64c3814
==== TABLE OF CONTENTS ====
Determination-of-a-metric-and-a-unitary-connection-from-one-boundary-patch//01-introduction.tex
Determination-of-a-metric-and-a-unitary-connection-from-one-boundary-patch//02-boundary.tex
Determination-of-a-metric-and-a-unitary-connection-from-one-boundary-patch//03-product.tex
Determination-of-a-metric-and-a-unitary-connection-from-one-boundary-patch//04-spheres.tex
Determination-of-a-metric-and-a-unitary-connection-from-one-boundary-patch//05-angular.tex
Determination-of-a-metric-and-a-unitary-connection-from-one-boundary-patch//06-matrix.tex
Determination-of-a-metric-and-a-unitary-connection-from-one-boundary-patch//07-global.tex
Determination-of-a-metric-and-a-unitary-connection-from-one-boundary-patch//paper.tex
Determination-of-a-metric-and-a-unitary-connection-from-one-boundary-patch//preamble.tex
Determination-of-a-metric-and-a-unitary-connection-from-one-boundary-patch//references.bib
===========================
==== BEGIN Determination-of-a-metric-and-a-unitary-connection-from-one-boundary-patch//01-introduction.tex ====
\section{Introduction}\label{sec:intro}
A connection Laplacian couples the geometry of a manifold to parallel
transport in a vector bundle. Its boundary measurements therefore pose
two simultaneous inverse problems: determining the metric that governs
its principal part and determining the connection that governs the
transport of vector-valued solutions. We consider both unknowns at zero
frequency, with all measurements confined to one boundary patch.
Let $M$ be a compact connected smooth manifold of dimension $n\geq3$,
with nonempty smooth boundary, and let $\Gamma\subset\partial M$ be a
nonempty relatively open proper subset. The bundle is the trivial
Hermitian bundle $M\times\C^2$. A smooth unitary connection is written
$d_A=d+A$, where
\[
A\in C^\infty(M;T^*M\otimes\mathfrak u(2)),\qquad
\mathfrak u(2)=\{B\in\C^{2\times2}:B^*=-B\}.
\]
For a smooth Riemannian metric $g$, write
$L_{g,A}=d_A^{*g}d_A$. Its zero-Dirichlet realization is invertible:
a section of zero energy is parallel, and a parallel section with zero
boundary trace is zero. Elliptic coercivity then gives, for each
$f\in C_c^\infty(\Gamma;\C^2)$ extended by zero to $\partial M$, a unique
solution $u_f^{g,A}$ of
\[
L_{g,A}u_f^{g,A}=0\quad\text{in }M,\qquad
u_f^{g,A}|_{\partial M}=f.
\]
The measured object is the sesquilinear energy form
\begin{equation}\label{eq:DN}
\langle\Lambda_{g,A,\Gamma}f,h\rangle
=\int_M\langle d_Au_f^{g,A},d_Au_h^{g,A}\rangle_g\,dV_g,
\qquad f,h\in C_c^\infty(\Gamma;\C^2).
\end{equation}
The Hermitian product is linear in its first argument. All integrations
use densities; no orientation of $M$ is required. In particular,
\eqref{eq:DN} specifies the boundary measurement without presupposing a
boundary metric or a boundary measure.
\begin{theorem}\label{thm:main}
Let $M$ and $\Gamma$ be as above. For $j=1,2$, let $g_j$ be a smooth
Riemannian metric on $M$ and let
$A_j\in C^\infty(M;T^*M\otimes\mathfrak u(2))$. If
\[
\langle\Lambda_{g_1,A_1,\Gamma}f,h\rangle
=\langle\Lambda_{g_2,A_2,\Gamma}f,h\rangle
\quad\text{for all }f,h\in C_c^\infty(\Gamma;\C^2),
\]
then there are a smooth diffeomorphism $\Phi:M\longrightarrow M$ and
a smooth map $U:M\longrightarrow U(2)$ such that
\begin{equation}\label{eq:main-conclusion}
\Phi|_\Gamma=\Id,\qquad U|_\Gamma=I,\qquad
g_2=\Phi^*g_1,\qquad
A_2=U^{-1}(\Phi^*A_1)U+U^{-1}dU.
\end{equation}
Here $I$ denotes the $2\times2$ identity matrix.
\end{theorem}
The two transformations in \eqref{eq:main-conclusion} preserve
\eqref{eq:DN}: the corresponding change of solution is
$u_2=U^{-1}(u_1\circ\Phi)$. Thus the conclusion describes precisely the
ambiguities inherent in these measurements. The patch need not be
connected, and the manifold may have several boundary components.
\subsection{Context and contribution}
Calder\'on's inverse conductivity problem asks whether electrical
measurements at the boundary determine an interior conductivity
\cite{Calderon1980}. For smooth isotropic conductivities in dimension
at least three, global uniqueness was established by Sylvester and
Uhlmann \cite{SylvesterUhlmann1987}. In the anisotropic formulation the
unknown is a metric, or equivalently a positive conductivity tensor
density, and boundary-fixing diffeomorphisms give an unavoidable
invariance. Lee and Uhlmann proved a boundary determination result and
uniqueness in the real-analytic setting under geometric and
topological hypotheses \cite{LeeUhlmann1989}.
Lassas and Uhlmann subsequently recovered real-analytic manifolds from
local boundary information \cite{LassasUhlmann2001}.
The Poisson embedding approach of Lassas, Liimatainen, and Salo
\cite{LLS20} expresses the geometry through the values of harmonic
solutions. We use the same broad principle, with source solutions of
a connection Laplacian providing both points and fiber identifications.
For connections, Albin, Guillarmou, Tzou, and Uhlmann recovered
Hermitian connections and matrix potentials from full Cauchy data on
fixed surfaces \cite{AlbinGuillarmouTzouUhlmann2013}. In higher
dimensions, Ceki\'c established recovery for line bundles on certain
known conformally transversally anisotropic geometries
\cite{Cekic2017Connections}, and for smooth unitary Yang--Mills
connections of arbitrary rank from same-patch measurements on an
arbitrary fixed smooth metric \cite[Theorem~1.3]{Cekic2020YM}.
His complex parallel transport method also recovers arbitrary-rank
connections and matrix potentials from full boundary data on fixed
geometries conformal to subdomains of $\R^2\times M_0$
\cite[Theorem~1.1]{Cekic2025Systems}. In the real-analytic category,
Gabdurakhmanov and Kokarev proved joint recovery of metric, Euclidean
bundle, and compatible connection from a boundary patch
\cite[Theorem~1.1]{GabdurakhmanovKokarev2025}. Their use of Green kernels
to identify the base and fibers is a close geometric antecedent.
