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\begin{document}

\section{Reducing first-order boundary data to a split
trace}\label{reducing-first-order-boundary-data-to-a-split-trace}

The index of a first-order elliptic boundary problem can be computed
after a sequence of explicit deformations. Each deformation has to
preserve two structures at once: the interior symbol must remain
invertible, and the boundary symbol must remain an isomorphism on the
decaying Cauchy data. This lesson constructs the whole sequence. It
normalizes the normal coefficient, freezes the collar symbol, collapses
all spectral rates while keeping the Calderón projection fixed, rotates
the boundary isomorphism after a zero-index stabilization, and reaches
the split trace model used by doubling.

The formulas keep every bundle map and every matrix factor in its typed
order. In particular, the inverse of the boundary symbol is defined only
on the stable bundle, the two off-diagonal blocks of the rotation have
different source and target bundles, and a quantized projection is never
treated as an exact operator projection.

The named prerequisites are \href{stable-boundary-models.pdf}{Stable
modes and the algebra of boundary data},
\href{calderon-cauchy-data.pdf}{Cauchy data from jumps and residues},
\href{generalized-collar-fredholm.pdf}{Fredholm boundary problems with
first-order Calderón defects}, \href{fredholm-stability.pdf}{Finite
defects under perturbation},
\href{global-elliptic-symbol-index.pdf}{Symbols, finite defects, and the
index on a closed manifold}, \href{bott-suspension.pdf}{The Bott
operator, suspension, and reduction of the index to Euclidean space},
and \href{split-doubling-boundary-index.pdf}{Doubling a boundary problem
and computing its index}. We use \(D_t=-i\partial_t\), an inward collar
coordinate \(t\geq0\), and complex bundles.

\subsection{1. The starting problem and the reduction
theorem}\label{the-starting-problem-and-the-reduction-theorem}

Let \(X\) be a compact smooth manifold with boundary \(Y\). Let
\(E,F\to X\) and \(G\to Y\) be smooth complex bundles. Consider a
first-order elliptic operator and an order-zero boundary operator

\[
 (P,B)_1:H^1(X;E)\longrightarrow
 L^2(X;F)\oplus H^{1/2}(Y;G),
 \qquad B=\mathcal B\gamma_0 .
 \tag{BR1}
\]

Here \(\mathcal B\in\Psi_{\mathrm{phg}}^0(Y;E|_Y,G)\). Write \(p\) for
the principal symbol of \(P\). Ellipticity at a nonzero conormal makes

\[
 c(y,t)=p(y,t,0,dt):E_{(y,t)}\longrightarrow F_{(y,t)}
 \quad\text{invertible on a sufficiently small collar.}
 \tag{BR2}
\]

After using \(c\) to identify the target with the source in that collar,
the boundary principal symbol has the form

\[
 \widetilde p(y,0,\eta,\zeta)=\zeta I_E-a(y,\eta),\qquad
 q(y,\eta)=\frac{1}{2\pi i}\int_{\Gamma_+}
 (zI_E-a(y,\eta))^{-1}\,dz .
 \tag{BR3}
\]

The contour surrounds the spectrum of \(a(y,\eta)\) in the upper
half-plane. The stable Cauchy bundle is \(L=\operatorname{ran}q\) over
\(T^*Y\setminus0\). If \(b=\sigma_0(\mathcal B)\), the complementing
condition is precisely that \(b|_L:L\to\pi^*G\) be a bundle isomorphism.

\textbf{Stable reduction theorem.} There is a bundle \(H\to X\), a
zero-index auxiliary first-order problem on \(H\), and a continuous
Fredholm deformation of the stabilized problem to a split bundle-trace
problem. If \((P_{\mathrm{sp}},B_{\mathrm{sp}})\) is the endpoint and
\(\widehat P_{\mathrm{sp}}\) is its geometric double, then

\[
 \boxed{
 \operatorname{ind}(P,B)
 =\operatorname{ind}(P_{\mathrm{sp}},B_{\mathrm{sp}})
 =\operatorname{ind}\widehat P_{\mathrm{sp}}
 =\operatorname{sind}(\widehat p_{\mathrm{sp}}).}
 \tag{BR4}
\]

The theorem assumes that the original complementing operator \(B\)
exists. Section 10 records the exact symbol-bundle obstruction exposed
when no such \(B\) exists.

\subsection{2. Normal coefficient normalization and collar
freezing}\label{normal-coefficient-normalization-and-collar-freezing}

Choose a collar \(Y\times[0,\delta)_t\). The isomorphism
\(c^{-1}:F|_{Y\times[0,\delta)}\to E|_{Y\times[0,\delta)}\) is used as a
collar frame for the target. This is a change of bundle coordinates, so
it does not alter the operator, its domain, its kernel, or its cokernel.
In this target frame the collar operator can be written

\[
 P^b=D_t-A(t)+R_0(t),
 \qquad
 \sigma_1(P^b)(y,t,\eta,\zeta)=\zeta I_E-a(y,t,\eta),
 \tag{BR5}
\]

where \(A(t)\) is tangential of order one and \(R_0(t)\) has order zero.
No factor of the original normal coefficient has been discarded: it
remains the target identification \(c\). Every normalized symbol
deformation below is transported back to \(F\) by left multiplication
with \(c\), so the global target bundle stays fixed.

Let \(\chi\in C_c^\infty([0,\delta))\), with \(\chi=1\) for
\(0\leq t\leq\delta/3\) and \(\chi=0\) for \(t\geq2\delta/3\). For
\(0\leq r\leq1\), define

\[
 a_r(y,t,\eta)
 =a(y,t,\eta)+r\chi(t)\bigl(a(y,0,\eta)-a(y,t,\eta)\bigr).
 \tag{BR6}
\]

This is a deformation of the full original principal symbol, not a
replacement by a rescaled symbol. To prove ellipticity, restrict to the
compact collar cosphere \(|\eta|^2+\zeta^2=1\). Since the original
symbol is invertible there, there is \(M<\infty\) such that

\[
 \sup\|(\zeta I_E-a(y,t,\eta))^{-1}\|\leq M .
 \tag{BR7}
\]

Uniform continuity gives, after decreasing \(\delta\),

\[
 \|r\chi(t)(a(y,0,\eta)-a(y,t,\eta))\|
 <\frac{1}{2M}
 \quad\text{on the collar cosphere.}
 \tag{BR8}
\]

Factor the deformed symbol on the right of the original inverse:

\[
 \zeta I_E-a_r
 =(\zeta I_E-a)
 \left[I_E-(\zeta I_E-a)^{-1}
 r\chi(t)(a(y,0,\eta)-a(y,t,\eta))\right].
 \tag{BR9}
\]

The bracket is invertible by its Neumann series. Thus every \(a_r\) is
elliptic. At \(t=0\), the symbol is always \(a(y,0,\eta)\), so the
stable bundle and \(b|_L\) do not change. Transporting this principal
deformation back by \(c\), and interpolating the lower-order terms,
gives an operator-norm continuous Fredholm path on the original source
and target bundles. At \(r=1\), the tangential operator is independent
of \(t\) on the smaller collar. All cutoff-transition terms are
supported where \(t\geq\delta/3\).

\subsection{3. The smooth stable projection and the fixed-projection
collapse}\label{the-smooth-stable-projection-and-the-fixed-projection-collapse}

For a fixed \((y_0,\eta_0)\), choose a contour \(\Gamma_+\) that
separates the upper and lower spectra of \(a(y_0,\eta_0)\).
Invertibility on this compact contour persists in a parameter
neighborhood. Differentiating the resolvent gives

\[
 \partial_\nu(zI-a)^{-1}
 =(zI-a)^{-1}(\partial_\nu a)(zI-a)^{-1}.
 \tag{BR10}
\]

Repeated differentiation under the contour integral proves that \(q\) is
smooth without choosing eigenvalue branches. Two local contours give the
same projection on their overlap because both enclose exactly the upper
spectrum. Positive homogeneity follows by scaling both the symbol and
the contour:

\[
 q(y,\rho\eta)
 =\frac{1}{2\pi i}\int_{\rho\Gamma_+}
 (zI-\rho a(y,\eta))^{-1}\,dz
 =q(y,\eta),\qquad \rho>0 .
 \tag{BR11}
\]

Choose \(Q\in\Psi_{\mathrm{phg}}^0(Y;E|_Y)\) with principal symbol
\(q\), and choose \(\Lambda\in\Psi_{\mathrm{phg}}^1(Y;E|_Y)\) with
positive scalar principal symbol \(\lambda(y,\eta)I_E\). The operator
\(Q\) need not satisfy \(Q^2=Q\); only its principal symbol enters the
following ellipticity calculation.

At the symbol level define, for \(0\leq r\leq1\),

\[
 a^{\langle r\rangle}
 =(1-r)a+i r\lambda(2q-I_E).
 \tag{BR12}
\]

The contour formula makes \(q\) a holomorphic function of \(a\), so
\(aq=qa\). On the primary summand for an eigenvalue \(\xi\), with
nilpotent part \(N_\xi\), the restriction of (BR12) is

\[
 \left((1-r)\xi+i r\lambda\,
 \operatorname{sgn}(\operatorname{Im}\xi)\right)I
 +(1-r)N_\xi .
 \tag{BR13}
\]

Its only eigenvalue stays strictly in the same half-plane as \(\xi\).
Therefore every \(a^{\langle r\rangle}\) has no real eigenvalue, and its
upper spectral projection is exactly \(q\). This primary-block
calculation is the needed proof; convexity of spectra of arbitrary
nonnormal matrices would not suffice.

Let \(\phi(t)=1\) on a smaller collar and have support inside the region
where \(A(t)=A(0)\). Quantize the deformation by

\[
 A_r=(1-r\phi)A
 +r\phi\bigl(i\Lambda Q-i\Lambda(I_E-Q)\bigr).
 \tag{BR14}
\]

Its principal tangential symbol is (BR12) with \(r\) replaced by
\(r\phi(t)\). Hence \(D_t-A_r\) remains elliptic, its stable projection
at the boundary remains \(q\), and the original \(\mathcal B\) remains
complementing throughout. The exact order \(\Lambda Q\) and
\(\Lambda(I_E-Q)\) has been retained.

\subsection{4. Replacing the interior partial
operators}\label{replacing-the-interior-partial-operators}

In the annulus where a collar cutoff varies, a tangential operator acts
in \(y\) while \(t\) is only a parameter. This is a partial
pseudodifferential operator. Its support stays away from \(Y\), so it
can be approximated by ordinary operators without changing any boundary
symbol.

For a tangential order-one symbol \(k(y,t,\eta)\), let
\(\omega(\eta,\varepsilon\zeta)\) be the angular cutoff from the
partial-operator approximation theorem, and set

\[
 k_\varepsilon(y,t,\eta,\zeta)
 =k(y,t,\eta)\omega(\eta,\varepsilon\zeta).
 \tag{BR15}
\]

After multiplying on both sides by the fixed annular cutoff,
\(K_\varepsilon\) is an ordinary full-variable operator and, for every
real \(s\),

\[
 \|K_\varepsilon-K\|_{H^s\to H^{s-1}}
 \leq C_s\varepsilon .
 \tag{BR16}
\]

The estimates are uniform in the compact deformation parameter because
only finitely many uniform symbol seminorms occur. The image of the
parameter interval is a compact subset of the Fredholm locus. A finite
operator-norm neighborhood cover therefore supplies one positive
tolerance for the entire path. Choose \(\varepsilon\) below that
tolerance. The ordinary approximating path is Fredholm and has the same
index at every parameter. This step changes no collar principal symbol,
no boundary map, and no stable bundle.

\subsection{5. The inverse boundary symbol and its
rotation}\label{the-inverse-boundary-symbol-and-its-rotation}

Let \(b:\pi^*E\to\pi^*G\) be the order-zero principal boundary symbol.
Its restriction \(b|_L:L\to\pi^*G\) is a smooth homogeneous bundle
isomorphism. Define

\[
 s:\pi^*G\longrightarrow\pi^*E,\qquad
 s=(b|_L)^{-1},
 \quad\text{with image in }L.
 \tag{BR17}
\]

Local frames of \(L\) turn this into the inverse of a smooth matrix with
nonzero determinant, proving smoothness. Since \(b\) and \(q\) have
degree zero, so does \(s\). Its complete typed identities are

\[
 bs=I_G,\qquad qs=s,\qquad sbq=q .
 \tag{BR18}
\]

The last identity follows because \(qe\in L\) and \(s\) is the inverse
of \(b\) there. Expressions such as \(qb\) or \(sq\) are not used
because their source and target bundles do not match.

Put \(c_\theta=\cos\theta\) and \(d_\theta=\sin\theta\). On
\(\pi^*(E\oplus G)\), define

\[
 \begin{aligned}
 b_\theta&=(c_\theta b,\ d_\theta I_G),\\
 s_\theta&=\binom{c_\theta s}{d_\theta I_G},\\
 q_\theta&=
 \begin{pmatrix}
 c_\theta^2q&c_\theta d_\theta s\\
 c_\theta d_\theta bq&d_\theta^2I_G
 \end{pmatrix},
 \qquad 0\leq\theta\leq\frac{\pi}{2}.
 \end{aligned}
 \tag{BR19}
\]

First,

\[
 b_\theta s_\theta
 =c_\theta^2bs+d_\theta^2I_G=I_G .
 \tag{BR20}
\]

Direct block multiplication using only (BR18) gives

\[
 q_\theta^2=q_\theta,\qquad
 q_\theta s_\theta=s_\theta,\qquad
 s_\theta b_\theta q_\theta=q_\theta .
 \tag{BR21}
\]

For example, the upper-left block of \(q_\theta^2\) is
\(c_\theta^4q+c_\theta^2d_\theta^2sbq=c_\theta^2q\), and the lower-left
block is
\(c_\theta^3d_\theta bq+c_\theta d_\theta^3bq=c_\theta d_\theta bq\).
The other two blocks follow in the same typed order. The last two
identities imply

\[
 \operatorname{ran}q_\theta=\operatorname{ran}s_\theta,\qquad
 q_0=q\oplus0,\qquad
 q_{\pi/2}=0\oplus I_G .
 \tag{BR22}
\]

Thus the stable bundle rotates from \(L\) to the pulled-back
\(G\)-summand while \(b_\theta\) remains its inverse measurement.

\subsection{6. Extending the boundary bundle and adding zero
index}\label{extending-the-boundary-bundle-and-adding-zero-index}

The bundle \(G\to Y\) has a smooth complement \(G'\) with
\(G\oplus G'\cong Y\times\mathbb C^N\). This follows from a finite
trivializing cover and the explicit embedding-and-orthogonal-complement
construction in the Bott lesson. Take

\[
 H=X\times\mathbb C^N,\qquad
 H|_Y\xrightarrow{\ \kappa\ }G\oplus G'.
 \tag{BR23}
\]

Let \(q_G:H|_Y\to G\) and \(\iota_G:G\to H|_Y\) be the projection and
inclusion supplied by this splitting, and put

\[
 \Pi_G=\iota_Gq_G,\qquad
 \Pi_G^2=\Pi_G,\qquad
 H|_Y=\operatorname{ran}\Pi_G\oplus\ker\Pi_G .
 \tag{BR24}
\]

Choose an order-one operator \(\Lambda_H\), block diagonal near \(Y\)
for \(G\oplus G'\), with the same positive scalar principal symbol
\(\lambda(y,\eta)I_H\) as \(\Lambda\). Form the split auxiliary problem
whose collar part is

\[
 P_H^b=D_t+i\Lambda_H,\qquad B_H=0 .
 \tag{BR25}
\]

Complete it elliptically in the interior as in the split-model theorem.
Its stable boundary space is zero, its boundary target is the zero
bundle, and that theorem gives

\[
 \operatorname{ind}(P_H,0)=0,\qquad
 \operatorname{ind}\bigl((P,B)\oplus(P_H,0)\bigr)
 =\operatorname{ind}(P,B).
 \tag{BR26}
\]

This is the only stabilization used below. It changes no index.

