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\begin{document}

\section{Dirac classes and the cotangent Dolbeault
element}\label{dirac-classes-and-the-cotangent-dolbeault-element}

\emph{Written by GPT-6.1 Sol (OpenAI). Public domain (CC0).}

Fix the Fourier convention in which the principal symbol of
\(-i\partial_j\) is \(\xi_j\). Clifford multiplication by a real
covector is self-adjoint and has square its squared norm. The compactly
supported smooth sections of a finite-rank Hermitian bundle are denoted
\(C_c^\infty\). All manifolds here are smooth and second countable.

\subsection{1. A local domain argument and the exhaustion
argument}\label{a-local-domain-argument-and-the-exhaustion-argument}

\textbf{Lemma CI.1.} Let \(Y\) be a Riemannian manifold, \(W\to Y\) a
finite-rank Hermitian bundle, and \(P\) a smooth formally symmetric
first-order operator on \(C_c^\infty(Y,W)\). Suppose its principal
symbol satisfies

\[
 \|\sigma_P(y,\zeta)\|\leq c\|\zeta\|
 \tag{CI.1}
\]

with one constant \(c\). Suppose either \(Y\) is compact or there is a
smooth proper function \(r:Y\to[1,\infty)\) with \(\|dr\|\leq c_r\).
Then \(P\) is essentially self-adjoint on \(C_c^\infty\). If \(P\) is
elliptic, its self-adjoint closure \(D\) satisfies

\[
 f(D\pm i)^{-1}\in\mathcal K(L^2(Y,W))
 \quad(f\in C_0(Y)).
 \tag{CI.2}
\]

\textbf{Proof: compactly supported vectors in the maximal domain.} The
adjoint domain is the maximal distributional domain
\(\{u\in L^2:Pu\in L^2\}\). Indeed testing the adjoint identity on
compactly supported smooth sections is exactly the distributional
identity. Work in a relatively compact coordinate chart and trivialize
the bundle; a smooth change of density and fiber metric gives the same
graph convergence problem in ordinary Euclidean \(L^2\). In this
trivialization write

\[
 P=\sum_j a_j(x)\partial_j+b(x).
\]

Let \(J_\varepsilon\) be convolution with a smooth compactly supported
approximate identity. For a vector supported inside the chart,
distributional integration by parts gives

\[
\begin{split}
 (PJ_\varepsilon-J_\varepsilon P)u(x)
 ={}&\sum_j\int\big(a_j(x)-a_j(y)\big)
       \partial_j k_\varepsilon(x-y)u(y)\,dy\\
 &+\sum_j\int k_\varepsilon(x-y)(\partial_j a_j)(y)u(y)\,dy\\
 &+\int k_\varepsilon(x-y)\big(b(x)-b(y)\big)u(y)\,dy.
\end{split}
 \tag{CI.3}
\]

The first two lines are uniformly bounded on \(L^2\): on the selected
compact neighborhood the coefficients are Lipschitz, and
\(\int |z|\,|\partial_jk_\varepsilon(z)|\,dz\) is independent of
\(\varepsilon\). The last line is uniformly bounded and tends to zero in
operator norm by uniform continuity of \(b\). On a smooth compactly
supported vector the entire commutator tends to zero, since both terms
before subtraction converge to \(Pu\). Smooth vectors are dense in
\(L^2\), and the uniform bound therefore proves convergence to zero for
every \(L^2\) vector. If additionally \(Pu\in L^2\), (CI.3) proves

\[
 J_\varepsilon u\longrightarrow u,\qquad
 PJ_\varepsilon u\longrightarrow Pu
 \quad\text{in }L^2.
 \tag{CI.4}
\]

For a general compactly supported maximal-domain vector take a finite
smooth partition in such charts. Multiplication by a partition function
preserves the maximal domain, because
\(P(\alpha u)=\alpha Pu+[P,\alpha]u\), and the commutator is a bounded
multiplication operator on its compact support. Mollify the finitely
many pieces and sum. Thus every compactly supported maximal-domain
vector belongs to the minimal closed domain. No ellipticity was needed
for this step.

\textbf{Proof: the deficiency spaces.} Choose a real smooth function
\(\rho\) equal to one on \([0,1]\), zero on \([2,\infty)\), and between
zero and one. Set \(\chi_R=\rho(r/R)\) in the noncompact case. It has
compact support, tends pointwise to one, and

\[
 \|[P,\chi_R]\|\leq cc_r\|\rho'\|_\infty/R.
 \tag{CI.5}
\]

Let \(u\in\ker(P^*\mp i)\). The preceding local argument places
\(\chi_Ru\) in the minimal domain. The imaginary part of its symmetry
identity gives

\[
 \|\chi_Ru\|^2
 \leq \|[P,\chi_R]u\|\,\|\chi_Ru\|.
 \tag{CI.6}
\]

Indeed \(P(\chi_Ru)=\pm i\chi_Ru+[P,\chi_R]u\), while
\(\langle P(\chi_Ru),\chi_Ru\rangle\) is real. Equations (CI.5)--(CI.6)
and dominated convergence imply \(u=0\). On a compact manifold use
\(\chi_R=1\) and the same argument immediately. The two ranges of the
minimal closure minus and plus \(i\) are dense because their orthogonal
complements are these deficiency spaces. They are closed because, for a
symmetric operator,

\[
 \|(P\pm i)v\|^2=\|Pv\|^2+\|v\|^2.
\]

They are therefore the whole Hilbert space. Their inverse resolvents are
adjoints of one another, and the closed operator is self-adjoint. This
proves essential self-adjointness and identifies its domain with the
maximal distributional domain.

\textbf{Proof: localized compactness.} For a compactly supported smooth
\(\chi\), local ellipticity gives

\[
 \|\chi u\|_{H^1}\leq C_\chi(\|Du\|_{L^2}+\|u\|_{L^2}).
 \tag{CI.7}
\]

Here the local symbol and Sobolev arguments are the actual proofs in
\href{KT-KK-11.html\#4-spectral-triples-and-the-circle}{\emph{Unbounded
Kasparov modules and spectral triples}, Lemmas 4.2a--4.2b}. In a chart,
invert the first-order principal symbol for large frequency, multiply
that inverse by a cutoff equal to one near \(\operatorname{supp}\chi\),
and quantize it. The proved composition expansion gives the identity
near that support modulo an operator of order minus one. Both the
inverse and this remainder map \(L^2\) to \(H^1\). Insert a second
cutoff equal to one near the first; its commutator with \(P\) is order
zero. The Sobolev bounds in that lemma give (CI.7). A finite chart
partition gives the displayed estimate for the entire support. It
applies to maximal-domain vectors by the compactly supported graph
approximation already established, or by the same distributional
parametrix identity.

Apply (CI.7) to \(u=(D\pm i)^{-1}v\). The inverse resolvent has norm at
most one, and \(D(D\pm i)^{-1}\) is bounded, so bounded subsets of
\(L^2\) are carried by \(\chi(D\pm i)^{-1}\) into a bounded \(H^1\) set
with fixed compact support. The compact Sobolev inclusion, proved by the
finite chart/Fourier-truncation argument in the same earlier lemma,
makes this operator compact. Approximate an arbitrary \(f\in C_0(Y)\)
uniformly by compactly supported smooth functions; such approximation
follows by scalar truncation, a finite chart partition and ordinary
convolution in each chart. The uniform resolvent bound passes
compactness to the limit and proves (CI.2). \(\square\)

\subsection{\texorpdfstring{2. Closed spin\(^{c}\) Dirac
classes}{2. Closed spin\^{}\{c\} Dirac classes}}\label{closed-spinc-dirac-classes}

\textbf{Proposition CI.2.} Let \(M\) be a closed Riemannian spin\(^{c}\)
manifold. The spin\(^{c}\) Dirac operator defines

\[
 [D_M]\in KK^{\dim M}(C(M),\mathbb C).
 \tag{CI.8}
\]

\textbf{Proof.} In even dimension let \(S=S^+\oplus S^-\) be the spinor
bundle and let \(\nabla\) be its Hermitian Clifford connection. For an
orthonormal frame put

\[
 D_M=-i\sum_j c(e_j^\flat)\nabla_{e_j}.
 \tag{CI.9}
\]

The connection's change-of-frame rule makes this a global operator. Its
principal symbol is \(c(\xi)\), hence its square is \(\|\xi\|^2\) and it
is elliptic. Metric compatibility and integration by parts show formal
symmetry: the additional divergence terms cancel the connection
derivatives of the frame, since
\(\sum_j\nabla_{e_j}e_j+\sum_j(\operatorname{div}e_j)e_j=0\). It is odd
for the spinor chirality. Lemma CI.1 gives its self-adjoint closure and
compact inverse resolvents. Its closed domain is \(H^1(M,S)\), by the
exact elliptic-domain proof in the earlier Lemma 4.2b.

Smooth scalar multiplication preserves this domain and

\[
 [D_M,f]=-i\,c(df),\qquad f\in C^\infty(M).
 \tag{CI.10}
\]

This commutator is bounded. Smooth functions are uniformly dense in
\(C(M)\), by the finite chart convolution argument. The Hilbert space
\(L^2(M,S)\) is separable, by finite bundle charts and the countable
Fourier dense families in their coordinate boxes, so it is a countably
generated Hilbert \(\mathbb C\)-module.
\href{KT-KK-11.html\#2-the-bounded-transform}{The Baaj--Julg
bounded-transform proof, Theorem 2.2 of the earlier lesson}, now proves
that

\[
 \big(L^2(M,S),\text{multiplication},
      D_M(1+D_M^2)^{-1/2}\big)
 \tag{CI.11}
\]

is the required even cycle. This constructs the class; no index theorem
is needed to obtain it.

In odd dimension use the ungraded spinor space and the same self-adjoint
operator, then the right-Clifford convention: tensor the Hilbert space
with \(Cl_1\), grade by that final factor, and replace \(D_M\) by
\(D_M\otimes\epsilon\). Its square, inverse-resolvent formula,
compactness and commutator checks are the same ones tensored with a
finite-dimensional algebra. The right module is countably generated. Its
bounded transform is an even cycle with coefficient \(Cl_1\), which is
exactly the definition of the odd group in (CI.8). \(\square\)

\subsection{\texorpdfstring{3. The \(J_+\) cotangent Dolbeault element
in every base
dimension}{3. The J\_+ cotangent Dolbeault element in every base dimension}}\label{the-j_-cotangent-dolbeault-element-in-every-base-dimension}

The following construction needs neither an orientation nor a
spin\(^{c}\) structure on the base manifold. Its explicit almost complex
convention is denoted \(J_+\); Section 5 compares it with the convention
compatible with the outward Thom elements. Let \(M\) be any closed
Riemannian manifold and \(Q=T^*M\). The Levi-Civita connection splits

\[
 T_{(x,\xi)}Q\cong T_xM\oplus T_x^*M.
\]

With the horizontal metric \(g\) and vertical metric \(g^{-1}\), define
the orthogonal almost complex structure

\[
 J(v,\eta)=(-g^{-1}\eta,gv).
 \tag{CI.12}
\]

Thus \(J^2=-1\); in a flat orthonormal coordinate system it gives the
complex coordinates \(x_j+i\xi_j\). Its spinor bundle is
\(\Lambda^{0,*}T^*Q\), graded by degree. The elementary complex exterior
spinor construction and determinant convention are the earlier
\href{KT-KK-13.html\#6-complex-bundles-and-relative-symbols}{\emph{Thom
isomorphisms and K-orientations in KK}, Section 6}; the Hermitian metric
identifies that model with the anti-holomorphic cotangent model here.

For a compactly supported \((0,q)\)-form let \(\bar\partial_J\) be the
\((0,q+1)\)-part of its exterior derivative, and take its formal adjoint
for this metric and density. Define

\[
 P_Q=\sqrt2\,(\bar\partial_J+\bar\partial_J^*).
 \tag{CI.13}
\]

Integrability of \(J\) is unnecessary: (CI.13) is an odd formally
symmetric first-order operator even when \(\bar\partial_J^2\ne0\). For a
real covector \(\zeta\), its principal symbol is

\[
 \gamma_J(\zeta)
 =i\sqrt2\big(\varepsilon(\zeta^{0,1})
                 -\varepsilon(\zeta^{0,1})^*\big).
 \tag{CI.14}
\]

The wedge/contraction relations give \(\gamma_J(\zeta)^2=\|\zeta\|^2\);
the symbol is self-adjoint and has norm \(\|\zeta\|\). This verifies
ellipticity and (CI.1) directly. The factor \(\sqrt2\) compensates for
\(\|\zeta^{0,1}\|^2=\|\zeta\|^2/2\).

The function

\[
 r(x,\xi)=\sqrt{1+\|\xi\|^2}
 \tag{CI.15}
\]

is smooth and proper because the base is compact and bounded closed ball
bundles are compact. Parallel transport preserves \(\|\xi\|\), so its
horizontal derivative is zero. Its vertical gradient has norm
\(\|\xi\|/\sqrt{1+\|\xi\|^2}\leq1\). Thus Lemma CI.1 applies, proving
essential self-adjointness and localized compactness for the closure
\(D_Q\). For \(f\in C_c^\infty(Q)\), its commutator is multiplication by
the symbol at \(df\), up to the fixed factor \(-i\); it is bounded and
preserves the closed domain. The earlier bounded-transform theorem
consequently constructs

\[
 [\bar\partial_{M,+}]
 =\big[L^2(Q,\Lambda^{0,*}T^*Q),\text{multiplication},
            D_Q(1+D_Q^2)^{-1/2}\big]
 \in KK(C_0(T^*M),\mathbb C).
 \tag{CI.16}
\]

Separable countable generation follows from the countable bundle chart
cover and compactly supported coordinate dense families, as before. This
proof applies when \(\dim M\) is odd and when \(M\) is nonorientable;
\(Q\) itself has the displayed almost complex structure and even real
dimension.

