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\begin{document}

\ReaderArmAliases{\ReaderOriginalAnchorStart{solving-an-elliptic-system-from-compatible-boundary-measurements}\ReaderOriginalAnchorEnd}
\section{Solving an elliptic system from compatible boundary
measurements}\label{solving-an-elliptic-system-from-compatible-boundary-measurements}

An elliptic equation leaves a finite-dimensional family of decaying
normal modes at each nonzero boundary frequency. Boundary measurements
must distinguish exactly that family. We first solve the resulting
projected operator problem, then prove the estimates that turn its
algebraic inverse into an inverse on Sobolev spaces. The last step
identifies every finite-dimensional obstruction as a smooth
density-valued section, including at the boundary.

The interior operator acts between arbitrary complex bundles of equal
finite rank. Its order is any positive integer, and the total orders of
the boundary measurements may exceed it. Neither self-adjointness,
positivity, scalar coefficients, nor a Dirichlet boundary condition is
assumed. All assertions use restriction Sobolev spaces on a compact
smooth manifold with boundary. The boundary realization below is taken
at \(s\geq m\); a separate potential estimate will hold at every real
\(s\).

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\subsection{1. The spaces, their weights, and the exact
interfaces}\label{the-spaces-their-weights-and-the-exact-interfaces}

Let \(X\) be a compact smooth manifold with boundary \(Y\), and let
\(E,F\) be smooth complex bundles of common finite rank. Choose a collar
\((y,t)\), with \(t\geq0\) pointing into \(X\), and identify its bundles
with pullbacks. Write \(D=-i\partial\). An elliptic differential
operator \(P:E\to F\) of order \(m\geq1\) has, in the collar, the form
\[
 P=\sum_{a=0}^m P_a(y,t,D_y)D_t^a,\qquad
 \operatorname{ord}P_a\leq m-a,\qquad P_m\text{ invertible}.
 \tag{BF1}
\] Boundary operators of transversal order below \(m\) have the form \[
 B_j u=\sum_{k=0}^{m-1}\mathcal B_{jk}\gamma_k u,
 \quad\gamma_k u=(D_t^ku)|_{t=0},\quad
 \operatorname{ord}\mathcal B_{jk}\leq m_j-k.
 \tag{BF2}
\] Here \(G_j\) is the target bundle, and zero entries are allowed.
Initially the \(\mathcal B_{jk}\) are differential operators. Every
proof also permits classical tangential pseudodifferential entries of
the displayed degrees. In that extension the \(m_j\) may be real. The
interior operator remains differential. We do not infer the whole
boundary calculus for arbitrary pseudodifferential interior operators
from this extension.

Put \(\mathcal E=E|_Y^{\oplus m}\), \(\mathcal G=\bigoplus_jG_j\), and
\[
 \mathcal C^s=\bigoplus_{k=0}^{m-1}H^{s-k-1/2}(Y,E),\quad
 \mathcal D^s=\bigoplus_jH^{s-m_j-1/2}(Y,G_j),\quad
 \mathcal Y^s=\bar H^{s-m}(X^\circ,F)\oplus\mathcal D^s.
 \tag{BF3}
\] The bar denotes the restrictions of whole-manifold Sobolev
distributions, with quotient norm; it does not impose zero boundary
jets. Our realization is \(A_s:\bar H^s(X^\circ,E)\to\mathcal Y^s\),
\(A_su=(Pu,Bu)\), for \(s\geq m\). The trace theorem gives
\(\gamma:\bar H^s\to\mathcal C^s\), since \(s>m-1+1/2\). Applying each
tangential entry in (BF2) then proves continuity of \(A_s\), including
negative boundary target orders when an \(m_j\) is large.

The proof uses the restriction and extension Sobolev scale, Fourier
inversion, the trace theorem at \(s>k+1/2\), smooth multiplication,
partitions of unity, compact Sobolev inclusions and interpolation of
restriction spaces. The \href{euclidean-symbol-calculus.pdf}{Euclidean
symbol calculus} and
\href{geometric-microlocal-calculus.pdf}{coordinate-free microlocal
calculus} supply asymptotic summation, composition, formal transpose,
Sobolev mapping and interior elliptic regularity.
\href{conormal-transmission.pdf}{Singularities along a submanifold and
smooth boundary passage}, Sections 10--13, supplies ordinary
transmission and the layer traces. \href{calderon-cauchy-data.pdf}{Cauchy
data from jumps and residues}, Sections 6--12, supplies the jump,
projection and smoothing interfaces displayed below. We use Hilbert
completeness, Cauchy--Schwarz, bounded adjoints and orthogonal
projections; the required functional representation is proved in Section
2. \href{fredholm-stability.pdf}{Fredholm stability}, Sections 3 and 5,
supplies norm stability, compact perturbations and the two-parametrix
criterion. Each earlier result is used with its stated hypotheses.

Here is the Calderón interface with signs fixed. Extend \(P\)
elliptically across \(Y\), and take a proper classical transmission
parametrix \(T\) with \(TP=I+R_E\), \(PT=I+R_F\), where both remainders
have smooth kernels. Define zero extension \(e^+\), interior restriction
\(r^+\), and \[
 V=r^+Te^+,\qquad K=r^+TP^c,\qquad
 P^cU=\frac1i\sum_{a=1}^m P_a\sum_{r=0}^{a-1}
             U_{a-1-r}\otimes D_t^r\delta_0.
 \tag{BF4}
\] For smooth inputs, the interface asserts \[
 u+R_E^+u=VPu+K\gamma u,\quad
 \gamma K=Q,\quad Q_{kl}\in\Psi^{k-l}_{\rm cl},\quad
 Q^2-Q\in\Psi^{-\infty},\qquad R_E^+=r^+R_Ee^+.
 \tag{BF5}
\] Moreover \(Q\gamma V:L^2(X,F)\to C^\infty(Y,\mathcal E)\) is
continuous. Notice which projection occurs here. The complementary
projection applied to \(\gamma V\) need not be smoothing. The principal
weighted symbol \(q\) projects onto all Cauchy jets of decaying
solutions of the frozen normal equation; its complementary space
consists of the opposite normal modes. Smoothness and multiplicities of
those spaces are the matrix spectral assertions in Sections 1--4 of
\href{stable-boundary-models.pdf}{Stable modes and the algebra of
boundary data}, without an eigenbasis assumption.

The original principal projection has the exact antipodal correspondence
proved in \href{calderon-cauchy-data.pdf}{Cauchy data from
jumps and residues, Section9}. Write
\(N=\operatorname{rank}E=\operatorname{rank}F\). In \(\dim X\geq3\), its
rank is necessarily \(mN/2\), and the complementing bijection therefore
forces
\(\operatorname{rank}\mathcal G=\sum_j\operatorname{rank}G_j=mN/2\).
This conclusion also applies to the real-order tangential boundary
extension here because the interior principal polynomial remains
differential and homogeneous. Equal target rank alone does not prove
complementing; the explicit three-dimensional scalar example in that
proof has equal rank and a zero measurement at a nonzero covector. In
dimension two a fixed differential target can be complementing on both
rays only if their complementary stable ranks agree, while interior
ellipticity alone need not give that agreement. No such tangential
assertion is imposed on dimension one.

All smoothing statements on a compact boundary mean a smooth kernel.
When one input is an interior \(L^2\) function we specify that domain
separately; a continuous map from \(L^2\) into \(C^\infty\) is not being
declared a smooth-kernel operator on arbitrary interior distributions.

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\subsection{2. Constructing order reductions without an index
assumption}\label{constructing-order-reductions-without-an-index-assumption}

Let \(H\) be a smooth Hermitian bundle on a closed compact manifold. For
each real \(a\) there is an invertible classical operator \[
 J_H^a:H^r(Y,H)\longrightarrow H^{r-a}(Y,H)
 \quad\text{for every real }r,
 \tag{BF6}
\] with positive scalar principal symbol \(|\eta|^a I_H\). Its inverse
is classical of order \(-a\). We need actual inverses so that changing
weights adds no undisclosed Fredholm index.

\subsubsection{2.1. The earlier construction and its exact
defect}\label{the-earlier-construction-and-its-exact-defect}

\textbf{Editorial correction.} The following earlier construction is
retained for comparison. Its claimed classical membership and classical
inverse fail at a positive noninteger order. The full norm and mapping
estimates do not remove that degree-zero defect. The corrected
construction in Section2.2 proves (BF6) at every original real exponent.

\begin{quote}
For \(a=2b>0\), choose an elliptic operator \(C\) of order \(b\) with
principal symbol \(|\eta|^b I_H\). This is obtained by patching that
symbol and quantizing. The operator \(D=I+C^*C\) has order \(2b\), is
formally self-adjoint and has the required positive principal symbol.
The elliptic parametrix gives
\(\|u\|_{H^b}\leq C_1(\|Cu\|_{L^2}+\|u\|_{L^2})\): apply a parametrix of
\(C\), and bound its smoothing remainder from \(L^2\) to \(H^b\). Thus
the quadratic form \(\|u\|_{L^2}^2+\|Cu\|_{L^2}^2\) defines an
equivalent Hilbert norm on \(H^b\). The Hilbert representation theorem
applied to that inner product solves \(Du=f\) uniquely for
\(f\in H^{-b}\). For \(r\geq b\) and \(f\in H^{r-2b}\), elliptic
regularity upgrades the same solution to \(H^r\), and the parametrix
estimate proves boundedness there. For \(r<b\), transpose the bounded
inverse at exponent \(2b-r>b\); self-adjointness yields a bounded
inverse from \(H^{r-2b}\) to \(H^r\), with both inverse identities
extending from smooth sections by density. Uniqueness follows from the
form identity after elliptic regularity, including for distributional
nullvectors. A two-sided parametrix \(D_0\) satisfies
\(D^{-1}-D_0=-D^{-1}(DD_0-I)\); this remainder has a smooth kernel
because elliptic regularity applies also to the transposed expression.
Hence \(D^{-1}\in\Psi^{-2b}_{\rm cl}\). Set \(J_H^a=D\). For \(a<0\),
invert the construction for \(-a\); set \(J_H^0=I\). No group law
between separately chosen \(J_H^a\)'s is needed.
\end{quote}

Here is the functional representation used in the form step. For a
nonzero bounded functional \(f\) on a Hilbert space, its kernel is
closed. Choose \(z\) outside that kernel and subtract its orthogonal
projection onto the kernel, obtaining \(z_0\ne0\) orthogonal to it. For
every \(u\), the vector \(u-f(u)z_0/f(z_0)\) lies in the kernel. Pairing
it with \(z_0\) expresses \(f(u)\) as a fixed inner product with \(u\),
with the conjugations determined by the chosen convention. The zero
functional is represented by zero. This proves the needed representation
from the Hilbert facts stated in Section 1. Apply it to the equivalent
form inner product on \(H^b\); it is complete because its norm is
equivalent to the Sobolev norm. This is not an invocation of a boundary
solvability theorem.

\subsubsection{2.2. The corrected operator, its inverse, and every
original
contribution}\label{the-corrected-operator-its-inverse-and-every-original-contribution}

This is a reconstruction of the local AN-03 course construction, not a
historical-source correction or novelty claim. Keep the original closed
compact smooth manifold \(Y\), Hermitian bundle \(H\), density,
cotangent norm, every real exponent and step-one classical expansion.
The receiving calculus, elliptic regularity, compact Sobolev inclusion
and Hilbert representation are the existing in-course providers. The
original \(D=I+C^*C\) is retained for comparison. The corrected operator
is separately named \(D_K\).

\paragraph{1. The obstruction for every noninteger positive
order}\label{the-obstruction-for-every-noninteger-positive-order}

Let \(a>0\) be noninteger, let \(q=\lfloor a\rfloor\), and suppose \(Y\)
has positive dimension and \(H\) has nonzero rank on the component under
consideration. For \(P\in\Psi^a_{\mathrm{cl}}(Y;H)\), its local symbol
has the step-one expansion \(p\sim\sum_{j\geq0}p_{a-j}\), with original
homogeneous coefficients. The first \(q+1\) coefficients of any putative
classical expansion of \(I+P\) must be these same coefficients:
successively divide their difference by each positive degree
\(|\eta|^{a-j}\) and take the ray limit. The identity has strictly
smaller degree than each of them. Uniqueness of homogeneous coefficients
therefore gives \[
 \sigma(I+P)-\sum_{j=0}^{q}p_{a-j}
 =I_H+r,\qquad r\in S^{a-q-1},\qquad a-q-1<0.
 \tag{OR1}
\] On each fixed nonzero ray \(r(y,t\eta)\to0\), whereas the left side
tends to \(I_H\ne0\). It cannot belong to \(S^{a-q-1}\), as a classical
order-\(a\) remainder would. Thus the obstruction applies to every such
noninteger \(a\), not only one example. It is the additional degree-zero
identity coefficient, lying outside \(\{a-j:j\in\mathbb N_0\}\). The
original full symbol still belongs to the ordinary order-\(a\) symbol
class; all its terms remain. For positive integer \(a\), degree zero is
already in that list and the identity can be added to that exact
coefficient. No alteration of the homogeneous step is used.