\subsection{7. The full two-by-two normal
homotopy}\label{the-full-two-by-two-normal-homotopy}

Quantize \(s\) by \(S\in\Psi_{\mathrm{phg}}^0(Y;G,E|_Y)\). Retain \(Q\)
with symbol \(q\). The product \(\mathcal BQ:E|_Y\to G\) has principal
symbol \(bq\). Choose a decreasing collar cutoff \(\vartheta(t)\), equal
to one near \(t=0\), with support where \(\phi=1\). For
\(0\leq\tau\leq\pi/2\), put

\[
 \alpha=\tau\vartheta(t),\qquad
 c_\alpha=\cos\alpha,\qquad d_\alpha=\sin\alpha .
 \tag{BR27}
\]

On \(\pi^*(E\oplus H)\), extend (BR19) by zero on the \(G'\)-summand:

\[
 \mathfrak q_\alpha=
 \begin{pmatrix}
 c_\alpha^2q&
 c_\alpha d_\alpha\,s q_G\\
 c_\alpha d_\alpha\,\iota_Gbq&
 d_\alpha^2\Pi_G
 \end{pmatrix}.
 \tag{BR28}
\]

The same multiplication as in Section 5, with \(q_G\iota_G=I_G\), proves

\[
 \mathfrak q_\alpha^2=\mathfrak q_\alpha,\qquad
 \operatorname{ran}\mathfrak q_\alpha
 =\left\{
 \binom{c_\alpha sg}{d_\alpha\iota_Gg}:g\in\pi^*G
 \right\}.
 \tag{BR29}
\]

The exact block expansion of \(I-2\mathfrak q_\alpha\) is

\[
 I-2\mathfrak q_\alpha=
 \begin{pmatrix}
 I_E-2c_\alpha^2q&-\sin(2\alpha)sq_G\\
 -\sin(2\alpha)\iota_Gbq&I_H-2d_\alpha^2\Pi_G
 \end{pmatrix}.
 \tag{BR30}
\]

Every sign and every inclusion in the normal homotopy comes from this
identity. With the convention that a \(G\)-valued term in the
\(H\)-component is inserted by \(\iota_G\), define the collar part

\[
 \begin{aligned}
 P_\tau^b\binom{u}{v}
 =\phi\Bigg[
 \binom{D_tu}{D_tv}
 +i\binom{
 \Lambda\bigl(u-2c_\alpha^2Qu-\sin(2\alpha)S q_Gv\bigr)}
 {\Lambda_H\bigl(v-\sin(2\alpha)\iota_G\mathcal BQu
                    -2d_\alpha^2\Pi_Gv\bigr)}
 \Bigg].
 \end{aligned}
 \tag{BR31}
\]

The two off-diagonal blocks are not interchangeable: \(Sq_G:H\to E\),
whereas \(\iota_G\mathcal BQ:E\to H\). Their principal symbols are
exactly the off-diagonal blocks in (BR30). Lower-order defects caused by
\(Q^2-Q\), quantization, and derivatives of the cutoffs do not enter the
principal ellipticity test.

The boundary operator is

\[
 B_\tau(u,v)
 =\cos\tau\,\mathcal B\gamma_0u
  +\sin\tau\,q_G\gamma_0v .
 \tag{BR32}
\]

At \(\tau=0\), (BR31) is the direct sum of the fixed-projection endpoint
(BR14) and \(P_H\), while (BR32) is \(B\) on the first summand.

\subsection{8. Ellipticity, boundary bijectivity, and index
tracking}\label{ellipticity-boundary-bijectivity-and-index-tracking}

The principal tangential generator in (BR31) is

\[
 a_{\tau,t}=i\lambda(2\mathfrak q_\alpha-I_{E\oplus H}) .
 \tag{BR33}
\]

It has eigenvalue \(i\lambda\) on
\(\operatorname{ran}\mathfrak q_\alpha\) and \(-i\lambda\) on its
kernel. Thus \(\zeta I-a_{\tau,t}\) is invertible for every real
\(\zeta\) when \((\eta,\zeta)\ne0\), and its upper spectral projection
is exactly \(\mathfrak q_\alpha\).

At the boundary, \(\alpha=\tau\). Define

\[
 \mathfrak s_\tau g
 =\binom{\cos\tau\,sg}{\sin\tau\,\iota_Gg}.
 \tag{BR34}
\]

Equations (BR29), (BR32), \(bs=I_G\), and \(q_G\iota_G=I_G\) give

\[
 \operatorname{ran}\mathfrak q_\tau
 =\operatorname{ran}\mathfrak s_\tau,\qquad
 b_\tau\mathfrak s_\tau
 =(\cos^2\tau+\sin^2\tau)I_G=I_G .
 \tag{BR35}
\]

Hence \(b_\tau\) is bijective on the stable Cauchy bundle for every
parameter. The generalized collar Fredholm theorem therefore gives

\[
 (P_\tau,B_\tau)_1:
 H^1(X;E\oplus H)\longrightarrow
 L^2(X;F\oplus H)\oplus H^{1/2}(Y;G)
 \quad\text{Fredholm}.
 \tag{BR36}
\]

Formula (BR31) is written in the collar target frame from Section 2;
multiplication by \(c\) returns its first component to \(F\). Thus the
displayed source and target spaces are fixed along the path. The
coefficients depend continuously on \(\tau\) in their operator norm.
Index local constancy, connectedness of \([0,\pi/2]\), and (BR26) now
give

\[
 \operatorname{ind}(P_\tau,B_\tau)
 =\operatorname{ind}(P,B)
 \quad\text{for every }0\leq\tau\leq\frac{\pi}{2}.
 \tag{BR37}
\]

The ordinary interior approximations from Section 4 can be made
uniformly along this path. They preserve (BR37), because they lie in one
fixed Fredholm neighborhood and are supported away from the boundary
symbol.

\subsection{9. The endpoint is the split trace
model}\label{the-endpoint-is-the-split-trace-model}

At \(\tau=\pi/2\), and where \(\vartheta=1\), the off-diagonal terms
vanish. Relative to \(H|_Y=G\oplus G'\), (BR31)--(BR32) become

\[
 \begin{aligned}
 P_{\pi/2}^b
 &=
 \bigl(D_t+i\Lambda_E\bigr)
 \oplus
 \bigl(D_t-i\Lambda_G\bigr)
 \oplus
 \bigl(D_t+i\Lambda_{G'}\bigr),\\
 B_{\pi/2}(u,v_G,v_{G'})
 &=\gamma_0v_G .
 \end{aligned}
 \tag{BR38}
\]

The \(G\)-summand is the complete stable space; the \(E\)- and
\(G'\)-summands are unstable. To match the split convention with normal
coefficient \(-I_G\) on the stable summand, apply the target-bundle
isomorphism

\[
 T_{\mathrm{sgn}}
 =I_E\oplus(-I_G)\oplus I_{G'} .
 \tag{BR39}
\]

On the collar it turns the \(G\)-block into \(-D_t+i\Lambda_G\), while
leaving the boundary trace unchanged. It extends to an invertible
target-bundle automorphism: keep the splitting on a smaller collar, join
the factor \(-1\) to \(1\) through the nonzero complex phases
\(e^{i\pi\rho(t)}\), and use the identity beyond the collar. After
reordering the summands,

\[
 E_{\mathrm{sp}}^+=E\oplus G',
 \qquad E_{\mathrm{sp}}^-=G,
 \qquad
 P_{\mathrm{sp}}^b=
 \begin{pmatrix}
 D_t+i\Lambda^+&0\\
 0&-D_t+i\Lambda^-
 \end{pmatrix},
 \qquad
 B_{\mathrm{sp}}=\gamma_0|_{E_{\mathrm{sp}}^-}.
 \tag{BR40}
\]

This accounts for the sign change rather than hiding it in a relabeling.
The split doubling theorem applies and yields

\[
 \operatorname{ind}(P,B)
 =\operatorname{ind}(P_{\mathrm{sp}},B_{\mathrm{sp}})
 =\operatorname{ind}\widehat P_{\mathrm{sp}}
 =\operatorname{sind}(\widehat p_{\mathrm{sp}}),
 \tag{BR41}
\]

which proves (BR4).

\begin{figure}
\centering
\pandocbounded{\includegraphics[keepaspectratio,alt={The fixed stable projection, its bundle rotation, and the split endpoint}]{../figures/stable_boundary_reduction.png}}
\caption{The fixed stable projection, its bundle rotation, and the split
endpoint}
\end{figure}

The left panel shows the two analytic deformations that leave \(q\)
fixed. The middle panel shows the exact stable vector and boundary
inverse during the rotation. The right panel records the endpoint
summands, the target sign map, and the index chain.

\subsection{10. The obstruction space and nonzero boundary
order}\label{the-obstruction-space-and-nonzero-boundary-order}

Let \(\pi:S^*Y\to Y\) and \(L(P)=\operatorname{ran}q\). The exact space
of order-zero complementing symbol data is the groupoid

\[
 \mathfrak C_0(P)
 =\left\{(G,\beta):
 G\to Y\text{ a complex bundle},\
 \beta:L(P)\xrightarrow{\sim}\pi^*G\right\}.
 \tag{BR42}
\]

\textbf{Editorial completion of the groupoid definition.} An arrow
\(h:(G,\beta)\to(G_1,\beta_1)\) is a smooth bundle isomorphism
\(h:G\to G_1\) over \(Y\) satisfying \[
 (\pi^*h)\beta=\beta_1.
 \tag{BR42a}
\] The identity of \(G\) satisfies this equation. If \(h\) and \(h_1\)
are composable arrows, then \((\pi^*(h_1h))\beta=(\pi^*h_1)\beta_1\),
the required equation for their composite. Multiplying the displayed
equation by \(\pi^*h^{-1}\) gives the equation for the inverse.
Associativity and the identity laws follow from composition of the
underlying bundle maps. This supplies every arrow and proves that the
stated collection is a groupoid. All pullbacks and symbol maps remain
over the same cosphere bundle.

Given \((G,\beta)\), the formula

\[
 b=\beta q:\pi^*E\longrightarrow\pi^*G
 \tag{BR43}
\]

extends \(\beta\) to a boundary symbol and is complementing. Conversely,
every complementing \(b\) restricts to such a \(\beta\). Thus failure of
\(L(P)\) to be pulled back is exactly the failure of this groupoid to
have an object.

\textbf{Editorial correction: the complete fiber of the restriction
map.} The canonical extension \(\beta q\) need not equal the original
\(b\). Put \(K(P)=\ker q\); the exact splitting is
\(\pi^*E=L(P)\oplus K(P)\), with projections \(q\) and \(I-q\), without
an orthogonality assumption. For fixed \((G,\beta)\), every
complementing symbol whose restriction is \(\beta\) is uniquely \[
 b=\beta q+c(I-q),\qquad
 c:K(P)\longrightarrow\pi^*G.
 \tag{BR43a}
\] Here \(c\) is any smooth bundle map on \(S^*Y\), extended
homogeneously with degree zero to \(T^*Y\setminus0\). To prove the
assertion, write every vector as \(e=qe+(I-q)e\). Then
\(b(qe)=\beta(qe)\), while \(b((I-q)e)=c((I-q)e)\) for \(c=b|_{K(P)}\).
This proves existence and the formula. Restriction to \(K(P)\) recovers
\(c\), proving uniqueness. Conversely the displayed formula restricts to
\(\beta\) on \(L(P)\), so it is complementing. The fiber is therefore
the affine space with origin \(\beta q\) and translation space
\(\Gamma^\infty(S^*Y;\operatorname{Hom}(K(P),\pi^*G))\). If \(S^*Y\) is
empty, there is one map on each empty pullback and the same assertion
has a one-element fiber; it imposes no condition on the
finite-dimensional boundary realization.

\begin{figure}
\centering
\pandocbounded{\includegraphics[keepaspectratio,alt={The full boundary symbol has independently specified maps on the stable and complementary bundles; groupoid arrows commute with the stable isomorphisms.}]{../figures/boundary-symbol-fiber.png}}
\caption{The full boundary symbol has independently specified maps on
the stable and complementary bundles; groupoid arrows commute with the
stable isomorphisms.}
\end{figure}

The left diagram gives the exact two summands in (BR43a). The right
diagram is (BR42a), with the identity on \(L(P)\). These are the precise
morphisms retained by the restriction map; the example (BR51) realizes
the free complementary entry as \(\kappa\).

The stable \(K\)-theory obstruction is the defined class

\[
 \operatorname{ob}_{\partial}(P)
 =[L(P)]\bmod
 \operatorname{im}\!\left(
 \pi^*:K^0(Y)\longrightarrow K^0(S^*Y)\right).
 \tag{BR44}
\]

If a complementing \(B\) exists, then \([L(P)]=\pi^*[G]\), so

\[
 \mathfrak C_0(P)\ne\varnothing
 \Longrightarrow
 \operatorname{ob}_{\partial}(P)=0 .
 \tag{BR45}
\]

Consequently a nonzero class in (BR44) forbids even a stably pulled-back
local boundary symbol. The converse classification, including
realization by a full operator problem, is not proved here and remains a
named topological dependency. The reduction theorem does not assert that
every elliptic interior operator admits \(B\).

If the boundary operator has total order \(\mu\), use the exact
order-changing isomorphism from the closed-manifold index lesson:

\[
 J_G^{-\mu}:
 H^{s-\mu-1/2}(Y;G)
 \xrightarrow{\ \sim\ }
 H^{s-1/2}(Y;G),
 \qquad
 J_G^{-\mu}\in\Psi_{\mathrm{cl}}^{-\mu}(Y;G).
 \tag{BR46}
\]

Then \(B^{(0)}=J_G^{-\mu}B\) has order zero and

\[
 (P,B^{(0)})
 =\bigl(I\oplus J_G^{-\mu}\bigr)(P,B),
 \qquad
 \ker(P,B^{(0)})=\ker(P,B),
 \qquad
 \operatorname{coker}(P,B^{(0)})\cong\operatorname{coker}(P,B).
 \tag{BR47}
\]

Hence the index is unchanged. If one instead uses an elliptic Fredholm
factor \(R\) of order \(-\mu\) that is not an isomorphism, the
composition law gives the required correction

\[
 \operatorname{ind}(P,RB)
 =\operatorname{ind}(P,B)+\operatorname{ind}R .
 \tag{BR48}
\]

The exact isomorphism (BR46) avoids this extra term.