Theorem~\ref{thm:main} concerns an unknown smooth metric and an arbitrary
smooth unitary connection on the fixed trivial rank-two bundle; it
imposes no Yang--Mills equation on that connection.
The scalar companion \cite{ScalarCompanion2026} develops a smooth
continuation argument based on separately harmonic Green kernels and
shrinking geodesic spheres. The present proof uses its product
continuation, sphere geometry, and scalar angular estimate
\cite[Sections~3--5]{ScalarCompanion2026}. We give the analytic arguments
needed here, including their extension to systems with scalar principal
part. The scalar uniqueness theorem itself is not applied to the
matrix-valued boundary data.
The additional difficulty appears in the first moments of the transfer
between local solution spaces. In scalar harmonic coordinates those
moments can be normalized away. For a connection, a harmonic frame
removes the zeroth-order term, but the transfer still has a
matrix-valued first moment. We represent it by matrices $D^1,\ldots,D^n$.
Their components complementary to the real scalar matrices satisfy a
first-order system whose principal
symbol is the conformal Killing symbol. In dimension at least three
this symbol is of finite type: two prolongations determine all third
derivatives from lower jets. The real scalar part of the displacement
is fixed by the coordinate normalization. Together these facts turn
an initial square-root estimate on $D$ into the stronger estimate
needed to continue the metric and connection across a frontier point.
This reduction is the main matrix-specific step of the proof.
\subsection{How the proof is organized}
Boundary symbol factorization first determines all metric and
connection jets on $\Gamma$, after imposing normal gauge. Attaching a
small exterior cap then converts the boundary form into equality of
Green matrices on an open interior set. Source solutions supported in
that set separate points, span fiber values, and realize the allowed
harmonic jets. They define a maximal local isometry with a unitary
bundle identification; Section~\ref{sec:boundary} prepares coordinates
and harmonic frames at any putative frontier point.
The local continuation mechanism has three stages. First,
Section~\ref{sec:product} continues a mixed Green matrix, initially
known when either variable lies in the matched region. Two
quantitative unique-continuation estimates, one in each variable,
allow continuation across suitable hypersurfaces in the product.
Second, Sections~\ref{sec:spheres} and~\ref{sec:angular} use this kernel
to transfer harmonic functions across small moving spheres. The
transfer is represented by a matrix density of total mass $I$.
An angular coercivity estimate bounds that density uniformly even as
the spheres shrink. The matrix lower-order terms are absorbed in the
same estimate.
Finally, Section~\ref{sec:matrix} extracts the first and second moments
of the transfer. The finite-type argument described above improves
the difference of the normalized inverse metrics from order $r$ to
order $r^{3/2}$, where $r$ is the sphere radius. A continuity argument
then lets the radii collapse, proving local agreement of the
normalized operators. Section~\ref{sec:global} removes the remaining
conformal factor and extends the resulting isometry and unitary
identification over the whole manifold, including every component of
the measured patch.
==== END Determination-of-a-metric-and-a-unitary-connection-from-one-boundary-patch//01-introduction.tex ====
==== BEGIN Determination-of-a-metric-and-a-unitary-connection-from-one-boundary-patch//02-boundary.tex ====
\section{Boundary data, exterior sources, and contact coordinates}
\label{bdy:section}\label{sec:boundary}
We first replace the measurements by a family of interior source
solutions. The boundary symbol determines jets of both the metric and
the connection, after a boundary-identity change of gauge. These jets
allow a common exterior cap, on which the Green matrices agree. Source
evaluations in that cap then identify points and fibers simultaneously.
The final part of the section prepares coordinates and harmonic frames
at a possible frontier of this identification. The cap and
source-evaluation construction follows the scalar construction in
\cite[Section~2]{ScalarCompanion2026}; we give the changes needed for
unitary connections and matrix-valued evaluations.
\subsection{Dirichlet solutions and boundary jets}
We use positive metric densities throughout. Write
\(L_{g,A}=d_A^{*g}d_A\), with positive principal symbol, and let
\(L_{g,A,D}\) denote its zero-Dirichlet realization. Its quadratic form
is coercive on \(H^1_0(M;\mathbb C^2)\). Indeed, the usual elliptic form
estimate and compactness reduce coercivity to the absence of a kernel.
A kernel element satisfies \(d_Au=0\); its norm is constant, and its
zero boundary trace makes it zero. Smooth sources and boundary data
therefore have unique solutions smooth up to the boundary. The same
facts hold on every smooth extension used below and on sufficiently
small coordinate balls; we use the standard smooth elliptic Dirichlet
theory of \cite[Chapter~5]{TaylorPDEI2011}.
We also use open-set unique continuation for systems with a smooth,
real scalar elliptic principal part and smooth matrix lower-order
terms; see \cite[Remark~3]{Aronszajn1957}. The usual scalar local
Carleman estimate, summed over components, proves this statement: after conjugation by the scalar
weight, a matrix first-order term is bounded by a constant times
\(\|\nabla v\|+\tau\|v\|\), and is absorbed for large Carleman
parameter \(\tau\). See \cite[Chapter~XXVIII]{HormanderIV2009} for the
underlying scalar estimates. Boundary Cauchy uniqueness follows from
the same interior statement. Extend the coefficients smoothly across
a boundary patch and extend a solution with zero trace and zero
covariant normal derivative by zero. Green's formula leaves no
boundary distribution, so the extension solves the equation
weakly. Interior regularity and unique continuation then give
vanishing near the patch and throughout the connected interior.
A unitary gauge \(V:M\to U(2)\) changes the connection to
\begin{equation}\label{bdy:gauge}
A^V=V^{-1}AV+V^{-1}dV,\qquad
d_{A^V}(V^{-1}u)=V^{-1}d_Au.
\end{equation}
A gauge equal to \(I\) on the boundary preserves the measured form.
In inward boundary-normal coordinates \((z,s)\), solve
\(\partial_sV=-A_sV\), \(V(z,0)=I\). This gives \(A^V_s=0\).
The gauge can be made global: in a collar replace the time argument
of this solution by a smooth function equal to \(s\) in a smaller
collar and equal to zero near the inner edge, and set \(V=I\)
farther inside. The ODE preserves unitarity. Apply this construction
separately to \(A_1,A_2\), using gauges \(V_1,V_2\), and retain their
names for the final return to the original trivializations. Until
then, \(A_j\) denotes the gauged connection.
\begin{lemma}[Boundary jets]\label{bdy:jets}
Suppose the local energy forms of \((g_1,A_1)\) and \((g_2,A_2)\)
agree on \(\Gamma\). In their respective boundary-normal coordinates,
based on the same boundary coordinates, and in the normal gauges
just constructed, the full Taylor jets of \(g_1,A_1\) and
\(g_2,A_2\) agree at every point of \(\Gamma\).