The normalization in (CI.13) has no effect on the class: positive scalar
rescaling gives the norm-continuous bounded-transform homotopy proved in
the earlier product-recognition section. These formulas construct the
analytic elements needed for the index comparison. Their construction
alone does not identify a symbol product or the embedding topological
index.

\subsection{4. Dependence on the geometric
choices}\label{dependence-on-the-geometric-choices}

\textbf{Lemma CI.4 (a smooth family is a cycle homotopy).} Suppose a
family \(P_s\), \(0\leq s\leq1\), satisfies Lemma CI.1, is elliptic, and
has smooth coefficients depending smoothly on \(s\). Identify its
Hermitian bundle and density spaces with one fixed graded \(L^2\) space
by smooth even pointwise unitary maps. Suppose the localized elliptic
estimates (CI.7) hold uniformly on compact sets and the scalar
multiplication source is fixed under these maps. Then the
bounded-transform cycles form a Hilbert-module homotopy. Norm continuity
of the unlocalized bounded transforms is not an additional hypothesis.

\textbf{Proof.} Write \(D_s\) for the self-adjoint closures. Every
compactly supported smooth vector is in every domain, and
\(s\mapsto D_su\) is norm-continuous for such a vector. The inverse
resolvents \(R_s^\pm=(D_s\pm i)^{-1}\) are strongly continuous. To prove
this at \(s_0\), first take \(v=(D_{s_0}\pm i)u\) with \(u\) compactly
supported and smooth. The resolvent identity on this common core gives

\[
 R_s^\pm v-u=R_s^\pm(D_{s_0}-D_s)u\longrightarrow0.
 \tag{CI.17}
\]

Such \(v\) are dense, by the core and surjectivity already proved. The
uniform bound \(\|R_s^\pm\|\leq1\) extends continuity to every vector.
The same argument for the opposite sign gives strong continuity of the
adjoints.

These fields therefore act adjointably on
\(\mathcal H=C([0,1],L^2(Y,W))\). Their pointwise resolvent identities
define a self-adjoint regular operator \(\mathcal D\), with domain
consisting of sections whose values lie in \(\operatorname{Dom}D_s\) and
whose images under \(D_s\) are continuous. For clarity, finite sums of
smooth compactly supported vectors with continuous scalar parameter
coefficients are a graph core. Given a continuous target section
\(v(s)\), select at each of finitely many parameters a smooth vector
\(u_j\) approximating \(R_{s_j}^\pm v(s_j)\) in its graph norm.
Continuity of \((D_s\pm i)u_j\) and of \(v(s)\) gives the same
approximation on a neighborhood. A finite parameter partition of unity
yields a section whose image under \(\mathcal D\pm i\) approximates
\(v\) uniformly. Applying the bounded inverse resolvent proves graph
convergence. This verifies the claimed closure and the dense domain; the
same resolvents prove regularity by the earlier resolvent lemma.

For a compactly supported smooth \(\chi\), the uniform estimate (CI.7)
makes the family \(\chi R_s^\pm\) collectively compact: its images of
unit balls lie in a uniformly bounded \(H^1\) set with fixed compact
support. Compact inclusion permits a finite-dimensional orthogonal
projection \(p\) such that

\[
 \sup_s\|(1-p)\chi R_s^\pm\|<\varepsilon.
\]

The operators \(p\chi R_s^\pm\) vary in norm, since their adjoints are
the strongly continuous operators \(R_s^\mp\chi^*p\) restricted to the
finite-dimensional range of \(p\). Uniform finite-rank approximation
proves that \(s\mapsto\chi R_s^\pm\) is a norm-continuous compact field.
Uniform approximation of \(f\in C_0(Y)\) by these smooth cutoffs proves
the same for \(fR_s^\pm\).

The smooth compactly supported source algebra preserves the core and has
bounded parameter-continuous commutators with \(\mathcal D\). The module
\(\mathcal H\) is countably generated, using constant vectors from a
countable Hilbert basis. Its localized inverse resolvents are the
compact fields just proved. Apply the earlier Hilbert-module
bounded-transform theorem to \(\mathcal D\). Evaluation at either
endpoint gives the original bounded-transform cycle by its resolvent
functional calculus. This is the required homotopy. \(\square\)

For a fixed spin\(^{c}\) structure, metrics are joined by their convex
positive-definite interpolation and Hermitian Clifford connections by
the affine interpolation of compatible connections, using the smooth
identification of the metric orthonormal frame bundles by the positive
square root of the relative metric matrix. The resulting spinor spaces
are identified evenly and pointwise. On a closed base all elliptic
estimates are uniform on the compact parameter interval. Lemma CI.4
proves independence of these metric and connection choices in (CI.8). In
odd dimension apply its resolvent and compact-field arguments after the
finite right-Clifford transfer: the inverse of
\(D_s\otimes\epsilon\pm i\) is
\((D_s\otimes\epsilon\mp i)(1+D_s^2)^{-1}\), so the same continuity and
compactness follow from the scalar resolvents and the fixed finite
Clifford factors.

For (CI.16), interpolate the base metrics. Their Levi-Civita
connections, the splittings (CI.12), and the exterior spinor bundles
vary smoothly on \(T^*M\times[0,1]\). An even Hermitian connection on
this parameter bundle identifies the fibers by parameter-direction
parallel transport; multiplication by base functions is preserved. Each
parameter has the proper radial function (CI.15), so essential
self-adjointness follows individually from CI.1. On any selected compact
support the parameter-dependent coefficients and ellipticity constants
are uniformly controlled, giving exactly the uniform local estimates
required in CI.4. It follows that the cotangent Dolbeault element is
independent of the chosen base metric. This argument does not require a
uniform global bound on the change between two Sasaki metrics.

\subsection{5. The Clifford inverse and the two cotangent
conventions}\label{the-clifford-inverse-and-the-two-cotangent-conventions}

Put \(V=T^*M\), \(R=C(M)\), \(B=C_0(V)\),
\(C=\Gamma(\operatorname{Cl}(V))\), and \(n=\dim M\); components of
different dimensions are treated separately. Write \[
 \epsilon_n=(-1)^{n(n-1)/2}.
 \tag{CI.18}
\] On \(\Lambda^*V_{\mathbb C}\), use \[
 c_j=\varepsilon_j+\iota_j,\qquad
 f_j=i(\varepsilon_j-\iota_j),\qquad
 \Gamma_0=(-1)^N.
 \tag{CI.19}
\] The Clifford Thom equivalence of \emph{Thom isomorphisms and
K-orientations in KK}, Theorem 4.2, has the concrete forward module \[
 \mathcal X=C_0(V,\pi^*\Lambda^*V_{\mathbb C}),\quad
 \Gamma_{\mathcal X}=\epsilon_n\Gamma_0,\quad
 e_j\longmapsto f_j,\quad X=c(\xi).
 \tag{CI.20}
\] Its bounded transform represents \(x_V\in KK_R(C,B)\). The
representation of a varying Clifford section is pointwise at the base
point. All algebras in this section are separable and the intermediates
are \(\sigma\)-unital: \(M\) is compact and second countable, \(C\) is
finite-dimensional over \(R\), and radial cutoffs give a countable
approximate identity of \(B\). Thus the actual preceding product
theorems apply.

\textbf{Lemma CI.5 (an explicit inverse with its grading).} The inverse
\(y_V\in KK_R(B,C)\) is represented on the Hilbert \(C\)-module \[
 \mathcal Y=\Gamma\bigl(x\longmapsto L^2(V_x)\widehat\otimes
                    \operatorname{Cl}(V_x)\bigr),\qquad
 \Gamma_{\mathcal Y}=\epsilon_n\Gamma_{\operatorname{Cl}},
 \qquad
 D_{\mathcal Y}=i\epsilon_n\sum_j e_j\partial_{\xi_j}.
 \tag{CI.21}
\] Here \(e_j\) acts by \textbf{left} multiplication on the coefficient
Clifford algebra. The right action is the ordinary right Clifford
action. The source \(B\) acts by scalar multiplication. Neither
orientation nor a spinor bundle is needed.

\textbf{Proof: the operator and its module.} Orthogonal changes of frame
act by the change of variable on \(L^2(V_x)\) and by the Clifford
algebra isomorphism. They preserve (CI.21), its grading and the fiber
Lebesgue measure. In a frame, Fourier transformation turns its symbol
into \(-\epsilon_n\sum e_j\zeta_j\), whose square is \(|\zeta|^2\). The
inverse resolvents are the Fourier multipliers \[
 \frac{D_{\mathcal Y}\mp i}{1+D_{\mathcal Y}^2}.
\] Their norms are at most one, their adjoints are the opposite
resolvents, and their ranges are the first vertical Sobolev domain. This
is exactly the Fourier/domain argument in the Bott lesson's Lemma 3.1,
with a finite Clifford coefficient. The resolvents glue under the frame
unitaries; their ranges contain the sections smooth and compactly
supported in the fiber in a chart. A finite partition proves density and
hence self-adjoint regularity by the resolvent lemma.

A finite cover and countably many smooth compactly supported fiber
vectors, times the finitely many coefficient Clifford frame vectors,
give countably many module generators. For a compactly supported source
in a chart, the localized resolvents are compact by the Bott lesson's
frequency cutoff and square-integrable-kernel argument. Tensoring with
the finite coefficient matrices and multiplying chart partitions on both
sides gives global finite-rank approximations. Summing over the finite
cover, then approximating an arbitrary \(B\)-source by fiber cutoffs,
proves local compactness. The smooth compactly supported source has
bounded commutators \(i\epsilon_n c(d_{\mathrm{vert}}b)\) and preserves
the Sobolev domain. The bounded-transform theorem supplies the cycle.

\textbf{Proof: the inverse identification.} On
\(\mathcal X\widehat\otimes_B\mathcal Y\), the proposed product is the
fiber operator \[
 Q=c(\xi)+i\Gamma_0\sum_j L_{e_j}\partial_{\xi_j},
 \qquad
 \Gamma_{\mathcal X\otimes\mathcal Y}
       =\Gamma_0\widehat\otimes\Gamma_{\operatorname{Cl}} .
 \tag{CI.22}
\] The two occurrences of \(\epsilon_n\) cancel in these expressions.
Its domain is the weighted vertical Sobolev domain. Squaring on smooth
compactly supported vectors gives the scalar harmonic oscillator plus
the \(n\) commuting self-adjoint involutions \[
 K_j=-i c_j\Gamma_0 L_{e_j}.
 \tag{CI.23}
\] The Clifford anticommutation relations give \(K_j^2=1\) and
\(K_jK_k=K_kK_j\). The scalar oscillator is bounded below by \(n\); its
ground vector is \(g(\xi)=\pi^{-n/4}e^{-|\xi|^2/2}\), and its other
Hermite levels are at least \(n+2\). These exact assertions, with the
complete graph domain and compact inclusion, are proved in the Bott
lesson's Lemma 4.0. Diagonalizing the finitely many matrices \(K_j\)
gives the bounded inverse of \(1+Q^2\) in the Hermite decomposition. The
operators \((Q\mp i)(1+Q^2)^{-1}\) are mutually adjoint bounded inverse
resolvents of \(Q\pm i\), with ranges the weighted graph domain: the
identity \(\|Qu\|^2=\langle u,(-\Delta+|\xi|^2+\sum K_j)u\rangle\)
identifies its graph norm with that weighted norm, and the smooth
Hermite core is dense for it. The resolvent argument proves self-adjoint
regularity, compact fiber resolvents and a gap at least \(\sqrt2\) off
the kernel. No unbounded-domain conclusion is inferred merely from
formal squaring.

The common \(-1\) eigenspace in (CI.23) is the right coefficient module
generated by \[
 w=2^{-n/2}\sum_{I=(i_1<\cdots<i_k)}
       (e_{i_1}\wedge\cdots\wedge e_{i_k})
             \otimes i^k e_{i_k}\cdots e_{i_1}.
 \tag{CI.24}
\] Indeed exterior creation and contraction, with the sign from moving a
generator through the ordered word, give \[
 (c_j-i\Gamma_0L_{e_j})w=0,\qquad
 f_jw=w e_j,\qquad
 \langle w,w\rangle=1_C.
 \tag{CI.25}
\] For the last identity each of the \(2^n\) coefficient words is
unitary. For completeness, the first identity determines every
coefficient recursively from the vacuum coefficient: its degree-\(k\)
coefficient is \(i^ke_{i_k}\cdots e_{i_1}\). Thus it describes the whole
simultaneous eigenspace, not just one vector in it. The generator \(w\)
is even, because its form degree and coefficient degree agree. Its
formula is orthogonally invariant: alternating Clifford multiplication
and the exterior inner product are invariant; reversal of the Clifford
word is invariant as well. Hence \(gw\) is a global even unit section.
Equations (CI.25) identify its source and coefficient actions with the
identity \(C\)-\(C\) correspondence.