\subsection{11. Three calculations that test the
construction}\label{three-calculations-that-test-the-construction}

\textbf{A nonnormal spectral block.} Let

\[
 a=
 \begin{pmatrix}i&1&0\\0&i&0\\0&0&-2i\end{pmatrix},
 \qquad
 q=\begin{pmatrix}1&0&0\\0&1&0\\0&0&0\end{pmatrix},
 \qquad\lambda=1 .
 \tag{BR49}
\]

The fixed-projection path is

\[
 a^{\langle r\rangle}
 =
 \begin{pmatrix}
 i&1-r&0\\
 0&i&0\\
 0&0&-i(2-r)
 \end{pmatrix}.
 \tag{BR50}
\]

The upper eigenvalue remains \(i\), its nilpotent part decreases to
zero, and the lower eigenvalue remains strictly negative imaginary. This
calculation shows why the full primary summand, rather than a moving
eigenbasis, is the correct object.

\textbf{A boundary map that differs from its stable restriction.} Take
\(E=\mathbb C^2\), \(G=\mathbb C\), \(q=\operatorname{diag}(1,0)\), and
\(b=(1,\kappa)\). Then \(s=(1,0)^t\) and \(bq=(1,0)\), while \(b\ne bq\)
when \(\kappa\ne0\). Formula (BR19) gives

\[
 q_\theta=
 \begin{pmatrix}
 c_\theta^2&0&c_\theta d_\theta\\
 0&0&0\\
 c_\theta d_\theta&0&d_\theta^2
 \end{pmatrix},
 \qquad
 s_\theta z=(c_\theta z,0,d_\theta z).
 \tag{BR51}
\]

It is a rank-one projection and \(b_\theta s_\theta z=z\). The unused
entry \(\kappa\) disappears only after the correctly typed factor
\(bq\); replacing it by \(b\) in the lower-left block would be wrong.

\textbf{Boundary order two.} If \(B:H^s(X;E)\to H^{s-5/2}(Y;G)\) has
total order two, then \(J_G^{-2}B\) maps to \(H^{s-1/2}(Y;G)\). Both
operators have the same kernel and canonically isomorphic cokernels by
(BR47). No boundary contribution is lost in the order change.

\subsection{12. Exercises with complete
solutions}\label{exercises-with-complete-solutions}

\textbf{1. Prove the homogeneous contour formula.} Starting from
\(a(y,\rho\eta)=\rho a(y,\eta)\), derive (BR11) with every contour
factor retained.

\textbf{Solution.} Use \(z=\rho w\), \(dz=\rho\,dw\), and

\[
 (zI-\rho a)^{-1}
 =\rho^{-1}(wI-a)^{-1}.
 \tag{BR52}
\]

The factors \(\rho\) and \(\rho^{-1}\) cancel, and the contour
\(\rho\Gamma_+\) becomes \(\Gamma_+\). Hence
\(q(y,\rho\eta)=q(y,\eta)\).

\textbf{2. Verify the remaining blocks of \(q_\theta^2\).} Use only the
identities in (BR18).

\textbf{Solution.} Besides the upper-left block computed after (BR21),
the three blocks are

\[
 \begin{aligned}
 (q_\theta^2)_{12}
 &=c_\theta^3d_\theta qs+c_\theta d_\theta^3s
 =c_\theta d_\theta s,\\
 (q_\theta^2)_{21}
 &=c_\theta^3d_\theta bq^2+c_\theta d_\theta^3bq
 =c_\theta d_\theta bq,\\
 (q_\theta^2)_{22}
 &=c_\theta^2d_\theta^2bqs+d_\theta^4I_G
 =d_\theta^2I_G .
 \end{aligned}
 \tag{BR53}
\]

Here \(qs=s\), \(q^2=q\), and \(bqs=bs=I_G\). Every composition is
type-correct.

\textbf{3. Recover the off-diagonal coefficient.} Why is the coefficient
in (BR30) \(-\sin(2\alpha)\)?

\textbf{Solution.} The off-diagonal entry of \(2\mathfrak q_\alpha\) is
\(2c_\alpha d_\alpha sq_G\). Since \(2c_\alpha d_\alpha=\sin(2\alpha)\),
subtracting from the identity gives

\[
 (I-2\mathfrak q_\alpha)_{12}
 =-\sin(2\alpha)sq_G ,
 \quad
 (I-2\mathfrak q_\alpha)_{21}
 =-\sin(2\alpha)\iota_Gbq .
 \tag{BR54}
\]

The minus sign is forced in both blocks.

\textbf{4. Check the endpoint stable modes.} Solve the three scalar
normal equations in (BR38) when every \(\Lambda\) has frozen eigenvalue
\(\lambda>0\).

\textbf{Solution.} The equations and their solutions are

\[
 \begin{array}{c|c|c}
 \text{block}&\text{solution}&\text{bounded for }t\geq0\\ \hline
 D_t+i\lambda&e^{\lambda t}a&a=0\\
 D_t-i\lambda&e^{-\lambda t}g&\text{every }g\\
 D_t+i\lambda&e^{\lambda t}a'&a'=0 .
 \end{array}
 \tag{BR55}
\]

Thus the stable space is exactly the \(G\)-summand and its trace is
measured by \(q_G\).

\textbf{5. Identify the symbol-level data space and its fiber.} Given an
isomorphism \(\beta:L(P)\to\pi^*G\), prove that (BR43) is complementing,
describe all symbols with that restriction, and explain why the
canonical extension need not recover their values on \(K(P)\).

\textbf{Solution.} For \(\ell\in L(P)\), \(q\ell=\ell\), so
\((\beta q)\ell=\beta\ell\); this restriction is bijective. Conversely,
if \(b\) is complementing, set \(\beta=b|_{L(P)}\). Then \(\beta\) is an
isomorphism. Writing \(e=qe+(I-q)e\) gives
\(b(e)=\beta qe+(b|_{K(P)})(I-q)e\). Thus the unique free map is
\(c=b|_{K(P)}\), and every choice of \(c\) gives a complementing symbol
by the same formula. The canonical extension chooses \(c=0\), so it need
not recover an original nonzero complementary entry. Therefore (BR42)
records exactly the symbol data relevant to the boundary condition.

\textbf{6. Track a noninvertible order reducer.} Suppose \(R\) in (BR48)
has index \(-2\) and \(\operatorname{ind}(P,B)=5\). Compute the new
index and state why the composition belongs to the declared order-zero
class.

\textbf{Solution.} The block target operator \(I\oplus R\) has index
\(-2\). Fredholm composition gives

\[
 \operatorname{ind}(P,RB)=5-2=3.
 \tag{BR56}
\]

The composition rule gives \(RB\in\Psi^0\) from \(R\in\Psi^{-\mu}\) and
\(B\in\Psi^\mu\). This is the stated upper order; a vanishing principal
symbol can lower the actual order. The index correction is analytic data
of \(R\), independent of that order count.

\subsection{13. Reading notes and
references}\label{reading-notes-and-references}

\href{stable-boundary-models.pdf}{Stable modes and the algebra of
boundary data}, Sections 4--5, proves smooth parameter dependence of the
Riesz projection and the primary-block deformation that keeps it fixed.
\href{calderon-cauchy-data.pdf}{Cauchy data from jumps and residues},
Sections 8--9, identifies the stable projection and the complementing
restriction without choosing roots.
\href{global-elliptic-symbol-index.pdf}{Symbols, finite defects, and the
index on a closed manifold}, Sections 3 and 10, supplies exact
order-changing isomorphisms and the operator-norm approximation of
partial operators. \href{bott-suspension.pdf}{The Bott operator,
suspension, and reduction of the index to Euclidean space}, Section 12,
supplies the explicit stable complement for \(G\).
\href{split-doubling-boundary-index.pdf}{Doubling a boundary problem and
computing its index} proves the zero index of the auxiliary split
problem and the final doubled symbol-index formula.

The construction, block verification, examples, figure, and exercises
here are independently written. Exact original-source correspondence
remains open because no original-author TeX source was available through
the checked literature indexes.

Written and dedicated to the public domain by Codex under CC0 1.0.

\subsection{14. Finite-product and receiving-map
proof}\label{finite-product-receiving-proof}

This supplement proves the actual finite collar product used by the
analytic receiving maps. It keeps the original normal coefficient,
complete tangential operators, source distributions, and every boundary
row order. The original reduction statements and formulas above remain
identifiable. Completing this finite-product proof alone does not close
every realization or deformation step of the stable reduction theorem.

The fixed-collar construction and exact symbol estimates are in
\href{generalized-collar-fredholm.pdf\#section*.90}{GI1--GI32
and AL1--AL18}. The
\href{generalized-collar-fredholm.pdf\#section*.208}{complete
composition proof} and
\href{generalized-collar-fredholm.pdf\#section*.249}{half-space
receiving maps} provide the analytic prerequisites explicitly used
below. The following calculation specializes those proofs to the literal
finite product, giving every complete coefficient and retaining both
error bundles.

\subsubsection{NP1. Original operators and actual
seed}\label{np1.-original-operators-and-actual-seed}

Let \(Y\) be the unchanged compact boundary and let \(E_Y,F_Y\) be the
separate original collar bundles, with their input densities. Let
\(m\ge 1\), \(D_r=-i\partial _r\), \(\delta =-i\partial _r\) on
coefficient families, and
\(P_c=\sum _{j=0}^m A_j(h(r))D_r^j:E_Y\to F_Y\), with
\(A_j\in \Psi _{\rm tan}^{m-j}\), \(A_m=M:E_Y\to F_Y\) invertible. In
subsequent formulas \(A_j(r)\) denotes exactly this composed family
\(A_j(h(r))\), so \(\delta \) differentiates the composed original
family. It never means its coefficient freeze. Boundary rows remain
\(B_j=\sum _{k=0}^{m-1}B_{jk}\gamma _k\),
\(B_{jk}\in \Psi _{\rm tan}^{m_j-k}(E_Y,G_j)\), with every original
\(m_j\) unrestricted. Retain \(R=1+|(\eta ,\kappa )|\) and
\(T=1+|\eta |\).

Use the actual GI24 seed
\(Q=\sum _i\vartheta _i T_i\vartheta _i:F_Y\to E_Y\), where
\(\sum _i\vartheta _i^2=1\) and the tangential cutoffs and bundle
transfers are the specified original ones. For each local exact symbol
polynomial \(p_i=\sum _{j=0}^m p_{ij}\kappa ^j\), set
\(b_{i\ell }=M_i^{-1}p_{i,m-\ell }:E_i\to E_i\), \(u_{i0}=I_{E_i}\), and

\[
 u_{in}=-\sum_{\ell=1}^{\min(m,n)}b_{i\ell}u_{i,n-\ell},
 \qquad c_{in}=u_{in}M_i^{-1}:F_i\to E_i,
 \qquad \mathcal Q_n=\sum_i\vartheta_i\operatorname{Op}_y(c_{in})\vartheta_i.
 \tag{NP1}
\]

Each product in the definition of \(u\) is a matrix-symbol product in
its stated order. Each product in the definition of \(\mathcal Q _n\) is
the complete tangential operator product, including its smoothing
contribution, its frame transfers, and its actual input density. They
are distinct operations. The local exact identity
\(p_i=M_i\kappa ^m(I+\sum _{\ell =1}^m b_{i\ell }\kappa ^{-\ell })\)
proves the recurrence: multiply the matrix inverse expansion by the
factor in parentheses and compare a fixed integer power. The high-normal
cone constant can be chosen so the sum has norm at most \(1/2\). The
finite geometric inverse remainder then proves, on both signed tails,
for every \(L\ge1\) and every \(\alpha ,\beta ,v\),

\[
 \left|\partial_{y,r}^{\beta}\partial_\eta^\alpha\partial_\kappa^v
  \left(q-\sum_{n=0}^{L-1}q_n\kappa^{-m-n}\right)\right|
 \le C_{L\alpha\beta v}|\kappa|^{-m-L-v}T^{L-|\alpha|},
 \qquad \mathcal Q_n\in\Psi_{\rm tan}^n(F_Y,E_Y).
 \tag{NP2}
\]

Here \(q_n\) is the complete left symbol of \(\mathcal Q _n\). The
statement for the actual cutoffs follows by the exact tangential
composition integral NC1: compact cutoff Fourier transforms have every
moment, and the complementary shift
\(|\zeta |\ge \varepsilon |\kappa |\) has arbitrarily rapid decay after
integration by parts in its compact base variable. The differentiated
local inverse estimate is AL5. For separated compact tangential
output/input supports \(\alpha (y),\beta (z)\), the original normal
amplitude has the same coefficient kernels:

\[
 \left|\partial_{y,z,r,s}^{\gamma}\partial_\kappa^v
  \left(k_Q-\sum_{n=0}^{L-1}
     \chi(r-s)\alpha(y)K_{\mathcal Q_n(r)}(y,z)\beta(z)\kappa^{-m-n}
  \right)\right|
 \le C_{\gamma Lv}|\kappa|^{-m-L-v}.
 \tag{NP3}
\]

The actual density and source/target frame factors are part of \(K\) and
are differentiated in \(\gamma \). To prove this estimate split
\(T\le \varepsilon |\kappa |\) and its complement. On the first region
insert NP2 and integrate by parts \(N\) times in \(\eta \), using the
nonzero tangential difference: \(N>L+D+d\) makes \(\int T^{L+D-N}d\eta \)
finite, where \(D\) is the requested external derivative cost. On the
complement use the original \(S^{-m,0}\) bound and choose \(N>d+L+D\).
Each subtracted \(c_{in}\) obeys the identical tail estimate after
restoring its full oscillatory integral. This is the AL8--AL10
calculation at the actual complete-cutoff scope. Properness changes away
from \(r=s\) are compact smooth full kernels and hence rapid in
\(\kappa \); the separated kernels on \(r=s\) are retained. Finally
\(\mathcal Q _0=\sum _i\vartheta _i M^{-1}\vartheta _i=M^{-1}:F_Y\to E_Y\)
exactly.

