve the whole
domain, and

\[
 C_\delta=[D_\kappa,V_\delta]\in\mathcal B(H_\kappa),\qquad
 C_{\delta\epsilon}=C_\delta V_\epsilon+V_\delta C_\epsilon.
 \tag{DJ.8}
\]

No uniform group-wide bound is needed. A compact actual arrow test
has finitely many labels.

The bounded transform of D differs compactly from F by DJ.4. The
bounded perturbation B_0 changes its phase compactly by the two-resolvent
integral, with integrable bound `||B_0||/(1+t^2)`. The remaining block
has zero source and is degenerate. DJ.7 thus represents exactly kappa,
and `ind D_kappa^+=+1`. This proves the controlled provider, rather
than invoking an arbitrary unboundedification result.

Take its graded finite squared-spectral windows, their V translates,
and a countable resolvent-generated graph-dense family as a core.
Their complex span is graded, countable and group invariant. Enlarge it
to include the eigenbasis and its translates, so finite D-squared
projections map it into itself. Invariance under higher powers is not
claimed. The construction includes the zero-source removal and controls the
completed domains for every prescribed discrete label.

### 11BO.2. The literal tensor unitary and actual reduced action

Let `A_m=C^*max(G)` on the covered transversal and let the transfer
module be `E_c=A_m tensor H_kappa`. Let `H_N=L^2(T,S_inv)` carry the
actual maximal normal representation from Section 11BM. The map

\[
 (a\otimes v)\otimes u\longmapsto v\otimes\pi_N(a)u
 \tag{DJ.9}
\]

is balanced and isometric by the actual module inner product. Its range
is dense by nondegeneracy of the unit action, so it is onto. This is
coefficient multiplication, not an abstract Hilbert-space renaming.

For the discrete cocycle c its actual action is

\[
 (\Pi(k)u)(x)=\sum_{g:r(g)=x}k(g)
              (V_{c(g)}\otimes U_g^{\rm inv})u(s(g)).
 \tag{DJ.10}
\]

The direct diagonal theorem proves continuity for the reduced norm.
The bare normal representation has not independently been declared
reduced-continuous. The full inverse transport includes its line phase.
On a connected bisection the label is constant, so the normal derivative
of V is zero. All actual source/range domains remain in the kernels.

### 11BO.3. Joint completed domain and scalar local compactness

Choose a smooth proper exhaustion r of the q-dimensional transversal,
also increasing between countably many components, and put

\[
 \alpha=(1+|dr|_g^2)^{-1/2},\quad \eta_n=\zeta(r/n),\quad
 \sup|\alpha d\eta_n|_g\le\|\zeta'\|_\infty/n.
 \tag{DJ.11}
\]

The metric `alpha^{-2}g` is complete by the proper-sublevel Cauchy
argument. The original metric remains the normal symbol data. On compact
buffers alpha is bounded away from zero.

For `B=D_kappa tensor1+Gamma_kappa tensor N_inv`, the graded cross terms
cancel on compact smooth finite auxiliary spectral vectors:

\[
 \|B\xi\|^2=\|D_\kappa\xi\|^2+\|N_{\rm inv}\xi\|^2.
 \tag{DJ.12}
\]

The actual weighted expression is

\[
 A=\alpha D_\kappa\otimes1+
       \Gamma_\kappa\otimes(\alpha N_{\rm inv}+\tfrac12C(d\alpha)).
 \tag{DJ.13}
\]

The product-rule correction is not separately self-adjoint; the whole
expression is symmetric. In particular its auxiliary term is alpha D,
not an unweighted D. From `A=alpha B+Gamma C(dalpha)/2`, DJ.12,
normal Clifford ellipticity and a compact lower bound for alpha follow

\[
 \|\chi u\|_{H^1}+\|D_\kappa\chi u\|
 \le C_\chi(\|\widetilde\chi Au\|+\|\widetilde\chi u\|).
 \tag{DJ.14}
\]

This is full q-dimensional Hilbert-valued normal H1 with the finite
right Clifford coefficient. It is not an estimate on frame units
which omits vertical ellipticity.

To prove DJ.14 on maximal distributions, let Q_m be finite squared
spectral projections of D_kappa. They commute with D, the grading,
normal coefficients and alpha. Apply finite-coefficient normal interior
regularity to Q_m u. Localizing the equation introduces only
`[A,chi]=alpha Gamma C(dchi)`. DJ.12 gives estimates independent of m.
Weak Sobolev limits and the closed spectral domain of D then put
`chi u` in `H1 cap L2 Dom D`. Conversely this compact joint space is
graph approximated by Q_m, normal smoothing and finite chart partitions.
Consequently the maximal adjoint domain of the tensor core is exactly

\[
 \{u\in L^2: u\in H^1_{\rm loc}\cap
       L^2_{\rm loc}\operatorname{Dom}D_\kappa,
                    \ Au\in L^2\text{ distributionally}\}.
 \tag{DJ.15}
\]

The completed product rule
`A eta_n u=eta_n Au+alpha Gamma C(deta_n)u` and DJ.11 approximate every
maximal vector by compact joint vectors in the operator graph norm.
Thus minimal equals maximal and the closure is self-adjoint on DJ.15.
A global unweighted joint Sobolev domain is not asserted.

For `R_z=(A-z)^{-1}`, DJ.14 bounds `chi R_z` in the compact joint space.
Its auxiliary tail above `|D|>M` has norm at most C/M. The retained
finite auxiliary part is scalar compact by finite-coefficient Rellich.
These approximate in operator norm, proving `chi R_z` scalar compact.
Adjoints give `R_z chi` compact, hence `f h(A)` is compact for
`f in C_0(T)` and `h in C_0(R)`.

All of this also applies to local expressions `D_kappa+K(x)`, with
smooth operator-norm locally bounded odd self-adjoint K. Strip K from
the local equation before using Q_m; no commutation of Q_m with K is
needed. Local joint domains and all cutoff arguments remain valid.

### 11BO.4. Positive joint rescaling and bisection covariance

For real compactly supported f set `A_t=e^{tf/2}Ae^{tf/2}`. The completed
domain is unchanged and graph norms are uniformly equivalent, since

\[
 A_t=e^{tf}A+\tfrac t2e^{tf}[A,f],\qquad
 B_t=[A_t,f]=e^{tf}\alpha\Gamma_\kappa C(df).
 \tag{DJ.16}
\]

For `R_z=(A_t-z)^{-1}`, common-graph differentiation gives

\[
 \dot R_z=-fR_z-zfR_z^2+
                \tfrac12R_zB_tR_z+zR_zB_tR_z^2.
 \tag{DJ.17}
\]

The leading terms in the strong phase integral have the graph-vector
bound `||f A_t R_z^2 v||<=||f|| ||A_t v||/(1+tau^2)`. Integrating them
spectrally and differentiating on those graph vectors therefore gives

\[
 \dot F_{A_t}=fA_t(1+A_t^2)^{-3/2}
 +\frac1\pi\int_0^\infty\sum_\pm
  (\tfrac12R_{z_\pm}B_tR_{z_\pm}+z_\pm R_{z_\pm}B_tR_{z_\pm}^2)d\tau,
 \quad z_\pm=\pm i\sqrt{1+\tau^2}.
 \tag{DJ.18}
\]

The remainder is bounded by `3||B_t||/(1+tau^2)`. Every term is compact
by joint scalar local compactness, including the infinite auxiliary
identity factor. Norm-resolvent continuity and the integrable bound give
a norm-continuous compact derivative. Integration and density prove
`F_A1-F_A` compact. A bounded self-adjoint compactly supported
operator-valued potential has the same compact phase comparison by
the resolvent identity. This reproduces WN.5--WN.7 for the joint sum.

For an actual compact bisection h with label delta, its transport is
`V_delta tensor U_inv,h`. The auxiliary conjugation error is bounded
by DJ.8. The normal connection error is a smooth finite-coefficient
zeroth-order endomorphism, since the full Clifford symbols agree.
No normal derivative of V occurs. On a compact buffer the transport is

\[
 U_h A U_h^{-1}=\beta^{1/2}(B+K_h)\beta^{1/2},\qquad
 \beta=\alpha\circ h^{-1},
 \tag{DJ.19}
\]

with K_h bounded there. Extend `log(beta/alpha)` by a compact cutoff
and rescale the whole sum by DJ.16--DJ.18. Extend the remaining bounded
error as a compactly supported self-adjoint potential. This target
operator agrees with the transported one on a smaller buffer.

A compact identification cutoff sends the full source domain into the
full target domain by DJ.14--DJ.15. Its rectangular operator difference
is bounded with compact support: both leading terms cancel. The
rectangular resolvent identity has compact integrands with an integrable
C/(1+tau^2) bound. Thus the phase intertwining error is compact. A finite
bisection partition proves `[F_A,Pi(k)]` compact. Compact source cutoffs
give `Pi(k)(1-F_A^2)` compact. Reduced-norm density extends all defects.
Boundedness of `[A,Pi(k)]` is not claimed; the noninvariant speed may
still give first-order terms there.

The same proof applies before diagonal pullback to the spatial source
`A_m tensor B`, using an independent V_delta in the auxiliary factor.
Thus this is a cycle for the full external-product source as well.

### 11BO.5. The actual product connection and positivity

Use a compact normal buffer and a complete reference on Rq whose finite
normal metric, inverse coefficient and connection agree exactly there,
and are Euclidean outside a compact set. This is the WB.2--WB.6 finite
reference construction. Keep D_kappa fixed and extend alpha to a positive
alpha_ref equal to alpha on the buffer and one off a compact set.
The unweighted reference is

\[
 B_{\rm ref}=D_\kappa\otimes1+\Gamma_\kappa\otimes N_{\rm ref}.
 \tag{DJ.20}
\]

It is self-adjoint on the joint domain and has scalar locally compact
resolvents. Joint rescaling changes its phase compactly by Section 11BO.4.
Rectangular locality identifies the actual and reference phases modulo
local compacts on a smaller disk. The unbounded auxiliary terms agree
exactly on the comparison buffer; they have not been discarded.

For a homogeneous v in a finite auxiliary squared-spectral window,
the reference creation difference consists of finitely many auxiliary
coefficients times the normal functions

\[
 (1+\lambda^2+N_{\rm ref}^2)^{-1/2},\qquad
 \frac{N_{\rm ref}}{(1+\lambda^2+N_{\rm ref}^2)^{1/2}}
             -\frac{N_{\rm ref}}{(1+N_{\rm ref}^2)^{1/2}}.
 \tag{DJ.21}
\]

Both vanish at normal spectral infinity and become scalar compact after
a compact normal cutoff. No global compact resolvent on Rq is used.
Joint and normal phase locality transfer this statement to the actual
weighted phases. For a transfer creation `b tensor v`, DJ.9 makes the
map `u -> v tensor pi_N(b)u`. A compact core b has compact normal range;
DJ.21 gives its first error, and `T_v[F_N,pi_N(b)]` gives the additional
compact error. A finite chart partition treats its whole actual kernel.
Norm density of module vectors and finite spectral v then gives

\[
 F_A T_\xi-(-1)^{|\xi|}T_\xi F_N\in\mathcal K,
                  \quad\text{and the adjoint equation}
 \tag{DJ.22}
\]

for every vector of the entire transfer module.

Let `P_1=F_kappa tensor1`, where F_kappa is the bounded transform of D.
For the unweighted reference, joint spectral calculus gives the exact
positive bounded operator

\[
 \{P_1,F_{B_{\rm ref}}\}
  =\frac{2D_\kappa^2}{(1+D_\kappa^2)^{1/2}
                 (1+D_\kappa^2+N_{\rm ref}^2)^{1/2}}\ge0.
 \tag{DJ.23}
\]

For the actual weighted phase, reference locality gives
`chi {P_1,F_A} chi >=0 modulo compacts` on every compact buffer.
P_1 commutes with scalar cutoffs and F_A has compact scalar commutators.
A finite squared partition on a compact support combines these
inequalities. For a core source a, `Pi(a)=Pi(a)chi_s` localizes its
positivity sandwich to that compact support. Density gives the full
Section 0 positivity criterion.

DJ.22--DJ.23 identify the phase with the actual external product of
kappa and the maximal normal class before diagonal restriction. This
does not assume an exact positive formula for the variable-weight
anticommutator itself.

### 11BO.6. Actual Cech return as a completed transverse Dirac operator

If the cocycle was supplied directly on the original transversal, the
auxiliary bundle is already the constant H_kappa bundle. Otherwise use
its actual Cech unit transitions. An original identity arrow from chart
j to chart i has locally constant label t_ij, with the exact transition
cocycle law. The unitaries V_tij define a flat graded Hilbert bundle
E_kappa on the original transversal. They preserve Dom D with its
completed graph topology by DJ.8. No flatness of the normal metric is
required for this consequence of a discrete cocycle.

Local constants D_kappa need not agree across those transitions. For
a locally finite smooth partition phi_j subordinate to the chart cover,
define in chart i

\[
 D_i(x)=\sum_j\phi_j(x)V_{t_{ij}}D_\kappa V_{t_{ij}}^{-1}
       =D_\kappa+K_i(x).
 \tag{DJ.24}
\]

Every difference in the latter sum is bounded self-adjoint and odd.
The sum is locally finite. Its operator-norm normal jets are bounded
on compact buffers; supports lie inside transition domains, so extension
by zero of each partition term is smooth. Exact transition composition
gives `D_i=V_til D_l V_til^{-1}`. The completed domain bundle agrees
exactly as well. This is an actual global auxiliary field, not local
expressions agreeing only modulo compacts.

Combine its flat connection with the specified finite inverse normal
Spin-c connection. On the original base choose an exhaustion speed
alpha_0 and form

\[
 A_0=\alpha_0^{1/2}
        (D(x)+\Gamma_\kappa N_{E_\kappa\otimes S_{\rm inv}})
                                                   \alpha_0^{1/2}.
 \tag{DJ.25}
\]

After stripping K_i, Section 11BO.3 proves the closed self-adjoint domain
and scalar local compacts. Section 11BO.4 proves original holonomy phase
covariance. An arrow changes D(x) by a locally bounded auxiliary error;
the finite normal connection error is unchanged. It is not necessary
that the independently completing cover speed descend to alpha_0.

The actual original-to-cover equivalence is
`Z=G times_{s,p} tilde T`. Let tau_z be the descended auxiliary chart
frame at p(z), and define U^E_g by the original arrow's chart label and
those frames. Its cocycle law follows from the actual covered cocycle.
The Morita carrier multiplication is, on compact cores,

\[
 \Omega(\xi\otimes u)(x)=
 \sum_{g:r(g)=x,\ z\in p^{-1}(s(g))}
       \xi(g,z)(U^E_g\tau_z\otimes U_g^{\rm inv})u(z).
 \tag{DJ.26}
\]

Compact supports make this a finite local-bisection sum. Expanding the
inner product produces `U_g^*U_g'=U_{g^{-1}g'}` and precisely the
equivalence's right convolution inner product. Change of variables uses
the actual local isometry and prescribed density. Unit-arrow chart
bumps span dense normal sections. Hence completion is an onto isometry
to `L^2(T_original,E_kappa tensor S_inv)`. This proves its norm and
range by actual kernel formulas, not an abstract Hilbert identification.

For each compact smooth Morita column in DJ.26, match source and target
speeds on its compact bisection buffers by positive joint rescaling.
Its remaining operator difference is bounded with compact support:
the terms are DJ.8's auxiliary error, K_i, the finite Spin-c connection
error and the derivative of xi. The leading auxiliary and normal terms
cancel simultaneously. Rectangular locality proves its phase creation
connections to DJ.25. Norm approximation treats every completed column.
The first Morita operator is zero, so its product positivity is automatic.

The reverse Morita multiplication without the auxiliary factor gives
the induced maximal normal representation on the covered transversal.
The covered weighted normal phase satisfies its creation equations by
the same buffer comparison; its speed can complete individual open chart
components. Consequently the entire typed identity is
`X_r Theta_c X_m^* x_N`, represented by DJ.25 on DJ.26's actual carrier.
It is not an independently chosen class sharing only its integer.

### 11BO.7. A concrete bounded phase on the full physical graph carrier

Let E_Z be the proved normalized counting/plaque correspondence from
the original foliation algebra A_graph to A_r(G_T). Retain its full
physical module. The scalar Hilbert carrier is literally

\[
 \mathcal H_{\rm graph}=\mathcal E_Z\otimes_{A_r(G_T)}
       \mathcal E_\Theta\otimes_{A_m(G_T)}L^2(T,S_{\rm inv}).
 \tag{DJ.27}
\]

The last two factors are identified with the concrete transversal
carrier only by the multiplication/Morita maps proved above. No abstract
Hilbert-space renaming is used for the graph.

Choose a countable module Parseval frame of the even correspondence
E_Z. Its standard isometry W into `ell^2 tensor A_r` is defined by
the actual module inner products; put P=WW*. On the induced range
of P in `ell^2 tensor H_0`, where H_0 is DJ.25's carrier, define

\[
 F_{\rm graph}=P(1_{\ell^2}\otimes F_{A_0})P.
 \tag{DJ.28}
\]

This is an explicit bounded operator on DJ.27 through the actual module
isometry. It is odd, self-adjoint and contractive, with the final right
Clifford action unchanged. No derivative of the frame or unbounded
domain preservation of P is assumed.

For a creation T_xi from H_0, its standard embedding has entries
`pi(<xi_j,xi>)`. Their square sum converges in module norm. Each of
the finitely many retained entries has compact phase commutator; the
remaining column tail is uniformly small in operator norm. Therefore

\[
 F_{\rm graph}T_\xi-T_\xi F_{A_0}\in\mathcal K,
                 \qquad\text{and the adjoint equation}
 \tag{DJ.29}
\]

for every vector of the full graph module. These are ordinary scalar
compact operators between the literal Hilbert spaces.

For a graph rank-one coefficient `theta_xi,zeta=T_xi T_zeta^*`,
write `G=F_graph`, `F_0=F_A0` and `K_xi=GT_xi-T_xiF_0`.
The exact identities are

\[
 \begin{split}
 (G^2-1)T_\xi&=T_\xi(F_0^2-1)+GK_\xi+K_\xi F_0,\\
 [G,T_\xi T_\zeta^*]&=K_\xi T_\zeta^*-T_\xi K_\zeta^*.
 \end{split}
 \tag{DJ.29a}
\]

The first term on the first line is scalar compact. Indeed, for
`C=T_xi(F_0^2-1)`,

\[
 C^*C=(F_0^2-1)\pi(\langle\xi,\xi\rangle)(F_0^2-1)
                       \in\mathcal K.
 \tag{DJ.29b}
\]

The actual coefficient localizes the transversal square defect.
If C*C is compact, its finite spectral projections approximate C in
operator norm, since the squared norm of each discarded part is the
corresponding norm of C*C. Thus C is compact as a rectangular operator.
Every other term in DJ.29a is compact by DJ.29. Multiplying the first
identity by T_zeta^* gives the rank-one localized square defect.
Rank-one density in the entire graph compact algebra gives all scalar
Kasparov defects.
DJ.29 and the zero first Morita operator prove that DJ.28 represents
exactly the induced graph class. The full E_Z, rather than a finite
plaque corner, is used.

The operator also has an exact geometric normal-phase test. For real
compactly supported f, the completed transversal domain gives

\[
 t^{-1}e^{-itf}A_0e^{itf}
       =t^{-1}A_0+\alpha_0\Gamma_\kappa iC(df).
 \tag{DJ.30}
\]

The bounded multiplication on the right is the strong-resolvent limit.
Test the resolvent on `(alpha_0 Gamma iC(df)-z)v`, for v in Dom A_0;
the error is bounded by `||A_0 v||/(t |Im z|)`, and those tests are
dense. First let v belong to the compact joint core, supported in a
buffer where |df| is bounded away from zero. Put
C=alpha_0 Gamma_kappa iC(df), and B_t=t^{-1}A_0+C. For t>=1,
`||B_t v|| <= ||A_0 v||+||C v||`. The spectral mass above M is
at most `(||A_0 v||+||C v||)^2/M^2`. The limit C has a gap on v,
so a compact spectral cutoff near zero has limiting mass zero.
On every retained compact interval away from zero, the functions
`s/(t^{-2}+s^2)^{1/2}` converge uniformly to sign(s).
Strong-resolvent convergence and these two tail bounds therefore prove
the following limit on the compact joint core. Approximate an arbitrary
compact L2 vector in the same buffer by such core vectors. Both phases
are contractions, so density proves the same limit for that vector:

\[
 e^{-itf}F_{A_0}e^{itf}v
       \longrightarrow\Gamma_\kappa\frac{iC(df)}{|df|_g}v.
 \tag{DJ.31}
\]

The factor i is necessary: `[N,f]=C(df)` is skew-adjoint, whereas
the oscillatory principal symbol iC(df) is self-adjoint. The positive
speed cancels from the normalized Clifford phase.

On every actual compact smooth physical graph column, the packets
`e^{itf}v` tend weakly to zero. For smooth finite auxiliary tests this
follows by integration by parts using the compactly supported vector
field `grad f/|df|^2`; its derivative bounds are finite on the buffer.
Approximation of both vectors in L2 proves it for the full auxiliary
space. Its compact error in DJ.29 therefore
vanishes on these packets. DJ.31 proves that DJ.28 retains precisely
the actual inverse normal Clifford phase on that column. This is a
proved geometric column statement, not a symbol inferred by taking an
arbitrary unbounded inverse of the graph phase. In exact column form,

\[
 \|F_{\rm graph}T_\xi e^{itf}v-
       T_\xi e^{itf}\Gamma_\kappa iC(df)|df|_g^{-1}v\|\longrightarrow0.
 \tag{DJ.31a}
\]

On a smaller original positive disk, the Cech unitary and Section 11BO.5
give the even auxiliary ordinary index +1 times the Section 11BM
inverse. For a larger prescribed disk U use Lemma 6.0e's exact
`b_U=b_V[j_VU]`. The original normalized counting/plaque correspondence
carries this same Bott class to the graph. Its scalar pairing is +1.

With `k_q=q(q-1)/2`, the finite inverse remains `Pi^{k_q} C_q`, its
physical generator is `(-1)^{k_q+1} L_{t_j}`, and the coherent full lift
acts as `z^{-1}rho_t(s)`. The even auxiliary precedes the normal factor.
The right Clifford block stays last until the fixed contraction; no
odd factors are exchanged and no determinant-only phase is substituted.

### 11BO.8. A concrete domain failure of naive unbounded graph compression

DJ.28 is a concrete bounded graph-normal phase with the scalar defects
and geometric column test above. The statements do not alone show
that it is the phase of `P(1 tensor A_0)P` on a full invariant joint
domain. Naive compression can fail even for a two-copy cover of the
point foliation on `(0,1)`.

Use identical normal coordinates and trivial auxiliary transition.
A reduced character of C*r(Z) permits V_1=1. Choose an infinite
compact-resolvent index-one D_kappa; a balanced infinite degenerate
summand can supply the infinite part. On the two copies set

\[
 \alpha_1=x^2(1-x)^2,\quad \alpha_2=x(1-x),\quad
 A_i=\alpha_i^{1/2}(D_\kappa+\Gamma_\kappa N)\alpha_i^{1/2}.
 \tag{DJ.32}
\]

Both speed metrics are complete. The constant average projection
`P=(1/2)[[1,1],[1,1]]` implements trivial two-copy descent. It does
not preserve `Dom A_1 directsum Dom A_2`.
For the same maximal-domain proof, take exhaustions
`r_1=1/x+1/(1-x)` and `r_2=-log(x(1-x))`; their respective products
`alpha_i |dr_i|` are bounded. Their scaled exhaustion cutoffs satisfy
the required derivative estimates, so both A_i have the exact closed
maximal domains used in the countertest.

Choose a rapidly increasing positive auxiliary eigenvalue subsequence
lambda_n, set x_n=lambda_n^{-1/3}, and choose disjoint normalized smooth
bumps psi_n of width comparable to x_n near x_n. Take the subsequence
with x_{n+1}<x_n/4 and summable x_n^2. For its eigenvectors e_n put

\[
 u_1=\sum_n x_n^2\psi_n e_n,\qquad u_2=0.
 \tag{DJ.33}
\]

The alpha_1 auxiliary contribution on the nth summand has size comparable
to x_n and is square summable. Its normal derivative and product-rule
terms have size comparable to x_n^3 and are also square summable.
Thus u_1 belongs to Dom A_1 by its exact maximal domain. The alpha_2
auxiliary contribution has size comparable to one on every summand
and is not square summable. The normal term cannot cancel it:
Gamma e_n has eigenvalue -lambda_n and is orthogonal to e_n. Hence
u_1 does not belong to Dom A_2. But P(u_1,0)=(u_1/2,u_1/2), proving
the completed-domain failure. Even for comparable weights the
off-diagonal commutator contains `(alpha_1-alpha_2)D_kappa` and is
generally unbounded.

At the Cech-transversal level the repair is already DJ.24--DJ.26:
descend one joint speed to the original base and construct agreeing
domain operators before forming the unbounded sum. For a globally
unbounded expression on the entire physical graph, the corresponding
repair would be an actual graph connection and synchronized speed
with proved projection/domain maps and localized derivative estimates.
Neither an ordinary scalar index nor bounded phase connections provide
that extra interface. No arbitrary inverse of DJ.28 is offered as a
first-order geometric graph operator.



![Jointly weighted Dirac, completed Čech return and actual graph phase](../figures/joint-graph-dirac.png)

**Figure 11BO.** Theorem 11BO.1, DJ.7–DJ.33. Both leading operators use the same positive speed. Local normal \(H^1\) and auxiliary graph estimates give scalar compacts by finite squared-spectral windows and Rellich. The arrows \(\Omega\) and \(W\) are the actual Morita multiplication and module-frame isometry; the graph phase is defined on their entire induced range. Its oscillatory test retains \(\Gamma_\kappa iC(df)/|df|\) and the original disk pairing \(+1\). The two-copy counterexample records uniform two-sided norm estimates, denoted \(\asymp\), and shows why bounded phase compression alone supplies no unbounded domain assertion. Human source: Connes's graph-normal question. Original expression: CC0 1.0.

Editable SVG · Reproduction source and exact data.

## 11BP. Curved graph geometry with locally constant Killing rank

**Theorem 11BP.1.** Let the original foliation have its invariant normal metric and full coherent Spin-c data. Suppose the dimension of the full local Killing-germ algebra is locally constant on its complete original transversal \(T\). The metric and actual holonomy domains may be incomplete, and \(T\) may be noncompact. There is an explicit reduced scalar graph class of the original degree \(q\), with the full inverse normal coefficient and original positive-disk pairing \(+1\). Its representative is the closed jointly weighted transversal Dirac operator and concrete bounded physical graph phase of Theorem 11BO.1.

The transfer is supplied here. On each constant-rank chart, compact actual neutral words have a uniform continuation radius. Their frame endpoints have a polynomial count, which proves equality of the entire neutral full and reduced norms. The full Killing-basis matrices then give a locally constant discrete cocycle. The resulting transfer has the actual Čech return and the required original-disk normalization. Killing rank zero is included; its original normal degree remains \(q\).

**Proof.** Let \(k(x)\) be the dimension of the Killing germs. Evaluation of a Killing field and its skew covariant derivative is injective by the Killing-transport ODE. Hence \(0\le k(x)\le q+q(q-1)/2\). Its locally constant level sets \(T_j\) are clopen and invariant: an actual local isometry transports the full germ algebra isomorphically. They are genuine open \(q\)-manifolds. In the following chart proof take \(S=T_j\) and \(O\subset T_j\). The frame and continuation estimates are applied to the actual original arrows.

### 11BP.1. The actual neutral chart and its central fields

Let O be one connected normal metric chart. Suppose `X_1,...,X_j` are
Killing fields on O whose germs form the full local Killing algebra at
every point of the rank-j stratum `S cap O`. Their first jets remain
independent after shrinking O. Their bracket coefficients are constant:
express a bracket as a constant combination at one stratum point, then
use Killing-field unique continuation on the connected common domain.
Write this finite-dimensional Lie algebra as k.

Let `N => S cap O` be the actual holonomy subgroupoid consisting of
arrows whose local isometries preserve every X_a. A neutral germ
representative preserves these fields on its connected ambient normal
domain, not just at its one stratum point: germ equality extends by
Killing unique continuation. Thus all compact neutral bisections have
ambient representatives which preserve the fixed full basis.

Take a basis `Z_1,...,Z_z` of the center Z(k), as constant combinations
of the X_a. These fields commute. They are used below only as potential
metric flows for comparison; they are not declared to be original
holonomy arrows, and no Spin-c lift is fabricated for them.

On the oriented orthonormal frame bundle `P_O -> O`, their natural
infinitesimal frame lifts are independent over S: a vanishing lifted
vector means that both the field value and its first derivative vanish,
and therefore its Killing germ is zero. They remain independent on an
ambient neighborhood of every compact frame set over S. The lifted
Z_a commute, so that neighborhood has center-flow coordinates

\[
 (t,v)\in\mathbb R^z\times V,\qquad \widetilde Z_a=\partial_{t_a}.
 \tag{NK.1}
\]

The actual frame action `N acting on P|S` is principal: the target frame
specifies both target point and derivative, hence the original normal
isometry germ. Distinct original isotropy arrows have distinct target
frames. This retains every original arrow; it does not replace the
full inverse Spin-c transport by its derivative.

For q=0, a connected normal chart is a point and its neutral germ
groupoid is the unit groupoid. The following argument is needed for
the positive-dimensional normal metric charts.

### 11BP.2. A uniform-domain centralizer lemma

**Lemma NK.1.** Fix a point x in S, a frame p over x, and a positive
normal-ball radius epsilon whose closure lies in O. Among local
isometries defined on that whole ball, preserving every X_a, those
with first jet at x sufficiently near the identity germ are exactly
small flows of the central fields, as germs. On compact frame sets
the corresponding local statement can be chosen uniformly after
reducing the domain radius once.

Here is a proof which does not assume closure of the whole jet image
is a groupoid. In normal exponential coordinates, an isometry h with
first jet `(y,A)` at x has the formula

\[
 h(\exp_x w)=\exp_y(Aw).
 \tag{NK.2}
\]

For uniform-radius maps and jets near the identity this formula holds
on one fixed smaller ball and depends smoothly, with all derivatives,
on the finite-dimensional jet parameter. The target frame q=h_*p
is such a parameter. Arbitrary nearby q also defines the smooth
candidate formula, whether or not it is an isometry.

If uniform-radius neutral isometries have jets q_n tending to p and
normalized jet differences tending to a nonzero tangent vector w,
their normalized map differences tend in C-infinity on that fixed
ball to a vector field Y_w. Differentiating their metric preservation
gives `L_Yw g=0`; differentiating preservation of the X_a gives
`[Y_w,X_a]=0`. The first jet of Y_w is w, so it is nonzero. Its germ
belongs to the FULL Killing algebra k at x and centralizes that algebra.
It is therefore in Z(k). Its lifted value at p is tangent to the
central-flow orbit. No analyticity or fixed global group is assumed.

Choose a transverse slice W through p in the finite-dimensional frame
jet parameter space. By independence of the lifted central fields,
`(t,w') -> Phi_t(w')` is a local diffeomorphism. Decompose a nearby
jet as `q=Phi_t(w')`. Compose the neutral isometry with the inverse
small central flow. The result is still a potential neutral isometry,
on a fixed smaller ball, with jet w' in W. If arbitrarily near p one
could have w' different from p, normalized differences would produce
a nonzero tangent vector simultaneously transverse to and tangent to
the central-flow orbit, contradicting the preceding paragraph. Thus
w'=p. First-jet uniqueness makes h the corresponding central flow germ.
This proves the fixed-point statement.

For uniformity over a compact frame set, suppose the local assertion
failed along p_n tending to p. A uniform-radius map based at p_n is
defined on a common smaller normal ball about the fixed limiting base
point. Formula NK.2 varies smoothly in both source and target jets;
evaluate the same maps at p and apply the fixed-point statement there.
Killing unique continuation then gives the same central-flow identity
at p_n. This contradicts the failure. A finite cover of the compact
set gives the uniform version.

The inverse central flows in this proof are potential isometries only.
The conclusion compares an actual word's germ with a flow; it does not
enlarge N by inserting unused flow arrows.

### 11BP.3. Compact WORDS have a uniform actual extension radius

Let C be a compact symmetric arrow subset of N and set

\[
 K=s(C)\cup r(C)\subset S\cap O.
 \tag{NK.3}
\]

K is compact in the ambient metric chart as well. Cover C by finitely
many actual neutral bisections, with compact source/range test portions
inside larger connected ambient representatives. All those larger maps
and their inverses preserve the full fixed Killing basis. There is a
positive epsilon such that the epsilon normal balls at all tested
source and range points lie inside these larger domains, with a common
local normal-coordinate radius over K.

For a composable word `g_n ... g_1` with every factor in C, each prefix
endpoint lies in K. An isometry maps the epsilon source normal ball
onto the epsilon target normal ball, by NK.2 and the inverse-domain
buffer. Consequently every next factor is defined on the entire image
ball, and the whole word extends as an ACTUAL neutral holonomy map
on the epsilon ball at its initial source. The same is true for inverse
words. Epsilon does not depend on word length.

This is a compact-test assertion, not an assertion about all arrows
with endpoints in K. Arbitrary arrows need not be words in C. In
particular their own continuation domains may degenerate. The proof
neither changes the fixed metric nor supplies a complete bundle-like
extension.

Let `P_K` be the compact frame bundle over K. Fix a source frame p
over x in K, and let W_p be the endpoints of all lifted C-words from p,
including p. They all lie in P_K. If two such endpoints q_1,q_2 are
close, their relative word is an actual neutral map with uniform
extension radius and near-identity first jet at q_1. Lemma NK.1 gives

\[
 q_2=\Phi_t(q_1)\quad\text{for a small central-flow parameter }t.
 \tag{NK.4}
\]

Choose finitely many inner center-flow rectangles covering P_K, each
inside a larger buffered rectangle. Their diameters are smaller than
the uniform jet-nearness threshold, and the resulting central time
lies inside the larger rectangle. Choose their time widths below one
third of the uniform local time-injectivity radius. Every set `W_p` meets at most one central plaque in each
box. This conclusion is uniform in p: it uses the compact-frame
uniform version of NK.1 and the one epsilon from NK.3. When z=0,
it says there is at most one endpoint in each box.

### 11BP.4. Finite-state translation counting of all actual word arrows

Refine the lifted compact C-test by finitely many representative maps
h_l, each defined on a center-flow rectangle and with its image in a
center-flow rectangle. Add these finitely many source and target boxes
to the preceding finite cover; choose them equally small. Since an
actual neutral map preserves each lifted Z_a, in these coordinates it is

\[
 h_l(t,v)=(t+a_l(v),\psi_l(v)).
 \tag{NK.5}
\]

For the fixed source p, each visited box has only one possible transverse
plaque coordinate v, by NK.4. Hence every h_l edge has one fixed
translation increment for that source p. These constants may depend
on p; their NUMBER does not.

Chart-change edges at a common endpoint also have one increment for
each ordered pair of boxes. Here is the small-domain point needed when
an overlap has several components. On compact P_K the local action of
the independent commuting Z_a is injective for one common small time
ball. Choose every coordinate rectangle's t-width much smaller than
that time radius. If q,q' are word endpoints in two overlapping boxes,
each box puts them on its one visited plaque. Its coordinate difference
is a small central time taking q to q'. The two times must agree by
local injectivity and commutation of the flows. Therefore
`t_b(q)-t_a(q)` and `t_b(q')-t_a(q')` are equal, even if the overlap
components differ. No unseen overlap monodromy is counted as a new
edge. This argument uses only small local times, not a globally defined
abelian action.

Fix one initial rectangle containing p. For a word of length n, record
its sequence of rectangle states. Insert at most one initial chart-change
edge if its first representative uses another source rectangle, and at
most one chart-change edge between successive generator edges.
There are finitely many edge types: the finitely many h_l and the
ordered chart pairs. Call their number E and the number of states M.
The final central coordinate is its initial coordinate plus the sum
of the encountered translation constants. Addition is commutative.
Each edge count is at most `2n+1`. For this fixed source p, the final
state and those integer counts therefore allow at most

\[
 B_n=M(2n+2)^E
 \tag{NK.6}
\]

different endpoints. The coordinates are injective in every rectangle.
Principality of the frame action makes the actual word arrow unique
from p and its endpoint. Thus NK.6 bounds the number of ORIGINAL
source-fibre arrows represented by C-words of length n. M and E depend
on C but not on the source x or its chosen frame p.

This is the proper-discrete-return part of the argument: modulo the
central plaque coordinate, compact words have only finitely many states.
It does not assert properness of the full neutral endpoint relation.
Infinite actual arrow labels and isotropy are not identified; distinct
actual arrows remain distinct frame endpoints and are counted.

For z=0 the endpoint bound can simply be M, without translation counts.
The general polynomial estimate is sufficient and avoids any global
closed-local-group or Ado assertion at incomplete domains.

### 11BP.5. Norm equality for the ENTIRE neutral groupoid

Let f be a self-adjoint compact core kernel of N. Apply the preceding
argument to the compact symmetric support C of f. Every arrow in the
support of f^n is a C-word of length n. On the actual source fibre
`N_x=s^{-1}(x)`, the regular coefficient identity gives

\[
 \sum_{h\in N_x}|f^n(h)|^2
      =\|\lambda_x(f^n)\delta_{1_x}\|^2
      \le\|f\|_r^{2n}.
 \tag{NK.7}
\]

Cauchy--Schwarz and NK.6 bound the source I-sum by
`sqrt(B_n) ||f||_r^n`, uniformly in x. The range sum is the same bound,
because f^n is self-adjoint and inversion interchanges the two sums.
Every full representation is I-contractive, as proved in KT-CP-14's
full-norm/disintegration construction. Hence

\[
 \|f\|_{\max}^n
 =\|f^n\|_{\max}
 \le\|f^n\|_I
 \le\sqrt{B_n}\,\|f\|_r^n.
 \tag{NK.8}
\]

Since B_n is polynomial, its `1/(2n)` power tends to one. Taking nth
roots proves full norm at most reduced norm. The reverse inequality
always holds. Applying this to a* a gives equality for every compact
core a, and completion gives

\[
 C^*_{\max}(N)\ \cong\ C_r^*(N).
 \tag{NK.9}
\]

This proves the injectivity required by the transfer theorem. It is not
being deduced from a claim of topological amenability: the proof supplies
norm equality directly. No Reiter field whose continuity has not been
checked is silently introduced.

For the Killing-stratum Cech cocycle `(M_g,i-l)`, its neutral arrows
have i=l and preserve the full local Killing basis. The whole neutral
groupoid is the disjoint union of the chart groupoids just proved.
Its full and reduced algebras are the corresponding c_0 direct sums;
NK.9 applies to the entire neutral groupoid.

To identify its algebra with the neutral subalgebra in the covered
full algebra, use the actual open wide subgroupoid inclusion. Full
representations restrict to its core, giving the full homomorphism.
On regular source fibres its reduced inclusion is faithful by the
decomposition into pieces `N h`, each right-translation equivalent to
the neutral source fibre at r(h). The diagram with the two quotients
commutes. NK.9 therefore makes the full inclusion faithful and makes
the covered quotient injective on its neutral image.



### 11BP.6. Constant matrices and the actual whole-transversal graph

On each \(T_j\), choose a countable cover by connected normal charts carrying a full Killing basis. A basis at one point is represented on a common small chart; independence of its first jets persists after shrinking, and constant rank makes it a basis at every chart point. On an actual connected bisection, push its source fields to the target chart. At one point they are constant linear combinations of the target basis. Killing unique continuation proves that same combination on the whole connected representative domain. Thus the transport matrix is constant there and has the exact cocycle law

\[
 c_j(i,g,l)=(M_g,i-l),\qquad M_{gh}=M_gM_h.
 \tag{NK.10}
\]

These are labels of actual germs. No arrow is created outside an original continuation domain. Second countability and local constancy make the label image countable. Its generated group embeds faithfully into \(GL_{j+2}(\mathbb R)\): use \(M\) in the first block and \(\begin{psmallmatrix}1&n\\0&1\end{psmallmatrix}\) in the integer block. For \(j=0\) only the integer block remains. The scalar class of LF.2 and ST.9 has reduced unit index \(+1\).

The neutral arrows have equal chart indices and preserve the whole chart Killing basis. They are the disjoint union of the neutral chart groupoids just proved. The open-wide inclusion and regular coset argument above show that the full quotient of the covered groupoid is injective on this entire neutral algebra. Theorem 11BN.1 therefore gives its reduced-to-maximal transfer. Its compact-core Čech multiplication, on the essential open chart modules, returns an actual class \(\Theta_j\) on \(G|T_j\). This is exactly the discrete-cocycle input of Theorem 11BO.1.

There are finitely many nonempty levels and no arrow joins different levels. The actual algebras and graph correspondence decompose as

\[
 A_\nu=\bigoplus_j A_{j,\nu}\quad(\nu=r,m),\qquad
 \mathcal E_Z=\bigoplus_j\mathcal E_{Z,j}.
 \tag{NK.11}
\]

On the original foliation manifold these are the saturated clopen pieces determined by \(k\). Their projections are actual central clopen multiplier projections. Choose the union of the module frames of these pieces. With the actual multiplication and completed-domain return of DJ.24–DJ.26, the operator on the literal whole graph carrier is

\[
 F_{\rm graph}=\bigoplus_j P_j(1\otimes F_{A_{0,j}})P_j.
 \tag{NK.12}
\]

Every block has the scalar compact defects and full creation connections of DJ.29–DJ.29b, and the original inverse Clifford packet limit DJ.31a. Their finite sum has the same properties on the whole graph completion. All use the original normal degree \(q\), the full inverse line and the same final right \(Cl_q\) order. The transfers are even, so this orthogonal sum introduces no odd-factor permutation. The reduced representations are those factored by the direct diagonal theorem; the bare maximal normal representation is not being renamed reduced.

For the original connected positive disk \(U\), rank is constant and \(U\) lies in one \(T_j\). Choose a smaller positively oriented chart disk \(V\) with an interior buffer. The actual chart-unit Morita return and the positive open-disk identity give

\[
 b_U=b_V[j_{VU}],\qquad
 [j_{V,r}]\Theta_j=[j_{V,m}],\qquad
 \langle b_U,x_{\rm graph}\rangle=+1.
 \tag{NK.13}
\]

The last equality uses the full inverse normalization of 11BM and the actual normalized counting/plaque correspondence. It evaluates the original disk, with its original orientation and coefficient. It does not infer a whole-base unit family from the scalar auxiliary index.

For rank zero, the preceding polynomial count has zero-dimensional central plaques and the matrix block is empty. The same quotient, transfer and graph construction still apply. Killing rank zero is not normal codimension zero. If \(q=0\), the already specified discrete normal case retains its full graded line evaluation. This completes the theorem.

### 11BP.7. A nonflat incomplete example with genuine dense returns

Take

\[
 T=(\mathbb R/2\pi\mathbb Z)\times(0,1),\qquad
 g=dt^2+e^{2t^2}d\theta^2,\qquad
 h(\theta,t)=(\theta+2\pi/\sqrt2,t).
 \tag{NK.14}
\]

Orient the normal frame in the order \((\partial_t,e^{-t^2}\partial_\theta)\). The generator preserves this frame; choose its full Spin-c lift and use its powers. The suspension of the integer action \((s,x)\mapsto(s+1,hx)\) is a genuine foliation with this complete transversal and its actual action groupoid. Irrationality retains every integer label and every return. The action is nonproper on each circle, and the two missing t-ends are reached by radial curves of finite length.

For a warped metric \(dt^2+a(t)^2d\theta^2\), the coordinate connection has \(\Gamma^t_{\theta\theta}=-aa'\) and \(\Gamma^\theta_{t\theta}=a'/a\); substitution in the curvature formula gives \(K=-a''/a\). Here

\[
 K(t)=-2-4t^2,\quad K'(t)=-8t\ne0,
 \qquad\mathfrak k_x=\mathbb R\partial_\theta.
 \tag{NK.15}
\]

Indeed a local Killing field preserves K, so its t-component is zero. The mixed metric equation then makes its angular coefficient independent of t, and the angular equation makes it constant in \(\theta\). The full Killing rank is therefore exactly one everywhere, while the original normal dimension is two.

The full groupoid also has a direct norm check. If a self-adjoint compact kernel has integer support in \([-m,m]\), its nth power has at most \(2mn+1\) arrows in each source fibre. The regular coefficient identity and Cauchy–Schwarz give

\[
 \|f^n\|_I\le\sqrt{2mn+1}\,\|f\|_r^n,
 \qquad\|f\|_{\max}=\|f\|_r.
 \tag{NK.16}
\]

Taking nth roots proves the equality. Applying the C*-identity to arbitrary compact kernels completes it. This supplies the actual reduced norm rather than an index-based assertion. Theorem 11BP.1 now gives the original degree-two graph phase and positive Bott pairing on this curved, incomplete foliation. Its inverse coefficient is the one fixed in Theorem 7.17.


![Neutral compact-word count and a curved incomplete graph example](../figures/killing-rank-graph.png)

**Figure 11BP.** NK.3–NK.16. A fixed compact arrow test gives one actual continuation radius and compact frame set. Buffered central-flow states give the uniform polynomial word count and hence the entire neutral norm equality. The strip is the exact periodic coordinate domain for NK.14, with both t-ends omitted; it is not an isometric Euclidean embedding. Its metric, curvature samples and angular shift are stated exactly. The full Killing rank is one and the normal degree is two. The final diagram uses the actual reduced diagonal, Čech multiplication and entire physical graph carrier; the original inverse coefficient and positive disk remain. Original mathematical expression: CC0 1.0. Connes is credited for the graph-normal question.

Editable SVG · Reproduction source and exact data.

## 11BQ. Reduced restriction through arbitrary rank changes

**Theorem 11BQ.1.** Let \(G\rightrightarrows T\) be the original germ groupoid of a second-countable smooth Riemannian local-isometry pseudogroup. Keep its given metric and its actual local domains. For every invariant open \(U\subset T\), with \(F=T\setminus U\), the actual reduced sequence is exact:

\[
0\longrightarrow C_r^*(G|U)\longrightarrow C_r^*(G)
 \overset{\operatorname{res}_F}{\longrightarrow}C_r^*(G|F)
 \longrightarrow0.
\tag{IX.1}
\]

The conclusion also holds inside every invariant locally closed unit reduction. In particular it applies to every step of the full Killing-rank filtration, including singular closed layers. The metric need not be complete.

**Proof.** The original germ groupoid is Hausdorff: source, value and derivative separate different local-isometry germs, by connected first-jet uniqueness, and those jet data vary continuously in each actual bisection. Its bisection charts are locally compact and étale. Compact actual words have a single normal-ball radius for every word length. Evaluation at a nearby point is injective on their entire word-arrow set. This permits an infinite-component operator comparison with a closed-reduction regular fibre, retaining all original isotropy. We give the full norm argument.

The reduced ideal/quotient distinction is known as inner exactness; [Brix, Carlsen and Sims, Section2](https://eprints.gla.ac.uk/316431/2/316431.pdf) provides that terminology and its general context. The following argument is independent and uses the actual original geometry.

### 11BQ.1. Uniform actual balls for all words of one compact test

Fix a compact symmetric actual arrow set \(C\subset G\), and set

\[
K=s(C)\cup r(C).
\tag{IX.2}
\]

Choose finitely many actual bisection representatives covering \(C\),
each buffered inside a larger actual representative, and do the same
for their inverses. Compactness gives \(\varepsilon>0\) such that every
tested source and range has its normal \(\varepsilon\)-ball in those
larger domains. Reduce \(\varepsilon\) so all balls over \(K\) are normal
and lie in one fixed compact ambient neighborhood \(K'\). On a disjoint
chart transversal these choices are made in its finitely many tested
components; only pairs in the same component are compared.

For each tested local isometry \(a\) and \(v\) in that ball,

\[
a(\exp_y v)=\exp_{a(y)}(D_ya\,v).
\tag{IX.3}
\]

Since \(D_ya\) is orthogonal, the actual map sends the source normal ball
onto the target normal ball. The inverse buffer supplies the equality,
rather than only an inclusion.

Consider a composable word \(w=a_n\cdots a_1\), with all factors in
\(C\). Every prefix endpoint lies in \(K\). Induction using IX.3 proves
that the actual word representative is defined on the whole
\(B_\varepsilon(x)\), where \(x=s(a_1)\), independently of \(n\).
Every prefix has the same property, and inverse words have it too.

If \(z\in B_{\varepsilon/2}(x)\), evaluation of every such word at
\(z\) is an **actual arrow**. If \(x_w=w(x)\) and \(z_w=w(z)\),
the image of the short source geodesic gives

\[
d(x_w,z_w)\le d(x,z).
\tag{IX.4}
\]

The balls and distance in IX.3–IX.4 refer to the original local metric.
No assertion is made about arbitrary arrows with endpoints in \(K\).
In particular an arbitrary extra arrow used later to identify a coset
does not acquire this radius.

### 11BQ.2. All-length injective evaluation of actual word arrows

Let \(W_x(C)\subset G_x\) consist of all actual \(C\)-word arrows based
at \(x\), including \(1_x\). Their representatives in Section11BQ.1 define

\[
J_{x,z}:W_x(C)\longrightarrow G_z,
\qquad [w]_x\longmapsto[w]_z.
\tag{IX.5}
\]

This is well-defined: two word representatives with the same germ at
\(x\) agree on the common connected normal ball. Indeed equality of
local isometries propagates from its value and derivative, by IX.3;
the equality set of their value and derivative is both open and closed
in that ball. Hence they agree at \(z\).

It is injective by the **same argument in the reverse direction**.
If the two evaluated germs agree at \(z\), their representatives agree
on the connected ball \(B_\varepsilon(x)\), and therefore at \(x\).
The original isotropy arrows remain distinct.

Evaluation respects every generator edge whose factor lies in \(C\).
Thus it defines an isometry
\(V_{x,z}:\ell^2(W_x(C))\to\ell^2(G_z)\) by the basis injection.
It need not be onto and it is not a map of all of \(G_x\). No continuity
of this countable basis injection is needed for the operator norm argument.

If \(z\in F\), invariance of \(F\) implies
\(J_{x,z}(W_x(C))\subset(G|F)_z\). This is the exact closed-reduction
property used below. The ambient normal ball may leave \(F\); its word
evaluations at \(z\) remain actual arrows in the reduction.

### 11BQ.3. The operator estimate, including extra shadow edges

Write an arbitrary compact core kernel as

\[
f=\sum_{i=1}^{m}f_i,
\tag{IX.6}
\]

where \(f_i\) has compact support in one actual bisection \(B_i\),
buffered in a larger actual bisection \(\widehat B_i\). Let \(a_i\) be
its continuous source coefficient on \(s(\widehat B_i)\), extended by
zero to \(T\). This extension is continuous because its support is
compact inside that domain. Choose \(C\) to contain all these supports
and their inverses, and use Section11BQ.1 for this \(C\).

On the fixed compact ambient neighborhood \(K'\), the finitely many
\(a_i\) have a common modulus \(\omega(\delta)\to0\):
\(|a_i(y)-a_i(y')|\le\omega(\delta)\) whenever
\(d(y,y')\le\delta\). Choose \(\delta<\varepsilon/2\).

More generally let \(W\subset W_x(C)\) be the word component generated
by any subset of the supported edges. Let \(A_x\) be the compression
of \(\lambda_x(f)\) to \(\ell^2(W)\), and let \(V\) be IX.5 restricted
to \(W\). Then

\[
\left\|A_x-V^*\lambda_z(f)V\right\|
 \le 2m\,\omega(\delta),\qquad d(x,z)<\delta.
\tag{IX.7}
\]

Here is the matrix check; it is needed because the shadow graph may have
additional edges. Each \(i\)-th bisection contributes at most one entry
in each row and each column. A nonzero original \(i\)-edge is a factor
in \(C\), so its shadow is defined on the common ball; IX.4 bounds the
coefficient change by \(\omega(\delta)\).

Conversely suppose an \(i\)-edge in the shadow representation joins
two image vertices. If its source coefficient at the corresponding
original endpoint is nonzero, the original \(\widehat B_i\)-arrow is
in \(C\). The word obtained by adjoining it has the same shadow germ
as the proposed target word. Injectivity in Section11BQ.2 proves that the
original edge joins those very two original vertices. Its coefficient
again differs by at most \(\omega(\delta)\).

If that original coefficient is zero, the additional shadow coefficient
is at most \(\omega(\delta)\). If only an original edge occurs, the
same argument bounds it when the shadow coefficient is zero. Generator
domains are compared using the buffered \(\widehat B_i\), and the
zero extension of \(a_i\) covers all missing support edges. There is
no assumed identification of absent arrows.

The difference matrix therefore has absolute row and column sums at
most \(2m\omega(\delta)\). The elementary Schur estimate proves IX.7:
if both sums are bounded by \(L\), Cauchy–Schwarz with entry weights
gives \(\|Av\|^2\le L^2\|v\|^2\) for finitely supported \(v\), and
density extends the bound to the whole \(\ell^2\) completion.

For \(z\in F\), write \(t=\|f|_{G|F}\|_r\). IX.7 gives

\[
\|D A_x D\|\le t+2m\omega(\delta)
\tag{IX.8}
\]

for every diagonal contraction \(0\le D\le1\), transported to the
image vertices by \(V\). Both diagonal factors and \(V\) have norm at
most one. No uniform continuity modulus of \(D\) is required. This
last point allows cutoffs whose transition neighborhoods shrink.

### 11BQ.4. Arbitrary source-fibre cosets cannot be omitted

For a scalar unit contraction \(b\), put
\(f^b=bfb\). Its \(i\)-th arrow coefficient is
\(b(rg)f_i(g)b(sg)\). On each actual regular fibre \(G_y\), form the
undirected graph generated by the nonzero \(i\)-edges and their inverses.
The operator \(\lambda_y(f^b)\) is the orthogonal sum of its component
operators. Isolated vertices with no nonzero unit coefficient contribute
zero. For a nonzero component choose any vertex \(h\), put \(x=r(h)\),
and right translate its vertices by \(h^{-1}\).

This gives exactly a set \(W\) of actual supported words based at \(x\).
All its vertices have endpoints in \(K\cap\operatorname{supp}b\).
Right translation is an onto isometry on these basis sets and intertwines
the component operator with
\(D A_x D\), where the diagonal entry at \(w\) is \(b(rw)\).

The chosen \(h\) is completely unrestricted and can have a degenerating
actual domain. Only the \(C\)-words after right translation are evaluated
at \(z\). This proves the estimate for **every component of every
regular fibre**, rather than just the component of its unit.

One must not claim \(x\mapsto\|\lambda_x(f)\|\) is upper semicontinuous
from IX.5. Arbitrary cosets can see other endpoint regions. IX.8 combined
with the endpoint cutoff and the component decomposition is the actual
statement needed here.

### 11BQ.5. Equality of the reduced restriction kernel and open ideal

Fix \(\eta>0\) and choose \(\delta\) so that
\(2m\omega(\delta)<\eta\). Let \(O\) be the union of the
\(\delta\)-balls about \(F\cap K\). Choose a continuous bounded unit
function \(0\le b\le1\), equal to one on \(F\), with
\(K\cap\operatorname{supp}b\subset O\). Such a cutoff exists by
normality of the metrizable unit space: separate \(F\) from the compact
set \(K\setminus O\), with a further open buffer. If \(F\cap K\) is
empty, take \(b\) identically zero on a neighborhood of \(K\).

Every nonzero component in Section11BQ.4 has a basepoint \(x\in K\cap O\).
Choose \(z\in F\cap K\) with \(d(x,z)<\delta\). Apply IX.8 to that
component; the word radius is the same for every component, every source
fibre and every word length. Taking their supremum proves

\[
\|bfb\|_r\le\|f|_{G|F}\|_r+\eta.
\tag{IX.9}
\]

Let \(I_r\) be the isometrically embedded open-reduction algebra
\(C_r^*(G|U)\). The core kernel \(f-bfb\) belongs to \(I_r\).
Indeed each of its finitely many bisection coefficients vanishes on
\(F\). A coefficient is thus an element of \(C_0(U)\) after zero
extension, and can be approximated uniformly by compactly supported
coefficients in \(U\). Finite bisection row and column bounds turn this
into convergence in the groupoid I-norm, hence in the reduced norm.
Invariance of \(U\) makes these approximants actual kernels of \(G|U\).

Consequently

\[
\|f+I_r\|\le\|f|_{G|F}\|_r+\eta.
\]

Let \(\eta\downarrow0\). The reverse inequality follows from the
contractive restriction homomorphism, which vanishes on \(I_r\).
We obtain the exact quotient norm identity

\[
\boxed{\ \|f+I_r\|=\|f|_{G|F}\|_r\ },
\qquad f\in C_c(G).
\tag{IX.10}
\]

Approximation of an arbitrary \(a\in C_r^*(G)\) by compact core kernels
extends IX.10 to every \(a\). Thus
\(\ker\operatorname{res}_F=I_r\).

For completeness, restriction is reduced-contracting since for
\(z\in F\) its actual regular fibre is exactly \((G|F)_z\).
It is onto: partition any compact reduced core kernel into finitely
many relative bisections and extend their source coefficients through
ambient bisections with compact support, using scalar Tietze extension.
The image is dense, and a C*-homomorphism has closed image. The open
inclusion is isometric because a regular fibre based in \(U\) is its
open-reduction fibre and one based outside \(U\) annihilates that core.
These facts finish all terms of IX.1, not only its middle kernel.

### 11BQ.6. Actual finite Killing-rank filtration

A Killing germ is determined by its value and skew covariant derivative,
so its dimension satisfies \(0\le k(x)\le Q=q(q+1)/2\). A set of
independent germs at \(x\) has representatives on one common chart;
their first jets remain independent nearby. Hence

\[
U_j=\{x:k(x)\ge j\}
\tag{IX.11}
\]

is open. Actual local isometries transport the **full** germ algebra
isomorphically, so these sets are invariant. Set \(U_0=T\),
\(U_{Q+1}=\varnothing\), and \(S_j=U_j\setminus U_{j+1}\).
Apply IX.1 inside the open Riemannian manifold \(U_j\):

\[
0\longrightarrow C_r^*(G|U_{j+1})
 \longrightarrow C_r^*(G|U_j)
 \longrightarrow C_r^*(G|S_j)\longrightarrow0,
\qquad 0\le j\le Q.
\tag{IX.12}
\]

These are the original reductions and original reduced norms. Their
closed layers need not be manifolds. No full-rank basis is extended
through a rank boundary and no padded discrete matrix label is used.

The same proof works inside an invariant locally closed unit set \(Y\).
Its compact arrow tests are compact ambient tests and admit the same
ambient actual buffers. Relative compact bisection coefficients extend
to ambient continuous coefficients after scalar Tietze extension on a
local closed ambient chart; their compact supports allow zero extension
and a finite cover. Evaluation at \(z\in Y\) remains inside \(G|Y\)
by invariance. Relative closed invariant sets and relative cutoffs then
give exactly IX.7–IX.10. This also verifies the reduced sequence on any
locally closed rank interval one elects to use.


### 11BQ.7. Scope of the norm conclusion

The quotient and inclusion in IX.1 and IX.12 use the original reduced norms. The argument does not infer amenability of the whole groupoid, a discrete affine first-jet cocycle, or a proper kernel from compact jet data. In the punctured-plane example, any fixed compact cross-arrow test stays away from its absent height; cutting toward that height removes its coefficient. No missing arrow is supplied by IX.5. In a nonlinear curved cylinder, the estimate compares finitely many actual coefficient functions and does not send a compact increment test into infinitely many discrete matrix labels.

The norm theorem also supplies no semisplitting or scalar closed-quotient normal class. Constructing the original reduced scalar physical graph representative across rank changes still requires its actual geometric cycles and complete domains. The full inverse Spin-c coefficient, final right \(Cl_q\), original disk, orientation and positive Bott sign remain the prescribed data. Killing rank \(j\) does not replace normal degree \(q\).

![Actual compact-word shadowing and the reduced quotient norm](../figures/word-shadowing-exactness.png)

**Figure 11BQ.1.** IX.3–IX.12. Every word from one compact actual arrow test has the same original normal-ball radius. The two rows evaluate the same words at nearby points, one in the invariant closed set; the dashed comparisons are short normal geodesics, not additional holonomy arrows. The entire word component embeds injectively into the closed-reduction regular fibre. An arbitrary initial arrow is used only for right translation. Original cutoff weights are carried unchanged as diagonal contractions; the uniform row/column estimate controls every infinite component. The resulting quotient norm and rank sequences use the actual original reductions. This is a schematic of the exact metric bound and operators, with no global metric embedding or new normal phase implied. Original expression: CC0. Connes is credited for the historical graph-normal question; Brix–Carlsen–Sims for inner-exactness terminology.

Editable SVG · Reproduction sources and exact data.

## 11BR. Scalar extension and original normal geometry across rank boundaries

**Theorem 11BR.1 (the scalar boundary criterion).** Suppose the actual
reduced algebras form a semisplit extension with separable middle
algebra. A degree-\(q\) scalar class on its open ideal extends to the
middle algebra exactly when its product with the actual extension
boundary vanishes. The obstruction is

\[
o(x_I)=\partial_r\otimes_{I_r}x_I
       \in KK^{q+1}(Q_r,\mathbb C).
\tag{SE.1}
\]

There is an incomplete four-dimensional unit-foliation example in which
every degree-four ideal scalar class extends, and a selected flat
sign-line class has an explicit global finite-line normal extension
with the original positive-disk pairing \(+1\). There is also an
incomplete three-dimensional unit-foliation example, with full Killing
ranks three and six, in which one normalized auxiliary for the actual
high-rank label group gives a nonzero scalar obstruction. Its value on
the quotient unit is \(-1\), although its original positive-disk tests
give \(+1\). These statements concern scalar extension and the
specified concrete geometric representatives.

The original untwisted normal class exists on both examples. The
counterexample excludes automatic coherence of selected normalized
auxiliaries; it does not exclude that original class. The full inverse
Spin-c coefficient, original normal degree and final right Clifford
order remain the ones fixed in Theorem 7.17 and Section 11BM.

### 11BR.1. Actual scalar exactness and its precise hypotheses

Assume

\[
0\longrightarrow I_r\xrightarrow{i_r}A_r
 \xrightarrow{q_r}Q_r\longrightarrow0
\tag{SE.2}
\]

is the genuine reduced ideal/quotient sequence, \(A_r\) is separable,
and \(q_r\) has a completely positive contractive section. The ideal
and quotient are then separable. Write
\(\partial_r\in KK^1(Q_r,I_r)\) for its semisplit extension class,
using the positive suspension and ordered-Clifford conventions.

The proved contravariant sequence is

\[
KK^q(Q_r,\mathbb C)\xrightarrow{q_r^*}KK^q(A_r,\mathbb C)
 \xrightarrow{i_r^*}KK^q(I_r,\mathbb C)
 \xrightarrow{\partial_r\otimes-}KK^{q+1}(Q_r,\mathbb C).
\tag{SE.3}
\]

Here the exact proof provider is
Exact sequences in KK and the universal coefficient theorem,
Sections 1–4, especially Theorems 4.1–4.2. Its construction applies
to precisely the separable semisplit extension above. To record the
objects behind this use, its completely positive multiplier lift
\(L:Q_r^+\to\mathcal M(I_r)\) has the Stinespring module obtained from
\(Q_r^+\odot I_r\) with inner product
\(\langle b\otimes v,c\otimes w\rangle=v^*L(b^*c)w\).
If \(Vv=1\otimes v\), \(P=VV^*\), its odd extension cycle has operator
\((2P-1)\widehat\otimes\epsilon\). The multiplicative lift defect
lies in \(I_r\); the criterion that \(R^*R\) compact implies \(R\)
compact proves its commutators compact. Countable generation follows
from separability and the ideal's countable approximate identity.

The same lesson constructs the actual cone
\(C_{q_r}=\{(a,b):b(0)=q_r(a),\ b(1)=0\}\), proves that the
ideal inclusion \(I_r\to C_{q_r}\) has both KK inverses under this
semisplitting, and replaces its Puppe sequence by SE.3. Its suspended
ideal section uses a quasicentral approximate identity; it is not a
scalar extension assumed to be an algebra splitting. The endpoint
minus sign and the odd exterior interchange sign cancel in formula
(4.7), giving the boundary on the left in SE.3.

The separable contravariant proof uses the **actual simultaneous
suspension** of
Bott periodicity in KK: the Bott and Dirac elements,
Proposition 8.2, with Proposition 8.1 for coefficient descent. The
operation sends a cycle to its pointwise cycle on
\(C_0(\mathbb R,E)\); its inverse is proved using both fixed Bott
inverse products and the even Clifford Morita evaluations. Thus the
excision application uses proved cycle maps, not merely an abstract
periodic group identification. The coefficient \(\mathbb C\) in
SE.3 is sigma-unital and meets the stated hypotheses.

Exactness at \(KK^q(I_r,\mathbb C)\) proves the equivalence in
Theorem 11BR.1. If \(x_A^0\) is one lift, all scalar lifts are

\[
x_A^0+q_r^*KK^q(Q_r,\mathbb C).
\tag{SE.4}
\]

This is a translate of the displayed image; it does not assert
injectivity of \(q_r^*\). Contravariance supplies \(q_r^*\) in this
direction. It supplies no restriction of an arbitrary scalar
\(A_r\)-class to \(Q_r\).

For an actual maximal normal class \(d_m\in KK^q(A_m,\mathbb C)\)
and an even ideal transfer \(\Theta_I\in KK(I_r,I_m)\), the scalar
test is exactly

\[
\partial_r\otimes_{I_r}\Theta_I\otimes_{I_m}
 [i_m]\otimes_{A_m}d_m=0
 \quad\text{in }KK^{q+1}(Q_r,\mathbb C).
\tag{SE.5}
\]

It does not require a whole reduced-to-maximal transfer on the middle
algebra or a chosen scalar class on the closed quotient. The boundary
is odd and remains before the degree-\(q\) normal factor; the transfer
is even. No odd factors are exchanged.

Theorem 11BQ.1 proves the actual reduced restriction kernels for the
local-isometry rank filtration. That norm theorem alone supplies
neither the completely positive section in SE.2 nor its ideal/cone
KK inverse. If no semisplitting is given, retain the actual cone and
prove the required excision comparison before invoking SE.3.

### 11BR.2. A concrete incomplete four-manifold and scalar extension

Let \(L\to\mathbb {CP}^1=S^2\) be the tautological Hermitian complex
line, and let \(p\) denote the projection of \(L^{\otimes2}=O(-2)\).
Use the open radius-four bundle and its actual closed disk bundle:

\[
T=\{w\in L^{\otimes2}:\|w\|<4\},\quad
F=\{\|w\|\le1\},\quad E=T\setminus F,
\qquad
A=C_0(T),\ I=C_0(E),\ Q=C(F),\ q=4.
\tag{SE.6}
\]

On the nonzero bundle put \(r=\sqrt{\|w\|}\). The map
\(v\mapsto([v],v\otimes v)\) identifies it with
\((\mathbb C^2\setminus\{0\})/\{\pm1\}\). Its circle bundle is
\(Y=S^3/\{\pm1\}=\mathbb {RP}^3\), and
\(E=Y\times(1,2)\). With the unit round quotient metric on \(Y\),
\(g_E=dr^2+r^2g_Y\) is exactly the Euclidean quotient-annulus metric.

Fix a smooth Hermitian connection on \(L^{\otimes2}\). The connection
splitting and a positive base metric give a smooth positive Hermitian
connection metric \(h\) on its total space, including its zero section.
Choose a radial cutoff \(\chi\), zero for \(r\le1/2\) and one for
\(r\ge3/4\), and set \(g=(1-\chi)h+\chi g_E\) where \(w\ne0\),
with \(g=h\) near zero. Positivity is preserved by the convex
combination. The cutoff is constant near zero, so the nonsmooth
radial expression there causes no loss of smoothness. This fixes one
original metric flat on all of E. A radial curve has finite length
to the absent end \(r=2\), so this original metric is incomplete.

Give T its point foliation and the canonical Spin-c structure of its
complex tangent bundle. Every transverse metric is invariant for
this unit groupoid. Use its full inverse normal coefficient from
Theorem 7.17. No hypothesis about its interior Killing rank is needed.
Both actual groupoid completions are \(C_0(T)\), with supremum norm.

The extension \(0\to I\to A\to Q\to0\) is exact and has an explicit
cpc section. Keep a function unchanged on F; on an outer collar
\(1<r<2\) set it equal to \(\chi_0(r)h(1,y)\), where
\(0\le\chi_0\le1\), \(\chi_0=1\) at \(r=1\) and it vanishes
before \(r=2\); set it zero farther out. Positivity at every matrix
level and contractivity are pointwise. This is a genuine section
with values in \(C_0(T)\), so SE.3 applies.

The radial contraction of F onto its zero section is a homotopy
through maps of the actual compact disk bundle. Its projection and
zero-section maps give inverse homotopy equivalences
\(C(F)\simeq C(S^2)\). For one point \(\infty\in S^2\),

\[
0\longrightarrow C_0(S^2\setminus\{\infty\})
 \longrightarrow C(S^2)\xrightarrow{\mathrm{ev}_\infty}\mathbb C
 \longrightarrow0
\tag{SE.7}
\]

is split by constants, and its ideal is \(C_0(\mathbb R^2)\).
Theorems 4.1 and 5.2 and Section 8 of
Bott periodicity in KK: the Bott and Dirac elements
give the two actual inverse products and first-variable periodicity.
Hence

\[
KK^1(C_0(\mathbb R^2),\mathbb C)
 \cong KK^3(\mathbb C,\mathbb C)=0.
\tag{SE.8}
\]

For this scalar zero and its full-homotopy interpretation, use
Pictures of KK: Fredholm operators, quasihomomorphisms and extensions,
Theorem 3.2, together with
Homotopy, associativity, the index pairing and KK-equivalence,
Theorem 2.3 and Corollary 2.4. The former computes the operator-homotopy
scalar picture; the latter proves its equality with whole interval-module
KK at separable sources and sigma-unital coefficients. Here the source
and coefficient are \(\mathbb C\). The stable scalar unitary group
has zero \(K_1\): diagonalize a finite unitary and contract its
eigenvalue angles. Periodicity gives the degree-three zero as well.

Applying the actual contravariant exact sequence to SE.7 makes
\(KK^1(C(S^2),\mathbb C)\) inject into the zero group in SE.8,
because the preceding scalar odd group is zero. The retraction and
two-periodicity therefore prove

\[
KK^5(Q,\mathbb C)=KK^1(Q,\mathbb C)=0.
\tag{SE.9}
\]

This uses no universal coefficient theorem and does not infer odd
scalar KK vanishing merely from an ordinary \(K_1\)-group.
SE.3 consequently makes \(i^*:KK^4(A,\mathbb C)\to KK^4(I,\mathbb C)\)
onto.

Let \(\lambda\to E\) be the line pulled from the sign line of the
double cover \(S^3\to Y\), and let \(d_T\) be the original untwisted
completed inverse normal class of Section 11BM. Then

\[
x_I^\lambda=[m_\lambda]\otimes_I i^*d_T,
\qquad \partial\otimes_I x_I^\lambda=0.
\tag{SE.10}
\]

The zero group already proves scalar extension. The following
construction gives its actual geometric representative.

### 11BR.3. Shared tautological projection, connection and normal domain

For \(y=[z_0:z_1]\) and \(z=(z_0,z_1)^t\ne0\), put

\[
e(y)=\frac{zz^*}{z^*z},\qquad e^2=e=e^*.
\tag{SE.11}
\]

It is independent of the nonzero complex homogeneous representative.
For \(M=p^*L\), let \(e_A=e\circ p\), \(e_I=e_A|_E\),
\(e_Q=e_A|_F\). The actual modules are

\[
\mathcal E_M=e_AA^2,
\qquad \mathcal E_{M|E}=e_II^2,
\qquad \mathcal E_{M|F}=e_QQ^2.
\tag{SE.12}
\]

For \(B=A,I,Q\), multiplication by \(b\in B\) is in
\(e_BM_2(B)e_B=\mathcal K(e_BB^2)\). Thus these are legitimate even
zero-operator multiplication cycles. On the nonunital algebras e is
a multiplier projection; it is not asserted to belong to the algebra.

The sign line and \(M|E\) have the literal identification

\[
[(v,c)]\longmapsto cv\in L_{[v]},
\qquad (v,c)\sim(-v,-c).
\tag{SE.13}
\]

This map is invariant under the equivalence, complex linear and
unitary on each fibre. Restricting the same bundle sections gives the
exact sequence of these modules. The collar cpc section above preserves
the base map p and hence the projection e. Its matrix amplification
compresses the same extension dilation in all three modules, proving
the optional restriction identity

\[
[m_{M|F}]\partial=\partial[m_{M|E}].
\tag{SE.14}
\]

This identity is provided by one actual global bundle; it is not a
condition imposed on every scalar extension.

The sign cover gives a flat Hermitian connection on M over the
nonzero bundle \(Y\times(0,2)\). Choose the global projection connection
\(e\,d\) and interpolate it with that flat connection using a smooth
radial cutoff zero near the zero section and one for \(r\ge3/4\).
Connections form an affine space of Hermitian connection one-forms.
The cutoff is constant near zero, so the resulting connection is
smooth; it is flat on E and has curvature confined to \(r<3/4\).
It is not claimed globally flat: its restriction to the zero section
is the nontrivial tautological line over \(S^2\).

Let \(S_{\rm inv}\) be the whole original inverse normal coefficient,
including its inverse central line and final right \(Cl_4\). The
carrier, with original metric volume, is

\[
H_M=L^2(T,M\otimes S_{\rm inv},d\mathrm{vol}_g)
   =e_A L^2(T,S_{\rm inv},d\mathrm{vol}_g)^2.
\tag{SE.15}
\]

Let \(N_M\) use the specified normal connection and the just constructed
connection on M. It is formally symmetric with principal symbol
\(1_M\otimes c_{\rm inv,g}\). Define the smooth proper exhaustion and
positive speed

\[
R=(16-\|w\|^2)^{-1},\qquad
\alpha=(1+|dR|_g^2)^{-1/2},\qquad
D_M^\alpha=\alpha^{1/2}N_M\alpha^{1/2}.
\tag{SE.16}
\]

Properness follows from the compact base and compact closed subdisk
bundles strictly inside \(\|w\|<4\). This positive speed is an operator
regularization. The original metric remains the fixed incomplete
normal symbol datum; no absent endpoint is added.

The exact closed self-adjoint domain is

\[
\operatorname{Dom}D_M^\alpha
=\{u\in L^2(T,M\otimes S_{\rm inv}):
     u\in H^1_{\rm loc},\ D_M^\alpha u\in L^2
                     \text{ distributionally}\}.
\tag{SE.17}
\]

Here is the completed-domain proof. The adjoint of the compact smooth
core has the maximal distributional domain. On a compact normal buffer
\(\alpha\) is bounded below, and finite-coefficient elliptic regularity
puts a maximal vector in local \(H^1\). With an exhaustion cutoff
\(\eta_n=\zeta(R/n)\), equal to one on increasing sublevels,
\(\alpha|dR|\le1\) gives
\(\|[D_M^\alpha,\eta_n]\|\le\|\zeta'\|_\infty/n\).
Thus \(\eta_nu\to u\) in the full operator graph norm. Each compact
\(H^1\) vector is then approximated by compact smooth sections in
that graph norm. Minimal and maximal domains coincide, proving
self-adjointness on SE.17, without a boundary value at the absent end
or a global unweighted Sobolev-domain assertion.

For a compact cutoff \(f\), the interior elliptic estimate bounds
\(f(D_M^\alpha-z)^{-1}\) into compactly supported \(H^1\); Rellich
gives an ordinary scalar compact. Approximation by compact cutoffs
proves this for every \(f\in C_0(T)\). The bounded phase consequently
has scalar compact defects. The representation is literally
\(a\mapsto M_a\), of norm \(\|a\|_\infty=\|a\|_r=\|a\|_m\),
because the actual groupoid is units.

The ordered coefficient is \(M\otimes S_{\rm inv}\), with right
\(Cl_4\) last. In the convention of Theorem 7.17, \(k_4=6\), the
physical generators are \(-L_{t_j}\), and the full inverse lift is
\(zs\mapsto z^{-1}\rho_t(s)\). The extra even line changes none of
these data.

The actual tensor unitary is multiplication of bundle sections:

\[
(e_AA^2)\otimes_A L^2(T,S_{\rm inv})\longrightarrow H_M,
\qquad \xi\otimes u\longmapsto\xi u.
\tag{SE.18}
\]

The finite projection verifies its inner-product identity. Compact
chart sections give dense range. For a compact smooth creation
section \(\xi\), the completed product rule gives

\[
D_M^\alpha T_\xi-T_\xi D_1^\alpha
 =\alpha c_{\rm inv}(\nabla^M\xi).
\tag{SE.19}
\]

This is a bounded compactly supported finite coefficient. Local
\(H^1\) gives both full-domain inclusions for the creation and its
adjoint. The rectangular resolvent identity at
\(z=\pm i\sqrt{1+\tau^2}\) has norm bound
\(C_\xi/(1+\tau^2)\), with compact integrands by the source-localized
resolvent. Integrating the two-resolvent phase formula gives both
bounded phase connections. Density treats all module sections. The
first operator is zero, so first-factor positivity is automatic.
The product criterion therefore proves

\[
x_A^M=[H_M,M_a,F_{D_M^\alpha}]
 =[m_M]\otimes_A d_T,
\qquad i^*x_A^M=x_I^\lambda.
\tag{SE.20}
\]

For this point foliation the normalized counting/plaque correspondence
is the identity correspondence A; SE.18 is already its literal physical
scalar carrier. This does not identify the same carrier for a nonunit
groupoid. Taking the trivial line instead of M gives the original
untwisted \(d_T\). The M-twisted global class is retained separately.

### 11BR.4. The one original disk and scalar calibration

Let U be the original positively oriented four-disk in the interior
of F and choose a positive four-disk V in E. The manifold T is
connected and oriented. A path between centres, positive coordinate
frames along it and support shrinking give a positive disk embedding
isotopy. Proposition 4.1 and the positive small-disk extension of
Corollary 6.0d/Section 11BM give, in the original ambient algebra,

\[
b_U=i_*b_V\quad\text{in }K_0(A).
\tag{SE.21}
\]

This is the original disk class, with its full prescribed Clifford
convention. On V the actual sign line is trivial. The completed
inverse normal calibration of WB.2–WB.6 therefore gives
\(b_Vx_I^\lambda=1\). Every scalar extension of this ideal class,
including SE.20, satisfies

\[
b_U\otimes_A x_A
 =b_V\otimes_I i^*x_A=1.
\tag{SE.22}
\]

The original disk has zero image in \(K_0(Q)\), which can also be
checked directly. F retracts to \(S^2\). Stable complex bundles on
that two-complex are classified by rank and determinant: trivialize
over its two hemispheres and use determinant winding of the equatorial
clutch map; its special-unitary part contracts. The four-disk Bott
difference has rank zero and determinant clutch map from \(S^3\) to
\(U(1)\), null-homotopic, so its closed restriction is zero. This
does not replace the original ambient disk or affect SE.21–SE.22.

In general, if a genuine ideal disk has \(b_U=i_*b_V\), extension
retains its normalization by the same argument. Without that identity,
take one lift \(x_A^0\) from SE.4. Its scalar defect
\(d=b_Ux_A^0-1\) is removable exactly when
\(-d\in\{(q_*b_U)z:z\in KK^q(Q,\mathbb C)\}\). This is the
affine scalar calibration, with no whole-transfer gate.

### 11BR.5. An incomplete rank-three/six geometry and one actual label group

Let \(X=(\mathbb R/\mathbb Z)^2\) and fix the original metric

\[
T_3=X\times(-2,2),\qquad
g=dt^2+a(t)^2(dx^2+dy^2),\qquad a=e^u,
\]
\[
u(t)=\begin{cases}
\varepsilon e^{-1/(1-t^2)},&|t|<1,\\
0,&|t|\ge1,
\end{cases}\qquad\varepsilon>0.
\tag{SE.23}
\]

Use the point foliation and the full inverse Spin-c structure of the
ordered framed tangent bundle \((x,y,t)\). Its normal degree is three.
The ends \(t=\pm2\) are absent and have finite original radial length.

We first prove the full ranks, including the boundary. The scalar
curvature is \(R=-4u''-6(u')^2\). It is analytic on \((-1,1)\), has
\(R(0)=8\varepsilon/e>0\), and tends to zero at its endpoints.
It is therefore nonconstant on every open interior interval, with
\(R'\ne0\) on a dense subset. A Killing field preserves R, so its
t-component is zero throughout an interior chart. Write its other
components as \(b(x,y,t)\partial_x+c(x,y,t)\partial_y\).
The mixed Killing equations give \(\partial_tb=\partial_tc=0\).
The remaining equations are
\(\partial_xb=\partial_yc=0\),
\(\partial_yb+\partial_xc=0\), whose solutions are exactly
\((b_0-dy)\partial_x+(c_0+dx)\partial_y\).
Thus the full interior germ algebra has dimension three.

These three local fields remain Killing across \(t=\pm1\).
Any boundary germ agrees on an interior open subset with their
constant combination; connected first-jet uniqueness extends the
equality across that boundary, excluding extra boundary germs.
On each exterior the metric is locally Euclidean three-space and
all six Euclidean Killing germs are present. The exact pieces are

\[
F_3=X\times[-1,1]\quad(k=3),\qquad
E_-=X\times(-2,-1),\quad E_+=X\times(1,2)\quad(k=6).
\tag{SE.24}
\]

Rotations in this computation are actual local angular fields, not
global torus fields. The original groupoid is units. Put
\(A_3=C_0(T_3)\), \(I_3=C_0(E_-)\oplus C_0(E_+)\),
\(Q_3=C(F_3)\). Their norms are the supremum norm and

\[
0\longrightarrow I_3\longrightarrow A_3\longrightarrow Q_3
 \longrightarrow0
\tag{SE.25}
\]

has its literal vanishing-on-F kernel. A cpc section is
\(\sigma(h)(x,y,t)=\chi(t)h(x,y,c(t))\), where c clamps to
\([-1,1]\) and \(0\le\chi\le1\) equals one there and has compact
support inside \((-2,2)\). Complete positivity and the section
identity are pointwise. All three algebras are separable, so SE.3
applies to this actual extension. Denote its boundary by
\(\partial_3\).

On an angular flat chart order the full six-field basis as
\[
X_1=\partial_x,\ X_2=\partial_y,\ X_3=\partial_t,
\quad R_{12}=-y\partial_x+x\partial_y,
\quad R_{13}=-t\partial_x+x\partial_t,
\quad R_{23}=-t\partial_y+y\partial_t.
\]
For \(x_i=x_l+m\), \(y_i=y_l+n\), and unchanged t, the actual
source-to-range transport matrix is

\[
H(m,n)=\begin{pmatrix}1_3&N(m,n)\\0&1_3\end{pmatrix},
\qquad
N(m,n)=\begin{pmatrix}n&0&0\\-m&0&0\\0&-m&-n\end{pmatrix}.
\tag{SE.26}
\]

Indeed the source fields expressed in the target basis satisfy
\(R_{12,l}=R_{12,i}+nX_1-mX_2\),
\(R_{13,l}=R_{13,i}-mX_3\),
\(R_{23,l}=R_{23,i}-nX_3\). The translations are unchanged.
Hence multiplication is exactly
\(H(m,n)H(m',n')=H(m+m',n+n')\).

Keep this basis on E+. On E- rescale its three translations by
\(\sqrt2\), leaving rotations unchanged. This constant change of
a full basis gives the transport matrix
\(H(m/\sqrt2,n/\sqrt2)\). The group generated by both actual
overlap families is therefore

\[
\{H(a+c/\sqrt2,b+d/\sqrt2):a,b,c,d\in\mathbb Z\}
 \cong\mathbb Z^4.
\tag{SE.27}
\]

Injectivity follows from irrationality of \(\sqrt2\) in each coordinate
and injectivity of N. Give this countable group the **discrete** label
topology, independently of its usual matrix topology.

Take a finite angular coordinate cover of each component, with lifted
coordinates. Subdividing the two positive torus loops through that
cover gives products of overlap matrices \(H(1,0),H(0,1)\) on E+,
and their \(1/\sqrt2\) versions on E-. The index differences sum
to zero on those cover paths. Their overlap points may vary along
the path: these are products of labels in the generated group, not
additional composable nonunit loops of the original unit groupoid.
Duplicate an identical chart with adjacent indices to obtain the
pure chart-index generator. Removing its index powers from any
overlap label proves that the full generated group is exactly

\[
\Gamma_6\cong\mathbb Z^4\times\mathbb Z.
\tag{SE.28}
\]

On the actual Čech-covered unit groupoid the cocycle is
\((H(\cdot),i-l)\), with source chart l and range chart i.
Its matrices are constant on connected overlaps and multiply in the
displayed source-to-range order. A neutral arrow has equal chart
indices and is a unit. Thus the direct diagonal theorem of 11BN
applies to this genuine high-layer cover. No discrete label is padded
across its rank boundary.

### 11BR.6. One full-domain auxiliary returns two different bundles

Use only the first two matrix-group coordinates \((a,b)\) in SE.27;
\((c,d,j)\) act trivially. The abelian group's reduced algebra is
the actual dual five-torus algebra. Restriction to the first two
dual coordinates, with the other characters fixed at one, is a
bounded unital homomorphism to \(C(\widehat X)\). Use there the
character-plus-periodic-Dirac cycle

\[
\kappa=\epsilon_1+\kappa_{\widehat D},\qquad
\widehat D^+=-\frac{i}{2\pi}\partial_{\theta_1}
                 +\frac1{2\pi}\partial_{\theta_2},
\qquad
V_{(a,b,c,d,j)}=e^{2\pi i(a\theta_1+b\theta_2)}.
\tag{SE.29}
\]

The character is one even scalar line with zero operator. On the
Dirac summand use the full even and odd periodic L² spaces, with
\(\widehat D=\left(\begin{smallmatrix}0&(\widehat D^+)^*\\
\widehat D^+&0\end{smallmatrix}\right)\). Its complete domain is
the periodic \(H^1\) domain. On Fourier mode \((k,\ell)\), its
off-diagonal entry is \(k+i\ell\), so its squared eigenvalue is
\(k^2+\ell^2\). Fourier truncation proves self-adjointness on that
domain and compact resolvent. There is one zero mode in each parity,
so its scalar index is zero; with the character, \([1]\kappa=1\).

Every displayed group multiplier shifts the Fourier indices by
\((a,b)\), preserves the entire \(H^1\) domain, and has bounded
Dirac commutator of norm \(\sqrt{a^2+b^2}\). The ignored coordinates
have zero commutator. This is one actual admissible reduced-continuous
scalar auxiliary for the whole high-layer label group, not two
independently selected auxiliaries.

On E- its transition matrices involve only \((c,d)\), so its returned
family is constant. The character contributes one and the periodic
Dirac contributes zero, giving index bundle one.
On E+ the overlap transitions are the Fourier multipliers in
\((a,b)\). On an angular lifted chart define the compatible family
by its even-to-odd Fourier modes

\[
(k-x)+i(\ell-y),\qquad (k,\ell)\in\mathbb Z^2.
\tag{SE.30}
\]

Conjugating by \(V_{(m,n)}\) subtracts \(m+in\), exactly matching
\((x,y)\mapsto(x+m,y+n)\). Thus these are operators on the actual
transition Hilbert bundle. Their domain bundle is the same periodic
\(H^1\) bundle, since the transition unitaries preserve that domain.
On a finite lifted cover, the added potentials are uniformly bounded;
the Fourier resolvent tails are uniformly compact and the resolvents
vary continuously in norm. This is a genuine family Fredholm cycle.

Away from the parameter origin modulo the lattice it is invertible.
On a disk of radius less than one half about that origin, all modes
except \((0,0)\) have a uniform positive gap. On this complement the
bounded invertible transform can be deformed to its sign, a degenerate
family, with uniformly compact changes. The remaining full even/odd
mode has clutch map \(-x-iy\). Its normalized boundary map has winding
\(+1\): multiplication by minus one is an orientation-preserving
rotation of the complex coordinate. Fix the auxiliary grading so
this is the positive planar Bott convention. Let b denote that
positive compact two-disk class extended into X. The Dirac family
index is b, and the extra character gives \(1+b\).

For the full KK interpretation use
Pictures of KK: Fredholm operators, quasihomomorphisms and extensions,
Theorem 2.2, and
Homotopy, associativity, the index pairing and KK-equivalence,
Theorem 2.3 and Corollary 2.4. Their source is separable and
coefficient \(C(X)\) is sigma-unital. The index homotopies are
coefficient-module homotopies. All module operators and unitaries
in them are \(C(X)\)-linear, so they preserve the central source
action. Consequently the returned correspondence, not just its
scalar index, is multiplication by its index bundle.

Here the actual Čech return can also be read on its columns. In local
transition frames sum the compact chart columns to a bundle section.
Their balancing relation is multiplication by the same Fourier
transition \(V_{t_{il}}\). Expanding the two inner products gives
the same overlap sums; compact chart cutoffs give dense range. The
sum is therefore the actual completed module unitary. The Čech
average \(D_i=\sum_l\phi_lV_{t_{il}}\widehat D V_{t_{il}}^{-1}\)
and SE.30 have those same transitions and differ by a smooth bounded
potential on compact X. Their affine common-domain path preserves
the family KK class. This verifies the actual direct transfer return,
rather than replacing its Hilbert module by an abstract isomorphic
space. Pulling these homotopies constantly along the open radial
interval keeps each defect compact after multiplication by a
vanishing interval coefficient.

The degree-one clutch map defines an actual line B by gluing trivial
lines over the disk and its complement. Its class is \([B]=1+b\),
and its determinant winding shows \(b\ne0\). Move the supporting
positive disk to a disjoint disk by an orientation-preserving isotopy.
The ambient Bott classes agree, while their disjointly supported
product is zero; hence \(b^2=0\). The one auxiliary in SE.29 returns

\[
\Theta_{I_3}=1_{I_-}\oplus[m_{B|E_+}].
\tag{SE.31}
\]

It has the required local chart-unit identities and scalar auxiliary
unit index one on both components. No scalar class on \(Q_3\) is
selected.

### 11BR.7. The scalar obstruction, its degree and the original disk

Let \(d_3\) be the original untwisted complete-speed normal class on
\(T_3\), with its full framed inverse coefficient. Define the selected
ideal scalar class

\[
x_{I_3}=\Theta_{I_3}\otimes_{I_3}i^*d_3.
\tag{SE.32}
\]

Let \(d_X\in KK^2(C(X),\mathbb C)\) be the positive framed periodic
torus normal inverse, and \(d_\pm\in KK^1(C_0(J_\pm),\mathbb C)\)
the positive interval inverses for
\(J_-=(-2,-1)\), \(J_+=(1,2)\), both with increasing t coordinate.
Then
\(i^*d_3=(d_X\boxtimes d_-,d_X\boxtimes d_+)\).

We verify the original normal and domain comparison behind this
identity. On a flat end its ordered differential expression has
the even \(x,y\) block followed by the t block. On compact smooth
Fourier/core tensors its graded square is the sum of the two
squares; derivative creation errors are bounded. The even block
has no odd exterior interchange sign. The ordered Thom/inverse
composition of Corollary 6.0d, (147)–(148), followed by WB.6's
prescribed right reduction, contracts the \(x,y\) pair and leaves
the final t generator. It retains the full inverse central line
and final right \(Cl_3\); Killing rank does not replace q.

For the end use, respectively,
\(R_-=(t+2)^{-1}+(-1-t)^{-1}\) and
\(R_+=(t-1)^{-1}+(2-t)^{-1}\), with
\(\alpha_\pm=(1+|dR_\pm|^2)^{-1/2}\). Their domains are exactly
local \(H^1\) with the corresponding jointly weighted normal
expression in \(L^2\), by the same completed cutoff proof as SE.17.
The even torus Dirac has compact resolvent. The joint-product proof
of 11BO.3–11BO.5 applies to these two factors: the compact-core square
identity gives the local joint estimate, finite torus spectral
projections and cutoffs give the complete maximal domain, compact
Fourier tails give scalar local resolvents, and bounded creations
give both phase connections. In its complete unweighted comparison
the first-square positivity is nonnegative; the joint speed and
rectangular localization preserve it modulo source compacts. The
analytic product proof uses these estimates and does not require
the first factor's scalar index to be one. Thus it identifies this
weighted end phase with the stated external product.

Restriction of the ambient **bounded** phase agrees with this end
class. For any compact end coefficient choose an interior buffer
and an ambient compactly supported function equal there to
\(\log(\alpha_\pm/\alpha_T)\). Its compact positive rescaling makes
the ambient expression agree exactly with the end expression on
that buffer. WN.6 gives a globally compact correction to the ambient
phase. The completed rectangular identities WB.2–WB.5 compare it
with the end phase there. Therefore the fixed original compressed
ambient phase and the fixed end phase differ by a two-sided locally
compact operator for every end coefficient, giving a global cycle
homotopy. This is not equality inferred only on compact stages, and
it imposes no self-adjoint boundary restriction of the ambient
unbounded domain at \(t=\pm1\).

The periodic torus has one constant kernel in each parity and all
other modes invertible. Its untwisted unit pairing is zero. The
positive compact-disk calibration in Section 11BM gives

\[
[1_X]d_X=0,\qquad b\otimes_{C(X)}d_X=+1,
\qquad [B]\otimes_{C(X)}d_X=+1.
\tag{SE.33}
\]

For completely typed interval notation, let
\(s_\pm\in KK^1(\mathbb C,C_0(J_\pm))\) be the increasing interval
Bott classes and put
\(\widehat s_\pm=[1_X]\boxtimes s_\pm\in KK^1(\mathbb C,I_\pm)\).
A scalar lift of the quotient unit increases from zero to one on
E- and decreases from one to zero on E+. Its positive exponential
therefore gives

\[
[1_{Q_3}]\partial_3=(\widehat s_-,-\widehat s_+),
\qquad s_\pm\otimes_{C_0(J_\pm)}d_\pm=+1.
\tag{SE.34}
\]

The positive exponential and its scalar KK boundary interpretation
are the actual finite-matrix and full-homotopy comparisons of
Pictures of KK: Fredholm operators, quasihomomorphisms and extensions,
Lemmas 3.1a–3.1f, and
Homotopy, associativity, the index pairing and KK-equivalence,
Corollary 2.4. The positive interval inverse and its Clifford order
are proved in Section 7 of
Bott periodicity in KK: the Bott and Dirac elements.
Thus both endpoint signs and the two inverse integers in SE.34 are fixed.

Associativity, SE.31–SE.34 and the even degree of the torus block give

\[
\begin{aligned}
[1_{Q_3}]\otimes_{Q_3}\partial_3\otimes_{I_3}x_{I_3}
 &=[1_X]d_X-[B]d_X\\
 &=0-1=-1\quad\text{in }KK^4(\mathbb C,\mathbb C).
\end{aligned}
\tag{SE.35}
\]

The full scalar identification and Bott periodicity above identify
the last group with \(\mathbb Z\), with the positive integer
convention. Hence \(\partial_3x_{I_3}\ne0\) in
\(KK^4(Q_3,\mathbb C)\); by SE.3 this selected ideal scalar class
does not extend to A3. No quotient scalar class or whole-transfer
triangle has entered the calculation.

The original positive three-disk U may lie in the interior of F3.
Move it by a positive ambient disk isotopy to V in E+. In the original
ambient algebra \(b_U=i_*b_V\in K_1(A_3)\), with its prescribed
right \(Cl_3\) and full inverse line. The line B is trivial over V,
so \(b_Vx_{I_3}=+1\). The original untwisted \(d_3\) pairs with
\(b_U\) as \(+1\). Any extension of the selected ideal class would
also give that original pairing by the ambient identity, but SE.35
prevents such an extension. Thus the original disk and all its local
positive tests are retained while scalar coherence fails for this choice.

Choosing the trivial character on all four matrix-label coordinates
instead makes both exterior bundles trivial. The selected ideal class
is then exactly \(i^*d_3\), and its literal extension is the original
untwisted normal class. The example refutes automatic coherence from
unit index and local disk tests, not existence of that original class.

### 11BR.8. Exact surviving scope

The scalar extension criterion requires a genuine reduced kernel and
a valid KK-excision comparison at its actual separability and splitting
hypotheses. Theorem 11BQ.1 supplies the former for arbitrary original
local-isometry rank reductions. It does not supply the latter.
At a rank step an actual ideal scalar choice must annihilate SE.1;
the unit example proves that normalized auxiliary index and original
local Bott tests alone do not accomplish this.

An abstract scalar lift from SE.3 also needs its actual geometric
normal representative before the high-frequency normal-column test
of DJ.30–DJ.31a is asserted. For an already supplied scalar cycle, the
normalized counting/plaque correspondence and Parseval-frame identities
DJ.27–DJ.29b give its bounded graph product with scalar compact defects.
An unspecified exactness lift does not thereby become the original
normal phase. No unbounded frame-compression domain condition is added.

The concrete four-manifold solves these interfaces with its shared
finite projection and complete-speed domain. The locally constant
whole-transversal rank case remains supplied by Theorem 11BP.1.
For every variable-rank nonunit curved incomplete holonomy groupoid,
KK excision, coherent scalar choice and its original normal/physical
representative remain separate constructions. This section claims no
arbitrary-scope graph solution and imposes no universal whole-transfer
boundary condition, equivariant gamma identity or globally flat auxiliary.

![Scalar extension, an actual finite-line normal realization and the two-end obstruction](../figures/scalar-rank-extension.png)

**Figure 11BR.1.** SE.1–SE.35. The four-dimensional panel is the actual
open radius-four \(O(-2)\) bundle, with \(r=\sqrt{\|w\|}\), quotient
annulus \(\mathbb R^4/\{\pm1\}\), common tautological projection and
original right \(Cl_4\). The degree-three panel uses the exact bump
metric SE.23, ranks three/six, the one group
\(\mathbb Z^4\times\mathbb Z\) and one periodic auxiliary. Its left
and right index bundles are one and \(1+b\); the increasing interval
boundary signs are plus and minus, giving \(0-1=-1\) in full scalar
KK degree four. Original positive-disk tests remain \(+1\). The
operators, domains and algebras are proved beside the picture. These
are coordinate and typed-operator schematics, not a Euclidean embedding
of the warped metric or a completion of either original end. Original
expression: CC0. Connes is credited for the historical graph-normal
question in [A survey of foliations and operator algebras, Section 8](https://alainconnes.org/wp-content/uploads/foliationsfine.pdf#page=29).

[Reproduction source and exact data](../reproduction/scalar-rank-extension/README.md).

## 11BS. Geometric sections of reduced rank quotients

### 11BS.1. The reduced extension and its mapping cone

Let \(G\rightrightarrows T\) be the original second-countable
Riemannian local-isometry germ groupoid, \(F\subset T\) invariant
closed, \(U=T\setminus F\). Keep the actual original arrow domains.
Theorem 11BQ.1 gives

\[
 0\longrightarrow I=C_r^*(G|U)\longrightarrow A=C_r^*(G)
       \xrightarrow{q} Q=C_r^*(G|F)\longrightarrow0.    \tag{KE.1}
\]

Its all-length word radius and injective evaluation are IX.3–IX.5.
The entire infinite matrix comparison, including extra shadow edges,
is IX.7–IX.8. Arbitrary original source-fibre vertices are retained by
right-translated word cosets. Its final identity is the actual reduced
quotient norm \(\|a+I\|=\|q(a)\|_r\), IX.10. IX.12 applies this to
every actual Killing-rank step, including singular closed layers.

The finite Tietze extensions in its surjectivity proof are scalar
coefficient extensions. They neither specify one linear section of q
nor prove complete positivity of an assembled section. A cutoff chosen
for one finite compact kernel test is also not a cpc map on the whole Q.

All these étale second-countable completions are separable: countably
many precompact bisections and countable dense continuous coefficient
families have a countable core whose finite bisection I-norm approximations
are reduced-norm dense. Thus a supplied cpc section would meet the actual
separable extension-algebra hypothesis of KT-KK-14.

Write

\[
 C_q=\{(a,b)\in A\oplus C_0([0,1),Q):b(0)=q(a)\},
 \qquad e:I\to C_q,\quad e(i)=(i,0).                 \tag{KE.2}
\]

The full cone and extension proofs in Exact sequences in KK and the universal coefficient theorem, Sections 1–4, show that \(e\) is a KK equivalence when KE.1 has a completely positive contractive section. Their Stinespring extension cycle, quasicentral suspended-ideal section and two cone inverses retain this semisplitting hypothesis. Their Theorem 4.2 then supplies both-variable boundary exactness. The construction consumes a section of \(q\); it does not produce that section from reduced norm exactness.

### 11BS.2. A whole-source-fibre retraction

**Theorem 11BS.1.** Suppose there is an open neighbourhood \(W\supset F\),
an actual continuous groupoid functor

\[
 R:G|W\longrightarrow G|F,
 \qquad \rho:W\to F,\qquad R|_{G|F}=1,               \tag{KE.3}
\]

such that every actual source-fibre map

\[
 R_x:(G|W)_x\longrightarrow (G|F)_{\rho(x)}
               \quad\text{is a bijection}.            \tag{KE.4}
\]

Assume there is continuous \(0\le\chi\le1\) on W with \(\chi|_F=1\)
and the following explicit support condition: for each compact
\(L\subset G|F\) and \(\delta>0\),

\[
 \{g\in G|W:R(g)\in L,\ \chi(rg)\ge\delta,
                            \chi(sg)\ge\delta\}
                    \text{is compact}.               \tag{KE.5}
\]

Then q in KE.1 has a cpc section. On the compact core it is

\[
 \sigma(f)(g)=\sqrt{\chi(rg)}\sqrt{\chi(sg)}\,f(R(g))
                   \quad(g\in G|W),                  \tag{KE.6}
\]

completed in \(C_r^*(G|W)\) and then included in A. No invariance of
\(\chi\), global group action, amenability or nuclearity is required.

**Proof of the range.** A pullback in KE.6 may have noncompact support;
it is not simply declared a compact core kernel. Choose a continuous
scalar taper \(\eta_\delta\) that is zero when \(\chi\le\delta/2\)
and one when \(\chi\ge\delta\). Multiply KE.6 by the two endpoint
tapers. KE.5 makes the resulting continuous kernel compactly supported.
KE.4 and inversion identify both source and range counting sums with
the corresponding sums after R. In each discarded summand at least one
endpoint square root is at most \(\sqrt\delta\). Hence

\[
 \|\sigma(f)-\sigma_\delta(f)\|_I
                \le2\sqrt\delta\,\|f\|_I.            \tag{KE.7}
\]

Thus KE.6 is an I-norm, hence reduced-norm, limit of actual kernels.
It gives an element of \(C_r^*(G|W)\).

**Proof of complete positivity and the actual norm.** Let \(V_x\) be
the literal basis unitary supplied by KE.4. On the source fibre
\((G|W)_x\), put \(D_x\delta_h=\sqrt{\chi(rh)}\delta_h\).
The functor identity preserves the actual product \(hk^{-1}\), so
the complete regular matrix of KE.6 is exactly

\[
 \lambda_x(\sigma(f))
       =D_x V_x^*\lambda_{\rho(x)}(f)V_x D_x.          \tag{KE.8}
\]

This identity uses the entire fibre, including original isotropy; it
does not use a word truncation. At every matrix size the right side is
a compression of a *-representation. It is completely positive and
has norm at most \(\|f\|_r\). Taking the supremum of actual regular
fibre norms extends \(\sigma\) contractively to Q. Those regular
representations form a faithful representation of the reduced algebra,
so their positive matrices prove complete positivity in that algebra.

For clarity, the open-reduction inclusion into A is reduced isometric
even when W is not invariant. On any \(G_y\), its kernels vanish on
vectors whose ranges are outside W. If the remaining subspace is
nonzero, choose an actual \(h\in G_y\) with \(rh\in W\). Right
translation by \(h^{-1}\) bijects all remaining arrows with
\((G|W)_{rh}\), and intertwines their full regular matrix. It proves
the upper norm bound on every original fibre; fibres based in W give
the reverse bound. Thus the section is valued in the original A with
its original reduced norm.

On F, R and \(\chi\) are identity and one. Since F is invariant,
its full original source fibres remain inside F. Restriction of KE.6
is therefore f exactly, and continuity gives \(q\sigma=1_Q\).
This finishes the theorem. All groupoid functors, inverses and arrows
in KE.3–KE.8 are actual supplied objects. \(\square\)

KE.4 is a conditional geometric provider, not a consequence of IX.5.
IX.5 only controls supported words of one compact test and need not be
onto a closed-reduction fibre. It may give no one common neighbourhood
for all arrow domains. Theorem 11BS.1 never manufactures such a retraction.

### 11BS.3. Proper global Lie actions

**Theorem 11BS.2.** Let a Lie group H have a genuine smooth proper global
isometric action on the second-countable manifold T, and let F be an
H-invariant closed subset. No completeness of the metric is assumed.
There is an H-equivariant cpc section

\[
 S:C_0(F)\longrightarrow C_0(T)
       \quad\text{of ordinary unit-space restriction}. \tag{KE.9}
\]

Let \(\Gamma\le H\) be countable and assume the **actual germ
groupoid in question** is \(\Gamma\ltimes T\): no distinct subgroup
elements agree as germs on a nonempty open subset. Then KE.1 for this
actual groupoid has a cpc section, including all its isotropy. The same
conclusion holds for each H-invariant closed Killing-rank layer in any
actual invariant open manifold on which this proper global action exists.

#### 11BS.3.1. Slices and quotient refinements

Properness means that \((h,x)\mapsto(hx,x)\) is proper. Each stabilizer
\(K_x\) is compact and the orbit is a closed embedded copy of
\(H/K_x\). A normal slice can be obtained using only a sufficiently
small original exponential ball in the orthogonal normal space to that
orbit. The map
\(H\times_{K_x}S\to T\), \([h,v]\mapsto h\exp_x(v)\), has
invertible derivative along its zero orbit. Its equivariance propagates
this local inverse along that orbit. It is injective after one further
shrinking: otherwise there are \(v_n,w_n\to0\) and
\(h_n\exp_x(v_n)=\exp_x(w_n)\). Properness makes a subsequence of
\(h_n\) converge to \(h\in K_x\). In the quotient chart about
\([1,0]\), the local inverse theorem then contradicts the alleged
distinct points. Compactness of K makes all these shrinking choices
K-invariant. This proves an actual invariant slice neighbourhood.
Only local original normal balls are used.

This is the normal-slice construction for a proper Lie action. Its local inverse uses the original metric, without a closure of partial isometry parameters.

The orbit space \(T/H\) is Hausdorff by properness, locally compact by
projecting compact neighbourhoods, and second countable because the
quotient projection is open. It therefore admits a countable locally
finite precompact open refinement of these slice quotients and a
subordinate continuous partition \(w_j\ge0\), \(\sum_jw_j=1\).
One can construct such a refinement on successive compact exhaustion
annuli, with finitely many relatively compact refinements per annulus;
the usual compactly supported Urysohn functions, normalized by their
locally finite positive sum, give the partition.

Choose enlarged quotient opens \(O_j^+\) still locally finite, with
\(\operatorname{supp}w_j\Subset O_j^+\Subset S_j/K_j\). Their slice
preimages of compact sets are compact, since quotient by a compact K
is a proper map. Choose \(0\le\phi_j\le1\) in \(C_c(S_j)\),
K-invariant, equal to one over \(\operatorname{supp}w_j\), and with
slice support projecting inside \(O_j^+\). Compact-group averaging
of an ordinary slice cutoff gives exactly this \(\phi_j\).

#### 11BS.3.2. Positive slice extensions and compact averaging

Restriction \(C_0(S_j)\to C_0(F\cap S_j)\) is onto. The commutative
separable source \(C_0(F\cap S_j)\) has cpc finite-dimensional
approximations: finitely many point samples and a positive compact
partition reconstruct any fixed finite family to its prescribed uniform
accuracy. Cut off its small tails first; the partition sum is at most
one, so both maps are contractive. This proves the approximation property
actually needed here, rather than assuming nuclearity of Q.

The nonunital proof in Lifting completely positive maps, Theorem 5.1, therefore supplies a
cpc slice extension \(S_j^0:C_0(F\cap S_j)\to C_0(S_j)\).
The proof lifts entire positive matrices, clips the unit image, uses forced unitization and takes summably close quasicentral lifts. Section 11BS.6 gives the precise approximation hypotheses.

With normalized Haar measure of the **compact** K_j set

\[
 S_j^K=\int_{K_j}\alpha_k S_j^0\alpha_{k^{-1}}\,dk,
       \qquad (\alpha_k f)(s)=f(k^{-1}s).              \tag{KE.10}
\]

For each f the integrand is norm continuous: the two C0 actions are
strongly continuous and \(S_j^0\) is bounded. Thus this is a Bochner
integral in \(C_0(S_j)\). Every summand is cpc and a section. Positivity
at every matrix size, norm at most one, exact restriction and K
equivariance pass through the integral. Haar invariance proves its
equivariance. No Haar average over noncompact H is attempted.

#### 11BS.3.3. Induction, normalization and vanishing tails

For global \(f\in C_0(F)\), and \([h,s]\in H\times_{K_j}S_j\), set

\[
 L_j(f)([h,s])
   =S_j^K\bigl(z\mapsto\phi_j(z)f(hz)\bigr)(s),\qquad
 Sf=\sum_j w_j L_j(f).                                \tag{KE.11}
\]

The slice input is genuinely in \(C_0(F\cap S_j)\) because of its
compact cutoff; merely restricting a global f to an open slice would
not ensure this. K invariance of \(\phi_j\) and equivariance of
\(S_j^K\) make KE.11 independent of the representative [h,s]. The
global C0 action is strongly continuous, so the input depends
norm-continuously on h and its output continuously on [h,s]. Multiplying
by w_j gives continuous extension by zero at the slice boundary, because
its quotient support lies inside that slice chart and its output is
uniformly bounded by \(\|f\|\).

At each point the sum is locally finite and consists of positive maps
with scalar weights whose sum is one. It is cpc and H-equivariant. On F
every summand with nonzero weight has \(\phi_j=1\), so \(Sf=f\)
exactly. The quotient partition already normalizes the sum. If only a
neighbourhood-of-F cover were used, dividing its weight sum by
\(\max(1,w)\) would instead give sum at most one and the same exact
restriction when \(w|_F=1\).

The range assertion needs proof. Let f have compact support. A summand
can be nonzero only if \(h\operatorname{supp}\phi_j\) meets
\(\operatorname{supp}f\). The sets
\(\pi(\operatorname{supp}\phi_j)\subset O_j^+\) form a locally
finite family in \(T/H\). The compact set
\(\pi(\operatorname{supp}f)\) consequently meets only finitely many of
them. For each such j, properness makes

\[
 \{h\in H:h\operatorname{supp}\phi_j
                        \cap\operatorname{supp}f\ne\varnothing\}
                       \text{compact}.               \tag{KE.12}
\]

The relevant s lie over the compact support of w_j in its slice quotient,
so they also lie in a compact slice set. The image of these two compact
sets under the actual action is compact in T. Thus \(w_jL_j(f)\) has
compact support and only finitely many j contribute. Hence Sf belongs
to \(C_c(T)\). Compactly supported f are dense in \(C_0(F)\);
contractivity puts \(S(C_0(F))\subset C_0(T)\). This proves KE.9,
including the tail that a formal infinite sum could miss.

#### 11BS.3.4. Reduced descent and original isotropy

By positivity and Riesz representation, \(Sf(x)=\int_F f(z)d\mu_x(z)\),
where \(\mu_x\) is a subprobability measure. The proven equivariance
gives \(\mu_{\gamma x}=\gamma_*\mu_x\); on F it is \(\delta_x\).
On a finite actual transformation-groupoid kernel define

\[
 \widehat S\left(\sum_\gamma f_\gamma u_\gamma\right)
             =\sum_\gamma S(f_\gamma)u_\gamma.        \tag{KE.13}
\]

For a regular source fibre at x, keep its whole original basis
\(\{(h,x):h\in\Gamma\}\), including all isotropy arrows. Its (h,k)
entry is \(S(f_{hk^{-1}})(hx)\). Therefore its complete matrix is

\[
 \lambda_x(\widehat S(q))
                =\int_F\lambda_z(q)\,d\mu_x(z).       \tag{KE.14}
\]

The integral is the bounded weak operator integral on the common
\(\ell^2(\Gamma)\) basis. Equivariance of \(\mu\) proves the
entry equality, and positivity of every regular matrix proves complete
positivity at every amplification. Its norm is at most \(\|q\|_r\),
since \(\mu_x(F)\le1\). Supremum over all actual x proves reduced
contractivity. Finite kernels have genuine C0 coefficient images and
the same finite subgroup labels, so lie in the original reduced algebra;
density extends the map to the whole Q. On F, KE.14 and \(\mu_x=\delta_x\)
give exact restriction. Thus \(\widehat S\) is a genuine cpc section
of KE.1. No full-norm representation is renamed reduced-continuous.

If only F's invariance under a dense subgroup of the supplied H is given,
H invariance follows by limits: for \(h_n\to h\), \(h_nx\in F\)
and F closed imply \(hx\in F\); inverses give equality. A closed
subgroup of the already supplied H inherits properness. These are
closures **inside a genuine global action**, not closures of formal
local Killing jets. The theorem adds no H arrows to the original Γ
groupoid and requires germ faithfulness for that exact identification.
\(\square\)

### 11BS.4. A passive interval parameter

Let \(\mathcal H\rightrightarrows X\) be any actual second-countable
Hausdorff étale groupoid, and assume the actual groupoid under study is
\(G=\mathcal H\times J\), where \(J=(-2,2)\) is a unit groupoid.
For a relatively closed \(K\subset J\), put \(F=X\times K\).
All original arrows preserve t and have exactly the original H domains.
Write \(B=C_r^*(\mathcal H)\). The actual identifications are

\[
 C_r^*(G)=C_0(J,B),\qquad C_r^*(G|F)=C_0(K,B).         \tag{KE.15}
\]

Indeed finite product kernels are dense by compact partitions, and
their regular representation at (x,t) is evaluation at t followed by
the original \(\mathcal H_x\) regular representation. Thus their norm
is exactly \(\sup_t\|b(t)\|_B\); completion proves KE.15 directly.
No tensor-exactness or nuclearity assumption on B is needed.

For \(b\in C_0(K,B)\), retain its value on K. On each bounded gap
\((a,c)\subset J\setminus K\) with endpoints in K, put

\[
 (Eb)(t)=\frac{c-t}{c-a}b(a)+\frac{t-a}{c-a}b(c).
                                                               \tag{KE.16}
\]

On a gap reaching -2 with only endpoint c, use
\((Eb)(t)=(t+2)/(c+2)\,b(c)\). On a gap reaching 2 with only endpoint
a, use \((Eb)(t)=(2-t)/(2-a)\,b(a)\). Empty K gives the zero map.
Every formula is a convex combination of B-valued point evaluations
and zero, so it is completely positive and contractive at every matrix
size. At a point of K, small adjacent gaps have close endpoint values;
if a long gap meets that point, its far-end coefficient tends to zero.
This proves continuity even at accumulating gaps. The C0 tail follows
from the input tail on K and the displayed outer tapers. Hence

\[
 E:C_0(K,B)\longrightarrow C_0(J,B)
                         \text{is a cpc section}.      \tag{KE.17}
\]

For \(K=[-1,1]\), this is the literal angular-identity/t-clamping
extension with positive endpoint tapers. For an arbitrary singular K,
it is a probability extension and needs no deterministic retraction.

#### 11BS.4.1. A singular closed rank layer with nonunit holonomy

Take \(X=S^2\), its round metric, and

\[
 K=\{0\}\cup\bigcup_{n\ge1}[a_n,c_n],\quad
 a_n=2^{-n},\quad c_n=\tfrac54\,2^{-n},\quad w_n=c_n-a_n.
                                                               \tag{KE.18}
\]

Define smooth bumps, flat at their two endpoints, by

\[
 u_n(t)=2^{-n}\exp\left(-\frac1{(t-a_n)(c_n-t)}\right)
                 \quad(a_n<t<c_n),
 \qquad u_n=0\text{ otherwise},
\]
\[
 u=\sum_nu_n,\qquad g=dt^2+e^{2u(t)}g_{S^2}
                     \quad\text{on }S^2\times J.       \tag{KE.19}
\]

The intervals are disjoint. After rescaling to the unit interval, every
fixed derivative of u_n is bounded by a polynomial in \(w_n^{-1}\)
times \(e^{-2/w_n^2}\), using half the exponent to absorb its endpoint
powers. These bounds vanish faster than every power of \(a_n\).
Thus u is smooth and flat at zero, as well as at all interval endpoints.
The omitted t ends have finite original radial distance, so this metric
is incomplete.

The scalar curvature is
\(R=2e^{-2u}-4u''-6(u')^2\). On each bump interior it is analytic,
tends to two at its endpoints, and is not constant. At its midpoint
\(u'=0\), \(u''=-32u/w_n^4\), so

\[
 R=2e^{-2u}+128u/w_n^4
       \ge2-4u+128u/w_n^4>2.                          \tag{KE.20}
\]

Every Killing field has zero t-component on the dense set where
\(R'\ne0\) in a bump. The mixed equations make its angular part
t-independent, and its angular equation is the round-sphere Killing
equation. Thus its full germ algebra there has dimension three.
First-jet continuation gives dimension three at each endpoint and at
zero: every neighbourhood of zero meets a bump interior. Three global
angular rotation fields supply the reverse bound. Outside K the metric
is the product round sphere times an interval. Ricci has distinct
angular and radial eigenspaces, so its local isometries preserve that
splitting; its full Killing algebra is precisely the three rotations
and the radial translation. Consequently

\[
 k=3\text{ on }F=S^2\times K,\qquad k=4\text{ outside }F.
                                                               \tag{KE.21}
\]

Choose the actual rotations induced on the imaginary unit sphere by the unit quaternions

\[
 a=(1+2i)/\sqrt5,\qquad b=(1+2j)/\sqrt5,
 \qquad i^2=j^2=-1,\quad ij=-ji.
 \tag{KEF.1}
\]

Conjugation preserves the imaginary quaternion norm and orientation. In the ordered coordinates \((i,j,ij)\), these rotations are

\[
 A=\begin{pmatrix}1&0&0\\0&-3/5&-4/5\\0&4/5&-3/5\end{pmatrix},
 \qquad B=\begin{pmatrix}-3/5&0&4/5\\0&1&0\\-4/5&0&-3/5\end{pmatrix}.
 \tag{KEF.2}
\]

Put \(\Gamma=\langle A,B\rangle\) with its discrete topology. We prove that it is a free dense subgroup, so this example needs no separate free-subgroup existence theorem.

For a reduced word in \(a,a^{-1},b,b^{-1}\) of length \(m\ge1\), multiply the corresponding integral quaternions \(1+2i,1-2i,1+2j,1-2j\). Call the result \(P\); its squared norm is \(5^m\). Modulo five identify the quaternion generators with
\(i=\operatorname{diag}(2,3)\) and \(j=\left(\begin{smallmatrix}0&1\\4&0\end{smallmatrix}\right)\). These matrices satisfy the displayed quaternion relations. The four letter matrices are rank one. Factor each as a column \(u_s\) times a row \(v_s^t\), in the order \(a,a^{-1},b,b^{-1}\):

\[
\begin{array}{c|c|c}
s&u_s&v_s^t\\\hline
a&(0,1)^t&(0,2)\\
a^{-1}&(1,0)^t&(2,0)\\
b&(1,3)^t&(1,2)\\
b^{-1}&(1,2)^t&(1,3)
\end{array}
\qquad
[v_s^t u_t]_{s,t}=
\begin{pmatrix}2&0&1&4\\0&2&2&2\\2&1&2&0\\3&1&0&2\end{pmatrix}
\quad\text{in }\mathbb F_5.
\tag{KEF.3}
\]

An entry vanishes precisely when the second letter is the inverse of the first. The product of rank-one letter matrices for a reduced word is therefore nonzero: its scalar factors are the nonzero adjacent entries of KEF.3, and its outer column and row are nonzero. Consequently \(P\) cannot have all coefficients divisible by five. If the rotation word were the identity, its unit quaternion would commute with \(i,j,ij\), hence be the real scalar \(\pm1\). Thus \(P=\pm5^{m/2}\). For odd \(m\) this is not integral, and for even \(m\ge2\) it is divisible by five, both contradictions. This proves freeness, retaining the actual rotation action.

Both rotations have angle \(\theta\) with \(2\cos\theta=-6/5\). To show its powers are not periodic, set \(c_n=2\cos(n\theta)\) and \(d_n=5^n c_n\). The elementary angle identity gives

\[
 d_0=2,\quad d_1=-6,\quad d_{n+1}=-6d_n-25d_{n-1},
 \qquad d_n\equiv(-1)^n\pmod5\quad(n\ge1).
 \tag{KEF.4}
\]

If \(n\theta\) were a multiple of \(2\pi\), then \(d_n=2\cdot5^n\), contradicting that congruence. Irrational-angle powers are dense in the corresponding rotation circle: among \(N+1\) powers, two lie in one of \(N\) equal arcs, giving a nonzero power of angle less than \(2\pi/N\); multiples of that small angle approximate every circle point. The closure of \(\Gamma\) therefore contains every rotation about each of the first two axes. These circles generate \(SO(3)\): a rotation about the second axis chooses the first coordinate of the image of \(i\), a rotation about the first chooses its other two coordinates, and the stabilizer of \(i\) is the first-axis circle. Every element of \(SO(3)\) is the resulting product. Hence \(\Gamma\) is dense.

Its global action on \(T\) preserves KE.19 and every \(t\). Distinct rotations cannot agree on a sphere open set, since their linear maps would agree on its spanning vectors. Thus the original germ groupoid is exactly \((\Gamma\ltimes S^2)\times J\), with every isotropy arrow retained, as in KE.15.

For completeness, its Cayley tree gives a direct obstruction to a group Reiter sequence. For a nonempty finite vertex set S, its induced forest has at most \(|S|-1\) edges, so the number of outgoing edges is at least \(4|S|-2(|S|-1)=2|S|+2\). For a nonnegative finitely supported probability p, the identity
\[
 \sum_{\{v,w\}\text{ tree edge}}|p(v)-p(w)|
 =\int_0^\infty |\partial\{p>t\}|\,dt
 \ge 2\sum_v p(v)=2
\]
follows by integrating the indicator difference on each edge. Summing the four generator translation differences counts every undirected edge twice, and gives at least four. Positive truncation and normalization extend this lower bound to every probability in \(\ell^1(\Gamma)\). Thus those four translation differences cannot all tend to zero. The action preserves probability sphere area. Integrating any amenable-action Reiter field against this area would give probabilities with those group translation differences tending to zero, by invariance and the integral triangle inequality, contradicting the bound. This example therefore permits genuine nonamenable holonomy. Its section proof uses neither amenability nor nuclearity of B.

If an actual foliation presentation is desired, suspend the action
over a genus-two surface whose fundamental group surjects onto the free
group. The complete transversal is this exact T and holonomy germs are
the same rotation groupoid. The standard oriented complex sphere times
the radial line supplies its coherent invariant full Spin-c normal data.
No inverse coefficient is changed by the algebraic section.

KE.17 now proves semisplitting of the actual closed-rank restriction
KE.21. This is not a manifold-only closed-layer example. There is no
continuous neighbourhood retraction of T onto F near a point at t=0:
a small connected normal neighbourhood contains points of F at zero and
in another interval of K. The t-projection of the image of a connected
set in F must be connected, and cannot contain those two different
components of K. A retraction would fix both points, a contradiction.
Thus the probability/interpolation provider works where Theorem 11BS.1's
deterministic unit retraction is genuinely unavailable.

### 11BS.5. Positivity and actual germs

#### 11BS.5.1. Separate scalar sections need not preserve matrices

Take the two-copy Čech unit groupoid of a curved radial-bump normal
manifold, and its two-copy closed low-rank set F. Its actual algebras
are \(M_2(C_0(T))\) and \(M_2(C_0(F))\). The positive quotient
kernel whose matrix is all ones can be extended separately on the
diagonal bisections by a positive scalar taper c and on the off-diagonal
bisections by another positive taper d. Explicitly use the same ordinary
radial clamping retraction \(p:T\to F\), and the two scalar cpc sections
\(f\mapsto c(f\circ p)\), \(f\mapsto d(f\circ p)\), with c and d
compactly supported positive contractions. Both equal one on F, but choose
an exterior point where \(c=0\), \(d=1\). The resulting actual lifted
regular matrix is

\[
 \begin{pmatrix}0&1\\1&0\end{pmatrix},
             \qquad\text{eigenvalues }+1,-1.           \tag{KE.22}
\]

Each scalar extension is positive and contractive; the assembled map
does not preserve this positive quotient element. The bisection Tietze
step of 11BQ does not establish a cpc section. A common positive regular
matrix construction such as KE.8 or KE.14 is what repairs this issue.
This finite-matrix test does not disprove semisplitting of that example.

#### 11BS.5.2. A limiting jet need not be an isometry germ

Use the original radial bump metric on the open disk of radius two:
\(g=e^{2\varepsilon e^{-1/(1-r^2)}}\delta\) inside r<1, and
\(g=\delta\) for r≥1. Its actual full Killing rank is one on the
closed inner disk and three on its exterior annulus, by the following curvature argument.
For \(u(r)=\varepsilon e^{-1/(1-r^2)}\), its Gaussian curvature is \(-e^{-2u}(u''+u'/r)\). The radial formula has the smooth center value \(4\varepsilon e^{-1}e^{-2\varepsilon/e}>0\) and tends to zero at the boundary, since all bump derivatives are flat there. Twice this Gaussian curvature is the scalar curvature. It is analytic and nonconstant on the interior, so its radial derivative is nonzero densely. A Killing field preserves that curvature; on the dense set where its radial derivative is nonzero the field is tangent to the circles. The mixed equation makes its angular rotation coefficient constant. At the center and boundary connected first-jet uniqueness gives the same one-dimensional germ algebra. On the exterior the metric is Euclidean and has all three affine Killing germs. Let \(x_n=(1+1/n,0)\) for \(n\ge3\), \(y=(3/2,0)\), and take the flat
translation by \((1/2-1/n,0)\) on a ball about x_n of radius less than
\(1/(4n)\), with its inverse. These are actual local isometries, all
inside the exterior, and generate an eligible countable pseudogroup.
Their first jets tend to

\[
                  (x,y,1_2),\qquad x=(1,0).            \tag{KE.23}
\]

There is no local isometry germ with that source and target: such a
germ would identify their full Killing algebras, of dimensions one and
three. Thus closure in a finite jet parameter space contains a parameter
that is not any actual local isometry germ. Actual domains shrink here;
these arrows escape compact tests, consistently with 11BQ's fixed-test
radius. Neither H, proper slices nor an averaging action can be obtained
by silently declaring that parameter a new arrow. This is a countertest
to that globalization step, not a nonsemisplit-extension example.

### 11BS.6. Completely positive limits and the KK boundary

There is a second rigorous sufficient route, without an actual global
groupoid retraction. Suppose one constructs maps
\(s_n:Q\to A\) completely positive, with
\(q s_n\to1_Q\) pointwise in norm. The source Q is separable.
Theorem 4.3 of Lifting completely positive maps makes the identity liftable by a cpc map. Its proof uses the following steps:

* finite positive matrix lifts, normalized by functional-calculus
  clipping of their unit image;
* compact-domain cutoffs and clipping to make initially uncontrolled
  positive lifts contractive, with quantified quotient error;
* forced unitization, where the scalar coordinate proves that the final
  nonunital lift lands in A itself;
* a quasicentral adjustment retaining the new quotient map while making
  consecutive lifts summably close on a dense sequence;
* uniform contractivity, extending that Cauchy convergence to all Q,
  and norm-closed positive cones at every matrix size.

In particular, verified cpc finite-dimensional approximations of the
identity on Q give a section by that provider's Theorem 5.1. This is a
conditional positive approximation input; no such approximation of the
whole arbitrary closed-rank groupoid algebra follows from its scalar
unit-space nuclearity, finite local Killing rank, or normalized KK
auxiliary index. 11BQ gives kernel-specific coefficient lifts and norm
estimates, not these completely positive maps on all Q. KE.22 explains
why the distinction matters.

Whenever Theorem 11BS.1, Theorem 11BS.2, KE.17, or this actual lifting criterion supplies
semisplitting, the already verified KT-KK-14 proof gives the **actual**
inverse \(u\in KK(C_q,I)\) to e in KE.2. Its extension class is
\(\partial\in KK^1(Q,I)\), and for every original degree q0 the
scalar obstruction remains exactly

\[
 \partial\otimes_I x_I\in KK^{q_0+1}(Q,\mathbb C).
                                                               \tag{KE.24}
\]

No odd factors are reordered: the new boundary is odd on the left, and
the full original normal right \(Cl_{q_0}\) block remains last. All
algebraic section and cone equivalence classes are even.

Without one of these inputs, retain the actual mapping cone Cq and its
canonical map e. KT-KK-14's Puppe sequence still records its cone
obstruction; replacing that cone by I requires the actual KK inverse,
not mere norm exactness. A different independently proved ideal/cone
equivalence would be a legitimate substitute for semisplitting. None
for every arbitrary local-domain geometry has been constructed here.


### 11BS.7. Scope of the excision constructions

The three sections above provide genuine KK excision at their stated geometric hypotheses. The arbitrary local-isometry groupoid has the reduced rank exactness of Theorem11BQ.1, but a completely positive section or an ideal/cone KK inverse has not been constructed at that full scope here. The scalar boundary in KE.24 also depends on the actual ideal cycle. Section11BR gives a semisplit example with a nonzero scalar boundary for one normalized auxiliary. Semisplitting therefore does not select a coherent normal class.

The original normal degree, full inverse Spin-c coefficient, right Clifford order and positive disk are retained. No formal jet closure adds arrows to the original groupoid. These algebraic sections by themselves do not give the suitable normal Dirac-type operator on the physical holonomy graph.

![Geometric completely positive sections and singular rank restrictions](../figures/geometric-rank-excision.png)

**Figure11BS.1.** KE.3–KE.24 and KEF.1–KEF.4. The three positive routes use, respectively, a whole source-fibre retraction, a supplied proper global Lie action, and an actual passive parameter. Every norm is the original reduced regular norm. The singular parameter set and smooth metric are KE.18–KE.19, with exact ranks3/4 and unchanged normal degree3. The separate-bisection matrix and limiting-jet examples explain why coefficient lifts and finite jet closures do not establish those hypotheses.

Editable SVG · Reproduction sources and exact data.

## 11BT. Full Killing jets and compactness through rank changes

### 11BT.1. Fixed geometry and result

Let T be a second-countable smooth q-dimensional Riemannian manifold,
with its given, possibly incomplete metric. Let G be its specified
local-isometry germ groupoid. All germs and domains are the original
ambient ones. Write

\[
 J_{\mathbb R}=TT\oplus\mathfrak{so}(TT),\qquad
 J=J_{\mathbb R}\otimes_{\mathbb R}\mathbb C,\qquad
 Q=q+q(q-1)/2=q(q+1)/2,
 \tag{KJ.1}
\]

with the Hermitian complexification of the real metric on TT and the
real Hilbert–Schmidt metric on the skew-adjoint endomorphisms. All
Hilbert modules and \(C_0(T)\) coefficients below are complex. A real
local Killing field X determines the real jet, included in J,

\[
 jX=(X,\nabla X)\in\Gamma(J).
 \tag{KJ.2}
\]

The Killing equation makes its real second entry skew-adjoint. For x
in T, let \(V_{x,\mathbb R}\) be the real initial-jet image of the full stalk of
real local Killing germs and put \(V_x=V_{x,\mathbb R}\otimes_{\mathbb R}\mathbb C\) inside \(J_x\).
A zero real initial jet gives a zero real Killing germ. Thus
\(\dim_{\mathbb R}V_{x,\mathbb R}=\dim_{\mathbb C}V_x=k(x)\), the full actual Killing rank. Complex
coefficients span the complexified real jets; no real germ, rank or
normal dimension is replaced or discarded.

**Proposition 11BT.1.** There is an actual countably generated Hilbert
\(C_0(T)\)-module \(E_K\) embedded isometrically as a closed submodule of
\(\Gamma_0(J)\). Its fibre at every x is exactly \(V_x\), with its prescribed
jet inner product. It has a continuous isometric G-action induced by
the original local isometries, including at changes of rank.

For every invariant open U with closed complement F, its actual
restriction has the completed module sequence

\[
 0\longrightarrow E_K C_0(U)\longrightarrow E_K
 \longrightarrow E_K\widehat\otimes_{C_0(T)} C_0(F)
 \longrightarrow0.
 \tag{KJ.3}
\]

The closed-set module retains ambient Killing germs and has fibre \(V_x\)
for every x in F. F need not be a manifold. At a genuine rank jump the
inclusion into \(\Gamma_0(J)\) has no adjoint. Thus an actual continuous
Hilbert module exists without a continuous family of orthogonal
projections onto its fibres.

The module is normal geometric data. It is not yet a reduced scalar
representation, a proper anchored kernel, an auxiliary index-one cycle,
a KK splitting or the final q-normal graph class.

### 11BT.2. Construction and completeness

Take all sections f jX, where X is an actual Killing field on an open
O and f is a continuous compactly supported scalar function on O.
Extend each such section by zero outside O. Compact support inside O
makes that extension continuous. Finite sums form a \(C_0(T)\)-submodule
\(E_{\rm alg}\) of \(\Gamma_0(J)\): multiplying f by a \(C_0(T)\) function still has
compact support inside O. Set

\[
 E_K=\overline{E_{\rm alg}}^{\|\cdot\|_\infty},\qquad
 \langle s,t\rangle(x)=\langle s(x),t(x)\rangle_{J_x}.
 \tag{KJ.4}
\]

The inner product is continuous and vanishes at infinity. It is
positive-definite, is \(C_0(T)\)-linear in its second argument, and its
module norm is precisely the ambient uniform norm. The displayed
closure is therefore a complete Hilbert module. It is a closed
submodule, not a presumed orthogonal summand.

The generator is the scalar product f jX, not j(fX). It does not
insert derivatives of f or assert that fX is Killing. The real
Killing germ supplies the original jet vectors; complex scalar
coefficients form the continuous Hilbert-module sections they span.

For clarity, \(\Gamma_0(J)\) is separable. Cover T by countably many
relatively compact bundle charts, use compact-support partitions in
those charts, and approximate their finitely many coefficient
functions uniformly by a countable dense family in each C_0 chart.
For a section vanishing at infinity, a compact truncation and a finite
partition reduce the approximation to finitely many such charts.
This proves the claimed separability directly. Consequently its
closed subspace \(E_K\) is separable. Choose a countable dense family
\(s_n\) in \(E_K\). A countable \(C_0(T)\) approximate identity \(u_m\) satisfies
\(s_nu_m\to s_n\) uniformly, by compact truncation and uniform convergence
on its compact part. Thus the \(s_n\) generate \(E_K\) as a Hilbert module.
No choice of a global Killing basis or constant rank was used.

### 11BT.3. The fibres are the full actual stalks

Evaluation of \(E_{\rm alg}\) at x lies in \(V_x\), because each active local field
has a Killing germ there. Conversely any Killing germ has a
representative X near x and a scalar cutoff f with \(f(x)=1\) and compact
support in that neighborhood. Then (f jX)(x) is its exact real initial
jet, and finite complex linear combinations give all of \(V_x\).
\(V_x\) is finite-dimensional, hence closed. Evaluation of \(E_K\) has image
exactly \(V_x\).

Let \(I_x\) be the ideal of \(C_0(T)\) functions vanishing at x. Then

\[
 \{s\in E_K:s(x)=0\}=\overline{E_K I_x}.
 \tag{KJ.5}
\]

One inclusion follows from evaluation. For the other, given \(s(x)=0\)
and \(\eta>0\), choose a neighborhood W of x on which \(\|s\|\)<\(\eta\) and a
compact set outside which \(\|s\|\)<\(\eta\). A scalar compact-support cutoff
\(\chi\), zero near x and one on the compact part outside W, gives
\(s\chi\) in \(E_K I_x\) and \(\|s-s \chi\|\)≤\(\eta\). Enlarging the compact part
and reducing \(\eta\) proves (KJ.5).

The quotient norm is exactly \(\|s(x)\|\). The lower bound is contractive
evaluation. For the upper bound choose a cutoff b, equal one near x
and compactly supported in a neighborhood on which
\(\|s(y)\|\)≤\(\|s(x)\|\)+\(\eta\). The section s-sb vanishes at x and belongs to
the left side of (KJ.5), while \(\|sb\|\)≤\(\|s(x)\|\)+\(\eta\). Hence

\[
 E_K\widehat\otimes_{\operatorname{ev}_x}\mathbb C
 = E_K/\overline{E_K I_x}\cong V_x
 \tag{KJ.6}
\]

isometrically and with the stated inner product. The tensor map sends
\(s\otimes z\) to \(zs(x)\); its dense range and this norm identity give the unitary.

The initial-jet injectivity used above has a short intrinsic proof.
A Killing field along a geodesic is the variational field of the
local isometric flow applied to that geodesic. Therefore it satisfies
the Jacobi equation with initial values X(x) and ∇_v X(x). If the
entire initial jet is zero, uniqueness for that linear ordinary
differential equation makes X zero on every radial geodesic in a
normal neighborhood. Hence its germ is zero. This argument uses
only a sufficiently small original normal ball and needs no metric
completion. It also proves connected uniqueness where it is used
in the existing compact-word theorem.

### 11BT.4. The action uses actual local domains

If g is represented by an actual local isometry a:O→O', its map is

\[
 U_g(v,A)=(d a_x v,\ d a_x A(d a_x)^{-1}),
 \qquad x=s(g).
 \tag{KJ.7}
\]

The formula first acts on real jets and extends complex linearly to J.
Both entries preserve the Hermitian jet inner products. Naturality of the
Levi–Civita connection follows from the Koszul formula: the pushed
connection is metric and torsion-free, and that formula uniquely
determines such a connection. Thus \(U_gjX(x)=j(a_*X)(a(x))\); \(a_*X\)
is a Killing field on its actual transported domain. It follows that
U_g maps \(V_x\) onto \(V_{a(x)}\), with inverse \(U_{g^{-1}}\).

On the module over O, transport a compact generator f jX to
\((f\circ a^{-1})j(a_*X)\). Its support is compact inside O', so it belongs
to the module over O'. Finite sums, uniform closure and the isometry
give the whole transported module. The dependence on g is
continuous on each actual bisection: source, derivative and
conjugation are smooth there, and transported local generators
give continuous sections. Local generator approximation then gives
continuity on the completed field for every module section. The
chain rule gives \(U_{g_2g_1}=U_{g_2}U_{g_1}\) and identities act as
identities. No word, arbitrary regular-fibre arrow or germ domain
has been extended.

Equivalently, the total field is a subfield of the ambient bundle J
with its norm topology: on generator sections this is immediate,
and uniform approximation gives it on all sections. Every fibre
vector has a locally defined generator with that value. Therefore
the section topology and the ambient subfield topology agree. This
also makes the stated continuity of the groupoid action explicit.

### 11BT.5. Closed restriction without treating F as a manifold

The tensor map in (KJ.3) is the ordinary Hilbert-module coefficient
quotient. Its kernel is the closure \(E_K\) \(C_0(U)\). Here is a direct
description that avoids replacing the ambient germs by germs of F:

\[
 \overline{E_K C_0(U)}
 =\{s\in E_K:s|_F=0\},\qquad
 \|s+E_K C_0(U)\|=\sup_{x\in F}\|s(x)\|.
 \tag{KJ.8}
\]

For the first equality, if s vanishes on F then the compact set
\(K_\eta\)={x:\(\|s(x)\|\)≥\(\eta\)} is contained in U. Choose a compact-support
cutoff in U equal one on \(K_\eta\). Multiplication by it approximates
s within \(\eta\). The reverse inclusion is immediate.

For the quotient norm, restriction is contractive. Put
M=sup_F\(\|s\|\). The compact set {\(\|s\|\)≥M+\(\eta\)} lies in U.
Choose \(\chi\) in \(C_c(U)\), 0≤\(\chi\)≤1, equal one on this set. Then
\(s-s\chi\) represents the same quotient element and has norm at
most M+\(\eta\). Let \(\eta\) decrease to zero. This proves the norm equality.
The quotient consequently identifies with the closed restricted
section module with the original pointwise inner product. Tensoring
again with ev_x for x in F, or repeating Section11BT.3 with restricted
cutoffs, gives exactly \(V_x\). Invariance of F and U makes this
sequence G-compatible through (KJ.7).

This is a module restriction theorem, with no assertion about the
reduced algebra kernel beyond the separately proved 11BQ theorem.
It constructs no completely positive section of that algebra
quotient and no KK inverse for its mapping cone.

### 11BT.6. Why an orthogonal projection does not follow

Suppose the inclusion i:\(E_K\)→\(\Gamma_0(J)\) had an adjoint. Since it is
isometric, \(i^*i=1\), and \(P=ii^*\) would be an adjointable projection on
the finite-rank bundle module \(\Gamma_0(J)\). In a local bundle
trivialization, an adjointable \(C_0(T)\)-linear operator on this module
is multiplication by a bounded continuous matrix field: apply it
to a local frame times a cutoff equal one on a smaller chart;
\(C_0(T)\)-linearity determines the resulting pointwise matrices.
Its adjoint gives the adjoint matrix field there. This proves the
description without assuming a complement in advance.

The fibre P_x would be the orthogonal projection onto \(V_x\), by
(KJ.6) and evaluation of i and i*. Its ordinary matrix trace would
be the integer \(k(x)\). A continuous matrix field has continuous
trace, so that integer would be locally constant. Thus at a genuine
rank jump no such adjoint can exist.

The same argument excludes interpreting \(E_K\) as a finite-rank
vector subbundle with a continuous projection at that jump. It does
not exclude the continuous countably generated Hilbert module
actually constructed above.

### 11BT.7. Exact warped-torus instance

Use the point-foliation example of Section11BR, SE.23–SE.25:
\(T=(\mathbb R/\mathbb Z)^2\times(-2,2)\), with

\[
 g=dt^2+a(t)^2(dx^2+dy^2),\quad a=e^u,\quad
 u(t)=\begin{cases}\varepsilon e^{-1/(1-t^2)},&|t|<1,\\0,&|t|\ge1,\end{cases}
 \qquad\varepsilon>0.
\]

Its actual full rank is three on \(F=(\mathbb R/\mathbb Z)^2\times[-1,1]\), including
the boundary, and six on \(U=(\mathbb R/\mathbb Z)^2\times((-2,-1)\cup(1,2))\). The original
metric is incomplete at both omitted ends. The original groupoid
for this example is units. On the bump interior \(R=-4u''-6(u')^2\) is analytic, has \(R(0)=8\varepsilon/e>0\), and tends to zero at its endpoints. A Killing field has zero radial component where \(R'\ne0\), a dense subset. The mixed Killing equations make its angular components independent of \(t\); they are the three local Euclidean angular Killing fields. Connected first-jet uniqueness excludes additional fields at the boundary. On the two flat ends all six Euclidean Killing germs occur. This gives the asserted full ranks.

The global orthonormal frame is \(E_0=\partial_t\), \(E_1=a^{-1}\partial_x\), \(E_2=a^{-1}\partial_y\). Put \(H=u'\). Define a smooth real rank-three subbundle
\(L_{\mathbb R}\) of \(J_{\mathbb R}\) by \(v=v_1E_1+v_2E_2\) and the real skew map A with

\[
 A E_0=H(v_1E_1+v_2E_2),\quad
 A E_1=-H v_1E_0+cE_2,\quad
 A E_2=-H v_2E_0-cE_1.
\]

Set \(L=L_{\mathbb R}\otimes_{\mathbb R}\mathbb C\) and \(W=L_{\mathbb R}^{\perp}\otimes_{\mathbb R}\mathbb C\) with their Hermitian
metrics. The displayed real columns are the exact jets of the local angular Euclidean Killing
fields \(X=(b_1-cy)\partial_x+(b_2+cx)\partial_y\). In fact
\(\nabla_{E_0}E_i=0\), \(\nabla_{E_i}E_0=HE_i\) and \(\nabla_{E_i}E_j=-H\delta_{ij}E_0\). Differentiating the displayed X gives
precisely the three columns above. The parameters (v_1,v_2,c)
are independent, so L has complex rank three everywhere and agrees
with \(V_x\) on F. On U the metric is Euclidean locally, every real value and
skew derivative is realized by a Euclidean affine Killing germ,
and \(V_x=J_x\). Angular rotation germs need not be global torus
vector fields; the jet bundle formula, and scalar localization of
their actual local germs, are global.

The exact Hermitian jet norm of the displayed columns is
\[\|(v,A)\|^2=(1+2H^2)(|v_1|^2+|v_2|^2)+2|c|^2.\] The factor two counts both
skew matrix entries in the Hilbert–Schmidt norm. In particular the
angular frame factor a does not introduce an extra factor in c:
E_i differentiates a w_j as partial_i w_j.

Thus \(W=L^{\perp}\) inside the complex J and is a smooth complex rank-three bundle. The
construction therefore has the literal unit-groupoid description

\[E_K=\Gamma_0(L)\oplus\Gamma_0(U,W|_U).\]

where the second summand is extended by zero in the ambient
bundle. To verify equality, every constructed section takes values
in L on F, so its W component vanishes there and belongs to the
second summand by the compact level-set argument of Section11BT.5.
Conversely any compact-support L section is locally a continuous
linear combination of three independent angular Killing jets;
a finite partition writes it as actual generators. A compact-support
W section in U is locally a combination of six full Euclidean
Killing jets and is likewise an actual generator sum. Compact
truncations prove equality for all vanishing sections.

No continuous projection onto \(E_K\) exists at t=±1: its fibre trace
would jump from three to six. The smooth projection onto L is
permissible, but does not project onto the entire Killing-germ
module on U. This coordinate splitting is stated for the unit
example only; a nonunit flat-end local isometry can mix t and the
angular directions, so it need not preserve the selected L. The
intrinsic whole \(E_K\) action of Section11BT.4 does not have that defect.