For the explicit circle example, use \(Y=\mathbb R/(2\pi\mathbb Z)\),
density \(dx\), scalar bundle and basis \(e_k(x)=(2\pi)^{-1/2}e^{ikx}\).
The self-adjoint elliptic multiplier \[
 Ce_k=(1+k^2)^{1/8}e_k,\qquad b=\tfrac14,
 \qquad De_k=[1+(1+k^2)^{1/4}]e_k,
 \quad a=2b=\tfrac12
 \tag{OR2}
\] retains the full original \(1+k^2\). Its classical status follows
directly from the finite Taylor formula for \((1+z)^{1/8}\): for each
positive integer \(L\), its coefficients are \(\binom{1/8}{l}\),
\(0\leq l<L\), and its integral remainder is \[
 \frac{z^L}{(L-1)!}\int_0^1(1-t)^{L-1}
 \left[\prod_{h=0}^{L-1}(\tfrac18-h)\right]
 (1+tz)^{1/8-L}\,dt .
 \tag{OR3}
\] Put \(z=\xi^{-2}\) and multiply by \(|\xi|^{1/4}\) on both frequency
rays. Every odd step-one homogeneous coefficient is zero; every even
coefficient and all derivatives of the integral remainder are retained.
The smooth full symbol at small frequency is \((1+\xi^2)^{1/8}\). The
forced leading symbol of \(D\) is \(|\xi|^{1/2}\), but \[
 1+(1+\xi^2)^{1/4}-|\xi|^{1/2}\longrightarrow1.
 \tag{OR4}
\] Taylor's formula at zero gives \((1+z)^{1/4}-1=O(z)\), so the extra
difference without the identity is \(O(|\xi|^{-3/2})\); it cannot cancel
the one. The classical first remainder would have order \(-1/2\). This
contradiction holds also on the integer frequency sequence defining the
circle multiplier.

The inverse has the same exact defect. With \(t=|\xi|\) and
\(d(\xi)=1+(1+\xi^2)^{1/4}\), \[
 d(\xi)^{-1}-t^{-1/2}
 =\frac{t^{1/2}-d(\xi)}{t^{1/2}d(\xi)},
 \qquad t[d(\xi)^{-1}-t^{-1/2}]\longrightarrow-1.
 \tag{OR5}
\] An order-\(-1/2\) classical inverse would have first remainder of
order \(-3/2\), which after multiplication by \(t\) would tend to zero.
Thus the inverse isomorphism does not establish the asserted classical
inverse.

\paragraph{2. The exact finite-rank
correction}\label{the-exact-finite-rank-correction}

For the original \(a=2b>0\), keep the chosen classical elliptic
\(C\in\Psi^b_{\mathrm{cl}}(Y;H)\) with principal symbol
\(|\eta|^b I_H\). Its original parametrix gives, for a fixed finite
\(C_1\), \[
 \|u\|_{H^b}\leq C_1(\|Cu\|_{L^2}+\|u\|_{L^2}),
 \qquad u\in H^b(Y,H).
 \tag{OR6}
\] Let \(K=\ker(C:H^b\to L^2)\). It is closed. A distributional kernel
section is smooth: if \(SC=I+R_C\) with smooth-kernel remainder, then
\(u=-R_Cu\) whenever \(Cu=0\). The \(H^b\) unit ball of \(K\) is
compact. Indeed compact \(H^b\hookrightarrow L^2\), valid because
\(b>0\), gives an \(L^2\)-convergent subsequence; (OR6) applied to its
differences upgrades it to an \(H^b\)-Cauchy subsequence in the closed
space \(K\).

For completeness this implies finite dimension. If a normed space were
infinite dimensional, start with any finite-dimensional subspace \(M\),
which is closed. Choose \(x\notin M\), write
\(\delta=\inf_{w\in M}\|x-w\|>0\), and choose \(w\in M\) with
\(\|x-w\|<2\delta\). The unit vector \((x-w)/\|x-w\|\) has distance
greater than \(1/2\) from \(M\). Repeating with the span of the
previously chosen vectors gives a sequence in the unit ball separated by
more than \(1/2\), contradicting compactness. This proves the needed
finite-dimensional claim without an index assumption on \(C\).

Choose an \(L^2\)-orthonormal smooth basis \(e_1,\ldots,e_d\) of \(K\),
using the fixed original metric and density. Write \((u,v)_{L^2}\)
linear in \(u\). The exact projection is \[
 \Pi_Ku=\sum_{j=1}^d(u,e_j)_{L^2}e_j,
 \qquad \Pi_K^*=\Pi_K,\quad\Pi_K^2=\Pi_K,
 \qquad D_K=C^*C+\Pi_K.
 \tag{OR7}
\] The empty sum is zero when \(d=0\). Its full kernel is
\(\sum_je_j(y)\otimes e_j(y')^*\), integrated against the original
density in \(y'\); it is smooth. Each coefficient is distributional
pairing with a fixed smooth section, so \(\Pi_K\) is bounded between
every pair of Sobolev spaces. In particular it is a smoothing operator,
not a discarded degree-zero identity. Thus
\(D_K\in\Psi^{2b}_{\mathrm{cl}}\), with principal symbol
\(|\eta|^{2b}I_H\).

On \(H^b\) define \[
 q_K(u,v)=(Cu,Cv)_{L^2}+(\Pi_Ku,\Pi_Kv)_{L^2},
 \quad q_K(u,u)=\|Cu\|_{L^2}^2+\|\Pi_Ku\|_{L^2}^2.
 \tag{OR8}
\] The upper bound is
\(q_K(u,u)\leq(\|C\|_{H^b\to L^2}^2+\|\Pi_K\|_{H^b\to L^2}^2)\|u\|_{H^b}^2\).
There is also \(\gamma>0\) such that
\(q_K(u,u)\geq\gamma\|u\|_{H^b}^2\). Otherwise, for each positive
integer \(n\), choose a nonzero vector with form-to-norm-squared ratio
less than \(1/n\), and divide it by its exact \(H^b\) norm to obtain
\(u_n\) of norm one. Then \(Cu_n\to0\) and \(\Pi_Ku_n\to0\) in \(L^2\).
Compact inclusion gives an \(L^2\)-Cauchy subsequence. Equation (OR6)
applied to differences makes it Cauchy in \(H^b\), with limit \(u\) of
norm one. Continuity gives \(Cu=0\) and \(\Pi_Ku=0\). But \(u\in K\)
entails \(\Pi_Ku=u\), a contradiction. If the section space is zero, any
positive \(\gamma\) gives the vacuous estimate. Every norm and both
original maps are retained in this argument.

\paragraph{3. The exact inverses on every Sobolev
space}\label{the-exact-inverses-on-every-sobolev-space}

Formal self-adjointness and projection idempotence give
\(\langle D_Ku,v\rangle=q_K(u,v)\), first on smooth sections and then on
\(H^b\) by density. The pairing is linear in its distribution and
conjugate linear in the test section. For \(f\in H^{-b}\), the
functional \(v\mapsto\langle f,v\rangle\) is bounded in the complete
equivalent form norm. The Hilbert representation proved in Section2 of
the lesson therefore gives unique \(u\in H^b\) satisfying this identity
for all \(v\in H^b\). Its exact estimate is \[
 \gamma\|u\|_{H^b}^2\leq q_K(u,u)
 =\langle f,u\rangle\leq\|f\|_{H^{-b}}\|u\|_{H^b},
 \qquad \|u\|_{H^b}\leq\gamma^{-1}\|f\|_{H^{-b}}.
 \tag{OR9}
\] For complex pairings the middle scalar is real nonnegative by the
form identity; its absolute value supplies the displayed inequality.
These are actual distributional solutions, and testing all smooth
sections gives \(D_Ku=f\).

For \(r\geq b\), \(f\in H^{r-2b}\) also belongs to \(H^{-b}\). Elliptic
regularity and the parametrix imply \(u\in H^r\) and a bounded inverse
there. More explicitly \(S_KD_K=I+R_K\) gives \(u=S_Kf-R_Ku\); the
mapping estimates yield \[
 \|u\|_{H^r}\leq\|S_K\|_{H^{r-2b}\to H^r}\|f\|_{H^{r-2b}}
 +\|R_K\|_{H^b\to H^r}\gamma^{-1}
 \|\iota\|_{H^{r-2b}\to H^{-b}}\|f\|_{H^{r-2b}}.
 \tag{OR10}
\] No smoothing contribution or embedding norm is removed.

For \(r<b\), set \(p=2b-r>b\). The adjoint of the already established
inverse \(T_p:H^{p-2b}\to H^p\) acts from \(H^{-p}\) to \(H^{2b-p}\),
which are exactly \(H^{r-2b}\) and \(H^r\). Formal self-adjointness of
\(D_K\) transposes both inverse identities. They extend from smooth
sections by their density and the bounded Sobolev maps. A distributional
nullvector of \(D_K\) is smooth by its parametrix and is zero by (OR8);
hence all these inverses agree on their common domains. Denote the
resulting compatible inverse by \(T=D_K^{-1}\).

Let \(D_0\in\Psi^{-2b}_{\mathrm{cl}}\) be a classical parametrix and
retain its whole right error \(R= D_KD_0-I\). Then \[
 T-D_0=-TR.
 \tag{OR11}
\] This is smoothing, with an explicit kernel argument. In a finite
collection of bundle charts let \(R(\cdot,y')\) be its smooth
output-variable kernel. For every input derivative \(\beta\) and every
real \(N\), \[
 \|\partial_{y'}^\beta T R(\cdot,y')\|_{H^N}
 \leq\|T\|_{H^{N-2b}\to H^N}
 \|\partial_{y'}^\beta R(\cdot,y')\|_{H^{N-2b}}.
 \tag{OR12}
\] The right side is continuous and locally uniformly bounded in \(y'\).
Difference quotients pass through the bounded map \(T\), so these are
its actual derivatives. Taking \(N\) above every requested output
derivative order plus \(\dim Y/2\) and using Sobolev embedding proves
all jointly continuous mixed kernel derivatives. All chart and density
factors remain those of the original smooth kernel. Thus
\(T\in\Psi^{-2b}_{\mathrm{cl}}\). Set \(J_H^a=D_K\) for \(a>0\),
\(J_H^a=(J_H^{-a})^{-1}\) using a separately constructed positive-order
operator for \(a<0\), and \(J_H^0=I\). These give precisely (BF6), with
its full original spaces and principal symbols, without an index
assumption or a group law.

If \(Y\) is zero dimensional, compactness makes it a finite set and all
bundle-section spaces finite dimensional. Take the actual identity for
each \(J_H^a\). All kernels are smooth and all operators belong to every
smoothing/classical order; the nonzero-covector symbol condition has
empty domain. The empty manifold and zero bundle also have their unique
invertible zero-space identity maps.

\paragraph{4. Comparison with the original operator and all receiving
maps}\label{comparison-with-the-original-operator-and-all-receiving-maps}

The original and corrected operators have the exact relation \[
 D-D_K=I-\Pi_K,
 \qquad \big[\|Cu\|_{L^2}^2+\|u\|_{L^2}^2\big]-q_K(u,u)
 =\|(I-\Pi_K)u\|_{L^2}^2.
 \tag{OR13}
\] The last identity uses the original orthogonal decomposition,
including the kernel component. On \(K\) both operators equal the
identity. The corrected form remains coercive on the entire original
space, including \(K^\perp\), by (OR6)--(OR9). In the explicit circle
example \(K=0\), so \(D_K=C^*C\) has full multiplier \((1+k^2)^{1/4}\),
and its actual inverse has multiplier \((1+k^2)^{-1/4}\); both retain
every endpoint factor and belong to the required classical classes. This
proves the mathematical bridge to the prior construction while repairing
its precise degree-zero defect.