The oscillator resolvents are compact on the global \(C\)-module, rather
than merely fiberwise: the compact resolvent field is norm-continuous
under the strongly continuous transition unitaries, first for rank ones
and then by approximation. A finite chart partition gives global
finite-rank approximations, as in the actual proof of the parametrized
Bott theorem. Creations from smooth compactly supported sections of
\(\mathcal X\) have bounded errors relative to \(D_{\mathcal Y}\): the
errors are multiplication by their position and vertical derivative,
both bounded on the selected support. The adjoint error follows by
vertical integration by parts; it gives the required adjoint domain
inclusion. Moreover \(\operatorname{Dom}Q\subset\operatorname{Dom}X\),
and \[
 2\operatorname{Re}\langle Q u,Xu\rangle
       \geq-n\langle u,u\rangle
\] on the graph domain, by the bounded matrix term in (CI.23). The
proved unbounded product criterion identifies (CI.22) with
\(x_V\otimes_B[\mathcal Y]\). Deforming the complementary oscillator
bounded transform to its sign leaves \(C\) on the even Gaussian summand
and an exactly degenerate complement: the source action is fiberwise
Clifford multiplication and intertwines (CI.25), and it graded commutes
with \(Q\). Thus \(x_V\otimes_B[\mathcal Y]=1_C\). The already proved
inverse pair of Theorem 4.2 now implies \[
 [\mathcal Y]
  =(y_V\otimes_C x_V)\otimes_B[\mathcal Y]=y_V .
\] This identifies the actual inverse, including every sign in (CI.21).
\(\square\)

Define the Clifford Dirac class by the closure of \[
 \mathscr D_M=-i\sum_j c_j\nabla_{e_j}
 \quad\hbox{on }\ L^2(M,\Lambda^*V_{\mathbb C}),\qquad
 \Gamma_M=\epsilon_n\Gamma_0,\qquad C\ni e_j\longmapsto f_j.
 \tag{CI.26}
\] The exterior connection is the metric connection. Clifford families
\(c\) and \(f\) graded commute. Consequently the graded commutator with
a smooth Clifford section is multiplication by its covariant derivative
contracted with \(c\), hence bounded. The scalar elliptic symbol is
\(c(\zeta)\). Lemma CI.1 and the closed-manifold elliptic domain proof
give self-adjointness, the domain \(H^1\), compact resolvents and the
bounded-transform cycle \[
 [\mathscr D_M]\in KK(C,\mathbb C).
 \tag{CI.27}
\] The formal symmetry verification is the same frame/divergence
cancellation as in CI.2. This construction applies in every dimension
and on nonorientable \(M\).

Set \[
 J_-(v,\eta)=(g^{-1}\eta,-gv).
 \tag{CI.28}
\] Construct \([\bar\partial_{M,-}]\) by Section 3 with \(J_-\) in place
of \(J_+\). Its proper radial function and ellipticity are unchanged. In
a local orthonormal identification of its anti-holomorphic forms with
\(\Lambda^*V_{\mathbb C}\), its horizontal and vertical symbols are
respectively \(f_j\) and \(c_j\).

\textbf{Theorem CI.6 (the full Clifford Thom--Dolbeault identities).}
For every closed manifold, including odd-dimensional and nonorientable
manifolds, \[
 y_V\otimes_C[\mathscr D_M]=[\bar\partial_{M,-}],
 \qquad
 x_V\otimes_B[\bar\partial_{M,-}]=[\mathscr D_M].
 \tag{CI.29}
\]

\textbf{Proof.} Give the fiber Hilbert bundle in (CI.21) its metric
parallel-transport connection. This transports the fiber Lebesgue
measure, Clifford multiplication and vertical derivative together, and
therefore preserves \(D_{\mathcal Y}\). The tensor evaluation identifies
\[
 \mathcal Y\widehat\otimes_C L^2(M,\Lambda^*V_{\mathbb C})
       \cong L^2(V,\pi^*\Lambda^*V_{\mathbb C})
\] with grading \(\Gamma_0\), because the two factors \(\epsilon_n\)
cancel. On a smooth core the connection tensor sum is \[
 L=\epsilon_n\left(-i\sum_jc_j\nabla_{H_j}
                         -\sum_j\widehat c_j\partial_{\xi_j}\right),
 \qquad \widehat c_j=\varepsilon_j-\iota_j .
 \tag{CI.30}
\] The horizontal lift \(H_j\) includes the variation of the fiber
coordinate under the connection; it is not a derivative at a fixed
arbitrary coordinate trivialization. Parallel transport preserves the
vertical operator, so its derivative along \(H_j\) is zero. In addition
\(c_j\) anticommutes with every \(f_k=i\widehat c_k\). These two
identities give exact anticommutation between the horizontal and
vertical summands of (CI.30). The horizontal summand is formally
symmetric: the fiber transport has zero infinitesimal divergence, and
the remaining frame/divergence cancellation is the one on \(M\). Thus,
on the core, \[
 \|Lu\|^2=\|D_{\mathrm{vert}}u\|^2+\|D_{\mathrm{hor}}u\|^2,\qquad
 2\operatorname{Re}\langle Lu,D_{\mathrm{vert}}u\rangle
          =2\|D_{\mathrm{vert}}u\|^2\geq0.
 \tag{CI.31}
\]

The principal symbol of \(L\) is an orthogonal Clifford symbol for the
connection metric. The proper function (CI.15) has bounded gradient for
it. Lemma CI.1 therefore proves essential self-adjointness and localized
compactness on the full cotangent space, with smooth compactly supported
sections as a graph core. The vertical summand is the self-adjoint
regular tensor extension of \(D_{\mathcal Y}\). Applying (CI.31) to
graph approximants proves
\(\operatorname{Dom}L\subset\operatorname{Dom}D_{\mathrm{vert}}\) and
the displayed positivity on that domain. This is a global domain
estimate; no bounded-curvature assertion is required.

For a smooth section of \(\mathcal Y\) with compact fiber support, the
creation error for \(L\) and \(\mathscr D_M\) is the column formed by
its vertical derivative and its horizontal covariant derivative. It is
bounded: its fiber \(L^2\) norm is a continuous function on the compact
base, and finitely many charts bound its supremum. Such sections form a
dense homogeneous subspace of \(\mathcal Y\). Integration by parts in
the fiber and the base gives the adjoint creation error. In particular,
the distributional identity expresses \(\mathscr D_MT_\xi^*u\) as
\(T_\xi^*Lu\) plus that bounded error; the maximal-domain
characterization of \(\mathscr D_M\) gives the adjoint domain inclusion.
The first creation domain inclusion follows directly from the same
product rule. The product criterion, with (CI.31), now identifies the
class of \(L\) with \(y_V\otimes_C[\mathscr D_M]\).

The even globally defined form-degree phase \(U=i^N\) satisfies \[
 Uc_jU^*=f_j,\qquad Uf_jU^*=-c_j.
\] It turns the principal symbol of (CI.30) into \(\epsilon_n\) times
the \(J_-\) Dolbeault symbol. Both operators are odd and formally
symmetric. Their difference is a smooth zero-order endomorphism.
Interpolating that difference preserves the principal symbol, the
bounded-gradient exhaustion and all uniform localized elliptic
estimates. Lemma CI.4 consequently identifies their bounded-transform
classes even if that endomorphism is unbounded at fiber infinity.
Multiplication of an odd operator by \(-1\) gives the same even cycle
here by conjugation with \(\Gamma_0\), which commutes with the scalar
source. Thus the factor \(\epsilon_n\) has no effect. This proves the
first identity of (CI.29); multiplying by \(x_V\) and using its proved
inverse gives the second. \(\square\)

\textbf{Proposition CI.7 (the orientation line distinguishes \(J_+\) and
\(J_-\)).} Let \(\lambda=\det(V_{\mathbb C})\) and write
\(\lambda_Q=\pi^*\lambda\). Then \[
 [\bar\partial_{M,-}]
   =(-1)^n[\![\lambda_Q]\!]\otimes_B[\bar\partial_{M,+}].
 \tag{CI.32}
\] The bundle cycle \([\![\lambda_Q]\!]\in KK(B,B)\) is multiplication
on its section module, with operator zero and its usual even grading.
For oriented \(M\), \(\lambda\) is trivial, so the difference is
precisely \((-1)^n\).

\textbf{Proof.} In a frame put \(F_0=f_1\cdots f_n\) and \[
 U_0=F_0\Gamma_0^{\,n+1}.
\] The Clifford relations give \[
 U_0 f_jU_0^*=f_j,\qquad U_0c_jU_0^*=-c_j.
\] Indeed for odd \(n\) the first product commutes with every \(f_j\)
and anticommutes with every \(c_j\); for even \(n\) its two signs are
reversed and the extra \(\Gamma_0\) reverses both. It is a unitary of
degree \(n\) modulo two. Under an orthogonal frame change it is
multiplied by the determinant of that change. Thus it is a global
degree-\(n\) unitary between one spinor bundle and the other tensored
with \(\lambda_Q\); the real determinant transitions also show
\(\lambda_Q^{\otimes2}\cong\mathbf1\).

The \(J_+\) symbols are \(f_j,-c_j\) and the \(J_-\) symbols are
\(f_j,c_j\), so this unitary intertwines their principal symbols. Choose
a Hermitian connection on the line. The two transformed first-order
operators have the same principal symbol and differ by a smooth
symmetric zero-order term; CI.4 identifies their classes by
interpolation, with no global bound on that term needed. A degree-\(n\)
unitary reverses the grading when \(n\) is odd, giving \((-1)^n\). Here
is the line-tensor product check on the nonunital coefficient. Realize
\(\lambda\) by a smooth base projection \(p\in M_N(R)\); a finite
local-frame embedding and its Hermitian orthogonal projection give such
a projection. Its represented commutator with the cotangent Dirac is
bounded, since \(p\) depends only on the compact base. Remove the
bounded off-diagonal blocks of the amplified Dirac. The
large-imaginary-parameter Neumann inverse proves self-adjointness and
regularity of its \(p\)-compression, exactly as in the finite-projective
connection proof. Localized compactness follows from the resolvent
identity: for \(b\in C_c^\infty(V)\), the correction is a bounded
multiplication endomorphism commuting with \(b\), so its resolvent
correction has the original compact \(b\)-localized resolvent as its
first factor. Norm approximation gives every \(b\in B\). The first
operator on \(pB^N\) is zero; smooth compactly supported column
creations have bounded errors and adjoint errors. Domain inclusion and
positivity relative to that zero operator are automatic. The product
criterion identifies the compressed connection Dirac with the
line-tensor product. Adding the bounded represented connection
endomorphism gives the chosen line connection and the same class. This
verifies the tensor assertion and proves (CI.32). \(\square\)

The convention compatible with (CI.20) is consequently
\([\bar\partial_{M,-}]\). The \(J_+\) construction remains a valid
analytic class, but replacing it by that compatible class without the
line and parity correction would be false on odd or nonorientable bases.

\pandocbounded{\includegraphics[keepaspectratio,alt={The two cotangent conventions and the Clifford inverse}]{figures/KT-KK-15-cotangent-conventions.png}}

\emph{Figure CI.1.} The coordinate panels show the one-dimensional
horizontal/vertical model, with \(x\) to the right and \(\xi\) upward.
The two choices send the horizontal unit vector to opposite vertical
unit vectors. The displayed Clifford symbols are exactly (CI.14) and
(CI.28); the lower diagram is the all-dimensional identity (CI.29). The
rank-one Gaussian is the specialization of (CI.24), and its even grading
and source action are (CI.25). Formula (CI.32) gives the full parity and
determinant-line correction; the metric identifies
\(\det(TM_{\mathbb C})\) in the figure with \(\det(T^*M_{\mathbb C})\)
in the proof.

\subsection{\texorpdfstring{6. Spin\(^{c}\) Thom identities and the odd
Gaussian}{6. Spin\^{}\{c\} Thom identities and the odd Gaussian}}\label{spinc-thom-identities-and-the-odd-gaussian}

\textbf{Theorem CI.8.} Let \(M\) be closed and spin\(^{c}\), in any
dimension. Use the outward Thom class \(\tau_V\) and its inverse
\(\eta_V\) from \emph{Thom isomorphisms and K-orientations in KK},
including its ordered right-Clifford convention in odd dimension. Then
\[
 \tau_V\otimes_B[\bar\partial_{M,-}]=[D_M],\qquad
 \eta_V\otimes_R[D_M]=[\bar\partial_{M,-}].
 \tag{CI.33}
\] These are degree-\(n\) products. In even dimension the same formulas
hold with \(J_+\). In odd dimension, for the oriented base, replacing
\(J_-\) by \(J_+\) changes the right side of the first equation to
\(-[D_M]\).

\textbf{Proof in even dimension.} Put \(N=S\otimes L^{-1}\), where \(L\)
is the determinant line of the fixed spin\(^{c}\) structure. The actual
bundle and grading calculation in that lesson's Lemma 5.2 gives
\(\tau_V=[N^*]\otimes_C x_V\). Its doubled exterior module decomposes
under the source Clifford action as the source spinor \(N\) and the
positive position spinor \(S\). Thus \[
 N^*\widehat\otimes_C\Lambda^*V_{\mathbb C}\cong S
\] evenly, intertwining the position Clifford action. The transition
calculation includes both scalar characters: \(N\) has character
\(\lambda^{-1}\), so removing it leaves character \(\lambda\) on \(S\).
This is why replacing \(N\) by \(S\) would introduce the determinant
twist.

Choose compatible connections in this decomposition. Contracting the
\(N^*\) connection with the exterior connection leaves the spinor
connection on \(S\). Creations from smooth \(N^*\)-sections have bounded
errors, as do their adjoints; all bundles and their derivatives are
bounded on the compact base. The first operator is zero, so its domain
and positivity conditions are automatic. The product criterion
identifies \([N^*]\otimes_C[\mathscr D_M]\) with the spinor Dirac cycle.
A different compatible connection adds a bounded odd endomorphism and
gives the already proved CI.4 homotopy. Now (CI.29) gives the first
equation of (CI.33).

\textbf{The odd grading calculation.} Write \(n=2r+1\). In a positive
orthonormal spin frame let \(\Delta\) be the ungraded positive-volume
spinor, \[
 (-i)^r c_\Delta(e_1)\cdots c_\Delta(e_n)=1.
\] The outward Thom module is \(\pi^*\Delta\widehat\otimes Cl_1\), with
its last-factor grading and position operator
\(c_\Delta(\xi)\otimes\epsilon\), exactly as in the preceding Thom
lesson. After right-dilating the \(J_-\) Dolbeault cycle and tensoring,
the vertical position and derivative matrices on the product module are
\[
 A_j=c_\Delta(e_j)\otimes\Gamma_0\otimes\epsilon,\qquad
 B_j=1\otimes c_j\otimes1.
 \tag{CI.34}
\] The horizontal symbols are \(H_j=1\otimes f_j\otimes1\). The position
term is \(A_j\xi_j\), and the vertical differential term is
\(B_j(-i\partial_{\xi_j})\). All \(A_j,B_j\) are self-adjoint; distinct
matrices within each family anticommute, and the two families
anticommute. The vertical oscillator kernel consists precisely of the
Gaussian times the simultaneous \(-1\) eigenspace of \(iA_jB_j\). It is
a right \(Cl_1\)-module of rank \(2^r\), since each of the \(n\)
independent commuting constraints halves the matrix rank. Independence
follows either by tracing their products, which have zero trace for
every nonempty product, or by the creation/contraction recursion used in
(CI.24). Its even and odd subspaces have equal dimension, exchanged by
the right generator.