\subsubsection{NP4. Both complete error coefficient
sequences}\label{np4.-both-complete-error-coefficient-sequences}

Define the two actual errors on their own bundles:

\[
 \mathcal R_F=P_cQ-I_{F_Y}:F_Y\to F_Y,
 \qquad \mathcal R_E=QP_c-I_{E_Y}:E_Y\to E_Y.
 \tag{NP4}
\]

For \(n\ge1\) their complete normal coefficients, at \(\kappa ^{-n}\),
are

\[
 \begin{aligned}
 \mathsf F_n&=
 \sum_{\substack{0\le j\le m,\ h\ge0,\ 0\le\ell\le j\\m-j+h+\ell=n}}
       \binom j\ell A_j\delta^\ell\mathcal Q_h:F_Y\to F_Y,\\
 \mathsf E_n&=
 \sum_{\substack{0\le j\le m,\ h\ge0,\ \ell\ge0\\m-j+h+\ell=n}}
       \binom{-m-h}\ell\mathcal Q_h\delta^\ell A_j:E_Y\to E_Y.
 \end{aligned}
 \tag{NP5}
\]

Both sums are finite because \(h+\ell \le n\). For the first identity
use the exact finite normal Leibniz formula for \(D_r^j\); for the
second use the restored normal moment NC5 and
\(1/\ell !\partial _\kappa ^\ell \kappa ^{-m-h}=\operatorname{binom}(-m-h,\ell )\kappa ^{-m-h-\ell }\).
All tangential compositions are complete. At \(n=0\) the two products
have the distinct coefficients \(M\mathcal Q _0=I_F\) and
\(\mathcal Q _0M=I_E\); subtracting these gives zero, not an omitted
boundary map. Thus the first error coefficients are

\[
 \mathsf F_1=M\mathcal Q_1+mM\delta(M^{-1})+A_{m-1}M^{-1},
 \qquad
 \mathsf E_1=\mathcal Q_1M-mM^{-1}\delta M+M^{-1}A_{m-1}.
 \tag{NP6}
\]

Each coefficient has the sharper order
\(\mathsf F _n\in \Psi _{\rm tan}^{n-1}(F_Y)\) and
\(\mathsf E _n\in \Psi _{\rm tan}^{n-1}(E_Y)\). For the top tangential
order \(n\), every term with \(\ell \ge 1\) already has order
\(n-\ell \le n-1\). The \(\ell =0\) terms have principal symbols
\(\sum _{m-j+h=n}\sigma _{m-j}(A_j)\sigma _h(\mathcal Q _h)\), or the
factors in the opposite order. NP1 implies that
\(\sigma _h(\mathcal Q _h)\) is the coefficient of the inverse of the
complete leading tangential homogeneous polynomial: the top homogeneous
component of each \(c_{ih}\) has that recurrence, scalar-cutoff
derivatives have lower tangential order, and \(\sum \vartheta _i^2=1\)
preserves it. Multiplication of that polynomial with its matrix inverse
makes these degree \(n\) sums zero in both orders for \(n\ge1\). The
principal tangential symbol characterization then gives the stated drop.
This proof retains the complete coefficient after cancellation; it does
not substitute its principal part for the coefficient.

The all-length sharpened remainders are the actual NC8/AC7 ones:

\[
 \begin{aligned}
 \mathcal R_F&\sim\sum_{n\ge0}\mathsf F_{n+1}\kappa^{-1-n},
 &\mathcal R_E&\sim\sum_{n\ge0}\mathsf E_{n+1}\kappa^{-1-n},\\
 \left|\partial_x^\beta\partial_\eta^\alpha\partial_\kappa^v
  \left(r_\bullet-\sum_{n<L}e_{\bullet,n+1}\kappa^{-1-n}\right)\right|
 &\le C|\kappa|^{-1-L-v}T^{L-|\alpha|},
 &\mathcal R_\bullet&\in\mathcal L^{-1}.
 \end{aligned}
 \tag{NP7}
\]

Here \(e_{F,n}=\mathsf F _n\) and \(e_{E,n}=\mathsf E _n\); no equality
of their sequences is asserted. The strengthening from a degree-zero
cancelled expansion to NP7 uses an actual frequency derivative in the
composition remainder. For a local \(p_i\) or \(\tau _i\), the leading
coefficient is the multiplication symbol \(M_i\) or \(M_i^{-1}\);
\(\partial _\eta \) kills it and lowers every subsequent tangential order
by one. The original enhanced GI5 estimates give the same free
derivative improvement globally. A \(\partial _\kappa \) derivative
lowers every normal power. MC20 expresses the exact local composition
error by the integral of \(C_t(\partial _{\xi}p_i,D_x\tau _i)\), or
\(C_t(\partial _{\xi}\tau _i,D_xp_i)\), plus the compact full-frequency
cutoff error. NC1--NC7 is uniform in \(0\le t\le1\), because scaled base
derivatives give factors \(t^a\le1\). Hence both exact local errors have
NP7. The GI27/GI28 global sums contain these errors and the complete
scalar-cutoff commutators; each commutator has the same free derivative
improvement. Their \(\Delta _i\) terms are zero on the specified actual
supports (GI26). Finite summation gives NP7 and its separated kernel
estimate with bound \(|\kappa |^{-1-L-v}\). This identifies the extra
analytic step; cancellation of one leading coefficient alone would not
prove it.

\subsubsection{\texorpdfstring{NP8. The literal finite product
\(Q\mathcal R _F\), at every normal
coefficient}{NP8. The literal finite product Q\textbackslash mathcal R \_F, at every normal coefficient}}\label{np8.-the-literal-finite-product-qmathcal-r-_f-at-every-normal-coefficient}

This is one actual finite operator product on the original bundles,

\[
 D:=Q\mathcal R_F=\mathcal R_EQ:F_Y\longrightarrow E_Y,
 \qquad D\in\mathcal L^{-m-1}.
 \tag{NP8}
\]

Equality follows by expanding \((QP_c-I_E)Q=Q(P_cQ-I_F)\) and using
associativity of the properly supported actual operators; it neither
commutes \(Q\) through an error nor identifies \(E_Y\) with \(F_Y\). Its
complete coefficient at \(\kappa ^{-m-1-n}\), for \(n\ge0\), is

\[
 \boxed{\displaystyle
 \mathcal D_n(r)=
 \sum_{\substack{h,k,\ell\ge0\\h+k+\ell=n}}
   \binom{-m-h}{\ell}\mathcal Q_h(r)\delta^\ell\mathsf F_{k+1}(r)
 :F_Y\longrightarrow E_Y,\qquad
 \mathcal D_n\in\Psi_{\rm tan}^{n}(F_Y,E_Y).}
 \tag{NP9}
\]

The independently typed expression from the other error side is

\[
 \mathcal D_n=
 \sum_{p+h+\ell=n}
    \binom{-1-p}{\ell}\mathsf E_{p+1}\delta^\ell\mathcal Q_h.
 \tag{NP10}
\]

The two expressions are equal because NP8 is an exact operator identity
and the common signed-tail coefficients are unique. For uniqueness, hold
\(\eta \) and the base parameters fixed in the cone and multiply the
first nonzero difference coefficient by its \(\kappa \) power as
\(|\kappa |\to \infty \). NP2/NP7 make the remainder tend to zero. The
complete tangential left symbols coincide at every \(\eta \), and hence
their actual operators coincide. The complete source and target remain
\(F_Y\to E_Y\) in both expressions. Since each summand of NP9 has
tangential order at most \(h+k=n-\ell \le n\), the stated order follows
without replacing a complete tangential factor by a Taylor polynomial.

For explicit initial coefficients the full formulas are

\[
 \begin{aligned}
 \mathcal D_0&=M^{-1}\mathsf F_1,\\
 \mathcal D_1&=\mathcal Q_1\mathsf F_1+M^{-1}\mathsf F_2
                -mM^{-1}\delta\mathsf F_1,\\
 \mathcal D_2&=\mathcal Q_2\mathsf F_1+\mathcal Q_1\mathsf F_2
      +M^{-1}\mathsf F_3
      -(m+1)\mathcal Q_1\delta\mathsf F_1
      -mM^{-1}\delta\mathsf F_2
      +\frac{m(m+1)}2M^{-1}\delta^2\mathsf F_1.
 \end{aligned}
 \tag{NP11}
\]

Every \(\delta \) differentiates the complete family immediately to its
right, with its variable normal dependence. No original tangential
coefficient was made constant.

Here is the analytic product proof at every finite length, specialized
to this actual \(D\). Its exact tangential composition in a compact
chart is the integral

\[
 (f\#_{\rm tan}g)(y,\eta,\kappa)
 =(2\pi)^{-d}\operatorname{Os}\iint
    e^{-iw\cdot\zeta}f(y,\eta+\zeta,\kappa)g(y+w,\eta,\kappa)
    \,dw\,d\zeta.
 \tag{NP12}
\]

The input density and frame transfers are included before taking this
integral. Integration by parts with \((1-\Delta _w)^J\) on the compact
second amplitude gives \((1+|\zeta |^2)^{-J}\). The two exact Peetre
inequalities \(T(\eta +\zeta )^c/T(\eta )^c\le (1+|\zeta |)^{|c|}\) and
their \(R\) counterparts control all positive and negative
differentiated weights. For any fixed length and derivative list their
cost is a finite polynomial degree \(K\). Choose \(2J>K+d\). On
\(|\zeta |\le \varepsilon |\kappa |\) and \(|\kappa |\ge C'T\) the two
original normal cones hold after increasing \(C'\) and decreasing
\(\varepsilon \). Inserting the complete coefficient sequences of \(Q\)
and \(\mathcal R _F\) gives the exact tangential products
\(\mathcal Q _h\mathsf F _{k+1}\). Terms with \(h+k\ge L\) and the
original cone remainders are bounded by
\(|\kappa |^{-m-1-L-v}T^{L-|\alpha |}\). On
\(|\zeta |\ge \varepsilon |\kappa |\) the absolute Fourier majorant has
integral at most \(C|\kappa |^{D+K-2J+d}T^{-|\alpha |}\); choose \(J\)
larger for any prescribed decay. Restore the high-shift parts of each
complete coefficient product by that rapid estimate. Derivatives of the
splitting cutoff live on \(|\zeta|\) comparable to \(|\kappa |\) and add
inverse powers of \(|\kappa |\). Thus the exact tangential composition
and its full differentiated remainder, including the smoothing
contributions, are established.

For the normal product, with the exact complete tangential composition
inside, use

\[
 d(r,\kappa)
 =(2\pi)^{-1}\operatorname{Os}\iint
     e^{-it\omega}Q(r,\kappa+\omega)\mathcal R_F(r+t,\kappa)
     \,dt\,d\omega.
 \tag{NP13}
\]

A compact \(t\) cutoff equal to one on the relevant proper intermediate
supports retains the actual kernel and has every derivative zero at
\(t=0\). On \(|\omega |\le \varepsilon |\kappa |\), finite Taylor
expansion of the first factor to length \(L\) has the integral remainder

\[
 \frac{\omega^L}{(L-1)!}
 \int_0^1(1-\theta)^{L-1}
       \partial_\kappa^LQ(r,\kappa+\theta\omega)
       \,d\theta.
 \tag{NP14}
\]

The exact moment
\((2\pi )^{-1}\operatorname{Os} \int \int e^{-it\omega }\omega ^\ell \mathcal R _F(r+t,\kappa )dtd\omega \)
is \(\delta ^\ell \mathcal R _F(r,\kappa )\): Fourier inversion gives
\(\delta _0(t)\), and integration by parts contributes
\((-i)^\ell \partial _t^\ell \). A truncated frequency moment is restored
to this full moment by retaining its complementary high-shift piece. To
bound that piece, integrate \((1-\partial _t^2)^J\) by parts; after
Peetre comparison its integrable majorant is
\(C|\kappa |^D T^{-|\alpha |}(1+|\omega |)^{K-2J}\). Its
\(|\omega |\ge \varepsilon |\kappa |\) integral is at most
\(C|\kappa |^{D+K-2J+1}T^{-|\alpha |}\). Choosing \(J\) after the
requested decay proves arbitrary rapidity, including the region
\(\kappa +\omega =0\) without using the high-normal expansion there.

For NP14 integrate \(\omega ^L\) by parts \(L\) times in \(t\). The
remaining amplitude is
\(\partial _\kappa ^LQ(r,\kappa +\theta \omega )\delta ^L\mathcal R _F(r+t,\kappa )\),
with the full proper cutoff derivatives retained. The original mixed
estimates and the exact tangential estimate already proved bound it
after all requested derivatives by
\(C|\kappa |^{-m-1-L-v}T^{-|\alpha |}\), uniformly in \(\theta \) on
\([0,1]\). Another \((1-\partial _t^2)^J\) integration gives an
integrable absolute \(\omega \) majorant. The original factor
\((1-\theta )^{L-1}/(L-1)!\) is integrated over its exact interval. This
proves the claimed remainder, and its stronger \(T^{-|\alpha |}\) bound
implies the requested \(T^{L-|\alpha |}\) bound. Substitution of NP2 and
NP7 into the restored moments, together with
\(\partial _\kappa ^\ell \kappa ^{-m-h}/\ell !=\operatorname{binom}(-m-h,\ell )\kappa ^{-m-h-\ell }\),
gives NP9. All longer terms are bounded by the same cone remainder using
\(T/|\kappa |\le 1/C'\). Thus

\[
 \left|\partial_x^\beta\partial_\eta^\alpha\partial_\kappa^v
 \left(d-\sum_{n=0}^{L-1}d_n\kappa^{-m-1-n}\right)\right|
 \le C_{L\alpha\beta v}|\kappa|^{-m-1-L-v}T^{L-|\alpha|}
 \tag{NP15}
\]

for every \(L\ge1\) on both real signed tails, with the same complete
coefficient symbols \(d_n\). This is an actual kernel calculation, not a
formal-series assertion. The mixed composition theorem gives the global
\(S^{-m-1,0}\) bound away from these cones.

For disjoint external tangential supports, partition the compact
intermediate tangential variable into finitely many sufficiently small
pieces. Each piece is separated from at least one external support. On
such a separated factor NP3 or the separated NP7 bound gives a smooth
coefficient kernel, and the exact remainder has every negative
tangential Fourier order after integration by parts in its smooth
intermediate variable. Choose that order below minus the tangential
dimension, the other factor's polynomial degree, and every external
derivative cost. The low tangential-frequency integral is absolutely
bounded by \(|\kappa |^{-m-1-L-v}\); its high-frequency tail is made
arbitrarily rapid by choosing that order further below any prescribed
exponent. Perform NP13--NP14 on these exact amplitudes. In the
configuration with the second factor separated, use its smooth output
variable and keep the transposed integral in the original factor order.
The intermediate density integral is precisely the kernel of the
complete \(\mathcal D _n\). This proves

\[
 \boxed{\displaystyle
 \left|\partial_{y,z,r,s}^\gamma\partial_\kappa^v
 \left(k_D-\sum_{n=0}^{L-1}
       \chi_D(r-s)\alpha(y)K_{\mathcal D_n(r)}(y,z)\beta(z)
                         \kappa^{-m-1-n}\right)\right|
 \le C_{\gamma Lv}|\kappa|^{-m-1-L-v}.}
 \tag{NP16}
\]

All coefficient kernels, input densities and frame factors remain actual
ones. Omitted properness pieces, when the final normal difference tends
to zero while the intermediate normal point stays away, have both
factors away from their own normal diagonals; repeated frequency
differentiation gives smooth compact full kernels and rapid normal
symbols. Changing the final properness cutoff changes a compact full
kernel away from \(r=s\) only. These terms are retained in the exact
remainder. On \(r=s\) the tangentially separated normal singularity is
present and NP16 does not call it a smooth full kernel.

\subsubsection{NP17. The finite corrected operator and both retained
error
squares}\label{np17.-the-finite-corrected-operator-and-both-retained-error-squares}

Define the literal GI30 correction with \(N=2\):

\[
 Q_2:=Q-Q\mathcal R_F=Q-\mathcal R_EQ:F_Y\longrightarrow E_Y.
 \tag{NP17}
\]

It is an actual finite product. The equality uses NP8. Its complete
normal coefficients, for every \(n\), are

\[
 \mathcal H_0=M^{-1},\qquad
 \mathcal H_n=\mathcal Q_n-\mathcal D_{n-1}\quad(n\ge1),
 \qquad \mathcal H_n\in\Psi^n(F_Y,E_Y).
 \tag{NP18}
\]

NP2 and NP15, with the product expanded one coefficient fewer, prove for
every \(L\ge1\) the two-tail cone remainder
\(|\kappa |^{-m-L-v}T^{L-|\alpha |}\). For \(L=1\) the complete product
NP8 itself has lower degree \(-m-1\). For \(L\ge2\), its remainder at
product length \(L-1\) has the stronger tangential factor
\(T^{L-1-|\alpha |}\), which is bounded by \(T^{L-|\alpha |}\). The
separated estimate is \(|\kappa |^{-m-L-v}\), with coefficient kernels
of these same \(\mathcal H _n\), by NP3 and NP16. Hence
\(Q_2\in \mathcal L ^{-m}\) at the actual complete-kernel scope.