For the original operator matrix entry orders \(b_i-a_j\), retain the
actual diagonal maps \(D_a=\operatorname{diag}(J^{a_j})\),
\(D_b=\operatorname{diag}(J^{b_i})\). The corrected \(J\)'s prove each
map \(H^r\to H^{r-a_j}\) and \(H^r\to H^{r-b_i}\). Therefore
\(D_b^{-1}MD_a\) is classical of order zero, and restoring variables
recovers every original entry order. For the boundary system, retain
\(a_k=k\), \(b_j=m_j\), \(r=s-1/2\) and both conjugations in (BF11). The
corrected operators preserve the scalar positive principal-symbol
isomorphisms and all smoothing ideals. Thus the order-minus-one errors,
projected series and (BF12)--(BF13) use the exact step-one calculus at
every real \(m_j\). The separate realization restriction \(s\geq m\),
all layer Fourier factors and every density-valued obstruction are
unchanged. This establishes the receiving proof, rather than inferring
it from a nonclassical isomorphism.

The general noninteger obstruction (OR1), the explicit symbol and
inverse tests (OR2)--(OR5), and the full repaired operator (OR6)--(OR13)
concern this course's local construction. This calculation does not
attribute the faulty construction to another author.

\paragraph{5. Every derivative of the full frequency
remainder}\label{every-derivative-of-the-full-frequency-remainder}

Write the integral factor after \(z^L\) in (OR3) as \(H_L(z)\), with
\(L\geq1\). Its \(p\)-th derivative on \([0,1]\) is obtained by
multiplying the integrand by \(t^p\prod_{h=0}^{p-1}(1/8-L-h)\) and
replacing the exponent by \(1/8-L-p\); these derivatives are bounded
because \(1+tz\geq1\). After substituting \(z=\xi^{-2}\), the full
differentiated frequency remainder is \[
 \partial_\xi^j\big[|\xi|^{1/4}\xi^{-2L}H_L(\xi^{-2})\big]
 =\sum_{h=0}^j\binom jh\partial_\xi^h(|\xi|^{1/4}\xi^{-2L})
 \sum_{\pi\in\mathcal P_{j-h}}H_L^{(|\pi|)}(\xi^{-2})
 \prod_{B\in\pi}(-1)^{|B|}(|B|+1)!\xi^{-|B|-2}.
 \tag{OR3a}
\] Here \(\mathcal P_l\) consists of all set partitions of
\(\{1,\ldots,l\}\); its empty partition at \(l=0\) gives \(H_L\). This
formula follows by the full product rule and the repeated chain rule: a
new derivative either creates a singleton block or joins exactly one
existing block, which generates every partition once. The exact
derivative \(\partial_\xi^l\xi^{-2}=(-1)^l(l+1)!\xi^{-l-2}\) supplies
every factor. On each frequency ray the first factor is the falling
product \(\prod_{v=0}^{h-1}(1/4-2L-v)\), times
\(\operatorname{sgn}(\xi)^h|\xi|^{1/4-2L-h}\). A partition contributes
\(|\xi|^{-(j-h)-2|\pi|}\). Thus each term is bounded by its complete
finite coefficient times \(|\xi|^{1/4-2L-j}\) for \(|\xi|\geq1\). This
proves every symbol derivative bound while retaining every zero
coefficient and ray sign.

\paragraph{6. The full original inverse and its exact
comparison}\label{the-full-original-inverse-and-its-exact-comparison}

The original form remains coercive, with the explicit bound \[
 \|Cu\|_{L^2}^2+\|u\|_{L^2}^2
 \geq(2C_1^2)^{-1}\|u\|_{H^b}^2.
 \tag{OR14}
\] It follows by squaring (OR6) and using \((x+y)^2\leq2(x^2+y^2)\).
Hilbert representation gives the original inverse \(T_D:H^{-b}\to H^b\).
Its every-real-order extension can be proved without assigning false
classical membership to \(D\). A parametrix \(S_A\) of the retained
classical elliptic \(A=C^*C\), with \(S_AA=I+R_A\), gives the full
identity \(u=S_Af-S_Au-R_Au\) when \(Du=f\). If \(u\in H^s\),
\(r\geq s\) and \(f\in H^{r-2b}\), it upgrades \(u\) to
\(H^{\min(r,s+2b)}\). Starting with \(s=b\) reaches every prescribed
\(r\geq b\) in finitely many steps since \(2b>0\). Each step retains the
bounded maps \(S_A:H^{r-2b}\to H^r\), \(S_A:H^s\to H^{s+2b}\),
\(R_A:H^s\to H^r\) and their embedding norms; their finite composition
yields a bounded inverse at that level. Transposing at \(2b-r\) gives
the smaller exponents as in Section3. A distributional nullvector
belongs to some Sobolev space on compact \(Y\). The same parametrix
bootstrap, with \(f=0\), successively raises its exponent by \(2b\),
makes it smooth, and the positive original form then forces it to
vanish. Thus the inverses are compatible.

Both actual inverses satisfy the exact comparison on every
distributional Sobolev domain: \[
 T_D-T=-T_D(I-\Pi_K)T:
 H^{r-2b}\longrightarrow H^{r+2b},\qquad r\in\mathbb R.
 \tag{OR15}
\] Indeed applying \(D\) gives \(- (D-D_K)T\); apply its compatible
inverse. The middle map acts on \(H^r\), and \(T_D\) at exponent
\(r+2b\) maps \(H^r\) into \(H^{r+2b}\), proving the whole stated gain.
No identity term or projection component has been suppressed. On \(K\)
both inverses are the identity; the difference is zero there. The exact
relation also retains the original nonkernel contribution and agrees
with the full circle multipliers in (OR2)--(OR5). These are proved
receiving consequences of the reconstruction, with no historical novelty
claim.

\subsubsection{2.3. Restoring every original entry
degree}\label{restoring-every-original-entry-degree}

More generally suppose an operator matrix \(M=(M_{ij})\) has entry
orders \(b_i-a_j\), so its domain factors have exponents \(r-a_j\) and
targets have exponents \(r-b_i\). Put \[
 D_a=\operatorname{diag}(J^{a_j}),\qquad
 D_b=\operatorname{diag}(J^{b_i}).
 \quad\widehat M=D_b^{-1}MD_a\in\Psi^0_{\rm cl}.
 \tag{BF7}
\] Each diagonal map identifies the equal-exponent space
\(\bigoplus H^r\) with the weighted one. Restoring the original
variables recovers each entry order individually. In particular weighted
smoothing means every entry is smoothing and is preserved by these
conjugations.

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\subsection{3. One-sided inversion on a projected
bundle}\label{one-sided-inversion-on-a-projected-bundle}

Let \(Q\in\Psi^0_{\rm cl}(Y;H,H)\) satisfy \(Q^2-Q\in\Psi^{-\infty}\),
and let \(B\in\Psi^\mu_{\rm cl}(Y;H,G)\), with real \(\mu\). Write
\(q,b\) for their principal symbols. The image of \(q\) is a smooth
bundle on \(T^*Y\setminus0\): the rank of a smooth idempotent is locally
constant, and a nonzero minor supplies local frames. This bundle need
not be the pullback of a bundle on \(Y\).

If \(b:qH\to G\) is onto at every nonzero covector, there exists
\(S\in\Psi^{-\mu}_{\rm cl}(Y;G,H)\) such that \[
 BS\equiv I_G,\qquad QS\equiv S.
 \tag{BF8}
\] Throughout this item \(\equiv\) denotes equality modulo smoothing. To
prove it, first reduce \(\mu\) to zero using (BF6). The symbol
\(c=bq:H\to G\) is onto. Give both bundles metrics. Then \(cc^*\) is
positive definite and \(t_0=c^*(cc^*)^{-1}\) is a smooth homogeneous
right inverse. Quantize it to \(T_0\); the error \(I-BQT_0\) has order
\(-1\). Asymptotically summing \(T_0\sum_{l\geq0}(I-BQT_0)^l\) gives
\(T_1\) with \(BQT_1\equiv I\), since after \(N\) terms the error is the
\(N\)-th power of an order \(-1\) operator. The complete calculus makes
the final error smoothing. Set \(S=QT_1\). Then \(BS\equiv I\) and
\(QS-S=(Q^2-Q)T_1\) is smoothing. Restoring weights gives order
\(-\mu\).

If instead \(b:qH\to G\) is one-to-one, there are
\(S'\in\Psi^{-\mu}_{\rm cl}(Y;G,H)\) and
\(S''\in\Psi^0_{\rm cl}(Y;H,H)\) with \[
 S'B+S''\equiv I_H,\qquad S''Q\equiv0.
 \tag{BF9}
\] After the same order reduction, consider the column \(C=(B,I-Q)^t\).
If \((I-q)v=0\) and \(bv=0\), then \(v\in qH\) and injectivity gives
\(v=0\). Thus its symbol \(c\) is injective; \((c^*c)^{-1}c^*\) is a
smooth left inverse. Quantization and the left version of the preceding
asymptotic series give a row \((T',T'')\) with \(T'B+T''(I-Q)\equiv I\).
Set \(S'=T'\), \(S''=T''(I-Q)\). Multiplying by \(Q\) gives the second
identity because \((I-Q)Q\) is smoothing. This proves both statements
for arbitrary ranks. Neither assertion alone promises a two-sided
inverse for the original boundary realization.

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\subsection{4. Uniqueness and restoration of every entry
degree}\label{uniqueness-and-restoration-of-every-entry-degree}

When \(b:qH\to G\) is bijective, choose the operators from both parts of
Section 3. In the quotient algebra, \[
 S'=S'BS=(S'B+S'')S=S,
 \tag{BF10}
\] where the added term vanishes because \(QS=S\) and \(S''Q=0\). Thus
the same \(S\) works in both identities. Comparing any other permitted
right inverse with this fixed left pair proves its uniqueness; comparing
a new left pair with the fixed right inverse proves uniqueness of
\(S'\), and then \(S''=I-SB\) is unique. This uniqueness is modulo
smoothing, not equality of particular quantizations, and it fails in
general if only one of the two symbol conditions holds.

Apply this to (BF2) and (BF5). The source weights are \(a_k=k\), the
target weights are \(b_j=m_j\), and the common exponent is \(r=s-1/2\).
The conjugated operators are \[
 \widehat Q=D_a^{-1}QD_a,\qquad
 \widehat{\mathcal B}=D_b^{-1}\mathcal B D_a.
 \tag{BF11}
\] They have order zero. The complementing condition is precisely that
the latter principal symbol maps \(\operatorname{ran}\widehat q\)
bijectively onto the whole target fiber. Indeed the original weighted
trace vector is the full Cauchy vector of the decaying normal solution,
and \(\mathcal B\) performs exactly the boundary measurements. The
scalar positive order reductions are fiberwise isomorphisms at every
nonzero covector.

Let \(\widehat S,\widehat S''\) be the inverses just constructed, and
undo (BF11). Then \[
 S_{kj}\in\Psi^{k-m_j}_{\rm cl}(Y;G_j,E),\qquad
 S''_{kl}\in\Psi^{k-l}_{\rm cl}(Y;E,E),
 \tag{BF12}
\] with \[
 \mathcal BS\equiv I,\quad QS\equiv S,\quad
 S\mathcal B+S''\equiv I,\quad S''Q\equiv0.
 \tag{BF13}
\] Consequently \(S:\mathcal D^s\to\mathcal C^s\) and
\(S'':\mathcal C^s\to\mathcal C^s\) are continuous for every real \(s\).
For example the \((k,j)\) entry raises \(s-m_j-1/2\) to \(s-k-1/2\),
since its order is \(k-m_j\). This arithmetic is valid even when
\(m_j\geq m\), and does not replace the separate restriction \(s\geq m\)
needed for the realization.

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\subsection{5. A quantitative one-sided layer
estimate}\label{a-quantitative-one-sided-layer-estimate}

This item proves the analytic point needed even when a transmission
operator is not an elliptic inverse. Let \(T\) be a proper classical
operator of integer order \(-m\), \(m\geq1\), having ordinary
transmission across \(t=0\), including every base and frequency jet. If
\(0\leq j<m\), define \(K_jv=r^+T(v\otimes D_t^j\delta_0)\). Then,
locally with fixed compact supports and globally after patching, \[
 \|K_jv\|_{\bar H^s(X^\circ)}
 \leq C_s\|v\|_{H^{s+j+1/2-m}(Y)}
 \qquad(s\in\mathbb R).
 \tag{BF14}
\] The same estimate holds for bundle matrices. A smooth off-diagonal
kernel contributes a smoothing operator and will be retained as such
throughout the proof.