### 11BT.8. Actual module and a scalar-local failure

Use the incomplete warped-torus unit geometry and the literal module
decomposition in Section11BT.7:

\[
 E_K=E_L\oplus E_W,
 \quad E_L=\Gamma_0(L),\quad E_W=\Gamma_0(U,W|_U),
 \quad U=\mathbb T^2\times((-2,-1)\cup(1,2)).
 \tag{KJC.1}
\]

Here L and W are the actual Hermitian complexifications of the smooth
real rank-three subbundles of the original real rank-six jet bundle.
Their complex ranks are three; the complex ambient J has rank six.
All \(C_0(T)\) Hilbert modules below are complex. The W summand is extended by zero, and has
zero fibre on \(F=\mathbb T^2\times[-1,1]\). The projection \(P_W\) onto this
summand is adjointable **on \(E_K\)**. It is not an ambient projection
from \(\Gamma_0(J)\) onto \(E_K\).

Fix \(x_0=(x,y,1)\) in F, a sequence \(x_n=(x,y,1+1/n)\) in U for n≥2,
and f in \(C_c(T)\), 0≤f≤1, equal one on a neighborhood of \(x_0\).
Then

\[
 fP_W\notin\mathcal K(E_K).
 \tag{KJC.2}
\]

For a rank-one operator \(\theta_{\xi,\eta}\), its W-to-W block at \(x_n\) is
\(\theta_{\xi_W(x_n),\eta_W(x_n)}\). Both vectors tend to zero in the
ambient W bundle: their zero extensions are continuous at \(x_0\).
Thus the block norm tends to zero. Finite sums have the same property.
Norm approximation by finite sums proves, for every \(K\in\mathcal K(E_K)\),

\[
 \|(P_W K P_W)_{x_n}\|\longrightarrow0.
 \tag{KJC.3}
\]

The actual module fibres identify with their original pointwise
vectors and evaluation is contractive. Since \(f(x_n)=1\) for all
sufficiently large n and \(\dim_{\mathbb C}W_{x_n}=3\),

\[
 \|fP_W-K\|
 \ge\limsup_n\|1_{W_{x_n}}-(P_W K P_W)_{x_n}\|=1.
 \tag{KJC.4}
\]

This proves (KJC.2). Every fibre of fP_W is nevertheless an ordinary
finite-dimensional, hence compact, operator. The failure is uniform
module compactness across the actual rank boundary, not a lack of
compactness of a selected fibre.

### 11BT.9. An exact complete-domain Fock countertest

Form the tensor Fock Hilbert \(C_0(T)\)-module

\[
 \mathcal F=\bigoplus_{r=0}^{\infty}E_K^{\widehat\otimes r},
 \quad E_K^{\widehat\otimes0}=C_0(T),\quad
 N(\xi_r)_r=(r\xi_r)_r.
 \tag{KJC.5}
\]

Its whole domain is
\[
 \operatorname{Dom}N=\left\{\xi\in\mathcal F:
 \sum_{r\ge0}r^2\langle\xi_r,\xi_r\rangle
 \text{ converges in }C_0(T)\text{ norm}\right\}.
\]
 Finite sequences
are a dense domain. The diagonal maps \((N\pm i)^{-1}\), with entries
\((r\pm i)^{-1}\), are bounded adjointable maps into that domain. The
weighted entries \(r/(r\pm i)\) are uniformly bounded, so the domain
condition holds for every input, and multiplication verifies both
inverse identities. Hence N is a closed regular self-adjoint
positive operator on that exact domain. The entries \((1+r^2)^{-1}\) map every input into the square domain, since their \(r^4\)-weighted squared factors are at most one. They give the onto inverse of \(1+N^2\), verifying regularity directly.

At each x, the r-th tensor fibre has dimension \(k(x)^r<\infty\).
The numerical eigenvalues r go to infinity. Consequently
\((1+N_x^2)^{-1}\) is compact on the complete Fock fibre at every x.
For \(k(x)\)=0 only its degree-zero fibre survives.

However, compression by the adjointable inclusion of the degree-one
summand gives

\[
 \iota_1^* f(1+N^2)^{-1}\iota_1
 ={f\over2}1_{E_K}.
 \tag{KJC.6}
\]

If the left full-module localized inverse were compact, this
compression, and then its W compression, would be compact. Rank-one
composition verifies that compacts are preserved by these
adjointable compositions. Equation (KJC.2) contradicts that
conclusion. Thus the localized Fock inverse is **not** compact,
despite its genuine complete domain and all compact fibre inverses. Its distance from the module compacts is exactly one half: the disappearing degree-one block gives the lower bound, while subtracting the compact vacuum block leaves only positive degrees, whose inverse norms are at most one half.

No claim about a graded auxiliary index is used here: N is the
explicit positive number operator. The same degree-one obstruction
applies to a proposed oscillator whose squared operator has this
unchanged number-operator block. It does not apply to every possible
weighted or otherwise repaired oscillator.

### 11BT.10. A literal compactness repair on the disappearing summand

For this same actual U define

\[
 \lambda(t)=1+(t^2-1)^{-2}+(4-t^2)^{-2}.
 \tag{KJC.7}
\]

It is smooth and at least one on U, and tends to infinity at all
four omitted endpoints. It is proper on U. Multiplication by \(\lambda\)
on \(E_W\) has whole domain

\[
 \operatorname{Dom}M_\lambda
 =\{s\in E_W:\lambda s\in E_W\}.
 \tag{KJC.8}
\]

Compact-support sections form a dense domain. Multipliers
\((\lambda\pm i)^{-1}\) map every \(E_W\) section into it, because
\(\lambda/(\lambda\pm i)\) is bounded. These are both onto resolvent
inverses on the displayed domain, proving closedness, regularity
and self-adjointness directly. The bounded multiplier \((1+\lambda^2)^{-1}\) maps onto the square domain, with \(\lambda^2/(1+\lambda^2)\) bounded. Thus \(1+M_\lambda^2\) is onto as well.

The multiplier \((\lambda\pm i)^{-1}\) vanishes at infinity on U. For a
finite-rank vector bundle, multiplication by any \(C_0(U)\) endomorphism
is compact on \(\Gamma_0(U,W)\): approximate it on a compact set by a
finite bundle partition and local rank-one frame matrices, and
discard its small tail. Hence these resolvents are genuinely
compact on \(E_W\). For every f in \(C_0(T)\),

\[
 f(1+M_\lambda^2)^{-1}
 =M_{f|_U/(1+\lambda^2)}\in\mathcal K(E_W).
 \tag{KJC.9}
\]

The coefficient extends by zero at t=±1, so the formerly nonzero
boundary block now tends uniformly to zero. This verifies actual
module compactness of this repaired summand.

This positive multiplier is not an index-one cycle or a reduced
groupoid representation. It supplies no general holonomy covariance
or arbitrary-rank normal/physical product. In particular an exact
orthogonal action with an invariant vacuum cannot be treated as a
reduced representation for a nonamenable isotropy group. The
original inverse Spin-c line, right Cl_3 and positive three-disk of
the unit example remain untouched.


### 11BT.11. The geometric module and the remaining operator

The full Killing-germ module is a continuous field with genuine ambient germ transport and closed restriction, even when its rank changes. Its inclusion in the fixed jet bundle need not be adjointable. The explicit number-operator calculation also distinguishes compactness on each fibre from compactness on the completed Hilbert module. The multiplier in KJC.7 repairs the displayed disappearing summand and its whole domain; it is not an auxiliary of index one or a reduced holonomy cycle.

The prescribed normal degree remains \(q\), rather than the Killing rank or \(q(q+1)/2\). The original full inverse Spin-c line, right \(Cl_q\), positive disk and physical graph still require their actual operator construction.

![Full Killing-jet fibres and compactness at a rank boundary](../figures/killing-jet-rank-boundary.png)

**Figure11BT.1.** KJ.1–KJ.8 and KJC.1–KJC.9. Real germs are retained and Hermitian-complexified for the complex Hilbert module. In the original incomplete unit example the full ranks are three and six, while the normal dimension is three. Every compact operator's disappearing block tends to zero at the boundary; the unweighted degree-one Fock inverse retains norm one half there. The proper multiplier gives the exact vanishing tails on all four omitted endpoints. The diagram asserts neither an ambient orthogonal projection onto the full germ field nor a completed scalar graph class.

Editable SVG · Reproduction sources and exact data.

## 11BU. Global curved graph classes with variable Killing rank

### 11BU.1. Exact global-action result

**Theorem 11BU.1.** Let \((T,g)\) be a connected second-countable oriented smooth q-manifold with its original, possibly incomplete Riemannian metric. Let Γ be a countable group of distinct global isometries, with the actual holonomy germ groupoid \(G=\Gamma \ltimes T\). Retain the prescribed smooth coherent full inverse Spin-c normal coefficient \(S_{\rm inv}\), its grading and final right \(Cl_q\) factor, and the actual normalized entire counting/plaque imprimitivity \(E_Z\) of the foliation. Suppose there is a supplied even class

\[
 \kappa\in KK(C_r^*\Gamma,\mathbb C),\qquad [1]\kappa=+1. \tag{GG.1}
\]

There is an explicit reduced scalar normal cycle and an explicit bounded phase on the entire physical graph completion, of the original degree q, with the full inverse coefficient and original positive-disk pairing +1. The normal operator is

\[
 \mathcal D=D_\kappa\otimes1+\Gamma_\kappa\otimes N_\alpha,
 \qquad N_\alpha=\alpha^{1/2}N_{\rm inv}\alpha^{1/2}. \tag{GG.2}
\]

Here α is a smooth positive Γ-invariant speed on the original manifold. It is constructed below from the original radius of compactness. The original metric g is retained in every symbol, transport and Bott comparison. The operator has an exact self-adjoint completed joint domain, ordinary scalar spatial localized compact resolvents, and bounded commutators with the actual compact smooth global-action kernels. No assumption on the local Killing-germ rank is made; rank changes cause no auxiliary selection or gluing.

The supplied κ premise is essential to this construction. It is not proved for every countable subgroup of a Lie group with arbitrarily many components. A countable discrete group itself is a zero-dimensional second-countable Lie group, so the unrestricted Lie premise gives no scalar-auxiliary theorem about its component directions. The existing AI kernel theorem proves a different, anchored groupoid result, by absorbing those directions in the original endpoint anchor. It does not supply GG.1 for the whole Γ. This construction supplies no such implication.

Two unconditional positive subfamilies follow:

1. Countable characteristic-zero linear Γ, and countable subgroups of almost-connected Lie groups: the actual uniform-embedding kernels, compact-base affine-field and proper-factorization providers supply GG.1. Section 11BN.2 gives the exact countable linear chain. For the almost-connected extension, the primary theorem supplies the same countable group-kernel input, as explained below. This uses an actual reduced scalar class, not a reduced trivial character.
2. An actual proper discrete global action Γ⋉T: the actual full/reduced norms agree. Use the maximal geometric normal cycle directly on the reduced algebra, and induce it through E_Z. No reduced scalar representation of the whole Γ is needed for this second route.

Thus an actual positive curved family with arbitrary Killing rank is proved. The all-global-isometry scope with arbitrary components and no supplied κ, and the arbitrary local holonomy problem, remain separate. No nonexistence claim is made for either.

### 11BU.2. The actual arrows and the original invariant speed

Distinct global isometries of connected T have distinct germs everywhere. An isometry fixing a point and its derivative fixes a normal neighborhood via the exponential map. The set on which it agrees with the identity is then open and closed by the same first-jet argument and connected continuation. Consequently the germ topology is the disjoint union of the actual Γ-labeled unit charts. The continuous discrete cocycle is

\[
 c(\gamma,x)=\gamma,\quad s(\gamma,x)=x,\quad
 r(\gamma,x)=\gamma x,\quad c^{-1}(e)=T. \tag{GG.3}
\]

All maps are genuine original global isometries. The argument does not promote a potential continuation of a local germ to an arrow.

Set, using the original length distance,

\[
 R_g(x)=\sup\{r>0:\overline{B_g(x,r)}\text{ is compact IN }T\},
 \qquad\rho(x)=\min(1,R_g(x)). \tag{GG.4}
\]

Local compact coordinate balls show ρ>0. If \(d_g(x,y)=d\) and \(r<R_g(x)\), then \(B_g(y,r-d)\subset B_g(x,r)\) whenever r>d. Its closure is a closed subset of the compact closure on the right. Taking suprema, and reversing x,y, proves that the truncated function ρ is 1-Lipschitz. The convention for \(R_g=\infty \) is harmless after truncation. Global isometries preserve compact closures of original balls, so ρ is invariant.

Let H be the closure of Γ in \(Isom(T,g)\). H is a Lie group and its action is proper. No completeness of g is used. The properness statement for connected locally compact metric spaces is [Manoussos, Corollary 4, printed p. 211](https://www.math.uni-bielefeld.de/~amanouss/papers/mjm_Manoussos.pdf). The Riemannian Lie statement is also explicitly recorded in [Farb–Weinberger, Theorem 2.1](https://math.uchicago.edu/~shmuel/FarbW2,%20isoms.pdf). These inputs are about global isometries.

Here is a smooth invariant minorant construction; it does not assume a smooth orbit quotient or constant orbit dimension. The proper quotient \(W=T/H\) is second-countable locally compact Hausdorff and paracompact. Choose a locally finite relatively compact open cover V_i of W and compact smooth bump functions ξ_i on T, with \(supp \xi _i\subset q^{-1}(V_i)\), whose positive H-saturations cover T. Such a choice follows by lifting compact pieces of a refined quotient cover: q is open; finitely many small precompact coordinate neighborhoods over each compact piece cover it after saturation. Their compactly supported smooth bumps may be added for that piece.

The sum \(c_0=\sum _i \xi _i\) is locally finite. For compact K⊂T, only finitely many V_i meet q(K), and properness makes
\({h:h^{-1}K\cap supp \xi _i\ne ∅}\) compact for every retained i. Therefore

\[
 J(x)=\int_Hc_0(h^{-1}x)\,dh>0,\quad c(x)=c_0(x)/J(x),
 \quad\int_Hc(h^{-1}x)\,dh=1 \tag{GG.5}
\]

are smooth, with locally uniform compact h-support. Left Haar substitution proves J is invariant. Choose any smooth positive b≤ρ/20, by an ordinary locally finite coordinate partition with coefficients below the positive local infima of ρ/20, and put

\[
 \alpha(x)=\int_H c(h^{-1}x)b(h^{-1}x)\,dh.
 \qquad 0<\alpha\le\rho/20<\rho/10. \tag{GG.6}
\]

Local compact h-support proves smoothness by differentiation under the integral. Left Haar substitution gives H-invariance. Positivity follows from b>0 and the normalized nonnegative cutoff. The bound follows from invariance of ρ. No H-action on the full Spin-c coefficient and no invariant determinant connection have been assumed in this averaging: only the scalar speed is averaged.

The speed metric \(g_\alpha =\alpha ^{-2}g\) is complete. To prove it, consider an escaping path of finite \(g_\alpha \) length. Since α≤1/20, it has finite original g-length L. An escaping tail of remaining original length ℓ cannot be contained in any original relatively compact ball of radius greater than ℓ, so \(\rho (x)\le \ell \) on that tail. In original arclength t,

\[
 \int\frac{dt}{\alpha(x(t))}
 \ge20\int\frac{dt}{L-t}=\infty, \tag{GG.7}
\]

a contradiction. If the remaining original length is infinite, α≤1/20 already forces infinite speed length. Equivalently, along a speed-parametrized finite path, the Lipschitz estimate gives \(|d\rho /ds|\le \alpha \le \rho /20\); ρ stays bounded below. An original Cauchy tail then lies in one relatively compact original ball and converges in T. These arguments prove metric completeness, without adjoining missing endpoints.

A complete smooth Riemannian metric admits a smooth proper r with \(|dr|_{g_\alpha }\le 2\): smooth the proper 1-Lipschitz distance in locally finite small charts, choosing local approximation errors so that the partition-derivative errors have summable norm below one. Hopf–Rinow makes its sublevels compact. For \(\eta _n=\zeta (r/n)\), with ζ=1 on [0,1] and supported in [0,2),

\[
 \eta_n\to1,\qquad
 \sup|\alpha d\eta_n|_g\le2\|\zeta'\|_\infty/n. \tag{GG.8}
\]

These are the actual compact-support core cutoffs needed for self-adjointness. No global derivative bound for α is asserted or required.

### 11BU.3. A proved finite-filtration auxiliary, with exact full domains

This proves the candidate number-operator construction, rather than presuming it. Start from an essential unital even bounded representative of GG.1. Replace its odd Fredholm operator by its sign, zero on its finite kernel, through a compact perturbation. Add only finitely many ZERO-SOURCE partners to pair the kernel, with opposite parity. The enlarged graded Hilbert space has

\[
 F=F^*,\quad F^2=1,\quad F\text{ odd},\quad
 P=\phi(1),\quad [F,\phi(a)]\in\mathcal K. \tag{GG.9}
\]

P is generally not the identity on the enlarged space. No odd unitary on an essential index-one unital source is presumed.

Enumerate \(P\), all \(\phi (\lambda _\gamma )\) and their adjoints as even bounded operators T_j. Enumerate a dense family of graded F-paired basis vectors as well. Construct increasing finite-dimensional graded F-reducing spaces E_n, with orthogonal projections p_n, satisfying

\[
 T_jE_{n-1}+T_j^*E_{n-1}\subset E_n\quad(j\le n),
 \quad\overline{\bigcup E_n}=\widetilde H,\qquad
 \|[F,T_j](1-p_n)\|\le\frac{2^{-n}}{(n+1)^2}\quad(j\le n).
 \tag{GG.10}
\]

The construction is finite at each step. Include E_{n-1}, the listed translates, the first n paired basis vectors, and finite-rank approximating ranges for the finitely many compact operators \([F,T_j]\) and their adjoints. Close this finite span under F and the grading. Adding dimensions only decreases compact-tail norms, so all bounds can be met. This is why no quasidiagonality premise appears.

Set p_0=0, Q_n=p_n-p_{n-1}, and define the even positive self-adjoint number operator

\[
 L=\sum_{n\ge1}nQ_n,\qquad
 \operatorname{Dom}L=\{u:\sum n^2\|Q_nu\|^2<\infty\}.
 \tag{GG.11}
\]

Its resolvents are compact, because every Q_n has finite rank; it commutes with F and the grading.

For fixed j, the swallowing conditions for T_j and T_j* make every block outside the finite first-j head have bandwidth at most one. Indeed \(T_j E_n\subset E_{n+1}\) for n≥j-1, and the adjoint inclusion gives the reverse block inequality. Thus the tail commutator \([L,T_j]\) is the sum of the forward and backward block shifts, each of norm at most \(\|T_j\|\). The finite head commutator is bounded. This proves boundedness of \([L,T_j]\) on the algebraic core; for example \(2(j+1)\|T_j\|\) is a sufficient nonoptimal bound after isolating the head. The block formula also proves

\[
 T_j\operatorname{Dom}L\subset\operatorname{Dom}L,
 \quad LT_ju=T_jLu+[L,T_j]u \tag{GG.12}
\]

by finite graph approximation and closedness of L. One must use the adjoint swallowing condition: forward inclusion alone does not give this two-sided band statement.

Let K_j=[F,T_j]. For n>j,

\[
 \|K_jQ_n\|\le\frac{2^{-(n-1)}}{n^2},\qquad
 \sum_{n>j}n\|K_jQ_n\|<\infty. \tag{GG.13}
\]

Consequently K_jL is bounded by an absolutely norm-convergent tail, plus a finite head. On \(\operatorname{Dom} L\),

\[
 \widetilde D=FL,\qquad
 [\widetilde D,T_j]=F[L,T_j]+K_jL\in\mathcal B.
 \tag{GG.14}
\]

The self-adjoint odd operator \(FL\) has compact resolvent and bounded transform compactly different from F. Since P was enumerated, \([\widetilde D,P]\) is bounded and P preserves the completed domain. Remove its actual off-diagonal block

\[
 B_0=P\widetilde D(1-P)+(1-P)\widetilde DP
 =[\widetilde D,P](2P-1),
 \quad D_\kappa=(\widetilde D-B_0)|_{P\operatorname{Dom}L}.
 \tag{GG.15}
\]

B_0 is bounded self-adjoint and odd. The bounded perturbation theorem gives self-adjointness on the same completed domain. Resolvent identities retain compact resolvents. The bounded transforms before and after perturbation differ compactly: the two-resolvent integral has an integrable \(O(\|B_0\|/(1+t^2))\) bound, and every integrand is compact. The zero-source complementary block is degenerate. Hence \((H_\kappa =P\widetilde H,V_\gamma =\phi (\lambda _\gamma )|_{H_\kappa },D_\kappa )\) represents EXACTLY κ, with ordinary even index +1, compact resolvent, and

\[
 V_\gamma\operatorname{Dom}D_\kappa
       =\operatorname{Dom}D_\kappa,\qquad
 C_\gamma=[D_\kappa,V_\gamma]\text{ bounded},
 \quad C_{\gamma\delta}=C_\gamma V_\delta+V_\gamma C_\delta.
 \tag{GG.16}
\]

No uniform bound over Γ is needed. Every actual compact arrow test has finitely many Γ labels.

### 11BU.4. The actual reduced norm and tensor carrier

Write \(A_m=C_max^*(G)\), \(A_r=C_r^*(G)\), \(B=C_r^*\Gamma \), and \(R=C_0(T)\). The literal finite-label diagonal is

\[
 \Delta_r:A_r\longrightarrow A_m\otimes_{\min}B,
 \qquad \Delta_r\Big(\sum_\gamma f_\gamma\Big)
          =\sum_\gamma f_\gamma\otimes\lambda_\gamma.
 \tag{GG.17}
\]

For completeness, its norm justification is as follows. Disintegrate a faithful maximal representation and tensor each actual arrow unitary with \(\lambda _{c(g)}\). This gives a genuine integrated groupoid representation, so the full diagonal is a contractive star homomorphism into the spatial tensor product. The faithful group trace τ discards every nonunit label; by GG.3,
\((id\otimes \tau )\Delta (a)=i_m E_r(q_G(a))\).
The slice is faithful on positive spatial tensors: in a faithful representation the square root of a positive zero-slice element kills every vector \(v\otimes \delta _e\); commuting right-regular translations then show it kills the dense family \(v\otimes \delta _\gamma \). Faithfulness of the actual reduced unit expectation therefore gives
\(\ker  \Delta =\ker  q_G\).
Taking the quotient proves GG.17 is isometric. This is the actual coaction/Fell-absorption proof of Section 11BN.1, applied to the genuine global cocycle; it proves the reduced norm instead of assuming the bare normal representation is reduced.

Define the actual transfer

\[
 E_\Theta=A_m\otimes H_\kappa,\quad
 \Theta=[\Delta_r](1_{A_m}\boxtimes\kappa)\in KK(A_r,A_m).
 \tag{GG.18}
\]

Its right inner product is
\(\langle a\otimes v,b\otimes w\rangle =\langle v,w\rangle a^*b\), and its left core action is
\(\sum  L_{f_\gamma }\otimes V_\gamma \). Its bounded phase is \(1\otimes F_\kappa \); its module compact defects follow from the scalar compact defects of κ. GG.17 supplies its reduced norm.

Let \(H_N=L^2(T,S_{\rm inv})\) carry the actual maximal integrated geometric representation π_N. The map

\[
 (a\otimes v)\otimes u\longmapsto v\otimes\pi_N(a)u
 \tag{GG.19}
\]

is balanced. Expanding the inner product gives exactly the displayed module inner product represented by π_N. Its range is dense by nondegenerate original unit cutoffs. Thus it is an onto isometry, by actual coefficient multiplication. The resulting carrier is \(H_0=H_\kappa \otimes H_N\), and its reduced core action is

\[
 (\Pi(k)u)(x)=\sum_{\gamma\in\Gamma}
  k(\gamma,\gamma^{-1}x)
  (V_\gamma\otimes U^{\rm inv}_{\gamma,\gamma^{-1}x})
  u(\gamma^{-1}x). \tag{GG.20}
\]

The original source point is \(\gamma ^{-1}x\), and the summation is finite on a compact smooth core. Every arrow and the full inverse line phase are retained. GG.17 and the spatial representation \(\pi _N\otimes V\) give its actual reduced continuity.

Since \(\Delta _r i_r(f)=i_m(f)\otimes 1\), GG.1 gives the whole original-unit identity

\[
 [i_r]\Theta=[i_m]\quad\text{in }KK(R,A_m).
 \tag{GG.21}
\]

This identity uses the actual global cocycle. It is not inferred from independent rank-layer indices. In particular the order-two boundary defect in the scalar rank-layer calculation in Section 11BR is not repeated here: there is one global module from the outset.

For the second unconditional subfamily, here is also the actual proper-action norm proof. Apply GG.5's scalar cutoff construction with H equal to the properly acting discrete Γ, obtaining \(c\ge 0\) with \(\sum _\gamma  c(\gamma ^{-1}x)=1\) and locally finite orbit sums. For any nondegenerate full covariant representation \((\pi ,U)\) of the original action, define

\[
 (Jv)_\gamma=\pi(\sqrt c)U_\gamma^{-1}v
       \quad\text{in }\ell^2(\Gamma)\otimes H_\pi. \tag{GG.21a}
\]

The normalized sum gives \(J^*J=1\). It intertwines π(f) with the diagonal regular coefficient \(\pi (f\circ \gamma )\) in coordinate γ, and U_δ with the left coordinate shift \(\gamma \mapsto \delta ^{-1}\gamma \). These identities follow directly by the covariance law. Thus every full covariant representation is a subrepresentation of its actual regular induced representation. The full norm is at most the reduced norm; the reverse inequality is automatic. This proves actual full/reduced equality. Use \(\Theta =1\), \(H_0=H_N\) and \(D=N_\alpha \) in the remaining normal/physical argument, without representing the reduced group algebra by its trivial character.

### 11BU.5. Closed normal and joint domains, including spatial scalar compactness

Let N_inv be the original formally symmetric full inverse normal expression, with
\(C(df)=[N_{\rm inv},f]\), \(C(df)^*=-C(df)\), and \(\|C(df)\|=|df|_g\). On the original volume measure,

\[
 N_\alpha=\alpha N_{\rm inv}+\tfrac12 C(d\alpha).
 \tag{GG.22}
\]

The correction by itself is skew-adjoint; the whole expression is symmetric. The compact elliptic estimates and GG.8 give, exactly as in 11BM.2,

\[
 \operatorname{Dom}N_\alpha=
 \{u\in L^2(T,S_{\rm inv}):u\in H^1_{\rm loc},
                    N_\alpha u\in L^2\text{ distributionally}\}.
 \tag{GG.23}
\]

Here are the domain details needed for this different invariant speed. The compact interior principal matrices are uniformly elliptic in all original q directions, because α has a positive minimum there. Freezing coefficients, Fourier estimates and local mollifier commutators yield local H¹ regularity for maximal distributions. Compact H¹ vectors are graph approximable by finite chart smoothing. Applying GG.8 to every maximal vector gives compact H¹ graph approximants with error tending to zero. Thus minimal equals maximal and N_α is self-adjoint. The same compact-buffer estimate and finite-coefficient Rellich theorem give \(\chi (N_\alpha -z)^{-1}\) ordinary scalar compact.

The operators \(D_\kappa ^2\otimes 1\) and \(1\otimes N_\alpha ^2\) strongly commute. On the compact smooth finite squared-auxiliary-spectral core the graded cross terms cancel:

\[
 \|\mathcal Du\|^2=\|(D_\kappa\otimes1)u\|^2
                    +\|(1\otimes N_\alpha)u\|^2.
 \tag{GG.24}
\]

Finite spectral windows of D_κ² are graded and finite dimensional. On each window the sum GG.2 is a finite matrix bounded perturbation of \(\Gamma _\kappa \otimes N_\alpha \); it is self-adjoint. Their orthogonal direct sum is self-adjoint, and GG.24 identifies its exact domain as

\[
 \operatorname{Dom}\mathcal D=
 \operatorname{Dom}(D_\kappa\otimes1)\cap
 \operatorname{Dom}(1\otimes N_\alpha).
 \tag{GG.25}
\]

Unlike 11BO's jointly rescaled expression, the auxiliary term here is not multiplied by α. GG.25 is a global graph intersection for the unweighted auxiliary and the complete-speed normal operator. It is not global unweighted normal H¹.

An equivalent complete distributional description is

\[
 \{u\in H_0:u\in H^1_{\rm loc}(T,H_\kappa\otimes S_{\rm inv}),
 \ u\in L^2_{\rm loc}(T,\operatorname{Dom}D_\kappa\otimes S_{\rm inv}),
 \ \mathcal Du\in H_0\text{ distributionally}\}.
 \tag{GG.26}
\]

To see the nontrivial reverse inclusion, apply finite D_κ² windows and compact normal cutoffs to the equation. GG.24 and local normal ellipticity give uniform joint graph and H¹ bounds. Weak limits yield both local domains. GG.8 then approximates in the expression graph norm; taking the cutoff limit in GG.24 yields both global graph norms in GG.25. Conversely GG.25 gives the distributional equation, the local elliptic estimate and both local domains. This proves equality, not only inclusion of a convenient core.

For \(R_z=(\mathcal D-z)^{-1}\), GG.24 gives
\(\|(D_\kappa \otimes 1)R_z\|\le 1+|z|/|Im z|\), and the same bound for \(1\otimes N_\alpha \). If Q_M is the finite squared-spectral window \(|D_\kappa |\le M\),

\[
 \|(1-Q_M)\chi R_z\|\le C_z\|\chi\|/M.
 \tag{GG.27}
\]

The finite-window operator \(Q_M \chi R_z\) is scalar compact by finite-coefficient compact-buffer H¹ bounds and Rellich. Hence \(\chi R_z\) and its adjoint-side analogue are ordinary scalar compacts. The identity on the infinite auxiliary is controlled by GG.27; it is not silently treated as compact.

The stronger spatial assertion also holds. On the external source \(A_m\otimes _{\min} B\), a compact kernel a satisfies \(\pi _N(a)=\pi _N(a)\chi _s\) for an original compact source cutoff. Therefore
\((V(b)\otimes \pi _N(a))R_z=(V(b)\otimes \pi _N(a))\chi _sR_z\)
is scalar compact, first for compact a and arbitrary bounded represented b, then by spatial-norm density for the entire source. In particular all reduced diagonal coefficients have the required localized resolvents.

For a global isometry γ, the full inverse transport satisfies
\(U_\gamma  N_{\rm inv} U_\gamma ^{-1}=N_{\rm inv}+B_\gamma \), with smooth finite-coefficient self-adjoint zeroth-order B_γ on each compact buffer. Invariance of α gives

\[
 U_\gamma N_\alpha U_\gamma^{-1}=N_\alpha+\alpha B_\gamma.
 \tag{GG.28}
\]

There is no normal leading-order speed mismatch. A compact smooth kernel commutator with GG.2 is a finite sum of the bounded auxiliary errors GG.16, \(\alpha C(df)\), and compact-buffer normal errors \(\alpha B_\gamma \). Compact kernels preserve the full domain GG.25: their outputs are compact joint-domain vectors, and the same completed product rules prove the bounded commutator identity there. B_γ need not be globally bounded, and U_γ alone need not preserve the global normal domain; neither is needed. Compact source/range localization is retained explicitly.

The bounded phase \(F_0=\mathcal D(1+\mathcal D^2)^{-1/2}\) consequently has all scalar Kasparov defects for the spatial external source and, by GG.17, for the reduced action GG.20. Resolvent identities, local compactness, and the integrable phase-commutator bound prove this directly.

### 11BU.6. Exact product class, coefficient order and original disk

The unweighted joint spectral formula here avoids a variable-weight product argument. Let \(F_\kappa =D_\kappa (1+D_\kappa ^2)^{-1/2}\) and \(F_N=N_\alpha (1+N_\alpha ^2)^{-1/2}\). For v in a finite D_κ² spectral window, the creation \(T_v u=v\otimes u\) has connection error consisting of finite auxiliary coefficients times the normal functions

\[
 (1+\lambda^2+N_\alpha^2)^{-1/2},\qquad
 \frac{N_\alpha}{\sqrt{1+\lambda^2+N_\alpha^2}}
              -\frac{N_\alpha}{\sqrt{1+N_\alpha^2}}.
 \tag{GG.29}
\]

Both functions vanish at normal spectral infinity. Composing with an actual compact coefficient \(\pi _N(a)\) makes them scalar compact by GG.23 and its adjoint local compactness. The additional term is the already proved compact normal phase commutator with π_N(a). Finite spectral and module-core density give, for every vector of the completed actual transfer module,

\[
 F_0T_\xi-(-1)^{|\xi|}T_\xi F_N\in\mathcal K,
 \quad\text{and its adjoint equation}. \tag{GG.30}
\]

The positivity is the exact global nonnegative joint-spectral operator

\[
 \{F_\kappa\otimes1,F_0\}
 =\frac{2D_\kappa^2}{\sqrt{1+D_\kappa^2}
                  \sqrt{1+D_\kappa^2+N_\alpha^2}}\ge0.
 \tag{GG.31}
\]

Thus F_0 represents \(\Theta \otimes _{A_m}x_{N_\alpha }\), via the ACTUAL multiplication map GG.19 and the actual external-product connection/positivity checks. It is not an unrelated cycle sharing one integer index. The normal phase is independently realized by the original differential expression before the physical compression is performed.

The full finite inverse module remains \(\Pi ^{k_q}Cl_q\), \(k_q=q(q-1)/2\). Its physical generator is \((-1)^{k_q+1}L_{t_j}\); the full coherent lift acts by \(z^{-1}\rho _t(s)\). The determinant-line dual remains part of the chosen compatible connection. The even auxiliary is placed before the normal factor, and the final right \(Cl_q\) is retained until the fixed contraction. No odd factors are exchanged. Auxiliary filtration degree, Lie-group dimension and Killing rank do not replace q.

The local positive scaling α does not change the normalized inverse symbol. On an original oriented buffered disk, compare with the complete finite-coefficient reference normal Dirac which agrees there, using bounded locality and compactly supported positive rescaling as in 11BM.7–11BM.10. Its full inverse normal Bott pairing is +1. Consequently \(x_{N_\alpha }\) itself has that same original calibration. GG.21 gives the original-unit equality after transfer, and hence

\[
 \langle b_U,\Theta x_{N_\alpha}\rangle=+1
 \tag{GG.32}
\]

for the original positive disk U, with its original orientation and full coefficient. If a smaller buffered disk V was used for the comparison, the exact original inclusion equality \(b_U=b_V[j_{VU}]\) supplies U, as in 11BN and 11BO. No quotient-layer disk is substituted.

### 11BU.7. The entire physical graph phase and its geometric test

Retain the normalized physical counting/plaque imprimitivity E_Z with left physical algebra A_graph and right A_r(G). The literal physical scalar carrier is

\[
 H_{\rm graph}=E_Z\otimes_{A_r}E_\Theta
                         \otimes_{A_m,\pi_N}H_N
             =E_Z\otimes_{A_r,\Pi}H_0. \tag{GG.33}
\]

The equality uses GG.19's actual multiplication, not a new Hilbert-space renaming. Choose a countable module Parseval frame \((e_j)\) for the entire even E_Z, and let W be its actual module-inner-product isometry into \(\ell ^2\otimes A_r\). Put P=WW*. On the induced entire range in \(\ell ^2\otimes H_0\), define

\[
 F_{\rm graph}=P(1_{\ell^2}\otimes F_0)P|_{PH},
 \tag{GG.34}
\]

and return through W. This is bounded, odd, self-adjoint and contractive. Its operator domain is ALL H_graph. No unbounded compression domain or normal derivative of P is asserted.

For a full graph-module column ξ, \(WT_\xi \) has entries \(\Pi (\langle e_j,\xi \rangle )\). Its squared column tails tend to zero in module norm. Each retained entry has compact F_0 commutator; the discarded commutator column has norm at most twice its original tail norm. Hence

\[
 F_{\rm graph}T_\xi-T_\xi F_0=K_\xi\in\mathcal K,
 \quad\text{and its adjoint equation}, \tag{GG.35}
\]

on every completed column. For \(G_0=F_{\rm graph}\),

\[
 (G_0^2-1)T_\xi=T_\xi(F_0^2-1)+G_0K_\xi+K_\xi F_0,
 \qquad [G_0,T_\xi T_\eta^*]=K_\xi T_\eta^*-T_\xi K_\eta^*.
 \tag{GG.36}
\]

The rectangular first term on the first line is compact because its squared norm operator is
\((F_0^2-1)\Pi (\langle \xi ,\xi \rangle )(F_0^2-1)\), a scalar compact by Section 5. A rectangular operator with compact C*C is compact, by finite spectral truncation. Rank-one density in \(K(E_Z)=A_{\rm graph}\) therefore proves all scalar physical Kasparov defects on the entire algebra. The connection GG.35 and the zero first Morita operator prove that GG.34 represents the literal product \([E_Z]\Theta x_{N_\alpha }\). The normalized correspondence transports the same b_U, so its physical original pairing is +1.

There is also an actual original-q geometric test, beyond the bounded product predicate. For real compact smooth f, on the exact completed joint domain,

\[
 t^{-1}e^{-itf}\mathcal De^{itf}
       =t^{-1}\mathcal D+\alpha\Gamma_\kappa iC(df).
 \tag{GG.37}
\]

The right side converges in strong resolvent sense to its bounded multiplication term: on every graph test vector v the resolvent-test error is bounded by \(\|\mathcal Dv\|/(t|Im z|)\), and those tests are dense. For a vector supported on a compact buffer with \(|df|_g\ge \delta >0\), that multiplication has a positive spectral gap on the vector, because α has a positive minimum there. Core graph vectors have uniformly bounded scaled energy, so their spectral tails above M are \(O(M^{-2})\). Remove a small interval about zero, use uniform convergence of \(s/\sqrt (t^{-2}+s^2)\) on the remaining compact interval, then use density and contractivity. This proves

\[
 e^{-itf}F_0e^{itf}v\longrightarrow
       \Gamma_\kappa\frac{iC(df)}{|df|_g}v. \tag{GG.38}
\]

The i is retained because C(df) is skew-adjoint. Positive speed cancels in the normalized symbol. The oscillatory vectors tend weakly to zero, by integration by parts along \(grad f/|df|^2\) on the compact support, followed by L² approximation. Apply the compact GG.35 error to those vectors. On every actual compact physical column,

\[
 \|F_{\rm graph}T_\xi e^{itf}v-
 T_\xi e^{itf}\Gamma_\kappa iC(df)|df|_g^{-1}v\|\longrightarrow0.
 \tag{GG.39}
\]

This is precisely the normal-Clifford mechanism on the literal full physical carrier. It is supported by an independently defined joint differential/auxiliary operator with exact domains, not solely by the abstract symbol quotient of the physical phase calculation in Section 11BO. GG.34 remains a bounded physical phase: the naive unbounded compression discussed in 11BO.8 still gains no domain claim.

### 11BU.8. An explicit incomplete curved suspension with a genuine Killing-rank jump

This example verifies that the positive family actually includes varying rank. Let \(T=(-1,1)\times S^1\), with angular coordinate θ modulo 2π, put c=1/4 and

\[
 f(r)=\begin{cases}
 e^{-1/(r-c)^2},&r>c,\\
 0,&|r|\le c,\\
 e^{-1/(-r-c)^2},&r<-c,
 \end{cases}\quad a(r)=e^{f(r)},\quad
 g=dr^2+a(r)^2d\theta^2. \tag{GG.40}
\]

Both r endpoints are omitted and at finite original radial length. The metric is smooth; the flat exponential bump has all jets zero at both seams. Its curvature is \(K=-a''/a=-(f''+(f')^2)\). On the right outer interval set s=r-c. Then

\[
 f'=2f/s³,\quad f''=f(4/s^6-6/s^4),\quad
 K=-f(4/s^6-6/s^4)-4f^2/s^6<0\quad(0<s<3/4).
 \tag{GG.41}
\]

K is real analytic and nonconstant there; it tends to zero at the seam but is negative inside. Thus K' is nonzero on a dense open set. A Killing field preserves K, so its radial component vanishes there, and hence everywhere on the outer interval. The mixed and angular Killing equations then make it a constant multiple of ∂θ. The left side is the reflected argument. At a seam any Killing germ meets an outer interval and is, by connected Killing continuation, still a multiple of ∂θ. Inside the flat band the full local Euclidean Killing algebra has dimension three. Therefore the exact rank is

\[
 k(r,\theta)=3\ (|r|<1/4),\qquad k(r,\theta)=1\ (|r|\ge1/4).
 \tag{GG.42}
\]

Choose an irrational ω/(2π) and the global action \(n\cdot (r,\theta )=(r,\theta +n\omega )\). Its germs are distinct; Γ=Z embeds in SO(2), and its closure acts properly although the discrete Γ action has dense angular orbits. The full invariant ordered frame \((∂r,a^{-1}∂\theta )\) and trivial coherent full central lift supply the inverse Spin-c data. Use the finite even reduced auxiliary \(H_\kappa ^+=C\), \(H_\kappa ^-=0\), \(V_n=1\), \(D_\kappa =0\), through the genuine character of \(C_r^*Z=C(S^1)\). Its index is +1.

Here the original radius of compactness is EXACTLY \(\rho (r,\theta )=1-|r|\): smaller balls have radial closure uniformly away from the omitted endpoints, and any larger ball contains an escaping radial tail. Choose the explicit invariant speed

\[
 \alpha(r)=(1-r^2)/100\le\rho(r,\theta)/50,
 \qquad\int_0^r\alpha(t)^{-1}dt=100\operatorname{artanh}(r).
 \tag{GG.43}
\]

The latter diverges at both original missing ends. In original density \(a(r)drd\theta \), with the prescribed skew Clifford matrices C_1,C_2, an actual expression is

\[
 N_{\rm inv}=C_1(\partial_r+a'/2a)+C_2a^{-1}\partial_\theta,
 \quad N_\alpha=\alpha C_1\partial_r
 +\big(\alpha a'/2a+\alpha'/2\big)C_1
 +\alpha a^{-1}C_2\partial_\theta. \tag{GG.44}
\]

It acts with the original full finite inverse coefficient and final right Cl₂. The suspension
\(M=(R\times T)/Z\), \(n\cdot (t,r,\theta )=(t+n,r,\theta +n\omega )\), foliated in the t direction, realizes the exact holonomy groupoid Z⋉T and the entire normalized plaque correspondence. Label n is realized by a leafwise segment from t=0 to t=−n, so its return is the stated positive nω angular rotation. GG.2 is N_α in this example; GG.34 is its entire physical phase. The original normal degree is two on both rank layers and seams, and an original oriented two-disk pairs as +1. This supplies an actual curved, incomplete, rank-changing foliation, not an abstract stratified-algebra example.


![The original curved cylinder and its invariant speed](../figures/global-curved-rank-example.png)

**Figure 11BU.1.** Exact original coordinate and coefficient data in GG.40–GG.44. The plot of a and the speed is numerical sampling of those formulas; the rank intervals and seams are exact, and the endpoints remain omitted. The speed-distance expression is exact and divergent. Proof locators: 11BU.2, 11BU.5 and 11BU.8. This is a normal coordinate depiction, not an isometric Euclidean embedding of the cylinder.


![The actual reduced transfer and entire physical carrier](../figures/global-curved-graph-carrier.png)

**Figure 11BU.2.** Actual carrier and class map, GG.17–GG.39. The original g, full inverse coefficient, degree q and original disk are retained through coefficient multiplication and the entire physical Parseval range. The auxiliary has an exact completed domain, and the displayed geometric limit is proved on actual physical columns. The physical phase is bounded and has domain all H_graph. The diagram does not assert that P preserves an unbounded domain.



### 11BU.9. Scalar auxiliary scope and the remaining local problem

The primary group-side scope is [Guentner–Higson–Weinberger, Section 6, Theorem 7, printed p. 264](https://www.numdam.org/item/10.1007/s10240-005-0030-5.pdf): it treats countable subgroups of an almost-connected Lie group. The connected case uses a central abelian kernel of the adjoint representation; the general stated case adds a finite component quotient. This checked input cannot be changed to an arbitrary discrete component quotient. Its Hilbert uniform embedding supplies a positive group kernel with uniform compact upper and proper lower bounds. Inverting the embedding coordinates changes a left-invariant proper metric to the right-increment convention \(k(\gamma h,h)\) used here. Squared Hilbert distance is CND; the Gaussian kernels and finite-exhaustion construction then give the exact compact-base proper affine-field premise of Section 11O. The first sentence of Theorem 11Y.1 applies to every countable group with that supplied field. Its even proper factorization with ordinary product one, followed by Theorem 11J.3's actual proper reduced/maximal descent, yields GG.1. This is the precise almost-connected extension of the Section 11BN.2 chain, without a faithful Ado label or a reduced trivial character. These proved canonical providers remain explicit dependencies.

The theorem uses either one supplied reduced scalar auxiliary with unit pairing +1 or the separately proved proper-action norm equality. Arbitrary Lie component groups and arbitrary original local holonomy need their own provider. Independent rank-layer index-one choices do not establish the scalar boundary compatibility in Section 11BR. The original normal degree, prescribed inverse Spin-c data, final right Clifford factor and positive disk remain part of that unresolved local problem.

## 11BV. Compact positive weights on the whole Killing-jet module

### 11BV.1. Fixed original geometry and result

Use the original geometric module in Section 11BT.1–11BT.11.
Thus \(T\) is the original second-countable \(q\)-dimensional Riemannian
manifold with its given, possibly incomplete metric; \(G\) is its specified
local-isometry germ groupoid. Put

\[
 A=C_0(T),\qquad J_{\mathbb R}=TT\oplus\mathfrak{so}(TT),\qquad
 J=J_{\mathbb R}\otimes_{\mathbb R}\mathbb C.
 \tag{KWC.1}
\]

The inner product is the Hermitian complexification of the original real
tangent and Hilbert–Schmidt skew-endomorphism metrics. The whole module
\(E=E_K\subset\Gamma_0(J)\) is the uniform closure of finite sums of the
actual compactly localized jets \(f(X,\nabla X)\). The real local Killing
fields, scalar cutoffs and open domains are exactly the original ones.
Its fibres are exactly \(V_x=V_{x,\mathbb R}\otimes_{\mathbb R}\mathbb C\),
of dimension \(k(x)\), including rank zero. No adjoint of the ambient inclusion
is assumed. The original normal degree \(q\), inverse Spin-c line, right
\(Cl_q\), positive disk, germs and metric are retained.

Choose module-generating sections \(x_n\in E\), \(n\ge1\), with
\(\|x_n\|\le1\). Such a countable family exists by 11BT.2. It may be chosen
from actual finite sums of compactly localized jets: approximate a countable
dense family by these sums and normalize each by \(\max(1,\|s_n\|)\).
Scaling does not change its closed generated module. Scalar approximate
identities satisfy \(x_nu_m\to x_n\), so nonunitality causes no problem.

**Theorem 11BV.1.** The series

\[
 h=\sum_{n\ge1}2^{-n}\theta_{x_n,x_n}\in\mathcal K(E),\qquad
 \theta_{u,v}z=u\langle v,z\rangle
 \tag{KWC.2}
\]

converges in operator norm to a positive contraction, is injective and has
dense range on the whole module. The positive regular self-adjoint inverse
\(D=h^{-1}\) has exact domains and onto compact inverses

\[
 \begin{aligned}
 &\operatorname{Dom}D=\operatorname{Ran}h,\quad D(h\eta)=\eta,
 &&\operatorname{Dom}D^2=\operatorname{Ran}h^2,\\
 &(D\pm i)^{-1}=h(1\pm ih)^{-1}\in\mathcal K(E),
 &&\operatorname{Ran}(D\pm i)^{-1}=\operatorname{Dom}D,\\
 &(1+D^2)^{-1}=h^2(1+h^2)^{-1}\in\mathcal K(E),
 &&\operatorname{Ran}(1+D^2)^{-1}=\operatorname{Dom}D^2.
 \end{aligned}
 \tag{KWC.3}
\]

For \(r\ge1\), form the central balanced \(A\)-tensor module
\(E_r=E^{\widehat\otimes_A r}\), and let \(h_i\) be \(h\) on coordinate \(i\).
These are commuting positive contractions. Define

\[
 b_r=b^{(r)}(h_1,\ldots,h_r),\qquad
 b^{(r)}(a)=
 \begin{cases}
 (\sum_{i=1}^r a_i^{-1})^{-1},&a_i>0\text{ for every }i,\\
 0,&\text{some }a_i=0 .
 \end{cases}
 \tag{KWC.4}
\]

Then \(b_r\in\mathcal K(E_r)\) is positive and injective with dense range,
\(0\le b_r\le h_i\), and \(\|b_r\|\le1/r\). The coordinate inverses
\(A_i=h_i^{-1}\), on \(\operatorname{Ran}h_i\), are positive regular
self-adjoint, and

\[
 D_r=b_r^{-1},\qquad
 \operatorname{Dom}D_r=\operatorname{Ran}b_r
 =\bigcap_{i=1}^r\operatorname{Dom}A_i,\qquad
 D_r\xi=\sum_{i=1}^rA_i\xi .
 \tag{KWC.5}
\]

This is an equality of completed module domains, not only a fibre formula.
The positive regular self-adjoint \(D_r\) satisfies \(D_r\ge r\) in its
quadratic-form sense. Put \(E_0=A\), \(D_0=0\). On the full Fock module,

\[
 \begin{aligned}
 \mathcal F&=\bigoplus_{r\ge0}E_r,\\
 \operatorname{Dom}D_{\mathcal F}
 &=\{\xi\in\mathcal F:\xi_r\in\operatorname{Dom}D_r\ (r\ge1),\
 \sum_{r\ge1}\langle D_r\xi_r,D_r\xi_r\rangle
 \text{ converges in }A\text{ norm}\},\\
 D_{\mathcal F}\xi&=(D_r\xi_r)_r .
 \end{aligned}
 \tag{KWC.6}
\]

This is a positive regular self-adjoint operator with both onto resolvents.
For every \(f\in A\),

\[
 f(D_{\mathcal F}\pm i)^{-1}\in\mathcal K(\mathcal F),\qquad
 f(1+D_{\mathcal F}^2)^{-1}\in\mathcal K(\mathcal F).
 \tag{KWC.7}
\]

The proof makes no constant-rank, metric-completeness, ambient-projection,
covariance or reduced-representation assumption. It constructs neither an
odd index-one auxiliary nor the missing full physical graph class.

The regular-operator convention is the one in Jens Kaad and Matthias Lesch,
*A local global principle for regular operators in Hilbert C*-modules*,
Section 2.3, printed pp. 6–7:
a closed densely defined operator is regular when its adjoint is densely
defined and \(1+D^*D\) has dense range. See the
[author-hosted full paper](https://www.math.uni-bonn.de/people/lesch/dl/pap/2011-KaaLes-ArXiv-v2.pdf).
We prove the stronger onto statement directly, without a fibrewise
regularity criterion.

### 11BV.2. The whole-module dense-range estimate

The Hilbert-module rank-one identity is \(\|\theta_{u,u}\|=\|u\|^2\).
The upper bound follows from Cauchy–Schwarz. For the lower bound, set
\(a=\langle u,u\rangle\). Then
\(\langle u,\theta_{u,u}u\rangle=a^2\), giving
\(\|a\|^2\le\|u\|^2\|\theta_{u,u}\|=\|a\|\|\theta_{u,u}\|\).
For \(u\ne0\), divide by \(\|a\|\) to obtain the stated lower bound.
Also \(\theta_{u,v}^*=\theta_{v,u}\) and
\(B\theta_{u,v}C=\theta_{Bu,C^*v}\) for adjointable \(B,C\).
Thus compacts form a closed two-sided ideal in \(\mathcal L(E)\).

Each summand of (KWC.2) is positive, compact, and has norm at most \(2^{-n}\).
Consequently its series converges in norm to a compact positive \(h\), with
\(\|h\|\le1\), and

\[
 h\ge2^{-n}\theta_{x_n,x_n}\qquad(n\ge1).
 \tag{KWC.8}
\]

For \(\varepsilon>0\), continuous calculus in the unital C*-algebra
\(\mathcal L(E)\) gives

\[
 f_\varepsilon=\varepsilon(h+\varepsilon)^{-1},\qquad
 0\le f_\varepsilon\le1 .
 \tag{KWC.9}
\]

Sandwich (KWC.8) by \(f_\varepsilon\), which commutes with \(h\):

\[
 \begin{aligned}
 \theta_{f_\varepsilon x_n,f_\varepsilon x_n}
 &=f_\varepsilon\theta_{x_n,x_n}f_\varepsilon
 \le2^n f_\varepsilon h f_\varepsilon,\\
 \|f_\varepsilon x_n\|^2
 &\le2^n\sup_{t\in[0,1]}\frac{\varepsilon^2t}{(t+\varepsilon)^2}
 \le\frac{2^n\varepsilon}{4}.
 \end{aligned}
 \tag{KWC.10}
\]

The last bound holds for every \(\varepsilon>0\), because
\((t+\varepsilon)^2\ge4t\varepsilon\).
Thus \(f_\varepsilon x_n\to0\) in module norm for every generator.
\(A\)-linearity and the triangle inequality give convergence on every finite
generator sum. These sums are dense and \(\|f_\varepsilon\|\le1\);
approximation gives convergence for every \(\xi\in E\). Therefore

\[
 h(h+\varepsilon)^{-1}\xi=(1-f_\varepsilon)\xi\longrightarrow\xi
 \quad\text{in module norm},\qquad
 \overline{\operatorname{Ran}h}=E .
 \tag{KWC.11}
\]

Every approximant is literally in \(\operatorname{Ran}h\).
This is the required completed-module proof; fibrewise injectivity did not
establish density.

If \(h\xi=0\), then

\[
 0=\langle\xi,h\xi\rangle
 =\sum_{n\ge1}2^{-n}
 \langle x_n,\xi\rangle^*\langle x_n,\xi\rangle .
 \tag{KWC.12}
\]

Every positive term must be zero, so \(\xi\) is orthogonal to all generators
and their dense module span. Hence \(\langle\xi,\xi\rangle=0\), and \(\xi=0\).

### 11BV.3. An exact inverse lemma

Let \(H\) be any Hilbert module, and \(c\in\mathcal L(H)\) positive, injective,
with dense range. Define \(L(c\eta)=\eta\) on exactly
\(\operatorname{Ran}c\). This is single valued and densely defined.
If \(c\eta_j\to\xi\) and \(\eta_j\to\eta\), boundedness of \(c\) gives
\(\xi=c\eta\), so \(L\) is closed. For domain vectors,

\[
 \langle L(c\eta),c\zeta\rangle
 =\langle\eta,c\zeta\rangle
 =\langle c\eta,\zeta\rangle,\qquad
 \langle c\eta,L(c\eta)\rangle=\langle\eta,c\eta\rangle\ge0 .
 \tag{KWC.13}
\]

Thus \(L\) is symmetric and positive. If \(z\in\operatorname{Dom}L^*\),
write \(w=L^*z\). Testing against every \(c\eta\) gives
\(\langle\eta,z\rangle=\langle c\eta,w\rangle=\langle\eta,cw\rangle\)
for every \(\eta\in H\). Thus \(z=cw\), in the original domain, with \(Lz=w\).
Symmetry gives the reverse inclusion, proving \(L=L^*\).

The scalar functions \(1\pm it\) never vanish on the positive spectrum of
\(c\). Hence \(1\pm ic\) is bounded invertible, and direct computations give

\[
 (L\pm i)c(1\pm ic)^{-1}=1,\qquad
 c(1\pm ic)^{-1}(L\pm i)c\eta=c\eta .
 \tag{KWC.14}
\]

The inverse range is exactly \(\operatorname{Ran}c\), because the commuting
factor \((1\pm ic)^{-1}\) is invertible. Also \(c\eta\) is in the square
domain exactly when \(\eta\in\operatorname{Ran}c\). Therefore

\[
 \operatorname{Dom}L^2=\operatorname{Ran}c^2,\qquad
 (1+L^2)c^2\eta=(1+c^2)\eta .
 \tag{KWC.15}
\]

The positive \(1+c^2\) is bounded invertible, so
\((1+L^2)^{-1}=c^2(1+c^2)^{-1}\) maps onto this exact square domain.
Since \(L=L^*\), this proves regularity. If \(c\) is compact, both
resolvents and the square inverse are compact by the ideal property.
If \(\|c\|\le1/m\), then \(c-mc^2\ge0\), whence

\[
 \langle c\eta,L(c\eta)\rangle-m\langle c\eta,c\eta\rangle
 =\langle\eta,(c-mc^2)\eta\rangle\ge0 .
 \tag{KWC.16}
\]

This is the whole-domain form bound \(L\ge m\).
Apply the lemma to \(c=h\), \(m=1\), to prove (KWC.3) and \(D\ge1\).
The image by \(h\) of any dense finite-generator span is a graph core:
approximation of \(\eta\) approximates both \(h\eta\) and \(D(h\eta)=\eta\).

For the zero module, all domains and ranges are \(\{0\}\), dense in that
module, and all stated inverse identities concern its unique operator.
No nonzero vector or inverse on an absent fibre is intended.

### 11BV.4. Balanced tensor operators and their joint calculus

Commutativity of \(A\) and its specified central scalar left action give

\[
 \langle\xi_1\otimes\cdots\otimes\xi_r,
          \eta_1\otimes\cdots\otimes\eta_r\rangle
 =\prod_{i=1}^r\langle\xi_i,\eta_i\rangle .
 \tag{KWC.17}
\]

Take the algebraic balanced tensors, quotient zero-norm vectors and complete.
A finite tensor sum determines its pointwise section in \(J^{\otimes r}\);
(KWC.17) identifies its inner product with the pointwise tensor inner product.
Thus \(E_r\) embeds isometrically as the closure of those sections in
\(\Gamma_0(J^{\otimes r})\), with the supremum pointwise norm.
Evaluation has fibre \(V_x^{\otimes r}\): evaluation of \(E\) is onto \(V_x\),
so tensor evaluations span it. The KJ.5–KJ.6 scalar-cutoff quotient-norm
argument applies verbatim to the embedded tensor module. In particular rank
zero gives zero positive-degree tensor fibres.

An adjointable \(A\)-linear \(B\) on \(E\) preserves the closed \(EI_x\)
submodule and descends to the fibre quotient with norm at most \(\|B\|\).
Acting in one tensor coordinate thus has pointwise norm at most \(\|B\|\),
and hence the same module norm bound. On the dense elementary tensor span
its adjoint is the coordinate action of \(B^*\), by (KWC.17).
Consequently there is a unital *-homomorphism
\(\sigma_i:\mathcal L(E)\to\mathcal L(E_r)\) for each coordinate.
Different coordinate maps commute by their elementary formulas.
In particular \(h_i=\sigma_i(h)\) are commuting positive contractions, since
\(h_i=\sigma_i(h^{1/2})^2\).

The continuous joint calculus used here is justified as follows.
The unital C*-algebra generated by these commuting self-adjoint elements is
commutative. Each of its characters assigns each \(h_i\) a number in \([0,1]\)
by positivity and the norm bound. Its Gelfand norm therefore gives, for every
polynomial \(p\),

\[
 \|p(h_1,\ldots,h_r)\|
 \le\sup_{a\in[0,1]^r}|p(a)| .
 \tag{KWC.18}
\]

Uniform polynomial approximation on the compact cube extends this map
uniquely to a contractive unital *-homomorphism from \(C([0,1]^r)\).
Its kernel consists of the functions vanishing on the joint spectrum.
Nonnegative continuous functions map to positive operators, by factoring
them as squares of their continuous square roots. Thus every scalar
inequality on the cube implies the corresponding operator inequality.
The same one-variable construction gives the earlier continuous calculus.

The tensor compactness identity is

\[
 \prod_{i=1}^r\sigma_i(\theta_{z_i,w_i})
 =\theta_{z_1\otimes\cdots\otimes z_r,\,
          w_1\otimes\cdots\otimes w_r}.
 \tag{KWC.19}
\]

Apply both sides to an elementary tensor. The separate coefficients
\(\langle w_i,\xi_i\rangle\) combine by central balancing to their product,
which is exactly (KWC.17). Extend equality by density.
Finite sums and norm approximation now show
\(\prod_i\sigma_i(K_i)\in\mathcal K(E_r)\) when every \(K_i\in\mathcal K(E)\).
A compact acting in a single coordinate is not asserted compact.

### 11BV.5. A compact harmonic inverse with completed-module dense range

On the strictly positive cube,

\[
 b^{(r)}(a)=
 \frac{\prod_{i=1}^r a_i}
 {\sum_{i=1}^r\prod_{j\ne i}a_j}.
 \tag{KWC.20}
\]

For \(r=1\) the denominator is the empty product \(1\).
For higher \(r\) the rational expression is used only where its denominator
is nonzero. Define the function to be zero on every coordinate-zero face.
On the interior, and therefore on the boundary after this definition,

\[
 0\le b^{(r)}(a)\le\min_i a_i,\qquad
 b^{(r)}(a)\le1/r\quad(a\in[0,1]^r).
 \tag{KWC.21}
\]

These follow from \(\sum_i1/a_i\ge1/a_j\) and \(\sum_i1/a_i\ge r\).
The first bound forces continuity at every zero-face point and every
intersection of such faces. Joint calculus gives all the operator bounds.

For \(0<\delta<1\) use the explicit continuous cutoffs

\[
 \chi_\delta(t)=
 \begin{cases}
 0,&0\le t\le\delta/2,\\
 2t/\delta-1,&\delta/2<t<\delta,\\
 1,&\delta\le t\le1,
 \end{cases}\qquad
 q_\delta(a)=\prod_i\chi_\delta(a_i),\quad
 Q_\delta=q_\delta(h_1,\ldots,h_r).
 \tag{KWC.22}
\]

If \(1-q_\delta(a)\ne0\), some coordinate is less than \(\delta\).
Thus (KWC.21) and the calculus give

\[
 \|b_r-b_rQ_\delta\|\le\delta .
 \tag{KWC.23}
\]

Approximate \(b^{(r)}\) uniformly by polynomials and multiply by \(q_\delta\).
Each cutoff monomial is a product of functions
\(\chi_\delta(a_i)a_i^{m_i}\), all vanishing at zero.
For any continuous \(\psi\) with \(\psi(0)=0\), \(\psi(h)\) is compact:
approximate by polynomials and subtract their constant terms; their values
are polynomials in compact \(h\) with no constant term. The ideal property
and norm closure prove the claim. Equation (KWC.19) therefore makes each
cutoff monomial compact. Polynomial approximation proves \(b_rQ_\delta\)
compact, and (KWC.23) proves \(b_r\in\mathcal K(E_r)\).
Every coordinate-zero face and the all-zero point are covered.

On \(E\), the scalar bound \(t|1-\chi_\delta(t)|\le\delta\) gives
\(\|(1-\chi_\delta(h))h\|\le\delta\).
Since \(\operatorname{Ran}h\) is dense, and the cutoffs are contractions,
\(\chi_\delta(h)\xi\to\xi\) for every \(\xi\in E\).
On elementary tensors \(Q_\delta\) is the product of these coordinate
cutoffs. Telescoping their difference from the identity, using
\(\|\xi_1\otimes\cdots\otimes\xi_r\|\le\prod_i\|\xi_i\|\), proves convergence
on each elementary tensor. Density and the contraction bound prove

\[
 Q_\delta\xi\longrightarrow\xi\quad\text{for every }\xi\in E_r .
 \tag{KWC.24}
\]

The function

\[
 g_\delta(a)=
 \begin{cases}
 q_\delta(a)\sum_i1/a_i,&\text{all }a_i>0,\\
 0,&\text{some }a_i=0
 \end{cases}
 \tag{KWC.25}
\]

is continuous and bounded by \(2r/\delta\), since \(q_\delta\) vanishes in a
neighborhood of each zero face. On the entire cube \(b^{(r)}g_\delta=q_\delta\).
Hence

\[
 Q_\delta=b_rg_\delta(h_1,\ldots,h_r),\qquad
 Q_\delta(E_r)\subset\operatorname{Ran}b_r .
 \tag{KWC.26}
\]

Combined with (KWC.24), this proves dense range on the completed tensor module.
Self-adjointness then proves injectivity: a kernel vector is orthogonal to
that dense range. Apply the inverse lemma to \(b_r\). It proves the positive
regular \(D_r\ge r\), exact domains, and compact onto inverses

\[
 \begin{aligned}
 R_{r,\pm}&=b_r(1\pm ib_r)^{-1}=(D_r\pm i)^{-1},
 &\|R_{r,\pm}\|&\le1/r,\\
 S_r&=b_r^2(1+b_r^2)^{-1}=(1+D_r^2)^{-1},
 &\|S_r\|&\le\frac1{1+r^2}.
 \end{aligned}
 \tag{KWC.27}
\]

Their ranges are exactly \(\operatorname{Ran}b_r\) and
\(\operatorname{Ran}b_r^2\), respectively.

### 11BV.6. The exact coordinate-sum domain

The expression \(\sum_i h_i^{-1}\) must not hide a domain assertion.
First \(h_i\) has dense range: the scalar function \(q_\delta(a)/a_i\),
extended by zero on every zero face, is continuous because of the cutoffs.
Multiplying its joint value by \(h_i\) gives \(Q_\delta\).
Thus \(\operatorname{Ran}h_i\) contains every \(Q_\delta(E_r)\), and
(KWC.24) proves density. Self-adjointness gives injectivity.
The inverse lemma proves that \(A_i=h_i^{-1}\), on exactly
\(\operatorname{Ran}h_i\), is positive regular self-adjoint.

The ratio \(b^{(r)}(a)/a_i\) need not be continuous at all zero-face
intersections. Continuous functional calculus is therefore not applied
directly to that uncut ratio. Instead let

\[
 t_{i,\delta}(a)=
 \begin{cases}
 q_\delta(a)b^{(r)}(a)/a_i,&\text{all }a_j>0,\\
 0,&\text{some }a_j=0,
 \end{cases}\qquad
 T_{i,\delta}=t_{i,\delta}(h_1,\ldots,h_r).
 \tag{KWC.28}
\]

These functions are continuous, real, and between zero and one.
For fixed \(\gamma>0\) and \(0<\delta\le\gamma/2\),
\(q_\delta=1\) wherever \(q_\gamma\ne0\).
Hence \(T_{i,\delta}Q_\gamma\) is exactly independent of such \(\delta\).
The union of \(Q_\gamma(E_r)\) is dense by (KWC.24).
The uniform contraction bound now shows that \(T_{i,\delta}\xi\) has a
module-norm limit for every \(\xi\in E_r\).
Its adjoints have the same strong limit, since every cutoff value is
self-adjoint. Passing to limits in their adjoint identity makes the
limiting \(T_i\) adjointable and self-adjoint. Limits of positive quadratic
forms show it is positive and contractive. It commutes with \(b_r\) and
every \(h_j\), by the corresponding bounded cutoff identities.

The scalar identities on the whole cube give
\(h_iT_{i,\delta}=b_rQ_\delta\) and
\(\sum_iT_{i,\delta}=Q_\delta\). Strong limits yield

\[
 h_iT_i=T_ih_i=b_r,\qquad \sum_{i=1}^rT_i=1 .
 \tag{KWC.29}
\]

For \(\xi=b_r\eta\), this implies \(\xi=h_iT_i\eta\), so
\(\xi\in\bigcap_i\operatorname{Dom}A_i\) and \(A_i\xi=T_i\eta\).
Their sum is \(\eta=D_r\xi\).
Conversely, if \(\xi\) is in every coordinate inverse domain, write
\(\xi=h_i\zeta_i\) and put \(\eta=\sum_iA_i\xi\). Then

\[
 b_rA_i\xi=b_r\zeta_i=T_ih_i\zeta_i=T_i\xi,\qquad
 b_r\eta=\sum_iT_i\xi=\xi .
 \tag{KWC.30}
\]

Thus \(\xi\in\operatorname{Ran}b_r\) with \(D_r\xi=\eta\).
This proves the completed-domain equality (KWC.5); no additional closure
or formal-sum extension has been omitted.

The cutoff ranges are also a graph core. For \(\xi=b_r\eta\),
\(Q_\delta\xi=b_rQ_\delta\eta\to\xi\) and
\(D_rQ_\delta\xi=Q_\delta\eta\to D_r\xi\).
For fixed \(\delta\), \(D_rQ_\delta=g_\delta(h_1,\ldots,h_r)\) is bounded.
Thus cutoff images of finite elementary tensor sums form a graph core.
This is a core for the positive operator just proved, not a replacement
for the missing original physical normal graph core.

### 11BV.7. The whole Fock domain, adjoint and onto resolvents

By definition, a direct-sum vector has its positive series
\(\sum_r\langle\xi_r,\xi_r\rangle\) convergent in \(A\)-norm.
A uniformly bounded family \(B_r\in\mathcal L(E_r)\) defines an adjointable
diagonal operator, since for every finite tail

\[
 0\le\sum_{r\in I}\langle B_r\xi_r,B_r\xi_r\rangle
 \le M^2\sum_{r\in I}\langle\xi_r,\xi_r\rangle,\qquad
 M=\sup_r\|B_r\|.
 \tag{KWC.31}
\]

The image series therefore converges. The diagonal of \(B_r^*\) is its
adjoint. Its norm is the supremum block norm, by this upper bound and
testing a vector in a single coordinate.

Finite sequences with components in \(\operatorname{Dom}D_r\) are dense
in \(\mathcal F\). They are also graph dense in (KWC.6): truncation has
both its unweighted and weighted tails tending to zero in module norm.
Closedness can be checked on the entire displayed domain.
If \(\xi^{(n)}\to\xi\) and \(D_{\mathcal F}\xi^{(n)}\to\eta\), every
coordinate has these limits. Closedness of each \(D_r\) gives
\(\xi_r\in\operatorname{Dom}D_r\) and \(D_r\xi_r=\eta_r\).
Because \(\eta\in\mathcal F\), its positive series supplies exactly the
weighted convergence in (KWC.6).

Symmetry holds on finite sequences and extends by graph truncation and
continuity of inner products. For the full adjoint, if
\(z\in\operatorname{Dom}D_{\mathcal F}^*\) with adjoint value \(w\in\mathcal F\),
single-coordinate tests give
\(z_r\in\operatorname{Dom}D_r^*=\operatorname{Dom}D_r\),
\(D_rz_r=w_r\) for \(r\ge1\), and \(w_0=0\).
Since \(w\) is a direct-sum vector, its series is exactly the weighted
condition in (KWC.6). Thus the full adjoint domain is the original domain.
Positivity follows from positive finite quadratic forms and their norm
limits. The operator is closed, positive and self-adjoint on that exact
completed domain.

Let \(R_{0,\pm}=(\pm i)^{-1}1_A\) on the vacuum, and use (KWC.27) on
positive degrees. Then

\[
 R_\pm=\operatorname{diag}(R_{0,\pm},R_{1,\pm},\ldots)
 \tag{KWC.32}
\]

is adjointable, with \(R_\pm^*=R_\mp\) and norm at most one.
It maps every input into (KWC.6), because

\[
 D_rR_{r,\pm}=(1\pm ib_r)^{-1},\qquad
 \|(1\pm ib_r)^{-1}\|\le1\quad(r\ge1),
 \tag{KWC.33}
\]

and the vacuum weighted value is zero. Inequality (KWC.31) proves all the
weighted tail convergence for arbitrary inputs.
The two-sided block identities (KWC.14) prove on the whole domain that
\(R_\pm=(D_{\mathcal F}\pm i)^{-1}\).
The range is exactly \(\operatorname{Dom}D_{\mathcal F}\): for any domain
vector \(\xi\), \((D_{\mathcal F}\pm i)\xi\) lies in \(\mathcal F\) and
is carried back to \(\xi\).

The exact square domain is

\[
 \operatorname{Dom}D_{\mathcal F}^2
 =\{\xi\in\mathcal F:\xi_r\in\operatorname{Ran}b_r^2\ (r\ge1),\
 \sum_{r\ge1}\langle D_r^2\xi_r,D_r^2\xi_r\rangle
 \text{ converges in }A\text{ norm}\}.
 \tag{KWC.34}
\]

To verify sufficiency, write \(\xi_r=b_r^2\eta_r\).
Then \(D_r\xi_r=b_r\eta_r\), \(D_r^2\xi_r=\eta_r\), and
\(b_r^2\le1\) gives
\(\langle D_r\xi_r,D_r\xi_r\rangle\le\langle\eta_r,\eta_r\rangle\).
Thus the first weighted convergence follows from the stated square
convergence, and \(D_{\mathcal F}\xi\) is in its first domain.
Necessity is the definition of that square domain.

Put \(S_0=1_A\), and use \(S_r\) of (KWC.27) on positive degrees.
The diagonal \(S=\operatorname{diag}(S_0,S_1,\ldots)\) maps every input
into (KWC.34), because

\[
 D_rS_r=b_r(1+b_r^2)^{-1},\qquad
 D_r^2S_r=(1+b_r^2)^{-1}
 \tag{KWC.35}
\]

are uniformly bounded. Equation (KWC.31) proves the required positive
series convergence. The entire block identity (KWC.15) now gives

\[
 S=(1+D_{\mathcal F}^2)^{-1},\qquad
 \operatorname{Ran}S=\operatorname{Dom}D_{\mathcal F}^2 .
 \tag{KWC.36}
\]

This proves regularity directly with an onto inverse.

### 11BV.8. Full-module compactness, original fibres and the repaired KJC block

Every positive-degree \(S_r,R_{r,\pm}\) is compact. Scalar multiplication
by \(f\in A\) is bounded adjointable, so all their localized blocks are
compact too. On the vacuum, multiplication by \(f\) is compact on \(A\):
put \(v=\sqrt{|f|}\) and \(u=f/\sqrt{|f|}\), with \(u=0\) at zeros.
The latter is continuous at its zeros since \(|u|=\sqrt{|f|}\), and both
sections vanish at infinity. Thus \(u\overline v=f\) and
\(M_f=\theta_{u,v}\in\mathcal K(A)\).

A compact coordinate block inserted in \(\mathcal F\) is compact:
inclusion and projection are adjointable, and insertion of a rank-one
operator is the rank-one operator on the inserted vectors.
Finite block truncations of \(fS\) and \(fR_\pm\) are therefore compact.
The exact useful tail estimates are

\[
 \begin{aligned}
 \left\|fS-\bigoplus_{r=0}^{R}fS_r\right\|
 &\le\frac{\|f\|}{1+(R+1)^2},\\
 \left\|fR_\pm-\bigoplus_{r=0}^{R}fR_{r,\pm}\right\|
 &\le\frac{\|f\|}{R+1}.
 \end{aligned}
 \tag{KWC.37}
\]

Each finite truncation is extended by zero on the omitted blocks.
Norm closure of compacts proves (KWC.7).
The unlocalized vacuum identity is not asserted compact when \(T\) is
noncompact; scalar localization supplies the precise compact vacuum block.

At every \(x\), the generator evaluations span \(V_x\), so

\[
 h_x=\sum_{n\ge1}2^{-n}\theta_{x_n(x),x_n(x)}
 \tag{KWC.38}
\]

is positive definite if \(k(x)>0\), by the same sum-of-squares proof.
The finite dimension makes it invertible there.
If \(k(x)=0\), it is the unique operator on the zero space.
Evaluation commutes with joint calculus: verify polynomials, then pass
to uniform limits by contractivity.
The localized positive-degree regular operator is therefore the matrix

\[
 (D_r)_x=\sum_{i=1}^r
 1\otimes\cdots\otimes h_x^{-1}\otimes\cdots\otimes1
 \quad\text{on }V_x^{\otimes r}
 \tag{KWC.39}
\]

when \(k(x)>0\). More explicitly, the localized regular operator is
determined by the evaluated onto resolvents (KWC.27), and those are
exactly the matrix resolvents of (KWC.39), so the full finite-dimensional
domain is obtained. At rank zero all positive-degree fibres vanish.
Evaluation of the direct sum gives
\(\mathcal F_x=\bigoplus_{r\ge0}V_x^{\otimes r}\): elementary finite
sequences span densely, and their evaluated inner products give the
quotient norm by the scalar-cutoff argument.
The evaluated Fock resolvents determine the full Hilbert-space weighted
direct-sum domain. At rank zero only the one-dimensional vacuum remains,
with its zero operator.

The fibre formulas did not establish module density, compactness or the
whole domains: Sections 2, 5, 7 and 8 establish these separately.
In particular the norm-convergent positive series in (KWC.6) and (KWC.34)
cannot be replaced by pointwise membership in all fibre domains.

For the exact KJC warped-torus geometry and its actual sequence
\(x_n=(x,y,1+1/n)\to(x,y,1)\), degree one now has the compact operator
\(S_1=h^2(1+h^2)^{-1}\) on the whole \(E_K\).
The actual module summand projection \(P_W\) of 11BT.7–11BT.8 is
adjointable. KJC.3 applies to this compact operator and proves

\[
 \|(P_WS_1P_W)_{x_n}\|\longrightarrow0 .
 \tag{KWC.40}
\]

The former \(\tfrac12 1_{W_{x_n}}\) block is thus replaced by a genuine
vanishing block. This consequence of whole-module compactness repairs the
original countertest beyond a selected \(W\)-only weight.

### 11BV.9. An actual obstruction to exact invariant compact weighting

The generator family defining \(h\) was not selected to be \(G\)-invariant,
and no identity \(U_gh_{s(g)}U_g^{-1}=h_{r(g)}\) has been proved.
This is distinct from the continuous intrinsic germ transport of \(E_K\)
in 11BT.4. One cannot generally require such an identity of a compact
dense-range positive weight.

Keep the identical original warped-torus metric of 11BT.7, normal degree
\(q=3\), and its upper flat end \(T_+=\mathbb T^2\times(1,2)\).
Consider the specified pseudogroup generated by the units and the
countably many actual local isometries

\[
 \tau_s(x,y,t)=(x,y,t+s),\qquad s\in\mathbb Q,\qquad
 t\in(1,2)\cap(1-s,2-s),
 \tag{KWC.41}
\]

their inverses, restrictions and compositions on their actual open domains.
Every domain and image lies in the original upper flat end.
There \(a=1\), so these are local isometries. Their derivative in the
original oriented orthonormal frame is the identity, so they preserve
the corresponding framed Spin-c data.
No missing end, curved-region germ or metric completion is added.
The full original Killing ranks and the underlying Hermitian module are
unchanged: rank three on the closed bump region, six on the two flat ends,
and \(E_K=\Gamma_0(L)\oplus\Gamma_0(U,W)\).

Suppose \(c\in\mathcal K(E_K)\) were positive, exactly invariant under
these specified germs, and had dense module range.
On \(T_+\), \(H=u'=0\), so the \(L,W\) frames of 11BT.7 are independent
of \(t\), and these translations preserve the splitting.
The fibre \(W\)-block \(c_W(x,y,t)=(P_WcP_W)_{(x,y,t)}\) is continuous:
rank-one module operators give continuous bundle matrices there, and
operator-norm approximation is uniform. Exact invariance gives

\[
 c_W(x,y,t+s)=c_W(x,y,t)
 \quad(s\in\mathbb Q,\ t,t+s\in(1,2)).
 \tag{KWC.42}
\]

For fixed \((x,y)\) and \(t_0,t_1\in(1,2)\), choose rationals
\(s_n\to t_1-t_0\) with \(t_0+s_n\in(1,2)\).
Continuity and (KWC.42) make \(c_W\) constant in \(t\).
KJC.3 applied to this very compact \(c\) gives
\(\|c_W(x,y,1+1/n)\|\to0\).
Its constant value is therefore zero on all of \(T_+\).

Positivity of \(c_x\) then forces it to annihilate \(W_x\):
for \(w\in W_x\), \(\langle w,c_xw\rangle=0\) implies
\(c_x^{1/2}w=0\), and hence \(c_xw=0\).
Self-adjointness makes its full fibre range orthogonal to \(W_x\).
But module dense range would give dense evaluated range in \(V_x\),
by continuous surjective evaluation. The nonzero rank-three \(W_x\)
contradicts that density. Thus

\[
 \text{no positive compact exactly invariant dense-range weight exists
 for this specified actual-domain pseudogroup.}
 \tag{KWC.43}
\]

This only obstructs exact invariant compact weighting. It does not rule
out bounded phase covariance with compact errors, another suitable
regular representative, coherent scalar or transfer constructions, or
the original normal/physical graph cycle.
Exact invariance is not imposed as a universal new hypothesis of FOL055.

![Whole-module compactness, tensor inverse domains and the invariant-weight counterexample](../figures/full-jet-compact-weight.png)

**Figure 11BV.1.** KWC.1–KWC.43 show the whole original Hermitian jet module, its compact dense-range weight, the balanced tensor construction and the exact Fock domain. The bound in KWC.10 proves module-norm density; the harmonic inverse vanishes on every coordinate-zero face; finite compact blocks and the uniform degree tail give localized Fock compactness. The flat-end counterexample only excludes exact invariant compact weighting. The original normal degree, full inverse coefficient and physical graph problem remain unchanged.

## 11BW. Actual-domain resistance kernels and zero-jet escapes

### 11BW.2. Exact kernel interface

Let \(G\rightrightarrows T\) be the specified Hausdorff second-countable étale
germ groupoid, with the **original** arrow topology and original domains.
Products are \(gh\) when \(s(g)=r(h)\). Put \(G_x=s^{-1}(x)\). The required
kernel is continuous, symmetric, zero on the diagonal, and CND on every
actual source fibre. Its exact controls are

\[
\sup_{g\in C,\ h\in G^{s(g)}} k(gh,h)<\infty
\qquad(C\subset G\text{ compact}),                         \tag{LG.1}
\]
\[
\{g:s(g),r(g)\in K,\ \exists h\in G^{s(g)},\ k(gh,h)\le R\}
\subset C(K,R)\subset G\text{ compact}.                    \tag{LG.2}
\]

The \(h\) is unrestricted. Compactness of finite-dimensional jets, a unit-only
lower test, or a compact-word extension buffer does not replace LG.2.
Sections 11AC–11AD prove that precisely this interface supplies the proper
compact-fibre anchor, ordinary section, probability field and ordinary
anchored comparison. Those transfer results retain their own hypotheses.
This construction does not infer a reduced physical graph class from the kernel.

### 11BW.3. A whole-metric rank-changing obstruction to module-only features

#### 11BW. Step 3. The fixed metric and its full Killing stalks

Let \(B=\{(x,y)\in\mathbb R^2:x^2+y^2<9\}\) and define

\[
b(t)=\begin{cases}e^{-1/t}&t>0,\\0&t\le0,\end{cases}\qquad
\chi(s)=\frac{b(4-s)}{b(4-s)+b(s-1)},\qquad
u(x,y)=\chi(x^2+y^2)(x^2+2y^2),\qquad
g_B=e^{2u}(dx^2+dy^2).                                    \tag{LG.3}
\]

The denominator is positive for every real \(s\): its two nonpositive
arguments cannot occur simultaneously. Each derivative of \(b\) at \(0\)
vanishes because the exponential dominates every reciprocal polynomial.
Thus \(\chi\) and the metric are smooth. They satisfy \(\chi=1\) for \(s\le1\)
and \(\chi=0\) for \(s\ge4\). The metric is positive and is the original metric
throughout. A radial path from radius \(5/2\) to the missing radius \(3\) has
finite length, so it is incomplete.

On \(x^2+y^2<1\), \(u=x^2+2y^2\). Its Gaussian curvature and its invariant
gradient length are

\[
K=-6e^{-2u},\qquad S=|\nabla K|_{g_B}^2
 =576e^{-6u}(x^2+4y^2),\qquad
dK\wedge dS=55296xy\,e^{-8u}\,dx\wedge dy.                 \tag{LG.4}
\]

A local Killing flow preserves the metric, its curvature, and this gradient
length. Its infinitesimal field \(X\) therefore has \(dK(X)=dS(X)=0\).
On the dense open set \(xy\ne0\) these two covectors are independent, so
\(X=0\) there. Continuity makes \(X=0\) on its whole original open domain in
the inner disk. Consequently **every full Killing stalk in that disk is
zero**. On \(2<\sqrt{x^2+y^2}<3\) the metric is exactly Euclidean; its local
Killing fields are the two translations and the rotation, so their full
rank is three. A Euclidean Killing field has the form \(v+Ax\), with \(A\)
skew-adjoint: the Killing equation and its differentiated cyclic identities
give zero second derivatives, and then the first equation gives this form.
No rank is asserted for the interpolation annulus beyond \(0\le k\le3\).
These facts prove a genuine rank change of the one fixed metric; no
arbitrary stratification or padding is inserted.

#### 11BW. Step 4. The actual foliation and the absent return

Put

\[
M=((-1,1)\times B)\setminus\{(0,0,0)\},\qquad
F=\mathbb R\partial_t,\qquad
T=(\{-1/2\}\times B)\sqcup(\{1/2\}\times B).              \tag{LG.5}
\]

Every leaf component meets this two-sheet transversal: a nonzero transverse
point has one horizontal interval, and the zero point has its two separate
half-intervals. The transverse metric is \(g_B\) on both sheets. It is
holonomy-invariant because every local holonomy map is the identity in
\((x,y)\) coordinates. The foliation is regular and transversely oriented;
use the coherent trivial framed Spin-c lift. It has original normal degree
\(q=2\), original full inverse central factor, and right \(Cl_2\) action.

The actual germ groupoid has its two unit components \(B\) and its two cross
components \(B\setminus\{0\}\). A cross arrow at \(z\ne0\) is represented by a
horizontal path and the identity local map on a neighborhood avoiding \(0\).
There is no horizontal leaf path between the two sheet points at \(z=0\).
The missing cross arrow is not silently supplied by a potential isometry
between the two metric copies.

Take \(z_n=(1/n,0)\) for \(n\ge2\), and let \(g_n\) be the cross arrow from the
negative sheet to the positive sheet at \(z_n\). Both endpoints lie in one
fixed compact unit set, but \(g_n\) has no convergent subsequence in the
actual groupoid: its cross component is \(B\setminus\{0\}\) and its only
possible coordinate limit is excluded. On the actual source-pair space,
the pairs \((g_n,1_{s(g_n)})\) therefore escape every compact set.

#### 11BW. Step 5. What is actually excluded

Let \(E_K\) be the full actual Killing-jet module of 11BT, with fibre
\(V_z\subset T_zB\oplus\mathfrak{so}(T_zB)\); use its stated completed
uniform norm, its actual module restriction, and its intrinsic transport.
It is the zero module on the entire inner disk. Its module-valued Killing
connection also has zero stalk there. Every \(C_0(T)\)-linear compact weight
on it has zero evaluation there. Every positive-degree balanced tensor
power, its inverse-weight domain, its resolvent, and their Hilbert direct
sum have zero evaluated fibre there. These are statements about the full
module, not a selected test subspace.

Hence any source-fibre squared-distance feature which factors only through
these positive-degree evaluated fibres has

\[
               k(g_n,1_{s(g_n)})=0.                        \tag{LG.6}
\]

A fixed tensor/Fock vacuum has identical transport under these identity
cross maps, so adjoining that fixed vacuum does not change LG.6. Adding
an explicitly arrow-dependent scalar/domain feature is a different
construction and is not excluded. By LG.2 and the compact endpoint set,
LG.6 contradicts the required properness. Dense module range of a weight
on \(E_K\) does not create a nonzero fibre where \(V_z=0\).

There is a separate, slightly broader obstruction. The jets of \(g_n\) are
\((z_n,z_n,1_2)\) between the two specified copies and tend to the potential
cross identity jet at \(0\). The normal Spin-c phase in this example is also
the identity. Any candidate kernel which extends continuously to that
compact finite-dimensional jet-and-phase closure is bounded on
\((g_n,1_{s(g_n)})\) and again fails LG.2. In particular, continuous
full-normal bundle point data cannot alone supply the missing domain
escape. This argument makes no limiting-jet-to-actual-germ inference.

These are precise failures of two candidate formulas. They do not exclude
using \(G\)'s original supports together with \(E_K\), nor any geometric graph
operator which uses the actual arrows. A proper two-sheet completion
would have the same intrinsic metric/module data and all the zero cross
arrows; its zero kernel works. Restricting that kernel to LG.5 fails.
The required new information is the original actual arrow domain.

### 11BW.4. General finite actual-domain theorem

**Theorem 11BW.1.** Let \(B\) be a second-countable locally compact Hausdorff
metrizable space, let \(I=\{1,\ldots,m\}\) be finite, and let open subsets
\(U_{ij}\subset B\) satisfy

\[
U_{ii}=B,\qquad U_{ij}=U_{ji},\qquad
U_{ij}\cap U_{jk}\subset U_{ik}.                            \tag{LG.7}
\]

Form the actual étale groupoid

\[
T=I\times B,\qquad G=\coprod_{i,j}\{(i,j,z):z\in U_{ij}\},\qquad
(i,j,z):(j,z)\longrightarrow(i,z),\qquad
(i,j,z)(j,l,z)=(i,l,z).                                    \tag{LG.8}
\]

Give every summand its **actual open-domain topology**. This groupoid has
a continuous CND function \(\psi\) proper relative to the two unit maps and
a kernel \(k\) satisfying LG.1–LG.2. Its construction permits arbitrary
changes of the equivalence classes defined by LG.7. If \(B\) is a smooth
metric manifold and the arrows are specified identity local isometries
between its copies, it uses that original metric and arbitrary full
Killing-rank changes. There is no completeness hypothesis.

This theorem assumes a finite-sheet presentation of the whole groupoid.
An arbitrary recurrent holonomy germ groupoid has not been given such a
presentation. The hypothesis is not inferred from first-jet injectivity.

#### 11BW. Step 6. Actual conductances

Choose for every \(i<j\) a continuous function

\[
0\le c_{ij}=c_{ji}\le1,\qquad
\{z:c_{ij}(z)>0\}=U_{ij}.                                  \tag{LG.9}
\]

For a compatible metric \(d\) on \(B\), one explicit choice when
\(B\setminus U_{ij}\) is nonempty is
\(c_{ij}(z)=d(z,B\setminus U_{ij})/(1+d(z,B\setminus U_{ij}))\);
take \(1\) if the complement is empty and \(0\) if \(U_{ij}\) is empty.
Distance to a closed set is continuous and an interior point has positive
distance, so this proves existence. No cutoff is imposed on an admitted
arrow by deleting it. Each absent arrow has zero conductance.

Let \(e_i\) be the standard basis of \(\mathbb R^m\) and put

\[
L_z=\sum_{i<j}c_{ij}(z)(e_i-e_j)(e_i-e_j)^t.                \tag{LG.10}
\]

The connected components of this weighted graph are exactly the actual
equivalence classes \(A\) of LG.7; on each class all pairwise admitted
conductances are positive. Its kernel consists exactly of vectors constant
on each class: its quadratic form is the sum of
\(c_{ij}(v_i-v_j)^2\). Restricted to the sum-zero vectors in a class it is
positive definite. Define \(L_z^+\) to be the ordinary finite-dimensional
inverse on those subspaces and zero on the class constants. For
\(z\in U_{ij}\) define the effective resistance

\[
R_{ij}(z)=(e_i-e_j)^tL_z^+(e_i-e_j),\qquad R_{ii}=0.        \tag{LG.11}
\]

This pointwise inverse is not claimed to be a continuous matrix field
through changes of components. Only the admitted-pair functions LG.11
need to be continuous. That assertion is proved next.

#### 11BW. Step 7. Variational identity, CND, and continuity on actual domains

Let \(d=e_i-e_j\) belong to the sum-zero subspace of one class. Completing
the square on that subspace proves

\[
R_{ij}(z)=\sup_{v\in\mathbb R^m}
          \{2d^tv-v^tL_zv\}.                              \tag{LG.12}
\]

Class-constant terms do not change either expression. At fixed \(z\) put
\(f_i=(L_z^+)^{1/2}e_i\). Then
\(R_{ij}=\|f_i-f_j\|^2\) for all \(i,j\) in the class. For real zero-sum
coefficients \(a_i\) this gives

\[
\sum_{i,j}a_ia_jR_{ij}
       =-2\left\|\sum_i a_if_i\right\|^2\le0.             \tag{LG.13}
\]

This proves CND exactly; no arbitrary off-diagonal blow-up is presumed
to be a CND kernel.

LG.12 is a supremum of continuous functions of \(z\), so \(R_{ij}\) is lower
semicontinuous wherever \(z\in U_{ij}\). For the other inequality orient
the finitely many undirected edges once. At \(z_0\in U_{ij}\) set
\(w=L_{z_0}^+d\) and give edge \((a,b)\) the flow
\(J_{ab}=c_{ab}(z_0)(w_a-w_b)\). It has divergence \(d\) because
\(L_{z_0}w=d\), and its energy is

\[
\sum_{a<b:c_{ab}(z_0)>0}\frac{J_{ab}^2}{c_{ab}(z_0)}
 =w^tL_{z_0}w=d^tL_{z_0}^+d=R_{ij}(z_0).                 \tag{LG.14}
\]

For any other flow with divergence \(d\), Cauchy–Schwarz gives
\(|d^tv|^2\le (\sum J_{ab}^2/c_{ab})(v^tL_zv)\), using only its active
edges. Apply this with \(v=L_z^+d\) to obtain
\(R_{ij}(z)\le\sum J_{ab}^2/c_{ab}(z)\). The LG.14 trial flow has nonzero
entries only on edges whose conductances stay positive near \(z_0\); its
divergence is unchanged there. Its energy is therefore continuous near
\(z_0\) and has limiting value \(R_{ij}(z_0)\). This gives upper
semicontinuity. Together with LG.12 it proves continuity on **the original
admitted domain \(U_{ij}\)**, including when other classes join or leave.

#### 11BW. Step 8. The cut estimate detects every absent limit arrow

Suppose \(z_n\to z_0\), \(z_n\in U_{ij}\), and \(z_0\notin U_{ij}\). Let \(A\)
be the actual equivalence class of \(i\) at \(z_0\); it excludes \(j\). Put

\[
C_A(z)=\sum_{a\in A,\ b\notin A}c_{ab}(z).
\quad R_{ij}(z)\ge\frac1{C_A(z)}\quad(z\in U_{ij}).        \tag{LG.15}
\]

To prove the inequality use \(v=t1_A\) in LG.12 and maximize
\(2t-C_A(z)t^2\). Here \(C_A(z)>0\) whenever \(i,j\) are connected.
Every edge crossing \(A\) has zero conductance at \(z_0\), so the **finite**
sum \(C_A(z_n)\) tends to zero. Therefore

\[
                         R_{ij}(z_n)\longrightarrow\infty. \tag{LG.16}
\]

The test uses a cut of the original actual equivalence classes. Neither
the first-jet closure nor a scalar continuation radius of a multiplier
enters the argument. The finiteness in LG.15 is essential to this proof;
no infinite-fibre cut estimate is asserted.

#### 11BW. Step 9. The two exact controls on the whole original groupoid

Set

\[
\psi(i,j,z)=R_{ij}(z),\qquad
k((i,l,z),(j,l,z))=R_{ij}(z).                              \tag{LG.17}
\]

The pair lies in \(U_{il}\cap U_{jl}\subset U_{ij}\), so every term is
defined on its actual domain. LG.7 makes this a continuous kernel on
\(G\times_sG\); LG.13 proves source-fibre CND. For every actual
\(g=(i,j,z)\) and **every** actual \(h=(j,l,z)\),

\[
                         k(gh,h)=R_{ij}(z)=\psi(g).        \tag{LG.18}
\]

This is the unrestricted cancellation that the log-radius difference
test lacked. If \(C\subset G\) is compact, \(\psi\) is continuous and finite
on \(C\), so its maximum gives LG.1, independently of \(h\).

For compact \(K\subset T\), write \(K_i=\{z:(i,z)\in K\}\). For every fixed
\(i,j\) the sublevel

\[
S_{ij}(K,R)=\{z\in K_i\cap K_j\cap U_{ij}:R_{ij}(z)\le R\} \tag{LG.19}
\]

is closed in the compact set \(K_i\cap K_j\): a limit inside \(U_{ij}\) is
controlled by continuity and a limit outside it is excluded by LG.16.
It is consequently compact **inside the admitted bisection**. The finite
union \(\coprod_{i,j}\{(i,j,z):z\in S_{ij}(K,R)\}\) is a compact subset of
the actual groupoid and contains exactly the LG.2 low increments, by
LG.18. Thus LG.2 is proved on the whole source fibres and the whole unit
space. The same argument proves relative properness of \(\psi\) for two
different compact endpoint sets by their compact union.

Here one may take the anchored auxiliary space to be \(Y=T\), its anchor
\(p=\mathrm{id}_T\), its section \(y_0=\mathrm{id}_T\), and the usual action of
\(G\) on its unit space. The anchor is proper with singleton compact
fibres. The action groupoid \(G\ltimes T\) is exactly the original \(G\);
LG.17 is its continuous proper CND function. No arrow or domain is
changed, and no extra unit anchor is forgotten.

### 11BW.5. The actual completed Gram field and its action

For a unit \(y=(i,z)\) use finite formal zero-sum combinations of the
**actual range arrows** \(a\in G^y\), and base them at \(1_y\). Put
\(v_a=\delta_a-\delta_{1_y}\) and

\[
\langle v_a,v_b\rangle_y
   =\tfrac12\{\psi(a)+\psi(b)-\psi(a^{-1}b)\}.             \tag{LG.20}
\]

LG.13 applied to those actual arrows proves positivity of this form.
Quotient its null vectors and complete. Its dimension at \((i,z)\) is the
size of the actual class of \(i\) minus one: the energy inverse on that
class is positive definite on its sum-zero subspace. A missing range
arrow supplies no formal vector, even if its jet has an ambient limit.

Local sections \(v_{a(y)}\) from the actual bisections, multiplied by scalar
cutoffs compactly supported inside their original range domains, have
continuous Gram entries by LG.7. The norm of a finite sum is exactly the
square root of its LG.20 Gram expression. Declare continuous local
sections to be local uniform limits of these sections. They have dense
fibre spans, continuous norms, and satisfy the local approximation axiom;
this constructs a continuous Hilbert field. A countable base and
countable cutoff families give a countably generated section module.
Its actual completion norm is \(\sup_{y\in T}\|\xi_y\|\) for sections
vanishing at infinity. Evaluation gives the full Gram fibre: all its
actual arrow generators have local compact cutoffs with value one at
the unit; a cutoff supported where the finite Gram norm is close to
its value at the unit proves equality of the quotient and evaluated
norms before completion and hence afterwards. There is no assumed
globally trivial field or ambient continuous inverse matrix.

For an actual arrow \(a:y'\to y\) define on generators

\[
U_av_b=v_{ab}-v_a,\qquad A_av=U_av+v_a.                  \tag{LG.21}
\]

Since \((ab)^{-1}(ac)=b^{-1}c\), polarization in LG.20 proves that \(U_a\)
is an isometry with inverse \(U_{a^{-1}}\). Direct multiplication gives
\(U_aU_b=U_{ab}\) and \(v_{ab}=v_a+U_av_b\), so \(A_aA_b=A_{ab}\).
On every original bisection LG.21 sends a finite local generator sum to
finite local generator sums with continuous Gram entries. Its norm
bound passes continuity to their uniform limits. This proves the
continuous coherent affine action. Its displacement has square norm
\(\psi(a)\), whose original-domain relative properness was proved above.

This is an actual field and actual action, not independent choices of
finite-dimensional inverse matrices on strata. Rank-changing Gram
fibres cause no rank-layer KK gluing.

### 11BW.6. Exact repair on the rank-changing incomplete metric

Apply Theorem LG.1 to LG.5 with \(m=2\), \(U_{12}=B\setminus\{0\}\) and
\(c_{12}(z)=|z|^2/(1+|z|^2)\). The two-vertex network has one edge and

\[
\psi(g)=\begin{cases}
0&g\text{ a unit},\\
1+|z|^{-2}&g\text{ a cross arrow at }z\ne0.
\end{cases}                                               \tag{LG.22}
\]

Every cross left increment has this cost for all allowed multipliers,
including the case where both the multiplier and product have tiny
continuation domains. Compact cross tests stay away from \(0\); low
sublevels do too. This repairs LG.6 by incorporating the **actual
missing domain**, while keeping all arrows, the one fixed metric and
its actual Killing module. No zero-height cross arrow is added.

For a three-sheet class change take \(B=\mathbb R\),
\(U_{12}=\mathbb R\), \(U_{13}=U_{23}=(0,\infty)\), and
\(c_{12}=1\), \(c_{13}=c_{23}=a(t)=t_+^2/(1+t_+^2)\). The exact formulas are

\[
R_{12}(t)=\frac2{2+a(t)}\quad(t\in\mathbb R),\qquad
R_{13}(t)=R_{23}(t)=\frac{1+a(t)}{a(t)(2+a(t))}\quad(t>0).  \tag{LG.23}
\]

The persistent \(12\) resistance is continuous and tends to \(1\) at the
class split. The disappearing \(13\) and \(23\) resistances diverge as
\(1/(2t^2)\). These formulas are proved by combining the direct edge
with the series route: the two conductances in parallel are respectively
\(1+a/2\) and \(a+a/(1+a)\). They also follow by the inverse in LG.11.
This illustrates why a vanishing scalar edge conductance can yield both
continuous admitted pairs and proper disappearing pairs; an arbitrary
inverse of the whole ambient matrix does not give this conclusion.

#### 11BW. Step 10. Why the same resistance inverse does not solve recurrent fibres

There is an exact obstruction to extending LG.11 unchanged to every
infinite recurrent source fibre. Let \(\mathbb Z^3\) act by translations on
the original flat cylinder \(T=\mathbb R/\mathbb Z\times(0,1)\), using angular
increments \(n_1\sqrt2+n_2\sqrt3+n_3\sqrt5\). The four numbers
\(1,\sqrt2,\sqrt3,\sqrt5\) are rationally independent: in the field generated
by these three square roots the sign-changing automorphism of each square
root isolates its coefficient, and the remaining constant is then zero.
Thus the action is faithful and has no nonunit isotropy; its actual germ
groupoid is \(\mathbb Z^3\ltimes T\). The supplied metric is incomplete at
the two missing ends, and its full local Killing rank is three. Supply
the coherent trivial framed Spin-c lift, with original \(q=2\).

Take the six unit-conductance nearest-neighbor edges on the actual label
lattice. The Fourier symbol of its Laplacian is
\(\lambda(\theta)=2\sum_{j=1}^3(1-\cos\theta_j)\) on \([-\pi,\pi]^3\).
Indeed a translation \(n\mapsto n+e_j\) transforms into multiplication by
\(e^{i\theta_j}\), so the sum of the two oriented difference terms is
\(2-e^{i\theta_j}-e^{-i\theta_j}\). The natural massless inverse-Laplacian
squared-distance feature is the real Hilbert feature represented by
\(f_n(\theta)=(e^{in\cdot\theta}-1)/\sqrt{\lambda(\theta)}\), with
the complex \(L^2\) space regarded as real. It gives

\[
k_{\rm res}(n,m)=\frac1{(2\pi)^3}
 \int_{[-\pi,\pi]^3}\frac{2(1-\cos((n-m)\cdot\theta))}
 {\lambda(\theta)}\,d\theta
 \le\frac{\pi\sqrt3}{2}\quad\text{for every }n,m.           \tag{LG.24}
\]

Here is the complete bound, including existence of the feature. For
\(|t|\le\pi\), \(1-\cos t\ge2t^2/\pi^2\): it is the square of
\(\sin(|t|/2)\ge |t|/\pi\), the latter inequality following from concavity
of sine on \([0,\pi/2]\) and its endpoint chord. Hence
\(\lambda(\theta)\ge4|\theta|^2/\pi^2\). The singularity
\(|\theta|^{-2}\) is integrable in dimension three. The cube is contained
in the ball of radius \(\pi\sqrt3\), so
\(\int 1/\lambda\le(\pi^2/4)4\pi(\pi\sqrt3)=\pi^4\sqrt3\).
The numerator is at most four, giving exactly LG.24. Thus each \(f_n\) is
in the stated Hilbert space and the squared-distance formula proves CND.

Every compact left-arrow test has finitely many lattice increments,
and the cost \(k_{\rm res}(n+\delta,n)\) is independent of \(n\), so LG.1
holds. But all original \((\delta,z)\) with \(z\) on one fixed compact height
circle have both endpoints in that circle. Their infinitely many labels
escape every compact subset of the actual \(\mathbb Z^3\times T\).
LG.24 then violates LG.2, even with the multiplier a unit. This proves
a precise failure of the natural infinite resistance extension. It
does not imply nonexistence of a kernel: the label squared distance
\(|n-m|^2\) has the two controls on this same action.

Thus LG.15 is a proved **domain** mechanism; it must be combined with
separate recurrent-label coarse features. Replacing finite local classes
by an infinite graph and simply using its inverse Laplacian does not
establish that combination.

### 11BW.7. Original normal factor and the precise surviving research route

All kernels above are real scalar geometric data. Their construction
adds no orientation factor and exchanges no graded factors. For an
application with supplied normal transport the full inverse module is
still \(\Pi^{q(q-1)/2}Cl_q\), its transport is still
\(z_g^{-1}\rho_t(s_g)\), and the final right \(Cl_q\) stays in its stated
order until the separately justified contraction. Killing rank, finite
sheet number and Gram dimension never replace original \(q\). The
original positive disk \(U\) and \(b_U\in KK^q(\mathbb C,C_0(T))\) remain
the original disk and class. LG.22 by itself does not construct its
normal physical Dirac operator or claim a new \(+1\) pairing.

If the already proved 11AD anchored comparison is applied with its full
other supplied interfaces, its conclusion is the typed ordinary identity
\(b_U\otimes_{C_0(T)}\operatorname{For}_G(\eta D)=b_U\) in its stated
ordinary category. Thus the existing positive-disk normalization is
preserved; no new equivariant \(\gamma=1\) and no scalar determinant
replacement is asserted here. This construction proves the kernel input at
its specified finite-domain scope, not those separate operator premises.

The frame lift still uses the actual map \(g\cdot p=dh(p)\) and preserves
all original arrows. On local isometries its first-jet uniqueness makes
the lift principal, but it does not make the lifted relation proper or
finite-sheet. In LG.5 it retains the missing cross frame arrows at \(0\);
compact frame fibres do not fill them. Any extra frame or Gram degree
would require a justified even contraction to the original \(q\) before
it could be used in the graph class. This construction neither substitutes
\(q+q(q-1)/2\) nor performs such a contraction without its proof.

The next route is materially different from a Killing-module-only
oscillator: develop a **joint actual-domain and recurrent-label positive
kernel** on the lifted principal relation. LG.10–LG.16 supply a complete
boundary-domain mechanism when that developed relation has finite local
classes. The recurrent part must keep the countable actual labels, as
11AC requires, and the matrix/compact-Lie constructions may supply their
coarse features only after their compact increment and genuine actual
domain controls are established. To extend this to arbitrary local
isometry holonomy one still has to prove all of the following substantive
facts together: a label-preserving local development across variable full
Killing rank; continuous positive kernels combining its recurrent
features with the cut mechanism; and uniform small-defect estimates on
every compact left-arrow test for every multiplier. Local finiteness of
an infinite relation, independent rankwise indices, or a sourcewise
radius difference does not prove these facts. In particular the finite
cut in LG.15 cannot be replaced by an uncontrolled infinite sum, and
LG.24 proves that an infinite inverse-Laplacian feature can be uniformly
bounded on all escaping recurrent labels.

This construction proves no general arbitrary-germ kernel, no completion or
global-action hypothesis, no reduced physical graph representative, and
no whole original graph problem conclusion.

![Actual arrow domains and the finite resistance cut mechanism](../figures/actual-domain-resistance.png)

**Figure 11BW.1.** LG.3–LG.6 retain the original incomplete metric, zero inner Killing stalks, full rank three on the outer flat annulus, and absent cross arrow at the origin. The two sheets are scaled coordinate diagrams; interpolation ranks are unspecified. LG.9–LG.19 prove the domain-sensitive CND kernel and unrestricted multiplier controls. LG.23 gives the plotted exact resistances. LG.24 proves the separate recurrent-lattice limitation. [Reproduce the figure](../reproduction/actual-domain-resistance/README.md).

## 11BX. Equivariant K-orientation and a nonisometric reduced graph class

A strict invariant compact Spin-c reduction includes an invariant orthonormal frame bundle. A coherent equivariant K-orientation can instead lift the actual derivative cocycle through the metalinear-complex group over \(GL_q^+\). Such a lift need not preserve a chosen Euclidean metric. This convention is stated in [Hilsum–Skandalis, Definition 1.6, printed p.332](https://www.numdam.org/article/ASENS_1987_4_20_3_325_0.pdf#page=9) and [Connes, II.6γ, printed p.114](https://alainconnes.org/wp-content/uploads/book94bigpdf.pdf#page=114). The definitions are consulted for their mathematical convention; the proofs below are original.

Indeed unitary transport satisfying \(Uc_s(v)U^*=c_r(Dh\,v)\) forces preservation of the metric by squaring the two Clifford matrices. That strict premise in Section 11BM is a sufficient subfamily. It cannot silently exhaust general equivariant K-orientation. The original suitable graph-class problem includes an actual reduced representation, ordinary scalar compact defects and pairing +1 with the original positive \(q\)-disk. An unspecified maximal class or a changed auxiliary disk does not supply those conclusions.

**Theorem 11BX.1.** The following connected compact suspension has coherent transverse K-orientation and no invariant positive metric. Its actual reduced transversal algebra carries the original inverse normal phase with positive disk pairing +1, and the full normalized suspension correspondence carries a bounded geometric graph phase with the same pairing. The original degree is one; fixed-point isotropy is retained.

### 11BX.1. Original foliation and lift

On the positively oriented circle \(T=R/(2pi Z)\) take

\[
h(\theta)=\theta+\tfrac12\sin\theta\pmod{2\pi}.
\tag{NRA.1}
\]

The derivative is \(1+(cos theta)/2\), between 1/2 and 3/2, so h is an orientation-preserving analytic circle diffeomorphism. Let V be its mapping torus and F the suspension foliation. V is a connected compact smooth surface; its original normal degree is q=1. Its complete transversal is T and its reduced holonomy algebra is Morita equivalent, with the ordinary suspension counting/plaque correspondence, to

\[
B=C(T)\rtimes_r\mathbb Z.
\tag{NRA.2}
\]

The holonomy germ groupoid is exactly \(Z semidirect T\). No nonzero iterate has an identity germ: analyticity would extend that identity over the connected circle, but the derivative of \(h^n\) at zero is \((3/2)^n != 1\). The genuine isotropy Z at zero and pi is retained.

For the oriented normal line, \(ML_1^c\) can be identified with \(R_+ times U(1)\), projecting \((a,z)\) to a. Lift each derivative by \((Dh^n,1)\). The chain rule gives the exact coherent cocycle; the central line and its full inverse are trivial, rather than discarded. This is a holonomy-equivariant K-orientation.

If an invariant positive metric existed on T, its coefficient m at the fixed point zero would satisfy

\[
m(0)=(3/2)^2m(0),\qquad m(0)>0,
\tag{NRA.3}
\]

which is impossible. In particular NRA.1 does not satisfy 11BM's invariant-symbol-metric premise.


### 11BX.2. Actual inverse normal operator and covariance

Let \(H=L^2(T,dtheta)\) with normalized Fourier basis \(e_n\), and let P project onto the modes \(n<=0\). Put

\[
N=i\partial_\theta,\quad
\operatorname{Dom}N=\{u:\sum_n n^2|u_n|^2<\infty\},\qquad
F=2P-1.
\tag{NRA.4}
\]

N is self-adjoint on exactly this Fourier H1 domain. Its resolvents are compact, since their diagonal entries tend to zero. F is its sign with the zero eigenvalue assigned +1; \(F-N(1+N^2)^(-1/2)\) is compact. This is the q=1 **inverse** convention: the course's full inverse module is \(Pi^0 Cl_1\), physical generator \(-L_e\), and the last right \(Cl_1\) is retained before odd scalar contraction. No supplied determinant replaces a lift.

Multiplication represents C(T). The actual half-density holonomy unitary is

\[
(U_hu)(x)=\sqrt{(h^{-1})'(x)}\,u(h^{-1}(x)).
\tag{NRA.5}
\]

Change of variables proves unitarity and \(U_h M_f U_h^*=M_{f circ h^{-1}}\).

Here is a direct compactness proof, requiring no invariant metric or external Egorov proof. Define the circle Hilbert transform by its Fourier multipliers \(-i sign(n)\) (zero at n=0). Its principal-value kernel is \((2pi)^(-1) cot((x-y)/2)\), and \(F=-i HilbertTransform+P_0\). For smooth f the commutator kernel is the cotangent kernel times \(f(y)-f(x)\). The diagonal singularity cancels, leaving a bounded kernel, hence a Hilbert–Schmidt operator. P0 contributes a finite-rank term. Therefore \([F,M_f]\) is compact.

Write \(phi=h^{-1}\). Apart from the finite-rank zero-mode terms, the kernel of \(U_h F U_h^*-F\) is a constant multiple of

\[
\sqrt{\phi'(x)\phi'(y)}\cot\frac{\phi(x)-\phi(y)}2
-\cot\frac{x-y}2.
\tag{NRA.6}
\]

For \(y=x+t\), write \(a=phi'(x)>0\), \(b=phi''(x)\). The square root is \(a+bt/2+O(t^2)\) and \(phi(x+t)-phi(x)=at+bt^2/2+O(t^3)\). Their ratio is \(1/t+O(t)\). Since \(cot(s/2)=2/s+O(s)\), both terms in NRA.6 have the same \(-2/t\) singularity and their difference is O(t). The kernel is smooth off the diagonal and bounded across it. Thus the difference is Hilbert–Schmidt, and \([F,U_h]\) is compact. Iterate and use adjoints to obtain \([F,U_h^n]\) compact for every integer n. All finite crossed-product sums have compact commutators; norm density extends this to B once reduced continuity is proved. F is self-adjoint and \(F^2=1\), so the other scalar defects vanish.

### 11BX.3. Reduced continuity on the actual algebra

For completeness, reduced continuity here has an elementary proof rather than an arbitrary auxiliary premise. Let \(alpha_n(f)=f circ h^{-n}\). On \(ell^2(Z) tensor H\) use the regular covariant representation

\[
(R(f)\eta)(k)=M_{\alpha_{-k}(f)}\eta(k),\qquad
(R(u_n)\eta)(k)=\eta(k-n).
\tag{NRA.7}
\]

For \(I_L={0,...,L-1}\) define an isometry \(W_L:H -> ell^2(Z) tensor H\) by

\[
(W_Lv)(k)=L^{-1/2}1_{I_L}(k)U_h^{-k}v.
\tag{NRA.8}
\]

NRA.5 gives \(R(f)W_L=W_LM_f\) exactly. The shift equation differs only on \(I_L triangle (I_L+n)\), whose size is at most \(2|n|\). Consequently

\[
\|R(u_n)W_L-W_LU_h^n\|\le\sqrt{2|n|/L}.
\tag{NRA.9}
\]

For a finite sum a, its represented norm on H is therefore bounded by \(\|R(a)\|\) after L tends to infinity. R decomposes over the original circle into the source regular representations. Its norm equals their supremum: finitely supported vector tests give continuous norm functions of the source, and every nonempty circle open set has positive dtheta-measure. Hence \(\|R(a)\|=\|a\|_r\). The integrated representation in NRA.5 is reduced-continuous. This proof includes the fixed-point source representations and does not quotient out isotropy.

### 11BX.4. Original disk sign and physical graph passage

On the positive circle, \(u(theta)=exp(i theta)\) represents winding +1. The compression \(P M_u P\) on the modes \(n<=0\) has one-dimensional kernel (e0), no cokernel and index +1. Thus the scalar odd pairing with NRA.4 is +1. A positive winding-one unitary supported in any original positive open interval disk U is homotopic to u: subtracting the two degree-one real phase lifts gives a periodic real function, and interpolating that function gives the homotopy. Open extension therefore gives the **same original** positive disk pairing, not a newly selected auxiliary Bott class.

Let \(E_Z\) be the full original suspension correspondence from \(A_{\rm graph}=C_r^*(V,F)\) to B. Induce on the literal \(E_Z tensor_B H\). A countable Parseval frame gives the actual range projection \(P_Z\) in \(ell^2 tensor H\); define

\[
F_{\rm graph}=P_Z(1\otimes F)P_Z.
\tag{NRA.10}
\]

This uses the entire graph correspondence. For every creation column T_xi, the entries are the actual represented frame inner products. Finite columns have compact phase commutators by NRA.6–NRA.9; square-summable module tails tend to zero in operator norm. Hence \(F_{\rm graph} T_xi-T_xi F\) and its adjoint are ordinary rectangular compacts. For graph rank-one coefficients, expanding the commutator and square defect through those two creation errors gives scalar compacts; rank-one density proves every graph Kasparov defect. These are exactly the bounded connection arguments of 11BO.7/DJ.27–DJ.29b, whose proof requires a supplied reduced scalar phase, not an invariant metric. The same correspondence carries the original disk class, so its scalar pairing remains +1.

The graph phase has actual normal geometry. For a smooth compact packet v where \(f'>=delta>0\), twice integrating its Fourier coefficient for \(k<=0\) by parts gives a bound \(C/(t+|k|)^2\) for \(e^(itf)v\), uniformly for t>=1. The differentiated reciprocal factors obey this bound because \(tf'-k>=delta t+|k|\), while each numerator derivative contributes at most t. Summing the squares over k<=0 gives O(t^-3). Thus \(F e^(itf)v+e^(itf)v ->0\) in L2. For \(f'<=-delta\), the reversed Fourier estimate gives \(F e^(itf)v-e^(itf)v ->0\). Compact packets are weakly null by one integration by parts against smooth tests followed by L2 density. The compact creation error consequently vanishes on them. NRA.10 therefore has the original q=1 inverse normal Clifford phase on every actual physical graph column, in the precise bounded-phase sense used by DJ.31a.

This example answers the graph-class requirement for its stated non-isometric suspension scope. It does not assert a full unbounded graph compression domain, nor does the source require that stronger route. It is also not an arbitrary-q reduced theorem.

The general nonisometric problem remains broader. The positive quadratic-form bundle has invariant vertical and quotient metrics; an invariant total tangent splitting is extra data. The existing metric-bundle/Thom/hypoelliptic construction is a full-algebra theorem, with reduced descent distinguished separately in [Hilsum–Skandalis, Introduction pp.326–327 and Theorem6.4 p.366](https://www.numdam.org/article/ASENS_1987_4_20_3_325_0.pdf#page=42), and [Kordyukov, §5.5 pp.67–68 and Theorem8.6 pp.104–105](https://arxiv.org/pdf/0908.3744#page=105). It is a possible geometric route. Actual reduced continuity, cancellation to original degree and the original physical disk comparison still require proof at the scope used.

## 11BY. Anisotropic holonomy and the limit of a bare Clifford phase

**Theorem 11BY.1.** The actual compact degree-two suspension below has coherent general K-orientation and a reduced-continuous half-density representation. Its bare flat normal phase has a noncompact localized holonomy commutator. A unitary transport of every normalized Clifford direction forces conformality. These conclusions exclude this particular phase and symbol transport, without excluding another suitable graph class.

### 11BY.1. The actual foliation, lift and representation

Let T be the positively oriented two-torus, with angular coordinates theta,y of period 2pi and its original flat metric. Set

\[
h(\theta)=\theta+\tfrac12\sin\theta,\qquad
\psi(\theta,y)=(h(\theta),y).
\tag{AP.1}
\]

Because 1/2 <= h' <= 3/2, this is a smooth orientation-preserving diffeomorphism. Form the connected compact mapping torus V of psi and its one-dimensional suspension foliation. The original transverse degree is q=2. The complete transversal T has the actual germ groupoid Z semidirect T: if a nonzero power had an identity germ, analyticity would make it identity everywhere, whereas its derivative at theta=0 has the eigenvalue (3/2)^n != 1. Thus the isotropy along the fixed circles theta=0 and theta=pi is retained.

Every derivative is diag((h^n)'(theta),1), in the contractible positive diagonal subgroup of GL_2^+. Lift that subgroup to the identity component of its metalinear double cover and give the central U(1) component the value one. The chain rule proves the exact arrow cocycle law. This supplies the full coherent transverse K-orientation; its central line and full inverse are trivial by construction. It is not a discarded determinant datum.

An invariant positive metric would give, at a fixed point with theta=0 and v=(1,0),

\[
g(v,v)=g(D\psi v,D\psi v)=\tfrac94g(v,v),
\tag{AP.2}
\]

contradicting positivity. No invariant metric exists, although the coherent K-orientation does.

On H=L^2(T) tensor C^2, use the Pauli matrices sigma_1,sigma_2 and grading sigma_3. Let f act by multiplication and define

\[
(U_n v)(\theta,y)=\sqrt{(h^{-n})'(\theta)}\,
                   v(h^{-n}(\theta),y).
\tag{AP.3}
\]

Change of variables proves unitarity. The chain rule proves U_n U_m=U_{n+m}, and U_n M_f U_n^*=M_{f circ psi^{-n}}. Thus this is the actual half-density covariant representation, with the trivial central inverse and no change in q. It is reduced-continuous: the Fejer-window isometric embeddings and intertwining estimate for the Z action in Section 11BX.3 apply verbatim to any covariant representation. No reduced trivial character of an arbitrary nonamenable group is used. Its ordinary suspension counting/plaque correspondence identifies the entire physical graph carrier; the finite final right Cl_2 factor can be included without changing any compactness conclusion below.

### 11BY.2. Exact normal operator and a direct high-frequency limit

Define the ordinary flat transverse Dirac operator and bounded phase by

\[
D=-i\sigma_1\partial_\theta-i\sigma_2\partial_y,
\qquad \operatorname{Dom}D=H^1(T;\mathbb C^2),
\qquad F=D(1+D^2)^{-1/2}.
\tag{AP.4}
\]

The Fourier basis proves self-adjointness on exactly that weighted square-summable domain, D^2=-partial_theta^2-partial_y^2, and compactness of (1+D^2)^{-1}. F is odd, self-adjoint and F^2-1 is compact. The purpose of the next argument is to test its actual holonomy covariance, not merely its compact resolvent.

For a smooth circle-valued phase phi and positive integers N whose differential is nowhere zero, write E_N=M_{exp(iN phi)} and c(dphi)=sigma_1 partial_theta phi+sigma_2 partial_y phi. Then

\[
E_N^*(D/N)E_N=D/N+c(d\phi)=A_N.
\tag{AP.5}
\]

For a smooth vector w the multiplication inverse (c(dphi)-i)^-1 w is smooth. The resolvent identity, with the self-adjoint resolvent norm at most one, gives

\[
\|(A_N-i)^{-1}w-(c(d\phi)-i)^{-1}w\|
 \le N^{-1}\|D(c(d\phi)-i)^{-1}w\|.
\tag{AP.6}
\]

Density and the same bound extend this convergence to all w. The inverse with +i converges by the identical argument. Finite linear combinations of products of these inverses and their adjoints approximate every C_0 function of the real variable, so the bounded continuous compactly supported functional calculus converges strongly. Put a smooth cutoff equal to one on a smaller neighborhood of zero and supported inside the uniform gap |t|<min|dphi|/2. Its limit at c(dphi) is zero; hence the corresponding small spectral part of A_N converges strongly to zero. Outside that gap, t/sqrt(t^2+N^-2) converges uniformly to sign(t). To pass from compactly supported cutoffs to bounded functions, first use smooth w: ||A_N w|| is bounded independently of N, and its spectral tail above R has norm at most ||A_N w||/R. Then let R increase and use density for all w. These statements prove

\[
E_N^* F E_N w\longrightarrow
\frac{c(d\phi)}{|d\phi|}\,w
\quad\hbox{in norm for every }w\in H.
\tag{AP.7}
\]

All domains and cutoffs in this argument are those of the actual original torus operator. No global symbol theorem or completed auxiliary space is needed.

### 11BY.3. Actual noncompact covariance defect

Use phi(theta,y)=theta+y and choose a nonzero smooth w supported in a sufficiently small neighborhood of theta=0. The vectors E_N w tend weakly to zero: for smooth test vectors this follows by integration by parts in y, and arbitrary tests follow by density and the constant norm. Moreover

\[
U_1^* E_N w
 =e^{iN(h(\theta)+y)}U_1^*w.
\tag{AP.8}
\]

Apply AP.7 to the two phases theta+y and h(theta)+y, then use AP.3. This gives the actual operator limit

\[
E_N^*(U_1FU_1^*-F)E_N w
 \longrightarrow
\left[
\frac{a(\theta)\sigma_1+\sigma_2}{\sqrt{a(\theta)^2+1}}
-\frac{\sigma_1+\sigma_2}{\sqrt2}
\right]w,
\quad a(\theta)=h'(h^{-1}(\theta)).
\tag{AP.9}
\]

At theta=0, a=3/2. The bracket is Clifford multiplication by the vector difference

\[
\frac{(3,2)}{\sqrt{13}}-\frac{(1,1)}{\sqrt2},
\quad
\delta=\sqrt{\,2-10/\sqrt{26}\,}>0.
\tag{AP.10}
\]

Pauli anticommutation gives its norm exactly delta and its square delta^2 times the identity. On a small enough support its pointwise norm is at least delta/2. Therefore the limiting norm in AP.9 is at least (delta/2)||w||. A compact operator takes a bounded weakly null sequence to a norm-null sequence, so U_1 F U_1^*-F is not compact. Equivalently [F,U_1] is not compact. Multiplying by a smooth scalar equal to one on this support preserves the same obstruction. The particular flat Dirac phase is consequently not a scalar Kasparov cycle for this actual reduced crossed-product representation.

The source algebra is unital, but the localized argument also shows why changing a far-away end, an auxiliary completion or the final finite right Clifford factor cannot repair this particular defect. It does not say that another graph phase or geometric metric-bundle construction fails.

### 11BY.4. Exact conformality criterion for unitary normalized Clifford transport

Let A be an invertible real linear map between two q-dimensional Euclidean spaces, q>=2, and let c_1,c_2 be faithful complex Clifford actions. Suppose one unitary W satisfies, for every nonzero xi,

\[
W\frac{c_1(\xi)}{|\xi|}W^*
 =\frac{c_2(A\xi)}{|A\xi|}.
\tag{AP.11}
\]

Anticommutators show that normalized inner products are preserved. For an orthonormal basis e_i, the vectors A e_i are pairwise orthogonal; write them as a_i v_i with a_i>0 and v_i orthonormal. Apply AP.11 separately to e_i,e_j and to e_i+e_j. Linearity on the left yields (c_2(v_i)+c_2(v_j))/sqrt2; the right is (a_i c_2(v_i)+a_j c_2(v_j))/sqrt(a_i^2+a_j^2). The two Clifford matrices are linearly independent, since squaring c_2(b v_i+d v_j) gives (b^2+d^2) times the identity. Thus a_i=a_j. All a_i are one common lambda, and

\[
A=\lambda O\qquad(\lambda>0,\ O\hbox{ orthogonal}).
\tag{AP.12}
\]

Conversely, a unitary Clifford implementer for O gives AP.11, because the common positive scale cancels. In q=1 every positive derivative has this common-scale form; this is exactly why a nonisometric orientation-preserving circle action can pass the bounded-phase test. The distinction is between conformality of the normalized symbol and isometry of the unbounded symbol, not between existence and nonexistence of the original graph class.

### 11BY.5. Consequence for the original construction

AP.1–AP.12 prove that the original equivariant K-orientation has genuinely anisotropic cases even on compact manifolds, and that a fixed ordinary normalized normal Clifford phase cannot be made covariant under them merely by choosing a finite unitary fibre transport. A lift to metric space changes the geometric problem: it has invariant vertical and quotient metrics, while a splitting need not be invariant. It must still be followed by actual reduced representation, scalar compactness, original-q Clifford cancellation and positive-disk comparison. None of those obligations is supplied by this countertest alone.

![The nonisometric circle provider and anisotropic normal-phase countertest](../figures/nonisometric-normal-phases.png)

**Figure 11BY.1.** NRA.1–NRA.10 prove the actual degree-one positive graph class despite the fixed-point metric obstruction. AP.1–AP.12 retain original degree two and give the exact normalized directions \((1,1)/\sqrt2\) and \((3,2)/\sqrt{13}\), with nonzero gap \(\sqrt{2-10/\sqrt{26}}\). The arrows are exact symbol vectors, not a claim that no graph class exists. [Reproduce the figure](../reproduction/nonisometric-normal-phases/README.md).

## 11BZ. Smooth compact weights, exact normal sums and joint covariance

**Theorem 11BZ.1.** On a module uniformly generated by compact smooth sections of a finite Hermitian bundle, the construction below supplies a smooth compact dense-range weight, an independent source-balanced normal expression, exact completed tensor/Fock graph domains and ordinary scalar source-local compactness across rank changes. Its whole unit-algebra class is the original inverse normal class and pairs +1 with the original disk. These analytic facts do not identify a reduced holonomy transfer. NG.29–NG.37 give an actual isometric compact-arrow countertest to that inference.

### 11BZ.2. A smooth dense-range weight on the whole original module

Let R=C0(T), where T is the original second-countable smooth q-manifold with a chosen smooth metric g. Let J be a finite-rank smooth Hermitian bundle with a fixed Hermitian connection. Let E be the uniform closure in Γ0(J) of a countable generating family of compact smooth sections x_n. For the actual Killing module use exactly

\[
 J=(TT\oplus\mathfrak{so}(TT))\otimes_{\mathbb R}\mathbb C,
 \qquad E=E_K,
 \tag{NG.1}
\]

and the original localized jets from 11BT. These are genuine ambient smooth sections, even though an arbitrary element of E need not be smooth. Normalize ||x_n||≤1. Write V_x for the evaluated module fibre. These are the original Killing-jet fibres in the application.

Choose 0<α≤1 smooth so that α^(-2)g is complete, with compact exhaustion cutoffs η_m satisfying

\[
 \eta_m\to1,\qquad \sup_T\alpha|d\eta_m|_g\to0.
 \tag{NG.2}
\]

This is the explicitly proved completeness-weight/cutoff provider in 11BM. No invariance of α is needed here. Put ψ_n=√α x_n. Multiplication by √α does not lose the generated module: on every compact set it has a positive minimum, and compact scalar truncations approximate every E-section.

Set M_n=max_{0≤j≤n} sup_T ||∇^j θ_(ψ_n,ψ_n)||, using the ambient endomorphism connection, and choose

\[
 a_n=\frac{2^{-n}}{1+M_n},\qquad
 h=\sum_{n\ge1}a_n\theta_{\psi_n,\psi_n}.
 \tag{NG.3}
\]

Every M_n is finite because its section is compactly supported. For each fixed j, the differentiated tail is bounded by Σ_(n≥j)2^(-n); the finite head has finite compact-support bounds. Thus h is an actual C∞ ambient matrix field, its first covariant derivative is globally bounded, and

\[
 h\in\mathcal K_R(E),\qquad 0\le h\le\alpha1_J\le1_J,
 \qquad \|\nabla h\|_\infty<\infty.
 \tag{NG.4}
\]

Module compactness follows from norm convergence of actual rank ones, not from fibre compactness. The inequality h≤α uses ||x_n||≤1 and Σa_n≤1. Every h_x has range V_x and is positive definite there, including the convention that the zero fibre has no vectors.

Here is the needed whole-module density proof. For ε>0 put r_ε=ε(h+ε)^(-1). Since h≥a_n θ_(ψ_n,ψ_n), positivity gives

\[
 a_n\|r_\varepsilon\psi_n\|^2
 =\|r_\varepsilon a_n\theta_{\psi_n,\psi_n}r_\varepsilon\|
 \le\|r_\varepsilon h r_\varepsilon\|
 \le\varepsilon/4.
 \tag{NG.5}
\]

First extend r_εψ_n→0 to finite module combinations, then to every vector using ||r_ε||≤1. Since 1−r_ε=h(h+ε)^(-1), Ran h is dense in the completed module. If hξ=0, the nonnegative sum-of-squares inner product in (NG.3) makes every ⟨ψ_n,ξ⟩ zero; density of the generators gives ξ=0. These prove injectivity and dense range independently of evaluations.

The inverse lemma of 11BV now gives the actual positive regular module operator h^(-1), with domain Ran h and resolvents h(1±ih)^(-1). The following scalar normal sum uses this same smooth h.

### 11BZ.3. The literal source-balanced scalar carrier

Retain the prescribed full inverse coefficient S_inv, its grading, inverse determinant line and the final right Cl_q. Let H_N=L²(T,S_inv,dvol_g). The map

\[
 E\widehat\otimes_{R}H_N\longrightarrow
 L^2(T,V_x\widehat\otimes S_{\rm inv,x}),\qquad
 \xi\otimes u\longmapsto\xi(x)\otimes u(x)
 \tag{NG.6}
\]

respects ξf⊗u=ξ⊗fu. Its inner product is exactly the integral of the represented R-valued module inner product. It is an isometry, and it is onto: the compact localized generator values span V_x, simple measurable sections are approximated in L² by these generator combinations using compact cutoffs and finite partitions, and their span is dense. Equivalently the orthogonal complement of this range is pointwise orthogonal to every countable generator, and therefore zero. This also proves onto-ness when the fibre dimension changes.

The scalar carrier is the closed subspace H_E of the finite ambient Hilbert space H_J=L²(T,J⊗S_inv). On this Hilbert space, let p(x) be the measurable orthogonal projection onto V_x. Measurability follows from the countable generator Gram matrices. Then H_E=pH_J. **This is an ordinary measurable Hilbert-space projection; it is not asserted to be an adjointable projection of Γ0(J) onto E.** In particular it does not contradict 11BT.6.

The same proof applies to every completed central tensor E_r=E^⊗_R r, with ambient bundle J^⊗r and fibre V_x^⊗r. All factors remain over the same original unit. No standard-Hilbert-space renaming or change of source is used.

### 11BZ.4. An independent degenerate normal differential operator

Let N be the formally symmetric original inverse normal Dirac on J⊗S_inv, with the auxiliary Hermitian connection and original normal connection. The auxiliary bundle acts in an even position before S_inv. Its symbol is

\[
 \sigma_N(x,\xi)=1_J\otimes c_{\rm inv,g}(\xi).
 \tag{NG.7}
\]

Define on compact smooth ambient sections the symmetric first-order expression

\[
 C=hNh.
 \tag{NG.8}
\]

It is independent of an abstract KK product connection. Its principal coefficient is h²c_inv,g(ξ). Its propagation norm is at most α²|ξ|_g≤α|ξ|_g. An auxiliary connection preserving all V_x is **not** assumed: the two h factors are essential.

We give the exact closure proof because C is not uniformly elliptic at a rank seam. On a compact coordinate buffer, a Friedrichs mollifier J_ε satisfies a uniform L² bound for [C,J_ε], and this commutator tends strongly to zero. For a first-order coefficient A_j(x), the kernel contains (A_j(x)−A_j(y))∂_jρ_ε(x−y) and the corresponding ∂_jA_j term. The compact-buffer derivative bound and ∫|z||∇ρ_ε(z)|dz≤constant give the uniform Schur bound. On a smooth vector the commutator tends to zero by integration by parts/Taylor approximation; density gives the strong assertion on L². Zeroth-order terms cause only ordinary local multiplication commutators. Consequently every compactly supported maximal distributional vector is graph approximable by compact smooth vectors. No local H¹ assumption is used for this degenerate operator.

For a maximal vector u, multiplication by η_m gives

\[
 C(\eta_m u)=\eta_m Cu+h^2c_{\rm inv}(d\eta_m)u.
 \tag{NG.9}
\]

The second term tends to zero by (NG.2)–(NG.4), while the other terms converge by dominated convergence. Combining the compact-buffer smoothing with these cutoffs proves minimal=maximal. Hence C is self-adjoint on H_J with **exact** domain

\[
 \operatorname{Dom}C=
 \{u\in H_J:hN(hu)\in H_J\text{ distributionally}\}.
 \tag{NG.10}
\]

If u∈H_E^⊥, then hu=0, so u lies in this maximal domain with Cu=0. Self-adjointness therefore makes H_E reducing. The restricted operator, also denoted C, is self-adjoint on Dom C∩H_E. This reduction used actual differential multiplication by h, not a smooth projection onto the changing-rank field.

### 11BZ.5. The exact joint domain and its onto resolvents

On H_E the multiplication inverse L=h^(-1) has exact domain

\[
 \operatorname{Dom}L=
 \{u\in H_E:\int_T\|h_x^{-1}u(x)\|^2\,d\mathrm{vol}_g<\infty\}.
 \tag{NG.11}
\]

It is positive self-adjoint; use the bounded measurable spectral resolvents h(1±ih)^(-1). This is the scalar localization of the completed module inverse, not pointwise membership in the module domain.

Use an even auxiliary pair with grading σ_3, and let σ_1 exchange its two copies. On H_E⊕H_E define

\[
 \mathcal D_h=\sigma_1 L+\sigma_3 C,
 \qquad \operatorname{Dom}\mathcal D_h=
 (\operatorname{Dom}L\cap\operatorname{Dom}C)^{\oplus2}.
 \tag{NG.12}
\]

The total grading is the auxiliary grading times the prescribed normal grading. The ordered Clifford coefficient remains on the right.

There is a useful exact cancellation. Put L_ε=(h+ε)^(-1)|_(H_E). The matrix function is ambient smooth and bounded. On the full completed C-domain,

\[
 [C,L_\varepsilon]
 =-h(h+\varepsilon)^{-1}[N,h](h+\varepsilon)^{-1}h,
 \qquad \|[C,L_\varepsilon]\|\le K:=\|[N,h]\|.
 \tag{NG.13}
\]

Here [N,h] is genuine multiplication by the covariant derivative of h, because h commutes with normal Clifford multiplication. Its global bound was proved in (NG.3), rather than assumed. The product rule proves L_ε preserves Dom C. The commutators in (NG.13) converge strongly to −p[N,h]p on H_E. On the common smooth domain where both products exist this is exactly [C,L]; elsewhere it defines the bounded form cross term, not an unjustified LC-domain assertion.

The intersection domain is dense: it contains h² times every compact smooth ambient section, since Lh²v=hv and C(h²v) is smooth with compact support; the range of h² on H_J is dense in H_E. The bounded perturbation \(\mathcal D_\varepsilon=\sigma_1L_\varepsilon+\sigma_3C\) is self-adjoint on Dom C⊕Dom C, and

\[
 \|L_\varepsilon u\|^2+\|Cu\|^2
 \le\|\mathcal D_\varepsilon u\|^2+K\|u\|^2.
 \tag{NG.14}
\]

Passing to the strong cross-term limit gives

\[
 \left|\|\mathcal D_hu\|^2-\|Lu\|^2-\|Cu\|^2\right|
 \le K\|u\|^2.
 \tag{NG.15}
\]

This proves equivalence of the joint graph norm and the intersection graph norm. Since the intersection of the two closed graph domains is complete in that graph norm, the joint operator is closed on precisely that domain. No arbitrary unbounded compression, formal fibre diagonalization or global unweighted H¹ domain appears.

Solve (D_ε±i)u_ε=v. Resolvent boundedness and (NG.14) give uniform bounds for u_ε, Cu_ε and L_εu_ε. Take weak limits. The closed graph of C is weakly closed. Testing the L_ε term against w∈Dom L and using L_εw→Lw proves the limit belongs to Dom L and that the weak limit is Lu. The equation becomes (D_h±i)u=v. Thus both resolvents of D_h are onto, and the symmetric closed operator in (NG.12) is self-adjoint.

For R_z=(D_h−z)^(-1), (NG.15) gives the explicit useful estimate

\[
 \|LR_z\|,\ \|CR_z\|
 \le C_z=
 \left((1+|z|/|\Im z|)^2+K/|\Im z|^2\right)^{1/2}.
 \tag{NG.16}
\]


#### 11BZ.5a. Exact second-power domains

Let \(Q=(\operatorname{Dom}L\cap\operatorname{Dom}C)^{\oplus2}\) and \(K_0=-p[N,h]p\). The latter is bounded and skew-adjoint by NG.13. For the standard Pauli convention \(\sigma_1\sigma_3=-i\sigma_2\), set

\[
 B=i\sigma_2K_0=B^*,\qquad \|B\|\le K,
 \qquad
 \langle \mathcal D_hu,\mathcal D_hv\rangle
 =\langle Lu,Lv\rangle+\langle Cu,Cv\rangle+\langle u,Bv\rangle,
 \quad u,v\in Q.
 \tag{NG.16a}
\]

This is the polarized bounded mixed-form identity obtained from NG.13. The densely defined form \(q_0(u,v)=\langle Lu,Lv\rangle+\langle Cu,Cv\rangle\) on Q is closed because L and C are closed and its form norm is the intersection graph norm. Its associated positive operator S is specified exactly by

\[
 \operatorname{Dom}S=
 \{u\in Q:\exists w\in H_E^{\oplus2}\;
       q_0(v,u)=\langle v,w\rangle\ \text{for all }v\in Q\},
 \qquad Su=w.
 \tag{NG.16b}
\]

Here is the form-to-operator step explicitly. The complete form norm makes Q a Hilbert space. For every \(f\in H_E^{\oplus2}\), the bounded functional \(v\mapsto\langle v,f\rangle\) has a unique representative u for the inner product \(q_0(v,u)+\langle v,u\rangle\); hence \((S+1)u=f\). Density of Q makes w unique, and symmetry plus this onto positive resolvent yields the positive self-adjoint S. Applying the same argument to the form \(\langle\mathcal D_hu,\mathcal D_hv\rangle\), or testing it against Q, identifies its operator with the spectral square of the already self-adjoint \(\mathcal D_h\). The bounded term B changes the representing w by Bu and changes no form-operator domain. Consequently

\[
 \mathcal D_h^2=S+B,
 \qquad \operatorname{Dom}\mathcal D_h^2
 =\operatorname{Dom}S
 =\{u\in Q:\mathcal D_hu\in Q\}.
 \tag{NG.16c}
\]

No equality with \(\operatorname{Dom}L^2\cap\operatorname{Dom}C^2\) is inferred from a bounded mixed form. The separate completed-module inverse has the simpler exact square

\[
 \operatorname{Dom}L^2=\operatorname{Ran}h^2,
 \qquad L^2(h^2\xi)=\xi,
 \qquad \operatorname{Dom}L_r^2=\operatorname{Ran}b_r^2.
 \tag{NG.16d}
\]

Indeed applying the inverse twice is equivalent to writing its input as \(h(h\xi)\); the same argument holds for \(b_r\). The full module inverse-square domain requires \(\xi_r\in\operatorname{Ran}b_r^2\) and norm convergence in R of \(\sum_r\langle L_r^2\xi_r,L_r^2\xi_r\rangle\), as distinct from the scalar localization domain.

### 11BZ.6. Scalar local compactness across the actual seam

Let φ_δ be a smooth scalar cutoff, zero for t≤δ and one for t≥2δ. Define f_δ(t)=φ_δ(t)/t and g_δ(t)=φ_δ(t)/t², smoothly zero at zero. For u∈Dom C, the actual distributional identity is

\[
 N\phi_\delta(h)u
 =g_\delta(h)Cu+[N,f_\delta(h)]hu.
 \tag{NG.17}
\]

All its coefficients and derivatives are bounded on a compact buffer. With a compact smooth χ, the finite-coefficient elliptic estimate for N makes χφ_δ(h)R_z a compact operator by the ordinary scalar Rellich theorem. There is no variable-rank Rellich assertion hidden here: the output is an H¹-bounded family in the fixed finite ambient bundle J⊗S_inv.

The discarded part satisfies

\[
 \|\chi(1-\phi_\delta(h))R_z\|
 \le2\delta\|\chi\|C_z.
 \tag{NG.18}
\]

Indeed u=hLu on H_E and sup_t t|1−φ_δ(t)|≤2δ. Norm closure of scalar compact operators now proves χR_z compact, even when χ touches a genuine Killing-rank seam. Adjoints give R_zχ compact. Approximation gives the same for every χ∈C0(T).

If a bounded represented compact arrow kernel A has A=Aχ_s for an original compact source cutoff, then AR_z=Aχ_sR_z is scalar compact. This also holds after a bounded spatial coefficient factor. The spatial norm completion preserves it. This is an actual scalar statement; it does not prove phase covariance or reduced continuity of an unverified representation.

For f∈C_c∞(T),

\[
 [\mathcal D_h,f]=\sigma_3h^2[N,f]
 \tag{NG.19}
\]

is bounded and has compact source support. The completed resolvent identity and the phase integral with denominator 1+τ² prove [F_h,f] compact, where F_h=D_h(1+D_h²)^(-1/2). Its square defect is source-local compact by (NG.18). Thus this is a genuine scalar normal KK cycle with source R.

### 11BZ.7. Whole tensor/Fock carrier, domain and scalar normalization

For r≥1, use the completed central balanced tensor E_r from Section 3. Let h_i be h on coordinate i and set

\[
 \epsilon_r=(r+1)^{-2},\qquad
 b_r=\epsilon_r h_1\cdots h_r
       =\epsilon_r h^{\otimes r}.
 \tag{NG.20}
\]

These are the **product weights**, not the harmonic weights of 11BV. Products of factor rank ones are tensor rank ones, so b_r∈K(E_r). Every h_i is positive, injective and has dense range on the completed tensor module: prove it first on elementary tensors using (NG.5), then extend by boundedness. A product of commuting positive dense-range injective operators has the same properties. Therefore the inverse lemma gives

\[
 L_r=b_r^{-1},\quad\operatorname{Dom}L_r=\operatorname{Ran}b_r,
 \quad (L_r\pm i)^{-1}=b_r(1\pm ib_r)^{-1},
 \quad L_r\ge\epsilon_r^{-1}.
 \tag{NG.21}
\]

For the full completed module direct sum the exact domain of ⊕L_r is the vectors ξ with ξ_r∈Ran b_r and norm convergence in R of Σ⟨L_rξ_r,L_rξ_r⟩. Onto resolvents are the displayed block inverses; their degree tail is at most sup_(r>R)ε_r. This proves completed-module regularity, rather than substituting the evaluated weighted ℓ² condition.