Associativity and the two original errors give the distinct exact
identities

\[
 \boxed{\displaystyle
 P_cQ_2=(I_F+\mathcal R_F)(I_F-\mathcal R_F)
          =I_F-\mathcal R_F^2:F_Y\to F_Y,
 \qquad
 Q_2P_c=I_E-\mathcal R_E^2:E_Y\to E_Y.}
 \tag{NP19}
\]

For the second identity expand
\((Q-\mathcal R _EQ)P_c=(I_E-\mathcal R _E)(I_E+\mathcal R _E)\); this
keeps the bundle types and factor order. These are exactly GI32 at
\(N=2\), with its sign
\(-(-\mathcal R _\bullet )^2=-\mathcal R _\bullet ^2\), rather than
replacing the errors by one untitled remainder.

Apply the proved actual product argument NP12--NP16 with \((-m,-1)\)
replaced by \((-1,-1)\), retaining the corresponding original bundles.
Both error powers belong to \(\mathcal L ^{-2}\). Their complete
coefficients at \(\kappa ^{-2-n}\) are

\[
 \begin{aligned}
 \mathcal J_{F,n}&=
   \sum_{p+q+\ell=n}
       \binom{-1-p}{\ell}\mathsf F_{p+1}\delta^\ell\mathsf F_{q+1}
       :F_Y\to F_Y,\\
 \mathcal J_{E,n}&=
   \sum_{p+q+\ell=n}
       \binom{-1-p}{\ell}\mathsf E_{p+1}\delta^\ell\mathsf E_{q+1}
       :E_Y\to E_Y.
 \end{aligned}
 \qquad
 \mathcal J_{\bullet,n}\in\Psi_{\rm tan}^{n}.
 \tag{NP20}
\]

For every \(L\) their cone remainders are
\(|\kappa |^{-2-L-v}T^{L-|\alpha |}\), and their separated remainders
are \(|\kappa |^{-2-L-v}\). The leading terms are the distinct full
products \(\mathcal J _{F,0}=\mathsf F _1^2\) and
\(\mathcal J _{E,0}=\mathsf E _1^2\). The next full terms are
\(\mathcal J _{F,1}=\mathsf F _1\mathsf F _2+\mathsf F _2\mathsf F _1-\mathsf F _1\delta \mathsf F _1\)
and
\(\mathcal J _{E,1}=\mathsf E _1\mathsf E _2+\mathsf E _2\mathsf E _1-\mathsf E _1\delta \mathsf E _1\);
the two products at the front do not commute.

The first corrected inverse coefficient is now obtained by actual
subtraction:

\[
 \mathcal H_1=\mathcal Q_1-M^{-1}\mathsf F_1
   =-M^{-1}A_{m-1}M^{-1}-m\delta(M^{-1})
   =-M^{-1}A_{m-1}M^{-1}+mM^{-1}(\delta M)M^{-1}.
 \tag{NP21}
\]

Differentiating \(MM^{-1}=I_F\) proves the last equality. In particular
every matrix factor and variable-normal derivative of the original \(M\)
is retained.

There is a useful exact correction to any overreading of the finite
inverse: its third normal coefficient is generally not the full inverse
coefficient. Let \(C_n\) be the complete coefficient sequence of the
unique formal two-sided inverse determined by the actual polynomial
\(P_c\). These are defined, not assumed, recursively by \(C_0=M^{-1}\)
and

\[
 C_n=-M^{-1}
 \sum_{\substack{0\le j\le m,\ 0\le\ell\le j\\h=n-m+j-\ell\ge0\\(j,\ell)\ne(m,0)}}
       \binom j\ell A_j\delta^\ell C_h.
 \tag{NP22}
\]

Every \(h<n\), every product is fully typed \(F_Y\to F_Y\) before
\(M^{-1}\), and its order is at most \(n-\ell \). This constructs each
\(C_n\). The analogous left recursion has
\(\operatorname{binom}(-m-h,\ell )C_h\delta ^\ell A_j\) followed by
\(M^{-1}\). The descending Laurent product is associative: at a fixed
derivative count \(n\), the generalized binomial identity
\(\sum _{a=u}^n \operatorname{binom}(p,a)\operatorname{binom}(q,n-a)\operatorname{binom}(a,u)=\operatorname{binom}(p,u)\operatorname{binom}(p+q-u,n-u)\)
follows by coefficients of \((1+x)^{p-u}(1+x)^q\), including negative
integer exponents. Thus the constructed right and left inverses coincide
by \(C^L\star (P_c\star C^R)=(C^L\star P_c)\star C^R\). This proves,
rather than postulates, the formal coefficient comparison; no actual
infinite parametrix is claimed.

The degree \(-2\) coefficient of NP19 gives \(M\mathcal H _2\) plus the
same lower-index expression as in NP22, with \(\mathcal H _0=C_0\) and
\(\mathcal H _1=C_1\). Subtracting the constructed full-inverse equation
therefore proves

\[
 \boxed{\displaystyle
 \mathcal H_2=C_2-M^{-1}\mathsf F_1^2
             =C_2-\mathsf E_1^2M^{-1}.}
 \tag{NP23}
\]

The two defect terms agree because the exact intertwining gives
\(\mathsf F _1M=M\mathsf E _1\), hence
\(M^{-1}\mathsf F _1^2=\mathsf E _1^2M^{-1}\). Equivalently, multiplying
NP19 coefficientwise by the constructed formal inverse proves the whole
comparison

\[
 \mathcal H_n=C_n-
  \sum_{p+q+\ell=n-2}
       \binom{-m-p}{\ell}C_p\delta^\ell\mathcal J_{F,q}
 \quad(n\ge2).
 \tag{NP24}
\]

Its other fully typed expression is the complete coefficient of
\(\mathcal R _E^2\star C\) at the same power. Every difference
\(\mathcal H _n-C_n\) has tangential order at most \(n-2\). NP23 is its
first nonzero candidate, not an assertion that it is nonzero for every
operator. This locates the finite error in the complete normal
coefficients instead of discarding it because the first two coefficients
are inverse coefficients.

\subsubsection{NP25. Exact domains and the arbitrary original boundary
orders}\label{np25.-exact-domains-and-the-arbitrary-original-boundary-orders}

All proper identities above first hold on
\(C_c^\infty(Y\times\mathbb R)\) in the displayed bundles and extend to
distributions by their actual properly supported kernels. The mixed
mapping theorem then gives, for every real \(s,t\),

\[
 \begin{aligned}
 Q &:H_{(s,t)}(F_Y)\to H_{(s+m,t)}(E_Y),\\
 Q\mathcal R_F=\mathcal R_EQ &:H_{(s,t)}(F_Y)\to H_{(s+m+1,t)}(E_Y),\\
 Q_2 &:H_{(s,t)}(F_Y)\to H_{(s+m,t)}(E_Y),\\
 \mathcal R_F^2 &:H_{(s,t)}(F_Y)\to H_{(s+2,t)}(F_Y),\\
 \mathcal R_E^2 &:H_{(s,t)}(E_Y)\to H_{(s+2,t)}(E_Y).
 \end{aligned}
 \tag{NP25}
\]

Continuity and density extend NP19 to these common Sobolev domains, with
\(P_c:H_{(s+m,t)}(E_Y)\to H_{(s,t)}(F_Y)\). The two source/target
identity bundles in NP19 are distinct throughout.

For the half-cylinder, MH3 gives the actual restriction/zero-extension
maps \(r^+Qe^+\), \(r^+De^+\), \(r^+Q_2e^+\), \(r^+\mathcal R _F^2e^+\)
and \(r^+\mathcal R _E^2e^+\) at the same gains as NP25 for \(s\ge0\)
and every real \(t\). The \(s\ge0\) restriction matters: the proof
embeds the original restriction space into normal \(L^2\) with
tangential Sobolev values before zero extension; no positive full normal
Sobolev bound for zero extension is inferred.

To retain every coefficient derivative in the original boundary source,
write the exact finite expression

\[
 C_PU=\sum_{a=0}^{m-1}\sum_{b=0}^{m-a-1}\sum_{q=0}^{m-a-b-1}
   i^{q-1}\binom{b+q}{q}
   (\partial_r^qA_{a+b+q+1})(0)U_a\otimes D_r^b\delta_0.
 \tag{NP26}
\]

It follows from
\(D_r^\ell u^0=(D_r^\ell u)^0+i^{-1}\sum _{k=0}^{\ell -1}U_{\ell -1-k}D_r^k\delta _0\)
and the exact distribution multiplication
\(A(r)(U D_r^k\delta _0)=\sum _{q=0}^k \operatorname{binom}(k,q)i^q(\partial _r^qA)(0)U D_r^{k-q}\delta _0\).
Reindex \(a=\ell -1-k\) and \(b=k-q\) to obtain exactly NP26. Thus the
\(q=0\) coefficient freeze and every \(q\ge1\) supported defect remain
distinguishable. All of them act \(E_Y\to F_Y\) before the
\(F_Y\)-valued delta source.

For any real \(\sigma \), put \(U_a\in H^{\sigma -a-1/2}(Y,E_Y)\). The
tangential coefficient of each \((a,b,q)\) term has order at most
\(m-a-b-q-1\), so its value belongs to \(H^{\sigma +b+q-m+1/2}(Y,F_Y)\).
The whole jet Gram calculation MH14--MH17, with all cross terms
retained, places the \(q\)-th part \(C_P^{[q]}U\) in
\(H_{(-m,\sigma +q)}\) supported at \(Y\). Define the actual finite
error-layer product

\[
 W:=Q_2\mathcal R_F^2=\mathcal R_E^2Q_2:F_Y\to E_Y,
 \qquad W\in\mathcal L^{-m-2}.
 \tag{NP27}
\]

The equality follows by two iterations of
\(\mathcal R _EQ=Q\mathcal R _F\) and the finite polynomial defining
\(Q_2\); it is not a commutation. The actual product proof NP12--NP16,
with orders \((-m,-2)\) or \((-2,-m)\), proves the class and full
separated normal expansions. MH4, applied with \(a=-m-2\) and the actual
supported data above, gives for every real \(\nu \)

\[
 r^+W C_P^{[q]}:
 \bigoplus_{a=0}^{m-1}H^{\sigma-a-1/2}(Y,E_Y)
 \longrightarrow\overline H_{(\nu,\sigma+q+2-\nu)}(Y\times\mathbb R_+,E_Y).
 \tag{NP28}
\]

No normal source term was dropped. Choose \(\nu >k+1/2\) for a fixed
original jet \(k<m\). The Fourier Cauchy--Schwarz trace bound, retaining
its factor \((2\pi )^{-1}\) and the integral
\(\int z^{2k}(1+z^2)^{-\nu }dz\), gives \(\gamma _k\) into
\(H^{\sigma +q+2-k-1/2}\). Applying the original complete \(B_{jk}\) of
order \(m_j-k\) therefore proves

\[
 B_j r^+W C_P^{[q]}:
 \bigoplus_{a=0}^{m-1}H^{\sigma-a-1/2}(Y,E_Y)
 \longrightarrow H^{\sigma+q+2-m_j-1/2}(Y,G_j)
 \qquad\text{for every original }m_j.
 \tag{NP29}
\]

Every row is a finite sum over the unchanged \(k\), and the same
\(\nu >m-1/2\) works for the whole row. When \(m_j>m\) no factor is
reassigned a smaller order. The \(q\ge1\) terms retain their additional
gain \(q\) separately. Equations NP27--NP29 apply to either retained
GI32 error side with its actual factor order; the equality makes their
common receiving map explicit.

\begin{figure}
\centering
\pandocbounded{\includegraphics[keepaspectratio,alt={Figure NP-F1. The actual finite product and both error bundles.}]{../figures/normal_product_261.png}}
\caption{Figure NP-F1. The actual finite product and both error
bundles.}
\end{figure}

Figure NP-F1 records the original typed operator product (NP8--NP16),
both distinct signed errors (NP19--NP20), the first unforced coefficient
defect (NP23--NP24), and the receiving map at every original boundary
order (NP26--NP29). These are exact map and coefficient identities,
without a numerical or scalar coefficient model. The full proof is
Section 14 above; the
\href{../figures/normal_product_261.py}{reproducible figure source}
retains every displayed type and factor. The finite-product derivation
here supplies its own proofs.

\subsection{15. Freezing, finite errors, and an actual
auxiliary}\label{stable-freezing-auxiliary-proof}

This editorial continuation supplies the complete realizations used in
Sections 2--8. It preserves every original paragraph and formula above.
Its collar class is the exact normal-polynomial class of the named
\href{generalized-collar-fredholm.pdf\#section*.90}{generalized
Fredholm theorem}, with degree one here. The full original normal
coefficient, separate target bundle, lower-order terms and every
boundary row remain explicit.

\subsubsection{BF1--BF3. Freezing the actual collar
family}\label{bf1bf3.-freezing-the-actual-collar-family}

The first-order object is the normal-polynomial collar class (GF1/GI1)
of the named generalized Fredholm prerequisite, with its original
interior operator supported away from the narrow collar. In particular
(BR5) is an exact collar representation, with its original target
identification. Write it in the original bundles as

\[
 P^b(t)=c(t)\bigl(D_t+K(t)\bigr),\qquad
 K(t)=-A(t)+R_0(t):E_Y\to E_Y.
 \tag{BF1}
\]

Here \(K(t)=-A(t)+R_0(t)\) is the complete tangential family. With the
original cutoff \(\chi\) of (BR6), the entire deformation is

\[
 P_u=P+u\,c(t)\chi(t)\bigl(K(0)-K(t)\bigr),\qquad0\le u\le1,
 \tag{BF2}
\]

The tangential added kernel has equal normal input and output, so its
cutoff confines both to the stated collar. Outside that collar the
original global operator is retained. At the endpoint the smaller-collar
expression is \(c(t)(D_t+K(0))\); the original variable \(c(t)\)
remains. On the original unit cosphere put
\(M_* =\sup\|(\kappa I-a(t))^{-1}\|\). Compactness makes this finite.
After shrinking the collar, uniform continuity gives
\(\|\chi(t)(a(0)-a(t))\|<1/(2M_*)\), including pure normal covectors.
The exact ordered factorization is

\[
 c(t)(\kappa I-a_u)
 =c(t)(\kappa I-a(t))
  \left[I-(\kappa I-a(t))^{-1}u\chi(t)(a(0)-a(t))\right]
 \tag{BF3}
\]

The bracket has norm-distance less than \(1/2\) from the identity and
its geometric inverse converges uniformly. This proves ellipticity for
every parameter. At the boundary \(a_u(0)=a(0)\), so the stable
projection and every original complementing boundary symbol remain
fixed. Finite symbol-seminorm mapping estimates prove continuity
\(P_u:H^s\to H^{s-1}\) for every real \(s\). Every original first-order
boundary row has target \(H^{s-m_j-1/2}(G_j)\) for \(s>1/2\). The
generalized Fredholm theorem applies for \(s\ge1\), and the
norm-homotopy theorem keeps the index constant.