We give the parameter estimate underlying (BF14). In a left-symbol chart
let \(a(y,t,\eta,\tau)\) be the symbol of \(T\), let
\(\lambda=\langle\eta\rangle\), and set \[
 k_j(y,t,\eta)=(2\pi)^{-1}\int e^{it\tau}
                  a(y,t,\eta,\tau)\tau^j\,d\tau,\quad t>0.
 \tag{BF15}
\] This is an oscillatory integral. It is not interpreted as an
absolutely convergent integral after arbitrary differentiation. For all
nonnegative integers \(A,N\) and multi-indices \(\alpha,\beta\), its
one-sided extension satisfies \[
 \big|t^N\partial_t^A\partial_\eta^\alpha\partial_y^\beta
                      k_j(y,t,\eta)\big|
 \leq C\lambda^{j+1-m+A-N-|\alpha|}
 \quad(t\geq0).
 \tag{BF16}
\] Equivalently its normal profile, after dividing by
\(\lambda^{j+1-m}\) and setting \(z=\lambda t\), is bounded in every
Schwartz seminorm on the closed half-line, with the usual loss
\(\lambda^{-|\alpha|}\) for tangential frequency derivatives. Constants
involve only finitely many classical symbol and transmission seminorms
for each displayed estimate.

Here are details of the normal-frequency argument. First freeze all
normal base derivatives at zero. Write \(d=j-m\), so initially
\(d\leq-1\), and rescale \(\tau=\lambda\sigma\). On \(|\sigma|\leq2\),
differentiated symbols have the bounds of a compactly supported smooth
function, times \(\lambda^d\). On the two tails, Taylor expansion of
each homogeneous component about the two normal rays gives integer
powers of \(\sigma\). Transmission makes their coefficients the same
Laurent coefficients on the positive and negative real tails. More
precisely, for any chosen number of tangential derivatives and any
integer \(L\), subtracting sufficiently many homogeneous components and
sufficiently many normal-ray Taylor terms leaves a remainder bounded
with those derivatives by \(C\lambda^d\langle\sigma\rangle^{-L}\). If a
tangential derivative is taken, include its extra factor
\(\lambda^{-1}\). This assertion follows directly from Taylor's integral
remainder on the compact normal-direction charts and the classical
symbol remainder; choosing both truncation lengths larger than \(L+|d|\)
supplies the bound.

For any finite set of these Laurent coefficients, there is a rational
function having exactly those coefficients: use the polynomial part and
a finite linear combination of \((\sigma-i)^{-h}\), \(h\geq1\).
Expanding \((\sigma-i)^{-h}\) at infinity gives a triangular system with
leading term \(\sigma^{-h}\); solve it successively to the chosen order.
Its inverse Fourier transform is a sum of derivatives of \(\delta_0\)
and, on \(z>0\), exponential polynomials \(c_hz^{h-1}e^{-z}\). Every
such polynomial is smooth and rapidly decreasing on the closed positive
half-line. The delta derivatives disappear after restriction to \(z>0\);
they are not incorrectly assigned point values at zero. The remainder
can be made to have as many integrable weighted derivatives as desired,
so its inverse Fourier transform and the desired \(z\)-derivatives and
weights are bounded. The same subtraction applied after multiplying by
powers of \(\sigma\) justifies every normal derivative. This proves the
frozen version of (BF16), including its limits at zero, by ordinary
absolutely convergent integrals after subtraction. It is the
quantitative use of all the transmission jets, not merely of the leading
parity.

For the actual symbol expand in the normal base variable to length
\(L\):

\[a(y,t,\eta,\tau)=\sum_{l<L}t^l\partial_t^la(y,0,\eta,\tau)/l!+t^La_L(y,t,\eta,\tau)\]

Each frozen term has the estimate just proved. The remainder has the
original symbol order, locally uniformly in \(t\). To bound its inverse
integral on \(0<z=\lambda t\leq1\), divide the \(\sigma\)-axis into
dyadic annuli. An annulus of radius \(R\) contributes at most
\(C R^{d+A+1}\) before integration by parts, and after \(M\)
integrations contributes at most this bound times \((zR)^{-M}\).
Splitting the sum at \(R=z^{-1}\) gives \(Cz^{-K}\), for an integer
\(K\) determined by the order and the requested derivatives. A
logarithmic borderline sum is bounded by increasing \(K\) by one. This
exponent does not depend on \(L\). Choose \(L>K\), also allowing for the
finitely many derivatives that hit \(t^L\). Its factor
\(t^L=\lambda^{-L}z^L\) then removes the singularity. For \(z\geq1\),
integrate more often in \(\sigma\); this gives any prescribed inverse
power of \(z\), and absorbs the polynomial \(z^L\). Derivatives of the
remainder in \(t,y,\eta\) obey the same symbol bounds. Thus all finite
sets of estimates (BF16) follow by choosing \(L\) sufficiently large.
This also proves smoothness up to zero without assuming that the symbol
at a positive normal base point itself has matching normal-ray tails.

We next turn this profile estimate into (BF14). Extend a smooth
half-line profile to the full line with bounded norms through any
prescribed finite order. One explicit finite-order construction on the
negative side is \(h_-(z)=\chi(z)\sum_{l=1}^{N+1}c_l h(-lz)\), where the
Vandermonde equations \(\sum_lc_l(-l)^a=1\), \(0\leq a\leq N\), match
the first \(N\) derivatives at zero. Away from zero use a cutoff and the
already rapidly decreasing profile. The resulting extension is bounded
in the finitely many weighted Sobolev norms in question; taking \(N\)
larger than their orders suffices. Apply this after normal rescaling,
rather than using a fixed-scale extension on each frequency. It
preserves the estimates in (BF16).

For completeness, the tangential operator bound needed here follows from
the ordinary Fourier-kernel proof also for these normal profiles as a
Hilbert-valued symbol. Localize to input tangential frequencies
\(|\eta|\asymp2^l\) and output frequencies \(|\theta|\asymp2^h\).
Fourier transformation in \(y\) of the compactly supported symbol, with
\(M\) integrations by parts, gives the factor
\(\langle\theta-\eta\rangle^{-M}\). For \(|h-l|\leq2\), change normal
variable to \(z=2^lt\). The \(L^2_t\) factor is \(2^{-l/2}\); each
normal or tangential derivative costs at most \(2^l\) by (BF16). Schur's
kernel bound and Plancherel therefore give, for any integer \(N\geq0\),
\[
 \|\Delta_h\widetilde K_j\Delta_l v\|_{H^N(\mathbb R^n)}
 \leq C_N 2^{l(N+j+1/2-m)}\|\Delta_l v\|_{L^2},
 \quad |h-l|\leq2.
 \tag{BF17}
\] Here \(\widetilde K_j\) is the chosen extension of the potential, and
the low-frequency blocks use \(2^l\geq1\). The estimate for a real
nonnegative \(s\leq N\) follows by weighted Fourier interpolation
between the \(N\) and zero estimates. For negative \(s\), in a fixed
output tangential annulus one has
\((1+|\theta|^2+\tau^2)^{s/2}\leq C2^{hs}\), so the zero estimate gives
the same exponent. When \(|h-l|>2\), the factor from the
\(y\)-integrations is bounded by any prescribed power of
\(2^{-\max(h,l)}\), after compensating the finite frequency volumes. The
profile estimates remain valid after those derivatives. Thus the
corresponding bound has an extra summable factor \(2^{-M|h-l|}\), with
\(M\) arbitrarily large and with any fixed Sobolev powers absorbed by
increasing \(M\).

The dyadic characterization of Fourier Sobolev norms and the elementary
convolution inequality for the summable sequence \(2^{-M|h-l|}\) now sum
(BF17). This gives a whole-space extension with norm at most the right
side of (BF14), hence also the restriction norm. Density extends the map
to every indicated boundary Sobolev space. Local trivializations and a
finite partition of unity prove the bundle and compact-manifold
versions. This completes the all-real estimate, including the negative
exponents; interpolation from nonnegative integers alone would not have
done so.

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\subsection{6. Interior forcing through a general transmission
operator}\label{interior-forcing-through-a-general-transmission-operator}

For the same general transmission \(T\) of order \(-m\), \[
 \|r^+Te^+f\|_{\bar H^{m+r}(X^\circ)}
 \leq C_r\|f\|_{\bar H^r(X^\circ)}\qquad(r\geq0).
 \tag{BF18}
\] We prove this without claiming that \(e^+:\bar H^r\to H^r\) is
bounded for all \(r\). At \(r=0\), zero extension is an \(L^2\)
isometry, and whole-space pseudodifferential continuity proves (BF18).

Induct on the nonnegative integer \(r\), simultaneously for all symbols
\(T\) with a bound controlled by finitely many seminorms. A tangential
derivative commutes with \(e^+\). For a normal derivative,
distributional differentiation gives \[
 D_t(r^+Te^+f)=r^+Te^+D_tf+r^+[D_t,T]e^+f
                  +i^{-1}r^+T((\gamma_0 f)\otimes\delta_0).
 \tag{BF19}
\] First prove this for smooth \(f\), by differentiating its zero
extension, and then use density. The commutator has order \(-m\) and
ordinary transmission, because its symbol is a normal base derivative of
that of \(T\). The induction estimate at \(r-1\) bounds the first term
in \(H^{m+r-1}\); the second is bounded there from \(f\in H^{r-1}\). The
trace theorem gives \(\gamma_0f\in H^{r-1/2}(Y)\), and (BF14) with
\(j=0\), \(s=m+r-1\), bounds the last term in the same space. Tangential
differentiation has the first two terms only. Since the integer Sobolev
norm of order \(m+r\) is controlled by the order-zero norm and these
first-derivative norms of order \(m+r-1\), the induction closes. Cutoff
commutators are handled by the same symbol estimates; smooth
off-diagonal kernels are harmless. All constants use finitely many
seminorms, so the simultaneous induction is valid.

Interpolation of the restriction Sobolev scale between consecutive
integers gives (BF18) for every real \(r\geq0\). In particular the
endpoint \(r=0\) is included. This proof, together with Section 5,
proves the two transmission estimates for arbitrary proper classical
transmission operators of integer order \(-m\). The operator need not
satisfy \(PT=I+R_F\). Its use in an actual boundary inverse will require
that additional identity.

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\subsection{7. The boundary source and its anisotropic
norm}\label{the-boundary-source-and-its-anisotropic-norm}

For the specific \(P^c\) in (BF4), normal coefficients must act on the
delta derivatives before they are evaluated. The identity
\(fD_t^r\delta=\sum_{j=0}^r\binom rj(-1)^{r-j}(D_t^{r-j}f)(0)D_t^j\delta\),
obtained by testing against a smooth function and applying Leibniz's
rule, gives the exact grouping \[
\begin{split}
 P^cU&=\sum_{j=0}^{m-1}v_j\otimes D_t^j\delta_0,\\
 v_j&=i^{-1}\sum_{\substack{l\geq0\\j+l+1\leq a\leq m}}
 \binom{a-1-l}{j}(-1)^{a-1-l-j}
 (D_t^{a-1-l-j}P_a)(y,0,D_y)U_l.
\end{split}
 \tag{BF20}
\] When the coefficients are independent of the normal variable, only
\(a=j+l+1\) remains, giving \(v_j=i^{-1}\sum_{l+j<m}P_{j+l+1}U_l\). For
variable coefficients the additional normal derivatives in (BF20) are
retained. This is precisely the same distributional source as (BF4),
with no coefficient jet removed.

Each contribution from \(U_l\) to the coefficient of \(D_t^j\delta\) has
tangential order at most \(m-j-l-1\). Hence
\(\|v_j\|_{H^{s-m+j+1/2}}\leq C\sum_l\|U_l\|_{H^{s-l-1/2}}\). Using
(BF14) in each term gives the full Poisson estimate \[
 \|KU\|_{\bar H^s(X^\circ)}
 \leq C_s\sum_{l=0}^{m-1}\|U_l\|_{H^{s-l-1/2}(Y)},
 \qquad s\in\mathbb R.
 \tag{BF21}
\] This is also valid for a general transmission \(T\) in place of the
parametrix in (BF4), with the same differential boundary source.