\end{lemma}
\begin{proof}
Write \(g=ds^2+k(s,z)\), with \(k\) a tangential positive matrix,
and put \(\rho=(k^{ab}\xi_a\xi_b)^{1/2}\). Green's formula identifies
the measured operator with the density
\(\nabla^A_\nu u\,dS_g\). Relative to the coordinate density, its
principal symbol is
\[
(\det k)^{1/2}\rho I.
\]
Its square determines \(C=(\det k)k^{-1}\). Since the tangential
dimension is \(m=n-1\),
\[
\det C=(\det k)^{m-1},\qquad
k=(\det C)^{1/(m-1)}C^{-1}.
\]
Thus the boundary metrics and densities agree. Division by their
common density gives equality of the ordinary covariant unit-normal
DN operators locally on \(\Gamma\). Localization on both sides by
cutoffs supported in \(\Gamma\) gives equality of their full symbols.
We spell out the symbol calculation, extending the metric
factorization of \cite{LeeUhlmann1989} to the present scalar-principal
system. Recovery of a connection together with the metric by this
even--odd symbol separation is also developed in
\cite[Theorem~1.1 and Section~3.3]{Gabdurakhmanov2021DN}; the
calculation below uses complex Hermitian fibers. In normal gauge,
\[
L_{g,A}=-\partial_s^2-\alpha\partial_s+P,
\qquad \alpha=\tfrac12\operatorname{tr}(k^{-1}\partial_s k),
\]
where \(P\) is the connection Laplacian on each tangential slice.
With \(D_z=-i\partial_z\), its degree-one symbol consists of the
scalar metric term and \(-2ik^{ab}A_a\xi_b\). Its degree-zero
symbol uses only slice coefficients and their tangential derivatives.
The local Dirichlet factorization has the form
\[
L_{g,A}=(-\partial_s+B-\alpha)(\partial_s+B)
\pmod{\Psi^{-\infty}},
\qquad
b\mathbin{\#}b-\alpha b-\partial_s b\sim\sigma(P),
\]
where \(b\sim\rho I+b_0+b_{-1}+\cdots\) is a left symbol and
\[
b\mathbin{\#}c\sim
\sum_{\mu}\frac{(1/i)^{|\mu|}}{\mu!}
(\partial_\xi^\mu b)(\partial_z^\mu c).
\]
The positive root \(\rho I\) selects the decaying Dirichlet branch.
The Poisson parametrix consequently identifies \(B(0)\), modulo a
smoothing operator, with the outward normal derivative. This
factorization is valid for smooth coefficients: its principal root
is scalar and each subsequent matrix coefficient is solved by
scalar division by \(2\rho\). The true inverse Dirichlet problem
changes only the local smoothing remainder.
The part of \(b_0\) involving the new jets is
\begin{equation}\label{bdy:first-jets}
\frac14\left(\operatorname{tr}_k\partial_s k
-\frac{(\partial_s k)(\xi^\sharp,\xi^\sharp)}{\rho^2}\right)I
-\frac{ik^{ab}A_a\xi_b}{\rho},
\qquad \xi^\sharp=k^{-1}\xi.
\end{equation}
This follows from
\(\partial_s\rho=-(\partial_s k)(\xi^\sharp,\xi^\sharp)/(2\rho)\)
and the degree-one factorization equation. At degree \(1-j\),
\(j\ge2\), the new normal derivatives arise only in
\(\partial_s b_{2-j}/(2\rho)\). Inductively their contribution is
\begin{equation}\label{bdy:highest-jets}
\frac{1}{(2\rho)^{j-1}}
\left[
\frac14\left(\operatorname{tr}_k\partial_s^j k
-\frac{(\partial_s^j k)(\xi^\sharp,\xi^\sharp)}{\rho^2}\right)I
-\frac{ik^{ab}(\partial_s^{j-1}A_a)\xi_b}{\rho}
\right].
\end{equation}
Every omitted term involves lower normal jets, with tangential
derivatives. Differentiating \(\rho\), a raised index, or a
coefficient multiplying the preceding highest jet produces only
such lower-jet expressions.
Subtract the two symbol recursions after their lower jets have
been identified. The metric term in
\eqref{bdy:highest-jets} is even in \(\xi\), whereas the connection
term is odd. The odd term determines all components of the new
connection jet. If the even term vanishes for a symmetric tensor
\(T\), then \(T(v,v)=\operatorname{tr}_k(T)\) for every unit vector
\(v\). Polarization gives \(T=\operatorname{tr}_k(T)k\), and taking
trace gives \((m-1)\operatorname{tr}_k(T)=0\). As \(m\ge2\),
\(T=0\). Beginning with the recovered boundary metric, this proves
all normal jets by induction; their tangential derivatives recover
all mixed jets. The remaining normal metric components and normal
connection component are fixed by the coordinate and gauge choices.
\end{proof}
\subsection{A common cap and its Green matrices}
Jet equality does not assert equality in an interior collar. It
provides instead a common smooth extension on the exterior side of a
smaller patch. The following variational argument turns that exterior
region into common interior data.
\begin{proposition}[Common exterior sources]\label{bdy:cap}
There are compact connected smooth extensions \(N_j\) of the original
manifold, with nonempty smooth boundary, and a common connected open
exterior cap \(E\subset N_j^\circ\), attached through a nonempty open
patch \(P\Subset\Gamma\), such that the extended metrics and
connections agree on \(E\). For each
\(f\in C_c^\infty(E;\mathbb C^2)\), the solutions
\(L_{j,D}^{-1}f\) agree on \(E\). The Dirichlet Green matrices,
normalized using metric volume, satisfy
\begin{equation}\label{bdy:common-green}
G_1(x,y)=G_2(x,y)\quad(x,y\in E,\ x\ne y),
\qquad G_j(x,y)=G_j(y,x)^*.
\end{equation}
\end{proposition}
\begin{proof}
Choose a boundary chart compactly contained in \(\Gamma\), and a
nonnegative smooth bump \(b\) whose support is compact in that chart
and whose positivity set \(P=\{b>0\}\) is nonempty and connected.