\subsubsection{BF4--BF6. The same parameterized finite inverse and both
errors}\label{bf4bf6.-the-same-parameterized-finite-inverse-and-both-errors}

Use the original extension \(h\), atlas, frames, square partition,
properness factors and a common sufficiently large full-frequency cutoff
for the compact parameter family. The uniform least singular value just
proved and finite suprema of the original symbol derivatives give the
uniform GI9--GI11 constants. The actual GI24 seed and its literal finite
correction are

\[
 S_u:=Q_{2,u}=Q_u-Q_u\mathcal R_{F,u},\qquad
 \mathcal R_{F,u}=P_{c,u}Q_u-I_F,\quad
 \mathcal R_{E,u}=Q_uP_{c,u}-I_E.
 \tag{BF4}
\]

The choices at parameter zero are the original choices of Section 14.
Differentiating the original matrix inverse gives
\(\partial_u p_u^{-1}=-p_u^{-1}(\partial_u p_u)p_u^{-1}\); repeated
differentiated Leibniz formulas and the fixed finite product estimates
prove smooth parameter dependence in every required mixed seminorm. Thus
\(S_u\in\mathcal L^{-1}\), both errors belong to \(\mathcal L^{-1}\),
and both squares belong to \(\mathcal L^{-2}\), with uniform all-length
signed-tail and separated-kernel bounds. Their complete coefficients are
those of NP1, NP5, NP9 and NP20 with the original parameter-dependent
families inserted. Retain the two different exact GI32 sides:

\[
 P_{c,u}S_u=I_F-\mathcal R_{F,u}^2,\qquad
 S_uP_{c,u}=I_E-\mathcal R_{E,u}^2.
 \tag{BF5}
\]

Their ordered algebra gives

\[
 \boxed{\displaystyle
 S_u-S_0=S_u(P_{c,0}-P_{c,u})S_0
           -\mathcal R_{E,u}^2S_0
           +S_u\mathcal R_{F,0}^2.}
 \tag{BF6}
\]

Indeed, the first term on the right before the two errors is
\(S_u(I_F-\mathcal R_{F,0}^2)-(I_E-\mathcal R_{E,u}^2)S_0\). Solving for
the difference gives (BF6), without identifying the error bundles or
commuting a factor.

\subsubsection{BF7--BF10. The exact boundary gain under
freezing}\label{bf7bf10.-the-exact-boundary-gain-under-freezing}

The original first-order supported source, including its normal
coefficient and sign, is

\[
 C_uU=i^{-1}c(0)U\otimes\delta_0=:CU,
 \qquad U\in H^{\sigma-1/2}(Y,E_Y),
 \qquad CU\in H_{(-1,\sigma)}(F_Y).
 \tag{BF7}
\]

Only the term \(a=b=q=0\) of NP26 occurs when \(m=1\). Put
\(\Delta_u=P_{c,u}-P_{c,0}\). This complete tangential order-one family
has \(\Delta_u(0)=0\). Define \(J_u(r)=\Delta_u(r)/r\) off zero and
\(J_u(0)=\partial_r\Delta_u(0)\). Near zero Taylor's integral identity
\(J_u(r)=\int_0^1\partial_r\Delta_u(vr)\,dv\) proves smoothness with the
full original factors. Away from zero bounded coefficient seminorms and
the nonzero denominator give bounded derivatives. Hence
\(\Delta_u=rJ_u\) exactly. The complete normal-symbol commutator is

\[
 [S_u,r]=-i\operatorname{Op}_r(\partial_\kappa s_u):F_Y\to E_Y,
 \qquad [S_u,r]\in\mathcal L^{-2}.
 \tag{BF8}
\]

For its actual kernel multiplication by the input coordinate minus the
output coordinate contributes \(s-r\). Integration by parts in
\(\kappa\) gives precisely \(-i\partial_\kappa s_u\). The normal
properness factors are retained; multiplication by \(s-r\)
differentiates the complete normal amplitude, including those factors.
Its coefficient at \(\kappa^{-2-n}\) is \(i(n+1)\mathcal H_n(u)\), and
the full remainders gain one normal order. Thus the stated class gain is
global. Define the actual finite Cauchy layer
\(\Pi_u U=\gamma_0r^+S_uCU\). Use (BF6), \(S_ur=rS_u+[S_u,r]\), and the
zero trace of output multiplication by \(r\). The resulting exact
identity is

\[
 \boxed{\displaystyle
 \Pi_u-\Pi_0
 =-\gamma_0r^+[S_u,r]J_uS_0C
   -\gamma_0r^+\mathcal R_{E,u}^2S_0C
   +\gamma_0r^+S_u\mathcal R_{F,0}^2C.}
 \tag{BF9}
\]

The zero trace first holds for smooth input by MH4 and then on every
displayed boundary Sobolev space by continuity. The first operator
before \(C\) has class \(\mathcal L^{-2}\): its ordered factors have
degrees \(-2,1,-1\). The tangential factor \(J_u\) has zero leading
normal coefficient. The two error operators before \(C\) separately have
degree \(-3\). MH4 with supported degree one gives outputs
\(\overline H_{(\nu,\sigma+1-\nu)}\) and
\(\overline H_{(\nu,\sigma+2-\nu)}\), respectively, for every real
\(\nu\). Taking \(\nu>1/2\) proves

\[
 \begin{aligned}
 \Pi_u-\Pi_0 &:H^{\sigma-1/2}(E_Y)\to H^{\sigma+1-1/2}(E_Y),\\
 \gamma_0r^+\mathcal R_{E,u}^2S_0C,
 \quad\gamma_0r^+S_u\mathcal R_{F,0}^2C
 &:H^{\sigma-1/2}(E_Y)\to H^{\sigma+2-1/2}(E_Y).
 \end{aligned}
 \tag{BF10}
\]

For every original \(B_{j0}\in\Psi^{m_j}(E_Y,G_j)\), composition gives
target \(H^{\sigma+1-m_j-1/2}(G_j)\); the two error terms retain the
additional target \(H^{\sigma+2-m_j-1/2}(G_j)\). This holds for
unrestricted \(m_j\). The full-cylinder difference need only have degree
\(-1\); its boundary gain is proved by the exact commutator.
Differentiated all-length remainders prove continuity of every displayed
receiver in the parameter.

\subsubsection{BF11--BF12. The principal symbol of the actual finite
layer}\label{bf11bf12.-the-principal-symbol-of-the-actual-finite-layer}

For any compact parameter family in this original collar class write
\(p_\theta=M_\theta(\kappa I-a_\theta)\), with the original bundle map
\(M_\theta:E_Y\to F_Y\). The original supported source has principal
factor \(i^{-1}M_\theta\). The upper normal integral for the actual
finite layer therefore uses the typed product

\[
 p_\theta^{-1}(i^{-1}M_\theta)
 =(\kappa I-a_\theta)^{-1}M_\theta^{-1}(i^{-1}M_\theta).
 \tag{BF11}
\]

Both original factors are present; their cancellation occurs only inside
this displayed product. For positive normal output the phase
\(e^{ir\kappa}\) closes the contour upward. Its real segment is
traversed from \(-R\) to \(R\), and its upper arc from \(R\) to \(-R\),
giving positive orientation. The upper analytic subtraction of MH4
retains the arc, including the \(\kappa^{-1}\) term and all higher
tails; polynomial normal terms are retained as supported distributions.
Fourier inversion and the upper residue, followed by the one-sided
limit, give

\[
 \sigma_0(\Pi_\theta)
 =\frac1{2\pi i}\int_{\Gamma_+}(zI-a_\theta)^{-1}dz
 =q_\theta.
 \tag{BF12}
\]

The AT upper-integral estimates justify every tangential derivative and
the leading-symbol extraction uniformly on compact parameter sets. All
lower complete coefficients and both finite errors remain in
(BF4)--(BF10). Consequently \(\Pi_\theta^2-\Pi_\theta\in\Psi^{-1}\),
since its leading symbol is \(q_\theta^2-q_\theta=0\); the complete
finite layer is not asserted to be an exact projection. For freezing
\(q_u=q_0\), as (BF10) also proves.

\subsubsection{BF13--BF14. A finite complement with its exact
maps}\label{bf13bf14.-a-finite-complement-with-its-exact-maps}

Choose a finite unitary trivializing cover of the original \(G\to Y\),
real smooth functions \(\theta_i\) supported inside its charts with
\(\sum_i\theta_i^2=1\), and unitary fiber maps
\(f_i:G|_{U_i}\to\mathbb C^{r_i}\). Gram--Schmidt constructs these
frames by division by positive smooth squared lengths. Set
\(N=\sum_i r_i\), using the appropriate ranks on distinct components.
The actual embedding is

\[
 e:G\longrightarrow Y\times\mathbb C^N,
 \qquad e(g)=(\theta_i f_i(g))_i,
 \qquad e^*e=I_G.
 \tag{BF13}
\]

Every component extends smoothly by zero. Evaluating the complete
quadratic form gives \(\sum_i\theta_i^2\|g\|^2=\|g\|^2\); polarization
proves the identity in (BF13). Thus \(ee^*\) is a smooth constant-rank
selfadjoint idempotent. Its complement
\(G'=\operatorname{ran}(I-ee^*)=\ker e^*\) is a smooth bundle: a nonzero
maximal minor supplies smooth image columns locally, and their
transition matrices are smooth. Take \(H=X\times\mathbb C^N\),
\(q_G=e^*\), \(\iota_G=e\), and

\[
 \kappa:H|_Y\xrightarrow{\sim}G\oplus G',
 \quad \kappa(v)=(e^*v,(I-ee^*)v),
 \quad\kappa^{-1}(g,g')=eg+g',
 \quad q_G\iota_G=I_G,
 \quad \Pi_G=\iota_Gq_G.
 \tag{BF14}
\]

Substituting in both directions proves both inverse identities. The
entire complementary component is retained. This directly proves
(BR23)--(BR24); the unfinished Bott lesson supplies no operative
assumption.

\subsubsection{BF15--BF20. The entire auxiliary operator and its zero
index}\label{bf15bf20.-the-entire-auxiliary-operator-and-its-zero-index}

Use the same original positive scalar symbol \(\lambda(y,\eta)\) as in
(BR12). Choose complete tangential operators \(\Lambda_G,\Lambda_{G'}\)
with that symbol and transport their block sum by \(\kappa\) to
\(H|_Y\), obtaining \(\Lambda_H\). Retain every lower-order term. Choose
a product metric and density for the auxiliary collar, extended into
\(X\); the original problem's metric and density are retained. Local
symbol quantization with a real square partition constructs an interior
\(\Lambda_X\) with positive scalar real leading symbol
\(\lambda_X(x,\xi)\) for every nonzero interior covector. With the real
cutoff \(0\le\phi\le1\), equal to one near the boundary, define the
actual global operator

\[
 P_H=\phi(D_t+i\Lambda_H)
        +i(1-\phi)\Lambda_X(1-\phi):H^1(X;H)\to L^2(X;H),
 \qquad B_H=0.
 \tag{BF15}
\]

Both interior cutoff factors are present. In the transition its complete
principal factors are

\[
 p_H=\phi(t)(\kappa+i\lambda(y,\eta))I_H
              +i(1-\phi(t))^2\lambda_X(y,t,\eta,\kappa)I_H.
 \tag{BF16}
\]

If \(0<\phi<1\), the final imaginary term is strictly positive at every
nonzero covector. If \(\phi=0\), it is the invertible positive interior
symbol. If \(\phi=1\), a nonzero tangential covector gives positive
imaginary part, while \(\eta=0,\kappa\ne0\) gives the nonzero real
normal coefficient. These cases prove ellipticity. The normal equation
\((D_t+i\lambda)v=0\) has solution \(e^{\lambda t}v(0)\), so its stable
space is zero and the zero boundary target is its isomorphic
measurement. The generalized Fredholm theorem applies.

For each complete \(A=\Lambda_H\) or \(\Lambda_X\), quantize the
positive degree-one-half leading symbol \(\sqrt{\lambda_A}I\) as
\(R_A\). The full composition and adjoint formulas retain the order-zero
remainder

\[
 H_A=\frac{A+A^*}{2}-R_A^*R_A\in\Psi^0,
 \qquad \operatorname{Re}(Aw,w)=\|R_Aw\|_2^2+(H_Aw,w),
 \qquad C_A=\|H_A\|_{L^2\to L^2}<\infty.
 \tag{BF17}
\]

Its order follows from the exact cancellation of the two degree-one
principal symbols. The order-zero mapping theorem bounds its displayed
norm. Use finite suprema over collar families; for the interior term
perform the same construction after localization of \((1-\phi)u\),
retaining every chart cutoff remainder. Product-density integration by
parts gives

\[
 2\operatorname{Im}(\phi D_tu,u)
 =\|\gamma_0u\|_Y^2+\int\phi'(t)|u|^2.
 \tag{BF18}
\]

Because \(\phi\ge0\), the tangential form may be integrated with this
weight and its lower bound remains \(-\sup C_{\Lambda_H}\|u\|^2\). Apply
the interior form to \((1-\phi)u\). Retaining the derivative term and
both interior factors gives

\[
 2\operatorname{Im}(P_Hu,u)
 \ge\|\gamma_0u\|_Y^2-C_H\|u\|_X^2,
 \qquad C_H=\|\phi'\|_\infty+2\sup C_{\Lambda_H}+2C_{\Lambda_X}.
 \tag{BF19}
\]

If \(T>C_H/2\), (BF19) applied to \((P_H+iT)u=0\) implies \(u=0\). The
generalized Fredholm kernel regularity first makes every kernel element
smooth. For a cokernel annihilator, dual regularity gives a smooth
\(v\). In the same product structure the complete adjoint and Green
equation are

\[
 \begin{aligned}
 P_H^*&=\phi D_t-i\phi'-i\phi\Lambda_H^*
            -i(1-\phi)\Lambda_X^*(1-\phi),\\
 (P_Hu,v)-(u,P_H^*v)&=i(\gamma_0u,\gamma_0v)_Y.
 \end{aligned}
 \tag{BF20}
\]

Interior tests and arbitrary boundary traces in the annihilation
equation give \((P_H^*-iT)v=0\) and \(\gamma_0v=0\). For \(-P_H^*\), the
term \(-\phi D_t\) contributes \(-\|\gamma_0v\|^2-\int\phi'|v|^2\),
while its additional multiplication \(+i\phi'\) contributes
\(+2\int\phi'|v|^2\). The full sum is
\(-\|\gamma_0v\|^2+\int\phi'|v|^2\). The adjoint tangential/interior
real forms equal the original real forms. Thus the same estimate with a
negative boundary term and a finite constant
\(C_Q=C_H+2\|\phi'\|_\infty\) proves \(v=0\) once \(2T>\max(C_H,C_Q)\).
Closed range and the zero annihilator prove surjectivity, so \(P_H+iT\)
is bijective. The path \(P_H+ivTI\), \(0\le v\le1\), leaves the
principal symbols and stable boundary space fixed. Index invariance
proves \(\operatorname{ind}(P_H,0)=0\). This is a fully specified
interior completion.