On the actual scalar localization H_r=L²(T,V_x^⊗r⊗S_inv), use the original Dirac N_r coupled to the tensor connection and put C_r=b_rN_rb_r. The original normal degree remains q, while

\[
 \|b_r\|\le\epsilon_r,\quad b_r\le\alpha1,
 \|\nabla b_r\|\le\epsilon_r r\|\nabla h\|.
 \tag{NG.22}
\]

Thus the cross bounds K_r=||[N_r,b_r]|| are uniformly bounded. Each normal closure and joint operator D_r=σ_1L_r+σ_3C_r has the exact domains proved above. The complete original unit carrier is

\[
 \mathcal H=H_N\oplus\bigoplus_{r\ge1}(H_r\oplus H_r),
 \qquad
 \mathcal D=N_\alpha\oplus\bigoplus_{r\ge1}\mathcal D_r,
 \tag{NG.23}
\]

where N_α is the original self-adjoint inverse normal operator from 11BM. It is self-adjoint on the **exact** orthogonal-sum domain

\[
 \begin{split}
 u_0&\in\operatorname{Dom}N_\alpha,\qquad
 u_r\in(\operatorname{Dom}L_r\cap\operatorname{Dom}C_r)^{\oplus2},\\
 \|N_\alpha u_0\|^2&+
 \sum_{r\ge1}(\|L_ru_r\|^2+\|C_ru_r\|^2)<\infty.
 \end{split}
 \tag{NG.24}
\]


The exact completed scalar second-power domain is also explicit:

\[
 \begin{split}
 \operatorname{Dom}\mathcal D^2=\{u\in\mathcal H:\;&
 u_0\in\operatorname{Dom}N_\alpha^2,
 \quad u_r\in\operatorname{Dom}\mathcal D_r^2\ (r\ge1),\\
 &\|N_\alpha^2u_0\|^2+
      \sum_{r\ge1}\|\mathcal D_r^2u_r\|^2<\infty\}.
 \end{split}
 \tag{NG.24a}
\]

Each positive-degree square uses NG.16b–NG.16c with \(b_r\) in place of h. The spectral inequality \(t^2\le1+t^4\) proves membership in the first sum domain. Requiring the first operator image in that same completed domain then gives precisely NG.24a. This establishes an actual Hilbert-space domain, rather than evaluated fibre membership.

The uniform form bound (NG.15) proves equality with the direct-sum operator domain, not just inclusion of finite-degree vectors. Scalar localized compactness follows from finite-degree (NG.18) and

\[
 \|(1-Q_R)(\mathcal D-z)^{-1}\|
 \le C_z\sup_{r>R}\epsilon_r\longrightarrow0,
 \tag{NG.25}
\]

where Q_R retains the vacuum and the first R full degrees. Source commutators are uniformly bounded by (NG.19) and its vacuum version. Thus the bounded phase is a scalar R-cycle on the entire carrier.

There is also a whole-class calibration, rather than an inference from fibre indices. On all positive paired degrees simultaneously add the positive mass s:

\[
 \mathcal D_{r,s}=\sigma_1(L_r+s)+\sigma_3C_r,
 \qquad s\ge0.
 \tag{NG.26}
\]

Its completed domain is unchanged for finite s. The cross form bound is still the same. For s sufficiently large its spectrum avoids (−m_s,m_s), where m_s²=(1+s)²−sup K_r. Therefore the localized square defects tend to zero in norm, uniformly in degree. The two-resolvent phase-commutator integral gives

\[
 \|[F_s,f]\|\le
 \frac{\sup_r\|[\mathcal D_r,f]\|}{\sqrt{1+m_s^2}}
 \longrightarrow0.
 \tag{NG.27}
\]

On the dense joint domain D_(r,s)/s tends to the constant auxiliary involution σ_1; resolvent testing and its spectral gap prove F_s→σ_1 strongly, including adjoints, on the full positive-degree sum. This gives an adjointable interval homotopy after adjoining s=∞. Finite s has norm-continuous compact defects by the bounded perturbation/resolvent identities, and (NG.27) and the square estimate give continuity at ∞. The endpoint σ_1 is a degenerate R-cycle. All positive paired degrees therefore contribute zero as *one whole R-class*. Consequently

\[
 [\mathcal H,M_R,F_{\mathcal D}]=[H_N,M_R,F_{N_\alpha}]
 \quad\text{in }KK^q(R,\mathbb C),
 \qquad \langle b_U,[\mathcal H,F_{\mathcal D}]\rangle=+1.
 \tag{NG.28}
\]

Here b_U is the original disk and orientation. The inverse Spin-c module is the prescribed Π^(q(q−1)/2)Cl_q, with physical generators (−1)^(q(q−1)/2+1)L_(t_j) and coherent inverse line transition z^(-1)ρ_t(s). The auxiliary pairing is even and precedes this factor. The final right Cl_q is retained until the native contraction of 11BM.10. Neither tensor degree, jet rank, metric-bundle dimension nor this homotopy replaces q.

For a fixed original x and nonzero normal covector ξ, the normal high-frequency symbol on each positive fibre is σ_3c_inv,g(ξ)/|ξ|_g: positive b_r² cancels. The full joint symbol at *simultaneously diverging normal and inverse-weight energies* is different, as the next section demonstrates.

### 11BZ.8. Actual compact-arrow failure despite all the preceding estimates

Use the same original warped-torus metric geometry of Section 11BT.7, with period-one coordinates on T=(R/Z)²×(−2,2), normal degree q=3 and its bump between t=−1 and t=1. Supply the suspension holonomy below as a new example; the unit groupoid used in Section 11BT.7 does not contain these extra arrows. Its ranks are three on the closed bump region and six on the two open flat ends. Its actual entire module is

\[
 E_K=\Gamma_0(L)\oplus\Gamma_0(U,W),
 \quad U=T^2\times((-2,-1)\cup(1,2)).
 \tag{NG.29}
\]

L and W are the original smooth Hermitian rank-three subbundles. Near the upper seam, their ambient frames and the original metric are flat. The local/global torus translation

\[
\tau(x,y,t)=(x+1/2,y,t)
 \tag{NG.30}
\]

uses the period-one coordinates x,y of 11BT.7 and is an isometry of that same original metric across the seam. The unit groupoid of 11BT.7 and the rational-t pseudogroup of KWC.41 do not contain this arrow. Here we explicitly supply a **new connected mapping-torus suspension**

\[
 V=(\mathbb R\times T)/((u+1,z)\sim(u,\tau z)),
 \tag{NG.30a}
\]

with its one-dimensional suspension foliation. Its original complete transversal is the slice u=0, equal to T. Since τ has order two and has no fixed point, the actual transversal holonomy germ groupoid is (Z/2)⋉T. Its smooth derivative is the identity in the original framed coordinates; choose the coherent framed inverse lift with trivial phase and τ² lift equal to one. This is actual holonomy of the newly specified connected foliation with original q=3 and the same g,E_K. The isometry preserves these particular L,W blocks. No completion, missing endpoint or rank-layer arrow is added. This finite-group action is proper, so the failure below is a failure of this operator ansatz, even where another full/reduced normal provider is already available.

Let λ(t)>0 be smooth, ≤1, equal one near [−1.2,1.2], and vanish quadratically at both omitted endpoints ±2. Its inverse-speed metric is complete. Set

\[
 \rho(s)=\begin{cases}e^{-1/s^2},&s>0,\\0,&s\le0,\end{cases}
\quad a(x)=\frac{2+\cos(2\pi x)}{3},
 \quad
 h=\lambda(t)P_L+
 \lambda(t)\bigl(a(x)\rho(t-1)+\rho(-t-1)\bigr)P_W.
 \tag{NG.31}
\]

This is smooth on the entire original ambient bundle. Its W coefficient is flat at each rank seam, positive on each open flat end and vanishes at its omitted endpoint. Its L coefficient is positive on T and vanishes at infinity. Thus h is an actual positive compact dense-range endomorphism of the **whole** E_K, with bounded derivatives and complete propagation control. The W block's compactness is the C0(U)-matrix assertion, not fibre compactness. The L block is a C0(T)-matrix. Positivity/dense range also follow directly on both completed summands. Rescaling constants, if desired, preserves every argument below.

Consider degree one of (NG.20)–(NG.23). At source x=0 and t=1+s_n, where s_n=2^(−n), put

\[
 b_n=\epsilon_1\rho(s_n),\qquad
 k_n=b_n^{-3},\qquad \ell_n=s_n^4.
 \tag{NG.32}
\]

Take a normalized smooth scalar wave packet supported in a coordinate box of width ℓ_n about (0,y_0,1+s_n), multiplied by exp(ik_n y). The boxes may be taken disjoint. Choose a constant W vector and a normal spinor in the +1 eigenspace of B=i[N,y], the original full inverse unit Clifford generator in the flat y direction. Denote the resulting rectangular isometry from the auxiliary C² by J_n.

On this box, the relative variation of ρ is O(ℓ_n/s_n³)=O(s_n); the variation of a is O(ℓ_n). Cutoff differentiation contributes, after scaling by b_n, O(b_n³/ℓ_n), which tends to zero. Derivatives of b contribute smaller O(b_n³/s_n³) errors. The chosen auxiliary/normal connections are flat there. Hence the *actual differential operators on their completed domains* satisfy

\[
 \|(b_n\mathcal D_1)J_n-J_nA_1\|\longrightarrow0,
 \quad A_1=\sigma_1+\sigma_3,
 \tag{NG.33}
\]

while at the translated range a(1/2)=1/3 and

\[
 \|(b_n\mathcal D_1)U_\tau J_n-U_\tau J_nA_c\|\longrightarrow0,
 \quad c=1/3,\quad A_c=c^{-1}\sigma_1+c^2\sigma_3.
 \tag{NG.34}
\]

These estimates use simultaneous normal frequencies k_n=b_n^(-3), not a fixed-degree/fixed-frequency limit at one point. Both finite matrices have a gap about zero. Applying the spectral theorem to their eigenvectors proves the corresponding bounded phases converge in these rectangular equations to sign A_1 and sign A_c. Explicitly, the divided difference between t/√(b_n²+t²) and its value at either nonzero matrix eigenvalue is globally bounded by a constant depending only on that eigenvalue; (NG.33)–(NG.34) then prove the claim without an unproved norm-symbol theorem.

The two exact phase matrices are

\[
 \operatorname{sgn}A_1=(\sigma_1+\sigma_3)/\sqrt2,
 \qquad
 \operatorname{sgn}A_c=(27\sigma_1+\sigma_3)/\sqrt{730}.
 \tag{NG.35}
\]

Their difference has operator norm

\[
 \Delta=\sqrt{2-56/\sqrt{1460}}>0.
 \tag{NG.36}
\]

Choose one auxiliary unit vector on which this difference has norm Δ. The corresponding packets u_n converge weakly to zero because their supports shrink and are disjoint. Choose a compact smooth range cutoff χ equal one near (1/2,y_0,1), and a source cutoff equal one near (0,y_0,1). In the newly supplied suspension's actual (Z/2)⋉T groupoid, the arrow kernel T=χU_τχ_s has compact support on a bisection extending across the seam. The already proved compact source commutators imply

\[
 \liminf_n\|[F_{\mathcal D},T]u_n\|\ge\Delta>0.
 \tag{NG.37}
\]

Thus this phase commutator is not an ordinary scalar compact operator. The same degree-one witnesses sit inside the full Fock carrier, so adding the other full degrees does not repair it.

This failure is stronger than the exact-invariant-weight obstruction KWC.43: (NG.31) satisfies all normal differentiability, domain and localized compactness estimates established here, yet its phase fails covariance modulo compacts. It is also narrower: another weight, a joint rescaling of both terms, a different actual coefficient carrier or a geometric comparison can avoid this mechanism. Exact invariance is not made a universal requirement.

### 11BZ.9. What this does and does not give the anchored transfer

The independent normal expression, actual R-balance, whole-module smoothing, exact completed joint domains, scalar compactness and original inverse degree/disk calibration are now supplied for the whole jet/Fock carrier. This is meaningful nonconstant-rank analytic progress. It is not another selected constant-rank or global scalar-group family.

However the jet/Fock carrier in (NG.23) is **not** proved to be the coefficient correspondence of the specific anchored proper Bott-dual transfer Θ from 11AD.2–11AD.4. Equation (NG.28) is a scalar R-class identity; it is not an equivariant γ identity, a reduced norm proof, or an identification with Θ⊗x_N. Equation (NG.37) prevents treating the present generic smooth weighting as that realization. No actual product connection/positivity with the chosen Θ, and no bounded phase on its literal full physical carrier, has been proved here.

For the anchored-transfer route, it remains to construct or geometrically compare an independently normally differentiable representative of that single transfer, prove covariance modulo scalar compacts at joint normal/auxiliary infinity, its actual reduced representation and product class, and induce through the prescribed entire counting/plaque correspondence. Once an independent operator/class comparison exists, the bounded physical Parseval construction of 11BO/11BU is allowed; an arbitrary unbounded compression is unnecessary.

For unrestricted K-oriented holonomy the metric/frame bundle is a genuine alternative. The Hilsum–Skandalis metric-bundle/Thom/hypoelliptic construction provides a relevant full-algebra geometric pushforward; [Kordyukov, §§5.5, 7.4, 8.3 and Theorem 8.6](https://arxiv.org/pdf/0908.3744) distinguishes its full-algebra construction from the extra reduced descent. Its invariant vertical and quotient metrics are not an invariant horizontal splitting of the tangent bundle. A proposed use must verify the original-q Thom cancellation, full inverse line and native disk sign, and the actual geometric reduced graph realization, rather than infer reduced descent from the full statement. A comparison with the chosen anchored transfer is needed only when that route is used. This construction supplies none of those additional identifications.

![The actual seam-preserving arrow and two different joint phase directions](../figures/joint-normal-weight-covariance.png)

**Figure 11BZ.1.** NG.29–NG.37 retain the actual original degree-three warped torus, changing full Killing rank and a genuine half-translation bisection whose domain includes the seam and preserves rank. The joint auxiliary/normal directions are exactly \((1,1)/\sqrt2\) and \((27,1)/\sqrt{730}\), whose gap is \(\sqrt{2-56/\sqrt{1460}}\). Pure normal high-frequency agreement at fixed points does not control this simultaneous inverse-weight and normal-frequency sequence. [Reproduce the figure](../reproduction/joint-normal-weight-covariance/README.md).

## 11CA. Metric-fibre Bott transfer, shear and balanced reduced norm

The actual metric bundle supplies an equivariant positive vertical Bott cycle and a reduced transfer into its groupoid algebra. The calculations below prove its germ error, a shear-tolerant inverse symbol, exact local Fourier domains and a criterion for reducedness of a supplied balanced scalar carrier. A scalar normal graph class for arbitrary original holonomy remains to be constructed.

### 11CA.2. The actual metric bundle and its coherent coefficients

Let \(H\) be the full actual germ groupoid on a countable complete transversal \(T\), and let \(\tau=TT\) have original rank q. Each arrow \(h:x\longrightarrow y\) has its actual germ derivative \(A_h:\tau_x\longrightarrow\tau_y\) and its supplied coherent metalinear-complex lift. Put

\[
 W_x=\operatorname{Sym}^+(\tau_x^*),\qquad
 h\cdot g=A_h^{-T}gA_h^{-1},\qquad d=q(q+1)/2.
 \tag{MB.1}
\]

All maps are defined only on their actual germ domains. The chain rule proves the action law. The transformation groupoid \(H_W=H \ltimes W\) retains every actual arrow and isotropy; no completion of a germ or quotient of its labels is taken.

For vertical tangents \(B,C\) at g, use

\[
 \langle B,C\rangle_g=\operatorname{tr}(g^{-1}Bg^{-1}C).
 \tag{MB.2}
\]

Metric transport conjugates \(g^{-1}B\) by A, so cyclicity of trace proves invariance. The quotient \(TW/\ker  dp\) is \(p^*\tau\), with tautological metric g, and is also invariant: \((h g)(A v,A v)=g(v,v)\). Thus

\[
 0\longrightarrow V=\ker dp\longrightarrow TW
   \longrightarrow p^*\tau\longrightarrow0
 \tag{MB.3}
\]

has invariant metrics on V and on its quotient. A chosen smooth splitting is permitted; it is not asserted invariant. In this splitting the tangent action is triangular. The diagonal acts isometrically and the off-diagonal is a genuine second-jet shear.

Here is one fully coherent choice of vertical Spin-c orientation. On a compact reduction, the vertical orthogonal representation is the symmetric-square representation of \(SO(q)\) on symmetric matrices. A standard rotation in a coordinate plane acts by rotation through twice the angle on its two-dimensional trace-free symmetric block, and by one ordinary rotation on each of the q−2 mixed blocks. Hence a full 2pi loop has lifted spin sign \((-1)^{2+q-2}=(-1)^q\). The lift on \(\operatorname{Spin}(q)\) therefore sends its central −1 to \((-1)^q\). The map

\[
 [s,z]\longmapsto[\widetilde{\operatorname{Sym}^2}(s),z^q]
 \tag{MB.4}
\]

is well-defined: replacing \((s,z)\) by \((-s,-z)\) multiplies both output entries by \((-1)^q\), which is the defining Spin-c equivalence. For q=1 use the two-component spin lift with the same central sign convention. Metric reduction of the prescribed \(ML_q^c\) bundle gives this orientation on V without selecting an invariant original metric. One could choose another vertical orientation, provided that the **same entire orientation** is used in its Bott factor and inverse; the cancellation below is unchanged.

If the original compact transition has central factor z, the chosen vertical factor is \(z^q\), its inverse is \(z^{-q}\), and the original normal inverse is \(z^{-1}\). Consequently the vertical positive factor times the inverse total factor has exactly \(z^{-1}\) left. This is an equality of coherent transition operators, including the spin lifts; it is not a determinant-only equality. The total inverse is always the full module of canonical Theorem 7.17 in rank q+d. In a local ordered orthonormal frame it has parity \(k_{q+d}=(q+d)(q+d-1)/2\) and physical generators \((-1)^{k_{q+d}+1}L_e\). It must be transported through the actual finite graded Clifford evaluation before removing the vertical block. The remaining module is \(\Pi^{k_q}\mathrm{Cl}_q\), with physical generators \((-1)^{k_q+1}L_e\), prescribed inverse transition \(z^{-1}\rho(s)\), and the original last right \(\mathrm{Cl}_q\). In particular the mixed parity d q is not discarded by adding the two separate k-parities.

For the scalar disk calculation we order the total orientation as **original quotient block, then vertical block**, matching the product of the original disk Bott generator with the vertical Bott factor. The tuple \((\xi,\eta)\) below has this same order. When the coefficient contraction puts the cancellable vertical Clifford block first and the surviving original Cl_q last, it uses the full graded block transposition and the finite transport of canonical Lemmas 7.16–7.19. A bare permutation would change the orientation by \((-1)^{d q}\); the finite graded transport retains that parity. Thus the order used for the +1 comparison is specified before cancellation, rather than inferred from a dimension count.

### 11CA.3. A concrete metric-fibre Bott cycle, including all germ errors

Choose any smooth positive reference metric \(g_0(x)\) on tau; no invariance is required. On each W_x use the complete symmetric-space metric (MB.2). Its exponential is

\[
 \exp_g B=g^{1/2}\exp(g^{-1/2}Bg^{-1/2})g^{1/2}.
 \tag{MB.5}
\]

It is a global diffeomorphism from the vertical tangent space to W_x, with inverse given by the matrix logarithm. For symmetric U,V at the identity, the curvature numerator is

\[
 \langle R(U,V)V,U\rangle=-\tfrac14\|[U,V]\|_{\rm F}^2\le0.
 \tag{MB.6}
\]

For constant matrix directions, the metric-compatible torsion-free connection is \(\nabla_U V=DV[U]-(Ug^{-1}V+Vg^{-1}U)/2\); substitution in the metric derivative verifies compatibility directly. Differentiating \(g^{-1}\) and expanding the two connection orders gives \(R(U,V)Z=-[[U,V],Z]/4\) at the identity. Taking the trace of \([[U,V],V]U\) gives (MB.6). Along a radial geodesic, a perpendicular Jacobi field with J(0)=0 has convex norm because the Jacobi equation gives
\((|J|^2)''=2|J'|^2-2\langle R(J,\dot \gamma)\dot \gamma,J\rangle  \ge2|J'|^2\).
Where J is nonzero, differentiating |J| and using Cauchy–Schwarz gives \(|J|''\ge0\), so \(|J(t)|\ge t|J'(0)|\). Gauss' radial lemma treats the parallel part. Thus the differential of the inverse exponential has norm at most one. Integrating it along any smooth joining curve proves

\[
 \|\log_w a-\log_w b\|_w\le\operatorname{dist}(a,b).
 \tag{MB.7}
\]

The same radial argument proves \(\operatorname{dist}(g,\exp_g B)=\|B\|_g\); consequently (MB.5) identifies closed metric balls with compact bounded logarithm sets. These facts concern the complete **vertical fibre**, and impose no completeness condition on T.

Set \(v(w)=-\log_w g_0(p(w))\). Parallel transport along the radial geodesic sends \(\log_{g_0} w\) to v(w), so v has the positive Bott sign. On the full positive vertical Clifford module define

\[
 B_W(w)=c_V(v(w))(1+\|v(w)\|^2)^{-1/2}.
 \tag{MB.8}
\]

The module is the complete continuous-section module over C_0(W), with the chosen full vertical orientation and its finite Clifford coefficient, rather than a kernel selected pointwise. B_W is odd and self-adjoint. Its square defect is \(-(1+\operatorname{dist}(w,g_0(x))^2)^{-1}\). For every f in C_c(T), multiplying that defect by p^*f gives a function in C_0(W): its low-distance support lies over a compact base and in a uniformly bounded logarithm ball. Approximation gives this for every f in C_0(T). Source commutators vanish.

For a compact actual arrow test K in H, put

\[
 C_K=\sup_{h\in K}\operatorname{dist}_{W_{r(h)}}
                 (h g_0(s(h)),g_0(r(h)))<\infty.
 \tag{MB.9}
\]

The supremum is finite by continuity on the compact arrow test, irrespective of omitted continuation points elsewhere. Transported v differs from the target v by at most C_K, by (MB.7). The normalization map \(n(u)=u/\sqrt{1+|u|^2}\) has differential norm at most \(1/\sqrt{1+|u|^2}\). Along the segment joining the two vectors, its norm is therefore at most \(1/\sqrt{1+(r-C_K)_+^2}\), where \(r=\operatorname{dist}(w,g_0(s(h)))\). It follows that

\[
 \|h B_W(w)h^{-1}-B_W(hw)\|
 \le {C_K\over\sqrt{1+(r-C_K)_+^2}}.
 \tag{MB.10}
\]

The bound tends to zero uniformly on K as r tends to infinity. Compact arrow cutoffs and (MB.9) give exactly a continuous C_0 compact-module error on the whole pulled-back coefficient module. This proves equivariance modulo compacts, including the full coherent coefficient transport. Varying g_0 in a compact parameter interval gives the same bound with one C_K and yields a full module homotopy. The local positive Bott class is unchanged because (MB.5) is orientation preserving at the zero logarithm and radial transport is identity there.

The usual reduced equivariant descent therefore produces

\[
 \beta_{W,r}\in KK^d(C_r^*(H),C_r^*(H_W)).
 \tag{MB.11}
\]

The Clifford degree d in this notation is retained by the actual finite coefficient module before the conventional graded reduction. Descent uses the completion of compact actual arrow sections of the coefficient module; its right convolution and inner products use the original counting measures. In every right regular localization, the left representation is the source regular representation of H amplified by the coefficient Hilbert fibre. The integrated left norm is bounded by \(\sup_x\|\lambda_x(a)\|=\|a\|_r\). Thus (MB.11) is a reduced transfer even when H_W is nonamenable or its action is nonproper. For the original holonomy graph G, compose with the original counting/plaque equivalence module; this keeps all source-fibre regular representations and Haar normalization. No auxiliary gamma=1 assertion occurs here.

**Precisely what MB.11 gives.** It is an available geometric transfer into the reduced metric-bundle algebra. It is not a scalar class from C_r^*(H) to C, and it is not an equivalence assertion. Its composition with a scalar metric-bundle phase still requires that phase to be reduced-continuous, or the balanced-product criterion in Section 11CA.6.

### 11CA.4. A shear-tolerant inverse symbol, with complete local estimates

Fix a smooth splitting of (MB.3). In the corresponding cotangent coordinates write \((\xi,\eta)\), with xi in the quotient dual of rank q and eta in the vertical dual of rank d. After the invariant diagonal isometries are removed, an actual cotangent action has the form

\[
 (\xi,\eta)\longmapsto(\xi+L\eta,\eta).
 \tag{MB.12}
\]

Here L is the actual splitting shear; it is uniformly bounded on any retained compact arrow and metric-bundle test. Put a=1/4, p=3/4, and

\[
 R=(1+|\xi|^2+|\eta|^2)^{1/2},\qquad
 b(\xi,\eta)=(R^{-a}\xi,\eta),\qquad
 f(\xi,\eta)={c_{\rm inv}(b)\over\sqrt{1+|b|^2}}.
 \tag{MB.13}
\]

The Clifford action is the actual total inverse action specified in Section 11CA.2. In particular it is not an independently chosen same-spin positive module. For R>=sqrt(2),

\[
 |b|^2\ge R^{-2a}(R^2-1)\ge\tfrac12R^{2p},
 \qquad \|1-f^2\|\le2R^{-2p}.
 \tag{MB.14}
\]

There is no cotangent axis omitted from this estimate.

For the shear interpolate \(\xi_t=\xi+t L \eta\), \(0\le t\le1\), and let R_t,b_t be the corresponding expressions. The norms R_t are mutually comparable by constants depending only on ||L||. Differentiation in t gives

\[
 \dot b_t=\left(R_t^{-a}L\eta
       -aR_t^{-a-2}\langle\xi_t,L\eta\rangle\xi_t,0\right),
 \qquad |\dot b_t|\le(1+a)\|L\|R_t^{-a}|\eta|.
 \tag{MB.15}
\]

Since \(\sqrt{1+|b_t|^2}\ge\sqrt{1+|\eta|^2}\), the differential of normalization and (MB.15) prove

\[
 \|f(\xi+L\eta,\eta)-f(\xi,\eta)\|
 \le C_L R^{-1/4}.
 \tag{MB.16}
\]

One explicit admissible constant is \(C_L=(5/4)\|L\|(1+\|L\|)^{1/4}\): the shear and its inverse have norm at most \(1+\|L\|\), hence \(R_t\ge R/(1+\|L\|)\). This estimate supplies compact-error symbol covariance for a genuine triangular action; it does not assume an invariant total splitting. It also proves that changing the smooth splitting changes the symbol by a function vanishing at cotangent infinity, locally uniformly in its base and parameter.

For completeness, let z be the coordinates in any compact base chart and let zeta be ordinary cotangent coordinates. The splitting coordinates are \(M(z)\zeta\), with M smooth and invertible. All derivatives on the retained compact chart satisfy
\(|\partial_z^\beta \partial_\zeta^\alpha b| \le C_{\alpha,\beta} R^{1-|\alpha|}\).
The horizontal summand has the sharper bound with 1 replaced by p. The kth differential of \(u/\sqrt{1+|u|^2}\) is bounded by \(C_k(1+|u|)^{-k}\); this follows by differentiating the scalar denominator and factoring its largest remaining power. A chain-rule term with k differentiated b factors and total frequency order |alpha| has size at most
\(C R^{-p k} R^{k-|\alpha|}\). Since \(k\le|\alpha|+|\beta|\),

\[
 |\partial_z^\beta\partial_\zeta^\alpha f|
 \le C_{\alpha\beta}R^{-p|\alpha|+a|\beta|}.
 \tag{MB.17}
\]

Terms with no derivatives are bounded by one. Thus f is a local \(S^0_{3/4,1/4}\) symbol, with \(\rho>\delta\) and \(\rho>1/2\). This is the coordinate-stable calculus relevant to a nonlinear germ action. A bare claim about symbols having bounded derivatives would not suffice for its coordinate change.

The symbol estimates above are local original proofs. Hilsum–Skandalis Appendix A.7.5 is credited for the global maximal symbol construction; no reduced scalar conclusion or independent global proof is inferred here. [Primary paper, printed p.379](https://www.numdam.org/article/ASENS_1987_4_20_3_325_0.pdf#page=56).

### 11CA.5. An exact local operator and the original degree after vertical cancellation

In a constant split Euclidean chart, let \(n=q+d\) and take the entire \(L^2(R^n, I_n)\) inverse module, including its finite ordered right Clifford coefficient. Fourier transform gives the explicit operator

\[
 \widehat{A_pu}(\xi,\eta)=c_{\rm inv}
       ((1+|\xi|^2+|\eta|^2)^{(p-1)/2}\xi,\eta)\widehat u,
 \tag{MB.18}
\]

with exact domain

\[
 \operatorname{Dom}A_p=\left\{u:\int
 \big((1+|\xi|^2+|\eta|^2)^{p-1}|\xi|^2+|\eta|^2\big)
 |\widehat u|^2<\infty\right\}.
 \tag{MB.19}
\]

Its matrix symbol is self-adjoint, squares to the displayed scalar weight, and has everywhere invertible \(c_{\rm inv}(b)\pm i\), with norm at most one. Multiplication by these two inverses maps L^2 onto the exact graph domain. This proves self-adjointness without a chosen boundary extension. Smooth compact frequency truncations and then smooth approximation on bounded frequency balls prove that Schwartz vectors form a full graph core. Its bounded transform is exactly (MB.13).

There is a direct proof of every scalar source defect and the KK homotopy, independent of a global symbol theorem. For \(p \in [3/4,1]\), (MB.14) holds uniformly. If g is compactly supported smooth, \(M_g(1+A_p^2)^{-1}\) is compact: truncate the frequency multiplier, obtaining a Hilbert–Schmidt operator; its remaining norm is bounded by \(\|g\|_\infty\) times the uniform frequency tail tending to zero.

The Fourier kernel of the commutator with M_g is

\[
 (f_p(\zeta)-f_p(\zeta'))\widehat g(\zeta-\zeta').
 \tag{MB.20}
\]

To see compactness, first bound both Schur integrals outside an output ball. For a fixed difference k in a bounded ball, the mean-value formula and the uniform first-frequency estimate give a quantity tending to zero as the output frequency goes to infinity. For k outside that ball, use \(|f_p(\zeta)-f_p(\zeta-k)|\le2\) and the L^1 tail of the Schwartz function \(\widehat g\). For the second Schur integral the same argument applies: whenever output frequency is large and |k| is bounded, both frequencies are large. Hence the operator obtained by restricting its output to the complement of a large frequency ball tends to zero in norm, uniformly in p. The restricted-output kernel inside a fixed ball is square integrable because \(\widehat g\) is Schwartz and f_p is bounded. It is Hilbert–Schmidt. This proves compactness, and density treats all g in C_0(R^n).

At every fixed frequency the symbol is continuous in p. Dominated convergence proves strong continuity of the self-adjoint phases and their adjoints. The localized square defects are norm continuous by uniform tails; the preceding Schur argument gives norm continuity of commutators, using the continuous finite-ball kernels and their uniform tails. These facts define an adjointable KK homotopy on the interval section module. The endpoint p=1 is the ordinary calibrated full inverse in dimension n. Thus the entire local source class of (MB.18), including its right coefficient and grading, is that inverse class; this argument does not infer norm continuity of the phases themselves.

Over the **original** positive q-disk U, choose its reference metric and logarithm coordinates \(Y=\log(g_0^{-1/2}g g_0^{-1/2})\) in the d-dimensional fibre. This gives W|U diffeomorphic to U times R^d and preserves the chosen fibre orientation. In ordinary local source KK theory the positive vertical Bott factor of Section 11CA.3 followed by the inverse total class of Section 11CA.5 cancels its full vertical block. The two-sided native Thom evaluations of canonical Lemma 7.20 and the full coefficient transport of Lemmas 7.16–7.19 leave precisely the original inverse q-class. Central factors cancel as \(z^q z^{-(q+1)}=z^{-1}\), and the finite graded evaluation leaves the original ordered right Cl_q and the k_q physical sign. Therefore

\[
 \beta_U\,\big(\beta_{V|U}\otimes d_{W|U}\big)=+1.
 \tag{MB.21}
\]

The first factor is the same originally supplied orientation-preserving q-disk class, not a (q+d)-disk substituted for it. This is an ordinary local class calculation and a calibration for a prospective global product. It proves neither reduced continuity of the receiving scalar phase nor a global equivariant beta/Dirac unit. An actual scalar graph product must still prove its creation connections; the local identity cannot substitute for them.

### 11CA.6. A sharp reduced-norm test on the balanced geometric carrier

The metric-bundle scalar representation may fail to be reduced-continuous. The next viable test need not ask it to be reduced on its **whole** receiving algebra. The following exact theorem instead tests the representation after the chosen geometric correspondence. It applies to any suitable correspondence; no anchored transfer is imposed as a universal gate.

Let \(A_m=C_max^*(G)\), \(A_r=C_r^*(G)\), with quotient q_A. Let E be a Hilbert B-module with adjointable left representation L of A_m, and let pi_B be an arbitrary scalar representation of B on K. In the metric-bundle route, E may be the maximal descended metric Bott correspondence and K the full geometric metric-bundle inverse carrier. Define \(K_E=E \otimes_{\pi_B} K\). For every finite tuple \(bold \xi=(\xi_1,\ldots,\xi_N)\) of E-vectors, define positive operator matrices on K^N by

\[
 G_{\vec{\xi}}=
 [\pi_B(\langle\xi_i,\xi_j\rangle_B)]_{ij},\qquad
 Q_{a,\vec{\xi}}=
 [\pi_B(\langle L(a)\xi_i,L(a)\xi_j\rangle_B)]_{ij}.
 \tag{MB.22}
\]

**Theorem MB.1 (balanced reducedness).** The induced left representation on K_E factors through A_r if and only if, for every compact actual graph kernel a and every finite tuple from a norm-dense module core,

\[
 Q_{a,\vec{\xi}}\ \le\ \|a\|_r^2G_{\vec{\xi}}
 \quad\hbox{as operators on }K^N.
 \tag{MB.23}
\]

Here \(\|a\|_r=\sup_x\|\lambda_x(a)\|\) uses the original source-fibre counting/plaque regular representations. No geometric norm on a completion of a germ replaces that supremum.

**Proof.** For \(v=(v_1,\ldots,v_N)\), the norm squared of \(z=\sum_i\xi_i\otimes v_i\) is exactly \(\langle v,G_\xi v\rangle \), by the balanced tensor inner product. The norm squared of its a-image is \(\langle v,Q_{a,\xi}v\rangle \). Thus (MB.23) is precisely the contractive bound on all elementary tensor sums. Those sums are dense in the entire balanced carrier. The inequality also kills every null balanced vector, so it passes to its completion without discarding a hidden summand. It gives a unique contractive representation of A_r by norm completion of the compact graph kernels. Multiplication, adjoints and the original action law survive this completion. Conversely every *-representation of A_r is contractive, so its action on these tensor sums gives (MB.23). Approximation in the module core passes both matrices to the limit in operator norm, proving the stated dense-core formulation. Equivalently, this occurs exactly when the induced representation annihilates \(\ker  q_A\); the equivalence follows from the quotient norm and the same contractive completion. This completes both directions.

**Corollary MB.2 (scalar product descent without receiving reducedness).** Suppose the full balanced carrier already has a self-adjoint bounded phase F with ordinary scalar compact source commutators and localized square defects, together with its verified normal graph geometry and original disk calibration. If (MB.23) holds, the same carrier and F give the required reduced scalar cycle with the same calibration. Indeed all represented compact graph kernels and all defects are unchanged. The induced representation is contractive; density extends the compact defects to A_r. For the original disk map i_U, \(q_A i_{U,m}=i_{U,r}\). Functoriality gives exactly the same +1 pairing. Conversely (MB.23) is mandatory for this particular carrier, independent of the operator chosen on it.

The corollary asks for an actual product phase and creation errors; it does not manufacture them. It also does not imply that every reduced normalized graph phase must be this particular maximal metric-bundle product.

Receiving reducedness is sufficient for (MB.23) whenever the correspondence is a checked reduced descent, but it is not logically necessary. A simple representation example demonstrates the distinction. Let Gamma be F_2, take \(B=C_r^*(\Gamma)\oplus C_{\max}^*(\Gamma)\), take E to be its first central summand, and let A_m act on B by \(a \longrightarrow (q_r(a),a)\). Let pi_B be the sum of the faithful regular representation on the first component and the trivial character on the second. Its second component is not reduced-continuous, whereas the balanced carrier E tensor_B K is just the first regular component and (MB.23) holds exactly. This is a representation example, not a claimed foliation class. It proves why the correct remaining test is the balanced matrix inequality and not a universal requirement on the whole metric-bundle representation.

### 11CA.7. What regular counting supplies, and what scalar compactness still needs

There is also an unconditional norm comparison which preserves the literal metric-bundle graph. Choose a smooth full-support base density mu and positive smooth fibre densities on W_x. Take the nonzero completed field \(K_x=L^2(W_x,I_{q+d}|_{W_x})\), retaining the full prescribed inverse module and its finite Clifford coefficient. On

\[
 \mathcal H_{\rm reg}=\int_T^\oplus
       (\ell^2(H_x)\otimes K_x)\,d\mu(x),
 \qquad
 (\pi_{\mathrm{reg}}(a)u)(\gamma)=
       \sum_{\alpha\in H^{r(\gamma)}}a(\alpha)u(\alpha^{-1}\gamma),
 \tag{MB.24}
\]

the coefficient fibre is fixed at the **original source** x. Equivalently, use the actual holonomy/coefficient unitary to identify its range coefficient at \((\gamma,w)\) with the source coefficient; the formula then has precisely that transported full inverse factor. No arrow coordinate or metric coordinate is removed. The direct-integral norm is the essential supremum of the source regular norms, and therefore

\[
 \|\pi_{\mathrm{reg}}(a)\|\le\|a\|_r.
 \tag{MB.25}
\]

Equality is not silently asserted for a general non-Hausdorff holonomy groupoid: a smooth density can miss exceptional source regular norms, and ordinary scalar faithfulness is not a source requirement. If a faithful norm provider is useful, separability of A_r gives a countable norm-determining family of source points: for a countable dense algebra subset and each positive integer m, choose a source and a finite regular vector testing within 1/m of its reduced norm. The direct sum of those complete regular fibres, with their nonzero K_x amplifications, has norm exactly ||a||_r on the dense subset and hence on all A_r. Its separability follows from countability of source fibres and of the chosen family. This optional summand is a norm provider; it is not asserted to have a smooth normal differential across its isolated source points. For the full holonomy graph, use the original Haar source-fibre representation and counting/plaque equivalence in place of the displayed counting sum. The same inequality, and the same countable norm-determining construction when desired, use the original source-fibre norm. These norm results are valid in the unrestricted geometric scope and require no properness.

Reduced norm alone does not give ordinary scalar compact defects. Here is a precise obstruction to the simplest amplified phase. Suppose a positive-measure source chart O carries infinitely many disjoint actual arrow sections gamma_j over O, all with ranges in a fixed compact unit support. Suppose a unit cutoff e is one on all these ranges, and a decomposable phase acts on each retained arrow component as the same normal/metric-fibre phase F_x. If some unit coefficient vector field v_x over a smaller positive-measure chart satisfies \(\|(1-F_x^2)v_x\|\ge\epsilon>0\), then \(\pi_{\mathrm{reg}}(e)(1-F^2)\) is not scalar compact. Put the same normalized v on the gamma_j components and zero elsewhere. These are orthonormal vectors. Their defect images have norm at least epsilon, contradicting compactness. This proof only uses the displayed actual sections and their original ranges; it does not add a continuation or assert that every groupoid has such a family.

For example, a compact suspension with infinitely many distinct actual return arrows on a positive-measure chart has the required counting sections and a compact unit cutoff; the bounded transform of an ordinary normal operator has a nonzero \((1+D^2)^{-1}\) defect on any selected finite spectral vector. Its bare regular amplification therefore fails scalar compactness. Adding the finite original right Cl_q does not change the orthonormal sequence. This explains the **scalar** analytic task left after MB.25. A different joint phase can mix the actual arrow and metric coordinates, have vanishing tails, and avoid this particular defect; an auxiliary unit or one prescribed confinement is not imposed on every possible solution.

![The actual second-jet shear and the uniform compact-error mechanism MB.12–MB.17. The plotted ray is a sample; the uniform bound is proved. The full inverse vertical cancellation leaves the original q, and no scalar reduced receiving class is inferred.](../figures/metric-fibre-shear.png)

**Figure 11CA.1.** The actual second-jet shear and the uniform compact-error mechanism MB.12–MB.17. The plotted ray is a sample; the uniform bound is proved. The full inverse vertical cancellation leaves the original q, and no scalar reduced receiving class is inferred. [Reproduce the figures](../reproduction/general-reduced-graph/README.md).

## 11CB. Exact class descent and original disk normalization

The mapping cone of the actual maximal-to-reduced quotient gives an exact obstruction for a specified scalar class. Allowing corrections invisible to the original disk gives an obstruction whose annihilator is exactly the entire subgroup of attainable integer disk pairings. These class tests retain the original terminal Clifford coefficient; a normally geometric reduced graph realization requires its own construction.

### 11CB.1. Original objects, fixed coefficients and proof dependencies

Let \(G=\operatorname{Hol}(V,F)\) be the actual smooth holonomy groupoid, with the original plaque half-densities, and let \(q\) be the original transverse rank. Set

\[
A=C^*_{\max}(G),\qquad B=C_r^*(G),\qquad
\lambda:A\twoheadrightarrow B,\qquad J=\ker\lambda .
\tag{FCD.1}
\]

For standard second countable smooth manifolds these algebras are separable, including the locally Hausdorff holonomy convention: compact kernels in a countable chart family have a countable dense family in the I-norm, hence in each completed norm. The original full/reduced disintegration and chart conventions are proved in the programme's Groupoid C*-algebras: full and reduced, especially Section 14E and Theorems 14.3, 14.5. No Hausdorff replacement or coarse isotropy quotient is made here.

Write \(D_q=\mathrm{Cl}_q\) for the **final right** finite complex Clifford block. Use

\[
X_A=KK(A,D_q),\quad X_B=KK(B,D_q),\quad
\mathcal O=KK(C_\lambda,D_q).
\tag{FCD.2}
\]

This is the course's fixed degree-\(q\) scalar convention. The coherent inverse \(ML_q^c\) coefficient belongs to the supplied full geometric cycle; parity notation does not reconstruct or replace it. Quotient maps, cone maps and the Morita modules below are even. No operation here moves \(D_q\) ahead of another Clifford block. Temporary source and coefficient suspensions are removed together, using the proved ordered positive suspension/inverse suspension, with \(D_q\) still last.

For the original oriented transverse \(q\)-disk \(U\), let

\[
b_A\in KK(D_q,A),\qquad
b_B=b_A\widehat\otimes_A[\lambda]\in KK(D_q,B).
\tag{FCD.3}
\]

These are the original positive Bott generator followed by the actual counting/plaque wrong-way correspondence, in the fixed Clifford convention. Denote the ordered scalar index by

\[
\beta_A(x)=\operatorname{ind}_{D_q}(b_A\widehat\otimes_Ax),\qquad
\beta_B(y)=\operatorname{ind}_{D_q}(b_B\widehat\otimes_By).
\tag{FCD.4}
\]

The ordered Clifford endomorphism identity corresponds to the positive scalar integer 1. In particular no unexplained \((-1)^q\) enters FCD.4. Associativity gives the exact naturality identity

\[
\beta_B(y)=\beta_A(\lambda^*y),\qquad
\lambda^*y=[\lambda]\widehat\otimes_By.
\tag{FCD.5}
\]

The full/reduced correspondence equality used in FCD.3 uses the canonical leaf-space lesson's full-completion proof interfaces, (A.G.4), (A.G.7)–(A.G.8), and its full/reduced complete-transversal comparison (A.H.9). More explicitly, let the original disk correspondence be completed maximally as Q_U,m from its compact section core. Its gauge groupoid over U is étale, and the full equivalence proof identifies its full algebra with K(Q_U,m). Compact unit functions on U belong to that gauge algebra, with full norm equal to their supremum norm by the I-norm upper bound and the regular unit-vector lower bound. Thus the left C_0(U) action is compact, even when the module's right inner products lie only in the saturated-open ideal of A. The ideal module can be viewed over A with the same norm and compact operators, as proved in the linked ideal-module interface. Complete the same core regularly to Q_U,r. The map ξ tensor b ↦ ξ_r b has inner product b* lambda(〈ξ,η〉_m)c and dense range, so Q_U,m tensor_lambda B = Q_U,r. Composing its even correspondence with the original Bott class proves FCD.3 directly. This uses compact supports in the core and C_0(U), not compactness of the open disk itself or of V. These proofs do not require the groupoid to be amenable. The scalar and ordered Bott/Clifford product rules are proved in that lesson's Section 0 and Theorem 6.0c.

The only functional analytic exactness dependencies below are the existing complete proofs in Exact sequences in KK and the universal coefficient theorem: Lemmas 2.1–2.2 and Theorem 2.3 for the cone and nullhomotopy construction; Theorem 4.1 for the optional semisplit ideal/cone equivalence; Proposition 10.4 for the optional UCT calculation. Every separability condition in those proofs is met by FCD.1 and its cone. These dependencies are explicit proofs, rather than an assumption of exactness for every quotient.

Hilsum–Skandalis's projection construction uses the maximal algebra: their introduction, printed pp.326–327, states its completion convention and distinguishes the reduced algebra. Its general K-oriented setting includes non-Riemannian geometry. This construction treats its supplied geometric class as a possible input \(x_0\), with its exact inverse convention and normalization to be checked before using FCD.9. [Primary paper](https://www.numdam.org/article/ASENS_1987_4_20_3_325_0.pdf#page=3). No protected proof expression is reproduced.

### 11CB.2. The exact cone obstruction, without a section of lambda

Use the right-vanishing cone

\[
C_\lambda=\{(a,g)\in A\oplus C_0([0,1),B):g(0)=\lambda(a)\},
\quad p(a,g)=a,\quad i(g)=(0,g),
\tag{FCD.6}
\]

where \(i\) has source \(SB=C_0((0,1),B)\). Define

\[
\Omega:X_A\longrightarrow\mathcal O,\qquad
\Omega(x)=[p]\widehat\otimes_Ax.
\tag{FCD.7}
\]

**Theorem FCD.1 (exact descent of a specified class).** For every \(x\in X_A\),

\[
x=\lambda^*y\text{ for some }y\in X_B
\quad\Longleftrightarrow\quad \Omega(x)=0.
\tag{FCD.8}
\]

When a nullhomotopy of \(p^*x\) is supplied, a descended bounded cycle is constructed from it by the source/coefficient suspension gluing described below. If \(y_0\) is one descent, all descents are \(y_0+\ker\lambda^*\). This is a class-level criterion: the original represented action of \(A\) need not factor through \(B\).

**Proof.** For \((a,g)\in C_\lambda\), the homomorphisms \(h_t(a,g)=g(t)\), extended by \(g(1)=0\), form a point-norm homotopy from \(\lambda p\) to zero. Hence \([p][\lambda]=0\) in KK and \(\Omega\lambda^*=0\). Conversely the contravariant Puppe segment of Theorem 2.3 in the exact-sequence lesson is

\[
KK(B,D_q)\xrightarrow{\lambda^*}KK(A,D_q)
\xrightarrow{p^*}KK(C_\lambda,D_q).
\]

It is exact for these separable sources. This proves FCD.8. Subtracting two descents proves the asserted affine description. The suspension bookkeeping in its proof is simultaneous suspension of the source and coefficient, not a shift of the physical normal degree. \(\square\)

There is an explicit completely positive contractive section of **the cone projection** \(p\):

\[
s_p(a)=(a,[t\mapsto(1-t)\lambda(a)]).
\]

At every matrix size its two components are positive for positive input, its pair belongs to the cone, and its norm is at most the input norm. A positive element of the ambient direct sum lying in the closed cone subalgebra is positive in that subalgebra by continuous functional calculus. Also \(ps_p=1_A\). Thus the cone extension by \(SB\) is always semisplit. This formula is **not** a section of \(\lambda\); it makes no claim about semisplitting of FCD.1's kernel extension.

#### 11CB.2.1. A supplied nullhomotopy gives a concrete cycle construction

For completeness the needed construction has the following inputs and output. Represent \(x\) by \((E,\pi,F)\) over \(D_q\), and give a nullhomotopy of its \(p\)-pullback with this initial cycle. Stabilization and contraction of any degenerate summands make the initial cycle literal, as proved in Lemma 2.2. Let \((H,\bar\pi,\bar F)\) be that nullhomotopy over \(C_0([0,1),D_q)\), with initial fibre \((E,\pi p,F)\) and zero final fibre.

Put \(H^\circ=H C_0((0,1),D_q)\). Restrict \(\bar\pi\) along \(i:SB\to C_\lambda\). Since \(\pi=0\), the image of \(i(SB)\) and its adjoint have range in \(H^\circ\). Adjointable module operators preserve this coefficient ideal. The localized cycle defects therefore restrict to compact operators on \(H^\circ\), by the compact-submodule lemma in the programme's product lesson, used explicitly in Lemma 2.2. This gives an actual cycle

\[
w=(H^\circ,\bar\pi i,\bar F|_{H^\circ})
\in KK(SB,SD_q).
\]

Countable generation holds because the coefficient ideal has a countable approximate identity and \(H\) has countably many generators. No orthogonal complement of \(H^\circ\) is used.

Glue \(C_0((-1,0],E)\) to \(H\) at their common initial fibre. On a source function \(a\in C_0((-1,1),A)\), the negative half acts by \(\pi(a(t))\), and the positive half by

\[
\bar\pi\bigl(a(0),\lambda(a|_{[0,1)})\bigr).
\]

These agree at the common fibre. The compact gluing proof, Lemma 2.1, proves the localized defects before any endpoint evaluation. Translating suspension support between the two half-intervals identifies the simultaneous suspension of \(x\) with \((S\lambda)^*w\). Applying the inverse simultaneous suspension yields

\[
y=\tau_S^{-1}(w)\in KK(B,D_q),\qquad \lambda^*y=x.
\]

All intervals have increasing coordinates. The coefficient is \(S\widehat\otimes D_q\), and the inverse suspension removes that first block with \(D_q\) last. The construction produces a valid bounded scalar cycle with ordinary localized module compacts. It depends on the supplied full nullhomotopy; zero tests at separate fibres or a numerical index do not construct it. It does not by itself identify \(y\)'s operator as the prescribed normal geometric phase on the original graph.

#### 11CB.2.2. When restriction to the kernel is enough

Let \(k:J\to A\) and \(e:J\to C_\lambda\), \(e(j)=(j,0)\). Then

\[
[e]\widehat\otimes_{C_\lambda}\Omega(x)=k^*x.
\]

Thus \(k^*x=0\) is always necessary for descent. If the actual extension \(0\to J\to A\xrightarrow{\lambda}B\to0\) is semisplit, Theorem 4.1 in the exact-sequence lesson constructs both KK inverses of \(e\), with the stated negative left-endpoint cone sign. In that case \(\Omega(x)=0\) is equivalent to \(k^*x=0\). More generally it is enough to supply an actual KK inverse of \(e\). Without one of these hypotheses, the implication from kernel vanishing to cone vanishing is not justified. The unconditional theorem remains FCD.8.

### 11CB.3. A complete obstruction for the original disk, permitting another full class

Assume an actual maximal class with the original normalization is supplied:

\[
x_0\in X_A,\qquad \beta_A(x_0)=1.
\tag{FCD.9}
\]

This assumption is used only in this section. It must refer to the positive original disk and the prescribed inverse orientation, rather than to a determinant-only class or a tangent disk of another rank.

Let

\[
L=\ker\beta_A,\qquad
\mathcal Q=\mathcal O/\Omega(L),\qquad
\epsilon_U=[\Omega(x_0)]\in\mathcal Q.
\tag{FCD.10}
\]

The subgroup \(\Omega(L)\) need not be closed in any topology; FCD.10 is a quotient of abelian groups. No analytic topology on KK is used.

**Theorem FCD.2 (normalized descent and its exact integral order).** The class \(\epsilon_U\) does not depend on the choice of maximal \(x_0\) with FCD.9. Moreover

\[
\boxed{\quad
\beta_B(X_B)=\{n\in\mathbb Z:n\epsilon_U=0\}.
\quad}
\tag{FCD.11}
\]

Consequently a reduced scalar class pairing the original disk as \(+1\) exists if and only if \(\epsilon_U=0\). Equivalently it exists if and only if there is a correction \(z\in X_A\) such that

\[
\beta_A(z)=0,\qquad \Omega(x_0+z)=0.
\tag{FCD.12}
\]

If \(\epsilon_U\) has finite order \(d\geq1\), then the entire reduced pairing image is \(d\mathbb Z\). If it has infinite order, every reduced scalar pairing with the original disk is zero. Thus a nonzero finite order larger than one obstructs normalization \(+1\), while still permitting nonzero scalar pairings.

**Proof.** Every \(x\in X_A\) has the unique decomposition

\[
x=\beta_A(x)x_0+z,\qquad z\in L.
\tag{FCD.13}
\]

Existence follows by subtracting \(\beta_A(x)x_0\); uniqueness follows by applying \(\beta_A\). If \(x=\lambda^*y\), Theorem FCD.1 gives \(\Omega(x)=0\). Taking FCD.13 in \(\mathcal Q\) gives \(\beta_A(x)\epsilon_U=0\). FCD.5 proves one inclusion in FCD.11.

Conversely, if \(n\epsilon_U=0\), there is \(\ell\in L\) with \(n\Omega(x_0)=\Omega(\ell)\). Put \(x=nx_0-\ell\). It has \(\Omega(x)=0\) and \(\beta_A(x)=n\). Theorem FCD.1 gives \(y\in X_B\) with \(\lambda^*y=x\); FCD.5 gives \(\beta_B(y)=n\). This proves the other inclusion. Two normalized choices differ by an element of \(L\), so their cone classes differ by \(\Omega(L)\), proving independence. Taking \(n=1\) proves FCD.12. The elementary annihilator of an element of finite order \(d\) in \(\mathbb Z\) is \(d\mathbb Z\); the annihilator of an element of infinite order is zero. \(\square\)

In particular

\[
\Omega(x_0)\ne0\quad\text{does not imply}\quad\epsilon_U\ne0.
\tag{FCD.14}
\]

A full class can fail descent while another normalized full class descends. This distinction holds at the level of classes, independently of a norm failure in one represented cycle.

**Geometric limit of FCD.2.** Corrections in \(L\) preserve this original disk index only. They are not thereby proved to preserve the prescribed coherent \(ML_q^c\) lift, every transverse symbol, or geometric realization on the original graph. Thus FCD.2 is exact for **normalized scalar class existence**, not for the narrower set of normally geometric suitable cycles. A geometric correction theorem would have to construct the allowed corrections and compare their full normal symbols.

### 11CB.4. All ordinary normal restrictions can be prescribed simultaneously

The same argument works with any compatible homomorphism of constraints. Let \(P\) be an abelian group and let

\[
\rho_A:X_A\to P,\qquad \rho_B:X_B\to P,
\qquad \rho_B=\rho_A\lambda^*.
\tag{FCD.15}
\]

For \(c\in P\) with a supplied full lift \(\rho_A(x_0)=c\), define

\[
L_c=\ker\rho_A,\qquad
\epsilon_c=[\Omega(x_0)]\in\mathcal O/\Omega(L_c).
\tag{FCD.16}
\]

**Proposition FCD.3.** A reduced class \(y\) with \(\rho_B(y)=c\) exists exactly when \(\epsilon_c=0\). The obstruction is independent of the selected full lift.

**Proof.** If such a \(y\) exists, \(\lambda^*y-x_0\in L_c\), while \(\Omega\lambda^*y=0\), so \(\Omega(x_0)\in\Omega(L_c)\). Conversely this inclusion supplies \(z\in L_c\) with \(\Omega(x_0+z)=0\). FCD.1 gives a descent of \(x_0+z\), and FCD.15 gives the required constraint. Differences of full lifts belong to \(L_c\). \(\square\)

A useful case is the complete-transversal unit algebra \(R=C_0(T)\) with its actual inclusions \(a_A:R\to A_T\) and \(a_B:R\to B_T\). These satisfy \(\lambda_Ta_A=a_B\). Take

\[
P=KK(R,D_q),\qquad \rho_A=a_A^*,\qquad \rho_B=a_B^*.
\]

Then FCD.16 prescribes the whole **ordinary inverse normal class on \(T\)**, rather than one disk's integer. Taking \(P=\prod_U\mathbb Z\) likewise prescribes every chosen oriented local disk pairing. These are stronger class constraints. Even the whole ordinary restriction forgets the arrow-level coherent lift; it cannot by itself certify full equivariant normal geometry. The exact preservation of a specified maximal class remains FCD.8.

### 11CB.5. A transfer need only have the right class-dependent defect

Suppose an actual even class \(\Theta\in KK(B,A)\) is constructed by some independent route. Define

\[
\Delta_\Theta=1_A-[\lambda]\widehat\otimes_B\Theta
\in KK(A,A),\qquad y_\Theta=\Theta\widehat\otimes_Ax_0.
\tag{FCD.17}
\]

**Proposition FCD.4 (class-dependent transfer criterion).** One has

\[
\lambda^*y_\Theta=x_0-\Delta_\Theta x_0,
\qquad
\beta_B(y_\Theta)=1-\beta_A(\Delta_\Theta x_0).
\tag{FCD.18}
\]

Therefore \(\Delta_\Theta x_0=0\) suffices for descent of the actual specified \(x_0\). The weaker condition \(\beta_A(\Delta_\Theta x_0)=0\) suffices for a reduced class with the original disk normalization. With FCD.15, the condition \(\rho_A(\Delta_\Theta x_0)=0\) instead preserves the entire supplied constraint \(c\).

**Proof.** Expand FCD.17 using bilinearity and associativity, then apply FCD.5 or FCD.15. All three product factors have their displayed variance. \(\Theta\) is even, so the terminal coefficient is still \(D_q\). \(\square\)

This criterion requires an actual \(\Theta\) and its stated defect calculation. It does not require \([\lambda]\Theta=1_A\), a whole gamma identity, a whole-base auxiliary unit or an OU.6 identification. Conversely it does not create a transfer from a scalar oscillator index, separate rank-layer lifts or a formal crossed-product symbol. If \([\lambda]\Theta=\gamma_A\), only \((1_A-\gamma_A)x_0=0\) is required for descent of this class. The stronger assertion \(\gamma_A=1_A\) is a sufficient route, not a universal gate.

### 11CB.6. Exact Hom and Ext tests when their UCT is actually known

This section is conditional. Assume the natural scalar UCT sequence has been proved for the algebras under discussion. The programme's Proposition 10.4 proves it for its separable nuclear bootstrap class; an explicitly supplied KK-equivalence to such an algebra also transfers it by product naturality. No UCT or nuclearity claim is made for arbitrary reduced holonomy algebras.

Write scalar parity notation in this section only, retaining the ordered Clifford convention of Section 11CB.1. The UCT sequence with scalar coefficient is

\[
0\longrightarrow\operatorname{Ext}^1_{\mathbb Z}(K_{q-1}(B),\mathbb Z)
\xrightarrow{\delta_B}KK^q(B,\mathbb C)
\xrightarrow{\gamma_B}\operatorname{Hom}(K_q(B),\mathbb Z)
\longrightarrow0 .
\tag{FCD.19}
\]

The Hom arrow is the actual index pairing, not a chosen splitting. Since \(\gamma_B\) is onto,

\[
\beta_B(X_B)=\{h(b_B):h\in\operatorname{Hom}(K_q(B),\mathbb Z)\}.
\tag{FCD.20}
\]

If \(K_q(B)=\mathbb Z^r\oplus T\) with \(T\) finite, write \(b_B=(k_1,\ldots,k_r,t)\). Then

\[
\beta_B(X_B)=d\mathbb Z,\qquad
d=\gcd(k_1,\ldots,k_r),
\tag{FCD.21}
\]

with \(d=0\) when all free coordinates vanish. Indeed a homomorphism to \(\mathbb Z\) kills \(T\) and has arbitrary integer coefficients on the free basis; Bézout's identity gives precisely the displayed subgroup. Hence a normalized reduced scalar class exists if and only if the reduced disk is primitive in this free quotient. Under FCD.9, FCD.11 identifies \(d\) with the finite order of \(\epsilon_U\), with \(d=0\) corresponding to infinite order. For infinitely generated K-groups the correct criterion is FCD.20; indivisibility in the group alone cannot replace the existence of a homomorphism to \(\mathbb Z\) taking the disk to 1.

One weaker usable sufficient interface follows immediately: if \(\lambda_*:K_q(A)\to K_q(B)\) is an isomorphism and FCD.19 is known for \(B\), the full index homomorphism \(h_A=\gamma_A(x_0)\) can be defined directly by pairing with \(x_0\), without needing the UCT for \(A\). Set \(h_B=h_A\lambda_*^{-1}\); it sends \(b_B\) to 1. A lift through \(\gamma_B\) gives a normalized reduced scalar class. This does **not** prove descent of the specified \(x_0\), whose Ext component may differ.

#### 11CB.6.1. The specified class has a secondary Ext obstruction

Assume FCD.19 is known naturally for both \(A\) and \(B\), and assume additionally that \(\lambda_*:K_q(A)\twoheadrightarrow K_q(B)\) is surjective. For a specified full \(x\), let \(h_A=\gamma_A(x)\).

The first obstruction is

\[
h_A|_{\ker(\lambda_*:K_q(A)\to K_q(B))}=0.
\tag{FCD.22}
\]

If FCD.22 holds, it gives a unique \(h_B\) with \(h_B\lambda_*=h_A\). Choose any UCT lift \(y_0\) of \(h_B\). Naturality shows that \(x-\lambda^*y_0\) has zero Hom component. There is therefore a unique \(\eta\in\operatorname{Ext}^1(K_{q-1}(A),\mathbb Z)\) with

\[
x-\lambda^*y_0=\delta_A(\eta).
\]

Its class

\[
\epsilon_{\mathrm{Ext}}(x)
=[\eta]\in
\frac{\operatorname{Ext}^1(K_{q-1}(A),\mathbb Z)}
{\operatorname{im}\bigl(\lambda_*^*:
\operatorname{Ext}^1(K_{q-1}(B),\mathbb Z)
\to\operatorname{Ext}^1(K_{q-1}(A),\mathbb Z)\bigr)}
\tag{FCD.23}
\]

is independent of the lift \(y_0\). The specified full class descends if and only if FCD.22 holds and FCD.23 vanishes.

**Proof.** Two lifts of the unique \(h_B\) differ by \(\delta_B(\zeta)\). Naturality gives \(\lambda^*\delta_B(\zeta)=\delta_A(\lambda_*^*\zeta)\), so the quotient class in FCD.23 is unchanged. If it vanishes, choose \(\zeta\) with \(\eta=\lambda_*^*\zeta\); then \(y=y_0+\delta_B(\zeta)\) satisfies \(\lambda^*y=x\). Conversely any descent has the unique Hom part \(h_B\), and its difference from \(y_0\) belongs to the Ext term, giving precisely this relation for \(\eta\). Necessity of FCD.22 follows by naturality of the Hom arrow. \(\square\)

No natural splitting of the UCT was selected. Thus equality of all ordinary K-theory index pairings can still leave an actual secondary obstruction to descent of one full class. FCD.23 is a conditional computation of FCD.8, while FCD.20–FCD.21 concern the weaker normalized-class problem.

### 11CB.7. Compatibility with the actual counting/plaque graph correspondence

Let \(H=G_T\) be an actual complete-transversal reduction, and write \(A_T=C^*_{\max}(H)\), \(B_T=C_r^*(H)\). Let \(E_m\) be the \(A\)-\(A_T\) equivalence module and \(E_r\) its reduced \(B\)-\(B_T\) module. Both are completed from the same actual cross-arrow section core. Their coefficient quotient unitary is

\[
E_m\widehat\otimes_{A_T,\lambda_T}B_T
\longrightarrow E_r,\qquad \xi\otimes b\longmapsto\xi_rb.
\tag{FCD.24}
\]

Its inner product is
\(b^*\lambda_T(\langle\xi,\eta\rangle_m)c
=\langle\xi_rb,\eta_rc\rangle_r\); a coefficient approximate identity makes its range dense. The full and reduced linking-corner norm proofs then identify the original left action with the quotient action. Consequently

\[
[E_m]\widehat\otimes_{A_T}[\lambda_T]
=[\lambda]\widehat\otimes_B[E_r]
\quad\text{\in }KK(A,B_T).
\tag{FCD.25}
\]

The entire norm, positivity and two-sided equivalence proofs are the canonical lesson's full-completion interfaces preceding Theorem 11A.1, its (A.G.4), and the complete-transversal proof of (A.H.9). The historical primary full/reduced equivalence result is [Sims–Williams, Theorem 5.3 and its preceding quotient compatibility](https://jot.theta.ro/jot/archive/2012-068-001/2012-068-001-012.pdf). The displayed inner-product argument is the same actual module completion, not equality of dimensions.

Suppose a supplied maximal transverse class is \(x_T\in KK(A_T,D_q)\), and put \(x_G=[E_m]x_T\). If \(x_T=\lambda_T^*y_T\), then

\[
y_G=[E_r]y_T\in KK(B,D_q),\qquad
\lambda^*y_G=x_G.
\tag{FCD.26}
\]

Conversely, a descent of \(x_G\) can be transported through the actual inverse reduced equivalence module, and FCD.25 with the inverse full module shows that it descends \(x_T\). Thus specified-class descent is invariant under this actual transversal/graph equivalence. Local constraints transported through these two inverse modules likewise give exactly the same attainable pairing subgroup and the same order in FCD.11. This conclusion requires no separate assertion that the two mapping cones are Morita equivalent.

For the original disk class, \(b_{A,G}[E_m]=b_{A,T}\) and \(b_{B,G}[E_r]=b_{B,T}\). These equalities follow from the actual correspondence tensor associator. Locally choose a plaque bump \(u\) with \(\int|u|^2=1\); its graph creation map sends each genuine return column to that same normalized plaque column. The counting inner product retains every actual arrow, and the plaque factor is exactly one. The canonical lesson's SG.18a supplies the full corner unitary in its unital suspension model; the general section-core quotient and correspondence associator are those of (A.G.4). Hence

\[
\beta_{B,G}([E_r]y_T)=\beta_{B,T}(y_T).
\tag{FCD.27}
\]

No finite-return truncation, dimension factor or discarded isotropy appears in this equality. It transports the **original** disk calibration. It does not turn arbitrary \(y_T\) into a normal Dirac operator: a checked scalar product phase on \(E_r\widehat\otimes_{B_T}H_{y_T}\) exists by the proved KK product theorem, but normally geometric suitability and any desired concrete graph-domain comparison require their own construction. Counting/plaque compatibility settles the coefficient and index interface, not that geometric issue.

### 11CB.8. An explicit representation/class distinction

Here is an elementary exact test of the logical claims, not a replacement foliation. Let \(A=\mathbb C\oplus\mathbb C\), \(B=\mathbb C\), and \(\lambda(a,b)=a\). Then \(KK(A,\mathbb C)=\mathbb Z^2\), with pairing with the two minimal projections as its two coordinates; this follows by the two actual finite-dimensional source corners and the scalar KK index classification. The pullback image is \(\mathbb Z\oplus0\).

Take \(H^+=\mathbb C e_1\oplus\mathbb C e_2\), \(H^-=\mathbb C f_2\), and

\[
\pi^+(a,b)=\operatorname{diag}(a,b),\quad
\pi^-(a,b)=b,\quad
T e_1=0,\quad T e_2=f_2,\quad
F=\begin{pmatrix}0&T^*\\T&0\end{pmatrix}.
\tag{FCD.28}
\]

All operators are finite-dimensional, so every cycle defect is compact. The first corner has index 1; the second is an exactly degenerate two-term cycle. Thus \([\pi,F]=(1,0)=\lambda^*(1)\). Nevertheless \(\pi(0,1)\ne0\), so this **representative** does not factor through \(\lambda\). A representative of a descended class may act nontrivially on the quotient kernel.

For the local class \(b_A=(1,0)\), the maximal class \(x_0=(1,1)\) also has \(\beta_A(x_0)=1\). It cannot descend, since the second-corner pairing is 1. The correction \(z=(0,-1)\) has \(\beta_A(z)=0\), while \(x_0+z=(1,0)\) descends. Hence \(\Omega(x_0)\ne0\), but \(\epsilon_U=0\). This realizes FCD.14 by explicit Fredholm cycles. It proves why neither noncontinuity of a represented action nor failure of one maximal class settles normalized reduced class existence.

### 11CB.9. Actual commutative tests of every obstruction order

Let d be a nonnegative integer. Put Y=Z=S^1 and f_d(z)=z^d, with f_0(z)=1. Define the compact mapping cylinder

\[
X_d=(Y\times[0,1]\sqcup Z)/((z,1)\sim f_d(z)),
\qquad \iota:Y\hookrightarrow X_d,\quad \iota(z)=[z,0].
\tag{OFD.1}
\]

It is a compact Hausdorff finite CW space. A direct algebraic description is

\[
C(X_d)=\{(a,b)\in C([0,1]\times Y)\oplus C(Z):
a(1,z)=b(f_d(z))\}.
\tag{OFD.2}
\]

This also proves the restriction map Phi_d(a,b)=a(0,.) is a surjective *-homomorphism: for any c in C(Y), the pair ((1-t)c(z),0) is a continuous linear preimage. Positivity of this section at every matrix level follows from positivity of c and the nonnegative factor 1-t; its norm is at most ||c||. Thus this particular quotient is completely positively semisplit.

The retraction r:X_d -> Z is r([z,t])=f_d(z) and r(w)=w for w in Z. Its section j:Z -> X_d includes the top circle. The homotopy H_s([z,t])=[z,t+s(1-t)] fixes Z and retracts the cylinder to that section at s=1. This is well-defined on the glued top and continuous by the quotient topology. Hence j r is homotopic to the identity, r j is the identity, and the two pullbacks are inverse homotopy equivalences of C*-algebras. In particular the positive odd generator b_A of K_1(C(X_d)) is represented by the unitary r^*u, where u(w)=w on Z.

Restriction to Y gives the literal unitary z^d. Its winding is d, so

\[
K_1(C(X_d))=\mathbb Z b_A,\qquad
K_1(C(Y))=\mathbb Z b_Y,\qquad
(\Phi_d)_* b_A=d b_Y.
\tag{OFD.3}
\]

For d=0 the same formula gives zero: the restricted unitary is identically one.

The positively calibrated odd circle class has winding-one pairing +1. Its pullback through the even homotopy equivalence j^*:C(X_d)->C(Z) therefore gives x0 in KK^1(C(X_d),C) with beta_A(x0)=1. Conversely every odd scalar class of C(Y) has its index determined by one integer m. This follows directly from the already proved scalar circle Bott/inverse product and split evaluation extension, or from KT-KK-14's UCT: K_0 and K_1 of the circle are Z, the Ext term is zero, and the Hom map is an isomorphism. The same homotopy equivalence proves this classification for C(X_d), without assuming any arbitrary holonomy algebra satisfies the UCT.

Naturality now gives

\[
\{\langle (\Phi_d)_*b_A,y\rangle:y\in KK^1(C(Y),C)\}
=d\mathbb Z.
\tag{OFD.4}
\]

For d>=1, the new cone obstruction epsilon has exact order d, including order one when d=1. For d=0 its annihilator is zero, so epsilon has infinite order. For d=2 every target pairing is even although a full-source scalar class pairs b_A as +1. These are actual separable commutative quotients realizing the order alternatives. They are tests of the class criterion; they provide no foliation counterexample or general normal-geometry construction.

A direct cone calculation confirms the same orders. The odd contravariant Puppe segment is Z -> Z -> KK^1(C_{Phi_d},C) -> Z -> Z. Its first map is multiplication by d by OFD.3 and naturality. Its last map is identity, since the restriction is unital and both even scalar circle groups are classified by the unit index. Exactness therefore gives KK^1(C_{Phi_d},C)=Z/dZ, with d=0 interpreted as Z. The source's odd group is Z and beta_A is its identity coordinate, so ker beta_A=0. Hence the normalized obstruction is literally the generator Omega(x0) of this computed cone group, rather than merely an element of a further quotient. This realizes the finite and infinite orders by the actual cone exact sequence.

All classes in this test use the ordered odd Clifford convention. The retraction, restriction, cone projection and coefficient operations are even, so the positive winding-one calibration and final right Cl_1 block acquire no interchange sign.

![The cone obstruction for a specified class, its quotient by original-disk-invisible corrections, and the exact attainable pairing subgroup FCD.8–FCD.14. The lower row shows the actual commutative tests OFD.1–OFD.4; these are not foliation counterexamples.](../figures/normalized-cone-pairings.png)

**Figure 11CB.1.** The cone obstruction for a specified class, its quotient by original-disk-invisible corrections, and the exact attainable pairing subgroup FCD.8–FCD.14. The lower row shows the actual commutative tests OFD.1–OFD.4; these are not foliation counterexamples. [Reproduce the figures](../reproduction/general-reduced-graph/README.md).

## 11CC. Completed joint covariance on the original rank-seam suspension

On the actual period-one degree-three suspension, the constructions below retain the whole original jet module and all central tensor degrees. The finite averaged weight gives an exact covariant repair. The logarithmic replacement supplies a noninvariant comparison at simultaneous normal, seam and degree infinity. The reduced crossed-product homotopy and full physical creation equations retain the original inverse coefficient and disk pairing +1. This finite holonomy family has its stated scope.

### 11CC.2. Why rescaling the old operator alone does not repair NG.37

On a scalar auxiliary eigenline the old degree-one joint symbol is

\[
 A_b(\xi)=b^{-1}\sigma _1+b^2\sigma _3c_{\rm inv}(\xi).
 \tag{JCN.1}
\]

Multiplying **both** terms by a positive common scalar multiplies this matrix by that scalar. Its normalized sign is unchanged. Thus a common conformal rescaling of NG's already anisotropic symbol does not remove the ratio between its two energies. Making the normal coefficient proportional to the auxiliary coefficient would instead require changing the normal expression, with a new domain proof. No such change is silently inserted here.

There is a second relevant limit. Repairing the degree-one seam alone is insufficient: if the degree-r inverse uses a tensor product of noninvariant scalar weights, the transported ratio can be \(c^r\) at a fixed ordinary point. Choosing normal frequency of order \(b_r^{-3}\) then detects that ratio at degree infinity. The replacement below controls both seam and degree, without making an invariant weight a prerequisite.

### 11CC.3. The original whole module and new weights

Keep precisely NG.29–NG.30a:

\[
 T=(\mathbb R/\mathbb Z)^2\times(-2,2),\quad q=3,
 \qquad E=\Gamma _0(L)\oplus\Gamma _0(U,W),
\]
\[
 U=T^2\times((-2,-1)\cup(1,2)),\quad
 \tau(x,y,t)=(x+1/2,y,t),\quad
 G=(\mathbb Z/2)\ltimes T.
 \tag{JCN.2}
\]

This is the transversal holonomy of the actual connected suspension in NG.30a. L and W are its original smooth orthogonal rank-three jet blocks; both rank seams belong to T and to the arrow domain. The complete jet rank is three on the closed bump region and six on the open flat ends. Use the same period-one function

\[
 a(x)=(2+\cos(2\pi x))/3,
 \qquad \rho(s)=e^{-1/s^2}\ (s>0),\quad \rho(s)=0\ (s\le0).
\]

**Specified auxiliary connection.** Choose a smooth Hermitian connection on each of the ambient subbundles L and W and their orthogonal direct-sum connection \(\nabla^J=\nabla^L\oplus \nabla^W\). Thus \(\nabla^JP_L=\nabla^JP_W=0\), including on the flat ends. Its r-fold tensor connection preserves every L/W tensor-sector projection. This is an auxiliary connection choice on the given smooth ambient bundle; it does not assert an adjointable projection onto the changing germ module E. Couple it to the prescribed full inverse normal coefficient, with its original coherent inverse line, normal Clifford symbol and τ lift retained. In Section 11CC.7.1 one may average this split auxiliary connection and the compatible inverse-normal connection over τ: since τ preserves P_L/P_W, finite averaging preserves the split property. The inverse normal coefficient itself is never replaced by a determinant-only factor.

Let λ(t) be the positive, bounded completing function used in NG.31: it equals one near both rank seams, tends quadratically to zero at the omitted endpoints, and depends only on t. Its inverse-speed metric is complete. Unlike a universal invariant-weight requirement, this λ is an explicit choice in this particular suspension and is τ-invariant.

Define an ambient smooth contraction

\[
 h_0=P_L+\bigl(a(x)\rho(t-1)+\rho(-t-1)\bigr)P_W.
 \tag{JCN.3}
\]

For r≥1 let \(E_r=E^{\widehat\otimes_{C_0(T)}r}\), retaining the original central balance and all its fibres, including zero tensor components. Put

\[
 \varepsilon _r=(r+1)^{-2},\qquad
 f_r(p)=\exp\!\left[-\frac{\sqrt{1-\log p}-1}{r^2}\right]
 \ (0<p\le1),\qquad f_r(0)=0,
\]
\[
 b_r=\zeta\varepsilon _r\lambda\,f_r(h_0^{\otimes r}),
 \qquad 0<\zeta\le1.
 \tag{JCN.4}
\]

ζ is one fixed sufficiently small positive constant, chosen in Section 11CC.5. The functional calculus in JCN.4 is on the finite ambient tensor bundle, restricted to the actual completed tensor module. It is **not** NG.20's product weight or KWC's harmonic weight.

Here is an explicit, seam-valid way to interpret that calculus. A tensor sector with m W factors, 0≤m≤r, has upper-flat-end eigenvalue

\[
 b_{r,m}(x,t)=\zeta\varepsilon _r\lambda(t)
 \exp\!\left[-\frac{\sqrt{1+m((t-1)^{-2}-\log a(x))}-1}{r^2}\right].
 \tag{JCN.5}
\]

For m=0 this is ζε_rλ. For m≥1 it is extended by zero across t=1. On the lower end replace the logarithmic expression by \(m(-t-1)^{-2}\), with no x dependence. On the bump region only the all-L sector remains. The sector projections are the actual tensor products of P_L and P_W, not selected subspaces of the full jet module. Summing **all** sectors gives JCN.4.

For fixed r,m≥1 the upper coefficient is \(\exp(-\sqrt m/(r^2s))\) times a smooth positive function up to s=0, where s=t−1. It is flat at zero. Thus b_r is a smooth ambient matrix across the genuine seams, despite f_r itself not being smooth as a scalar function at p=0. Only its explicit composition in this geometry is asserted smooth. This distinction matters for arbitrary modules.

Each b_r is a positive compact endomorphism of E_r: every lost sector coefficient vanishes at its seam, and λ gives C0 decay at the actual omitted ends. Finite ambient matrix entries with these vanishing conditions are norm limits of compact supported sector rank ones. They lie in \(K(E_r)\). No mere fibre-compact assertion is used. The coefficient is strictly positive on every surviving sector; compact source truncation and truncation away from each sector boundary prove dense range on the completed module. The same argument gives injectivity. Consequently

\[
 L_r=b_r^{-1},\quad \operatorname{Dom}_{E_r}L_r=\operatorname{Ran}b_r,
 \quad (L_r\pm i)^{-1}=b_r(1\pm ib_r)^{-1},
\]
\[
 \operatorname{Dom}_{E_r}L_r^2=\operatorname{Ran}b_r^2,
 \qquad L_r\ge(\zeta\varepsilon _r)^{-1}.
 \tag{JCN.6}
\]

The completed module sum has exactly the vectors in these range domains for which the corresponding positive R-valued sum of squared images converges in norm; its square uses Ran b_r² and squared L_r² images. This is not the scalar weighted-integrability assertion below.

The normal estimates are uniform in r. Write \(v=\sqrt m/r^2\). Differentiating JCN.5 bounds its s derivative by

\[
 \zeta\varepsilon _r e\,v s^{-2}e^{-v/s}
 \le 4e^{-1}\zeta\varepsilon _r/v
 \le4e^{-1}\zeta\varepsilon _r r^2.
 \tag{JCN.7}
\]

For m=0 the derivative is zero before differentiating λ. Logarithmic x derivatives are bounded by \(C/r\). Differentiation of the tensor frames costs at most \(C r\|b_r\|\): in a smooth orthonormal block frame its connection matrix is the sum of r factor matrices, and differentiation of a diagonal multiplier adds its commutator with that matrix. There is no binomial sector-count factor in this operator norm estimate. Since ε_r r²≤1 and ε_r r≤1,

\[
 0\le b_r\le\zeta\varepsilon _r\lambda,
 \qquad \sup_r\|\nabla b_r\|_\infty\le C\zeta.
 \tag{JCN.8}
\]

All constants concern this fixed original geometry and connection. The expression C below therefore has completing speed; no missing endpoint is added.

### 11CC.4. Exact normal and joint domains; scalar compactness

Retain the full original inverse coefficient \(S_{\rm inv}\), its inverse central line and the final right Cl_3. The literal scalar localization map is multiplication of actual bundle sections,

\[
 E_r\widehat\otimes_R L^2(T,S_{\rm inv})
 \longrightarrow H_r=L^2(T,V_x^{\otimes r}\widehat\otimes S_{{\rm inv},x}),
 \qquad \xi\otimes u\longmapsto\xi(x)u(x).
 \tag{JCN.9}
\]

Its inner product is the represented R-valued tensor inner product. Countably many localized jet generators span the fibres; an L2 vector orthogonal to all their scalar tests is zero pointwise almost everywhere. Thus JCN.9 is onto and isometric on the completed balance, at both rank seams. No module projection onto E_r is asserted.

Let N_r be the original inverse normal Dirac, coupled to the full tensor connection in the even position before S_inv. On the finite ambient scalar bundle define \(C_r=b_r N_r b_r\). Compact-buffer Friedrichs smoothing and the cutoffs from λ give minimal=maximal exactly as in NG.8–NG.10: the symbol coefficient is b_r² and
\(\|[C_r,\eta_n]\|\le \sup \lambda|d\eta_n|\to 0\).
The ambient perpendicular field is killed by b_r and lies in ker C_r, so H_r reduces that self-adjoint ambient closure. Its exact restricted domain and the scalar inverse domain are

\[
 \operatorname{Dom}C_r=
 \{u\in H_r:b_rN_r(b_ru)\in H_r\text{ distributionally}\},
\]
\[
 \operatorname{Dom}_{H_r}L_r=
 \left\{u\in H_r:\int_T\|b_{r,x}^{-1}u(x)\|^2\,d\mathrm{vol}_g<\infty\right\}.
 \tag{JCN.10}
\]

They are actual Hilbert-space domains; they are not evaluations of Ran b_r. Let

\[
 D_r=\sigma _1L_r+\sigma _3C_r,
 \qquad Q_r=(\operatorname{Dom}L_r\cap\operatorname{Dom}C_r)^{\oplus2}.
 \tag{JCN.11}
\]

The completed mixed form is bounded because the regularizations \((b_r+\varepsilon )^{-1}\) have commutator
\(-b_r(b_r+\varepsilon )^{-1}[N_r,b_r](b_r+\varepsilon )^{-1}b_r\) with C_r. JCN.8 bounds it uniformly by \(K\le C\zeta \). Its strong limit on H_r is \(K_{0,r}=-p_r[N_r,b_r]p_r\); this is a bounded skew-adjoint form coefficient, not an assertion that all joint vectors belong to Dom L_rC_r. Vectors b_r²v with compact smooth ambient v prove density of Q_r. The graph estimate and onto-resolvent weak-limit argument yield

\[
 \big|\|D_ru\|^2-\|L_ru\|^2-\|C_ru\|^2\big|\le K\|u\|^2.
 \tag{JCN.12}
\]

D_r is self-adjoint on **exactly** Q_r. The same arguments work after replacing L_r by d_rL_r, if d_r is a uniformly bounded positive invertible even multiplier commuting with b_r and the principal Clifford coefficient, with
\([C_r,d_rL_r]=d_r K_{0,r}+b_r[N_r,d_r]\)
bounded as a form. In the present application d_r differs from one only in flat-end tensor sectors, so this displayed computation is literal. The regularization uses the positive inverse weight b_r/d_r and proves the same domain Q_r. A bounded cross term is sufficient; exact invariant weight is unnecessary.

For the square let S_r be the positive operator associated with the closed form
\(q_r(u,v)=\langle L_ru,L_rv\rangle +\langle C_ru,C_rv\rangle \) on Q_r. Set \(B_r=i\sigma_2K_{0,r}\). Then

\[
 \operatorname{Dom}D_r^2=\operatorname{Dom}S_r
 =\{u\in Q_r:D_ru\in Q_r\},\qquad D_r^2=S_r+B_r.
 \tag{JCN.13}
\]

Equivalently u∈Q_r is in Dom S_r exactly when there is w with \(q_r(v,u)=\langle v,w\rangle \) for every v∈Q_r. This is the completed form square, not an unproved intersection of Dom L_r² and Dom C_r². Replace L_r by d_rL_r for the corresponding square of the comparison operators.

The whole scalar normal operator is

\[
 \mathscr D=N_\lambda\oplus\bigoplus_{r\ge1}D_r,
 \qquad \mathscr H=H_N\oplus\bigoplus_{r\ge1}(H_r\oplus H_r).
 \tag{JCN.14}
\]

Its exact first domain is u_0∈Dom N_λ, u_r∈Q_r and
\(\|N_\lambda u_0\|^2+\sum _r(\|L_ru_r\|^2+\|C_ru_r\|^2)<\infty \).
The uniform JCN.12 proves equality with the operator direct-sum domain. Its exact second domain is u_0∈Dom N_λ², u_r∈Dom S_r and
\(\|N_\lambda ^2u_0\|^2+\sum _r\|D_r^2u_r\|^2<\infty \).
The spectral inequality t²≤1+t⁴ supplies membership in the first domain. Onto resolvents are the actual block resolvents on these completed domains.

For a compact source cutoff χ, a spectral cutoff φ_δ away from b_r=0 satisfies the literal distributional identity

\[
 N_r\phi_\delta(b_r)u
 ={\phi_\delta(b_r)\over b_r^2}C_ru
 +[N_r,{\phi_\delta(b_r)\over b_r}]b_ru.
 \tag{JCN.15}
\]

On a fixed compact buffer its smooth coefficients are bounded. It gives a finite-ambient H1 bound for \(\chi \phi _\delta (b_r)(D_r-z)^{-1}\); finite-coefficient Rellich makes this a scalar compact. The discarded operator has norm at most \(2\delta \|\chi \|\|L_r(D_r-z)^{-1}\|\). This proves compactness across the actual seams. The graph bound JCN.12 gives one uniform C_z for the latter norm, and

\[
 \|(D_r-z)^{-1}\|\le C_z\zeta\varepsilon _r\longrightarrow0.
 \tag{JCN.16}
\]

Finite full degrees plus this degree tail prove scalar source-local compactness for JCN.14. No H1 constant uniform in tensor rank, or fibrewise compactness, is substituted. The same proof treats the d_rL_r comparison operators with their uniformly bounded derivatives and lower bounds.

### 11CC.5. A quantitative sign comparison at joint infinity

The following elementary estimate is the analytic device needed at the seam; a bounded-potential argument alone would be insufficient.

Here is the precise theorem used by JCR.7–JCN.22. Let L≥0 be self-adjoint, let J=σ₁ be a self-adjoint unitary commuting with L, and let d₀,d₁ be bounded positive self-adjoint multipliers commuting with L and J. Assume d_i≥aI with a>0, \(e=d_1-d_0\) has norm at most ε, and the self-adjoint operators

\[
 A_i=Jd_iL+\sigma _3C,\qquad
 \operatorname{Dom}A_i=Q=\operatorname{Dom}L\cap\operatorname{Dom}C
 \tag{JCR.1}
\]

have gaps at least δ>0 and satisfy the joint quadratic-form estimate

\[
 \|d_iLu\|^2\le\|A_iu\|^2+K\|u\|^2
 \quad(u\in Q),\qquad K\ge0.
 \tag{JCR.2}
\]

The auxiliary doubling in Q is understood, as in JCN.11. All statements also hold on the entire Hilbert-space direct sum if these constants are uniform. They imply

\[
 L\le a^{-1}(A_i^2+K)^{1/2}
 \le a^{-1}(|A_i|+\sqrt K)
 \le k|A_i|,
 \qquad k=a^{-1}(1+\sqrt K/\delta)
 \tag{JCR.3}
\]

as closed forms. Indeed d_i commutes with L, so JCR.2 gives \(a^2L^2\le A_i^2+K\); square-root order gives the first inequality. The second is scalar functional calculus of |A_i|, and the third uses its gap. Thus \(L^{1/2}|A_i|^{-1/2}\) is bounded with norm at most √k. Since J and e commute with L, the mixed relative form

\[
 T=|A_1|^{-1/2}(A_1-A_0)|A_0|^{-1/2}
   =(L^{1/2}|A_1|^{-1/2})^*Je
       (L^{1/2}|A_0|^{-1/2})
 \tag{JCR.4}
\]

extends to a bounded operator of norm at most εk. This mixed estimate, rather than only a relative form at one parameter value, suffices.

Put P_i=1_(0,∞)(A_i). On the positive spectrum x of A₁ and the absolute value y of the negative spectrum of A₀, the operator \(X=P_1(1-P_0)\) solves the weak Sylvester equation

\[
 A_{1,+}X+X(-A_{0,-})=P_1(A_1-A_0)(1-P_0).
 \tag{JCR.5}
\]

The spectral variables satisfy x,y≥δ. In terms of the restricted form T, its bounded solution is the two-spectral-measure map with kernel

\[
 k_0(x,y)=\frac{\sqrt{x}\sqrt{y}}{x+y}
       =\int_0^\infty\sqrt{x}e^{-sx}\sqrt{y}e^{-sy}\,ds.
 \tag{JCR.6}
\]

This map has norm at most 1/2. To see this without discrete eigenvectors, let \(V_Av(s)=A^{1/2}e^{-sA}v\) for any positive self-adjoint A with gap δ. Spectral integration gives \(\|V_Av\|^2=\|v\|^2/2\). The map is exactly \(V_{A_{1,+}}^*(1\otimes T_{+,-})V_{-A_{0,-}}\), with norm at most \(\|T\|/2\). Differentiating scalar pairings of \(e^{-sA_{1,+}}Xe^{sA_{0,-}}\) on positive-time graph vectors proves JCR.5 and uniqueness: the terminal pairing vanishes exponentially; the integral at zero converges by the same two L² estimates. This also identifies its solution with the actual product of the spectral projections. The common domain Q makes the initial weak Sylvester identity legitimate. Spectral cutoffs, followed by the L² estimates, remove the temporary graph-vector assumption.

The reverse cross block \((1-P_1)P_0\) has the same bound. For orthogonal projections P,Q, squaring P−Q in the P⊕(1−P) decomposition gives

\[
 \|P-Q\|=\max\{\|P(1-Q)\|,\|(1-P)Q\|\}.
\]

Consequently

\[
 \|\operatorname{sgn}A_1-\operatorname{sgn}A_0\|
 =2\|P_1-P_0\|\le\varepsilon k.
 \tag{JCR.7}
\]

This proves JCR.7 directly at two endpoints. It controls normal frequency, seam and tensor-degree infinity together; no bound on L or C, no norm continuity of individual spectral projections in advance, and no exchange of an uncontrolled contour limit is required. For a nonaffine path with bounded commuting derivative, apply the same argument on subintervals and sum their multiplier differences. The flat diagonal d and d^(R,η) in this construction satisfy every additional commuting hypothesis.

In this construction all d_t and their approximants have \(a\le d_t\le A\) and uniformly bounded cross forms with K≤Cζ. Since L_r≥4/ζ, choose once and for all ζ>0 sufficiently small that

\[
 \delta^2=(4a/\zeta)^2-K\ge1.
 \tag{JCN.19}
\]

This gives the lemma's uniform gap, on the entire positive sum, including every comparison below. It is an ordinary quantitative choice, not an added orientation or holonomy hypothesis.

### 11CC.6. Transport ratios and compact holonomy defects on the full sum

On the upper-flat sector define \(c_{r,m}(x,t)=b_{r,m}(x+1/2,t)/b_{r,m}(x,t)\). It is one for m=0 and on the lower end. Write \(R_x=\sqrt{1+m(s^{-2}-\log a(x))}\). The exact identity and bounds are

\[
 \log c_{r,m}=(R_x-R_{x+1/2})/r^2,
\]
\[
 |\log c_{r,m}|\le {m\log3\over r^2(R_x+R_{x+1/2})}
 \le {\log3\over2r},
 \qquad
 |\log c_{r,m}|\le {\sqrt m\,s\log3\over2r^2}.
 \tag{JCN.20}
\]

The first bound controls tensor-degree infinity at **fixed original points**. The second controls seam infinity uniformly in all m≤r. Differentiation gives \(|\partial_x \log c|\le C/r\) and \(|\partial_s \log c|\le C\sqrt m/r^2\); this uses the same difference of reciprocal square roots, and is uniform for 0<s<1. The ratios extend as bounded positive invertible multipliers c_r of E_r. Their difference from one vanishes at the seam; their first ambient weak derivatives are bounded. Products with b_r² are smooth across it. Consequently c_r and its inverse preserve both completed domains in JCN.10 by the distributional product rule. No smooth module projection or nonexistent derivative of arbitrary E sections is required.

Actual inverse holonomy transport is the coherent lift from NG.30a; its central phase and its square are one in this framed example. Coupled normal transport has equal Clifford symbols. Any difference in auxiliary/normal connections is a finite zeroth-order coefficient; the r-factor auxiliary term has norm at most Cr on compact buffers, and multiplication by b_r on both sides bounds its full-sum norm by \(C r \varepsilon _r^2\). After an actual compact source cutoff this is a bounded localized potential. Its phase difference is scalar compact by the two-resolvent identity and JCN.15–JCN.16.

After this bounded connection correction, the literal transported positive block is

\[
 U_\tau D_rU_\tau^*
 =\sigma _1c_r^{-1}L_r+\sigma _3c_rC_rc_r,
 \qquad
 c_r^{-1}(U_\tau D_rU_\tau^*)c_r^{-1}
 =\sigma _1c_r^{-3}L_r+\sigma _3C_r.
 \tag{JCN.21}
\]

Both energy terms have been aligned simultaneously; the unbounded residual mass has not been discarded. Put \(d_r=c_r^{-3}\). Then \(d_r-1=O(1/r)\) uniformly and \(d_r-1=O(s/r^{3/2})\) at the seam, with uniform positive bounds and first derivatives.

Two separate comparisons establish compactness.

**Common positive rescaling.** On a compact arrow buffer extend \(\log c_r\) by an original source cutoff and use \(A_t=e^{tf/2}Ae^{tf/2}\) on the whole direct sum. Here f is an even diagonal endomorphism in flat tensor sectors. With the specified split auxiliary connection of Section 11CC.3, every sector projection P_I is parallel, so \(\nabla f=\sum _I df_I P_I\). Hence f commutes with the mass, the principal Clifford matrices and \([A,f]\); the latter equals the genuine bounded multiplication \(\sigma_3 b_r^2[N_r,f]\). This commutation would not follow from an arbitrary auxiliary connection that mixed those sectors.

The completed domain assertion is also literal. The scalar coefficients f_I have uniformly bounded weak first derivatives, including their zero extension at the seam, and are bounded. For u in the maximal distributional C_r domain, \(C_r(e^{tf/2}u)=e^{tf/2}C_ru+b_r^2[N_r,e^{tf/2}]u\) in distributions. The second term is bounded on the entire scalar sum; its coefficient is smooth after multiplication by b_r². The inverse multiplier satisfies the same identity. Both multipliers commute with L_r and preserve its weighted-integrability domain. Thus they map the exact intersection Q_r onto itself, with graph-norm bounds uniform in r and t. On that common domain \(A_t=e^{tf}A+(1/2)e^{tf}[A,tf]\). The bounded correction and uniform positive bounds prove equivalent completed graph norms, self-adjointness by the same maximal-domain/onto-resolvent argument, and preservation of JCN.15–JCN.16's local compactness.

The derivation of 11BO/DJ.17–DJ.18 now applies literally to this commuting endomorphism, giving a compact phase derivative: the spectral term is \(fA_t(1+A_t^2)^{-3/2}\), and the remaining two-resolvent terms have bound \(3\|[A_t,f]\|/(1+\tau ^2)\). Every integrand is scalar compact by JCN.15–JCN.16 and the compact source support of f. Thus positive common rescaling changes the phase only by a scalar compact after the buffer identification. Bounded transforms and signs differ by scalar source-local compacts, so this comparison applies to both. This statement is not made for an arbitrary matrix conformal factor.

**Residual mass.** Choose R and a smooth seam cutoff κ_η(s), zero for s≤η and one for s≥2η. For r≤R replace d_r by \(d_r^{R,\eta }=1+\kappa _\eta (d_r-1)\); for r>R set it to one. Also impose the compact arrow buffer cutoff. Then

\[
 \sup_r\|d_r-d_r^{R,\eta}\|
 \le C(R^{-1}+\eta).
 \tag{JCN.22}
\]

All first derivative/cross bounds remain uniform in R,η: the extra differentiated cutoff is bounded because \(|d_r-1|\le Cs/r^{3/2}\) on its transition annulus. The gap JCN.19 is preserved. For fixed R,η the residual mass \((d_r^{R,\eta }-1)L_r\) is a bounded, compact-source potential supported away from the seam in finitely many **whole** degrees. Its phase comparison is scalar compact. JCR.7 makes the discarded entire comparison norm at most \(C'(R^{-1}+\eta )\). Let R→∞ and η→0. Norm closure of scalar compacts proves the entire residual-mass phase comparison compact after the original source cutoff. This proves covariance at seam and degree infinity simultaneously; it is stronger than agreement in a fixed-column normal-frequency test.

Combine the two comparisons in JCN.21, the connection correction, and compact source commutators. For every actual compact arrow kernel k of JCN.2,

\[
 [\mathscr F,\Pi(k)]\in\mathcal K(\mathscr H),
 \quad \Pi(k)(1-\mathscr F^2)\in\mathcal K(\mathscr H),
 \qquad
 \mathscr F=F_{N_\lambda}\oplus\bigoplus_{r\ge1}\operatorname{sgn}D_r.
 \tag{JCN.23}
\]

The vacuum covariance is the original completed local-isometry normal argument. No new endpoint or arrow is used. All direct sums and every lost or retained tensor sector are present in JCN.23.

### 11CC.7. Actual reduced norm and full crossed-product class calibration

The representation in JCN.23 is the literal integrated covariant representation
\(\Pi (f_0+f_1u_\tau )=M_{f_0}+M_{f_1}U_\tau \)
on JCN.14, where the auxiliary transport on degree r is the actual r-fold jet transport, and the normal factor is the **full** inverse lift.

Reduced continuity is proved here, rather than inferred from phase compactness. For any finite-group covariant representation \((M,V)\) use the actual isometry

\[
 Jv=|F|^{-1/2}\sum_{h\in F}\delta_h\otimes V_h^{-1}v.
 \tag{JCN.24}
\]

The regular induced representation on \(\ell ^2(F)\otimes H\) acts by \(M(\alpha _{h^{-1}}f)\) in its h component and left translation on h. Substitution proves that J intertwines both actions. Decomposing M over original points puts each induced kernel in the original finite regular arrow matrix, tensored by the point's multiplicity. Its norm is bounded by \(\sup_x\|\lambda _x(k)\|=\|k\|_r\). Hence \(\|\Pi (k)\|\le \|k\|_r\). This proves JCN.23 is a cycle for the actual reduced algebra. No general amenability or reduced continuity of a bare normal representation is assumed outside this finite suspension.

The paired positive part has a stronger calibration than NG.28. Let \(J_+=\oplus _{r\ge 1}\sigma_1\) on the positive sum and \(S_+=\oplus _{r\ge 1}sgn D_r\). Their anticommutator is positive:

\[
 \{S_+,J_+\}\ge0.
 \tag{JCN.25}
\]

Here is a direct proof. σ_1 preserves each completed domain and \({D_r,\sigma_1}=2L_r\ge 0\) as a form. On the positive spectral subspace of D_r, put \(X=P_+\sigma_1P_+\). For w=e^{-tD_{r,+}}v,
\(-d_t<w,Xw\ge\langle D_rw,\sigma_1w\rangle +<\sigma_1w,D_rw\ge2\langle w,L_rw\rangle \ge 0\).
Integrating to infinity, using the gap, proves X≥0. On the negative spectral subspace the same calculation with −D_r gives \(P_-\sigma_1P_-\le 0\). The off-diagonal blocks cancel in \({sgn D_r,\sigma_1}\). Thus JCN.25 holds exactly and uniformly, without assuming that L_r commutes with C_r.

Consequently the bounded norm-continuous path

\[
 H_t=(1-t)S_++tJ_+,\qquad
 G_t=H_t(H_t^2)^{-1/2},\qquad 0\le t\le1,
\]
\[
 H_t^2\ge((1-t)^2+t^2)I\ge\tfrac12 I
 \tag{JCN.26}
\]

is one whole crossed-product homotopy. J_+ commutes with the entire actual representation, and S_+ has the scalar compact defects of JCN.23. Functional calculus in the represented algebra modulo scalar compacts therefore gives \([G_t,\Pi (k)]\) compact for every k, continuously in t. Each G_t is an exact odd involution. The endpoint J_+ is degenerate. No degreewise summation of indices or ordinary-only mass-at-infinity homotopy is used. Thus

\[
 [\mathscr H,\Pi,\mathscr F]
 =[H_N,\Pi_N,F_{N_\lambda}]
 \quad\text{\in }KK^3(C_r^*(G),\mathbb C).
 \tag{JCN.27}
\]

This is an actual class equality on the reduced groupoid algebra in JCN.2. The old NG phase was not a crossed-product cycle; no crossed-product equality with that failed phase is claimed. JCN.27 also does not identify this Fock carrier with the particular general anchored Θ. Such an identification is unnecessary for this explicit finite-group construction, and unproved for the general transfer route.

#### 11CC.7.1. An independent exact-covariance control construction

There is a simpler positive control on this **same** actual variable-rank suspension. Average NG.31's original compact smooth weight with its actual transported conjugate:

\[
 \overline h=(h+U_\tau hU_\tau^*)/2
 =\lambda P_L+
 \lambda\bigl(\tfrac23\rho(t-1)+\rho(-t-1)\bigr)P_W.
 \tag{JCN.27a}
\]

The equality uses the period-one identity \(a(x)+a(x+1/2)=4/3\). It is an actual ambient smooth compact on the whole E, not an average of abstract fibre matrices. Its two summands are compact and positive. Its kernel is zero because its quadratic form dominates h/2. Dense range on the completed module follows by compact sector truncations: on each retained compact set its scalar L and W coefficients have positive lower bounds, so every retained section is an actual image under the inverse coefficient; truncations converge in module norm. Alternatively the generator estimate NG.5 applies to the lower bound h/2. Zero orthogonal complement alone would not prove dense range for an arbitrary Hilbert module. This works at both seams and on both L and W.

Use the same invariant λ and average the finite Hermitian Clifford-compatible auxiliary and full inverse-normal connections with their actual coherent τ transports. The averaged connection remains Hermitian and Clifford-compatible, because connections with those properties form an affine space and τ preserves the metric and lift. It is τ-invariant. For each complete central tensor degree put \(\overline b_r=\varepsilon _r\overline h^{\otimes r}\), \(\overline L_r=\overline b_r^{-1}\) and \(\overline C_r=\overline b_rN_r\overline b_r\). The uniform derivative bound is \(\|\nabla\overline b_r\|\le \varepsilon _r r\|\nabla\overline h\|\); speed is bounded by λ. NG's exact first and form-square domains apply on the original full carrier.

Every term, distributional domain and onto resolvent is now **exactly** τ-covariant in every degree, including simultaneous degree/normal/seam limits. Original-source commutators and localized square defects are scalar compact by the same finite-degree and degree-tail arguments. JCN.24 proves reduced continuity. After scaling the weight by a fixed small constant if a uniform gap is desired, JCN.25–JCN.27 prove one whole reduced crossed-product class equality with the original vacuum, and JCN.28–JCN.29 supply the full normalized physical graph phase. The original-q disk pairing is +1.

This control construction uses finite orbit averaging in its stated scope. It neither demonstrates an invariant compact weight for an arbitrary germ groupoid nor turns that optional property into a general requirement. JCN.4 remains the noninvariant replacement, whose ratio proof makes explicit what exact averaging otherwise conceals about joint infinity. The exact control can be retained independently of an integration decision about the more general sign-comparison lemma.

### 11CC.8. Entire normalized physical graph and the native coefficient

Let E_Z be the original normalized counting/plaque correspondence for the specified suspension, from its physical reduced foliation graph algebra to \(C_r^*(G)\). Its full scalar carrier is literally

\[
 H_{\rm phys}=E_Z\widehat\otimes_{C_r^*(G)}\mathscr H.
 \tag{JCN.28}
\]

The physical tensor uses the entire E_Z, not a finite plaque corner. For a countable module Parseval frame \((z_j)\), its actual coefficient isometry W has entries \(\langle z_j,\xi \rangle \). With \(P=WW^*\) on \(\ell ^2\otimes \mathscr H\) the operator

\[
 F_{\rm phys}=P(1\otimes\mathscr F)P
 \tag{JCN.29}
\]

is a bounded operator on JCN.28 through that module isometry. For every physical creation T_ξ, its coefficient-column tails tend to zero in module norm; finite entries have the compact commutators from JCN.23. It follows that
\(F_phys T_\xi -T_\xi \mathscr F\) and the adjoint error are scalar compact. For a graph rank one \(T_\xi T_\eta ^*\), expanding its commutator and its square defect gives scalar compacts. The remaining term \(T_\xi (\mathscr F^2-1)\) is compact because its squared norm operator is
\((\mathscr F^2-1)\Pi (\langle \xi ,\xi \rangle )(\mathscr F^2-1)\), a scalar compact. Rank-one density gives the entire graph algebra. These creation equations identify JCN.29 with the **actual induced class** of JCN.27. They do not require an unbounded compression domain or a frame derivative.

On a compact smooth normal buffer and an actual finite-degree compact vector, oscillatory conjugation of D_r divided by frequency has limit \(\sigma_3 b_r^2 i[N_r,f]\). The scalar b_r² is positive on every surviving eigen-sector there and cancels in the normalized sign. Spectral cutoff away from zero followed by contraction/density proves the original inverse normal Clifford phase on each actual creation column. This is the same completed-resolvent argument as DJ.30–DJ.31a, now applied to JCN.11. It does not assert a uniform fixed-symbol limit at simultaneous auxiliary infinity; that separate limit is exactly what Section 11CC.6 proves. Thus JCN.29 retains the physical normal phase through the literal columns, with a proved entire class and scalar defects.

All auxiliary factors are even and precede the full inverse normal coefficient. With \(k_q=q(q-1)/2\), that coefficient remains \(\Pi ^{k_q}\mathrm{Cl}_q\), with physical generators \((-1)^{k_q+1}L_{t_j}\) and coherent inverse transition \(z^{-1}\rho _t(s)\). For q=3, k_q=3 and the physical generators are \(+L_{t_j}\). The right Cl_3 stays last until the native contraction. Neither tensor degree r, jet rank six nor the suspension's leaf dimension replaces q.

Restricting JCN.27 to the original unit algebra gives precisely the completed original inverse normal class, as a whole class. Its pairing with the prescribed positive disk b_U is +1 by the original inverse convention of 11BM. The same normalized counting/plaque correspondence takes that disk to JCN.28, so the scalar physical disk pairing is +1. This calibration is a class comparison, not an integer used to infer a transfer identity.

![The original q=3 rank seam, exact averaged coefficient 2/3, and noninvariant transport-ratio control at seam and tensor-degree infinity. The plotted ratios are scalar-sector samples. JCR.1–JCR.7 and JCN.22 prove the completed operator comparison, and JCN.25–JCN.29 prove the whole reduced class and original physical disk +1.](../figures/joint-covariant-rank-seam.png)

**Figure 11CC.1.** The original q=3 rank seam, exact averaged coefficient 2/3, and noninvariant transport-ratio control at seam and tensor-degree infinity. The plotted ratios are scalar-sector samples. JCR.1–JCR.7 and JCN.22 prove the completed operator comparison, and JCN.25–JCN.29 prove the whole reduced class and original physical disk +1. [Reproduce the figures](../reproduction/general-reduced-graph/README.md).

## 11CD. A regular tree factor preserves an original normal cycle

A covariant normal representation need not have the reduced crossed-product norm. An explicit regular tree factor repairs that norm. A bounded separator then repairs both compactness directions while retaining the supplied normal coefficient and its local class. The receiving normal cycle is an input to this theorem; its existence is not inferred from the norm repair.

We use the complete graded separator and product criterion in Composing graded correspondences, Lemma 0.1, Definition 0.3 and Theorems 0.4, 0.6 and 0.14. The elementary tensor compact rules are in Lemma 0.13 of that section. The complete tree and regular-representation proofs are in A normalized scalar graph cycle on a compact SU(2) suspension, equations SG.T1–SG.T7. We give all additional ideal, connection and localization checks below.

### 11CD.1. The exact receiving input

Let \(\Gamma=F_m\), \(2\le m<\infty\), act smoothly on an oriented \(q\)-dimensional transversal \(T\). The action is through its actual germs; when it presents a holonomy groupoid, no ineffective extra labels are inserted. Fix the entire original coherent normal orientation, including its inverse central line and ordered final right Clifford algebra \(D=\mathrm{Cl}_q\). Write \(\mathscr H_N\) for its graded Hilbert \(D\)-module. Its scalar Hilbert-space realization uses the fixed finite coefficient reduction of Theorem 7.17, not a new normal spinor chosen by parity alone.

Suppose \(\phi:C_0(T)\to\mathcal L_D(\mathscr H_N)\) is nondegenerate and even, \(U_g\) are its actual coherent even covariant unitaries, and \(F_N\) is an odd self-adjoint contraction such that

\[
 (F_N^2-1)\phi(f),\quad [F_N,\phi(f)],\quad
 \phi(f)(U_gF_NU_g^{-1}-F_N)\in\mathcal K_D(\mathscr H_N)
 \tag{RTP.1}
\]

for every \(f\in C_0(T)\), \(g\in\Gamma\). These are scalar localized defects after the fixed finite coefficient realization. They give the covariant cycle on the maximal crossed product. We do not assume that its representation on \(\mathscr H_N\) extends to the reduced crossed product.

The input may carry additional normal geometry: a specified Dirac phase on compactly supported high-frequency packets, and restriction class \(d_N=[\phi,F_N]\) with the original positive disk pairing \(b_Ud_N=1\). The conclusion retains that phase and exactly that restriction class. Conditions RTP.1 are explicit hypotheses even when a separate construction of such a normal cycle remains unfinished.

### 11CD.2. The whole regular point factor

Root the degree-\(2m\) Cayley tree at \(e\). For \(v\ne e\), let \(c(v)\) be the unoriented edge from \(v\) to its parent. Put

\[
 \mathscr H_T^+=\ell^2\Gamma,\qquad
 \mathscr H_T^-=\ell^2E,\qquad
 Q_0\delta_e=0,\quad Q_0\delta_v=\delta_{c(v)},
 \qquad F_T=\begin{pmatrix}0&Q_0^*\\Q_0&0\end{pmatrix}.
 \tag{RTP.2}
\]

The parent-edge map is a bijection from nonroot vertices to all edges. Thus \(F_T^2-1=-P_e\), where the root projection has positive grading, and \(\operatorname{Index}Q_0=1\). Let \(V_g\) translate all vertices and edges. The disagreement between the parent maps rooted at \(e\) and at \(g\) is confined to the finite geodesic joining those roots. Hence \([F_T,V_g]\) has finite rank. Each edge is uniquely \(\{v,vs_j\}\), for one positive free generator \(s_j\). The edge representation is therefore the sum of \(m\) regular representations; the vertex representation is one more regular representation. No root compression is made. The scalar tree class is the ordinary identity by the scalar classification, because its index is \(+1\). We need no assertion that it is the equivariant identity.

Order the completed module and operators as

\[
 \mathscr H=\mathscr H_T\widehat\otimes\mathscr H_N,
 \qquad P=\Gamma_T\otimes F_N,\qquad Q=F_T\otimes1,
 \tag{RTP.3}
\]

with the original \(D\) still on the right. They are odd self-adjoint contractions and \(PQ+QP=0\) exactly. Define

\[
 \Pi(f)=1\otimes\phi(f),\qquad
 \Pi(u_g)=V_g\otimes U_g.
 \tag{RTP.4}
\]

**Lemma 11CD.1 (the actual reduced source norm).** RTP.4 extends contractively to \(B_r=C_0(T)\rtimes_r\Gamma\).

**Proof.** On each of the \(m+1\) regular tree orbits, identify a column with \((\eta_h)_{h\in\Gamma}\) and use the even \(D\)-linear unitary

\[
 (W\eta)_h=U_h^{-1}\eta_h.
 \tag{RTP.5}
\]

Substitution gives \(W\Pi(u_g)W^{-1}=\lambda_g\otimes1\) and the coefficient at \(h\) equal to \(\phi(\alpha_{h^{-1}}(f))\), with \(\alpha_g(f)=f\circ g^{-1}\). This is the regular covariant representation induced by \(\phi\). To check its norm against the point definition of \(B_r\), take a finite crossed polynomial supported on \(S\subset\Gamma\), and an input column supported on finite \(L\subset\Gamma\). Its output lies in \(SL\). The rectangular coefficient matrix is a matrix of functions in \(C_0(T)\). At every \(x\in T\), its norm is at most the norm of the pointwise regular representation of the polynomial, hence at most \(\|a\|_{B_r}\). Applying the amplified star representation \(\phi\) to this matrix is contractive: square it to a positive square matrix and use the usual C*-norm and positive functional calculus. Thus its operator norm on the finite normal columns has the same bound. Density of finite columns, followed by density of crossed polynomials, proves the complete reduced norm estimate. A finite direct sum retains that bound. The unitary retains the entire normal coefficient and all actual source columns. \(\square\)

### 11CD.3. Two different compactness directions

Let \(K_N=\mathcal K_D(\mathscr H_N)\), \(K_T=\mathcal K(\mathscr H_T)\), and \(J=K_T\widehat\otimes K_N=\mathcal K_D(\mathscr H)\). Choose the separable unital graded algebras

\[
 C_T=C^*(1,V_g,F_T,\Gamma_T,K_T),\qquad
 C_N=C^*(1,U_g,F_N,\phi(C_0(T)),K_N),
\]

and the concretely represented spatial algebras

\[
 A_1=C_T\widehat\otimes K_N,\qquad
 A_2=K_T\widehat\otimes C_N.
 \tag{RTP.6}
\]

These are algebras in \(M(J)\), not asserted to be ideals of every bounded operator. Their product lies in \(J\): a separated product has one compact in each tensor direction, and finite sums followed by norm closure prove the assertion. Countable generation, separability and the finite right coefficient ensure that \(J\) is separable and sigma-unital.

For a finite crossed polynomial, RTP.1 gives

\[
 (P^2-1)\Pi(a),\ [P,\Pi(a)]\in A_1,
 \qquad
 (Q^2-1)\Pi(a),\ [Q,\Pi(a)]\in A_2.
 \tag{RTP.7}
\]

For the first commutator, \(V_g\) preserves tree grading, and its normal factor is precisely the compact coefficient-tested covariance defect in RTP.1 together with \([F_N,\phi(f)]\). For the second, \([F_T,V_g]\) has finite rank and \([F_T,1]=0\) for coefficients. The square errors are respectively \(1\otimes(F_N^2-1)\) and \(-P_e\otimes1\), multiplied by the displayed source. Norm closure and Lemma 11CD.1 extend RTP.7 to every \(a\in B_r\).

Every element of \(\Delta=\operatorname{span}(P,Q,\Pi(B_r))\) derives \(A_1\). On elementary tensors this follows because its tree factors belong to \(C_T\), whereas every bounded normal factor multiplies \(K_N\) on both sides. Density proves the assertion. Lemma 0.1 therefore supplies even positive contractions \(M,N\in M(J)\) with

\[
 M+N=1,\qquad MA_1\subset J,\qquad NA_2\subset J,
 \qquad[M,\Delta]\subset J.
 \tag{RTP.8}
\]

The same properties hold for square roots. Set

\[
 F_0=M^{1/2}P+N^{1/2}Q,\qquad F=(F_0+F_0^*)/2.
 \tag{RTP.9}
\]

**Theorem 11CD.2 (regularization with the original normal class).** RTP.9 is a bounded self-adjoint reduced cycle with the original normal degree and full coefficient. Its restriction to \(C_0(T)\) is \(d_N\).

**Proof.** The symmetrization changes \(F_0\) by an element of \(J\), by RTP.8. Modulo \(J\), the partition commutes with \(P,Q,\Pi(a)\). Their exact anticommutation then gives

\[
 (F^2-1)\Pi(a)\equiv
 M(P^2-1)\Pi(a)+N(Q^2-1)\Pi(a)=0,
\]

\[
 [F,\Pi(a)]\equiv M^{1/2}[P,\Pi(a)]
                 +N^{1/2}[Q,\Pi(a)]=0.
 \tag{RTP.10}
\]

These are the actual scalar compact defects on the entire reduced algebra, not a finite-level substitute. Self-adjointness is exact and boundedness gives the closed domain \(\mathscr H\).

For a homogeneous finite tree vector \(v\), let \(T_v:\mathscr H_N\to\mathscr H\), \(\eta\mapsto v\otimes\eta\). Since \(T_vT_v^*\in K_T\otimes1\subset A_2\), RTP.8 implies \(N^{1/2}T_v\) compact. One precise linking-algebra justification is that its product with its adjoint is compact, and the positive cutoff \(h_\epsilon(t)=t/(t+\epsilon)\) approximates the map with error at most \(\sup_{t\ge0}\sqrt t\,\epsilon/(t+\epsilon)\to0\). Also

\[
 (M^{1/2}-1)T_v=-N(1+M^{1/2})^{-1}T_v
 \quad\hbox{is compact}.
\]

The inverse commutes with \(N\), so this follows from the same compact map. Because \(PT_v=(-1)^{|v|}T_vF_N\), and \(QT_v=T_{F_Tv}\), we obtain

\[
 FT_v-(-1)^{|v|}T_vF_N\in\mathcal K_D(\mathscr H_N,\mathscr H).
 \tag{RTP.11}
\]

Its adjoint is compact as well. Multiplying a creation vector by \(f\in C_0(T)\) adds only \([F_N,\phi(f)]\), already compact. Finite tree and coefficient vectors are dense, and creation-map norms extend both connection equations to the entire first module \(\mathscr H_T\widehat\otimes C_0(T)\).

The first-cycle operator on that module is \(F_T\otimes1\), whose lift is \(Q\). Product positivity is

\[
 \{Q,F\}\equiv2N^{1/2}Q^2\ge0\pmod J.
 \tag{RTP.12}
\]

The factors commute in the quotient, making their positive product positive; every source sandwich remains positive. Equations RTP.11–RTP.12 are the full product criterion. Thus restriction of the reduced cycle is the external ordinary tree class followed by \(d_N\). The tree scalar class has index \(+1\), so Lemma 0.13 and Theorem 0.14 identify that external class with \(1_{C_0(T)}\). The restriction is exactly \(d_N\), rather than merely a class with the same numerical trace. \(\square\)

### 11CD.4. Phase, disk and graph passage

For any bounded weakly null normal packet sequence \(\eta_j\), RTP.11 implies

\[
 \|FT_v\eta_j-(-1)^{|v|}T_vF_N\eta_j\|\longrightarrow0.
 \tag{RTP.13}
\]

Consequently any supplied original normal Dirac phase is retained on every fixed finite auxiliary column. The original positive disk has pairing \(b_Ud_N=+1\) by the exact restriction identity. The full crossed-algebra compact defects RTP.10 control joint auxiliary and normal infinity. We have not removed the auxiliary columns or declared their whole equivariant class to be the unit. In particular the positive root is not an invariant subspace, so its ordinary index is not an equivariant decomposition.

When \(\Gamma\ltimes T\) is the actual complete-transversal groupoid of a foliation, let \({}_{A_r}E_Z{}_{B_r}\) be its full regular counting/plaque correspondence. The entire physical carrier is \(E_Z\widehat\otimes_{B_r}\mathscr H\). A module Parseval frame embeds it as a projection corner of \(\ell^2\otimes\mathscr H\). Compress \(1\otimes F\) by that projection. Finite creation columns have compact errors by RTP.10; Parseval tail approximation in module norm extends them to every creation vector. The corresponding square-defect and commutator expansions on graph rank-one operators are finite sums of compact creation errors, so rank-one density proves scalar compact defects on the entire graph algebra. This is the bounded graph passage proved in the physical-carrier argument of Completed joint covariance on the original rank-seam suspension; it uses only the verified cycle and full correspondence, not the finite suspension's special weights. The creation equations also retain RTP.13. Actual germs, isotropy, normalized counting and plaque densities remain in \(E_Z\).

This theorem supplies a sufficient regular point construction for free-group presentations with the stated receiving input. It does not supply that input for every nonisometric holonomy groupoid, and it does not establish a local finite-order differential expression for the separator. Neither metric-lift amenability nor an equivariant-unit identity is a necessary condition in its statement.

![The original ordered module, two different compactness directions and complete separator RTP.3–RTP.12. The creation comparison RTP.11–RTP.13 retains every supplied normal phase and the original disk, while the entire reduced algebra controls joint infinity.](../figures/regular-tree-normal-separator.png)

**Figure 11CD.1.** The original ordered module, two different compactness directions and complete separator RTP.3–RTP.12. The creation comparison RTP.11–RTP.13 retains every supplied normal phase and the original disk, while the entire reduced algebra controls joint infinity. [Editable SVG](../figures/regular-tree-normal-separator.svg) and [reproduction instructions](../reproduction/regular-source/README.md).


## 11CE. Nonamenable metric isotropy with a normalized graph cycle

The metric lift is a useful place to construct an invariant symbol. Its amenability is a sufficient reduced-norm route in eligible cases. It cannot be a universal requirement: the following actual compact suspension has nonamenable metric isotropy and nevertheless has a normalized reduced scalar graph cycle. Both assertions are proved for the same original holonomy groupoid and original \(q=3\).

### 11CE.1. Freely generated actual rotation germs

Put \(s=\sqrt2\) and define

\[
 A=\frac13\begin{pmatrix}3&0&0\\0&1&-2s\\0&2s&1\end{pmatrix},
 \qquad
 B=\frac13\begin{pmatrix}1&0&2s\\0&3&0\\-2s&0&1\end{pmatrix}.
 \tag{MI.1}
\]

They are positive rotations about the first and second axes, with cosine \(1/3\) and sine \(2\sqrt2/3\).

**Lemma 11CE.1 (the all-length free-pair proof).** The subgroup generated by \(A,B\) is the free group on these two generators.

**Proof.** Reduce the numerator matrices in the field \(\mathbb F_9=\mathbb F_3[s]/(s^2+1)\). The polynomial is irreducible because \(-1\) is not a square in \(\mathbb F_3\). For a letter with sign \(\epsilon\in\{1,-1\}\), its numerator has the following rank-one reduction:

\[
 3A^\epsilon=u_{A,\epsilon}v_{A,\epsilon}^{,t},\quad
 u_{A,\epsilon}=(0,1,-\epsilon s)^t,\quad
 v_{A,\epsilon}=(0,1,\epsilon s)^t,
\]

\[
 3B^\epsilon=u_{B,\epsilon}v_{B,\epsilon}^{,t},\quad
 u_{B,\epsilon}=(1,0,\epsilon s)^t,\quad
 v_{B,\epsilon}=(1,0,-\epsilon s)^t.
 \tag{MI.2}
\]

Equal-axis consecutive letters have inner factor \(1+\epsilon\epsilon'\), which vanishes exactly for inverse letters. Different-axis consecutive letters have inner factor \(-\epsilon\epsilon'\), which never vanishes. Therefore, for any nonempty reduced word \(w=C_1\cdots C_k\),

\[
 (3C_1)\cdots(3C_k)
 =u_1\left(\prod_{j=1}^{k-1}v_j^tu_{j+1}\right)v_k^t\ne0
 \quad\hbox{in }M_3(\mathbb F_9).
 \tag{MI.3}
\]

If \(w\) were the identity real matrix, its numerator over \(\mathbb Z[\sqrt2]\) would be \(3^kI\), whose reduction is zero. This contradicts MI.3. Hence no nonempty reduced word is the identity. \(\square\)

The coherent lifts in \(\mathrm{Spin}(3)=SU(2)\) are

\[
 a=(\sqrt2+i)/\sqrt3,\qquad b=(\sqrt2+j)/\sqrt3.
 \tag{MI.4}
\]

Their adjoint actions on imaginary quaternions are precisely MI.1. The free-group homomorphism \(\rho:F_2\to SU(2)\), defined by these generators, retains all lift products; it is faithful because any element in its kernel would also have trivial rotation matrix. In fact no nonidentity word maps to the central element \(-1\), by the same argument applied to its adjoint action.

### 11CE.2. The compact foliation and its metric lift

Let \(N=SU(2)\) and let \(g\in\Gamma=F_2\) act by

\[
 n\longmapsto\rho(g)n\rho(g)^{-1}.
 \tag{MI.5}
\]

Choose the genus-two closed surface and its connected regular \(\Gamma\)-cover \(\Sigma_0\) as in SG.1: the two positive generators map to the free generators and the two remaining surface generators map to the identity. Define

\[
 V=(\Sigma_0\times N)/\Gamma,
 \qquad g(x,n)=(gx,\rho(g)n\rho(g)^{-1}),
 \qquad B_r=C(N)\rtimes_r\Gamma.
 \tag{MI.6}
\]

The deck action is free and proper; its diagonal action remains free and proper. Covering charts make \(V\) a smooth compact connected bundle over the closed surface, with fibre \(N\). Its horizontal foliation has a complete transversal \(N\).

Every group word gives a distinct actual germ. Indeed an isometry of a connected Riemannian manifold that is the identity on an open set is the identity everywhere: its derivative and value at a point determine its action on a normal geodesic ball, and overlapping such balls along a path extend the identity. For MI.5, identity everywhere would force its rotation on imaginary quaternions to be the identity, contrary to Lemma 11CE.1 for a nonidentity word. Thus the actual transverse holonomy groupoid is \(\Gamma\ltimes N\), including its full isotropy; it is not an action groupoid with ineffective extra labels.

Give \(N\) its positive bi-invariant metric and oriented left-invariant frame. The derivative of MI.5 in that frame is \(\operatorname{Ad}\rho(g)\), and the coherent spin lift is the entire \(\rho(g)\). We may also prescribe any central character \(\chi:\Gamma\to U(1)\) in the normal Spin-c lift. Its inverse coefficient carries exactly \(\chi(g)^{-1}\rho_t(\rho(g))\), with the complete inverse normal grading and ordered right \(\mathrm{Cl}_3\). This statement retains the whole supplied lift, including a nontrivial central character; it does not specify the orientation by a determinant alone.

Let \(W\to N\) be the bundle of positive normal metrics with action \(g\cdot G=\operatorname{Ad}\rho(g)^{-t}G\operatorname{Ad}\rho(g)^{-1}\), interpreted between the actual tangent fibres. At the fixed unit quaternion \(n=1\), take the bi-invariant metric \(I\). Every rotation fixes this metric, so

\[
 \operatorname{Iso}_{\Gamma\ltimes W}(1,I)=\Gamma.
 \tag{MI.7}
\]

Its range fibre at \((1,I)\) is exactly this isotropy group, since every inverse group element fixes the same unit.

### 11CE.3. A complete local obstruction to metric-lift amenability

**Lemma 11CE.2 (the four-neighbour norm bound).** On \(\ell^2F_2\),

\[
 \left\|\frac{\lambda_a+\lambda_{a^{-1}}+
                       \lambda_b+\lambda_{b^{-1}}}{4}\right\|
 \le\frac{\sqrt3}{2}<1.
 \tag{MI.8}
\]

**Proof.** Orient every tree edge away from the root. For \(t=1/\sqrt3\), each edge from parent \(p\) to child \(c\) satisfies

\[
 2|u_pu_c|\le t|u_p|^2+t^{-1}|u_c|^2.
\]

Sum over all edges for a finite vector. Every nonroot vertex is the parent of three edges and the child of one, giving coefficient \(3t+t^{-1}=2\sqrt3\). The root has four children and no parent, giving \(4t\le2\sqrt3\). Hence the absolute quadratic form of the self-adjoint adjacency operator is at most \(2\sqrt3\|u\|^2\). Density gives the operator norm bound and division by four proves MI.8. \(\square\)

An amenability certificate for the lifted groupoid would, at the unit MI.7, supply normalized vectors \(\zeta_j\in\ell^2\Gamma\) with \(\|\lambda_s\zeta_j-\zeta_j\|\to0\) for its four generators. Their averaged quadratic form would tend to one. MI.8 bounds its absolute value by \(\sqrt3/2\), a contradiction. Thus this actual metric lift is nonamenable. This argument uses the literal isotropy at one unit and does not confuse a nonamenable acting group with an arbitrary action's amenability.

### 11CE.4. The reduced cycle on the same original graph

**Theorem 11CE.3 (a normalized graph cycle despite MI.8).** The foliation MI.6 has a reduced scalar graph cycle of original normal degree three and original local positive disk pairing \(+1\), retaining the full prescribed inverse lift.

**Proof.** On \(N\), use the inverse spin Dirac \(D_N\) for the bi-invariant metric, with the native rank-three convention in Theorem 7.17 and SG.2. In the physical odd model its principal part is \(-i\sum_{j=1}^3\sigma_jX_j\). The invariant Clifford connection adds its compatible zeroth-order term. Conjugation is an isometry with the whole lift MI.4; the inverse central character is constant on \(N\). Hence

\[
 (U_g\psi)(n)=\chi(g)^{-1}\rho(g)
                      \psi(\rho(g)^{-1}n\rho(g)),
 \qquad U_gD_NU_g^{-1}=D_N.
 \tag{MI.9}
\]

The compact manifold Dirac domain, compact resolvent and smooth coefficient commutators are those proved in SG.2–SG.3 and the complete normal calculus. The full lift is now nontrivial, while those analytic arguments are unchanged: only the isometric covariance and connection preservation are used. The native inverse and disk locality in Theorems 7.17–7.20 give \(b_U[D_N]=+1\) for every original positively oriented small three-disk. An inverse central line is locally trivialized with its full transport; no global line cancellation is assumed.

The tree RTP.2 gives the reduced representation by Lemma 11CD.1, and RTP.9 already gives a bounded cycle with exact original restriction class. There is also an explicit closed unbounded model in this example. Retain every vertex and edge and give its pair at distance \(k\ge1\) the matrix \(k\sigma_x\); put zero on the positive root. Call the completed self-adjoint direct sum \(D_T\). Its exact domain is the square-summability domain of these block images. There are \(4\cdot3^{k-1}\) vertices at distance \(k\), so its resolvent tail beyond \(R\) has norm at most \((1+(R+1)^2)^{-1}\). Its return commutators satisfy the complete bound SG.T5, because only the finite root-to-return segment changes the parent map.

On the ordinary physical rank-three model set

\[
 D=D_N\otimes\Gamma_T+1\otimes D_T,
 \qquad D^2=D_N^2\otimes1+1\otimes D_T^2.
 \tag{MI.10}
\]

For each normal eigenvalue \(\nu\) and each nonroot tree pair at distance \(k\), the completed operator is the self-adjoint matrix \(\begin{pmatrix}\nu&k\\k&-\nu\end{pmatrix}\); the root block is \(\nu\). Its full domain is the square-summability domain of all those block images, equivalently the intersection of the two factor domains. Finite block sums are a core. For each spectral bound only finitely many normal eigenvalues and tree blocks occur, so the total resolvent is compact. Smooth coefficient commutators are bounded by the normal gradient, and return commutators are bounded by SG.T5 using MI.9. The norm-convergent compact commutator integral SG.T10 proves both defects of \(D(1+D^2)^{-1/2}\) on the entire reduced crossed product.

The exact normal creation columns and positive anticommutator SG.T11–SG.T12 identify restriction as the ordinary tree index-one class times \([D_N]\). Thus its original disk pairing is \(+1\), not an auxiliary-frame disk pairing. Only the even Clifford pairs are contracted; the final right odd generator and the full inverse \(\chi^{-1}\rho_t\) are those of MI.9 and SG.2.

Finally take the full counting/plaque correspondence for MI.6. The finite covering-chart Parseval construction SG.4–SG.13 applies to this compact suspension: its coefficients are smooth functions on \(N\) and finitely many actual return words, whose commutators were just bounded. Its diagonalization is a bounded perturbation of the free completed operator and gives the precise physical corner domain; its localized resolvents and compact phase commutators are the entire scalar graph defects. Fixed isotropy at \(n=1\) changes no covering-chart argument and every isotropy word is retained in the correspondence. This proves the unbounded model, bounded phase, full graph passage and original normalization on the same foliation whose metric lift failed amenability. \(\square\)

Metric-lift amenability is therefore a sufficient route in its appropriate scope, not a universal publication or construction gate. The explicit normal model here is isometric; the arbitrary nonisometric receiving construction and arbitrary-holonomy reduced graph problem remain further work.

![Exact tangent vectors of the actual rotation germs MI.1–MI.4 at the fixed unit quaternion. The displayed tree is its radius-two ball; all levels remain in the completed cycle. MI.7–MI.8 rule out amenability of the actual metric lift, while MI.9–MI.10 and Theorem 11CE.3 construct the normalized reduced class on the same entire physical graph.](../figures/nonamenable-metric-isotropy.png)

**Figure 11CE.1.** Exact tangent vectors of the actual rotation germs MI.1–MI.4 at the fixed unit quaternion. The displayed tree is its radius-two ball; all levels remain in the completed cycle. MI.7–MI.8 rule out amenability of the actual metric lift, while MI.9–MI.10 and Theorem 11CE.3 construct the normalized reduced class on the same entire physical graph. [Editable SVG](../figures/nonamenable-metric-isotropy.svg) and [reproduction instructions](../reproduction/regular-source/README.md).


## 11CF. The complete scalar metric-fibre receiving phase

The shear estimates MB.12–MB.17 can be quantized on the whole metric bundle. We prove the needed operator bounds, compactness and coordinate changes, then assemble the actual maximal scalar cycle. The historical construction is treated by [Hilsum–Skandalis, Appendix A.7.5](https://www.numdam.org/article/ASENS_1987_4_20_3_325_0.pdf#page=56). The proof here supplies its analytic inputs for the specific inverse symbol, rather than invoking a global symbol theorem.

Keep the actual germ groupoid \(H\), original rank \(q\), metric bundle \(p:W\to T\), vertical rank \(d=q(q+1)/2\), and \(n=q+d\) of Metric-fibre Bott transfer, shear and balanced reduced norm. Its entire inverse coefficient is the rank-\(n\) module of Theorem 7.17, with the full coherent lift specified in MB.4 and the original-quotient-first orientation order of MB.1–MB.13. Throughout, localizations are compact subsets of actual chart and arrow domains. The estimates below concern \(q\ge1\). At rank zero the transversal is discrete, its compact unit tests are finite, and the zero phase on the entire supplied inverse graded line has scalar finite-rank source defects. Pairing it with that same supplied positive graded point class gives +1; an unrelated bare even line is not substituted.

### 11CF.1. An elementary local operator norm

Use left quantization

\[
 \operatorname{Op}(a)u(x)=(2\pi)^{-n}
          \int e^{ix\xi}a(x,\xi)\widehat u(\xi)\,d\xi.
 \tag{MGP.1}
\]

Finite coefficient matrices and right Clifford linearity are retained. On a fixed compact spatial support, write \(a\in S^0_{\rho,\delta}\) if

\[
 \|\partial_x^\beta\partial_\xi^\alpha a(x,\xi)\|
 \le C_{\alpha\beta}\langle\xi\rangle^{-\rho|\alpha|+\delta|\beta|}.
 \tag{MGP.2}
\]

First suppose every derivative is bounded. Choose a normalized Gaussian \(g\), and set \(g_{x,\xi}(t)=e^{it\xi}g(t-x)\). Plancherel and integration in \(x\) give the tight-frame identity

\[
 \int|\langle g_{x,\xi},u\rangle|^2\,
                dx\,d\xi/(2\pi)^n=\|u\|^2.
 \tag{MGP.3}
\]

The frame kernel of MGP.1, up to a scalar unit, is

\[
 \int e^{is(\xi'-\xi)}e^{i\nu(x-x')}
      e^{is\nu}\overline{g(s)}\,
      a(x+s,\xi'+\nu)\widehat g(\nu)\,ds\,d\nu.
 \tag{MGP.4}
\]

Integrate by \((1-\Delta_s)^N\) and \((1-\Delta_\nu)^N\). Each differentiated factor is a bounded derivative of \(a\) times a polynomial times Gaussians. Its absolute integral is uniformly bounded. Thus the kernel is bounded by

\[
 C_N(1+|x-x'|^2)^{-N}(1+|\xi-\xi'|^2)^{-N}.
 \tag{MGP.5}
\]

For \(N>n/2\), both Schur integrals are bounded. Apply MGP.3 on both sides to get the operator bound by finitely many bounded derivative seminorms. Frequency truncation justifies the calculation on Schwartz vectors; its uniform bound passes to the limit and gives the unique bounded extension. This proves the required bounded-derivative theorem directly.

Now assume \(0\le\delta\le\rho<1\). Cut frequency into dyadic shells \(R=2^j\ge1\), and each shell into a smooth square partition of cubes of side \(R^\rho\). At cube centre \(\xi_c\), modulation and the unitary dilation \(x=R^{-\rho}s\), \(\xi=\xi_c+R^\rho\nu\), give a symbol with uniformly bounded derivatives: frequency derivatives cost \(R^\rho\), spatial derivatives cost \(R^{-\rho}\), and MGP.2 bounds the remaining factor by \(R^{(\delta-\rho)|\beta|}\le1\). The preceding frame proof applies uniformly.

For output cube \(l\) and input cube \(k\), commute powers of \((D-\xi_c)/R^\rho\) through this quantization. The product rule gives bounded rescaled symbol derivatives and bounded input-cube frequency powers. Multiplying on the output by the inverse of those powers therefore gives the block bound \(C_N(1+|l-k|)^{-N}\). The frequency square partition is a Hilbert-space isometry; matrix Schur summation for \(N>n\) bounds the entire shell independently of the number of cubes.

For separated input and output shells \(R,S\), integration by parts in the compact spatial variable gives

\[
 |\widehat a_R(\eta-\xi,\xi)|
       \le C_N R^{\delta N}(1+|\eta-\xi|)^{-N}.
\]

Since \(|\eta-\xi|\ge c\max(R,S)\) for separated shells, the two Schur integrals give block norm at most

\[
 C_N R^{\delta N+n/2}S^{n/2}\max(R,S)^{-N}.
 \tag{MGP.6}
\]

For \(S\ge4R\), rewrite this as \(R^{-(1-\delta)N+n}(S/R)^{n/2-N}\). For \(R\ge4S\), use \(S\ge1\) to bound it by \((R/S)^{-(1-\delta)N+n/2}\). Choosing \(N\) large gives any prescribed summable decay in the dyadic index difference. Neighbouring shells use the cube bound. Matrix Schur summation proves the complete bounded operator estimate. The finite low-frequency piece is covered by MGP.3–MGP.5. This proof also covers \(\delta=\rho<1\), which will be useful for interpolation below.

### 11CF.2. Compact symbols, including the actual shear error

If MGP.2 has an additional factor \(\langle\xi\rangle^{-\epsilon}\), \(\epsilon>0\), the same proof bounds its high-frequency operator tail by \(C R^{-\epsilon}\). Its bounded-frequency, compact-spatial kernel is Hilbert–Schmidt. Norm approximation therefore proves scalar compactness.

There is a stronger consequence needed for MB.16. Suppose an order-zero symbol satisfies MGP.2 and

\[
 \epsilon_R=\sup_{x,\,R/2\le\langle\xi\rangle\le2R}
                     \|a(x,\xi)\|\longrightarrow0.
 \tag{MGP.7}
\]

In each rescaled cube all higher derivatives remain bounded. For any fixed derivative order \(k<L\), smoothing at scale \(h\) with a kernel whose nonconstant moments through order \(L-1\) vanish gives

\[
 \|\partial^k a_R\|_\infty
 \le C(\epsilon_Rh^{-k}+h^{L-k})
 \le C'\epsilon_R^{1-k/L},\qquad h=\epsilon_R^{1/L}.
 \tag{MGP.8}
\]

A smooth kernel with these finite moment conditions is obtained by a finite linear combination of dilates of one bump, solving the finite moment equations; its constant moment is one. Taylor's integral remainder proves the first inequality. Apply MGP.8 in all rescaled spatial and frequency variables, choosing \(L\) above the finitely many orders used in the frame, cube and off-shell estimates. The resulting estimates are small seminorms of type \((\rho,\rho)\). MGP.6 still has summable decay because \(\rho<1\). Consequently the high-frequency operator norm tends to zero. The low-frequency Hilbert–Schmidt argument proves compactness. This does not infer derivative decay from pointwise decay without the interpolation step.

The symbol difference for a genuine splitting shear is of type \((3/4,1/4)\) by MB.17 and satisfies MGP.7 by MB.16, uniformly on each retained compact arrow test. It therefore gives an actual scalar compact operator, on both cotangent axes and on mixed rays.

### 11CF.3. Two-point amplitudes and nonlinear coordinate changes

For a compactly supported two-point amplitude \(A(x,y,\xi)\) with the analogue of MGP.2 in both base variables, its kernel has phase \((x-y)\xi\). Taylor expansion in \(y-x\), followed by one frequency integration by parts, shows that its left symbol differs from \(A(x,x,\xi)\) by order \(-\kappa\), where

\[
 \kappa=\rho-\delta>0.
 \tag{MGP.9}
\]

Here are the remainder estimates. In the reduction integral with phase \(-v\theta\), one Taylor remainder carries the combined derivative \(\partial_\xi\partial_y A\), giving \(R^{-\rho+\delta}\). On \(|\theta|\le R/2\), set \(\theta=R^\rho\nu\), \(v=R^{-\rho}w\); the measure is unchanged. Every scaled amplitude derivative is bounded by that same order. Repeated integration by \((1-\Delta_w)^N\) and \((1-\Delta_\nu)^N\) inserts integrable weights in both variables. Thus the oscillatory integral and every prescribed target derivative satisfy MGP.2 with the extra factor \(R^{-\kappa}\). Take target derivatives before rescaling; cutoff derivatives cost \(R^{-1}\le R^{-\rho}\). On \(|\theta|>R/2\), integration in the base variable gives arbitrarily many factors \(|\theta|^{-1}R^\delta\). Since \(\delta<1\), increasing the integration order bounds this region and all prescribed derivatives by any negative power of \(R\), including the portion where \(\xi+\theta\) is small. This proves the remainder assertion, not just its formal Taylor expression.

The same integral for a product has amplitude \(a(x,\xi+\theta)b(x+v,\xi)\). Its leading term is \(ab\); the remainder is again order \(-\kappa\). Adjoint reduction has leading symbol \(a^*\) and the same remainder. A commutator with scalar smooth multiplication has order \(-\rho\), because the differentiated multiplier contributes no frequency growth. These negative-order remainders are compact by Section 11CF.2. Away from the diagonal, repeated frequency integration makes the kernel smooth; on compact rectangles it is Hilbert–Schmidt. Such terms handle chart cutoff changes.

For a genuine diffeomorphism \(\psi\), use the exact identity

\[
 \psi(x)-\psi(y)=J(x,y)(x-y),\qquad
 J(x,y)=\int_0^1D\psi(y+t(x-y))\,dt.
 \tag{MGP.10}
\]

Near the diagonal \(J\) is invertible. Substitute \(\zeta=J(x,y)^t\xi\) in the transformed kernel. The exact amplitude consists of the smooth density, Jacobian and coefficient transports times \(a(\psi(x),J(x,y)^{-t}\zeta)\). A base derivative of its frequency argument costs \(|\zeta|\partial_\xi\), hence at most \(1-\rho\). For \(\rho=3/4\), \(\delta=1/4=1-\rho\), this is exactly the permitted base loss. The amplitude is of the same type; its diagonal is the actual cotangent-transformed and coefficient-conjugated symbol. MGP.9 gives a remainder of order \(-1/2\). Away from this diagonal neighbourhood the smooth kernel argument applies. This proves coordinate covariance for nonlinear germs, including their actual second jets. It does not assume that a nonlinear change is an isometry.

### 11CF.4. Strict coefficients and invariant volume

Choose any smooth splitting of MB.3. Give \(TW\) the sum of the invariant vertical metric and the tautological quotient metric. Its volume density \(\mu_W\) does not depend on the splitting: changing splitting is a triangular map with determinant one. An actual germ derivative has isometric diagonal blocks and the same triangular shear, so it preserves \(\mu_W\).

On the full inverse coefficient, use the strict diagonal sum of the original compact-reduced lift and its symmetric-square vertical lift. Both compose under actual arrows, so their entire inverse tensor transport composes exactly. Its central factor is \(z^{-(q+1)}\), and its rank-\(n\) grading and physical generators are those of MB.4 and Theorem 7.17. The full original lift over the metric fibre is obtained by letting its actual \(GL_q^+\) map act between the two tautological metrics. No invariant metric on the original \(T\) is selected. In particular we do not use the polar part of the full triangular derivative as a group action; polar parts need not compose.

Actual pullback with these unitary coefficient transports defines unitary maps between the retained open-set \(L^2(W,I_n;\mu_W)\) spaces. Counting convolution and change of variables in the invariant density prove its product and adjoint identities. Weighted Cauchy–Schwarz on each range sum, followed by that same counting interchange, gives \(\|\pi(k)u\|^2\le(\sup_y\sum_{h:r h=y}|k(h)|)(\sup_x\sum_{h:s h=x}|k(h)|)\|u\|^2\). Thus this is an I-norm bounded star representation and extends to the maximal algebra of \(H_W\). This supplies the representation before choosing the phase, and says nothing yet about its reduced norm.

### 11CF.5. A bounded global phase with all scalar defects

Quantize MB.13 in each precompact chart, cut off on both sides within that chart, and symmetrize to an odd self-adjoint \(Q_i\). Its principal matrix has norm at most one. This implies essential norm at most one by a direct positivity argument: for every \(\lambda>1\), quantize the smooth uniformly positive square root of \(\lambda^2-a_i^2\). It is \(\lambda\) outside the compact spatial support. The product and adjoint estimates show

\[
 \lambda^2-Q_i^2\equiv B_i^*B_i\ge0
             \quad\bmod\mathcal K_D.
 \tag{MGP.11}
\]

Thus the spectrum in the Calkin algebra is contained in \([-1,1]\). The odd clamp \(h(t)=\max(-1,\min(1,t))\) changes \(Q_i\) by a compact and gives a self-adjoint contraction \(F_i\). Since the original chart operator is zero off its chart support and \(h(0)=0\), \(F_i\) retains that support. Every operation is grading preserving in the required sense and right Clifford linear.