We verify separately the anisotropic estimate that exposes the
half-order and the restriction \(j<m\). In a chart of dimension
\(n=\dim X\), define \[
 \|w\|_{a,b}^2=(2\pi)^{-n}\int
 (1+|\eta|^2+\tau^2)^a(1+|\eta|^2)^b
                    |\widehat w(\eta,\tau)|^2\,d\eta\,d\tau.
 \tag{BF22}
\] For a boundary layer,
\(\widehat{v\otimes D_t^j\delta}=\widehat v(\eta)\tau^j\). Substitution
\(\tau=\langle\eta\rangle\sigma\) gives exactly \[
 \int_{\mathbb R}\tau^{2j}(1+|\eta|^2+\tau^2)^{-m}d\tau
 =c_{jm}\langle\eta\rangle^{2j-2m+1},\quad
 c_{jm}=\int_{\mathbb R}\sigma^{2j}(1+\sigma^2)^{-m}d\sigma<\infty.
 \tag{BF23}
\] At infinity the exponent is \(2j-2m\leq-2\), and at zero it is
nonnegative. Thus \textbf{Editorial restoration of the Fourier factors.}
Comparing the \(n\)-dimensional measure in (BF22) with the
\((n-1)\)-dimensional boundary Sobolev measure gives the exact identity
\[
 \|v\otimes D_t^j\delta\|_{-m,s}^2
 =\frac{c_{jm}}{2\pi}\|v\|_{H^{s+j-m+1/2}}^2,
 \qquad c'_{jm}=\left(\frac{c_{jm}}{2\pi}\right)^{1/2}.
 \tag{BF23a}
\] This also holds in dimension one, using measure one on
\(\mathbb R^0\). The integral \(c_{jm}\) remains exactly the one in
(BF23). If \(k\) is an integer with \(k\geq\max(s,0)\), then \[
 \|w\|_{s-m-k,k}\leq\|w\|_{-m,s},
 \tag{BF24}
\] because \((1+|\eta|^2+\tau^2)^{s-k}\leq\langle\eta\rangle^{2(s-k)}\).
Commuting each of the finitely many tangential derivatives of order at
most \(k\) through a proper \(T\in\Psi^{-m}\), and using whole-space
Sobolev continuity on each resulting commutator, proves
\(T:H^{a-m,k}\to H^{a,k}\). The integer \(k\geq0\) norm is equivalent to
the sum of the \(H^a\) norms of those derivatives, as is immediate from
its Fourier weight. Therefore \[
 \|KU\|_{\bar H^{s-k,k}}\leq C_s
                 \sum_l\|U_l\|_{H^{s-l-1/2}}.
 \tag{BF25}
\] This derivation holds at negative as well as positive \(s\). For a
parametrix, \(PKU=r^+R_FP^cU\) is smooth, and elliptic normal recovery
gives another route from (BF25) to (BF21). Our proof of (BF21) already
supplied that recovery through the transmission profiles, so it does not
use an unproved normal-regularity assertion as an extra assumption.

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\subsection{8. An explicit inverse and its left
error}\label{an-explicit-inverse-and-its-left-error}

Assume now the complementing condition, so that (BF13) holds, and use
the actual elliptic parametrix \(T\) in (BF4). Define \[
 L(f,g)=(I+KS''\gamma)Vf+KSg.
 \tag{BF26}
\] This formula produces an interior section. Each occurrence of a trace
is after applying \(V\), so the expression is meaningful at the lowest
permitted forcing regularity \(f\in L^2\).

Set \(R=I-S\mathcal B-S''(I-Q)\); it is a smoothing matrix by (BF13).
For a smooth \(u\), write \(U=\gamma u\), \(f=Pu\),
\(g=Bu=\mathcal BU\). Taking traces in (BF5) yields
\((I-Q)U=\gamma Vf-\gamma R_E^+u\). Substitution into the definition of
\(R\) gives the exact identity \[
 U=Sg+S''\gamma Vf-S''\gamma R_E^+u+RU.
 \tag{BF27}
\] Use (BF5) once more to obtain \[
 u=L(Pu,Bu)+\mathcal R u,
 \quad
 \mathcal R=K(R\gamma-S''\gamma R_E^+)-R_E^+.
 \tag{BF28}
\] The map \(\mathcal R:\bar H^m(X^\circ,E)\to C^\infty(X,E)\) is
continuous. Indeed \(\gamma\) maps that space into the finite sum of
boundary Sobolev spaces in (BF3); \(R\) makes the result smooth. The map
\(R_E^+\) takes interior \(L^2\) continuously into smooth sections on a
neighborhood of \(X\). Applying \(S''\) preserves smoothness of its
trace, and \(K\) maps smooth boundary sections continuously into
\(C^\infty(X)\) by (BF21) at all high orders and Sobolev embedding.
These observations prove every term of the assertion, including its
actual input domain. In particular the \(u\)-dependent smoothing term in
(BF27) is accounted for; it has not been dropped in constructing \(L\).

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\subsection{9. Both rows of the right
error}\label{both-rows-of-the-right-error}

Let \(R_F^+=r^+R_Fe^+\) and \(H=r^+R_FP^c\). From \(PT=I+R_F\) and
restriction away from the delta support, \(PV=I+R_F^+\) and \(PK=H\).
Therefore \[
 PL(f,g)=f+K_1f+K_2g,
 \quad K_1=R_F^++HS''\gamma V,\quad K_2=HS.
 \tag{BF29}
\] Here \(K_1:L^2(X,F)\to C^\infty(X,F)\) continuously, and
\(K_2:\mathcal D'(Y,\mathcal G)\to C^\infty(X,F)\) continuously. The
latter has a kernel smooth in the boundary input and the interior output
up to the boundary: differentiating the smooth kernel of \(R_F\),
restricting its second variable to \(Y\), and transposing the boundary
operator \(S\) in that variable preserves smoothness. This direct kernel
observation also checks continuity in the strong distribution topology,
rather than only at one fixed Sobolev order.

Taking the trace of (BF26) gives
\(BL(f,g)=\mathcal B(I+QS'')\gamma Vf+\mathcal BQSg\). The quotient
algebra (BF13) yields \[
 \mathcal B(I+QS'')
 \equiv\mathcal B+\mathcal BQ-\mathcal BQS\mathcal B
 \equiv\mathcal BQ.
 \tag{BF30}
\] The second step uses both \(QS\equiv S\) and \(\mathcal BS\equiv I\),
with their specified domains. Define the actual operators \[
 F_0=\mathcal B(I+QS'')-\mathcal BQ,\quad
 K_3=(\mathcal BQ+F_0)\gamma V,\quad
 K_4=\mathcal BQS-I.
 \tag{BF31}
\] Both \(F_0\) and \(K_4\) are smoothing matrices. The Calderón
interface makes \(Q\gamma V\) continuous from \(L^2\) to \(C^\infty\);
the other term \(F_0\gamma V\) has that property by the trace bound
following (BF18). Thus \[
 BL(f,g)=g+K_3f+K_4g,
 \quad K_3:L^2(X,F)\to C^\infty(Y,\mathcal G),
 \tag{BF32}
\] continuously, and \(K_4\) has a smooth kernel on \(Y\times Y\).
Equations (BF29) and (BF32) specify the entire right error; neither row
is inferred merely from the left error.

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\subsection{10. The Fredholm realization and regularity of
solutions}\label{the-fredholm-realization-and-regularity-of-solutions}

For every real \(s\geq m\), (BF18) gives \(V:\bar H^{s-m}\to\bar H^s\),
hence \(\gamma V:\bar H^{s-m}\to\mathcal C^s\). Equations (BF12) and
(BF21) show that the two remaining terms in (BF26) have the same mapping
property. Therefore \[
 L:\mathcal Y^s\longrightarrow\bar H^s(X^\circ,E)
 \quad\text{is bounded for every }s\geq m.
 \tag{BF33}
\] The identities (BF28), (BF29) and (BF32), first proved on smooth
sections, extend by density to these spaces. Their errors are compact at
this fixed level: each maps bounded sets into a higher Sobolev space on
a compact manifold, and the inclusion back to the target level is
compact. Section 5 of \href{fredholm-stability.pdf}{Finite defects under
perturbation} now proves \[
 A_s:\bar H^s(X^\circ,E)\longrightarrow\mathcal Y^s
 \text{ is Fredholm},\qquad s\geq m.
 \tag{BF34}
\] The result includes closed range, finite-dimensional kernel and
finite-dimensional cokernel. An approximate inverse is not being
confused with exact solvability of arbitrary data.

There is also a regularity statement with the same exact spaces. If
\(u\in\bar H^m\), \(Pu\in\bar H^{s-m}\) and \(Bu\in\mathcal D^s\) for
some \(s\geq m\), then (BF28) and (BF33) give \(u\in\bar H^s\). In
particular a nullvector is smooth up to \(Y\), because the right side of
(BF28) is \(\mathcal R u\). Thus the kernel is the same
finite-dimensional smooth space at every permitted exponent. Nothing
here replaces a trace theorem by evaluating an arbitrary distribution at
the boundary.

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\subsection{11. Why Fredholmness forces the full symbol
condition}\label{why-fredholmness-forces-the-full-symbol-condition}

Conversely, start with a smooth differential operator of order \(m\)
between equal-rank bundles and boundary operators (BF2), without
assuming interior ellipticity or the complementing condition.
Fredholmness of \(A_s\) at one \(s\geq m\) forces both interior
ellipticity and the bijectivity of the boundary symbol on the decaying
normal space. The argument also distinguishes the two separate failures
of the boundary condition.

We recall the elementary high-frequency symbol test with its
normalization. In a coordinate ball of dimension \(d\), fix
\(\xi_0\ne0\), a fiber vector \(v\), and \(\chi\in C_c^\infty\) with
\(L^2\) norm one. The functions
\(w_\lambda(x)=\lambda^{d/4}e^{i\lambda\xi_0\cdot x}\chi(\lambda^{1/2}(x-x_0))v\)
are bounded in \(L^2\), converge weakly to zero, and for an order-zero
classical operator \(C\), \[
 \|Cw_\lambda-\lambda^{d/4}e^{i\lambda\xi_0\cdot x}
       \chi(\lambda^{1/2}(x-x_0))c_0(x_0,\xi_0)v\|_{L^2}
 \longrightarrow0.
 \tag{BF35}
\] To verify the estimate, write the Fourier transform as a packet
centered at \(\lambda\xi_0\) of width \(\lambda^{1/2}\). In the region
of that width, Taylor's formula in \(x\) and direction \(\xi/|\xi|\)
makes the leading-symbol error \(O(\lambda^{-1/2})\), and the lower
symbol has size \(O(\lambda^{-1})\). Outside a fixed enlarging multiple
of the packet width, the Fourier transform of \(\chi\) decays faster
than any power. Splitting into those two regions and using the
finite-seminorm \(L^2\) bound proves (BF35). Applying the identical
argument to the adjoint gives its dual version. Smooth bundle frames and
cutoffs change none of these limits.

A Fredholm operator has a bounded inverse modulo finite-rank operators
by Section 5 of \href{fredholm-stability.pdf}{Finite defects under
perturbation}. Thus it cannot take a weakly null sequence of unit norm
to zero in norm: a finite-rank remainder tends to zero in norm on such a
sequence. If \(p(x_0,\xi_0)\) has a nullvector at an interior point,
normalize the above packet in \(H^s\) by a factor asymptotic to
\(\lambda^{-s}\). Differentiating the differential operator gives
\(\|Pu_\lambda\|_{H^{s-m}}\to0\), while all boundary traces vanish. This
contradicts the preceding Fredholm fact. Since the interior bundles have
equal rank, injectivity of the square symbol is equivalent to
invertibility. The same test proves the symbol bound at a boundary
point: first choose interior points approaching that point, where the
smallest singular value approaches zero, then choose the frequency and
the support of the packet inside that point's coordinate ball so that
all frozen-symbol errors and all compact remainders tend to zero. A
diagonal sequence supplies the contradiction. Consequently \(P\) is
elliptic up to the boundary, so the Calderón construction and the
estimates already proved apply even before any assumption on \(B\).