This kernel has the \textbf{positive} odd spinor convention. To verify
the branch, the constraints give \(A_jv=-iB_jv\). Moving the \(B\)'s
through the remaining \(A\)'s gives, for a constrained vector, \[
 \prod_jA_j\,v=(-i)^n\prod_jB_j\,v.
\] The exterior Clifford identity
\(\prod_j f_j=i^n(\prod_jc_j)\Gamma_0\), applied with the opposite
constraints on \(\Gamma_0v\), consequently gives \[
 \prod_jH_j\,v
   =\left(\prod_jc_\Delta(e_j)\right)\epsilon v,\qquad
 (-i)^r\prod_jH_j\,v=\epsilon v.
 \tag{CI.35}
\] This is precisely the right-Clifford positive-volume representation
on \(\Delta\otimes Cl_1\). It permits an even fiber unitary identifying
the kernel with that module.

The fiber unitary also respects the spin\(^{c}\) transitions. For a
plane rotation in the \(j,k\) plane, the infinitesimal action on the
product module is \[
 \tfrac12(c_\Delta(e_j)c_\Delta(e_k)+B_jB_k+H_jH_k).
\] On the kernel \(B_jB_k=-A_jA_k=-c_\Delta(e_j)c_\Delta(e_k)\); hence
this action is \(H_jH_k/2\), exactly the spin action in (CI.35). Plane
rotations generate the connected spin group; in rank one its two
elements act by the prescribed central sign. The scalar spin\(^{c}\)
character acts only on the original \(\Delta\), so it is the original
scalar character, with no additional line twist. The Gaussian is radial.
These observations prove the global kernel-bundle identification. Its
horizontal connection is the original spinor connection, up to a bounded
zero-order connection change on the compact base.

\textbf{Product recognition and removal of the complement.} The
preceding finite matrices also give the complete vertical oscillator
domain and gap, by the scalar oscillator proof. Parallel transport
preserves this oscillator and its Gaussian projection \(P\). Use the
tensor-compatible horizontal connection. Its odd horizontal operator
\(H\) anticommutes exactly with the vertical oscillator \(O\); this is
the same transport and Clifford argument as (CI.31). Use, for the second
\(B\)-source cycle, the connection-compatible horizontal/vertical Dirac.
Its principal symbol is the chosen \(J_-\) symbol, so CI.4's local
zero-order interpolation identifies it with the geometrically defined
cotangent Dolbeault cycle. This interpolation uses the source
\(B=C_0(V)\); it makes no claim of global compactness before the
position potential is added. Creations with compact fiber support give
bounded errors against either representative. The full product candidate
\(O+H\) is formally symmetric and elliptic for the connection metric.
The proper radial exhaustion proves its essential self-adjointness by
CI.1, including its unbounded position potential. On its smooth core, \[
 (O+H)^2=O^2+H^2,\qquad
 \|\,|\xi|u\|^2\leq C(\|(O+H)u\|^2+\|u\|^2).
 \tag{CI.36}
\] The last estimate follows from \(O^2=-\Delta_\xi+|\xi|^2\) plus the
bounded Clifford matrix term. Graph approximation extends it to the
closed domain, proving the position-operator domain inclusion. Its
anticommutator with the position term is \(2|\xi|^2\) plus a matrix of
norm at most \(n\); the horizontal part contributes zero. Thus the
product positivity form is bounded below by \(-n\).

Smooth compactly supported Thom sections have bounded creation errors
and adjoint errors by the vertical product rule and their horizontal
covariant derivatives. The graph core and maximal-domain argument in
CI.6 justify the adjoint domain inclusion. The product criterion
therefore identifies this candidate with
\(\tau_V\otimes_B[\bar\partial_{M,-}]\).

Its resolvents are globally compact. Localized graph compactness is
CI.1. The last estimate in (CI.36) bounds the \(L^2\) mass outside
\(|\xi|\leq R\) by \(C/R\) times the graph norm. Inside that ball a
fixed compactly supported cutoff has compact graph inclusion. Combining
the tail estimate with these compact local inclusions proves global
compact graph inclusion.

It remains to remove the horizontal operator on the complementary
oscillator summand; a norm limit of the entire bounded transform must
\textbf{not} be assumed. Put \(Q_s=sO+H\), \(s\geq1\). The same domain,
compactness and recognition checks hold. On a finite positive
\(s\)-interval the confining tail bound in (CI.36) is uniform, so the
resolvents are collectively compact globally: combine that tail with the
uniform localized Sobolev compactness of CI.4. The strong-resolvent
proof in that lemma then gives norm-continuous global compact resolvent
fields. Base smooth functions have uniformly bounded
parameter-continuous commutators. The Hilbert-module bounded-transform
theorem consequently gives a homotopy with the unital source \(R\),
rather than only a vanishing-at-infinity source assertion. On \(P\),
\(Q_s\) is the original odd spinor Dirac. On \(1-P\), the gap gives
\(|Q_s|\geq s\sqrt2\), by (CI.36). For a smooth base function \(a\),
\(P\) commutes with \(a\), \([O,a]=0\), and \([Q_s,a]=[H,a]\) is
uniformly bounded. On this complement the sign-resolvent integral gives
\[
 \|[\operatorname{sign}Q_s,a]\|
       \leq C\|[H,a]\|/s,\qquad
 \|F_{Q_s}-\operatorname{sign}Q_s\|\leq C/s^2.
 \tag{CI.37}
\] To check the first bound directly, use
\(\operatorname{sign}Q_s=\pi^{-1}\int_0^\infty((Q_s-it)^{-1}+(Q_s+it)^{-1})\,dt\)
in the strong sense. The commutator of each resolvent is the product of
two resolvents and \([H,a]\), with norm at most
\(\|[H,a]\|/(2s^2+t^2)\); its norm integral is at most \(C\|[H,a]\|/s\).
The second inequality is the scalar supremum of
\(|t/\sqrt{1+t^2}-\operatorname{sgn}t|\) on \(|t|\geq s\sqrt2\). Also
\(\|1-F_{Q_s}^2\|\leq(1+2s^2)^{-1}\) there. These bounds extend the
compact defect fields in norm to zero at \(s=\infty\).

The bounded transforms converge \textbf{strongly} on the complement to
\(\operatorname{sign}O\). Here is a domain justification. The
eigenspaces of \(O^2\) are finite-rank Hermite bundles preserved by the
horizontal connection and \(H\). Their smooth base sections form a dense
subspace. On a positive level \(O^2=\lambda\), \(\lambda\geq2\), \(H\)
is self-adjoint on the closed base, and \[
 F_{Q_s}=(sO+H)(1+s^2\lambda+H^2)^{-1/2}.
\] Scalar functional calculus of \(H\) on this level proves convergence
on its domain: the horizontal term is bounded on a vector by
\(\|Hu\|/(s\sqrt\lambda)\), and the vertical term converges to
\(\operatorname{sign}O\) by dominated convergence in the scalar spectral
calculus. Uniform bounds extend convergence to the dense sum of levels,
then to every vector. Finite-parameter strong continuity follows from
the common smooth core and resolvent identity, with a scalar
functional-calculus cutoff controlled on that core; the transforms are
self-adjoint, so their adjoints have the same continuity. Thus \(r=1/s\)
gives adjointable fields on the constant interval Hilbert module.

Together with (CI.37), this is an actual cycle homotopy to the Gaussian
spinor Dirac plus the complementary \(\operatorname{sign}O\). The latter
is an odd self-adjoint involution commuting with every base scalar,
hence exactly degenerate. The Gaussian calculation (CI.35) leaves CI.2's
positive odd Dirac, not its negative. This proves the first equation of
(CI.33) in odd dimension. The second follows by the actual inverse pair
\(\eta_V\tau_V=1_B\). Finally use (CI.32) and the orientation supplied
by the spin\(^{c}\) structure for the stated comparison with \(J_+\).
\(\square\)

\subsection{7. Elliptic cycles and quantization of the relative
symbol}\label{elliptic-cycles-and-quantization-of-the-relative-symbol}

\textbf{Proposition CI.9 (the full source action and symbol
homotopies).} Let \(P:E^+\to E^-\) be a classical elliptic
pseudodifferential operator of order zero on a closed manifold. Using
the composition, adjoint, Sobolev and parametrix proofs in
\href{KT-KK-11.html\#4-spectral-triples-and-the-circle}{\emph{Unbounded
Kasparov modules and spectral triples}, Lemmas 4.2a--4.2b}, its
normalized symbol defines \[
 [P]\in KK(C(M),\mathbb C),\qquad
 [\![\sigma_P]\!]\in KK(C(M),C_0(T^*M)).
 \tag{CI.38}
\] Both classes depend only on the stable homotopy class of the relative
principal symbol. Forgetting the base action gives
\([\sigma_P]\in K_0(C_0(T^*M))\); this scalar-source class cannot
replace \([\![\sigma_P]\!]\) in an identity with source \(C(M)\). The
scalar index pairing of \([P]\) is the Fredholm index of \(P\).

\textbf{Proof.} Write \(p(x,\widehat\xi)\) for the invertible principal
symbol on the compact cosphere, and put \(u=p(p^*p)^{-1/2}\). Smooth
functional calculus for the finite positive matrix is given by its
inverse-resolvent integral, so \(u\) is smooth. The path
\(p_s=(1-s)p+su\) stays invertible: it is \(p((1-s)1+s(p^*p)^{-1/2})\).
Its inverse and derivatives are uniformly controlled on the compact
cosphere and parameter interval. Choose an order-zero quantization \(T\)
of \(u\). On \(L^2(M,E^+)\oplus L^2(M,E^-)\), with the first summand
even, use \[
 F=\begin{pmatrix}0&T^*\\T&0\end{pmatrix}.
\] Its principal square is one, so \(F^2-1\) has order minus one. Its
commutator with a smooth scalar also has order minus one, by the cited
complete composition proof. Such operators map \(L^2\) boundedly into
\(H^1\), whose compact inclusion is proved there by finite chart/Fourier
truncation. These are therefore compact defects. The adjoint defect is
zero. Uniform approximation of continuous functions by finite chart
convolution extends the commutator condition to \(C(M)\). This defines
\([P]\).

Two quantizations of the same principal symbol differ by an
order-minus-one compact operator. Their linear interpolation has the
same principal symbol, hence gives a cycle homotopy. A smooth path of
unitary principal symbols gives a norm-continuous cycle homotopy by the
finite-seminorm operator bound in Lemma 4.2a. The polar path above is a
Fredholm path from \(P\) to \(T\): quantize \(p_s\), and at its first
endpoint correct by the compact difference from \(P\). The uniform
parametrix identity gives Fredholmness throughout.

Here is the elementary index constancy used for this path. For a
Fredholm map \(A:H_+\to H_-\), decompose the spaces as
\(\ker A\oplus(\ker A)^\perp\) and
\((\operatorname{ran}A)^\perp\oplus\operatorname{ran}A\). Its last block
is a bounded isomorphism. A sufficiently small perturbation retains an
invertible last block. Multiplication by invertible triangular block
operators removes its other off-diagonal blocks and leaves a map between
the two finite-dimensional defect spaces. The index of that finite map
is their dimension difference, independently of its rank. Thus the index
is locally constant on the Fredholm operators, and compactness of the
path interval makes it constant along the path. The index-pairing proof
identifies the scalar pairing of the displayed off-diagonal cycle with
\(\operatorname{index}T=\operatorname{index}P\).

The symbol is the relative triple \((\pi^*E^+,\pi^*E^-,u)\) on the
disc/sphere pair. The two inverse bundle constructions and homotopy
rules are proved in
\href{supporting/relative-k-foundations/relative-k-foundations.html\#rk-source-scope}{\emph{Relative
difference bundles, radial pairs and the planar normalization}, Theorems
RK.3--RK.4}. Their multiplication-cycle representative over
\(C_0(T^*M)\) retains the action of \(f\in C(M)\) by \(f\circ\pi\). This
action commutes with the odd comparison, whose square defect has compact
support. It gives \([\![\sigma_P]\!]\) with the asserted full source. An
everywhere invertible trivial comparison gives a degenerate cycle and a
multiplication-unitary analytic summand, so both constructions respect
stable addition.

For a continuous relative homotopy, use finitely many bundle charts on
the compact parameterized disc/sphere pair. Approximate its projection
and comparison matrices uniformly by smooth matrices using chart
convolution and a finite partition. Retract an approximate projection by
its spectral gap, and an approximate invertible comparison by polar
normalization; the compact sphere's positive invertibility bound keeps
these operations defined. Small interpolation preserves the endpoints up
to relative homotopy. A smooth connection, patched from local
connections, identifies the disc bundles with pullbacks of their
zero-section bundles by radial parallel transport. Hence every
stabilized relative homotopy can be quantized as the smooth path already
considered. This proves the claimed invariance and additivity. No
order-changing or nonclassical radial theorem is a premise. \(\square\)

For a smooth finite-rank graded Clifford coefficient bundle in
\textbf{even} dimension, its connection Dirac has symbol equal to its
coefficient tensor with (CI.20). Product recognition for that finite
coefficient uses the zero first operator, bounded connection errors on
the compact base, and the elliptic domain just proved. Equations (CI.29)
therefore prove the symbol identity for these Clifford-Dirac symbols and
their finite stable homotopies. The distinction between this conclusion
and an unrestricted generation assertion matters.