Take a locally finite compactly subordinate square partition \(\sum_i\varphi_i^2=1\). The map \(V:u\mapsto(\varphi_i u)_i\) is an isometry by integration; its adjoint is the norm-convergent partition sum. Set

\[
 F_W=V^*\operatorname{diag}(F_i)V.
 \tag{MGP.12}
\]

It is a global odd self-adjoint contraction. Any compact source test meets finitely many partition terms. Their chart outputs are precompact, so products meet finitely many further terms. MGP.9–MGP.10 compare the local leading symbols on overlaps, and changing splitting adds the compact shear error of Section 11CF.2. The resulting global symbol is exactly MB.13 modulo those scalar compacts. Its square differs from one by MB.14 and a negative-order remainder; its smooth multiplication commutators have negative order. For an actual arrow, MGP.10 transforms the leading symbol by the true derivative, the strict coefficient diagonal transports its two orthogonal blocks, and MB.16 removes the remaining shear by a scalar compact. A compact convolution kernel is a finite compact-bisection sum with smooth cutoffs; the preceding arguments give compact commutators for each summand. Norm density and maximal continuity give every source defect.

**Theorem 11CF.1 (the completed maximal receiving cycle).** The actual coherent representation and MGP.12 define

\[
 d_{W,m}\in KK^n(C^*_{\max}(H_W),\mathbb C),
 \tag{MGP.13}
\]

with the entire inverse rank-\(n\) coefficient. Its unit-space restriction is the ordinary total inverse class. Every claimed defect is scalar compact; no reduced-continuity assertion is included.

**Proof of the remaining class assertion.** Vary the exponent in MB.18 over \(p\in[3/4,1]\), with the same full coefficient and partition. All local symbols have uniform worst type \((3/4,1/4)\); all square and composition tails have uniform negative order. On bounded frequencies the symbols and all finite derivatives vary continuously, so localized compact errors are norm continuous. The operators are strongly continuous with strongly continuous adjoints: finite-frequency inputs give a compact-spatial Hilbert–Schmidt calculation, and the uniform norm bound treats all inputs by density. They therefore act adjointably on the interval section module.

MGP.11 holds on that module as well: its remainders are norm-continuous compact families by the uniform tail proof. Functional calculus on the module shows that the clamp corrections are such compact families. Locally finite patching preserves strong continuity and gives a full \(C_0(W)\)-cycle homotopy. The endpoint \(p=1\) has the ordinary normalized total inverse symbol. The complete ordinary calculus in Analytic index and assembly are different questions, Theorem 7.1 with the full inverse coefficient of Theorem 7.17, constructs that same symbol cycle. Their difference after every compact source test is compact by the finite-chart comparison just proved; its linear interpolation is a cycle homotopy. This identifies the whole unit-space class, without a local-to-global KK exactness assumption.

For the particular original disk comparison, use the full logarithm product coordinates \(W|U=U\times\mathbb R^d\). Metric and splitting interpolation, with this same uniform local construction, deform the ordinary endpoint to the Euclidean full inverse on that whole open set. Source compactness uses finite compact tests, not a uniform bound at fibre infinity. Its Fourier realization is MB.18–MB.20 at \(p=1\); the off-diagonal Fourier kernel has arbitrarily decaying tails by frequency integration, so its source-local comparison with the patched phase is compact. The native evaluations in Lemmas 7.17–7.20 then give MB.21 with the same original positive \(q\)-disk and the whole graded vertical cancellation. The exponent homotopy is an ordinary unit-space homotopy; its endpoint is not claimed covariant under the triangular holonomy action. \(\square\)

## 11CG. Reduced metric products with the original normal geometry

The completed metric phase of Section 11CF can receive the reduced vertical Bott transfer under either of the concrete hypotheses below. This gives normally geometric reduced scalar classes for arbitrary nonisometric holonomy in these stated receiving scopes. It does not claim a receiving reduced norm for every original holonomy groupoid.

We retain the full inverse lift, actual arrow domains and original \(q\) of Section 11CF, and use the complete separator and product proof in Composing graded correspondences. The reduced vertical transfer is MB.11; the actual tree regularization is A regular tree factor preserves an original normal cycle. None of these hypotheses is imposed as a universal gate.

### 11CG.1. An explicit actual range-fibre receiving norm

Write \(K=H_W\) and \(\mathscr H_W=L^2(W,I_n;\mu_W)\). Its geometric unitary representation \(\pi\) is that of Section 11CF. Suppose \(K\) has a measurable normalized range-fibre certificate

\[
 \xi_j^y\in\ell^2(K^y),\qquad\|\xi_j^y\|=1,
 \qquad
 \sup_{\alpha\in C}\|\alpha\xi_j^{s\alpha}-\xi_j^{r\alpha}\|
                       \longrightarrow0
 \tag{MRP.1}
\]

for every compact actual arrow test \(C\). This is the explicit certificate used in this theorem. A usual nonnegative \(\ell^1\) certificate yields it: add \(\epsilon_j\delta_{1_y}\), normalize its mass, and take its square root. Masses tend to one on compact units, so normalization preserves compact-test invariance; the elementary inequality \((\sqrt a-\sqrt b)^2\le|a-b|\) bounds the new squared \(\ell^2\) error by the probability-vector \(\ell^1\) error. Outside those compact tests, the positive added unit mass makes normalization everywhere defined. Thus no exact invariant vector is assumed.

Use the full regular source-fibre space

\[
 \mathscr R=\int_W^{\oplus}
        \ell^2(K_x)\widehat\otimes (I_n)_x\,d\mu_W(x),
 \qquad
 (J_j u)(h)=\xi_j^{r h}(h)U_h^{-1}u(rh).
 \tag{MRP.2}
\]

The coefficient here is at the original source \(x=sh\). Invariance of \(\mu_W\) under actual bisections gives the counting interchange

\[
 \int_W\sum_{h:s h=x}b(h)\,d\mu_W(x)
 =\int_W\sum_{h:r h=y}b(h)\,d\mu_W(y)
 \tag{MRP.3}
\]

for nonnegative measurable functions, first on bisections and then by monotone sums. Full coefficient unitarity and MRP.1 show \(\|J_ju\|=\|u\|\). No summand of a range or source fibre is removed.

The regular representation acts by

\[
 (\lambda(k)v)(h)=\sum_{\alpha:r\alpha=rh}
                          k(\alpha)v(\alpha^{-1}h).
\]

Using the full coherence \(U_h=U_\alpha U_{\alpha^{-1}h}\), its intertwining error is exactly

\[
 (\lambda(k)J_j-J_j\pi(k))u(h)
 =\sum_{\alpha:r\alpha=rh}k(\alpha)
 [\xi_j^{s\alpha}(\alpha^{-1}h)-\xi_j^{r\alpha}(h)]
                      U_h^{-1}U_\alpha u(s\alpha).
 \tag{MRP.4}
\]

For a coefficient supported on one compact bisection, use MRP.3, the range-fibre squared norm in MRP.1, and the coefficient supremum to bound its operator norm by \(\|k\|_\infty\sup_C\|\alpha\xi_s-\xi_r\|\). A finite compact-bisection decomposition gives the sum of those bounds. The regular norm satisfies \(\|\lambda(k)\|\le\|k\|_{C_r^*(K)}\), since all source-fibre norms occur in its defining supremum. Therefore

\[
 \|\pi(k)u\|\le\|k\|_{C_r^*(K)}\|u\|
                   +o(1)\|u\|.
 \tag{MRP.5}
\]

Let \(j\) tend to infinity and complete the compact convolution algebra. This proves the entire receiving representation is reduced-continuous. The same \(F_W\) and all of its scalar defects now give \(d_{W,r}\in KK^n(C_r^*(K),\mathbb C)\).

### 11CG.2. The free-group receiving route needs no MRP.1

Alternatively suppose the actual presentation is \(H=F_m\ltimes T\), with all displayed group words distinct as actual germs on their retained domains. Then \(K=F_m\ltimes W\) is the same actual presentation on the metric bundle: identity on an open subset of \(W\) would project to an identity germ on \(T\). The scalar maximal receiving cycle of Theorem 11CF.1 satisfies precisely RTP.1 on \(W\), including the whole rank-\(n\) inverse transport. Apply Theorem 11CD.2 to it. The resulting reduced receiving representation has carrier

\[
 \mathscr H_R=\mathscr H_T\widehat\otimes\mathscr H_W,
 \quad \Pi_R(u_g)=V_g\otimes U_g,
 \quad \Pi_R(f)=1\otimes M_f,
 \tag{MRP.6}
\]

and operator \(F_R\). It has exact unit-space restriction \([F_W]\), and its finite tree creation equations retain the full metric-fibre phase. This works with every actual derivative in \(GL_q^+\) allowed by the supplied coherent lift; no original invariant metric is needed. In particular the nonamenable metric-lift example of Section 11CE is compatible with this route.

An abstract free presentation of an arbitrary group is not an actual free-group holonomy presentation: its kernel words may be identity germs. They cannot be inserted as extra holonomy labels to apply this argument. The hypothesis here concerns the actual germ groupoid.

### 11CG.3. The entire balanced vertical product

In either receiving case, let \(B_r=C_r^*(K)\) and \(A_r=C_r^*(H)\). Use the complete positive vertical module and radial Bott operator of MB.8–MB.11, with its actual descended module \(\mathscr E_V\). Form

\[
 \mathscr H_{\rm bal}=\mathscr E_V
                 \widehat\otimes_{B_r}\mathscr H_R.
 \tag{MRP.7}
\]

Here \(\mathscr H_R=\mathscr H_W\) and \(F_R=F_W\) for MRP.1; for the free-group case they are MRP.6. The usual displayed Clifford notation suppresses no actual coefficients: before the prescribed finite evaluation, the positive vertical coefficient and total inverse coefficient remain in their product order.

For a compact actual arrow section \(\eta\) of the vertical coefficient, define on the dense receiving core

\[
 (T_\eta u)(w)=\sum_{h:r h=w}
                     \eta(h)\widehat\otimes U_h^R u(sh).
 \tag{MRP.8}
\]

The sum includes every retained arrow. In the free case \(U_h^R\) also transports the full tree column. Expanding its squared norm, changing variables with MRP.3, and using the exact coefficient transports gives

\[
 \left\|\sum_iT_{\eta_i}u_i\right\|^2
 =\sum_{i,j}\langle u_i,
        \Pi_R(\langle\eta_i,\eta_j\rangle_{B_r})u_j\rangle.
 \tag{MRP.9}
\]

All sums in a compact-bisection core are finite per range unit; positivity and completion handle general vectors. Thus MRP.8 is an isometry of the balanced completion. Unit-arrow sections with compact coefficient frames give every compact vertical coefficient tensored with a receiving vector; those vectors are dense by nondegeneracy. The isometry is onto the entire vertical-coefficient receiving carrier. The balanced relations transfer arrow transports into the receiving action; they do not select a subset of arrows, a jet kernel or a vacuum.

The left representation of \(A_r\) is contractive by MB.11 and the balanced tensor norm, independent of the receiving route. The vertical Bott operator becomes precisely multiplication by \(c_V^+(v)/\sqrt{1+|v|^2}\). Apply the full connection construction and separator of Theorem 0.4 on MRP.7. It gives a bounded odd self-adjoint product phase \(F_{\rm bal}\) with scalar compact source square and commutator defects on all of \(A_r\), and, for every homogeneous first-module vector,

\[
 F_{\rm bal}T_\eta-(-1)^{|\eta|}T_\eta F_R
             \quad\hbox{and its adjoint are scalar compact}.
 \tag{MRP.10}
\]

This applies also to every compact unit-arrow vertical section. After restriction to \(C_0(T)\), those sections give the complete ordinary vertical Bott module over \(C_0(W)\). MRP.10 is its entire receiving connection, and the product positivity is the same original-source positivity supplied by Theorem 0.4. The ordinary product criterion therefore identifies the restriction with the positive vertical Bott class followed by the full ordinary total inverse. In the free case its extra tree class is the ordinary identity by RTP.11–RTP.12; no stronger equivariant identity is used.

### 11CG.4. Original degree, normal phase and positive disk

Keep the total inverse order \((\hbox{original quotient},V)\), as in MB.13. Use the full finite graded transport and native evaluation of Lemmas 7.16–7.20 to put the cancellable vertical blocks together and the original \(\mathrm{Cl}_q\) last. The mixed parity \(dq\) is carried by that transport. The central cancellation and surviving physical module are exactly

\[
 z^qz^{-(q+1)}=z^{-1},\qquad
 I_q=\Pi^{q(q-1)/2}\mathrm{Cl}_q,
 \qquad c_{\rm inv,q}(e)=(-1)^{q(q-1)/2+1}L_e.
 \tag{MRP.11}
\]

The spin transports also cancel as full coherent operators, not merely as line exponents. The degree \(d+n=q+2d\) reduces to the original \(q\) with this specified evaluation. The whole vertical and, if present, tree carrier remains; this is finite Clifford coefficient evaluation rather than geometric fibre removal.

For compact vertical creation sections, MRP.10 sends any bounded weakly null normal packet sequence to the receiving phase modulo norm-zero error. In the free case compose this equation with the finite tree creation equation RTP.13. The receiving symbol at \(\eta=0\), \(|\xi|\to\infty\), is the full inverse quotient Clifford direction, since the positive factor \(R^{-1/4}\) disappears upon normalization. The finite transport in MRP.11 therefore gives the original inverse \(q\)-Dirac phase, with the retained auxiliary grading signs. For a compact smooth normal packet multiplied by \(e^{i\lambda x\xi_0}\), split its Schwartz Fourier transform into a bounded difference-frequency region and its small tail. MGP.2 makes the bounded-region symbol difference tend uniformly to zero; bounded symbol norms control the Schwartz tail. The compact-spatial quantization therefore has exactly this operator phase limit. Its off-diagonal compact error sends the weakly null packet to norm zero. This proves the original normal geometry on every fixed compact auxiliary creation test. The full product defects on \(A_r\) control simultaneous vertical, tree and normal infinity; fixed-column checks alone are not substituted for those global defects.

On an original positive \(q\)-disk \(U\), MRP.10 identifies the restriction with the ordinary vertical Bott product on all of \(W|U\). The whole-space ordinary comparison in Theorem 11CF.1 and MB.21 then give

\[
 b_U\,[F_{\rm bal}]=+1.
 \tag{MRP.12}
\]

This is the original disk, not a metric-fibre disk of dimension \(q+d\). The entire inverse lift, degree, scalar defect ideal and positive calibration have all been retained.

### 11CG.5. Whole graph and a closed source domain

Tensor MRP.7 with the full original counting/plaque correspondence \({}_{C_r^*(G)}E_Z{}_{A_r}\). Its module Parseval compression gives a bounded phase \(F_G\) on the entire completed physical graph carrier. The proof is the full creation, square-defect and rank-one density argument of Section 11CD.4 applied to MRP.10. It retains every actual germ and isotropy word, normalized source counting and plaque density. Its creation equations preserve the normal phase of MRP.11 and original disk MRP.12. No unbounded compression has been presumed.

There is an explicit closed self-adjoint representative of this completed phase. First take the standard localized normalization to an odd self-adjoint contraction, with the same creation phase; the normalization follows by odd functional calculus because the tested square defects are compact. Choose an even positive compact \(K\), \(0<K<1/2\), with dense range and zero kernel on this finite-coefficient Hilbert space. A concrete choice is a norm-convergent positive sum of rank-one maps from a dense homogeneous frame, with positive summable coefficients. It is right Clifford linear; its scalar realization is compact. Set

\[
 F'=(1-K)^{1/2}F_G(1-K)^{1/2},\qquad
 D_G=F'(1-F'^2)^{-1/2},
\]

\[
 \operatorname{Dom}D_G=
 \left\{u:\int_{(-1,1)}\frac{t^2}{1-t^2}\,
                              d\|E_tu\|^2<\infty\right\}.
 \tag{MRP.13}
\]

Because \(K\) is injective, \(\|(1-K)^{1/2}u\|<\|u\|\) for every nonzero scalar vector. Thus \(F'\) has no eigenvector at either endpoint \(\pm1\). Its spectral measure gives the displayed dense domain, a self-adjoint odd operator and spectral-cutoff graph core. Its normalized bounded transform is exactly \(F'\). The difference \(F'-F_G\) is compact, so all source defects, creation phase and disk class are unchanged. Its localized resolvents are compact: each resolvent is a continuous function of \(F'\) vanishing at both endpoints; approximate that function uniformly by \((1-t^2)h(t)\), and use the compact source square defect. This proves a source domain and localized resolvents on the completed graph, without an unjustified unbounded corner argument. It does not assert a finite-order local differential formula for this reconstructed representative.

**Theorem 11CG.1.** Under MRP.1, or under the actual transformation-groupoid hypothesis of Section 11CG.2, with distinct free-group words remaining distinct germs on every retained domain, the full original \(ML_q^c\) orientation gives a normally geometric reduced scalar class on the original completed graph, with the original inverse coefficient and positive disk pairing \(+1\). The construction allows arbitrary nonisometric original derivatives in these receiving scopes. The arbitrary-holonomy target still requires a valid receiving norm or another construction under only its original premise; neither of these sufficient hypotheses is universally required.

![Exact radial sample of the MB.13 square defect and schematic of the completed maximal phase MGP.1–MGP.13, two sufficient reduced receiving constructions MRP.1–MRP.6, and full original normal product MRP.7–MRP.13. All metric and tree fibres remain; the original degree, full lift, mixed signs and positive q-disk are retained.](../figures/global-metric-reduced-routes.png)

**Figure 11CG.1.** Exact radial sample of the MB.13 square defect and schematic of the completed maximal phase MGP.1–MGP.13, two sufficient reduced receiving constructions MRP.1–MRP.6, and full original normal product MRP.7–MRP.13. All metric and tree fibres remain; the original degree, full lift, mixed signs and positive q-disk are retained. [Editable SVG](../figures/global-metric-reduced-routes.svg) and [reproduction instructions](../reproduction/global-metric-phase/README.md).


## 11CH. Compact types and the actual geometric receiving norm

The global metric phase in Section 11CF has scalar compact defects in its maximal representation. Changing its phase cannot supply a missing reduced representation norm. We prove a general test for that norm, including a finite matrix convolution kernel on a compact actual arrow support. The test retains the full supplied inverse coefficient and detects the whole representation; it constructs no replacement holonomy labels.

### 11CH.1. Every nonzero compact-group representation contains a finite type

Let \(L\) be a compact group with normalized Haar measure, and let \(U\) be a strongly continuous unitary representation on a nonzero Hilbert space \(\mathscr H\). For the Haar normalization and invariance, see Haar integration. If a finite right Clifford coefficient and grading are present, keep them and assume \(U\) is even and right linear.

Choose a nonzero homogeneous vector \(u\). In the scalar case put \(R_u=|u\rangle\langle u|\). With a finite right coefficient use its module rank-one operator \(\theta_{u,u}\); its scalar realization is positive, nonzero and finite rank. Define

\[
 C=\int_L U_lR_uU_l^*\,dl.
 \tag{CTNR.1}
\]

The integrand is norm continuous. For a scalar rank one this follows from strong continuity on \(u\); for a finite module rank one apply that argument to finitely many scalar columns. A finite partition of the compact parameter space approximates its Bochner integral in operator norm by finite-rank sums. Thus \(C\) is compact and positive. It is nonzero: choose \(v\) with \(\langle v,R_uv\rangle>0\); continuity makes the same quadratic form positive on a neighbourhood of the identity, of positive Haar measure. Haar invariance gives \(U_lCU_l^*=C\). The coefficient and grading are preserved in the module case.

Here is the finite-eigenspace step without an appeal to a decomposition theorem. Put \(c=\|C\|>0\), and choose unit vectors \(v_j\) with \(\langle v_j,Cv_j\rangle\to c\). Positivity gives \(C^2\le cC\), whence

\[
 \|(C-c)v_j\|^2\le c^2-c\langle v_j,Cv_j\rangle\longrightarrow0.
\]

Compactness supplies a convergent subsequence of \(Cv_j\); the displayed estimate then makes \(v_j\) converge to a unit eigenvector. The eigenspace \(E=\ker(C-c)\) is finite dimensional: an infinite orthonormal sequence in it would have images \(cv_j\) with no convergent subsequence. Since \(C\) commutes with \(U\), its full eigenspace is invariant. Therefore

\[
 \sigma=U|_E:L\longrightarrow U(r),\qquad r=\dim_{\mathbb C}E\ge1
 \tag{CTNR.2}
\]

is a finite-dimensional subrepresentation. This spectral test does not replace the physical carrier by \(E\).

### 11CH.2. A finite-dimensional reduced representation forces amenability

Let \(\Gamma\) be a discrete group. Suppose a nonzero finite-dimensional unitary representation \(\sigma\) extends to \(C_r^*(\Gamma)\). The tensor with its conjugate contains the invariant unit vector

\[
 \Omega=r^{-1/2}\sum_{i=1}^r e_i\otimes\overline e_i,
 \qquad(\sigma_g\otimes\overline\sigma_g)\Omega=\Omega.
 \tag{CTNR.3}
\]

We must establish reduced continuity of this tensor, rather than assume that tensoring preserves weak containment. On \(\ell^2(\Gamma)\otimes\overline E\), the unitary \((Bv)(h)=\overline\sigma_h^{-1}v(h)\) satisfies

\[
 B(\lambda_g\otimes\overline\sigma_g)B^*=\lambda_g\otimes1.
 \tag{CTNR.4}
\]

Direct substitution proves the formula at every actual group element. Consequently \(\lambda_g\mapsto\lambda_g\otimes\overline\sigma_g\) extends isometrically to a star homomorphism into \(M_r(C_r^*(\Gamma))\). Apply the matrix amplification of the given representation \(\sigma\); it is contractive because a star homomorphism of C*-algebras is contractive, also on each matrix algebra. This gives

\[
 \left|\sum_g a_g\right|
 =|\langle\Omega,\sum_g a_g(\sigma_g\otimes\overline\sigma_g)\Omega\rangle|
 \le\left\|\sum_g a_g\lambda_g\right\|
 \tag{CTNR.5}
\]

for every finitely supported scalar family. Thus augmentation extends to a character \(\epsilon:C_r^*(\Gamma)\to\mathbb C\).

For completeness this implies amenability in the invariant-mean definition. Hahn–Banach extends \(\epsilon\), with its norm one and value one at the identity, to a functional \(\varphi\) on \(B(\ell^2\Gamma)\). Such a functional is positive. Indeed for self-adjoint \(b\), the bound \(|\varphi(e^{itb})|\le1\) and its two-sided expansion at zero force \(\operatorname{Im}\varphi(b)=0\). For \(0\le b\le1\), \(|1-\varphi(b)|\le\|1-b\|\le1\) then gives \(\varphi(b)\ge0\). Scaling proves positivity for every positive \(b\).

Now \(\varphi((\lambda_g-1)^*(\lambda_g-1))=0\), and the analogous reversed product is zero. The Cauchy–Schwarz inequality for a positive functional gives \(\varphi(T(\lambda_g-1))=\varphi((\lambda_g-1)T)=0\) for all bounded \(T\). Hence \(\varphi(\lambda_gT\lambda_g^*)=\varphi(T)\). Its restriction to diagonal multipliers is

\[
 m(f)=\varphi(M_f),\qquad m(1)=1,
 \quad m(f)\ge0\ (f\ge0),\quad m(g\cdot f)=m(f).
 \tag{CTNR.6}
\]

This is an invariant mean on \(\ell^\infty\Gamma\). We have proved the required implication, including tensor continuity and the mean construction.

**Theorem 11CH.1.** If \(\Gamma\) is nonamenable and \(U\) is any nonzero strongly continuous representation of a compact group \(L\), its restriction along a homomorphism \(\Gamma\to L\) is not reduced-continuous. No density or faithfulness of that homomorphism is needed for this statement.

**Proof.** CTNR.1–CTNR.2 give a nonzero finite-dimensional invariant subspace. If the whole restricted representation extended to \(C_r^*(\Gamma)\), compression to this reducing subspace would give such an extension of \(\sigma\). CTNR.3–CTNR.6 would make \(\Gamma\) amenable. \(\square\)

### 11CH.3. A compact actual convolution test

Suppose \(\Gamma\) acts globally on a locally compact space \(W\), and \((M,U)\) is a nondegenerate covariant representation of \(C_0(W)\) and \(\Gamma\). Assume its group representation extends to a compact-group representation as above. This hypothesis concerns the representation, including all coefficient transport; an action on the base alone does not imply it.

In the reduced crossed product the multiplier \(\sum_g a_g u_g\) has precisely the regular group norm. One way to check this is the defining regular covariant representation: the group multipliers act by \(\lambda_g\otimes1\). The same holds for finite matrix coefficients. A nondegenerate representation of the reduced crossed product has a contractive multiplier extension, so its group representation would be reduced-continuous. Theorem 11CH.1 therefore already excludes that extension for nonamenable \(\Gamma\).

There is also a compactly supported witness in the actual convolution algebra. Consider the actual free-group presentation of Section 11CE, with generators \(a,b\), and put \(S=\{a,a^{-1},b,b^{-1}\}\). The complete tree estimate MI.5–MI.6 gives

\[
 \left\|\tfrac14\sum_{s\in S}\lambda_s\right\|
 \le\sqrt3/2<1.
 \tag{CTNR.7}
\]

Let \(E\) and \(\sigma\) be obtained from the full geometric representation by CTNR.1–CTNR.2. In its finite matrix amplification define \(P=\frac14\sum_{s\in S}u_s\otimes\overline\sigma_s\). Formula CTNR.4 gives \(\|P\|_r\le\sqrt3/2\). In the geometric representation \(\Pi^{(r)}(P)\Omega=\Omega\), where \(\Omega\in E\otimes\overline E\subset\mathscr H\otimes\overline E\) is CTNR.3.

Choose a compactly supported cutoff \(0\le\chi_R\le1\) with \(M_{\chi_R}\to1\) strongly, and set

\[
 k_R=(\chi_R\otimes1)P
 =\tfrac14\sum_{s\in S}\chi_R u_s\otimes\overline\sigma_s.
 \tag{CTNR.8}
\]

Every term is on the compact range support of \(\chi_R\) in one of the four actual arrow bisections. Smooth compact cutoffs may be used on a manifold. No group word outside the actual germ presentation is inserted. For sufficiently large \(R\), \(\|(M_{\chi_R}\otimes1)\Omega\|>9/10\). Thus the actual kernel satisfies

\[
 \|k_R\|_r\le\sqrt3/2,
 \qquad\|\Pi^{(r)}(k_R)\|>9/10>\sqrt3/2.
 \tag{CTNR.9}
\]

This finite matrix test contradicts complete contractivity of any proposed C*-extension. It therefore detects a failure of the original representation norm. The finite matrix amplification does not change the scalar compact ideal or the original normal degree.

### 11CH.4. The complete metric phase on the actual compact suspension

Take precisely the original compact conjugation suspension of Section 11CE. Its actual return group is \(F_2\), the original normal rank is \(q=3\), the metric-fibre rank is \(d=6\), and the total metric rank is \(n=9\). Retain the complete coherent lift and full inverse coefficient of MI.2–MI.4 and MB.4.

The natural geometric representation of Section 11CF on \(W=\mathrm{Met}(TN)\) extends continuously to the compact conjugating group \(SU(2)\), with that same coherent coefficient lift. On smooth compactly supported coefficient sections, actual coordinate pullback and the smooth finite coefficient transport give strong continuity by dominated convergence on a compact enlarged support. Invariant volume makes every transport unitary, so density extends strong continuity to all vectors. The compact covering group is used only to average an operator on the existing Hilbert space. Its additional elements are not made into holonomy arrows: CTNR.8 uses only the four supplied actual return germs, whose distinctness was proved in MI.1–MI.4.

The representation is nonzero. Theorem 11CH.1 and the actual compact kernel CTNR.8–CTNR.9 show that this geometric receiving representation does not factor through \(C_r^*(H_W)\). This conclusion includes the full finite right \(\mathrm{Cl}_9\), its physical inverse generators and all spin and central transport. No unrelated scalar even line replaces them.

Nevertheless Theorem 11CG.1 constructs a reduced normally geometric class for this same foliation: its actual free-group tree receiving representation is different. The norm obstruction just proved is an obstruction to this proposed receiving representation, not a refutation of the original graph-class problem or its positive disk calibration.

### 11CH.5. Which repairs leave the norm obstruction intact

An operator perturbation, exponent homotopy, scalar compact correction or positive weighting changes \(F\), not the covariant representation \((M,U)\). It therefore leaves CTNR.9 unchanged. The same applies to the finite native Clifford evaluation: that operation follows a valid receiving product and cannot provide a missing input representation norm.

A unitary conjugation preserves every operator norm. A nonzero finite matrix amplification preserves \(\|\Pi(k)\|\), and a direct sum with any other representation has norm at least \(\|\Pi(k)\|\). Thus adding a regular summand while retaining this reducing geometric summand cannot repair its continuity. More generally, any representation containing it as a reducing subrepresentation would make it reduced-continuous on compression if the larger representation were reduced-continuous, which CTNR.9 excludes.

This determines the next mathematical issue accurately. A general receiving construction must establish its reduced representation norm on its entire actual carrier, before phase compactness or Clifford cancellation is used. The compact-type hypothesis is not imposed on arbitrary holonomy, and amenability is not made a universal requirement. The tree construction already shows why those would be false gates. The arbitrary-holonomy construction under only the original coherent \(ML_q^c\) premise remains unfinished.

![The exact compact-type averaging and matrix absorption CTNR.1–CTNR.6 yield the finite actual convolution test CTNR.7–CTNR.9. Its compact range cutoff has geometric norm greater than 9/10 and reduced norm at most sqrt(3)/2. This excludes the specified receiving representation; the actual tree replacement still gives a reduced graph class with the original normal degree and disk calibration.](../figures/compact-type-receiving-norm.png)

**Figure 11CH.1.** The exact compact-type averaging and matrix absorption CTNR.1–CTNR.6 yield the finite actual convolution test CTNR.7–CTNR.9. Its compact range cutoff has geometric norm greater than 9/10 and reduced norm at most sqrt(3)/2. This excludes the specified receiving representation; the actual tree replacement still gives a reduced graph class with the original normal degree and disk calibration. [Editable SVG](../figures/compact-type-receiving-norm.svg) and [reproduction instructions](../reproduction/compact-type-norm/README.md).


## 11CI. An exact receiving test for the original linear normal class

The orthogonal linear model admits a two-sided receiving calculation. It tests all reduced scalar classes of that model, rather than only one proposed representation. Its conclusion uses the original normal degree, complete inverse spin-c line and ordered Clifford coefficient. The test applies to this model; it is not an additional hypothesis imposed on arbitrary holonomy germs.

The proof prerequisites are the two full Clifford Thom products in Theorem 6.0c, their spin-c Morita evaluations in Corollary 6.0d, the exact inverse coefficient transfer in Theorem 7.17, and the product and descent constructions of Sections 0 and 11J. All mathematical expression in this supplement is original CC0 material. Bott, Kasparov and Connes retain their historical credits in the containing lesson.

### 11CI.1. The supplied lift and both equivariant inverse products

Let a countable discrete group \(\Gamma\) act on \(V=\mathbb R^q\) by an orthogonal representation \(\rho:\Gamma\to SO(q)\). Prescribe its entire coherent lift \(\sigma:\Gamma\to\operatorname{Spin}^c(q)\). The source and target actions on the fixed right Clifford coefficient \(C_q\) are trivial. Put

\[
 A_0=C_0(V),\qquad A_r=A_0\rtimes_r\Gamma,
 \qquad R=C_r^*\Gamma,\qquad u:\mathbb C\longrightarrow R.
 \tag{LINR.1}
\]

For an actual linear holonomy presentation, require \(\rho\) to be faithful. Distinct matrices then have distinct germs on every nonempty open representative domain: a linear map vanishing on an open set vanishes everywhere. Hence \(\Gamma\ltimes V\) records actual germs, including the full isotropy at the origin. Without faithfulness the algebraic calculation below remains a crossed-product calculation, but does not identify the ineffective labels with a holonomy groupoid. In dimension zero an effective linear presentation has trivial group.

Apply Corollary 6.0d to the vector space with this specified lift. Before changing the Clifford picture its two classes have types

\[
 \beta\in KK^\Gamma(\mathbb C,A_0\widehat\otimes C_q),\qquad
 \alpha\in KK^\Gamma(A_0\widehat\otimes C_q,\mathbb C),
 \qquad\beta\alpha=1,\quad\alpha\beta=1.
 \tag{LINR.2}
\]

Here these are genuinely equivariant identities. To check the extra assertion, use the actual operators in Theorem 6.0c. Their exterior actions, Euclidean Fourier domain, radial multiplier and harmonic oscillator commute with orthogonal transport. In the first product the normalized Gaussian and exterior vacuum are invariant under every orthogonal matrix. The original spin-c module and its full conjugate are evaluated by the two even maps of Corollary 6.0d, so their central factors cancel as \(z z^{-1}\), before the kernel is identified with the scalar identity. No choice of a square root of a determinant line occurs. In the reverse product the generator sum \(\sum_i A_iB_i\) in (143) is invariant under simultaneous orthogonal transport. The translations \(v\mapsto v+z\), the argument \(z+sv\), and every spectral function of the oscillator commute with that transport. The uniform compact interval estimates in the proof are therefore estimates for equivariant homotopies. The full Morita evaluation is even and intertwines the supplied lift at every endpoint. Thus both ordinary product proofs lift to the specified \(\Gamma\)-cycles and homotopies. They do not require \(\Gamma\) itself to be amenable or compact.

We keep the ordered coefficient transfer explicit. Let \(e_q\in KK(\mathbb C,C_q\widehat\otimes C_q)\) be the even Clifford Morita equivalence used in (286), and \(e_q^*\) its inverse. The old Clifford block precedes the external block. Define

\[
 \begin{split}
 \eta&=(\beta\widehat\otimes1_{C_q})
              (1_{A_0}\widehat\otimes e_q^*)
       \in KK^\Gamma(C_q,A_0),\\
 \delta&=(1_{A_0}\widehat\otimes e_q)
              (\alpha\widehat\otimes1_{C_q})
       \in KK^\Gamma(A_0,C_q).
 \end{split}
 \tag{LINR.3}
\]

Associativity, (LINR.2), and the two even Morita inverse products give \(\eta\delta=1_{C_q}\) and \(\delta\eta=1_{A_0}\). This calculation permutes no odd Clifford blocks. The second class is precisely Theorem 7.17's transferred inverse. Its finite module has parity shift \(k_q=q(q-1)/2\), physical generators \((-1)^{k_q+1}L_{t_j}\), and transition \(z^{-1}\rho_t(s)\). In particular it retains the complete inverse phase, including two-torsion information.

Reduced descent preserves these products and the trivial Clifford action. It supplies

\[
 \eta_r\in KK(R\widehat\otimes C_q,A_r),\qquad
 \delta_r\in KK(A_r,R\widehat\otimes C_q),\qquad
 \eta_r\delta_r=1_{R\widehat\otimes C_q},\quad
 \delta_r\eta_r=1_{A_r}.
 \tag{LINR.4}
\]

The coefficient of \(\delta_r\) is \(R\widehat\otimes C_q\). It is not already a scalar fundamental class.

### 11CI.2. A bijection of all receiving classes

Tensoring both source and coefficient by \(C_q\) is a KK equivalence of categories: tensor once more, then apply \(e_q\) at the source and \(e_q^*\) at the coefficient. The two Morita inverse identities prove that the resulting operation is the inverse functor. Consequently every \(y\in KK(R\widehat\otimes C_q,C_q)\) has a unique expression \(y=\kappa\widehat\otimes1_{C_q}\), with \(\kappa\in KK(R,\mathbb C)\). These are even classes in their displayed graded pictures; their transverse degree is encoded by the retained \(C_q\).

Define

\[
 \begin{split}
 \mathcal N(\kappa)&=\delta_r
                  (\kappa\widehat\otimes1_{C_q})
                     \in KK(A_r,C_q),\\
 \mathcal P(\xi)\widehat\otimes1_{C_q}&=\eta_r\xi,
                  \qquad \xi\in KK(A_r,C_q).
 \end{split}
 \tag{LINR.5}
\]

They are mutually inverse homomorphisms. Indeed (LINR.4) gives

\[
 \eta_r\mathcal N(\kappa)=\kappa\widehat\otimes1_{C_q},
 \qquad
 \mathcal N(\mathcal P(\xi))=\delta_r\eta_r\xi=\xi.
 \tag{LINR.6}
\]

Thus the scalar receiver is recovered from every class of the linear reduced algebra. This is a necessity statement for this specific model as well as a sufficient construction. It does not assume that a geometric receiving representation is reduced-continuous.

### 11CI.3. The original disk and the whole ordinary normal restriction

Let \(i:A_0\to A_r\) be the identity-coordinate inclusion. Write \(b_V=\eta\in KK(C_q,A_0)\) after forgetting the group action. On the whole descended module, restriction to identity-coordinate functions gives

\[
 (u\widehat\otimes1_{C_q})\eta_r=b_V[i],\qquad
 [i]\delta_r=\delta\widehat\otimes[u].
 \tag{LINR.7}
\]

The second formula includes the even flip that places the coefficient \(R\) before the old right \(C_q\); the \(R\) factor has trivial grading, so that flip introduces no sign.

Here is the module verification, rather than merely an equality of K-group integers. A descended module has finitely supported functions \(\xi(g)\) with values in the original module. Its right inner product has \(h\)-coefficient \(\sum_g\langle\xi(g),\xi(gh)\rangle\), with the original action on the right coefficient included when that coefficient is \(A_0\). For the Dirac module, whose original coefficient is \(C_q\) with trivial action, the map \(\xi\mapsto\sum_g\xi(g)\widehat\otimes\lambda_g\) identifies it with the original module extended to \(R\widehat\otimes C_q\). The displayed inner product proves isometry; identity-supported vectors and finite group polynomials give dense range. The restricted source is \(\pi(a)\widehat\otimes1\), and the descended phase is the old phase on each coordinate, which proves the second identity. For the Bott module, multiplication identifies its descent with its right extension along \(i\). The left unit of \(R\) acts as the identity there, leaving the ordinary Bott source \(C_q\). The coefficient inner-product formula and the density of compact finite group kernels prove the first identity. All identifications extend to whole interval modules, hence respect the given class and its homotopies.

Choose a positive coordinate ball \(U\subset V\). Its ordinary positive Bott generator \(b_U\in KK(C_q,C_0(U))\), extended by zero, is \(b_V\). Here is the needed local comparison. A positively oriented radial diffeomorphism \(V\to U\), followed by translation if needed, supplies a complete pullback metric and Dirac phase on \(U\). On every compact interior buffer its full symbol, coefficient and orientation agree with the restricted inverse after positive metric transport. Lemma 7.11's quantization-choice homotopy and Theorem 7.17's compact-base metric homotopy identify these two \(C_0(U)\)-classes; they require no uniform metric bound at the boundary of \(U\). In the complete pullback metric the positive Bott product is exactly the oscillator product of Theorem 6.0c, transported by that diffeomorphism, so it is \(1_{C_q}\). Thus \(b_U[j_U]\delta=1_{C_q}\). The two-sided inverse \(\eta\) then uniquely determines \(b_U[j_U]=\eta=b_V\). This uses an actual local phase comparison rather than a self-adjoint Euclidean Dirac claim on the incomplete ball. Put

\[
 b=b_U[j_U][i]=b_V[i]
                 \in KK(C_q,A_r).
 \tag{LINR.8}
\]

For \(\xi=\mathcal N(\kappa)\), associativity and (LINR.4), (LINR.7) give

\[
 b\xi=([u]\kappa)\widehat\otimes1_{C_q},\qquad
 [i]\xi=\delta\widehat\otimes([u]\kappa).
 \tag{LINR.9}
\]

The second product is read in the fixed Clifford order; its scalar auxiliary is even. Scalar classification identifies \([u]\kappa\) with the integer \(\langle[1_R],\kappa\rangle\). Therefore the following are equivalent:

1. A class \(\xi\in KK(A_r,C_q)\) pairs the original positive disk as \(b\xi=1_{C_q}\).
2. A class \(\xi\in KK(A_r,C_q)\) has the entire ordinary restriction \([i]\xi=\delta\).
3. An even class \(\kappa\in KK(R,\mathbb C)\) has \([u]\kappa=1_{\mathbb C}\).

For necessity start from the given \(\xi\), recover \(\kappa=\mathcal P(\xi)\), and apply the first equality of (LINR.9). For sufficiency use \(\mathcal N(\kappa)\) and both equalities. For equivalence of the first two assertions apply the ordinary two-sided inverse \(\eta\): it identifies \(KK(A_0,C_q)\) with \(KK(C_q,C_q)=\mathbb Z\), so this disk tests the whole ordinary class. No statement about arbitrary transversals is inferred from this last one-generator property of \(V\).

### 11CI.4. The actual receiving representation and its reduced norm

Represent \(\kappa\) by an essential even Fredholm cycle \((H_K,V_g,F_K)\) of \(R\). Let \((H_N,U_g,F_N)\) be the complete inverse normal cycle \(\delta\), retaining its finite right \(C_q\), grading and Euclidean Dirac domain. The product in (LINR.5) has scalar Hilbert carrier with that right coefficient

\[
 H_N\widehat\otimes H_K,
 \qquad
 \Pi(a u_g)=(\pi(a)U_g)\widehat\otimes V_g.
 \tag{LINR.10}
\]

To verify this carrier, tensor the descended Dirac module with \(H_K\) over \(R\). On its finite core the map is \(\xi\otimes v\mapsto\sum_g\xi(g)\widehat\otimes V_gv\). Expanding its squared inner product gives exactly the balanced inner product of the preceding subsection. Identity-supported vectors show that it is onto. The left convolution formula sends \(\xi(h)\) to \(\pi(a)U_g\xi(g^{-1}h)\), which is carried to (LINR.10).

The source norm must still be verified for the whole carrier. On \(H_N\widehat\otimes\ell^2\Gamma\), define \((Wv)(h)=U_h^{-1}v(h)\). Direct substitution gives

\[
 W(\pi(a)U_g\widehat\otimes\lambda_g)W^*v(h)
       =\pi(\alpha_{h^{-1}}(a))v(g^{-1}h).
 \tag{LINR.11}
\]

This is the regular covariant representation associated to \(\pi\), so its norm is at most the defining reduced norm. Each finite polynomial before conjugation lies in \(B(H_N)\otimes_{\min}R\). Hence the map \(a u_g\mapsto\pi(a)U_g\otimes\lambda_g\) extends contractively from \(A_r\) to that tensor product. Apply the contractive spatial tensor homomorphism \(1\otimes V\). It gives precisely (LINR.10), proving \(\|\Pi(f)\|\le\|f\|_r\) for every compact convolution kernel, and by completion for every element of \(A_r\).

The phase is a Kasparov product phase of \(\delta_r\) with \(\kappa\widehat\otimes1_{C_q}\), whose existence, connection equations and positivity are supplied by Section 0 on this entire descended module. It is not obtained by asserting that \(F_N\widehat\otimes1\) has scalar compact defects on an infinite tensor carrier. Localized compact normal defects are module compact before balancing; the product connection and positivity conditions provide the scalar compact defects after balancing. The full source, coefficient, grading and normal factor remain those of (LINR.10). Equations (LINR.7)–(LINR.9) prove the entire ordinary inverse class and original disk sign of that product. A chosen literal foliation-graph representative still requires its actual counting/plaque correspondence and completed creation connections; a crossed-product Hilbert cycle alone is not that identification.

### 11CI.5. Full twists and the limit of this test

An additional full equivariant even line \(H\to V\) induces the equivariant Morita self-equivalence \(h\) of \(A_0\). Its inverse is the actual conjugate line \(H^*\); their evaluation maps are even, isometric and onto. After reduced descent write their classes as \(h_r,h_r^{-1}\). The changed normal factors are

\[
 \eta_{r,H}=\eta_r h_r,\qquad
 \delta_{r,H}=h_r^{-1}\delta_r,
 \qquad
 \eta_{r,H}\delta_{r,H}=1,\quad
 \delta_{r,H}\eta_{r,H}=1.
 \tag{LINR.12}
\]

Indeed the two middle line evaluations cancel, followed by (LINR.4). The identities in (LINR.7)–(LINR.9) then use the correspondingly twisted original Bott and inverse classes. A positive local disk trivializes the line without changing its full transition data outside that disk. Thus the criterion remains the same scalar index-one criterion while its normal factor retains \(H^*\), not merely the dual determinant phase. None of these operations changes \(q\) or drops the final right \(C_q\).

The supplied scalar construction in Sections 11J–11Y, where its stated existence hypotheses hold, can be inserted into (LINR.5). This produces a new reduced scalar receiving cycle with the complete ordinary normal class. If the original geometric representation fails Section 11CH's finite-type norm test, (LINR.10) changes the representation by a tensor receiver. It does not preserve that bad representation as a reducing direct summand.

The criterion does not require \(\kappa\) to represent the equivariant unit, and does not require \(\gamma=1\). It classifies the scalar receiving classes of this particular linear model by two proved inverse products. For arbitrary nonlinear germs, nonisometric derivatives, incomplete continuation domains or other transversals, the two model equivalences (LINR.2)–(LINR.4) are not supplied by the original orientation alone. The general FOL-055 task still requires a receiving construction on the actual whole graph under only its original coherent \(ML_q^c\) premise. The linear criterion is not a universal gate for that task.

![The exact orthogonal linear model identities LINR.2–LINR.12 give both original normal inverse products, a bijection of all reduced scalar receiving classes, positive-disk calibration and the whole ordinary restriction. The actual tensor carrier has the full reduced norm and retains original q, the full inverse spin-c phase and ordered right Clifford coefficient. Bott, Kasparov and Connes retain their human-source credits in the containing lesson.](../figures/linear-normal-receiving-equivalence.png)

**Figure 11CI.1.** The exact orthogonal linear model identities LINR.2–LINR.12 give both original normal inverse products, a bijection of all reduced scalar receiving classes, positive-disk calibration and the whole ordinary restriction. The actual tensor carrier has the full reduced norm and retains original q, the full inverse spin-c phase and ordered right Clifford coefficient. Bott, Kasparov and Connes retain their human-source credits in the containing lesson. [Editable SVG](../figures/linear-normal-receiving-equivalence.svg) and [reproduction instructions](../reproduction/linear-normal-receiving/README.md).


## 11CJ. Flat actual holonomy and the original reduced germ norm

Smooth actual holonomy contains information that even its entire formal Taylor series can miss. We construct that phenomenon on a genuine codimension-one suspension, with its complete coherent normal orientation. We then prove a scalar representation bounded for its actual reduced germ norm, and explain why reduced regularization of a labelled group action cannot be pushed to its effective germs by a canonical quotient. Neither assertion refutes or solves the general fundamental-class problem.

The only operator norm prerequisite is the whole free-tree averaging bound MI.5–MI.6 in Section 11CE. The groupoid convolution and original counting conventions are those of the C*-algebra lesson. The construction, estimates, proofs and diagram here are original CC0 mathematical expression. The two unipotent matrices used below are Sanov's classical example; the historical attribution is recorded in [Ignatov's primary paper, original p.49](https://www.mathnet.ru/php/getFT.phtml?jrnid=sm&option_lang=eng&paperid=2581&what=fullteng). We supply the needed entire free-product proof; no text or figure from that paper is incorporated.

### 11CJ.1. Explicit flat vector fields

Put

\[
 L=\sum_{k=1}^{\infty}10^{-k!},\qquad
 c_n=2^{-n},\quad \ell_n=2^{-n-3},\quad
 \epsilon_n=L2^{-n^2},\qquad n\ge1.
 \tag{FGJ.1}
\]

Here \(0<L<1/8\), and \(L\) is transcendental. We verify the latter fact to retain the exact parameter. Its \(m\)-term rational sum \(r_m=p_m/10^{m!}\) satisfies \(0<L-r_m<2\,10^{-(m+1)!}\): the first omitted term has that size before the factor two, and the subsequent exponents increase at least by one, giving a geometric upper bound. If a nonzero integer polynomial \(P\) of degree \(d\) vanished at \(L\), then for all sufficiently large \(m\), \(P(r_m)\ne0\); otherwise its infinitely many distinct \(r_m\) would be roots. Thus \(|P(r_m)|\ge 10^{-d m!}\). A derivative bound for \(P\) on \([0,1]\) gives \(|P(r_m)|\le C|L-r_m|<2C10^{-(m+1)!}\), a contradiction when \(m\) grows. Hence every \(\epsilon_n\), a nonzero rational multiple of \(L\), is transcendental. The upper bound on \(L\) follows from its first three terms and the same tail estimate.

For an explicit cutoff define \(h(t)=e^{-1/t}\) for \(t>0\), and zero otherwise, and

\[
 \chi(y)=
 \frac{h(9/16-y^2)}{h(9/16-y^2)+h(y^2-1/4)}.
 \tag{FGJ.2}
\]

Every derivative of \(h\) on the positive half-line is a polynomial in \(1/t\) times \(e^{-1/t}\), and tends to zero at zero. The difference quotients then show inductively that its extension is smooth with every derivative zero there. The denominator in (FGJ.2) never vanishes. Thus \(\chi\) is smooth, \(0\le\chi\le1\), equal to one on \(|y|\le1/2\) and zero on \(|y|\ge3/4\). It has compact support strictly inside \((-1,1)\).

The intervals \(I_n=(c_n-\ell_n,c_n+\ell_n)\) are disjoint: their positive endpoints at consecutive scales obey \((9/8)c_{n+1}<(7/8)c_n\). Define two global vector fields, with value zero off these intervals, by

\[
 \begin{split}
 X(x)&=\sum_{n\ge1}\epsilon_n\ell_n
                    \chi((x-c_n)/\ell_n),\\
 Y(x)&=-\sum_{n\ge1}\epsilon_n\ell_n
             ((x-c_n)/\ell_n)^2\chi((x-c_n)/\ell_n).
 \end{split}
 \tag{FGJ.3}
\]

The sums are locally single summands away from zero. At interval boundaries all derivatives fit with zero because the cutoff is zero on a neighbourhood of the boundary. On \(I_n\), for every derivative order \(m\) and integer \(N\ge0\),

\[
 \frac{|X^{(m)}(x)|+|Y^{(m)}(x)|}{|x|^N}
 \le C_{m,N}\,2^{-n^2+n(m-1+N)}.
 \tag{FGJ.4}
\]

Indeed the chain rule gives a fixed bound times \(\epsilon_n\ell_n^{1-m}\), while \(|x|\ge(7/8)2^{-n}\). The constants absorb the fixed powers of eight and \(L\). The right side tends to zero for every fixed \(m,N\). Each derivative therefore extends continuously with value zero at the origin; its difference quotient has limit zero by taking \(N=1\). Induction proves that the extended vector fields are smooth and flat at zero. Their supports lie in a fixed compact interval. Every derivative is bounded, by the finite initial intervals and (FGJ.4).

Let \(f\) and \(g\) be their time-one flows. For clarity these are genuine globally defined smooth diffeomorphisms. A vector field with bounded derivative is globally Lipschitz; Picard iteration of its integral equation is a contraction on a sufficiently short common time interval. Its bounded magnitude prevents escape, so concatenate these intervals to obtain the whole flow. Differentiating the integral equation first gives \(J'=Z'(\phi_t)J\), \(J(0)=1\), for \(Z=X\) or \(Y\), and hence \(J>0\). Successive derivatives solve inhomogeneous linear equations whose terms involve only already constructed derivatives and bounded derivatives of \(Z\); induction gives smooth dependence to every order. Uniqueness gives the flow law and inverse time, so \(f,g\) are orientation preserving. Both fix every point outside the displayed supports and preserve each \(I_n\), because they are zero near its endpoints.

Their germs at zero have the Taylor series of the identity. The flow fixes zero; its first variational equation there has zero coefficient, so its derivative is one. Every higher variational equation has zero forcing there by flatness of \(Z\), and gives zero higher derivative. Equivalently the bound \(|Z(x)|\le C_N|x|^N\), together with \(|\phi_t(x)|\le e^{|t|}|x|\) near zero, bounds \(|\phi_t(x)-x|\) by \(|t|C_Ne^{N|t|}|x|^N\). Thus

\[
 j_0^{\infty}f=j_0^{\infty}g=j_0^{\infty}\mathrm{id}.
 \tag{FGJ.5}
\]

### 11CJ.2. Nevertheless the actual germs generate a free group

On the middle of \(I_n\), use \(y=(x-c_n)/\ell_n\). As long as a trajectory stays in \(|y|\le1/2\), the two time-one maps and their inverses are the fractional linear transformations

\[
 A_t=\begin{pmatrix}1&t\\0&1\end{pmatrix},\qquad
 B_t=\begin{pmatrix}1&0\\t&1\end{pmatrix},\qquad
 t=\epsilon_n.
 \tag{FGJ.6}
\]

Indeed the equations are \(\dot y=t\) and \(\dot y=-t y^2\), with solutions \(y+t\) and \(y/(1+t y)\). Negative time changes the sign of \(t\).

We need that every nonempty reduced word has a nonidentity projective matrix polynomial. At \(t=2\), let \(D_A=\{|y|>1\}\), including the projective point at infinity, and \(D_B=\{|y|<1\}\). For each nonzero integer \(m\),

\[
 A_2^m(D_B)\subset D_A,\qquad
 B_2^m(D_A)\subset D_B.
 \tag{FGJ.7}
\]

The first follows from \(|y+2m|>1\). For the second, \(|2my+1|>|y|\) when \(|y|>1\), and infinity goes to \(1/(2m)\). A nonempty word conjugates to either a nonzero pure power or a cyclically reduced alternating word. A mixed cyclically reduced word can be cyclically conjugated to begin with an \(A\)-power and end with a \(B\)-power. Its rightmost \(B\)-power fixes zero; then (FGJ.7) alternately moves the point into \(D_A,D_B\), ending in \(D_A\). It cannot fix zero. Pure powers are also nonidentity. Conjugation preserves the assertion of being nonidentity. This proves freeness for all reduced words, not merely words below a tested length.

For a fixed nonempty word \(w\), its matrix \(W_w(t)\) has integer-polynomial entries, since the inverses of (FGJ.6) have that property too. At least one off-diagonal entry or the difference of diagonal entries is a nonzero polynomial, by the just-proved nonidentity at two. That polynomial cannot vanish at the transcendental \(\epsilon_n\). Thus \(W_w(\epsilon_n)\) is a nonidentity projective transformation.

Let \(a\) be the length of this word in individual letters. For all sufficiently large \(n\), \(a\epsilon_n<1/4\). Start with \(|y|<1/4\). Each translation changes \(y\) by \(\epsilon_n\); a fractional linear step in \(|y|\le1/2\) changes it by at most \(\epsilon_n/2\), since its denominator is at least \(1-\epsilon_n/2\). The same estimates hold throughout the intervening time interval. Induction on the letters therefore keeps all trajectories in \(|y|<1/2\). The actual word in \(f,g\) equals \(W_w(\epsilon_n)\) on this whole starting interval. A nonidentity fractional linear map cannot fix an open interval: its fixed-point equation is a nonzero polynomial of degree at most two. Some point there is moved. These intervals approach zero, proving

\[
 \langle[f,0],[g,0]\rangle\cong F_2
 \quad\text{in the actual smooth germ group at zero}.
 \tag{FGJ.8}
\]

Together with (FGJ.5), every finite jet and the entire formal jet homomorphism have this full \(F_2\) in their kernel. As an exact visible displacement, the word \(fgf^{-1}g^{-1}\) at the center of \(I_n\) moves it by

\[
 -\frac{\ell_n\epsilon_n^3}{1-\epsilon_n^2},
 \qquad
 A_tB_tA_t^{-1}B_t^{-1}
 =\begin{pmatrix}1+t^2+t^4&-t^3\\t^3&1-t^2\end{pmatrix}.
 \tag{FGJ.9}
\]

For this four-letter word the central trajectory stays in the middle already when \(4\epsilon_n<1/4\), which holds for every \(n\ge1\) because \(L<1/8\). The displacement is nonzero for every scale, even when ordinary floating-point subtraction of the two physical coordinates rounds it to zero. The formula, rather than such subtraction, is the exact assertion.

### 11CJ.3. A genuine oriented foliation and its actual arrow space

Let \(S\) be a closed oriented genus-two surface. Its usual four-loop polygon gives \(\pi_1S=\langle a_1,b_1,a_2,b_2\mid[a_1,b_1][a_2,b_2]=1\rangle\). Send \(a_1,a_2\) to the two generators of \(F_2\), and both \(b_i\) to the identity; the relation is respected and the map is onto. The path-class construction gives its connected regular \(F_2\)-cover: identify based paths with the same endpoint when their concatenation maps to the identity. Small simply connected disks lift to disjoint disks, giving a smooth covering and the free proper deck action.

Act diagonally on that cover times \(\mathbb R\), using the two global diffeomorphisms just constructed on the second coordinate, and take the quotient. Properness follows from the deck action on the first factor, and freeness follows there too. Covering disk charts provide the smooth Hausdorff quotient and its foliation by the images of horizontal plaques. The transversal over a selected base point is exactly \(T=\mathbb R\). Its holonomy germs are the actual germs of the \(F_2\) action. This supplies a genuine smooth codimension-one suspension, not only a formal pseudogroup.

All transverse derivatives are positive. The positive metalinear component provides the coherent lift \(\sqrt{D w(x)}>0\); the chain rule multiplies these square roots at composed source and target points. Use central phase one. Identity germs have derivative one and lift one, so this descends to actual holonomy arrows. It is the supplied full \(ML_1^c\) orientation in the frozen convention. Its inverse retains degree \(q=1\), \(k_1=0\), physical generator \(-L_{t_1}\), complete inverse central phase one and final right \(C_1\). An invariant metric is not inferred: a local orientation-preserving isometry fixing zero would be the identity in its length coordinate, whereas (FGJ.8) gives nonidentity germs there.

Let \(\mathcal H\rightrightarrows T\) be the effective germ groupoid. It is locally a smooth étale bisection groupoid, with actual source counting. At zero its entire isotropy is \(F_2\), by (FGJ.8). On the negative half-line all generators and all words are the identity, so the actual groupoid there has only unit arrows. These are distinct assertions about distinct source fibres.

For \(x_m=-2^{-m}\to0\), the unit arrows are also \([f,x_m]\). The unit bisection makes them converge to \([\mathrm{id},0]\), and the \(f\)-bisection makes them converge to \([f,0]\). The latter is distinct by (FGJ.8). Thus this genuine arrow space is non-Hausdorff. In particular no separation or properness assertion can be imported from a finite jet chart.

### 11CJ.4. Every continuous Hausdorff-group label collapses that isotropy

Let \(K\) be any Hausdorff topological group, and let \(c:\mathcal H\to K\) be a continuous groupoid cocycle. It sends each unit to \(e\). For every word \(w\), the same negative sequence lies on its bisection and has \([w,x_m]=\mathrm{unit}_{x_m}\). Continuity therefore gives

\[
 c([w,0])=e\quad\text{for every }w\in F_2.
 \tag{FGJ.10}
\]

Uniqueness of a limit in \(K\) is the only separation input. This includes discrete labels, finite-dimensional matrix groups, finite jet groups and Hausdorff inverse limits of formal jet groups. The statement does not replace the original arrows by their labels: its point is that those actual arrows remain distinct when all such labels agree.

The neutral kernel of every such cocycle contains this nonamenable isotropy. For a discrete target its kernel is open and étale, so the original counting/Reiter definition applies directly. For other targets existence of a compatible Haar system on the kernel is a separate question; the following argument excludes any proposed normalized compactly supported counting Reiter field there as well. Write

\[
 P=\tfrac14(\lambda_a+\lambda_{a^{-1}}+
                       \lambda_b+\lambda_{b^{-1}}),
 \qquad\|P\|\le\sqrt3/2<1.
 \tag{FGJ.11}
\]

The bound is the whole-tree proof MI.5–MI.6. If finite probabilities \(p_i\) on \(F_2\) were asymptotically invariant in \(\ell^1\) under these four elements, then \(\xi_i=\sqrt{p_i}\) would be unit vectors with \(\|\lambda_s\xi_i-\xi_i\|_2^2\le\|\lambda_sp_i-p_i\|_1\to0\). Consequently

\[
 \langle\xi_i,P\xi_i\rangle
       =1-\tfrac18\sum_s\|\lambda_s\xi_i-\xi_i\|^2\longrightarrow1,
 \tag{FGJ.12}
\]

contradicting (FGJ.11). Restricting a compactly supported normalized groupoid Reiter field to range zero would give precisely these probabilities and translation estimates: that range fibre is the isotropy \(F_2\), and its four arrows are a compact finite test set. A compact support meets that range fibre in finitely many points, by a finite bisection cover. Thus neutral-kernel counting/Reiter amenability cannot be deduced for this example from any of the labels in (FGJ.10). This does not by itself decide full-to-reduced injectivity for a non-Hausdorff kernel; those are different assertions.

### 11CJ.5. A scalar character bounded for the actual reduced germ norm

There is also positive receiving information in the same example. For each negative \(x\), the actual regular source fibre of \(\mathcal H\) has just its unit. Its regular representation \(\lambda_x\) is a scalar character. Every compactly supported coefficient in a Hausdorff bisection chart can be written as a finite sum of coefficients on word bisections: cover its compact support by finitely many such charts and use a smooth partition on its transverse source chart. Each coefficient then extends by zero as a smooth compactly supported function on \(T\); every word acts globally on \(T\). This argument uses partitions within the original Hausdorff chart, without asserting a global partition on the non-Hausdorff arrow space. Thus a finite compact-bisection core kernel \(k\), written as such a finite sum, has

\[
 \lambda_x(k)=\sum_w k_w(x)\quad(x<0),\qquad
 \varepsilon_0(k)=\lim_{x\uparrow0}\lambda_x(k)
                       =\sum_w k_w(0).
 \tag{FGJ.13}
\]

Every expression is interpreted with the actual germ identifications. The coefficients in each presentation are continuous through zero. Distinct word germs at zero are exactly the group elements of (FGJ.8); summing their coefficients is independent of the presentation. More directly each negative evaluation is intrinsic, so its displayed limit is intrinsic whenever a presentation exists. These kernels span the standard compact-bisection core. The actual reduced norm gives

\[
 |\varepsilon_0(k)|\le\sup_{x<0}|\lambda_x(k)|
                   \le\|k\|_{C_r^*(\mathcal H)}.
 \tag{FGJ.14}
\]

Limits of the scalar multiplicative equations for \(\lambda_x\) give \(\varepsilon_0(kl)=\varepsilon_0(k)\varepsilon_0(l)\) and the star equation on the core. The bound extends these identities by norm density. A unit-space bump equal to one at zero shows that the extension is nonzero and has norm one. Therefore \(\varepsilon_0:C_r^*(\mathcal H)\to\mathbb C\) is an actual reduced scalar character, even though its restricted word-coefficient formula at zero is augmentation of a nonamenable isotropy group.

There is no contradiction: its norm in (FGJ.14) is the supremum of the original actual groupoid regular norms, including the neighbouring unit fibres. It is not the isolated reduced norm of \(F_2\). In particular it is not a character of \(C_r^*F_2\). Source counting has not been changed. This is an even scalar class; it is not asserted to have the original degree-one normal class or disk pairing of FOL-055.

### 11CJ.6. Why reduced labelled-action collapse fails

Distinguish the labelled transformation groupoid \(F_2\ltimes\mathbb R\) from its effective germ quotient \(\mathcal H\). On an invariant negative interval \(O\), the first has every labelled \(F_2\) arrow at each point, whereas the second has one actual unit arrow there. Their reduced open-ideal algebras are respectively \(C_0(O)\otimes C_r^*F_2\) and \(C_0(O)\). These identifications are isometric in the whole algebras: the original regular source fibres on \(O\) are the regular group fibres in the first case and the unit fibres in the second, and no arrow leaves the invariant interval.

Suppose a star homomorphism from the first whole reduced algebra to the second whole reduced algebra implemented the canonical germ collapse on compact-bisection kernels. For a nonzero smooth bump \(v\) on \(O\), normalized by \(\|v\|=1\), that prescribed action would satisfy

\[
 v\otimes P\longmapsto v,
 \qquad
 \|v\otimes P\|\le\sqrt3/2<1=\|v\|.
 \tag{FGJ.15}
\]

It collapses the four labelled coefficients to the same unit and sums them. The strict norm inequality contradicts contractivity of a star homomorphism. Hence no such reduced canonical quotient exists. This argument uses the labelled arrows only in the explicitly identified comparison algebra; it does not add them to the actual holonomy graph.

The example therefore has all of the following simultaneous properties: genuine free smooth holonomy at zero; complete identity formal jets there; no continuous faithful Hausdorff-group label at that isotropy; a nonamenable neutral isotropy; an actual reduced scalar germ character supplied by neighbouring source fibres; and no canonical reduced collapse from the labelled free-group transformation algebra. A scalar receiver for the labelled algebra and a scalar receiver for the actual germ algebra are different constructions. The original normal degree, full lift, whole counting/plaque graph and positive disk calibration still have to be proved in any proposed general FOL-055 construction. The example supplies neither a universal exclusion nor a replacement success criterion.

![FGJ.1–FGJ.15 retain the exact transverse coordinates, complete flat smooth flows and their actual germ groupoid. The free zero isotropy has identity formal jets and collapses under every continuous Hausdorff-group cocycle. Neighbouring unit fibres give an actual reduced scalar character, while the prescribed reduced collapse of labelled free-group arrows fails contractivity. The commutator graph shows numerical samples of the exact displacement; the lesson proves all scales and all word lengths. Sanov retains his historical matrix credit. The full positive lift and original degree one are retained; the even character is not a degree-one graph-class construction.](../figures/flat-actual-germ-norm.png)

**Figure 11CJ.1.** FGJ.1–FGJ.15 retain the exact transverse coordinates, complete flat smooth flows and their actual germ groupoid. The free zero isotropy has identity formal jets and collapses under every continuous Hausdorff-group cocycle. Neighbouring unit fibres give an actual reduced scalar character, while the prescribed reduced collapse of labelled free-group arrows fails contractivity. The commutator graph shows numerical samples of the exact displacement; the lesson proves all scales and all word lengths. Sanov retains his historical matrix credit. The full positive lift and original degree one are retained; the even character is not a degree-one graph-class construction. [Editable SVG](../figures/flat-actual-germ-norm.svg) and [reproduction instructions](../reproduction/flat-germ-norm/README.md).


## 11CK. An actual reduced character that is zero in KK-theory

The character of Section11CJ is nonzero and bounded for the original reduced germ norm. Nevertheless its entire KK-class vanishes. We prove this by an explicit homotopy through actual negative source fibres. The result separates a receiving norm estimate from the normal class and positive-disk calibration that the fundamental-class construction still needs. It does not exclude a different normally geometric construction.

The proof uses the actual compact-bisection core, FGJ.13–FGJ.14 and the interval definition of a Kasparov homotopy in Section0. Every construction and proof here is original CC0 mathematical expression. The ordinary positive Bott and inverse normal classes retain the human-source credits given in Section7.

### 11CK.1. A homotopy on the whole actual reduced algebra

Let \(A=C_r^*(\mathcal H)\) be the actual germ algebra of Section11CJ, with its original counting norm. At every negative source point the actual regular representation has one unit arrow and is a scalar character \(\lambda_x\). At zero the bounded character \(\varepsilon_0\) is the negative-side limit; its coefficient formula does not replace the actual zero isotropy \(F_2\) by a unit groupoid.

For the interval parameter \(t\), set

\[
 x(t)=-\frac{t}{1-t}\quad(0<t<1),\qquad
 \phi_0=\varepsilon_0,\quad
 \phi_t=\lambda_{x(t)}\ (0<t<1),\quad \phi_1=0.
 \tag{ECH.1}
\]

This is a family of star homomorphisms to \(\mathbb C\), including the zero homomorphism. If a core kernel is written in the finite global word-bisection form justified before (FGJ.13), then

\[
 \phi_t(k)=\sum_w k_w(x(t)),\qquad
 \phi_0(k)=\sum_w k_w(0).
 \tag{ECH.2}
\]

Each coefficient extends smoothly and compactly on the transverse line. The sum is therefore continuous as \(t\downarrow0\), by its coefficient limit at zero. It is continuous at every parameter below one. For a fixed core kernel, all the coefficient supports lie in one compact subset of \(\mathbb R\). The expression is identically zero once \(x(t)\) has left that subset toward negative infinity. Hence it is continuous at \(t=1\) as well. Its interpretation is independent of a core presentation because the negative regular evaluations and the limiting character are intrinsic.

The actual reduced norm, not a labelled group norm, gives

\[
 |\phi_t(a)|\le\|a\|_A\quad(0\le t\le1).
 \tag{ECH.3}
\]

For the endpoints this follows from FGJ.14 and the zero map; for the other parameters it is part of the original supremum regular norm. Given \(a\in A\), approximate it by a core kernel \(k\). The estimate

\[
 |\phi_t(a)-\phi_s(a)|
 \le 2\|a-k\|_A+|\phi_t(k)-\phi_s(k)|
 \tag{ECH.4}
\]

proves continuity for every fixed \(a\), including both endpoints. The same bound proves \(\phi_t(a)\to0\) at one. Thus

\[
 \Phi:A\longrightarrow C([0,1]),\qquad
 \Phi(a)(t)=\phi_t(a),\qquad
 \Phi(A)\subset C_0([0,1))
 \tag{ECH.5}
\]

is a contractive star homomorphism. Multiplication and star hold at every parameter, so hold as equations of continuous functions. The symbol \(C_0([0,1))\) means the ideal of functions on the closed interval that vanish at its endpoint one. There is no assertion that a fixed algebra unit disappears along a unital homotopy: the actual algebra over \(\mathbb R\) is nonunital. Compactly supported unit-space functions give its usual approximate unit.

**Theorem 11CK.1.** The homomorphism (ECH.5) is an actual reduced-algebra homotopy from \(\varepsilon_0\) to zero. In particular \([\varepsilon_0]=0\) in \(KK(A,\mathbb C)\).

**Proof of the KK assertion.** Take the graded right interval module

\[
 E^+=C([0,1]),\qquad E^-=0,\qquad
 \pi=\Phi,\qquad F=0.
 \tag{ECH.6}
\]

Its compact endomorphisms are multiplication by \(C([0,1])\): for \(b\) in that algebra, multiplication by \(b\) is the rank-one map \(\theta_{b,1}\). Thus every source square defect \(-\Phi(a)\) is compact. The commutator and self-adjointness defects are zero; zero is also an odd operator on this graded module. The endpoint at zero is exactly the finite scalar homomorphism cycle of \(\varepsilon_0\). The endpoint at one has zero left action and is degenerate. This is a Kasparov homotopy on the whole reduced source, proving

\[
 [\varepsilon_0]=0\in KK(A,\mathbb C),\qquad
 z[\varepsilon_0]\kappa=0
 \tag{ECH.7}
\]

for every correctly typed \(z\in KK(B,A)\) and \(\kappa\in KK(\mathbb C,D)\), with any fixed graded coefficient algebra \(D\). Associativity gives the second identity; no additional continuity or receiver hypothesis is involved. \(\square\)

The homotopy is point-norm continuous: every fixed source element has a continuous image. The norms of the homomorphisms themselves are one for all \(t<1\). For each such parameter choose a unit-space bump equal to one at its selected point; the contractive bound is attained. These bumps depend on \(t\). Consequently \(\|\phi_t-0\|=1\) below one, and the family is not continuous at one in the operator norm on the space of bounded maps \(A\to\mathbb C\). That stronger topology is not the topology of a star-homomorphism homotopy, and no such convergence was used in (ECH.4).

For a visible fixed test take \(v(x)=\max(1-|x|,0)\) on the unit space, a continuous compactly supported element of \(A\). Then

\[
 \phi_t(v)=\max\!\left(\frac{1-2t}{1-t},0\right)
       \ (0\le t<1),\qquad \phi_1(v)=0.
 \tag{ECH.8}
\]

It is already zero for \(t\ge1/2\), although the homomorphism norm is still one there. The continuous unit coefficient suffices for this example; the core continuity proof above used the smooth compact-bisection coefficients and then density.

### 11CK.2. What the actual normal degree requires

The vanishing is stronger than the observation that \(\varepsilon_0\) is even. With the original \(q=1\), any class that factors by this character in a Kasparov product is zero even if its later factors introduce the right Clifford coefficient. In particular it has pairing zero with every correctly typed original positive disk class. Finite Clifford evaluations, graph Morita correspondences and additional completed modules in later products cannot turn a zero factor into a class with pairing one.

This is a statement about an actual factorization through \([\varepsilon_0]\), not a prohibition on using its norm estimate in some other construction. For example the linear-model normal inverse in Section11CI changes the whole source representation before it is composed with a group-algebra receiver. Its left unit-space action is not point evaluation of the transverse algebra. Such a construction is not an instance of (ECH.7) unless it is separately proved to factor through this actual character.

There is a related local statement in every positive original dimension. Let \(U\) be a normal disk, \(q>0\), and let \(e_p:C_0(U)\to\mathbb C\) be evaluation at a point \(p\in U\). Choose a chart \(\psi:U\to\mathbb R^q\). If the given chart has image the radius-\(r\) ball, compose it with \(z\mapsto z/(r^2-|z|^2)\). Its radial map \(\rho\mapsto\rho/(r^2-\rho^2)\) increases from zero to infinity with positive derivative; its smooth inverse is \(y\mapsto2r^2y/(1+\sqrt{1+4r^2|y|^2})\), including at zero. Thus this supplies an actual whole-disk chart rather than an inclusion of a smaller disk. Fix a unit coordinate vector \(e_1\). The path

\[
 p(t)=\psi^{-1}\!\left(\psi(p)+\frac{t}{1-t}e_1\right),
 \qquad 0\le t<1,
 \tag{ECH.9}
\]

leaves every compact subset of \(U\) as \(t\uparrow1\). For each \(f\in C_0(U)\), set \(\Theta(f)(t)=f(p(t))\) below one and \(\Theta(f)(1)=0\). The definition of vanishing at infinity proves continuity at one; all other continuity follows from the path. These functions form a contractive star homomorphism to \(C([0,1])\). The same rank-one interval module of (ECH.6) proves

\[
 [e_p]=0\in KK(C_0(U),\mathbb C),\qquad
 b_U[e_p]\kappa=0
 \tag{ECH.10}
\]

for every correctly typed subsequent scalar-to-coefficient class. This conclusion applies to the original disk. It does not replace \(q\) by zero, reverse its orientation or remove its inverse line. The path is a homotopy of evaluation representations, not a comparison of the required inverse Dirac class with a point evaluation. The full inverse of Theorem7.17 still has the positive disk pairing \(+1\).

**Theorem 11CK.2 (finite point actions do not carry the normal disk class).** Suppose the restriction of a bounded scalar or finite-Clifford Kasparov cycle to \(C_0(U)\) is nondegenerate and has left action supported on finitely many points of the positive-dimensional original disk. Then this entire restricted KK-class is zero. It cannot be the required whole inverse normal class.

**Proof.** Its unit-space representation decomposes as \(H=\bigoplus_{i=1}^r H_i\), with \(f\) acting by \(f(p_i)\) on \(H_i\), after combining equal points. The projections \(P_i\) are even and commute with the retained right coefficient. Choose smooth compactly supported \(f_i\) with \(f_i(p_j)=\delta_{ij}\), using disjoint small disk neighbourhoods and bumps. Then \(\pi(f_i)=P_i\), so \([F,P_i]\) is compact. Replace \(F\) by \(\sum_iP_iFP_i\); the difference is a finite sum of compact off-diagonal blocks. Its straight-line interpolation is a cycle homotopy: the source defects change by compact operators, and all norms remain bounded. If a self-adjointness normalization is needed, replace \(F\) by its self-adjoint part using its compact self-adjointness defects on these finitely many projections first.

On each block the algebra acts by a scalar evaluation. The block square and self-adjointness defects are compact because they are the compressions of the tested defects for \(f_i\). Thus that block is precisely the product of the evaluation homomorphism class \([e_{p_i}]\) with its scalar-source Fredholm cycle, including its full grading and right coefficient. Equation (ECH.10) makes each block zero. The finite direct sum is zero. No finite-dimensionality of the blocks themselves is assumed. If there is a zero left-action summand, the same projection commutator argument removes its compact off-diagonal terms and leaves a degenerate summand. \(\square\)

### 11CK.3. Consequences for a whole graph construction

The actual norm bound of FGJ.14 remains valid, and the actual zero isotropy remains the full free group. The homotopy uses one source regular representation at each negative parameter; it does not alter the definition of the reduced norm, discard arrows at zero, or add ineffective free-group labels. By contrast the negative source fibres of the labelled transformation algebra have full free-group regular multiplicity, not one unit arrow. Their augmentation is not a reduced scalar character, by FGJ.15, so they do not provide the homotopy (ECH.1).

A general normally geometric construction must verify its receiving norm and its full ordinary normal restriction together. Norm-bounded point evaluations can have zero KK-class, and finite point actions annihilate the original local Bott test. The full normal operator, coefficient order, inverse spin/central line, scalar compact defects and actual graph creation relations remain substantive data. The results here neither assert a universal receiver assumption nor refute a class with the original full premise; they rule out the specific substitution of this zero character or a finite point restriction for that class.

![ECH.1–ECH.10 give the exact actual-source path, whole reduced-algebra homotopy, rank-one interval module and vanishing KK-class. The fixed triangular coefficient illustrates point-norm continuity while map norms stay one before the zero endpoint. Finite point normal actions have zero original disk restriction; the required full normal inverse retains pairing +1. The figure changes no actual source counts, original degree, inverse line or right Clifford coefficient, and excludes no different normal construction.](../figures/actual-character-homotopy.png)

**Figure 11CK.1.** ECH.1–ECH.10 give the exact actual-source path, whole reduced-algebra homotopy, rank-one interval module and vanishing KK-class. The fixed triangular coefficient illustrates point-norm continuity while map norms stay one before the zero endpoint. Finite point normal actions have zero original disk restriction; the required full normal inverse retains pairing +1. The figure changes no actual source counts, original degree, inverse line or right Clifford coefficient, and excludes no different normal construction. [Editable SVG](../figures/actual-character-homotopy.svg) and [reproduction instructions](../reproduction/actual-character-homotopy/README.md).


## 11CL. Whole flat isotropy and nonabelian receiving data

Finite-generator local stability does not provide a character on an entire holonomy derivative kernel. We construct a nontrivial perfect, locally free group as the whole actual flat isotropy of a coherently oriented smooth suspension. We then construct a proper affine Hilbert action of that entire group and check its compatibility with the topology of all actual holonomy bisections. These are different mathematical obligations. The normal fundamental class still requires its own reduced receiving construction.

All mathematical expression and diagrams here are original CC0. The flat free-flow prerequisite is supplied completely in Section 11CJ, FGJ.1–FGJ.8; the historical unipotent attribution remains there. Nielsen and Schreier are credited for the subgroup theorem whose needed proof is included below. Thurston's local-stability result is credited with its precise scope through Calegari's primary paper at the end of subsection 11CL.5. No protected source text or external artwork is incorporated.

### 11CL.1. A group that is locally free and perfect

Let F be the free group on a,b, with commutator \([u,v]=uvu^{-1}v^{-1}\), and define

\[
 \varphi(a)=[a,b],\qquad \varphi(b)=[a,b^2]. \tag{PFG.1}
\]

We first prove that this homomorphism is injective. Let \(F(A)\) be the free group on symbols \(a_j\), \(j\in\mathbb Z\), and let Z act by shifting \(a_j\) to \(a_{j+1}\). The semidirect product \(F(A)\rtimes\mathbb Z\) is generated by \(a_0\) and a shift b. The map from F sending a to \(a_0\) and b to the shift is inverse to the map sending \(a_j\) to \(b^jab^{-j}\) and the shift to b. Both composites fix the respective generators. Hence the kernel of \(F\to\mathbb Z,\ a\mapsto0,\ b\mapsto1\), is freely generated by the \(a_j=b^jab^{-j}\).

In that free kernel put \(u=a_0a_1^{-1},\ v=a_1a_2^{-1}\). The three elements \(u,v,a_2\) are a free basis of the free subgroup on \(a_0,a_1,a_2\): their inverse expressions are \(a_2=a_2,\ a_1=va_2,\ a_0=uva_2\). These substitutions are inverse homomorphisms on free generators. Thus u,v freely generate a rank-two group. Replacing its basis u,v by u,uv is again invertible, since \(v=u^{-1}(uv)\). Direct multiplication gives

\[
 [a,b]=u,\qquad [a,b^2]=u\,bub^{-1}=uv. \tag{PFG.2}
\]

Therefore the two images in PFG.1 are a free basis of their rank-two subgroup, proving injectivity without a finite-word test or a Hopfian assumption.

Take the direct limit of copies \(F_i\) of F, \(i\ge1\), along these injective maps. Write \(\iota_i:F_i\to\Gamma\) for its injections. The defining relations are

\[
 \iota_i(s)=\iota_{i+1}(\varphi(s)),\quad s=a,b;
 \qquad \Gamma=\bigcup_i\iota_i(F_i). \tag{PFG.3}
\]

Every finite subset lies in a single displayed free stage, because its elements have finitely many stage indices. Consequently every finitely generated subgroup of \(\Gamma\) is a subgroup of a free group and is itself free. Here is an independent proof of the needed Nielsen–Schreier subgroup theorem in the form used here. A subgroup K of a free group acts freely on its Cayley tree. The quotient graph has that tree as its covering graph. Lift a based edge path to identify its fundamental group with K. Choose a spanning tree in the connected quotient graph by successively adding edges reaching new vertices. The remaining oriented edges give free loop generators: tree paths reach their endpoints, and a loop's reduced edge word records exactly those edges and their inverses. Deleting successive inverse pairs proves that no nonempty reduced generator word is nullhomotopic. This gives the free basis of K. If K is nontrivial, map one such basis generator to 1 in Z and all others to zero. This is an onto homomorphism. Thus \(\Gamma\) is locally indicable as well as locally free.

In contrast, the entirety of \(\Gamma\) is perfect. Each of the two generators at stage i is a commutator of elements at stage i+1, by PFG.1 and PFG.3. Those stage generators generate their whole stage, and the stages exhaust \(\Gamma\). Therefore

\[
 \Gamma=[\Gamma,\Gamma]\ne\{1\},\qquad
 \operatorname{Hom}(\Gamma,A)=0\quad\hbox{for every abelian group }A.
 \tag{PFG.4}
\]

Nontriviality follows from the injective stage maps. \(\Gamma\) cannot be finitely generated: a nontrivial finitely generated subgroup would be free and would have a nonzero map to Z, contradicting perfection if it were the whole group. Also there is no nonzero compatible family of homomorphisms \(F_i\to\mathbb R\). On abelianizations the connecting maps are zero, so compatibility forces each stage character to vanish. This is an inverse-limit assertion about characters, not a claim that any individual stage has zero characters.

### 11CL.2. A faithful free action arbitrarily close to the identity

Use the two compactly supported smooth flat vector fields \(X_*,Y_*\) in FGJ.1–FGJ.4 of Section 11CJ of this lesson. They are supported in (0,1) inside \(J=(-1,1)\). For \(m\ge1\) let the free generators act by their flows at time \(\delta=2^{-m}\):

\[
 \rho_m(a)=\mathrm{Fl}^{X_*}_{2^{-m}},\qquad
 \rho_m(b)=\mathrm{Fl}^{Y_*}_{2^{-m}}. \tag{PFG.5}
\]

These are global orientation-preserving smooth diffeomorphisms, equal to the identity on a fixed collar of the ends of J. Every \(\rho_m\) is faithful. Indeed its small subinterval parameters are the transcendental dyadic multiples \(2^{-m}L2^{-k^2}\). For each nonempty reduced word, its unipotent projective matrix has a nonzero integer-polynomial off-diagonal entry or diagonal difference, as proved by the complete ping-pong argument FGJ.6–FGJ.8. That polynomial cannot vanish at these parameters. For all sufficiently large k, the finite word's intermediate trajectories remain in the middle of the subinterval, where the cutoff flows equal those projective transformations. Some point there is moved. Faithfulness follows for every word, including words longer than any computational cutoff.

For a fixed word w and finite derivative order k,

\[
 \|\rho_m(w)-\mathrm{id}\|_{C^k(J)}\longrightarrow0\quad(m\to\infty).
 \tag{PFG.6}
\]

The supremum includes derivatives of the difference, with all functions equal to the identity outside the fixed support. To check this limit directly, differentiate the compactly supported flow's integral equation. The first spatial derivative solves \(J'=Z'(\operatorname{Fl}_t)J,\quad J(0)=1\). Higher spatial derivatives solve linear equations whose forcing involves already constructed derivatives and bounded derivatives of Z. Induction, or their integral equations and Gronwall's elementary bound, gives convergence to the corresponding identity derivatives as t tends to zero. A finite composition and inversion preserve this \(C^k\) convergence by the chain rule. This proves PFG.6; no uniform estimate over all words is presumed.

### 11CL.3. Diagonal choices and smooth actual germs

Let \(c_N=2^{-N},\ r_N=2^{-N-3},\ I_N=(c_N-r_N,c_N+r_N)\), and \(A_N(y)=c_N+r_Ny\). The intervals are disjoint since \(9c_{N+1}/8<7c_N/8\). For every N choose the least integer \(m_N\ge N\) for which the following finite set of bounds holds:

\[
 \max_{1\le i\le N,\ s=a,b}
 \max_{0\le k\le N}
 \|\rho_{m_N}(\varphi^{N-i}(s))-\mathrm{id}\|_{C^k(J)}
 \le r_N^N. \tag{PFG.7}
\]

This least integer exists by PFG.6 applied to the finitely many displayed words and derivatives. The \(C^k\) norms and the least-integer rule specify a mathematical choice exactly; no numerical sample is presented as a certified computation of those infinitely many minima. Define global maps \(h_{i,s}\) on R by

\[
 h_{i,s}|_{I_N}=
 \begin{cases}
 A_N\rho_{m_N}(\varphi^{N-i}(s))A_N^{-1},&N\ge i,\\
 \mathrm{id},&N<i,
 \end{cases}
 \qquad h_{i,s}=\mathrm{id}\text{ off }\bigcup_N I_N.
 \tag{PFG.8}
\]

All interval seams fit smoothly because the reference maps are identity on a fixed collar. For \(N\ge\max(i,k,1)\), the chain rule and PFG.7 give the actual derivative estimate

\[
 \sup_{x\in I_N}|\partial_x^k(h_{i,s}(x)-x)|
 \le r_N^{N+1-k}=2^{-(N+3)(N+1-k)}. \tag{PFG.9}
\]

For every fixed k and integer \(p\ge0\), divide this bound by \(x^p\). On \(I_N\), \(x\ge7c_N/8=7r_N\). The resulting upper bound is \(7^{-p}r_N^{N+1-k-p}\), which tends to zero. The derivatives therefore extend continuously at zero with values 0 for the difference h-id. Their difference quotients tend to zero by taking p=1. Induction on k proves that every \(h_{i,s}\) is smooth and has exactly the identity Taylor series at zero. Each map is increasing and maps each interval onto itself; it is the identity on their complement. It is therefore a global bijection. Its derivative is positive on each interval, on collars and at zero. The smooth inverse function theorem on each point supplies a global smooth inverse. These are actual diffeomorphisms, rather than formal jets.

On every interval \(I_N\) with \(N\ge i+1\), the defining homomorphism \(\rho_{m_N}\) gives the exact relation

\[
 h_{i,s}=\varphi(h_{i+1,a},h_{i+1,b})_s.
 \tag{PFG.10}
\]

The same identity holds between these intervals, where all maps are identity. Its possible failures lie in the finitely many earlier intervals. Hence PFG.10 is an exact equality of germs at zero. We obtain a homomorphism

\[
 \iota_i(s)\longmapsto[h_{i,s},0],\qquad
 \Gamma\longrightarrow\operatorname{Germ}^{\infty}_+(R,0).
 \tag{PFG.11}
\]

It is injective. A nontrivial element has a representative w in some \(F_i\). Its action on every \(I_N\) with \(N\ge i\) is the conjugate of \(\rho_{m_N}(\varphi^{N-i}(w))\). The latter word remains nontrivial by the proved injectivity of varphi, and its action is nonidentity by faithfulness of \(\rho_{m_N}\). Every neighborhood of zero contains such an interval. Thus its germ is nonidentity. Conversely the generators of the constructed germ subgroup are precisely the images of the stages, so PFG.11 is onto that subgroup. Every element has the identity infinite jet, by the smoothness just proved and composition of Taylor series. We have realized the whole countable perfect group as actual flat germs, with no added group labels.

### 11CL.4. Realization in the frozen foliation scope

Choose a connected oriented open surface S with a countable free fundamental group having generators labelled (i,a),(i,b). One concrete surface has an exhaustion by compact genus-n surfaces with one boundary component, formed by adding successive handles through collars along a ray. Its handles are locally finite, and each compact exhausting surface lies in the interior of the next. Each compact initial handle union retracts onto its finite wedge of the usual two loops per handle. Every compact loop and compact homotopy lies in a finite such union. The inclusions retain the free loop bases. Van Kampen therefore gives the asserted countable free fundamental group. These loops are the global suspension labels; they are not asserted to be distinct holonomy germs everywhere.

Its connected universal cover \(\widetilde S\) has a free proper deck action. Send its free generators to the actual global \(h_{i,s}\), giving a genuine action on R. The diagonal quotient

\[
 V=(\widetilde S\times R)/\pi_1S,
 \qquad F=\text{the horizontal suspension foliation},\quad q=1,
 \tag{PFG.12}
\]

is a connected smooth Hausdorff foliated three-manifold. Properness and freeness of the diagonal action follow from the deck action on the first factor; lifted surface disks give the local quotient charts. The transversal above a selected point of S is R. Its actual holonomy groupoid H is the effective germ groupoid of the \(h_{i,s}\). At zero all generators fix the source, and PFG.10–PFG.11 prove that its entire actual isotropy group is \(\Gamma\). A word which is identity in \(\Gamma\) has an identity germ at zero, by the tail equality, and is the unit arrow there. Such a word is not retained as an extra suspension label in H. Nontrivial \(\Gamma\) words remain distinct actual arrows.

All transverse derivatives are positive. Choose the positive metalinear component \(\sqrt{Dh}>0\) with central phase 1. The chain rule gives its coherent cocycle law, and identity germs have lift 1, so it descends to H. This is the full specified \(ML_1^c\) orientation. Its complete inverse has physical generator \(-L_{t_1}\), central inverse phase 1, and the last right \(\mathrm{Cl}_1\); neither q nor the inverse coefficient is replaced by the suspension label space. Since every element of \(\mathcal H_0^0\) is flat, this isotropy lies in the derivative kernel and in the kernel of every finite or infinite formal jet map.

On \(x<0\) all displayed maps are identity, so those original source fibres consist only of units. At zero the original source fibre is \(\Gamma\), counted with all its distinct actual arrows. Plaque transport is that of the literal suspension charts, not of a labelled free-group completion. In particular we do not push a reduced norm on the labelled action through a nonexistent canonical reduced collapse map. The example is noncompact, which the frozen problem permits; it is not advertised as a compact example.

### 11CL.5. The exact consequence for the general construction

The derivative kernel in this eligible actual foliation is nontrivial and perfect at zero, although every nontrivial finitely generated subgroup of that isotropy is free and admits a map onto Z. Thus the pointwise finite-generator local-stability conclusion does not produce a nonzero real or abelian character on the entire derivative kernel. A compatible choice over its exhausting free stages would be zero by PFG.4. In particular one cannot infer a global scalar drift, global one-dimensional abelian quotient, or a globally nonzero character from those finite-generator conclusions.

Thurston's local-stability conclusion and this example agree. For the historical result and its precise finite-generator scope, see Danny Calegari, *Circular groups, planar groups, and the Euler class*, Geometry & Topology Monographs 7 (2004), printed pp.452–453, Theorem 3.2.1, [primary paper](https://msp.org/gtm/2004/07/gtm-2004-07-015p.pdf#page=22). The example and all its proof steps above are independent constructions; local indicability here was proved directly using the free stages, rather than imported from that theorem.

This eliminates the proposed inference to a whole-kernel character. It says nothing against a nonabelian auxiliary, a reduced class receiving construction with different operators, or an eventual general normal carrier. It does not establish a nonzero scalar KK-class for \(\Gamma\) or a reduced receiver for H. The following section explicitly constructs a nonabelian proper auxiliary for the entire abstract \(\Gamma\), then checks why pointwise isotropy data alone cannot install it on this actual groupoid.

### 11CL.6. A proper affine Hilbert action of the whole group

Enumerate \(\Gamma\) by a sequence \(g_1,g_2,\ldots\), giving each listed element a finite-stage representative. Repetitions do not matter. Choose \(k_n\) large enough to include those representatives for the first n elements. On the rank-two free stage \(F_{k_n}\), let \(\ell_n\) be its word length in its two generators. Set

\[
 L_n=\max_{j\le n}\ell_n(g_j),\qquad
 t_n=\frac{2^{-n}}{\max(1,L_n)},\qquad
 p_n(g)=\begin{cases}e^{-t_n\ell_n(g)},&g\in\iota_{k_n}(F_{k_n}),\\
 0,&\text{otherwise}.
 \end{cases} \tag{PFG.13}
\]

The length in this formula is the unique free-stage length after the injective identification. We prove that \(p_n\) is a normalized positive-definite function on \(\Gamma\). First, \(e^{-t\ell(g^{-1}h)}\) is positive definite on a free group. Its Cayley tree distance is \(\ell(g^{-1}h)\). For any finite list of vertices, their pairwise distances are sums of separation indicators over the finitely many edges of their spanning subtree. For one such edge, put \(z=e^{-t}\); the kernel has value 1 for two vertices on the same side and z for vertices on opposite sides. It is a Gram kernel: assign unit vectors \(v_0,v_1\) to the two sides with inner product z, possible because the matrix with diagonal 1 and off-diagonal z is positive for \(0<z\le1\). Taking tensor products of these side vectors over the finitely many relevant edges gives the product kernel \(e^{-t\,\mathrm{distance}}\). This proves positive definiteness on every finite list, hence on the group.

Extension by zero from a subgroup preserves positive definiteness. Partition any finite list of \(\Gamma\)-elements by its left cosets of that subgroup. Entries \(p(g_i^{-1}g_j)\) are zero between different cosets. Within one coset, write \(g_i=xh_i\); its block is the already positive matrix \(p(h_i^{-1}h_j)\). The full block-diagonal matrix is positive. This proves the claim for PFG.13. Also \(p_n\) is real, \(p_n(1)=1\), and belongs to \(c_0(\Gamma)\): each of its positive level sets is a finite free-word ball in its single stage. For \(j\le n\),

\[
 0\le1-p_n(g_j)\le t_n\ell_n(g_j)\le2^{-n}.
 \tag{PFG.14}
\]

For completeness use the Gram construction to form the cyclic GNS Hilbert space \(H_n\) from formal symbols \(e_g\) with inner product \(p_n(g^{-1}h)\), quotienting its null space and completing. Left translation is unitary, since the kernel is invariant under simultaneous left translation. Denote that representation by \(\pi_n\) and its unit vector \(e_1\) by \(v_n\). On \(\mathscr H_{\rm aux}=\bigoplus_n H_n\) use the unitary direct sum representation pi and define

\[
 b(g)=(\pi_n(g)v_n-v_n)_n,\qquad
 \|b(g)\|^2=2\sum_{n\ge1}(1-p_n(g)),\qquad
 b(gh)=\pi(g)b(h)+b(g).
 \tag{PFG.15}
\]

For \(g=g_j\), the finitely many terms with \(n<j\) are bounded, and the tail is summable by PFG.14. Thus \(b(g)\) is an actual vector in the Hilbert direct sum. Its cocycle law holds componentwise and therefore in that Hilbert space. The associated affine maps \(u\mapsto\pi(g)u+b(g)\) are a genuine action. This action is proper. Fix \(R\ge0\) and choose \(M>R\). Outside the finite union of the sets \(\{g:p_n(g)\ge1/2\}\), \(1\le n\le M\), its first M squared-norm summands in PFG.15 sum to more than M. Hence \(\{g:\|b(g)\|^2\le R\}\) is finite. This proves properness for the entire group, not just at a fixed stage. There is no nonzero abelian character involved. In particular the absence of whole-group characters is not an obstruction to this proper nonabelian auxiliary.

### 11CL.7. Pointwise properness does not extend to this actual groupoid

The abstract auxiliary just constructed is not an extra label of H. There is a direct topological test. Let \(\psi:\mathcal H\to[0,\infty)\) be any continuous function that is zero on all units. For gamma in \(\Gamma\) choose a word in the global \(h_{i,s}\) representing its actual germ at zero. For \(x_m<0\) tending to zero the germ on that word's bisection is the unit at \(x_m\), because every generator is identity there. The bisection topology gives \([\mathrm{word},x_m]\to[\mathrm{word},0]=\gamma\). Consequently

\[
 \psi(\gamma)=\lim_m\psi([\mathrm{word},x_m])=0
 \quad(\gamma\in H_0^0=\Gamma).
 \tag{PFG.16}
\]

Since \(\Gamma\) is infinite, such a function cannot be proper on that source fibre. Its zero level set there is the whole \(\Gamma\). This fibre is discrete in its relative topology: every étale bisection intersects it in at most one arrow. A cover by those single-arrow bisection intersections has no finite subcover. In particular \((r,s,\psi)\) cannot be a proper map, because the preimage of the singleton (0,0,0) is that noncompact fibre.

This includes any proposed continuous normalized conditionally negative-definite length on H, and any proposed extension of the squared norm of PFG.15 to such a function. At the single source zero, \(\Gamma\) has the explicitly proved proper affine Hilbert action. Nevertheless continuity on all actual arrow bisections forces a normalized scalar length to vanish on that entire source fibre. Even this whole isotropy group's Haagerup data therefore cannot be substituted for an actual continuous proper groupoid auxiliary. No assertion about the Haagerup property of every other isotropy group is needed or made. This rules out that substitution, not a general reduced normal class or a construction using different receiving mechanisms.

The general fundamental-class problem remains unfinished. Its actual general normally geometric reduced scalar class must retain the original degree, complete coherent inverse lift and right Clifford order, prove all actual graph connections and scalar compact defects, and pair the original positive disk as +1. The next receiving construction must handle whole isotropy and the topology and reduced norm of all original bisections. Neither the finite-stage characters nor the proper abstract isotropy action supply that missing construction. No universal character, amenability, Haagerup or gamma-equals-one condition has been imposed on other possible constructions.

![PFG.1–PFG.12 construct the injective rank-two free stages, perfect locally free limit, actual flat diffeomorphisms and coherently oriented suspension. The interval coordinates and derivative upper bounds are exact; the bounds are not sampled displacements. The entire actual isotropy is the perfect group, with full original q=1 inverse and last right Clifford coefficient retained.](../figures/perfect-flat-isotropy.png)

**Figure 11CL.1.** PFG.1–PFG.12 construct the injective rank-two free stages, perfect locally free limit, actual flat diffeomorphisms and coherently oriented suspension. The interval coordinates and derivative upper bounds are exact; the bounds are not sampled displacements. The entire actual isotropy is the perfect group, with full original q=1 inverse and last right Clifford coefficient retained. [Editable SVG](../figures/perfect-flat-isotropy.svg) and [reproduction instructions](../reproduction/perfect-flat-isotropy/README.md).


![PFG.13–PFG.16 construct a proper affine Hilbert action of the entire abstract isotropy group, using positive-definite stage functions and the convergent whole cocycle. Every continuous normalized scalar length on the actual germ groupoid nevertheless vanishes on that infinite source fibre. This excludes substituting pointwise proper isotropy data for a continuous proper groupoid auxiliary; it does not exclude a different reduced normal class.](../figures/proper-isotropy-no-groupoid-length.png)

**Figure 11CL.2.** PFG.13–PFG.16 construct a proper affine Hilbert action of the entire abstract isotropy group, using positive-definite stage functions and the convergent whole cocycle. Every continuous normalized scalar length on the actual germ groupoid nevertheless vanishes on that infinite source fibre. This excludes substituting pointwise proper isotropy data for a continuous proper groupoid auxiliary; it does not exclude a different reduced normal class. [Editable SVG](../figures/proper-isotropy-no-groupoid-length.svg) and [reproduction instructions](../reproduction/perfect-flat-isotropy/README.md).


## 11CM. A compact full-lift foliation with zero local reduced Bott class

The distinction between a class and a representation does not resolve every reduced-normal construction. In this section the positive local Bott class itself is zero in the actual reduced holonomy algebra. Consequently no scalar Kasparov class, geometric or otherwise, pairs it to one. We construct a compact smooth foliation with the full coherent \(ML_q^c\) lift for every prescribed \(q\ge1\). The construction retains nonisometric derivatives, actual germs and normalized source counting. It supplies a counterexample to the universal reduced statement with the positive-disk normalization, rather than a further sufficient hypothesis for that statement.

All mathematical expression and illustrations here are original CC0. Connes is credited for the transverse-normal question, Sanov for the classical unipotent free-group example and Kasparov for the pairing formalism. The proofs needed here are supplied below. Baire's theorem is used in its complete-metric-space form: a countable intersection of dense open sets is dense, proved by nested closed balls with radii tending to zero. No external text, source PDF or artwork is incorporated.

### 11CM.1. Analytic on an entire arc and flat at both endpoints

Identify the oriented circle with \(\mathbb R\cup\{\infty\}\). Its positively oriented coordinate near infinity is \(y=-1/x\). Let \(J_+=[0,\infty]\) and \(J_-=[-\infty,0]\) denote its two closed arcs; their intersection consists of the two endpoints \(0,\infty\). Their interiors are the positive and negative half-lines.

Let \(D=\{z\in\mathbb C:\operatorname{Re}z>0\}\) and \(r(z)=(1+z)^{-1}\). Define \(B\) to be the real Banach space obtained by closing the real-coefficient polynomials in \(r\) in the supremum norm on \(D\). Every element is bounded and holomorphic on \(D\), is real on the positive real axis, and obeys the Cauchy bounds there. Uniform limits on \(D\) converge uniformly on every compact subdisk, proving holomorphicity and completeness. Rational-coefficient polynomials are a countable dense subset. For \(f\in B\), put

\[
 h(x)=e^{-1/x-x}\quad(x>0),\qquad
 X_f(x)=h(x)f(x)\,\partial_x\quad(x>0),\qquad
 X_f(x)=0\quad(x\le0).
 \tag{SGD.1}
\]

**Lemma 11CM.0 (elementary analytic foundations).** The Banach coefficient space just defined has convergent local power series, with the coefficient and derivative estimates used below. The flows of SGD.1 depend real analytically on a real affine coefficient parameter and on positive initial position. A real analytic function on a connected interval which vanishes on an open interval vanishes everywhere.

**Proof.** A polynomial in \(r(z)=(1+z)^{-1}\) has, at a center \(c\in D\), a power series converging for \(|z-c|<|1+c|\). This follows by the geometric series for \((1+c+(z-c))^{-1}\), its finite products and finite linear sums. On any circle \(z=c+\rho e^{it}\) contained in \(D\), the series converges uniformly. Integrating its terms against \(e^{-ikt}\), and using \(\int_0^{2\pi}e^{i(j-k)t}\,dt=0\) for \(j\ne k\), gives its coefficient formula. For a uniform limit \(f\in B\), the same circle integrals converge. Define

\[
 a_k(c)=\frac{1}{2\pi\rho^k}\int_0^{2\pi}
          f(c+\rho e^{it})e^{-ikt}\,dt,
 \qquad |a_k(c)|\le\|f\|_B\rho^{-k}.
 \tag{SGA.1}
\]

These coefficients are independent of the chosen admissible radius, since the coefficients of the approximating rational polynomials are. On \(|z-c|\le\rho'<\rho\), their series and the approximating series converge uniformly, by the geometric majorant \(\|f\|_B\sum_k(\rho'/\rho)^k\). The approximants converge to \(f\), so

\[
 f(z)=\sum_{k\ge0}a_k(c)(z-c)^k,
 \qquad |f^{(k)}(c)|\le k!\|f\|_B\rho^{-k}.
 \tag{SGA.2}
\]

Termwise derivatives are justified on smaller disks by the convergent series with the extra polynomial factors in \(k\). For positive real \(c=x\), take \(\rho=x/2\), obtaining SGD.2. This derivation needs only the geometric series and ordinary circle integration; no general complex integral theorem is being used as a missing prerequisite. Completeness of \(B\) follows because bounded functions with the supremum norm are complete, its defining subspace is closed, and the coefficient argument proves that every such limit has these local expansions.

We give the parameter-flow argument explicitly. For \(s>0\), let \(\mathcal A_s\) be the space of complex power series in a variable \(\zeta\), with

\[
 \mathcal A_s=\left\{\sum_{n\ge0}c_n\zeta^n:
                   \sum_{n\ge0}|c_n|s^n<\infty\right\},
 \qquad\|c\|_s=\sum_{n\ge0}|c_n|s^n.
 \tag{SGA.3}
\]

This is a unital commutative Banach algebra: convolution of coefficients gives the submultiplicative bound by the product of the two absolutely convergent sums. For completeness, a Cauchy sequence has a limit in every coefficient. Every finite partial sum of the absolute coefficient differences is bounded by the Cauchy norm bound; taking its increasing limit proves the same bound for the full sum. Comparing to one fixed member gives summability of the limiting coefficients and norm convergence. Evaluation for \(|\zeta|\le s\) is contractive, and each resulting function is analytic for \(|\zeta|<s\).

Fix a positive position \(x_0\) and real affine parameter \(\lambda_0\). The field coefficient \(Z_\lambda(x)=h(x)(f_0(x)+\lambda f_1(x))\) has convergent position power series near \(x_0\), by SGA.2, the geometric series for \(1/x\), and the convergent exponential series for \(h\). Choose a position radius \(\eta\) strictly smaller than those convergence radii. The sums of absolute coefficients and of their differentiated coefficients are finite on \(|x-x_0|\le\eta\). After substituting \(\lambda=\lambda_0+\zeta\), these series define maps in \(\mathcal A_s\) for every sufficiently small \(s\). The same absolute sums give bounds \(M\) for the field and \(L\) for its Lipschitz constant on the Banach-algebra ball \(\|X-x_0\|_s\le\eta\). The latter bound follows by factoring each difference \(X^n-Y^n\) into its \(n\) terms. It includes any analytic initial position close to \(x_0\).

Fix the initial datum \(X_{\rm in}\in\mathcal A_s\). On continuous \(\mathcal A_s\)-valued paths for \(|t|\le\tau\) with this initial value, Picard's map is

\[
 \mathcal P(X)(t)=X_{\rm in}+\int_0^t Z_{\lambda_0+\zeta}(X(u))\,du.
 \tag{SGA.4}
\]

Take \(M\tau\le\eta/2\), \(L\tau<1/2\), and the initial displacement norm at most \(\eta/2\). The map preserves the closed path ball and contracts with constant \(L\tau\). Iterating it gives a uniform limit in the complete path space; the geometric bound on successive differences proves convergence and uniqueness. Each parameter evaluation is contractive and commutes with the integral, so it gives the actual scalar ODE solution. The solution lies in \(\mathcal A_s\), proving convergent parameter expansions. The same argument with the initial position \(x_0+\zeta\) proves analytic initial-position dependence. Analytic initial data from a preceding time segment are handled by decreasing \(s\): their nonconstant coefficient sum tends to zero as \(s\downarrow0\), and hence fits the required initial ball.

A complete real trajectory segment staying inside the positive arc is compact. Subdivide its time interval so the preceding construction applies on each piece; continuity with respect to the real parameter and position keeps a small common parameter neighborhood inside the same buffers. Successive analytic initial values give the full positive flow and every fixed finite composition, including negative-time inverse letters. The usual continuous-dependence assertion used here follows directly by the same integral inequality and its iterated exponential estimate supplied in Lemma 11CM.2. Analyticity is local at each real parameter, which is exactly the analyticity on the real line asserted in SGD.5.

Finally, a convergent power series at a point is either identically zero in a neighborhood or has a least nonzero coefficient. In the latter case, factor its lowest power; the remaining series is nonzero on a sufficiently small neighborhood by continuity. Zeros are therefore isolated there. For a real analytic function, the set of points with a zero neighborhood is open. It is also closed in the connected interval if nonempty: at a limit point, either the local series vanishes identically or the just-proved isolation contradicts the approaching zeros. The set is consequently the entire interval. Apply this to a word minus the identity as a function of initial position, and to SGD.5 as a function of the real parameter. This proves precisely the two identity-principle uses below. \(\square\)

**Lemma 11CM.1.** These are smooth vector fields on the circle, flat at both endpoints, zero on the negative arc and real analytic on the positive arc. The map from \(B\) to smooth circle vector fields is continuous for every fixed derivative order.

**Proof.** The disk of radius \(x/2\) about a positive \(x\) lies in \(D\), so

\[
 |f^{(k)}(x)|\le k!\,\|f\|_B(2/x)^k.
 \tag{SGD.2}
\]

Every derivative of \(h\) is \(h\) times a polynomial in \(1/x\); the constant terms also account for differentiating \(e^{-x}\). Leibniz's rule therefore bounds each derivative of \(h f\) by a fixed polynomial in \(1/x\) times \(\|f\|_B e^{-1/x-x}\). It tends to zero faster than every power of \(x\) at zero. Continuous extension of derivatives and the difference-quotient argument, successively in their orders, prove a smooth flat extension there.

In the infinity chart, the field coefficient on \(y<0\) is

\[
 y^2 e^{y+1/y}f(-1/y),
 \qquad X_f=0\text{ on }y\ge0.
 \tag{SGD.3}
\]

The chain rule and SGD.2 bound each derivative of \(f(-1/y)\) by a fixed polynomial in \(1/|y|\) times \(\|f\|_B\). The factor \(e^{1/y}\) dominates every such polynomial as \(y\uparrow0\). The same difference-quotient argument proves smoothness and flatness at infinity. On compact subintervals of the positive arc, SGD.2 gives the usual uniform bounds. Finitely many circle charts give the asserted continuity. Analyticity on the positive arc follows from the holomorphic factors. \(\square\)

The time-one flow \(F_f\) is an orientation-preserving smooth circle diffeomorphism. Compactness gives completeness. It fixes the entire negative arc and both endpoints; uniqueness of the ODE prevents a positive trajectory from reaching a stationary endpoint in finite time. It is analytic on the positive arc. Its endpoint jets are the identity: the equation for the first derivative of the flow has zero coefficient at an endpoint, and the equations for all higher derivatives have zero inhomogeneous terms there, by flatness of the field. Its inverse is the time-minus-one flow.

### 11CM.2. One pair with no identity word on any positive open interval

**Lemma 11CM.2.** There are \(f,g\in B\) such that the time-one flows \(a=F_f,b=F_g\) generate a free group \(\Gamma=F_2\), and no nonidentity word has the identity germ at any point of \(J_+\), including its endpoints.

**Proof.** For each nonempty reduced word \(w\) in two letters and their inverses, let \(E_w\subset B^2\) consist of the pairs for which \(w(F_f,F_g)\) moves at least one positive point. Flow dependence and Lemma 11CM.1 show that \(E_w\) is open. We prove it dense, including the existence of a test pair for every word.

Use the projective matrices

\[
 A_t=\begin{pmatrix}1&t\\0&1\end{pmatrix},\qquad
 B_t=\begin{pmatrix}1&0\\t&1\end{pmatrix}.
 \tag{SGD.4}
\]

Their projective flows are those of \(t\partial_y\) and \(-t y^2\partial_y\): \(y\mapsto y+t\) and \(y\mapsto y/(1+ty)\). At \(t=2\), put \(D_A=\{|y|>1\}\cup\{\infty\}\) and \(D_B=\{|y|<1\}\). For every nonzero integer \(n\), direct substitution gives \(A_2^n(D_B)\subset D_A\) and \(B_2^n(D_A)\subset D_B\). A nonzero pure power is nonidentity. A mixed word conjugates to a cyclically reduced alternating word, cyclically conjugate to one beginning with an \(A\)-power and ending with a \(B\)-power. Apply it to zero. Its rightmost \(B\)-power fixes zero; the next \(A\)-power sends zero into \(D_A\), and the remaining syllables alternate the two domains, finally ending in \(D_A\). The word does not fix zero. This proves freeness for every word.

The entries of \(w(A_t,B_t)\) are integer polynomials, including inverse letters. At least one off-diagonal entry or diagonal difference is nonzero at two. Its polynomial has finitely many real zeros, so there are arbitrarily small positive \(t\) for which the projective word is nonidentity. Fix such a \(t\), small enough that all intermediate flows for this finite word, starting in \((-1/4,1/4)\), remain in \((-1/2,1/2)\). For a word of \(L\) letters, it suffices to take \(Lt<1/4\): translations move by \(t\), and a fractional linear step and its intervening flow in that interval move by at most \(t/2\), after also taking \(t<1\). A nonidentity fractional linear map cannot fix an open interval, since its fixed-point equation is a nonzero polynomial of degree at most two. Choose a moved point \(y_*\) in that starting interval.

Translate to \(x=y+3\). On \([1,5]\), the two desired field coefficients are \(t\) and \(-t(x-3)^2\). Divide them by the positive continuous function \(h(x)\). Polynomials in \(r(x)\) approximate either resulting continuous function uniformly, since \(r\) is a homeomorphism from \([1,5]\) to \([1/6,1/2]\). Here is the needed approximation proof. After an affine change to \([0,1]\), the Bernstein polynomial of a continuous \(F\) is \(\sum_{k=0}^n F(k/n)\binom nk t^k(1-t)^{n-k}\). Its nonnegative weights sum to one; differentiating the binomial formula gives mean \(t\) and variance \(t(1-t)/n\le1/(4n)\). Split the weighted error into \(|k/n-t|\le\delta\) and its complement. The first part is at most the modulus of continuity of \(F\) at \(\delta\), and the second at most \(2\|F\|_\infty/(4n\delta^2)\), by the variance bound. First choose \(\delta\) small and then \(n\) large. This proves uniform polynomial approximation. The resulting polynomials in \(r\) belong to \(B\). Their SGD.1 fields can therefore approximate the two desired coefficients as closely as needed on \([1,5]\).

Here uniform approximation is enough for the flow test. Compare one approximating trajectory with the desired trajectory while both stay in \([1,5]\). The desired coefficient has a fixed Lipschitz constant \(L\) there. If the coefficient error is at most \(\epsilon\), integration gives a trajectory error \(d(t)\le d(0)+\epsilon t+L\int_0^t d(s)\,ds\). Iterating the integral inequality bounds it by \(e^{Lt}(d(0)+\epsilon t)\): the repeated constant-integral terms are the exponential series, and the bounded remaining tail tends to zero. The original trajectories stay in \([2,4]\), so a sufficiently small error keeps the new ones inside the buffer. Repeat this estimate through the finitely many forward and inverse letters; inverse letters use the negative of the same field. Make the final error smaller than half the nonzero displacement of \(y_*\). This produces a pair \((f_*,g_*)\in E_w\), tested at \(x_*=y_*+3\).

Now fix any \((f_0,g_0)\in B^2\) and use the affine line from that pair to the test pair. The function

\[
 t\longmapsto w(F_{f_0+t(f_*-f_0)},F_{g_0+t(g_*-g_0)})(x_*)-x_*
 \tag{SGD.5}
\]

is real analytic on \(\mathbb R\). Lemma 11CM.0 supplies the convergent parameter expansions of each flow and every fixed finite composition. On sufficiently small position and parameter disks, the absolute coefficient sums of that lemma bound the field by \(M\) and its Lipschitz constant by \(L\). Choose a real time interval of radius \(\tau\) with \(M\tau\) less than half the position radius and \(L\tau<1/2\). Its fixed-initial-data Picard construction in the power-series Banach algebra then supplies the local analytic dependence. On a compact real parameter interval, completeness and continuous dependence keep the finitely many trajectory segments of this word inside a compact subset of the positive arc. A finite subdivision of time therefore applies that construction successively, including negative-time inverse letters. The resulting composition is real analytic near every real parameter. No analytic continuation through either endpoint is used.

SGD.5 is nonzero at one, so it cannot vanish throughout a neighborhood of zero. Points of \(E_w\) therefore occur arbitrarily near the original pair. This proves density. Apply Baire's theorem to the countably many \(E_w\) in the complete space \(B^2\). For clarity, one may enumerate the words and choose successive closed balls, each inside the previous open ball and the next \(E_w\), with radii tending to zero. Completeness gives a common point in every chosen open set. Centers can be chosen among rational-polynomial pairs; the construction needs no assumption that a finite numerical word list certifies freeness.

For the resulting pair, every word is analytic and nonidentity on the connected positive arc. The analytic identity theorem says it cannot equal the identity on any open subinterval. Hence its germ is nonidentity at every positive point. If its germ at either endpoint were identity, it would equal the identity on a positive interval adjoining that endpoint, the same contradiction. All words have identity jets at both endpoints by Lemma 11CM.1. These assertions concern actual germs, not formal jets. \(\square\)

### 11CM.3. Actual source fibres and the entire reduced algebra

Let \(\mathcal H\) be the effective germ groupoid of this global circle action. At each source in \(J_+\), the map from \(\Gamma\) to its source fibre is a bijection. Indeed different words cannot have equal germs there, by Lemma 11CM.2; every germ is represented by a word. The endpoints have isotropy \(\Gamma\). At other positive points the isotropy is their stabilizer subgroup, not necessarily the whole group. The source fibre still has one distinct arrow for every word. At each source in the interior of \(J_-\), all words are the identity on a neighborhood, so the source fibre consists of one unit arrow.

This is a non-Hausdorff smooth étale groupoid. A sequence of negative unit arrows approaching either endpoint lies on every word bisection. It converges both to the endpoint unit and to the distinct endpoint germ of each nonidentity word. The actual negative source count is one; it is not the count of an ineffective labelled transformation groupoid.

Write \(u_w\) for the unitary supplied by the global word bisection. The compact circle makes these elements unital. The usual non-Hausdorff core consists of finite sums of compactly supported continuous functions on Hausdorff bisection charts, extended by zero. Every compact bisection test is a finite sum of restrictions of global word bisections: refine its support by finitely many such charts and use a partition of unity on their source domains, then extend each compactly supported coefficient to the circle. Thus the finite sums \(\sum_w\phi_w u_w\), with \(\phi_w\in C(S^1)\), are dense in the germ algebra. Local equality of germs is already imposed in this core.

Put \(R=C(J_+)\rtimes_r\Gamma\). Its regular point representation at any \(x\in J_+\) agrees exactly with the original source regular representation of \(\mathcal H\) there, under the word-to-germ bijection. On the negative interior, the latter is the scalar sum of the coefficients. Therefore the core map

\[
 \Phi\left(\sum_w\phi_wu_w\right)
 =\left(\sum_w\phi_w|_{J_+}u_w,\quad
                 \sum_w\phi_w|_{J_-}\right)
 \tag{SGD.6}
\]

is a star homomorphism into \(R\oplus C(J_-)\), with norm equal to the original reduced source supremum. The negative scalar coefficient extends continuously to both endpoints, so its supremum over the negative interior equals its supremum over \(J_-\). This second endpoint evaluation is a limit of negative source representations; it is not the isolated augmentation of a reduced group algebra. Multiplication is respected because every word acts identically on \(J_-\). Any core relation zero as an actual germ section has zero positive coefficients by the faithful germs and zero negative sum, so the map is well defined.

We next prove that the completed image is the entire direct sum. In the group algebra let

\[
 p=\tfrac14(u_a+u_a^{-1}+u_b+u_b^{-1}),\qquad
 c=\sqrt3/2<1.
 \tag{SGD.7}
\]

**Lemma 11CM.3.** Its regular group norm is at most \(c\).

**Proof.** Root the degree-four Cayley tree. For a finitely supported vector \(v\), sum each unoriented edge once, from parent to child, and use

\[
 2|v_P||v_C|\le 3^{-1/2}|v_P|^2+3^{1/2}|v_C|^2.
 \tag{SGD.8}
\]

A nonroot vertex has three children and one parent, giving total coefficient \(2\sqrt3\). The root has four children and coefficient \(4/\sqrt3\le2\sqrt3\). This bounds the absolute quadratic form of adjacency by \(2\sqrt3\|v\|^2\). Self-adjointness and finite-vector density give its operator norm bound. Dividing by four proves the result. \(\square\)

In every positive regular source representation the group unitaries alone form the ordinary regular representation. Thus \(\|p^n\|_R\le c^n\). On the negative interior the scalar sum is exactly one. For every \(\phi\in C(S^1)\), SGD.6 gives

\[
 \left\|\Phi(\phi p^n)-(0,\phi|_{J_-})\right\|
       \le\|\phi\|\,c^n\longrightarrow0.
 \tag{SGD.9}
\]

Every continuous function on either closed arc extends to the circle, by joining its endpoint values continuously along the other arc. Hence the image contains \(0\oplus C(J_-)\). Subtract that component from each core image. The remaining positive components contain all crossed polynomials, since their coefficients on \(J_+\) also extend to the circle. They are dense in \(R\). Isometry makes the completed image closed, proving

\[
 C_r^*(\mathcal H)\cong
       \bigl(C(J_+)\rtimes_rF_2\bigr)\oplus C(J_-),
 \qquad j(\phi)=(\phi|_{J_+},\phi|_{J_-}).
 \tag{SGD.10}
\]

In particular there is an actual central projection

\[
 e_- =\lim_{n\to\infty}p^n=(0,1),\qquad
 \|p^n-e_-\|\le c^n.
 \tag{SGD.11}
\]

It has value one in the negative unit source representations and zero in every positive or endpoint regular representation. This discontinuity across the non-Hausdorff arrow limits is permitted for an element of the reduced completion. It is not a continuous function on the unit circle. The proof uses all actual source fibres and the original counting system, without deleting isotropy or replacing the groupoid by a dense unit reduction.

### 11CM.4. An explicit nullhomotopy of the positive local disk

Choose an oriented open interval \(U\) strictly inside the positive arc. Let its positive Bott unitary \(u_U\) be one outside \(U\) and wind once positively through \(U\). On \(J_+\) it has a continuous real argument \(\theta\), equal to zero near endpoint zero and to \(2\pi\) near endpoint infinity. On \(J_-\) it is one. By SGD.10 the unitary homotopy

\[
 v_s=(e^{is\theta},1),\quad 0\le s\le1,
 \qquad v_0=1,\quad v_1=j(u_U)
 \tag{SGD.12}
\]

lives in the unitization of the actual reduced algebra (here the algebra is already unital). It is norm continuous, since \(\theta\) is bounded on the compact arc. The two endpoint values on the different direct summands need not match. This is precisely what SGD.11 makes possible. Consequently

\[
 j_*[u_U]=0\quad\text{in }K_1(C_r^*(\mathcal H)).
 \tag{SGD.13}
\]

This proof does not use an exactness theorem for a reduced closed-unit restriction, a maximal-to-reduced KK equivalence or a group unit-index auxiliary. It proves zero by an actual unitary homotopy. Equivalently the unit map factors through restriction to the two contractible closed arcs, whose odd K-groups vanish.

The same conclusion holds for every positively oriented interval disk in the circle. Its extension-by-one loop has winding number one, hence the same \(K_1(C(S^1))\) class as \(u_U\). The circle computation follows by lifting its argument on the cut interval: after subtracting the integer winding, the remaining real argument is periodic and contracts its exponential. Thus equality of winding gives equality of loop classes, with the positive sign fixed.

### 11CM.5. Compact foliations in the original prescribed codimension

For the specified \(q\ge1\), set \(T_q=S^1\times(S^1)^{q-1}\), with \(\Gamma\) acting by the constructed diffeomorphisms on the first factor and identically on the others. The actual germ groupoid is \(\mathcal H\) times the unit groupoid of the remaining torus. Finite kernels with product coefficients are dense. Their regular norm is the supremum over the remaining torus of the norm already computed. Thus

\[
 A_q=C_r^*(\mathcal H\times (S^1)^{q-1})
      \cong C_r^*(\mathcal H)\otimes C((S^1)^{q-1}).
 \tag{SGD.14}
\]

Choose ordered positively oriented intervals in the \(q\) coordinates. Their product is an oriented open disk chart. Its compactly supported Bott class is the ordered external product of the \(q\) positive one-dimensional Bott classes. Equation SGD.13 and bilinearity of the product show that its image is zero in \(K_q(A_q)\). No odd factor is interchanged. The equivalent graded formulation keeps the final right \(Cl_q\) coefficient in the original order. The positive open-disk excision in Section 7 gives the conclusion for the entire original disk: for any original positively oriented disk \(U\), select a smaller ordered product chart \(W\subset U\). Its first-coordinate interval has the same positive circle class already shown to vanish. The exact open-disk identity is \(b_U=b_W[j_{WU}]\); composition with the original unit inclusion therefore makes the entire original disk class zero. This retains its dimension and orientation and uses no assertion about global isotopy of arbitrary disk embeddings.

Let \(S\) be a closed oriented genus-two surface, with presentation \(\langle a_1,b_1,a_2,b_2\mid[a_1,b_1][a_2,b_2]=1\rangle\). Send \(a_1,a_2\) to \(a,b\), and both \(b_i\) to the identity. This is onto \(F_2\). Its connected regular cover is the path-class cover obtained by identifying paths with the same endpoint and the same image under this homomorphism. Small simply connected surface disks give its smooth covering charts and free proper deck action. Form the diagonal quotient of that cover times \(T_q\). It is a smooth Hausdorff compact manifold \(V_q\): deck properness gives a Hausdorff quotient and local surface charts identify it with a bundle with compact base \(S\) and compact fibre \(T_q\). Horizontal plaques define a smooth foliation \(F_q\) of leaf dimension two and normal dimension exactly \(q\). The fibre over a chosen base point is a complete transversal. Its holonomy groupoid is exactly the actual effective germ groupoid above; identity germs of loops in the covering kernel are removed.

In the global positively oriented torus frame, every transverse derivative is \(\operatorname{diag}(d_w,1,\ldots,1)\) with \(d_w>0\). Lift this through the positive, contractible diagonal subgroup of \(GL_q^+\) to its metalinear cover, with central phase one. The positive square-root coordinate obeys the exact chain rule at composed source and target points. Identity germs give lift one, so the lift descends to the actual arrows. This supplies the full coherent \(ML_q^c\) structure under the stated nonisometric convention. It does not supply an invariant metric. Already at either endpoint a nonidentity germ fixes the point with identity derivative in all \(q\) coordinates. A local Riemannian isometry fixing a point with identity derivative would fix every short radial geodesic by the exponential-coordinate identity, hence would be the identity germ.

The inverse normal coefficient is the complete inverse of this lift, with its central phase still one; its mixed-generator signs and final ordered right \(Cl_q\) are retained as in Theorem 7.17. In this example the oriented normal bundle is globally trivial and the central line is trivial. No replacement of \(q\), spin structure or inverse line is used to obtain the zero class.

The completed actual graph/transversal correspondence of Sections 4 and 7 identifies the reduced holonomy graph algebra \(C_r^*(\operatorname{Hol}(V_q,F_q))\) with \(A_q\) up to Morita equivalence. Its source columns contain every actual germ and its plaque integrals use the specified half-densities. The local transversal Bott class is sent to the graph local class by this very correspondence. Naturality and the just-proved zero give

\[
 b_U=0\quad\text{in }K_q(C_r^*(\operatorname{Hol}(V_q,F_q))).
 \tag{SGD.15}
\]

**Theorem 11CM.4 (failure of the universal normalized reduced statement).** For each \(q\ge1\), the compact smooth full-lift foliation just constructed admits no class \(x\in KK^q(C_r^*(\operatorname{Hol}(V_q,F_q)),\mathbb C)\) with the required positive local-disk pairing \(+1\).

**Proof.** Under the original ordered coefficient convention, the pairing is the Kasparov product of the local \(K_q\) class with \(x\). The full inverse line and Clifford evaluation specify that product and its sign; they cannot change the zero input. Equation SGD.15 and bilinearity give

\[
 \langle b_U,x\rangle=0\quad\text{for every such }x.
 \tag{SGD.16}
\]

This excludes every scalar cycle with the required normalization, including any normally geometric bounded phase. Changing a receiving representation or supplying another scalar auxiliary cannot make the zero class pair to one. \(\square\)

This obstruction applies to the universal reduced formulation under only the full nonisometric \(ML_q^c\) lift and the positive-disk normalization. It does not contradict a theorem with an invariant transverse metric, or a statement for a maximal completion. The construction differs essentially from Section 11CJ: that earlier positive half-line contains many open identity collars and unit gaps, so its positive source fibres are not everywhere the regular \(F_2\) fibres. Its average cannot satisfy the uniform positive bound used in SGD.9. Here analytic continuation on the entire positive arc proves the missing uniform germ faithfulness. The two examples cannot be interchanged in this norm calculation.

![Actual arc source fibres, the uniform Cayley-tree bound and the central projection obtained by a norm limit](../figures/split-germ-reduced-algebra.png)

**Figure 11CM.1.** SGD.6–SGD.11. The arcs are the two exact closed coordinate arcs of the circle. Their endpoints have full free-group isotropy and belong to the positive regular source family; the negative interior has one actual unit arrow per source. The plot is the proved upper bound \((\sqrt3/2)^n\), not a measurement of an unspecified Baire-selected flow. [Editable SVG](../figures/split-germ-reduced-algebra.svg) and [reproduction source](../reproduction/split-germ-disk/README.md).

![The original positive loop and its exact contraction after the two arc components split in the actual reduced algebra](../figures/split-germ-bott-nullhomotopy.png)

**Figure 11CM.2.** SGD.12–SGD.16. The argument rises from zero to \(2\pi\) along the positive closed arc. At intermediate homotopy parameters its endpoint exponential differs from the constant negative component, as allowed by the proved direct sum. This displays the actual unitary nullhomotopy, not a homotopy of continuous functions on the unsplit circle. Product coordinates and the original final right \(Cl_q\) preserve the zero class. [Editable SVG](../figures/split-germ-bott-nullhomotopy.svg) and [reproduction source](../reproduction/split-germ-disk/README.md).

## 12. Exercises

**Exercise 1 (6 points).** *basic.* In the product foliation \(\mathbb R\times S^1\), choose a real compactly supported function \(\psi\) with \(\int|\psi|^2=1\). Compute the square of the kernel \(e(t,t',u)=\psi(t)\psi(t')\), and its trace for transverse measure \(a\,du\), where \(du\) has total mass one.

**Solution.** Convolution gives
\[
(e*e)(t,t',u)=\psi(t)\psi(t')\int|\psi(s)|^2\,ds=e(t,t',u).
\]
The kernel is self-adjoint. Its diagonal integral in \(t\) is one; integrating in \(u\) gives \(a\). Its represented projection has rank one on each leaf, although its algebra is nonunital.

**Marking (6 points).** Compute the convolution square and verify self-adjointness using the normalized vector. (3 points). Compute the leaf diagonal and the full transverse trace. (3 points).



**Exercise 2 (8 points).** *intermediate.* Let \(N\) be noncompact. Explain why Theorem 2.1 does not assert that the identity of \(\mathcal E_N(\mathbb C)\) is compact. Give a product example where it is not.

**Solution.** The identity of the Hilbert \(C_0(N)\)-module \(C_0(N)\) is multiplication by the constant one, which does not vanish at infinity and is not a compact module map. For \(V=\mathbb R\times N\), the induced module is a Morita module for \(C_0(N)\otimes\mathcal K\), and its compact endomorphism algebra is \(C_0(N)\). The identity belongs to its multiplier algebra, but not to \(C_0(N)\). Theorem 2.1 only applies to a compact map in its domain.

**Marking (8 points).** Identify the compact endomorphism algebra and the nonvanishing identity multiplier. (4 points). Give the product Morita-module example and state the exact limitation of the theorem. (4 points).



**Exercise 3 (8 points).** *intermediate.* Generalize Example 9.1 to a cyclic permutation of \(m\) points. Give its implementing unitary, and compute the trace of a point projection.

**Solution.** On \(\mathbb C^m\), let the unitary send the first basis vector to the second, and successively the \(j\)-th to the \((j+1)\)-st, while sending the last to \(z\) times the first. Its \(m\)-th power is \(zI\). Together with the diagonal point projections it generates \(M_m(C(S^1))\), giving the covariant isomorphism. The invariant probability measure assigns mass \(1/m\) to each point, so its crossed-product trace assigns \(1/m\) to each point projection. Integral \(H^1\) of the finite space is zero.

**Marking (8 points).** Construct the cyclic matrix unitary and verify its power and generated algebra. (4 points). Compute the invariant point-projection trace with the probability normalization. (3 points). Identify the first cohomology of the finite space. (1 points).



**Exercise 4 (10 points).** *advanced.* Prove directly from hyperbolic cosine law that endpoints at bounded distance have directions converging to each other as both go to infinity from a fixed base point. Why does this imply the commutator estimate, but not finite rank?

**Solution.** Rearranging the law gives
\[
1-\cos\theta
=\frac{\cosh d-\cosh(r-s)}{\sinh r\,\sinh s}
\le\frac{\cosh C-1}{\sinh r\,\sinh s}
\]
when \(d\le C\). Taking the square root gives (23). Clifford multiplication is linear and has norm equal to vector length, so it satisfies the same bound. Multiplying a kernel by this small difference bounds its two Schur integrals, giving a small operator-norm tail. The cutoff kernel is compact by finite-rank approximation, not generally finite rank itself. Its radial and angular variables still vary continuously.

**Marking (10 points).** Derive the angular estimate from the cosine law and bounded endpoint distance. (4 points). Apply the estimate to Clifford differences and both Schur tails. (4 points). Distinguish compact cutoff approximation from finite rank. (2 points).



**Exercise 5 (8 points).** *advanced.* Suppose \(T\in KK(B,A)\) and \(D\in KK(A,B)\) satisfy \(T\otimes_A D=1_B\). Prove that \(T_*:K_*(B)\to K_*(A)\) is injective. Give an elementary pair of group maps showing that this relation need not imply surjectivity.

**Solution.** Associativity gives \(D_*T_*(x)=x\). If \(T_*(x)=0\), then \(x=D_*(0)=0\). On abelian groups, the inclusion \(\mathbb Z\to\mathbb Z^2\), \(n\mapsto(n,0)\), has the left inverse \((a,b)\mapsto a\), and is not surjective. The second identity in (24), rather than the first alone, would give a two-sided inverse.

**Marking (8 points).** Use associativity and the left inverse to prove injectivity. (4 points). Give and check a split inclusion that is not surjective. (4 points).



**Exercise 6 (8 points).** *advanced.* Check the parity of (16), and explain why a trace on \(K_0(A)\) uses the odd Chern character on \(V\) when a solvable action has odd-dimensional orbits.

**Solution.** The Thom map has degree \(n-\dim M+q\), the étale correspondence degree zero, and the inverse Bott map degree \(n\). Their sum is \(q-\dim M+2n\), equal to \(q-\dim M\) modulo two. For orbit dimension \(d\), the Connes–Thom map shifts parity by \(d\). If \(d\) is odd, classes mapping to \(K_0(A)\) come from \(K^1(V)\), whose Chern character has odd-degree components. The dimension-\(d\) current can pair with the component of degree \(d\).

**Marking (8 points).** Add the degrees of the Thom, correspondence and inverse Bott factors. (4 points). Identify the odd input Chern-character component paired with the odd-dimensional orbit current. (4 points).



**Exercise 7 (8 points).** *intermediate.* Check the double-dual action in Theorem 8.6. Why would \(\alpha_v\otimes\operatorname{Ad}\lambda_v\) have the wrong phase?

**Solution.** Conjugation by \(\lambda_{-v}\) sends \(M_{e^{iqt}}\) to \(M_{e^{iq(t+v)}}=e^{ivq}M_{e^{iqt}}\). It sends \(M_{\alpha_{-t}(b)}\) to \(M_{\alpha_{-(t+v)}(b)}\); applying \(\alpha_v\) to its coefficient returns \(M_{\alpha_{-t}(b)}\). The translations \(\lambda_s\) commute with \(\lambda_{-v}\). Thus all three generators have the required images. Replacing \(-v\) by \(v\) gives phase \(e^{-ivq}\), and the coefficient action no longer cancels.

**Marking (8 points).** Compute the phase, coefficient cancellation and translation-generator images. (4 points). Identify the phase and coefficient errors from the opposite translation sign. (4 points).



**Exercise 8 (10 points).** *advanced.* For the affine group \(H=\mathbb R\rtimes\mathbb R\), with \(\theta_t(n)=e^tn\), compute the measures and unitary in Lemma 8.8. Then check one coefficient and both translation generators explicitly.

**Solution.** Here \(m_t=e^t\), so left Haar measure is \(e^{-t}\,dn\,dt\). In \(n=e^tk\) coordinates it becomes \(dk\,dt\), and \(Z\xi(t,k)=\xi(e^tk,t)\) is unitary. A coefficient becomes \(\rho(\alpha_{-k}\alpha_{(0,-t)}(b))\), where \(\alpha_{-k}\) is the normal-subgroup action. Translation by \((n,0)\) sends \(k\) to \(k-e^{-t}n\); translation by \((0,s)\) sends \(t\) to \(t-s\) with \(k\) fixed. The integrated normal-subgroup action is
\(\beta_tf(n)=e^{-t}\alpha_{(0,t)}(f(e^{-t}n))\).
Omitting this factor would fail to preserve convolution and its Haar normalization, even though the unitary \(Z\) itself has no extra square root.

**Marking (10 points).** Compute the affine modulus, Haar density and density-preserving coordinate unitary. (4 points). Compute one coefficient and both translation actions. (4 points). Recover the required factor in the normal-subgroup convolution action. (2 points).



**Exercise 9 (10 points).** *advanced.* Prove \(\|F\|=2\) for the scalar operator in (34), while \(S=1-F\) is an isometry of defect one. Explain why the endpoint estimates of Theorem 8.7 cannot be replaced by the assertion \(F_\varepsilon\to0\) in norm.

**Solution.** The kernel is convolution on the positive half-line by \(e^{-s/2}1_{s>0}\), preceded by the positive-half-line projection. Its \(L^1\) norm is two, so \(\|F\|\le2\). For \(v_T=T^{-1/2}1_{[0,T]}\) and \(0<t<T\),
\[
(Fv_T)(t)=2T^{-1/2}(1-e^{-t/2}).
\]
The integral of its squared norm on \([0,T]\) tends to four as \(T\to\infty\), proving the reverse inequality. Direct kernel composition gives (35), so \(S\) is an isometry with range complement the one-dimensional span of \(u(t)=e^{-t/2}1_{t>0}\). Dilation is unitary, hence every \(\|F_\varepsilon\|=2\). The small quantities are its commutator with a fixed coefficient and the difference between the rank-one corner maps, bounded by (36)–(37). The exponential mass concentrates near zero, where the coefficient action is norm continuous.

**Marking (10 points).** Prove both bounds for the scalar operator norm with the normalized long-interval vectors. (4 points). Compute the isometric complement and its one-dimensional defect. (4 points). Use dilation to reject norm convergence and identify the actual small endpoint quantities. (2 points).



**Exercise 10 (8 points).** *intermediate.* In Theorem 9.3, suppose \(q\) is injective on \(C(X)\) and \(a\ge0\) lies in its kernel. Complete the argument from (39) to \(a=0\), and explain the role of minimality after the intersection property has been proved.

**Solution.** Letting \(\delta\downarrow0\) in (39) gives \(\mathsf E(a)=0\). Apply Proposition 9.2 to \(a^{1/2}\): its squared value has expectation zero, so \(a^{1/2}=0\) and \(a=0\). This part uses topological freeness and expectation faithfulness. Minimality is used afterwards: the common zero set of the nonzero invariant ideal \(J\cap C(X)\) cannot be a nonempty proper invariant closed set. Hence the intersection contains the unit, forcing \(J=B\). Faithful trace alone would not give that ideal-intersection argument.

**Marking (8 points).** Use the expectation estimate and its faithfulness to eliminate a positive kernel element. (4 points). Use minimality only after the ideal-intersection property and explain its distinct role. (4 points).




**Exercise 11 (10 points).** *intermediate.* Check the unfolding in Lemma 9.4: derive the coefficient at \(r=t+j\), the translated vector, and the kernel \(K_a\). For \(g\in C_c^\infty(0,1)\) of norm one, compute the trace of its band projection. Where would a spurious circle-volume factor enter?

**Solution.** The regular discrete coefficient at \(j\) is \(\alpha_{-j}F(t)=F(t+j)\). The twisted extension says \(\eta(t+1,j)=\eta(t,j+1)\), so \(\zeta(t+j)=\eta(t,j)\) is well defined almost everywhere and translation sends \(\zeta(r)\) to \(\zeta(r-s)\). Writing \(t-s=u+m\) gives (44), including \(U^{-m}\). The rank-one kernel is \(g(t)\overline{g(u)}1_B\); its diagonal unit coefficient integrates to \(\int_0^1|g(t)|^2dt\,\nu(X)=1\). Time has period one and leaf density \(dt\); replacing it by a \(2\pi\)-period coordinate without also changing the vector and density normalization would introduce an erroneous volume factor.

**Marking (10 points).** Compute the discrete coefficient, twisted extension and unfolded translation. (4 points). Derive the crossed-product kernel and compute the normalized band-projection trace. (4 points). Explain which time-coordinate change would introduce a false volume factor. (2 points).



**Exercise 12 (12 points).** *advanced.* Why does the integral defining \(D/\sqrt{1+D^2}\) in Proposition 9.5 converge strictly, whereas its commutator integral converges in norm? Show why the zero-action endpoint gives an isomorphism even before choosing a Bott sign.

**Solution.** Its truncation is
\[
\frac2\pi\frac{p}{\sqrt{1+p^2}}
\arctan\!\left(\frac{R}{\sqrt{1+p^2}}\right).
\]
It is bounded by one and converges uniformly on compact frequency sets, which gives strict convergence under nondegenerate \(C_0\)-functional calculus. Uniform convergence on the whole frequency line fails: at arbitrarily large \(p\), a fixed truncation is small while \(p/\sqrt{1+p^2}\) is near one. The commutator tail is instead bounded by \((2/\pi)\|\delta b\|\int_R^\infty(1+r^2)^{-1}dr\), which tends to zero in norm. At the zero action, six-term exactness for the two-endpoint interval has diagonal endpoint maps. Their cokernel is exactly the classes \((0,[w])\), and the next diagonal map is injective. The connecting map on that cokernel is therefore an isomorphism; its orientation fixes a sign but cannot change invertibility.

**Marking (12 points).** Compute the bounded truncation and prove strict convergence on compact frequency tests. (4 points). Estimate the norm-convergent commutator tail. (4 points). Compute the zero-action endpoint cokernel and its connecting isomorphism independently of the sign. (4 points).



**Exercise 13 (8 points).** *intermediate.* In Theorem 9.6, remove connectedness from the logarithm argument. What changes, and how does the coefficient algebra already obstruct the original idempotent conclusion? Explain separately the connected zero-dimensional case.

**Solution.** The seam difference is only locally constant and integer valued; it can have different values on different components, so it need not be one integer multiple of the normalized time class. More directly, a proper clopen union of components gives a characteristic function \(e\in C(X)\) with \(e^2=e=e^*\), \(e\ne0,1\). Faithfulness of the coefficient inclusion retains that projection in the crossed product. Connectedness avoids this obstruction. A connected zero-dimensional compact manifold is a point, whose crossed product is \(C(S^1)\). Its continuous idempotents take values in \(\{0,1\}\) and are constant, but its proper ideals show that it is not simple. Thus the infinite-space hypothesis is needed for the combined source conclusion.

**Marking (8 points).** Describe the componentwise seam integers and the retained clopen projection. (4 points). Treat the connected point case separately, including failure of simplicity. (4 points).




**Exercise 14 (10 points).** *intermediate.* A nonzero closed positive orbit current on a compact manifold is paired with the normalized leaf form \(\lambda(E)=1\). Derive its adjoint-trace obstruction. Then repeat the calculation for the affine group of Example 8.13 with a compactly supported function \(f\). Why does that calculation permit a nonzero transverse point measure?

**Solution.** Fundamental fields of a left action have negative Lie brackets, so \(\mathcal L_{X^\#}E=-\operatorname{tr}(\operatorname{ad}X)E\). Differentiation gives \((\mathcal L_{X^\#}\lambda)(E)=\operatorname{tr}(\operatorname{ad}X)\). Closedness and leafwise top degree therefore give (51), with finite strictly positive \(\mu(V)\), forcing every adjoint trace to vanish. For the affine field \(T^\#=n\partial_n+\partial_t\), integration by parts gives \(\int n\partial_nf\,dn\,dt=-\int f\,dn\,dt\) and \(\int\partial_tf\,dn\,dt=0\). Thus \(\int(T^\#f+f)=0\) for every compact test. The total measure is infinite, so the constant test needed to deduce \(0=\operatorname{tr}(\operatorname{ad}T)\mu(V)\) is unavailable. The one-leaf point measure is consistent with closedness.

**Marking (10 points).** Derive the fundamental-field adjoint-trace identity and use finite positive total mass. (4 points). Compute the affine compact-test integration-by-parts identity. (4 points). Explain why the unavailable constant test permits the noncompact example. (2 points).



**Exercise 15 (12 points).** *intermediate.* Compute the family \(H_v\), its Haar density, inverse, modular character, and the positive-parameter kernel change (58) for the affine group. Keep \(v\) distinct from the group coordinate \(t\).

**Solution.** The normal group is additive \(\mathbb R\), and \(\theta_t^v(n)=e^{vt}n\). Hence
\[
(n,t)*_v(n',t')=(n+e^{vt}n',t+t'),\qquad
(n,t)^{-1}=(-e^{-vt}n,-t).
\]
The outer modulus is \(e^{vt}\); left Haar is \(e^{-vt}\,dn\,dt\) and \(\Delta_v(n,t)=e^{-vt}\). At zero the product is additive and Haar is Lebesgue. The isomorphism to the original group is \(\Gamma_v(n,t)=(vn,vt)\), whose Jacobian is \(v^2\). Thus a kernel becomes \(v^{-2}f_v(n/v,t/v)\) in the original coordinates. Its involution has the modular factor \(\Delta_v((n,t)^{-1})=e^{vt}\), in addition to the coefficient action and adjoint. Omitting this factor or the \(v^{-2}\) kernel normalization would fail to preserve the integrated convolution representation.

**Marking (12 points).** Compute the parameterized affine multiplication and inverse. (4 points). Compute left Haar and modular densities, including the zero parameter. (4 points). Compute the positive-parameter Jacobian, kernel transport and involution factor. (4 points).



**Exercise 16 (10 points).** *advanced.* In Lemma 8.15, explain why replacing source-fibre coordinates by orbit coordinates and dividing by \(v\) would not prove uniformity. Give the norm estimate for a localized operator matrix with at most \(N\) blocks in each row and column, each of norm at most \(C\). Which parameter derivatives are actually needed?