To test boundary injectivity, use the order reductions of (BF11), which
turn \(\mathcal C^s\) and \(\mathcal D^s\) into equal-exponent spaces;
follow them by invertible reductions to \(L^2\). These further
reductions have scalar principal symbols and leave the principal
projection and measurement symbols unchanged up to conjugacy. Call the
resulting order-zero operators \(Q_0,B_0\). If
\(0\ne v\in\operatorname{ran}q_0(y_0,\eta_0)\) and \(b_0v=0\), choose
normalized boundary packets \(w_\lambda\) with that vector and undo the
reductions to obtain \(U_\lambda\) bounded and weakly null in
\(\mathcal C^s\). Formula (BF35) gives
\(\|QU_\lambda-U_\lambda\|_{\mathcal C^s}\to0\) and
\(\|\mathcal BQU_\lambda\|_{\mathcal D^s}\to0\). Set
\(u_\lambda=KU_\lambda\). By (BF21) it is bounded and weakly null in
\(\bar H^s\), and its norm is bounded away from zero since
\(\gamma u_\lambda=QU_\lambda\) and the trace map is bounded. Moreover
\(Pu_\lambda=r^+R_FP^cU_\lambda\to0\) in \(\bar H^{s-m}\), since this
map is smoothing and therefore compact on the fixed input Sobolev
spaces. The boundary output is \(\mathcal BQU_\lambda\to0\). After
normalizing the input norms, this contradicts Fredholmness.

Finally suppose the restricted symbol is not onto. Choose a nonzero dual
target vector \(h_0\) annihilating \(b_0\operatorname{ran}q_0\). The
dual packet construction gives target vectors
\(g_\lambda\in\mathcal D^s\) and dual vectors
\(h_\lambda\in(\mathcal D^s)'\), both bounded and weakly null, with \[
 \langle g_\lambda,h_\lambda\rangle=1,
 \qquad\|(\mathcal BQ)'h_\lambda\|_{(\mathcal C^s)'}\to0.
 \tag{BF36}
\] For example choose the same scalar packet for both after the \(L^2\)
reduction and dual unit fiber vectors with pairing one. The last limit
is (BF35) for the transposed order-zero symbol. Let \(Z\) be a bounded
generalized inverse of the Fredholm realization and put
\(u_\lambda=Z(0,g_\lambda)\). Then \(u_\lambda\) is bounded and weakly
null in \(\bar H^s\), and the finite-rank error gives
\(Pu_\lambda\to0\), \(Bu_\lambda-g_\lambda\to0\) in the respective
target norms. The trace of (BF5) implies
\((I-Q)\gamma u_\lambda=\gamma VPu_\lambda-\gamma R_E^+u_\lambda\to0\)
in \(\mathcal C^s\): the first term tends to zero by (BF18), the second
by compactness of the smoothing map. Hence
\(g_\lambda-\mathcal BQ\gamma u_\lambda\to0\). Pairing with
\(h_\lambda\) contradicts (BF36), since the bounded Cauchy vectors
paired with \((\mathcal BQ)'h_\lambda\) give a limit zero. This proves
surjectivity separately, without a boundary-rank assumption that would
make it automatic. Together the two tests prove necessity of the full
complementing condition.

When \(Y\) is zero-dimensional, its cotangent bundle has no nonzero
vectors, so the boundary symbol tests are vacuous. Boundary spaces are
finite-dimensional, and the one-dimensional interior elliptic ODE has
finite-dimensional Cauchy data and closed-range estimates. The same
parametrix proof applies with all boundary matrices smoothing. No
nonexistent tangential high-frequency sequence is used in that case.

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\subsection{12. Recovering smooth interior dual sections at the
boundary}\label{recovering-smooth-interior-dual-sections-at-the-boundary}

Fix a smooth positive density only to identify local distribution
coordinates. Intrinsically the continuous bilinear dual of an
\(F\)-section is an \(F^*\otimes\Omega_X\)-valued distribution, where
\(\Omega_X\) is the density bundle. We write \(P^\dagger\) for the
formal transpose defined by
\(\langle Pu,v\rangle=\langle u,P^\dagger v\rangle\) for compactly
supported interior test sections. For the bilinear convention
\(D_t^\dagger=-D_t\). Using Hermitian anti-duals instead gives the
equivalent usual Hilbert adjoint formulation; none of the solvability
conditions depend on that choice.

At level \(s=m\), a continuous relation on the range has the form \[
 \langle Pu,v\rangle+\sum_j\langle B_ju,h_j\rangle=0
 \quad(u\in C^\infty(X,E)),
 \quad v\in L^2(X,F^*\otimes\Omega_X),\quad
 h_j\in H^{m_j+1/2-m}(Y,G_j^*\otimes\Omega_Y).
 \tag{BF37}
\] All these sections are smooth up to the relevant boundary. We give
the complete proof of the interior assertion, which does not follow from
interior elliptic regularity alone.

First insert \(u=L(0,g)=KSg\), with arbitrary smooth \(g\). Equations
(BF29) and (BF32) imply \[
 \sum_j\langle g_j,h_j\rangle
 =-\langle K_2g,v\rangle-\langle K_4g,h\rangle.
 \tag{BF38}
\] The kernels of \(K_2\) and \(K_4\) are smooth in their boundary input
variables. Differentiation in those variables may be moved onto the
kernel, and the fixed distributions \(v,h\) applied in the other
variables; the resulting derivatives are continuous of every order. Thus
their transposes applied to \(v,h\) are smooth boundary sections.
Formula (BF38) proves \(h\in C^\infty(Y,\mathcal G^*\otimes\Omega_Y)\).

Tests supported away from \(Y\) now give \(P^\dagger v=0\) in
\(X^\circ\), hence \(v\) is smooth in that open set. To reach the
boundary, use (BF37) as an identity on the doubled collar: \[
 P^\dagger e^+v=J_h,
 \qquad
 J_h=-\sum_{k=0}^{m-1}(-1)^k
       \left(\sum_j\mathcal B_{jk}^\dagger h_j\right)
                   \otimes D_t^k\delta_0.
 \tag{BF39}
\] The sign \((-1)^k\) is the transpose of the trace \(\gamma_k\). The
tangential formal transposes act on density-valued duals, and therefore
include the derivative of any local density or frame coefficient. The
coefficients of \(J_h\) are smooth, including when the
\(\mathcal B_{jk}\) are pseudodifferential. There are no delta
derivatives of order \(m\) or higher because the transversal order was
reduced below \(m\).

Write \(P^\dagger=\sum_{a=0}^mQ_a(y,t,D_y)D_t^a\), after moving normal
derivatives to the right. The normal leading coefficient is
\(Q_m=(-1)^mP_m^t\), with the appropriate dual bundle identifications,
and is invertible. For a smooth section \(w\) on the exterior side
\(t\leq0\), direct distributional differentiation gives \[
 P^\dagger e^-w=e^-P^\dagger w
 -i^{-1}\sum_{a=1}^m Q_a\sum_{r=0}^{a-1}
                    (\gamma_{a-1-r}w)\otimes D_t^r\delta_0.
 \tag{BF40}
\] Multiplication of the coefficients after the delta derivatives is
exactly as in (BF20). In the coefficient of \(D_t^{m-1-l}\delta_0\), the
not-yet-chosen jet \(\gamma_lw\) occurs as \(-i^{-1}Q_m(y,0)\gamma_lw\);
every other term depends only on the jets already chosen at smaller
\(l\). Start with the coefficient of \(D_t^{m-1}\delta\) and proceed
downwards. Invertibility of \(Q_m\) therefore prescribes unique smooth
jets \(\gamma_0w,\ldots,\gamma_{m-1}w\) so that the boundary part of
(BF40) is \(-J_h\). This triangular computation supplies the generalized
Cauchy data of \(v\) without presupposing their classical existence.

Choose the remaining exterior jets recursively so that \(P^\dagger w\)
has every normal jet zero at \(Y\). At step \(r\geq0\), the equation
\(\gamma_r(P^\dagger w)=0\) contains \(Q_m\gamma_{r+m}w\) with
invertible coefficient and only previously specified jets otherwise.
Thus it has a unique smooth solution for that next jet. These jet fields
are consistent in overlapping charts because the equations and the
initially determined boundary distribution are intrinsic; equivalently
perform the recursion using a fixed collar connection on the dual
bundle.

We spell out their smooth realization. Convert the prescribed
\(D_t\)-jets to ordinary derivative jets \(a_k(y)\). With a cutoff
\(\chi\) equal to one near zero, form on the negative side \[
 w(y,t)=\sum_{k=0}^\infty\chi(t/\epsilon_k)\frac{t^k}{k!}a_k(y).
 \tag{BF41}
\] Choose \(\epsilon_k\downarrow0\) so fast that the \(k\)-th term has
all \(C^r\) seminorms on the compact boundary bounded by \(2^{-k}\) for
\(r\leq k/2\), away from its prescribed order-\(k\) jet. This is
possible because each derivative of order \(r<k\) leaves a positive
power of \(\epsilon_k\). For each fixed \(r\), only finitely many
earlier terms are excluded, so the tail and its derivatives through
order \(r\) converge uniformly. At zero the \(k\)-th term has precisely
its order-\(k\) jet, since its cutoff is constant there. The sum is
smooth and has every assigned jet. A fixed exterior cutoff makes it
supported in the collar without changing those jets. This is the needed
Borel construction with its convergence argument.

The distribution \(W=e^+v+e^-w\) is locally \(L^2\) across the boundary.
By (BF39)--(BF40), \(P^\dagger W=e^-P^\dagger w\) near \(Y\). The latter
is smooth across \(t=0\) because all jets of \(P^\dagger w\) vanish
there. Elliptic interior regularity on the doubled collar implies that
\(W\) is smooth near \(Y\). Its restriction to \(t\geq0\) is \(v\),
which proves the asserted smoothness up to the boundary. This proof
keeps both the boundary distribution and the exterior extension terms;
setting \(v=0\) outside without matching its jets would not yield a
smooth forcing term.

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\subsection{13. All solvability obstructions and the stable
index}\label{all-solvability-obstructions-and-the-stable-index}

Let \(Z\) be the space of smooth pairs \((v,h)\) satisfying (BF37).
Fredholmness at \(s=m\) says the continuous annihilator of its range is
finite-dimensional, and Section 12 identifies it exactly with this
\(Z\). Consequently, for every \(s\geq m\), \[
\begin{split}
 \operatorname{ran}A_s
 =\{(f,g)\in\mathcal Y^s:
 \langle f,v\rangle+\sum_j\langle g_j,h_j\rangle=0
 \text{ for every }(v,h)\in Z\}.
\end{split}
 \tag{BF42}
\] To prove the equality at higher \(s\), necessity follows by
continuity from smooth sections. Conversely
\(\mathcal Y^s\subset\mathcal Y^m\). Data in the displayed annihilator
have a solution \(u\in\bar H^m\) by the closed-range alternative at
level \(m\); Section 10 upgrades that solution to \(\bar H^s\). The
smooth functionals remain linearly independent on \(\mathcal Y^s\),
since smooth data belong to every \(\mathcal Y^s\) and are dense in
\(\mathcal Y^m\). Thus both the kernel and the cokernel dimensions are
independent of \(s\), and \[
 \operatorname{ind}A_s=\dim\ker A_m-\dim Z
 \quad(s\geq m).
 \tag{BF43}
\] This includes every interior and boundary obstruction; the absence of
a nonzero homogeneous solution does not by itself remove \(Z\).

Changing terms of order below \(m\) in \(P\), or below \(m_j\) in each
total-order boundary operator, gives a compact change in \(A_s\). The
interior change maps \(H^s\) to \(H^{s-m+1}\). A boundary entry of
degree at most \(m_j-1-k\), composed with \(\gamma_k\), maps into
\(H^{s-m_j+1/2}\), one full order above its target. Compact inclusion on
\(X\) or \(Y\) proves the assertion. The same reasoning uses any
strictly positive order gap in a classical real-order tangential
extension. Section 5 of \href{fredholm-stability.pdf}{Finite defects
under perturbation} shows that the index is unchanged by all these
lower-order terms.