\subsubsection{BF21--BF26. All-real-order approximation of the complete
annular
terms}\label{bf21bf26.-all-real-order-approximation-of-the-complete-annular-terms}

Let \(\Theta\) be any compact freezing, collapse or rotation parameter
set, and let \(k_\theta(x,\eta)\) be one complete original tangential
order-one annular symbol, including \(x=(y,t)\), its frame factors and
its compact output localization. Every base derivative has bound
\(C_\beta T(\eta)\), uniform in the parameter. Put
\(\lambda_0(\eta)=(1+|\eta|^2)^{1/2}\), retain \(T=1+|\eta|\) and
\(R=1+|(\eta,\kappa)|\), and choose \(\chi_0\) equal to one on \([0,1]\)
and zero on \([4,\infty)\). For \(0<\varepsilon\le1\) set

\[
 \omega_\varepsilon(\eta,\kappa)
       =\chi_0\left(\frac{\varepsilon^2\kappa^2}{1+|\eta|^2}\right),
 \qquad k_{\varepsilon,\theta}=k_\theta\omega_\varepsilon.
 \tag{BF21}
\]

This multiplies the complete original symbol. For fixed \(\varepsilon\),
its support satisfies \(|\kappa|\le2\lambda_0/\varepsilon\), where the
full and tangential weights are comparable. Tangential derivatives cost
\(T^{-1}\), and normal derivatives of the cutoff cost
\(\varepsilon\lambda_0^{-1}\) with their exact transition support.
Repeated derivatives prove the ordinary full-variable \(S^1\) estimates.
Expanding the original classical homogeneous terms and the full
\(1+|\eta|^2\) denominator supplies the classical expansion. Its
homogeneous cutoff vanishes on an open pure-normal cap, so each
component extends smoothly there. Quantization with the original annular
input and output factors gives an ordinary full-variable operator for
every fixed positive \(\varepsilon\).

Let \(n=d+1\) and
\(f_{\varepsilon,\theta}=k_\theta(1-\omega_\varepsilon)\), retaining its
compact output localization. Its support has
\(|\kappa|\ge\lambda_0/\varepsilon\), and \(T\le\sqrt2\lambda_0\).
Therefore every actual base derivative obeys

\[
 |\partial_x^\beta f_{\varepsilon,\theta}(x,\xi)|
 \le\sqrt2C_\beta\varepsilon\langle\xi\rangle,
 \qquad \xi=(\eta,\kappa),\quad\langle\xi\rangle=(1+|\xi|^2)^{1/2}.
 \tag{BF22}
\]

Apply \((1-\Delta_x)^J\) under the compact-base Fourier transform. The
finite actual derivative norms, with all output factors differentiated,
give

\[
 |\widehat f_{\varepsilon,\theta}(\nu,\xi)|
 \le C_J\varepsilon\langle\xi\rangle\langle\nu\rangle^{-2J}.
 \tag{BF23}
\]

Use the original Fourier convention:

\[
 \widehat{\operatorname{Op}(f)u}(\zeta)
 =(2\pi)^{-n}\int\widehat f(\zeta-\xi,\xi)\widehat u(\xi)d\xi.
 \tag{BF24}
\]

The weighted kernel for the exact map \(H^s\to H^{s-1}\) is bounded by

\[
 (2\pi)^{-n}C_J\varepsilon\,
 \langle\zeta\rangle^{s-1}\langle\xi\rangle^{1-s}
 \langle\zeta-\xi\rangle^{-2J}
 \le (2\pi)^{-n}2^{|s-1|/2}C_J\varepsilon
       \langle\zeta-\xi\rangle^{-2J+|s-1|}.
 \tag{BF25}
\]

Both comparisons
\(\langle\zeta\rangle/\langle\xi\rangle\le\sqrt2\langle\zeta-\xi\rangle\)
and its inverse follow from the triangle inequality. They justify the
displayed bound also for negative \(s-1\). Choose \(2J>n+|s-1|\). The
two Schur integrals are at most
\((2\pi)^{-n}2^{|s-1|/2}C_J\varepsilon\int\langle\nu\rangle^{-2J+|s-1|}d\nu\),
a finite number. The Schur estimate proves

\[
 \sup_{\theta\in\Theta}
 \|K_{\varepsilon,\theta}-K_\theta\|_{H^s\to H^{s-1}}
 \le C_s\varepsilon\qquad(s\in\mathbb R).
 \tag{BF26}
\]

Multiply each exact difference on both sides by the original annular
cutoffs. Their bounded multiplier norms multiply the estimate; no factor
is commuted. Smooth frame and density transfers remain in the complete
kernels and have the same finite multiplier bounds. Finite chart
summation proves the global estimate. Applying the same proof to
parameter differences proves continuity. On the unit full cosphere the
leading-symbol difference is supported in
\(|\eta|\le\varepsilon|\kappa|\) and bounded by
\(C|\eta|\le C\varepsilon\), so the original continuous mixed symbol is
approximated uniformly too.

For the compact continuous family of original Fredholm operators, use a
finite cover by half-radius Fredholm balls. Their finite minimum
positive radius and (BF26) provide one positive \(\varepsilon\) for the
entire path. Every approximant and the short segment from the original
operator to its approximant lie in the appropriate Fredholm ball, so
their indices agree. All approximated terms have both supports away from
the boundary. Consequently the complete collar operator, the normal
coefficient, every boundary map, and both finite collar errors (BF5) are
exactly retained.

\subsection{16. The complete global rotation and split
endpoint}\label{stable-global-rotation-proof}

\subsubsection{BF27--BF29. Retaining the original lower-order terms and
boundary
orders}\label{bf27bf29.-retaining-the-original-lower-order-terms-and-boundary-orders}

Let \(P_{\mathrm f}\) denote the actual endpoint of (BF2). In the
smaller collar it is \(c(t)(D_t-A(0)+R_0(0))\). Choose a real cutoff
\(\rho\), equal to one near the boundary and supported where that
equality holds. The full lower-order path is

\[
 P_{\mathrm f,v}=P_{\mathrm f}-v c(t)\rho(t)R_0(0),
 \qquad 0\le v\le1.
 \tag{BF27}
\]

Every original lower-order coefficient remains in this explicit formula.
The added operator has mixed order zero and is bounded \(H^s\to H^s\);
inclusion into \(H^{s-1}\) is compact. Its principal interior and
boundary symbols remain exactly those of the original problem. The
generalized Fredholm theorem and compact-perturbation invariance prove
that every member is Fredholm with the original index. At the endpoint,
put \(P_{\mathrm a}=P_{\mathrm f,1}\). Choose the cutoff \(\phi\)
supported where \(\rho=1\) and the original frozen equality holds. Let
\(Q\) be the actual complete quantization of the original stable symbol
\(q\), and retain the full \(\Lambda\) of (BR14). Define

\[
 \begin{split}
 A_v(t)&=(1-v\phi(t))A(0)
            +v\phi(t)\bigl(i\Lambda Q-i\Lambda(I_E-Q)\bigr),\\
 P_{\mathrm a,v}&=P_{\mathrm a}
       -v c(t)\phi(t)\bigl(i\Lambda Q-i\Lambda(I_E-Q)-A(0)\bigr).
 \end{split}
 \tag{BF28}
\]

On the support of the added term the exact collar operator is
\(c(t)(D_t-A_v(t))\); elsewhere the original full \(P_{\mathrm a}\) is
retained. On each primary summand the original nilpotent part becomes
\((1-v\phi)N_\xi\), and its eigenvalue is
\((1-v\phi)\xi+iv\phi\lambda\operatorname{sgn}(\operatorname{Im}\xi)\).
The imaginary part has the original strict sign, including both endpoint
parameters. Hence the original stable symbol remains \(q\) throughout.
The complete quantized defect \(Q^2-Q\) is retained in the operator and
does not enter that eigenvalue computation. All original boundary maps
remain complementing. Continuity follows from the finite complete symbol
bounds. Thus the entire collapse path is Fredholm with the original
index. Put \(P_{\mathrm c}=P_{\mathrm a,1}\). Where \(\phi=1\), its
exact original-bundle formula is \(c(t)[D_t+i\Lambda(I_E-2Q)]\).

For multiple original rows with arbitrary orders \(m_j\), use the exact
isomorphisms of (I9) in the closed-manifold index prerequisite, without
changing the original rows:

\[
 \begin{split}
 G&=\bigoplus_jG_j,\qquad
 R=\operatorname{diag}(J_{G_j}^{-m_j}),\\
 R &: \bigoplus_j H^{s-m_j-1/2}(G_j)
             \xrightarrow{\sim} H^{s-1/2}(G),\\
 \mathcal B^{(0)}&=R\mathcal B,\qquad
 (P,B^{(0)})=(I\oplus R)(P,B).
 \end{split}
 \tag{BF29}
\]

The complete inverse of \(R\) and every original factor of \(B\) are
retained. Restriction of the symbol
\(b^{(0)}=\sigma_0(\mathcal B^{(0)})\) to the original stable space is a
bijection, because the diagonal leading symbols of the exact order
reducers are invertible. Denote its inverse by \(s\), with image in
\(L\), exactly as in (BR17). In the original order-zero setting \(R=I\)
and \(\mathcal B^{(0)}=\mathcal B\). The target isomorphism identifies
kernels and cokernels at every \(s\ge1\); it contributes no index term.
The noninvertible alternative retains precisely the correction (BR48).

\subsubsection{BF30--BF35. The literal global two-by-two
family}\label{bf30bf35.-the-literal-global-two-by-two-family}

Use the actual finite complement (BF13)--(BF14) and the full zero-index
auxiliary (BF15). Quantize \(s\) by an actual \(S\in\Psi^0(G,E_Y)\). Let
\(\vartheta\) be supported where \(\phi=1\), equal to one on a smaller
collar, and put \(\alpha=\tau\vartheta(t)\), \(0\le\tau\le\pi/2\). The
actual operator matrix is

\[
 \mathcal Q_\alpha^{\rm op}=
 \begin{pmatrix}
  \cos^2\alpha\,Q&\cos\alpha\sin\alpha\,S q_G\\
  \cos\alpha\sin\alpha\,\iota_G\mathcal B^{(0)}Q
                    &\sin^2\alpha\,\Pi_G
 \end{pmatrix},\qquad
 \mathcal Q_0^{\rm op}=\begin{pmatrix}Q&0\\0&0\end{pmatrix}.
 \tag{BF30}
\]

No entry of this complete matrix is asserted to be an operator
projection. The principal symbol is the exact projection
\(\mathfrak q_\alpha\) of (BR28), with \(b\) replaced by \(b^{(0)}\) and
the corresponding inverse \(s\). Every ordered block product is
retained. Put \(\mathcal C(t)=\operatorname{diag}(c(t),I_H)\) and
\(\mathcal L=\operatorname{diag}(\Lambda,\Lambda_H)\). The entire global
rotation is

\[
 \begin{split}
 \mathcal P_\tau
 &=P_{\mathrm c}\oplus P_H
   -2i\mathcal C(t)\phi(t)\mathcal L
                  (\mathcal Q_\alpha^{\rm op}-\mathcal Q_0^{\rm op}),\\
 \mathcal B_\tau^{(0)}(u,v)
 &=\cos\tau\,\mathcal B^{(0)}\gamma_0u
                      +\sin\tau\,q_G\gamma_0v,\\
 \mathcal B_\tau^{\rm orig}&=R^{-1}\mathcal B_\tau^{(0)}.
 \end{split}
 \tag{BF31}
\]

The difference term has kernel confined to the stated collar and
vanishes exactly outside the support of \(\vartheta\). At \(\tau=0\) it
is zero, giving exactly the stabilized original collapse endpoint, with
\(\mathcal B_0^{\rm orig}=B\). Where \(\vartheta\) is nonzero,
\(\phi=1\), the auxiliary interior term vanishes, and (BF31) is exactly
the original-target form of (BR31). Outside that support the two full
original summands remain, including both auxiliary interior cutoff
factors. This specifies every global operator that the original collar
formula left implicit.

On this rotation region, the principal symbol and its exact inverse are

\[
 \begin{split}
 p_\tau&=\mathcal C\left[
      (\kappa-i\lambda)\mathfrak q_\alpha
       +(\kappa+i\lambda)(I-\mathfrak q_\alpha)\right],\\
 p_\tau^{-1}&=\left[
      (\kappa-i\lambda)^{-1}\mathfrak q_\alpha
       +(\kappa+i\lambda)^{-1}(I-\mathfrak q_\alpha)\right]
       \mathcal C^{-1}.
 \end{split}
 \tag{BF32}
\]

Multiply the two displayed expressions in both orders. The mixed
products vanish by
\(\mathfrak q_\alpha(I-\mathfrak q_\alpha)=0=(I-\mathfrak q_\alpha)\mathfrak q_\alpha\),
and the remaining products give their respective identity bundles. For
\(\eta\ne0\), the positive real \(\lambda\) makes both scalar
denominators nonzero at real \(\kappa\). At a pure normal covector, the
products \(\lambda\mathfrak q_\alpha\) extend continuously by zero and
the symbol is exactly \(\mathcal C\kappa\), which is invertible. Finite
compact parameter bounds give a common inverse bound on the unit mixed
cosphere. In the region where \(\phi\) varies the rotation term is zero,
so ellipticity is that already proved for (BF28) and (BF16). Outside the
collar the original elliptic operator is retained. This proves global
ellipticity at every parameter, including both cutoffs' transition
regions.

At the boundary define

\[
 \mathfrak s_\tau g=
   \binom{\cos\tau\,sg}{\sin\tau\,\iota_Gg},\qquad
 b_\tau^{(0)}=(\cos\tau\,b^{(0)},\sin\tau\,q_G).
 \tag{BF33}
\]

The identities \(b^{(0)}s=I_G\), \(qs=s\), \(sb^{(0)}q=q\), and
\(q_G\iota_G=I_G\) give all four idempotence blocks in (BR21)/(BR53).
They prove

\[
 \mathfrak q_\tau\mathfrak s_\tau=\mathfrak s_\tau,
 \qquad \mathfrak s_\tau b_\tau^{(0)}\mathfrak q_\tau=\mathfrak q_\tau,
 \qquad b_\tau^{(0)}\mathfrak s_\tau=I_G,
 \qquad \operatorname{ran}\mathfrak q_\tau
          =\operatorname{ran}\mathfrak s_\tau.
 \tag{BF34}
\]

The first identity puts the image in the stable range; the second writes
every stable vector as such an image; the third proves injectivity and
the inverse measurement. Thus the boundary map is bijective on the
complete stable bundle at every parameter. The exact original target map
\(R^{-1}\) preserves that bijectivity while retaining all row orders.

The generalized Fredholm theorem now applies to (BF31) on the fixed
domain \(H^s(E\oplus H)\), the fixed interior target
\(H^{s-1}(F\oplus H)\), and the original row targets
\(\bigoplus_jH^{s-m_j-1/2}(G_j)\), for every \(s\ge1\). The complete
operators depend continuously on \(\tau\) in these norms. Norm-homotopy
invariance and (BF19)--(BF20) give

\[
 \operatorname{ind}(\mathcal P_\tau,\mathcal B_\tau^{\rm orig})
   =\operatorname{ind}(P,B)+\operatorname{ind}(P_H,0)
   =\operatorname{ind}(P,B).
 \tag{BF35}
\]

Every annular partial term has the compact-parameter approximation
(BF21)--(BF26), with its full variable matrix symbols. The uniformly
chosen approximation and its short Fredholm segment preserve (BF35).
Near the boundary the exact finite cylinder family and both GI32 squares
remain unchanged.