Using the separate normal collars, move the boundary from \(s=0\)
to the graph \(s=-b(z)\), with \(b\) small enough to remain inside
an extended collar. The common open cap is
\[
E=\{(z,s):-b(z)<s<0\}.
\]
Extend the first coefficient fields smoothly to negative \(s\),
retaining positive definiteness of the metric and skew-Hermitian
values of the connection. Use the same exterior extension for the
second pair. Lemma~\ref{bdy:jets} makes the resulting pasting smooth
to all orders on a neighborhood of \(\operatorname{supp}b\).
Restricting it to \(s\ge-b(z)\) gives smoothness also at the flat
edges of the bump. The bundles remain trivial in the extended
normal gauges.
For the energy comparison only, identify the extensions by the
identity on the original \(M\) and by cap coordinates on \(E\).
In collar charts this map and its inverse are continuous and
piecewise smooth with bounded first derivatives, and hence
bi-Lipschitz. They identify the relevant \(H^1_0\) spaces. Fix a
vector source \(f\) compactly supported in \(E\). The solution
\(w_j=L_{j,D}^{-1}f\) is the unique minimizer of
\[
\mathcal J_j(w)=\frac12\int_{N_j}|d_{A_j}w|_{g_j}^2\,dV_{g_j}
-\operatorname{Re}\int_E\langle f,w\rangle\,dV_{g_j}
\quad\text{on }H^1_0(N_j;\mathbb C^2).
\]
Its trace on the original boundary is smooth and supported in
\(\operatorname{supp}b\Subset\Gamma\), because the original and
extended boundaries coincide wherever \(b=0\). Transport \(w_1\)
to \(N_2\), and inside the original \(M\) replace it by the
second harmonic extension of this trace. The replacing difference
has zero trace on \(\partial M\), so its zero extension is in
\(H^1_0(N_2)\). The competitor is therefore admissible even at the
narrowing edges of the cap.
On \(M\), the source is zero and \(w_1\) is the first harmonic
extension of its trace. Equality of the measured quadratic forms
makes its interior energy equal to that of the replacement. On
\(E\), both coefficients and the source pairing are unchanged.
Thus \(\min\mathcal J_2\le\min\mathcal J_1\). Reversing the
argument gives equality, and uniqueness of the minimizer proves
\(w_1=w_2\) on \(E\).
Normalize the Green matrix by
\[
L_{j,x}G_j(x,y)=\delta_y^{g_j}(x)I,\qquad
G_j(\cdot,y)|_{\partial N_j}=0,\qquad
L_{j,D}^{-1}f(x)=\int_{N_j}G_j(x,y)f(y)\,dV_{g_j}(y).
\]
The source densities on \(E\) coincide. Varying the source gives
equality of the distribution kernels on \(E\times E\), hence
pointwise equality off the diagonal. Self-adjointness of the
Dirichlet inverse gives the adjoint symmetry in
\eqref{bdy:common-green}.
\end{proof}
\subsection{Evaluations identify points and fibers}
Fix a nonempty open set \(U_0\Subset E\). For
\(f\in C_c^\infty(U_0;\mathbb C^2)\), set
\begin{equation}\label{bdy:source-evaluations}
w_{j,f}=L_{j,D}^{-1}f,\qquad I_j(p)f=w_{j,f}(p).
\end{equation}
Thus \(I_j(p)\) is a linear map from the common source space to the
fiber at \(p\). The maps agree on \(E\). This construction is
related to the source and Poisson embeddings of
\cite[Section~6.1 and Appendix~A]{LLS20}. The following
point-distribution proof, in the spirit of
\cite[Chapter~III, Section~3]{Malgrange1956}, supplies the precise
system-valued separation needed here.
\begin{lemma}[Values and harmonic two-jets]\label{bdy:evaluations}
Let \(N\) be a compact connected smooth manifold of dimension
\(n\ge2\) with nonempty boundary, let \(L=d_A^{*g}d_A\), and let
\(U\Subset N^\circ\) be a nonempty open source set.
\begin{enumerate}
\item At each interior point, \(f\mapsto L_D^{-1}f(p)\) is onto
\(\mathbb C^2\). At any two distinct interior points the two
values are jointly onto \(\mathbb C^2\oplus\mathbb C^2\).
\item If \(p\notin\overline U\), the second jets of \(L_D^{-1}f\)
at \(p\) fill exactly the linear space of jets satisfying
\(Lu(p)=0\).
\item Each evaluation is zero on \(\partial N\), and for fixed
\(f\) varies continuously up to that boundary.
\end{enumerate}
\end{lemma}
\begin{proof}
The last assertion is smooth Dirichlet regularity. For the first
two, a complex-linear functional on values or jets is represented
by a dual-vector distribution \(T\), supported at one or two points.
Suppose it annihilates all source solutions, and define
\(v=(L_D^{-1})^tT\) by distributional transposition. Then
\[
\langle v,f\rangle=\langle T,L_D^{-1}f\rangle=0
\quad(f\in C_c^\infty(U;\mathbb C^2)),
\qquad L^tv=T.
\]
Here the transpose and pairing are bilinear on the dual bundle;
no complex conjugation convention is needed. Elliptic regularity
makes \(v\) smooth off the support of \(T\). The interior with
one or two points removed is connected: a path can be diverted
around each point in a punctured coordinate ball when \(n\ge2\).
Unique continuation from the nonempty set obtained by removing
these points from \(U\) makes \(v\) supported at those points in
the interior. If a point belongs to \(U\), the asserted initial
vanishing is distributional there as well.
At each support point a distribution is a finite sum of derivatives
of the point mass. If its highest derivative order is \(k\), its
image under \(L^t\) has exact highest order \(k+2\). Indeed, the
highest homogeneous coefficient vector polynomial is multiplied
by the scalar elliptic quadratic polynomial; this multiplication
is injective. Consequently no nonzero order-zero distribution
can be the image of a point-supported distribution. There are
therefore no value annihilators, proving both surjectivity claims.
For a two-jet annihilator at \(p\notin\overline U\), the image
\(T\) has order at most two. Thus \(v=c\delta_p\), with \(c\)
a dual-fiber vector. Its image is precisely the functional
\(u\mapsto c(Lu(p))\). Conversely all such functionals
annihilate the source solutions. Finite-dimensional duality gives
the exact stated jet space. In particular, values and gradients
may be prescribed freely, since the Hessian trace can be chosen
to satisfy the equation.
\end{proof}
We can now define the correspondence whose continuation will prove
the theorem. A \emph{match} consists of a local isometry
\(\Phi:W\to N_1^\circ\), with \(W\subset N_2^\circ\) connected,
open, and containing \(E\), and a smooth unitary bundle map \(J\)
from the second fiber at \(p\) to the first fiber at \(\Phi(p)\).