Finally fix \(s\). In finitely many coordinate and bundle charts, a
differential coefficient change in a sufficiently high \(C^N\) norm
gives a small operator norm from \(H^s\) to \(H^{s-m}\); the
multiplication estimate followed by the finite differential operators
proves this. Boundary changes are controlled by the same multiplication
estimate and trace theorem. For tangential pseudodifferential boundary
entries, finitely many symbol seminorms replace the coefficient norms.
Thus the map from coefficients to \(A_s\) is continuous in these
explicit topologies. The norm stability theorem Section 3 of
\href{fredholm-stability.pdf}{Finite defects under perturbation} gives an
open neighborhood of the given elliptic problem on which the Fredholm
index is constant. Uniform invertibility of the principal interior
symbol and of the restricted boundary symbol on the compact cosphere
bundle also shows directly that sufficiently small principal-coefficient
changes remain elliptic; no global labeling of normal roots is needed.
Index independence of \(s\) then identifies this integer at every
permitted level.

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\subsection{13.1. The closed operator and the three
cokernels}\label{AN03-BF-CLOSED-001}

For a prescribed boundary value, (BF42) tests an interior datum and a
boundary datum together. If the boundary value is zero, the same
differential expression instead defines a densely defined closed
operator on interior \(L^2\). Its cokernel and index need not equal
those of the combined data operator. We construct the connecting maps
and retain the entire boundary contribution.

Use the original bundles, density, collar, differential expression and
boundary rows. At the original level \(s=m\), set \[
 \mathsf H=\bar H^m(X^\circ,E),\quad
 \mathsf F=L^2(X,F),\quad \mathsf G=\mathcal D^m,
 \quad N_B=\{u\in\mathsf H:Bu=0\}.
 \tag{BF46}
\] The subspace \(N_B\) is closed in \(\mathsf H\), because
\(B:\mathsf H\to\mathsf G\) is bounded. Define \(P_Bu=Pu\), with domain
\(N_B\), as an operator from \(L^2(X,E)\) to \(\mathsf F\). Its domain
contains every smooth section compactly supported in the interior: every
trace in (BF2) of such a section is zero. These sections are dense in
\(L^2(X,E)\). One may verify density by discarding successively smaller
boundary collars, whose integrals tend to zero, and then smoothing in
finitely many interior charts. Thus \(P_B\) is densely defined with
exactly the domain stated here.

\textbf{The graph norm retains the full Sobolev norm.} For every
\(\epsilon>0\), the compact operator
\(\mathcal R:\mathsf H\to\mathsf H\) in (BF28) satisfies, with some
finite \(C_\epsilon\), \[
 \|\mathcal R u\|_{\mathsf H}
 \leq\epsilon\|u\|_{\mathsf H}
       +C_\epsilon\|u\|_{L^2(X,E)}.
 \tag{BF47}
\] Here compactness follows from its bounded map into \(\bar H^{m+1}\)
and compact Sobolev inclusion. To prove (BF47), suppose a fixed positive
\(\epsilon\) admits no such constant. For each positive integer \(n\),
choose a nonzero vector violating the proposed inequality with
\(C_\epsilon=n\), and divide by its \(\mathsf H\) norm. The resulting
\(u_n\) satisfy \(\|u_n\|_{\mathsf H}=1\),
\(\|\mathcal R u_n\|_{\mathsf H}>\epsilon\), and
\(\|u_n\|_{L^2}<\|\mathcal R\|/n\).

A bounded sequence in this separable Hilbert space has a weakly
convergent subsequence. For completeness, expand in a countable
orthonormal basis, successively extract convergent scalar coordinates,
and take the diagonal subsequence. The finite sums of the squared
limiting coordinates are bounded by one, so the coordinates define a
Hilbert vector. Approximation by finite sums then proves weak
convergence against every vector. The continuous injection into \(L^2\)
shows that its weak limit must be zero, because the \(L^2\) norms here
tend to zero. Compactness of \(\mathcal R\) now gives norm convergence
of its images to zero along that subsequence: every norm-convergent
image subsequence has weak limit zero, and compactness excludes a
subsequence remaining a fixed distance from zero. This contradicts the
lower bound \(\epsilon\), proving (BF47).

Let \(C_L\) be a bound for \(f\mapsto L(f,0):\mathsf F\to\mathsf H\),
and use (BF47) with \(\epsilon=1/2\). For \(u\in N_B\), the exact
identity (BF28) gives \[
 \|u\|_{\mathsf H}
 \leq 2C_L\|Pu\|_{\mathsf F}
       +2C_{1/2}\|u\|_{L^2(X,E)}.
 \tag{BF48}
\] No kernel is discarded in this estimate. Write \(C_e\) for the norm
of \(\mathsf H\hookrightarrow L^2(X,E)\) and \(C_P\) for the norm of
\(P:\mathsf H\to\mathsf F\). With
\(\|u\|_{\mathrm{gr}}=(\|u\|_{L^2}^2+\|Pu\|_{\mathsf F}^2)^{1/2}\),
Cauchy--Schwarz gives both full comparisons \[
 \|u\|_{\mathsf H}
 \leq2(C_L^2+C_{1/2}^2)^{1/2}\|u\|_{\mathrm{gr}},
 \qquad
 \|u\|_{\mathrm{gr}}
 \leq(C_e^2+C_P^2)^{1/2}\|u\|_{\mathsf H}.
 \tag{BF49}
\] If \(u_n\in N_B\), \(u_n\to u\) in \(L^2\), and \(Pu_n\to f\) in
\(\mathsf F\), (BF48) on differences makes \(u_n\) Cauchy in
\(\mathsf H\). Its limit belongs to the closed subspace \(N_B\), agrees
with \(u\) under the injection into \(L^2\), and has image \(f\) under
the bounded map \(P\). This proves that \(P_B\) is closed. The identity
on its domain is a bounded isomorphism between \(N_B\) with its
\(\mathsf H\) norm and \(\operatorname{dom}P_B\) with its graph norm.
Since \(m\geq1\), compact Sobolev inclusion also makes the graph-domain
injection into \(L^2\) compact.

\textbf{The boundary range and every quotient.} Equation (BF32) for
\(f=0\) gives the actual bounded right approximate inverse \[
 BK S=I_{\mathsf G}+K_4:\mathsf G\longrightarrow\mathsf G,
 \qquad KS:\mathsf G\longrightarrow\mathsf H.
 \tag{BF50}
\] The compact operator \(K_4\) makes \(I+K_4\) Fredholm by Section 5 of
\href{fredholm-stability.pdf}{Finite defects under perturbation}. Its
closed finite-codimensional range is contained in
\(\operatorname{ran}B\). Any linear subspace containing that range is
closed: it is the inverse image of a subspace of the finite-dimensional
quotient by \(\operatorname{ran}(I+K_4)\). The same observation proves
finite codimension of \(\operatorname{ran}B\). It does not prove that
\(B\) is onto.

Define the three quotients \(C_P=\mathsf F/\operatorname{ran}P_B\),
\(C_A=(\mathsf F\oplus\mathsf G)/\operatorname{ran}A_m\), and
\(C_B=\mathsf G/\operatorname{ran}B\). Then the actual maps are \[
 0\longrightarrow C_P\xrightarrow{\ j\ }C_A
 \xrightarrow{\ \pi\ }C_B\longrightarrow0,
 \qquad j([f])=[(f,0)],\quad \pi([(f,g)])=[g].
 \tag{BF51}
\] The first map is well-defined because adding \(P_Bu\) with
\(u\in N_B\) adds \(A_mu=(Pu,0)\). It is injective: if \((f,0)=A_mu\),
then \(Bu=0\) and \(f=P_Bu\). The second map is well-defined because
adding \(A_mu\) changes \(g\) by \(Bu\); it is onto by choosing
\((0,g)\). If \([g]=0\) in \(C_B\), choose \(u\in\mathsf H\) with
\(Bu=g\). Subtracting \(A_mu\) gives \([(f,g)]=[(f-Pu,0)]\), which
belongs to the image of \(j\). Conversely every image of \(j\) has zero
image under \(\pi\). This proves exactness with all original targets.

The range of \(P_B\) is closed, since it is the inverse image of the
closed range of \(A_m\) under the continuous map \(f\mapsto(f,0)\). Its
cokernel is finite-dimensional by the injection \(j\) into the
finite-dimensional \(C_A\). Its kernel equals \(\ker A_m\), which is
finite-dimensional and smooth by Section 10. Thus \(P_B\), considered as
a bounded operator from its graph domain, is Fredholm. Exactness and
equality of kernels yield \[
 \operatorname{ind}P_B
 =\operatorname{ind}A_m+\dim C_B.
 \tag{BF52}
\] The boundary codimension has a plus sign. The two indices agree
exactly when \(B:\mathsf H\to\mathsf G\) is onto. No boundary equation
or exceptional finite-dimensional direction was removed to obtain this
conclusion.

\textbf{The dual maps and smooth obstructions.} Let \(Z\) retain its
meaning from Section 13. Let \(Z_P\) be the continuous annihilator of
\(\operatorname{ran}P_B\) in the density-valued dual of \(\mathsf F\),
and \(Z_B\) the continuous annihilator of \(\operatorname{ran}B\) in the
density-valued dual of \(\mathsf G\). The dual connecting maps form \[
 0\longrightarrow Z_B\longrightarrow Z
 \longrightarrow Z_P\longrightarrow0,
 \qquad h\longmapsto(0,h),\quad (v,h)\longmapsto v.
 \tag{BF53}
\] The first map is injective and well-defined because \(h\) annihilates
every \(Bu\). For the second, inserting \(u\in N_B\) into (BF37) shows
that \(v\) annihilates \(P_Bu\). Its kernel is precisely the pairs
\((0,h)\) with \(h\in Z_B\). To prove surjectivity, identify
\(v\in Z_P\) with a functional on the finite-dimensional \(C_P\),
transfer it through the injection \(j\), and extend it to \(C_A\) by
extending a basis of \(j(C_P)\). This extension is continuous on the
finite-dimensional quotient and hence on \(\mathsf F\oplus\mathsf G\).
It is represented by a pair \((\widetilde v,h)\) annihilating
\(\operatorname{ran}A_m\), with \(\widetilde v=v\) by restriction to
\((f,0)\). Section 12 makes both members of every such pair smooth. It
therefore also proves that all elements of \(Z_P\) and \(Z_B\) are
smooth density-valued sections, with the original boundary degrees in
(BF37). A representative \(h\) for a given \(v\) is unique only modulo
\(Z_B\).

The full homogeneous solvability condition is consequently \[
 \operatorname{ran}P_B
 =\{f\in\mathsf F:\langle f,v\rangle=0
                 \text{ for every }v\in Z_P\}.
 \tag{BF54}
\] This follows either from (BF42) with \(g=0\) and surjectivity in
(BF53), or directly by the closed-range annihilator argument. The
conclusion includes boundary-induced restrictions on the allowable
interior data; smoothness in the open interior alone would not have
proved it. If \(Y\) is zero-dimensional, the boundary spaces are
finite-dimensional and the same maps and estimates apply. If \(Y\) is
empty, \(\mathsf G=0\) and \(C_B=0\), and the displayed sequences reduce
to the identity between the two remaining cokernels.

\textbf{An endpoint example in which the indices differ.} Keep the
compact interval \(X=[0,1]\), scalar bundles, density \(dt\), and
\(Pu=-u''\) of order two. On the two-point boundary, take \[
 Bu=(u(0)-u(1),\ u(0)-u(1)),\qquad
 \mathsf H=H^2([0,1]),\quad\mathsf F=L^2([0,1]),
 \quad\mathsf G=\mathbb C^2.
 \tag{BF55}
\] This is a boundary measurement of transversal order zero: its matrix
on \(\gamma_0u=(u(0),u(1))\) has rows \((1,-1)\) and \((1,-1)\), and its
entries on \(\gamma_1u\) are zero. Operators on the finite boundary have
smooth kernels; this matrix is therefore permitted by the tangential
pseudodifferential extension in (BF2). The differential interior
principal symbol is \(\xi^2\), and the boundary cosphere is empty. The
dimension-one case established in Section 11 applies.

For every \(f\in L^2([0,1])\) and \(d\in\mathbb C\), every solution of
\(Pu=f\), \(Bu=(d,d)\) is \[
 u(t)=c+\left[\int_0^1(1-r)f(r)\,dr-d\right]t
              -\int_0^t(t-r)f(r)\,dr,
 \qquad c\in\mathbb C.
 \tag{BF56}
\] The integral and its first derivative are absolutely continuous, its
second weak derivative is \(f\), and Cauchy--Schwarz bounds all
displayed integrals; thus \(u\in H^2\). Differentiating proves
\(-u''=f\), and substituting the two endpoints proves \(u(0)-u(1)=d\).
Conversely the difference of two solutions with the same data is affine,
and equality of its two endpoint values makes its slope zero. These
observations prove the displayed complete solution family.