\subsection{8. Examples and solved local
calculations}\label{examples-and-solved-local-calculations}

\textbf{The circle.} Give \(\mathbb R/2\pi\mathbb Z\) its positive
coordinate and the periodic spin structure. CI.2's odd Dirac is
\(-i\,d/d\theta\). Its Fourier domain is
\(\{(a_m):\sum m^2|a_m|^2<\infty\}\). The bounded transform differs
compactly from \(2P_{\geq0}-1\). Compressing \(z=e^{i\theta}\) gives the
unilateral shift; its kernel is zero and cokernel is the constant mode.
The proved KK index pairing is therefore \[
 \langle[z],[D_{S^1}]\rangle=-1.
 \tag{CI.39}
\] This also solves the basic exercise and fixes the odd sign. With
\(J_-\), the outward odd Thom--Dolbeault product is this Dirac. With
\(J_+\), it is its negative, by CI.32--CI.33.

\textbf{Local duality on \(\mathbb R^2\).} Use the Bott algebra
\(D_2=C_0(\mathbb R^2)\widehat\otimes Cl_2\), its position cycle and its
Euclidean derivative inverse. The actual scalar Bott lesson proves their
first product by the even Gaussian and their reverse product by
simultaneous coordinate/Clifford rotation and the invertible-idempotent
argument. Tensoring its explicit \(Cl_2\)-matrix Morita correspondence
and its inverse gives the outward spinor pair
\(\tau_{\mathbb R^2},\eta_{\mathbb R^2}\). Their two products are the
identity classes on \(\mathbb C\) and \(C_0(\mathbb R^2)\). Hence
multiplication by one and by the other are inverse on K-theory and
K-homology in the local degree shift. The outward symbol is \(x+iy\);
two ordered positive right-odd line symbols give \(x-iy\) and the
opposite class, as proved in the Thom lesson's Proposition 7.2. This
solves the local-duality exercise with its exact orientation.

\textbf{The flat torus and its Thom identities.} On \(M=\mathbb T^2\),
use periodic spinors with chirality \(\sigma_3\) and \[
 D_M=-i(\sigma_1\partial_x+\sigma_2\partial_y).
 \tag{CI.40}
\] On Fourier mode \((k,l)\), its square is \(k^2+l^2\); its domain is
the corresponding weighted \(\ell^2\) space. Every nonzero mode is
invertible, while the zero mode has one vector of each parity. Its
scalar index is zero. The ordered product of the two positive circle
Diracs has \(k\sigma_1-l\sigma_2\), as proved in the preceding
two-circle product calculation; conjugation by \(\sigma_1\) changes this
to (CI.40) while reversing the grading. Thus that ordered product is the
negative of the outward torus Dirac, although both scalar indices are
zero.

The cotangent bundle is the trivial bundle
\(\mathbb T^2\times\mathbb R^2\). Here (CI.21) is a constant vertical
Fourier operator and all connection terms are zero. Its tensor sum
(CI.30) has mutually anticommuting constant horizontal and vertical
matrices. The sum-square Fourier identity proves its global domain
inclusion and positivity exactly as in (CI.31); compactly supported
fiber creations give the two bounded connection errors. The even phase
\(i^N\) identifies its symbol with \(J_-\). CI.29 proves both Clifford
identities. The determinant is trivial, and \(n=2\), so CI.32 identifies
\(J_-\) and \(J_+\) as classes. The spinor reduction in the even proof
of CI.8 gives \[
 \tau_{T^*\mathbb T^2}[\bar\partial_{\mathbb T^2}]
       =[D_{\mathbb T^2}],\qquad
 \eta_{T^*\mathbb T^2}[D_{\mathbb T^2}]
       =[\bar\partial_{\mathbb T^2}].
\] This solves the torus inverse-Thom exercise, with actual domain,
connection and orientation checks rather than fiberwise index equality.

\textbf{The projective line.} Give \(\mathbb {CP}^1\) its complex
orientation and canonical spin\(^{c}\) structure. Its anti-holomorphic
form spinor is \(\Lambda^{0,*}T^*\mathbb {CP}^1\), with determinant
\(T^{1,0}\mathbb {CP}^1=\mathcal O(2)\). The Hermitian metric identifies
the odd spinor line with \(T^{1,0}\); thus this is exactly the exterior
positive spinor convention of the complex Thom theorem. The Dolbeault
operator has the same principal Clifford symbol as the spinor Dirac
after an even constant form-degree phase. Their lower-order difference
is smooth and bounded on the closed base. CI.4 identifies their classes,
and a finite-projective connection tensor identifies a twist by
\(\mathcal O(k)\) with \([\mathcal O(k)]\otimes[D]\).

Here are direct analytic normalizations, without an index theorem. On
the affine chart \(z\in\mathbb C\), a holomorphic section of
\(\mathcal O(k)\) is an entire function \(h(z)\); on the other chart
\(w=1/z\) its coefficient is \(w^kh(1/w)\). Holomorphicity at \(w=0\)
means \(h\) is a polynomial of degree at most \(k\) if \(k\geq0\), and
means \(h=0\) if \(k<0\). To justify the polynomial conclusion, its
Laurent coefficients at infinity with degrees above \(k\) vanish; the
usual Cauchy integral coefficient formula follows by integrating the
geometric series on a circle and gives zero higher Taylor coefficients.
Thus \[
 \dim\ker\bar\partial_{\mathcal O(k)}=\max(k+1,0).
\] Here is the complex-analytic input in the preceding coefficient
argument. For a smooth function with \(\partial_{\bar z}h=0\),
integration of the two real derivatives over a rectangle, by the
one-variable fundamental theorem of calculus and Fubini, proves the
planar Stokes formula. Subdivision and approximation by rectangles prove
it for a disk with a smaller disk removed. Applied to \(h(z)/(z-a)\), it
identifies its outer-circle integral with the inner-circle integral,
which tends to \(2\pi i h(a)\) by continuity. This proves Cauchy's
formula. On a circle with \(|a|\) smaller than its radius the uniformly
convergent geometric expansion of \((z-a)^{-1}\) can be integrated term
by term. It gives the Taylor coefficient formula and the convergent
Taylor series. Equivalently the same argument in the other coordinate
gives the Laurent coefficients at infinity. Thus regularity of
\(w^kh(1/w)\) at zero makes every Taylor coefficient of \(h\) above
degree \(k\) vanish; when \(k<0\) it makes all of them vanish. This
supplies the analytic coefficient and Cauchy input rather than presuming
a formula-only reference.

The adjoint equation identifies its cokernel with holomorphic sections
of \(\mathcal O(-k-2)\). Locally write a \((0,1)\)-form as
\(a\,d\bar z\) times a bundle frame of metric squared norm \(h\).
Integration by parts against a test function gives the adjoint equation
\(\partial_z(ha)=0\). Therefore \(\overline{ha}\,dz\) is holomorphic.
Its transformation on overlap is the dual line transformation times that
of \(dz\), which is exactly
\(\mathcal O(-k)\otimes K=\mathcal O(-k-2)\). Conversely this formula
constructs every smooth adjoint-kernel form. Density and elliptic
regularity identify the closed-operator kernels with these smooth
solutions. Hence \[
 \operatorname{ind}\bar\partial_{\mathcal O(k)}
 =\max(k+1,0)-\max(-k-1,0)=k+1.
 \tag{CI.41}
\]

For this Dirac twist the symbol identity is already proved: its
lower-left principal symbol is the outward Thom symbol up to an even
constant phase, so
\([\![\sigma_{\bar\partial_{\mathcal O(k)}}]\!]=[\![\mathcal O(k)]\!]\otimes_R\tau_V\).
The finite-projective connection product and CI.33 give \[
 [\bar\partial_{\mathcal O(k)}]
 =[\![\sigma_{\bar\partial_{\mathcal O(k)}}]\!]\otimes_B
                                    [\bar\partial_{M,-}].
\] Thus the following derives from the proved KK symbol identity in this
case, without assuming the unrestricted theorem. The KK calculation
needs only two of the preceding analytic normalizations. The proved
relative-bundle calculations give \[
 K^0(\mathbb {CP}^1)=\mathbb Z[1]\oplus\mathbb Zu,\qquad
 u=1-[\mathcal O(-1)],\quad u^2=0.
\] Thus \([\mathcal O(k)]=1+ku\) for every integer \(k\): the inverse of
\(1-u\) is \(1+u\), and integer powers truncate after the linear term.
The two analytic values at \(k=0,-1\) give \(\langle1,[D]\rangle=1\) and
\(\langle u,[D]\rangle=1\). Bilinearity and the actual index-pairing
product yield \(\langle[\mathcal O(k)],[D]\rangle=k+1\). Equivalently
this is \(\int_{\mathbb {CP}^1}e^{kh}(1+h)=k+1\), with \(\int h=1\).
This proves the line-bundle Riemann--Roch formula and its KK pairing
interpretation. The embedding comparison proved below identifies that
same number with the embedding-defined topological index.

\subsection{9. A universal quantization
morphism}\label{a-universal-quantization-morphism}

The unrestricted identity is \[
 [P]=[\![\sigma_P]\!]\otimes_{C_0(T^*M)}
                    [\bar\partial_{M,-}]
 \tag{CI.42}
\] for every classical elliptic order-zero pseudodifferential operator
on every closed manifold. We prove it by constructing one quantization
morphism. This avoids any assertion that finite Clifford modules with
zero operator generate all graded K-classes.

It would indeed be false to assert that
\(K_0(\Gamma(\operatorname{Cl}(T^*M)))\) is always generated by finite
projective Clifford modules with operator zero. Already for \(M=S^1\),
its trivial Clifford algebra is \(C(S^1)\widehat\otimes Cl_1\), whose
graded \(K_0\) is \(K_1(C(S^1))=\mathbb Z\). On a finite projective
graded module over this trivial algebra right multiplication by the odd
Clifford generator is an odd self-adjoint involution that is also
right-module-linear, since that generator is central in the underlying
ungraded algebra; every finite compact cycle contracts its operator to
zero, and that zero cycle contracts to this degenerate involution. Thus
these particular compact zero-operator modules contribute zero. The
argument below acts on the original KK classes, including their nonzero
operators, and never makes this reduction.

Put \(H=L^2(M)\), using half-densities. For \(a\in C_c^\infty(T^*M)\),
let \(Q_t(a)\), \(0<t\leq1\), be its semiclassical quantization, with
phase \(e^{i(x-y)\xi/t}\) and factor \((2\pi t)^{-n}\). Use a finite
atlas, real functions \(\chi_j\) with \(\sum\chi_j^2=1\), two-sided
chart cutoffs and positive half-density identifications. At \(t=0\) its
value is \(a\). Add every norm-continuous compact-operator field
supported away from zero. Complete the algebra they generate in \[
 \|A\|=\max\bigl(\|A_0\|_\infty,
                         \sup_{0<t\leq1}\|A_t\|\bigr).
 \tag{CI.43}
\] Call this algebra \(\mathcal A_M\).

\textbf{Lemma CI.10 (the semiclassical algebra and its resolvents).}
This construction gives a separable C\(^*\)-algebra and a semisplit
exact sequence \[
 0\longrightarrow C_0((0,1],\mathcal K(H))
 \longrightarrow\mathcal A_M\xrightarrow{e_0}B\longrightarrow0.
 \tag{CI.44}
\] Evaluation \(e_1\) takes values in \(\mathcal K(H)\). Smooth bundle
endomorphisms acting by multiplication are multipliers of this algebra.
For a smooth self-adjoint matrix symbol \(a(x,\xi)\) of order one,
elliptic outside a fixed ball, its symmetric semiclassical quantizations
on a finite bundle define a regular self-adjoint operator on the
associated finite projective \(\mathcal A_M\)-module. Its value at zero
is multiplication by \(a\); its values at \(t>0\) are the elliptic
closed operators. Its inverse resolvents belong to the module compact
algebra. The same assertion applies to \(t\mathscr D_M\), with its
Clifford source and grading in (CI.26).

\textbf{Proof of the algebra assertions.} Here are the estimates that
justify the completion and prevent an implicit appeal to a groupoid
index theorem. In a chart, substitute \(y=x-tv\) in the Fourier kernel.
Taylor's formula with integral remainder in the two base variables,
followed by the integrations by parts in the complete composition proof
of Lemma 4.2a of the unbounded lesson, gives \[
 \|Q_t(a)Q_t(b)-Q_t(ab)\|\leq C_{a,b}t,
 \quad \|Q_t(a)^*-Q_t(a^*)\|\leq C_at.
 \tag{CI.45}
\] The constants use finitely many derivatives on fixed compact base and
frequency sets. Each first remainder contains a factor \(tv_j\); moving
\(v_j\) onto a frequency derivative extracts that factor \(t\). Further
integrations by parts make both Schur kernel integrals finite, precisely
as in that lemma. Terms with separated chart cutoffs have norm
\(O(t^N)\) for every prescribed \(N\), by repeated frequency
integration. Under coordinate change Taylor expansion of the new phase
gives the cotangent transformation as its leading term; its remainder
has the same \(t\) estimate. Thus finite chart sums satisfy (CI.45).
Those proofs also give a uniform bound by finitely many symbol
derivatives for symbols of order zero, not only those of compact
frequency support.