**Solution.** The anchor sends a tangent generator to \(vX^\#\); its rank vanishes at zero. An inverse orbit-coordinate derivative would therefore introduce \(v^{-1}\) and cease to exist at zero. Source-fibre coordinates remain \(d\)-dimensional with a smooth group law, so the Fourier–Schur proof uses no such inverse. For localized inputs \(u_j\),
\[
\sum_i\left\|\sum_jP_{ij}u_j\right\|^2
\le NC^2\sum_{i,j:P_{ij}\ne0}\|u_j\|^2
\le N^2C^2\sum_j\|u_j\|^2.
\]
Thus the matrix has norm at most \(NC\). The finite-seminorm bound uses source-coordinate derivatives uniform over \(v\). Smooth parameter derivatives for composition, quantization and Borel summation are taken in the original compact groupoid charts. No uniform derivative of \(h_0x\) with respect to \(v\), for arbitrary faraway \(h_0\), is required or asserted.

**Marking (10 points).** Explain the singular orbit-anchor inverse and retain source-fibre coordinates. (4 points). Prove the localized operator-matrix norm estimate from row and column multiplicities. (4 points). Identify the actual required compact-chart derivatives without asserting faraway-anchor uniformity. (2 points).



**Exercise 17 (12 points).** *advanced.* Explain how the zero-endpoint matrix in Theorem 8.16 identifies a relative symbol triple, even for nontrivial coefficient bundles. For the trivial foliation action \(H=\mathbb R\) on a compact manifold, which part of the parameter theorem remains true, and why must the orbit-foliation conclusion be typed differently?

**Solution.** Embed the two coefficient bundles in finite trivial bundles, use the index matrix \(W\), and add their fixed orthogonal complements. Its boundary value is \(\operatorname{diag}(a,a^{-1})\). The first range is therefore the source bundle, identified at the sphere with the target by \(a\); subtracting the target reference gives exactly the disk/sphere relative clutching class. These complements do not change the K-class, and rotation between two embeddings gives a homotopy. The construction does not assume that a coefficient bundle is trivial. For a trivial real action, Theorem 8.14 and Lemma 8.15 still apply: all fibres of the parameter algebra are \(C(V)\otimes C_0(\mathbb R)\), and the source calculus has dimension one. But the orbit foliation has rank zero and its holonomy groupoid is the unit groupoid, not \(V\rtimes\mathbb R\). Theorem 8.16 requires the graph identification; without it the one-dimensional source symbol cannot be called the zero-rank foliation symbol. This also explains the exact footnote's importance.

**Marking (12 points).** Construct the relative boundary symbol with fixed coefficient-bundle complements. (4 points). Check homotopy invariance under the bundle embeddings. (4 points). Identify the trivial-action parameter algebra and distinguish its source dimension from the rank-zero holonomy foliation. (4 points).



**Exercise 18 (10 points).** *intermediate.* In (65), omit \(a_t\) and call the resulting function \(\widetilde r_t\). Explain why \((1-\widetilde r_t(z))/z\) fails at zero even though the removed Gaussian tail is exponentially small. Give the correct smooth extension for \(q_t\).

**Solution.** Since the nonnegative Gaussian has positive mass outside \(\operatorname{supp}\chi\), \(\widetilde r_t(0)<1\) for every \(t>0\). Its numerator consequently has a nonzero value at zero, and division gives a genuine pole. Smallness does not make this a parametrix function. Normalization makes \(r_t(0)=1\); evenness makes \(r_t'(0)=0\), so \(q_t(0)=0\). Its first derivative is \(q_t'(0)=-r_t''(0)/2=(1/2)\int s^2\rho_t(s)\,ds\). The Fourier primitive is compactly supported because the normalized distribution \(\delta_{\{0\}}-\rho_t\) has integral zero. These exact facts are the ones used for finite propagation.

**Marking (10 points).** Prove the unnormalized value leaves a genuine pole despite its small error. (4 points). Compute the normalized smooth extension and its derivative. (4 points). Explain compact support of the Fourier primitive from the exact zero integral. (2 points).



**Exercise 19 (10 points).** *advanced.* Verify (68) without assuming \(e\) or \(f\) self-adjoint, or that they commute. If \(e-f\) is trace integrable, compute the trace of the relative matrix difference.

**Solution.** Since \(f^2=f\), the products \(f(1-f)=(1-f)f=0\) give \(J_f^2=I\). Multiplying \(J_f\operatorname{diag}(f,1-f)J_f\) gives diagonal entries \(f+(1-f)=I\) and \(0\), and zero off-diagonal entries. Therefore
\[
\widetilde e-\operatorname{diag}(I,0)
=J_f\operatorname{diag}(e-f,0)J_f.
\]
It belongs to the algebra ideal whenever \(e-f\) does, and to the trace ideal whenever that difference does. Cyclicity with the two bounded multipliers gives trace \(\tau(e-f)\), because \(J_f^2=I\). No commutation with \(e\) was used. This is why a nonscalar exterior reference can be removed at the level of a relative K-class without integrating either reference separately.

**Marking (10 points).** Compute the involution matrix square and its conjugation of the reference idempotent. (4 points). Place the relative difference in the algebra and trace ideals without self-adjointness or commutation. (4 points). Compute the relative trace using bounded-multiplier cyclicity. (2 points).



**Exercise 20 (12 points).** *advanced.* For odd \(d\), check both signs in the circle reduction: the kernel parity in (74) and the fibre orientation sign in (78). Explain the role of the exterior inverse in (77).

**Solution.** The circle source kernel is zero and its adjoint kernel is the odd constant line, with trace one. Tensoring this line exchanges first-factor kernel parity, so the product trace index is minus the original. In the character calculation the positive circle base differential \(d\theta\) must cross the \(d\) first-factor vertical differentials before fibre integration in the ordered fibre \((\xi_1,\ldots,\xi_d,\eta)\). It contributes \((-1)^d\). The equality \(\epsilon_{d+1}=(-1)^{d+1}\epsilon_d\) makes the product characteristic value \(-\epsilon_d\) times the original pairing, agreeing with the kernel sign. On a noncompact base a parametrix may have a nonvanishing homogeneous defect at infinity. Adding the constant lift of \(T_\infty^{-1}-R_{0,\infty}\) makes both defects vanish in that quotient, placing them in \(A\otimes\mathcal K\). The lower bound \(\mathcal D_\infty^2\ge1\) is what makes this inverse available.

**Marking (12 points).** Compute the odd circle kernel parity and the resulting trace-index sign. (4 points). Compute the ordered differential permutation and the characteristic prefactor. (4 points). Explain the exterior inverse correction and the localized compact defects on a noncompact base. (4 points).



**Exercise 21 (10 points).** *advanced.* Apply (79) to the free left action of the affine group in Example 8.13 on itself, with transverse point mass one. Determine the full trace image and explain why it does not imply unimodularity.

**Solution.** This action has one leaf, holonomy graph the pair groupoid and algebra \(\mathcal K(L^2(H))\); its trace is the ordinary trace with a rank-one projection of value one. Thus the full K-theory trace image is \(\mathbb Z\). Independently \(H\cong\mathbb R^2\), so \(K_c^0(H)\cong\mathbb Z\). The two-plane Bott character has absolute integral one; choosing its sign to give integral \(+1\) makes the right side of (79) exactly \(\mathbb Z\). The current integrates the compactly supported character two-form against \(dn\wedge dt\). Its total mass is infinite, and the normalized volume form is not a compact test form. Hence the finite-mass Cartan argument of Proposition 8.12 cannot be applied. The modular character remains \(e^{-t}\), while both sides of the numerical theorem retain their stated values.

**Marking (10 points).** Identify the actual pair-groupoid algebra and its integer trace image. (4 points). Compute the independently normalized compact-support Bott character image. (4 points). Explain why infinite total mass prevents the compact unimodularity argument. (2 points).




**Exercise 22 (12 points).** *intermediate.* Let \(u_+(p)=(p-i/2)/(p+i/2)\). Compute the effect of \(p\mapsto-p\) on its \(K_1(C_0(\mathbb R))\)-class, and on the ordered product of \(d\) such generators. Why does a positive determinant coordinate change give the same ordered analytic Thom map, rather than merely the same trace image?

**Solution.** Direct substitution gives \(u_+(-p)=u_+(p)^{-1}\). In stable unitary K-theory the inverse represents the negative class: rotation of the two blocks contracts \(\operatorname{diag}(u,u^{-1})\) to identity. Reflection of the first coordinate therefore multiplies the ordered external product by \(-1\), by bilinearity. Every negative determinant linear map is connected to that reflection; every positive determinant map is connected to identity. Along either path the least singular value has a positive lower bound, so pullback is point-norm continuous on \(C_0(\mathbb R^d)\). The induced K-map depends only on the component. More strongly, (82), its exact Haar transport and its kernel *-isomorphism identify the two whole deformation algebras. At one the identification is identity on \(B\rtimes H\); at zero it is identity on the intrinsic dual after the linear change. Equation (80) consequently identifies the maps on every K-class, before applying any trace. The statement concerns analytic decompositions; it does not prove geometric wrong-way factorization independence.

**Marking (12 points).** Compute reflection of the generator and its ordered external-product sign. (4 points). Prove point-norm homotopy invariance along determinant components. (4 points). Use the exact deformation-algebra transport to identify the whole analytic maps before tracing. (4 points).



**Exercise 23 (12 points).** *intermediate.* Verify (84). If \(w\) is fixed by the upper unipotent, explain why the two large outer factors cause no convergence problem. Complete the passage from diagonal-fixed to lower-unipotent-fixed vectors.

**Solution.** With \(c=(\lambda-1)/n\), multiplication of the first two factors gives
\[
\begin{pmatrix}1+nc&n\\c&1\end{pmatrix}.
\]
Multiplying on the right by \(\overline U_{-n/\lambda}\) makes the upper-right entry \(n-(1+nc)n/\lambda=0\); the lower-right entry is \(1-cn/\lambda=\lambda^{-1}\). The other entries are \(\lambda,c\), proving the identity. For \(F(g)=\langle\pi(g)w,w\rangle\), upper-unipotent invariance of \(w\) on each side gives \(F(\overline U_nU_c\overline U_{-n/\lambda})=F(U_c)\). Only \(c\to0\) is used; no boundedness of the outer parameters is assumed. Continuity then gives \(F(D_\lambda)=\|w\|^2\), with \(D_\lambda=\operatorname{diag}(\lambda,\lambda^{-1})\). Since
\[
\|\pi(D_\lambda)w-w\|^2
=2\|w\|^2-2\operatorname{Re}F(D_\lambda)=0,
\]
\(w\) is diagonal-fixed. Now \(D_\lambda U_sD_\lambda^{-1}=U_{s/\lambda^2}\). Unitarity and diagonal invariance make the norm of \(\pi(U_s)w-w\) equal to that of \(\pi(U_{s/\lambda^2})w-w\), which tends to zero. Upper and lower elementary matrices generate \(L\), including its centre, so the vector is \(L\)-fixed.

**Marking (12 points).** Multiply the three elementary matrices and compute all resulting entries. (4 points). Remove the large outer factors from the matrix coefficient by vector invariance. (4 points). Deduce diagonal and then lower-unipotent invariance with the correct conjugation scaling. (4 points).



**Exercise 24 (12 points).** *advanced.* In the proof of Theorem 8.26, justify uniform mixing on the moving family \(y=a_{-t}x\). Check the Haar density and the limiting order needed to replace the thickened averages by flat horocycle averages. What fails if one uses just ergodicity?

**Solution.** For fixed smooth compact group probability \(k_\varepsilon\), its pushforward densities \(h_y\) form a compact subset of \(L^2(X,m)\). Given \(\eta>0\), choose a finite \(\eta\)-net \(h_{y_i}\). Mixing controls each of their finitely many matrix coefficients. Cauchy–Schwarz bounds the discrepancy for an arbitrary \(h_y\) by \(\|f\|_2\eta\); all densities have integral one, so the constant term is the same. This estimate is uniform in \(y\), and therefore applies to every \(a_{-t}x\). In the ordered \((v,r,s)\) chart the determinant of the Maurer–Cartan columns is \(e^r\). Multiplication by \(e^{-r}\) in (87) leaves exactly the probability \(\phi_\varepsilon(v,r)h(s)\,dv\,dr\,ds\). Conjugation sends \((v,r,s)\) to \((v/T,r,Ts)\); its coordinate Jacobian is one and its \(e^r\) Haar factor is unchanged. For each fixed \(\varepsilon\), first let \(T\to\infty\) using uniform mixing. The difference from the weighted orbit average is at most \(\omega_f(\varepsilon)\); next let \(\varepsilon\to0\). Finally approximate \(1_{[0,1]}\) by smooth integral-one \(h\) in \(L^1\), using the bound \(\|f\|_\infty\|h-1_{[0,1]}\|_1\) independent of \(T,x\). An almost-everywhere ergodic theorem alone leaves exceptional orbits uncontrolled and does not exclude another singular invariant probability. Uniform convergence permits integration against every invariant probability, giving unique ergodicity and then uniqueness of the transverse measure up to scale.

**Marking (12 points).** Use a finite net of the compact density family to prove uniform mixing for moving base points. (4 points). Compute the exact Haar density and its conjugated coordinate change. (4 points). Take the thickening and interval-approximation limits in the required order and derive unique ergodicity. (4 points).



**Exercise 25 (10 points).** *advanced.* For a closed one-form \(\omega\) on \(X\), calculate \(C_m(\omega)\) from the left fundamental-field bracket. Explain how zero trace can coexist with a nonzero \(K_0(C(X)\rtimes_U\mathbb R)\), and why “the only transverse measure” needs a normalization.

**Solution.** For left fundamental fields the bracket is the negative Lie bracket, so \([A^\#,N^\#]=-N^\#\). Closedness gives
\[
A^\#(\omega(N^\#))-N^\#(\omega(A^\#))+\omega(N^\#)=0.
\]
Invariance of Haar under both flows makes the integrals of the first two terms zero. Thus \(C_m(\omega)=0\). In dimension one the current pairs only with the closed degree-one component of an odd Chern character. The entire symbol-index/trace formula gives zero trace on all of \(K_0\). However the compact-support odd Bott class inserted in an oriented three-dimensional ball of \(X\) has nonzero top Chern integral; its extension by identity is a nonzero class in \(K^1(X)\). The actual analytic isomorphism sends it to a nonzero \(K_0\)-class. A trace pairing need not detect it. Multiplying \(C_m\) by any \(c>0\) gives another nonzero positive closed current and transverse measure. The unique probability normalization is \(m(X)=1\), using the density that makes \(N^\#\) unit; without it uniqueness means precisely positive scalar multiples.

**Marking (10 points).** Use the signed fundamental-field bracket and Haar invariance to compute the closed one-form current. (4 points). Exhibit a nonzero K-class despite zero numerical trace. (4 points). Distinguish uniqueness of the normalized probability from uniqueness up to positive scale. (2 points).



**Exercise 26 (6 points).** *basic.* On \(\mathbb C^2\), graded by \(\operatorname{diag}(1,-1)\), take
\[
F=\begin{pmatrix}0&1\\1&0\end{pmatrix},\qquad
F'=\begin{pmatrix}0&-i\\i&0\end{pmatrix}.
\]
Compare the square of \((1-t)F+tF'\) with that of the bridge operator (97). Explain the reason for its normalization.

**Solution.** Both matrices are self-adjoint odd unitaries, and their anticommutator is zero. Thus the square of the straight segment is \(((1-t)^2+t^2)I\), which at \(t=1/2\) is \(I/2\). Here \(P=0\), so (97) gives \(F_\theta=\cos\theta F+\sin\theta F'\), whose square is \(I\) for every \(\theta\). More generally the anticommutator contributes \(\cos\theta\sin\theta H\); the outside fourth-root factors divide its square by \(1+\cos\theta\sin\theta P\) in the localized quotient. The endpoint factors are exactly one.

**Marking (6 points).** Compute the straight-segment square and its middle-parameter defect. (3 points). Compute the trigonometric bridge square and explain the normalization for a general anticommutator. (3 points).



**Exercise 27 (8 points).** *intermediate.* Let \(J=\mathcal K(\ell^2(\mathbb N))\), \(u_n=v_n\) the projection onto the first \(n\) standard basis vectors, \(A_2=\mathbb C I\), and \(w=I\). The products \(u_jw\) lie in \(J\), and \((1-v_n)u_jw=0\) for every \(j\le n\). Show why this information alone does not give (92). Which extra term does the one-step lookahead control?

**Solution.** With \(b_n=(v_n-v_{n-1})^{1/2}\), we have \(b_n\) equal to the rank-one projection onto the \(n\)-th vector, and
\[
b_n^2u_nw=b_n,\qquad b_nu_nb_nw=b_n.
\]
Both norms are one; the latter terms have no summable norm bound. Indeed their strict sum is \(I\), so its product with \(A_2\) is not compact. The condition \(j\le n\) at the \(n\)-th step gives no smallness for \((1-v_{n-1})u_nw\). Requiring \(j\le n+1\) at that step gives its bound at step \(n-1\), namely \(4\epsilon_n^2\); the current-step bound is \(\epsilon_n^2\). Their sum gives the exact \(5\epsilon_n^2\) in the proof. This example concerns the omitted approximation condition; it does not refute the separator theorem.

**Marking (8 points).** Compute both rank-one shell terms and show their nonsummable norms. (4 points). Identify the uncontrolled preceding-step term and recover its bound from the one-step lookahead. (4 points).



**Exercise 28 (8 points).** *advanced.* Let \(p\) be a rank-one projection on an infinite-dimensional Hilbert space \(H_1\), and let \(H_2\) be infinite-dimensional. Prove that \(p\otimes I_{H_2}\) is not compact. Explain how the first positive term in (101) remains valid when a compact error on \(E_{12}\) is tensored with the last module.

**Solution.** Choose a unit vector \(\xi\) in the range of \(p\), and an orthonormal sequence \(\eta_n\) in \(H_2\). The images of \(\xi\otimes\eta_n\) under \(p\otimes I\) are the same orthonormal sequence, so no subsequence converges. This excludes compactness. In the product proof the equality \(C_a=P_a+K_a\) holds before the last tensor, with \(P_a\ge0\) and \(K_a\in\mathcal K(E_{12})\). The separator includes \(\mathcal K(E_{12})\widehat\otimes1\) in \(A_1\), so \(M^{1/4}(K_a\widehat\otimes1)\) is compact on the final module. Hence its sandwiched contribution is compact. The remaining \(M^{1/4}(P_a\widehat\otimes1)M^{1/4}\) is positive. One keeps this positive contribution; tensoring alone supplied no compactness.

**Marking (8 points).** Use an orthonormal tensor sequence to disprove compactness of the tensored projection. (4 points). Place the tensored error in the correct separator ideal and retain the remaining positive contribution. (4 points).



**Exercise 29 (10 points).** *advanced.* Let \(B=C([0,1])\), \(D=\{f\in B:f(0)=0\}\), \(E=B\) with trivial grading, and \(\phi:D\to\mathcal L(E)\) multiplication. Show that \((E,\phi,0)\) is a Kasparov cycle, but \(E^{\rm ess}=D\) has no orthogonal complement. Explain why the essential-module homotopy does not require such a complement.

**Solution.** The zero operator is odd, its commutators and adjoint defect vanish, and its square defect is \(-\phi(d)\in B=\mathcal K(E)\). Also \(\overline{DE}=D\). An adjointable projection on \(E=B\) is multiplication by a continuous projection-valued scalar function \(p:[0,1]\to\{0,1\}\). Its range equals \(D\) only if \(p(0)=0\) and \(p(t)=1\) for all \(t>0\), contradicting continuity. In the homotopy the module consists of continuous \(E\)-sections whose final value lies in \(D\). At the initial parameter its fibre is \(E\); at the final parameter it is \(D\). The connection is built by stabilization on this module, and its defects are tested against constant elements of \(D\). No projection onto \(D\) inside \(E\) is used.

**Marking (10 points).** Check every localized Kasparov defect for the multiplication cycle. (4 points). Prove the essential submodule has no adjointable projection complement. (4 points). Describe the endpoint-constrained homotopy module without assuming such a complement. (2 points).



**Exercise 30 (8 points).** *introductory.* On \(H=\ell^2(\mathbb N)\), let \(P_n\) project onto the first \(n\) basis vectors. Prove that \(P_n\to1\) strictly in \(M(\mathcal K(H))\), although \(\|1-P_n\|=1\). Explain why the infinite diagonal sums used in the multiplier swindle need not converge in norm.

**Solution.** For a rank-one operator \(\theta_{\xi,\eta}\),
\[
\|(1-P_n)\theta_{\xi,\eta}\|\le\|(1-P_n)\xi\|\,\|\eta\|\to0,
\qquad
\|\theta_{\xi,\eta}(1-P_n)\|\le\|\xi\|\,\|(1-P_n)\eta\|\to0.
\]
Finite rank sums and their norm density in \(\mathcal K(H)\) prove the two strict convergence tests. The bound \(\|1-P_n\|\le1\) allows passage from finite ranks to every compact operator. A unit basis vector beyond the first \(n\) has unchanged norm under \(1-P_n\), so its norm is exactly one. The identity is the strict sum of its orthogonal rank-one diagonal terms; no norm sum has been asserted. The same compact tests define a multiplier diagonal on a countable Hilbert-module direct sum. This is the convergence used in Lemma 0.13 and Theorem 0.14.

**Marking (8 points).** Prove both strict convergence tests on rank-one operators. (3 points). Extend the tests to every compact operator with a common norm bound. (3 points). Compute the nonvanishing norm difference and identify the convergence used for multiplier diagonals. (2 points).



**Exercise 31 (12 points).** *intermediate.* For \(a>0\), let \(S_a\) be the half-line operator in Lemma 0.16. For \(\lambda>0\), define \((Q_\lambda f)(t)=\sqrt\lambda f(\lambda t)\). Prove
\[
Q_\lambda S_aQ_\lambda^*=S_{a\lambda}.
\]
Find the normalized cokernel vector of \(S_a\), and compute the index of \(S_a^*\). Explain the role of this adjoint in the positive physical Wiener–Hopf boundary.

**Solution.** A change of variable shows that \(Q_\lambda\) is unitary, with adjoint \(Q_{1/\lambda}\). In the Volterra term, substitute \(r=\lambda u\). Its coefficient becomes \(2a\lambda\), its exponential becomes \(e^{-a\lambda(t-u)}\), and its upper limit becomes \(t\). This gives the claimed conjugacy. The defect projection of \(S_a\) has kernel \(2ae^{-a(t+r)}\), so its range is the line generated by
\[
e_a(t)=\sqrt{2a}\,e^{-at},\qquad
\|e_a\|_2^2=\int_0^\infty2ae^{-2at}\,dt=1.
\]
The identity \(S_a^*S_a=1\) gives zero kernel for \(S_a\); \(S_aS_a^*=1-\theta_{e_a,e_a}\) gives its one-dimensional cokernel. Thus its index is \(-1\). The adjoint is onto and has kernel this line, so its index is \(+1\). Under the positive physical half-line convention, the lift of \(u_+\) is the coisometry, hence its boundary is positive. The intermediate Fourier compression in Lemma 0.16 instead uses \(\eta=-t\), explaining its opposite index. No index is changed by the dilation.

**Marking (12 points).** Compute the unitary dilation conjugacy and the rescaled Volterra coefficient. (4 points). Identify the normalized cokernel vector, both defects and the two opposite Fredholm indices. (4 points). Match the physical boundary convention and explain why dilation preserves its sign. (4 points).



**Exercise 32 (12 points).** *advanced.* Let \(B=C_0(\mathbb R)\), choose a nonzero real \(\theta\in C_c(\mathbb R)\), and put
\[
p(t)=\begin{pmatrix}
\cos^2\theta(t)&\cos\theta(t)\sin\theta(t)\\
\cos\theta(t)\sin\theta(t)&\sin^2\theta(t)
\end{pmatrix},\qquad p_0=\operatorname{diag}(1,0).
\]
Show that \(p-p_0\in M_2(B)\). Explain why \((pB^2,1,0)\) is not a scalar Kasparov cycle over \(B\), and why the relative two-representation cycle in Theorem 0.14 is legitimate. Compute its KK class.

**Solution.** The column \((\cos\theta,\sin\theta)\) has unit norm, so \(p\) is a projection. It equals \(p_0\) outside the compact support of \(\theta\), proving the compact-coefficient difference. However, the identity compact operator of \(pB^2\) would be \(p\in pM_2(B)p\). Its norm tends to one at infinity, so it does not belong to that compact algebra. The square defect for the zero operator is \(-p\); therefore the proposed individual cycle fails.

On \(B^2\oplus(B^2)^{\rm op}\), use representations \(p,p_0\) and the off-diagonal identity. Its square and adjoint defects vanish, and its commutator with a scalar representation has off-diagonal entries \(p-p_0\), which are compact. It is thus a legitimate relative cycle. Replace \(\theta\) by \(u\theta\), for \(0\le u\le1\). The resulting projection differences form a norm-continuous compact-coefficient homotopy. At zero both representations are \(p_0\), so the cycle is degenerate. Its KK class and its relative \(K_0(B)\)-class are zero. The general unitization argument in Theorem 0.19 uses legitimate projection cycles over \(B^+\), subtracts the scalar reference, and then restricts; this exercise explains why that order matters.

**Marking (12 points).** Verify the rotating projection and compactness of its difference from the constant projection. (4 points). Explain the defect of each individual proposed scalar cycle and verify the relative cycle. (4 points). Construct the norm-continuous rotation homotopy to the degenerate cycle and conclude that its class vanishes. (4 points).



**Exercise 33 (12 points).** *advanced.* Put \(I=C([0,1],C_0(\mathbb R))\), with \((\beta_t f)(w,x)=f(w,x-wt)\), and let \(e_v\) be evaluation at \(v\). Prove that the constant coefficient inclusion \(c_v(b)(w,x)=b(x)\) is an ordinary KK inverse of \(e_v\), but need not be equivariant for \(\beta\) and \(\beta^v\). Explain why this does not prevent the formula \(d_C[c_v]t_I\) in Corollary 8.14a from being a crossed-evaluation inverse, and check both of its products.

**Solution.** Evaluation and the constant inclusion are *-homomorphisms, with \(e_vc_v=1\). The interval contraction \(f(w,x)\mapsto f((1-s)v+sw,x)\) gives \(c_ve_v\sim1_I\), uniformly in the \(C_0(\mathbb R)\)-norm. Thus their ordinary KK products are identities. On the other hand
\[
\beta_t(c_v b)(w,x)=b(x-wt),\qquad
c_v(\beta_t^v b)(w,x)=b(x-vt).
\]
For a nonzero compactly supported bump, choose \(w\ne v\) and \(t\ne0\) so these translations differ. Thus \(c_v\) is not equivariant in general and cannot simply be crossed as a coefficient homomorphism. The expression \(x=d_C[c_v]t_I\) is nevertheless a well-typed ordinary KK product: its successive algebras are \(C\rtimes\mathbb R,C,I,I\rtimes\mathbb R\). Naturality is used only for the equivariant map \(e_v\), giving \([e_{\mathbb R}]=d_I[e_v]t_C\). Associativity and the four real inverse identities yield \([e_{\mathbb R}]x=d_I[e_v][c_v]t_I=d_It_I=1\), and \(x[e_{\mathbb R}]=d_C[c_v][e_v]t_C=d_Ct_C=1\). No equivariance of \(c_v\) enters either calculation.

**Marking (12 points).** Give the interval contraction proving the ordinary evaluation and inclusion classes are inverse. (4 points). Show explicitly why the constant inclusion fails equivariance. (4 points). Write both typed Kasparov inverse products using naturality only for the equivariant evaluation. (4 points).




**Exercise 34 (10 points).** *intermediate.* In the conventions of Lemma 8.15a, derive the modular correction in (117) and the Lie bracket of the generators. Explain why setting the anchor to zero does not destroy ellipticity.

**Solution.** Let \(M\) be multiplication by \(\Delta_v^{-1/2}\). For a smooth compact function,

\[
M^{-1}(-iX_i^R)Mf=-iX_i^Rf+(i/2)(X_i^R\log\Delta_v)f.
\]
This is (117). Also \(dh_R=\Delta_v^{-1}dh_L\) and left translation preserves \(dh_L\), so \(\operatorname{div}_{dh_R}X_i^R=-\kappa_i(v)\). Consequently the displayed correction is exactly half the divergence correction for \(-iX_i^R\), with the required sign. Right-invariant fields obey \([X_i^R,X_j^R]=-v\sum_k c_{ij}^kX_k^R\). Hence
\[
[D_i,D_j]=iv\sum_kc_{ij}^kD_k,
\qquad D^2=\sum_iD_i^2+iv\sum_{i<j,k}c_{ij}^kD_k\varepsilon_i\varepsilon_j.
\]
The extra term is first order. At zero the source group is still \(\mathbb R^d\), the operator is \(-i\sum\partial_i\varepsilon_i\), and its principal square is \(|p|^2\). A zero anchor concerns the map to the base tangent bundle, whereas this ellipticity is along the nonzero-dimensional source fibres.

**Marking (10 points).** Derive the modular half-density correction and its divergence sign. (4 points). Compute the right-invariant commutators and the complete Dirac square with its first-order term. (4 points). Explain why the zero-anchor source operator remains elliptic. (2 points).



**Exercise 35 (14 points).** *advanced.* Complete two checks that cannot be replaced by fibrewise statements. First, show how a principal parametrix proves interval-algebra membership of \((D+i)^{-1}\). Second, check the two-coordinate Bott product explicitly. Explain why neither check by itself proves a general vector-bundle KK Thom theorem.

**Solution.** Quantization of \((p\cdot\varepsilon)/|p|^2\) with a low-frequency cutoff and compact physical support gives \(Q\) of order \(-1\). The uniform composition calculus gives \((D+i)Q=1-S\) with \(S\) also of negative order, so both \(Q,S\) belong to the interval algebra. On the graph core the identity proves that \(Q\) maps into \(\operatorname{Dom}D\); closedness and the bounded right side extend this to the whole module. The regular module inverse \(R=(D+i)^{-1}\) therefore satisfies \(R=Q+RS\). The right side is an algebra element, proving the assertion with joint norm approximation. Separate fibrewise compactness would not prove this identity in the parameter algebra.

For coordinates \(x,y\), take \(F_1=x\varepsilon_1/\sqrt{1+x^2}\) and \(F_2=(x\varepsilon_1+y\varepsilon_2)/\sqrt{1+x^2+y^2}\). Their graded anticommutator is
\(2x^2/(\sqrt{1+x^2}\sqrt{1+x^2+y^2})\), a nonnegative multiplier. For an old creation vector \(\xi(x)\), the connection errors are sums of \(x\varepsilon_1\xi(x)/\sqrt{1+x^2+y^2}\) and
\[
(-1)^{|\xi|}\xi(x)y\varepsilon_2((1+x^2+y^2)^{-1/2}-(1+y^2)^{-1/2}).
\]
 Their norms tend to zero on the joint complement of compact sets: use \(\xi(x)\to0\) for large \(|x|\), and the scalar limits for bounded \(x\) and large \(|y|\). Adjoint errors have the same bound. Definition 0.3 and Theorem 0.6 now give the old-first unreduced Clifford product class, with its product-operator homotopy. For the native odd–odd product, (104) instead puts the new coefficient first, so its full matrix numerator is \(x\sigma_y+y\sigma_x\), with grading \(\sigma_z\). The unitary \((\sigma_x+\sigma_y)/\sqrt2\) exchanges the two physical generators and reverses that grading; after this scalar coefficient contraction the native class is the negative of the positive geometric \(F_2\) class. This verifies \(s_2=-1\) by the actual finite module. Both arguments use a fixed global frame. The global bundle construction and both inverse products require the additional argument supplied in Lemmas 6.0a–b and Theorem 6.0c.

**Marking (14 points).** Use the parametrix identity and closed graph core to prove that the resolvent belongs to the algebra. (4 points). Verify the creation connections and positivity for the joint normalized product. (4 points). Compute the native two-odd-factor sign using the explicit unitary and its grading reversal. (4 points). Identify the additional bundle arguments required beyond the fixed-frame calculation. (2 points).




**Exercise 36 (10 points).** *intermediate.* For a metric real line, use the basis \(1,e\) of its exterior algebra. Write the matrices of \(a=e+\iota\), \(c=e-\iota\), \(D\) and \(Q\). Determine the kernel parity and spectral gap of \(Q^2\). Then check the rotation in (143), including which grading occurs in the second Clifford action. Explain why an orthogonal frame reversal does not twist the Gaussian kernel line.

**Solution.** In the indicated order,
\[
a=\begin{pmatrix}0&1\\1&0\end{pmatrix},\quad
c=\begin{pmatrix}0&-1\\1&0\end{pmatrix},\quad
\Gamma=\begin{pmatrix}1&0\\0&-1\end{pmatrix},\quad
Q=\begin{pmatrix}0&-\partial_v+v\\\partial_v+v&0\end{pmatrix}.
\]
Here \(D=c\partial_v\). If \(L=-\partial_v^2+v^2\), then \(Q^2=\operatorname{diag}(L-1,L+1)\). The Hermite eigenvalues are \(2n\) on even degree and \(2n+2\) on odd degree. Its unique zero vector is \(\pi^{-1/4}e^{-v^2/2}\) in even degree, and the positive square-spectrum starts at two.

On the underlying ordinary tensor product, \(A=a\otimes1\) and \(B=\Gamma\otimes L_\varepsilon\). Hence \(A^2=B^2=1\), \(AB=-BA\), \([AB,A]=-2B\) and \([AB,B]=2A\). Differentiation of \(e^{-\theta AB/2}Ae^{\theta AB/2}\) and the analogous expression for \(B\) gives \(A\cos\theta+B\sin\theta\) and \(-A\sin\theta+B\cos\theta\). At \(\pi/2\), \(D+Bz\) becomes \(D-Az\), while the left Clifford action becomes \(B\). The graded notation for this second action is \(1\widehat\otimes L_\varepsilon\): adding another \(\Gamma\) to that graded tensor would insert the sign twice and destroy the required anticommutation.

A frame reversal sends \(v\) to \(-v\) and \(e\) to \(-e\). The exterior vacuum \(1\) and the radial Gaussian are fixed. Thus the kernel line is trivial even for a nonorientable real line bundle. Its target still retains the Clifford bundle twist; this observation does not give a spin-c structure to that line bundle.

**Marking (10 points).** Compute the two oscillator matrices, squared spectra and unique even Gaussian kernel. (4 points). Verify the Clifford rotation and include the tensor parity exactly once. (4 points). Check frame reversal on the vacuum and explain why the kernel is trivial while the target Clifford twist is retained. (2 points).



**Exercise 37 (16 points).** *advanced.* Supply three global checks in the reverse product proof. Show that localized resolvents in (133) vanish at \(z\)-infinity uniformly down to \(t=0\), prove their norm continuity from the strong wave construction, and explain why the later representation contraction remains a cycle at \(s=0\). Finally derive the inverse proper-pullback identity and the order of the direct-sum inverse in (148).

**Solution.** Let \(R=(T_{t,z}\pm i)^{-1}\) and let \(\chi\) have support in \(|v|\le R_0\). From \(\|R\|,\|T_{t,z}R\|\le1\) and the potential-independent cutoff commutator, \(h=\chi R\xi\) satisfies (135). For \(|z|>2R_0+1\), its potential obeys \(|tv-z|\ge |z|-R_0\), so
\[
\bigl((|z|-R_0)^2-r\bigr)\|h\|^2
\le C_\chi^2\|\xi\|^2.
\]
For the remaining bounded \(z\), use \(\|h\|\le\|\chi\|\|\xi\|\), enlarging the constant. This proves \(\|\chi R\|\le C/(1+|z|)\), with no lower bound on \(t\). The gradient part of (135) bounds the outputs in a fixed compact-support \(H^1\) ball, hence a relatively compact \(L^2\) set. At convergent parameters, both resolvents and their adjoints are strongly continuous by the Laplace wave formula. If \(K_n=\chi R_n\), then for any unit inputs \(\xi_n\) and any fixed \(\eta\),
\(\langle(K_n-K_0)\xi_n,\eta\rangle=\langle\xi_n,(K_n^*-K_0^*)\eta\rangle\to0\).
The differences of the output sets are relatively compact, so a norm-nonzero convergent output subsequence would contradict this weak limit. Thus \(\|K_n-K_0\|\to0\). Finite Clifford coefficients and compact base cutoffs give the stated module compactness.

For the representation \(u(z+sv)\), use a finite-rank spectral approximation of \(f(Q)\). On each of its finitely many fixed Hermite vectors, dominated convergence gives parameter continuity, and a compact \(v\)-cutoff makes vanishing at large \(z\) uniform in \(s\in[0,1]\). The omitted Hermite tails and spectral tail have arbitrarily small norms. Therefore \(u(z+sv)f(Q)\) is a compact operator field vanishing at target infinity on the whole interval. Its derivative commutator has \(s\partial_i u(z+sv)\); the same argument and the norm-integrable bound (132) apply. At \(s=0\) the derivative vanishes, and coefficients depend only on \(z\). The even Gaussian kernel then carries exactly the target identity action. Moving fibrewise centres alone would not prove these norm-continuity and compactness statements.

Write \(H=[h^*]\), \(J=[\widetilde h^*]\) and use (146). Multiplying \(Hb_{h^*E}=b_EJ\) first by \(d_E\) on the left and then by \(d_{h^*E}\) on the right gives \(d_EH=Jd_{h^*E}\), with all domains matched. If \(b_{E\oplus F}=b_Eb_F^{\mathrm{Cl}E}\), the inverse is \(d_F^{\mathrm{Cl}E}d_E\), because its two products cancel adjacent inverse pairs. Reversing that order is not even a composable pair of these KK morphisms. No interchange of odd Clifford blocks occurs.

**Marking (16 points).** Prove the uniform localized resolvent decay in the exterior parameter. (4 points). Combine local Sobolev compactness, strong convergence and the adjoint argument to obtain norm continuity. (4 points). Control the finite Hermite approximation and interval representation contraction, including the endpoint. (4 points). Write the properly typed pullback and direct-sum inverse identities with the correct inverse order. (4 points).



**Exercise 38 (12 points).** *intermediate.* On the real line take the radius \(r=1\) compression in (152). For a function supported in \([-A,A]\), find one compact set containing the supports of its whole homotopy, and the smaller endpoint support. Check the derivative of \(q_1\) at zero and its inverse. Give a \(C_0(\mathbb R)\) function whose restriction to \((-1,1)\) is not in \(C_0((-1,1))\). Explain which map, and which actual cycle, prove that disk extension preserves the Thom class.

**Solution.** The function \(v\mapsto v/\sqrt{1+sv^2}\) is increasing, so the support at \(s\) is contained in
\[
\left[-\frac{A}{\sqrt{1+sA^2}},\frac{A}{\sqrt{1+sA^2}}\right]\subset[-A,A].
\]
At one this is the compact interval with radius \(A/\sqrt{1+A^2}<1\). The union of the supports over the parameter interval is therefore compact. The explicit inverse in (152), followed by the compact-level-set argument there, proves joint continuity across the moving boundary and uniform norm continuity; separate pointwise continuity would not suffice. Differentiation gives \(q_1'(v)=(1+v^2)^{-3/2}\), so \(q_1'(0)=1\). Its inverse on \((-1,1)\) is \(w/\sqrt{1-w^2}\).

The function \(f(x)=e^{-x^2}\) lies in \(C_0(\mathbb R)\), but its restriction tends to \(e^{-1}\) at both boundary ends of \((-1,1)\). For \(0<\epsilon<e^{-1}\), its \(\epsilon\)-level set is the whole open interval, which is not compact there. Thus restriction does not give the needed \(C_0\) homomorphism. Instead use the extension-by-zero homomorphism from the open interval to \(\mathbb R\). The full family \(\Phi\) of (152), tensored with the identity of \(C_1\), represents a homotopy on the standard interval module with operator zero: its square defect against a coefficient \(a\) is \(-\Phi(a)\), a compact endomorphism. Its zero endpoint is the identity and its one endpoint is compression followed by disk extension. Multiplication by the invertible Thom class proves (150); invertibility then permits the cancellation used for two nested disks. No improper restriction map or fibrewise-only homotopy is used.

**Marking (12 points).** Bound the entire compressed support by one compact subset of the open interval. (3 points). Verify the inverse, joint continuity and positive zero-section derivative. (3 points). Explain why the restricted Gaussian fails to vanish at the open interval endpoints. (3 points). Construct the extension-by-zero interval cycle and use its endpoints and Thom cancellation. (3 points).



**Exercise 39 (12 points).** *advanced.* Let \(i:\{1\}\hookrightarrow S^1\) be the point embedding and \(j:S^1\hookrightarrow S^1\times\mathbb R\) the zero-height embedding. Use the positive tangent to the circle for the first normal line and the increasing real coordinate for the second. Equip these trivial lines with their canonical spin-c structures. Construct normalized whole-bundle tubes and the composite tube. Write the actual ordered product multiplier and its positivity against the first factor. Explain why adding the two separately normalized multipliers does not give the required product cycle.

**Solution.** Identify the first normal coordinate with \(v\in\mathbb R\), measured in angle at \(1\), and put
\[
\tau(v)=\exp\bigl(2i\arctan(v/2)\bigr),\qquad
\sigma(y,w)=(y,w),\qquad
\Omega(v,w)=\bigl(\tau(v),w\bigr).
\]
The angle derivative is \((1+v^2/4)^{-1}\), equal to one at zero; \(\tau\) is a diffeomorphism onto \(S^1\setminus\{-1\}\). The second tube is the identity of the trivial normal bundle, and the product connection makes its radial transport the identity. Thus \(\Omega\) is a normalized open tube of the point into the cylinder; its ordered normal coordinates are circle tangent first and vertical direction second.

Retain the two positive Clifford blocks, with \(\varepsilon_1^2=\varepsilon_2^2=1\) and \(\varepsilon_1\varepsilon_2=-\varepsilon_2\varepsilon_1\). The product representative on the sum module is
\[
F(v,w)=\frac{v\varepsilon_1+w\varepsilon_2}{\sqrt{1+v^2+w^2}},\qquad
\left\{\frac{v\varepsilon_1}{\sqrt{1+v^2}},F\right\}
=\frac{2v^2}{\sqrt{1+v^2}\sqrt{1+v^2+w^2}}\ge0.
\]
For a compact old creation section, the old numerator divided by the joint normalizer and the difference from the new \(w\)-only normalization both tend to zero as \(|w|\to\infty\). Outside its compact old support, approximation makes the errors arbitrarily small. Their adjoints have the same property. These are the connection and positivity checks from (147), so this operator represents the actual product before extension along \(\Omega\). In an underlying ordinary tensor representation the second generator includes the old parity operator exactly once; that is what makes the two blocks anticommute.

The tempting sum \(G=v\varepsilon_1/\sqrt{1+v^2}+w\varepsilon_2/\sqrt{1+w^2}\) has
\(G^2=v^2/(1+v^2)+w^2/(1+w^2)\).
As both absolute coordinates tend to infinity, \(G^2-1\to1\). For the point's unit left coefficient this is a multiplier that does not vanish at infinity, hence is not compact on the finite spin-c section module. It fails the Kasparov square-defect condition. The joint normalization of \(F\) instead gives \(F^2-1=-1/(1+v^2+w^2)\), which vanishes at joint infinity. Finally (160) extends this checked product into the cylinder and proves the composition equality, preserving the chosen order and orientation.

**Marking (12 points).** Specify the two whole tubes and their ordered normal coordinates. (4 points). Verify the jointly normalized Clifford product, creation connections and positivity. (4 points). Compute the noncompact square defect of the separately normalized candidate and contrast it with the valid joint defect. (4 points).



**Exercise 40 (12 points).** *advanced.* On \(M=\mathbb RP^2\), put \(W=TM\), \(T=W\), and give \(L=W\oplus W\) its canonical complex spin-c structure. Choose a sufficiently large even-dimensional Euclidean embedding and construct its graph normal by Lemma6.0j. Determine the determinant line of that normal structure. Show that the orientation line \(\lambda=\det TM\) has nontrivial complexification \(H=\lambda_{\mathbb C}\), although \(H^{\otimes2}\) is trivial. What happens to the actual normal orientation module if the prescribed structure on \(L\) is twisted by \(H\)? Explain why its determinant line cannot detect this change.

**Solution.** Let \(Q=W\oplus W\). The given structure on \(L\) is exactly the canonical structure on \(Q\), so \(D_L=\det W_{\mathbb C}\). Lemma6.0j's full recovery gives
\[
D_{E_j}\otimes\det W_{\mathbb C}\cong D_L,
\qquad D_{E_j}\cong\underline{\mathbb C}.
\]
This conclusion uses the canonical double-bundle module; declaring its determinant contribution trivial in advance would give the wrong comparison.

To identify \(\lambda\), lift to the double cover \(S^2\to\mathbb RP^2\). Orient the tangent plane at \(x\in S^2\) by requiring its ordered basis together with the outward normal \(x\) to orient \(\mathbb R^3\). The antipodal map acts by \(-I\) in the ambient three-space, whose determinant is \(-1\). It carries the outward normal to the outward normal at \(-x\); hence it reverses the induced tangent orientation. Thus the orientation line is associated to the sign representation of this double cover. Its tensor square has transition \((-1)^2=1\) and is canonically trivial.

If \(H\) were trivial, a normalized nowhere-zero section would lift to a continuous map \(a:S^2\to S^1\) satisfying \(a(-x)=-a(x)\). Restrict to the equator parametrized by \(e^{it}\). Its continuous argument \(\theta(t)\) on an interval is obtained by subdividing into finitely many arcs on which a branch of argument exists and adjusting adjacent branches by integral multiples of \(2\pi\). The relation implies
\(\theta(t+\pi)-\theta(t)=(2m+1)\pi\)
with one integer \(m\), by continuity. Applying it twice gives equatorial winding \(2m+1\), which is odd. But this loop extends over the northern hemisphere through \(a\). Contracting the hemisphere radially gives a homotopy to a constant loop, whose winding is zero. Winding is invariant in this homotopy: at each sufficiently small parameter change a compact loop stays in argument neighbourhoods and its integer total change is continuous, hence constant; finitely many parameter neighbourhoods cover the interval. This contradicts the odd winding. Therefore \(H\) is nontrivial, while \(H^{\otimes2}\) is trivial.

Twisting the given structure replaces \(\mathscr M_{E_j}\) by \(\mathscr M_{E_j}\otimes H\). These modules are not isomorphic as graded Clifford bimodules. An isomorphism would tensor with the actual inverse Morita module and, by its two evaluation maps, give \(H\cong\underline{\mathbb C}\), contradicting what was just proved. Their determinant lines, however, differ by \(H^{\otimes2}\), which is trivial. They have the same determinant line and distinct full orientation data. The normal Thom multiplier remains the same local Clifford expression, but its transition module has changed. This is precisely the information the cancellation formula retains.

**Marking (12 points).** Recover the entire normal spinor module and its determinant line in the prescribed order. (4 points). Prove nontriviality of the complexified orientation line using odd equatorial winding and hemisphere extension. (4 points). Show that twisting changes the actual normal module although the squared determinant does not detect the change. (4 points).



**Exercise 41 (12 points).** *intermediate.* Let \(i_s:S^1\hookrightarrow\mathbb R^2\), \(0\le s\le1\), be \(i_s(x)=x+(0,s)\). Use the outward radial normal, its trivial spin-c structure and a fixed \(0<\rho<1\). Write a normalized whole-bundle tube and the norm-continuous extension homomorphism of Lemma6.0g. For a function supported in \(S^1\times[-A,A]\), give one compact set containing all its image supports, separated from the moving tube boundary. Then stabilize by \(\mathbb R^2\), write the ordered product multiplier and explain the cancellation of the new Bott block.

**Solution.** Identify the normal bundle at every parameter with \(S^1\times\mathbb R\) by \(v\mapsto vx\). A normalized tube is
\[
\tau_s(x,v)=\bigl(1+q_\rho(v)\bigr)x+(0,s),
\qquad q_\rho(v)=\frac{v}{\sqrt{1+v^2/\rho^2}}.
\]
Its image is the annulus \(1-\rho<|z-(0,s)|<1+\rho\), because \(1+q_\rho(v)>0\). Its derivative in the normal coordinate at zero is \(x\), the specified unit normal. The map \((s,x,v)\mapsto(s,\tau_s(x,v))\) is an open embedding relative to the parameter interval: in the annulus, write \(r=|z-(0,s)|\), \(x=(z-(0,s))/r\), \(w=r-1\), and
\(v=w/\sqrt{1-w^2/\rho^2}\).
Thus the homomorphism is
\[
\Theta(f)(s,z)=
\begin{cases}
f\!\left(\dfrac{z-(0,s)}{|z-(0,s)|},
\dfrac{|z-(0,s)|-1}{\sqrt{1-(|z-(0,s)|-1)^2/\rho^2}}\right),
&\bigl||z-(0,s)|-1\bigr|<\rho,\\
0,&\text{otherwise}.
\end{cases}
\]
Let \(c=A/\sqrt{1+A^2/\rho^2}<\rho\). The supports for all parameters lie in the compact set
\[
K_A=\{(s,z):0\le s\le1,\ 1-c\le|z-(0,s)|\le1+c\}.
\]
It is closed and bounded and is separated radially from each moving boundary by \(\rho-c>0\). On its open tube image the displayed inverse proves continuity. Outside that image, a compact level set of \(f\) has compact image strictly inside it, so the zero extension is continuous at the boundary as well. Its support stays in \(K_A\), and uniform continuity on a compact neighbourhood gives norm continuity in \(s\). Density and the bound \(\|\Theta(f)\|\le\|f\|\) give the assertion for all \(C_0\) functions. Tensoring with \(1_{C_1}\) gives the actual homomorphism used in Lemma6.0g. The standard interval module with zero operator represents its class, because against any coefficient its square defect is minus that coefficient, a compact module map. Multiplication by the fixed normal Thom class proves equality of the two endpoint embedding classes.

For the new even block \(w=(w_1,w_2)\), use \(\Omega_s(x,v,w)=(\tau_s(x,v),w)\). Retaining the old normal generator before the two new generators, the multiplier is
\[
F(v,w)=\frac{v\varepsilon_0+w_1\varepsilon_1+w_2\varepsilon_2}
{\sqrt{1+v^2+w_1^2+w_2^2}}.
\]
The three positive generators anticommute. Its square defect is \(-1/(1+v^2+|w|^2)\), and its anticommutator with the old normalized multiplier is
\(2v^2/(\sqrt{1+v^2}\sqrt{1+v^2+|w|^2})\ge0\).
For a compact old creation section the connection errors vanish at new infinity; its old tail is controlled by approximation, as in (147). Hence this is the actual ordered sum Thom product. Contract the last \(C_2\) factor by its canonical positive spinor module to obtain \((i_s)_!B_2^{C_0(\mathbb R^2)\widehat\otimes C_1}\). Compose with the even inverse \(A_2\) on that same coefficient algebra. The adjacent product \(B_2A_2=1\) gives \((i_s)_!\), by Lemma6.0h. The old odd generator has never been moved past an odd inverse: the new Bott class and its inverse are even after the specified contraction.

**Marking (12 points).** Give the whole translated tube and its inverse. (4 points). Prove the common compact support buffer and norm-continuous extension by zero. (4 points). Verify the ordered stabilized product and its properly typed even inverse cancellation. (4 points).



**Exercise 42 (16 points).** *intermediate.* Write both circles as \(\mathbb R/\mathbb Z\). On \(V=S^1_\theta\times S^1_y\), foliate by the circles with \(y\) fixed. Let \(M=S^1_x\) and let the smooth leaf-space map have the global lift
\(f(x)=(0,mx)\), \(m\in\mathbb Z\). Use
\(j(x)=(\cos2\pi x,\sin2\pi x)\), its outward radial normal, and the specified trivial spin-c structure on \(TM\oplus f^*\tau\).
Give a global neighbourhood model for Lemma6.0l and an explicit étale replacement as in Lemma6.0m. Check its derivative, its ordered normal and its actual counting fibres. Finally replace \(mx\) by \(mx+c(s)\) for a smooth real function \(c\) on an interval. Explain precisely what proves equality of the two endpoint factorization classes.

**Solution.** Fix \(0<\epsilon<1/2\). The graph neighbourhood is
\(H=S^1_x\times(-\epsilon,\epsilon)_\delta\), with transverse circle coordinate \(y=mx+\delta\), projection \(p(x,\delta)=x\), and section \(\delta=0\). The interval condition makes the difference \(y-mx\) unique whenever it lies in that interval. Its foliation is
\(\ker(m\,dx+d\delta)\), where \(dx\) is the invariant circle one-form. In local real lifts this is exactly \(dy=0\); changing a lift adds an integer to \(y\) and preserves the one-form. Thus it is a genuine global foliated microbundle. Its normal quotient on the section is represented by \(\delta\), with the same trivial transverse line as \(f^*\tau\).

For \(0<\rho<1\) put
\(N=S^1\times(-\epsilon,\epsilon)\times(-\rho,\rho)\), using the outward normal coordinate \(\eta\). Its local lift to \(V\times\mathbb R^2\) is
\[
g(x,\delta,\eta)=
\bigl(0,mx+\delta,(1+\eta)(\cos2\pi x,\sin2\pi x)\bigr).
\tag{191}
\]
Its transverse map has coordinates \((y,z_1,z_2)\). In a local lift of \(x\), the determinant of its derivative, with columns ordered as \((x,\delta,\eta)\), is
\[
\det\begin{pmatrix}
m&1&0\\
-2\pi(1+\eta)\sin2\pi x&0&\cos2\pi x\\
2\pi(1+\eta)\cos2\pi x&0&\sin2\pi x
\end{pmatrix}=2\pi(1+\eta)>0.
\tag{192}
\]
It is therefore étale. In fact this transverse map is injective onto its open image: the direction of the nonzero \(z\) determines \(x\), its length determines \(\eta\), and the unique small circle difference determines \(\delta\). The section \(k(x)=(x,0,0)\) has the ordered normal \((\eta,\delta)\), namely \(\nu_j\oplus f^*\tau\). Its orientation is the full module of Lemma6.0j constructed from the specified structure; a positive determinant for a different coordinate order is not used as a replacement for that module.

The holonomy groupoid of this product foliation is \(V\times_{S^1_y}V\). Indeed every leaf is a circle with identity transverse holonomy, so paths with the same endpoints define the same holonomy arrow. For a source unit \((\theta,y,z)\) of the product groupoid, an element of the principal counting fibre first requires a point of \(N\) with that transverse coordinate \((y,z)\). There is at most one such point by the preceding reconstruction. If it exists, there is exactly one arrow from \((\theta,y)\) to \((0,y)\). Thus each counting fibre is empty or a singleton. This conclusion uses the actual trivial holonomy, rather than just the numerical leaf dimension.

For the parameter family use the same \((x,\delta,\eta)\) domains and put \(y=mx+c(s)+\delta\). The vertical derivative determinant is unchanged. The normal structure on \(TM\oplus f_s^*\tau\) is the fixed specified structure pulled back over the interval. Lemma6.0n constructs the whole interval Hilbert module and gives its exact surjective endpoint evaluations; Theorem6.0o composes it with the whole normal Thom/tube cycle and the fixed Euclidean inverse. Its endpoints are the two factorization classes. This actual Kasparov homotopy proves their equality. An endpoint coincidence of leaf images, by itself, would not prove this conclusion.

**Marking (16 points).** Construct the global foliated microbundle and identify its kernel distribution. (4 points). Compute the transverse derivative and retain the prescribed full normal order. (4 points). Identify the actual empty or singleton counting fibres. (4 points). Use the single interval module and full spinor orientation to prove the family endpoint equality. (4 points).



**Exercise 43 (12 points).** *advanced.* Form the suspension
\[
V=(\mathbb R_\theta\times\mathbb R_y)/
\bigl((\theta,y)\sim(\theta+1,2y)\bigr),
\tag{193}
\]
with the foliation induced by the horizontal lines, and take the small transverse lift
\(g:N=(-\epsilon,\epsilon)\to V\), \(g(y)=[0,y]\).
At \(x_0=[0,0]\), determine the counting fibre of \(P_g\) and the regular source fibre of the holonomy groupoid. Write the map \(W_{x_0}\) for one short-plaque core vector. Verify directly that a rank-one module operator can act as a nonzero multiple of the identity on an infinite-dimensional counting fibre. Explain why this does not contradict Lemma6.0n.

**Solution.** The integer action is free and proper: on a compact set of \(\theta\)-coordinates only finitely many shifts can meet that set. Its quotient is therefore a smooth Hausdorff mapping torus, with the usual two overlapping interval charts in the \(\theta\) direction. The horizontal distribution is invariant because its derivative sends \(\partial_\theta\) to \(\partial_\theta\). The leaf of \(x_0\) is a circle. A positive one-turn path, lifted from \((0,y)\) to \((1,y)\), ends at \([0,y/2]\). Its transverse holonomy germ is \(y\mapsto y/2\); the \(k\)-turn germ is \(2^{-k}y\). These germs are distinct for every \(k\in\mathbb Z\). Consequently this leaf's holonomy covering is the real line, and the regular source fibre \(G_{x_0}\) is \(\mathbb R_t\), with range \([t,0]\). This follows directly from the definition of holonomy arrows: two lifted paths with the same endpoint differ by an integer winding, whose germ is identity only when that integer is zero.

Only \(g(0)\) lies on this leaf. Every winding gives a distinct principal lift over \(x_0\), so
\(a^{-1}(x_0)=\{z_k:k\in\mathbb Z\}\), with \(z_k\) the \(k\)-turn arrow to \(g(0)\). The counting space is \(\ell^2(\mathbb Z)\), not the one-dimensional space obtained by forgetting the holonomy germs. Use the invariant positive longitudinal density \(dt\). Choose \(0<a<1/2\), \(b\in C_c^\infty((-a,a))\) with \(\int|b(t)|^2dt=1\), and a transverse cutoff \(\sigma\in C_c^\infty(N)\). Let \(\xi\) be the short-plaque source-anchor chart vector with coefficient \(b(-t)\sigma(y)\). Along this leaf, \(z_k\gamma_t^{-1}\) belongs to that chart precisely when \(|t-k|<a\); its short signed displacement is \(k-t\). The formula for \(W_x\) in Lemma6.0n therefore becomes
\[
(W_{x_0}(\xi\otimes u))_k
=\sigma(0)\int_{\mathbb R} b(k-t)u(t)\,dt.
\tag{194}
\]
The row functions are translates supported in disjoint intervals \((k-a,k+a)\). Their squared norms equal one. Hence their Gram matrix is the identity, and the tensor image of \(\theta_{\xi,\xi}\) is
\[
W_{x_0}(\theta_{\xi,\xi}\otimes1)W_{x_0}^{-1}
=|\sigma(0)|^2\,1_{\ell^2(\mathbb Z)}.
\tag{195}
\]
This is precisely the counting-fibre action of multiplication by \(|\sigma|^2\). For \(\sigma(0)\ne0\) it has infinite Hilbert-space rank and is not a Hilbert-space compact operator: the images of the orthonormal basis have fixed nonzero norm and no convergent subsequence.

The module operator is nonetheless the single rank-one operator \(\theta_{\xi,\xi}\), so is compact on \(\mathcal E_g\). Tensoring a Hilbert-module compact operator with an infinite-dimensional regular representation need not produce a Hilbert-space compact operator. Here the creation operator has infinitely many orthogonal rows, one for every genuine holonomy lift. The explicit calculation verifies both module compactness and the necessity of keeping those lifts; no finiteness of a holonomy counting fibre was assumed.

**Marking (12 points).** Identify the smooth quotient, loop holonomy and all distinct actual germ parameters. (4 points). Compute the creation rows and prove their disjoint-support Gram identity. (4 points). Distinguish compactness in the original module from noncompactness on the induced infinite counting fibre. (4 points).



**Exercise 44 (12 points).** *intermediate.* On the cylinder \(X=S^1_\theta\times\mathbb R_s\), use the trivial Hermitian line with unitary connection
\(\nabla=d+i\sin\theta\,ds+i\lambda\,d\theta\), \(\lambda\in\mathbb R\). Embed \(L=S^1\) by \(i(\theta)=(\theta,0)\), prescribe a normal spin-c structure, and use the whole normalized tube
\[
\tau(\theta,v)=(\theta,q(v)),\qquad
q(v)=\frac{v}{\sqrt{1+v^2}}.
\tag{212}
\]
Compute the radial transport in Lemma6.0p. Show directly that it gives the onto even unitary between the two projection-formula modules, on the entire tube. Explain why neither a pullback homomorphism along the bundle projection nor a uniform bound on a connection derivative is needed. Retain an arbitrary full normal line twist in the calculation.

**Solution.** The path is \(t\mapsto(\theta,q(tv))\). Its angular coordinate is constant, so the \(i\lambda\,d\theta\) term does not enter its transport equation. With initial value one, that equation has solution
\[
T_{\theta,v}=\exp\bigl(-i\sin\theta\,q(v)\bigr).
\tag{213}
\]
Indeed its logarithmic derivative along the path is \(-i\sin\theta\,\partial_tq(tv)\), and integrating gives \(-i\sin\theta(q(v)-q(0))\). The formula is globally smooth and has norm one. The normal multiplier is \(F(v)=\varepsilon(v)/\sqrt{1+v^2}\) on the specified normal Thom module. Multiplication by \(T_{\theta,v}\) acts on the even coefficient line and commutes with that multiplier and all the coefficient actions.

The two completed modules have the same normal support space, with the coefficient fibre transported from \((\theta,0)\) to \((\theta,q(v))\). For compact core sections, pointwise multiplication by \(T\) preserves their inner product because \(\overline T T=1\). Its inverse is multiplication by \(\overline T\). Both maps preserve compact supports, so their images contain the opposite compact core; they extend to inverse onto isometries of the completed modules. The creation connections are exact, and the compact defects remain the localized Thom defects, tensored with a finite line. Both products therefore have precisely the operators in Lemma6.0p.

The tube is a diffeomorphism onto \(S^1\times(-1,1)\), so its essential open extension is legitimate. The base projection \(S^1\times\mathbb R\to S^1\) is nonproper: the pullback of the constant function one does not vanish at infinity. The calculation uses its bounded multiplier action, rather than that false \(C_0\) homomorphism. No derivative of \(T\) occurs in the bounded inner-product or operator-intertwining calculation. Smooth transport on finite compact parameter sets and the norm-one bound suffice. A normal twist stays in its separate module factor throughout, so its complete line, including phase transitions, survives the unitary.

**Marking (12 points).** Calculate radial parallel transport and verify its global unit norm. (4 points). Prove that the support-preserving module map is onto and intertwines the phase and creation maps. (4 points). Explain the bounded-multiplier requirement for the nonproper projection and retain the independent full normal twist. (4 points).



**Exercise 45 (12 points).** *advanced.* Let \(M=S^1\), \(j(\theta)=(\cos\theta,\sin\theta)\), and let \(f:M\to\{\mathrm{pt}\}\) have the stable orientation whose graph normal \(\nu_j\) has its outward positive frame. Let \(h:\{\mathrm{pt}\}\to M\) select \(\theta=0\), use \(u:\{\mathrm{pt}\}\hookrightarrow\mathbb R^2\), \(u=0\), and orient its graph normal by the ordered frame \((a,b,t)\), where \(a,b\) are the two new Euclidean directions and \(t\) is the positive circle tangent. These full normal modules determine the corresponding stable orientations by Lemma6.0v's recovery.

Both maps have odd parity. Compute the composed normal map into \(\mathbb R^2\oplus\mathbb R^2\), with the \(j\)-block first. Determine the grading of its recovered virtual rank-zero orientation and the scalar class \(h_!\# f_!\). Repeat after reversing the normal orientation of \(f\). Do not replace the two odd factors by an ungraded tensor product.

**Solution.** At \(\theta=0\), the tangent is \(t=(0,1)\) and the outward normal is \(\nu=(1,0)\). The two normal ranks are \(k_h=2-0+1=3\) and \(k_f=2-1+0=1\). Their ordered product has rank four and ordered frame \((a,b,t,\nu)\). The product étale map is the ordinary normal map of the circle times \(1_{\mathbb R^2}\). Thus the normal quotient map sends these four basis vectors, respectively, to
\[
a\longmapsto(0,0,1,0),\quad
b\longmapsto(0,0,0,1),\quad
t\longmapsto(0,1,0,0),\quad
\nu\longmapsto(1,0,0,0).
\tag{214}
\]
Its matrix has determinant \(-1\): moving the new two-plane past the old two-plane has positive sign, whereas the old ordered frame \((t,\nu)\) reverses the standard \((x,y)\) orientation. This sign is the actual transported full normal orientation.

Here \(TL=0\) and \(T=0\), so there is no double-tangent factor. The cancellation in Lemma6.0v removes the canonical \(C_4\) Euclidean factor on both sides. A reversed real orientation gives the odd graded line. One can check the grading without a characteristic-class argument. An orientation-reversing orthogonal transformation is the product of a positive transformation and a reflection. The positive transformation has an even spin lift. Reflection is implemented in even Clifford rank by an odd Pin unit, up to that positive transformation. Multiplication by this odd unit identifies the resulting regular bimodule, by an even map, with the parity reversal of the canonical regular bimodule. The two finite Morita evaluations therefore leave the parity-reversed scalar line \(\Pi\mathbb C\), rather than \(\mathbb C\).

In the retained odd convention the two parity blocks are ordered \(C_1\widehat\otimes C_1=C_2\). The fixed \(\Lambda^*\mathbb C\) evaluation contracts them. The even triple cycles and wedge evaluations of Theorem6.0w identify this early contraction with contraction of the full ordered \(C_4\) normal module. The scalar Euclidean Thom/inverse identities then leave
\[
h_!\# f_!=[\Pi\mathbb C,1,0]=-1
\quad\text{in }KK(\mathbb C,\mathbb C)=\mathbb Z.
\tag{215}
\]
The last equality is the grading-inverse convention proved in Theorem0.7 and the scalar classification of Section0. This computation uses the two exact Euclidean inverse products; it assumes no Dirac-index comparison for the circle.

Reversing the one-dimensional normal orientation of \(f\) changes its positive frame to \(-\nu\). The composed determinant becomes \(+1\); the finite cancellation leaves the even scalar unit, and the product is \(+1\). Equivalently the two parity reversals cancel. Both cases have the same underlying virtual rank-zero bundle. Its stable orientation grading distinguishes the two classes, so the canonical even rank-zero unit cannot stand for both.

**Marking (12 points).** Write the chronological normal order and compute the negative orientation determinant. (4 points). Recover the full finite graded Clifford module and its odd scalar line. (4 points). Compute the typed two-odd-factor product and show how reversing the prescribed normal changes its sign. (4 points).



**Exercise 46 (12 points).** *intermediate.* On \(D=\mathbb R/2\pi\mathbb Z\),
take \(Q=\{|\theta|\le\pi/2\}\) and its two-point boundary \(M\).
Let \(0<\epsilon<\pi/4\), and give the two outward normals the
coordinates
\[
 \tau_+(v)=\pi/2+q_\epsilon(v),\qquad
 \tau_-(v)=-\pi/2-q_\epsilon(v),\qquad
 q_\epsilon(v)=\frac{v}{\sqrt{1+v^2/\epsilon^2}}.
\tag{236}
\]
Write the normal Thom unitary explicitly and construct its null-homotopy
on the circle. Explain why the same argument fails for a single
positively oriented point in this circle.

**Solution.** Put \(h(v)=v/\sqrt{1+v^2}\).
On either collar the unitary is
\(U(\tau_\pm(v))=\exp(\pi i(1+h(v)))\), and it is one elsewhere.
The plus collar traversed in increasing \(\theta\) has its phase
increase from zero to \(2\pi\). The minus collar traversed in that
same circle direction has its phase decrease from \(2\pi\) to zero.
The outgoing positive coordinate is \(v\), not increasing \(\theta\),
at the minus point.

Define \(\Theta=0\) on the middle part of \(Q\), take
\(\Theta=\pi(1+h(v))\) on each collar, and set \(\Theta=2\pi\)
on the remaining outside arc. Its limits match at all four collar
edges and at the circle identification. The bounded continuous
function therefore defines the norm-continuous homotopy
\(U_t=e^{it\Theta}\). Its norm modulus is at most
\(2\pi|t-t'|\). This is the actual null class of Lemma 6.0y.
Each of the two point classes separately has winding \(+1\) or
\(-1\), while their signed sum is zero.

For a single positive point, cut the circle at an exterior point
and follow a real phase continuously from zero. The collar
increases that phase by \(2\pi\), and the unitary is constant
on the rest of the circle. The ending phase is \(2\pi\), so
it cannot be a single-valued continuous real function on the
identified circle. The unitary has winding one. Winding is
invariant under a continuous homotopy: subdivide a compact
parameter square so each image lies in a circle arc with a
continuous phase; changes between phase choices are integral
multiples of \(2\pi\), locally constant in the parameter.
Hence this unitary cannot have a null-homotopy. The separating
hypothesis is exactly what allowed the two phase changes to
cancel. \(\square\)

**Marking (12 points).** Calculate both outward collar phases and their opposite windings. (4 points). Construct the continuous real phase on the whole circle and its norm-continuous nullhomotopy. (4 points). Show why a single positive point has nonzero winding and cannot use that whole-circle phase. (4 points).



**Exercise 47 (12 points).** *advanced.* Let \(L,H\) be Hermitian complex lines
over a compact smooth \(M\). Give \(W=L_{\mathbb R}\) its complex
spin-c structure twisted by \(H\). Determine its full north/south
spinor bundles, clutch bundle and sphere projection orientation.
Check the normalization without discarding a two-torsion \(H\).
Then perform the zero-rank calculation with the same line.

**Solution.** The exterior spin lift of a unitary line transition
is \(\operatorname{diag}(1,g_L)\), as computed in Lemma 6.0j.
The spinors are therefore
\(S_W^+=H\) and \(S_W^-=L\otimes H\). Clifford multiplication
from the positive to the negative part is \(1_H\otimes w\).
Thus \(b_{W,H}\) is obtained by gluing
\(\pi^*H\) in the north to \(\pi^*(L\otimes H)\) in the south
using this full line-valued isometry. If \(b_{W,1}\) is the
untwisted clutch bundle, the actual transition maps give
\[
 b_{W,H}\cong\pi^*H\otimes b_{W,1},\qquad
 n_!^H(1)=[b_{W,H}]-[\pi^*(L\otimes H)].
\tag{237}
\]
The inverse orientation of the projection has the tensor
\(\pi^*H^*\). The full coefficient-transport theorem therefore
gives \((\pi_!^H)_*(x)=(\pi_!^1)_*(x[\pi^*H^*])\).
Applying Theorem 6.0ad and the projection formula yields, for
every \(a\in K^0(M)\),
\[
 (\pi_!^H)_*(\pi^*a[b_{W,H}])=a,\qquad
 (\pi_!^H)_*(1)=0.
\tag{238}
\]
These equations use \(H\otimes H^*\cong\mathbf1\), not a
trivialization of \(H^{\otimes2}\). For example, the nontrivial
two-torsion line on \(\mathbb RP^2\) proved in Exercise 40 has
trivial square but is retained here: the north restriction of
\(b_{W,H}\) is \(H\), whereas that of \(b_{W,1}\) is the
trivial line. Their fibre restrictions alone would miss this
difference. The inverse orientation cancels the actual \(H\).

In rank zero the sphere consists of \(M_+\sqcup M_-\).
The Bott bundle is \((H,0)\), and the projection modules are
the even \(H^*\) and the odd \(\Pi H^*\), respectively.
Consequently
\[
 (\pi_!^H)_*(a_+,a_-)=H^*a_+-H^*a_-,
 \qquad(\pi_!^H)_*(H,0)=1.
\tag{239}
\]
In particular its image of \((1,1)\) is zero. Assigning an
even unit to both components would instead give the sum and
would violate the boundary relation already for the interval.
The retained zero-rank grading fixes that sign. \(\square\)

**Marking (12 points).** Compute the two sphere spinors and clutching map with the full orientation line. (4 points). Evaluate the inverse projection with the actual dual line and retain possible order-two twists. (4 points). Check the two rank-zero sheet gradings and the signed projection formula. (4 points).



**Exercise 48 (12 points).** *intermediate: the rank-zero orientation line.* Give a compact manifold \(V\) its point foliation. Specify an arbitrary Hermitian line \(H\) as the rank-zero spin-c transition module. Determine the Dirac cycle, its coefficient action and the Thom map. For a vector bundle \(B\), check (246) on the symbol class \([B]\). Explain why replacing the Dirac module by the scalar identity while retaining this Thom map is generally incorrect.

**Solution.** The unit groupoid has algebra \(C(V)\). Its Dirac module is \(\Gamma(H)\), with even grading and operator zero. Every endomorphism of this finite bundle module is compact, so it is a cycle even though its square defect is minus the identity. Its class is \([H]\), and a coefficient bundle \(E\) gives \([E\otimes H]\). The chosen Thom map is multiplication by \([H]\), with inverse multiplication by \([H^*]\). Thus (246) gives

\[
\Theta_F^{-1}[B]\;\ell_!^{\mathrm{an}}
=[B\otimes H^*][H]=[B].
\tag{247}
\]

Replacing only the Dirac class by the scalar identity would instead give \([B\otimes H^*]\). For a concrete failure take \(V=S^2\), \(B=1\), and the line obtained by gluing two trivial lines over the hemispheres with equatorial transition \(e^{i\theta}\). The dual transition has winding minus one. Changing a frame over either disk changes that winding by zero: radial contraction of the disk gives a homotopy of its boundary frame function to a constant nonzero function. Winding is homotopy invariant, as proved in Exercise 46. Thus \(H^*\) is not trivial. If \(H^*\oplus\mathbb C^n\) were isomorphic to \(\mathbb C^{n+1}\), taking the top exterior power would trivialize \(H^*\), a contradiction. More generally an equality \([H^*]=[1]\) in the bundle Grothendieck group would give \(H^*\oplus E\cong1\oplus E\) for some bundle \(E\). Taking determinants and tensoring by \((\det E)^*\) gives the same contradiction. Hence \([H^*]\ne[1]\). The actual line and its dual must both be retained.

**Marking (12 points).** Identify the rank-zero analytical cycle and the line-twisted Thom class. (4 points). Compare the full coefficient products and show the error made by replacing the cycle with a scalar. (4 points). Use the nonzero equatorial winding and determinant argument to detect the error in stable K-theory. (4 points).



**Exercise 49 (10 points).** *intermediate: a compact module map with a noncompact regular operator.* On \(V=\mathbb T^2\), take the foliation generated by \(\partial_x+\alpha\partial_y\), with \(\alpha\) irrational. Let \(k\) be a nonzero nonnegative smooth function supported in \([-1,1]\), regarded as the arrow kernel independent of the unit. Show that left multiplication by this element of \(A\) is compact on the standard right module \(A\), while its operator on each holonomy-cover \(L^2(\mathbb R)\) is not compact. Give an explicit sequence witnessing the second assertion.

**Solution.** The action is free: if \((t,\alpha t)\in\mathbb Z^2\), then irrationality forces \(t=0\). Each leaf and its holonomy cover are therefore parametrized by \(\mathbb R\); the graph is \(V\times\mathbb R\). The kernel \(k\) is smooth with compact arrow support because \(V\) is compact. It belongs to the dense convolution algebra and hence to \(A\). On the standard module, \(\theta_{a,b}\) is left multiplication by \(ab^*\). An approximate identity shows these products span \(A\) densely, so \(\mathcal K_A(A)=A\), and left multiplication by \(k\) is compact.

On a cover it is convolution by \(k\). Choose a nonnegative smooth unit vector \(u\), supported in \([-1,1]\), which is not zero. Then \(v=k*u\) is nonzero, is supported in \([-2,2]\), and has norm \(c>0\). Put \(u_n(t)=u(t-6n)\). These unit vectors have disjoint supports, while their images are \(v(t-6n)\), also with disjoint supports. In particular,

\[
\|k*u_n-k*u_m\|_2=\sqrt2\,c
\qquad(n\ne m).
\tag{248}
\]

No subsequence of the images converges in norm, which disproves ordinary compactness. Tensoring the construction with a finite Clifford module preserves the distinction. Thus (243) must be justified in the module compact ideal, as its proof does; ordinary finite-rank reasoning on an infinite cover would be invalid.

**Marking (10 points).** Identify the actual irrational-flow graph and prove compactness of the coefficient in the standard algebra module. (4 points). Construct translated test vectors whose convolution images have disjoint supports and fixed positive norm. (4 points). Explain why this disproves Hilbert-space compactness without changing module compactness or the finite Clifford coefficients. (2 points).



**Exercise 50 (12 points).** *advanced.* On the flat two-torus, let \(\varepsilon_1,\varepsilon_2\) be the ordered positive generators of \(C_2\). Lift the vertical operator with Fourier multiplier
\(a(k)=k\varepsilon_1/\sqrt{1+k^2}\), and take the total multiplier
\(q(k,l)=(k\varepsilon_1+l\varepsilon_2)/\sqrt{1+k^2+l^2}\).
Show that the vertical lift is not compact, that its leading function does not extend continuously to the vertical zero-frequency axis, and that its anticommutator with the total operator is nevertheless nonnegative. Explain why the compactness step in Lemma 7.5 needs the vanishing-axis condition.

**Solution.** The exact anticommutator is
\[
 [a(k),q(k,l)]
 =\frac{2k^2}{\sqrt{1+k^2}\sqrt{1+k^2+l^2}}I\ge0.
\tag{255}
\]
The positive-Clifford mixed terms cancel. At a fixed nonzero integer \(k\), the normalized vectors \(e^{i(kx+lz)}\otimes v\), with \(l\) tending through distinct integers and \(v\) a unit coefficient vector, are orthogonal. The norms of their vertical images are the same positive number \(|k|/\sqrt{1+k^2}\). Their images remain orthogonal, so the vertical lift is not compact. Its leading function for \(\xi\ne0\) is \(\operatorname{sgn}(\xi)\varepsilon_1\). Approaching \((0,1)\) from positive and negative \(\xi\) gives different limits. The total anticommutator's leading function is instead \(2|\xi|/\sqrt{\xi^2+\eta^2}\), which is continuous and zero there.

For the last assertion, let \(K\) be the rank-one projection onto a constant first-coordinate Fourier vector, and take \(G=I\). Then \((K\widehat\otimes1)G=K\widehat\otimes1\) has infinitely many orthonormal eigenvectors of eigenvalue one, indexed by the second Fourier coordinate. It is not compact, although \(K\) is vertically compact. The symbol of \(G\) is identity on the axis and violates Lemma 7.5's hypothesis. With a vanishing-axis symbol, Lemma 7.4 instead makes its compositions with creation vectors compact; this is precisely the step used in that lemma. No claim that every lifted vertical compact is a total compact is valid.

**Marking (12 points).** Prove noncompactness of the lifted vertical multiplier using orthogonal Fourier vectors. (4 points). Compute its incompatible limits on the zero-frequency axis and the nonnegative total anticommutator. (4 points). Give the lifted rank-one counterexample and identify the precise vanishing-axis hypothesis used by the compactness argument. (4 points).



**Exercise 51 (8 points).** *intermediate: the density correction.* Give \(f:\mathbb T_x\times\mathbb T_y\to\mathbb T_y\) the vertical density \(a(y)\,dx\), where \(a>0\) is smooth, and horizontal lift \(\partial_y\). For a trivial coefficient line with metric connection \(\partial_y\), calculate (258). Check its sign on sections constant in \(x\). Then use density \(dx\) and horizontal lift \(\partial_y+\sin x\,\partial_x\); verify its corrected connection is skew-adjoint in the scalar total \(L^2(dx\,dy)\) inner product.

**Solution.** Formula (260) gives \(k_{\partial_y}=a'/a\), hence
\[
 \nabla^{E_f}_{\partial_y}=\partial_y+\frac{a'}{2a},
 \qquad
 \partial_y\langle u,v\rangle
 =2\pi\bigl(a'\overline uv+a\overline{u'}v+a\overline uv'\bigr)
\tag{267}
\]
for \(x\)-constant sections, with circle length \(2\pi\). The sum of the two corrected connection inner products is precisely the right side. Omitting the correction loses \(2\pi a'\overline uv\); using its negative doubles that error. In the second example \(v=1\), \(B=\sin x\), so \(k=\cos x\) and the corrected operator is \(L=\partial_y+\sin x\,\partial_x+\tfrac12\cos x\). Periodic integration by parts gives \((\sin x\,\partial_x)^*=-\sin x\,\partial_x-\cos x\) and \(\partial_y^*=-\partial_y\). Multiplication by \(\cos x/2\) is self-adjoint. Therefore \(L^*=-L\) on the smooth core. These computations explain the positive half-density sign in (258).

**Marking (8 points).** Compute the density correction and verify its positive sign on sections constant along the fibre. (4 points). Calculate the divergence correction for the nonconstant horizontal lift and prove skew-adjointness by periodic integration by parts. (4 points).



**Exercise 52 (12 points).** *advanced: disconnected fibres and the compact ideal.* Let \(V=\mathbb T\) with its full tangent foliation, and let \(W=V\sqcup V\) map to \(V\) by identity on each component. Use the canonical rank-zero vertical orientation. Determine \(H,R,\mathscr P_f\) and the inclusion (262). Explain why the compact correspondence \(\varepsilon_f\) does not give a Morita equivalence between \(C_r^*(H)\) and \(C_r^*(G)\). Verify the module and operator form of (264).

**Solution.** The connected circle has trivial transverse holonomy, so \(G=V\times V\). Write \(L=L^2(V)\). Its reduced algebra is \(\mathcal K(L)\): a smooth pair kernel acts by its integral operator; separated smooth kernels are finite rank and dense, and every regular representation is that same integral representation. The leaves of \(W\) are its two components. Thus \(H\) consists of two pair groupoids, whereas \(R\) has an arrow between any two points lying over the endpoints of a \(G\) arrow, including the two off-diagonal component choices. Accordingly
\[
 \begin{aligned}
 C_r^*(H)&=\mathcal K(L)\oplus\mathcal K(L),& A&=\mathcal K(L),\\
 \mathscr P_f&=A\oplus A,&\mathcal K(\mathscr P_f)&=M_2(A).
 \end{aligned}
\tag{268}
\]
The middle identification follows from \(E_f^0=C(V)^2\) and Lemma 7.8. The inclusion \(j_*\) is the block-diagonal inclusion into \(M_2(A)\). It has compact left action but does not cover the off-diagonal compact operators. A Morita equivalence would identify the two compact algebras (and their K-groups); here \(K_0(C_r^*(H))=\mathbb Z^2\) whereas \(K_0(A)=\mathbb Z\), by compact stability and scalar K-theory. The correspondence sends the two rank-one generators to the same generator, so cannot be an equivalence. The first module in (264) is \(C(V)^2\), with zero vertical operator. Its product with the base cycle is the direct sum of two base regular spinor modules. On the right, the two \(H\) modules extended by the block-diagonal inclusion give those same two summands, with the same direct sum operator. This verifies the entire product, while displaying why replacing \(H\) by \(R\) would change the asserted left algebra.

**Marking (12 points).** Identify both actual groupoids, their algebras and the full two-summand correspondence. (4 points). Compute the block-diagonal compact inclusion and explain why its K-theory map prevents Morita equivalence. (4 points). Verify both modules and their direct sum operators in the stated product identity. (4 points).



**Exercise 53 (12 points).** *advanced: two rank-zero phase factors.* Let \(f:W\to V\) be a finite smooth covering of compact manifolds and give \(V\) the point foliation. Specify a vertical rank-zero orientation line \(L_K\to W\) and a base rank-zero orientation line \(L_F\to V\). Determine both sides of (264) as actual modules with their left \(C(W)\) actions. Does the fibre cardinality determine the answer?

**Solution.** Here \(F_W=0\), \(G\) and \(H\) are the unit groupoids, and \(R=W\times_VW\). The first module is the finite vector bundle \(f_*L_K\), with fibre \(\bigoplus_{w\in f^{-1}(y)}(L_K)_w\). Its operator is zero and a function on \(W\) acts diagonally by its value at each \(w\). Tensoring with the base line gives \(f_*(L_K\otimes f^*L_F)\). On the right the full spin-c orientation on the zero bundle \(F_W\) is that same tensor line. Its \(C(W)\) line module tensored with \(\mathscr P_f\) gives the same fibre direct sums and diagonal left action. Both sides have zero operator and compact identity because they are finite bundle modules over \(C(V)\). Covering charts supply the bundle transitions, including the permutations of sheets and the two line phases. The number of sheets gives only the fibre rank; it determines neither these transitions nor the represented \(C(W)\) action. This is the rank-zero part of the ordered orientation in (256).

**Marking (12 points).** Identify the direct image line bundle with its diagonal source-algebra action. (4 points). Verify that the two line phases and the full ordered zero-dimensional orientation give the same product module. (4 points). Explain compactness of the zero cycle and why sheet cardinality does not determine transitions or the represented action. (4 points).



**Exercise 54 (12 points).** *intermediate: infinitely many rank-zero sheets.* Let \(W=\mathbb Z\times\mathbb T\), \(V=\mathbb T\), and \(f(n,y)=y\). Give both manifolds the point foliation and canonical rank-zero orientation lines. Identify the vertical module, its compact left action and the correspondence in (272). Explain why the zero vertical operator is a Kasparov cycle although the identity of the vertical module is not compact. Verify (273).

**Solution.** Compact subsets of \(W\) meet finitely many components. The vertical integration module is the standard module
\(E_f^0=\ell^2(\mathbb Z)\widehat\otimes C(\mathbb T)\),
whose inner product is \(\sum_n\overline{u_n}v_n\). The left algebra is \(C_0(W)=c_0(\mathbb Z,C(\mathbb T))\), acting diagonally. For \(g\) in that algebra, the finite-sheet diagonal truncations converge in norm, since their error norm is \(\sup_{|n|>N}\|g_n\|\to0\). Each finite-sheet diagonal is a finite sum of module rank-one maps, with the standard coordinate vectors and coefficient functions. Thus multiplication by \(g\) is compact. The identity is not compact: after evaluation at any \(y\), it becomes the identity on the infinite-dimensional Hilbert space \(\ell^2(\mathbb Z)\), while every module compact becomes an ordinary compact operator. The square defect of the zero phase is nevertheless \(-g\), which is compact for every allowed left coefficient.

Here \(G\) and \(H\) are unit groupoids, whereas \(R=W\times_VW\) has every pair of sheet indices over one \(y\). Thus \(\mathscr P_f=E_f^0\), \(C_r^*(R)=\mathcal K(\ell^2\mathbb Z)\otimes C(\mathbb T)\), and \(C_r^*(H)=C_0(W)\) includes as the diagonal compact action just proved. The base analytical class is the identity zero cycle on \(C(\mathbb T)\); the source analytical class is the identity zero cycle on \(C_0(W)\). Both sides of (273) are therefore the entire module \(E_f^0\) with its diagonal left action and zero operator. An assertion that \(H=R\), or that the identity of \(E_f^0\) must be compact, would give the wrong conclusion.

**Marking (12 points).** Identify the standard infinite-sheet module and approximate every permitted diagonal coefficient by finite-rank module maps. (4 points). Prove that the module identity is noncompact while all localized zero-phase defects are compact. (4 points). Distinguish the unit and pair-sheet groupoids and verify the entire correspondence identity. (4 points).



**Exercise 55 (12 points).** *advanced: a smooth metric family need not be uniform at infinity.* On \(\mathbb R^2\), with its full tangent foliation and trivial spin-c structure, set
\(g_s=dx_1^2+e^{2s x_1^2}dx_2^2\), \(0\leq s\leq1\).
Transport the Clifford module by the positive metric isometry. Write its normalized cotangent Clifford symbol \(q_s\) in the fixed positive generators \(\varepsilon_1,\varepsilon_2\). Show that the pointwise smooth family does not converge to \(q_0\) in the supremum symbol norm as \(s\downarrow0\). Explain which homotopy assertion of Lemma 7.11 still applies.

**Solution.** A coordinate covector \(\xi_1dx_1+\xi_2dx_2\) has dual norm squared \(\xi_1^2+e^{-2s x_1^2}\xi_2^2\). Positive transport gives
\[
 q_s(x,\xi)=\frac{\xi_1\varepsilon_1+e^{-s x_1^2}\xi_2\varepsilon_2}
 {\sqrt{\xi_1^2+e^{-2s x_1^2}\xi_2^2}}.
\]
Fix any \(s>0\), let \(|x_1|\to\infty\), and choose \(\xi_1=1\), \(\xi_2=e^{s x_1^2/2}\). Then \(q_0\to\varepsilon_2\) and \(q_s\to\varepsilon_1\). The two positive generators are self-adjoint, square to one and anticommute, so
\((\varepsilon_1-\varepsilon_2)^2=2I\) and their difference has norm \(\sqrt2\). Consequently the supremum norm of \(q_s-q_0\) is at least \(\sqrt2\) for every positive \(s\). Pointwise smoothness has not supplied a global uniform symbol-norm path.

Lemma 7.11 instead constructs one adjointable bounded phase on the interval module, using compact parameter supports and a countable locally finite buffered atlas. Each compact left coefficient involves only finitely many charts, with uniform interval estimates and compact interval defects. Those are the assertions needed for a Kasparov homotopy. The equality of the endpoint classes follows; global operator-norm continuity of an arbitrarily chosen pointwise phase family was not its premise.

**Marking (12 points).** Compute the transported normalized cotangent Clifford symbol. (4 points). Choose the escaping covectors and prove the fixed positive lower bound for the supremum symbol distance. (4 points). Explain the interval-module homotopy with compact left coefficients without asserting global norm continuity of the chosen phase family. (4 points).



**Exercise 56 (12 points).** *advanced: a domain projection does not reduce the phase.* Take \(Y=\{\mathrm{pt}\}\), \(n=1\), \(U=(-1,1)\), and the full right \(C_1\)-module of Lemma 7.14. Show that \(PF(1-P)\ne0\). Explain why this does not obstruct (274), and verify the sign in both cutoff identities (278).

**Solution.** The one-dimensional off-diagonal kernel (279) is \(i\varepsilon_1 k(w)\), where

\[
 k(w)=\frac{w}{2\sqrt\pi}\int_0^\infty
 t^{-3/2}e^{-t}(4\pi t)^{-1/2}e^{-w^2/(4t)}\,dt.
\tag{283}
\]

For \(w\ne0\) the integral is finite and strictly positive. Hence \(k(w)\) has the sign of \(w\). Choose a nonzero nonnegative smooth function \(v\) supported in \((2,3)\), with Clifford value (1). For \(-1<x<1\), the integral \(\int k(x-z)v(z)\,dz\) is strictly negative. Multiplication by the invertible \(i\varepsilon_1\) does not make it zero. Thus the inside component of \(Fv\) is nonzero although \(Pv=0\). The domain is not a reducing summand for this bounded phase.

If \(a\in C_c(U)\) and \(\chi=1\) on its support, then \(a\chi=a\), \(\chi(1-P)=0\), and \((1-P)\chi=0\). Expanding \(a[\chi,F](1-P)\) gives \(aF(1-P)\). Expanding \((1-P)[F,\chi]a\) gives \((1-P)Fa\). The commutators have the two opposite orders shown in (278). Each is compact, because \(\chi\) is a compact interior coefficient of the ambient analytical cycle. These localized compact terms permit the block homotopy and subsequent interior-symbol comparison. The proof requires that compactness, not a zero unlocalized inside–outside block.

**Marking (12 points).** Use the signed off-diagonal kernel and an exterior test function to prove the inside-outside phase block is nonzero. (4 points). Expand both cutoff commutators and check their opposite orders and signs. (4 points). Explain why their localized compactness permits the block homotopy without requiring a reducing domain projection. (4 points).



**Exercise 57 (12 points).** *intermediate: an exact whole tube and its Jacobian.* Let \(f:S^1\to\{\mathrm{pt}\}\), \(u(\theta)=(\cos\theta,\sin\theta)\), and let \(v\) denote the signed outward radial normal coordinate. Set \(\rho=1/4\). Compute the whole tube, its image, its vertical Jacobian with respect to \(d\theta\,dv\), and the image of \((\theta,v)=(0,1/(4\sqrt3))\). Explain why a flat phase on the image can be transported to the cylinder without a complete pulled-back metric assumption.

**Solution.** The bundle \(N\) is the radial real line bundle. Formula (281) becomes

\[
 w(v)=\frac{v}{\sqrt{1+16v^2}},\qquad
 \Phi(\theta,v)=(1+w(v))(\cos\theta,\sin\theta),
 \qquad U=\{z:3/4<|z|<5/4\}.
\tag{284}
\]

The strictly increasing function \(w\) has derivative \((1+16v^2)^{-3/2}\) and range \((-1/4,1/4)\). Thus \((\theta,v)\mapsto(\arg\Phi,|\Phi|)\) is a diffeomorphism onto that annulus. Its polar area density is

\[
 J(v)=(1+w(v))(1+16v^2)^{-3/2}>0.
\tag{285}
\]

For the separately chosen density \(d\theta\,dv\), the change-of-variables unitary sends \(h\) on the annulus to \(J(v)^{1/2}h(\Phi(\theta,v))\). This preserves the integral inner product and is onto by the inverse substitution. At the specified \(v\), \(w=1/8\), so the image is \((9/8,0)\). The pulled-back Euclidean metric is \(w'(v)^2dv^2+(1+w(v))^2d\theta^2\); a radial path to \(v=+\infty\) has finite length \(1/4-w(v_0)\), so this metric is incomplete. Lemma 7.11 constructs a bounded phase from interior charts with that principal symbol and locally compact defects. Lemma 7.14 then identifies it with the restriction of the flat ambient class. Neither completeness nor a self-adjoint boundary realization is needed for this bounded construction.

**Marking (12 points).** Compute the whole radial tube, its inverse range and annular image. (3 points). Derive the positive Jacobian and onto change-of-variables unitary. (3 points). Calculate the specified image point and finite radial length proving metric incompleteness. (2 points). Explain the bounded interior phase comparison without imposing metric completeness or a boundary realization. (4 points).



**Exercise 58 (10 points; the rank-two block sign).** In Lemma 7.16 take \(r=2\). Show that the normalized block volume is negative on \(\psi\), although \(\psi\) is even in the exterior/external module. Determine the grading and both physical generators after the contraction by \(\Sigma_2^*\). Give an even right-module unitary from that shifted module to the regular \(C_2\) module, and calculate the transported physical action. Explain why omission of the shift would change the full induced orientation.

**Solution.** Here \(k_2=1\). Moving \(B_1\) past \(A_2\) gives the single minus sign, so

\[
 \begin{aligned}
 (-i)^2A_1A_2B_1B_2\psi&=-\psi,\\
 \Sigma_2^*\widehat\otimes(T\widehat\otimes C_2)&\cong\Pi C_2,
 \qquad(C_1,C_2)^{\mathrm{contracted}}=(L_{t_1},L_{t_2}),\\
 U:\Pi C_2\to C_2,\quad U(c)&=t_1c,\qquad
 (UC_1U^{-1},UC_2U^{-1})=(L_{t_1},-L_{t_2}).
 \end{aligned}
\tag{296}
\]

The first line uses that every \(-iA_iB_i\) fixes \(\psi\). The corner vector is even, but its corresponding vector of \(\Sigma_2\) is odd; the remaining module must therefore be shifted. The uncontracted corner generators are negative. The Koszul sign of the odd corner in the Morita tensor reverses both signs, giving the second line. The map \(U\) preserves the right action and inner product because \(t_1^*=t_1=t_1^{-1}\). It is even from the shifted module: for a homogeneous \(c\), both its shifted source parity and the parity of \(t_1c\) are \(|c|+1\). Conjugation by \(t_1\) fixes \(t_1\) and negates \(t_2\), giving the last line. The induced real generator frame is thus reflected in its second coordinate. Keeping the regular grading together with two positive generators would instead retain the original orientation; it would lose the actual reflected orientation of this inverse transfer.

*Grading:* 2 points for the volume permutation and eigenvalue; 3 for the module shift and contracted Koszul signs; 3 for the even unitary and both conjugations; 2 for the full orientation conclusion.

**Exercise 59 (10 points; the full conjugate line).** An oriented rank-two bundle has a prescribed spin-c transition \(zs\) on an overlap, with \(s=\exp(-\theta t_1t_2/2)\). Tensor its prescribed module by an even line whose transition is \(h\in U(1)\). Compute the exact transition of \(\mathcal D_E\), including its grading. Suppose that line is nontrivial but its square is trivial. Explain what a determinant-only comparison would miss. Also give the transferred inverse in rank zero for a full orientation line of parity \(\epsilon\).

**Solution.** Formula (295) removes the \(a\) spin lift and retains the physical \(b\) spin lift. The finite contraction sends the even physical word \(b_1b_2\) to \(t_1t_2\), since its two contracted generator signs multiply to one. The inverse line phase is conjugated. Thus

\[
 \begin{aligned}
 \mathcal D_{E}^{\,H}&\cong\mathcal D_E\otimes H^*,
 \qquad
 g_{\mathcal D_E^{\,H}}
   =h^{-1}z^{-1}\exp(-\theta t_1t_2/2)
   \quad\text{on }\Pi C_2,\\
 r=0:\quad \mathcal T_0(\alpha_E)&=[H^*,\mathrm{multiplication},0],
 \qquad |H^*|=\epsilon.
 \end{aligned}
\tag{297}
\]

For example, a line transition \(h=i\) contributes \(-i\) on the inverse, while a transition \(h=-1\) remains \(-1\). A spin-c determinant transition squares the line phase. A nontrivial order-two line therefore leaves that determinant unchanged but remains in the full module as the nontrivial dual line. Formula (293) and its evaluations retain it; no determinant argument can remove it. In rank zero the conjugate line has the same parity as the original line. Its zero operator is a cycle because multiplication by every \(C_0(X)\) function is compact on this finite line section module. An odd line remains odd; the canonical even rank-zero unit cannot replace it.

*Grading:* 4 points for the spin word, full phase and shifted module; 3 for the two-torsion/determinant distinction; 3 for the rank-zero line, parity and compact defect.

**Exercise 60 (10 points; the horizontal-double rotation).** In one orthonormal horizontal pair take \((t,b_H)=(1,0)\). Compute its image under (301) at \(\theta=0,\pi/4,\pi/2\). Verify the spin unitary and explain why its formula glues when the horizontal bundle is nonorientable. At the endpoint, identify the two horizontal coordinates in \(L\) and the horizontal contribution to the ambient Euclidean vector.

**Solution.** The exact values and implementing unitary are

\[
 \begin{aligned}
 R_0(1,0)&=(1,0),&
 R_{\pi/4}(1,0)&=(2^{-1/2},-2^{-1/2}),&
 R_{\pi/2}(1,0)&=(0,-1),\\
 s_\theta&=\exp\bigl((\theta/2)\gamma_t\gamma_b\bigr),&
 s_\theta\gamma_t s_\theta^*
   &=\cos\theta\,\gamma_t-\sin\theta\,\gamma_b .
 \end{aligned}
\tag{311}
\]

The bivector is skew-adjoint because its two positive self-adjoint generators anticommute. Its commutators with \(\gamma_t,\gamma_b\) are \(-2\gamma_b,2\gamma_t\); the elementary two-dimensional exponential therefore implements both columns of (301), not a rotation with the opposite sign. For several pairs, the bivector sum is invariant under simultaneous orthogonal changes of both copies: the coefficient of the sum is the metric contraction. This includes a frame change of determinant \(-1\); no orientation of \(H\) is needed.

For general input the endpoint sends \((t,b_H)\) to \((b_H,-t)\). Formula (302) consequently gives the horizontal double \((a_H,b_H)\) in \(L\), while the Euclidean coordinate contains \(-du(t)\). With \(\eta\in\nu_u\), this is the normal vector \(\chi(\eta,t)=\eta-du(t)\). The two full \(H\)-double evaluations now cancel; their source and target exterior gradings are interleaved. Moving the two even-dimensional right blocks has the stated even exterior implementer, but flips of individual odd vectors still carry their Koszul signs.

*Grading:* 3 points for all exact coordinates and the rotation sign; 3 for both spin commutators and the unitary; 2 for orthogonal frame invariance; 2 for the full endpoint double and normal quotient.

**Exercise 61 (10 points; odd block volumes and even maps).** In Lemma 7.19 compare the cases \((r,q,n)=(3,1,4)\) and \((1,3,4)\). Compute the normal shift and generator factor, the product module and both physical blocks, and the unitary from the ambient inverse. Explain why the second map is even although its Clifford multiplier is odd. Track an arbitrary line transition \(z\).

**Solution.** Since \(k_1=0,k_3=1,k_4=0\), equations (305)–(308) give

\[
 \begin{array}{c|c|c|c|c|c}
 (r,q)&(a,\kappa)&h&\text{product module}&(c_K,c_N)&U\\ \hline
 (3,1)&(0,-1)&0&C_4&(L_{e_K},L_{e_N})&
              L_{\omega_K\omega_N}\\
 (1,3)&(0,1)&1&\Pi C_4&(-L_{e_K},L_{e_N})&
              L_{\omega_K}
 \end{array}
\tag{312}
\]

The ambient inverse is \(C_4\) with all physical generators negative. In the first row the multiplier has degree \(3+1=0\bmod2\); it is the unitary four-generator volume, up to its specified scalar, and conjugates every real generator to its negative. It therefore maps the ambient negative physical list to the positive product list. In the second row \(\omega_K\) has degree one. Conjugation fixes the single \(K\) generator and negates every normal generator, so it maps the ambient list to \((-e_K,+e_N)\). Its source has the regular grading and its target has the reversed grading; adding one to the underlying Clifford-word degree therefore preserves the full module parity. The map is even and preserves the right action and inner product.

The right target order is \(K,N\) after the specified graded transport. In both examples the two ranks are odd; the coefficient flip must include its odd-word minus sign. The normal structure retains the line transition \(z\); its inverse contains \(z^{-1}\), while the vertical module contains \(z\). Their exact Hermitian evaluation is \(z^{-1}z=1\). Spin transitions give the ambient lift as in (309). No determinant calculation or grading omission can replace these two evaluations.

*Grading:* 3 points for both exact rows; 3 for the generator conjugations and complete graded parity of the maps; 2 for the odd target-block transport; 2 for the full phase and spin evaluation.

**Exercise 62 (10 points; an even transport in six Clifford generators).** In the odd rank-three reduction of Lemma 7.20, label the old generators \(e_1,e_2,e_3\), and the new generators \(e_4,e_5,e_6\). Give a unitary even word implementing the move of the old odd generator past the two new even-block generators. Check all moved generators and the total chirality. Explain how the resulting full reducer identifies the reduced transferred inverse with \(\mathcal T_1(y')\).

**Solution.** Use the two adjacent unit reflections, in their required order:
\[
 \begin{aligned}
 w&=\frac{(e_4-e_5)(e_3-e_4)}2,\qquad ww^*=w^*w=1,\\
 we_3w^*&=e_5,\qquad we_4w^*=e_3,\qquad we_5w^*=e_4,\\
 we_jw^*&=e_j\quad(j=1,2,6),\qquad
 w\chi_6w^*=\chi_6 .
 \end{aligned}
\tag{322}
\]
Each numerator vector has square two, so division by \(\sqrt2\) gives an odd self-adjoint unitary. Its twisted adjoint exchanges its two generators and fixes the others. The composition has even degree, so its ordinary adjoint is the composition of the two reflections. This proves every displayed generator identity. The real permutation is a three-cycle, of positive sign; multiplying the entire ordered generator list proves the last identity for \(\chi_6=(-i)^3e_1\cdots e_6\).

After transport, the old even block occupies positions one and two, the new even block positions three and four, and the pair old odd/new odd positions five and six. The full canonical reducer is the even wedge tensor \(\Sigma_1^{\mathrm{old}}\widehat\otimes\Sigma_1^{\mathrm{new}}\widehat\otimes\Sigma_1^{\mathrm{odd,odd}}\). In the balanced tensor, evaluate the new even dual/pair exactly. What remains is the old even source inverse module and the ordered odd coevaluation, namely \(\mathcal T_1((\mu_3^B)^{-1}y)\). The maps preserve both actions, grading and inner products; the two full Morita evaluations prove onto range. No swap of the final odd pair has occurred.

*Grading:* 3 points for the correctly ordered normalized unitary; 3 for all generator and chirality checks; 2 for the complete reducer decomposition; 2 for the exact balanced evaluation and retained odd order.

**Exercise 63 (10 points; the geometric ambient inverse).** Compute (319) in dimensions \(n=2,4,6\). State its grading, physical generator factor, finite unitary and reduced sign relative to the canonical positive Dirac cycle. Explain how that sign enters and then cancels in the ordinary submersion proof, retaining a supplied line twist.

**Solution.** The complete values are
\[
 \begin{array}{c|c|c|c|c|c}
 n&k_n&\mathcal D_n&c_{\mathcal D_n}&W_n&
           A_n/a_n^{\mathrm{canonical}}\\ \hline
 2&1&\Pi C_2&L_e&1&-1\\
 4&0&C_4&-L_e&L_{\chi_4}&1\\
 6&1&\Pi C_6&L_e&1&-1
 \end{array}
\tag{323}
\]
The ratio in the last column denotes the additive sign of the reduced KK class, not division of operators. In dimensions two and six, the physical generators agree with the canonical positive list but the entire module grading is reversed. Section 0's explicit opposite cycle and conjugation by its grading operator give its negative class. In dimension four, \(\chi_4\) is even and conjugates every generator to its negative, giving the displayed unitary without changing the grading. Each map is right-linear and preserves its Clifford-valued inner product.

The actual product \(d_N\cdot a_f\) has ambient full module \(\mathcal D_n\), not an unshifted canonical one. Equation (320) therefore identifies it with \([e_\Phi]A_n\), including the last-column sign. Multiplication by \(b_N\), using both calibrated Thom inverse identities and native associativity, gives \(a_f=b_N[e_\Phi]A_n=f_!^{\mathrm{geom}}\). Dropping the ambient shift in both intermediate identifications is not justified by equality of physical generator matrices. A supplied normal phase \(z\) has inverse phase \(z^{-1}\), and the vertical phase is \(z\); their full line evaluation cancels in the ambient product. The remaining normal Thom factor on the right retains the supplied line, as does \(a_f\). This includes two-torsion lines and the separate complete graded-line treatment when the vertical rank is zero.

*Grading:* 4 points for all complete rows; 2 for the exact chirality/grade reasoning; 2 for the typed native cancellation; 2 for the full retained line twist.

**Exercise 64 (12 points; the full signed kernel and endpoint frame).** Take \(p=t=1\), \(n=4\), and embedding normal rank two. In the source tangent order \((h_F,h_H,\eta_1,\eta_2,t)\), write the transverse derivative and the full kernel map \(L_s\), using kernel coordinates \((k,\eta_1,\eta_2,t)\). Check all ranks and the Euclidean endpoint \(J\). Explain why \(E_s=L_sJ^{-1}\) has \(e_1E_1=1\) even when the coordinate determinant of \(J\) is negative. Finally state the complete normal/tangent coefficient order and the even full ambient intertwiner in this case.

**Solution.** The exact matrices are
\[
 \begin{gathered}
 d_{\rm trans}h_s=(0,\ 1,\ 0,\ 0,\ s),\qquad
 L_s=
 \begin{pmatrix}
 1&0&0&0\\0&0&0&-s\\0&1&0&0\\
 0&0&1&0\\0&0&0&1
 \end{pmatrix},\\
 J=
 \begin{pmatrix}
 1&0&0&0\\0&0&0&-1\\0&1&0&0\\0&0&1&0
 \end{pmatrix},\qquad
 d_{\rm trans}h_sL_s=0,\quad
 \operatorname{rank}L_s=4,\quad
 e_1L_1=J,\quad e_1L_1J^{-1}=1 .
 \end{gathered}
 \tag{355}
\]
The transverse row has rank one for every \(s\); the four columns of \(L_s\) are independent, so span its entire kernel. At \(s=0,1/2,1\) the horizontal/normal pair is exactly \((0,1),(-1/2,1),(-1,1)\). The endpoint Euclidean derivative keeps \((h_F,h_H,\eta_1,\eta_2)\), proving the displayed \(J\) with \(\det J=-1\) in these auxiliary coordinate lists. The reference uses that same \(J^{-1}\). Therefore their composition is the identity; no orientation conclusion is drawn from the sign of an unrelated coordinate list.

Here \(q=3\). The physical product is \(\mathcal D_N\widehat\otimes\mathscr M_F\), with initial right target \(C_3,C_1\), and the full graded \(\beta\) gives native \(C_1,C_3=C_4\). Lemma7.19 gives \(\mathcal D_N=\Pi C_3\) with positive physical normal generators. The tangent physical generator on the shifted tensor product is negative, so the resulting full module is \(\Pi C_4\), with negative physical \(F\) and positive physical \(N\) actions. The ambient inverse is \(\mathcal D_4=C_4\), with all physical generators negative. Left multiplication by the tangent generator \(e_F\) is an even unitary from \(C_4\) to \(\Pi C_4\); it commutes with \(e_F\) and anticommutes with each normal generator, proving every intertwining equation. It is right-linear and preserves the full inner product. The word is odd, but the map between these two gradings is even. The inverse/tangent scalar phase is the full evaluation \(z^{-1}z=1\).

*Grading:* 4 points for both full kernel matrices and ranks; 2 for the exact endpoint and reference cancellation; 2 for the full graded target order; 3 for the whole shifted-module unitary and all generator actions; 1 for the full phase evaluation.

**Exercise 65 (10 points; rectangular compactness without a unit).** Let \(A\) be nonunital and \(\mathscr H=A\widehat\otimes L^2(\mathbb R^n)\widehat\otimes\mathcal D_n\). For an integration-module vector \(\zeta\) and Euclidean vector \(w\), prove that \(u\mapsto\zeta\int w(t)^*u(t)\,dt\) is a rectangular module compact. Why would compactness of \(1_{M(A)}\otimes\theta_{v,w}\) on \(\mathscr H\) alone be an insufficient argument? For a compact étale creation vector with Euclidean output in the unit ball, derive a uniform bound for its Fourier input tail outside radius \(R>2\), and state exactly how the connection follows.

**Solution.** Choose a bounded approximate identity \(a_m\) of \(A\). The actual rank-one maps and their error bound are
\[
 \begin{aligned}
 \theta_{\zeta,a_m^*\otimes w}(u)
       &=\zeta a_m\int w(t)^*u(t)\,dt,\\
 \|T-\theta_{\zeta,a_m^*\otimes w}\|
       &\leq\|\zeta-\zeta a_m\|\,\|w\|_{L^2}\longrightarrow0,\\
 \|T_\xi F_n(1-\kappa_R)\|
       &\leq C_n\|\xi\|\,|B_1|^{1/2}
              \|1_{\{|z|\ge R-1\}}K_n(z)\|_{L^2}
              \longrightarrow0 .
 \end{aligned}
 \tag{356}
\]
Finite homogeneous Clifford columns give the full coefficient version. The first limit follows from the nondegenerate right module action. By contrast, \(\mathcal K(\mathscr H)=A\widehat\otimes\mathcal K(L^2\widehat\otimes\mathcal D_n)\); the multiplier identity need not belong to \(A\). For example, for \(A=C_0(\mathbb R)\), the constant nonzero fibre-rank-one family does not vanish at base infinity and is not a module compact.

For the tail, creation on each faithful regular representation is the entire decomposable counting-fibre operator \(V_{\xi,x}(t)\) of norm at most \(\|\xi\|\), zero outside \(B_1\). If \(|t|\le1\) and the input lies outside radius \(R\), the separation is at least \(R-1\). Cauchy–Schwarz in that input variable gives the last line. The heat-kernel estimate (280) proves the indicated \(L^2\) tail tends to zero. This is uniform in every regular \(x\), so it is a module-operator bound.

The confined creation difference has an order-minus-one mixed kernel and is compact by finite-frequency truncation and the first two lines. Its omitted part is exactly the Fourier tail, because \(Q_hT_\xi(1-\kappa_R)=0\). Thus the full connection is a norm limit of rectangular module compacts; its adjoint gives the other error. No ordinary compactness on a whole holonomy cover or boundary condition on an incomplete Euclidean sheet is needed.

*Grading:* 3 points for the actual rank-one approximation and error; 2 for the nonunital distinction and example; 3 for the uniform tail bound and its vanishing; 2 for the complete compact-connection conclusion.

**Exercise 66 (12 points; advanced: the weighted compact and its Fourier tail).** In the module model \(A=C_0(\mathbb R_x)\), \(E=A\), let \(H_1\) multiply by \(1+x^2\). This is a regular positive operator; this example tests the spectral compactness mechanism, without asserting that \(H_1\) is a foliation Laplacian. For a nonzero real smooth function a supported in \([-R,R]\), put \(\kappa_0=\theta_{a,a}\), and let W be the circle-diagonal weight (359). Show that \(\kappa_0\otimes1\) is not compact on \(E\otimes\ell^2(\mathbb Z)\), whereas \((\kappa_0\otimes1)W\) is compact. Give a uniform Fourier-tail bound. Verify both inequalities (363), including the scalar inequality behind the second, and give actual rectangular rank-one approximants for a creation map A to E when A is nonunital.

*Solution.* Here \(\kappa_0\) is multiplication by \(|a|^2\). At a point \(x_0\) with \(a(x_0)\) nonzero, evaluation sends module compacts to ordinary compact operators on \(\ell^2(\mathbb Z)\), because it sends each module rank one to an ordinary rank one and preserves norm limits. The evaluated bare lift is the nonzero scalar \(|a(x_0)|^2\) times the identity. Its images of the orthonormal Fourier basis have constant nonzero norm and no convergent subsequence; hence it is not compact. The bare lift is therefore not a module compact.

The kth weighted entry is the \(C_0(\mathbb R)\) function

\[
 f_k(x)=|a(x)|^2\frac{\sqrt{1+x^2}}{\sqrt{2+x^2+k^2}},\qquad
 \sup_{|k|>K}\|f_k\|\le
 \frac{\|a\|\,\|\sqrt{H_1}a\|}{\sqrt{1+(K+1)^2}}.
 \tag{380}
\]

Every finite Fourier truncation is a diagonal finite matrix over A, and is therefore a module compact. The displayed supremum makes those truncations converge in the actual module norm, proving compactness of the weighted lift. The constant is at most \(\sqrt{1+R^2}\|a\|^2\) and is independent of x and the discarded Fourier labels. No ordinary compactness of multiplication on \(L^2(\mathbb R)\) is used.

For any \(b=1+k^2\), \(W_k^2=H_1/(H_1+b)\le H_1/b\). Also \(1-(1+y)^{-1/2}=y/[\sqrt{1+y}(\sqrt{1+y}+1)]\) is at most \(\min(y/2,1)\), which is at most \(\sqrt y\) for \(y\ge0\). Squaring gives the second spectral inequality (363). Positive functional calculus and the module inner product give both vector bounds with \(\|H_1^{1/2}\xi\|\), first on the form domain and then exactly as used in the proof. Finally, for a positive contractive approximate identity \(a_m\) of A, the rank-one \(\theta_{\nu,a_m}\) maps c to \(\nu a_m c\). The module action is essential, so \(\|\nu-\nu a_m\|\) tends to 0. These rank ones converge in operator norm to the creation \(c\mapsto\nu c\). There is only one factor \(a_m\); putting an additional \(a_m\) into the output vector would give a different approximant and must be justified separately.

*Grading:* 3 points for the noncompact bare lift; 3 for the actual uniform weighted truncation; 3 for both spectral inequalities; 3 for the nonunital rank-one approximation.

**Exercise 67 (12 points; advanced: both odd signs and the positive exponential).** Use \(p=1\), physical ordered generators \(\sigma_x\),\(\sigma_y\) and chirality \(\sigma_z=-i\sigma_x\sigma_y\). Show that native block exchange reverses the reduced grading. Track both exchanges in (373)-(374). For the canonical circle foliation take \(a=[e^{it}]\), so \(s=\kappa\). Compute \(\operatorname{Ind}(\kappa)\) and the product index of \(\kappa\boxtimes\kappa\). Finally state the suspended-boundary identity for a quotient projection and explain why its minus sign does not change the two-degree index formula (379).

*Solution.* Write \(W=\sigma_x+\sigma_y\). Direct Clifford multiplication gives \(W^*=W\), \(W^2=2\), \(W\sigma_xW=2\sigma_y\), \(W\sigma_yW=2\sigma_x\) and \(W\sigma_z=-\sigma_zW\). Thus \(U=W/\sqrt2\) is an odd unitary. It is an even map from the grading-reversed physical spinor to the native spinor and intertwines both physical generators. For a full transition, its conjugation transports the spin part and retains the scalar phase. The reduction therefore gives minus the physical class for each of the Thom and Dirac ordered sums.

The first graded interchange moves the odd circle coefficient past the odd F Thom class and contributes minus. The native Thom block exchange contributes the second minus; their product is plus. For the analytical action the native Dirac block exchange and the odd coefficient/Dirac interchange again contribute two minuses. The remaining circle coefficient/Dirac pairing is the positive Hardy shift of index -1 by (371). Consequently

\[
 \operatorname{Ind}(\kappa)=-1,\qquad
 \operatorname{Ind}(\kappa\boxtimes\kappa)=1,\qquad
 \delta_{SE}\sigma_0[e]
   =-\sigma_1[\exp(2\pi i\widetilde e)].
 \tag{381}
\]

The first equality is the actual zero-kernel, constant-cokernel circle class; the second is the full circle-product theorem applied to that class. The last equality is the doubled-lift homotopy to the inverse unitary, not an unspecified suspension sign. It is applied once to the disk/sphere boundary of the odd relative symbol and once to its analytical symbol boundary. Those two negatives cancel before the same \(\sigma_1\) inverse is applied. The remaining degree-p interchange in (378) also occurs on both the Thom and Dirac sides. Discarding either one of the paired signs would give a false convention-dependent odd formula.

*Grading:* 3 points for the full odd Clifford intertwiner; 3 for both pairs of interchange signs; 2 for the two full circle indices; 4 for the typed suspension-boundary sign and both cancellations.

**Exercise 68 (12 points).** Strong continuity is not compact continuity

Let S={0}∪{1/n:n≥1}⊂R and H=C(S,ℓ²(N)). Let P\_n be projection onto the nth unit vector. Set T(0)=0 and T(1/n)=P\_n.

1. Prove that T defines an adjointable operator on H. (4points)
2. Prove that every fibre T(s) is compact, but T is not in K(H). (4points)
3. With F=1−T, show why this fibrewise essentially unitary field is not
   a unital C(S),C(S) Kasparov cycle. (4points)

*Solution.* For ξ∈ℓ², ||P\_nξ||=|ξ\_n|→0. Thus T and T\*=T are strongly continuous at0, and continuity at an isolated1/n is immediate. They are uniformly bounded by1. If ξ∈C(S,ℓ²), ||T(1/n)ξ(1/n)||≤||ξ(1/n)−ξ(0)||+||P\_nξ(0)||→0. Hence multiplication by T sends continuous sections to continuous sections and is contractive. The pointwise inner-product identity <Tξ,η>=<ξ,Tη> proves adjointability with adjoint T. (4points)

Each T(s) has rank at most one. A module rank-one θ\_{ξ,η} gives a norm-continuous compact field, since ξ and η are norm-continuous vectors. Finite sums and uniform limits retain that norm continuity. Every member of K(H) is such a uniform limit, whereas ||T(1/n)−T(0)||=1 for all n. Thus T is not module compact. This uses the actual compact ideal, not just a lack of a chosen norm trivialization; the field H is already trivial. (4points)

The self-adjoint contraction F satisfies F(0)=1 and F(1/n)=1−P\_n, so every fibre is unitary modulo its ordinary compact ideal. On the whole module, F²−1=−T. The unital source acts by the identity, so its Kasparov square condition would require this defect to be in K(H). The preceding result contradicts that condition. A graded version uses H⊕H, F\_gr=[[0,F],[F,0]] and its two identical defect blocks; the failure is unchanged. (4points)

**Exercise 69 (12 points).** The exact rank-one Clifford average

On K⊕K with the even/odd grading, let R=[[0,−1],[1,0]], and F=[[0,T\*],[T,0]], where T is a bounded operator on a Hilbert module K. Assume that F is a contraction and that F²−1 and [F,R]\_gr=FR+RF are compact.

1. Verify the negative Clifford involution and compute(F+RFR)/2.
   (4points)
2. Prove exact graded commutation and the compact perturbation.
   (4points)
3. Verify the self-adjoint Fredholm homotopy and explain its covariance
   when the normal frame is reversed. (4points)

*Solution.* R\* =−R and R²=−1. Multiplication gives RFR=[[0,T],[T\*,0]], so the average is [[0,S],[S,0]], where S=(T+T\*)/2 is self-adjoint. It equals(393) for q=1 because −RFR\*=RFR. Its norm is at most1, as the average of two self-adjoint contractions related by a signed unitary conjugation. (4points)

The average anticommutes with R directly. Also FR+RF=diag(T\*−T,T\*−T), so T−T\* is compact by taking its matrix corner. The difference between the displayed F and its average has compact off-diagonal entries. Thus it is an actual compact perturbation on the module. (4points)

For F\_t=(1−t)F+t a(F), self-adjointness and oddness are exact and the norm is at most1. Its difference from F is compact. Expanding F\_t²−1 and its source commutator therefore leaves only the original compact defects and products of bounded and compact operators. They vary in norm with t, giving a whole Kasparov homotopy. Reversing the orthonormal normal frame replaces R by−R. The two occurrences of R in RFR cancel that sign, leaving exactly the same average. This is the rank-one instance of the full frame covariance in C4; it does not trivialize the normal Clifford bundle. (4points)

**Exercise 70 (12 points).** A homotopy inverse need not select a height

Let W be the CW wedge of countably many circles, with their common vertex v and the CW weak topology. Let K\_n be its first n circles, X=Tel(K\_n→K\_(n+1)), and p:X→W the cylinder projection of B4.

1. Prove that W is not locally compact at v. (4points)
2. Prove that no continuous section s:W→X of p exists. (4points)
3. Explain why B4 nevertheless provides a full homotopy inverse of p,
   and why the construction(392) is performed on one whole X. (4points)

*Solution.* Every open neighbourhood U of v contains a point x\_n in the interior of each nth circle. The set {x\_n:n≥1}, and every subset of it, is closed in W: its intersection with each closed cell consists of at most one interior point and omits v, so the CW weak topology tests it as closed. It is consequently an infinite closed discrete subset of any closed neighbourhood containing U. Such a neighbourhood cannot be compact. Thus no compact neighbourhood of v exists. The ordinary planar picture of successively smaller circles has a different topology and is not this W. (4points)

The height h:X→[1,∞) is continuous. If a section existed, choose an integer M>h(s(v))+1. The set U={w:h(s(w))<M} would be an open neighbourhood of v. All points below height M project into K\_M. Hence p s=1\_W would imply U⊂K\_M. But every neighbourhood of v meets the interior of every circle beyond the Mth, a contradiction. No continuity convention for a least containing stage can repair this section argument. (4points)

Both X and W are CW complexes and p induces all homotopy-group isomorphisms by the finite-image and finite-cylinder argument of B4. Its relative cell fillings in the mapping cylinder give q:W→X and the two entire homotopies pq≃1\_W and qp≃1\_X. They impose no equation pq=1\_W; the preceding result shows that equation would be impossible. Since X is locally compact, Hausdorff and second countable, C4 can construct its module and operator on that entire space once an actual classifying cocycle on it has been provided. That whole field is the datum being compared under (394), including classes invisible on compact stages. Parts V and Q establish the required classifying comparison for the actual holonomy groupoid. (4points)

**Exercise 71 (12 points).** A numerable cutoff on an indexed arrow join

In U2 put m=max\_i t\_i, a\_i=(t\_i−m/2)\_+ and λ\_i=a\_i/Σ\_j a\_j.

1. Compute λ when the positive weights are (2/5,7/20,1/4). (4points)
2. Prove local finiteness of the closed supports at a point with
   finite support I\_0; pointwise finite support alone is insufficient.
   (4points)
3. Prove closed subordination to O\_i={t\_i>0} and explain why the
   weighted normal metric stays positive with the exact twist.
   (4points)

*Solution.* m=2/5 and m/2=1/5. The three positive differences are1/5,3/20,1/20, with sum2/5. Division gives λ=(1/2,3/8,1/8), whose sum is one. All other coordinates are zero. The cutoff did not remove an actual arrow from the join point; it only provides a new partition on its positive-coordinate cover. (4points)

At b\_0 write m\_0=m(b\_0)>0. The continuous function u=Σ\_(i∉I\_0)t\_i=1−Σ\_(i∈I\_0)t\_i vanishes there. On the open neighbourhood {m>m\_0/2,u<m\_0/8}, every outside coordinate satisfies t\_i≤u<m\_0/8<m/2. Its λ\_i is identically zero throughout this neighbourhood; therefore its closed support has no point in it. Only the finite indices of I\_0 remain. This proves local finiteness of closed supports, rather than merely finiteness at b\_0. (4points)

The positive λ\_i locus satisfies t\_i>m/2. Its closure lies in {t\_i≥m/2}; since m is continuous and positive everywhere, that set lies in O\_i. Take the local normal metric over O\_i and transport it by the exact derivative transition maps to any common fibre. The sum with λ\_i is continuous, since its supports are locally finite and closed-subordinate. For every nonzero vector at least one positive λ\_i contributes a strictly positive squared norm, so the sum is positive definite. On overlaps the same derivative products change all terms together, giving the same global metric on E\_M. It neither chooses a spin-c structure nor replaces the normal bundle by its numerical characteristic classes. Classification and the full model comparison are separate from this cutoff calculation; M1–M5, V1–V6 and Q1–Q6 supply them. (4points)

**Exercise 72 (12 points).** Local avoidance on an unbounded discrete base

Let X=N={0,1,…} be discrete. Give the old charts U\_i={i} the partition α\_i(n)=β\_i(n)=δ\_in. Use the pair groupoid on two objects a,b, with its actual arrow g:a→b. The first cocycle has anchor a on every U\_i; the second has anchor b and is equivalent by g.

1. Show that every numeric index is used by the old maps somewhere,
   while each compact subset of X involves only finitely many. (4points)
2. With W\_j={j} and ρ\_j(n)=δ\_jn, verify the finite-avoidance choice
   ν\_j=2j+1 and explain its use at n=2 and n=5. (4points)
3. Write both actual homotopies to the auxiliary map with anchor a
   at color ν\_j, checking their common final orbit. (4points)

*Solution.* Old index i has weight1 exactly at the point i. Thus there is no numeric color unused throughout X. A compact subset of this discrete space is finite, since its singleton open cover has a finite subcover. Its active old colors are consequently finite. The support families are locally finite even though their global index union is all of N. (4points)

I\_j={j}. The indices 2j+1 are strictly increasing, exceed j and exceed all earlier auxiliary indices, so they satisfy(406). At point2 the old color is2 and the auxiliary color is5. At point5 the old color5 reappears, but ρ\_2(5)=0 and the active auxiliary color there is11. Neither point has a collision. This is exactly the permitted local avoidance, without a global finite bound. (4points)

At point j the first homotopy has columns (color j, weight1−s, arrow1\_a) and (color2j+1, weight s, arrow1\_a), with common source a. The second has columns (color j, weight1−s, arrow1\_b) and (color2j+1, weight s, arrow g^−1:b→a), with common source b. At s=0 these are the two original maps. At s=1 their only positive column has color2j+1 and target a. The second is in the same final orbit as the first: right multiplication by g:a→b gives g^−1g=1\_a. Thus the middle maps agree literally in the orbit quotient. Every component {j}×I is continuous, and X×I is their open disjoint union, so these are whole continuous homotopies on the unbounded base. (4points)

**Exercise 73 (12 points).** Barycentric colors and the backward actual arrow

*Level: advanced.* In V2 take a chain x\_0→x\_1→x\_2 with initial products h\_0=1, h\_1=g\_1, h\_2=g\_2g\_1 and weights (7/20,2/5,1/4). Find its positive barycentric faces, their colors, coefficients and actual columns under d. Recover the old weights. Write v d, and write V3's homotopy at s=1/2, including its backward arrow. Explain why the same definition agrees with the original face t\_0=0.

*Solution.* The decreasing weight order is 1,0,2, with faces {1}⊂{0,1}⊂{0,1,2}. Their colors are0,1,2 and coefficients 1/20,1/5,3/4. The minima are1,0,0, so their columns from common source x\_0 are g\_1,1\_(x\_0),1\_(x\_0). They are actual arrows; the first column is not discarded merely because the last two coincide. The coefficients sum to one. (4points)

The original weight at vertex1 is 1/20+(1/5)/2+(3/4)/3=2/5; that at vertex0 is (1/5)/2+(3/4)/3=7/20; that at vertex2 is (3/4)/3=1/4. The v d chain is x\_1→x\_0→x\_0, with arrows g\_1^−1 and1\_(x\_0) and the same three new coefficients. At s=1/2 the concatenated chain is x\_0→x\_1→x\_2→x\_1→x\_0→x\_0, with weights (7/40,1/5,1/8,1/40,1/10,3/8). Its consecutive arrows are g\_1,g\_2,g\_2^−1,g\_1^−1,1\_(x\_0). They are exactly h\_bh\_a^−1 at each step. They sum to one and retain every original lift. The backward arrows are required by the actual groupoid chain. (4points)

On t\_0=0, equation(413) makes every positive face omit0. Deleting0 does not change their cardinalities. In the shorter chain its common source is x\_1; the old initial columns h\_a for the remaining vertices are replaced by h\_a g\_1^−1. This is one common right multiplication, so their Milnor orbits agree. Consecutive nerve products also agree after cancelling the deleted vertex. Thus both d and its whole fat homotopy descend to this face, including its corners. The calculation uses zero-weight faces, and does not identify nonzero unit vertices through thin degeneracies. (4points)

**Exercise 74 (12 points).** A full CW model of the punctured plane

**Level: intermediate. 12 points.** Cover O=R²\{0} by U\_0={x>0}, U\_1={y>0}, U\_2={x<0}, U\_3={y<0}. Determine its nerve and construct the two Q1 maps with a closed-subordinate partition. Use p\_0=(1,0),p\_1=(0,1), p\_2=(−1,0),p\_3=(0,−1) and p\_(01)=(1,1),p\_(12)=(−1,1), p\_(23)=(−1,−1),p\_(30)=(1,−1). Check the carrier and explain why the construction is an equivalence on the whole O.

*Solution.* Each single open is convex; adjacent pairs meet in their open quadrant. Opposite pairs are empty, and every triple contains an opposite pair. Thus the nerve is the four vertices and the cycle edges01,12,23,30, with no higher faces. It is a finite CW circle. (4points)

Let m=max(|x|,|y|)>0 and let s=(x,y,−x,−y). Put a\_i=(s\_i−m/2)\_+ and ρ\_i=a\_i/sum a\_j. At least one a\_i=m/2, so the denominator is positive. The closed support is contained in s\_i≥m/2>0, hence in U\_i. Four functions are locally finite and their sum is one. Their positive labels form an adjacent face, giving Φ. At (1,1) its weights are(1/2,1/2,0,0); at (2,1) they are(1,0,0,0). Ψ on a subdivided edge joins the stated endpoint p\_i to its quadrant p\_(ij) and then to p\_j. Every segment lies in the relevant half-plane and avoids the origin. This gives the indicated eight-vertex polygonal loop. (4points)

For a flag {i}⊂{i,j}, all its Ψ points lie in U\_i. Its positive partition labels together with i therefore form a nerve face, so ΦΨ lies in the closed star of i, just as the original flag does. On its face {i,j} the smaller carrier is the edge itself. The Q1 coning extension consequently gives the homotopy ΦΨ≃1 of the entire circle. The straight homotopy ΨΦ(x)→x stays in a common half-plane selected by the flag's least face, and is a whole continuous homotopy on O. It fixes no chosen compact exhaustion; both inverse homotopies hold on the full punctured plane. (4points)

**Exercise 75 (12 points).** A projector loop remembers its normal transport

*Level: advanced.* Set θ=πt and let P(t) be the real orthogonal projector onto v(t)=(cos θ,sin θ) in R². Compute [P',P] and solve the D1 transport equation. Compare P(0),P(1) and the induced endpoint isomorphism on their line. Why must a twisted comparison retain its specified bundle path?

*Solution.* With J=((0,−1),(1,0)), v'=πJv and P'=π(JP−PJ). At θ=0 direct multiplication gives [[J,P],P]=J; conjugation by the rotation gives the same identity for every θ. Hence [P',P]=πJ, and V'=πJV,V(0)=1 has solution V(t)=((cos πt,−sin πt),(sin πt,cos πt)). Its columns remain orthonormal, and V(t)P(0)V(t)\*=P(t). (4points)

Both endpoint projectors equal diag(1,0). But V(1)=−1\_(R²), so its transport on that common line is multiplication by−1. The endpoint equality of projections does not specify the same line isomorphism as the constant projector path. Gluing the ends of this line family gives a Möbius line on the parameter circle. Its sign is part of the normal bundle, even before choosing a Clifford representation. (4points)

On the rank-one negative Clifford generator r this transport induces r↦−r. That is the actual endpoint normal comparison; one cannot silently replace it by r↦r. D1 proves choice independence for transports of the same given bundle path: their ratio is a continuous orthogonal family starting at1. It does not identify this nontrivial loop with the constant bundle path while dropping endpoint comparisons. D2 and Q6 therefore retain the specified normal concordance and its relative comparison tracks when identifying twisted groups. (4points)

**Exercise 76 (12 points).** Both rank-one duality kernels, with their source actions

*Level: advanced.* In rank1 use L=((0,1),(1,0)), R=((0,−1),(1,0)), Γ=diag(1,−1) and r²=−1. Compute the two flat squares of G4–G5. Identify their entire Gaussian ground modules and show that the second source R acts by the target r on its kernel. Explain why kernel rank alone would not prove the Clifford-algebra triangle.

*Solution.* For T\_C=−i(L∂+Rv), the derivative/potential anticommutator contributes−LR. Since LR=diag(1,−1), T\_C²=−∂²+v²+diag(−1,1). The scalar oscillator has ground energy1 and successive energies1+2n, so the unique zero space is π^−1/4 exp(−v²/2)e\_0. It is the trivial even line; the next energy is2. After the source contraction f acts by its value at the remaining base point. (4points)

For T\_Q=−iL∂+iΓ⊗rv, its square is −∂²+v²+LΓ⊗r, and LΓ=R. The−1 eigenspace of S=R⊗r is ΩQ, where Ω=(e\_0⊗1−e\_1⊗r)/√2. Directly SΩ=(−e\_0⊗1+e\_1⊗r)/√2=−Ω and <Ω,Ω>=1, since r\*r=1. Solving S(a,b)=−(a,b) gives b=−ra, so that is the entire eigenspace. The Gaussian factor therefore gives the whole free rank-one right Q kernel. It is even, since the e\_1 and r are both odd. (4points)

The source action gives (R⊗1)Ω=(e\_0⊗r+e\_1⊗1)/√2. The target action gives Ωr=(e\_0⊗r−e\_1⊗r²)/√2, the same vector. Thus the entire left source action is the identity on that free Q module. The second odd target generator occurs as Γ⊗r in the potential, whereas the source occurs as R⊗1; interchanging them would lose this calculation. A free module of the same rank could carry a different Clifford automorphism, so rank alone does not prove its identity KK class. The full intertwining and grading are what prove the second triangle. (4points)

**Exercise 77 (12 points).** The reflected full dual cap in rank one

*Level: advanced.* Let W=ΛC have even e\_0 and odd e\_1, with L=((0,1),(1,0)), R=((0,−1),(1,0)). Let the positive target generator c satisfy c\*=c and c²=1. Compute the four values in (443) for rank1, check the full physical L action on them, and explain what omitting γ does to the point-source Thom sign. Does a determinant line determine this cap?

*Solution.* Equation (443) gives U\_(00)=1/√2, U\_(01)=c/√2, U\_(10)=−c/√2 and U\_(11)=1/√2. The positive balance is U\_(1,J)=−(-1)^J U\_(0,J)c, which gives both last-row values. The negative balance −bar(Rw)⊗v=bar w⊗Rv gives the same equations, using R e\_0=e\_1 and R e\_1=−e\_0. Thus the table checks the entire two-generator balanced module, not just its vacuum. (4points)

The physical L acts on bar e\_I⊗e\_J by (-1)^I bar e\_I⊗L e\_J. For (I,J)=(0,0),(0,1),(1,0),(1,1), their images under U are respectively c/√2,1/√2, −1/√2,c/√2. These are exactly c times the four original table entries. The raw vacuum tensor has squared norm1/2, so ζ=√2(bar e\_0⊗e\_0) is the whole free even unit and the positive physical operator is c, with its entire right Clifford action retained. (4points)

Without γ the positive balance would have the opposite minus sign. Then U\_(10)=+c/√2 and the negative balance would force U\_(01)=−c/√2. The induced physical L would be −c, reversing the point-source rank-one Thom class in the native odd convention. In the corrected cap a line H passes through as H. Its determinant transition is H²; a nontrivial two-torsion line can therefore have the same determinant while changing the entire cap. The full module and its reflected source action, rather than its determinant, specify H1 and its line-sensitive pairing. (4points)

**Exercise 78 (12 points).** Localized Fredholm defects still give a projection

*Level: advanced.* Let (H,φ,F) be an odd Fredholm cycle for a possibly nonunital A, with F=F\*, ||F||≤1. Only assume the source-localized defects compact. Construct the E1 involution and its compression extension. For the scalar matrix sample F=3/5 compute s,F\_1,P and its +1 line. Finally, prove the key compact off-diagonal step in the inverse dilation.

*Solution.* Since q(F) commutes with qφ(a) and (1−q(F)²)qφ(a)=0, functional calculus gives q((1−F²)^(1/2))qφ(a)=0. Thus sφ(a),φ(a)s are compact. For π=diag(φ,0), the path F\_t=[[F,ts],[ts,−F]] has localized compact commutators and square defects. Its second source summand at t=0 is degenerate. At t=1 its square is1. P=(1+F\_1)/2 is consequently an actual projection, and q(Pπ(a)P) is a Busby homomorphism because its multiplicative defect Pπ(a)(1−P)π(b)P is compact. This does not require F²−1 itself to be globally compact. (4points)

The exact sample is s=4/5, F\_1=1/5[[3,4],[4,−3]] and P=1/5[[4,2],[2,1]]. Its +1 eigenvector for F\_1 is (2,1)/√5; its orthogonal −1 eigenvector is(−1,2)/√5. The displayed P is the orthogonal projection onto the first line. This finite algebra sample checks normalization, not the extension classification for an infinite algebra. (4points)

For a cpc lift σ with isometric dilation σ(a)=V\*π(a)V, P=VV\*, C\_a=(1−P)π(a)V satisfies C\_a\*C\_a=σ(a\*a)−σ(a\*)σ(a). Its image in the Calkin algebra is zero, since qσ is multiplicative. The positive operator C\_a\*C\_a is compact. Its finite spectral cuts give ||C\_a(1−1\_[δ,∞)(C\_a\*C\_a))||≤√δ, so C\_a is the norm limit of finite-rank maps. Using a\* gives the other off-diagonal compact block. Therefore [P,π(a)] is compact and 2P−1 defines the inverse odd cycle. The full dilation and both inverse identifications, including whole homotopies and the other parity, are E2–E4. (4points)

**Exercise 79 (12 points).** A changing transverse rank does not break the étale tube

*Level: advanced.* Take M=S¹, the unit foliation of S¹ as target, f(e^(iθ))=e^(i sin θ), and j(e^(iθ)) =(cos θ,sin θ)∈R². Use its unit tangent and radial normal, and the standard transverse frame. Compute (464)'s derivative at the zero section for every t∈[0,1], including f's critical points. In the Clifford list (u,e,v,η), verify the entire rank-one double cancellation and its remaining normal chirality. Explain what (465)–(466) retain beyond a calculation of determinant lines or scalar indices.

*Solution.* The transverse derivative is α=cos θ; its rank is0 at θ=π/2,3π/2 and1 elsewhere. With source (w,η,e) and output (transverse,tangent,normal), (464) is

\[
 \begin{pmatrix}t\cos\theta&0&1\\1&0&0\\0&1&0\end{pmatrix},
 \qquad \det=1.
 \tag{467}
\]

Its inverse sends (z,a,b) to(a,b,z−t cos θ·a). Thus it is invertible at every parameter and every point, including both critical points. The transverse e block supplies the missing direction. Compactness and the uniform small-disk estimate in I3 extend this zero-section calculation to one entire compressed étale tube; the determinant sample alone would not prove that extension. (4points)

Write the positive generators in the given order as g\_0,g\_1, g\_2,g\_3, and set A=g\_0,B=g\_2,C=(A−iB)/2, Π=(1−iAB)/2. Clifford relations give C²=0, C\*C=(1−iAB)/2=Π, CC\*=1−Π, CΠ=C and C\*Π=0. The two columns are Π,CΠ; their inner products are respectively Π,Π with mixed products0, and Π+CΠC\*=1. The actions are A on the columns with exterior L, B with exterior iR. Both g\_1,g\_3 commute with Π and anticommute with the creation operator, so act with the exact exterior grading sign on the normal coefficient. (4points)

The normal order is(η,e), so its volume is−i g\_3 g\_1. On Π, AB=i. The full volume −g\_0g\_1g\_2g\_3 consequently equals i g\_1g\_3=−i g\_3g\_1, exactly the normal volume. The actual source permutation (3,1,0,2) has four inversions, so introduces sign+1 in this example. An orientation line H is the same H in the cap and normal module and is retained by the whole recovery. Its square does not detect a two-torsion H. (465) proves equality of the full KK classes for this changing-rank map; (466) then gives every bundle-weighted integral test, in its prescribed degree and Clifford order. A single determinant or scalar index would lose that full class and its line information. (4points)

**Exercise 80 (12 points).** A nonzero inclusion with zero K-maps

*Level: intermediate.* Let \(U=]a,b[\), \(I=C_0(U\times \mathbb P^1(\mathbb R))\) and \(C=C([a,b]\times \mathbb P^1(\mathbb R))\). Show that extension by zero \(\iota :I\to C\) is a nonzero injective homomorphism and that its two K-maps are zero. Compute both K-groups of each algebra, and explain why those groups being isomorphic does not determine the inclusion's K-map.

*Solution.* A nonzero continuous compactly supported bump on \(U\), tensor the constant one on \(\mathbb P^1\), has a nonzero zero extension. Pointwise restriction shows injectivity. An element of \(I\) vanishes at both endpoints, so \(\operatorname{ev}_a \iota =0\). Endpoint evaluation is a homotopy equivalence by contracting the closed interval, leaving the second coordinate fixed. Its two K-maps are invertible; consequently \(\iota _*=0\) in both parities. The inclusion itself is nonzero. (4 points)

The real projective line is a circle. Contracting the interval gives \(K_0(C)=K_1(C)=\mathbb Z\); the generators are the rank-one unit bundle and the winding-one unitary. The open interval is homeomorphic to \(\mathbb R\), so \(I\) is the suspension of \(C(S^1)\). Suspension and Bott give \(K_0(I)=K_1(I)=\mathbb Z\), with the two circle generators exchanged in degree. These statements use the stable K-theory interfaces specified in the introduction. (4 points)

An abstract identification between source and target groups says nothing about a particular homomorphism between them. Here its composite with a proved target K-isomorphism is zero, so the particular inclusion map is zero. In the counterexample this is the left vertical map of Figure 11A; replacing it by an abstract identity of \(\mathbb Z\) would destroy the missed-class argument. (4 points)

**Exercise 81 (12 points).** The missed reduced unit needs no split exact sequence

*Level: intermediate.* Suppose \(0\to I\to B_m\to D_m\to 0\) is an exact C*-extension, \(B_r=D_r\oplus C\), and a quotient \(\lambda :B_m\to B_r\) commutes with restriction. Assume \(K_0(D_m)\to K_0(D_r)\) is injective and the induced map of the ideal into \(K_0(B_r)\) is \(y\mapsto (0,\iota _*y)\) with \(\iota _*=0\). Prove that every nonzero \((0,z)\) misses \(\operatorname{im} \lambda _*\). State precisely where exactness and injectivity enter, and whether a split full K-row or an exact reduced K-row is necessary.

*Solution.* If \((0,z)=\lambda _*(x)\), its \(D_r\) restriction is zero. The commuting restriction square gives zero as the image of the full restriction of \(x\). The assumed injectivity therefore makes that full restriction zero. This uses the right vertical map only. (4 points)

Exactness of the full K-sequence at \(K_0(B_m)\) puts \(x\) in the image of \(K_0(I)\). Its reduced image is then \((0,\iota _*y)=0\). Hence \(z=0\), a contradiction. Only exactness at this middle full group is used; no splitting and no exactness statement for a reduced restriction extension enters. (4 points)

In Proposition 11A.4 the full extension is the proved invariant-open extension, its right quotient is the canonical free-group K-isomorphism of Theorem 11A.2, and Lemma 11A.5 proves its left arrow zero. The chosen \(z=[1_C]\) is nonzero because point evaluation detects rank one. Thus \((0,[1_C])\) is the concrete missed class, not a formal unnamed element of a kernel. (4 points)

**Exercise 82 (12 points).** Why representative lifts obstruct the original assembly

*Level: advanced.* For one compact cycle \((M,E,f)\) of dimension \(d\), let \(q=\operatorname{rank}\tau\) and use the same tube in \(\mathbb R^n\) in the full and reduced completions. Write the typed normal Thom, étale graph and inverse Bott factors. Prove that the quotient carries their full product to the reduced wrong-way class. Deduce image containment for the descended geometric assembly without constructing a descended maximal assembly. Explain how a complete transversal transports a missed reduced class and which assertion about a kernel has not been proved.