We require
\begin{equation}\label{bdy:matching}
\Phi|_E=\operatorname{Id},\qquad J|_E=I,\qquad
d_{\Phi^*A_1}(Ju)=Jd_{A_2}u,\qquad
J(p)I_2(p)=I_1(\Phi(p)).
\end{equation}
In the connection identity the first connection is pulled back to
\(W\). The identity on \(E\) is a match.
\begin{lemma}[The maximal match]\label{bdy:matches}
All matches agree on overlaps and are injective on base points.
Their union is a unique maximal match \((\Phi,J)\) on a connected
open set \(W\subset N_2^\circ\). On this set,
\begin{equation}\label{bdy:matched-green}
G_1(\Phi(p),\Phi(q))
=J(p)G_2(p,q)J(q)^*\qquad(p,q\in W,\ p\ne q).
\end{equation}
\end{lemma}
\begin{proof}
If two possible images of the same point were distinct, their
evaluation identities would impose a fixed invertible linear
relation between the evaluations at two distinct points of
\(N_1^\circ\). This contradicts joint surjectivity in
Lemma~\ref{bdy:evaluations}. Once the image agrees, surjectivity
at the source point determines \(J\) uniquely. The same argument
using the evaluations in \(N_2\) rules out two base points with
the same image. Thus the maps glue on every overlap. Their
union is connected because every domain contains \(E\), and
retains all the properties in \eqref{bdy:matching}.
For fixed \(p\), the evaluation identity and the common source
density give \eqref{bdy:matched-green} when \(q\in U_0\), away
from the pole. As a function of \(q\), the adjoint of each side
solves the same pulled-back connection equation after the bundle
identification. Injectivity of \(\Phi\) ensures that \(q=p\)
is the only possible pole. Unique continuation on the connected
set \(W\setminus\{p\}\) proves the identity everywhere claimed.
\end{proof}
\subsection{Coordinates at a possible interior frontier}
It remains to show that the maximal domain is the entire interior.
We next prepare the local data needed to enlarge it. The use of
harmonic \emph{frames} is essential: constant coefficient columns
in these frames solve the equation, even when the connection has
no nonzero parallel sections.
\begin{proposition}[Contact data]\label{bdy:contact}
If the maximal domain \(W\) has an interior frontier, there are a
point \(p\in\partial W\cap N_2^\circ\), a point \(q\in N_1^\circ\),
and charts about \(p,q\), both with coordinate range a neighborhood
of \(0\in\mathbb R^n\), with the following properties.
\begin{enumerate}
\item Both charts avoid \(\overline{U_0}\). The old match is
coordinate identity on its domain in these charts. That domain
contains a region \(x_n<\kappa(x')\), where \(\kappa\) is smooth,
\(\kappa(0)=0\), and \(d\kappa(0)=0\).
\item In smooth local unitary frames, the metrics, connection
matrices, and Green matrices agree on the known region (for the
Green matrices, on its square off the diagonal), and
\(g_1(0)=g_2(0)=I_n\).
\item There are invertible matrix functions \(F_j\) with harmonic
columns, chosen as corresponding source solutions. In these
harmonic frames there are corresponding harmonic matrix functions
\(X_j^l\), \(1\le l\le n\), satisfying
\begin{equation}\label{bdy:trace-coordinates}
\tfrac12\operatorname{Re}\operatorname{tr}X_j^l(x)=x^l,
\qquad X_j^l(0)=0,
\qquad \partial_iX_j^l(0)=\delta_i^l I.
\end{equation}
\item There are finitely many further corresponding harmonic
column functions with zero value and gradient at \(0\), whose
Hessians at \(0\) span the space of symmetric
\(\mathbb C^2\)-valued tensors \(H\) satisfying
\(\sum_i H_{ii}=0\). All these paired functions, including the
frames, agree on the known region and have identical jets at
\(0\).
\end{enumerate}
\end{proposition}
\begin{proof}
Choose a small coordinate ball near an interior frontier and a
point of \(W\) close enough to that frontier that its distance
to the complement is attained inside the chart. The Euclidean
ball with that center and radius lies in \(W\) and touches its
complement at an interior point \(p\). Its boundary gives the
required smooth one-sided contact hypersurface.
If \(p_k\in W\) tends to \(p\), compactness of \(N_1\) and
\(U(2)\), using the fixed global trivializations, gives a
subsequence for which \(\Phi(p_k)\to q\) and \(J(p_k)\to J_0\).
Continuity of every source solution gives
\begin{equation}\label{bdy:limit-match}
J_0 I_2(p)=I_1(q).
\end{equation}
The point \(q\) cannot lie on the boundary, where the right side
would be zero, since \(I_2(p)\) is onto. Two distinct possible
limits \(q\) contradict joint surjectivity on \(N_1\); at a fixed
\(q\), surjectivity of \(I_2(p)\) determines \(J_0\). The full
limits therefore exist. Also \(q\notin E\): otherwise its
identity match on \(E\) and \eqref{bdy:limit-match} would relate
the evaluations at the distinct points \(q,p\) of \(N_2\).
Both \(p\) and \(q\) consequently avoid \(\overline{U_0}\).
Choose two source solutions whose columns give an invertible
matrix \(F_1(q)\). Their corresponding second-side columns give
\(F_2(p)\), invertible by \eqref{bdy:limit-match}. Shrink the
charts so both matrices are invertible. In a harmonic frame,
\(v=F_j^{-1}w\) satisfies an equation with scalar elliptic
principal part and no zeroth-order term, since every column of
\(F_j\) is harmonic. Lemma~\ref{bdy:evaluations} is invariant
under this smooth frame change. It therefore supplies matrices
\(X_1^l\) with value zero and gradients equal to any chosen real
coordinate covectors times \(I\). Choose those covectors
independently, and let \(X_2^l\) be the corresponding source
matrices expressed in \(F_2\).
Set \(x^l=\frac12\operatorname{Re}\operatorname{tr}X_1^l\) and
\(y^l=\frac12\operatorname{Re}\operatorname{tr}X_2^l\). The first
functions form coordinates at \(q\). On matched points, the
source identity implies \(JF_2=F_1\circ\Phi\) and
\(X_2^l=X_1^l\circ\Phi\). Thus the scalar coordinates and the
metric Gram matrices of their differentials agree under the
isometry. Passing to \(p,q\), their Gram matrices agree and are
positive definite. Hence the \(y^l\) are coordinates at \(p\).