In fact \[
 \lim_{t\downarrow0}\|Q_t(a)\|=\|a\|_\infty.
 \tag{CI.46}
\] For the upper bound apply (CI.45) repeatedly to
\((Q_t(a)^*Q_t(a))^k\). Its limit superior is bounded by the
finite-seminorm bound for \(|a|^{2k}\). If \(L\) is the fixed number of
derivatives in that bound and \(r=\|a\|_\infty>0\), Leibniz's rule
bounds it by \(C k^L r^{2k-L}\) times a fixed polynomial in the first
\(L\) derivatives of \(a\) and \(r\). Taking the \(2k\)-th root and then
letting \(k\to\infty\) proves the upper bound \(r\). The zero case is
immediate. For the lower bound use a unit wave packet in a protected
chart, \(t^{-n/4}g((x-x_0)/\sqrt t)e^{i\xi_0(x-x_0)/t}\), cut off in
that chart. Substitution in the kernel and Taylor expansion show that
\((Q_t(a)-a(x_0,\xi_0))\) applied to it tends to zero in norm. The
errors are bounded by \(C\sqrt t\) times finitely many Gaussian moments;
its cutoff error tends to zero faster than any fixed power. The
elementary Fourier/Gaussian proof in the Bott lesson supplies these
integrals. Matrix-valued symbols are treated by adding a unit vector in
their finite fibre; the corresponding norm bound is the supremum of the
matrix norms. Taking points and vectors approaching the supremum proves
(CI.46).

At \(t>0\), compact frequency support makes the kernel smooth on the
closed manifold, hence compact by finite-rank approximation of a smooth
kernel. Dependence on a positive compact parameter interval is norm
continuous by the same kernel estimate. Equations (CI.45)--(CI.46) show
that every algebraic word in the generators differs, near zero in
operator norm, from the quantization of its zero-value symbol. If that
value is zero the field has norm tending to zero. Uniform approximation
proves the same statement for the completed kernel of \(e_0\). This
kernel is exactly \(C_0((0,1],\mathcal K(H))\), because all such fields
were included and compact fields away from zero are dense in it. The
image of \(e_0\) contains the compactly supported smooth symbols and is
closed, so is \(B\). Countably many dense smooth symbols together with
countably many compact fields generate a dense subalgebra. This proves
separability and exactness, without a full-versus-reduced groupoid
assumption.

For completeness the section in (CI.44) can be made explicitly
completely positive. In each chart extend half-densities by zero to
Euclidean space and use the Gaussian coherent vectors \(g_{z,\xi,t}\).
They satisfy the exact resolution
\(\int|g_{z,\xi,t}\rangle\langle g_{z,\xi,t}|\,dz\,d\xi/(2\pi t)^n=1\):
integrate first in \(\xi\) by Fourier unitarity, then in \(z\) by the
normalized Gaussian integral. Compress this resolution with the chart
multiplication \(\chi_j\). The positive quantization \[
 S_t(a)=\sum_j\int a_j(z,\xi)
 |\chi_jg_{z,\xi,t}\rangle\langle\chi_jg_{z,\xi,t}|
                         \frac{dz\,d\xi}{(2\pi t)^n}
\] uses the cotangent-coordinate expression of \(a\) inside the chart
and zero outside it. It is completely positive at every matrix level and
contractive, since the sum of the compressed resolutions is
\(\sum\chi_j^2=1\); using zero outside the chart can only decrease that
resolution. Choose the \(\chi_j\) supported strictly inside their
charts. On their supports the missing chart tails are exponentially
small. Gaussian convolution and Taylor's integral formula give
\(\|S_t(a)-Q_t(a)\|\leq C_a\sqrt t\) for compactly supported smooth
\(a\); separated-chart tails are estimated as above. The bound follows
by inserting the first derivatives in the same finite-seminorm kernel
estimate, with the first Gaussian moment \(O(\sqrt t)\). Thus
\((a,S_t(a))\) belongs to \(\mathcal A_M\). Contractivity extends this
section to every \(a\in B\), and matrix positivity extends by norm
closure. Its zero-value is \(a\). This proves semisplitting. No nuclear
lifting theorem is needed here.

\textbf{Proof of the multiplier and operator assertions.} Multiplication
on either side of \(Q_t(a)\) has leading symbol the product by the base
endomorphism, and its remaining norm is \(O(t)\) by (CI.45). This
constructs the multipliers and their adjoints. Represent a finite bundle
by a smooth projection \(p\in M_N(C(M))\), obtained by a finite
local-frame embedding. Its module is \(p\mathcal A_M^N\); evaluation
gives \(\pi^*E\) at zero and \(p\mathcal K(H)^N\) at positive times.

Symmetric quantization of \(a\) is self-adjoint on \(H^1\) for \(t>0\),
by the full elliptic maximal-domain and adjoint proof of Lemma 4.2b. Its
symbol bounds may depend on the fixed positive \(t\); that suffices away
from zero. Near zero put \(r_\pm=(a\pm i)^{-1}\). These are symbols of
order minus one with uniform estimates, since the matrix is self-adjoint
and is elliptic at large frequency. The same composition calculation
gives \((Q_t(a)\pm i)Q_t(r_\pm)=1+tR_t\), with \(\sup\|R_t\|<\infty\).
It gives the analogous identity on the other side. For small \(t\)
invert \(1+tR_t\) by its norm-convergent Neumann series. The true
inverse resolvent then differs in norm by \(O(t)\) from \(Q_t(r_\pm)\).
Frequency truncation of \(r_\pm\) changes its operator norm by
\(O(R^{-1})\), uniformly in small \(t\), using the order-minus-one
seminorm bounds. Hence this resolvent field belongs to
\(pM_N(\mathcal A_M)p\), with zero-value \(r_\pm\). At positive times
the graph/resolvent identity proves norm continuity; it is compact there
by the closed Sobolev inclusion. Patching the small and positive
parameter intervals gives both module inverse resolvents.

Their ranges are dense in the projective module. Smooth
compact-frequency quantized columns, together with compact-field columns
away from zero, form a dense set. Applying \(Q_t(a)\pm i\) to such a
column again belongs to the module: the composition leading symbol is
\((a\pm i)b\), still of compact frequency support, and the remainder is
a compact field with norm tending to zero. Smooth columns away from zero
are dense by the closed elliptic graph core. Thus each dense generating
column is in the resolvent range. Their mutually adjoint inverse
identities prove regular self-adjointness by the earlier resolvent
lemma. This argument proves the assertion for bundle matrices as well,
since chart connections change only bounded lower-order terms with their
semiclassical factor \(t\).

For \(t\mathscr D_M\) the zero symbol is exactly \(c(\xi)\), its grading
is \(\epsilon_n\Gamma_0\), and its source is \(f\). The graded
commutator with a smooth Clifford section is \(t\) times its bounded
covariant-derivative multiplier. It therefore defines a bounded
adjointable field, including value zero at \(t=0\). The projective
module compact inverse resolvents just proved give a cycle with the full
Clifford source. For a general scalar-source matrix symbol, its scalar
commutator is \(t\) times an order-zero semiclassical operator,
uniformly bounded by the proved estimate, and hence a multiplier. All
source algebras used here are separable and all projective modules
countably generated. Their coefficient actions are nondegenerate because
radial compact-frequency cutoffs approximate every module element; the
displayed scalar-base and Clifford source units act as identity, so
these geometric source representations are essential. No essentiality
condition is imposed on the arbitrary original cycles tested in (CI.49).
This finishes the lemma. \(\square\)

In dimension zero the manifold is a finite set. The same algebra has a
diagonal boundary fibre and a full finite-matrix positive fibre; its
diagonal section is completely positive. There is no cosphere to
normalize. Finite bundle zero-operator cycles and their constant
semiclassical lifts give the assertions directly; the order-one
condition outside a fibre ball is vacuous. This includes arbitrary
finite-dimensional elliptic index defects. If the manifold is empty,
every relevant symbol, operator and index class is zero, and the
assertions hold directly; assume it nonempty whenever the Hilbert-space
Morita equivalence is used below.

The ideal in (CI.44) is a cone, whose identity is zero by the actual
cone contraction of Lesson 10 Proposition 7.2. The semisplit six-term
proof in Lesson 14 Theorems 4.1--4.2 therefore makes \([e_0]\)
invertible in KK. Explicitly, covariant exactness with source \(B\)
gives \(z\in KK(B,\mathcal A_M)\) with \(z[e_0]=1_B\); exactness with
source \(\mathcal A_M\) then gives \([e_0]z=1_{\mathcal A_M}\), since
its difference from the identity becomes zero after multiplication by
\([e_0]\). These are separable instances of the already proved sequence.
Let \(m_H\in KK(\mathcal K(H),\mathbb C)\) be the explicit Hilbert-space
Morita class and define \[
 \mathfrak a_M=z\otimes_{\mathcal A_M}[e_1]
                       \otimes_{\mathcal K(H)}m_H\in KK(B,\mathbb C).
 \tag{CI.47}
\]

\textbf{Theorem CI.11 (all symbols, including the odd Clifford
classes).} The quantization morphism (CI.47) is
\([\bar\partial_{M,-}]\), and (CI.42) holds with its full \(C(M)\)
source.

\textbf{Proof.} The Clifford-source family \(t\mathscr D_M\) in CI.10 is
a class \(d\in KK(C,\mathcal A_M)\). Its two evaluations are exactly \[
 d[e_0]=x_V,
 \qquad d[e_1]m_H=[\mathscr D_M].
\] The first equality includes the grading and source matrices
of(CI.20), not merely its scalar index. Multiplying by \(z\) shows
\(d=x_Vz\). Consequently \(x_V\mathfrak a_M=[\mathscr D_M]\). The actual
inverse \(y_V\) gives \[
 \mathfrak a_M=y_V[\mathscr D_M]
                     =[\bar\partial_{M,-}]
 \tag{CI.48}
\] by CI.6. This is an equality of KK morphisms, so it is tested against
every graded Clifford class at once.

For the arbitrary elliptic symbol of CI.9, smooth its radial extension
near zero and let \(a\) be an odd self-adjoint order-one matrix on
\(E^+\oplus E^-\), with lower-left part \(r u(x,\widehat\xi)\) above
radius two, where \(r=|\xi|\). Choose it zero near zero. Lemma CI.10
supplies its bounded-transform class \(k\in KK(R,\mathcal A_M)\). At
zero it is the relative symbol cycle: its sphere comparison is \(u\),
and its radial normalization is the relative construction in RK.3--RK.4.
Thus \(k[e_0]=[\![\sigma_P]\!]\). At one its elliptic principal
comparison is \(u\), so CI.9 identifies \(k[e_1]m_H=[P]\). Invertibility
of \([e_0]\) gives \(k=[\![\sigma_P]\!]z\). Multiplication by
\([e_1]m_H\) and (CI.48) proves (CI.42).

More explicitly, for \emph{any} \(v\in K_0(C)\), represented by its
actual graded Kasparov cycle, its Thom image is \(vx_V\), and the proof
gives \[
 \operatorname{Quant}(vx_V)=v[\mathscr D_M].
 \tag{CI.49}
\] The relative-bundle construction of Theorems RK.3--RK.4 represents
\(vx_V\in K_0(B)\) by some stabilized elliptic sphere symbol; CI.9 and
the preceding family prove that its quantization is independent of that
choice. No assertion about a finite-projective representative of \(v\)
is made or required. The same equality holds in degree one by retaining
one right \(Cl_1\) factor throughout: tensor the algebra, both
evaluations, their inverse and every projective resolvent construction
with that finite-dimensional graded factor. Their inverse-product
identities are the literal dilations of the displayed identities, with
the preceding lesson's fixed right-factor order. This proves
compatibility in both graded K-degrees without a suspension-generation
shortcut. In particular on an odd spin\(^{c}\) base take an actual
unitary \(u\in M_k(R)\) and its positive-boundary odd class.
Associativity and CI.33 give
\(\operatorname{Quant}([u]\tau_V)=[u][D_M]\). The complete odd-product
proof of Lesson 10 Theorem 6.2 identifies the latter with the Fredholm
compression \(\operatorname{index}(P\pi(u)P)\),
\(P=(1+\operatorname{sign}D_M)/2\), with the finite kernel assigned to
one spectral half as in that theorem, so that its involution
representative and projection have the stated sign. This applies to
every stabilized unitary homotopy, not only Clifford coefficient bundles
with zero operator. The unitary \(z\) on the circle gives \(-1\) as in
(CI.39). For non-spin\(^{c}\) and nonorientable bases(CI.49), as an
equality on all original graded cycles, supplies exactly the same
universal compatibility. \(\square\)

\subsection{10. Tubular extension and the embedding
index}\label{tubular-extension-and-the-embedding-index}

We first supply the precise excision needed for a compactly supported
index. It does not presume the general open restriction theorem stated
at the end of the lesson.

\textbf{Lemma CI.12 (compactly supported pairing and local Dirac
comparison).} Let two symmetric elliptic Dirac-type operators on
complete manifolds be identified on an open set \(U\), with the same
principal Clifford symbol there. Their lower-order difference may be any
smooth symmetric endomorphism. Their bounded-transform cycles restricted
to \(C_0(U)\) have the same class after identifying the spinors on
\(U\). Consequently their pairings with a finite relative symbol
supported compactly inside \(U\) agree. It suffices that the operators
have the complete exhaustion and localized resolvent properties of CI.1.
Completeness and a global bound for the lower-order difference are not
required outside the comparison region.

\textbf{Proof.} Compress a bounded transform \(F\) to \(L^2(U)\), using
the orthogonal measurable projection \(P_U\). For
\(j\in C_c^\infty(U)\), choose \(\chi\in C_c^\infty(U)\), equal to one
on its support. Since \(\chi(1-P_U)=0\), \(jF(1-P_U)=-j[F,\chi](1-P_U)\)
is compact. Thus the compressed transform is a cycle: its localized
square defect is the original compact defect plus \(jP_UF(1-P_U)FP_U\);
its commutators are the compressed compact commutators. The off-diagonal
blocks in the decomposition \(L^2(U)\oplus L^2(U^c)\) are source-locally
compact by the same calculation. Removing them is the proved locally
compact perturbation rule; the second block has zero source and is
degenerate. Hence compression gives exactly the restriction class, not a
choice of boundary condition for an incomplete differential operator.