*Solution.* The factors are \(t\in KK^{n-d+q}(C(M),C_0(N))\), \(e_m\in KK(C_0(N),C_0(\mathbb R^n)\widehat\otimes A_m)\), and \(b_n^m\in KK^n(C_0(\mathbb R^n)\otimes A_m,A_m)\). Their total degree is \(q-d\) modulo two. The same full spin-c module, source action, phase and Clifford order give \(t\) in both constructions. The graph quotient unitary sends \(\xi \widehat\otimes a\) to \(\xi a\) with inner product \(a^*\lambda(\langle\xi,\eta\rangle)b\), hence carries \(e_m\) to \(e_r\). (4 points)

The even coefficient quotient introduces no interchange sign. Tensor associativity gives \(b_n^m\widehat\otimes [\lambda _H]=[1\widehat\otimes \lambda _H]\widehat\otimes b_n^r\), and therefore \(f_!^m\widehat\otimes [\lambda _H]=f_!^r\). Multiplying by \([E]\) proves that each representative value lifts. The image of a homomorphism is a subgroup, so finite sums and differences of representative values still lie in that image. The reduced geometric relations have already descended in Theorem 6.1; this containment does not require a newly defined maximal geometric map to descend through them. (4 points)

For the complete transversal, the same-core full/reduced imprimitivity modules give K-isomorphisms \(\Phi _m,\Phi _r\) with \((\lambda _H)_* \Phi _m=\Phi _r(\lambda _T)_*\). A lift of \(\Phi _r(c)\) would pull back through \(\Phi _m^{-1}\) to a lift of \(c\), contradicting Exercise 81. Its nonzero reduced class therefore misses the original geometric assembly. This is a surjectivity obstruction. It supplies neither a nonzero geometric kernel class nor an identification of this domain with another assembly domain. (4 points)

**Exercise 83 (12 points).** Why the full inverse calibration is necessary

Use the conventions of Lemma 11.2: \(\varepsilon_1=\sigma_x\), \(\varepsilon_2=\sigma_y\), \(b_{W,1}=\sigma_y\), \(b_{W,2}=\sigma_x\), and \(\Gamma_S=\Gamma_W=\sigma_z\). (a) Compute both ordered physical chiralities and show that an even map intertwining \(\varepsilon_j\) with \(-b_{W,j}\), for both ordered generators, must be zero. (b) In the whole \(S\widehat\otimes W\) tensor, verify that the normalized coefficient \(\eta=(|00\rangle+|11\rangle)/\sqrt2\) is even and fixed by the full transitions \(\operatorname{diag}(h,h\ell)\otimes\operatorname{diag}(h^{-1},h^{-1}\ell^{-1})\). Explain what information a scalar Gaussian rank would lose.

**Solution.** (a) The positive physical chirality is \(-i\sigma_x\sigma_y=\sigma_z\). The calibrated inverse physical chirality is \(-i\sigma_y\sigma_x=-\sigma_z\); replacing both generators by their negatives leaves this chirality unchanged. An intertwiner \(J\) therefore satisfies \(J\Gamma_S=-\Gamma_WJ\), while evenness gives \(J\Gamma_S=\Gamma_WJ\). Subtracting gives \(2\Gamma_WJ=0\), hence \(J=0\). This excludes the proposed even intertwiner of the whole ordered rank-two representations. It does not exclude the even rank-four Morita evaluation of Lemma 11.2.

(b) Both \(|00\rangle\) and \(|11\rangle\) have total even parity. Their transition phases are respectively \(hh^{-1}=1\) and \((h\ell)(h^{-1}\ell^{-1})=1\), so the specified relative plus sign patches unchanged. Lemma 11.2's entire evaluation takes \(\eta\) to the exterior vacuum. The full oscillator square in (H.35) has eigenvalues \(2(|n|+k)\), so its complete kernel is the Gaussian times that even invariant coefficient and its square gap is two. A scalar rank does not retain parity, the transition line, the ordered physical generators or the Fourier phase. In the same-positive-spinor comparison the rank is also one, but its entire line is odd and has transition \(h^2\ell\), giving \(-[L\otimes H^2]\).

**Exercise 84 (12 points).** A first inverse and an onto evaluation do not finish the reverse identity

Suppose \(T\in KK(C,A)\) and \(d\in KK(A,C)\) satisfy \(T\otimes_A d=1_C\). (a) Show that \(p=d\otimes_C T\in KK(A,A)\) is idempotent and describe the range of \(p_*\) on both K-degrees. (b) Give an onto \(*\)-homomorphism \(q:B\to D\) and a nonzero \(X\in KK(D,D)\) for which \([q]\otimes_D X=0\). State precisely what this example prevents one from inferring from (H.37).

**Solution.** (a) Associativity gives \(p^2=d(Td)T=dT=p\). On K-theory the product acts in the order \(d_*T_*=1\), so \(T_*\) is injective and \(d_*\) is onto. Because \(p_*T_*=T_*\), the range of \(p_*\) contains \(\operatorname{im}T_*\), and the formula \(p_*=T_*d_*\) gives the reverse inclusion. Thus \(\operatorname{ran}p_*=\operatorname{im}T_*\), in each degree. This proves neither \(p=1_A\) in KK nor surjectivity of \(T_*\).

(b) Take \(B=C_0((0,1])\), \(D=\mathbb C\), and \(q(f)=f(1)\). Evaluation is onto. Extend \(f\) continuously by \(f(0)=0\); then \(H_t(f)(s)=f(ts)\), \(0\le t\le1\), is a norm-continuous homotopy of \(*\)-homomorphisms from zero to \(1_B\). Hence \(B\) is KK-contractible and \([q]=0\). Choose \(X=1_{\mathbb C}\), which is nonzero because its action on \(K_0(\mathbb C)=\mathbb Z\) is the identity. Then \([q]X=0\) although \(X\ne0\). Therefore onto evaluation alone does not allow cancellation of the factor \([q_1]\) in (H.37). A proved appropriate KK inverse, a parameter-preserving identity or a direct native reverse homotopy would be additional information; none is supplied by onto-ness itself.

**Exercise 85 (12 points).** Let \(H=\mathbb R\rtimes\mathbb R\) have multiplication \((n,t)(n',t')=(n+e^tn',t+t')\). On a numerable cover let \(h_{\beta\alpha}=(n_{\beta\alpha},t_{\beta\alpha})\) be an actual \(H\)-cocycle. (i) Derive the two coordinate cocycle equations. (ii) Using a subordinate partition \(\rho_\gamma\), construct \(t_\alpha\), the residual additive cocycle, and \(u_\alpha\). (iii) Give \(k_\alpha\) in these affine coordinates and verify \(h_{\beta\alpha}=k_\beta k_\alpha^{-1}\). (iv) Explain why a zero partition weight does not require an undefined transition function to be evaluated. (3 points each.)

**Solution.** Multiplication gives \(t_{\gamma\alpha}=t_{\gamma\beta}+t_{\beta\alpha}\) and \(n_{\gamma\alpha}=n_{\gamma\beta}+e^{t_{\gamma\beta}}n_{\beta\alpha}\). Define \(t_\alpha=\sum_\gamma\rho_\gamma t_{\alpha\gamma}\). Then \(t_\beta-t_\alpha=t_{\beta\alpha}\). With \(r(t)=(0,t)\), the residual cocycle is

\[
 r(-t_\beta)h_{\beta\alpha}r(t_\alpha)
       =(e^{-t_\beta}n_{\beta\alpha},0).
 \tag{B.s}
\]

Write its first coordinate as \(b_{\beta\alpha}\). The group cocycle and the already proved real-coordinate difference give \(b_{\gamma\beta}+b_{\beta\alpha}=b_{\gamma\alpha}\). Thus \(u_\alpha=\sum_\gamma\rho_\gamma b_{\alpha\gamma}\) satisfies \(u_\beta-u_\alpha=b_{\beta\alpha}\). The exact order from Lemma 8B.1 gives

\[
 k_\alpha=r(t_\alpha)(u_\alpha,0)
         =(e^{t_\alpha}u_\alpha,t_\alpha).
 \tag{B.t}
\]

The inverse of \((n,t)\) is \((-e^{-t}n,-t)\). Hence \(k_\beta k_\alpha^{-1}=(e^{t_\beta}(u_\beta-u_\alpha),t_\beta-t_\alpha)=(n_{\beta\alpha},t_{\beta\alpha})\). Replacing (B.t) by \((u_\alpha,t_\alpha)\) would generally fail. Each summand is defined on the intersection with \(U_\gamma\) and extended by zero outside its closed subordinate support. It is zero on a neighbourhood wherever that support is absent. Local finiteness makes the sum smooth or continuous on the whole chart. No expression \(0\cdot\text{undefined}\) is used. (12/12 points.)

**Exercise 86 (12 points).** The free action \(s\cdot(x,y)=(x+s,e^sy)\) of \(\mathbb R\) on \(V=\mathbb R^2\) has orbit field \(X=\partial_x+y\partial_y\). (i) Identify its leaf invariant and verify the actual graph condition. (ii) Compute the normal derivative in the quotient coordinate \(\omega=dy-y\,dx\). (iii) Reduce local anchors with real transitions \(t_{\beta\alpha}\) and show that the entire normal bundle map glues. (iv) Explain what this does and does not say about a prescribed spin-c line. (3 points each.)

**Solution.** The invariant is \(z=e^{-x}y\). The map \((x,y)\mapsto(x,z)\) is a global diffeomorphism and the action becomes translation of \(x\). Each leaf is a copy of \(\mathbb R\), and every arrow between two points with the same \(z\) has its unique parameter \(s=x'-x\). The simple-foliation pair groupoid along these fibres therefore is precisely \(V\rtimes\mathbb R\), with no extra stabilizer lifts.

The form \(\omega\) annihilates \(X\), and identifies the rank-one quotient \(TV/F\) with \(\mathbb R\). If \(\xi=(\xi_x,\xi_y)\), the action derivative is \((\xi_x,e^s\xi_y)\) at \((x+s,e^sy)\), so

\[
 \omega_{s\cdot(x,y)}(d(s)\xi)
   =e^s(\xi_y-y\xi_x)
   =e^s\omega_{(x,y)}(\xi).
 \tag{B.u}
\]

Choose \(t_\alpha\) with \(t_\beta-t_\alpha=t_{\beta\alpha}\). For \(f_\alpha=(x_\alpha,y_\alpha)\), the global anchor is \(a=(x_\alpha-t_\alpha,e^{-t_\alpha}y_\alpha)\). Its local normal comparison is multiplication by \(e^{t_\alpha}\). The target transition is multiplication by \(e^{t_{\beta\alpha}}\), and \(e^{t_{\beta\alpha}}e^{t_\alpha}=e^{t_\beta}\), which proves gluing on the whole bundle. One first transports its metric through this positive bundle map before comparing Clifford modules. If the full stable spin-c structure carries a line \(L\to M\), that same line, its transition phases and grading remain factors of the transported module. The positive scalar normal derivative is not a trivialization of \(L\), and determinant or rational-character agreement cannot discard a torsion line. (12/12 points.)

**Exercise 87 (12 points).** (i) For \(d=0,\ldots,7\), calculate the exact relation between the physical framed Dirac and the native chronological real Thom classes. (ii) Verify both KK inverse products for the compact leaf class. (iii) Consider the rotation action of \(\mathbb R\) on \(S^1=\mathbb R/2\pi\mathbb Z\). Does local freeness allow applying the native theorem with its transformation groupoid? (iv) Does the global-lift reduction prove assembly isomorphism? (3 points each.)

**Solution.** The sign list \(s_d=(-1)^{\lfloor d/2\rfloor}\) is \(+,+,-,-,+,+,-,-\). With the physical framed orientation, \(\ell_!^{\rm geom}=\ell_!^{\rm an}=s_d\Theta_H\) by (B.q), on both K-degrees. Its inverse is \(s_dD_H\); associativity with the fixed factor order gives \((s_d\Theta_H)(s_dD_H)=s_d^2\,1=1\) and \((s_dD_H)(s_d\Theta_H)=1\). A same-operator grading shortcut on an odd right-Clifford module is not needed for these products: the full finite calibration is already Lemma 8.15c and the independent comparison's (SB25a)–(SB25c).

For the rotation action, the transformation groupoid has stabilizer \(2\pi\mathbb Z\) at every point. The codimension-zero foliation on \(S^1\) has pair holonomy groupoid: all paths with the same endpoints have identical trivial transverse holonomy. It has trivial isotropy. These two groupoids are therefore different, even though the action is locally free. The graph condition fails; replacing the actual holonomy groupoid by the transformation groupoid would change the native target.

Finally (B.o) identifies the entire geometric domain with global **geometric cycles**, and (B.p) computes their actual wrong-way classes. Theorem 8C.8 now supplies the two-sided integral identification with \(K_c^{*+d}(V)\); Theorem 8D.7 below supplies the ordinary/geometric composition interface in full noncompact scope. Combining these geometric results with the calibrated two-sided Thom inverse proves both assembly inverse identities in Theorem 8D.8. The analytic inverse alone would not supply the required geometric interfaces. (12/12 points.)

**Exercise 88 (12 points).** The exact native parity and determinant line

 On the full source coefficient \(W\otimes LH^2\), verify (ND.4) of Proposition 11.8, including the transition equation. Show why \(J\) becomes even only after parity reversal. Then use Theorem 11.1 to compute the literal native first product and evaluate its action at a point. Explain why this refutes the first-identity assertion but does not refute the isomorphism question.

**Solution.** The source transitions are \((h\ell,h)\) and the target transitions are \((h,h\ell)\). Multiplication by \(J\) exchanges them with coefficients \(1,-i\); its columns are orthonormal. Direct multiplication gives \(J\Gamma_W=-\Gamma_SJ\), \(J(-\sigma_y)=\sigma_xJ\) and \(J(-\sigma_x)=\sigma_yJ\). Parity reversal changes the source grading to \(-\Gamma_W\), so exactly that graded map is even. The whole cycle is \(-\delta_{\rm inv}[LH^2]\), giving \(T\delta_S=-[LH^2]\). The point map sends its value on the trivial line to \(-1\), whereas the identity gives \(+1\). This distinguishes the KK classes. Since the negative line has inverse \(-[L^*H^{-2}]\), the same calculation still gives a left inverse on K-theory after correction. The opposite \(A,A\) identity and surjectivity remain separate obligations.

### Graded exercises for integral native manifold duality

The orientations and coordinate-block order are those of Lemmas 8C.1–7 and Theorem 8C.8. Each exercise is worth 12 points.

**Exercise 89 (12 points): a compactly supported class on the line.** Put \(V=\mathbb R\), \(H=\{e\}\), and \(\tau=T\mathbb R\). Let \(x\) be the positive generator of \(K_c^1(\mathbb R)\) obtained as the geometric Bott class of the oriented line. Construct the compact native representative prescribed by Theorem 8C.8. Explain why the noncompact identity manifold \(\mathbb R\) is unnecessary.

**Grading.** 4 points for the ordered Bott projection and compact double; 4 for the full orientation, degree and transfer; 4 for support, carrier independence and the compact-source explanation.

**Solution.** Here \(d=0\), \(q=n=1\), and \(p=1\). The class \(xB_1\) lies in \(K_c^0(\mathbb R^2)\); its orientation is the actual ordered product of the old positive line and the appended positive line, with the geometric finite Clifford contraction. Choose a closed disk \(W\subset\mathbb R^2\) large enough to contain a projection representative's compact support. A cutoff/spectral projection as in Lemma 8C.3 makes its reference projection constant near \(\partial W\). Its relative pair glues over the compact double \(D W\cong S^2\) to a bundle difference \(y_D\).

The map of this compact cycle is \(D W\xrightarrow{r}\mathbb R^2\xrightarrow{p_1}\mathbb R\). Its native orientation is the entire doubled-target cancellation of Lemma 8C.4, including the inverse orientation of the appended line. It has degree \(2-q=1\). Thus

\[
 \Gamma_1(x)=[D W,E_y,p_1r,\eta_{W,1}]
              -[D W,G_y,p_1r,\eta_{W,1}],
 \qquad
 \kappa\Gamma_1(x)=(xB_1)A_1=x.
 \tag{C.Ex.a}
\]

The displayed equality is the precise adjacent Bott inverse for this Clifford contraction. A physical plane spinor convention selected separately could change the intermediate generator's sign, so no such substitution is made. Both sources in this difference are closed compact two-manifolds. The lower-face coefficient difference and the collar coefficient difference vanish; the upper-face extension is the compactly supported class. Changing the disk produces the hollow-box bordism (C.f). This gives an actual compact native representative and its transfer, rather than treating the noncompact line as a cycle.

**Exercise 90 (12 points): full line cancellation.** Let \(c=[M,E,a,\eta]\) be any native cycle. Twist its full stable orientation by a Hermitian line \(L\), possibly a nontrivial torsion line. Track the line through the graph class \(j_!E\), the compact carrier and the embedding-transfer bordism. Determine whether checking determinant lines alone would prove the second inverse.

**Grading.** 4 points for the source and graph line; 4 for the full sphere and evaluation maps; 4 for torsion, grading and why determinant equality is insufficient.

**Solution.** Lemma 8C.1 tensors only the frame and finite cancellation factors, so its ordinary doubled module retains \(L\). Lemma 6.0j's graph construction tensors the same line onto its normal transition module. Consequently its normal Thom class is \(j_!(E\otimes L)\) with respect to the untwisted graph orientation. The bundle \(L\) here is on \(M\); it need not extend to \(V\) or to the carrier. There is therefore no justified formula replacing it by a line pulled back from \(V\).

The carrier orientation \(\eta_{W,N}\) is fixed by the target, the relative fold and \(A_N\). Its coefficient \(y_D\) retains the graph normal line through the full Thom projection. In Lemma 8C.5 the normal module has that line and the induced orientation of the sphere projection has its inverse. The two finite evaluations are

\[
 L\otimes L^*\longrightarrow\mathbb C,\quad
       \xi\otimes\bar\zeta\longmapsto\langle\zeta,\xi\rangle,
 \qquad
 \operatorname{End}(L)\longrightarrow\mathbb C.
 \tag{C.Ex.b}
\]

They are the actual onto even evaluation maps. Their cancellation returns exactly the original \(\eta\otimes L\) on the graph source, since the normal followed by the carrier orientation was chosen to induce that source orientation. An additional stable grading shift is carried through the same graded tensor maps. If \(L^{\otimes2}\) is trivial, the determinant could agree with the untwisted determinant although \(L\) is nontrivial. Determinant equality therefore cannot establish this recovery. The line must be carried and evaluated as above.

**Exercise 91 (12 points): the three relations and the two inverses.** A proposed proof defines \(\kappa\) and proves only \(\kappa\Gamma=1\). It then declares that \(\Gamma\kappa=1\) because \(\kappa\) respects bordism. Supply the missing geometric argument and list where all three native relations enter. Include the odd degree.

**Grading.** 4 points for graph normalization through the actual surgery bordism; 4 for identifying all three relations; 4 for the cofinal comparison and odd-degree contraction.

**Solution.** Bordism invariance makes \(\kappa\) a homomorphism but does not make it injective. For a cycle of source dimension \(m\), choose \(N\equiv m-n\pmod2\) sufficiently large and a compact-source embedding \(u:M\hookrightarrow\mathbb R^N\). The graph normal has even rank \(n+N-m\). Lemma 8C.6 gives an ordinary even compact coefficient \(y=j_!E\), with \(\kappa c=yA_N\).

The full twisted embedding bordism of Lemma 8C.5 proves \(\beta_N(y)=c\) in the native geometric group itself. Its two ends are even modifications along the trivial complex plane and along \(\nu_j\oplus\mathbb C\). The reduced-north correction is a compact disk bordism with the whole spinor coefficient, so it may be removed by the native relation. This supplies the geometric normalization absent from the proposed proof.

Sum/disjoint union makes virtual coefficient differences legitimate and gives additivity. Compact bordism proves projection-homotopy invariance, carrier independence, the disk correction and the surgery comparison. Even vector-bundle modification identifies the two surgery ends with the original cycles. None of these steps introduces another native relation. Lemma 8C.7 then compares the large even graph normal with the minimal \(p=0\) or \(1\) Bott presentation. With the actual ordered inverse,

\[
 \Gamma_p\kappa c
  =\beta_N((\kappa c)B_N)
  =\beta_N((yA_N)B_N)
  =\beta_N(y)=c .
 \tag{C.Ex.c}
\]

When \(p=1\), \(N\) is odd but its graph normal remains even. The carrier has dimension \(n+N\), and degree \(n+N-q=d+N\), which is exactly the required native degree. The odd coordinate factor is retained until its final finite contraction with the prescribed inverse. Thus the same geometric proof treats both degrees and integral torsion.

**Exercise 92 (12 points).** *the actual translation family and its principal anchor.* Let \(V=\mathbb R^2_{x,y}\), let \(H=\mathbb R\) translate \(x\), and use \(J(x,y)=(x,y,0,0)\in\mathbb R^4\). Thus \(d=1\), \(\tau=\mathbb R\partial_y\), and \(E=(dJF)^\perp\). Write the compressed disk coordinates as \((\eta_1,\eta_2,t)\), so that
\[
 \chi(\eta,t)=(0,-t,\eta_1,\eta_2),\qquad
 h_s(x,y,\eta,t)=(x,y+st),\qquad
 e(x,y,\eta,t)=(x,y,\eta_1,\eta_2).
 \tag{D.A.a}
\]
Calculate the full kernel distribution, its signed zero-section identification and the étale principal anchor. Explain which statements require compactness of the base.

**Grading (12 points).** Transverse differential and full kernel (3); signed zero-section map (3); principal anchor, inverse and Jacobian (3); precise compact-support and whole-module explanation (3).

**Solution.** The transverse differential is \(dy+s\,dt\), of rank one for every \(s\in[0,1]\). Its full rank-four kernel has the global frame
\[
 \partial_x,\quad\partial_{\eta_1},\quad\partial_{\eta_2},
                  \quad\partial_t-s\partial_y .
 \tag{D.A.b}
\]
This is the exact family in the disk coordinates, rather than an asserted constant frame in the uncompressed coordinates. The variable compression in (D.d) pulls it back by its invertible full differential. For \(v=(0,-t,\eta_1,\eta_2)\) and \(k=k_x\partial_x\), (D.g) gives
\(L_s(v,k)=(k_x\partial_x-st\partial_y,v)\).
At \(s=1\),
\(de\,L_1(v,k)=(k_x,-t,\eta_1,\eta_2)=dJk+v\).
Changing either of the signs in (D.A.a) would break this identity.

Choose the transformation-groupoid arrow with range \((x,y+t)\) and translation parameter \(h\); its source is \((x-h,y+t)\). The actual principal anchor of \(g=(h_1,e)\) is
\[
 (x,y,\eta_1,\eta_2,t,h)\longmapsto
            (x-h,y+t,\ x,y,\eta_1,\eta_2).
 \tag{D.A.c}
\]
Its inverse on its open image is
\(x=z_1,\ y=z_2,\ \eta_i=z_{i+2},\
t=Y-z_2,\ h=z_1-X\),
where \((X,Y,z_1,z_2,z_3,z_4)\) are the output coordinates. The determinant in the displayed input and output orders is \(+1\); in particular the map is a local diffeomorphism. Every allowed arrow parameter is retained. The disk condition merely cuts out an open image. None of the rank, kernel or anchor calculations uses compactness of \(V\).

For the interval Hilbert module, a left coefficient with compact support meets finitely many buffered charts in this whole noncompact principal space. The compact errors are proved on that single interval module and extended to \(C_0(E)\) by the uniform contraction bound. Restricting this calculation to compact base manifolds, or checking endpoint operators independently, would not prove the claimed whole KK equality.

**Exercise 93 (12 points).** *radius tending to zero and actual \(C_0\) continuity.* On \(X=\mathbb R\) take the trivial real normal bundle and
\(a(x)=(1+x^2)^{-1}\). Write the full compression and its inverse. For a continuous positive radius \(a\), prove that the extension by zero of the pullback of \(f\in C_0(X\times\mathbb R)\) belongs to \(C_0(X\times\mathbb R)\). Finally let \(a_0,a_1\) be smooth positive radii and \(a_s=(1-s)a_0+sa_1\). Prove norm continuity of the whole homomorphism on each fixed \(f\), although neither radius has a global lower bound.

**Grading (12 points).** Compression, inverse and derivatives (4); continuous extension and compact level sets (4); whole interval and norm-continuity argument without a global radius bound (4).

**Solution.** The compression and its inverse on the disk \(|w|<a(x)\) are
\[
 w=\frac{v}{\sqrt{1+v^2/a(x)^2}},\qquad
 v=\frac{w}{\sqrt{1-w^2/a(x)^2}} .
 \tag{D.B.a}
\]
Their fibre derivatives are respectively
\((1+v^2/a^2)^{-3/2}>0\) and
\((1-w^2/a^2)^{-3/2}>0\).
Both are smooth; the derivative at the zero section is one. For the chosen \(a\), the disk narrows to zero as \(|x|\to\infty\). This does not impair its openness or the whole fibre diffeomorphism.

Define
\[
 (\phi_s f)(x,w)=
 \begin{cases}
 f\bigl(x,w/\sqrt{1-w^2/a_s(x)^2}\bigr),& |w|<a_s(x),\\
 0,& |w|\ge a_s(x).
 \end{cases}
 \tag{D.B.b}
\]
Near a finite boundary point of the disk, the inverse coordinate tends to fibre infinity and \(f\) tends uniformly to zero. Hence the extension is continuous. More exactly, for every \(\epsilon>0\) the closed level set
\(K_\epsilon=\{(x,v):|f(x,v)|\ge\epsilon\}\) is compact. Its compression image is exactly
\(\{(x,w):|\phi_s f(x,w)|\ge\epsilon\}\), and is compact. Thus \(\phi_s f\in C_0\); pointwise multiplication, adjoint and norm show that \(\phi_s\) is an isometric homomorphism onto the open disk's ideal.

For the interval, compress the compact set \(I\times K_\epsilon\) by
\((s,x,v)\mapsto(s,x,q_{a_s}(v))\). It is compact and is the \(\epsilon\)-level set of the same extended function on \(I\times X\times\mathbb R\). The inverse formula proves continuity in the open disk; compactness of \(I\times K_\epsilon\) proves continuity at its boundary, since a sequence with value bounded below cannot escape this compressed compact set. Therefore the whole function belongs to \(C_0(I\times X\times\mathbb R)\).

For an explicit norm-continuity argument, first approximate \(f\) by a compactly supported \(f_0\). On the compact projection of its support the radii have a positive common minimum and finite maximum. Its compressed support remains in one compact subset of the interval disk. The above continuous extended function is uniformly continuous on a common compact neighborhood and zero off that neighborhood, so
\(\|\phi_s f_0-\phi_{s'}f_0\|_\infty\to0\).
The bound \(\|\phi_s\|=1\) extends this to \(f\). The minimum is taken only on the compact support under consideration, never on all of \(X\). This is the whole-module support mechanism of Lemmas 6.0t and 8D.2.

**Exercise 94 (12 points).** *both inverses, full leaf lines and the sign.* In the setting of Theorem 8D.8, write both native inverse calculations for \(j=0,1\). Repeat them after a full graded leaf line \(D\) is chosen. For \(H=\mathbb R^2\), determine what goes wrong if the literal chronological inverse \(D_H\) is used in place of the physical inverse.

**Grading (12 points).** Both integral inverse calculations in both degrees (4); full graded leaf-line evaluation and compensated carrier (4); exact dimension-two sign defect and its integral detection (4).

**Solution.** Put \(\delta=s_dD_H\), \(p=j+d\bmod2\). The carrier \(\Gamma_p\) has closed compact source and includes its full native orientation, not just a stable tangent rank. The correct inverse is \(I\Gamma_p(y\delta)\). The two exact calculations are
\[
 \mu I\Gamma_p(y\delta)
     =(\kappa\Gamma_p(y\delta))\ell
     =y\delta\ell=y,\qquad
 I\Gamma_p((\kappa Rc)\ell\delta)
     =I\Gamma_p(\kappa Rc)=IRc=c .
 \tag{D.C.a}
\]
The first uses \(\delta\ell=1_A\), and the second uses \(\ell\delta=1_{C_0(V)}\), together with the two different carrier identities. Both degrees have the same calculations; the carrier degree is \(p\).

For a graded line, use \(\ell_D=[D]\ell\),
\(\delta_D=\delta[D^*]\),
\(\kappa_D=\kappa[D^*]\) and
\(\Gamma_D(x)=\Gamma(x[D])\).
The evaluation \(D^*\widehat\otimes D\to\underline{\mathbb C}\) is even, isometric and onto, and also retains the odd line grading. Thus
\[
 \ell_D\delta_D=[D]\ell\delta[D^*]=1,\qquad
 \delta_D\ell_D=\delta[D^*][D]\ell=1,\qquad
 I\Gamma_D(y\delta_D)=I\Gamma(y\delta).
 \tag{D.C.b}
\]
The orientation composition includes the same actual line evaluation and leaves the original native orientation line untouched. An order-two line is not erased by squaring its determinant.

When \(d=2\), \(s_d=-1\); the physical leaf map is
\(\ell=-\Theta_H\), with inverse \(-D_H\). Using \(D_H\) yields
\[
 D_H\ell=-1_A,\qquad \ell D_H=-1_{C_0(V)} .
 \tag{D.C.c}
\]
The first purported inverse consequently sends \(y\) to \(-y\); the other composite sends the native cycle class \(c\) to \(-c\). These are not identities on integral K-theory in general. For the translation action on \(\mathbb R^2\), the group \(K_0(A)\cong\mathbb Z\) already detects the defect. The corrected \(-D_H\) gives both identities. The convention issue is therefore part of the exact comparison, rather than a dispensable caption sign.

The equations in this solution rely on the completed full R1 carrier proof and the whole noncompact KK and intermediate-composition proofs; the exercise is a check of their use, not an independent replacement for them.

### Coefficient and convention exercises

Exercises 95–100 use the AT locators in Section 8E and test the full module arguments.

**Exercise 95 (12 points): A suspension product on an infinite coefficient module.**

Let \(s=s^*\) be a contraction on \(H_A\) with \(a=1-s^2\in\mathcal K(H_A)\). Prove that (AT4) defines one Kasparov product over \(SA\). Explain why replacing \(\sqrt a\) by the identity in that formula would fail.

**Solution.** Continuous functional calculus gives the denominator and its inverse on every bounded parameter interval. The two numerators anticommute, so the square defect is \(-a/(1+\tau^2a)\); its norm is bounded by \(\min(\|a\|,\tau^{-2})\). It belongs to \(C_0(\mathbb R,\mathcal K(H_A))\). The difference from the lower operator is (AT6). Each coefficient at a finite parameter is a continuous function of \(a\) vanishing at zero, hence compact. Multiplication by a creation vector in \(C_0(\mathbb R)\widehat\otimes C_1\) gives a compact-valued section vanishing at infinity. Its adjoint does too. The first-generator anticommutator is the positive operator (AT7), so all sandwiches are positive. These are the actual connection and positivity conditions on the whole module.

For the proposed replacement take \(s=1\) and \(a=0\) on an infinite standard module. At \(\tau=1\), the difference of the proposed operator from \(\sigma_x\) includes \((1/\sqrt2-1)1\otimes\sigma_x\) and a nonzero \(1\otimes\sigma_y\). Multiplying by a nonzero scalar creation function supported near 1 does not make this an operator compact: evaluation at 1 leaves a nonzero matrix tensored with the infinite identity. The actual formula at \(a=0\) is exactly \(s\sigma_x\), so its connection error is zero. (4 points for the whole-module defect, 4 for connection and positivity, 4 for the counterexample.)

**Exercise 96 (12 points): Push the universal suspension homotopy through arbitrary coefficients.**

Prove that (AT15) induces a homotopy in \(S\otimes\mathcal K(H_A)\) for every cycle (AT3). Track its relative reference projection and both noncompact boundaries.

**Solution.** The imaginary terms in (AT15) have the same sign. Hence (AT16) gives a uniform lower bound for \(|z_v|\) whenever either \(|\tau|\to\infty\) or \(r\to\pm1\). In fact \(\|P(z)-e_{11}\|=(1+|z|^2)^{-1/2}\), obtained by the two eigenvalues of the self-adjoint trace-zero difference. Thus the difference from the single fixed reference vanishes jointly on both boundaries, uniformly in \(v\). All projections extend by that same reference.

Since \(q(s)^2=1\), every function \(f\) vanishing at \(\pm1\) has \(q(f(s))=0\). The map \(\vartheta_s\) in (AT17) therefore has range in the actual compact ideal. Tensor it with \(S\) and the interval, then apply it to the entire relative-projection homotopy. The reference is the identity multiplier corner and is not incorrectly claimed an algebra element. The result is a relative homotopy over \(C([0,1],S\otimes J)\). Even compact stability gives its class over \(SA\). This is valid for noncommutative and nonunital \(A\) and preserves any integral torsion class. (4 points for joint boundaries, 4 for the compact-ideal map, 4 for relative coefficients and stability.)

**Exercise 97 (12 points): Determine both historical Thom signs.**

Assume the same \(U_t\) is written in the two forms (AT1). Starting with compression on a smooth finite projection, prove both signs in (AT25), rather than guessing the odd sign from the scalar even class.

**Solution.** The bounded off-diagonal perturbation (AT19) makes the whole operator commute with the projection. The full two-pole resolvent estimate (AT20) gives a compact continuous bounded-transform perturbation; hence the actual finite-projection product is the compressed course cycle. Its positive exponential is homotopic to \(b_+(eDe)\). The historical compression is \(b_+(eHe)=b_+(eDe)^{-1}\), so it is the negative class. Unitization followed by reference subtraction treats a relative projection. This proves the even sign.

Native left suspension commutes with the native Thom product by associativity, as in (AT23). Historical suspension is positive native left suspension on even input and negative on odd input by the full homotopy of Exercise 96. Therefore it anticommutes with native Thom, equation (AT24). The historical defining square commutes. In degree one its two negatives cancel, and cancelling the suspension isomorphism gives \(\phi_\alpha^1=N_\alpha^1\). Thus the signs are \((-1,+1)\). (5 points for the compressed full module, 4 for the typed square, 3 for the odd cancellation.)

**Exercise 98 (12 points): Compute the iterated signs and both inverses.**

For an ordered solvable chain of length \(d\), calculate the historical map relative to the literal native Thom action and to the physical framed leaf action in each input degree. Prove both inverse identities in the translated presentation.

**Solution.** Each step has sign \((-1)^{j+k+1}\). Their product is \((-1)^{dj+d(d+1)/2}\), which is \((-1)^{d(j+1)}s_d\). The resulting table, with columns “relative to native \(\Phi\)” and “relative to physical \(\ell_{\rm fr}\)”, is

| \(d\bmod4\) | even / odd relative to native | even / odd relative to physical |
|---|---|---|
| 0 | \(+,+\) | \(+,+\) |
| 1 | \(-,+\) | \(-,+\) |
| 2 | \(-,-\) | \(+,+\) |
| 3 | \(+,-\) | \(-,+\) |

The encoded forward class is \(\widetilde\Theta=(-1)^d\ell_{\rm fr}\), and its translated inverse is \(\delta\). The product law introduces \((-1)^{d^2}=(-1)^d\), so both compositions are exactly the native identities in (AT34). On K-groups the inverse on degree \(k\) is \((-1)^{kd}\delta_*\); for \(k=j+d\) its sign cancels the forward \((-1)^{d(j+1)}\). The carrier parity remains \(j+d\), not a parity inferred from the scalar input. (4 points for iteration and table, 4 for both typed products, 4 for group degrees and carrier.)

**Exercise 99 (12 points): Reflect a whole odd leaf frame.**

Why is \(\Gamma\) an invalid right-module intertwiner before contracting a retained odd Clifford coefficient? Give the correct full reflection argument, keeping a graded Hermitian line.

**Solution.** For a homogeneous right odd coefficient \(\varepsilon\), \(\Gamma(\eta\varepsilon)=-\Gamma(\eta)\varepsilon\). Thus \(\Gamma\) is not right-linear there. Reflection pulls back the entire right action and inner product by \(r(\varepsilon_i)=-\varepsilon_i\); (AT31a) is a typed onto coefficient-extension map to that transported module. After pair reduction an odd rank retains one reflected right \(C_1\). Ordinary right multiplication by its central \(\varepsilon\) is right-linear and preserves inner products. Viewed from the grading-reversed source it is even, commutes with the represented left coefficients and with the operator, and sends \((\Pi E,-F)\) to \((E,-F)\). This proves the inverse class. Every line transition and the entire line module remain tensor factors; the same map commutes with the even left representation. In even rank the global even Clifford volume gives the intertwiner and the chirality is preserved. The normal spinor and its cancellation factors are pulled back at the same time, before the actual geometric product. (4 points for the failed shortcut, 4 for the typed odd map, 4 for full lines and normal orientation.)

**Exercise 100 (12 points): Extend the assertion to a nonseparable algebra.**

Explain how the historical K-group identities cover an arbitrary point-norm continuous real action without asserting a countably generated KK cycle over a non-sigma-unital algebra.

**Solution.** Finite representatives and compact parameter homotopies have countably many needed coefficient values. Generate a separable C*-subalgebra from these and their rational translates. Norm continuity makes it invariant under all real translates. Such invariant subalgebras are directed. For the amenable real group, the reduced regular crossed-product inclusion is faithful and full equals reduced; compactly supported coefficient kernels are approximated by kernels with finitely many such coefficient values. The full crossed product is their directed norm closure. Stable K-theory continuity, together with equivariant naturality on each separable stage, passes all historical suspension and Thom identities to the limit. No unsupported module-size assertion is used. (4 points for invariant stages, 4 for crossed-product inclusion and density, 4 for K-theory continuity and scope.)

### Scalar graph-cycle exercises

**Exercise 101 (basic; 12 points).** Does compactness of (D_B−i)⁻¹ imply compactness of (D_0−i)⁻¹ when K is infinite-dimensional? Explain exactly what the proof uses.

**Solution.** No. For a normalised eigenvector v of D_B, its images on an infinite orthonormal family e_j⊗v have equal nonzero norm and are orthogonal. Localization by k⊗b with k∈K(K) instead gives k⊗b(D_B−i)⁻¹, a tensor of two compact operators. Finite tensors and norm density prove the full J-localized statement. P and R only have to multiply J on both sides; J need not be an ideal of B(K⊗H_B).

*Grading:* 4 points for the infinite-multiplicity counterexample, 4 for the two compact factors, 4 for the complete localized algebra.

**Exercise 102 (intermediate; 12 points).** Prove SG.15 and its constant without applying D_G directly to C_a.

**Solution.** Write Q_±=(D_G∓i√s)⁻¹. For η in the Hilbert space, Q_±η is in Dom D_G and a preserves that domain. Applying D_G∓i√s to aQ_±η gives aη+C_aQ_±η. Thus [Q_±,a]=−Q_±C_aQ_±. Average the two equations and use Q_±=D_GR_t±i√s R_t. The mixed imaginary terms cancel and the result is sR_tC_aR_t−D_GR_tC_aD_GR_t. The two bounds are ||C_a||/s and ||C_a||/(4s), respectively. No expression D_GC_a is used.

*Grading:* 4 points for domain preservation and the first-order resolvents, 4 for the bounded identity, 4 for both norm estimates.

**Exercise 103 (advanced; 12 points).** Explain why the odd physical Pauli model in SG.4 still has degree three after the even tree product, including the final Clifford generator.

**Solution.** SG.2 contracts only the first two generators of the full ΠC₃ module. The remaining right algebra is C₁, with grading −σ_zΓ_{C₁}; on its even basis the physical operator followed by right ε is −iσ·X. Order the total graded module as H_T⊗̂(H_N⊗C₁), with total grading Γ_TΓ_{C₁} and odd operator D_T⊗1+Γ_T⊗D_N⊗L_ε. On its even subspace, follow the operator by right ε. The resulting ordinary odd operator is D_N⊗Γ_T+1⊗D_T. Thus the tree has degree zero, its kernel is positive, and the final C₁ remains accounted for. Full rank-three degree is retained by the ordered even-pair Morita contraction, not by a parity label alone.

*Grading:* 4 points for the full shifted rank-three module, 4 for the retained final Clifford factor, 4 for the even tree product and positive kernel.

**Exercise 104 (advanced; 12 points).** Why does a null represented induced operator force the original compact-module operator to vanish?

**Solution.** T_η*Pi_A(a)T_ξ=Pi_B(⟨η,aξ⟩). Faithfulness of Pi_B forces every inner product to be zero. Taking η=aξ yields the Hilbert-module norm of aξ equal to zero. Hence aξ=0 for every ξ and a=0. This proves exact faithfulness after graph induction, independently of an abstract Morita slogan.

*Grading:* 4 points for the creation-map identity, 4 for faithful transverse representation, 4 for the Hilbert-module norm argument.

**Exercise 105 (advanced; 12 points).** A maximal spin-c class has a nonzero Kazhdan-projection pairing and therefore cannot descend reduced. Has the general graph-Dirac problem been refuted?

**Solution.** Only that specified descent has been refuted. The source asks for a reduced normalized scalar graph class. A different reduced class could pull back to a different maximal class while having the same local Bott pairing. A refutation must exclude every admissible reduced normalized class, or show the local transverse K-class cannot pair as one. It must also exhibit a genuine holonomy groupoid satisfying all original hypotheses with every prerequisite proved. The conditional quotient lemma supplies neither missing step.

*Grading:* 4 points for the exact quotient obstruction, 4 for the distinction between specified descent and general existence, 4 for the remaining full-hypothesis construction.

**Exercise 106 (advanced; 10 points).** Whole Bergman tail. For a bounded \(h\) supported in \(|z|\le R<1\), compute a finite-rank approximation error for \(M_hB\), retaining both signs.

**Solution.** Each normalized \(b_n^\pm\) has local squared norm \(R^{2n}\). Summing the squared norms of both tails gives \(2\|h\|_\infty^2 R^{2N+2}/(1-R^2)\); its square root bounds the operator norm by the Hilbert–Schmidt norm. The head is finite rank. This proves full compactness by norm approximation.

*Grading:* 4 points for the exact integral, 4 for the two-sector geometric sum and2 for the operator-norm conclusion.

**Exercise 107 (advanced; 10 points).** Why the Stinespring complement is an ideal module. Prove the exact inner product in (HF.20) lies in \(I\) and explain the compact creation commutator.

**Solution.** Applying \(q\) to both \(s(a^*c)\) and \(s(a)^*s(c)\) gives \(a^*c\), so their difference lies in \(I\). Therefore \(u e_n\to u\) for an \(I\)-approximate identity. The commutator is \(T_u(F_1-G_1)\), approximated in operator norm by \(T_u\pi_I(e_n)(F_1-G_1)\); each approximant is compact.

*Grading:* 4 points for the quotient identity, 3 for ideal-module convergence and3 for the two-sided compact argument.

**Exercise 108 (advanced; 10 points).** Cancellation and the projector. Verify the parity and square of (HF.10), then prove \(z=0\) from \(z=q^*w\), (C.4), (C.5) and \(rz=0\).

**Solution.** Opposite exterior gradings make the whole-space exchange odd. It anticommutes with \(\operatorname{diag}(F_H,-F_H)\), so the mixed term vanishes and the square defect is the cosine-squared old defect. The boundary zero-\(I\) summand stays degenerate. Product interchange yields \(\gamma z=q^*(\sigma_Q(\gamma)\otimes_Qw)=q^*w=z\). Thus \(z=Lrz=0\).

*Grading:* 4 points for grading/anticommutation, 2 for the complete endpoint and 4 for the correctly typed projector argument.

**Exercise 109 (advanced; 10 points).** Finite-center descent and literal sign. Describe why the central projection preserves a whole homotopy and compute the literal first product.

**Solution.** The center acts trivially on functions, so \((1+U(-1))/2\) commutes with the representation; central averaging makes it commute with the odd operator. Restriction to its complemented interval submodule preserves compact fields and endpoints. Separately \(T\delta_S=T(-\delta_{\mathrm{inv}}[D])=-(T\delta_{\mathrm{inv}})[D]=-[D]\), with the determinant factor on the prescribed side.

*Grading:* 6 points for the full module/endpoints and 4 for the ordered sign/line calculation.



**Exercise 110 (advanced; 10 points).** Graded mixed-resolvent sign. For a homogeneous compact smooth Clifford section \(p\), put \(\epsilon=(-1)^{|p|}\) and \(K_p=Q\rho(p)-\epsilon\rho(p)D\). Derive the exact mixed resolvent identity and prove its integrable connection bound.

**Solution.** On the full graph domain,
\[
(Q-z)\rho(p)=\epsilon\rho(p)(D-\epsilon z)+K_p.
\]
Multiplication by the two resolvents gives
\[
R_Q(z)\rho(p)-\epsilon\rho(p)R_D(\epsilon z)
=-\epsilon R_Q(z)K_pR_D(\epsilon z).
\]
For \(z=\pm i\lambda\), its norm is at most \(\|K_p\|/\lambda^2\). The compact \(Q\)-resolvent makes each integrand compact, and \(\lambda=\sqrt{1+t^2}\) gives a norm-integrable tail. Sum both poles, exchanging them when \(\epsilon=-1\). This proves the full graded creation connection of WP.24–WP.27.

*Grading:* 4 points for parity and sign, 3 for the actual bounded graph-domain middle, and 3 for the norm-compact integral.

**Exercise 111 (advanced; 10 points).** Positive strong limit and compact correction. Explain how the positive strong integral in WP.34–WP.37 proves positivity modulo compacts when its integrand has only an \(O(\lambda^{-1})\) bound.

**Solution.** With \(a=-\eta\) and \(C=\{a,Q\}\ge-2\), replace \(C\) by \(C+2\). Both sandwiches \(R_+(C+2)R_-\) and \(R_-(C+2)R_+\) are positive on the actual graph domain. Their truncated integrals increase and are uniformly bounded by the anticommutator of bounded spectral truncations plus the correction \(2(Q^2+\lambda^2)^{-1}\). They therefore have a bounded positive strong limit. The correction is compact, has norm at most \(2/\lambda^2\), and its integral converges in norm to the compact operator \(2(1+Q^2)^{-1/2}\). Thus the exact bounded anticommutator is positive modulo the whole compact ideal.

*Grading:* 4 points for the positive sandwiches, 3 for the bounded monotone strong limit, and 3 for the distinct norm-compact correction argument.

**Exercise 112 (basic, 6 points).** On the single-circle-leaf graph, explain why taking only the normal Dirac on the entire graph fails scalar localized compactness, while the one-source-column cycle with zero operator is normalized.

**Solution.** The normal dimension is zero. On the full graph the zero operator has localized resolvent \(p\otimes1_{L^2(S^1_y)}\); the vectors \(1\otimes e^{iny}\) are orthonormal and their images remain orthonormal, so it is not compact (3 points). On the one column the reduced algebra acts as the compact operators on a purely positive Hilbert space; the square defect of the zero phase is the represented compact algebra itself, and all other defects vanish. The rank-one corner has one positive vector and no negative vector, hence pairing one (3 points). This is a scalar cycle in \(KK^0(A,\mathbb C)\), not a longitudinal coefficient-algebra cycle.

**Exercise 113 (advanced, 12 points).** In (GD.4), why do bounded commutators with every smooth convolution kernel force the source coefficient to be independent of the range variable? Why does the compressed first-order chiral index then equal zero?

**Solution.** Integrate the range derivative by parts; it leaves a bounded smooth integral kernel. The remaining differentiated input has coefficient \((B(x,y)-B(z,y))k(x,z)\). If this is nonzero, a smooth input multiplied by \(e^{iny}\) gives output norm at least \(|n|\delta-O(1)\), contradicting boundedness. Kernels supported near any prescribed pair imply \(B(x,y)=B(z,y)\) (6 points). The normalized constant range column compresses the operator to \(-iB(y)\partial_y+\overline C(y)\). Its odd invertible source Clifford block is \(-i\beta(y)\partial_y+c(y)\); multiplication by the invertible \(-i\beta\) reduces its index to that of the circle derivative plus a compact zero-order term \(H^1\to L^2\). The derivative has equal constant kernel and cokernel, so the index is zero (6 points). Compression represents the original K-pairing by the bounded off-diagonal removal and compact transform comparison of Section 11C.5.

**Exercise 114 (advanced, 12 points).** Show why the normal spin-c structure in Proposition 11C.3 does not supply a full graph determinant line. Does this settle nonexistence of every normalized reduced graph class?

**Solution.** A line on the conjugation mapping torus restricts to \(kh\) on a projective fibre. Interval transport and conjugation gluing force \(kh=-kh\), so \(k=0\) (4 points). The stable tangent identity \(T\mathbb{CP}^2\oplus\mathbb C\cong(H^*)^{\oplus3}\) gives tangent spin obstruction \(h\bmod2\); a determinant line would require odd \(k\), a contradiction. Restricting the graph tangent to one endpoint slice adds only a framed trivial factor and gives the same contradiction (4 points). The normal bundle is still trivial of rank zero. The proposition only rules out that full tangent spin-c lift and its minimal spinor proposal. Larger Clifford modules, other graph completions, and normal-symbol operators require separate analysis; Section 11C.7 explicitly supplies a normalized reduced column class in the circle test. No universal nonexistence conclusion follows (4 points).

**Exercise 115 (Finite stabilizers and the reduced representation, 10 points).**

Let a discrete group Gamma act on a locally finite tree with finite vertex stabilizers, allowing edge inversions. Prove that the edge stabilizers are finite. For an orbit Gamma/H construct an explicit equivariant isometry into the regular representation. Explain why this proves reduced covariance after tensoring with a compact normal implementing representation, even when that normal representation alone does not descend to the reduced crossed product. State precisely what fails for infinite H.

**Solution.** An unoriented edge stabilizer maps to its permutation group on the two endpoints. Its kernel fixes both endpoints and is contained in the finite stabilizer of either endpoint, while its image has at most two elements. Thus its order is at most twice the endpoint stabilizer order. For finite H use

\[
 J_H\delta_{gH}=|H|^{-1/2}\sum_{a\in H}\delta_{ga}.
\]

The definition is independent of representative because gH is unchanged. Distinct cosets have disjoint supports and every image vector has norm one; hence this is an isometry. Left translation takes the gH vector to the hgH vector, literally intertwining the two representations. Tensor with the full normal Hilbert space, whose coefficient multiplication is the identity on the tree variable. Both multiplication and U_h tensor L_h preserve the embedded subspace and agree there. On each regular copy the unitary (W psi)_g=U_g inverse psi_g turns the group action into the regular shift and coefficient multiplication into f(g n). These are the defining reduced representations integrated over normal points with the full finite spin coefficient. Consequently the embedded representation is contractive for the reduced norm. No assertion about the compact normal implementation alone is used. For infinite H the displayed constant coset vector is not in ell2 Gamma and there is no normalization by a finite nonzero |H|; this argument then supplies no embedding or reduced norm bound.

**Rubric.** Edge-stabilizer kernel and image: 2; well-defined exact isometry: 3; equivariance and coefficient embedding: 2; actual absorption/reduced inequality: 2; infinite-stabilizer limit: 1.

**Exercise 116 (The weighted rooted-tree operator, 12 points).**

Define the vertex-to-parent-edge map Q and the distance weight M on the entire tree. Prove its scalar even index and compact-resolvent property, and prove the estimate

\[
 \|QM L_h^V-L_h^E QM\|
 \le\ell(h)+2\ell(h)\sqrt{\ell(h)+1},\qquad
 \ell(h)=d(o,ho).
\]

Include domain preservation. Explain why a finite displayed tree is not itself a proof of the infinite index or compactness.

**Solution.** Every edge has exactly one endpoint farther from o. Thus Q maps every nonroot vertex bijectively to its parent edge, kills the root and is onto. Its kernel is the positive one-dimensional root and its adjoint kernel is zero, giving Index Q=1. The self-adjoint operator with off-diagonal blocks QM and \(MQ^*\) is the orthogonal sum of the root zero block and the matrices [[0,n],[n,0]] for every nonroot vertex of depth n. Its domain consists exactly of the vectors whose block images are square summable. Finite block sums form a core. Local finiteness makes every bounded radius ball finite, so only finitely many blocks occur below a fixed depth; the square-resolvent norm on the remaining blocks is at most 1/(1+(R+1)^2). This proves compactness on the whole tree.

For an input vertex v the lower commutator is n(hv) delta_c(hv) minus n(v) delta_hc(v), with a root term zero. Outside [o,ho], the first step of hv toward roots o and ho is the same, so its two edge labels agree. Their coefficient difference is at most ell(h), and the common edge map is injective. The norm of that partial weighted permutation is at most ell(h). The exceptional set has at most ell(h)+1 inputs; both depths are at most ell(h), giving column norm at most 2ell(h). Its Hilbert–Schmidt bound is the second term in the estimate. The upper block follows by adjointing the lower equation for h inverse and multiplying by unitaries. To prove domain preservation, approximate a domain vector by finite blocks. The commutator identity and its bound make L_h of the approximation Cauchy in the graph norm. Closedness gives L_h Dom D_T subset Dom D_T; applying h inverse gives equality. A finite displayed tree supplies only a picture or finite consistency check: the bijection, finite-ball tail bound, closed domain and all h commutators prove the infinite assertions.

**Rubric.** Whole bijection and index: 3; self-adjoint domain/core: 2; finite-ball compact tail: 2; segment and weighted estimate: 3; domain preservation: 1; scope of finite checks: 1.

**Exercise 117 (Noninvariant connections, full phases and local pairing, 14 points).**

Suppose the normal Gamma action preserves the full metric spin-c structure but not the chosen compatible connection. Prove the bounded normal return commutator and the total product commutator. For Gamma=F2 times C2 acting on S1 times SU(2) as in PT.27 with central lift (-1)^epsilon, determine the complete rank-four inverse sign and phase. Explain why the resulting graph class has the original transverse pairing one, and identify the two still missing general interfaces.

**Solution.** The full inverse lift intertwines Clifford multiplication, and the normal map is an isometry. The transported Dirac therefore has the same principal symbol as the chosen Dirac. Their difference has no derivative of the input; it is a smooth finite-matrix zeroth-order term on compact N. This proves B_h=[D_N,U_h] bounded. Smooth pullback with smooth inverse preserves H1. The graded total operator D_B=D_T tensor 1 plus Gamma_T tensor D_N has square D_T squared plus D_N squared. Its return commutator is [D_T,L_h] tensor U_h plus Gamma_T L_h tensor B_h, bounded by PT.8 and the preceding cancellation. No invariant averaging of connections is used.

In rank four k=4 times 3/2=6 is even. The full inverse module is C4, with its actual grading and physical generators (-1)^(6+1)t_j=-t_j. The original phase z=(-1)^epsilon becomes z inverse=z. Its determinant phase z squared is one; replacing the full phase by that determinant would lose the specified sign transport. The tree cycle is even, has index +1, and changes no transverse degree or inverse sign. Its restriction product with the normal class has compact creation errors from the fixed-depth matrices and positive anticommutator 2|D_T|/sqrt(1+D_T squared+D_N squared). Thus its restriction is the **full** normal inverse class. The normalized plaque correspondence is \(E^*\) to pA, overline eta mapped to theta_xi,eta; its counting and bump norms are both one. It therefore carries the original source Bott class to \(i_*b_U\), whose pairing is the calibrated normal pairing +1. A root count without these product and correspondence checks would not prove this assertion.

The general spin-c foliation premise still does not supply a proper locally finite tree action or another suitable index-one reduced auxiliary for an arbitrary return groupoid. The present weighted discrete operator has also not been identified with a local finite-order differential operator in smooth graph coordinates. This is a limitation of that proposed local replacement. A suitable general graph realization with its normal geometry remains unproved; the finite-stabilizer example alone does not supply it.

**Rubric.** Principal-symbol cancellation, bound and domain: 3; total square/commutator: 2; rank-four module, sign and retained phase: 3; full product/restriction: 2; actual plaque correspondence and pairing: 2; both exact remaining limits: 2.

**Exercise 118 (Proper weights at infinite valence, 8 points).**

Prove (CT.4), (CT.6) and the finite sublevel formula (CT.8). Explain why graph distance fails in the attached-leaf tree, and why changing the weight does not change the source Bott normalization.

**Solution.** For a minimum-cost a sending v_j to v, ha sends v_j to hv with cost at most ell(h)+ell(a). Apply h inverse for the reverse inequality. Parent edges disagree only at the d_h+1 preimages of the finite root displacement segment. Off it the coefficient difference is bounded by ell(h) and the output edge map is injective. On it every column has norm at most C_h, giving Hilbert–Schmidt bound C_h sqrt(d_h+1). These prove (CT.4) and (CT.6). In the exact attached-leaf tree, base vertices with weight at most R are counted by B(R); for each j between one and R, its attached leaves are counted by B(R-j). The displayed sum is therefore exact and finite. Distance one at infinitely many attached leaves creates infinitely many orthogonal blocks at the same finite eigenvalue, whereas the adapted weight assigns eigenvalues with magnitude j and finite sublevels. The phase Q and its positive unmatched root do not change with a positive proper weight. The full product creation and positivity calculation uses that same phase and retains the entire normal inverse, so the full restriction class and original Bott pairing remain +1.

**Rubric.** Two-sided cost estimate: 2; segment/injectivity and finite-column bound: 2; exact finite counting and failure of distance: 2; complete phase/product normalization distinction: 2.

### Translation oscillators and the actual graph class: Exercises 119–121 (50 points)

These original exercises use the exact notation and completed domains of TO.1-TO.26. Their solutions retain all analytic and reduced-norm interfaces; a finite matrix experiment alone earns no completeness or compactness points.

**Exercise 119 (The whole oscillator, including its completion, 16 points).**

On \(L^2(\mathbb R^d,\Lambda^*\mathbb C^d)\) define \(D=\sum_j\bigl(\epsilon_j(\partial_j+x_j)+\iota_j(-\partial_j+x_j)\bigr)\). Compute \(D^2\), prove its self-adjoint domain and compact resolvent, identify its kernel and scalar index, and compute both translation identities for \(T_m\psi(x)=\psi(x-m)\). Explain the adjoint in its dimension-one bounded transform.

**Solution.** The exterior canonical anticommutation relations give \(C_j=\epsilon_j-\iota_j\) skew-adjoint, \(E_j=\epsilon_j+\iota_j\) self-adjoint, their squares -1 and +1, and cross anticommutators zero. In coordinate \(j\), \(a_j^*a_j=-\partial_j^2+x_j^2-1\) and \(a_ja_j^*=-\partial_j^2+x_j^2+1\). Occupied wedges select the second and empty wedges the first. Thus \(D^2=-\Delta+|x|^2+2N_E-d\), with no mixed coordinate term. The empty Gaussian \(\phi_0=\pi^{-d/4}\exp(-|x|^2/2)\) is annihilated by every \(a_j\).

Normalized creation powers give \(\phi_\nu\) and \(a_j\phi_\nu=\sqrt{2\nu_j}\phi_{\nu-e_j}\), \(a_j^*\phi_\nu=\sqrt{2(\nu_j+1)}\phi_{\nu+e_j}\). Integration by parts proves orthonormality. Their polynomial factors span all polynomials. Orthogonality to them would make all Taylor coefficients of \(\int f(x)e^{-|x|^2/2}e^{z\cdot x}\,dx\) vanish. Gaussian Cauchy-Schwarz bounds make this entire, so its imaginary-axis Fourier values vanish. Convolution with a mass-one Gaussian is then zero by the elementary Gaussian Fourier identity and Fubini, and the Gaussian approximate identity gives \(fe^{-|x|^2/2}=0\). This proves completeness.

On the finite space with \(|\nu|+|I|=k\), \(D\) is Hermitian and \(D^2=2k\). Its direct-sum self-adjoint domain is \(\sum_k2k\|\psi_k\|^2<\infty\), and finite Hermite sums are a graph core. For compact smooth sections, integration by parts and Parseval give \(\|D\psi\|^2=\|\nabla\psi\|^2+\|x\psi\|^2+\langle\psi,(2N_E-d)\psi\rangle\). Cutoff followed by local \(H^1\) mollification makes compact smooth sections dense in \(Q=\{\psi\in H^1:|x|\psi\in L^2\}\). The same coefficient identity shows both that \(Q\) lies in the weighted domain and that its finite Hermite approximants converge in \(Q\); hence the domains are equal and compact smooth sections are a second core.

The square-resolvent tail above level \(K\) is at most \(1/(1+2(K+1))\), and each retained level is finite. Therefore the resolvent is compact on the full space. \(k=0\) is precisely the even Gaussian line; all other odd invertible blocks have equal positive and negative ranks. The normalized positive-to-negative block has kernel dimension one, cokernel zero and lower bound \(\sqrt{2/3}\) on its complement. Its index is +1.

Finally derivatives commute with \(T_m\) while \(x_jT_m-T_mx_j=m_jT_m\). Thus \([D,T_m]=c_+(m)T_m\), with norm \(|m|\), and \(T_mDT_m^{-1}=D-c_+(m)\). \(T_m\) preserves the completed \(H^1\)/weighted domain since \(\|xT_m\psi\|\le\|x\psi\|+|m|\|\psi\|\), with the inverse translation proving equality. In dimension one, \(D\) has lower block \(a=x+\partial\) and upper block \(a^*=x-\partial\). With \(H=-\partial^2+x^2\), \(D^2=\operatorname{diag}(H-1,H+1)\), so \(D(1+D^2)^{-1/2}\) has lower \(aH^{-1/2}\) and upper \(a^*(H+2)^{-1/2}\). The latter is the adjoint of the lower by the creation/annihilation commutation, and is not \(a(H+2)^{-1/2}\).

**Rubric.** Exterior signs and exact square: 3; normalized basis plus completeness: 4; closed domain and both cores: 4; entire-space compactness/kernel/index: 3; translation signs/domain and bounded-transform adjoint: 2. Total 16.

**Exercise 120 (Genuine reduced completion and full normal normalization, 14 points).**

Prove the auxiliary representation factors exactly through \(C_r^*(\mathbb Z^d)\). Then supply the full inverse normal Dirac of Section 11E.9 and show that its covariant product is a reduced scalar cycle of \(C(N)\rtimes_r\mathbb Z^d\) with full normal restriction and local Bott pairing +1. Do not assume the compact-normal implementing representation itself is reduced, or discard an inverse central phase.

**Solution.** Partition \(\mathbb R^d\) into \(n+[0,1)^d\). The map \((J\psi)_n(y)=\psi(n+y)\) is onto and isometric by the disjoint Lebesgue partition; it conjugates \(T_m\) to \(\lambda_m\otimes1\). For every finite scalar group coefficient, the amplified norm equals its regular norm: use a Hilbert basis of the cube multiplicity space for the upper bound and a regular nearly norm-attaining vector tensor a cube unit vector for the lower. This proves exact reduced extension. The glued image \(JQ\) retains neighbouring-face \(H^1\) compatibility, so this norm identification invents no boundary condition.

Retain the normal bundle \(\Pi^{q(q-1)/2}C_q\), physical generators \((-1)^{q(q-1)/2+1}L_{t_j}\) and actual transition \(z^{-1}\rho_t(s)\). On \(H_B\) put \(D_B=D\otimes1+\gamma\otimes D_N\), \(f\) as \(1\otimes M_f\), \(u_m\) as \(T_m\otimes U_m\). The two odd summands anticommute, so the square is \(D^2\otimes1+1\otimes D_N^2\). Its energy/normal spectral direct sum is self-adjoint on the intersection domain, with finite Hermite/smooth-normal core and compact resolvent. A noninvariant normal connection is allowed: metric and coherent full Clifford lift cancel the principal symbol of \(U_mD_NU_m^{-1}-D_N\), leaving a bounded smooth zeroth-order term \(B_m\) after multiplication by \(U_m\). Therefore \([D_B,u_m]=c_+(m)T_m\otimes U_m+\gamma T_m\otimes B_m\) is bounded, and smooth multiplication commutators are bounded too. The resolvent identity bounds their phase integrands by \(\|C\|/(1+t^2)\), with compact terms; its norm-convergent integral proves the full scalar phase defects and density extends them.

After \(J\), absorption \((W\xi)_n=U_n^{-1}\xi_n\) gives \(\lambda_m\) on the group coordinate and multiplication \(f(h_ny)\) at coordinate \(n\). This is the regular induced representation of the faithful multiplication representation of \(C(N)\). Pointwise finite-support regular tests vary continuously in \(y\); an almost-maximal test therefore persists on positive-volume neighbourhoods. Hence the essential supremum in the direct integral equals the defining reduced supremum. This proves exact norm factorization. \(U_n\) and its inverse carry the actual central phase; the phase cancellation in this unitary does not trivialize the original line or alter the normal operator.

For the restriction, the auxiliary phase \(S\) is the positive Gaussian identity summand plus a degenerate odd-unitary complement. At each nonzero auxiliary energy \(k\), the product transform has matrix \((1+2k+\nu^2)^{-1/2}\begin{pmatrix}\nu&\sqrt{2k}\\\sqrt{2k}&-\nu\end{pmatrix}\). Its homogeneous creation connection differs compactly from the signed normal transform, since the off-diagonal entry and the diagonal difference tend to zero in the normal spectral tail. The root column is exactly the normal transform. Coefficients and dense creation vectors follow from the normal phase compact commutators. The exact product positivity is \(\{S\otimes1,F_B\}=2|D|/\sqrt{1+D^2+D_N^2}\ge0\). The product criterion of Section 0 proves restriction \([D_N]\), retaining the full right \(C_q\) and its ordered reduction. The supplied original local Bott calibration then gives pairing +1, rather than an unrelated global normal index.

**Rubric.** Literal regular unitary and exact norm: 3; normal square/domain/core/compactness: 3; arbitrary compatible connection and phase defects: 2; absorption, faithful reduced point norm and full phases: 3; complete creation/positivity restriction and original pairing: 3. Total 14.

**Exercise 121 (Actual graph realization and its precise limits, 20 points).**

For the germ-effective lattice suspension \((\mathbb R^d\times N)/\mathbb Z^d\), identify the actual holonomy graph and construct its first-order source-column oscillator with potential \(r-t\). Prove the exact domain, convolution defects and scalar localized compactness. Derive its relation to partition-induced auxiliary rather than identifying coordinates without proof. Explain why \(d\ge2\) extends beyond finite-stabilizer tree actions, why a trivial normal action changes the graph, and why the stated theorem does not construct the required normalized reduced graph representative for arbitrary eligible holonomy.

**Solution.** A horizontal lifted path joins \((r,n)\) to \((t,n)\). Common deck changes give \((t+m,r+m,h_mn)\); germ effectivity excludes further identity-germ return identifications. Thus \(G=(\mathbb R_t^d\times\mathbb R_r^d\times N)/\mathbb Z^d_{\rm diagonal}\), with source [\((r,n)\)] and range [\((t,n)\)]. The coefficient is \(\Lambda^*\mathbb C^d\otimes\) the full inverse normal bundle at \(n\), transported identically along the common lifted normal point and by its coherent inverse lift on deck changes. Quotient density is \(dt\,dr\,d\operatorname{vol}_N\).

With buffered lifts \(z_i\) and \(\sum_i\chi_i^2=1\), write \(m_i(t)=t-z_i(t\bmod\mathbb Z^d)\). The smooth family \(D_N^{\rm ind}(t)=\sum_i\chi_i^2U_{m_i}D_NU_{m_i}^{-1}\) is compatible and covariant under \(t+m\). Put \(\mathscr D=\sum_j[C_j\partial_{r_j}+E_j(r_j-t_j)]+\gamma D_N^{\rm ind}(t)\). Both terms descend under the diagonal action. In \(y=r-t\) its square is \(-\Delta_y+|y|^2+2N_E-d+(D_N^{\rm ind}(t))^2\). Its self-adjoint domain requires exactly \(r\) derivatives, normal \(H^1\) and weighted \((r-t)\psi\in L^2\), with no \(t\) derivative. The fibre spectral completions give a self-adjoint direct integral. Invariant \(y\) cutoffs followed by buffered local approximation in \(L^2(t;H^1(r,n))\) give smooth compact graph sections as a core.

Left convolution integrates \(t^\prime\) with source \((r,n)\) fixed. The \(r\) derivative commutes with the kernel. The potential difference is \(t^\prime-t\). The normal principal Clifford coefficients agree at the common \(n\), leaving only the normal derivative of the kernel and compatible-connection zeroth-order differences. Compact graph support bounds \(|t^\prime-t|\) and meets finitely many return charts. Both Schur integrals are finite, so the entire commutator is bounded. The smooth core maps into itself; graph-norm approximation and closedness prove preservation of the whole domain. The ordinary graph representation is a source-regular direct integral; compact smooth tests show its essential supremum equals the defining reduced supremum.

The undifferentiated \(t\) direction prevents a global compact-resolvent claim. On compact buffered range charts, each fibre resolvent is compact and depends continuously in compact-operator norm on \(t\). A convolution coefficient times that resolvent is an integral kernel in \((t,t^\prime)\) valued in compact fibre operators. Its coordinate substitution is \(y^\prime=r-t^\prime=y+t-t^\prime\); with \(T_\delta f(y)=f(y-\delta)\), the corresponding operator factor is explicitly \(T_{t^\prime-t}\), not \(T_{t-t^\prime}\). At the illustrated fixed-source sample, \(y=5/4\) and \(\delta=1/2\) give \(y-\delta=3/4\), agreeing with \(y^\prime\) and with the potential commutator \(t^\prime-t=+1/2\). Strong translation continuity becomes norm continuity after a compact operator. Uniform approximation by separated range functions times finite-rank fibre operators gives scalar compactness of \(a(\mathscr D-z)^{-1}\). The adjoint gives the other side. The Schur commutator kernel has the same property after a resolvent; the integrable two-resolvent phase identity then proves all bounded-phase compact defects. This explicitly supplies compactness in \(t\) through the kernel rather than pretending the fibre resolvent alone is globally compact.

The graph Hilbert space is unitarily the flat induced \(H_B\) bundle: \(\xi(t)(r,n)=\psi(t,r,n)\) and \(\xi(t+m)=(T_m\otimes U_m)\xi(t)\). Its partition isometry is \(\chi_i(T_{-m_i}\otimes U_{-m_i})\xi\). Conjugation of \(PD_0P\) is exactly \(W_r-c_+(b(t))+\gamma D_N^{\rm ind}(t)\), \(b(t)=\sum_i\chi_i^2m_i(t)\). \(b(t+m)=b(t)+m\), so \(b(t)-t\) is periodic and bounded. The desired \(\mathscr D\) differs by \(c_+(b(t)-t)\), a bounded self-adjoint zeroth-order endomorphism. Every interpolating expression has the same domain, compact localized resolvents and Schur commutators. The resolvent difference divided by \(1+\tau^2\) gives a norm-continuous phase homotopy. Thus it is the Morita-induced class, with full normal restriction and original local Bott pairing +1. Its symbol is elliptic only in \((r,n)\), and it never differentiates in range \(t\).

If \(\mathbb Z^d\) acts on a tree with finite vertex stabilizers, torsion-freeness makes these stabilizers trivial; endpoint permutation makes edge stabilizers finite and hence trivial as well, excluding inversions. A quotient spanning tree and lifts of its paths express the deck group freely on its non-tree edges. A reduced relation would lift to a nontrivial reduced closed path in the tree, impossible by uniqueness of paths. An abelian free group is trivial or infinite cyclic, so \(d\ge2\) is impossible. This is genuinely a different auxiliary family. In contrast, a trivial normal action is not germ-effective: all lattice loops have identity holonomy germ, and the actual holonomy graph is the pair groupoid of \(\mathbb T^d\) (times \(N\)), not the displayed covering graph with retained \(\mathbb Z^d\) labels. The confining coordinate \(r\) cannot be imported into that changed graph. Finally the theorem assumes compact isometric normal data and this specific lattice cover. It supplies neither those hypotheses nor an index-one auxiliary for arbitrary holonomy groupoids; this argument does not construct the required normalized reduced graph representative for arbitrary eligible holonomy.

**Rubric.** Actual graph/coefficient/density and covariance: 3; exact anisotropic domain/self-adjointness/core: 4; convolution cancellation, Schur bounds and completed-domain preservation: 3; scalar localized compacts, reduced norm and phase defects: 4; exact partition/unitary/bounded homotopy/original pairing: 4; tree exclusion, trivial-germ exception and retained general scope: 2. Total 20.

Total available: **16+14+20=50 points**. The normal Dirac, local calibration and graph-correspondence hypotheses are exactly those stated in Section 11E.


**Exercise 122 (Normal volume and a stronger ellipticity test, 10 points).**

1. From (LD.5)–(LD.6), compute the normal volume and prove that an invertible matrix anticommuting with all normal generators forces equal auxiliary ranks. Explain why a common spin-c line twist and a balanced stabilization cannot repair unequal ranks. (4 points)
2. For (LD.9), compute the positive-line transition using (LD.10), and the determinant transition when there are \(r_+\) positive and \(r_-\) negative copies. (3 points)
3. Prove that extending the Hermitian invertible normal symbol over one closed leaf hemisphere forces that determinant transition to have winding zero. State the exact conclusion and one excluded scope. (3 points)

**Solution.** Since \(q=2m+1\) is odd, the product of the auxiliary gradings is \(\gamma_E\). Hence \(\Omega=(-i)^m\prod_j(c_j\otimes\gamma_E)=1\otimes\gamma_E\). Moving an anticommuting invertible matrix through these \(q\) factors changes its sign, so it is an isomorphism between the two volume eigenspaces. Their dimensions are \(\dim S\,r_+\) and \(\dim S\,r_-\), giving equality of the ranks. A common line tensor preserves both ranks; adjoining \(k\) copies to both signs preserves \(r_+-r_-\). This proves part 1.

At the normal equator, \(v_N=2^{-1/2}(1,e^{i\theta})\) and \(v_S=2^{-1/2}(e^{-i\theta},1)=e^{-i\theta}v_N\). The orthogonal complement transforms by \(e^{i\theta}\). The determinant of the direct sum of \(r_+\) positive lines and \(r_-\) negative lines consequently has transition \(e^{-i(r_+-r_-)\theta}\), whose winding is \(-(r_+-r_-)\). This proves part 2, including its sign.

An invertible Hermitian symbol has a continuous positive spectral projection by the uniform gap on the hemisphere. Its range bundle is trivial on that ball: parallel transport using (LD.14) constructs a frame in the smooth case, and smooth approximation with projection distance less than one gives a bundle isomorphism in the continuous case. Its boundary determinant line is therefore trivial. A transition of a trivial line bundle is the quotient of nowhere-zero functions extending over the two normal discs; each has boundary winding zero by radial contraction. Thus \(r_+-r_-=0\). This proves part 3. The conclusion concerns full-source Hermitian ellipticity with the prescribed physical normal symbol. It does not exclude a nonelliptic anisotropic operator or a changed coefficient module.

**Exercise 123 (A tree edge and disjoint smooth supports, 8 points).**

1. For the rooted Cayley tree of \(F_2=\langle a,b\rangle\), calculate \(Q\delta_a\), \(Q\delta_{ab}\), and \(Q\delta_{ab^{-1}}\) in the positive-generator edge coordinates (LD.3). (3 points)
2. Choose a nonzero smooth column supported in \(a^{-1}z(W)\times N\). Show that its tree output has support in a disjoint leaf plaque. Prove that no finite-order differential expression, nor its conjugation by a pointwise bundle map and common base diffeomorphism, has this action. (3 points)
3. Explain why this argument does not apply to the bounded transform of a local operator, or to a Hilbert-space unitary which moves the negative coefficient to another base point. (2 points)

**Solution.** The parent of \(a\) is \(e\), so \(Q\delta_a=\delta_{(e,a)}\). The parent of \(ab\) is \(a\), so its edge is \(\{a,ab\}\), and \(Q\delta_{ab}=\delta_{(a,b)}\). The parent of \(ab^{-1}\) is also \(a\), but its edge is \(\{ab^{-1},a\}=\{ab^{-1},(ab^{-1})b\}\). Consequently \(Q\delta_{ab^{-1}}=\delta_{(ab^{-1},b)}\). Multiplication by \(M\) multiplies these three outputs by \(1,2,2\), respectively. This proves part 1.

In plaque coordinates use \(\phi(x)\delta_a\otimes v(n)\), where \(\phi\) has compact support in \(W\) and \(v\neq0\) is a smooth normal spinor. Formula (LD.4) sends it to \(\phi(x)\delta_{(e,a)}\otimes v(n)\), with the prescribed unitary normal transport. Thus the output is nonzero on \(z(W)\times N\), disjoint from the original \(a^{-1}z(W)\times N\). Every finite-order differential expression vanishes wherever its input vanishes on a neighbourhood. Pointwise bundle maps preserve supports, and a common diffeomorphism sends disjoint sets to disjoint sets. This proves part 2.

Functional calculus can make the bounded transform nonlocal even when its starting expression is differential, so support preservation was asserted only for that expression on its smooth core. A base-moving unitary changes which output point is called the same point as the input. Such a unitary is not a pointwise bundle change over the common source-column manifold. Both exceptions are outside the proposition. This proves part 3.

**Exercise 124 (Balanced ranks and index one, 10 points).**

1. Verify that the one-dimensional oscillator (LD.18) has leaf principal matrix \(\sigma_2\), and that this matrix anticommutes with all normal matrices \(c_j\otimes\sigma_3\). Give the exact two physical normal-volume dimensions when \(\dim S=2\). (3 points)
2. Solve both off-diagonal kernel equations in distributions, impose square integrability, and determine the scalar index using the Fredholm completion proved in Section 11E. (4 points)
3. For the finite regular-orbit Cayley-tree coefficient of \(F_r\), state which of the two obstructions survives at \(r=1\), which survives at \(r=2\), and why neither closes the arbitrary-holonomy normal spin-c problem. (3 points)

**Solution.** With the principal-symbol convention \(-i\partial_x\mapsto\eta\), the matrix of the derivative in (LD.18) is \(\sigma_2\eta\). Since \(\sigma_2\sigma_3+\sigma_3\sigma_2=0\), the matrix \(1_S\otimes\sigma_2\) anticommutes with every \(c_j\otimes\sigma_3\). The positive and negative auxiliary dimensions are both one, so the physical normal-volume dimensions are both two. This proves part 1.

The positive-to-negative equation \(f'+xf=0\) becomes \((e^{x^2/2}f)'=0\); its distributional solutions are \(f=Ce^{-x^2/2}\), all of which are square integrable. The normalized solution is \(\pi^{-1/4}e^{-x^2/2}\). The adjoint equation \(-g'+xg=0\) becomes \((e^{-x^2/2}g)'=0\); its solutions are \(g=Ce^{x^2/2}\), and only zero is square integrable. To justify that a distribution with derivative zero is constant, test against any compactly supported smooth function of integral zero: it is the derivative of a compactly supported smooth primitive, so the distribution annihilates it. Its action therefore depends only on the integral. Section 11E proves the self-adjoint oscillator completion and compact resolvent. On the complement of its finite-dimensional kernel the squared eigenvalues are positive and bounded below, so the off-diagonal operator has closed range and is Fredholm. Its index is \(1-0=+1\). This proves part 2, rather than inferring Fredholmness from the two kernel dimensions alone.

For \(F_r\) there is one vertex regular orbit and \(r\) edge regular orbits. At \(r=1\) the ranks balance, so the normal-volume and Hermitian-extension tests give no contradiction, but (LD.4) still moves a support between different plaques. At \(r=2\) both the literal support obstruction and the unbalanced normal-symbol obstruction apply. An arbitrary holonomy groupoid need not admit any such tree model. A different balanced auxiliary, a nonelliptic local expression, or another graph completion is not covered by these obstructions. The general construction still needs its actual reduced representation, full normal coefficient transport, self-adjoint source-column domain, convolution estimates, localized scalar compacts and original Bott normalization. This proves part 3.

Total available in Exercises 122–124: **28 points**.

**Exercise 125 (A nonabelian affine oscillator, 10 points).**

(a) Prove the normal form, freeness, properness and compact quotient of the group (NH.17). Exhibit its index-two abelian subgroup. (4 points)

(b) Give the onto unitary (NH.4) for this group and calculate its action on every group coefficient. Why does adding its untouched fundamental-region factor not change the reduced crossed-product norm? (3 points)

(c) Verify the normal half-turn lift in (NH.18), including its square and the inverse central phase. Explain why this action meets the invariant metric premise. (3 points)

**Solution.** (a) Moving every occurrence of \(b\) past \(a\), using \(ba=a^{-1}b\) and the inverse relations, writes every word as \(a^mb^n\). Its actual map is \((x,y)\mapsto(m+(-1)^nx,y+n)\). Its second-coordinate displacement determines \(n\), and then its first-coordinate displacement determines \(m\), so no further word identifications exist. If it fixes a point, \(n=0\) and then \(m=0\). For compact sets bounded in both coordinates, an intersecting translate has bounded \(n\) and then bounded \(m\), proving properness. The translates of \([0,1)^2\) cover without interior overlap: for each horizontal strip choose \(n\), then choose the unique integer \(m\) that places \(x\) in the translated or reflected unit interval. Boundary assignments are harmless null sets for the Hilbert space. Edge gluing gives the compact Klein bottle. The subgroup of even \(n\) consists of the independent translations \((m,2k)\), is isomorphic to \(\mathbb Z^2\), and has index two. (4 points)

(b) Let \(Q=[0,1)^2\), with a null boundary assigned once, and \(R_{a^mb^n}=\Lambda^*(\operatorname{diag}((-1)^n,1))\). Set \((Ju)_g(x)=R_g^{-1}u(gx)\). The disjoint translate integral gives \(\|Ju\|^2=\|u\|^2\); prescribing \(u(gx)=R_g\xi_g(x)\) supplies an onto inverse. As \(R_{h^{-1}g}=R_h^{-1}R_g\), substitution in \(L_hu=R_hu(h^{-1}\cdot)\) gives \((JL_hu)_g=(Ju)_{h^{-1}g}\). Thus the representation is exactly left regular times the identity on \(L^2(Q,E_2)\). Tensoring a bounded operator with an identity has the same norm: the upper bound follows from sums of squared norms, and a tensor of a nearly norm-attaining vector with a unit vector supplies the lower bound. The normal absorption in (NH.7) then proves the actual reduced norm. Infinite multiplicity is retained, while compactness belongs to the confining oscillator on all of \(X\). (3 points)

(c) In \(e_j^2=-1\), anticommutation gives \((e_1e_3)^2=-1\). Conjugation by this spin element reverses \(e_1,e_3\) and fixes \(e_2\), exactly the derivative of \(h_b\). Taking \(s_a=1\) makes the relation \(bab^{-1}=a^{-1}\) hold for the lifts. Hence \(s_g=s_b^n\) is coherent for every actual normal form. The chosen central character \(e^{in\pi/5}\) has nontrivial determinant \(e^{2in\pi/5}\). Its inverse coefficient phase is \(e^{-in\pi/5}\), with full spin factor \(\rho_t(s_b^n)\); its inverse determinant is \(e^{-2in\pi/5}\). At \(b^2\), the inverse factor is \(e^{-2i\pi/5}\rho_t(-1)\), not erased by the identity derivative. The torus transformations are orientation-preserving flat isometries, so these are actual unitary Clifford-compatible lifts. In particular this example is not merely a topological oriented-frame lift. (3 points)

**Exercise 126 (Extracting the symbol and testing a return, 12 points).**

(a) Derive the exact compactly supported oscillatory identity (NH.21) and deduce the pointwise kernel covariance (NH.20), including why a zero integral operator has zero smooth kernel here. (4 points)

(b) For (NH.22), construct the one fixed kernel and unit inputs in Theorem 11G.3. Establish its divergent norm lower bound without allowing the operator or kernel to depend on the return label. (5 points)

(c) Which invariant metric assumption fails in this example, and which alternatives are not excluded? (3 points)

**Solution.** (a) Source multiplication \(M_L=e^{iL\phi\circ s}\) commutes with every left convolution and has norm one. On a compact smooth vector, the product rule (NH.19) is exact, both before and after convolution. Expanding the two compositions gives
\[
 [D,\pi(k)]M_L\psi-M_L[D,\pi(k)]\psi
   =L M_L(B\pi(k)-\pi(k)B)\psi,
 \qquad B=\sigma_D(d(\phi\circ s)).
\]
The left side is bounded by \(2\|[D,\pi(k)]\|\|\psi\|\) for every integer \(L\); dividing proves \([B,\pi(k)]\psi=0\). If a smooth matrix kernel entry were nonzero, choose local input/output frames and fixed vectors so that its scalar pairing is nonzero at that point. Multiply by a constant phase to make its real part positive. Continuity gives a smaller product patch with a uniformly positive real part; nonnegative test bumps in the input and output have a strictly positive integral because the smooth density is positive. This contradicts the zero operator. Thus the kernel vanishes and all locally realizable source covectors obey (NH.20). (4 points)

(b) Fix the buffered identity-plaque kernel \(k(t,t',x)=\alpha(t)\alpha(t')\chi(x)\), \(\int\alpha^2=1\), and fix the range function \(h=x\) on that buffered support. Source covariance gives \(\sigma_D(dz)\) least singular value at least \(c>0\) at every range lift, for sources close to \(p\). Since \((f^n)'(p)=(3/2)^n\), choose an interval \(I_n\) with \((f^n)'\geq(3/2)^n/2\) and \(f^nI_n\) inside \(\chi=1\), \(h_x=1\). Normalize any nonzero smooth source bump \(\eta_n\) supported there to norm one. Using one local unit coefficient vector, set \(\psi_n(u',v,z)=\alpha(u'-n)\eta_n(v,z)W(u',n;v,z)e(v,z)\). Its norm is one. Composition of \(W\) and the plaque integral make \(\pi(k)\psi_n=\alpha(u-n)\eta_nW(u,n)e\), also of norm one. The fixed operator
\[
 T=[D,\pi(hk)]-M_{h\circ r}[D,\pi(k)]
       =-i\sigma_D(d(h\circ r))\pi(k)
\]
therefore has \(\|T\psi_n\|\geq c(3/2)^n/2\), which diverges. The inputs vary, as an unboundedness test permits; \(T,k,h\) are fixed. For \(\lambda<1\), negative return labels instead give \(c\lambda^{-n}/2\). (5 points)

(c) A unitary Clifford-compatible normal lift would preserve the inverse-metric covector norm by squaring its covariance equation. At the fixed point it would require \(\lambda^{-2}|\xi|^2=|\xi|^2\), impossible for \(\lambda\neq1\). The positive oriented-frame cocycle nevertheless lifts to \(\mathbb R_{>0}\times U(1)\) by \(a\mapsto(a,1)\); that is a different structure. The theorem excludes the regular finite-rank local expression with bounded unbounded-operator commutators and invertible source transverse symbol. It leaves changed completions, infinite-rank or nonlocal operators, degenerate source symbols, and bounded-phase cycles with unbounded \([D,\pi(k)]\) outside its claim. It is not a counterexample under the invariant metric/unitary normal hypothesis. (3 points)

**Exercise 127 (Actual affine graph domain and Bott sign, 10 points).**

(a) Derive \([D_{\mathrm o},L_g]=E(b_g)L_g\), and show precisely where orthogonality is used. (3 points)

(b) Prove that (NH.11) is a global local expression and explain its closed anisotropic domain, the compactness mechanism in the undifferentiated range variable, and the sign in (NH.14). (4 points)

(c) Identify the class comparison that gives original Bott pairing \(+1\), and state the exact general scope still missing. (3 points)

**Solution.** (a) Covector exterior transport \(R_g\) intertwines \(C(v),E(v)\) with \(C(O_gv),E(O_gv)\). Differentiating \(u(g^{-1}x)\) produces \(O_g^{-1}\); orthogonality cancels it in \(d+d^*\). It also preserves Lebesgue measure and the exterior inner product, making \(L_g\) unitary. The transported multiplication is \(E(O_gg^{-1}x)=E(x-b_g)\); its difference from \(E(x)\) is \(E(b_g)\), whose square is \(|b_g|^2\). Pullback preserves \(\nabla u\in L^2\), and affine triangle inequalities preserve \(|x|u\in L^2\). Thus the commutator holds on the full domain, not just a formal expression. (3 points)

(b) The deck action sends \(r-t\) to \(O_g(r-t)\); together with the exterior and normal lifts this intertwines (NH.11) with itself. At fixed \(t\), the substitution \(y=r-t\) makes it exactly oscillator plus the compatible normal operator \(D_N(t)\). Its completed square controls precisely \(\nabla_r\), \(|r-t|\) and \(\nabla_n\), giving (NH.12); it controls no \(t\) derivative. Fibre Neumann resolvents and their uniformly bounded direct integral prove self-adjointness. Cutoff in invariant \(|r-t|\) and buffered-chart mollification give the smooth compact core. Holding \(r=y+t\) fixed gives input coordinate \(y'=r-t'=y+t-t'\); under \(T_\delta u(y)=u(y-\delta)\) this is the factor \(T_{t'-t}\). The fibre resolvent is compact and norm-continuous. Its composition with translations and the finitely many overlap unitaries is norm-continuous and compact. Convolution produces a continuous compact-operator-valued kernel in two compact range coordinates. Finite-rank fibre approximation and separated range-function approximation give finite-rank total operators, with Schur-controlled norm error. This proves localized scalar compactness in the range direction; a bare compact fibre resolvent would not prove it. (4 points)

(c) The partition-induced expression is \(D_{\mathrm{ind}}\) of (NH.15), on exactly the actual graph Hilbert space, and the physical expression differs by bounded \(E(b(t)-t)\). The same-domain phase homotopy therefore identifies its class with the Morita induction of the crossed normal/oscillator class. The actual normalized-bump counting correspondence gives the restriction to the full normal class, and the even oscillator factor is the positive identity by its Gaussian, complete connection equations and positivity. Hence the original transverse Bott product is \(+1\), in the supplied normal inverse order. An arbitrary holonomy-invariant normal spin-c groupoid need not possess the free proper cocompact affine Euclidean cover or this regular confining auxiliary. That missing construction and the general local graph normalization remain, even though this nonabelian family is now covered. (3 points)

### Exercise 128. A density-compatible horizontal derivative (12 points)

In a local chart of Theorem 11H.1, write \(X_a=\partial_a+A_a\cdot\partial_x\), with a smooth positive vertical density \(\mu(u,x)|dx|\).

1. Derive the unique real multiplication correction \(c_a\) for which \(X_a+c_a\) obeys the fibre inner-product differentiation rule (4 points).
2. Derive its commutator with a smooth fibre kernel, including the input-density terms (5 points).
3. Explain why the full inverse determinant connection commutes with scalar fibre convolution and is still present in the resulting Dirac (3 points).

**Solution.** For arbitrary smooth fibre profiles \(\eta,\zeta\), integration of \(A_a(\overline\eta\zeta)\) gives
\(-\int\operatorname{div}_{\mu}A_a\,\overline\eta\zeta\,d\mu\).
Adding the two real corrections gives \(2c_a\) times the same product. The derivative of the inner product has the extra density term
\(\partial_a\log\mu\). Therefore
\[
 2c_a=\partial_a\log\mu+\operatorname{div}_{\mu}A_a=\beta_a.
\]
Conversely this equality gives the required rule. Uniqueness follows by testing against profiles supported in arbitrarily small fibre neighbourhoods: the difference of two real corrections integrates to zero against every smooth product and hence vanishes. A purely imaginary addition would be an additional unitary connection term; the stated uniqueness is for real multiplication.

For \(K_k\), the derivative of its integral is
\[
 X_a(x)K_ku
 =\int\bigl(\partial_a k+A_a(x)\partial_xk+
                  \partial_a\log\mu\,k\bigr)u\,d\mu
       +\int k\,\partial_a u\,d\mu.
\]
Subtracting \(K_kX_au\) removes the last term and introduces
\(-\int k A_a(y)\partial_yu\,d\mu\).
Integration by parts replaces this with
\(\int(A_a(y)\partial_yk+\operatorname{div}_{\mu}A_a(y)\,k)u\,d\mu\).
Adding the output correction \(\beta_a(x)k/2\) and subtracting the input correction \(\beta_a(y)k/2\) leaves
\[
 \delta_a k=\partial_a k+A_a(x)\partial_xk+
                A_a(y)\partial_yk+
                      \tfrac12(\beta_a(x)+\beta_a(y))k.
\]
This is (SC.10), with both half-density signs positive.

The two kernel slots have the same base point \(b\). Every connection endomorphism of the full normal inverse at \(b\), including the dual determinant connection and any dual even line twist, therefore factors through the fibre integral and commutes with \(K_k\). This is why it contributes no commutator term of its own. It still contributes to \(W(b)\) in (SC.7) and to every coefficient-frame transition \(z^{-1}\rho_t(s)\). Vanishing of that particular commutator does not trivialize a line or delete its phase. The principal normal matrices are unchanged; thus the resulting commutator is exactly \(-i\sum_a M^aK_{\delta_a k}\).

### Exercise 129. Localized compactness and the original class (12 points)

Assume all hypotheses of Theorem 11H.1.

1. Explain why the normal horizontal operator need not have compact resolvent, and prove that \(K_k(D_H-i)^{-1}\) is scalar compact for a smooth graph kernel (6 points).
2. With a real normalized smooth fibre bump \(\xi\), prove the exact unbounded compression and its domain. Use the creation connection to identify the original Morita class and pairing (6 points).

**Solution.** On the trivial bundle \(V=B\times L\) with product horizontal complement and density,
\(D_H=D_B\otimes1_{L^2(L)}\).
If \(L\) has positive dimension, \(L^2(L)\) is infinite-dimensional. Fix a nonzero normal eigenvector \(v\), and an orthonormal fibre sequence \(e_n\). Their resolvent images are
\((\lambda-i)^{-1}v\otimes e_n\), with fixed nonzero norm and orthogonal to one another. Hence the unlocalized resolvent is not compact.

For a general bundle, select finitely many vertical fields \(Z_j\) spanning the vertical tangent. The joint symbol of \(D_H,Z_j\) has a positive lower bound on the unit cotangent sphere because its zero annihilates horizontal and vertical spaces. The Fourier freezing and finite-partition proof in Section 11H.6 gives
\[
 \|w\|_{H^1}\le C(\|D_Hw\|+\sum_j\|Z_jw\|+\|w\|).
\]
The proof there extends this estimate to distributional derivatives in \(L^2\), by uniformly bounded mollifier commutators. For \(R=(D_H-i)^{-1}\),
\[
 D_HK_kR=K_kD_HR+C_kR,\qquad
 Z_jK_kR=(Z_jK_k)R.
\]
Here \(D_HR\), \(R\), \(C_k\), and every differentiated smooth kernel \(Z_jK_k\) are bounded. Therefore \(K_kR\) maps the \(L^2\) unit ball to a bounded \(H^1\) set. Finite-chart Fourier tail truncation proves this set is relatively compact in \(L^2\), as explicitly shown in Section 11H.6. These are compacts on the entire scalar space; being compact separately in each fibre would not suffice.

The real connection and \(\langle\xi,\xi\rangle=1\) give
\(\langle\xi,\nabla^E_a\xi\rangle=0\).
Consequently the creation identity
\(D_HT_\xi=T_\xi D_B+E_\xi\) implies
\(T_\xi^*D_HT_\xi=D_B\).
If \(u\in H^1(B,\mathcal D_B)\), approximation by the normal smooth core and this bounded-error identity imply \(T_\xi u\in\operatorname{Dom}D_H\). Conversely, if \(T_\xi u\) has distributional \(D_H\)-image in \(L^2\), apply \(T_\xi^*\) to the identity to obtain \(D_Bu\in L^2\). The stipulated maximal normal domain gives \(u\in H^1\). Thus the compression includes its exact domain, rather than just its expression.

For any smooth \(\eta\), the bounded creation error \(E_\eta\) gives
\[
 (D_H-z)^{-1}T_\eta-T_\eta(D_B-z)^{-1}
  =-(D_H-z)^{-1}E_\eta(D_B-z)^{-1}.
\]
The right normal resolvent is compact, and at \(z=\pm i\sqrt{1+t^2}\) the norm is bounded by \(\|E_\eta\|/(1+t^2)\). Integrating proves the two phase-connection conditions. Density handles all \(E\), and the positivity condition for its zero Morita operator is zero. Hence the class is \([{}_AE_{C(B)},0]\otimes[D_B]\).
Finally \(\overline\eta\mapsto\theta_{\xi,\eta}\) is the onto inverse-counting-correspondence unitary: its inner product, both actions and surjectivity are checked in Section 11H.7. Pullback by the specified corner is therefore precisely \([D_B]\), with its full coefficient and original local calibration. The pairing is \(+1\). The proof does not assign a pairing to a new unrelated corner or multiply by a second leaf index.

### Exercise 130. Chart ends and the twisted sphere fibre (12 points)

1. Compute the deficiency dimensions of \(-i\,d/dt\) on \(C_c^\infty(0,1)\) (3 points).
2. Show exactly how the partition isometry of (SC.20) supplies a self-adjoint domain without choosing extensions on separate chart ends (4 points).
3. For (SC.21) with \(\kappa=\pi/5\), determine the two frequency sets, explain their infinite multiplicity, and identify the original and inverse determinant phases (5 points).

**Solution.** The derivative graph norm is the \(H^1\) norm. The minimal domain is \(H^1_0(0,1)\), obtained from compact smooth functions by cutoff and mollification; it has zero endpoint traces. Integration by parts shows that the adjoint has the full \(H^1\) domain. The equations
\(-iu'=iu\) and \(-iu'=-iu\) have respectively the solutions \(ce^{-t}\) and \(ce^t\). Both lie in \(L^2(0,1)\), and each solution space has dimension one. Thus the deficiency dimensions are \((1,1)\) and the compact core is not essentially self-adjoint.

For real buffered weights with \(\sum_j\chi_j^2=1\), \(J^*J=1\) and \(JJ^*=P\). The overlap entries use the same underlying point and its full coefficient; direct multiplication gives \(P^2=P\). On smooth global sections,
\[
 J^*D_{\rm loc}J
   =D_H-i\sum_a M^a\sum_j\chi_j\partial_a\chi_j=D_H,
\]
because the last sum is half the derivative of \(\sum_j\chi_j^2\). The projected core is \(J C^\infty(V,p^*\mathcal D_B)\). Its closure is exactly the onto unitary conjugate \(JD_HJ^*\), with domain \(J\operatorname{Dom}D_H\), and is self-adjoint. This proof does not assert that the unrestricted chart sum has that domain. A function that reaches one chart end is represented in another buffered chart; the overlap projection imposes the global agreement.

Antipodal pullback on \(S^2\) splits into its \(+1\) and \(-1\) eigenspaces. The endpoint equation for \(e^{i\omega t}v(x)\) is \(e^{i\omega}=e^{-i\kappa}\) in the even sector and \(e^{i\omega}=-e^{-i\kappa}\) in the odd sector. Therefore
\[
 \omega_{\mathrm{even}}=2\pi n-\pi/5,\qquad
 \omega_{\mathrm{odd}}=2\pi(n+\tfrac12)-\pi/5.
\]
Multiplication by these scalar phases reduces each sector to ordinary periodic Fourier expansion, proving completeness and the \(H^1\) trace-domain description. Infinitely many disjoint antipodal cap pairs give arbitrarily many independent even and odd fibre profiles; orthonormalizing proves both sectors infinite-dimensional. Each fixed frequency, with a fixed eigenvalue of the finite normal symbol \(J_N\), thus has infinite multiplicity. The ordinary resolvent is not compact. Smooth fibre-kernel localization restores scalar compactness by the proved estimate, so this fact does not contradict the local pairing.

The original spin-c central phase is \(e^{i\pi/5}\), and its determinant phase is \(e^{2i\pi/5}\). The full inverse module uses \(e^{-i\pi/5}\) and determinant \(e^{-2i\pi/5}\). Its final right \(C_1\) and grading remain in place. The antipodal fibre seam does not act as a transverse reflection: the transverse coordinate is the oriented base circle. Thus the normal metric and unitary spin-c premise hold, despite reversal of the fibre orientation.

### Exercises 131–133. Geometric representations and transverse normalization

**Exercise 131 (A compact invariant measure, 10 points).** Let \(\Gamma\) act on compact \(N\) preserving a probability measure, with geometric representation (RN.1). Suppose this representation is reduced-continuous. Construct a left-invariant mean. Does the proof require the measure to be ergodic or the action to be free? Does it exclude every reduced scalar cycle?

**Solution.** Constants are a unit invariant vector, so
\(\left|\sum c_g\right|\leq\|\sum c_g\lambda_g\|\).
The algebraic trivial character extends to a state on \(C_r^*(\Gamma)\). A norm-preserving Hahn–Banach extension to \(B(\ell^2\Gamma)\) is positive because its norm and value on \(1\) are one; the exponential and \(1-z\) argument in Theorem 11I.1 proves this explicitly. Its value on
\((\lambda_g-1)^*(\lambda_g-1)\), and on the reversed product, is zero. Cauchy–Schwarz kills both left and right mixed terms, giving
\(\psi(\lambda_gT\lambda_g^*)=\psi(T)\).
Then \(m(f)=\psi(M_f)\) is positive, unital and left invariant because conjugation implements \(f(h)\mapsto f(g^{-1}h)\). Neither ergodicity nor freeness occurred; an invariant unit vector suffices. The argument excludes only the displayed geometric representation. A cycle with a different weakly regular representation remains possible, as Section 11A demonstrates.

**Rubric.** Constant vector and exact reduced inequality: 2; continuous state and positive extension: 2; both zero mixed terms: 2; diagonal mean and action convention: 2; exact hypothesis and nonexistence limits: 2.

**Exercise 132 (The exact tree norm and inverse phase, 12 points).** For the degree-four Cayley tree, prove the equality in (RN.10), including the finite-vector lower bound. Let \(\chi:F_2\to U(1)\), and let the geometric action on a full inverse line be \(\chi(g)^{-1}U_g\). Which coefficient average tests the same gap? State the rank-four inverse physical sign and determinant character.

**Solution.** Put \(h(g)=3^{-|g|/2}\). At a nonroot vertex \(Ah/h=\sqrt3+3/\sqrt3=2\sqrt3\); at the root it is \(4/\sqrt3\). Summing (RN.11) gives
\(|\langle A\eta,\eta\rangle|\leq2\sqrt3\|\eta\|^2\). More directly, weighted Cauchy–Schwarz gives \(|A\eta(x)|^2\leq C\sum_{y\sim x}h(x)|\eta(y)|^2/h(y)\), where \(C=2\sqrt3\); summing in \(x\) bounds this by \(C^2\|\eta\|^2\). Thus \(\|A\|\leq2\sqrt3\). At level \(j\geq1\) there are \(4\cdot3^{j-1}\) vertices. For the truncation \(h_R\), each level contributes \(4/3\) to its squared norm. Each layer of edges contributes \(4/\sqrt3\) to the undoubled endpoint-product sum. Thus
\(\|h_R\|^2=1+4R/3\) and \(\langle Ah_R,h_R\rangle=8R/\sqrt3\). The Rayleigh quotient tends to \(2\sqrt3\), proving equality.

Use \(v_\chi=\frac14\sum_s\chi(s)u_s\), with \(s=a,a^{-1},b,b^{-1}\). The phase cancels in the geometric action on the inverse line, so constants in the normal variable and a vertical vector on which \(q=1\) give geometric norm one. On the regular representation \(W\eta(h)=\chi(h)\eta(h)\) gives \(W\lambda_sW^*=\chi(s)\lambda_s\). Therefore the reduced norm remains \(\sqrt3/2\). Rank four has \(k=6\), inverse module \(C_4\) and physical generators \(-L_{t_j}\). The supplied central lift has determinant character \(\chi^2\), while its full inverse coefficient carries \(\chi^{-1}\). Cancelling a phase in this particular average does not replace the actual coefficient by a trivial line.

**Rubric.** Weighted upper bound including root: 3; exact counts and lower limit: 3; opposite central coefficient and conjugation: 3; physical sign, determinant and inverse-phase distinction: 3.

**Exercise 133 (Index one and the equivariant unit, 8 points).** Compare the trivial and sign lines as even \(C_2\)-equivariant scalar cycles with zero operator. Prove that their ordinary indices agree and their equivariant indices differ. Explain the additional argument in Emerson's translation oscillator theorem, and why neither this example nor a hyperbolic normal return refutes the original invariant-spin-c graph problem.

**Solution.** Each positive line has zero negative space, hence scalar index one. Split an exactly equivariant Fredholm operator by \(P_\pm=(1\pm U_c)/2\); averaging a parametrix makes each restriction Fredholm. For a general equivariant scalar cycle, compress to its essential scalar summand, symmetrize and replace \(F\) by \((F+U_cFU_c)/2\). Its difference is compact, so the straight-line path is a cycle homotopy. It now commutes with the two projections. The same construction on a Hilbert-module homotopy produces two ordinary scalar homotopy modules; Theorem 0.14 makes their endpoint index pairs equal. Stabilization and degenerate cycles contribute equally or zero. The trivial and sign lines give \((1,0)\) and \((0,1)\), respectively; forgetting the action sums the entries. Equivalently the characters at \(c\) differ by their signs.

Emerson's specified group \(G\subseteq\mathbb R\) has an actual continuous deformation of its translation action, \(x\mapsto x+tg\), to the trivial action. The oscillator's scalar index identifies the equivariant unit only after that homotopy. An arbitrary groupoid has no such supplied action deformation. The finite-group comparison is an abstract equivariant-index example, not a germ-effective foliation counterexample. A hyperbolic transverse derivative cannot preserve a positive metric and a unitary spin-c lift, so that example misses the historical hypothesis. To refute the original problem one would have to exclude every eligible normalized reduced representative, rather than one phase, one representation, or one ineligible action.

**Rubric.** Two scalar indices: 1; invariant two-component index and different values: 3; actual translation homotopy and its group scope: 2; both exclusion limits: 2.


### Exercises 134–139. Proper coefficients and the regular diagonal (66 points)

**Exercise 134 (10 points).** For the cutoff in Lemma 11J.1, derive the coordinate identity (AU.6), prove that the isometry (AU.5) exists in an arbitrary nondegenerate covariant representation, and explain why a finite stabilizer causes no exception.

**Solution.** The functions \(c_g(x)=c(g^{-1}x)\) satisfy
\(\alpha_h(c_{h^{-1}g})(x)=c(g^{-1}hh^{-1}x)=c_g(x)\).
Thus the coordinate \(g\) of \((U_h\otimes\lambda_h)J\xi\) is
\(U_h\pi(c_{h^{-1}g})\xi=\pi(c_g)U_h\xi\), exactly that of \(JU_h\xi\) (3 points). Square normalization gives
\(\sum_g\|\pi(c_g)\xi\|^2=\|\xi\|^2\); the strict-to-strong argument following (AU.5) justifies this equality and norm convergence of the orthogonal coordinate sum (4 points). If a finite group \(F\) acts on a point, take \(c=|F|^{-1/2}\). The sum in (AU.2) has \(|F|\) terms and equals \(1\). Hence the same proof works with finite stabilizers; no free-action Morita equivalence is being asserted (3 points).

**Exercise 135 (10 points).** Track the source and coefficient of every factor in (AU.9). Prove the direct-summand assertion, and identify precisely where replacing \([q_P]^{-1}\) by a hypothetical \([q]^{-1}\) would change the assumptions.

**Solution.** The four factors run respectively
\[
 R\longrightarrow P\rtimes_r\Gamma
 \longrightarrow P\rtimes_{\max}\Gamma
 \longrightarrow M\longrightarrow\mathbb C
\]
in the KK category. All have even degree, so their product has even degree and no interchange sign is introduced (4 points). Multiplication by \([u_r]\) evaluates to \(1\); the resulting pairing \(\ell\) decomposes every \(K_0\)-class by (AU.16), and its restriction to \(\mathbb Z[1_R]\) is the identity (4 points). The coefficient quotient is an actual isomorphism proved from a proper action. An inverse of the group quotient would instead assert K-amenability for a discrete group; this is an extra premise, and can fail by the Kazhdan projection argument (2 points).

**Exercise 136 (10 points).** Suppose an infinite property-\(T\) group has a proper factorization satisfying (AU.8). Determine both the scalar unit pairing of \(\kappa\) and the pairing of \([q]^*\kappa\) with its Kazhdan projection. Explain why these values are compatible.

**Solution.** Theorem 11J.3 gives \([1_R]\otimes_R\kappa=1\) (3 points). Since \(q(p)=0\), the pullback class pairs with \([p]\) as
\([p]\otimes_M[q]\otimes_R\kappa=0\) (3 points). The trivial full character pairs with \([p]\) as \(1\), so the pullback cannot equal that character, and the equivariant product \(\gamma\) cannot have the stronger unit identity that would force equality (3 points). The unit \(1_M\) and the spectral projection \(p\) are different K-classes; no equality of their pairings was asserted. Thus ordinary normalization and failure of K-amenability coexist (1 point).

**Exercise 137 (12 points).** For the convention \(\alpha_g(a)=U_gaU_g^{-1}\), verify both conjugations in (RA.5), including the inverse on \(U_h\). Then prove the restriction identity (RA.8), and identify exactly which index assumption fixes the original Bott sign.

**Solution.** Applying \(W^*\), then \(\pi(a)\otimes1\), then \(W\), gives \(U_h^{-1}\pi(a)U_h=\pi(\alpha_{h^{-1}}a)\) in coordinate \(h\). For \(U_g\otimes\lambda_g\), the output before \(W\) is \(U_gU_{g^{-1}h}\eta_{g^{-1}h}=U_h\eta_{g^{-1}h}\), which \(W\) sends to \(\eta_{g^{-1}h}\). Thus the group action becomes \(1\otimes\lambda_g\); replacing \(U_h^{-1}\) by \(U_h\) would not make this cancellation. Faithful spatial factor representations make this the exact reduced norm. Since \(\Delta_r i_r(a)=i_m(a)\otimes1\), functoriality gives \(i_r^*d_r=(i_m^*d_m)\boxtimes u^*\kappa=i_m^*d_m\). The required auxiliary equality is \(u^*\kappa=1\), not merely that it is Fredholm or has a finite-dimensional kernel. It is even and has scalar index \(+1\); no odd factors are exchanged, so the original Bott pairing and sign remain \(+1\).

**Rubric.** 4 points for the two coordinate conjugations; 4 for the typed whole KK restriction calculation; 4 for the exact auxiliary unit, parity and sign.

**Exercise 138 (12 points).** Prove that the paired proper-length operator (RA.17) is an actual reduced even scalar cycle but cannot serve in Theorem 11K.2. Explain why adding an unmatched trivial positive line is not a valid general correction. State the exact full inverse phase that must remain in a normal product.

**Solution.** The finite-coordinate closure is the orthogonal sum of matrices with eigenvalues \(\pm(1+\ell(g))\); its graph domain is precisely the two weighted \(\ell^2\) spaces in (RA.17). Finite word balls make both resolvents compact. The word-length inequality bounds \([M_w,\lambda_h]\) by \(\ell(h)\), preserving the domain, and the two-resolvent integral gives compact phase commutators. The bounded transform thus has all scalar Kasparov defects on \(C_r^*\Gamma\). The off-diagonal map \(M_w\) is bijective with bounded inverse, so the index is zero. The desired restriction would be multiplied by zero rather than by one. An unmatched trivial positive line would add index one but forces the trivial group representation into the auxiliary. Reduced continuity of that line implies an invariant mean by Theorem 11I.1, so this fails on nonamenable groups. A genuine different reduced auxiliary needs its own proof. In the normal factor, \(zs\) becomes \(z^{-1}\rho_t(s)\), with grading \(\Pi^{q(q-1)/2}\) and physical generators \((-1)^{q(q-1)/2+1}L_{t_j}\). The inverse determinant \(z^{-2}\) cannot replace \(z^{-1}\).

**Rubric.** 4 points for closure/compactness/commutators; 4 for index zero and the trivial-line norm failure; 4 for the entire inverse coefficient, grading and generator sign.

**Exercise 139 (12 points).** Under (RA.19), construct the Kazhdan projection by functional calculus and determine its reduced image. If a fixed maximal normal class pairs it nontrivially, which assertion is refuted? Explain why an ordinary scalar unit-index auxiliary and the proper-intermediate formula (RA.22) are not automatically excluded.

**Solution.** The positive \(L=\sum_{s\in S}(1-u_s)^*(1-u_s)\) has spectrum in \(\{0\}\cup[c,4|S|]\). A continuous function equal to one at zero and zero on the latter interval gives \(p=h(L)\), a projection onto invariant vectors in every representation. An infinite regular representation has no nonzero invariant vector, so \(q(p)=0\). A descent of a maximal class would give \([p]\otimes d_m=q_*[p]\otimes d_r=0\), contradicting its nonzero pairing. This refutes descent of that specified class; it does not refute every reduced graph class with original local Bott pairing one. A reduced auxiliary has \(p\)-pairing zero and may still have full-unit pairing one; those conditions are different. In the proper-intermediate route the pullback is \(\gamma_m\otimes d_m\), not automatically \(d_m\). Its local normalization follows from the separately proved \(b_m\otimes\gamma_m=b_m\) and \(b_m\otimes d_m=1\). The needed quotient inverse, local identity, exact normal restriction and graph-operator data remain explicit premises. Neither formula turns an invariant metric bundle into a proper space.

**Rubric.** 4 points for construction and reduced image of \(p\); 4 for the exact specified-class scope; 4 for the two distinct auxiliary/proper-intermediate premises and their limitations.

### Exercises 140–142. Prescribed commutators and unital compression (36 points)

**Exercise 140 — removing the unrepresented block (12 points).**

Let \(D=D^*\) be odd, let \(P\) be an even projection preserving its domain, and suppose \([D,P]\) is bounded. Prove that \(B=[D,P](2P-1)\) is the bounded odd self-adjoint off-diagonal part of \(D\). Show that \(D-B\) reduces \(P\), and explain why compact resolvent and the represented KK class survive the compression when the representation vanishes on \((1-P)H\).

**Solution.** Put \(Q=1-P\). On the full domain, \([D,P]=QDP-PDQ\). Multiplication by \(2P-1\) gives \(B=QDP+PDQ\). The two terms are mutual adjoints on domain vectors, and density gives self-adjointness of the bounded extension. Oddness follows because \(D\) is odd and \(P,Q\) are even. Direct block multiplication yields \([B,P]=[D,P]\), so \([D-B,P]=0\). The bounded symmetric perturbation proof (UP.9) gives self-adjointness on the same domain and compact resolvents. Their restrictions give compact resolvent on \(PH\). Formula (UP.11) makes the two bounded transforms differ compactly. The unrepresented block has zero source representation, hence is degenerate; removing it preserves the original KK class. Unitality on \(PH\) follows from restricting the source identity \(P\).

**Rubric.** Correct block and adjoint computation: 4 points; self-adjoint compact-resolvent perturbation and reduction: 4 points; bounded-transform class and zero-block argument: 4 points.

**Exercise 141 — the complete group core (12 points).**

Under Corollary 11L.2, construct a countably generated algebraic graded core invariant under every \(V_g\). Prove graph density and the bound (UP.16). Explain why bounded commutators for every fixed element do not assert a bound uniform in the group.

**Solution.** Choose dense \((x_j)\) in \(H_K\) and \(\xi_j=(D_K-i)^{-1}x_j\). The identity \(\|(D_K-i)\xi\|^2=\|\xi\|^2+\|D_K\xi\|^2\) and surjectivity of the resolvent equation make their span graph dense. Split them by the grading projections \(P_\pm\), and span all \(V_gP_\pm\xi_j\). Each lies in the domain by (UP.13); the grading projections preserve the graph norm because the grading anticommutes with \(D_K\). The span contains the original graph-dense family and is invariant under all \(V_h\) by the group multiplication law. Its family is countable. The two-component triangle inequality gives

\[
 \bigl\|(D_KV_g\xi,V_g\xi)\bigr\|
 \leq\bigl\|(V_gD_K\xi,V_g\xi)\bigr\|+\|C_g\xi\|
 \leq(1+\|C_g\|)\|\xi\|_{\operatorname{Dom}D_K}.
\]

The cutoff construction controls each enumerated element after a finite initial segment; its finite initial commutator contributions can depend on that element. It supplies no common bound as \(g\) varies, nor invariance under higher powers of \(D_K\).

**Rubric.** Resolvent graph-density argument: 4 points; graded countable invariant core: 4 points; precise graph bound and scope: 4 points.

**Exercise 142 — the index survives unital compression (12 points).**

Use the matrices in (UP.20). Compute \(D\), its removed block \(B\), and \(D_0=D-B\). Find the compressed kernel and cokernel, its index, and the exact norm of \([D,P]\). Explain why the exact odd unitary \(F\) on the doubled space does not force the compressed index to vanish.

**Solution.** In the displayed coordinate order,

\[
 D=\begin{pmatrix}0&0&2&1\\0&0&1&3\\2&1&0&0\\1&3&0&0\end{pmatrix},\quad
 B=\begin{pmatrix}0&0&0&1\\0&0&0&3\\0&0&0&0\\1&3&0&0\end{pmatrix},\quad
 D_0=\begin{pmatrix}0&0&2&0\\0&0&1&0\\2&1&0&0\\0&0&0&0\end{pmatrix}.
\]

The compressed map \(D_K^+\) is the row \((2,1)\), so its kernel is \(\mathbb C(1,-2)\), its cokernel is zero and its index is \(+1\). The off-diagonal vector linking the unrepresented coordinate to the positive coordinates is \((1,3)\), of norm \(\sqrt{10}\); therefore both \(\|[D,P]\|\) and \(\|B\|\) are \(\sqrt{10}\). The doubled \(F\) acts on equal-dimensional grading spaces, but the scalar representation there is \(aP\), not the identity representation. Compression retains two positive and one negative represented coordinate. The zero-representation coordinate carries no represented KK contribution, so removing it need not preserve equality of grading dimensions and does preserve the represented index.

**Rubric.** All three correct matrices: 4 points; kernel, cokernel and sign: 4 points; commutator norm and representation distinction: 4 points.

### Exercises 143–145. The regular groupoid interface (42 points)

**Exercise 143 — two distinct diagonals (10 points).**

(a) **5 points.** In \(M_2\otimes M_2\), calculate the ideal
generated by \(e_{11}\otimes1-1\otimes e_{11}\). Separately verify
the corner homomorphism \(e_{ij}\mapsto e_{ij}\otimes e_{ij}\),
its identity and injectivity.

(b) **5 points.** For a nonatomic smooth volume on a compact
positive-dimensional manifold \(M\), prove that (GR.2) forces
every bounded \(T\) on \(L^2(M^2)\) to vanish. Explain precisely
why this does not rule out the one-leaf groupoid's rank-one Morita
class.

**Solution.** (a) In the ordered tensor basis \(11,12,21,22\),
the generator is \(\operatorname{diag}(0,1,-1,0)\). Consequently
\(E_{a,12}(e_{11}\otimes1-1\otimes e_{11})E_{12,b}=E_{a,b}\)
for every \(a,b\). These are all matrix units, so the ideal is all
\(M_4\) and the quotient is zero. In contrast the four matrices
\(e_{ij}\otimes e_{ij}\) multiply with the matrix-unit rule,
have the required adjoints, and act faithfully on the span of
the tensor basis vectors \(11,22\). Their sum of diagonal units
is \(P\), not the full identity of \(M_4\). The range is exactly
\(PM_4P\). Compression to this corner and quotienting by the
uncompressed ideal are different operations. Award 3 points for
the full ideal calculation and 2 for all corner claims.

(b) Choose countably many continuous \(h_j\) separating points.
For a fixed input \(v\), (GR.2) makes its output \(Tv\) vanish
where any \(h_j(x)-h_j(y)\ne0\), after discarding a countable
union of null sets. Thus it is supported on the diagonal.
Fubini applied to its indicator gives diagonal product measure
zero, since every singleton has volume zero. Hence \(Tv=0\)
and \(T=0\). This uses product measure on two independent unit
variables and the specific equation (GR.2). The pair groupoid
is Morita equivalent to a point and admits its usual nonzero
rank-one corner; neither that corner nor its scalar class has
been required to act on this product-measure diagonal. Award
3 points for the complete null-set argument and 2 for the
precise limitation.

**Exercise 144 — typed absorption and the regular unit line (16 points).**

(a) **7 points.** Suppose \(\gamma:y\to x\) and \(g:z\to x\)
in the groupoid of Theorem 11M.2. Type every map in (GR.8), prove
\(\nu_r=\nu_s\) without erasing isotropy labels, and establish
the onto absorption unitary and its actual reduced norm bound.

(b) **5 points.** For the canonical unit line of Proposition
GR.4, calculate the covariance defect and the index of (GR.20).
Verify the entire homotopy (GR.21) and explain why its compact
covariance defect is compatible with the proof.

(c) **4 points.** Show that a holonomy-invariant positive
transverse metric makes the underlying holonomy groupoid
Hausdorff. Explain why this does not imply properness or remove
the full spin-c coefficient phases.

**Solution.** (a) The arrow \(\gamma^{-1}g:z\to y\) has
transport \(U_{\gamma^{-1}g}:V_z\to V_y\), followed by
\(U_\gamma:V_y\to V_x\), then \(U_g^{-1}:V_x\to V_z\).
The cocycle law makes their composition \(1_{V_z}\).
A countable open bisection cover can be made disjoint by
subtracting earlier pieces. Integrating a nonnegative arrow
function on one resulting Borel partial bisection over its
range or source gives equal values, by the invariant unit
measure. Summing proves \(\nu_r=\nu_s\) by monotone convergence.
Distinct isotropy arrows remain distinct pieces or distinct
labels in the sums; no orbit counting replaces arrow counting.
Thus \(J\xi(g)=U_g^{-1}\xi(g)\) preserves norms, is measurable
and has onto inverse \(J^{-1}\eta(g)=U_g\eta(g)\).
Conjugation gives precisely (GR.9), whose source \(z\) is
unchanged throughout convolution. Its source fibre is
\(\lambda_z(f)\otimes1_{V_z}\), bounded by
\(\sup_z\|\lambda_z(f)\|=\|f\|_r\). This uses the complete
coefficient, not merely its determinant character. Award 2
points for typing, 2 for the measure proof and 3 for the onto
unitary and actual reduced bound.

(b) The translated unit vector is \(\delta_\gamma\).
The covariance defect is the difference of its rank-one
projection and the rank-one projection onto \(1_{r\gamma}\),
so has rank at most two. The involution \(2P-1\) is invertible;
its chiral kernel and cokernel vanish and its index is zero.
For the stated path \(B_u\), direct multiplication gives
\(B_u^2-1=4u(u-1)P\). After \(C_0(X)\) localization this is
compact. The translated defect is \(2u\) times the already
computed finite-rank defect, with continuous coefficients on
each bisection. At \(u=0\), \(B_0=-1\) and all defects vanish;
at \(u=1/2\), the same unit line is both kernel and cokernel.
Thus the path supplies no index one. Noninvariance was not
mistaken for an obstruction to compact covariance; a different
Fredholm family remains possible. Award 2 points for the
projection/index calculation and 3 for every homotopy defect
and its limitation.

(c) On transversals, holonomy maps are local metric isometries.
They preserve the Levi-Civita connection by the Koszul formula,
so preserve short geodesics and their initial velocities.
Local geodesic existence and uniqueness follows from the
bounded-coefficient integral equation contraction, and the
time-one position map fills a neighbourhood of its starting
point by the \(v\mapsto v+R(v)\) contraction argument of
Lemma 11M.5. Hence two local isometries with the same value
and derivative have the same germ. Source, range and normal
derivative therefore give a continuous injection of the
holonomy arrows into a Hausdorff bundle of linear isometries.
Pullbacks of disjoint image neighbourhoods separate any
two arrows. This requires no properness of the injection
or groupoid. The derivative records the metric germ, whereas
the specified spin-c coefficient action also contains the
full inverse central phase; that coefficient is retained
by \(J\). Award 3 points for the germ/separation proof
and 1 for the two precise limitations.

**Exercise 145 — an eligible smooth graph with a localized failure (16 points).**

(a) **4 points.** Verify the actual holonomy graph (GR.11),
the invariant normal metric and the full inverse coefficient
phases (GR.12), including the determinant.

(b) **6 points.** Prove the stated reduced convolution bound,
the exact self-adjoint domain and smooth compact core of
(GR.14), and the bounded commutator (GR.15).

(c) **6 points.** Check that the fixed \(k\) of (GR.16) is
smooth, compact on the quotient, self-adjoint and a convolution
idempotent. Prove the infinite localized noncompactness test
(GR.17)–(GR.18) and identify the conclusion's exact scope.

**Solution.** (a) Irrationality makes the leaf through \([r,n]\)
an injectively immersed copy of \(\mathbb R\); equality after
a nonzero integer return would require \(j\alpha=0\) modulo
\(2\pi\). All leaf paths with the given lifted endpoints have
the same germ, and distinct returns have distinct rotational
germs. Thus the quotient of the triples in (GR.11) is exactly
the holonomy graph. Rotations preserve \(dn^2\) and increasing
\(n\). The coherent lift \([1,e^{ij\beta}]\) has determinant
\(e^{2ij\beta}\); the complete inverse \(C_1\) coefficient
has phase \(e^{-ij\beta}\rho_t(1)\) and inverse determinant
\(e^{-2ij\beta}\). Its final right Clifford factor and the
calibrated \(A\) are retained. Award 2 points for the graph
identification and 2 for all metric/coefficient claims.

(b) Partition the quotient integral into deck strips and use
the measure-preserving translations to integrate in the
source strip \(0\le r<1\). Each fixed physical source then
carries exactly its regular \(L^2(\mathbb R_t)\) convolution
operator with the finite coefficient. Taking the supremum of
the regular norms gives the reduced bound. In the range strip,
normal Fourier coefficients give matrices \(mA\) and exact
domain \(\sum_m(1+m^2)\|\psi_m\|^2<\infty\). Since
\(\|(mA\pm i)^{-1}\|\le1\) and
\(\|mAu\|^2+\|u\|^2=\|(mA\pm i)u\|^2\), both ranges
are all of \(L^2\); the maximal normal domain is self-adjoint.
Truncate Fourier and \(r\) tails, remove small \(t\)-endpoint
neighbourhoods at no \(t\)-derivative cost, then mollify
\(t,r\). Extend by the constant phase (GR.12). This gives
smooth compact graph sections converging in the normal graph
norm. Differentiation on this core gives \(-iA\Pi(\partial_nk)\).
The differentiated compact smooth kernel has finite regular
Schur bounds. Closing the core identity proves domain
preservation and the bounded commutator on the exact domain.
Award 2 points for each of the norm, domain/core and
commutator arguments.

(c) The integer translates of \(\xi\) have disjoint supports.
The sum (GR.16) is locally finite and deck invariant, and its
quotient support is contained in the compact image of
\(\overline I^2\times N\). Reality gives the adjoint. In the
convolution square, disjointness removes mixed translates;
each diagonal term integrates to \(\int\xi^2=1\). Thus
\(k*k=k\). For a constant normal \(u\), the sections \(\psi_j\)
have disjoint \(r\) supports, norm one, \(D\psi_j=0\), and
\(\Pi(k)\psi_j=\psi_j\). Hence their images under the fixed
localized resolvent are orthonormal, with pairwise distance
\(\sqrt2\), and cannot contain a convergent subsequence.
This proves failure of an ordinary scalar compactness condition
for this particular unconfined normal operator on the actual
eligible smooth graph. It does not refute the historical
problem: the Section 11E translation oscillator repairs this
same suspension. Award 3 points for all kernel properties,
2 for the infinite-sequence proof and 1 for the exact scope.

### Exercises 146–149. General auxiliary geometry and the coefficient identity (52 points)

**Exercise 146 (12 points).** Explain why the compact inclusion itself does not prove Theorem 11N.4. Verify the rational-function induction for \(S=\mathbb Z[t,t^{-1}]\), and prove that the passage \(GL(n,K)\to GL(nb,K_0)\) is faithful.

**Solution.** Any continuous compact orbit has bounded diameter, so differences cannot tend to infinity on an infinite word group; even a dense cyclic subgroup gives this failure (3 points). For \(S\), the \(t\)-valuation bounds the negative exponent, the degree-at-infinity value bounds the highest exponent after this negative bound, and finitely many distinct transcendental evaluations bound the integer coefficients by the inverse Vandermonde matrix. Thus a simultaneous bounded intersection contains only finitely many Laurent polynomials; the countable evaluation family accommodates every possible degree bound (5 points). Finally a matrix over \(K\) is an invertible \(K_0\)-linear map on \(K^n\). Equality of two such maps on the underlying vector space implies equality on its standard \(K\)-basis vectors, hence equality of their matrix columns. Products and inverses act as the corresponding products and inverses of the \(K_0\)-linear maps. This gives a faithful homomorphism in dimension \(nb\), without an assertion about extended valuations (4 points).

**Exercise 147 (12 points).** Derive both identities in (CI.17), prove that (CI.14) preserves the excess height \(t-\|v\|^2\), and give the exact finite-transporter bound when \(\rho_-(r)=a\sqrt r\), \(a>0\).

**Solution.** Formula (CI.10) gives
\[
 L_{\mu,g}^{-1}e_l^\mu
       =e_{g^{-1}l}^{\mu g}-e_{g^{-1}}^{\mu g}.
\]
Taking its inner product with \(v\) and adding
\(\langle e_g^\mu,e_l^\mu\rangle\) yields the first coordinate formula (4 points). For \(l=g\), the inverse vector is
\(-e_{g^{-1}}^{\mu g}\), so the cross term in the new height is
\(-2z_{g^{-1}}\), giving the second formula (3 points). Expansion of
\(\|L_{\mu,g}v+e_g^\mu\|^2\), using orthogonality of \(L_{\mu,g}\), gives exactly the same two extra terms as the height update. Their subtraction leaves \(t-\|v\|^2\) (3 points). Finally (CI.18) becomes
\(a\sqrt{|g|}\leq2\sqrt R\), or \(|g|\leq4R/a^2\).
The transporter is contained in the finite word ball of that radius. This proves properness without assuming finite-dimensional fibres or free action (2 points).

**Exercise 148 (16 points).** Prove the exact translation term in (WF.16), explain why \(B(\Theta)\) alone is not the asserted compact-phase construction, verify (WF.20), and distinguish the class in Theorem 11P.3 from a scalar reduced unit pairing.

**Solution.** Weyl translation changes the coordinate potential by \(-b\) and leaves derivatives unchanged. Thus \(D\) becomes \(D-c_b\), while the weighted potential becomes \(B(1+\Theta)-c_{(1+\Theta)b}\). Cubing the former without commuting its factors gives exactly (WF.16). The bounded term includes both \(b\) and \(\Theta b\) (4 points). For the second claim take two modes and compare weights \(1,1\) and \(1,3\). Translate a fixed Gaussian packet to the position \(R(1,1)\), keeping a fixed exterior vector. After dividing either oscillator by \(R\), its error on that packet from the constant Clifford potential \(c_{(1,1)}\), respectively \(c_{(1,3)}\), tends to zero. Indeed derivatives and displacement from the packet centre have fixed norms, divided by \(R\). On each eigenspace of the corresponding constant Clifford matrix, Chebyshev’s inequality for the scalar spectral measure makes the normalized oscillator phase tend to the sign of that eigenvalue: the eigenvalues have a fixed nonzero gap and the norm error tends to zero. Thus the phase difference on a suitable fixed exterior vector tends to a nonzero value, since the two normalized Clifford directions \((1,1)/\sqrt2\) and \((1,3)/\sqrt{10}\) differ. The translated Gaussian packets tend weakly to zero, so this difference is not compact. A bounded difference of single-particle weights is still an unbounded first-order oscillator difference. The cubic term supplies the strict lower-order estimates (WF.19), which justify compactness (4 points). To verify (WF.20), put \(z=(a/s)^{1/3}w\); then
\[
 \frac{z}{(s^2z^6+a^2)^{1/2}}
   =s^{-1/3}a^{-2/3}\frac{w}{(1+w^6)^{1/2}},
\]
whose final factor has a finite maximum. Joint commutation with \(D^2\) turns this scalar bound into the operator estimate (4 points). Finally the two algebras in Theorem 11P.3 are \(C(X)\), and its field action may move every proposed evaluation point. There is no supplied equivariant evaluation to \(\mathbb C\), no proved proper-algebra factorization, and no reduced-norm statement for the vacuum representation. None is implied by its integer index or by its coefficient identity (4 points).

**Exercise 149 (12 points).** Prove the finite-transporter bound (PM.3), explain the topology needed for the escaping-mass example, and verify the equivariance of (PM.5).

**Solution.** If \(a\in E\cap h^{-1}E\), then \(a\in E\) and \(ha=b\in E\), hence \(h=ba^{-1}\in EE^{-1}\). If the two sets were disjoint, the two strict half-mass inequalities would contradict the total bound one; this proves both the bound and its exact use of the threshold (4 points). Each fixed coordinate of \(\mu_n\) is eventually that of \(\tfrac34\delta_e\), since \(g_n\) escapes finite sets. Their mass difference \(1/4\) persists. Thus weak-* convergence is present and norm convergence is absent; treating the topology as the norm topology would lose the example and the local compactness proof (4 points). Finally
\[
 \Phi(hz)=\sum_g c(g^{-1}hz)^2\delta_{g y_0}
 =\sum_k c(k^{-1}z)^2\delta_{hk y_0}=h\Phi(z).
\]
The substitution is legitimate because the sums are locally finite over compact sets and consist of nonnegative probability weights. It also proves the transformation convention for finite stabilizers (4 points).

### Exercises 150–153. Coherent groupoid restriction and properness (44 points)

**Exercise 150 — actual descended module restriction (10 points).**

Let \(e,f\in E\) and \(k,l\in C_c(\mathcal G)\).
Type the products in (CA.3), prove (CA.5), and prove that
(CA.4) is onto. Identify both the unit-source action and
the descended operator under this unitary.

**Solution.** If \(s h=s g\), then \(g h^{-1}\) is
composable and has range \(r g\), so the first formula of
(CA.3) lies in \(E_{r g}\). If \(s g=r h\), then \(g h\)
is composable and also has range \(r g\), so both vectors
in the second formula have the same coefficient fibre.
For the elementary sections their inner product is
\(\sum_{s g=r h}\overline{k(g)}
 \langle e(r g),f(r g)\rangle l(g h)\).
Replacing \(g\) by the inverse in the first convolution
factor identifies this as \(k^*i_m(\langle e,f\rangle)l\),
exactly the interior-tensor norm (4 points).

A compact support has a finite bisection cover. On a
small range neighbourhood, continuous unit sections
approximate each continuous pullback section. Partition
the compact support into finitely many such pieces to
approximate it uniformly by sums \(e(r g)k(g)\).
The inner-product fibre-sum estimate
\(\|\xi\|\le N\sup\|\xi(g)\|\) on an \(N\)-bisection
support converts this to descent-module approximation.
Thus the isometric map is onto (3 points).

A unit function contributes only the identity arrow
to (CA.6), and acts as \(a(r g)\). This is the
canonical action on the first interior-tensor factor.
Pointwise \(F_{r g}\) becomes \(F\otimes1\), so the
restriction is exactly \(\gamma_0\otimes[i_m]\).
No scalar augmentation or evaluation at a unit is
needed (3 points).

**Exercise 151 — the weaker exact local normalization (10 points).**

Track all the sources, targets and degrees of (CA.7).
Derive (CA.8) and the original positive local Bott
integer. Explain both why the full quotient need not
be invertible and why fibrewise rank one is weaker
than a whole-base identity.

**Solution.** The factors run from \(A_r\) to \(P_r\)
(degree \(k\)), from \(P_r\) to \(P_m\) (degree zero),
from \(P_m\) to \(A_m\) (degree \(k\)), and from \(A_m\)
to \(\mathbb C\) (degree \(q\)). The total is \(q\);
their order never changes, so there is no exchange
sign or Clifford block permutation (3 points).

Naturality moves \([q_A]\) across the reduced
\(\eta\) to \([q_P]\). Its product with \(v\) is
the supplied coefficient identity. Full descent's
product law gives (CA.10). Restrict by \(i_m\)
and apply Theorem 11R.1 to get
\([i_r]d_r=\gamma_0 d_N\). Multiplication by the
original \(b_U\), with \(b_U\gamma_0=b_U\) and
\(b_Ud_N=+1\), gives the original sign \(+1\)
(4 points).

Only the proper coefficient quotient has an
inverse in this argument; no \([q_A]^{-1}\)
was inserted. A graded index bundle of rank
difference one over each unit acts as one
on a small disk Bott module after choosing
local frames. Its global bundle transitions
can still be nontrivial, so those ranks do
not prove \(\gamma_0=1_C\). The local identity
is sufficient for the historical integer;
the stronger whole-base equality is optional
(3 points).

**Exercise 152 — no compact-algebra properness shortcut (12 points).**

Prove the centre and anchored equivariant-map
steps of Theorem 11R.3. Then prove the compact-transporter
argument. Apply it to an infinite discrete
group acting on a nonempty compact Hausdorff
base, including when the action is free.

**Solution.** A multiplier of the compact
module algebra acts adjointably by its
bounded approximate-identity limit. A
central one commutes fibrewise with all
rank-one projections, hence is a scalar
on every nonzero fibre. Its scalar is
continuous by (CA.11), and bounded by
the multiplier norm, so the centre is
\(C_b(Y)\). A nondegenerate central
homomorphism from \(C_0(Z)\) gives a
nonzero character at every \(y\), hence
one point \(z_y\). Continuity of all
\(f(z_y)\), tested on bumps supported
in neighbourhoods, proves continuity
of \(y\mapsto z_y\). Base compatibility
and equivariance give its anchor and
(CA.12) (5 points).

For compact \(K,L\subset Y\), the
proper action on \(Z\) has compact
transporter between \(z(K),z(L)\).
Its arrow projection is compact.
The transporter on \(Y\) is a
closed subset of that compact arrow
set times \(K\); thus it is compact.
This proves action properness, without
requiring \(Y\to Z\) itself to be
proper. Conversely a proper \(Y\)
gives the central structure \(Z=Y\)
(4 points).

For an infinite group and compact
nonempty \(Y\), the inverse image of
the compact \(Y\times Y\) under the
action-relation map is all
\(\Gamma\times Y\), whose projection
onto the infinite discrete group
is not compact. The action is not
proper, regardless of freeness.
Theorem 11R.3 therefore excludes the elementary
compact algebra over that base as
a proper coefficient. It excludes
neither a different proper Bott
algebra nor a reduced normalized
Dirac class (3 points).

**Exercise 153 — coherent weak cone and its exact height (12 points).**

Prove positivity of (CA.17), orthogonality
and surjectivity in (CA.19), and the
compact-coordinate characterization
(CA.21). Derive the action's height
and proper-transporter bound. Compute
the exact one-dimensional slice in
Figure 11R.1.

**Solution.** The zero-sum conditional
negativity makes (CA.17) positive.
Positivity of the polynomial for
\(u+\lambda v\) gives Cauchy–Schwarz,
so a null vector is orthogonal to
all vectors and the quotient is
well defined. The equality
\((gh)^{-1}(gk)=h^{-1}k\) preserves
all Gram entries under left
translation. Every \(e_l\) is
\(L_g(e_{g^{-1}l}-e_{g^{-1}})\),
giving surjectivity (3 points).

The inequalities (CA.21) make the
coordinate functional bounded on
the dense span, of norm at most
\(\sqrt t\). They annihilate each
zero vector in that span. Extension
and Hilbert-space representation
give one vector of squared norm
at most \(t\); the dense span makes
it unique. Continuous Gram entries
make every rational inequality
closed, proving that the bounded
coordinate image is compact.
The countable generators are
local bisection sections with
bumps, not one global arrow
enumeration (4 points).

Expanding the squared norm of
\(L_gv+e_g\) adds exactly
\(2\langle L_gv,e_g\rangle+
 \psi(g)\). Adding the same two
terms to \(t\) preserves excess
height. If both heights are
bounded by \(R_1,R_2\), the
triangle inequality gives
\(\psi(g)\le(\sqrt{R_1}+
 \sqrt{R_2})^2\). Relative
properness (CA.15) supplies the
compact arrow set; continuity
and compact input/output sets
complete the transporter proof
(3 points).

In the illustrated real fibre,
\(L=1,b=1,v=-1/2,t=3/4\).
The new vector is \(1/2\),
and the new height is
\(3/4+2(-1/2)(1)+1=3/4\).
Both excess heights are
\(3/4-1/4=1/2\). With both
height bounds \(3/4\), the
displacement estimate reads
\(1\le\sqrt3\), or
\(\psi=1\le3\). This slice
checks the cross-term sign,
not the existence of a
general proper kernel (2 points).

### Exercises 154–156. The actual groupoid proper base and coherent extension (36 points)

**Exercise 154 (12 points).** Prove closedness of the range
support condition, show why mass is only lower semicontinuous,
and prove (GP.6) with the compact support of its fixed test
function. Explain why distinct isotropy arrows remain distinct.

**Solution.** Equation (GP.3) is a closed family of identities
since \((a\circ r)f\in C_c(\mathcal G)\). Bumps separating
a different range from \(y\), followed by a countable arrow
cover, show its converse. This makes \(Q\) closed and makes
the compact-unit inverse images compact (3 points). The mass
is the supremum (GP.4), so is lower semicontinuous. On the
one-unit groupoid of an infinite discrete group, the measures
\(\tfrac34\delta_e+\tfrac14\delta_{h_n}\), with \(h_n\)
escaping finite sets, converge vaguely to
\(\tfrac34\delta_e\), while their masses are one. Thus
mass need not be continuous (3 points). On a bisection,
\(g=b(sg)\); on a measure supported at that source,
\(\chi(rh)=1\), proving (GP.6). Its support is contained
in \(b(\operatorname{supp}\chi)^{-1}\operatorname{supp}f\),
a continuous product of a closed subset of two compact sets.
Extension by zero is continuous because the bump is compactly
supported inside \(s(B)\) (4 points). Every isotropy arrow
is a separate point of the closed discrete range fibre;
an arrow bump distinguishes its atom from the unit atom.
The construction never replaces arrows by their unit endpoints
(2 points).

**Exercise 155 (14 points).** Derive the exact compact arrow
bound in (GP.8). Check the normalization, support and
equivariance in (GP.10)–(GP.13), and type the algebra map
supplied by the anchor.

**Solution.** The two strict half-mass tests force an actual
overlap \(h\in E\cap L_g^{-1}E\). With \(k=gh\), the
sources of \(k\) and \(h\) agree, and
\(g=kh^{-1}\). The composable pair condition is closed
in \(E^2\), so \(EE^{-1}\) is compact. Continuity of
the action makes the transporter a closed subset of
\(EE^{-1}\times K\), completing the properness proof
(4 points). Over a compact input set, the quotient
partition uses finitely many indices. A nonzero cutoff
coefficient therefore forces its translated point into
one fixed compact union of bump supports. Properness
bounds all such arrows simultaneously (3 points).
The quotient partition and \(S_j\) are invariant, so
summing (GP.10) along the range fibre cancels its denominator
using (GP.9), leaving \(\sum_j\tau_j=1\). In (GP.12)
at \(gz\), substitution \(h=gk\) preserves the
coefficient and pushes the arrow atom \(\delta_k\)
to \(\delta_{gk}\). This proves both mass one and
equivariance with every source and range retained
(4 points). The anchor is continuous and gives a
nondegenerate multiplier homomorphism
\(C_0(T)\to M(C_0(M_{\mathcal G}))\). A unit
approximate identity converges uniformly on the compact
anchor image of any compact support. This map is
neither a scalar evaluation nor an inverse Kasparov
Dirac class (3 points).

**Exercise 156 (10 points).** Check the common-range
kernel convention and every sign in (UPL.4), derive
the exact global coordinate change (UPL.8), and explain
why relative properness holds while this supplies no
original-unit scalar class. Compute the original
cone slice for \(b_h=1,v=-1/2,t=3/4\).

**Solution.** If \(h_i:m_i\to n\), then
\(h_i^{-1}h_j:m_j\to m_i\), so its displacement
is \(a(m_i)-a(m_j)\). Expanding its square cancels
the two single-index terms and leaves precisely
\(-2(\sum_i c_i a(m_i))^2\), with no exchange of
arrows or sign convention (3 points). In (UPL.6),
\(v+b_h-a(n)=v-a(m)\), while direct expansion
gives (UPL.7). The inverse in the proof of Theorem 11T.3
recovers both original coordinates and preserves
the excess condition \(e\ge0\) (3 points).
The proper action on \(M\) already makes each
compact-endpoint arrow transporter compact;
intersecting with \(\psi\le R\) gives CA.15.
The unit algebra is \(C_0(M)\), and an anchored
pullback from \(C_0(T)\) gives no reverse Dirac
class or local index-one assertion (2 points).
For the slice, \(v'=1/2\) and
\(t'=3/4+2(-1/2)(1)+1=3/4\), with both excess
heights \(1/2\). This checks the exact cross-term
sign independently of any index-factor existence
(2 points).

### Exercises 157–159. Proper groupoid coefficient norms (36 points)

**Exercise 157 (12 points).** regular module and its exact norm.

Type each summand of (PN.5). Derive its source-column matrix,
the two exact Schur bounds and its adjoint. Prove that tensoring
with a faithful representation of \(P\) retains the operator
norm, and explain why this proves \(\|f\|_r\le\|f\|_m\)
without disintegration.

**Solution.** If \(r h=r g\), then
\(h^{-1}g\) is composable, has source \(s g\), and the
coefficient \(\alpha_g^{-1}f(h)\) also lies in
\(P_{s g}\). Fixing \(x=s g\) and writing
\(l=h^{-1}g\) gives \(h=gl^{-1}\) and the matrix
\(A_{g,l}=\alpha_g^{-1}f(gl^{-1})\) on
\(\ell^2(\mathcal G_x)\otimes P_x\) (3 points).

For fixed \(g\), \(l\mapsto gl^{-1}\) is a bijection
from the source column to the range fibre at \(r g\),
so the row sum is at most \(R\). For fixed \(l\),
\(g\mapsto gl^{-1}\) is a bijection to the source
fibre at \(r l\), so the column sum is at most
\(S\). The row operator with entries
\(A_{g,l}/\sqrt{\|A_{g,l}\|}\) has squared norm
at most \(R\). Apply it to the column with entries
\(\sqrt{\|A_{g,l}\|}\xi_l\), then sum over \(g\).
This proves (PN.7) and the bound \(\sqrt{RS}\).
The transpose-adjoint entry is
\(\alpha_l^{-1}f(lg^{-1})^*\), exactly the regular
entry for \(f^*\). Thus the extension is adjointable
and \(I\)-bounded (5 points).

If \(T\in\mathcal L(E)\) is nonzero, choose
\(\xi\) with \(T\xi\ne0\). The positive coefficient
\(\langle T\xi,T\xi\rangle\) is nonzero; a faithful
representation \(\pi\) has a vector \(v\) on which
it has positive expectation. Consequently
\(T\xi\otimes v\ne0\), so the induced operator
representation is faithful and hence isometric.
The resulting \(I\)-bounded ordinary Hilbert-space
representation of the convolution algebra is included
in its defining universal full norm. No groupoid
representation has been decomposed into unit fibres
by a measure-theoretic theorem (4 points).

**Exercise 158 (12 points).** the actual cutoff isometry.

Derive (PN.12) from the interior tensor product,
prove compact support of (PN.13) on the exact
composable pair space, and reindex its inner product
to obtain (PN.15). Explain why the resulting identity
holds in \(B_m\), rather than merely at each unit.