Consequently \(\operatorname{ran}B=\{(d,d):d\in\mathbb C\}\),
\(\operatorname{ran}A_2=\mathsf F\oplus\operatorname{ran}B\), and both
kernels are the one-dimensional constant functions. For the homogeneous
operator choose \(d=0\) in (BF56), which proves
\(\operatorname{ran}P_B=\mathsf F\). Thus \(\dim C_B=\dim C_A=1\),
\(C_P=0\), \(\operatorname{ind}A_2=0\), and \(\operatorname{ind}P_B=1\),
exactly as (BF52) requires. The sole combined-data dual obstruction is
\((v,h)=(0,(1,-1))\) up to scalar multiplication; its interior member
vanishes. Both boundary coordinates remain throughout the calculation.

\begin{figure}
\centering
\pandocbounded{\includegraphics[keepaspectratio]{../figures/boundary-cokernel-sequence.png}}
\caption{The quotient maps in (BF51) and the index identity (BF52). The
middle representative changes by subtracting the entire pair
\((Pu,Bu)\). Its projection retains the boundary class, and its kernel
consists exactly of classes represented by \((f,0)\).}
\end{figure}

The freely accessible treatment by Lashi Bandara, Magnus Goffeng and
Hemanth Saratchandran,
\href{https://arxiv.org/abs/2104.01919v2}{\emph{Realisations of elliptic
operators on compact manifolds with boundary}, arXiv:2104.01919v2},
Section 2.2, describes closed realizations through the maximal graph
domain and generalized traces, and defines semi-regularity by
containment of the operator domain in the order-\(m\) Sobolev space.
Here the domain \(N_B\), its graph norm, both cokernel sequences and
every connecting map are constructed directly from the completed
combined-data proof. Identifying this domain with a kernel of
generalized traces on the entire maximal domain requires the further
trace extension and kernel statements used in that source; those
statements are separate from the domain and maps proved here.

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\subsection{14. Four calculations that expose the distinct
conditions}\label{four-calculations-that-expose-the-distinct-conditions}

\textbf{A measurement can mix both normal halves.} Take the scalar
frozen equation \((D_t^2+a^2)v=0\), \(a>0\), on \(t\geq0\). Its decaying
solution is \(ce^{-at}\), with Cauchy vector \((c,ia c)\); the growing
solution has vector \((c,-ia c)\). The projection onto the first along
the second is \[
 q=\frac12\begin{pmatrix}1&(ia)^{-1}\\ia&1\end{pmatrix}.
 \tag{BF44}
\] For the measurement \(b(u_0,u_1)=u_1+\alpha a u_0\), the restriction
to the decaying space is multiplication by \(a(i+\alpha)\). It is
bijective exactly when \(\alpha\ne-i\), even though \(b\) acts on both
normal halves. For that parameter the inverse into the decaying space is
\(sg=(g/[a(i+\alpha)],ig/(i+\alpha))\). Direct multiplication gives
\(bs=1\), \(qs=s\), and \(s''=I-sb\) satisfies \(s''q=0\). At
\(\alpha=-i\), an entire decaying mode is invisible. Complex
coefficients cause no change in the test.

\textbf{Extra target equations produce an independent obstruction.} For
two decaying scalar amplitudes \((c_1,c_2)\), take boundary measurements
\((c_1,c_2,c_1-c_2)\). The restricted symbol is injective but maps
\(\mathbb C^2\) into a two-dimensional subspace of \(\mathbb C^3\). The
dual vector \((-1,1,1)\) annihilates its image. At each high tangential
frequency it gives an incompatible target direction, exactly the
situation detected by (BF36). A left symbol inverse cannot justify
finite codimension of the boundary realization.

\textbf{High total boundary order changes the space, not the trace
count.} For an order-four system, let two boundary rows have total
orders \(m_1=1\) and \(m_2=5\). Both still use only
\(\gamma_0,\ldots,\gamma_3\), with entries of degree \(m_j-k\). At
\(s=4\) the target exponents are \(5/2\) and \(-3/2\). The inverse
entries in the second column have degrees \(-5,-4,-3,-2\), so data of
order \(-3/2\) are sent respectively to Cauchy exponents
\(7/2,5/2,3/2,1/2\). Every exponent agrees with (BF3). Imposing
\(m_j<m\) would unnecessarily exclude this allowed system.

\textbf{Uniqueness and solvability have separate finite defects.} On the
interval \([0,1]\), consider \(Pu=-u''\) and the two endpoint derivative
measurements \(g_0=u'(0)\), \(g_1=u'(1)\). Integration gives the
necessary relation \(\int_0^1 f(t)dt+g_1-g_0=0\). Conversely, given that
relation, define \(u(t)=c+g_0t-\int_0^t(t-r)f(r)dr\). It satisfies the
equation and both measurements, and all solutions differ by the constant
\(c\). The kernel and the obstruction space both have dimension one, so
the index is zero at every \(s\geq2\). The smooth obstruction is the
density pair \((dt,-1,1)\), displaying the interior and boundary parts
together.

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\subsection{15. Six problems with complete
solutions}\label{six-problems-with-complete-solutions}

\textbf{1. Correct a normal coefficient before grouping a layer.} Let
\(P=a(t)D_t^2+b(t)D_t+c(t)\), with scalar smooth coefficients and
\(a(0)\ne0\). Compute \(P^c(U_0,U_1)\) as \(v_0\delta+v_1D_t\delta\),
retaining every coefficient derivative.

\textbf{Solution.} The ungrouped identity is
\(P^cU=i^{-1}[a(t)(U_1\delta+U_0D_t\delta)+b(t)U_0\delta]\). Since
\(aD_t\delta=a(0)D_t\delta-(D_ta)(0)\delta\), \[
 v_1=i^{-1}a(0)U_0,\qquad
 v_0=i^{-1}\big[a(0)U_1+(b(0)-(D_ta)(0))U_0\big].
 \tag{BF45}
\] The coefficient \(c\) produces no jump because multiplication alone
commutes with zero extension. Omitting \(D_ta\) would change the
distributional equation whenever the normal leading coefficient varies.

\textbf{2. Compute the endpoint integral and identify the first excluded
layer.} Evaluate \(c_{1,2}\) in (BF23), and decide whether the same
integral is finite for \(j=2,m=2\).

\textbf{Solution.} With \(\sigma=\tan\theta\),
\(\sigma^2(1+\sigma^2)^{-2}d\sigma=\sin^2\theta\,d\theta\), so its
integral from \(-\pi/2\) to \(\pi/2\) is \(\pi/2\). Hence the normal
Fourier integral for \(m=2,j=1\) is \((\pi/2)\langle\eta\rangle^{-1}\).
For \(j=2,m=2\), the integrand tends to one at infinity and is not
integrable. This failure concerns the preliminary whole-space \(H^{-m}\)
layer bound; it does not license declaring every higher layer undefined.
Such layers require their own order bookkeeping and transmission
analysis, outside the range used by \(P^c\).

\textbf{3. Find the freedom in a merely surjective projected inverse.}
Let \(q=\operatorname{diag}(1,1,0)\) on \(\mathbb C^3\) and
\(b(x_1,x_2,x_3)=x_1\). Find all vectors \(s:\mathbb C\to\mathbb C^3\)
with \(bs=1\), \(qs=s\). Explain why the uniqueness argument (BF10)
cannot apply.

\textbf{Solution.} The first equation fixes the first component to one
and the second sets the third to zero. Thus \(s(z)=(z,cz,0)\), with
arbitrary \(c\in\mathbb C\). On \(\operatorname{ran}q\), the measurement
has kernel spanned by \((0,1,0)\). If a pair \(s',s''\) in (BF9)
existed, applying \(s'b+s''=I\) to that vector would give zero on the
left, because \(s''q=0\), and a nonzero vector on the right. The missing
injectivity is exactly what prevents uniqueness.

\textbf{4. Derive the boundary row identity without assuming \(BQ=B\).}
In the quotient algebra assume \(BS=1\), \(QS=S\), \(SB+S''=1\),
\(S''Q=0\). Prove \(B(1+QS'')=BQ\). Is \(BQ=B\) needed?

\textbf{Solution.} Replace \(S''\) by \(1-SB\): \(B(1+QS'')=B+BQ-BQSB\).
Now \(BQSB=BSB=B\), using \(QS=S\) and then \(BS=1\). The result is
\(BQ\). There is no need for \(BQ=B\), and that equality can fail: in
the first worked calculation the boundary measurement generally has a
nonzero value on the growing normal mode. Only its restriction to the
decaying space governs complementing ellipticity.

\textbf{5. Show why a smooth obstruction is the same at two Sobolev
levels.} Suppose \(s_2>s_1\geq m\), the range at \(s_1\) is the
annihilator of a finite-dimensional smooth space \(Z\), and
(BF28)--(BF33) hold. Prove that the cokernel dimension at \(s_2\) is
also \(\dim Z\).

\textbf{Solution.} A datum at level \(s_2\) annihilating \(Z\) belongs
to the range at level \(s_1\), so it equals \(A_{s_1}u\) with
\(u\in H^{s_1}\subset H^m\). The left parametrix identity writes
\(u=L(f,g)+\mathcal R u\), and both terms lie in \(H^{s_2}\). Thus this
datum belongs to the range at \(s_2\). The reverse inclusion is the
defining relation of \(Z\). Its independent linear functionals stay
independent on the higher space because they are already independent on
smooth data, which are contained in it. Therefore that annihilator has
codimension \(\dim Z\), proving the claim without identifying Hilbert
adjoints at two different Sobolev exponents.

\textbf{6. Test a perturbation at its correct order.} For an elliptic
order-three system, a boundary row has total order four and contains
\(C(y,D_y)\gamma_2u\) with \(C\in\Psi^2\). Replace \(C\) by \(C+R\),
where \(R\in\Psi^{2-\epsilon}\), \(\epsilon>0\). Show that this change
preserves the index. Then explain what is needed if instead
\(R\in\Psi^2\).

\textbf{Solution.} At level \(s\geq3\), the trace \(\gamma_2u\) belongs
to \(H^{s-5/2}\). The new term belongs to \(H^{s-9/2+\epsilon}\), while
the declared row target is \(H^{s-9/2}\). Compact inclusion on the
closed boundary makes the perturbation compact, so the index is
unchanged by Section 5 of \href{fredholm-stability.pdf}{Finite defects
under perturbation}. If \(R\) has the full order two, no positive
compactness gain follows. A sufficiently small finite collection of its
order-two symbol seminorms makes the realization norm-small, so local
Fredholm stability still preserves the index. A large change of
principal boundary symbol can destroy the restricted symbol's
injectivity or surjectivity and has no such guarantee.

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\subsection{16. Further questions and
references}\label{further-questions-and-references}

The full proofs above start with the original projected boundary
algebra, construct both error rows, and identify smooth dual
obstructions by matching every exterior jet. Bandara, Goffeng and
Saratchandran,
\href{https://arxiv.org/abs/2104.01919v2}{\emph{Realisations of elliptic
operators on compact manifolds with boundary}, arXiv:2104.01919v2},
Section 2.2, provide the freely accessible closed-realization comparison
used in Section 13.1. The original operator, all total boundary orders,
both remainder rows and every earlier estimate remain in the receiving
proof.

For comparison, Gerd Grubb's author-hosted
\href{https://web.math.ku.dk/~grubb/dist10n.pdf}{chapter on boundary
operators} develops transmission symbols, Poisson operators and mapping
estimates. The
\href{https://web.math.ku.dk/~grubb/dist11n.pdf}{following chapter on
boundary problems} develops parametrices and regularity for systems in
that calculus.

Three directions emerge from the proofs. First, the decaying-mode bundle
over the boundary cosphere need not descend to a bundle over the
boundary. Determining when local boundary measurements exist is a
topological problem beyond the assertion that a given measurement is
complementing. Second, replacing the interior differential operator by a
general transmission operator introduces boundary symbols on function
spaces, with Poisson and singular Green terms; proving the two estimates
here does not construct that entire calculus. Third, rough coefficients
or nonsmooth boundaries require replacing the classical symbol
expansion, smooth collar jets and compact smooth remainders used in this
proof. Each direction needs additional theorems. These are further
mathematical questions beyond the hypotheses proved here.

\end{document}