\subsubsection{BF36--BF38. Endpoint, target morphism, and the
double}\label{bf36bf38.-endpoint-target-morphism-and-the-double}

At \(\tau=\pi/2\) and \(\vartheta=1\), the full matrix (BF30) is exactly
\(\operatorname{diag}(0,\Pi_G)\), since both factors \(\cos(\pi/2)\)
vanish. Thus on the original bundles

\[
 \begin{split}
 \mathcal P_{\pi/2}^{b}
 &=c(t)(D_t+i\Lambda_E)
       \oplus(D_t-i\Lambda_G)\oplus(D_t+i\Lambda_{G'}),\\
 \mathcal B_{\pi/2}^{\rm orig}(u,v_G,v_{G'})
 &=R^{-1}\gamma_0v_G.
 \end{split}
 \tag{BF36}
\]

This is an equality of complete operators, including all retained lower
terms of the three \(\Lambda\)'s. The exact block diagonal choice of
\(\Lambda_H\) makes its \(G\) and \(G'\) factors equal to those shown.
Choose a smooth \(\rho_1\), equal to one on this smaller collar and zero
outside a slightly larger collar carrying the same auxiliary splitting.
Define on the original target bundle \(F\oplus H\)

\[
 T(t)=I_F\oplus
     \left[I_H+(e^{i\pi\rho_1(t)}-1)\Pi_G\right],
 \qquad
 T(t)^{-1}=I_F\oplus
     \left[I_H+(e^{-i\pi\rho_1(t)}-1)\Pi_G\right].
 \tag{BF37}
\]

The two products equal the identity by \(\Pi_G^2=\Pi_G\). Near the
boundary this is exactly (BR39) on the auxiliary target and the identity
on \(F\); outside the collar it is the identity. The globally invertible
target morphism \(T\), the boundary target morphism \(R\), and the
original collar frame \(c^{-1}\) preserve kernels and identify
cokernels. No global identification of \(E\) with \(F\), or global
extension of \(G\) itself, is needed.

In that original collar target frame, after reordering only the
displayed boundary summands, the complete endpoint has
\(E_{\mathrm{sp}}^+=E_Y\oplus G'\), \(E_{\mathrm{sp}}^-=G\), normal
blocks \(D_t+i\Lambda^+\) and \(-D_t+i\Lambda^-\), and boundary map
\(\gamma_0|_{G}\). The global source is still \(E\oplus H\) and global
target still \(F\oplus H\). For its Hilbert realization choose product
collar metrics and density. The identity on sections is a bounded
isomorphism between the original and these new Sobolev norms, because
smooth positive densities and positive definite fiber metrics are
uniformly comparable on the compact manifold. This changes no
coefficient or bundle map of the actual operator; the same kernel and
range vectors have canonically identified quotient spaces. The
split-double prerequisite (DI19)--(DI24) explicitly permits distinct
global source and target bundles and an arbitrary elliptic interior on
the first half. Glue the second half's source by the boundary identity
and its target by this precise collar target frame. Its complementary
split half has index zero by (DI6)--(DI18), including all adjoint cutoff
terms (DA1)--(DA5). The coupling path and exact seam distribution prove
the index identity at \(H^1\). The full annular norm approximation
(DI35)--(DI42) then gives the closed continuous-symbol index.
Consequently

\[
 \operatorname{ind}(P,B)
 =\operatorname{ind}(P_{\mathrm{sp}},B_{\mathrm{sp}})
 =\operatorname{ind}\widehat P_{\mathrm{sp}}
 =\operatorname{sind}(\widehat p_{\mathrm{sp}}).
 \tag{BF38}
\]

The double in this formula has the specified zero-index complementary
half. No factor of two enters. On a component with empty boundary,
retain the original closed operator and the actual auxiliary bundle
already chosen in (BF13). There the cutoff is zero and the actual
auxiliary is \(i\Lambda_X\); the same kernel and cokernel energy proof
(BF17)--(BF20), with no boundary term, gives its zero index. The
original closed symbol-index theorem applies directly and gives the same
index equality. On zero-dimensional boundary components the boundary
spaces are finite-dimensional, so all tangential operators and boundary
maps are finite-rank and the symbol conditions on the empty cosphere
impose no restriction. The same actual operator paths, elliptic normal
coefficient, and finite-dimensional target maps remain Fredholm by the
generalized theorem; the auxiliary kernel/cokernel proof uses the actual
boundary traces and still gives zero index. No symbol argument on an
empty cosphere is used to identify a finite-dimensional trace.

\subsubsection{BF39--BF41. The stable obstruction also gives stable
realization}\label{bf39bf41.-the-stable-obstruction-also-gives-stable-realization}

Here is an editorial strengthening of (BR44)--(BR45), preserving their
source statement and exact groupoid. A class
\(\operatorname{ob}_{\partial}(P)=0\) is equivalent to the existence of
bundles \(G_1,G_2\to Y\) and an integer \(N\ge0\) with an actual smooth
isomorphism

\[
 L(P)\oplus\pi^*(G_2\oplus\underline{\mathbb C}^{N})
       \xrightarrow{\sim}
       \pi^*(G_1\oplus\underline{\mathbb C}^{N}).
 \tag{BF39}
\]

To prove the forward direction, write the virtual class on \(Y\) as
\([G_1]-[G_2]\). Equality in the Grothendieck group of bundles on
\(S^*Y\) means that there is a finite bundle \(K\) there such that
\(L(P)\oplus\pi^*G_2\oplus K\cong\pi^*G_1\oplus K\). Apply the finite
complement construction (BF13) on the compact base \(S^*Y\) to choose
\(K'\) with \(K\oplus K'\cong\underline{\mathbb C}^{N}\). Add \(K'\) to
the isomorphism and obtain exactly (BF39). Conversely the displayed
isomorphism gives \([L(P)]=\pi^*([G_1]-[G_2])\) after cancelling the
same trivial summand in the group, proving the zero class. This is an
equality of stable classes with a proved actual stable isomorphism; it
does not assert that the original unstabilized groupoid has an object.

Set \(W=G_2\oplus\underline{\mathbb C}^{N}\),
\(Z=G_1\oplus\underline{\mathbb C}^{N}\), and let
\(\beta:L(P)\oplus\pi^*W\to\pi^*Z\) be the actual isomorphism just
constructed. Choose \(W'\) with
\(W\oplus W'\cong Y\times\mathbb C^{M}\), and extend that trivial bundle
as \(H_0=X\times\mathbb C^{M}\). On its collar put the stable summand
\(H_0^-=W\), the unstable summand \(H_0^+=W'\), and use the full
positive-interior split model (DI2). Its boundary map is the actual
trace onto \(W\), and its index is zero by (DI6)--(DI18). Denote its
complete interior operator by \(P_{H_0}\) and its stable principal
projection by \(q_{H_0}\). On the stabilized original problem define the
full degree-zero boundary symbol

\[
 b_{\mathrm{st}}=
    \beta(q\oplus q_{H_0}):
       \pi^*(E_Y\oplus H_0|_Y)\longrightarrow\pi^*Z.
 \tag{BF40}
\]

Its restriction to \(L(P)\oplus\pi^*W\) is exactly \(\beta\), hence
bijective. Quantize this smooth homogeneous symbol by the original
finite chart/partition construction, retaining the complete resulting
operator \(\mathcal B_{\mathrm{st}}\), and put
\(B_{\mathrm{st}}=\mathcal B_{\mathrm{st}}\gamma_0\). The generalized
Fredholm theorem proves that \((P\oplus P_{H_0},B_{\mathrm{st}})\) is an
actual Fredholm realization. Its symbol restriction has exactly the
specified isomorphism; no inverse of an unspecified operator is assumed.
Conversely a complementing boundary symbol after such pulled-back stable
addition gives a stable isomorphism of the form (BF39), and therefore
forces the original obstruction class to vanish.

The added interior operator has a specified zero-index boundary
realization, but \(B_{\mathrm{st}}\) need not split into an original
boundary condition and that auxiliary trace. Thus this construction
asserts stable existence, without assigning an undefined index to an
original operator lacking a boundary condition. The exact affine fiber
in (BR43a) still describes every complementary symbol entry. Together
these calculations prove

\[
 \operatorname{ob}_{\partial}(P)=0
 \quad\Longleftrightarrow\quad
 \text{a pulled-back stable addition admits a complementing realization}.
 \tag{BF41}
\]

This closes the stable existence statement named after (BR45); unstable
existence remains exactly the existence of an object of (BR42).

\subsubsection{BF42--BF43. The collar representation has an exact
defect}\label{bf42bf43.-the-collar-representation-has-an-exact-defect}

The collar class used throughout this lesson is the exact
normal-polynomial class (GF1), whose degree-one formula is (BR5). If the
words ``first-order elliptic operator'' are read as allowing an
arbitrary full-variable pseudodifferential symbol, conormal
invertibility alone does not prove (BR5). A degree-one positively
homogeneous continuous symbol extends by zero to the zero covector: its
bounded unit-cosphere norm gives the exact bound
\(\|p(x,\xi)\|\le C|\xi|\), which proves continuity of that extension.
Use this extension in the tangential restriction below. The exact
principal defect for that wider reading is

\[
 d_p(y,\eta,\kappa)
   =p(y,\eta,\kappa)-c(y)\kappa-p(y,\eta,0),
 \qquad c(y)=p(y,0,1).
 \tag{BF42}
\]

Every term is a map from the original source fiber to its original
target fiber. The defect vanishes exactly when the principal symbol is
normal-linear with that specified coefficient and tangential term. This
is a necessary equality for (BR5), and that equality is an explicit
datum of the named collar prerequisite, rather than a consequence of
conormal invertibility.

For a concrete wider symbol, take one tangential variable, scalar
bundles, and a nonnegative smooth compactly supported function \(\psi\)
on \(\mathbb R\) with \(\psi(0)=1\). Define the homogeneous
tangential-normal term to be zero on \(\eta=0\), and set

\[
 \begin{split}
 p(\eta,\kappa)&=\kappa+i|\eta|\psi(\kappa/|\eta|),\qquad c=1,\\
 d_p(\eta,\kappa)&=i|\eta|\bigl(\psi(\kappa/|\eta|)-1\bigr),\\
 p_v(\eta,\kappa)&=\kappa+i|\eta|
       \bigl((1-v)\psi(\kappa/|\eta|)+v\bigr),\quad0\le v\le1.
 \end{split}
 \tag{BF43}
\]

The original \(p\) is smooth off the zero covector: near either
pure-normal direction the compact support of \(\psi\) makes the added
term identically zero. It is elliptic because a nonzero \(\kappa\) gives
a nonzero real part, while \(\kappa=0,\eta\ne0\) gives the nonzero
imaginary part \(i|\eta|\). Its conormal coefficient is the identity,
but its displayed defect is nonzero wherever \(\psi\ne1\) and
\(\eta\ne0\). Hence a general conormal argument cannot supply the
claimed representation. The last displayed formula proves the exact
bridge for this example: the entire original symbol is the
parameter-zero member, every member remains elliptic by the same
real/imaginary argument, the conormal coefficient stays fixed, and the
endpoint is the normal-linear mixed symbol \(\kappa+i|\eta|\). Quantize
the original full symbol with a fixed full-frequency cutoff near zero,
and its endpoint tangential term with a fixed tangential-frequency
cutoff near zero. Their linear combination gives a continuous family of
complete mixed symbols with exactly the displayed leading symbols; the
two complete cutoff terms remain in the actual operators. This proves a
specific deformation rather than asserting that all wider symbols have
already been reduced. The complete stable-reduction proof (BF1)--(BF41)
applies to the original declared normal-polynomial collar object.

The defect also defines an exact symbol space with its connecting maps.
Let \(\mathcal V\) be the vector space of the degree-one positively
homogeneous continuous principal symbols in the original source and
target bundles, smooth in the base and wherever \(\eta\ne0\), with
smooth conormal restriction at \((0,1)\). Let \(\mathcal W\) be its
normal-linear subspace. Define the linear map
\(N:\mathcal V\to\mathcal W\) by

\[
 \begin{split}
 (Np)(y,\eta,\kappa)&=p(y,0,1)\kappa+p(y,\eta,0),\\
 \mathcal D&=\{d\in\mathcal V:d(y,0,1)=0,\ d(y,\eta,0)=0\},\\
 N^2&=N,\qquad d_p=(I-N)p\in\mathcal D,\qquad
 \mathcal V=\mathcal W\oplus\mathcal D.
 \end{split}
 \tag{BF44}
\]

Evaluation of \(Np\) at the positive conormal recovers exactly
\(p(y,0,1)\), and evaluation at \(\kappa=0\) recovers exactly the
original tangential restriction. These two identities prove \(N^2=N\).
They also prove that \(N\) is the identity on \(\mathcal W\), and that
its kernel is exactly \(\mathcal D\). Thus the inclusion
\(\mathcal W\hookrightarrow\mathcal V\), the retraction \(N\), and the
defect projection \(I-N\) give the displayed direct sum, with every
original factor recovered by \(p=Np+d_p\). This is a proved relationship
between the two symbol presentations. Projection onto \(\mathcal W\)
does not assert preservation of ellipticity for an arbitrary matrix
symbol; the actual elliptic interpolation for the example remains the
fully calculated (BF43).

\subsection{17. Proof sources and the completed
scope}\label{proof-sources-and-the-completed-scope}

The full finite inverse and both errors are proved in Section 14; the
exact freezing receiver, uniform annular approximation and complete
auxiliary are proved in Section 15; the original-target global path, all
boundary orders, split double, stable existence criterion and collar
defect are proved in Section 16. The original reduction formulas,
examples and exercise solutions remain identifiable. The finite
complement is proved here, so this reduction uses no unproved result of
the Bott lesson.

The operative Fredholm, composition, supported-source and norm-homotopy
proofs are the linked complete course prerequisites, with their original
domains. The closed order isomorphisms are (I9), and the exact
distinct-target double is (DI19)--(DI42). The proof establishes the
stable reduction theorem for the declared normal-polynomial collar
object at every Sobolev level \(s\ge1\), including every original
boundary row order. The broader normal-nonlinear presentation defect is
explicitly calculated in (BF42)--(BF44).

Severino T. Melo, Thomas Schick and Elmar Schrohe,
\href{https://arxiv.org/abs/math/0403059v3}{\emph{A K-Theoretic Proof of
Boutet de Monvel's Index Theorem for Boundary Value Problems}}, version
3 (2005), Section 1.1, provides a related primary treatment of
invertible order changes, stable complements and zero-index boundary
realizations in Boutet de Monvel's full operator algebra. The comparison
here used their original-author TeX. Their cited existence of auxiliary
operators is not the proof of the actual auxiliary (BF15):
(BF17)--(BF20) supplies its complete kernel and cokernel argument. Their
larger operator algebra is not identified with the exact collar object
in (BR5). The calculations and editorial strengthenings here assert no
novelty.

\end{document}