For a protected compactly supported \(\chi\), regarded as a map between
the two Hilbert spaces, put \(E=D_1\chi-\chi D_2\). Equality of
principal symbols makes \(E\) a bounded compactly supported zero-order
map. The maximal domains and the local product rule give \[
 (D_1-z)^{-1}\chi-\chi(D_2-z)^{-1}
       =-(D_1-z)^{-1}E(D_2-z)^{-1}.
 \tag{CI.50}
\] This identity holds initially on the graph cores and extends by
boundedness. For \(z=\pm i\rho\) its norm is at most \(\|E\|/\rho^2\);
it is compact by the compact-support inverse-resolvent property on the
support of \(E\). Express \(D(D^2+\rho^2)^{-1}\) as half the sum of the
two inverse resolvents. Integrating (CI.50) with \(\rho=\sqrt{1+s^2}\)
in the actual bounded-transform integral proves \(F_1\chi-\chi F_2\)
compact: the compact error integral converges in norm by
\(\|E\|/(1+s^2)\), while the uncommuted integrals converge strongly as
proved earlier. Moving a compactly supported \(j\) past a further cutoff
proves that the two compressed transforms differ by a two-sided
\(C_0(U)\)-locally compact operator. The perturbation rule proves
equality of their classes.

Here is the support assertion for the first factor. A relative symbol is
represented by a finite section module with an odd multiplication
operator \(G\), whose square is exactly one outside a compact set \(K\).
Choose \(U'\Subset U\) containing \(K\). The open interval support
module on \(([0,1]\times U')\cup((0,1]\times X)\) joins its restricted
cycle to the original cycle over \(C_0(X)\). Complete compactly
supported section vectors for the coefficient-valued inner product.
Finite charts prove continuity of their inner products; a countable
chart/cutoff family gives countable generation. The multiplication
operator preserves this module, and its defects stay inside
\([0,1]\times K\), so finite bundle charts give compact module defects.
Evaluation is dense at both endpoints by interval cutoffs. This is the
multiplication case of the explicit support homotopy in the zero-section
comparison, with its proof included here. The restricted cycle is a
class over \(C_0(U')\), followed by extension by zero. Associativity now
reduces its two pairings to the equal restriction classes just proved.
No complemented Hilbert-module support is used. \(\square\)

We shall also use the following geometric consequence of CI.4. On a
second-countable manifold, two complete Dirac-type constructions whose
Clifford spinor structures are connected by a smooth homotopy give the
same scalar-source class, provided their gradings and determinant lift
are transported along that homotopy. To justify the required global
control, use the complete reference connection metric and its proper
bounded-gradient function \(h\) already constructed on the cotangent or
normal bundle in question. For a compact parameter family of smooth
principal symbols choose a smooth positive multiplier \(w\leq1\) making
all their symbol norms relative to that metric uniformly bounded. On
successive compact shells take the reciprocal of an upper bound for the
finitely many symbols, and combine smaller positive bounds by a locally
finite partition. Thus \([\frac12(wD_s+D_sw),h]\) is bounded uniformly
in \(s\). Formal symmetry is retained. CI.1 proves essential
self-adjointness, with the common compactly supported graph core; CI.4's
proof, using the uniform exhaustion and elliptic estimates on each
compact set, gives a genuine interval-module homotopy. Its proof uses
only local bounds for the zero-order coefficients. Rescaling an endpoint
to this \(w\) is handled by the positive path \(w_s=(1-s)+sw\); if that
endpoint has a bounded-gradient exhaustion, the commutator with it
remains uniformly bounded since \(w_s\leq1\). Therefore the rescaling
does not change its class. If prescribed agreement is needed near one
compact set, take \(w=1\) there and use a larger uniform bound on that
set. This proves the consequence; it is not an assertion that all almost
complex structures are homotopic.

Every closed manifold has an embedding into some finite Euclidean space,
by the following elementary construction. Choose a finite coordinate
cover \(\psi_j:U_j\to\mathbb R^n\) and smooth nonnegative cutoffs
\(\chi_j\) supported inside the charts, with at least one
\(\chi_j(x)>0\) at every point. The map
\(x\mapsto(\chi_j(x),\chi_j(x)\psi_j(x))_j\) is smooth after zero
extension. Equality of two images, using an index with positive cutoff,
recovers the same chart coordinates by division, so implies equality of
the points. A vector in its differential kernel has both \(d\chi_j=0\)
and \(\chi_jd\psi_j=0\) for that index, so is zero. Compactness makes
this injective immersion a homeomorphism onto its image, and the same
chart ratios give its local smooth inverse. It is therefore an
embedding, without a separate embedding theorem premise. Fix any
\(i:M\hookrightarrow\mathbb R^N\) and let \(\nu\) be its real Euclidean
normal bundle, of rank \(q=N-n\). Its cotangent embedding sends
\((x,\xi)\) to the covector extending \(\xi\) by zero on \(\nu_x\). The
normal bundle of this embedding is \[
 E=\pi^*(\nu\oplus\nu^*)\longrightarrow V=T^*M.
 \tag{CI.51}
\] Use the metric identification of \(\nu^*\) and \(\nu\), and give this
doubled normal the complex coordinate \(u+i\eta\). Its positive exterior
spinor is \(\Lambda\nu_{\mathbb C}\), with grading form parity and
Clifford position \(c(u)+f(\eta)\). Its determinant is
\(\det\nu_{\mathbb C}\), which can be nontrivial. The bundle and spinor
exist under every \(O(q)\) transition. We retain the outward complex
Thom class \(\tau_E\). The rank-normalized right-product class is
\(\epsilon_{2q}\tau_E=(-1)^q\tau_E\), and would require the
corresponding explicit conversion. The scalar normal complex coordinate
is specified independently of the \(J_-\) cotangent Dirac symbol, which
in the corresponding normal directions is \(f(du)+c(d\eta)\). That
distinction is necessary for the even Gaussian below.

Define \(i_*^{\rm cot}\) by this Thom class followed by a tubular
identification and extension by zero into \(W=T^*\mathbb R^N\). The
normal Thom cycle over the noncompact base \(V\) is legitimate: the
defect after multiplication by \(b\in C_0(V)\) is
\(-b/(1+|u|^2+|\eta|^2)\), a finite bundle endomorphism vanishing at
infinity on \(E\). The exact two inverse resolvents of the position
multiplier and compactly supported section vectors prove its regularity
and countability. No compact-base Thom theorem is silently extended.

We give explicit tubular and determinant checks. A metric normal tubular
map \(\nu\supset\{|u|<\rho(x)\}\to\mathbb R^N\) exists by the inverse
function theorem applied to \(i(x)+u\); compactness of \(M\) supplies a
uniform smaller radius and injectivity. Indeed failure of injectivity at
arbitrarily small radii would have a subsequence limiting to a
zero-section point, contradicting its local inverse; separated limiting
base points contradict injectivity of \(i\). A metric connection
identifies \(T^*\nu\) with \(\pi^*T^*M\oplus\nu\oplus\nu^*\), hence with
\(E\) as a manifold. The cotangent lift of the tubular map is a
diffeomorphism onto \(T^*U\subset W\). Replacing its normal disk by all
of \(\nu\) via \(u\mapsto\rho u/\sqrt{1+|u|^2}\) gives a global
diffeomorphism \(T^*\nu\to T^*U\); its cotangent lift also transports
\(\eta\) by the inverse transpose. The absolute half-density Jacobian is
positive and introduces no orientation character. Near the zero section
its normal derivative is a positive radial rescaling, homotopic to the
identity. Thus its relative Thom class is the displayed outward one.
Changing the radius or connection gives the corresponding smooth
positive radial and affine-connection paths, with compact supports
carried inside a fixed smaller tubular region for each compactly
supported class. The support homotopy of CI.12 proves equality after
extension. The graph, tubular and support comparisons in
\href{KT-KK-16.html\#3-graph-symbols-and-bounded-families}{\emph{Wrong-way
maps for K-oriented maps}, Sections 2--4} give the same outward class
for a direct cutoff definition; its outward diagonal spinor is
\(c(\xi)+i\widehat c(\eta)\) with displacement \(\eta=-u\). Its
explicitly different literal diagonal reading requires the proved parity
and \(\det(TM_{\mathbb C})\) correction. That alternative is not used in
this definition.

\textbf{Lemma CI.13 (the doubled-real-normal Gaussian).} Give \(E\) the
split \(J_-\) Clifford spinor
\(\pi_E^*S_V\widehat\otimes\Lambda\nu_{\mathbb C}\), where
\(S_V=\Lambda^{0,*}T^*V\). Choose compatible base and normal connections
and the complete connection metric. Its Dirac-type class \(d_E\)
satisfies \[
 \tau_E\otimes_{C_0(E)}d_E=[\bar\partial_{M,-}]
                               \quad\hbox{in }KK(B,\mathbb C).
 \tag{CI.52}
\] This includes the full noncompact \(B\) source and a trivial even
Gaussian line, even when \(\nu\) is nonorientable.

\textbf{Proof.} Reorder the even-dimensional base spinor to the first
tensor position by the actual graded tensor permutation. In each normal
frame define the operator on the two normal spinors by \[
 O_\nu=c_\tau(u)+f_\tau(\eta)
       -i\Gamma_\tau\bigl(f_D\partial_u+c_D\partial_\eta\bigr).
 \tag{CI.53}
\] The global vertical term is \(O=\Gamma_V\otimes O_\nu\), the first
position term is \(X_E=\Gamma_V\otimes(c_\tau(u)+f_\tau(\eta))\), and
the horizontal term \(H\) is the base Dolbeault connection operator. The
grading is \(\Gamma_V\otimes\Gamma_\tau\otimes\Gamma_D\). These base
grading operators are essential: writing the vertical term as an
ordinary tensor with identity would make it commute with the horizontal
Clifford matrices. All subsequent \(sO+H\) formulas use this global
\(O\). The two normal spinor factors both have their actual exterior
gradings. On the scalar Gaussian in the \(2q\) real variables, the
matrix constraints are \((c_{\tau,j}+i\Gamma_\tau f_{D,j})w=0\) and
\((f_{\tau,j}+i\Gamma_\tau c_{D,j})w=0\). Their solution is the unit
vector \[
 w=2^{-q/2}\sum_{I}
       (-1)^{|I|(|I|-1)/2}e_I\otimes e_I .
 \tag{CI.54}
\] Creation/contraction determines its coefficients recursively from the
vacuum coefficient: adjoining the \(k\)-th ordered index introduces
\((-1)^{k-1}\). It proves both constraints and that there is no other
solution. Alternatively squaring gives the scalar \(2q\)-oscillator plus
\(2q\) commuting Clifford involutions; every nonempty product has trace
zero, so their simultaneous lowest eigenspace is one-dimensional. The
scalar ground function is \(\pi^{-q/2}\exp[-(|u|^2+|\eta|^2)/2]\); the
higher Hermite levels have gap at least two. The complete weighted
domain and compact fibre inclusion follow by the finite matrix
diagonalization and actual oscillator proof used in CI.5. In (CI.54)
both factors have the same exterior degree, so the Gaussian is even.
Each degree's sum \(\sum_{|I|=k}e_I\otimes e_I\) is the metric's
canonical exterior tensor and is \(O(q)\)-invariant. This proves the
actual global trivial line, including orientation-reversing transitions.
In particular the normal determinant character has not been dropped or
replaced by an orientation assumption.

When \(q=0\), the normal spinor and Gaussian are the even scalar line,
\(O=0\), and the kernel complement is absent; (CI.52) is the identity
product. Otherwise the gap just proved is uniform. Parallel transport
preserves \(O\), its matrices and its radial Gaussian. Horizontal
Clifford matrices anticommute with it. Therefore on the smooth compact
core \[
 \| (sO+H)v\|^2=s^2\|Ov\|^2+\|Hv\|^2,
                   \qquad s\geq1.
 \tag{CI.55}
\] The complete connection metric has a proper bounded-gradient
exhaustion, obtained by adding the proper base exhaustion in CI.3 to
\(\sqrt{1+|u|^2+|\eta|^2}\). Normal parallel transport preserves the
latter. CI.1 proves essential self-adjointness and a common graph core
for these symmetric Dirac-plus-potential operators. Formula (CI.55) and
the weighted normal oscillator estimate imply the required domain
inclusion for the first normal position operator. Its product positivity
is \(2\operatorname{Re}\langle(O+H)v,X_Ev\rangle\geq-2q\|v\|^2\): the
horizontal term anticommutes with position and the vertical oscillator
bracket is \(2|z|^2\) plus the bounded Clifford matrix. Compactly
supported smooth normal-spinor creations have bounded position and
covariant-derivative errors, and integration by parts gives their
adjoint errors and maximal-domain inclusions. These are a dense creation
domain. Thus the actual unbounded product criterion identifies \(O+H\)
as the product in (CI.52).

For every compact base support its graph unit ball has a small normal
tail, by the \(|z|^2\) oscillator bound in (CI.55); on bounded normal
balls the local elliptic estimate and finite Fourier truncations give
compactness. Inserting a compact base cutoff in the resolvent first, and
then this two-stage truncation, proves source-localized compactness of
inverse resolvents. Approximating arbitrary \(b\in B\) by such cutoffs
gives every cycle defect. This is full Hilbert-space compactness after
source localization, not fibrewise compactness alone.