Fix a target chart neighborhood \(U_1\) on which the \(x^l\)
are coordinates. The full convergence of partner points permits
a source chart neighborhood \(U_2\) so small that
\(\Phi(W\cap U_2)\subset U_1\). Choose a small coordinate ball
\(\Omega\) about zero contained in both \(x(U_1)\) and
\(y(U_2)\), and restrict the two charts to their inverse images
of \(\Omega\). For a matched point in the restricted source
chart, \(x(\Phi(p'))=y(p')\in\Omega\); its partner therefore
lies in the restricted target chart. Thus the old map is
coordinate identity throughout its domain in these charts. The
matrix gradient identity at
\(q\) now reads \(\partial_iX_1^l(0)=\delta_i^lI\); equality on
the one-sided region and smoothness give the same identity for
\(X_2^l\).
Because the harmonic-frame equation has no zeroth-order term,
a jet with zero value and gradient is constrained only by
\(g_1^{ij}(0)H_{ij}=0\). The jet lemma supplies a finite basis
of these Hessians and corresponding source pairs. They agree
on the known region, so their complete jets agree at the contact
point. The same conclusion holds for every smooth coefficient
or paired function already identified there.
Finally polar-orthonormalize the original harmonic frames:
write \(F_j=H_jS_j\), with \(H_j\) unitary and \(S_j\) positive
Hermitian. On the matched set, \(JF_2=F_1\) implies
\(S_2=S_1\) and \(JH_2=H_1\). In the unitary frames \(H_j\),
the old bundle identification is therefore \(I\); connection
matrices and Green matrices agree there. We continue to write
\(F_j\) for the harmonic-frame matrices in these unitary frames.
A common real linear coordinate change makes the metric at zero
Euclidean and the contact hyperplane horizontal. Apply the same
linear change to the list of \(X_j^l\); this preserves all
identities in \eqref{bdy:trace-coordinates}. Reverse the final
normal coordinate if necessary to put the known side below its
graph. In these coordinates the Hessian constraint is the one
in the statement.
\end{proof}
\subsection{Dilation of the contact data}\label{bdy:dilation}
The local analysis uses the preceding charts on a small physical
scale and a fixed coordinate range. For \(\lambda>0\), set
\begin{equation}\label{bdy:dilation-formulas}
\begin{aligned}
g_{j,\lambda}(x)&=g_j(\lambda x),&
A_{j,\lambda}(x)&=\lambda A_j(\lambda x),\\
G_{j,\lambda}(x,y)&=\lambda^{n-2}G_j(\lambda x,\lambda y),&
F_{j,\lambda}(x)&=F_j(\lambda x).
\end{aligned}
\end{equation}
Here the metric normalization is \(\lambda^{-2}\) times the
pullback metric. Thus the pulled-back connection Laplacian is
multiplied by \(\lambda^2\), and the factor
\(\lambda^{n-2}\) in the kernel exactly compensates for metric
volume. In particular the displayed kernel still has unit
matrix point source for \(g_{j,\lambda}\). Harmonic functions
and the harmonic frames dilate by composition, without a
multiplicative factor.
On every fixed bounded coordinate range, these metrics converge
smoothly to \(I_n\), the connection matrices converge smoothly
to zero, and all fixed finite orders of their coefficients and
frames remain bounded. Source-free harmonic-frame operators
have zero zeroth-order terms; their ordinary-coordinate first-order
coefficients tend to zero. The known side becomes
\(x_n<\lambda^{-1}\kappa(\lambda x')\), which contains every
fixed bounded region lying strictly below \(x_n=0\) for all
sufficiently small \(\lambda\).
We also have uniform kernel bounds on fixed separated sets.
An interior parametrix of order \(-2\) for the original smooth
scalar-principal system gives
\(|G_j(x,y)|\le C|x-y|^{2-n}\) in a sufficiently small fixed
chart, with a smooth remainder; interior estimates give the
corresponding bounds for derivatives away from the diagonal.
After \eqref{bdy:dilation-formulas}, every fixed positive lower
bound for \(|x-y|\) therefore gives bounds of any prescribed
finite order, uniformly for small \(\lambda\). The local
constructions below first specify their bounded ranges and
positive separation margins and then restrict \(\lambda\).
They never demand a bound uniform as the separation tends to
zero from this initialization alone.
For the local argument, our convention for a normalized equation in
a harmonic frame is
\begin{equation}\label{bdy:harmonic-normalization}
\mathcal L_jv=-c_jF_j^{-1}L_{g_j,A_j}(F_jv),\qquad c_j>0.
\end{equation}
The scalar multiplier acts outside the differential operator.
Its second-derivative coefficient matrix is \(c_jg_j^{-1}\),
and its zeroth-order term is zero. This convention applies to
both the original and the dilated data; the functions \(c_j\)
will be chosen in Section~\ref{sec:spheres}.
The remaining local task is to continue the match through zero.
Sections~\ref{sec:product}--\ref{sec:matrix} will prove that the
metrics are conformal and that suitable normalizations
\eqref{bdy:harmonic-normalization} give equal operators on a
neighborhood of zero. Section~\ref{glob:section} will then use
the vanishing zeroth-order terms to remove the conformal factor
and prove that \(F_1F_2^{-1}\) is a unitary connection
identification. Unique continuation extends every source
evaluation, so the local identification enlarges the maximal match.
==== END Determination-of-a-metric-and-a-unitary-connection-from-one-boundary-patch//02-boundary.tex ====
==== BEGIN Determination-of-a-metric-and-a-unitary-connection-from-one-boundary-patch//03-product.tex ====
\section{Continuation of solutions in two variables}
\label{cross:section}\label{sec:product}
The local data give a matrix Green kernel when at least one variable lies
in the region where the two structures already agree. We must continue
this kernel to separated pairs outside that region. This section proves
the required continuation for elliptic systems acting on separate tensor
indices. The geometric construction and its scalar analytic foundation
are those of \cite[Section~3]{ScalarCompanion2026}; we reproduce the
argument, including the interpolation and overlap constructions, and
make the extension to systems explicit. No scalar inverse-problem
uniqueness theorem is used.
\subsection{Separate equations and extension from a cross}
Let $V_i$ be finite-dimensional Hermitian vector spaces. In a coordinate
domain in $\mathbb R^n$, consider an operator
\[
\mathsf L_i=-\Delta_{g_i}I_{V_i}
+C_i^a\partial_a+D_i,
\]
where $g_i$ is a smooth Riemannian metric and $C_i^a,D_i$ are smooth
endomorphism-valued coefficients. A \emph{product solution} is a
$V_1\otimes V_2$-valued function $F(x,y)$ satisfying
\[
(\mathsf L_1^x\otimes I)F=0,
\qquad (I\otimes\mathsf L_2^y)F=0.