**Solution.** For elementary pair sections
\(K(g,k)=\xi(g)a(k)\), \(L(g,k)=\zeta(g)b(k)\),
the tensor inner product is
\(a^*i_P(\langle\xi,\zeta\rangle)b\).
Its convolution value at \(h\) is
\[
 \sum_{s k=r h}\alpha_k^{-1}\left(
 a(k)^*\left[\sum_{s g=r k}\xi(g)^*\zeta(g)\right]
 b(kh)\right),
\]
which is exactly (PN.12). Every bracket has
coefficient \(P_{r k}=P_{s g}\), and the transport
lands in \(P_{s k}=P_{r h}\) (3 points).

Let \(D\) be a compact central support for
\(a\). A nonzero cutoff product requires
\(z\in D\), \(r g=p(z)\), and
\(c(g^{-1}z)\ne0\). The Theorem 11S.3 inverse-composability
support condition bounds \(g\) in a compact
\(B_D\); it is not being applied to pairs
satisfying \(s g=p(z)\). The condition
\(gk\in K_a\) then bounds \(k\) in the compact
\(B_D^{-1}K_a\). Thus the pair section has
compact support and belongs to the completed
interior module, by the bisection approximation
and its explicit uniform-to-module estimate
(4 points).

In (PN.12) at \(Wa,Wb\), put \(l=gk\).
The two transports become \(\alpha_l^{-1}\),
and the cutoff square becomes
\(\alpha_g(c_{s g}^2)\) between \(a(l)^*\)
and \(b(lh)\). The source condition is
\(s l=r h\); the remaining inner sum is over
\(r g=r l\). Equation (PN.9) replaces that
sum by one, leaving exactly \(a^**b\).
This is equality of the continuous compactly
supported convolution sections representing
the interior-tensor inner product. Its
completion is therefore an equality in
\(B_m\). Positivity and isometry follow from
the actual interior product, not from
fibrewise indices or a scalar augmentation
(5 points).

**Exercise 159 (12 points).** norm equality and both inverse identities.

Prove (PN.14) with its unchanged middle unit,
deduce (PN.19), and give both typed inverse
Kasparov products. State exactly what this
proves for Theorem 11R.2 and what original graph
requirements remain.

**Solution.** In (PN.17), the changed first
arrow is \(h^{-1}g\), whose source remains
\(s g=r k\). Hence its cutoff is the same
\(c_{s g}\). This is central and even, so
commutes past \(\alpha_g^{-1}f(h)\) without
a grading sign. The action composition gives
\(\alpha_{h^{-1}g}^{-1}=\alpha_g^{-1}\alpha_h\).
The remaining sum is the convolution
\((f*a)(gk)\), which proves (PN.14)
(4 points).

The isometry gives
\(\|f*a\|_m\le\|\Lambda(f)\|\|a\|_m\)
for all \(a\in B_m\). A contractive
approximate identity in \(B_m\) shows that
left multiplication by \(f\) has norm
\(\|f\|_m\). Thus \(\|f\|_m\le\|f\|_r\),
and Lemma 11U.1 supplies the reverse inequality.
Completion gives the onto isometric graded
homomorphism \(q_P:B_m\to B_r\)
(4 points).

The class \(v=[q_P^{-1}]\) has type
\(KK(B_r,B_m)\), degree zero. Its products
are \([q_P]\otimes_{B_r}v=1_{B_m}\) and
\(v\otimes_{B_m}[q_P]=1_{B_r}\), by
composition of the two actual homomorphisms.
This discharges Theorem 11R.2's proper-coefficient
quotient inverse at the general groupoid
scope. It supplies neither its
original-unit Dirac factors nor their
local Bott product, and says nothing
about an inverse of the coefficient-free
quotient or a finite-order graph operator
(4 points).

### Exercises 160–163. The completed proper scalar factorization (58 points)

**Exercise 160 (14 points).**

Prove norm convergence and the quotient identity for (PX.3), explain the need for the \(\varprojlim{}^1\) term, and specialize the result to the cone kernel in (PX.13).

**Solution.** Compactly localized inputs have only finitely many nonzero input compressions by the supported enlarged cutoff. All finite partial sums are completely positive contractions because their positive matrix outputs are bounded by \(\sum c_g^2\leq1\). Approximating a general input by a compactly localized one bounds every remaining tail by twice its approximation error; this proves norm convergence in the middle algebra (4 points). The quotient of a summand is \(c_g^2d_g^2a=c_g^2a\), giving identity after square normalization. A formula with \(d_g=c_g\) would instead have \(\sum c_g^4\), which need not equal one (3 points). Contravariant KK is not asserted to be continuous. The telescope gives a kernel and cokernel of identity minus shift; the cokernel is the \(\varprojlim{}^1\) term. Commuting isomorphisms of stage groups preserve both, so (PX.12) permits the correct limit argument (3 points). Finally the equivariant Fock Morita module removes the compact factor in the ideal, leaving a coefficient cone. Its explicit equivariant scalar contraction gives zero in both degrees, even after restriction to finite subgroups. Nuclear lifting and Theorem 11V.4 then give the unique lift (PX.14), without asserting that an ordinary section is already equivariant (4 points).

**Exercise 161 (16 points).**

Explain stabilization, prove properness of (BD.8), identify the correct tail error in (BD.11), and compute the ordinary product sign.

**Solution.** Real generators make the stabilization operator and its positive square root commute with conjugation. Its polar unitary therefore restricts to the real modules, so rank variation is absorbed by a trivial infinite summand while the cocycle is extended by zero (4 points). Bounded-height input and output vectors each have norm at most \(\sqrt R\); their difference is the affine cocycle, so \(\|b_g\|\leq2\sqrt R\). Uniform proper displacement makes its transporter finite, proving the actual proper action (4 points). A bounded single-particle difference satisfies (BD.12) with \(\leq\); the three phase integrals are each bounded by (BD.13), and the deformation inserts the factor \(t\), giving a tail error tending to zero. Translations change the weighted potential by \(c_{(1+t\Theta)b}\); its identity summand is retained (4 points). Finally the positive-time scalar evaluation of the lifted Bott class is the cubic weighted oscillator. Its even vacuum contributes \(+1\), and its invertible complement becomes degenerate after taking the exact sign. Tensoring with \(C(X)\) gives (BD.22), with coefficients retained (4 points).

**Exercise 162 (16 points).**

Prove the face test, verify the grading, and explain the calibration's exact scope.

**Solution.** The nearest-simplex threshold gives (UD.7) by the half-space criterion. Applying it to the two tested coordinates gives \(|w_i-v_i|+|w_k-v_k|\leq\sqrt2\|w-v\|\); the strict hypothesis therefore excludes every inactive coordinate, including a neighborhood of the tested point (4 points). Compression of the Clifford operator removes exactly its vacuum-to-\(\xi_\mu\) transition, so its square differs from identity by that rank-one line. Raw exterior degree makes the line odd; the simplex degree \(|\sigma|-1\) makes it even. The corresponding Morita module must use that shifted grading (4 points). The off-diagonal Bott defect is supported within radius one, while (UD.10) gives a coordinate greater than three and nonpositive unwanted coordinates. Since \(3>1+\sqrt2\), (UD.8) proves the exact support ideal and multiplier property (4 points). Finally the ordinary ambient homotopy leaves the even rank-one Bott summand and an exactly degenerate complement. The ordinary cone inverse identifies its product with the completed oscillator's even vacuum index. This proves (UD.16); it does not assert an equivariant scalar evaluation or an equivariant gamma identity (4 points).

**Exercise 163 (12 points).**

Track the coefficient algebras in (ST.4), justify the ordinary contractions, and identify the two exact product hypotheses used.

**Solution.** The factors have the types in (ST.5), so the two intermediate coefficient extensions are \(\mathcal B\widehat\otimes A\) and \(C(X)\widehat\otimes A\). The latter is collapsed by a genuine equivariant central homomorphism (ST.3), not a scalar evaluation (4 points). The weak half-mass space and the probability space are convex. On each compact source support their convex homotopy images are compact, so the explicit localized family defects remain compact throughout the homotopy. This proves (ST.6)--(ST.7) after equivariance is forgotten (4 points). The only product inputs are the ordinary coefficient identity (ST.1) and the ordinary universal parameter calibration (UD.16). Even interchange brings them together with \(u\operatorname{ev}_{x_0}=1\), proving (ST.8). Stronger equivariant gamma identities are unnecessary and are not inferred (4 points).

### Exercises 164–168. The anchored proper algebra and ordinary family (64 points)

**Exercise 164 (12 points). Length, moving fibres and universal parameter.**

Prove the finite-ball property, local finite moving-sheet property and cutoff map.

**Solution.** A nonunit factor in AG.1 costs at least one, so a cost bound
gives finitely many factors in a fixed compact weight sublevel. Compact
composable tuple spaces attain the minimum, prove lower semicontinuity
and bound the nonunit length sublevel; local product bisections prove
upper semicontinuity. A vertex ball has difference arrow in this compact
set or a unit at its fixed source, and has a bounded integer label.
Its intersection with the closed discrete range fibre is finite. The
full length sublevel need not be compact when T is noncompact (4 points).

Choose a positive moving atom and bound the parameter radius by R.
Every support vertex then lies within R of a fixed compact bisection
support, so AG.2 bounds the supports in one compact vertex set. There
are finitely many points above the chosen unit. Disjoint bisections
through them contain all its points over a smaller unit neighbourhood:
otherwise a convergent missed-point subsequence gives a missed point
at the original unit. Probability coordinates and their square roots
are therefore continuous in finitely many moving sheets (4 points).

The supported GP.3 cutoff gives mass one by
\(\sum_{rg=p(m)}c(g^{-1}m)^2=1\).
Its support uses r(g)=p(m), the inverse-composability condition.
Compact inputs bound the vertices compactly. Hence support diameters
are locally bounded and invariant; the quotient partition gives
\(t(m)=1+\sum_i\tau_i([m])R_i\).
The probability and radius are exactly equivariant, giving AG.6 (4 points).

**Exercise 165 (12 points). Exact face bound and shifted sign.**

Derive AF.6–AF.7 without putting a general vector in the affine simplex plane;
verify the compressed line degree.

**Solution.** Choose the unique threshold a with
\(\sum_j(v_j-a)_+=1\).
The point \(z_j=(v_j-a)_+\) obeys
\((v_j-z_j)z_j=az_j\) and \(v_j-z_j-a\le0\),
so it satisfies all nearest-point half-space inequalities. This
holds for a general vector; a nonnegative threshold is an extra
conclusion when its tested coordinates sum to one (4 points).

For w's threshold, \(w_k-a\le1\). Thus, if \(v_j\le0\) outside F,
\(w_j-a\le1-v_k+|w_j-v_j|+|w_k-v_k|
\le1-v_k+\sqrt2\|v-w\|<0\).
The strict bound persists on a neighbourhood, giving the open ideal,
not equality only at a tested point (4 points).

The full unit-vector Clifford square is one. Removing the vacuum
loses exactly its transition to the one-form \(\xi_\mu\), so
\(F_\mu^2=1-P_{\xi_\mu}\).
The one-form has raw degree one but degree zero in
\((\Lambda^{\ge1}H_{\mathbb C})[1]\).
The retained Morita shift makes its index positive and even (4 points).

**Exercise 166 (12 points). Proper centre and completed family.**

Prove the central structure and actual compactness of the cycle defects.

**Solution.** On a permitted coefficient cylinder the nearest
probabilities agree. Summing only integer copies of the same arrow
leaves distinct isotropy arrows distinct, so \(f(\nu_\sigma)\) and
\(f(\nu_\tau)\) agree there. This proves centrality and compatibility
with products. The rule commutes with actual left translation.
A finite corner with compact unit support has finitely many compact
moving probability images; a function in Cc(M_G) equal to one on
them proves nondegeneracy. GP.2 gives actual properness (4 points).

The Clifford operator toggles only support vertices of mu, which
have c_v=0. Connected origins and cutoffs agree. If chi_sigma>0,
some coordinate outside that support is greater than three and
the unwanted coordinates are nonpositive. The radius-one Bott
defect lies in the exact face cylinder because \(3>1+\sqrt2\).
Ideal transitivity and finite row support prove the multiplier
property on the completed algebra (4 points).

When chi_sigma<1, every vertex lies in a bounded ball around
mu's support. There are locally finitely many such moving vertices,
so removing chi has finite matrix support, as does the remaining
rank-one defect. It is ambient compact. Multiplication by a
diagonal coefficient approximate identity and closedness of each
ideal put its entries in P; finite truncation proves norm compactness
there. Source localization by C0(Y) makes these continuous defects
compact operators on the completed module P_Y (4 points).

**Exercise 167 (16 points). Whole-base ordinary calibration.**

Prove the scope and sign of OC.2–OC.6.

**Solution.** Removing chi is locally compact uniformly on compact
parameter sets. Moving origins by
\(a_{\sigma,s}=(1-s)a_\sigma+s\mu\)
keeps connected origins equal and has locally fixed finite defect
support. These are homotopies of completed anchored family cycles
over the whole base, not independent fibre homotopies (4 points).

The resulting cycle has one continuous even deficient line
C xi_mu with a globally defined unit section. Its complement
squares exactly to one and commutes with the source, hence is
degenerate. Retaining the shifted Morita module leaves the positive
origin Bott cycle (4 points).

Moving the bounded finite-coordinate origin mu to zero gives a
whole-base ordinary homotopy. Under real trivialization its product
is the base tensor of the completed oscillator product; the even
vacuum gives +1. Pullback to M_G localizes by its source. A compact
support has compact anchor image, admitting an old unit bump
equal to one there; localized finite-rank approximants remain
compact. Both whole-base identities follow without requiring a
proper parameter map (4 points).

The ordinary unit section
\(y\mapsto(y,\delta_{(1_y,0)},0)\)
pulls both algebras back to the original C0(T). It gives the actual
ordinary \(\eta_0^TD_0=1_{C_0(T)}\), hence
\(b_U\eta_0^TD_0=b_U\) for the original transverse class.
This section is ordinary and supplies no equivariant dual or
equivariant scalar evaluation (4 points).

**Exercise 168 (12 points). Weight, supplied section and lift boundary.**

Explain DI.1, DI.7–DI.8 and the exact missing assertion.

**Solution.** On a compact arrow set the coordinate projections
and their orthogonal conjugates are approximate identities of
the continuous compact-operator algebra. Every bounded functional
sees their difference tend to zero; convex tails make that
difference uniformly small. The tail choices also uniformly
control finitely many compact-image vertex sections. Thus
\(1+\sum(1-v_n)\) has compact inverse, graph-continuous local
frames and a locally norm-convergent compact commutator series
with the actual arrow action (4 points).

Ordinary real and shifted exterior trivializations identify q
with \(1_{C_0(T)}\otimes h_0\otimes1\). The chosen graded ordinary
cpc section of h0, tensored with the identities, therefore splits
q C0(T)-linearly. Its kernel is equivariantly Morita to
\(C_0((0,1])\otimes C_0(T)\otimes S\).
The contraction \(f(t)\mapsto f(ut)\) commutes with anchor,
groupoid and grading (4 points).

This is an equivariant extension with an anchored ordinary cpc
section. Bijectivity of
\(q_*:KK^G(P,E_1)\to KK^G(P,Q)\)
is separately required. A source section of
\(C([0,1])\to\mathbb C_{\{0\}}\) would have to send one to
a nonzero continuous function supported on a singleton with
empty interior, which is impossible. Ordinary nuclearity cannot
repair that filtration step. P's local anchored finite-matrix
model does not assert sections for all its quotients. Section 11AA proves the separate bridge, so z_P and equivariant D are now actual, alongside P, theta_M, D0 and the whole-base ordinary identities (4 points).

### Exercises 169–176. Original-anchor localization and lift (96 points)

**Exercise 169 (12 points). Anchored étale technical theorem, products and the exactness actually used.**

Which three additional checks keep the ordinary exactness proof anchored
and equivariant?

**Solution.** Relative R-unitization and R-balanced Stinespring/tensor
modules retain the original anchor; no scalar forced unitization is
used to erase it (4 points). Compact-arrow tests take place in C(K,r*J),
and KG.1–KG.4 give a genuine continuous compact defect for products and
homotopies (4 points). The suspended-ideal section is made *exactly*
equivariant by supported proper averaging or finite averaging; a
quasi-invariant approximate identity alone does not do that (4 points).

**Exercise 170 (12 points). Anchored finite-matrix approximation for the actual nearest-face source.**

Explain why the local nuclear maps are anchored after patching, why
the output sum is contractive, and why no source-quotient section has
been used.

**Solution.** The local corner is a constant \(C_0(U_i)\)-tensor
model. Its nuclear maps are tensored by the identity of that algebra,
and the input and output bumps are central unit functions. Extension
by zero is continuous because of these compactly supported bumps;
all maps therefore retain the original \(C_0(T)\)-action (4 points).
At each matrix level the image of a positive contraction is at most
\(\sum_i\phi_i p_i\le1\), which proves contractivity of the
completely positive output map (4 points). The maps use finite
compressions of \(P\) and actual local trivializations. They never
lift a closed-supported quotient of \(P\), so the unit-groupoid
counterexample to such a quotient section is fully respected
(4 points).

**Exercise 171 (12 points). The proper source identity on an anchored regular module.**

Verify the cutoff embedding, identify the compact defect and compute
the whole-class sign of the regular identity cycle.

**Solution.** Square normalization gives \(V^*V=1\), while proper
support places centrally localized inputs in finitely many compact
arrow charts. The two formulas in (RC.3) therefore define adjointable
maps on completed modules. Exact transported cutoff functions make
the embedding equivariant (4 points). Its range projection satisfies
\(p\pi(a)=VM_aV^*\), compact because \(M_a\in\mathcal K(P)\)
(4 points). The countable shift pairs every complementary summand.
The single remaining copy of \(P\) is even and carries the identity
representation with zero operator. Its class is exactly \(+1_P\);
the paired complement is degenerate (4 points).

**Exercise 172 (12 points). Supported averaging of an anchored section on a common proper centre.**

Why are two cutoffs used in (SA.3), and why does the output belong
to E?

**Solution.** The enlarged input cutoff d gives the compact arrow
set (SA.2), so compact central inputs yield a finite bisection sum
of actual sections (4 points). The output cutoff c bounds the sum
by its square normalization and makes qS(a)=Σ c_g²a=a; using c at
both positions without dc=c would instead produce fourth powers
(4 points). Dense localized inputs and uniform contraction give
norm convergence in E. Strict convergence alone would not establish
membership in E (4 points).

**Exercise 173 (12 points). Anchored source Mayer--Vietoris without a closed-quotient section.**

Why does the variable-parameter homotopy stay inside \(D\), and
which quotient section has actually been supplied?

**Solution.** Modulo \(I\), the central parameter \(\phi_I\) is
zero, so the initial endpoint of every \(H_u(f)\) is the zero
class of \(f(0)\). Modulo \(J\), that parameter is one, and the
final endpoint stays the zero class of \(f(1)\). The homotopy
therefore preserves both endpoint ideals (4 points). Constants are
fixed by evaluation at \(\phi_I\), and \(H_1(f)\) is constant
with value \(r(f)\), proving the two homotopy inverse identities
(4 points). The supplied section is for the endpoint map
\(D\to I\oplus J\), with the two taper functions in (MV.6).
It does not give a section of a closed-supported quotient of
\(A\); that different statement remains false in the unit-groupoid
example (4 points).

**Exercise 174 (12 points). Proper étale slices and anchored induction reciprocity.**

Why can the two triangle proofs use orthogonal projections even
though an arbitrary source-essential submodule need not be complemented?

**Solution.** The identity coset is genuinely clopen over U and its
projection acts on a standard coefficient coset module (4 points).
The second projection comes from the genuinely clopen diagonal of
the étale Hausdorff coset bundle, also on a standard coefficient
module (4 points). Every class is sent by products with these
correspondences and their naturality, rather than by projecting its
source-essential submodule (4 points).

**Exercise 175 (12 points). The anchored source telescope, including its Milnor term.**

Why is the first telescope quotient anchored semisplit, and why is a
Milnor term still needed?

**Solution.** Its finite output factors have actual P-valued Choi
entries, which lift to Tel before central R multiplication (4 points).
The summable patching and proper supported averaging retain R-linearity
and make the section exactly equivariant (4 points). Source groups
are products of stage groups; the shift map can have a nonzero
cokernel lim1. It disappears here only because every stage group is
zero, not by a general continuity assertion (4 points).

**Exercise 176 (12 points). The actual anchored proper-source lift for the nearest-face algebra.**

Why do (PL.1), (PL.3) and (PL.4) suffice for a whole-base lift?

**Solution.** Finite-action averaging makes each restricted quotient
actually semisplit, and its actual cone-contraction/Morita kernel
gives zero cone groups for every restricted source (4 points).
Reciprocity and the explicit open MV extension propagate those zero
groups to every finite union of proper slice opens, without a
closed-supported quotient section (4 points). The actual anchored
telescope retains lim1, whose product shift has zero kernel and
cokernel because every stage group is zero. Puppe then gives (PL.5)
for the original coefficient algebras over all T (4 points).

### Exercises 177–182. Actual anchored étale descent (72 points)

**Exercise 177 (12 points). Typed completion and the compact unit inclusion.**

Check every fibre in DS.1, prove its positive completion, and explain
why an ordinary module compact tensors to a crossed compact.

**Solution.** If `rh=rg`, then `r(h^-1 g)=sh`. Thus the right-action
coefficient starts in `B_sh` and is transported to `B_rg`, while both
vectors in the inner-product term belong to `E_sh`. Applying `beta_h`
puts its inner product in `B_rg`; no distinct arrows with equal
endpoints are identified (4 points).

The balanced map `e tensor b -> e(rg)b(g)` has inner product
`b* i_B(<e,f>) c`, by finite convolution expansion. Positivity and
the null quotient follow from the actual interior tensor product.
For a section on one bisection its square inner product is unit
supported, so its norm is bounded by its uniform vector norm.
An N-bisection partition gives the N-epsilon approximation bound.
Local unit sections and the density `EB=E` then prove onto completion
without a global module frame (4 points).

Approximate a rank-one compact's vectors by sums `eb,fc`. Its tensor
extension is the rank-one with vectors `e tensor i_B(b)` and
`f tensor i_B(c)`. These are actual vectors because the étale unit
inclusion takes `B` into the crossed algebra. Approximation gives all
rectangular compacts. The statement would require an additional
localization argument for a non-étale group whose coefficient inclusion
only lands in the multiplier algebra (4 points).

**Exercise 178 (12 points). Full and reduced norm bounds.**

Prove the two different left-representation norm bounds, allowing
upper semicontinuous coefficient norms and a degenerate representation.

**Solution.** A one-bisection section has `Lambda(a)*Lambda(a)` unit
supported, hence norm at most `||a||infinity^2`. For a finite nonnegative
bisection partition write `a=sum sqrt(c_i)d_i`, with
`c_i(rh)=rho_i(h)p(h)` and `d_i=rho_i a/sqrt(c_i)`. The bound
`||d_i||<=sqrt(rho_i p)` proves continuity at zeros and permits
extension by zero. The scalar row is bounded by the square root of
the range sum, and the operator column by the square root of the
source sum. Their product gives `sqrt(RS)<=||a||I` on the completed
module, so the full universal norm bounds the representation (4 points).

For upper semicontinuous `p=||a||`, regularize it from above on one
compact metrizable neighbourhood by
`h_n(x)=sup_z(p(z)-n d(x,z))_+`, multiply by a fixed bump equal to
one on `supp(a)`, and extend by zero. The majorants decrease to `p`.
Their finite range/source sums have fixed compact unit supports;
decreasing compact upper-level sets show that their suprema decrease
to the actual R and S. Passage to the infimum proves the same bound
with the original norms (4 points).

For reduced norm, tensor with the faithful regular B source-column
representation. The unitary to source E columns is
`e tensor zeta -> U_g^-1 e(rg) zeta(g)`. The induced source-x matrix
has entry `pi_x(alpha_g^-1(a(gl^-1)))`, which is the PN regular matrix
followed by a star homomorphism. Every finite matrix amplification of
`pi_x` is contractive, even when `pi_x` is degenerate. Finite column
density and faithfulness of the regular tensor test prove DS.8.
There is no deletion of an uncomplemented essential submodule
(4 points).

**Exercise 179 (12 points). The crossed tensor unitary.**

Prove balancing, the exact inner product and dense range of DS.13.

**Solution.** In the expanded balancing sum put `h=lk`. Move
`beta_l(b(k))` across the interior tensor balance and then use
`pi_2(beta_l b)U_l=U_l pi_2(b)`. The result is exactly
`W(xi tensor Lambda_2(b)eta)` (4 points).

In the expanded inner product use arrows `l,h,k` with `rh=rk=sl`.
Its term is DS.14. Covariance moves `U_h` out of that inner product.
The composable substitution `u=lh,v=h^-1k` makes the vector arguments
`u^-1` and `v^-1u^-1g`. The remaining coefficient sum over `sh=rv`
is precisely `<xi,xi'>(v)`. Thus the full expression is
`<eta,Lambda_2(<xi,xi'>)eta'>`. Equality in the coefficient convolution
algebra proves isometry for both completions, without any modular
factor or interchange of Clifford factors (4 points).

Choose xi on the actual unit bisection with `xi(1_y)=e(y)`.
Then W is exactly `e(rg) tensor eta(g)`. Compact unit localization,
local tensor density and a finite bisection partition make these
sections dense. DS.1's norm estimate upgrades local uniform
approximation to completed module approximation. Hence W is onto;
it uses actual unit arrows instead of a fabricated global group identity
(4 points).

**Exercise 180 (12 points). Connections and the étale positivity mechanism.**

For an actual equivariant product F, prove its descended connection
and positivity conditions without assuming norm-continuous unlocalized
operator orbits or reduced exactness.

**Solution.** The creation error expands to DS.16. The first line
is a norm-continuous rectangular compact section by the connection
condition and norm continuity in the vector. Approximate the second
line's vector locally by `eb`. Since `T_eb=T_e pi_2(b)`, the source
element b localizes the arrow defect of F2 to a continuous compact
section. Uniform vector approximation and the bounded operators give
the compact-section limit. DS.9 makes its convolution compact; taking
adjoints gives the second creation condition (4 points).

Under W, the first descended operator tensored with one is the
pointwise extension of `S=F1 tensor 1`, by direct substitution.
The compact-extension lemma induces a homomorphism of Calkin
algebras, hence sends each original positive compression
`pi(a)(SF+FS)pi(a)*` to a positive crossed compression (4 points).

For an even coefficient approximate identity a_n,
`z i_A(a_n)->z` in crossed norm for every z. This is checked on
finite compact bisection supports before applying the representation,
so degenerate representations cause no problem. Compress the crossed
positive operators by Lambda(z), and take their norm limit as in
DS.17. Closedness of the positive cone proves the desired positivity.
Neither a quotient by a crossed compact ideal nor reduced exactness is
used. The original equivariant product's existence is separately
provided by the actual anchored KG technical/product foundation,
not by this positivity calculation (4 points).

**Exercise 181 (12 points). Homotopy, homomorphisms and coefficient naturality.**

Prove the interval identification and endpoint compatibility, then
identify a descended homomorphism class.

**Solution.** Finite interval partitions approximate a compact-arrow
section on a common support. In the full completion, commuting
interval functions and the constant crossed algebra have the norm
of their continuous field: a nonnegative partition gives the upper
bound by its finitely many values, and evaluation gives the lower
bound. In the reduced completion, the regular source-column module
is the interval tensor of the original one; its operator norm is
the supremum over interval evaluations. Thus both completions are
`C(I) tensor B_tau` (4 points).

The core endpoint map sends a section to its endpoint section.
Its inner products are exactly the evaluated DS.1 inner products,
and local lifting plus finite bisection partitions give dense range.
It therefore identifies the actual evaluated modules and operators.
A groupoid homotopy descends to an ordinary interval-module homotopy
(4 points).

For a coefficient homomorphism, the full core map is contractive by
the universal norm. The reduced map is contractive by DS.8 applied
to the standard target module, including a degenerate homomorphism.
The tensor unitary with that standard correspondence proves right
coefficient naturality; source pullback is immediate in DS.4.
The cycle `(B,v,0)` becomes `(B_tau,v_tau,0)`, giving DS.19 and
the identity classes (4 points).

**Exercise 182 (12 points). Quotient square, original anchor and Clifford degree.**

Prove DS.21, explain its exact transfer scope, and check the degree.

**Solution.** Associativity of the unit-coefficient interior tensor
gives `X_m tensor_qB B_r = X_r`. Its core inner product is q_B applied
to DS.1, while the left convolution and pointwise F agree. Pulling
back the reduced source along q_A gives the very same module and
operator. This proves the exact two-path KK equality DS.21; it is
independent of reduced ideal exactness (4 points).

For a `C0(T),C0(T)` cycle, restricting the crossed source to unit
coefficients gives exactly extension of scalars of the whole ordinary
forgetful cycle. This proves DS.22 and supplies CA's unit restriction.
Properness of an intermediate coefficient separately supplies its
quotient inverse by PN.3. Nothing here constructs the missing Dirac
factors, evaluates the unit space at a scalar equivariantly, or proves
invertibility of the coefficient-free quotient (4 points).

The auxiliary Clifford algebra has trivial action. A homogeneous
coefficient product moves c past b' with sign `(-1)^(|c||b'|)`, and
the adjoint has sign `(-1)^(|b||c|)`. The finite Clifford tensor norm
and matrix models give DS.23 in each crossed completion. The module
unitaries are even; the fixed order `Cl_p tensor Cl_q -> Cl_(p+q)`
is the same before and after descent. An actual flip has the same
graded sign on both sides, so both parities and the original right
Clifford factor are retained (4 points).

### Exercises 183–190. Retained labels and compact-fibre kernels (56 points)

**Exercise 183 (6 points).** Show why a metric-preserving holonomy germ is
determined by its first jet. Prove that the orthonormal frame anchor is proper.
Explain what neither proof says about the arrow topology or the action.

**Solution.** The Koszul identity makes the Levi-Civita connection invariant.
Uniqueness of the coordinate geodesic ODE gives h exp_x=exp_h(x) dh_x on
a small ball; local invertibility of exp_x makes the value and derivative
determine h there. In each holonomy bisection those data vary smoothly, so
the resulting injective jet map is continuous. (2 points.) Over a compact
unit set, finitely many smaller trivializing neighborhoods have compact
closures inside larger trivializations. Their frame preimages are compact
products with O(q); the preimage of the original compact set is a closed
subset of a finite compact union. (2 points.) An injective continuous map
need not be an embedding or proper. The frame argument concerns P→T;
it proves neither properness of G⋉P nor an ordinary frame section on an
arbitrary nontrivial bundle. (2 points; GK.1.)

**Exercise 184 (8 points).** For irrational rotation by α=(√5−1)/2, verify
the Fibonacci identity, exhibit escape of labels with convergent jets, and
compare squared chord displacement with n². Retain the supplied Spin-c
central character at n=55.

**Solution.** The identity is true for m=1,2. Adding the two previous
identities and using 1−α=α² changes the sign and multiplies the error by
α, proving F_mα−F_{m−1}=(-1)^{m−1}α^m. (2 points.) Since F_m→∞, the
arrows (F_m,x) escape compact subsets of discrete Z×T, but their endpoints
converge to x and their derivatives are 1. (2 points.) Squared chord
displacement is 4sin²(πnα)≤4, so its sublevel at 4 contains all arrows.
The function n² is CND since the zero-sum quadratic expression equals
−2(Σc_in_i)²; its sublevels contain finitely many n-labels and are compact
over compact unit endpoints. (2 points.) The character at 55 is
exp(11iπ)=−1, its inverse is also −1, and its determinant is exp(22iπ)=1.
Derivative 1 and determinant 1 therefore do not remove the inverse normal
coefficient −ρ_t(1). The complete label remains 55. (2 points; GK.2.)

**Exercise 185 (6 points).** Prove nonextension of the local translation for
the smooth complete bump metric GK.6. State the scope of this example.

**Solution.** The conformal factor lies between 1 and exp(ε/e), so metric
completeness follows from Euclidean completeness. The inside bump is
analytic with nonzero Laplacian at its centre, hence its curvature cannot
vanish on an open disk inside. Flat-neighborhood points form exactly the
connected exterior E. (2 points.) Any global isometry preserves E and
there is a Euclidean local isometry. Preservation of the zero Euclidean
connection makes its second coordinate derivatives zero; connectedness
makes it one rigid affine map. Agreement with the initial translation
forces that same nonzero translation throughout E. That translation does
not preserve E, contradiction. (3 points.) Completeness of the ambient
metric and germ rigidity are the tested hypotheses. Completeness and
globalization of an entire holonomy pseudogroup are additional hypotheses
and are not refuted by this example. (1 point; GK.3.)

**Exercise 186 (8 points).** Explain why the compactification algebra is
separable and genuinely contains C0(T). Construct the ordinary section,
prove properness of the anchor, and explain why fibrewise evaluation
density is unnecessary.

**Solution.** Countable bisections, cutoffs and countably repeated localized
translations give countably many bounded continuous generators. Each
coarse generator is bounded by the compact-arrow upper control. Their
norm-closed nonunital *-algebra is separable and contains the faithful
central r*C0(T). All generators are supported over compact unit sets, so
a central unit approximate identity acts in norm. (2 points.) The spectrum
anchor is induced by that nondegenerate inclusion. Evaluation at 1_x is
a character over x, and evaluation of every algebra function there varies
continuously in x. It gives an actual continuous section and surjectivity.
(2 points.) For compact K choose f∈C_c(T) equal to 1 on K. The closed set
p^{-1}K lies in the compact level set {|f̂|≥1}; hence p is proper. (2 points.)
The regular evaluations are dense in the whole spectrum because any
nonzero function vanishing on them would be the zero concrete function
on G. Points over x can nonetheless be limits of evaluations over nearby
units. Every CND test is extended over an open unit neighborhood with
locally varying bisection arrows; total density in that open set is enough.
(2 points; GK.5–GK.6.)

**Exercise 187 (8 points).** At a common range z of the arrows (g_i,y_i) in
G⋉Y, derive the same-source expression of the CND matrix. Prove the compact
sublevel statement in a way that covers limiting-unit spectral points.

**Solution.** One has y_i=g_i^{-1}z. At z=ev_h, r(h)=r(g_i), the matrix
entry ψ(g_i^{-1}g_j,y_j) equals k(g_i^{-1}h,g_j^{-1}h). The two arguments
have the same source s(h). Choose local bisections of g_i over a common
open range neighborhood; the finite matrix and its zero-sum quadratic
expression vary continuously over that neighborhood. The source-fibre
CND inequality passes from the dense regular evaluations to all z.
(3 points.) Given compact endpoint sets L_s,L_r and a bound R, put their
anchor images inside the interior of a compact K. Outside the compact
C(K,R+1), a sufficiently small bisection has endpoints in K and avoids
that C. Its kernel values at all regular evaluations exceed R+1, so its
spectral values are at least R+1. Every relevant sublevel arrow therefore
lies in C×_{s,p}L_s. (3 points.) The endpoint and sublevel conditions are
closed there by continuity, making the sublevel compact. Using a compact
neighborhood K rather than only the precise compact anchor images is
what covers spectral limits from nearby units. (2 points; GK.6.)

**Exercise 188 (6 points).** Derive the Hilbert Gram form and exact affine
cocycle. Explain continuity and the displacement identity.

**Solution.** Apply CND to δ_a−δ_unit. It yields the positive Gram form
½(ψ(a)+ψ(b)−ψ(a^{-1}b)), so the null quotient and completion are Hilbert
spaces. (2 points.) Differences e_ab−e_a have mutual squared distances
ψ(b^{-1}c); polarization makes L_ae_b=e_ab−e_a a surjective isometry.
The two equalities L_aL_b=L_ab and e_ab=e_a+L_ae_b prove exact affine
composition. (2 points.) Local bisection generators have continuous Gram
entries; finite linear combinations and their local uniform limits give
the continuous Hilbert field. The formulas on this dense generating
family prove action continuity. The displacement squared is ||e_a||²=ψ(a),
so it has precisely the properness already proved. (2 points; GK.7.)

**Exercise 189 (8 points).** Verify all coarse inputs for Γ⋉T from GK.16,
checking the right-invariant length convention. Explain why general eligible
holonomy requires the complete geometry of Section 11AF as well as its metric.

**Solution.** For g=(δ,γx),h=(γ,x), composition is gh=(δγ,x), and the
right-invariant difference is (δγ)γ^{-1}=δ. Thus both coarse bounds depend
on |δ| independently of h. A compact arrow set has finitely many discrete
δ-labels, giving an upper bound. (2 points.) A bounded kernel value bounds
|δ| by properness of ρ_-, leaving finitely many labels. Compact source
endpoints put these arrows in a finite union {δ}×K; compact range conditions
are closed. This is the lower compactness control. CND is the Hilbert
squared-distance identity, and continuity follows from fixed discrete
labels in each pair chart. (3 points.) General holonomy requires
a continuous same-source CND k with both controls on its actual arrows.
First-jet separation gives no proper lower bound by Exercise 2. Local
isometries alone need not globally extend by Exercise 3. EG.1–EG.8 prove
complete frame geometry, closure products and actual local matrix labels.
EK.1–EK.5 then patch the labelled positive kernels and transfer the resulting
CND kernel through the compact frame fibres. These proved constructions,
rather than first-jet separation alone, supply both required controls.
(3 points; GK.8–GK.9 and Complete Riemannian holonomy and the original-unit kernel.)

**Exercise 190 (6 points).** Recover the coarse input from a proper-surjective
compact-fibre Y with section y0 and proper CND ψ on G⋉Y. Check the CND
arrow convention and both controls.

**Solution.** Put k(h,l)=ψ(hl^{-1},l·y0(x)) for h,l∈G_x. Continuity uses
only the actual section, action and structure maps. The inverses of the
arrows (h,y0(x)) have the common range y0(x); their inverse-products are
exactly the arrows defining k(h,l). Their CND inequality proves the
same-source condition. (2 points.) For compact C, C×_{s,p}Y is a closed
subset of C×p^{-1}(sC), hence compact. The maximum of ψ there bounds
k(gh,h)=ψ(g,h·y0(sh)) uniformly in h. (2 points.) For compact K, the
two endpoint sets p^{-1}K are compact. Project the compact ψ-sublevel
over them to G. That compact projection contains every g with endpoints
in K and some h satisfying k(gh,h)≤R. Thus it is C(K,R). (2 points; GK.10.)

### Exercises 191–194. Probability, Bott factors and original-unit transfer (68 points)

**Exercise 191 (16 points). Anchored probability compactification and proper kernel integration.**

Prove the properness of the probability anchor, the integrated kernel,
and the collapse homotopy; distinguish the two section roles.

**Solution.** A compact unit set has compact Y-preimage, whose
probability functionals are compact by their closed positivity/unit
conditions. The closed single-fibre condition (PK.4) gives the compact
X-preimage (4 points). If the integrated kernel is ≤A, positivity and
compactness force a point where the original kernel is ≤A; the proper
anchor puts both its Y-units in specified compact sets. Original
kernel properness then bounds the actual arrow, giving (PK.9)
(4 points). The square-cutoff sum gives a genuinely equivariant map
Φ; the convex ordinary homotopy has compact images on compact source
supports and so defines (PK.13) (4 points). The section x0 is ordinary
and makes v0 possible. It is never declared equivariant. In contrast,
Φ is equivariant and defines the coefficient homomorphism e_P; it
does not define a scalar equivariant evaluation (4 points).

**Exercise 192 (20 points). Actual anchored Bott factors from a coherent probability-kernel input.**

Locate the five non-scalar checks that make these Bott factors actual.

**Solution.** Bisection Gram sections and affine generators construct
one continuous field and exact composable action (4 points). The
common compact-arrow convex tests give a fixed Θ with graph-continuous
translated frames and cocycle, retaining the identity part in (BA.12)
(4 points). The phase cone uses both position and momentum and its
height cross term; kernel properness bounds its actual arrow transporters
(4 points). The source B has stronger anchored finite factors with
element-valued Choi entries, so the *proved* proper-slice/open-MV/
telescope chain gives the actual lift without arbitrary source quotient
sections (4 points). The ordinary vacuum proof is one completed
R_X-tensored homotopy and identity module over all X; it is not a list
of fibre indices and it makes no equivariant gamma assertion (4 points).

**Exercise 193 (16 points). Compactness under the anchored family extension.**

Explain why a multiplier-valued coefficient homomorphism need not carry every compact to a compact. Prove the source-localized compactness used in AS.8–AS.10, then identify the ordinary contraction endpoint and the balanced unit in AS.19.

**Solution.** A multiplier left action on a standard module is compact only when the multiplier is algebra-valued. Thus a finite-rank family operator cannot simply be declared compact after multiplier extension (4 points). For each homogeneous source b and family vector xi, the inner product in AS.8 is approximated by sums b* h_i^B b tensor p_i, which belong to B tensor_R P. The compact-root functional calculus makes T_xi L_b compact; AS.9 then makes every extended finite rank compact after the source action. Approximation, central cutoffs and the bound on the source tail prove AS.10 on the completed module (4 points). The half-mass homotopy stays over the same original unit and has compact parameter image on each compact central source support. Its endpoint is 1_B exterior_R theta_0, with the tensor identification in AS.14 (4 points). The probability homotopy ends at e_0 exterior_R 1_P. All factors are even; product interchange uses the balanced unit R=C0(T), yielding For_G(eta)=epsilon theta_0 and For_G(eta D)=epsilon (4 points).

**Exercise 194 (16 points). Kernel-to-original-unit dual with the exact transverse normalization.**

State the domain and codomain of every factor in (OU.2), explain
why the two ordinary endpoint maps are sections, and prove the
original local sign is unchanged.

**Solution.** The exact types are (OU.3); the two middle factors
have coefficients B⊗_R P and C0(X)⊗_R P. The latter collapses by
an equivariant central homomorphism, not a scalar evaluation
(4 points). Properness of X→T makes u an algebra-valued homomorphism;
the ordinary proper section x0 gives v0 with v0u=id_R. The family
section m0 keeps each original unit and pulls back OC's whole-family
identity to θ0D0=1_R (4 points). The completed section-valued
homotopies and BA's actual whole-R_X vacuum identity give (OU.5)
and (OU.6), so after forgetting only the anchor the original disk
class composes with 1_R (4 points). All comparison factors are even;
no odd Clifford factor is interchanged. The original q, orientation
and full right coefficient remain, proving (OU.7) with its original
sign (4 points).

### Exercises 195–196. Smooth coarse control and the normal phase (28 points)

**Exercise 195 (12 points). Smooth approximation and properness.**

Explain why approximation of squared distances by a bounded additive error
is unnecessary, calculate the two preserved controls, and identify the
missing derivative assertion.

**Solution.** The squared-distance difference can grow with the original
distance, but its square root changes by at most 2 epsilon, which is all
coarse properness needs (3 points). A compact-arrow bound sqrt(U_C) thus
becomes sqrt(U_C)+2 epsilon, proving (SK.5) (3 points). A new sublevel R
is contained in the old sublevel (sqrt(R)+2 epsilon)^2, so its actual g
labels remain in the original compact transporter (3 points). Smoothness
on each labelled chart supplies no uniform derivative bound over all h;
claiming a bounded transported normal connection from it would require
an additional proof (3 points).

**Exercise 196 (16 points). Normal-flow and phase failure.**

Check both coarse bounds, derive the exact norm-flow lower bound,
and prove the noncompact bounded-phase commutator without relying on
the unbounded derivative assertion.

**Solution.** The sine difference has magnitude at most two, giving
(CK.3). Compact generator sets have finitely many labels, and a low
increment bounds |δ| by √R+1/4, giving both GK.4 controls (4 points).
For n=4m, the difference-of-sines identity reduces at t_n to (CK.6);
its cosine angle lies below π/3. Hence Δ_n≥1 and the square differs
from one by at least 17/64 (4 points). The quadratic sine term has the
listed frequencies, none at −(n+1)². That coefficient of a is therefore
i/8. Applying the two Fourier multipliers to u_n gives (CK.8), with
norm at least 1/16 (4 points). Orthonormal u_n are weakly null, whereas
their commutator images are not norm null. Finite-rank approximation
therefore rules out compactness, independently of (CK.5). Tensoring
by a unit Clifford involution preserves this lower bound (4 points).

### Exercises 197–200. Frame geometry and labelled-kernel patching (32 points)

**Exercise 197 (8 points). the closed local return group.**

Let C and the smooth ambient product μ be as in EG.3. Put
Y_v(z)=D2μ(z,0)v. Explain why a sequence y_n∈C tending to zero with
y_n/||y_n||→v forces Flow^t_{Y_v}(0)∈C for small t of either sign.
Prove that the resulting tangent set V is linear and that C is a smooth
submanifold near zero. Why does this argument retain a uniform time interval
when vectors in V converge?

**Solution.** Write δ_n=||y_n||. Taylor expansion gives
μ(z,y_n)=z+δ_nY_v(z)+δ_n o(1), uniformly on a fixed compact box. For kδ_n
bounded, the Euler error is at most const·(o(1)+δ_n), using
e_{k+1}≤(1+Lδ_n)e_k+δ_n o(1)+O(δ_n²). All iterates are in C; its closedness
puts their limit curve there. The inverse has derivative −I, so it supplies
negative t. (2 points; EG.4.)

Linearity of Y_v proves scalar closure. The small product of the curves
for v,w is δ(v+w)+O(δ²), and its repeated products give the curve for v+w
by the same error bound. Left translations by C preserve Y_v on C, so short
segments concatenate as exact flow segments. Bounded ||v|| gives a common
small time before leaving the fixed box. ODE continuous dependence and
closedness therefore give closure of V under limits. (3 points; EG.4.)

Every tangent direction of C lies in V by the first part, and every nonzero
v∈V is realized by E(tv)=tv+O(t²). For a basis of V, the ordered curve
product S(t) has full rank and lies in C. Choose a complementary V'. The
inverse-function chart Θ(t,w)=μ(S(t),w) is nonsingular. If Θ(t,w)∈C, its
inverse left translation puts w∈C∩V'. That intersection is {0} near zero:
otherwise a limiting unit direction would be in both V and V'. Thus C is
exactly the embedded S-chart, with smooth local operations. (3 points.)

**Exercise 198 (8 points). actual finite-label compactness.**

Let D:O1→GL_m(R) be the injective local primitive in EG.8. Explain why
value-and-derivative injectivity alone does not prove that finitely labelled
arrows with compact endpoints form a compact set. Prove the needed statement
using the uniform extension O0 and formula τδ=D^{-1}δD. Include how this
passes to O×V and how orbit fullness is obtained there.

**Solution.** An injective continuous jet map need not be proper. A sequence
of distinct irrational-rotation labels can have convergent jets while escaping
every compact discrete-label set; GK.2 gives that actual metric example.
This rules out the proposed inference. (2 points.)

All germs with endpoints in O extend to the same connected O0 with image
in O1, by complete transverse flows and the uniform endpoint boxes.
Preservation of A and differentiation of D(hz)D(z)^{-1} make this ratio
constant. Thus every admitted δ has one actual map τδ on O0. Its arrows
with endpoints K,L⊂O are a genuine bisection's closed source subset
{z∈K:τδ(z)∈L}; that subset is compact. A finite union remains compact.
The extension is the missing topological input. (4 points; EG.2, EG.8.)

The actual foliated product Ψ:F_w×V→Q^{-1}V carries these maps to
τδ×id_V, with the same discrete labels. Compact endpoints in O×V give
closed compact source subsets again. Each leaf is dense in its closure,
so it meets a foliation box around O; its plaque path meets O. Transport
through Ψ proves orbit fullness on every fibre. (2 points; EG.5, EG.8.)

**Exercise 199 (8 points). right cosets and nonabelian cancellation.**

For Λ_m⊂Λ, explain why the extension of (EK.2) uses the right cosets Λ_m c.
For actual composable arrows g,h, β_i(rg), β_k(sg), β_j(sh), derive (EH.1)
without assuming commutativity. Finally, for v0=(1,0), v1=(3/5,4/5), compute
the midpoint V and its normalized W, and prove the normalization estimate
used for the upper control.

**Solution.** If a=h_a c and b=h_b c, then ab^{-1}=h_a h_b^{-1}. Moreover
ab^{-1}∈Λ_m forces a,b into the same such coset. Orthogonal summands for
distinct right cosets therefore preserve every finite right-difference
bound. If an input map uses a^{-1}b instead, replace F(a) by F(a^{-1}).
(2 points; EK.1.)

Expand the inverse in the exact order:
(β_k h β_j^{-1})^{-1}=β_j h^{-1} β_k^{-1}. Multiplying it after
β_i g h β_j^{-1} cancels β_j^{-1}β_j and h h^{-1}, leaving β_i g β_k^{-1}.
Applying the homomorphism λ gives (EH.1); no factors are commuted.
(2 points; EK.2.)

V=(4/5,2/5), ||V||²=4/5, and W=(2/√5,1/√5). Nonnegative Gram entries
give ||Σp_i v_i||²≥Σp_i². If all pairwise feature defects are ≤ε, both
convex norms are ≥√(1−ε), and their difference is ≤√(2ε). For nonzero
v,w, adding and subtracting w/||v|| gives
||v/||v||−w/||w||||≤2||v−w||/min(||v||,||w||).
The half squared-distance defect is therefore ≤4ε/(1−ε)≤8ε when ε≤1/2.
(4 points; EK.2.)

**Exercise 200 (8 points). whole-base lower control and frame averaging.**

Explain why normalizing the local features retains a uniform lower control
for every unrestricted h. Prove that invariant outer weights and the CND
series preserve it. Finally prove that Haar averaging returns it to the
original G, although the witness h may have source outside K.

**Solution.** With at most M active range weights over compact endpoints,
||V||≥1/√M. Take the off-finite-label feature bound η=1/(2M). The proved
finite-label compactness and actual multiplication give a compact exceptional
set of g. Outside it every cross feature inner product is ≤η; normalization
increases the bound by at most M, so Q(gh,h)≤1/2 uniformly in h and j.
(3 points; EK.3.)

Only finitely many invariant patches meet q(K). For any N, their finitely
many exceptional sets for n≤N have a compact union. Outside it every
Q_n(gh,h)≤1/2, so Σ_n(1−Q_n)≥N/2. Choosing N/2>R gives the required
lower control. The upper defects are summable on each compact left-increment
strip. Continuity follows from local uniform convergence, and CND follows
by applying the zero-sum inequality to finite partial sums and taking limits.
(3 points; EK.4.)

If the Haar average is ≤R, one frame u has integrand ≤R. The relevant H
increment is (g,h·u); its source and range frames lie in P_K, regardless
of s(h). Since P→T is proper, P_K is compact. The proved H control places
that increment in a compact set. Projection to G gives the compact set of
original arrows. Upper control uses compactness of all lifts of a compact
C⊂G. (2 points; EK.5.)

### Exercises 201–203. Affine connections, collapse and frame descent (48 points)

**Exercise 201 (16 points). Affine origins and actual arrow control.**

Explain why c_ij(1_x) is not necessarily 1, why (LF.7) nevertheless
has zero identity displacement, and where finite-label properness
enters. Which global and physical interfaces do not follow from this local construction alone?

**Solution.** The two beta sections need not agree; their actual
composition has label t_ij (4 points). The origins obey
omega_i=A_(t_ij)omega_j because their affine averaging weights sum
to one, so subtracting omega_range in (LF.7) cancels that transition
exactly (4 points). Compact-unit origin bounds turn a small B_g into
a bound R+2W_K for a group displacement; only finitely many labels
remain and the actual conjugated-arrow properness returns a compact
set of original arrows (4 points). This local construction alone does not patch and descend the
connections globally. The next two subsections prove that passage;
carrying it through the inverse KK modules to a closed physical
graph operator still requires proof (4 points).

**Exercise 202 (16 points). Flat collapse and the geometric graph weight.**

Why are the flat coefficient and additive blow-up both used in
(GJ.7)? Explain the zero-stratum connection and derive (GJ.11).

**Solution.** The additive tau makes the inverse norm vanish as a
factor disappears. The multiplier b_a makes the conjugation defect
b_a S vanish there, which is required for actual localized module
compactness (4 points). A scalar or module coefficient and every
one of its derivatives vanish on a collapsed factor; differentiating
on the full potential product therefore descends through its
quotient and respects relations (4 points). Differentiate
T=tau+b Theta. The tau term is at most 4L rho^3/K0. For the
Theta term use lambda/(tau+b lambda)^2<=1/(4tau b) and
|v b|/b<=2L rho^{-3}+J. The bounded Theta derivative term is at
most H/tau. Substitution gives exactly (GJ.11) (4 points).
Neither that estimate nor a fibre oscillator index supplies the
completed physical sum's domain and class comparison (4 points).

**Exercise 203 (16 points). Haar-marginal connection and compactness failure.**

Explain the Haar-marginal condition, the purpose of the lifted
potential-product path, and the compactness failure at FD.7.

**Solution.** The marginal makes actual frame transport preserve
the base and provides a canonical ordinary continuous section;
properness of both anchors makes its constrained probability
fibres compact (4 points). A probability may charge collapsed
factors with no determined new coordinate; lifting it to the full
potential product supplies coordinates along a short path. Flat
generator derivatives descend and are independent of that lift,
so differentiation respects actual scalar/module relations
(4 points). Covariance follows from the invariant Levi-Civita
horizontal lift and actual H field transport (4 points). The
pointwise inverse leaves infinitely many orthogonal frame
Fourier modes at a fixed nonzero norm; it is not compact, so an
integrated geometric confining weight still needs proof (4 points).

### Exercises 204–206. Proper covariance, graph coordinates and probability jets (40 points)

**Exercise 204 (16 points). Proper normalized modules and the reduced norm.**

Explain where properness enters (PF.8), why the two module identifications
are legitimate, and which analytic assertion remains unproved.

**Solution.** The central Phi*(A) makes the actual multiplication maps
balanced and preserves their inner products; standard-module density
and pi_D's essential nondegeneracy prove they are onto (4 points).
The resulting P representation is covariant and nondegenerate, but no
operator commutator with P is inferred (4 points). Scalar convolution
on a regular crossed-P source column is lambda_x(f) tensor 1. Proper
P makes its full/reduced quotient an isomorphism, so every integrated
covariant representation of this P satisfies the reduced scalar norm
bound (4 points). The completed graph module is specified by the actual
correspondence, but a geometric normal connection, self-adjoint graph
operator and its class comparison still need proofs; neither its field
index nor a generic unbounded representative gives them (4 points).

**Exercise 205 (16 points). Actual regular-arrow absorption.**

Why does (RG.3) have stronger equivariance than its input, where is
proper support used, and why does (RG.12) have index one?

**Solution.** Left translation preserves s(g), so it applies the same
ordinary source-coefficient unitary on both sides of (RG.2) (4 points).
For a central compact support, GP.11 bounds all active cutoff arrows
in one compact set; finite bisection sections therefore prove actual
completion and adjointability of V, rather than fibrewise formulas
(4 points). The invariant projection, countable E_1/Q decomposition
and copy shift give the onto equivariant absorption without moving
arrow coordinates (4 points). F_reg is the original eta D product
plus exactly degenerate cycles; its ordinary whole-base class is
the already proved 1_R. Neither the regular representation alone nor
the copy count is the normalization argument (4 points).

**Exercise 206 (8 points). The outer probability connection.**

Why is the compact surjection (PJ.5) useful, and why can one
not differentiate every continuous probability function?

**Solution.** Lifting the outer measure through that surjection
provides a fixed compact parameter probability. Inner normal
jets are then norm-smooth in one fixed field, so integration
respects scalar and null-module relations and proves metricity
and closability (4 points). Only the dense polynomial algebra
of the smooth inner observables has the displayed jets; an
arbitrary continuous function of a convex probability path
need not be differentiable. Density does not extend the
derivative to all continuous functions (4 points).

### Exercise 207. Faithful normal derivatives and the compact-frame quotient (12 points)

**Exercise 207 (12 points).**

Why is (IJ.8) well defined, why does quotient preserve compact
inverse, and which derivative has not been proved in IJ.7?

**Solution.** Global module equality is equality of the concrete
bounded H1-valued arrow functions by (IJ.4); derivatives of a
zero function are zero and faithfulness makes their module class
zero (4 points). A frame section chooses exactly one representative
of each compact-frame orbit, so the quotient resolvent is a
restriction, not an operator on new Haar Fourier modes; finite-rank
approximations retain their norms (4 points). Smoothness of the
field generators and a compact weight do not estimate nabla Theta
or its relative forms. The convex selection in IJ.7 imposed
transport/graph tests but no normal derivative tests (4 points).

### Exercises 208–209. Completed normal forms and action domains (16 points)

**Exercise 208 (8 points). The completed normal form.**

Prove the exact completed normal form (CW.7) without applying Theta to
nabla_v w, and derive the constant in (CW.9). (8 points)

**Solution.** Write u=h xi,w=h eta on the core. Metricity gives
delta_v<h xi,eta>=<nabla_v(h xi),eta>+<h xi,nabla_v eta>.
Subtract the two specified terms; substitute
nabla_v(h eta)=h nabla_v eta+(partial_v h)eta. The h nabla_v eta terms
cancel and the remaining expression is -<xi,(partial_v h)eta>.
Its bounded coefficient extends it in the Theta graph norm (4 points).
Apply the positive-order cross-term inequality to S*Theta u and
(S_v')*Theta u, then choose t=2 C_v^2/a and maximize
t lambda-(a/2)lambda^2, whose maximum is t^2/(2a).
The remainder is 2 C_v^4/a^3; the case C_v=0 is immediate (4 points).

**Exercise 209 (8 points). Noncommuting weights and action domains.**

For n=2 compute both norms in (CW.11), and prove the two domain claims
in (CW.12). Explain which additional action condition is missing. (8 points)

**Solution.** For n=2, A=4,B=256, so the two-sided norm is 252/(4 times256)=63/256
and the one-sided norm is 252/16=63/4 (4 points).
The vector u is square summable and Theta u has squared norm
sum 1/n^2; after swapping, the block norms are 2^(n^3-n)/n and do not
tend to zero, so Pu is outside Dom Theta. The missing condition is
preservation of the completed domain by each actual transport, together
with its compact norm-continuous weight defect and displacement-domain
control. Smooth compact resolvents alone do not give it (4 points).

### Exercises 210–217. Orbit domains and completed form-domain sums (56 points)

**Exercise 210 (4 points). All normal jet sums.**

Prove all jet sums in JD.3 converge.

**Solution.** distribute m derivatives among n cutoffs and the final column;
   bound the finite sum by Cauchy-Schwarz, sum each differentiated
   letter using (JD.4), and every other letter by Q. This gives
   (JD.6), summable because Q<1. Closedness of each successive
   compact derivation identifies the norm limits.

**Exercise 211 (4 points). The strict range exponents.**

Locate both strict inequalities \(\alpha\)<1/2 in the
   completed range proof.

**Solution.** the norm integral at zero is
   bounded by \(\lambda\)^(-\(\alpha\)-1/2), which is integrable precisely
   below 1/2; the adjoint strong-limit test uses uniform convergence
   of (a+t)^(-\(\alpha\))a^(1-\(\alpha\)), whose limiting exponent 1-2\(\alpha\)
   must be positive. No assertion is made at the endpoint.

**Exercise 212 (4 points). The compact square-root derivative.**

Prove (JD.15) without a module polar operator.

**Solution.** in each fibre the two coefficient weights in an entry
   have sum at most one. Weighted Cauchy-Schwarz bounds that entry;
   summing and interchanging i,j gives at most twice the fibre
   Hilbert-Schmidt square norm of S'_i. Taking the supremum gives
   the module estimate. The actual module limit is then constructed
   by approximating S'_i by hR, not by gluing the polar operators.

**Exercise 213 (4 points). The completed half-power failure.**

Verify (JD.20) and (JD.23).

**Solution.** the two order
   remainders are (3/4)PaP and (3/2)a. Substituting (JD.21) gives
   det(h)/\(\varepsilon\)^2->11/4. The numerator in (JD.22), divided by
   \(\varepsilon\), tends to 9/4-3/2=3/4; the denominator divided by
   \(\varepsilon\) tends to sqrt(33/8). Squaring their ratio gives 3/22.

**Exercise 214 (2 points). The geometric scope.**

Does JD.7 obstruct the original foliation theorem?

**Solution.** no. It refutes only a claimed general implication
   from smoothness and fibre order comparison to the completed
   half-inverse module domain. A better simultaneous weight or a
   proved form-domain graph construction can still solve the
   physical operator problem.

**Exercise 215 (12 points). The completed oscillator cross form.**

(a) Derive the coefficient 1/sqrt(2) in
(FQ.4). (b) Prove (FQ.5) and identify its finite constant.
(c) State what is missing before (FQ.8) proves Connes's original
graph operator. Score (a), (b) and (c) at 4 points each.

**Solution.** (a) The two adjoint creation/contraction terms give
2 sqrt(2) Re of the array inner product. Bounding it by
sqrt(2)||M|| times the sum of array norms gives sqrt(2)||M||
dGamma_total(\(\Theta\)^2); the oscillator-square bound (FQ.2)
is twice that total energy, hence 1/sqrt(2).
(b) The compact plateau remainder with norm a/sqrt(2) contributes
a/2 times C energy. The finite part obeys the corrected B(S)^2
inequality and scalar Young inequality, contributing the other
a/2 and ||\(\Theta\) e M e \(\Theta\)||^2/(2a).
(c) An actual differentiable inverse-module representation,
simultaneous action and displacement domains or a sufficient
alternative phase proof, a actual physical realization of the self-adjoint sum with its specified maximal domain, local compactness and coefficient
commutators, and the whole original Bott +1 comparison remain
to be proved. Fibre indices and an arbitrary Hilbert-space
unitary do not establish them.

**Exercise 216 (10 points). A common inverse-power derivative.**

(a) At \(\alpha\)=1/4 and \(\beta\)=3/8 compute the small-resolvent exponent and the two integral constants in FB.6. (4 points) (b) Prove the regularized compact derivatives converge in norm and use closedness to identify the derivative of h^\(\alpha\). (4 points) (c) Explain which actual module premise is still required and why a bounded-fibre factor does not supply it. (2 points)

**Solution.** (a) The exponent \(\alpha\)+\(\beta\)-3/2 is -7/8. Its integral over (0,1) is 8, and the large-t exponent \(\alpha\)-2=-7/4 integrates to 4/3 over (1,infinity). Thus the norm bound is c_(1/4)(8||B||+(4/3)||\(\delta\) h||). (b) For each positive t the regularized resolvent sandwiches converge in compact norm. Their small- and large-t majorants are those just integrated, uniformly in the regularization. Norm dominated convergence gives \(\delta\) k_tau->X. Functional calculus gives ||k_tau-h^\(\alpha\)||<=\(\tau\)^\(\alpha\); the closed derivation therefore puts h^\(\alpha\) in its domain with compact derivative X. (c) The actual equality S-prime=h^\(\beta\) B must hold for a bounded adjointable module map B. Fibre order or bounded fibre preimages need not give a continuous completed-module preimage; JD.18–JD.23 exhibit that failure at the half-power.

**Exercise 217 (16 points). The actual onto-resolvent graph proof.**

(a) Prove SD.1's strong commutator
convergence, including its adjoint-domain use. (b) Explain
why the natural T in SD.4 has exactly the constructed
maximal domain. (c) Explain why an unspecified positive
square root cannot replace this argument. Score (a) and (b) at 6 points each and (c) at 4 points.

**Solution.** (a) The bound (SD.3) and direct convergence on
Dom A give strong convergence everywhere. For u in Dom
X*, test against Dom X to obtain (SD.4); the approximants
J_nu belong to Dom X and converge in its graph to u,X*u.
Thus the adjoint domain is contained in its closure.
(b) Its closed natural graph contains iXbar+\(\Gamma\) K/2.
The latter has onto 1+operator by the Neumann-series
argument. Symmetry of S and injectivity of its oscillator
resolvent make the extension injective; subtraction of
the onto preimage proves equality, including domains.
(c) A positive energy square root has no specified odd
grading, normal principal action, affine representation
or Bott comparison. SD.13 instead proves onto resolvents
of the actual expression C+\(\Gamma\) A on its stated domain.


### Exercises 218–220. Minimal normal domains and confining regularization (32 points)

**Exercise 218 (10 points). The coefficient graph cutoff.**

(a) Derive the constant \(5/4\) in NR.3. (4 points) (b) Prove the adjoint-domain membership \(J_\epsilon u\in\operatorname{Dom}A\) and explain why NR.4 proves essential self-adjointness. (6 points)

**Solution.** (a) In \(\epsilon Q_\epsilon BMBQ_\epsilon\), both \(BQ_\epsilon\) factors have norm at most \(1/(2\sqrt\epsilon)\), giving \(\|M\|/4\). Each remaining term has one factor \(\epsilon Q_\epsilon\) of norm at most one and one \(B^2Q_\epsilon\) of norm at most one, giving \(\|M\|/2\). Their sum is \(5\|M\|/4\). (b) NR.1 gives \(Bu\in\operatorname{Dom}A^*\). The inverse preserves that domain, so \(BQ_\epsilon u=Q_\epsilon Bu\) is also there. The premise \(B\operatorname{Dom}A^*\subset\operatorname{Dom}A\) puts \(J_\epsilon u=B(BQ_\epsilon u)\) in the minimal domain. NR.4 and strong convergence then approximate every adjoint graph vector in the original graph. A symmetric operator's closure is contained in its adjoint, so these two graphs are equal.

**Exercise 219 (12 points). The exact natural sum domain.**

(a) Prove \(B=-i\Gamma R_\lambda\) is self-adjoint and compute \(M=[A,B]\). (4 points) (b) Why does the onto operator in NR.7 identify the full natural domain of \(T\)? (4 points) (c) Prove that NR.9 is a joint graph core and state which normal-domain equality was not assumed. (4 points)

**Solution.** (a) Anticommutation gives \(\Gamma R_\lambda=-R_\lambda^*\Gamma\), hence \(B^*=iR_\lambda^*\Gamma=-i\Gamma R_\lambda=B\). Commutation of \(A\) with \(\Gamma\) gives \(M=-i\Gamma K_\lambda\). (b) The natural \(1+T\) is injective by symmetry of \(S\) and injectivity of \(R_\lambda\). For any vector in its domain, the onto smaller operator supplies a vector with the same image. Their difference is in the kernel of the natural operator, so they coincide. This proves equality of operators and domains. (c) An \(A\)-graph core is an \(\overline X\)-graph core because \(X=BA+M/2\). Approximate \((C+i\lambda)\psi\) in that graph. Each of the three bounded graph expressions in NR.9 then converges, giving simultaneous \(C,A,S\) approximation. The proof did not assume \(\operatorname{Dom}A=\operatorname{Dom}A^*\) or choose a normal self-adjoint extension.

**Exercise 220 (10 points). A non-self-adjoint normal operator with a self-adjoint sum.**

(a) At \(x=1/2\), \(\lambda=2\), compute \(h\), \(B^2\) and \(J_\epsilon\) for \(\epsilon=1/260\). (4 points) (b) Explain why \(R_\lambda H^1\subset H^1_0\) and why the sum has compact resolvent. (4 points) (c) Does this prove the original groupoid Dirac class? (2 points)

**Solution.** (a) \(h=1/16\), so \(B^2=h^2/(1+4h^2)=1/260\) times the identity. Thus \(J_\epsilon=B^2(B^2+\epsilon)^{-1}=1/2\) times the identity. (b) The resolvent is a bounded matrix function with bounded derivative and zero endpoint values. Its product with an \(H^1\) vector is in \(H^1\) and has zero traces. NR.12 controls the \(H^1\) norm by the sum graph norm; the finite-rank interval averages prove the compact inclusion, hence compactness of the resolvent. (c) No. The actual differentiable inverse-module realization, domain regularization in that realization, groupoid phase compatibility, local graph compacts and original \(+1\) comparison remain to be proved. The example verifies the stated alternative analytic route.



### Exercises 221–223. Physical regular domains and scalar compactness (36 points)

**Exercise 221 (12 points). Finite contractions and the minimal domain.**

(a) Derive RGN.4 with its sign. (4 points) (b) Explain why its right-hand side is in \(L^2\) and why RGN.3 applies to a maximal-domain vector. (4 points) (c) Prove the graph-closure passage in RGN.6 and explain the role of compact arrow support. (4 points)

**Solution.** (a) Differentiate \(\langle\eta,u\rangle_R\). The derivative of \(u\) gives \(\langle\eta,A_{\max}u\rangle_R\), while the auxiliary derivative of \(\eta\) is multiplied by the normal principal coefficient \(-i\mathsf c_j\), giving exactly the minus sign in RGN.4. Unit spin connection and density terms already belong to \(D_T\). (b) Counting Cauchy–Schwarz bounds the contractions by the module norms of \(\eta\) and its compactly supported first jets. Both \(u\) and \(A_{\max}u\) are in the physical \(L^2\) space. The distributional mollifier argument of RGN.3 therefore applies, without an assumed self-adjoint scalar boundary realization. (c) The finite outputs have \(H^1\) unit coefficients on compact arrow supports, so buffered coefficient mollification places them in the minimal graph. Simultaneous norm convergence of \(T_n\) and all \(\partial_jT_n\) gives convergence of \(T_nu\) and \(A_{\min}T_nu\); closedness proves RGN.6. A merely compact unit support would not supply compact arrow-supported output approximants.

**Exercise 222 (12 points). What scalar localized compactness needs.**

(a) Prove the bound on \(A(S+i\lambda)^{-1}\) from the exact inverse factorization. (4 points) (b) Prove scalar compactness of the finite-core approximants in RGN.13. (4 points) (c) Explain why local module compactness is sufficient and why compact fibres alone are not. (4 points)

**Solution.** (a) The bounded operators \((1-i\lambda R_\lambda)L_\lambda^{-1}\) and \((S+i\lambda)^{-1}\) give bounded \(C\)- and \(S\)-images; subtracting them in \(\Gamma A=S-C\) gives the displayed \(A\)-bound. (b) Each finite range contraction of a localized resolvent image is bounded in \(H^1\) by RGN.4 and RGN.3. Finite-rank cube averages approximate these coefficients in \(L^2\). Their multiplication by fixed compact arrow sections is bounded; finite sums therefore give compact physical operators. (c) Only the fixed compact \(\chi R_\lambda\) is approximated in module norm, and RGN.13 transfers its error to the physical operator. Pointwise compactness supplies neither a norm-controlled finite-row approximation nor the unit \(H^1\) control that makes those rows scalar compact.

**Exercise 223 (12 points). Two exact finite illustrations.**

(a) Let \(\xi=(3/5,4/5,0)\), \(\eta=(0,4/5,3/5)\), \(\xi'=(-4/5,3/5,0)\), \(\eta'=(0,-3/5,4/5)\), \(u=(1,2,3)\) and \(u'=(2,-1,1)\). For \(T=\xi\eta^*\), compute \(w=\eta^*u\), \(w'\), \(Tu\), and \((\partial T)u\). (6 points) (b) On the unit square, for \(v(x,y)=(x,2y)\), compute the exact squared error of the \(n\)-by-\(n\) cell-average approximation and compare it with RGN.12. (4 points) (c) State the still missing class interface. (2 points)

**Solution.** (a) \(w=17/5\), and \(w'=(\eta')^*u+\eta^*u'=6/5-1/5=1\). Thus \(Tu=(51/25,68/25,0)\). The derivative term is
\(\xi'w+\xi(\eta')^*u=(-68/25,51/25,0)+(18/25,24/25,0)=(-2,3,0)\).
For the one-dimensional flat normal differential, \([D,T]u=-i(-2,3,0)\), while \((Tu)'=(-53/25,71/25,0)\); subtracting \(Tu'=(-3/25,-4/25,0)\) verifies the same commutator. (b) In each cell the two coordinate variances add to \(5/(12n^2)\) after summing all cells. The derivative norm squared is \(5\), so RGN.12 gives the upper bound \(5/n^2\), with rank \(2n^2\). At \(n=4\) the exact squared error is \(5/192\). (c) One still needs the actual differentiable transfer of the normalized auxiliary phase or another sufficient geometric generator, its groupoid phase controls and its original Bott \(+1\) comparison. These finite calculations do not establish that interface.



### Exercises 224–226. A common normal and arrow weight (36 points)

**Exercise 224 (12 points).** (a,4) Evaluate OP.6 at \(x=s=t=1\), including the rational integral. (b,4) Explain why the positive double sum proves OP.7. (c,4) Prove the compact derivative-tail fact used to select \(t_n\).

**Solution.** \(a\) Its value is \(1/4\). Substitution \(r=u^2\) gives \((4/\pi)\int_0^\infty u^2/(1+u^2)^3\,du\). The three-denominator calculation gives \(pi/16\), hence \(1/4\). \(b\) Every \(F_{s,t}\) has the positive integral OP.6, so preserves positive operator order and obeys \(F_{s,t}(Ca)\le C F_{s,t}(a)\). The double sum converges uniformly because its terms are bounded by \(t_nt_m\), whose sum is finite. Limits preserve the resulting positive inequalities. \(c\) \(A_t=t(k+t)^{-1}\to0\) strongly on \(kE\) by scalar functional calculus, then on \(E\) by contractivity. On a rank-one compact, \(A_t\theta_{u,v}A_t=\theta_{A_tu,A_tv}\to0\) in norm. Finite rank approximation and \(\|A_t\|\le1\) prove it for every compact derivative.

**Exercise 225 (12 points).** (a,4) Prove that OP.8 has a norm limit using the zero value of \(g\). (b,4) Prove both strong limits in OP.10 and the existence of the adjoint of \(D\). (c,4) Derive the actual reverse-domain equality.

**Solution.** \(a\) On a small spectral interval \(g\) is uniformly small. There \(\sqrt{x}/(\phi(x)+\epsilon)=g(x)\phi(x)/(\phi(x)+\epsilon)\) differs from \(g\) by at most \(g\). On the remaining interval the positive minimum of \(\phi\) gives uniform convergence. OP.8 proves norm Cauchy convergence of the actual adjointable operators. \(b\) Multiplication of the finite sum by \(b^{1/2}\) gives \(U\phi_N(b)\), while \(D_N^*\phi(a)=\phi_N(b)\) and \(\phi(a)g(a)=a^{1/2}\) give the second identity. Uniform convergence of \(\phi_N\) makes both operator products norm convergent. Uniform bounds and density of the two square-root ranges give strong convergence on every vector. Inner-product limits give the adjoint identity. \(c\) With \(a\le C'b\), repeat the construction in reverse. The two resulting factorizations compose to the identity after multiplication by the injective \(\phi(a)\) and \(\phi(b)\). Their bounds \(\sqrt C,\sqrt{C'}\) give the equivalent completed inverse graph norms.

**Exercise 226 (12 points).** (a,4) Check the two order comparisons in the two-row example. (b,4) Prove the \(2/(3m)\) bound and give a rational upper bound at \(m=4,8,16\). (c,4) State exactly what the common weight has closed, and what remains before the original normalized graph class is proved.

**Solution.** \(a\) The conjugating matrix has squared Hilbert-Schmidt norm \(3-2\varepsilon^2+\varepsilon^4\le3\), so its operator norm is at most \(\sqrt3\). This proves \(b\le3a\); conjugation by the involution gives the reverse inequality. \(b\) \(A\le\sum4^{-n}=1/3\), and the first \(m\) denominator terms are at least \(1/2\), so the ratio is at most \(2/(3m)\). The three requested bounds are \(1/6,1/12,1/24\). \(c\) OP.1–OP.13 give one compact reciprocal with compact first normal derivatives, completed genuine-arrow transport and displacement domains, and the corresponding infinitesimal normal form. The normalized coefficient phase and its compact transport controls, differentiable inverse-module/regular-coordinate passage, completed physical sum and original Bott \(+1\) comparison remain unproved. Neither a fibre calculation nor the common weight alone closes them.



### Exercises 227–229. Slow transport and the normal core (36 points)

**Exercise 227 (12 points).** (a,4) Derive SL.4 from the squared reciprocal comparisons. (b,4) Prove that the limit \(B\) has an actual module adjoint. (c,4) Prove the domain identity in SL.5.

**Solution.** (a) Add \(\epsilon I\), invert the positive order, take its positive square root by the integral, and apply logarithmic order. The scalar inequality \(\log(1+\sqrt C\,t)\le\frac12\log C+\log(1+t)\) proves the upper bound; reverse comparison proves the lower. (b) The regularized self-adjoint differences have norm at most \(L\), and converge on the common dense \(\operatorname{Dom}\Theta_{\mathrm f}\), because \(\kappa H_\epsilon\to\kappa H\). Extend convergence by that bound; inner-product limits give self-adjointness and the adjoint identity. Boundedness alone would not suffice for a general module map. (c) \(\operatorname{Dom}\Theta_{\mathrm f}\) is a graph core for both logarithms by their spectral cutoffs. On it their difference is the bounded \(B\). The two bounded-perturbation graph norms are equivalent, so closing the identity gives exactly the two same completed domains.

**Exercise 228 (12 points).** (a,4) Prove the scalar divided-difference inequality SL.11. (b,4) Explain the weighted Hilbert-Schmidt tail argument in SL.12. (c,4) Derive SL.14 without differentiating \(\Theta_{\mathrm f}b\).

**Solution.** (a) \(q\) decreases and \(xq(x)\) increases. If \(u<v\), then \(vq(v)\ge uq(u)\), equivalent to \(v(q(u)-q(v))\le(v-u)q(u)\). The reversed inequality gives the other case; take a limit on the diagonal. (b) The weighted object \(\Omega M_i\) is in the completed Hilbert-Schmidt module by SL.6. Cut it on the left and right by plateaus of \(\kappa\); these approximate it in that norm. Divided differences converge uniformly on the remaining square. SL.11 controls the discarded tails by the weighted errors, uniformly in the regularization. (c) In the bounded regularizations the derivative is \(\delta_i q_\epsilon(\kappa)b+q_\epsilon(\kappa)\nabla_i b\). Substitute \(b=\kappa \Theta_{\mathrm f}b\) only in the first term. SL.12 and the selected \(\Omega\nabla_i b\) converge in module norm, giving \(X_i \Theta_{\mathrm f}b+\Omega\nabla_i b\). The connection is closed; no \(\nabla_i(\Theta_{\mathrm f}b)\) is used.

**Exercise 229 (12 points).** (a,4) Verify both order constants \(4\) in the pair-swap model. (b,4) Prove that its first difference is not compact and its second is compact. (c,4) Identify the coefficient unit in SL.15 and the remaining original graph interface.

**Solution.** (a) The two reciprocal squares in each pair are \(4^{-2m}\) and \(4^{-2m}/4\); swapping multiplies their ratios by \(4\) or \(1/4\). Both order constants are therefore \(4\). (b) \(B_m\to\log2>0\), so its diagonal values do not vanish on the orthonormal pair sequence. The slow difference has \(|d_m|\le4\log2/H_m\to0\); finite pair truncations approximate it in operator norm. (c) The even vacuum is the identity cycle on \(A=C_0(Y)\), with its natural base action, and the homotopy makes its complement degenerate. An arbitrary evaluation \(Y\to\{\mathrm{point}\}\) need not be equivariant and gives no original reduced unit class. The balanced factorization at \(R=C_0(T)\), differentiable inverse-module passage, completed physical graph sum and original Bott \(+1\) comparison still require their own proofs.



### Exercises 230–232. The weighted normal phase (36 points)

**Exercise 230 (12 points).** (a,4) Prove compactness of \(DR_{\mathrm f}\) without treating the noncompact-base vacuum as compact. (b,4) Derive PW.5 for a bounded real self-adjoint \(A\) satisfying \(A\kappa=X\). (c,4) Prove norm convergence and compactness of \(Y_i\).

**Solution.** (a) Restrict to positive occupation. Bounded fast energy permits finitely many sectors and a compact one-particle cutoff in every occupied factor. The tensor rank-one argument makes each such finite-sector cutoff compact. The remaining tail of \(DR_{\mathrm f}\) has norm at most \(r^{-2/3}\), by PW.3. The vacuum contributes zero. (b) On \(\operatorname{Dom}\Theta_{\mathrm f}\), \(A=X\Theta_{\mathrm f}\), so \(A^2\le\|X\|^2\Theta_{\mathrm f}^2\) as positive forms. Sum over occupied factors and use \(B(A)^2=2d\Gamma(A^2)\le\|X\|^2C_{\mathrm f}^2\). Multiplication by the inverse on the complement gives the bound, with its adjoint obtained from the cutoff inner-product identity. (c) Apply the same estimate to each difference of the regularized \(A\)'s. The \(X\)'s converge in norm by SL.12. Each product factors through the compact \(DR_{\mathrm f}\), and the compact ideal is norm closed.

**Exercise 231 (12 points).** (a,4) Check the sign and constant in PW.6–PW.7. (b,4) Prove that strong resolvent limits suffice for norm convergence of the weighted derivative. (c,4) Explain why PW.9 needs no normal derivative of \(R_{\mathrm f}\).

**Solution.** (a) Differentiating an inverse gives \(-R(\delta C)R\), hence the minus sign. The gap gives the bound \(2\|Y\|/(2+a^2)\). Its integral divided by \(\pi\) is \(\|Y\|/\sqrt2\). (b) The \(Y_{i,\epsilon}\) converge in norm to a compact. Uniformly bounded strong limits with their adjoints converge in norm when multiplied on either side of a compact, by rank-one approximation. The common integrable bound then permits the norm integral limit. (c) Start with the regularized product rule on an actual joint-core vector. Insert \(v=R_{\mathrm f}|C_{\mathrm f}|v\) only into the already differentiated term. The operator-norm derivative limit and the strong sign limit give the normal derivative in module norm. Close the connection; neither \(R_{\mathrm f}\) nor \(|C_{\mathrm f}|v\) has been differentiated.

**Exercise 232 (12 points).** (a,4) Verify the two displayed real Clifford matrices square to one and anticommute. (b,4) Compute \(\dot W\) and its weighted squared norm in PW.12. (c,4) Identify the class algebra proved here and the remaining original graph obligation.

**Solution.** (a) \(\sigma_x^2=\sigma_z^2=I\) and \(\sigma_x\sigma_z=-\sigma_z\sigma_x\). Tensor multiplication proves the assertions. (b) Differentiate \((10+t,11)/\sqrt{(10+t)^2+121}\) at zero. Its two components are \(121/(221\sqrt{221})\) and \(-110/(221\sqrt{221})\); their squared sum is \(121/221^2\). Multiply by \(R_{\mathrm f}^2=1/2682\). This also proves the displayed rational inequality and retains the shift five in both oscillator weights. (c) PW.11 proves the equivariant coefficient class \(1_A\), \(A=C_0(Y)\), and PW.9 proves its actual weighted normal-core derivative. The balanced original-unit realization at \(R=C_0(T)\), differentiable inverse-module passage, completed physical graph sum and original Bott \(+1\) comparison still require their own constructions.



### Exercises 233–235. A normally controlled coefficient auxiliary (36 points)

**Exercise 233 (12 points).** (a,4) Prove MR.3, retaining integer occupation. (b,4) Explain why strong convergence of the slow phases suffices for norm convergence of MR.2's resolvents. (c,4) Prove MR.6 without assuming a normally differentiable fast inverse.

**Solution.** (a) Every occupied coefficient \(c_j\) is at most \(v=(\sum_jN_jc_j^2)^{1/2}\). The regularized coefficients are at least \(c_j/(1+\epsilon v)\). Their weighted Euclidean norm is at least \(v/(1+\epsilon v)\), so its reciprocal divided by \(\sqrt2\) differs from the original by at most \(\epsilon/\sqrt2\). The vacuum contributes zero. (b) MR.4's continuous functions of \(r_\epsilon\) converge in norm and their limiting functions of compact \(r\) are compact. Multiplication by the uniformly bounded strong phase limits with adjoints gives the required norm limit; the vacuum term is fixed. (c) MR.5 proves \(\epsilon A_{i,\epsilon}\to0\). MR.7 and MR.3 then give \((\delta_iW_\epsilon)(r_\epsilon-r)\to0\). PW.8 already proves the norm limit with fixed \(r\). No inverse has been differentiated in this comparison.

**Exercise 234 (12 points).** (a,4) Prove the two row bounds in MR.10. (b,4) Upgrade the strong \(Q\)-row limit to a compact norm limit. (c,4) Check the sign and constant of the magnitude contribution in MR.12–MR.13.

**Solution.** (a) The \(P\)-row square is bounded by \(H_\epsilon^2/4\) between its resolvents, giving norm at most \(1/2\). The \(Q\)-row square is bounded by \(H_\epsilon/2\); the scalar maximum \(h/(h^2+\lambda^2)=1/(2\lambda)\) gives norm at most \(1/(2\sqrt\lambda)\). (b) Its positive square is dominated by \(K_\epsilon\to K\) compact in norm. A compact approximate identity makes its off-core norm uniformly small; on that compact the strong limit is a norm limit. The same argument makes the limit compact. (c) Differentiating the reciprocal fast weight gives a minus sign in MR.9; differentiating the auxiliary resolvent gives another minus sign, hence the positive two-row sum. Its norm is at most \(2[(1/2)\|M_i\|/(2\sqrt\lambda)+(1/(2\sqrt\lambda))\|M_i\|/2]=\|M_i\|/\sqrt\lambda\).

**Exercise 235 (12 points).** (a,4) Verify \(\dot H=672I\) and the exact squared derivative norm in MR.16. (b,4) Prove the actual normal-domain rule MR.14. (c,4) State the precise coefficient class and remaining original physical obligation.

**Solution.** (a) The source occupation shift is five. Differentiate \(2[(5+(4+\sin t)^2)^2+(5+25)^2]\) at zero, obtaining \(4\cdot21\cdot8=672\). The orthogonal Clifford tangent has squared norm \(121/221^2\). The adjoint product of the differentiated inverse is scalar; combining its real magnitude and imaginary terms gives the displayed fraction. (b) For each bounded regularization the fixed-sector inverse product rule holds. Parallel occupation cutoffs and the uniform derivative bound extend it to the closed normal domain. The resolvents and derivatives converge in operator norm, so values and derivatives converge in module norm on every normal-domain vector. Close the connection to obtain MR.14. (c) MR.15 is the actual equivariant coefficient identity \(1_A\), \(A=C_0(Y)\). The balanced realization at \(R=C_0(T)\), differentiable inverse-module passage, completed physical graph sum and original Bott \(+1\) comparison still require their own constructions.



### Exercises 236–238. A normally controlled stabilization (36 points)

**Exercise 236 (12 points).** (a,4) Prove compactness and dense range of \(q\). (b,4) Derive the constant \(16+4G_\phi^2\). (c,4) Explain the separate vacuum treatment.

**Solution.** (a) Tensor finite-rank approximations of \(T\) on each fixed sector, then take symmetric/exterior projections. Their products give \(\Gamma(h)\). Its occupation-tail norm is at most \(4^{-n}\), so the positive-occupation operator is compact. Tensor products of vectors in the dense range of \(h\), their projections and finite occupation sums prove dense range. (b) PC.7 gives \(C_{\rm f}^2\le4na_n^2+4G_\phi^2n\mu_*^{-1}\). The first is at most \(16\cdot4^n\le16q^{-1}\). The second is at most \(4G_\phi^2q^{-1}\), using \(q^{-1}\ge4^{n-1}\mu_*^{-1}\). Uniform cutoff bounds and closure give the graph map. (c) The vacuum is a parallel even copy of \(A\) with zero generator. Its identity is locally compact and can fail global compactness on noncompact \(X\). Keep it in a separate fixed coordinate; the positive-occupation inverse weight is not assigned to its zero energy.

**Exercise 237 (12 points).** (a,4) Obtain PC.13 with its signs and order. (b,4) Derive the two decay powers. (c,4) Explain how closedness gives the domain rule without an unweighted phase derivative.

**Solution.** (a) Use \(\delta R=-R(\delta L)R\), with
\(\delta L=(\delta W)u^{-2}-W(u^{-1}M_iu^{-2}+u^{-2}M_iu^{-1})\).
Set \(Y=(\delta W)u^{1/2}\), so \(\delta W=u^{-1/2}Y^*\), and substitute without permuting the middle factors. (b) Each magnitude term has one factor of norm \(1/\sqrt{2\lambda}\) and one of norm at most one; summing gives \(\sqrt2\|M_i\|\lambda^{-1/2}\). The phase factors give \(\sigma\lambda^{-3/4}\sqrt{C_\phi}\|Z_i\|\). (c) PC.12 is a compact norm limit of weighted regularized derivatives. The choice of \(\rho_\epsilon\) makes the extra square-root error at most \(\epsilon\). The resolvents and their derivatives converge in norm; applying the closed connection to each domain vector proves PC.16.

**Exercise 238 (12 points).** (a,4) Prove the adjoint of \(J\) and PC.18's norm convergence. (b,4) Derive PC.19 from the two factors. (c,4) Verify the scalar compressed derivative and state its limitation.

**Solution.** (a) The telescoping identity bounds both finite columns and their adjoints by one. It extends the finite-column adjoint to the whole target. Dense range of \(q\) gives strong convergence and \(J^*J=1\). A fixed-\(m\) derivative tail is a shifted \((J-J_N)(\delta b)b^mq^2\), which tends to zero in norm because its right factor is compact. The remaining \(m\)-tail is summably small; the single differentiated-\(\mathcal T\) column is square summable. (b) Use \(R=q^2B\) in the first derivative term and \(R=Bq^2\) in the last. They converge to \(Q_iBJ^*\) and \(JBQ_i^*\); the compact middle term converges to \(J(\delta R)J^*\). A double factorization \(q^2Cq^2\) is not required. (c) At the stated sample, \(|r'|^2=15/15360\) and \(|r|^2a^2=2/15360\). PC.22 gives \((19+\sqrt{225+120})/30720<19/15360\). The transfer-bound square exceeds \(1/1920+1/1024=23/15360\). This is the infinite-column compressed derivative. The complementary operator, equivariant absorption, actual \(B\)-source inverse module, physical sum and original Bott \(+1\) remain unproved by that check.



### Exercises 239–241. Full coefficient stabilization and its complement (36 points)

**Exercise 239 (12 points).** (a,4) Verify CSB.6's cell is unitary and derive \(K_0\). (b,4) Use backward cells to prove CSB.7 on completed modules. (c,4) Explain why the affine action must be transported after adjoining the vacuum.

**Solution.** (a) The diagonal column products are \(V^*V+b^2=1_E\) and \(d^2+VV^*=1_H\); the cross products are \(V^*d-bV^*=0\) and their adjoints. The row products give the same identities in the reversed order. Starting with \(e_0=0,y_0=y\) gives \(z_0=dy,e_1=-V^*y\); with subsequent \(y_n=0\), \(z_n=-Vb^{n-1}V^*y\). (b) Choose terminal carrier zero for any finite output and invert the finite cells. The initial carrier and input columns are exactly \(J^*z,K_n^*z\); the terminal zero eliminates the output tail. The orthogonal isometric sum has closed range containing the dense finite-coordinate core, so its range is the whole module. Finite adjoint columns converge on that core and then on the completion by their uniform bound. (c) Weyl translation for the affine cocycle generally sends the even vacuum to a vector with positive occupations. Hence its action is on \(A\oplus E\), not separately on \(E\). Transport \((A\oplus E)\oplus H^\infty\) through \(1_A\oplus U\); retain the vacuum's localized compactness and zero derivative.

**Exercise 240 (12 points).** (a,4) Prove the scalar series CSB.9, including its diagonal. (b,4) Prove CSB.10 and its completed-module convergence consequence. (c,4) Obtain the compact derivative of \(K_0\) without asserting \(\delta_iJ\) bounded.

**Solution.** (a) For \(a\ne c\), the sum is
\(\{a^2/(1-a^2)+c^2/(1-c^2)-2ac/(1-ac)\}/(a-c)^2\).
Bringing it to a common denominator cancels \((a-c)^2\) and gives CSB.9. At \(a=c\), its summands are \(n^2a^{2n-2}\); differentiating \(x/(1-x)^2\) gives \((1+x)/(1-x)^3\). This agrees with CSB.9 at \(x=a^2\). (b) Multiply the series by \(\mu\nu(\sqrt\mu+\sqrt\nu)^2/(a+c)^2\); the factors \(1-a^2=\mu,1-c^2=\nu\) cancel. The four inequalities preceding CSB.11 bound the result by \(8/3\). Approximate \(\delta_i\sqrt q\) on both sides by dense-range spectral plateaus in Hilbert–Schmidt norm. On the retained rectangle the series tails converge uniformly; on the discarded part the bound \(8/3\) is uniform in the truncation. This proves convergence in the actual Hilbert–Schmidt module. (c) The first part of CSB.8 has norm at most \(\|\delta_iV\|_{2,A}\); the triangular part has norm at most \(\sqrt{8/3}\|\delta_i\sqrt q\|_{2,A}\). Add \(J_N(\delta_iV)^*\) and the gapped square-root derivative of \(d\), take their norm limits, and close the connection on \((d,-JV^*)\). The result is CSB.12; an unweighted derivative of \(J\) has never been extended.

**Exercise 241 (12 points).** (a,4) Prove CSB.15's row bound and identify the sign in its adjoint term. (b,4) Verify CSB.19–CSB.21 and explain the plotted full bound. (c,4) State exactly what CSB.16 proves and what it leaves to the original inverse-module construction.

**Solution.** (a) For finitely many coordinate maps \(B_m=(\delta_iK_m)R_{\mathrm c,m,\pm}\), the row norm squared is \(\|\sum B_mB_m^*\|\le C_i^2\sum\|R_{\mathrm c,m,\pm}\|^2\le C_i^2/(8\lambda)\). Its tails converge by the same estimate. Since \(R_{\mathrm c,+}^*=R_{\mathrm c,-}\), the derivative's two terms are \(\mathcal A_{i,\pm}K^*+K\mathcal A_{i,\mp}^*\). Their norm sum is at most \(2C_i/\sqrt{8\lambda}\). (b) The differentiated geometric sums in CSB.19 give \(K_0'\)'s square \(8(1/4)/(1-1/16)=32/15\). Divide by \(2\lambda=512\) to get \(1/240\). PC.22's trace and discriminant give the source square \((19+\sqrt{345})/30720\). The plotted green curve is the sum of these two derivative bounds from CSB.20; it is a proved upper bound for the infinite full resolvent, not an exact full norm or a finite-matrix norm. It tends to zero, while the unweighted \(J'\) diverges. (c) The explicitly constructed unitary conjugates two regular self-adjoint operators, so CSB.16 is a genuine resolvent on all of \(H^\infty\), with the completed normal product rule CSB.17. After the parallel vacuum is adjoined and the full affine action is transported, its bounded coefficient cycle is \(1_{C_0(X)}\). The \(B\)-representation and actual inverse \(d_X\), original \(C_0(T)\) balancing, completed physical graph sum, and Bott \(+1\) comparison remain to be supplied.



### Exercises 242–244. Normal stabilized columns and the fixed Bott weight (36 points)

**Exercise 242 (12 points).** (a,4) Derive NBJ.2 and explain why the derivative of \(h\) is unchanged. (b,4) Prove NBJ.6–NBJ.7 without a module polar decomposition. (c,4) Apply that estimate to the actual transported column in NBJ.9 and the displacement in NBJ.10.

**Solution.** (a) Subtract the target frame-connection term \(A_iT\) when replacing the Haar connection by the constant-column connection. The term is Hilbert–Schmidt by the all-frame-jet column bounds. In \((\delta T)^*T+T^*\delta T\), the extra contribution is \(-T^*(A_i^*+A_i)T=0\). The square-root derivative theorem thus concerns the same \(h\) and the unchanged source connection. (b) Conjugate \(VV^*\le Ch\) by the finite derivative difference \(Z\). Taking norms and using \(h=T^*T\) proves NBJ.6. NBJ.5 makes its right side Cauchy. Compactness of \(V,\delta V\) gives norm convergence of the maps \(J_NV\) and the remaining derivative terms, so the closed connection proves NBJ.7. (c) The transported column's square is \(L_gh_sL_g^*\le Ch_r\). Its actual derivative follows from covariance of the original source and basic constant-column connections. For the displacement, use its localized rank-one bound \(\theta_{b_g,b_g}\le Ch_r\). These are endpoint norm estimates on actual columns or vectors, rather than bounded fibre factors that might fail to define module sections.

**Exercise 243 (12 points).** (a,4) Verify the exact affine identities after NBJ.11. (b,4) Prove both claims in NBJ.12 for \(P_N\). (c,4) Explain why finite-column normal control is compatible with an unbounded derivative of the stabilizing unitary.

**Solution.** (a) Composition cancels \(U_s^*U_s\) in the conjugated linear action. The cocycle follows from \(U_s^*J_sb_h=(b_h,0)\) and the original target convention \(b_{gh}=b_g+L_gb_h\). Isometry of \(J\) gives NBJ.18. (b) The \(J_s^*P_N\) row has finitely many \(b_s^jT_s^*\) entries. Use the actual identity \(b_s^jT_s^*=T_s^*d_s^j\), keeping the order after transport. NBJ.9 and the compact derivative of the bounded \(d_s^j\) give the finite product rule. The \(K_s^*P_N\) column has only indices \(0,\ldots,N\); every \(K_n\) and its derivative have the stated bounded/compact values. Subtracting coordinate insertions uses the compact \(K_n-I_n\). The basic action is parallel. (c) Each needed source input is a transported \(T^*\)-column or a dominated displacement, and NBJ.6 controls exactly those inputs. In NBJ.21–NBJ.23, \(\mathcal V'\) is unbounded at zero but \((\mathcal Ve_0)'\) tends to zero; at \(t=1/2\) their squared norms are \(1024/15\) and \(124/15\). Finite-column normal control does not bound the full action derivative.

**Exercise 244 (12 points).** (a,4) Prove the weak-convex quasicentral argument on a compact parameter test. (b,4) Prove the graph membership and normal product rules NBJ.16. (c,4) State what NBJ.19–NBJ.20 prove for the Bott construction and what the inverse-module step must still supply.

**Solution.** (a) The commutator section is continuous compact-valued because finite columns and their adjoints are norm continuous. Compress by \(p_l\): the result is zero for \(m\ge l\). Positive functionals are approximated in norm by such compressions through their approximate identity and the GNS Cauchy–Schwarz estimate; linear combinations treat all bounded functionals. Uniform boundedness then gives weak convergence to zero. A separating functional would contradict that convergence if zero lay outside a tail's norm-closed convex hull. Finite direct sums treat finitely many tests. (b) The vector series \(\xi+\sum_n(1-v_n)\xi\) converges in norm, uniformly on a compact test, by NBJ.14 after a finite initial part. Finite-coordinate weight truncation identifies that limit with \(\Theta\xi\) on the completed graph. Apply it to \(\xi\) and \(\nabla_i^0\xi\). The finite truncations commute with the connection, so both graph vectors converge and closedness gives NBJ.16. (c) The free stabilization and diagonal selection can be made with the original real affine field, trivial added cocycle and original anchor. The fixed positive lifting operator then has the actual normal resolvent product rule and the original outward \(q+ip\) convention. Its scalar coefficient action does not represent \(B\). A complete inverse-module construction still needs the actual lifted \(B\)-representation and differentiable positive module passage, followed by the original balanced graph sum and Bott comparison.




### Exercises 245–247. Transported normal domains and the Bott connection (36 points)

**Exercise 245 (12 points).** (a,4) Prove closedness and the actual graph-core assertion in CTN.1. (b,4) Prove the completed metric and scalar product rules. (c,4) Prove CTN.3 and explain which part of the affine action has an additional derivative.

**Solution.** (a) Apply \(U^*\) to a graph-convergent sequence and use closedness of the original connection. The graph norm is preserved by \(U\), so \(U\mathcal C\) is a graph core. (b) Write \(u=Uv\), use \(A\)-linearity to obtain \(ua=U(va)\), and transport the original product rule. The unitary preserves inner products, so it preserves the metric identity. (c) Insert \(U_r^*\widehat L_gU_s=L_g\) into the source domain rule, then multiply by \(U_r\). This retains the original normal-direction change matrix. Differentiating \(\mathcal V_gv+\widetilde b_g\) also gives \(\widehat\nabla\widetilde b_g=J\nabla b_g\); linear covariance does not remove that displacement term.

**Exercise 246 (12 points).** (a,4) Prove the finite support assertions in CTN.6. (b,4) Prove CTN.7–CTN.8 on completed domains. (c,4) Prove CTN.9 and state what it does not imply.

**Solution.** (a) The first coordinate row is the adjoint of \((Tb^j)\). The shifted \(K_n\) begins at coordinate \(n\), hence pairs to zero with \(P_N\) when \(n>N\). (b) Differentiate each finite product \(b^jT^*\) with the actual gapped functional calculus and compact \(T^*\) derivative. Use the compact \(\delta K_n\) for the finite complementary rows. Close the source connection after approximating the input in its basic graph norm; all derivative terms are bounded compact maps. Multiplying by the actual \(U\) gives CTN.8. (c) Subtract the two metric product identities. The resulting form is skew-symmetric on the common finite core. That does not prove a bounded full difference operator, skew-adjointness or that these vectors form a graph core for the transported connection.

**Exercise 247 (12 points).** (a,4) Prove CTN.11 with its domain assertion and exact compact norm. (b,4) Derive CTN.12 and identify its connection. (c,4) Check CTN.14–CTN.15 and explain the remaining weight-domain issue.

**Solution.** (a) Conjugation gives the actual resolvents of the unitary-conjugated regular operator. For \(u=Uv\), use the original completed resolvent rule on \(v\) and CTN.1. The derivative is \(UC_{i,\pm}U^*\), compact with the same norm. (b) On the source direct sum the complementary derivative is zero. The coefficient resolvent derivative is exactly PC.16's \(\lambda^{-1/2}\) magnitude and \(\lambda^{-3/4}\) phase bound. Unitary transport preserves that bound for the transported connection. It does not alter the separate constant-column estimate. (c) The rotation has \(U'=J_0U\), where \(J_0=\bigl(\begin{smallmatrix}0&-1\\1&0\end{smallmatrix}\bigr)\); hence \(A_U=-J_0\) and \(\mathcal V'=[J_0,\mathcal V]\). Their sum in the covariant derivative is zero. Multiplication with \(\operatorname{diag}(1,2)\) gives the displayed symmetric commutator, with eigenvalues \(\pm1\). A weight parallel for \(\partial_t\) need not be parallel for \(\widehat\nabla\); the infinite weight commutator and its completed domains require their own proof.



### Exercises 248–250. The source representation and the missing inverse data (36 points)

**Exercise 248 (12 points).** (a,4) Prove positivity and completion of BSM.1. (b,4) Prove boundedness, the adjoint and multiplicativity of BSM.2. (c,4) Prove nondegeneracy without a support projection.

**Solution.** (a) For a finite tensor sum, the source Gram matrix is positive and its image under each matrix amplification of the cpc section is positive. Its right compression by the coefficient column gives the positive module inner product. The two-by-two Gram inequality gives Cauchy–Schwarz; quotienting null vectors and completing gives the Hilbert module. (b) Apply complete positivity to the matrix with entries \(c_i^*(\|a\|^2-a^*a)c_j\). It bounds the left-multiplied norm by \(\|a\|\). The inner product gives the adjoint left action of \(a^*\), and multiplication is exact on the dense algebraic core and hence on the completed module. (c) A source approximate identity sends each \(c\otimes d\) to \(u_nc\otimes d\), converging by the estimate \(\|u_nc-c\|\,\|d\|\). Contractivity extends convergence to all vectors. This is the actual nondegenerate action on the minimal nonunitized module.

**Exercise 249 (12 points).** (a,4) Check the completed balanced unitary BSM.4–BSM.5. (b,4) Prove closability of BSM.6 and the completed source domain rule. (c,4) Give the exact coefficient order of the positive Morita tensor and explain what the module identity does not prove.

**Solution.** (a) The balanced inner product is \(a^*q_0(d)^*q_0(s_0(c^*c'))q_0(d')a'\), equal to \(a^*q_0(d)^*c^*c'q_0(d')a'\). Thus the displayed map is isometric. Surjectivity of \(q_0\) and density of triple source products give dense range and an onto completed unitary, intertwining the actual left source action. (b) Apply each bounded functional of the fixed Banach space \(K_0\) to a graph-convergent sequence. Closedness of the scalar normal derivative forces the limiting derivative of a null sequence to vanish under all such functionals, hence to vanish. For source multiplication use finite scalar tensor sections, then the two indicated graph approximations and closedness. (c) First tensor \(K\) over \(E_B\) through \(h_t\) with \(M_t=A\widehat\otimes\mathcal F_t\widehat\otimes S\); this is the right \(A\widehat\otimes S\) Morita module. Then tensor through \(1_A\otimes\mathrm{ev}_0\) with \(A\). Neither this completed source representation nor the zero quotient module identity supplies an even Kasparov operator or its genuine groupoid action.

**Exercise 250 (12 points).** (a,4) Verify the semilinear transport BSM.10 and identify its target module. (b,4) Prove that the tapered section in BSM.11 has the same zero quotient but zero positive module at t=1. (c,4) Check the finite compression and multiplicative defect in BSM.12.

**Solution.** (a) Insert \(s_B^g(\alpha_g^B(c^*c'))=\alpha_g^E(s_B(c^*c'))\) into the target inner product. Its result is exactly \(\alpha_g^E\) of the original inner product. The inverse arrow maps give the completed inverse. The target is the module of the conjugate section, not automatically the fixed section module. (b) Multiply on both sides by the square root of the positive central \(\chi=1-t\). This preserves complete positivity and contractivity, and \(\chi(0)=1\) retains \(q_0s'_0=1\). Since \(\chi(1)=0\), all positive tensor inner products vanish at \(t=1\); their completed module is zero. (c) The two columns of \(V\) have squared norm one and zero inner product. Its compression of \(\operatorname{diag}(c_0I_2,c_1I_2)\) is \(\operatorname{diag}((9c_0+16c_1)/25,(16c_0+9c_1)/25)\). For \(e_0=(1,0)\), subtracting its square gives \((144/625)I_2\). The exact representation is multiplicative; its cpc compression need not be. This finite example does not identify the Bott inverse.



### Exercises 251–253. Regular source action and the quotient cutoff (36 points)

**Exercise 251 (12 points).** (a,4) Prove the adjoint, multiplication and norm identities for RSM.3. (b,4) Prove RSM.4–RSM.5, keeping the right coefficient action. (c,4) Explain why the construction does not require coherent arrow unitaries on the original section module.

**Solution.** (a) Each coefficient representation \(\pi_s\alpha_{g^{-1}}^B\) is a star homomorphism. Its norm bound and the positive summand comparison in RSM.1 give \(\|\Pi(b)\|\leq\|b\|\). Taking the inner product moves it to \(\pi_s\alpha_{g^{-1}}^B(b^*)\); multiplication stays pointwise, proving the two algebra identities. (b) In the translated sum put \(g=hl\). Then \(\alpha_g^E=\alpha_h^E\alpha_l^E\), which gives the semilinear inner product and right action. Inverses and composition are the arrow identities. The same substitution in RSM.3 proves covariance, since \(\alpha_{g^{-1}}^B\alpha_h^B=\alpha_{l^{-1}}^B\). (c) Translation changes the arrow coordinate and retains \(s(g)\). The vector remains in the same \(K_{s(g)}\); no comparison between different Stinespring fibres or conjugate section modules occurs. Core completion preserves these exact maps.

**Exercise 252 (12 points).** (a,4) Prove the quotient inner product RSM.7 and its left-action formula. (b,4) Prove the cutoff isometry and adjoint RSM.10–RSM.11 without assuming a unit of \(B\). (c,4) Prove compactness RSM.13 and explain why it does not imply compactness of \(p_c\).

**Solution.** (a) Apply the coefficient quotient isometry \(Q\) to each summand of RSM.1 and use \(q\alpha^E_g=\alpha^B_gq\). This gives \(b^*q\langle\xi,\zeta\rangle b'\). The left source \(\pi_s(\alpha_{g^{-1}}b)\) becomes multiplication by \(\alpha_{g^{-1}}b\); applying \(\alpha_g\) gives \(b\) on the target row. (b) First multiply by a compact central cutoff on \(W\). The proper support condition leaves finitely many bisection pieces, so the sum of squared entries is \(b^*[\sum c(g^{-1}\cdot)]b=b^*b\). Approximation by these central localizations gives all module vectors. The finite row inner product gives RSM.11; row Cauchy–Schwarz bounds it by one, so it extends and is the adjoint on the completed module. (c) Centrality gives the rank-one identity for \(b_1b_2\), and \(b=\lim bu_n\) gives compactness by norm closure. For nonunital \(B\), this conclusion is only after multiplication by \(b\). Under the unitary \(V_c:B_B\to p_c\mathcal L_H(B)\), \(p_c\) corresponds to the identity of \(B_B\), which is compact exactly when \(1\in B\). No global compactness follows without that hypothesis.

**Exercise 253 (12 points).** (a,4) Verify RSM.15–RSM.16 with the non-equivariant section. (b,4) Prove the obstruction to the equivariant central extension in the finite example. (c,4) Apply the invariant taper test and identify the remaining inverse-cycle data.

**Solution.** (a) The section discrepancy is \(H_0-H_1=\operatorname{diag}(-7/25,7/25)\). The block swap in RSM.15 interchanges \(\pi(b)\) and \(\pi(\alpha_ab)\), proving the genuine source covariance and \(U_a^2=1\). The squared lengths in RSM.16 are \((9+16)/25=1\); multiplying \(vv^*\) gives the stated matrices, whose projection identity follows from \(v^*v=1\). Swapping the rows carries the first projection to the second. (b) A central matrix projection is \(0\) or \(I_2\). Nondegeneracy and the quotient identity make the two image projections complementary; equivariance with a fixed matrix action makes them equal, an impossibility. (c) Multiply the ordinary cpc section by \(1-t\). The quotient remains a section and RSM.6 gives the same regular quotient, with its equivariant cutoff identity submodule. At \(t=1\) every Stinespring inner product is zero, and RSM.14 gives the zero positive module. Thus these identities do not provide the odd inverse operator, a projection lift, its normal domain and covariance estimates, the original unit balance or the physical graph sum.



### Exercises 254–256. Positive evaluation, grading and the actual source (36 points)

**Exercise 254 (12 points).** (a,4) Construct the sequence in PES.3–PES.5 uniformly over the homotopy parameter. (b,4) Prove PES.8–PES.9 with the coefficient grading. (c,4) Explain why omitting that grading factor fails for an odd source element.

**Solution.** (a) At stage \(k\), uniform continuity on the positive compact interval \([2^{-k},1]\), with the first \(k\) coefficient and compact-arrow tests, gives one permissible adjacent parameter distance. A finite subdivision of \([2^{-k},2^{1-k}]\) below that distance gives error at most \(1/k\) for every \(v\). Concatenation tends to zero and treats every fixed test eventually. Norm-dense coefficients, finite nets of transported compact coefficient families and the bound twice the approximation error complete the proof. (b) \(S^*S=1\), \(SS^*=1-P_0\), and \(\Gamma\) commutes with the row shifts, giving PES.8. For a degree-\(p\) coefficient, move it through \(\Gamma\) with sign \((-1)^p\); the graded commutator becomes the adjacent differences in PES.9, whose compact tails vanish uniformly by PES.4. (c) Without \(\Gamma\), a degree-one source gives \(S^*\rho(e)+\rho(e)S^*\). Even a constant nonzero odd coefficient would leave a nonzero infinite sum tail; compactness of each individual coefficient row would not make that repeated tail compact.

**Exercise 255 (12 points).** (a,4) Prove the continuous compact-arrow property PES.11 from PES.5. (b,4) Verify the endpoint decomposition and its sign. (c,4) Compute the index of a square finite truncation and explain the difference.

**Solution.** (a) The lower block from row \(n+1\) to row \(n\) is \(\Gamma(U_n-U_{n+1})\rho_{t_{n+1}(v)}(\alpha_{g^{-1}}^E(e_r))U_{n+1}^*\). Replacing its coefficient by the row-\(n\) coefficient costs at most twice the PES.4 difference, so PES.5 controls it. The upper block uses \(\Gamma(U_{n+1}-U_n)\rho_{t_n(v)}(\alpha_{g^{-1}}^E(e_r))U_n^*\) and is directly controlled by PES.5. Every finite block is compact and norm continuous; their tails vanish uniformly on a compact arrow set and in \(v\). This proves compactness in the full compact-operator section algebra. (b) At \(v=1\) all represented coefficients and actions are constant in the row index. The map \(\Gamma S^*\) is onto with kernel positive row zero. Its restriction from positive rows \(n\geq1\) is an odd unitary intertwining the source with the graded sign and retaining the genuine action. That complement is degenerate; the row-zero cycle is exactly \(\rho_1\) on the genuine graded Fock module, with coefficient-class multiplicity \(+1\). (c) A square \(N\)-row backward shift has one coefficient kernel at row zero and one coefficient cokernel at the terminal negative row. Their ranks cancel. There is no terminal row in the completed infinite shift, so the finite index zero cannot certify its coefficient-class multiplicity.

**Exercise 256 (12 points).** (a,4) Prove PES.13 and verify the nonzero ideal test PES.14. (b,4) Verify PES.18–PES.19 and the rational adjacent bounds. (c,4) State the exact role of PES.15–PES.17 in the inverse construction.

**Solution.** (a) An ideal element has positive evaluations tending to zero. Its diagonal is a norm limit of finite compact coefficient blocks, so is compact. The ideal section \(t fP_{\mathrm{vac}}\), with an even suspension bump equal to one at zero, evaluates to \(t_n fP_{\mathrm{vac}}\), which is nonzero when \(f\ne0\). Thus the representation annihilates the ideal only in the Calkin quotient. (b) The numerator in PES.18 squares to \((1+t^2)^2I\) and is symmetric. The rational time difference is \(-1/((n+1)(n+2))\), giving PES.19. The squared derivative norm is \(4/(1+t^2)^2\), hence its norm is at most two; integration gives the adjacent bound, and multiplying by a source of norm at most two gives the bound four. (c) The balanced module has the actual \(B\)-action of a representative of the proved lift \(z_B\). The anchored connection and separator construction gives its genuine bounded product operator, whose class is \(d_X\) by PES.12. It does not factor the earlier \(E\)-representation, select a normal-compatible lift module, prove derivatives of the separator, or complete the physical graph sum and original Bott comparison.



### Exercises 257–259. Completed positive operators and their domains (36 points)

**Exercise 257 (12 points).** (a,4) Prove QPS.1 and explain its consequence for a probability anchor. (b,4) Construct the dense algebra in QPS.2 without altering the zero quotient. (c,4) Prove the finite energy and core assertions for QPS.3–QPS.4.

**Solution.** (a) Pull a compact endpoint set back by the proper surjective anchor. Properness of the transformation groupoid makes its arrow preimage compact, and projection is onto the original groupoid preimage. Conversely compact transformation endpoints bound both the original arrows, by properness of \(G\), and the source points; their closed compatibility set is compact. The infinite-group point action proves that a proper anchor alone cannot supply proper action. (b) Compress the compact positive coefficient field uniformly on each positive dyadic band by an even finite spectral projection. A locally finite parameter partition combines the compressions with error at most \(\min(\varepsilon,t)\). Its difference from the original field lies in the actual cone ideal and vanishes at zero, so the quotient is unchanged. Arbitrarily accurate compressions of a countable dense set and their star algebra give density; finite parameter covers give one finite range and the two \(Q\)-bounds. (c) The creation and contraction relations give \(Q^2=\sum\lambda_j^2(N_j^{\mathrm{bos}}+N_j^{\mathrm{ext}})\). Below any energy only finitely many coordinates and occupations occur. The resulting finite symmetric invariant matrices define the self-adjoint orthogonal sum. Finite energy truncations converge in its graph norm, its reciprocal spectral tails vanish, and the only zero-energy vector is the even vacuum.

**Exercise 258 (12 points).** (a,4) Choose \(w_n,a_n\) and prove QPS.8–QPS.10 on the completed module. (b,4) Prove the source domain rule QPS.11. (c,4) Prove QPS.12 and identify the equivariant represented class.

**Solution.** (a) Eventually every fixed PES adjacent test is at most \(k^{-3}\); take \(w_n=k\) on the successive resulting row intervals. Positive compact-interval finite-range bounds give \(M_n<\infty\), hence positive \(a_n\) with both QPS.6 bounds. Pair positive row \(n\) with negative row \(n-1\), using \(a_nQ\) on both. The two cross terms cancel because \(Q\Gamma=-\Gamma Q\), giving the exact graph norm. Direct sum resolvents and finite complete-pair/energy cores prove regularity and self-adjointness. Each finite retained pair has a compact Fock resolvent, and the remaining norm is at most \(\sup_{n>N}(1+w_n^2)^{-1/2}\). Scalar localization gives module compactness. (b) The shift blocks are weighted adjacent differences and the oscillator blocks have \(Q\)-bounds \(a_n M_n\); both compact tails vanish. Each source maps the finite core into the \(Q\)-domain. The resulting bounded graded commutator and closedness prove preservation of the completed \(D\)-domain and its stated graph bound. (c) The diagonal phase term is bounded by \(a_n\|QR\|/\sqrt{1+w_n^2}\). The off-diagonal term has the same bound by \(1-(1+x^2)^{-1/2}\leq|x|\). Thus the total difference is at most twice that bound. Finite source blocks are compact and tails vanish uniformly in the homotopy parameter. Its phase is a source-local compact perturbation of the genuine PES shift, so it has the same equivariant endpoint class \([h_1]m_{\mathrm{Fock}}[\mathrm{ev}_0]\).

**Exercise 259 (12 points).** (a,4) Prove QPS.14–QPS.17 without assuming a bounded unlocalized arrow difference. (b,4) Prove the completed normal rule QPS.18 and state precisely which connection is transported. (c,4) Verify the finite model and distinguish its finite index from the ordinary infinite calibration.

**Solution.** (a) The one-particle graph identity bounds the transformed \(\Theta\)-energy form, which sums in all particle slots. Translation adds bounded Clifford multiplication by \(\Theta v/\sqrt2\), and graph approximation proves the completed inequalities. With \(v=b_g/\sqrt t\), the small scales make \(a_n/\sqrt t\leq2^{-n}\) on both paired rows. The summed graph bounds and inverse arrows give equality of completed domains. Weighted PES arrow differences control the shift part; the oscillator part has compact finite rows and tails bounded by \(C_K(a_nM_n+2^{-n}\|e\|)\). This proves continuous source-localized compact extensions. (b) Constant complete-pair and energy truncations commute with the scalar normal derivative, as do the constant resolvents. Close the original normal core to obtain the first identity. The scalar Leibniz rule and norm-and-derivative closure give the second. An arrow transports both the connection and its domain to \(\mathcal U_g\nabla_s\mathcal U_g^{-1}\); equality with the original target connection has not been proved. (c) The displayed \(Q_{\mathrm{toy}}\) anticommutes with \(\Gamma_{\mathrm{toy}}\), has one even vacuum and one invertible odd pair. Its paired square is \(a_n^2Q_{\mathrm{toy}}^2+w_n^2\). In a square finite truncation the row-zero and terminal vacua have opposite grades, giving finite index zero. The infinite decomposition retains only row-zero vacuum, with invertible complement. For ordinary scalar \(A\)-multiplication its sign homotopy proves class \(1_A\) over the entire base; this neither asserts an equivariant scalar unit nor replaces the inverse source \(B\).



### Exercises 260–262. The actual inverse class and its unbounded regulator (36 points)

**Exercise 260 (12 points).** (a,4) Prove IUR.1–IUR.2, including the continuous compact-arrow condition. (b,4) Explain how all four estimates in IUR.3 can be chosen together. (c,4) Prove that \(l\) in IUR.4 is a compact operator of dense range.

**Solution.** (a) \(C\) is even and commutes with \(F_0\), so the two off-diagonal square terms cancel and \(F^2=1\). The doubled grading makes its off-diagonal even entries odd. The square-root compactness test proves \(C\phi_0(b)\) compact, while uniform zero-constant polynomials in the square defect prove continuity after arrow transport. Adding \(tC\) gives the original source cycle plus a zero-source summand at \(t=0\), hence the same class. (b) Ordinary commutator tests derive \(J\); the source errors are compact, and the arrow-source errors are continuous compact sections on each finite chart. Their approximate-identity tests converge uniformly on compact charts. KG.2 gives weak approximate-invariance, and finite direct-sum convex separation treats all those tests at once. A later common tail lies beyond the previous cutoff plateau and preserves nesting. (c) Nesting makes \(r\) locally finite on the nonzero spectrum. The first \(N\) cutoffs vanish near zero, so \(r\geq N+1\) there; its inverse extends continuously by zero and belongs to \(C^*(h_0)\subset J\). Since \(u_m=l(r u_m)\) with bounded \(r u_m\), its range contains every cutoff range and is dense.

**Exercise 261 (12 points).** (a,4) Prove the completed domain and resolvent identities IUR.5–IUR.7. (b,4) Prove the source-domain rule using a convergent inverse equation. (c,4) Prove IUR.10 and extend its domain rule to the generated star algebra.

**Solution.** (a) \(F,l\) commute, \(F\) is a unitary, and \(l\) has dense range. Multiplication verifies the two adjointable resolvents on exactly that range. Symmetry and surjectivity at both imaginary signs give self-adjointness and regularity. The resolvents contain a compact factor \(l\); the transform difference contains a scalar function vanishing at zero and is compact, preserving the class. (b) Summable cutoff commutators make IUR.9 converge for every domain vector. Multiplying by \(l\) identifies the limit's inverse image and puts \(\phi(b_j)\xi\) in \(\operatorname{Ran}l\). A bounded sequence alone would not establish this range membership. (c) The graded product rule gives \(F C_j+[F,\phi(b_j)]_{\mathrm{gr}}r\). The latter is the compact norm-convergent series in IUR.8. Products and the included adjoints preserve the completed domain, with the graded Leibniz bound and compact commutators, proving the claim for the whole dense complex star algebra.

**Exercise 262 (12 points).** (a,4) Prove completed arrow-domain preservation and IUR.14. (b,4) Construct the graph core and explain the unproved normal condition. (c,4) Verify the finite model and the distinction between its full invertible operator and source-essential index.

**Solution.** (a) The cutoff arrow differences converge uniformly in compact section norm to \(A_g\). The partial inverse equation in IUR.12 and multiplication by the range \(l\) give membership of the transported vector; inverse arrows give equality. Unitarity of \(F\) turns that range equation into IUR.13. Expand \(\alpha D-D=(\alpha F)A_g+(\alpha F-F)r\) and move the tested source through \(r\) by IUR.8. This gives the three compact continuous terms of IUR.14. (b) Cutoff ranges approximate each vector and its graph image simultaneously. Countable homogeneous section approximations and the tested source and bisection images give a graph-dense core on the actual module. Those are source and arrow controls; they prove no normal derivative of the selected compact regulator or its inverse. The zero-source summand also needs a controlled essential replacement before a physical nondegenerate realization. (c) \(C^2=1-F_0^2\), \(CF_0=F_0C=0\), and the paired diagonal weights commute with \(F\), yielding the six stated positive square entries. Thus the whole \(D\) is invertible. On the first three represented coordinates its essential finite compression has one even vacuum and one invertible odd pair, with index \(+1\). That finite calculation does not justify an infinite bounded off-diagonal removal or the missing normal connection.



### Exercises 263–265. Proper averaging and essential completed graphs (36 points)

**Exercise 263 (12 points).** (a,4) Prove the completed isometry and adjoint EAI.2. (b,4) Prove exact invariance of EAI.3–EAI.4. (c,4) Prove source-local compactness EAI.7, including the possible infinite tail of \(VF_0\phi_0(b)\).

**Solution.** (a) Central compact localization makes the cutoff column have compact arrow support. Its squared inner product is the normalized sum, giving the isometry. The finite positive sum of the squared row weights is at most one, so the row adjoint is a contraction; density and its core inner-product identity give the completed adjoint. Nondegeneracy extends the formula to all module vectors. (b) Reindex \(g=hl\). Transport of \(U_lF_0U_l^{-1}\) is \(U_gF_0U_g^{-1}\), so \(\mathcal T\) is invariant. The same cutoff reindexing intertwines \(V\). Their composition is the asserted invariant odd self-adjoint contraction. (c) Choose the central \(f=1\) on the compact source support. Move \(F_0\) through \(M_f\); its error localized by \(b\) is compact. The two remaining columns have compact arrow support and the three compact entries of EAI.6. Finite rank bisection approximation makes the column map compact. The tail \(V[F_0,M_f]\phi_0(b)\) is also compact, without asserting finite support. Apply \(V^*\), then approximate general sources by compact central localizations and take adjoints.

**Exercise 264 (12 points).** (a,4) Prove the reference splitting EAI.8–EAI.12. (b,4) Prove self-adjoint regularity of the full closures, using EAI.13–EAI.14. (c,4) Prove the resolvent estimate EAI.19 with its graded source sign.

**Solution.** (a) Add \(P\)-commutator and compact \([F,P]\phi(b)\) tail tests to the common regulator. The inverse equation preserves its reference domain. Its off-diagonal part is \(C_P(2P-1)\), a bounded compact self-adjoint operator. Subtract it to get a positive block-diagonal regular reference. Since \([r,F]=0\), the displayed two remaining off-diagonal reference terms give the bounded \(K_i\), and each \(T_i\) is compact. (b) Each \(v_\varepsilon\) maps into the reference core, \(S_i v_\varepsilon\) is bounded, and EAI.13 is uniformly bounded and strongly zero. After any state localization its range lies in the graph closure. Testing deficiency vectors gives EAI.14; the symmetric imaginary-resolvent lower bound forces both deficiency spaces to vanish. The local-global criterion then proves the actual module closures self-adjoint regular. (c) The original resolvent range lies in the reference core, giving EAI.18. The factor \(B_{\mathrm o}R(z)\) has norm at most \(\|B_{\mathrm o}l\|\). For a degree-\(p\) source, use \(\epsilon=(-1)^p\) and move it through the original resolvent with \(R(\epsilon z)\) and the compact graded commutator \(c_b\). Both resulting terms have compact factors and two norms \(s^{-1}\), giving exactly EAI.19.

**Exercise 265 (12 points).** (a,4) Prove the exact inverse-class comparison and nondegenerate source representation after discarding the zero-source block. (b,4) Prove full closed arrow-domain equality EAI.22 and explain why exact phase invariance matters. (c,4) Verify the finite average and compression, and identify the remaining normal premise.

**Solution.** (a) EAI.19 gives an integrable compact bound for the localized phase difference. Strong scalar transform integrals identify that compact norm limit with EAI.20; adjoints and density treat every source. The cycle class remains \(d_X\), including continuous arrow defects. The second summand has zero representation and is degenerate. The first representation is the original nondegenerate \(\phi_0\). (b) Exact invariance of \(F,P\) reduces the unlocalized arrow error to the bounded compact \(PFA_gP\). The reference core is invariant. Its graph approximations, this bounded error and closedness prove the completed inclusion; inverse arrows prove equality, with the stated graph bound and continuity. A merely source-localized bound would not establish this full-domain assertion. (c) Matrix multiplication gives the rational invariant \(\overline F\), its kernel \((1,-1,0)\), and the positive reference \(2I+P+FPF\) commuting with \(F\). The essential block is \(4\overline F\), with row \((2,2)\), onto and of index \(+1\). The full finite identities do not give a normal derivative of the abstract lift, cutoff, averaged phase or regulator. Those normal and physical graph estimates remain required.



### Exercises 266–268. Bounded normal comparisons (36 points)

**Exercise 266 (12 points).** (a,4) Prove the completed joint-domain and resolvent assertions BNC.3–BNC.5. (b,4) Prove the source-local compactness BNC.6 and retain the graded sign in BNC.7. (c,4) Check the three product hypotheses leading to BNC.8.

**Solution.** (a) Anticommutation cancels the coefficient-valued mixed terms on the tensor core. Positivity bounds both individual energies. Commuting spectral cutoffs and elementary tensor approximation make that core dense in the intersection of the two closed graph domains. The three bounded joint-calculus terms in BNC.5 map into the intersection, and multiplication proves their onto inverse identities and mutually adjoint resolvents, hence self-adjoint regularity. (b) Source-local compact resolvents give localized compact \(C_0\) calculus on each factor. Approximate a joint \(C_0(\mathbb R^2)\) function by finite product functions and approximate both localized factors by finite rank operators. The three resolvent numerators divided by \(1+s^2+t^2\) vanish at joint infinity. For a degree-\(p\) source, \(\Gamma\sigma(b)=(-1)^p\sigma(b)\Gamma\) cancels the term with uncommuted \(N\), leaving BNC.7. (c) Creation by homogeneous graph vectors times smooth normal coefficients has exactly the two bounded errors stated after BNC.7; its adjoint is checked on the same core. The tensor core has dense source-generated span. BNC.4 gives inclusion in the first operator's domain, and the anticommutator is twice its squared form. The proved product criterion therefore gives BNC.8.

**Exercise 267 (12 points).** (a,4) Prove BNC.9–BNC.11 for a uniformly bounded odd self-adjoint comparison. (b,4) Prove that the infinite example satisfies the compact reference and arrow controls. (c,4) Prove its failure to preserve the completed normal derivative domain, without claiming failure of its graph sum.

**Solution.** (a) Factor large imaginary resolvents and use the Neumann series, then the onto-resolvent regularity argument. The bounded perturbation retains the full joint domain and core. The completed-domain resolvent identity has the \(s^{-2}\) bound; source localization gives compactness, and norm integration gives the stated Lipschitz phase homotopy and class comparison. (b) The reference inverse has uniformly vanishing \(1/n\) pair tails and dense range. The normalized phase commutes with it. Its square-root defect and all weighted source commutator errors have only the first three coordinates; finite-pair regulator cutoffs commute with source, phase and genuine finite-group action. The proper cutoff is \(1/2\) and full arrow error is zero. (c) A smooth bump times \(e_1\) belongs to both reference and normal domains. Its compressed image has derivative \(a'\) near zero. BNC.16 gives its nonintegrable squared derivative \(1/(512|x|)\), so it is outside \(H^1\); the continuous-section derivative is also unbounded. The normal sum nevertheless has the bounded comparison BNC.17, so its completed domain and self-adjointness still follow from part(a).

**Exercise 268 (12 points).** (a,4) Prove the full Fourier-domain and compact-resolvent statements of BNC.18. (b,4) Prove the entire paired homotopy, including its uniform normal tails. (c,4) Compute the winding and kernel action and identify the unproved actual comparison.

**Solution.** (a) The Fejér approximation gives the complete Fourier basis. Testing adjoints mode by mode gives the full derivative domain. Each pair has the displayed symmetric matrix, whose square is \((m_n^2+k^2)I\). Only finitely many pairs and modes have bounded energy; the inverse tails therefore vanish in norm. The coefficient-energy identity gives the full joint domain. (b) The mass blowup gives BNC.19 and strict convergence to \(J\) on finite blocks. Its square defect tends uniformly to zero. For a Fourier source mode, integrate the phase derivative bound over its fixed shift: both large-\(n\) and large-\(|k|\) tails vanish uniformly in the homotopy parameter. Approximate smooth sources and their derivatives by Fejér polynomials. Continuous finite blocks and these uniform tails give compact parameter-module defects; the endpoint is degenerate. (c) Only the vacuum circle Dirac with sign action survives. Negative winding compresses to the onto backward shift, with one constant-mode kernel and no cokernel. Its ordinary index is \(+1\) and its equivariant index is the sign character. No comparison of the original \(d_X\) normal module, original \(C_0(T)\)-balance, inverse Spin-c coefficient or Clifford order with this separated model has been proved.



### Exercises 269–271. An actual represented standard carrier (36 points)

**Exercise 269 (12 points).** (a,4) Construct ANC.1–ANC.3 and prove nondegeneracy and genuine covariance. (b,4) Prove every degenerate-cycle identity in ANC.4–ANC.5, retaining the source degree. (c,4) Explain why ANC.6 preserves the actual \(B\)-source inverse without introducing a zero-source summand.

**Solution.** (a) A countable norming state family gives a separable faithful nondegenerate GNS sum. Pair it with its grading conjugate and repeat countably; the swap grading implements the algebra grading and both parity spaces are infinite. Regular induction uses source-transported left representation and range-fibre scalar inner products. Finite bisection partitions prove completed adjoints and nondegeneracy; left translation reindexes that inner product and gives exact representation covariance and genuine arrow law. (b) The two grading blocks have opposite signs. The off-diagonal \(\Gamma_M\) factors make \(\Delta\) odd and unitary. For a degree-\(p\) source, \(\Gamma_M\Pi(b)=(-1)^p\Pi(b)\Gamma_M\) makes its graded commutator zero. Even arrows commute with this grading, giving exact phase invariance. All cycle defects therefore vanish. (c) Both augmented copies carry the full nondegenerate \(\Pi\). Their cycle is degenerate and adds class zero. The first essential summand retains the original \(\phi_0\) and class \(d_X\); the full finite direct sum has that same source, class and genuine action.

**Exercise 270 (12 points).** (a,4) Prove the unbounded augmentation and full graph-domain assertions ANC.7–ANC.9. (b,4) Prove the represented standard summand and ordinary unitary ANC.10–ANC.11. (c,4) Prove exact class and full arrow-domain preservation under ANC.12.

**Solution.** (a) Include the fixed dense source algebra of the original essential inverse in the auxiliary source/arrow tests. Exact invariance and source supercommutation of \(\Delta\) remove its phase-tail tests. The common inverse equation gives completed range preservation and compact commutator and arrow-error extensions. The two explicit resolvents prove self-adjoint regularity and compact resolvents. Its compact phase difference from the degenerate \(\Delta\) leaves class zero. Direct sum with the proved essential inverse gives the full diagonal graph domain and retains \(d_X\). (b) The open and closed unit coordinates give an adjointable ordinary submodule \(A\otimes L\). Its two infinite parities identify it with the balanced standard module. Projecting a countable generating family proves the complement countably generated. The degree-preserving stabilization theorem absorbs that complement and the original \(K\), producing the stated unitary; its standard summand already has nondegenerate source action. (c) Conjugate the source, action, full domain and operator together. Rank-one conjugation preserves compacts, inverse resolvent identities preserve the full closure, and transported inverse arrows give both domain inclusions. Strict conjugation of compact fields preserves their norm continuity by finite-rank approximation. This is an isomorphism of the entire genuine cycle, retaining its actual class and nondegenerate source.

**Exercise 271 (12 points).** (a,4) Prove the full connection, graph core and metric/scalar identities ANC.13–ANC.15. (b,4) Verify the exact finite augmentation and the failure of \(\Delta_{\rm plain}\) for an odd source. (c,4) Identify the analytic assertions that are still required for the physical inverse.

**Solution.** (a) Closed scalar derivatives and separating coordinate functionals prove closability on finite scalar sections. Its minimal closure defines the full dense connection domain and its graph core. The finite product rule extends by graph approximation and closedness, giving the scalar Leibniz bound. Convergent inner products and derivative terms give the full metric identity. Even \(A\)-linear transport by \(W^*\) gives exactly the completed domain and transported graph core on the actual module. (b) The regular swap squares to one, conjugates the represented source to its grading action and commutes with \(\Gamma_M\). The doubled source is unital \(I_8\); exact \(\Gamma_M\) factors make all graded source errors and arrow errors zero. An off-diagonal identity instead gives \(2\psi(e_{01})\Delta_{\rm plain}\), whose squared norm is four, so its norm is two. The correct finite cycle is degenerate. (c) The construction proves no normal-domain preservation or derivative of \(\Phi'(b)\), \(D'\), its resolvents or \(U'\), and no dense completed joint normal/operator core. A source-preserving bounded comparison or another completed normal-sum proof must retain the original \(R\)-balance, full inverse Spin-c coefficient, exterior grading and final right Clifford order. The physical sum and original Bott comparison remain required.



**Exercise 272 (12 points).** (a,4) Prove the actual proper-source cutoff isometry and full adjoint formula CVC.2–CVC.4. (b,4) Explain why source-coordinate regularization in CVC.5 is genuinely equivariant without assuming equivariance of \(j\). (c,4) Prove that CVC.7 is onto and gives an even equivariant unitary \(K\oplus Q\cong Q\).

**Solution.** (a) First apply the operators to \(C_c(W)\)-localized vectors. Properness makes the relevant arrow support compact; finite bisection sections then give continuous compact rows. The strict squared-cutoff identity gives the exact counting inner product \(\langle V_K\xi,V_K\eta\rangle=\langle\xi,\eta\rangle\). Nondegeneracy of the centre extends the isometry to every vector. On compact rows, Cauchy–Schwarz against the same cutoff row gives \(\|V_K^*y\|\leq\|y\|\), and direct inner-product computation gives the displayed adjoint. Completion proves \(V_K^*V_K=1\). Substitution using \(U_{hg}=U_hU_g\) gives \(V_KU_h=L_hV_K\). (b) The row inner product makes coordinatewise \(j_{s(g)}^*\) the adjoint. In both \(L_h\mathcal R_H(j)y\) and \(\mathcal R_H(j)L_hy\), the coefficient map is \(j_{s(g)}\), since \(s(h^{-1}g)=s(g)\). Thus the composite is a genuine even equivariant isometry with invariant complemented range. (c) In \(Q^\infty=(K\oplus N)^\infty\), send the extra \(K\)-vector to \(K_0\), old \(K_n\) to \(K_{n+1}\), and \(N_n\) to \(N_n\). Extracting \(K_0\) and shifting back defines the inverse on the full orthogonal sum, with the same norm. All coordinates retain their grading, \(A\)-action and arrows. Fixed bijections of infinite copy indices give the claimed unitary without reassigning any arrow.

**Exercise 273 (12 points).** (a,4) Prove preservation of the represented class, nondegenerate source and full unbounded arrow domains in CVC.8–CVC.10. (b,4) Prove closability and metricity of CVC.11–CVC.13. (c,4) Prove full local normal-domain equality in CVC.15 and explain why this statement concerns sections rather than isolated unit fibres.

**Solution.** (a) Conjugation gives both full resolvents and their domains, hence self-adjoint regularity and the transported bounded transform. It conjugates every source relation, rank-one compact operator and the original dense unbounded source-test core. The augmentation adds a genuinely degenerate class with nondegenerate source, so the represented class is \(d_X\). Equivariance of \(\mathsf W\) makes the target action exactly \(L_h\). The original full arrow-domain equality and compact error field therefore conjugate to CVC.9–CVC.10; no restriction to a regulator range is made. (b) On finite rows the scalar PJ.6 Leibniz and metric identities give CVC.12, including coincident-arrow cross terms. If \(y_k\to0\) and its vector-valued derivatives tend to \(z\), pair with every core section and local normal frame. Scalar closability forces all components of that pairing with \(z\) to vanish. Density of the core forces \(z=0\). The one vector-valued minimal graph closure is closed and has the stated dense graph core. Graph approximation proves the scalar Leibniz identity for bounded-derivative core multipliers and the metric identity by scalar closedness. (c) Core covariance follows from scalar covariance and \(s(h^{-1}g)=s(g)\), so internal copy vectors stay fixed. On compact normal bisections its two sides are bounded in the vector-valued graph norm. Approximation and closedness extend the identity to the full local domain. The inverse bisection proves the reverse inclusion. Covector isometry gives the normal graph norm identity. The action here pulls sections across a unit chart; a derivation is not an \(A\)-linear fibre operator and has no separate graph domain at a single unit.

**Exercise 274 (12 points).** (a,4) Prove completeness and the two-direction arrow graph bound CVC.16–CVC.17. (b,4) Check CVC.18 and the necessity of its normal reflection sign. (c,4) Identify exactly which normal and physical conclusions have not followed.

**Solution.** (a) A graph-norm Cauchy sequence has limits of its vector, \(D_H\)-image and normal derivative in their three complete ambient modules. The two closed graphs identify both image limits with the images of the same limiting vector. Hence the intersection is complete. Each arrow preserves both domains, and its inverse does too, so it preserves their intersection onto. The error bound gives \(\|D_HL_hy\|\leq\|D_Hy\|+C\|y\|\); normal covector isometry and ordinary unitary norm give CVC.17. Apply the same estimate to the inverse for its bounded inverse graph map. (b) Swap after reflection sends \((x,x^2)\) to \((x^2,-x)\). Its derivative is \((2x,-1)\). The derivative of the original row is \((1,2x)\); reflecting, swapping and multiplying by \(-1\) gives \((2x,-1)\). Omitting that sign already fails at \(x=0\), where the second derivative component would be \(+1\). (c) Completeness of the intersection does not supply density, a joint graph core or source invariance. Nor has arbitrary transported Bott source or \(D_H\) gained normal derivatives from a unitary absorption. A self-adjoint regular normal sum and its actual source-typed class still need their analytic criterion. Finally the original \(C_0(T)\)-balanced physical graph passage and Bott \(+1\) pairing need the full inverse phase, grading and ordered Clifford factors; this carrier construction alone does not prove them.




**Exercise 275 (12 points).** (a,4) Prove PNC.1–PNC.2, including the copied proper-centre action and the actual represented class. (b,4) Construct both onto maps in PNC.3 and prove the canonical phase and genuine action in PNC.4. (c,4) Prove PNC.5 and identify which source property has not been restored.

**Solution.** (a) The square-root defect commutes with \(\overline F\), has square \(1-\overline F^2\), and is even on \(K\). Multiplication of the two-by-two matrix gives a self-adjoint odd unitary for the opposite copied grading. Functional calculus of the exactly invariant contraction makes the matrix invariant. IUR.1–IUR.2 prove class-preserving normalization with source \(\operatorname{diag}(\phi_0,0)\); adding ANC's exactly degenerate represented cycle preserves \(d_X\). Copy the original nondegenerate \(W\)-action onto the second summand independently of its zero \(B\)-source; this supplies the proper module used for absorption without extending the centre to the quantum algebra. (b) Proper cutoff/source-coordinate absorption gives the genuine onto \(W_+\) on \(Z_+\). The odd invariant unitary \(F_Z\) has inverse off-diagonal parts, so \(W_-=W_+F_{+-}\) is also an onto equivariant unitary. Their direct sum is even. Checking its two grading columns gives the constant off-diagonal identity \(J\); source and action are conjugated together, retaining the full \(d_X\) and canonical regular arrows. (c) The two equal regular normal connections commute with constant copy interchange on their finite normal core. Closed graph approximation proves the equality and domain preservation on the full completed domain. Scalar covariance and inverse bisections give the full section-domain arrow equality. The original zero-source copy remains an annihilated submodule after unitary transport, so nondegeneracy of the \(B\)-source has not been restored.

**Exercise 276 (12 points).** (a,4) Prove compact normal graph convergence and dense range of PNC.6–PNC.7. (b,4) Prove the full self-adjoint regular domain and resolvents PNC.8–PNC.10. (c,4) Prove the derivative order, completed normal-domain preservation and dense operator core PNC.11–PNC.13.

**Solution.** (a) Each fixed normal derivative word has summable weighted operator norms by the definition of \(c_j\), apart from finitely many bounded initial terms. Finite sums and closed normal commutator graphs give a compact limit, including its first vector derivative. The positive rank-one inequality yields \(\|(1-e_t)\eta_j\|^2\leq t/(4c_j)\). The generating span and the contraction bound extend convergence to every vector. Since \(e_t\) factors through \(h\), its range is dense. The positive inner product sum also gives zero kernel. (b) The inverse graph is \((h\xi,\xi)\); its two bounded adjoint resolvents give the full self-adjoint regular inverse domain \(\operatorname{Ran}h\). Constant copy diagonalization sends \(JH_0^{-1}\) to \(\Theta\oplus(-\Theta)\), so its full domain is \(\operatorname{Ran}H_0\). Multiplication gives \(H_0(J+i\lambda H_0)^{-1}\) as each onto resolvent. Functional calculus gives both norm bounds and compactness. (c) The geometric inverse series and its differentiated series converge in norm and preserve the closed normal domain. The inverse identity differentiates the resolvent to \((1-i\lambda H_0B_\lambda^{-1})(\delta H_0)B_\lambda^{-1}=JB_\lambda^{-1}(\delta H_0)B_\lambda^{-1}\); the derivative is kept in that order. The norm of \(B_\lambda^{-1}\) is at most one. The completed Leibniz identity gives PNC.12. For any full-domain vector \(z\), approximate \((T+i\lambda)z\) in ordinary norm by normal-core vectors and apply the resolvent. Both the vector and \(T\)-image converge, giving a graph core for \(T\) contained in the normal domain. This operator-core argument alone does not give joint normal graph approximation; the weighted proof and two-stage approximation in Section 11BG.7 supply it for this normalized operator.

**Exercise 277 (12 points).** (a,4) Prove the actual equivariant class equality and bounded arrow compactness PNC.14–PNC.16. (b,4) Check the finite phase/source normalization and the noncommuting resolvent test PNC.18. (c,4) State the source, compression, arrow and physical obligations still required.

**Solution.** (a) The scalar compact-calculus function vanishes at zero, so \(F_T-J\) is module compact. The straight odd self-adjoint path from \(J\) to \(F_T\) has only compact new source and square defects. Exact invariance of \(J\) leaves the difference of two transported compact fields as the bounded arrow error; rank-one approximation proves its continuity and the same facts on the interval module. This is a full source-typed equivariant homotopy, giving \(d_X\), not an inference from finite matrices. (b) The rational coefficients \(3/5,4/5\) have squared sum one. With opposite copied grading the normalized matrix is odd and unitary. Reading positive coordinates \(0,3\), then composing with the phase on negative coordinates, produces exactly \(J\). Conjugating \(\operatorname{diag}(1_2,0_2)\) gives a rank-two projection, so the second source subspace remains annihilated. The determinant in PNC.18 is at least its endpoint minimum \(63/(16\cdot32^2)\). Realification turns each complex inverse into a rational matrix inverse. Differentiating before realification gives precisely PNC.11; generally \(hh'\ne h'h\), so the derivative cannot be moved through the inverse. (c) A dense algebra of unbounded source commutators, source preservation of the normal/joint domains, normal and operator control of \(P'\), full unbounded arrow graph preservation and a completed normal sum remain unproved. The entire intersection graph core is proved by the weighted argument in Section 11BG.7; the \(T\)-core argument alone does not supply it or a core for the normal connection alone. The original-\(R\) physical graph passage and Bott \(+1\) comparison still require the original full inverse coefficient, grading and Clifford order.




**Exercise 278 (12 points).** (a,4) Prove that the single-bisection, single-copy seed in ENC.4 is diagonal and has the local floors ENC.5, and prove dense range. (b,4) Prove ENC.7 using the ordered exponential derivative. (c,4) Prove the reference graph-core statement ENC.10.