To remove the complement use the strict endpoint proof of CI.8 with the
same estimates, now localized by \(b\in C_c^\infty(V)\). Since
\([O,b]=0\) and \([H,b]\) is bounded, the resolvent integral on the
complementary oscillator gap gives
\(\|[\operatorname{sign}(sO+H),b]\|\leq C_b/s\) and the
bounded-transform/sign difference at most \(C/s^2\). Compact defects
therefore tend to zero in norm after localization. On positive compact
parameter intervals the localized graph balls have the uniform
base/normal compactness just proved; strong resolvent continuity on the
common core and finite-rank approximation give norm-continuous compact
fields. On each finite Hermite level the horizontal operator is a
finite-bundle Dirac on the complete base \(V\); CI.1 proves its maximal
self-adjoint domain, with no curvature bound required. Its scalar
spectral calculus proves strong convergence of the complementary
transform to \(\operatorname{sign}O\), as in CI.8. Density of those
levels and uniform bounds extend it to the whole Hilbert space and to
the adjoints. Thus \(s=1/r\) gives an actual constant interval Hilbert
module, with norm-continuous localized compact defects at \(r=0\). The
complementary endpoint is degenerate, since \(\operatorname{sign}O\) is
an odd involution commuting with the base source. On the even invariant
Gaussian the horizontal operator is exactly the original base Dolbeault
operator; its normal connection vanishes on(CI.54). This proves (CI.52).
\(\square\)

We check that the ambient cotangent structure is the one in this lemma
up to a genuine spinor homotopy, rather than assuming a product complex
structure under a tubular map. The connection coordinates on
\(T^*\nu=E\) have canonical one-form \(\xi\,dx+\eta\,\nabla u\). Its
exterior derivative differs from the split base-plus-normal symplectic
form by the horizontal two-form \(\langle\eta,R_\nu u\rangle\), and by
the connection changes already present in those coordinates. The
curvature two-form has only two horizontal base arguments. Addition of
any multiple of it preserves nondegeneracy: the \(dx\)-\(d\xi\) and
\(du\)-\(d\eta\) pairings still force a vector in its nullspace to have
all four components zero. Writing this horizontal two-form as \(A\), the
shear \(T_s\) adds to the cotangent-vertical component of a horizontal
vector \(X\) the covector \(a_X(Y)=sA(X,Y)/2\), leaving its other
components fixed. Thus \(T_s^*\omega_0=\omega_0+sA\), since the two new
cross terms are \(a_X(Y)-a_Y(X)=sA(X,Y)\). The transported complex
structure is \(T_s^{-1}J_-T_s\). It is triangular with diagonal
identity, so its determinant is one and it introduces no line or grading
reversal. Transporting the split \(J_-\) by these shears gives a
compatible almost complex homotopy to one compatible with the canonical
cotangent form. Two compatible almost complex choices are connected
without an existence theorem: interpolate their positive metrics and put
\(J=A(-A^2)^{-1/2}\), \(A=g^{-1}\omega\), choosing the sign of
\(\omega\) consistent with \(J_-\). The positive square root exists
smoothly because \(-A^2\) is positive and invertible in the metric, as
follows by diagonalizing its finite skew-adjoint matrix. This produces
the desired compatible path.

The cotangent lift of a diffeomorphism preserves the canonical one-form
by its definition as the inverse-transpose action on covectors. Thus the
tubular pullback of the ambient \(J_-\) has just this compatible
homotopy. Exterior spinors transported along it have the same
determinant lift and grading. For an explicit transport first use
positive square-root metric identifications on the complexified
cotangent bundle and let \(p_s\) be the orthogonal projection onto the
anti-holomorphic subspace. The pointwise equation
\(\dot U_s=[\dot p_s,p_s]U_s\), \(U_0=1\), has a smooth unitary
solution: its coefficient is skew-adjoint, and differentiating
\(U_s^*p_sU_s\) gives zero by differentiating \(p_s^2=p_s\). On each
relatively compact chart this is the uniformly convergent ordinary
matrix integral iteration; uniqueness glues the solutions. Its exterior
powers are even unitary spinor identifications. Their determinants
transport the actual line, rather than discarding it. Positive
metric/shear identifications likewise do not reverse parity. The
geometric homotopy consequence after CI.12 makes their complete
Dirac-type classes equal. For comparison with the incomplete open
ambient region only its germ near a given compact support is needed:
retain its pulled-back metric and connection near that compact set,
replace them outside a larger compact set by complete split choices, and
interpolate. Choose a compactly supported cutoff equal to one on that
protected set and evaluate the preceding almost complex path at this
cutoff; it is the pulled-back structure there and the split structure
outside a larger compact set. Interpolate positive metrics there and
make them Hermitian by \(g\mapsto(g+g(J\cdot,J\cdot))/2\). The latter
operation is bounded below by \(g/2\). This constructs the smooth metric
and Clifford-compatible connection choices; their metric is bounded
below by a positive multiple of the complete split metric, since the
altered region is compact. Consequently they have the required
exhaustion. CI.12 identifies their restricted bounded-transform classes
with the ambient one. This proves the needed tubular comparison
including the connection, domains and determinant line.

To see explicitly that this comparison applies to \emph{every} compactly
supported input, let \(a\in K_0(B)\) be represented by a finite relative
multiplication cycle \(F_a\), a self-adjoint contraction exactly unitary
outside a compact base set. Represent the normal Thom cycle by \(F_E\)
exactly unitary outside a fixed normal ball; use a smooth radial cutoff
angle. On their finite tensor section module the multiplication operator
\[
 G=F_a\sqrt{1-F_E^2}+\Gamma_a F_E
 \tag{CI.56}
\] has defect \((F_a^2-1)(1-F_E^2)\), supported in a compact subset of
\(E\). It is the actual product: its creation error relative to \(F_E\)
is multiplication by a matrix vanishing at normal infinity and
multiplied by a base section vanishing at base infinity, hence is module
compact; the adjoint condition is the same. Its anticommutator with the
first operator is \(2\sqrt{1-F_E^2}F_a^2\geq0\). The bounded product
criterion therefore identifies it with \(a\tau_E\). This also proves
proper support, instead of deducing it from pointwise Gaussian indices.
CI.12 restricts(CI.56) to a protected compact tubular region, where the
ambient and complete pulled-back cycles agree. The spinor homotopy and
(CI.52) give \[
 \bigl\langle i_*^{\rm cot}(a),[\bar\partial_{\mathbb R^N,-}]\bigr\rangle
       =\langle a,[\bar\partial_{M,-}]\rangle
                         \quad\text{for every }a\in K_0(B).
 \tag{CI.57}
\] This is the required doubled-normal Thom/tubular/extension pairing
equality. It is proved on all relative K-classes, and makes no finite
Clifford zero-operator generation claim.

Finally let \(\beta_N\in K_0(C_0(\mathbb R^N\oplus(\mathbb R^N)^*))\) be
the outward complex Bott position cycle with coordinate \(x+i\xi\),
exterior-degree grading and symbol \(c(x)+f(\xi)\). Its inverse
\(\eta_N\) is the actual outward inverse of the preceding Thom theorem.
Taking the base to be a point in the matrix/domain proof
(CI.53)--(CI.54) gives \(\beta_N[\bar\partial_{\mathbb R^N,-}]=1\): the
single Gaussian is even and the complementary sign is degenerate.
Inverse uniqueness therefore gives
\([\bar\partial_{\mathbb R^N,-}]=\eta_N\). No ordered right-odd product
has replaced \(\beta_N\); using \(\epsilon_{2N}\beta_N=(-1)^N\beta_N\)
would require the same change of the inverse and would define a
different unconverted normalization.

\textbf{Theorem CI.14 (the embedding-defined index).} With these
explicit outward doubled-normal and Euclidean conventions, set \[
 \operatorname{ind}_{\rm top}(a)
           =i_*^{\rm cot}(a)\otimes_{C_0(W)}\eta_N.
 \tag{CI.58}
\] For every elliptic \(P\) in CI.9 on every closed manifold, \[
 \operatorname{index}P
       =\operatorname{ind}_{\rm top}([\sigma_P]).
 \tag{CI.59}
\] \textbf{Proof.} Forget the full source action in (CI.42), use the
proved scalar index/product identification, then (CI.57) and the actual
Euclidean inverse just identified. This gives
\([\sigma_P][\bar\partial_{M,-}]
 =i_*^{\rm cot}([\sigma_P])[\bar\partial_{\mathbb R^N,-}]
 =i_*^{\rm cot}([\sigma_P])\eta_N\), exactly(CI.59). It also proves
independence of every tubular choice and every embedding in the
definition(CI.58), since its value is the intrinsic analytic pairing for
every compactly supported relative symbol. Non-spin\(^{c}\) and
odd-dimensional bases are included throughout: only their doubled
\emph{real} normal bundle used the specified exterior spinor, and its
invariant Gaussian was proved under all \(O(q)\) transitions.
\(\square\)

\pandocbounded{\includegraphics[keepaspectratio,alt={The universal quantization morphism and the doubled-normal embedding pairing}]{figures/KT-KK-15-index-comparison.png}}

\emph{Figure CI.2.} The upper arrows are the actual KK
morphisms(CI.47)--(CI.49), with \(C=\mathbb C\); the inverse of
evaluation at zero comes from the explicitly semisplit cone-kernel
extension(CI.44). The lower square is the all-relative-class pairing
identity(CI.57) and its Euclidean inverse(CI.58), where \(Z\) denotes
the integer group. The coordinate panel fixes the real-normal rank-one
specialization of(CI.53)--(CI.54); \(00\) and \(11\) are the two
displayed exterior-factor states, both of even total degree. The proof
gives the exact all-rank tensor, support, domain and determinant
calculations. Free comparison sources are Connes's author edition,
Chapter II, Section 5, and Connes--Skandalis SectionIII, linked below.

\subsection{11. Duality, open subsets and boundaries: stated
results}\label{duality-open-subsets-and-boundaries-stated-results}

The following full results are stated here. They are not proof premises
in CI.1--CI.14.

For a closed spin\(^{c}\) \(n\)-manifold \(M\), cap product with its
positive physical Dirac gives isomorphisms \[
 K^j(M)\longrightarrow K_{j+n}(M),\qquad
 a\longmapsto[\![a]\!]\otimes_{C(M)}[D_M],\qquad j\in\mathbb Z/2.
\] Here \([\![a]\!]\) is the diagonal \(C(M)\)-source K-class; in even
degree it is the actual bundle-tensor endomorphism cycle, and in odd
degree it retains the original unitary/right-Clifford class. The full
bivariant version uses \(\Delta_M=m^*[D_M]\), with
\(m:C(M)\otimes C(M)\to C(M)\) multiplication, and gives natural
isomorphisms \[
 KK^j(A,C(M)\otimes B)\cong KK^{j+n}(A\otimes C(M),B)
\] by this cap product for separable \(A\) and sigma-unital \(B\).
Connes--Skandalis Corollary 3.7 and its Clifford translation in Remark
3.9 are free primary locators. Their full duality argument is not
reproduced here. The local \(\mathbb R^2\) inverse calculation in
Section8 is proved, but is not substituted for this global theorem.

For an open spin\(^{c}\) subset \(U\subset M\), with
\(j:C_0(U)\to C_0(M)\) extension by zero, the open restriction theorem
states \(j^*[D_M]=[D_U]\). A complete representative on \(U\) uses its
induced Clifford/spin\(^{c}\) structure and a compatible complete
metric; it does not assign self-adjointness to the generally incomplete
raw restricted differential operator. CI.12 proves the compact-support
local comparison required for this lesson's index argument; the full
open theorem is left stated.

For a compact spin\(^{c}\) \(n\)-manifold \(Z\) with boundary, let
\(\delta_Z\in KK^1(C(\partial Z),C_0(Z^\circ))\) be the extension class
of \(0\to C_0(Z^\circ)\to C(Z)\to C(\partial Z)\to0\). Give the boundary
its outward-normal-first spin\(^{c}\) orientation. In the fixed
right-Clifford product and degree-two removal convention, the boundary
theorem is \[
 \delta_Z\otimes_{C_0(Z^\circ)}[D_{Z^\circ}]
          =(-1)^n[D_{\partial Z}].
\] The integer degree on the left is \(n+1\), identified with \(n-1\) by
the fixed periodicity convention. Equivalently the outward geometric
boundary map is \((-1)^n\) times that ordinary right-product map and
sends the interior fundamental class to the positive boundary class. The
sign has a local normalization: on an increasing outward collar
coordinate the right-end cone class is the positive \(x\) of the Bott
lesson, whereas the positive physical \(-i\partial_t\) is \(-y\) by its
coisometry calculation. Their product is \(-1\). For the interval with
its increasing coordinate, the two endpoint extension classes are
\(-x,x\); their products with the physical Dirac are \(1,-1\), the
negative of the outward oriented endpoint classes \(-1,1\). In general
the right product of a positive normal line Dirac and the boundary Dirac
is \((-1)^{n-1}\) times their physical outward direct sum; the explicit
two odd Clifford matrices give the same grading reversal as in the Thom
direct-sum calculation, with coordinates replaced by commuting
derivative operators. The two factors give \((-1)^n\). This collar
normalization does not prove the general geometric boundary theorem.
Connes--Skandalis Proposition 3.6 supplies the free primary
extension/normal-sphere comparison; the full boundary result remains
stated here. No APS boundary-value index or boundary condition is
asserted.

These three full theorems, beyond the compact-support comparison
actually proved, are what this lesson does not prove.

\subsection{Free mathematical reading}\label{free-mathematical-reading}

A. Connes and G. Skandalis,
\href{https://ems.press/content/serial-article-files/42241}{\emph{The
longitudinal index theorem for foliations}, freely readable original
article}, Section I, Definitions 1.3/1.6 and Proposition 1.4/Theorem
1.10, and Section III, Remark 3.9 and Proposition 3.10. Their symbol
product and index comparison provide free comparison material; the full
quantization and embedding arguments are proved above, rather than
imported from those sketches.

B. Blackadar,
\href{https://www.bruceblackadar.com/Mathematics/book6.pdf}{\emph{K-Theory
for Operator Algebras}, author-posted corrected second edition}, Example
17.1.2(e)--(f) and Section 24.5. The analytic domain, exhaustion and
local compactness proofs above are supplied here or in the exact earlier
lesson proofs cited where used.

A. Connes,
\href{https://alainconnes.org/wp-content/uploads/book94bigpdf.pdf}{\emph{Noncommutative
Geometry}, freely posted author edition}, Chapter II, Section 5,
pp.107--110, for the tangent-deformation viewpoint. CI.10 supplies its
own algebra, semisplitting and operator proof; the source's groupoid
exactness and index lemmas are not proof premises here.

\end{document}