\]
We suppress the identity factors below. The coefficients of one equation
act only on its indicated tensor index and depend only on its indicated
variable. In particular, the two equations commute. Their sum has
scalar real elliptic principal part on the product; ordinary unique
continuation therefore identifies product solutions that agree on a
nonempty open subset of a connected common domain. Multiplication of
either equation by a positive scalar function does not change its
solutions or any continuation constructed from them.
For matrix Green kernels the first tensor factor is the output fiber.
The second is the conjugate of the input fiber: the equation in $y$ is
obtained by conjugating the coefficients of the second connection
Laplacian. Equivalently, the adjoint columns of the kernel solve the
second equation. This convention places the two matrix actions on
separate indices, as required here.
The first construction extends compatible data on
$(D_1\times S_2)\cup(S_1\times D_2)$. Its mechanism is a basis that is
orthogonal both on a large domain and after restriction to the observed
set. Common-basis methods have a classical antecedent in separately
harmonic extension \cite{Zeriahi1982}; singular systems also enter
quantitative Runge approximation \cite{RulandSalo2019}. The interpolation
threshold and the construction below follow the scalar companion.
\begin{lemma}[Extension from a cross]\label{cross:series}
Let $D_i$ be bounded connected coordinate domains, and let
$S_i\Subset D_i$ be nonempty open sets. Suppose a product solution $F$
is given consistently on
\[
(D_1\times S_2)\cup(S_1\times D_2),
\]
with the sum of its two $L^2$ norms at most $M$. Suppose
$O_i\Subset D_i$ are open sets and every $V_i$-valued solution $w$ of
$\mathsf L_iw=0$ on $D_i$ satisfies
\begin{equation}\label{cross:interpolation-input}
\|w\|_{L^2(O_i)}
\le C\|w\|_{L^2(S_i)}^{\gamma_i}
\|w\|_{L^2(D_i)}^{1-\gamma_i},
\qquad 0<\gamma_i<1,\qquad \gamma_1+\gamma_2>1.
\end{equation}
Then $F$ extends as a product solution to $O_1\times O_2$, agreeing with
the two given pieces on their intersections. Its $C^m$ norm on every
compact subset of the output product is at most $C_mM$.
These constants are uniform when geometric margins, ellipticity, the
required coefficient bounds, interpolation constants, and a positive
lower bound for $\gamma_1+\gamma_2-1$ are uniform. A specified output
order requires only finitely many coefficient bounds.
\end{lemma}
\begin{proof}
Let $\mathcal H_1(D_1)$ be the closed subspace of
$L^2(D_1;V_1)$ consisting of distributional $\mathsf L_1$-solutions.
Restriction $R:\mathcal H_1(D_1)\to L^2(S_1;V_1)$ is compact by interior
elliptic estimates and is injective by unique continuation. The spectral
theorem for $R^*R$ gives an orthonormal basis $(w_j)$ of
$\mathcal H_1(D_1)$ such that
\begin{equation}\label{cross:singular-values}
(w_i,w_j)_{L^2(S_1)}=\mu_j^2\delta_{ij},\qquad 0<\mu_j\le1.
\end{equation}
For every $N$ there is a uniform constant $C_N$ with
\begin{equation}\label{cross:singular-decay}
\mu_j\le C_N(1+j)^{-N}.
\end{equation}
Indeed, multiply a solution by a cutoff equal to one near
$\overline{S_1}$ and compactly supported in $D_1$. Interior estimates
bound its $H^k$ norm by its $L^2(D_1)$ norm. Fourier truncation in a
containing cube, component by component, gives an approximation of rank
$O(A^n)$ and error $O(A^{-k})$. The minimax characterization of singular
values yields $\mu_j\le C_kj^{-k/n}$. The fixed fiber dimension changes
only the constants.
For $y\in S_2$, expansion in the first variable gives
\[
F(x,y)=\sum_j w_j(x)\otimes b_j(y),\qquad
\|b_j\|_{L^2(S_2;V_2)}\le M.
\]
Contracting the first tensor index against $\overline{w_j}$ extends the
coefficients to $D_2$:
\begin{equation}\label{cross:coefficient-extension}
b_j(y)=\mu_j^{-2}\int_{S_1}
\sum_\alpha\overline{w_{j,\alpha}(x)}F_{\alpha}(x,y)\,dx,
\qquad \|b_j\|_{L^2(D_2;V_2)}\le M\mu_j^{-1}.
\end{equation}
Here $F_\alpha(x,y)\in V_2$ are the first-index components in an
orthonormal basis of $V_1$; the displayed contraction leaves a vector
in $V_2$. Because the
second system acts on that remaining index, $\mathsf L_2b_j=0$.
Restriction orthogonality proves that this formula agrees with the
original coefficient on $S_2$. The assumed interpolation gives
\[
\|w_j\|_{L^2(O_1)}\le C\mu_j^{\gamma_1},\qquad
\|b_j\|_{L^2(O_2)}\le CM\mu_j^{\gamma_2-1}.
\]
Thus the tensor series converges absolutely in $L^2(O_1\times O_2)$:
the norm of its $j$th term is at most
$CM\mu_j^{\gamma_1+\gamma_2-1}$, and
\eqref{cross:singular-decay} makes these bounds summable. The limit
solves both equations distributionally. Interior estimates for their
elliptic sum give smoothness and the claimed bounds.
Agreement on the second arm requires a completeness check. Take
$z\in L^2(S_1;V_1)$ orthogonal to the closure of
$R\mathcal H_1(D_1)$. The $V_2$-valued function
\[
y\longmapsto\int_{S_1}\sum_\alpha\overline{z_\alpha(x)}F_\alpha(x,y)\,dx
\]
solves the second system and vanishes on $S_2$, hence vanishes on
connected $D_2$. Every second-arm slice therefore belongs to the
closed restricted range. Its expansion in the orthonormal basis
$w_j/\mu_j$ is complete, with coefficients $\mu_jb_j(y)$.
Consequently the series equals $F$ on $S_1\times O_2$; it converges
there by the positive exponent $\gamma_2$. On $O_1\times S_2$ it is the
original expansion, converging with exponent $\gamma_1$. These establish
both required agreements. Only upper bounds for $\mu_j$ have been
used, so uniformity does not require continuous choices of singular
bases or lower bounds on singular values.