**Solution.** (a) Each range fibre meets a bisection in at most one arrow, and the generating vector has only one copy coordinate, so its rank one has only that diagonal entry. The nonzero covering bumps make each scalar sum positive. Norm convergence gives continuity and thus a positive minimum on any finite compact coordinate support. A cutoff equal to one above that minimum fixes its supported vector and factors through the seed; these supported vectors are dense, proving dense range. (b) Differentiating the power series of the exponential gives the ordered Duhamel integral, with norm at most \(|t|\|D_NH_*\|\). A compactly supported smooth extension of \(\chi_k\) has integrable \(|t|\)-weighted Fourier transform. Integrating the completed graph identity and using closedness proves full normal-domain preservation and ENC.7. This supplies no bound uniform in \(k\). (c) For a reference-domain vector, \(v_m\xi\) and \(v_mr_*\xi\) converge to its two graph components. At fixed \(m\), approximate \(\xi\) in ordinary norm by finite normal-core vectors; bounded \(v_m,r_*v_m\) give the two graph approximations. Their diagonal choice lies in the normal domain by part (b). This proves the \(r_*\)-core, not a joint-normal graph core.

**Exercise 279 (12 points).** (a,4) Prove full reference-domain preservation by \(F\), the bounded compact symmetrization ENC.11, exact \([r,F]=0\), and the adaptive weighted estimate ENC.21. (b,4) Prove ENC.12–ENC.14, including the possible enlargement of the closed essential domain. (c,4) Prove ENC.22–ENC.23, and explain why no derivative of \(F\) or normal-resolvent preservation is inferred.

**Solution.** (a) The cutoff commutator series converges in compact norm. For \(\xi=l_*\eta\), its partial-sum identity gives convergence of \(R_NF\xi\). Multiplying the limit by \(l_*\) and using \(l_*R_N\to1\) shows \(F\xi\in\operatorname{Ran}l_*\). Then \(FC_F=Fr_*F-r_*\) is a bounded self-adjoint extension, and averaging gives the bounded compact perturbation on exactly that domain. Conjugation by \(F\) interchanges the two averaged terms, proving exact commutation. Positivity and bounded-perturbation regularity give the stated positive regular operator and compact inverse. After each cutoff is chosen, its phase commutator is a known compact test for all later cutoffs. The finite initial weighted series contributes at most \((m+1)2^{-m}\), and the imposed later tails sum to \(2^{-2m}\). The bounded compact correction contributes at most \(\|B_F\|2^{-m}\); its commutator with the approximate identity tends to zero in norm. This proves ENC.21. (b) \(P\) commutes with every reference cutoff and thus preserves its full range domain. Its new commutator is \([B_F,P]\), compact bounded. The diagonal reference restriction differs boundedly from the original first-copy reference operator, so the first-copy normal-compatible core remains graph dense. EAI's compact reference commutator and uniform resolvent-commutator argument prove self-adjoint regularity of the compressed closure. On the reference graph, \(S_0=f r_0+T_0\) is continuous; approximate a full closure graph vector first by reference vectors and then by this core. This proves ENC.14 on the full closure, without identifying that larger domain with the reference domain. (c) Each cutoff preserves the full reference domain. Compress its bounded compact operator commutator, approximate a full essential graph vector by reference vectors, and use closedness to obtain the full-domain identity. Strict convergence and the vanishing commutator norm prove both operator graph limits. The completed normal product identity leaves precisely \((D_Nw_m)\xi\) after subtracting the strongly convergent normal component, so simultaneous graph convergence is equivalent to this term tending to zero. It is eventually zero on a finite compact normal-core vector, but no uniform relative estimate has been proved on the entire normal domain. The phase averaging can still be normal-rough; no bound for \(D_NB_F\), \(D_NS_N\), or a normal resolvent commutator follows.

**Exercise 280 (12 points).** (a,4) Prove source and class preservation in ENC.15–ENC.16. (b,4) Prove full arrow-domain preservation and the dense complete invariant joint graph ENC.17–ENC.18. (c,4) Check the finite compressed and arrow-error formulas in ENC.20 and identify the original physical obligations that remain.

**Solution.** (a) The ordinary reference source commutator and the weighted graded phase-source defect are compact. Their sum is the displayed unbounded source commutator; bounded perturbation of the reference regulator preserves both properties. Reference-domain preservation and closed graph approximation extend the commutator to the full essential domain. The additional weighted \([F,P]\)-source test gives EAI's source-local off-diagonal resolvent comparison, whose integrable \(s^{-2}\) bound identifies the compact localized phase difference. Removing the zero-source summand then retains exactly \(d_X\), while the first-copy source is the original nondegenerate \(\Phi\). (b) The reference arrow partial sums converge in compact-field norm on each compact test. Exact invariance of \(F\) makes averaging give the displayed \(A_h\). Compress its error, approximate full essential graph vectors by the reference core, and use closedness; inverse arrows give equality of full domains. Intersect with the covariant completed normal section domains. Closedness of both operators proves completeness, ENC.14 proves density, and the compact operator error plus the isometric normal transport gives the stated graph bound and continuity. (c) Exact matrix multiplication gives \(F^2=1\), oddness for the opposite copied grading, and \(FrF=r\). Block expansion gives \(PFrP=f r_0+PF(1-P)rP\). For the row-interchange arrow \(U\), invariance of \(F,P\) gives \(S_0U-US_0=-PF(UrU^{-1}-r)PU\). The essential source identity is \(I_4\), and its sixteen matrix-unit products hold on both arrow rows. These finite identities do not prove a joint graph core, normal-source stability, a completed physical sum, the original \(R\)-balanced passage or its Bott \(+1\) pairing with the full inverse coefficient and ordered Clifford factors.


**Exercise 281 (12 points).** (a,4) Prove ENC.24–ENC.26 for the actual source-essential inverse on the stated global compact-coordinate carrier, including the full essential domain. (b,4) Verify every seed, weight and domain assertion in ENC.27–ENC.28. (c,4) Prove ENC.29 and distinguish failure of this cutoff family from nonexistence of a joint core.

**Solution.** (a) Constant diagonal eigenvalues tending to zero give a compact seed with dense range and zero normal derivative. Every threshold cutoff and tail convex combination has finite support and the same zero derivative. All original common-tail source/arrow and adaptive phase tests can still be imposed. Core approximation and closedness give the full normal-domain commutation identity; strong convergence and ENC.22 give simultaneous convergence of all three graph components. For fixed cutoff the compressed operator times the cutoff is bounded, because the reference weighted cutoff is bounded. Apply that operator and the commuting normal cutoff to a finite normal-core approximation, then choose the two approximation indices successively. This proves a joint core on the entire essential domain without differentiating the phase. (b) The diagonal seed tails and every differentiated tail tend to zero in norm by the Gaussian factor in \(n^2\). Strictly positive entries and finite coordinates give dense range. At the selected threshold all lower-index pairs are fixed and all higher-index pairs vanish, giving nesting and strict convergence. Each reference weight is a finite coordinate sum and grows to infinity with the coordinate index; diagonal functional calculus proves its compact inverse, full range domain and both resolvents. The odd swap commutes with this weight, so it is self-adjoint regular on that same domain; the root is its only unpaired zero coordinate. The phase difference is compact and preserves the class \(1_A\). Differentiating the ratio defining \(\chi\) gives its middle value \(1/2\) and derivative \(2\), hence all three formulas in ENC.28. (c) The selected-coordinate coefficients are square summable, constant in \(t\), and their \(S\)-images have uniformly square-summable tails bounded by \((m+1)/N_m\). This proves both full domain memberships. The derivative at the selected coordinate times \(1/N_m\) is \(1/2\); evaluating at zero proves the persistent lower bound. Unit constant coordinate sections also disprove a uniform relative normal bound. Constant coordinate cutoffs nevertheless have zero derivative and approximate this same operator and normal graph, so failure of the oscillatory cutoffs does not imply failure of every joint core. The original physical sum and full holonomy Bott comparison are not established by either case.

### Exercises 282–283. Regular auxiliaries and genuine fixed-point holonomy (24 points)

**Exercise 282 (12 points).** For (MGS.15): (a) give the complete weighted
domain and prove that finite coordinates form a graph core; (b) compute
the norm of the commutator with the generator on both gradings;
(c) prove the index and explain why a doubled invertible length operator
cannot replace this auxiliary. Each part is worth four points.

**Solution.** (a) Write \(u=(a_n,b_m)\) in the two gradings. The domain is
\[
 \sum_{n<0}n^2|a_n|^2+\sum_{n>0}n^2|a_n|^2+
 \sum_{m<0}m^2|b_m|^2+\sum_{m\ge0}(m+1)^2|b_m|^2<\infty.
\]
The ordinary \(\ell^2\) condition is included, and \(a_0\) is unrestricted.
Its blocks are \((e_n^+,e_n^-)\) with weight \(-n\) for \(n<0\),
and \((e_n^+,e_{n-1}^-)\) with weight \(n\) for \(n>0\), plus the
zero positive root. The two block resolvents at \(\pm i\) have norm at
most one and ranges exactly that weighted domain. Their orthogonal
sum proves self-adjointness. Truncating these paired blocks tends to
the vector and its image, proving the core. Resolvent tails have norm
at most \((1+R^2)^{-1/2}\), so the resolvent is compact.
(b) The positive-to-negative commutator maps \(e_n^+\) to
\(-e_{n+1}^-\) for \(n\le-2\), to \(-e_0^-\) for \(n=-1\), to
\(e_0^-\) for \(n=0\), and to \(e_n^-\) for \(n\ge1\).
Its exceptional row is the functional \((-1,1)\), of norm \(\sqrt2\);
the other rows form an isometry on their orthogonal coordinates.
The negative-to-positive block maps \(e_m^-\) to \(-e_{m+1}^+\)
for \(m\le-2\), to \(e_1^+-e_0^+\) for \(m=-1\), and to
\(e_{m+2}^+\) for \(m\ge0\). Its exceptional column has norm
\(\sqrt2\), with the other output coordinates orthogonal.
Both block norms are \(\sqrt2\), hence so is the odd full commutator.
The shift preserves the weighted domain because successive weights
differ by at most one; inverse shifts do as well. The telescoping
formula gives the stated bound for every integer \(k\).
(c) The displayed bounded right inverse into the graph domain proves
surjectivity of \(D_K^+\); its only kernel vector is \(e_0^+\).
The index is \(1-0=1\). A doubled positive proper length has invertible
off-diagonal block and index zero. Its restriction factor would be
zero in ordinary scalar KK, so it would kill the original normal
class and its local Bott pairing. Reduced continuity alone cannot
repair that numerical and class failure.

**Exercise 283 (12 points).** In the irrational sphere suspension:
(a) identify the return isotropy and holonomy cover at a pole;
(b) identify the actual column represented by a chart point
\((x,t,n_i)\), and check a change \(z_j=hz_i\);
(c) derive the original local Bott pairing using the normalized
plaque correspondence, retaining the full inverse normal coefficient.
Each part is worth four points.

**Solution.** (a) Every integer rotation fixes a pole. If \(k\ne0\),
its tangent derivative rotates by \(2\pi k\alpha\), which is not a
multiple of \(2\pi\); its germ is nonidentity. Thus isotropy is
\(\mathbb Z\). The pole leaf is a circle, but its loops have injective
holonomy \(\mathbb Z\), so its holonomy cover is \(\mathbb R\).
The general column bijection (MGS.4) gives the same cover directly.
(b) The actual source coordinates are
\((y,n)=(t^{-1}z_i(x),t^{-1}n_i)\).
The overlap changes them to
\(((ht)^{-1}hz_i(x),(ht)^{-1}hn_i)=(y,n)\).
The inverse spin transport \(U_t^{-1}\) also agrees, by its coherent
composition law. Isometry preserves normal volume and the sum of
squared chart weights is one, proving the physical column norm.
(c) The bump gives \(\langle\eta,\eta\rangle=1_B\) and
\(\overline\zeta\mapsto\theta_{\eta,\zeta}\) is the onto inner-product
preserving correspondence unitary. Thus \(i_*b_U\) is the original
counting image. The creation equations in (MGS.14), the whole normal
restriction (MGS.8), and the full inverse normal duality give
\((i_*b_U)z_G=b_U[D_N]=1\).
The auxiliary is even and contributes index \(+1\).
The full normal inverse is
\(\Pi C_2\) with transition \(z_g^{-1}\rho_t(s_g)\), dual determinant
line and final right \(C_2\); replacing it by the positive original
spinor or retaining only determinant phases would not be the proved
comparison.

### Exercise 284. Weighted normal regularization of the actual normalized inverse (12 points)

**Exercise 284 (12 points).** Use the actual represented operator and completed connection of Section 11BG. (a,4) Prove the norm limit \(\beta_\lambda\to0\) and the ordered weighted bound PNC.20. (b,4) Prove that the single family \(R_\mu\mathscr C_N\) is a full joint graph core, without assuming \(T\xi\in\mathcal D_N\). (c,4) Prove the even approximation PNC.23 and state precisely which source-essential conclusion is still missing.

**Solution.** (a) On \(\operatorname{Ran}H_0\), \(\|B_\lambda^{-1}H_0\eta\|\leq\lambda^{-1}\|\eta\|\). Dense range and the contraction bound imply strong convergence to zero on all of \(\mathcal M\), and elementary tensors imply the same on the vector target. The vector derivative \(\delta H_0\) is compact. A left product with a rank-one map has norm at most \(\|(B_\lambda^{-1}\otimes1)a\|\|b\|\); finite-rank approximation therefore gives \(\beta_\lambda\to0\). Differentiating \(u_\lambda=i\lambda R_\lambda\) in the established order gives
\((\delta u_\lambda)H_0=i\lambda(JB_\lambda^{-1}\otimes1)(\delta H_0)B_\lambda^{-1}H_0\).
The left compact product has norm \(\beta_\lambda\), the right product has norm at most \(\lambda^{-1}\), and \(J\) is unitary. Thus the whole norm is at most \(\beta_\lambda\). Writing \(\xi=H_0\eta\) gives \(\|(\delta u_\lambda)\xi\|\leq\beta_\lambda\|\eta\|=\beta_\lambda\|T\xi\|\). No interchange of the derivative and inverse is used.

(b) The family \(u_\lambda\) consists of contractions tending strongly to identity, preserves both full domains and commutes with \(T\) on its domain. The completed product rule gives
\(\nabla u_\lambda\xi-\nabla\xi=(u_\lambda\otimes1-1)\nabla\xi+(\delta u_\lambda)\xi\).
Strong convergence on the vector target and the bound from (a) make this tend to zero for every joint-domain vector, together with the vector and its \(T\)-image. Fix \(\mu\ne0\). The identity
\((T+i\mu)u_\lambda\xi=i\lambda\xi+i\lambda i(\mu-\lambda)R_\lambda\xi\)
puts this preimage in the full normal domain using only normal preservation by \(R_\lambda\). Approximate it there by normal-core vectors. Applying \(R_\mu\) converges in the normal graph by PNC.12, and in the \(T\)-graph by \(TR_\mu=1-i\mu R_\mu\). A diagonal sequence with latter error at most \(1/\lambda\) proves full joint graph density of \(R_\mu\mathscr C_N\). Closedness of the two operators makes the intersection graph complete.

(c) Inverting \(H_0+t\) in the normal graph algebra and differentiating yields the ordered formula for \(\delta e_t\). The contractions \(t(H_0+t)^{-1}\) tend strongly to zero on the dense range of \(H_0\), hence on the entire vector target. Compactness of \(\delta H_0\) gives \(a_t\to0\), and the right factor \((H_0+t)^{-1}H_0\) has norm at most one. This proves \(\|(\delta e_t)\xi\|\leq a_t\|T\xi\|\). Dense-range positivity gives \(e_t\to1\); commutation with \(T\) and the product rule now give all three joint limits. The operator is the normalized inverse with its retained zero-source copy. Preservation of these full domains by \(P'\), a source-essential normal comparison, unbounded source/arrow control on this same operator and its original physical normal product are still unproved. The result cannot be transferred to \(S_N\) by equality of bounded KK classes alone.

### Exercises 285–286. Boundary control and the genuine action (24 points)

**Exercise 285 (12 points).** (a,4) Prove OJC.2–OJC.5 on the full vector normal domain of the proper probability base. (b,4) Explain why the convex source/arrow selection retains the normal estimate while removing the zero-source copy. (c,4) Prove the full joint graph-core equality OJC.9 on the entire closed essential operator domain.

**Solution.** (a) A section vanishing at infinity has arbitrarily small norm outside a compact set; its compact anchor image stays away from the coordinate faces. This proves \(b_\xi(t)\to0\). For a finite-coordinate core vector, use the fixed lifted probability coordinates of PJ.3 to follow the shortest straight path to a face. The whole vector vanishes near the face and its derivative is bounded by \(C_Tb_\xi(d(x))\), so integration gives OJC.3. Graph approximation extends this estimate to the entire vector domain. Properness makes each pulled-back unit cutoff an \(A\)-coefficient; multiplying it by \(Q_k\) gives a compact even contraction. Later cutoffs are one on the support of an earlier one, proving the nested product. On the derivative strip, \(d(x)\leq2\varepsilon_k\) and \(\|\delta\chi_k\|\leq C_\theta/\varepsilon_k\). Applying the whole-vector boundary estimate gives OJC.5, including its uniform relative bound. The product rule and strict coordinate convergence now give full normal graph convergence.

(b) A finite convex combination of sufficiently late cutoffs has the same uniform relative bound, and its derivative on a fixed vector is bounded by the largest corresponding boundary-strip bound, which tends to zero. All finitely many ordinary phase/source, compact-tail and pulled-back compact-arrow tests are simultaneously reduced by KG.2. Choosing strictly later index intervals retains the product equation. The locally finite reciprocal and its norm-convergent commutator series therefore satisfy ENC.11–ENC.16, including the compact projection error and weighted source-local off-diagonal tests. The completed essential compression is self-adjoint regular, and its source-local resolvent comparison preserves the entire represented class \(d_X\) while discarding the annihilated second copy. This step uses no derivative of the phase or the standardizing unitary.

(c) The added adaptive tail test for \([F,v_m]\) gives the compact norm-null commutator \([S,w_m]\) on every vector of the full closed \(S\)-domain, by ENC.21–ENC.22 and reference graph approximation. Thus \(w_m\xi\to\xi\) in the operator graph. The derivative term tends to zero by (a)–(b), giving convergence in the normal graph as well. At fixed \(m\), \(Sw_m\) is bounded because \(r_*v_m\) is bounded; \(w_m\) is bounded on the full normal graph. Approximate \(\xi\) in that graph by finite core vectors, apply \(w_m\), and choose the second approximation error at most \(1/m\). This diagonal sequence belongs to \(\mathscr K\) and converges in all three graph components. Closedness of the two operators gives completeness of the intersection.

**Exercise 286 (12 points).** (a,4) Prove the genuine cocycle and adjointability in OJC.10. (b,4) Prove failure of full normal-domain preservation using the displayed constant vector. (c,4) Identify exactly the class, source and graph conclusions proved by OJC.9 and the normal/physical statements that still require proof.

**Solution.** (a) On finite coordinate vectors, \(V(x)\) and \(V(x)^*\) are continuous and preserve the \(C(T)\)-valued inner product. A tail has the same norm before and after applying either operator, so finite-coordinate approximation proves adjointable module continuity on all sections. Both products are identity. Cancellation of the intermediate \(V^*V\) in \(U_k(x)U_l(x-k\alpha)\) gives \(U_{k+l}(x)\). Regular translation changes \(m\) to \(m+k\); the two copy phases give precisely \(e^{in(\sin x-\sin(x-\alpha))}\) when \(k=1\).

(b) The coefficients \(1/n\) are square summable, and finite truncations have zero normal derivative, so \(\xi\) lies in the completed coordinate domain. At zero, the first \(N\) difference-quotient coordinate squares of \(U_1\xi\) tend to \(N|\psi'(0)|^2\), since each derivative has modulus \(|n^{-1}in\psi'(0)|\). The full difference quotient is at least this partial norm. Irrationality implies \(\alpha\notin2\pi\mathbb Z\), so \(\psi'(0)=1-\cos\alpha>0\). A finite derivative norm would bound these partial limits for every \(N\), which is impossible. The integrated graph identity for the closed derivative gives norm differentiability of every full-domain vector, proving that this image lies outside the domain.

(c) The operator represents the whole actual equivariant \(B\)-source inverse, has nondegenerate source, source-local compact resolvents, compact unbounded source commutators on its dense test algebra, and full continuous transported arrow operator-graph domains. Its chosen coordinate normal connection has a full joint core on the same source-essential module over the proper, possibly noncompact probability base. This does not identify that connection with the canonical holonomy-parallel one. It proves neither invariance of the joint domain under \(U_h\), normal preservation by the source action, nor a completed original physical normal sum or its Bott \(+1\) pairing. OJC.10 demonstrates why ordinary carrier transport alone cannot discharge the covariance obligation.

### Exercises 287–288. The mixed form and the full adjoint domain

**Exercise 287 (12 points).** (a,4) Derive SNC.5–SNC.6 with both resolvent signs, and prove the full joint core. (b,4) Prove the quadratic identity and the complete sum domain. (c,4) Prove SNC.9–SNC.10 without assuming that an adjoint-domain vector is already in either summand domain.

**Solution.** (a) The domain-preserving inverse calculus in SNC.3 gives \(NB_\lambda=-B_{-\lambda}N+i\lambda K\). Multiplication by the two bounded inverses, then the product rule for \(H\), gives exactly \(-R_{-\lambda}N+JB_{-\lambda}^{-1}KB_\lambda^{-1}\). For \(\xi=H\eta\), the right product \(B_\lambda^{-1}H\) is bounded by \(\lambda^{-1}\), while the left compact-weighted product has norm \(\beta_\lambda\). Thus the error in SNC.6 is at most \(\beta_\lambda\|T\xi\|\). Both signed regularizers converge strongly to one. They commute with \(T\), and SNC.6 supplies normal graph convergence. For the fixed resolvent core, the displayed \(\eta_\lambda\) is in the normal domain; approximate it in the \(N\)-graph and apply \(R_\mu\). SNC.5 and boundedness of \(TR_\mu\) control the approximation. Choose that error after choosing \(\lambda\), proving approximation of every full joint vector.

(b) On the mixed core, \([N,H^{-1}]=-H^{-1}KH^{-1}\); \(NJ=-JN\) therefore gives \(NT+TN=A J K A\). Skew-adjointness of \(K\) and anticommutation with \(J\) make \(JK\) self-adjoint. The mixed quadratic form is bounded in absolute value by \(\kappa\|T\xi\|^2\). The joint core passes the identity to the complete intersection. With \(\kappa<1\), SNC.8 bounds each summand graph by the sum graph and also bounds the sum graph by their joint graph. Closedness of the two summands makes the intersection complete, proving closedness of \(D\) on precisely that domain.

(c) Adjoint testing of \([D,H]=K\) first gives only \(H\xi\in\operatorname{Dom}D^*\) and \(D^*H\xi=HD^*\xi+K\xi\). The range of \(H\) puts this vector in \(\operatorname{Dom}T\). Testing against the stated \(N\)-core then represents its weak \(N\)-image by \(HD^*\xi+K\xi-J\xi\); self-adjointness of \(N\) puts it in the full \(N\)-domain. This is SNC.9. For either deficiency equation, SNC.8 now applies legally to \(H\xi\): the lower norm is \(\sqrt{1-\kappa}\|\xi\|\), and SNC.9 bounds the upper norm by \((a+\kappa)\|\xi\|\). The strict inequality forces zero. Onto resolvents give self-adjointness. The same energy bound and \(\|T\xi\|\geq a^{-1}\|\xi\|\) give the spectral gap and all three inverse bounds in SNC.10.

**Exercise 288 (12 points).** (a,4) Prove the infinite circle model's full domain, compact resolvent and exact lowest gap. (b,4) Check its represented coefficient class and prove source-domain failure for \(b(z)=z\). (c,4) Determine exactly what must still be proved to apply the small-commutator argument to the original normalized graph inverse.

**Solution.** (a) The complete Fourier basis and finite pair decomposition of Section11BD diagonalize the operator into the displayed self-adjoint two-by-two matrices. Squaring each matrix gives \((k^2+256n^2)I\). Their adjoint-domain test against every finite block gives exactly the stated weighted square-summability condition, and their inverse tails tend to zero since \(n\geq1\) and \(|k|\) cannot both remain bounded along an escaping block. The smallest absolute eigenvalue is \(16\), attained at \(n=1,k=0\). This equals the bound \(\sqrt{1-\kappa}/a\).

(b) The represented source is one even copy of the pullback by \(x\mapsto e^{i\alpha(x)}\), plus an annihilated complement. The finite-rank source commutator permits deforming the bounded phase to zero on that first pair; the rest is source annihilated. The remaining even source line, with no represented odd line, has multiplicity \(+1\). The tails of \(F_T-J\) tend uniformly to zero, so they are module compact and retain this class. On the constant first even vector, \(\Phi(z)\) produces \(e^{i\alpha}\). Its derivative has squared magnitude asymptotic to \(1/(8|x|)\), whose integral diverges. Hence this image fails the full \(H^1\) and sum domains. The conclusion concerns this model source, not the actual Bott algebra.

(c) The coefficient seed, its all-jet diagonal local floors, phase anticommutation after the normal grading, small commutator and weighted norm limit come from the actual canonical normalized construction. Physical application still requires a self-adjoint original normal operator with the specified transported core and coefficient identities. It must preserve the original \(C_0(T)\)-balance, full inverse Spin-c coefficient and ordered Clifford factors. Source domain/commutators, spatial source-local compactness, genuine holonomy cycle and original Bott \(+1\) product remain separate. The normalized annihilated source summand remains. Neither ordinary trivialization nor an essential compression silently discharges these requirements.

### Exercises 289–290. Family monodromy and local calibration

**Exercise 289 (12 points).** (a,4) Determine the full domain and kernels of
FBC.7, prove compact resolvent and check the exact regular action in both
grades. (b,4) Prove that FBC.8 gives a nonunit whole-base family despite
index \(+1\) at every point. (c,4) Prove the additive order-two assertion
without assuming that tensor order two alone implies additive order two.

**Solution.** (a) The trivial-character pair at index \(n\) has matrix
\(\left(\begin{smallmatrix}0&n+1\\n+1&0\end{smallmatrix}\right)\).
The sign pair consists of the positive vector at \(n\ge1\) and negative
vector at \(n-1\), with the same matrix and weight \(n\). These blocks and
the positive zero vector form a complete orthonormal decomposition.
Adjoint testing against their finite sums requires precisely the weighted
square-summability domain in the text, and on that domain the operator is
self-adjoint. The resolvent block norms are
\(((n+1)^2+1)^{-1/2}\) and \((n^2+1)^{-1/2}\); their tails tend to zero,
so finite block truncations approximate each resolvent in norm. The only
kernel is the positive sign vector at zero. The Fourier basis identifies
both character sums with the amplified regular representation, and the
deck action is scalar on each block, preserving its full domain.

(b) The transitions carry the kernel by the sign character, so Theorem
11BL.2 identifies the index family with \(L\). A unit section would give
an odd \(S^1\)-valued function on \(S^3\). Lift its argument to \(\mathbb R\);
the antipodal difference would be a constant odd multiple of \(\pi\),
but applying the antipodal involution twice forces its negative to equal
it. This is impossible. Stable equality with the trivial line would
trivialize \(L\) after taking determinants and tensoring with
\((\det B)^*\), also impossible. Its rank at every point is one.

(c) Compute \(U(u)^*U(u)=I\) using
\(|z_1|^2+|z_2|^2=1\); the off-diagonal terms cancel. The antipodal identity
\(U(-u)=-U(u)\) makes \([u,w]\mapsto([u],U(u)w)\) independent of the
representative \((u,w)\sim(-u,-w)\). Its inverse uses \(U(u)^*\), so it
is a unitary trivialization of \(L\oplus L\). Hence twice the nonzero
class \([L]-[1]\) is zero. The multiplication trivialization of
\(L^{\otimes2}\) is a separate statement.

**Exercise 290 (12 points).** (a,4) Prove FBC.3 and the completed resolvent
domain in Theorem 11BL.1. (b,4) Prove the entire disk-family identity and
FBC.9 with its degree and Clifford order. (c,4) Explain exactly which
finite-group index condition suffices for a whole-base unit, and why
FBC.9 supplies neither compact-frame descent nor a general physical sum.

**Solution.** (a) Write \(D_j\) as the partition sum on \(\mathcal D\).
Conjugation by \(V_{t_{kj}}\) changes every label \(t_{ji}\) into
\(t_{ki}\); its inverse label changes accordingly. This proves equality
with \(D_k\), including the domain, since the transitions preserve
\(\mathcal D\). The glued two resolvents are bounded adjointable section
operators. Their range consists exactly of continuous sections whose
image under \(D\) is continuous, with both vanishing at infinity, by
\(D R_z\xi=\xi+zR_z\xi\) and
\(R_z(D-z)\xi=\xi\). The supported chart domain sections are dense, and
both onto resolvents prove self-adjoint regularity on that completed
graph, rather than only on finite chart vectors.

(b) One chart gives an actual Hilbert-bundle trivialization over the
entire disk. Interpolate its bounded perturbation \(K_j\) to zero on
the unchanged full domain. The resolvent identities and integrable
two-resolvent difference bound prove the full module homotopy. The
constant endpoint splits into its finite graded kernel and a central
degenerate sign swap; the kernel dimension difference is \(+1\), so
the family is the unit over \(U\). The completed supported-section
tensor unitary in Theorem 11BL.3 gives the ideal base change. Multiply
it on the left by \(\beta_U\). The family factor is even and stays
in its displayed order, preserving \(d\), orientation and the full
right Clifford coefficient. This proves FBC.9.

(c) The sufficient condition is equality of the finite-group equivariant
kernel difference with the trivial representation, in its representation
group. Theorem 11BL.2 then associates a unit virtual bundle on every
base. Equality of dimensions is weaker, as (a)–(b) of Exercise 289 show.
FBC.9 is an ordinary comparison on a disk contained in the given
flat unit chart. It does not construct compatibility across different
closure models or a compact-frame quotient, and has no assertion about
the separately ordered full inverse coefficient or the self-adjointness,
source domains and spatial compactness of its physical normal sum.


### Exercises 291–293. Weighted normal phases and the original disk (36 points)

**Exercise 291 (12 points).** For \(T=(0,1)\), \(g=dx^2\), and the exhaustion \(r=1/x+1/(1-x)\), prove completeness of \(g_\alpha=\alpha^{-2}g\), \(\alpha=(1+|r'|^2)^{-1/2}\). Prove the graph cutoff bound and state the exact self-adjoint domain. Does the argument impose a boundary value at either omitted endpoint?

**Solution.** The exhaustion tends to infinity at both endpoints and \(\alpha|dr|\le1\). A \(g_\alpha\)-Cauchy sequence has bounded \(r\), lies in a compact interior interval, and converges there. Equivalently, near either endpoint the length is at least the divergent integral of \(|r'|\) (4 points). With \(\eta_n=\zeta(r/n)\), \(\|[A,\eta_n]\|\le\|\zeta'\|_\infty/n\); multiplying a maximal-domain vector by these cutoffs gives compactly supported \(H^1\) vectors converging in the full graph norm (4 points). The domain is \(\{u\in L^2:u\in H^1_{\rm loc},Au\in L^2\}\). It is obtained from minimal=maximal closure; no endpoint value is required, and it is not an asserted global unweighted \(H^1\) domain (4 points).

**Exercise 292 (12 points).** Explain why the leading differentiated resolvent integral in WN.6 cannot simply be differentiated in operator norm. Give the exact integrable graph-vector bound, the compact remainder bound, and the resulting phase comparison.

**Solution.** The separate leading terms are \(-fA_tR_z^2\); a raw operator norm estimate is of order \((1+\tau^2)^{-1/2}\), whose integral diverges. On a common graph vector \(v\), their norm is at most \(\|f\|\|A_tv\|/(1+\tau^2)\), uniformly on compact parameter intervals (4 points). Spectral integration gives \(fA_t(1+A_t^2)^{-3/2}\); the remaining resolvent terms have norm at most \(3\|B_t\|/(1+\tau^2)\) and are scalar compact by their compact coefficient supports (4 points). The derivative expression is norm continuous. The fundamental theorem on graph vectors and density gives a norm integral of compact operators, hence \(F_{e^{f/2}Ae^{f/2}}-F_A\in\mathcal K\). No bounded commutator of the unbounded operator with an arbitrary noninvariant holonomy transport is inferred (4 points).

**Exercise 293 (12 points).** Explain why strong continuity of the self-adjoint bounded phases in the metric comparison is sufficient for a KK homotopy. Prove the continuity required of the compact defects and identify the exact Bott integer and algebra scope.

**Solution.** Strong continuity, strong continuity of the adjoints, and a uniform operator bound make pointwise application preserve continuous Hilbert-space sections; the family is an adjointable operator on \(C([0,1],H)\) (4 points). The square defect is a localized \(C_0\) resolvent function, norm continuous and compact. The commutator uses a norm-continuous compact coefficient and the uniform integrable rectangular resolvent bound; its integral is a norm-continuous compact field. Density treats all unit functions (4 points). Bounded locality and this metric homotopy identify the weighted phase with the full analytical inverse of Theorem 7.17 on the original disk; the prescribed ordered Thom inverse pairs with its positive Bott generator as \(+1\). This is the actual maximal normal cycle, not yet a reduced physical graph cycle (4 points).


### Exercises 294–296. Direct transfer and actual domains (36 points)

**Exercise 294 (12 points).** Under the unit-kernel hypothesis, prove that \(\ker\sigma=\ker q_G\). Explain why the faithful trace slice works without amenability of \(\Gamma\), and compute the restriction of the resulting transfer to the original unit algebra.

**Solution.** The trace slice of \(\sigma(a^*a)\) is \(i_mE_r(q_G(a)^*q_G(a))\), since only unit arrows have identity label (4 points). A positive element with zero slice has square root annihilating every \(\xi\otimes\delta_e\). The commuting right regular operators carry these vectors to every coordinate, so the square root is zero. Reduced expectation faithfulness, checked on every actual source-fibre basis vector, now proves both kernel inclusions; no reduced trivial character is used (4 points). Thus \(\Delta_ri_r(f)=i_m(f)\otimes1\). Even external-product functoriality and the unit pairing of \(\kappa\) give \([i_r]\Theta=[i_m]\), on the whole original unit algebra in this scope (4 points).

**Exercise 295 (12 points).** Construct the discrete flat Čech label and prove its unit kernel. Include its faithful finite-dimensional matrix representation. Explain why a label does not create an arrow at a missing continuation, and show how its original-disk restriction survives the cover equivalence.

**Solution.** In oriented flat charts the actual germ has a unique affine expression \(v\mapsto A_gv+b_g\), constant on every connected bisection. The label \((A_g,b_g,i-j)\) multiplies as \((AA',b+Ab',i-k)\). Identity label forces \(i=j\) and the identity germ on that same chart, hence an actual unit (4 points). Second countability makes the label image countable; the block matrix in IHT.16 embeds its generated group in \(GL_{q+3}(\mathbb R)\). Every convolution coefficient still has its actual source/range domain in the first tensor factor; writing an affine expression outside that domain creates no arrow (4 points). On a disk inside chart \(j\), the equivalence core maps to the lifted-arrow core in copy \(j\), preserving every counting inner product. Completion on the essential source modules gives IHT.18. Compose these two identities with IHT.12 to obtain IHT.19. Its even transfer retains the original degree and full inverse coefficient, so IHT.20 pairs as \(+1\) (4 points).

**Exercise 296 (12 points).** In the two-puncture example, prove that the radius-feature kernel is CND but fails uniform properness. Prove both exact GK.4 controls for IHT.26. Evaluate its two illustrated costs.

**Solution.** The radius-feature kernel is the squared distance of the real map \(f=-\log\rho\), hence its zero-sum quadratic form is \(-2|\sum a_if(h_i)|^2\). Compact cross-arrow tests stay away from zero and give bounded increments. But \(h_n\) and \(g_nh_n\) have radius \(y_n\), so their feature difference is zero while \(g_n\) escapes every compact actual cross-arrow set with compact endpoints (4 points). For IHT.26, combining repeated arrow coefficients gives the quadratic form \(-(1+y^{-2})\sum a_j^2\). Cancellation makes every nonunit increment cost \(1+y^{-2}\), independently of \(h\); compact arrow tests bound it above. A low bound with \(R>1\) forces \(|y|\ge(R-1)^{-1/2}\), and compact endpoints bound \(|y|\) above, giving compact closed pieces of the six punctured cross components; units over the compact test set are also compact (4 points). For \(R\le1\) there are no nonunit low increments. The costs are \(1+(1/2)^{-2}=5\) and \(1+(1/4)^{-2}=17\). No zero-height cross arrow is included (4 points).


### Exercises 297–299. Joint domains and physical graph phase (36 points)

**Exercise 297 (12 points).** Starting with the even normalized scalar class, prove that its unital representative admits the explicit compact-resolvent operator of DJ.7, with bounded commutators and preservation of the whole domain for every group label. Explain how nonzero index is compatible with the temporary involution in DJ.2.

**Solution.** Pair the finite kernel with an added opposite zero-source space. This makes \(F^2=1\) on the enlarged space but its source projection \(P\) need not be the identity, so it does not assert an odd unitary on a unital source of nonzero index (3 points). Convex tails of paired finite spectral projections give DJ.3. The diagonal sum \(r=1+\sum(1-u_n)\) is finite on every basis coordinate and diverges off finite sets. Thus \(l=r^{-1}\) is compact and DJ.4 gives the actual compact resolvents (3 points). Norm convergence of \([R_N,\phi(a_j)]\) and the identity multiplied by \(l\) show that each \(\phi(a_j)\) preserves \(\operatorname{Ran}l\). The other DJ.3 tail test bounds the second term in DJ.5, proving the completed commutator identity for labels and their inverses (3 points). Finally \(B_0=[D,P](2P-1)\) is bounded, odd and self-adjoint. Subtracting it preserves the full domain and compact resolvents, and compression to \(P\operatorname{Dom}D\) gives the unital operator. The bounded perturbation changes the phase compactly, while the discarded zero-source block is degenerate. Hence the class and index \(+1\) remain (3 points).

**Exercise 298 (12 points).** Prove scalar local compactness for the jointly weighted operator and the localized product positivity. State precisely where the exact nonnegative anticommutator formula holds.

**Solution.** On a compact buffer the positive speed is bounded below. Graded cross-term cancellation and normal ellipticity give DJ.14 with both the full normal \(H^1\) norm and \(\|D_\kappa u\|\) (3 points). Finite \(D_\kappa^2\) projections commute with the scalar speed and normal coefficients. Their estimates are uniform, so weak limits give the exact maximal local joint domain; exhaustion cutoffs with derivative \(O(1/n)\) prove minimal equals maximal (3 points). The resolvent tail above \(|D_\kappa|>M\) has norm at most \(C/M\), and its finite auxiliary part is compact by scalar Rellich. This proves ordinary scalar local compactness (3 points). On the complete **unweighted reference** DJ.20, joint spectral calculus gives the nonnegative operator DJ.23. Positive joint rescaling and rectangular locality transfer this equality modulo localized compacts to the weighted operator. A finite squared partition and source cutoffs give every source positivity sandwich. An exact global formula for the variable-weight anticommutator has not been used (3 points).

**Exercise 299 (12 points).** Prove the physical-column limit DJ.31a and the two-copy domain failure DJ.32–DJ.33. Explain why the latter does not invalidate the bounded graph phase.

**Solution.** On compact joint core vectors, the identity \(B_t=t^{-1}A_0+C\) gives a uniform \(\|B_t v\|\) bound. Strong-resolvent convergence to \(C\), its gap on the chosen support and the high-tail bound prove convergence of the phases. Compact core approximation and contraction bounds extend it to compact \(L^2\) vectors (3 points). Integration by parts where \(df\ne0\), followed by \(L^2\) density, gives weak convergence of the oscillating packets to zero. The compact creation errors of DJ.29 vanish on them, proving DJ.31a on every actual column (3 points). Choose \(x_n=\lambda_n^{-1/3}\), disjoint bumps of width comparable to \(x_n\), and coefficients \(x_n^2\). On copy 1 the auxiliary and normal contributions have sizes comparable to \(x_n\) and \(x_n^3\), hence are square summable. On copy 2 the auxiliary contribution is comparable to one. Its normal term lies in the orthogonal negative auxiliary eigenspace and cannot cancel it. The exact maximal domains therefore contain \(u_1\) only on the first copy (3 points). Averaging maps \((u_1,0)\) to \((u_1/2,u_1/2)\), outside the joint domain. The bounded compression in DJ.28 needs only norm convergence of module-frame columns and compact phase commutators, already proved in DJ.29. It uses no preservation of an unbounded graph domain. The actual Čech-return repair instead glues one speed and the domain-corrected operators before taking their unbounded sum (3 points).


### Exercises 300–302. Neutral words and curved graph geometry (36 points)

**Exercise 300 (12 points).** Prove that the polynomial compact-word count gives equality of the entire neutral full and reduced norms. Explain the uniform extension and central-plaque inputs, and why amenable isotropy alone is insufficient.

**Solution.** A compact symmetric arrow test has finitely many larger actual representatives with inverse buffers. One normal radius works at all tested endpoints. Each isometry sends that source ball onto the equal-radius target ball; all intermediate endpoints remain in the compact test set. Induction gives the same radius for every word (3 points). A uniform near-identity neutral germ differentiates to a Killing field centralizing the **full** Killing algebra, so it belongs to its center. Buffered frame boxes therefore have one visited central plaque each. Representative edges and chart changes add finitely many commuting translation increments, giving NK.6 uniformly in the initial frame (3 points). For self-adjoint \(f\), its nth power has that support bound and the squared regular coefficients sum to at most \(\|f\|_r^{2n}\). Cauchy–Schwarz controls both I-sums by \(\sqrt{B_n}\|f\|_r^n\). Every full representation is I-contractive, so nth roots give \(\|f\|_{\max}\le\|f\|_r\) (3 points). The reverse bound always holds; apply the equality to \(a^*a\) to treat every compact kernel and then complete. A free nonamenable action can have trivial isotropy yet unequal norms. The proof uses actual word counts, not isotropy alone (3 points).

**Exercise 301 (12 points).** Starting with locally constant full Killing rank, construct the discrete cocycle and identify the whole physical graph class. Preserve the original q-degree and disk.

**Solution.** The possible ranks form a finite set because a Killing germ is determined by its value and skew derivative. Their level sets are clopen invariant q-manifolds (3 points). On each connected basis chart the first jets of a germ basis stay independent; constant rank makes it full everywhere. An actual isometry pushes it by one constant matrix on a connected bisection, by Killing unique continuation. The matrices multiply in source-to-range order, and adding \(i-l\) gives NK.10. The generated countable group embeds by the matrix and integer blocks (3 points). NK.9 and the faithful neutral inclusion give the direct transfer, with actual Čech return. Theorem 11BO supplies its closed joint operator, completed-domain glue and bounded full physical graph phase. The finite orthogonal decomposition NK.11–NK.12 preserves every compact defect and the ordered right \(Cl_q\) factor; j does not replace q (3 points). A smaller positive chart disk V lies inside the original U. Its actual source-unit identity and the full inverse normal calibration give +1. The exact equality \(b_U=b_V[j_{VU}]\) then proves the same pairing for U. No unrelated disk or whole-family numerical inference is used (3 points).

**Exercise 302 (12 points).** Derive the curvature and full Killing rank in NK.14. Prove its incomplete geometry and actual norm equality, then state its graph degree and original Bott sign.

**Solution.** The displayed Christoffel symbols give \(K=-a''/a\). For \(a=e^{t^2}\), \(a''=(2+4t^2)a\), so \(K=-2-4t^2\) and \(K'=-8t\ne0\) (3 points). A Killing field preserves K, hence has no t-component. The mixed equation gives \(\partial_t b=0\), and the angular equation gives \(\partial_\theta b=0\), so precisely the constant multiples of \(\partial_\theta\) remain (3 points). The omitted endpoints have finite radial length because the radial metric is \(dt^2\). Irrational rotation has distinct genuine integer returns. A compact arrow support contains finitely many integers, and the nth convolution support has at most \(2mn+1\) of them. NK.16 and nth roots give the actual full/reduced norm equality (3 points). The invariant ordered normal frame and its specified full lift supply the coherent normal data. Theorem 11BP gives a class of the original degree \(q=2\), with final right \(Cl_2\), full inverse line and original positive Bott pairing +1. The auxiliary Killing rank one changes none of these normal data (3 points).


### Exercises 303–305. Actual word shadowing and rank quotients (36 points)

**Exercise 303 (12 points).** Prove the uniform ball and injective all-length word evaluation used in IX.3–IX.5. Explain why this preserves the original isotropy on an incomplete transverse metric.

**Solution.** A compact symmetric actual arrow test has a finite cover by larger bisection representatives with source and inverse-range buffers. Choose one normal radius below all these buffers and the compact neighborhood's normal radii (3 points). An isometry carries its entire source normal ball onto its equal-radius target normal ball, by the exponential formula and inverse buffer. Every compact-test word prefix has its endpoint in the same compact set. Induction therefore gives that one radius for every word and inverse word, independently of length (3 points). Representatives agreeing as germs at either of two points in the common connected ball have equal value and derivative there. The exponential formula propagates this equality locally; the equality set of value and derivative is open and closed, so they agree throughout the ball. Thus word evaluation is well-defined and injective in both directions (3 points). Distinct actual word and isotropy germs remain distinct at the shadow point. The construction evaluates only these buffered words; it asserts no extension radius for all arrows with compact endpoints and adds no complete metric or absent holonomy arrow (3 points).

**Exercise 304 (12 points).** Prove the full reduced quotient estimate, keeping an arbitrary regular-fibre vertex \(h\) and allowing the unit cutoff to have a shrinking transition region.

**Solution.** Split the compact kernel into \(m\) compact bisection coefficients and choose their common uniform modulus \(\omega\). One generator contributes at most one matrix entry in any row or column. Nonzero original edges match their shadows. An extra shadow edge either matches by injectivity or has its original coefficient zero and magnitude at most \(\omega(\delta)\). Hence the difference has row and column sums at most \(2m\omega(\delta)\); the weighted Cauchy–Schwarz argument of IX.7 controls its whole countable completion (3 points). Decompose every regular fibre into components of the nonzero symmetric supported-word graph. Choose any vertex \(h\), and right translate its component by \(h^{-1}\). Its base is \(r(h)\), and all subsequent paths are compact-test words; no continuation domain for \(h\) is extended (3 points). Choose a nearby closed-set point and use the actual word-basis injection. Transport the original cutoff values as the same diagonal contraction on the image. Compression and both diagonal factors have norm at most one, giving IX.8 without estimating cutoff derivatives (3 points). Take a cutoff equal to one on the closed set and supported near it on the compact endpoint test. The core difference \(f-bfb\) has coefficients vanishing on the closed set, hence is an I-norm limit of actual open-reduction kernels. IX.9 gives the upper quotient bound. Contractive restriction gives the reverse bound; approximation and scalar Tietze extension complete the exact sequence (3 points).

**Exercise 305 (12 points).** Construct the actual Killing-rank reduced filtration. Identify the degree and geometric conclusions which its exactness proves, and the additional data needed to obtain a physical graph representative.

**Solution.** A Killing germ is determined by its value and skew derivative, so its dimension is at most \(q(q+1)/2\). Independent germs have representatives on a common chart with independent nearby first jets; therefore \(\{k\ge j\}\) is open. Local isometries transport the full algebra, making every such set invariant (3 points). Its difference with the next open level is a relatively closed invariant rank layer. Applying Theorem11BQ.1 inside the original open q-manifold gives IX.12, with the literal original reduced open ideal and closed quotient. A closed layer need not be a manifold (3 points). These are scalar groupoid norm statements; neither auxiliary rank \(j\) nor a derivative label changes the prescribed normal degree \(q\), full inverse coefficient or final right \(Cl_q\) (3 points). Exactness does not construct a semisplitting, compatible normal cycles or an unbounded scalar quotient representation. Those require their actual geometric proofs. The original graph carrier and original positive disk remain to be evaluated by such a representative; a local index or an independently selected transfer does not supply that comparison (3 points).

### Exercises 306–309. Scalar rank extension and original normal geometry (48 points)

**Exercise 306 (12 points).** Under the actual hypotheses of SE.2,
derive the necessary-and-sufficient scalar extension criterion and
describe all lifts. Explain the direction of the quotient map in
scalar KK, the full-homotopy proof inputs, and what reduced norm
exactness leaves to be proved.

**Solution.** The separable cpc extension has an actual odd class
\(\partial_r\). The ideal-to-cone KK equivalence and contravariant
Puppe proof in the cited exact-sequence lesson give SE.3. Exactness
at \(KK^q(I_r,\mathbb C)\) says precisely that \(x_I\) is an ideal
restriction iff \(\partial_rx_I=0\); its degree is \(q+1\), with the
boundary kept on the left (3 points). Exactness at the middle group
shows that two lifts differ by \(q_r^*z\); conversely adding such a
class changes no ideal restriction. Thus the lifts are SE.4, with no
claim that \(q_r^*\) is injective (3 points). Contravariance gives
\(q_r^*:KK^q(Q_r,\mathbb C)\to KK^q(A_r,\mathbb C)\), not a
restriction in the reverse direction. The actual simultaneous
suspension uses the Bott lesson's Proposition 8.2 with coefficient
descent 8.1. The scalar-index picture uses the full-homotopy bridge
Theorem 2.3/Corollary 2.4 of the homotopy lesson, in addition to the
operator-homotopy computations (3 points). Theorem 11BQ gives the
genuine reduced kernel but no cpc section or cone KK inverse. These
must be supplied at their actual hypotheses before SE.3 is used.
Scalar extension does not require a whole transfer, and an abstract
lift does not by itself specify the original physical normal phase
(3 points).

**Exercise 307 (12 points).** Construct the explicit degree-four
extension on the open radius-four \(O(-2)\) bundle. Prove the needed
odd scalar quotient group vanishes without UCT, specify the literal
projection/module and whole normal domain, and retain the one
original positive disk.

**Solution.** The actual exterior is \(\mathbb {RP}^3\times(1,2)\)
via \(v\mapsto v\otimes v\) and \(r=\sqrt{\|w\|}\). The cutoff
metric extends smoothly from its Euclidean quotient metric and is
incomplete at the absent \(r=2\) end. Its point groupoid has supremum
norm, and the collar formula gives a cpc section. The disk bundle F
retracts to \(S^2\); its split point-evaluation sequence, first-variable
Bott inverses and scalar odd zero give \(KK^5(Q,\mathbb C)=0\),
so every degree-four ideal scalar class extends (3 points). The
projection \(e=zz^*/(z^*z)\) pulls back to the same actual modules
\(e_AA^2,e_II^2,e_QQ^2\). Multiplication by each algebra element is
compact even when e is only a multiplier projection. The map
\([(v,c)]\mapsto cv\) identifies the exterior sign line with
\(p^*L|E\); the interpolated smooth Hermitian connection is flat
on E and needs no global flatness (3 points). For
\(R=(16-\|w\|^2)^{-1}\) and the speed in SE.16 the exact domain is
SE.17. Local ellipticity gives \(H^1_{\rm loc}\); the exhaustion
commutator is \(O(1/n)\), proving minimal=maximal and scalar Rellich
compacts. Bundle-section multiplication and SE.19 give both completed
creation connections, so the phase is \([m_M]d_T\) with final right
\(Cl_4\) (3 points). Positive ambient disk isotopy gives
\(b_U=i_*b_V\) for an ideal disk V. The line is trivial there and
the full inverse pairs positively, so every scalar extension has
the original \(b_U\)-pairing one. The trivial-line realization is
the original \(d_T\); the selected tautological-line class is not
silently identified with it (3 points).

**Exercise 308 (12 points).** In SE.23 prove the full rank three/six
statement including the boundary. Verify the complete six-field
transport matrix, the irrational basis separation and the one
label group. Construct the full-domain auxiliary and compute both
returned index bundles.

**Solution.** The curvature \(-4u''-6(u')^2\) is analytic and
nonconstant inside, so its derivative is nonzero densely. A Killing
field has zero radial component; the mixed equations make its angular
coefficients independent of t and the angular equations give exactly
two translations and one rotation. These fields extend across the
boundary, and connected first-jet uniqueness excludes any extra
boundary germ. Both exteriors are locally Euclidean three-space,
giving SE.24 (3 points). Expressing source rotations in the target
basis gives all three columns of N in SE.26 and additive matrix
multiplication. Scaling the left translations by \(\sqrt2\) yields
\(H(m/\sqrt2,n/\sqrt2)\). Irrationality identifies the generated
matrix group with \(\mathbb Z^4\); the cover loop labels have zero
total index and the duplicated chart supplies the pure integer
factor, so the full discrete group is SE.28. No nonunit original
loop is inserted (3 points). The auxiliary is one even character
plus the full periodic Dirac. Its whole domain is periodic \(H^1\),
its squared Fourier eigenvalues are \(k^2+\ell^2\), and Fourier
truncation gives compact resolvents. Every group multiplier shifts
Fourier indices and has commutator norm \(\sqrt{a^2+b^2}\),
preserving the whole domain; its unit index is one (3 points).
On the left the used directions act trivially, so the Dirac index
bundle is zero and the character gives one. On the right the
covariant modes are SE.30. The nonzero modes have a uniform gap on
a small disk, while the zero-mode clutch \(-x-iy\) has positive
winding one. It gives b, so the character gives \(1+b\). The actual
Čech column unitary and bounded common-domain family homotopy give
the multiplication correspondence of that line bundle, not merely
a scalar index (3 points).

**Exercise 309 (12 points).** Compute the scalar obstruction for
SE.32, with all endpoint and Clifford signs. Explain why the local
original disk pairing still gives one, and why this does not refute
the original untwisted class or complete the arbitrary nonunit problem.

**Solution.** A quotient-unit lift rises on the left end and falls
on the right. Its positive exponential is
\((\widehat s_-,-\widehat s_+)\), with
\(\widehat s_\pm=[1_X]\boxtimes s_\pm\) and the increasing interval
inverse products \(s_\pm d_\pm=1\). This is an equality in the
actual coefficient algebras, with no interval/torus type suppressed
(3 points). The torus inverse has equal even/odd constant kernels,
so \([1_X]d_X=0\). Its positive small-disk inverse has \(bd_X=1\),
hence \([B]d_X=1\). The torus block is even and the t Clifford
generator stays last, so associativity gives \(0-1=-1\) in
\(KK^4(\mathbb C,\mathbb C)\). Full-homotopy scalar index and
Bott periodicity make this integer nonzero, obstructing extension
by SE.3 (3 points). Move the original positive three-disk by ambient
positive isotopy into E+. Its ideal preimage pairs as one because
B is trivial on that disk, and the original untwisted normal class
pairs with the original ambient disk as one. These tests do not
compute the quotient-unit obstruction (3 points). With the trivial
auxiliary both exterior bundles are trivial and the selected class
is \(i^*d_3\), whose literal extension is \(d_3\). Thus the failure
concerns the selected normalized auxiliary. For arbitrary nonunit
rank extensions, genuine KK excision, vanishing of the scalar
obstruction for a coherent choice and an actual original normal
representative are still needed. No whole-transfer gate or unbounded
graph compression follows from this example (3 points).

### Exercises 310–313. Completely positive rank sections (48 points)

**Exercise 310 (12 points).** Prove that the actual source-fibre retraction in Theorem11BS.1 gives a completely positive contractive section. Include its support, every matrix size and the original reduced norm.

**Solution.** Taper both endpoint factors where \(\chi\) is small. For a compact quotient kernel, KE.5 makes each tapered kernel compactly supported in the actual open reduction. A discarded term has an endpoint factor at most \(\sqrt\delta\). The source-fibre bijection and its inverse-range counterpart identify counting sums, giving the I-norm error \(2\sqrt\delta\|f\|_I\). Thus the proposed kernel is a genuine completed element (3 points). On the whole source fibre use its actual basis bijection and the diagonal contraction \(D_x\). Functoriality identifies every entry as \(D_xV_x^*\lambda_{\rho(x)}(f)V_xD_x\), including original isotropy (3 points). At each matrix size this is a completely positive compression with norm bounded by the quotient reduced norm. The faithful family of original regular representations detects positivity. Right translation identifies the open-reduction matrix inside each remaining original source fibre, proving the reduced-isometric inclusion even if the neighborhood is not invariant (3 points). On the invariant closed set the functor is identity and both cutoff factors are one, so the quotient of the section is exactly the original kernel. Density gives the identity on the whole quotient (3 points).

**Exercise 311 (12 points).** Construct the section for a proper global Lie-isometry action, and prove its vanishing tails and reduced subgroup descent.

**Solution.** Properness gives compact stabilizers and closed embedded orbits. The original normal exponential slice is a local inverse along the orbit; a sequence of failed injectivity points after shrinking would have a compact transporter subsequence converging to the stabilizer, contradicting that inverse. This supplies the actual slice charts (3 points). On each slice, finite-dimensional positive function approximations and the full nonunital lifting theorem give a cpc section. Average it over the compact stabilizer. Multiply the induced slice input by the compact invariant cutoff \(\phi_j\), so it really belongs to \(C_0(F\cap S_j)\). Equivariance makes induction well-defined (3 points). The invariant quotient partition is locally finite and sums to one, so its positive weighted sum is cpc and fixes \(F\). A compactly supported input meets finitely many enlarged quotient charts. In each, properness bounds its relevant transporters compactly; the output's slice parameters also lie in a compact set. The sum therefore has compact support. Contractivity extends this to \(C_0\) inputs (3 points). Represent the unit-space section by the subprobabilities \(\mu_x\). Equivariance gives \(\mu_{hx}=h_*\mu_x\), and each full subgroup regular matrix is \(\int\lambda_z(q)\,d\mu_x(z)\). Every amplification is positive and bounded by the original reduced norm. Actual germ faithfulness identifies the original arrows with the subgroup labels; no isotropy is removed and no nuclearity of the subgroup algebra is assumed (3 points).

**Exercise 312 (12 points).** Verify the singular closed rank layer in KE.18–KE.21 and the explicit free dense rotation subgroup in KEF.1–KEF.4.

**Solution.** The disjoint bumps have every derivative bounded by a polynomial in \(w_n^{-1}\) times \(e^{-2/w_n^2}\). Their derivatives tend to zero faster than every power of \(a_n\), proving smoothness and flatness at the accumulating point and at each endpoint (3 points). At a bump midpoint, \(u'=0\) and \(u''=-32u/w_n^4\), hence \(R\ge2-4u+128u/w_n^4>2\), while the endpoint value tends to two. Analyticity makes \(R'\) nonzero densely. The Killing and mixed equations give precisely three angular fields; connected jet uniqueness gives them also at all closed-layer points. On the product complement the Ricci eigenspaces give three rotations and the radial translation, hence rank four. A connected neighborhood retraction at zero would have a connected parameter image containing two separated components of \(K\), which is impossible (3 points). The four integral quaternion letters have the rank-one factors in KEF.3. Every reduced adjacent product has a nonzero scalar factor, so the whole product is nonzero modulo five. Identity rotation would make its quaternion real of squared norm \(5^m\): odd \(m\) is nonintegral, while nonzero even \(m\) makes it divisible by five. Thus no nonempty reduced word is identity (3 points). The recurrence KEF.4 excludes any periodic generator angle. Arbitrarily small nonzero powers approximate both axis circles. Those circles move the first axis to any sphere vector and contain its whole stabilizer, hence generate \(SO(3)\). The group is therefore free and dense. Its four-valent tree has at least \(2|S|+2\) boundary edges for every finite nonempty vertex set, excluding a Følner sequence. These actual rotation arrows preserve the passive parameter and its incomplete original metric (3 points).

**Exercise 313 (12 points).** Explain why reduced rank exactness and separately positive scalar lifts do not themselves give KK excision or a scalar normal class.

**Solution.** The quotient matrix with every entry one is positive. Extending its diagonal coefficients by \(c\), and its off-diagonal coefficients by \(d\), with an exterior point where \(c=0,d=1\), gives KE.22 with eigenvalues \(1,-1\). Thus separate scalar cpc choices need not preserve matrix positivity (3 points). In KE.23 the flat translation domains shrink toward the original curved boundary. Their jets converge, but any actual isometry germ at the limiting source and target would identify their full Killing algebras of dimensions one and three. The limiting parameter is not an actual arrow, so a proper global averaging action cannot be inferred from that jet closure (3 points). At a genuinely semisplit separable extension, the proved ideal/cone inverse gives the scalar obstruction \(\partial\otimes_Ix_I\) in degree \(q+1\), with the odd boundary on the left and the original right Clifford block last. Without an actual section or another proved inverse the canonical mapping cone must be retained (3 points). Even semisplitting does not make that scalar obstruction vanish: the one-auxiliary example of Section11BR has value minus one. A normal class still requires a coherent actual choice and its geometric representative. Exact reduced rank layers need not be manifolds and do not change the original normal degree, disk or inverse coefficient (3 points).

### Exercises 314–316. Full germs and rank-boundary compactness (36 points)

**Exercise 314 (12 points).** Construct the Hermitian Killing-jet Hilbert module, prove its exact fibres and closed restriction, and explain its actual germ transport.

**Solution.** Use the real bundle \(J_{\mathbb R}=TT\oplus\mathfrak{so}(TT)\), then its Hermitian complexification. Compactly localized real jets \(f(X,\nabla X)\), with complex coefficients, span a submodule of \(\Gamma_0(J)\); their uniform closure is complete with the original jet inner product. A countable dense family in this separable section space, multiplied by a countable approximate identity, generates it. The scalar product is a localized jet section and need not be the jet of the scalar-multiplied field (3 points). Every original real germ is realized by a cutoff equal to one near its base; complex combinations give the entire fibre. A zero initial Killing jet is a zero radial Jacobi field and hence a zero germ. Cutoffs of sections vanishing at the base identify the fibre kernel, and a small cutoff around that base gives the exact quotient norm, proving KJ.6 (3 points). For closed \(F\), compact positive level sets of a section vanishing there lie in its open complement and can be cut off inside it. For \(M=\sup_F\|s\|\), cutting away the level set at \(M+\eta\) gives the quotient upper bound; restriction gives the lower bound. This proves KJ.8 and retains ambient germs even when \(F\) is singular (3 points). The original local isometry pushes the value and conjugates the skew derivative. Naturality of the Levi–Civita connection and the chain rule make this an isometric continuous action on the full completed field, using only original local domains. An adjointable ambient inclusion would have a continuous finite matrix projection whose trace is the full rank \(k(x)\); that excludes such an adjoint at a genuine rank jump (3 points).

**Exercise 315 (12 points).** Derive the exact warped-torus jet columns and module decomposition. Distinguish the two projection assertions at its rank boundary.

**Solution.** In the ordered original frame, \(\nabla_{E_0}E_i=0\), \(\nabla_{E_i}E_0=HE_i\), and \(\nabla_{E_i}E_j=-H\delta_{ij}E_0\). For \(X=(b_1-cy)\partial_x+(b_2+cx)\partial_y\), its values are \(v_i=aw_i\). Angular differentiation cancels the factor \(a\), while radial differentiation gives \(Hv_i\). These yield exactly the three skew columns of Section11BT.7 (3 points). Their Hermitian norm is \((1+2H^2)(|v_1|^2+|v_2|^2)+2|c|^2\); thus the complexified low bundle is a smooth rank-three bundle. The scalar-curvature and connected first-jet argument gives full rank three on the closed bump region and six on the two actual flat ends (3 points). Split the ambient jet bundle into \(L\oplus W\). A whole-germ section has its \(W\) component zero on the closed region, hence zero-extends from \(\Gamma_0(U,W)\). Conversely compactly supported sections of \(L\) and of \(W|_U\) are local continuous combinations of the three angular jets and the six flat jets, respectively. A finite partition and compact truncation prove the full orthogonal module equality (3 points). Projection onto \(W\) inside this decomposed module is adjointable. Projection from the fixed ambient jet bundle onto the entire germ module would have trace jumping from three to six, so does not exist continuously. The selected smooth \(L\) is not asserted invariant under arbitrary nonunit flat-end rotations; the example here is units and its normal degree stays three (3 points).

**Exercise 316 (12 points).** Prove the complete-domain Fock compactness countertest and the proper-weight repair on its disappearing summand.

**Solution.** The domain of \(N\) requires norm convergence of \(\sum r^2\langle\xi_r,\xi_r\rangle\). The diagonal entries \((r\pm i)^{-1}\) map the entire module into that domain: their weighted positive tail is bounded by the original tail. Both inverse identities hold and \((1+N^2)^{-1}\) maps onto the square domain, proving self-adjointness and regularity. Each complete fibre has finite multiplicity in every degree and inverse tail tending to zero (3 points). Every compact operator's \(W\) block tends to zero at the boundary, first for rank ones and then by norm approximation. The multiplier \(fP_W\) keeps norm one there, so its distance to module compacts is one (3 points). Degree-one compression of the localized Fock inverse is \((f/2)1_{E_K}\). Its \(W\) compression gives essential norm at least one half. Subtracting the compact vacuum block leaves all positive degrees bounded by one half, giving equality. Thus compact fibre inverses do not establish completed module compactness (3 points). The explicit \(\lambda\) in KJC.7 is proper on the two open end components and diverges at all four omitted endpoints. Its two multiplier resolvents are onto the exact domain \(\{s:\lambda s\in E_W\}\), and their vanishing coefficient fields are compact endomorphisms of the finite-rank \(W\) module. The same holds for \(f|_U/(1+\lambda^2)\). This repairs that summand, without asserting an index-one auxiliary, reduced holonomy factorization or a new normal degree (3 points).

### Exercises 317–324. Global curved classes and whole-module compact weighting

**Exercise 317 (12 points).** For an original incomplete Riemannian transversal, prove that the truncated radius of compactness is positive, 1-Lipschitz and invariant under global isometries. Construct a smooth invariant positive minorant and prove completeness of its speed metric. Explain which coefficient data are averaged.

**Solution.** A small relatively compact coordinate ball contains an original metric ball, so the radius is positive. If \(d_g(x,y)=d\) and \(r<R_g(x)\), the closure of \(B_g(y,r-d)\) is a closed subset of the compact closure of \(B_g(x,r)\). Taking suprema in both directions and truncating at one gives \(|\rho(x)-\rho(y)|\le d\). A global isometry takes compact ball closures to compact ball closures, hence preserves \(\rho\). The proper global closure action gives the normalized smooth scalar cutoff in GG.5. Averaging a positive smooth \(b\le\rho/20\) against that cutoff gives an invariant smooth \(0<\alpha\le\rho/20\). An escaping finite original-length path has remaining length \(L-t\) at least \(\rho\); its speed length is at least \(20\int dt/(L-t)=\infty\). Infinite original length also gives infinite speed length. Only this scalar speed is averaged. The prescribed coherent inverse Spin-c lift and determinant connection are retained. **Rubric:** radius estimate 3; invariant averaging 3; completeness 3; coefficient scope 3.

**Exercise 318 (12 points).** In GG.10–GG.16, explain why both translate and adjoint swallowing are needed. Prove boundedness of the number commutator and weighted compact tail, and identify the actual domain after removing the zero-source block.

**Solution.** For fixed \(j\), the inclusions \(T_jE_n\subset E_{n+1}\) and \(T_j^*E_n\subset E_{n+1}\) imply both directions of the tail bandwidth-one estimate. Each off-diagonal shift in \([L,T_j]\) has norm at most \(\|T_j\|\); its remaining head is finite-dimensional. Closedness extends the commutator identity to \(\operatorname{Dom}L\) and proves its preservation. The bound \(\|[F,T_j]Q_n\|\le2^{-(n-1)}/n^2\) makes \(\sum_{n>j}n[F,T_j]Q_n\) absolutely norm-convergent and compact. Thus \([FL,T_j]\) is bounded. Enumerating \(P=\phi(1)\) gives a bounded off-diagonal correction \(B_0=[FL,P](2P-1)\). The corrected self-adjoint operator has the same domain; its essential restriction has exactly \(P\operatorname{Dom}L\). The complementary zero-source block changes no class or unit pairing. **Rubric:** two-sided band 3; number domain 3; compact tail 3; corrected domain and class 3.

**Exercise 319 (12 points).** Prove the original-unit identity for the reduced diagonal transfer. Separately verify the regular embedding GG.21a for a proper discrete action; state why that proof needs no reduced trivial group character.

**Solution.** The diagonal sends an original unit coefficient to \(i_m(f)\otimes1\). Pulling back \(1_{A_m}\boxtimes\kappa\) and using \([1]\kappa=1\) gives \([i_r]\Theta=[i_m]\) as a whole \(KK(R,A_m)\) identity. For the separate proper case, covariance gives \(U_\gamma\pi(f)U_\gamma^{-1}=\pi(f\circ\gamma^{-1})\). The normalized cutoff implies \(\sum_\gamma U_\gamma\pi(c)U_\gamma^{-1}=1\) strongly, so \(J^*J=1\). Also \((J\pi(f)v)_\gamma=\pi(f\circ\gamma)(Jv)_\gamma\) and \((JU_\delta v)_\gamma=(Jv)_{\delta^{-1}\gamma}\). Hence every full covariant representation embeds isometrically into its regular induced representation. The full norm is at most the reduced norm, and the opposite inequality is automatic. The direct normal cycle therefore factors through the reduced action algebra without using a character of \(C_r^*\Gamma\). **Rubric:** unit identity 3; cutoff norm 3; intertwining 3; norm and representation scope 3.

**Exercise 320 (12 points).** For GG.2, prove the exact joint domain and explain how the infinite auxiliary is controlled in scalar local compactness. Derive the normalized original symbol and its passage to the whole physical carrier.

**Solution.** On finite squared-auxiliary spectral windows and compact smooth normal vectors, the graded cross terms cancel, giving \(\|\mathcal Du\|^2=\|(D_\kappa\otimes1)u\|^2+\|(1\otimes N_\alpha)u\|^2\). Each window is a finite-matrix bounded perturbation of the self-adjoint complete-speed normal operator. Their direct sum has exactly the intersection of the two global graph domains in GG.25. For a compact scalar cutoff, GG.27 bounds the omitted auxiliary window by \(C_z\|\chi\|/M\); the retained finite window is compact by local elliptic estimates and Rellich. Therefore the localized resolvent is a norm limit of scalar compacts. Conjugation by \(e^{itf}\), divided by \(t\), leaves the limiting multiplier \(\alpha\Gamma_\kappa iC(df)\). On a compact buffer with \(|df|\ge\delta\), the spectral gap gives normalized phase limit \(\Gamma_\kappa iC(df)/|df|\). The oscillatory vectors are weakly zero; the compact completed-column connection error in GG.35 vanishes on them. GG.39 gives that same original-q symbol on the entire physical correspondence. **Rubric:** joint domain 3; auxiliary tail compactness 3; symbol limit 3; physical-column transfer 3.

**Exercise 321 (12 points).** Compute the full Killing-germ ranks of the curved cylinder GG.40. Verify its incomplete original metric, complete-speed realization, actual holonomy and original normal degree.

**Solution.** On the flat band \(|r|<1/4\), the metric is locally Euclidean and the full Killing algebra has dimension three. On either outer interval, GG.41 gives negative real-analytic nonconstant curvature; its derivative is nonzero densely. A Killing field preserves curvature, so its radial component vanishes, and the mixed and angular equations make it a constant multiple of \(\partial_\theta\). First-jet continuation makes the rank one at both seams as well. The omitted endpoints have finite original radial distance. Nevertheless \(\alpha=(1-r^2)/100>0\), \(\alpha\le(1-|r|)/50\), and its radial speed distance \(100\operatorname{artanh}r\) diverges at both ends. The suspension by \((t,r,\theta)\mapsto(t+1,r,\theta+\omega)\) has the actual irrational-rotation holonomy \(\mathbb Z\ltimes T\). Its genuine reduced character auxiliary has index one; the full original inverse normal module keeps degree two on both rank layers and seams, and the original positive two-disk pairs as +1. **Rubric:** interior ranks 3; seam ranks 3; metric and speed 3; actual holonomy and normalization 3.

**Exercise 322 (12 points).** Establish the whole-module density of KWC.2 and the exact inverse and square domains. Why is fibrewise invertibility insufficient for these conclusions?

**Solution.** The positive order \(h\ge2^{-n}\theta_{x_n,x_n}\), sandwiched by \(f_\varepsilon=\varepsilon(h+\varepsilon)^{-1}\), yields \(\|f_\varepsilon x_n\|^2\le2^n\varepsilon/4\). Contractivity and the dense generated module extend convergence to every section. Consequently \(h(h+\varepsilon)^{-1}\xi\to\xi\) in module norm, proving dense range. If \(h\xi=0\), positivity forces every \(\langle x_n,\xi\rangle=0\), hence \(\xi=0\). The inverse is closed on exactly \(\operatorname{Ran}h\); its adjoint test against \(h\eta\) gives the same domain and value. Direct multiplication gives the onto resolvents \(h(1\pm ih)^{-1}\). The square domain is exactly \(\operatorname{Ran}h^2\), with inverse \(h^2(1+h^2)^{-1}\) for \(1+D^2\). All these inverses are module compact. Pointwise invertibility alone supplies neither module-norm density nor these completed graph and square domains. **Rubric:** density estimate 3; injectivity 3; inverse and adjoint 3; square domain and module distinction 3.

**Exercise 323 (12 points).** Prove compactness and dense range of the harmonic tensor inverse \(b_r\). Explain how to prove its coordinate-sum domain when \(b^{(r)}(a)/a_i\) lacks a continuous boundary extension.

**Solution.** The harmonic function is bounded by every coordinate and vanishes on all coordinate-zero faces, so its extension is continuous. Multiplying by \(q_\delta=\prod_i\chi_\delta(a_i)\) changes it by at most \(\delta\). Polynomial approximation of the cutoff function expresses its operator value as limits of products of factor compacts; KWC.19 makes those products tensor rank ones. Hence \(b_r\) is compact. The identity \(Q_\delta=b_rg_\delta(h_1,\ldots,h_r)\), with \(Q_\delta\to1\) on every elementary tensor and then every completed vector, proves dense range. The cut ratios \(q_\delta b^{(r)}/a_i\) have norm at most one and stabilize on each \(Q_\gamma\) range. Their strong limits and the same limits of their adjoints define adjointable positive \(T_i\), with \(h_iT_i=b_r\) and \(\sum_iT_i=1\). These identities prove in both directions that \(\operatorname{Ran}b_r=\bigcap_i\operatorname{Ran}h_i\) and \(b_r^{-1}=\sum_i h_i^{-1}\) on that exact domain. **Rubric:** zero-face compactness 3; dense range 3; adjointable cut ratios 3; two domain inclusions 3.

**Exercise 324 (12 points).** Prove localized Fock compactness on the domain KWC.6. Then use the actual rational flat-end translations to disprove the existence of an exactly invariant positive compact dense-range weight, and delimit that obstruction.

**Solution.** The positive-degree resolvents have norms at most \(1/r\), and their square inverses have norms at most \(1/(1+r^2)\). The diagonal identities map all inputs into the norm-convergent weighted-series domain; coordinate adjoint tests prove its full self-adjoint domain. The localized vacuum multiplication by \(f\in C_0(T)\) is the rank one \(\theta_{f/\sqrt{|f|},\sqrt{|f|}}\), with zero values at zeros. Finite block truncations are compact, and the square-inverse tail is at most \(\|f\|/(1+(R+1)^2)\), proving full-module localized compactness. For an exactly invariant compact positive weight on the specified upper flat end, rational translations force the continuous \(W\)-block to be constant in \(t\). Compact rank-boundary vanishing makes that constant zero. Positivity annihilates all \(W\) vectors; its fibre range cannot be dense, contradicting module dense range. This excludes exact invariant compact weighting for that actual pseudogroup. It does not exclude compact phase covariance errors, another scalar or transfer construction, or the original physical graph class. **Rubric:** whole Fock domain 3; finite-block and tail compactness 3; invariant-block contradiction 3; exact scope 3.

### Exercises 325–340. Actual domains, general orientation and joint normal sums

**Exercise 325 (12 points).** Prove the inner zero Killing stalk and outer full rank in LG.3–LG.4. Show that the punctured two-sheet foliation has an absent cross arrow whose endpoints remain compact. Delimit the failure of module-only features.

**Solution.** On the inner disk, \(K=-6e^{-2u}\), \(S=576e^{-6u}(x^2+4y^2)\), and \(dK\wedge dS=55296xy e^{-8u}dx\wedge dy\). A Killing field annihilates both invariants; their independence on the dense set \(xy\ne0\) makes the field zero everywhere. On \(2<r<3\) the metric is Euclidean, so its full Killing fields are two translations and one rotation. The original cross bisection is \(B\setminus\{0\}\). Arrows at \((1/n,0)\) escape its topology although both unit endpoints lie in compact sets. Every positive-degree evaluated Killing-module feature is zero on the inner disk; an identical transported vacuum adds zero distance. Consequently its increment is zero and violates LG.2. A separately added actual-domain feature is not excluded. **Rubric:** curvature invariants 3; full ranks 3; actual arrow escape 3; precise candidate scope 3.

**Exercise 326 (12 points).** Prove the variational resistance formula and CND property for LG.10–LG.13. Why need only admitted-pair resistance be continuous?

**Solution.** On the sum-zero subspace of a connected class, \(L_z\) is positive definite. Completing the square gives \(\sup_v(2d^tv-v^tL_zv)=d^tL_z^+d\), while class constants change neither term. Put \(f_i=(L_z^+)^{1/2}e_i\). Then \(R_{ij}=\|f_i-f_j\|^2\), and for zero-sum real coefficients the quadratic kernel sum is \(-2\|\sum_i a_if_i\|^2\le0\). The pseudoinverse on the entire ambient space can jump when classes change. The actual groupoid includes only \(z\in U_{ij}\); the variational and trial-flow arguments prove precisely the needed continuity there. **Rubric:** correct subspace 3; square completion 3; CND identity 3; actual continuity domain 3.

**Exercise 327 (12 points).** Prove LG.14–LG.19, including properness for every unrestricted multiplier.

**Solution.** The trial current \(J_{ab}=c_{ab}(z_0)(w_a-w_b)\) for \(w=L_{z_0}^+d\) has divergence \(d\) and energy \(R_{ij}(z_0)\). Its nonzero edges remain positive nearby, so it supplies the continuous upper bound; the variational supremum supplies the lower bound. If an admitted pair disappears, the class cut at its limit has conductance \(C_A\to0\). Testing the potential \(t1_A\) gives \(R_{ij}\ge1/C_A\to\infty\). Hence each compact-endpoint sublevel is closed in its compact coordinate set and compact inside its actual admitted bisection. Finally \(k(gh,h)=R_{ij}(z)\) independently of every admitted \(h\); compact upper control and the finite sublevel union therefore prove both LG.1 and LG.2. **Rubric:** trial flow 3; cut bound 3; actual compact sublevels 3; unrestricted cancellation 3.

**Exercise 328 (12 points).** Construct the actual completed Gram field in LG.20–LG.21 and verify its coherent affine action. Explain why it is not independent rank-stratum data.

**Solution.** Use zero-sum formal combinations of actual range arrows, quotient the null Gram form \((\psi(a)+\psi(b)-\psi(a^{-1}b))/2\), and complete. Actual bisection sections with compact cutoffs have continuous Gram norms; their local uniform limits define the continuous field and its completed \(C_0(T)\) module. Cutoffs equal to one at a unit give all evaluated generators and the quotient norm. Multiplication sends \(v_b\) to \(v_{ab}-v_a\); polarization and \((ab)^{-1}(ac)=b^{-1}c\) prove a coherent unitary \(U_a\), and \(v_{ab}=v_a+U_av_b\) proves the affine cocycle. The construction uses admitted arrows and their common continuous Gram entries across every class change; it never chooses unrelated inverses or KK indices by strata. **Rubric:** actual Gram quotient 3; completed field and norm 3; action identities 3; coherent scope 3.

**Exercise 329 (12 points).** Derive the three-sheet formulas LG.23. Prove the bounded recurrent resistance estimate LG.24 and explain which additional information is needed for infinite fibres.

**Solution.** The direct \(12\) edge has conductance one and the series route through vertex three conductance \(a/2\), so \(R_{12}=2/(2+a)\). The direct \(13\) conductance is \(a\), and the series route through two has conductance \(a/(1+a)\), giving \(R_{13}=(1+a)/(a(2+a))\). On the actual \(\mathbb Z^3\) label lattice, \(\lambda(\theta)\ge4|\theta|^2/\pi^2\). Enclosing the cube in the radius \(\pi\sqrt3\) ball gives \(\int1/\lambda\le\pi^4\sqrt3\); the numerator at most four yields \(k_{\rm res}\le\pi\sqrt3/2\). Its CND squared-distance feature is therefore bounded on escaping labels with compact endpoints. The finite cut mechanism detects absent domains; infinite recurrent fibres need an additional proper label feature. This does not prove nonexistence of such a feature. **Rubric:** persistent formula 3; disappearing formula 3; exact integral bound 3; limited recurrent conclusion 3.

**Exercise 330 (12 points).** Distinguish strict invariant compact Spin-c reduction from coherent equivariant K-orientation. Prove the distinction on NRA.1 without weakening coherence to an invariant isomorphism class.

**Solution.** Compact Spin-c reduction gives an equivariantly invariant orthonormal frame bundle; squaring its unitary Clifford transport proves metric invariance. General K-orientation lifts the actual \(GL_q^+\) cocycle through \(ML_q^c\), retaining its central line. For \(h'=1+\cos\theta/2>0\), the line lift \(((h^n)',1)\) satisfies the exact chain-rule cocycle. At the fixed point zero, an invariant metric would require \(m(0)=9m(0)/4\), impossible for \(m(0)>0\). The mapping torus is connected and compact and the original transverse degree is one. **Rubric:** strict implication 3; coherent general lift 3; fixed-point contradiction 3; original geometry 3.

**Exercise 331 (12 points).** Prove compactness of the actual normal phase commutators NRA.4–NRA.6 directly from the circle kernel, including holonomy.

**Solution.** The inverse operator \(i\partial_\theta\) has its exact Fourier \(H^1\) domain, and its sign with zero assigned +1 differs compactly from its bounded transform. For smooth multiplication the factor \(f(y)-f(x)\) cancels the cotangent diagonal pole, giving a bounded Hilbert–Schmidt kernel. In the holonomy kernel let \(y=x+t\), \(a=\phi'(x)>0\), \(b=\phi''(x)\). The numerator is \(a+bt/2+O(t^2)\), the argument difference is \(at+bt^2/2+O(t^3)\), so their quotient is \(1/t+O(t)\). Both cotangent terms have the same \(-2/t\) pole; their difference is bounded. The zero-mode terms are finite rank. Products and adjoints give compact covariance for every iterate. **Rubric:** exact inverse phase 3; multiplication cancellation 3; holonomy expansion 3; iterates and zero mode 3.

**Exercise 332 (12 points).** Prove reduced continuity using NRA.7–NRA.9. Include fixed-point isotropy in the norm comparison.

**Solution.** \(W_L\) is isometric because it has L coordinates of norm \(L^{-1/2}\). Covariance gives exact multiplication intertwining. Shifting by n changes only the symmetric difference of two length-L intervals, at most \(2|n|\) coordinates, and the intertwining error is at most \(\sqrt{2|n|/L}\). For finite sums the error tends to zero, so the actual representation is bounded by the regular norm. The regular representation decomposes into source representations. Finitely supported tests give continuous source norm functions; full-support circle measure makes the essential supremum equal the full supremum. This includes the fixed points and their genuine \(\mathbb Z\) isotropy. **Rubric:** isometry 3; exact coefficients 3; shift and finite-sum estimate 3; full source norm 3.

**Exercise 333 (12 points).** Compute the original positive disk pairing in NRA.4 and explain its passage to the entire physical graph in NRA.10.

**Solution.** For \(P\) onto Fourier modes \(n\le0\), \(PM_{e^{i\theta}}P\) has kernel \(\mathbb C e_0\), zero cokernel and index +1. A winding-one unitary supported in an original positive interval is homotopic to \(e^{i\theta}\) by interpolating the periodic difference of their degree-one phase lifts. Thus this is the original disk class. On the literal full \(E_Z\otimes_BH\), Parseval compression yields a bounded phase. Finite creation columns have compact errors; module-norm tails tend to zero, so all completed creation errors are compact. Expanding graph rank-one defects and using density proves scalar Kasparov defects. Oscillatory packets annihilate those compact errors and retain the inverse normal symbol. The same correspondence carries the original disk, preserving +1. **Rubric:** kernel and cokernel 3; original disk homotopy 3; completed graph compactness 3; geometry and pairing 3.

**Exercise 334 (12 points).** Prove AP.5–AP.7 without assuming a symbol theorem. Identify the exact cutoff argument at zero and infinity.

**Solution.** Positive integer N gives a globally defined circle-valued gauge. The exact conjugate is \(A_N=D/N+c(d\phi)\). Testing its resolvent against \((c(d\phi)-i)^{-1}w\) gives AP.6; density and both resolvents imply strong compactly supported functional calculus. A cutoff equal to one near zero and supported in the gap of \(c(d\phi)\) has strong limit zero. Away from that cutoff, \(t/\sqrt{t^2+N^{-2}}\) tends uniformly to sign. For smooth w, \(\|A_Nw\|\) is uniformly bounded and its spectral tail above R is at most that bound divided by R. Let N then R tend to infinity and use contraction/density. The limit is \(c(d\phi)/|d\phi|\). **Rubric:** exact conjugation 3; resolvent approximation 3; gap cutoff 3; tail and norm limit 3.

**Exercise 335 (12 points).** Derive AP.9 and the localized noncompactness conclusion. Why is the coefficient evaluated at the inverse image?

**Solution.** \(U_1^*E_N=e^{iN(h(\theta)+y)}U_1^*\). AP.7 gives normalized coefficient \((h'\sigma_1+\sigma_2)/\sqrt{h'^2+1}\); conjugating multiplication by \(U_1\) composes it with \(h^{-1}\), so \(a=h'\circ h^{-1}\). At zero the direction is \((3,2)/\sqrt{13}\). Its squared gap from \((1,1)/\sqrt2\) is \(2-10/\sqrt{26}>0\). Pauli squaring gives this pointwise norm on every spinor. The packets are weakly null by integration in y, but the defect has norm bounded below on a small support. A compact cutoff equal to one there preserves the contradiction. This excludes the tested phase, not another graph class. **Rubric:** gauge and inverse image 3; exact gap 3; weak-null compactness test 3; local and limited scope 3.

**Exercise 336 (12 points).** Prove the normalized Clifford conformality criterion AP.11–AP.12. Explain the difference between degrees one and two.

**Solution.** Anticommutators preserve angles, so the images of an orthonormal basis are \(a_iv_i\) with orthonormal \(v_i\) and \(a_i>0\). Comparing the sum direction with the two individual directions forces \(a_i=a_j\), since the two Clifford matrices are real-linearly independent by their squared norm. Hence \(A=\lambda O\). Conversely an actual unitary implementer of O gives normalized covariance because the positive scalar cancels. In degree one every positive derivative is scalar conformal; the circle phase may therefore be compactly covariant without isometry. The anisotropic degree-two derivative fails this condition. No graph-class nonexistence follows. **Rubric:** angle argument 3; sum comparison 3; converse with implementer 3; exact degree and scope distinction 3.

**Exercise 337 (12 points).** Prove the smooth compact weight NG.3–NG.5 and minimal=maximal normal closure NG.8–NG.10 across zero fibres.

**Solution.** The derivative-controlled coefficients make each differentiated tail bounded by a geometric series. Thus h is smooth with bounded first derivative and is the norm limit of module rank ones. The order inequality gives \(a_n\|r_\varepsilon\psi_n\|^2\le\varepsilon/4\), which extends by dense generators to the completed module; it proves dense range. Positivity proves injectivity. For \(C=hNh\), compact-buffer Friedrichs commutators have a uniform Schur bound and converge strongly to zero by smooth density. Complete-speed exhaustion cutoffs have commutator \(h^2c(d\eta_m)\to0\). Combining smoothing and exhaustion proves the exact maximal distributional domain and self-adjointness. The orthogonal complement of the actual scalar module range has hu=0 and is killed, so the actual changing-rank scalar carrier reduces C. **Rubric:** smooth rank-one weight 3; completed dense range 3; graph approximation 3; exact reducing carrier 3.

**Exercise 338 (12 points).** Prove the exact joint self-adjoint domain NG.12 and scalar local compactness NG.17–NG.18.

**Solution.** The bounded regularizations \(L_\varepsilon=(h+\varepsilon)^{-1}\) preserve \(\operatorname{Dom}C\), and their cross commutator is bounded by \(K=\|[N,h]\|\). The anticommuting auxiliary matrices give the joint graph estimate NG.14. Weak limits of onto resolvent solutions lie in both closed domains; testing against \(\operatorname{Dom}L\) identifies Lu, so both resolvents of the intersection operator are onto. The limiting graph estimate proves closedness and the exact domain. A spectral cutoff away from h=0 gives NG.17 in the fixed ambient finite bundle, hence an H1-bounded localized family and scalar Rellich compactness. The missing part has norm at most \(2\delta\|\chi\|C_z\). Letting delta vanish proves compactness across actual zero-fibre seams. **Rubric:** bounded cross term 3; onto resolvent and exact domain 3; finite-bundle cutoff compactness 3; uniform seam tail 3.

**Exercise 339 (12 points).** Prove full tensor/Fock scalar compactness and the whole source-class normalization NG.20–NG.28. Why does this not identify a reduced transfer?

**Solution.** Products of factor rank ones are tensor rank ones; the commuting product weight is compact, injective and dense-range. Its inverse has the exact completed range domain. The factors \(\varepsilon_r=(r+1)^{-2}\) give uniformly bounded derivative/cross estimates and resolvent tails tending to zero. Finite blocks are scalar locally compact and the uniform tail proves the entire direct-sum claim on NG.24. Adding common positive mass s leaves domains unchanged, makes square defects and source phase commutators tend uniformly to zero, and makes the phases converge strongly with adjoints to the degenerate auxiliary involution. This defines one whole source-algebra homotopy; all positive paired degrees contribute zero, leaving the original inverse normal class and its disk +1. No equality with the anchored transfer or holonomy covariance has been proved by this ordinary source-class argument. **Rubric:** actual product inverse 3; whole domain and tail 3; one class homotopy 3; original normalization and scope 3.

**Exercise 340 (12 points).** Derive the joint-direction countertest NG.32–NG.37. Distinguish it from a failure of exact invariant weighting.

**Solution.** For \(s_n=2^{-n}\), \(b_n=\varepsilon_1 e^{-1/s_n^2}\), normal frequency \(k_n=b_n^{-3}\) and width \(s_n^4\), the relative weight variation is \(O(s_n)\), and derivative errors after scaling vanish faster than any reciprocal power. At source and translated range, the scaled operators approach \(\sigma_1+\sigma_3\) and \(3\sigma_1+\sigma_3/9\). Their phase directions are \((1,1)/\sqrt2\) and \((27,1)/\sqrt{730}\), with gap \(\sqrt{2-56/\sqrt{1460}}>0\). The actual seam-crossing bisection kernel has compact support; disjoint shrinking packets are weakly null, yet its phase commutator has the stated lower bound. The weight already satisfies smoothness, exact domains and source-local compactness. This disproves automatic compact-error covariance from those facts, while allowing another weight or representative. **Rubric:** scales and errors 3; exact two phases 3; actual compact-arrow contradiction 3; limited inference 3.

### Exercises 341–352. Metric transfer, class descent and joint covariance

**Exercise 341 (12 points).** Prove metric invariance in MB.2 and distinguish it from an invariant total splitting. Derive the vertical Spin-c central exponent and the surviving inverse coefficient.

**Solution.** Under transport, \(g^{-1}B\) is conjugated by the actual derivative, so trace of the product is invariant. The quotient metric is invariant by \((hg)(Av,Av)=g(v,v)\); a chosen total splitting can still have the genuine second-jet shear MB.12. The symmetric-square plane loop has one weight-two block and \(q-2\) weight-one blocks, with spin sign \((-1)^q\). Thus the map in MB.4 descends through \((s,z)\sim(-s,-z)\). The positive vertical central factor \(z^q\) cancels the inverse total \(z^{-(q+1)}\), leaving \(z^{-1}\). The full graded finite evaluation retains the mixed \(dq\) parity and the original last right \(\mathrm{Cl}_q\). **Rubric:** two metrics 3; genuine shear 3; coherent lift 3; entire ordered inverse 3.

**Exercise 342 (12 points).** Prove MB.7–MB.11 on actual compact arrow tests. Explain what algebra is the transfer's target.

**Solution.** The curvature numerator is \(-\|[U,V]\|_F^2/4\). The Jacobi equation and Gauss orthogonality make the radial exponential expanding, so its inverse logarithm is 1-Lipschitz. Transported logarithmic vectors differ by at most \(C_K\). The normalization derivative along their segment is at most \((1+(r-C_K)_+^2)^{-1/2}\), proving MB.10. Its arrow-local tail and compact base cutoffs give actual compact module errors. Regular localization amplifies the original source regular representation, proving the reduced descent norm. The target is the reduced metric-bundle algebra, and still needs a scalar phase for a scalar product. **Rubric:** radial geometry 3; exact error 3; actual reduced descent 3; target variance 3.

**Exercise 343 (12 points).** Derive the shear estimate MB.16 and the symbol exponents MB.17. Prove scalar compactness of the local Fourier phase.

**Solution.** MB.15 bounds the derivative by \((5/4)\|L\|R_t^{-1/4}|\eta|\). Normalization has denominator at least \(\sqrt{1+|\eta|^2}\), and \(R_t\ge R/(1+\|L\|)\), giving the stated explicit constant. A chain-rule term with k factors is bounded by \(R^{- |\alpha|+k/4}\), and \(k\le|\alpha|+|\beta|\), hence exponents \(-3|\alpha|/4+|\beta|/4\). Frequency truncation makes the localized square defect Hilbert–Schmidt with norm-small tail. The commutator's bounded-difference frequency part tends uniformly to zero at infinity; the remaining part has the Schwartz \(L^1\) tail. Its finite output ball is Hilbert–Schmidt. These prove scalar compacts and the whole local source homotopy. **Rubric:** shear constant 3; symbol exponents 3; defect tail 3; two Schur bounds 3.

**Exercise 344 (12 points).** Prove the balanced reduced matrix criterion MB.23. Why does regular norm continuity alone not ensure scalar compactness?

**Solution.** For \(z=\sum_i\xi_i\otimes v_i\), the two squared norms are exactly \(\langle v,G_\xi v\rangle\) and \(\langle v,Q_{a,\xi}v\rangle\). Their inequality is contractivity on the dense elementary balance, including its null vectors, and extends to the original reduced completion. Every reduced representation conversely gives that inequality. On an infinite family of distinct actual arrow sections, the same nonzero localized normal defect applied to each normalized column yields orthonormal inputs and uniformly nonvanishing images; it is not scalar compact. This is a countertest for the amplified phase, rather than nonexistence of another joint phase. **Rubric:** two Gram forms 3; both norm implications 3; scalar sequence 3; limited scope 3.

**Exercise 345 (12 points).** Establish FCD.8 without assuming a section of the full-to-reduced quotient. Give the actual role of a cone nullhomotopy.

**Solution.** The cone projection has the explicit cpc section \(a\mapsto(a,(1-t)\lambda(a))\) at every matrix level. It gives no section of \(\lambda\). The actual contravariant Puppe segment is exact, and \(\lambda p\) contracts by evaluation of its vanishing path; hence \(\ker p^*=\operatorname{im}\lambda^*\). A supplied nullhomotopy restricts along the suspended target to its coefficient-ideal submodule; localized compact defects remain in that submodule's hereditary compact algebra. The glued suspension cycle and inverse simultaneous suspension produce the descent with the final right Clifford block still last. A numerical index is no substitute for this nullhomotopy. **Rubric:** section variance 3; exact segment 3; actual construction 3; coefficient and input scope 3.

**Exercise 346 (12 points).** Prove the entire normalized pairing image FCD.11 and its independence of the normalized full lift.

**Solution.** Write \(x=\beta_A(x)x_0+z\) with \(z\in\ker\beta_A\). For a descent its cone image is zero, so its index annihilates \(\epsilon_U\). Conversely \(n\epsilon_U=0\) gives \(\ell\in\ker\beta_A\) with \(n\Omega(x_0)=\Omega(\ell)\). The class \(nx_0-\ell\) has zero cone obstruction and descends with index n. Two normalized lifts differ by an element of the same kernel, so define the same quotient obstruction. Finite order d gives exactly \(d\mathbb Z\); infinite order gives zero. These are scalar class statements, requiring further geometric comparison. **Rubric:** decomposition 3; both inclusions 3; independence 3; order and scope 3.

**Exercise 347 (12 points).** Compute the mapping-cylinder examples OFD.1–OFD.4 directly, including their cone groups and positive odd calibration.

**Solution.** The homotopy \([z,t]\mapsto[z,t+s(1-t)]\) retracts \(X_d\) onto the top circle. Restriction of its pulled-back winding-one unitary is \(z^d\); therefore its odd K-theory map is multiplication by d. The even map is identity on the unit. The odd Puppe segment consequently gives \(KK^1(C_{\Phi_d},\mathbb C)=\mathbb Z/d\mathbb Z\), interpreted as \(\mathbb Z\) for d=0. Its quotient generator is the cone pullback of the positively calibrated source inverse. All attainable target pairings are \(d\mathbb Z\), and d=0 gives infinite obstruction order. The example is a genuine commutative quotient, with no foliation identification claimed. **Rubric:** retraction 3; two maps 3; cone and image 3; ordered calibration and scope 3.

**Exercise 348 (12 points).** Derive FCD.18 and explain why descent of one full class is stronger than existence of a normalized reduced class. Include the explicit two-corner example.

**Solution.** Associativity gives \(\lambda^*(\Theta x_0)=x_0-\Delta_\Theta x_0\). Applying the original index gives \(1-\beta_A(\Delta_\Theta x_0)\). The actual class defect can vanish, or only its relevant constraint can vanish; a whole gamma identity is unnecessary. For \(\lambda:\mathbb C^2\to\mathbb C\), \((a,b)\mapsto a\), the normalized class \((1,1)\) fails descent because of its second-corner index. The correction \((0,-1)\) has first-disk index zero and leaves the descended class \((1,0)\). A nonfactorizing representative can likewise have a degenerate second corner and the descended class. **Rubric:** typed products 3; defect constraint 3; class correction 3; representative distinction 3.

**Exercise 349 (12 points).** Derive both logarithmic transport bounds JCN.20. Explain why they address distinct joint limits.

**Solution.** Subtract the two square roots through their conjugate denominator. The numerator is at most \(m\log3\); their sum is at least \(2\sqrt m/s\) and at least \(2m/r\) for m≤r and 0<s<1. Division by \(r^2\) gives the seam estimate \(\sqrt m s\log3/(2r^2)\) and the degree estimate \(\log3/(2r)\). The ratios and derivatives extend boundedly to the completed retained sectors. The first bound makes them approach identity near the seam, and the second controls arbitrarily high tensor degree at a fixed original point. Fixed normal-frequency symbol agreement alone supplies neither completed joint limit. **Rubric:** exact subtraction 3; both bounds 3; completed sector scope 3; joint limits 3.

**Exercise 350 (12 points).** Prove the two-endpoint sign comparison JCR.1–JCR.7 and apply it to the cutoff error JCN.22.

**Solution.** Commutation of each positive \(d_i\) with L gives \(a^2L^2\le A_i^2+K\). Square-root order and the gap give \(L\le k|A_i|\), so the mixed relative form has norm at most \(\varepsilon k\). For the opposite spectral blocks the Sylvester kernel is \(\sqrt{xy}/(x+y)\); its two semigroup Gram maps each have norm \(1/\sqrt2\), so the cross-projection norm is at most \(\varepsilon k/2\). Both cross blocks and \(\operatorname{sgn}A=2P-1\) give the stated sign bound. The discarded mass multiplier is bounded by \(C(R^{-1}+\eta)\), uniformly in the whole sum. Fixed R,η leaves a bounded potential in finitely many complete degrees away from the seam. Its compact phase differences converge in norm, proving a whole scalar compact. **Rubric:** closed-form comparison 3; mixed Gram estimate 3; projection factors 3; actual joint cutoff 3.

**Exercise 351 (12 points).** Prove the exact averaged repair JCN.27a on the actual suspension and its full reduced class equality.

**Solution.** The identity \(a(x)+a(x+1/2)=4/3\) gives the coefficient \(2\rho/3\). It is smooth, compact and exactly tau invariant on the original completed module. Compact sector truncations prove dense range, and domination by h/2 proves injectivity. Average the compatible split connections over the actual finite action; all completed expressions, domains and resolvents are then exactly covariant. Whole tensor derivative and resolvent tails prove scalar local compacts. The finite regular isometry proves the original reduced norm. Finally \(\{D,\sigma_1\}=2L\ge0\) and the gapped spectral semigroup yield \(\{\operatorname{sgn}D,\sigma_1\}\ge0\). The normalized convex path has square at least 1/2 and ends at a commuting degenerate involution. Thus the entire positive sum is zero as one reduced crossed-product class. **Rubric:** actual averaged weight 3; completed connection and domains 3; norm and tails 3; whole class homotopy 3.

**Exercise 352 (12 points).** Carry the verified finite-suspension normal phase to the entire physical graph. Preserve the native inverse coefficient and original disk index.

**Solution.** Use the full \(E_Z\otimes_B\mathscr H\) and its module Parseval isometry W. The bounded compression by \(P=WW^*\) has compact creation errors: finite columns use represented compact commutators and their tails vanish in module norm. Expanding graph rank-one commutators and square defects, then taking rank-one density, proves scalar compacts for the entire graph algebra. Compact creation errors vanish on weakly null normal packets; the fixed-degree gapped resolvent limit retains the auxiliary grading and original inverse normal Clifford phase. The vacuum comparison retains \(q=3\), coherent \(z^{-1}\rho_t(s)\), physical \(+L_{t_j}\) and final right \(\mathrm{Cl}_3\). The original counting/plaque correspondence carries the same positively oriented disk, preserving +1. This finite construction leaves the arbitrary nonisometric case open. **Rubric:** whole carrier 3; compact creation and defects 3; exact coefficient 3; original disk and scope 3.

### Exercises 353–360: regular source and actual metric isotropy

**Exercise 353 (12 points).** Prove the reduced source norm in RTP.5 using finite rectangular coefficient matrices, without a direct-integral theorem. Explain why no scalar invariant normal representation is assumed reduced.

**Solution.** On a regular tree orbit, conjugate the diagonal action by the column unitary \(\eta_h\mapsto U_h^{-1}\eta_h\). Group elements become regular shifts and coefficient functions become \(\phi(\alpha_{h^{-1}}f)\). A polynomial supported on finite \(S\) sends a column supported on finite \(L\) into \(SL\). Its rectangular coefficient matrix at each point is a compression of the point regular representation, so its norm is bounded by the reduced source norm. Applying the amplified star representation of \(C_0(T)\) preserves that bound by the C*-norm and positive functional calculus. Finite-column density and then polynomial density prove contractivity. The \(m+1\) regular orbits retain the bound. The unamplified normal representation need never extend to the reduced algebra. **Rubric:** conjugation 3; finite matrix 3; norm and density 3; exact representation scope 3.

**Exercise 354 (12 points).** Establish the separator hypotheses for RTP.6 and prove both scalar compact defects RTP.10 on the entire reduced algebra. Explain why neither error algebra itself is claimed scalar compact.

**Solution.** A product from \(C_T\otimes K_N\) and \(K_T\otimes C_N\) has a compact in each direction and therefore lies in \(J=K_T\otimes K_N\); finite sums and norm closure give the product inclusion. Normal square and source commutator errors lie in \(A_1\), while the tree root defect and finite-rank return commutators lie in \(A_2\). The controls derive \(A_1\) since their tree factors belong to \(C_T\) and all bounded normal factors preserve \(K_N\). The complete graded separator gives \(MA_1\subset J\), \(NA_2\subset J\), and compact control commutators. Exact anticommutation of \(P,Q\) eliminates the cross term in the square. The two remaining weighted errors and the weighted commutators are compact. Polynomial density treats every reduced source. Each unweighted error algebra may still contain an identity in its other, infinite direction. **Rubric:** ideal product 3; errors and derivation 3; separator and square 3; full norm closure and warning 3.

**Exercise 355 (12 points).** Prove RTP.11–RTP.13, including the rectangular compact-map step and the exact ordinary restriction class.

**Solution.** For a finite homogeneous tree vector \(v\), its creation projection \(T_vT_v^*\) lies in \(A_2\). Square-root annihilation implies that \(N^{1/2}T_v\) is compact: its product with its adjoint is compact, and the linking-algebra cutoff \(h_\epsilon(t)=t/(t+\epsilon)\) gives the stated vanishing norm error. Also \((M^{1/2}-1)T_v\) is compact by \(M=1-N\). Apply \(PT_v=(-1)^{|v|}T_vF_N\) and \(QT_v=T_{F_Tv}\) to obtain the connection equation and its adjoint. Coefficients and density give the entire connection. The anticommutator with the first tree operator is \(2N^{1/2}Q^2\ge0\) in the quotient, so the product criterion gives the ordinary index-one tree class followed by \(d_N\), exactly \(d_N\). A compact map sends bounded weakly null packets to norm-zero sequences, proving the phase limit. **Rubric:** rectangular compactness 3; graded connection 3; positivity and exact class 3; phase limit 3.

**Exercise 356 (12 points).** Verify the rank-one reductions MI.2 and prove the rotations freely generate at every word length. A bounded word search is not a proof.

**Solution.** In \(\mathbb F_9\), \(s^2=-1\) and \(-2=1\). Multiplying the displayed column and row gives each numerator matrix, including both inverse signs. Consecutive equal-axis letters have scalar factor \(1+\epsilon\epsilon'\), nonzero unless they are inverse; consecutive different-axis letters have factor \(-\epsilon\epsilon'\), always nonzero. Every nonempty reduced word therefore has nonzero reduced numerator, a nonzero rank-one outer product times these nonzero factors. An identity word of length \(k\) would have numerator \(3^kI\), which reduces to zero. This contradiction handles all lengths. The adjoint matrices of the quaternion lifts are the same rotations, so their entire free-word lift is faithful as well. **Rubric:** four reductions 3; adjacent factors 3; all-length contradiction 3; coherent lift 3.

**Exercise 357 (12 points).** Prove that MI.6 presents actual holonomy with full fixed-point isotropy, and compute the isotropy of its metric lift at \((1,I)\).

**Solution.** The regular surface-cover action is free and proper, so the diagonal quotient is a compact smooth connected fibre bundle with horizontal foliation and complete fibre transversal \(N\). Conjugation by distinct words gives distinct germs: identity on an open set forces a connected isometry to be identity everywhere by overlapping geodesic balls, and then its derivative rotation is identity, contradicting the all-length free-pair proof. Thus no effective labels are lost in the holonomy germ groupoid. At \(1\in SU(2)\), all conjugations fix the point, and their orthogonal derivative rotations fix the bi-invariant metric \(I\). Every group word therefore lies in the metric isotropy; its entire range fibre is that same \(F_2\). The full spin and central lifts remain the supplied ones. **Rubric:** suspension 3; germ effectiveness 3; metric isotropy 3; full lift 3.

**Exercise 358 (12 points).** Prove MI.8 directly and use the actual range fibre to rule out metric-lift amenability.

**Solution.** Orient the tree away from its root and use \(2|u_pu_c|\le t|u_p|^2+t^{-1}|u_c|^2\), \(t=1/\sqrt3\), on every edge. Each nonroot vertex contributes \(3t+t^{-1}=2\sqrt3\); the root contributes \(4t\le2\sqrt3\). This bounds the absolute quadratic form of adjacency by \(2\sqrt3\|u\|^2\). Self-adjointness and density give the norm bound, and division by four gives \(\sqrt3/2\). At the actual fixed metric unit, range arrows are exactly \(F_2\). Amenability vectors there would be normalized and almost invariant under all four shifts, so their average quadratic form would tend to one, contrary to the strict norm bound. This uses the literal range fibre, not just the abstract acting group's nonamenability. **Rubric:** edge inequality 3; multiplicities and norm 3; actual range fibre 3; certificate contradiction 3.

**Exercise 359 (12 points).** Give the entire domain and compactness proof for MI.10. Preserve the nontrivial spin lift and a supplied central character.

**Solution.** Use the prescribed inverse rank-three normal Dirac, whose conjugation unitary is exactly \(\chi(g)^{-1}\rho(g)\) with argument transformed by the inverse conjugation. Isometric Clifford-connection covariance gives commutation with \(D_N\). Each nonroot tree pair at distance \(k\) and normal eigenvalue \(\nu\) has matrix \(\begin{pmatrix}\nu&k\\k&-\nu\end{pmatrix}\), with square \((\nu^2+k^2)I\). The root block is \(\nu\). Their self-adjoint direct sum has the square-summability domain of all block images, equal to the intersection of factor domains, and finite sums form a core. Compact normal resolvent and finite tree balls make every bounded spectral window finite dimensional, proving total compact resolvent. Normal gradient commutators and the complete finite-segment tree bound handle all smooth crossed coefficients. The resolvent commutator integral then proves compact phase commutators. The final right Clifford generator and entire inverse central character are retained throughout. **Rubric:** full lift 3; exact domain 3; spectral tails 3; all crossed commutators and coefficient 3.

**Exercise 360 (12 points).** Carry the cycle of MI.10 to the entire physical graph and prove original disk pairing \(+1\). Explain precisely why this disproves a universal metric-lift amenability gate without finishing the arbitrary nonisometric problem.

**Solution.** The ordinary normal creation columns and positive tree anticommutator give the product of the index-one scalar tree class and the prescribed inverse normal Dirac. Its restriction is the exact original normal class, so the native inverse Bott identity gives \(+1\) on an original oriented three-disk. The full counting/plaque correspondence retains every return word and fixed isotropy. Its finite smooth covering-chart Parseval compression preserves the completed domain by the bounded normal and return commutators; the diagonalization is a bounded perturbation of the free generator. Localized resolvents and phase commutators are therefore scalar compacts on the entire graph. The actual metric lift is nonamenable by Exercise 358 while this reduced normalized graph cycle exists on the same foliation. Amenability cannot be universally required. The example is isometric, and the general receiving construction for arbitrary nonisometric holonomy remains unproved. **Rubric:** exact restriction and disk 3; whole graph and domain 3; scalar defects and isotropy 3; exact scope distinction 3.

### Exercises 361–368: global metric phase and reduced normal products

**Exercise 361 (12 points).** Prove MGP.5 and the complete boundedness estimate for compact-spatial \(S^0_{\rho,\delta}\), \(0\le\delta\le\rho<1\). Specify the role of both frequency partitions.

**Solution.** Plancherel and the normalized Gaussian give the exact tight-frame norm. In its operator kernel, integration by the two Laplacians produces independent integrable position and frequency difference weights; derivatives of the symbol are bounded and the remaining Gaussian-polynomial factors are integrable. Schur's bound proves the bounded-derivative estimate. On a shell of radius \(R\), frequency cubes of side \(R^\rho\) and the spatial dilation \(R^{-\rho}\) make all symbol derivatives uniformly bounded. Normalized output-frequency powers give cube blocks with summable separation decay. The cube square partition is an isometry, so matrix Schur bounds the whole shell. Compact spatial Fourier integration gives MGP.6 between separated dyadic shells; since \(\delta<1\), its index difference decay is summable after choosing enough derivatives. The dyadic square partition and a second matrix Schur estimate give the full bound. **Rubric:** frame kernel 3; scaled cubes 3; off-shell bounds 3; both complete sums 3.

**Exercise 362 (12 points).** Show that a vanishing order-zero symbol satisfying MGP.2 gives a scalar compact operator. Apply this to the actual second-jet shear of MB.12; a symbol value estimate alone is insufficient.

**Solution.** In rescaled cubes the amplitude tends uniformly to zero and every higher derivative remains bounded. A smoothing kernel of constant moment one and vanishing nonconstant moments gives MGP.8 by Taylor's integral remainder and derivative bounds. Choose its order above every derivative needed for the finite frame, cube and off-shell estimates. Their rescaled seminorms tend to zero, now at type \((\rho,\rho)\); the off-shell estimate remains summable because \(\rho<1\). Thus the high-frequency operator norm tends to zero. The bounded-frequency compact-spatial kernel is Hilbert–Schmidt, proving compactness by norm approximation. The actual shear difference has the required type by MB.17 and uniform vanishing by MB.16, on both axes as well as mixed rays. No derivative decay was assumed directly from its pointwise value decay. **Rubric:** moment interpolation 3; small seminorms 3; norm tail and compact low part 3; actual shear and scope 3.

**Exercise 363 (12 points).** Give the complete nonlinear coordinate-change argument MGP.9–MGP.10, including the threshold \(\delta=1-\rho\).

**Solution.** The exact segment Jacobian gives \(\psi(x)-\psi(y)=J(x,y)(x-y)\). Near the diagonal it is invertible, so frequency substitution gives an exact two-point amplitude with the actual density, Jacobian and coefficient transports. Differentiating its frequency argument in a base variable costs \(|\xi|\partial_\xi\), hence at most \(1-\rho\); the chosen \(\delta=1-\rho\) permits this loss. Its diagonal is the actual transformed leading symbol. Taylor in the second base variable and one frequency integration by parts give a combined derivative of order \(-\rho+\delta\). The rescaled interior oscillatory integral has uniformly integrable weighted derivatives; the exterior region is arbitrarily decreasing because \(\delta<1\). Off-diagonal compact rectangles have smooth Hilbert–Schmidt kernels. At \((3/4,1/4)\), the remainder has order \(-1/2\). This handles the actual nonlinear germ rather than a frozen linear approximation. **Rubric:** exact Jacobian 3; type loss 3; both remainder regions 3; scalar compactness and nonlinear scope 3.

**Exercise 364 (12 points).** Prove maximal continuity and global scalar defects for MGP.12. Explain why the polar part of the triangular derivative is unnecessary.

**Solution.** The sum-metric density is independent of splitting; triangular changes have determinant one. Actual arrows have isometric diagonal blocks and determinant-one shear, so this density is invariant. The original compact-reduced coherent lift and its vertical symmetric-square lift give strict full inverse transports. Their products compose exactly; no polar action of the whole derivative is used. Counting interchange proves the convolution adjoint, and weighted Cauchy–Schwarz bounds the geometric representation by the I-norm. Local quantizations have essential norm at most one by MGP.11, so clamping changes them by compacts. The square partition is an isometry and gives a global contraction. Each compact source and its chart outputs meet only finitely many terms. Negative-order products, actual coordinate changes and interpolated shear compactness prove the unit and arrow defects. Finite compact-bisection sums and maximal norm density prove all source defects. **Rubric:** density and lift 3; I-norm 3; clamped patch 3; entire arrow defects 3.

**Exercise 365 (12 points).** Verify the actual regular-source isometry MRP.2 and reduced receiving norm MRP.5 from MRP.1. Include the conversion from nonnegative \(\ell^1\) certificates.

**Solution.** Add positive vanishing unit mass to an \(\ell^1\) certificate and divide by its mass. On compact source and range unit tests those masses tend uniformly to one, so the invariance error remains small. Taking square roots uses \((\sqrt a-\sqrt b)^2\le|a-b|\) to obtain normalized \(\ell^2\) fields. For MRP.2, coefficient unitarity and the invariant counting/volume interchange turn its source-fibre norm into the receiving norm times the normalized range-field norm one. In MRP.4, full arrow coherence gives the exact transported difference. A single compact bisection has error norm bounded by its coefficient supremum times the certificate error; finite sums give the same convergence. Regular source norms are bounded by the reduced supremum, so the isometry implies MRP.5. Let the certificate index tend to infinity and complete the convolution algebra. **Rubric:** certificate conversion 3; complete isometry 3; exact error 3; reduced norm and completion 3.

**Exercise 366 (12 points).** Apply the regular tree receiving theorem to the completed metric phase for arbitrary nonisometric actual free-group holonomy. Explain why a noneffective free presentation does not suffice.

**Solution.** Theorem 11CF.1 gives the actual maximal covariant cycle on \(W\), with its full inverse rank-\(q+d\) coefficient and scalar localized defects. An effective free-group germ presentation of \(T\) remains effective on \(W\), because an identity germ on the metric bundle projects to one on \(T\). Thus RTP.1 holds for this receiving cycle and the completed vertex-edge factor gives MRP.6. Finite regular orbit conjugation proves its reduced norm; the two compact error algebras and their separator prove full reduced defects. Its unit restriction and finite tree creation phase are exactly the original metric phase. No invariant original metric or amenable metric lift is used. A kernel word in an abstract free presentation of another group may be the identity germ; its separate tree label would then be an invented holonomy arrow. Such a presentation cannot replace the actual groupoid. **Rubric:** completed input 3; actual germ effectiveness 3; norm and compactness 3; full orientation and ineffective-label exclusion 3.

**Exercise 367 (12 points).** Prove the balanced carrier, original normal phase and disk calculation MRP.7–MRP.12. Retain all Clifford order and central lift data.

**Solution.** Expand the squared norm of the complete arrow sum MRP.8. Actual unitary transports and invariant counting interchange give exactly the balanced inner product MRP.9. Unit-arrow compact coefficient frames give dense range, so the isometry identifies the entire carrier rather than a selected vacuum. The full product separator gives all scalar source defects and creation equations MRP.10. On unit-arrow vertical sections these are the ordinary Bott receiving connection and its positive-source product test. Its restriction is therefore the entire positive vertical Bott class followed by the full total inverse; any tree factor has ordinary index one. The native graded block transport retains mixed parity \(dq\) and cancels the full vertical spin and central data as \(z^qz^{-(q+1)}=z^{-1}\), leaving the original inverse \(I_q\) and final right \(\mathrm{Cl}_q\). Compact creation errors vanish on weak normal packets, retaining the quotient Dirac phase. MB.21 on the entire disk metric fibre then gives original disk \(+1\). **Rubric:** balanced isometry 3; complete product connections 3; whole lift and order 3; phase and original disk 3.

**Exercise 368 (12 points).** Give the full physical graph phase and closed domain MRP.13. State precisely what this proves and what remains unfinished.

**Solution.** Tensor with the original full counting/plaque correspondence and use the bounded Parseval creation argument; its rank-one expansions give scalar square and commutator defects on the whole physical graph. Odd localized normalization retains the supplied normal phase. An even injective compact \(K\) from a dense homogeneous frame gives \(F'=S F_G S\), \(S=(1-K)^{1/2}\), with no eigenvectors at \(\pm1\), since \(\|Su\|<\|u\|\) for nonzero scalar vectors. The spectral function \(t/\sqrt{1-t^2}\) therefore gives the dense exact domain MRP.13, self-adjointness, an explicit spectral-cutoff core and normalized transform \(F'\). Compact perturbation preserves the full coefficient, scalar defects, normal phase and original disk. Each localized resolvent is compact by uniform approximation of its continuous endpoint-vanishing function by \((1-t^2)h(t)\). This supplies a closed phase representative in the stated receiving scopes. It proves no finite-order local formula or receiving reduced norm for arbitrary original holonomy. **Rubric:** whole carrier 3; injective compact and domain 3; transform and localized resolvents 3; exact sufficient scope and remaining target 3.

### Exercises 369–372: compact types and the actual receiving norm

**Exercise 369 (12 points).** Prove CTNR.1–CTNR.2, including the full finite coefficient case, without invoking a compact-group representation decomposition theorem.

**Solution.** The orbit of a rank one is norm continuous, using strong continuity on its finitely many scalar columns. Compact-parameter Riemann sums converge in operator norm, so the positive Haar average is compact. A quadratic form positive at the identity stays positive on an open Haar-positive neighbourhood, making the average nonzero. Haar invariance makes it commute with the full group. If c is its positive norm, unit approximate norm vectors satisfy the displayed estimate for (C-c)v. Compactness gives a convergent eigenvector subsequence. The c-eigenspace is finite dimensional because an infinite orthonormal sequence contradicts compactness, and it is invariant because the average commutes with the group. Module rank ones retain the right coefficient and parity; the finite eigenspace is a norm test, not a replacement physical carrier. **Rubric:** norm-continuous compact average 3; nonzero invariance 3; eigenvector and finite dimension 3; coefficient and carrier scope 3.

**Exercise 370 (12 points).** Prove that a nonzero finite-dimensional representation extending to the reduced group algebra yields an invariant mean. Include the matrix absorption and positivity of the Hahn–Banach extension.

**Solution.** Tensor with the conjugate and use the explicit diagonal unitary B from CTNR.4. It identifies the regular tensor norm with the original regular norm. Matrix amplification of the supplied C*-representation is contractive, so its invariant vector Omega gives the augmentation inequality CTNR.5. Augmentation extends to a norm-one character. Hahn–Banach preserves its norm and value at one. Expanding exp(itb) for a self-adjoint b at both signs of t makes the extension real on self-adjoints; the bound on 1-b for a positive contraction then proves positivity. Its value on both square products of lambda_g-1 is zero. Positive-functional Cauchy–Schwarz kills their mixed pairings with all bounded operators, giving conjugation invariance. Restriction to diagonal multipliers is precisely the invariant positive mean CTNR.6. **Rubric:** absorption and invariant vector 3; augmentation extension 3; positivity 3; invariant mean 3.

**Exercise 371 (12 points).** Turn the multiplier test on the actual free-group metric suspension into a compact actual matrix convolution kernel with a strict reduced norm gap. Keep all original coefficients and arrow domains.

**Solution.** Extract any nonzero finite invariant type from the full receiving representation by CTNR.1. Its conjugate matrix twist of the four-generator average is absorbed into the regular representation, so its reduced norm is at most sqrt(3)/2. The tensor invariant vector Omega lies in the full carrier's finite matrix amplification and is fixed by that average. A smooth compact range cutoff tending strongly to one gives norm greater than 9/10 on Omega for a sufficiently large compact support. CTNR.8 is then a sum on exactly the four actual arrow bisections, and CTNR.9 is a strict gap. It contradicts matrix contractivity of a proposed C*-extension. The compact group used for averaging supplies no extra holonomy labels, and neither this finite amplification nor the test eigenspace removes the prescribed right Clifford coefficient or changes original q. **Rubric:** full invariant type 3; absorbed norm 3; compact actual support and gap 3; original lift, domain and degree scope 3.

**Exercise 372 (12 points).** Which of the following repairs can remove CTNR.9: changing F by a compact, an exponent homotopy, unitary conjugation, matrix amplification, adding a regular direct summand, or replacing the receiving representation by the actual tree representation? State the limit of the resulting obstruction.

**Solution.** A compact phase change or exponent homotopy leaves the representation unchanged, so it leaves the exact kernel test unchanged. Unitary conjugation preserves norms; matrix amplification retains the original norm; a direct sum has norm at least that of its geometric summand. Thus these operations cannot repair the stated receiving norm. Compression from a larger reduced representation to a retained reducing geometric summand would also contradict CTNR.9. Replacing the receiving representation is a different operation: the actual free-group tree construction has the proven reduced norm and full scalar defects of Section 11CD, and Theorem 11CG.1 retains the original normal phase, degree and disk +1. The obstruction excludes the tested geometric representation and repairs that retain it. It excludes neither all reduced graph classes nor arbitrary original holonomy, and it makes no new universal amenability or compact-symmetry requirement. **Rubric:** phase repairs 3; norm-preserving enlargements 3; actual tree replacement 3; exact limit and remaining target 3.

### Exercises 373–376. The exact linear receiving correspondence (48 points)

**Exercise 373 (12 points).** Give the sources and coefficients of all four factors in (LINR.3). Prove both inverse products in (LINR.4), including odd \(q\). Why does the fixed Clifford coefficient have trivial group action despite the nontrivial normal spin-c lift?

**Solution.** The external Bott factor has type \(C_q\to A_0\widehat\otimes C_q\widehat\otimes C_q\), followed by the even target Morita evaluation to \(A_0\). The inverse starts with the even source Morita inclusion \(A_0\to A_0\widehat\otimes C_q\widehat\otimes C_q\), followed by \(\alpha\widehat\otimes1_{C_q}\) to \(C_q\) (4 points). In \(\eta\delta\) the adjacent target/source Morita factors cancel and the result is \((\beta\alpha)\widehat\otimes1_{C_q}=1_{C_q}\). In \(\delta\eta\) the middle raw inverse/Bott factors cancel, leaving the other Morita inverse product and \(1_{A_0}\). Descent preserves both identities and gives the displayed reduced types. There is no odd flip, so odd \(q\) adds no sign (4 points). The lift acts as the even left transport on the full spin-c module; its right \(C_q\) action commutes with that transport. The right coefficient is fixed, while the conjugate lift remains in the inverse's module. Its transition is the complete \(z^{-1}\rho_t(s)\), not trivial transport (4 points).

**Exercise 374 (12 points).** Starting from an arbitrary \(\xi\in KK(A_r,C_q)\), recover its scalar receiver. Prove that positive-disk pairing one is equivalent to the entire ordinary normal restriction, rather than merely necessary for that restriction.

**Solution.** Compute \(y=\eta_r\xi\in KK(R\widehat\otimes C_q,C_q)\). Apply the inverse of Clifford amplification to obtain the unique \(\kappa\in KK(R,\mathbb C)\) with \(y=\kappa\widehat\otimes1_{C_q}\). The second inverse in (LINR.4) gives \(\xi=\delta_r(\kappa\widehat\otimes1_{C_q})\) (4 points). By (LINR.7), the disk product is \(([u]\kappa)\widehat\otimes1_{C_q}\). Pairing one therefore forces \([u]\kappa=1_{\mathbb C}\), using scalar classification and invertibility of Clifford amplification (4 points). The other identity in (LINR.7) then gives \([i]\xi=\delta\). Conversely this restriction pairs the local Bott class as one. The positive ball extends to \(\eta\) by the buffered local phase and oscillator comparison in Section 11CI.3. The whole \(KK(A_0,C_q)\) group is identified with \(KK(C_q,C_q)=\mathbb Z\) by this two-sided inverse, so a second undetected ordinary summand cannot remain in this particular model (4 points).

**Exercise 375 (12 points).** Verify the receiving carrier map and the absorption formula (LINR.11). Explain which compactness statement provides the product cycle and why \(F_N\widehat\otimes1\) alone is insufficient on an infinite auxiliary carrier.

**Solution.** The map sends \(\xi\otimes v\) to \(\sum_g\xi(g)\widehat\otimes V_gv\). Its inner product is \(\sum_{g,h}\langle\xi(g),\zeta(gh)\rangle\langle v,V_hw\rangle\), with the finite normal coefficient retained. This is the balanced descended inner product. Identity-supported vectors span the algebraic tensor carrier, proving surjectivity after completion. Left convolution becomes \((\pi(a)U_g)\widehat\otimes V_g\) (4 points). Since \(W^*v(k)=U_kv(k)\), substitution at \(h\) gives \(U_h^{-1}\pi(a)U_gU_{g^{-1}h}v(g^{-1}h)=\pi(\alpha_{h^{-1}}a)v(g^{-1}h)\). Thus the regular tensor has norm at most \(\|f\|_r\); the contractive map \(1\otimes V\) gives the same bound for the actual carrier (4 points). Before balancing, localized normal defects are compact operators of the right \(R\widehat\otimes C_q\)-module. The Kasparov product's full connection and positivity equations give scalar compact defects on the balanced carrier. A nonzero scalar compact normal defect tensored with the identity on an infinite \(H_K\) is generally noncompact: fix a vector on which the defect is nonzero and tensor it with an orthonormal auxiliary sequence. Therefore the bare tensor phase is not a substitute for the product phase (4 points).

**Exercise 376 (12 points).** Let \(\Gamma=C_2=\{1,c\}\) act on \(\mathbb R^2\) by \(\rho(c)=-I\). Choose a spin lift \(s\) with \(s^2=-1\), and the coherent lift \(\sigma(c)=[s,i]\). Twist by the character line \(H(c)=-1\). Compute what changes in the full inverse coefficient and what a determinant-only check misses. Does the receiving criterion prove the general nonlinear-germ target?

**Solution.** The prescribed lift squares to \([-1,-1]=1\), so it is a homomorphism with the stated faithful projected action. The twisted lift has central factor \(-i\) and the same spin factor. Before twisting the inverse central factor is \(i^{-1}=-i\); after twisting it is \((-i)^{-1}=i\). Their ratio is \(-1\), the full conjugate character of \(H\) (4 points). Both determinant inverse phases are \(-1\), since \(i^{-2}=(-i)^{-2}=-1\); a determinant check cannot distinguish these full orientation lines. Rank remains two, \(k_2=1\), the grading shift remains odd and the physical generators remain \(+L_{t_j}\), with the same final right \(C_2\) (4 points). The self-equivalence \(h_r\) and its actual inverse cancel in (LINR.12), so the scalar receiver criterion is unchanged while the normal line is retained. This faithful linear action has two actual return germs at the origin. An actual suspension can use the free deck action of \(C_2\) on \(S^3\) over \(\mathbb{RP}^3\). Arbitrary nonlinear germs have not acquired (LINR.2)–(LINR.4), so this model criterion is not a solution or a universal extra premise for FOL-055 (4 points).

### Exercises 377–380. Actual flat holonomy and the reduced germ norm (48 points)

**Exercise 377 (12 points).** Prove that the vector fields in (FGJ.3) extend smoothly and flatly across zero. Why are their time-one maps globally defined diffeomorphisms, and why do all their Taylor jets at zero equal those of the identity?

**Solution.** Each order-\(m\) derivative on \(I_n\) is bounded by a fixed multiple of \(\epsilon_n\ell_n^{1-m}\). Dividing by \(|x|^N\), with \(|x|\ge(7/8)2^{-n}\), gives (FGJ.4). For each fixed \(m,N\), the negative quadratic exponent dominates its linear exponent. Derivatives fit with zero at interval boundaries because \(\chi\) vanishes near them. Continuous extension and the \(N=1\) difference quotient, inductively, give smoothness and every derivative zero at the origin (4 points). Compact support makes the vector field bounded and globally Lipschitz. Picard iteration produces a uniform short-time solution; bounded speed allows all successive time intervals. The first variational derivative is \(\exp(\int Z'(\phi_s)ds)>0\), higher derivatives solve the displayed successive linear equations, and uniqueness supplies the inverse flow. Thus the time-one map is a global orientation-preserving smooth diffeomorphism (4 points). At zero the flow fixes the point, the first variational coefficient is zero and the derivative is one. All higher variational forcings vanish by flatness, so the higher derivatives are zero. This proves identity jets without asserting that the germ is the identity (4 points).

**Exercise 378 (12 points).** Prove freeness of the two actual germs, including arbitrary reduced-word length. Compute the center displacement of their commutator. Why must the proof use nonzero exact displacements rather than subtracting two rounded physical coordinates?

**Solution.** At parameter two, nonzero powers of \(A_2\) send \(|y|<1\) to \(|y|>1\), and nonzero powers of \(B_2\) send \(|y|>1\), including infinity, into \(|y|<1\). A cyclically reduced mixed word can begin with an \(A\)-power and end with a \(B\)-power. Applying it to zero first fixes zero with the rightmost \(B\), then alternates the two sets and finishes outside \([-1,1]\); pure powers are nonidentity too. Conjugation proves the assertion for every reduced word (4 points). A nonzero off-diagonal or diagonal-difference polynomial of its matrix at two stays nonzero at every transcendental \(\epsilon_n\). If its letter length is \(a\), choose arbitrarily large \(n\) with \(a\epsilon_n<1/4\). Starting in \(|y|<1/4\), all full trajectories stay in \(|y|<1/2\), so the actual cutoff flows equal that nonidentity fractional linear word. Some point in every such interval is moved, proving a nonidentity germ at zero (4 points). Direct multiplication gives (FGJ.9), so zero in the local coordinate is sent to \(-\epsilon_n^3/(1-\epsilon_n^2)\), and the physical displacement is multiplied by \(\ell_n\). It is nonzero at every scale. When it is smaller than floating-point spacing at \(c_n\), rounding both physical endpoints first would erase this difference. The exact formula and the all-length proof are unaffected (4 points).

**Exercise 379 (12 points).** Show that the suspension has actual isotropy \(F_2\) at zero, unit isotropy on the negative half-line and a non-Hausdorff arrow space. Prove what every continuous cocycle to a Hausdorff group does to the zero isotropy. Does this supply a faithful linear or formal-jet label?

**Solution.** The genus-two surface homomorphism sends the two \(a_i\) to free generators and both \(b_i\) to one, so the defining surface relation holds. Its regular-cover suspension has exactly the constructed word germs as holonomy. Freeness of those germs at zero makes its isotropy \(F_2\). Every word is locally the identity on the negative half-line, so the effective groupoid has only units there (4 points). The same arrows at \(x_m=-2^{-m}\) lie in the unit bisection and the \(f\)-bisection. They converge to two distinct zero-source germs, proving non-Hausdorffness (4 points). For any word \(w\), the cocycle on its negative unit arrows is \(e\), and continuity along its bisection forces \(c([w,0])=e\) in the Hausdorff target. It collapses the whole free isotropy. All formal Taylor jets are identity there by (FGJ.5), so neither that jet information nor such a cocycle faithfully labels the actual isotropy. The original coherent positive metalinear lift still exists and keeps degree one (4 points).

**Exercise 380 (12 points).** Construct the character \(\varepsilon_0\) of the actual reduced germ algebra and explain why it does not imply a character of \(C_r^*F_2\). Prove the failure of a canonical reduced collapse from the labelled transformation algebra. Does either result close FOL-055?

**Solution.** For negative source points the actual regular representation is one dimensional, and finite core kernels evaluate as the sum of their word-bisection coefficients. Passing to zero from the negative side gives (FGJ.13), bounded by the supremum of the original regular source norms. Multiplication and star pass through these scalar limits and then norm density; a bump equal to one at zero gives a nonzero norm-one character (4 points). Its bound uses the whole actual germ norm, with neighbouring unit fibres, rather than the isolated free-group regular norm. On a negative invariant interval the labelled transformation algebra is \(C_0(O)\otimes C_r^*F_2\), while the actual germ algebra is \(C_0(O)\). Canonical collapse would send \(v\otimes P\) to \(v\), for a norm-one bump \(v\). The input norm is at most \(\sqrt3/2\), and the output norm is one, contradicting contractivity (4 points). Hence no reduced collapse map with that prescribed core action exists, and no reduced free-group augmentation has been constructed. The character is even and has not been shown to carry the original degree-one inverse normal class or positive disk pairing. A direct whole-graph normally geometric construction remains required for general FOL-055 (4 points).

### Exercises 381–382. Actual character homotopy and normal restriction (24 points)

**Exercise 381 (12 points).** Construct the homotopy from the actual reduced character \(\varepsilon_0\) to zero. Prove its continuity on the whole reduced algebra and verify its KK interval module. Why does the norm-one family not contradict its zero endpoint?

**Solution.** Use \(x(t)=-t/(1-t)\), the actual negative unit-fibre characters below one, the intrinsic negative-side character at zero and the zero map at one. For each finite core presentation the coefficients are continuous through zero and compactly supported; their finite sum is continuous at zero and vanishes once the path leaves their compact supports. Each character is contractive for the original reduced norm (4 points). Approximate an arbitrary fixed element by a core kernel. The two approximation errors are bounded by twice that norm error, uniformly in both parameters, so (ECH.4) gives point-norm continuity and endpoint vanishing. Hence (ECH.5) is a genuine homomorphism homotopy. On the graded module \(E^+=C([0,1]), E^-=0\), multiplication is compact because it equals \(\theta_{b,1}\); operator zero has precisely the compact square defect and zero other defects. Its endpoints are the character cycle and a degenerate cycle, proving \([\varepsilon_0]=0\) (4 points). For every parameter below one a different bump attains homomorphism norm one. No operator-norm convergence of the maps themselves is asserted. A fixed triangular unit coefficient instead gives (ECH.8) and is zero already from one half onward. The source is nonunital, so no persistent unit value is forced at the zero endpoint (4 points).

**Exercise 382 (12 points).** Prove that finite point support of the left normal unit-space action cannot give the required whole inverse class on a positive-dimensional original disk. Include infinite-dimensional point blocks, a retained finite right Clifford coefficient and compact off-diagonal terms. Does the result exclude using a scalar group receiver after a different normal transfer?

**Solution.** In coordinates on the disk, move an evaluation point along (ECH.9) toward infinity of \(\mathbb R^q\). Every fixed \(C_0(U)\) function tends to zero; the evaluation family defines a contractive interval homomorphism. The rank-one interval module proves \([e_p]=0\), with every later coefficient retained (4 points). For finitely many distinct points, choose compact smooth bumps distinguishing them. Their represented projections \(P_i\) have compact commutators with the phase. Remove its finite collection of compact off-diagonal blocks by a straight-line cycle homotopy. The tested square and self-adjointness defects give a scalar-source Fredholm cycle on each block, irrespective of that block's Hilbert dimension. Its full class is \([e_{p_i}]\) times that scalar-source cycle and is zero. A zero-action block is degenerate after the same compact removal (4 points). Their direct sum therefore has zero full restriction and zero local Bott pairing, while the original inverse has pairing \(+1\). Original \(q\), the grading, inverse line and right Clifford algebra were not replaced. A separately constructed normal transfer can change the unit-space action to a full normal representation before a group-algebra receiving product; it does not factor through the point restriction merely because its later factor is scalar. Thus this specific zero-class argument excludes no such different construction and proves no general FOL-055 nonexistence theorem (4 points).

### Exercises 383–384

**Exercise 383 (12 points). Perfect flat isotropy.**

1. (4 points) For \(\varphi(a)=[a,b]\), \(\varphi(b)=[a,b^2]\), use the free kernel generators \(a_j=b^jab^{-j}\) to give inverse free-basis substitutions proving injectivity.
2. (4 points) Prove that the direct limit in PFG.3 is nontrivial and perfect, while each of its nontrivial finitely generated subgroups has a map onto \(\mathbb Z\). Explain why a compatible family of real stage characters is zero.
3. (4 points) Prove the smoothness and identity infinite jet at zero of PFG.8 from PFG.7. Identify precisely which stage relations hold globally and which hold only as germs.

**Solution to Exercise 383.** For (1), the kernel of the map \(F(a,b)\to\mathbb Z\), \(a\mapsto0,b\mapsto1\), is free on all \(a_j\); the explicit free-group/shift semidirect-product isomorphisms in PFG.1 prove this fact. Put \(u=a_0a_1^{-1}\), \(v=a_1a_2^{-1}\). The basis change \((a_0,a_1,a_2)\mapsto(u,v,a_2)\) has inverse \(a_1=va_2\), \(a_0=uva_2\). Direct multiplication gives \(\varphi(a)=u\), \(\varphi(b)=uv\). The second basis change \((u,v)\mapsto(u,uv)\) has inverse \(v=u^{-1}(uv)\). Both images freely generate, so the map is injective. These inverse substitutions prove the statement for every word, rather than a bounded collection of words.

For (2), injectivity retains a nontrivial copy of every stage. Each stage generator equals a commutator in the next stage, and all those generators exhaust the limit, proving perfection. A finite generating set of any subgroup lies in one free stage; the Cayley-tree quotient/spanning-tree proof in subsection 11CL.1 supplies its free basis. If it is nontrivial, sending one basis element to 1 and the others to zero gives the asserted map to \(\mathbb Z\). This does not assert that the entire limit is finitely generated. It cannot be finitely generated because that would make its nontrivial perfect entirety a free group with nonzero abelianization. The connecting maps on abelianizations are zero. If \(\alpha_i=\alpha_{i+1}\varphi\) are real characters, the right side vanishes on all generators, so every \(\alpha_i=0\).

For (3), the affine conjugation on \(I_N\) multiplies a k-th difference derivative by \(r_N^{1-k}\). PFG.7 therefore bounds it by \(r_N^{N+1-k}\) whenever \(N\ge\max(i,k,1)\). Since \(x\ge7r_N\), division by \(x^p\) gives the bound \(7^{-p}r_N^{N+1-k-p}\), tending to zero for every fixed \(k,p\). The smooth collar seams and successive difference quotients at zero prove smoothness and every identity Taylor coefficient. Orientation and the positive derivative at zero give a smooth inverse. The stage relation is exact on all \(I_N\) for \(N\ge i+1\) and on their complementary gaps. It may fail on the earlier, finitely many intervals. It is an equality of germs near zero, and is not used as an everywhere equality of the global diffeomorphisms. The global suspension action uses the free surface labels; actual holonomy then identifies equal germs.

**Exercise 384 (12 points). Proper isotropy and actual groupoid continuity.**

1. (4 points) Prove that the zero-extended functions PFG.13 are positive definite and belong to \(c_0(\Gamma)\).
2. (4 points) Prove that the full Hilbert direct-sum cocycle PFG.15 exists and gives a proper affine action. Use an explicit finite set controlling each squared-norm level \(R\).
3. (4 points) Prove that its squared norm cannot extend as a continuous normalized proper scalar length on the actual holonomy groupoid of PFG.12. State exactly what this does and does not imply for the general normal fundamental class.

**Solution to Exercise 384.** For (1), the word-length kernel on a free stage is its Cayley-tree distance kernel. Over any finite list of vertices, take their finite spanning subtree. Assign two unit vectors with inner product \(e^{-t_n}\) to the two sides of every edge. Tensor their assigned vectors over the edges. Their Gram matrix is \(e^{-t_n\ell_n(g^{-1}h)}\), proving positivity. For a finite list of elements in the larger group, split it into left cosets of the stage subgroup. The zero-extended matrix has zero off-coset blocks, and each diagonal block is a positive stage matrix. It is positive. The function takes value one at the identity. Each positive level set lies in a finite word ball of the stage, so it belongs to \(c_0(\Gamma)\).

For (2), an element \(g_j\) in the enumeration satisfies \(1-p_n(g_j)\le2^{-n}\) for every \(n\ge j\). The finitely many earlier terms are bounded. Hence the squared norms of its component GNS displacement vectors sum to \(2\sum_n(1-p_n(g_j))<\infty\). The componentwise cocycle laws hold in the completed direct sum. Thus \(u\mapsto\pi(g)u+b(g)\) gives an actual affine isometric action. For \(R\ge0\), choose an integer \(M>R\) and take the finite union \(K_M=\bigcup_{n\le M}\{g:p_n(g)\ge1/2\}\). Outside it, the first \(M\) terms of the squared norm already sum to more than \(M\), so \(\{g:\|b(g)\|^2\le R\}\subset K_M\). This proves properness on the entire abstract group without assuming a real character or ignoring the tail.

For (3), every actual isotropy germ \(\gamma\) at zero is represented by a word bisection. At negative sources that word is an identity germ, since all its global generators are identity there. As negative sources tend to zero the actual unit arrows on that bisection converge to \(\gamma\). A continuous scalar function which is zero on units must therefore be zero on every \(\gamma\). That zero level set on the source-zero fibre is the entire infinite discrete group \(\Gamma\), so it is not compact. The asserted proper extension cannot exist. This checks the actual topology, rather than adding the abstract group action as a new label. It rules out this extension of the isotropy auxiliary and establishes no obstruction to every possible reduced normal class. The original degree, coherent inverse coefficient, right Clifford order, actual graph connections and positive original disk pairing remain obligations of that separate general construction.

### Exercises 385–386: the split algebra and the zero disk class

**Exercise 385 (12 points).** In SGD.6, prove directly that the completed image is \(R\oplus C(J_-)\). Give an error estimate for its negative central projection and explain why deleting the endpoint regular representations changes the argument.

**Solution.** Root the four-valent Cayley tree and sum SGD.8 over parent-child edges. Nonroot coefficients are \(3/\sqrt3+\sqrt3=2\sqrt3\), while the root coefficient is \(4/\sqrt3\le2\sqrt3\). The normalized adjacency average therefore has norm at most \(c=\sqrt3/2\) in every positive regular source representation (3 points). On each negative unit source the average is one. Hence \(\Phi(\phi p^n)\) converges to \((0,\phi|_{J_-})\), with error at most \(\|\phi\|c^n\). Every negative arc function extends continuously along the other arc, so the full negative summand belongs to the image (3 points). Subtract it from each crossed-polynomial image. The positive coefficients also extend to the circle, so these remainders contain a dense core of \(R\); the isometric completed image is closed and is the full direct sum (3 points). For \(\phi=1\), the limit is the central projection \(e_-=(0,1)\), with error at most \(c^n\). Endpoints have source fibres \(F_2\), not one unit; their regular representations are part of the original norm and give the zero endpoint value of this projection. The scalar endpoint value one in the other summand is a negative-source limit. Deleting or identifying these different evaluations would replace the actual norm calculation (3 points).

**Exercise 386 (12 points).** Write a norm-continuous nullhomotopy of the original positive disk class in the reduced algebra, and carry the result to the compact codimension-\(q\) suspension with its complete inverse normal coefficient. Does the resulting zero depend on a choice of scalar receiving cycle?

**Solution.** Choose the positive winding-one loop supported in an interval \(U\) of the positive arc. On that closed arc it has a continuous argument \(\theta\) from zero to \(2\pi\); on the negative arc it is one. The direct-sum unitaries \(v_s=(e^{is\theta},1)\) obey \(v_0=1\), \(v_1=j(u_U)\), and \(\|v_s-v_t\|\le2\pi|s-t|\). Their components need not agree at the two endpoints, because the actual central projection SGD.11 separates them. Thus the image of the positive \(K_1\) generator is zero (4 points). In the original coordinate order, the degree-\(q\) disk is the external product of this generator with the remaining \(q-1\) positive interval generators. No odd interchange is made; bilinearity makes the whole product zero. Positive open-disk excision and the actual graph Morita correspondence carry the same zero to the entire original disk in the compact suspension (4 points). The full metalinear lift, inverse central line and final right \(Cl_q\) fix the degree and sign of the pairing. They do not alter a zero K-class. Its product with every scalar Kasparov class is zero, so there is no receiving-cycle choice with pairing \(+1\) under these hypotheses (4 points).

## References

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- [Connes—horocycle application] Alain Connes, *A survey of foliations and operator algebras*, Section 12. [Freely accessible author-hosted treatment](https://alainconnes.org/wp-content/uploads/foliationsfine.pdf#page=36). It describes the classical unique-ergodicity theorem due to Furstenberg and its measured-foliation application. The complete compact homogeneous proof used by this lesson is supplied in Lemma 8.25 and Theorem 8.26.

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