\end{proof}
The remaining issue is geometric: we must place two observed sets so
that the interpolation exponents add to more than one. Strict Carleman
weights will supply those exponents from a curvature condition on the
boundary of the region where the product solution is known.
\subsection{A geometric criterion for local extension}
We next build the interpolation estimates needed in
Lemma~\ref{cross:series}. For a metric $g$, call a smooth real function
$\theta$ a \emph{strict weight} at a point where $d\theta\ne0$ if
\begin{equation}\label{cross:strict-weight}
\operatorname{Hess}_g\theta(e,e)
+\operatorname{Hess}_g\theta(v,v)>0
\quad\text{for all }v\perp e,\ |v|_g=1,
\qquad e=\frac{\nabla^g\theta}{|\nabla^g\theta|_g}.
\end{equation}
For the system $\mathsf L=-\Delta_gI+C^a\partial_a+D$, the strict
Carleman estimate gives, after shrinking the coordinate neighborhood,
\begin{equation}\label{cross:carleman}
\tau^3\|e^{\tau\theta}w\|_{L^2}^2
+\tau\|\nabla(e^{\tau\theta}w)\|_{L^2}^2
\le C\|e^{\tau\theta}\mathsf Lw\|_{L^2}^2,
\qquad \tau\ge\tau_0,
\end{equation}
for compactly supported vector-valued $w$. All norms sum over the
fiber components. The scalar form is the elliptic strict-weight estimate
of the classical
Carleman theory; see \cite[Chapter~XXVIII]{HormanderIV2009} and
\cite[Definition~8.3 and Theorem~9(a)]{KochTataru2005}.
We recall its energy proof to record the system dependence. First take
one scalar component and $\mathsf L=-\Delta_g$,
write $v=e^{\tau\theta}w$ and
\[
e^{\tau\theta}\mathsf Le^{-\tau\theta}=A_\tau+B_\tau,\qquad
A_\tau=-\Delta_g-\tau^2|d\theta|_g^2,\quad
B_\tau=\tau(2\nabla^g\theta\cdot\nabla+\Delta_g\theta).
\]
Here $A_\tau$ is self-adjoint and $B_\tau$ is skew-adjoint for
metric volume. Integration by parts gives
\begin{align*}
\|(A_\tau+B_\tau)v\|^2
&=\|A_\tau v\|^2+\|B_\tau v\|^2
+([A_\tau,B_\tau]v,v),\\
([A_\tau,B_\tau]v,v)
&=4\tau\int\operatorname{Hess}_g\theta(\nabla v,\nabla\overline v)
+4\tau^3\int\operatorname{Hess}_g\theta
(\nabla\theta,\nabla\theta)|v|^2
-\tau\int(\Delta_g^2\theta)|v|^2.
\end{align*}
Choose a real constant $m$ strictly between the negative of the least
unit tangential Hessian value and the unit normal Hessian value.
After shrinking the patch, these inequalities have a positive margin.
Add $4m\tau(A_\tau v,v)$ to the commutator form. Its gradient
coefficient becomes $4\tau(\operatorname{Hess}_g\theta+mg)$,
and its leading zero-order coefficient is
\[
4\tau^3|d\theta|_g^2
\big(\operatorname{Hess}_g\theta(e,e)-m\big)>c\tau^3.
\]
The tangential gradient form is positive. Its mixed and possibly
negative normal terms cost at most
$C\tau\|\nabla_e v\|^2$, and
\[
\|\nabla_e v\|^2
\le C\tau^{-2}\|B_\tau v\|^2+C\|v\|^2.
\]
Finally
$|4m\tau(A_\tau v,v)|\le\tfrac12\|A_\tau v\|^2+
C\tau^2\|v\|^2$.
For large $\tau$, the squared parts absorb the normal derivative
cost and the positive zero-order term absorbs the remaining errors.
This yields the scalar estimate. Summing it over components gives the
estimate for $-\Delta_gI$. The conjugated matrix lower-order terms have
norm at most $C(\|\nabla v\|+\tau\|v\|)$, so their squared norm is
absorbed by the $\tau$ and $\tau^3$ terms after increasing $\tau_0$.
This proves \eqref{cross:carleman} for the stated systems. The constants
are uniform on compact sets of strict weights, with fixed positive
strictness margins and bounded matrix coefficients. The scalar cutoff
commutators used below act componentwise, and interior estimates for
these systems have the same orders as in the scalar case.
The existence criterion below is the geometric criterion of
\cite[Section~3, ``A geometric criterion for local extension'']{ScalarCompanion2026}.
Its geometric proof depends only on the principal metrics. The system
estimate just proved and Lemma~\ref{cross:series} provide the analytic
steps in the present setting. All gradients, lengths, orthogonality
relations, and Hessians in its statement use the principal metrics
$g_1,g_2$ and their product connection.
\begin{proposition}[Product extension criterion]\label{cross:criterion}
Let $\rho$ be smooth near $z_0=(x_0,y_0)$, with $\rho(z_0)=0$ and
$d_x\rho,d_y\rho\ne0$. Put
\[
a_i=\frac{\nabla_i\rho}{|\nabla_i\rho|^2},\qquad H=\operatorname{Hess}\rho.
\]
Suppose that at $z_0$
\begin{equation}\label{cross:hessian-test}
H((a_1,-a_2),(a_1,-a_2))
+H((v_1,v_2),(v_1,v_2))>0
\quad
\left(v_i\perp a_i,\ |v_i|=|a_i|\right).
\end{equation}
Every product solution given on $\{\rho>0\}$ near $z_0$ extends to
a neighborhood of $z_0$, agreeing on a smaller part of that side.
In fact, the construction uses only two compact product sets in
$\{\rho>\varepsilon\}$ for some $\varepsilon>0$.
The neighborhoods and estimates on smaller neighborhoods are uniform
under small smooth perturbations preserving the strict hypotheses,
provided the data on those compact sets are bounded.
\end{proposition}
\begin{proof}
We first split the product Hessian condition into a strict weight in
each factor. Their sum will lie below a defining function for the given
side, with a quadratic gap. That gap places two intersecting product
sets in the given side; the weights then yield interpolation exponents
greater than one half, allowing Lemma~\ref{cross:series} to apply.
\paragraph{Splitting the weight between the factors.}
We seek symmetric forms $P_i$ satisfying
\[
P_i(a_i,a_i)+P_i(v_i,v_i)>0,
\qquad H+K\,d\rho^{\otimes2}>P_1\oplus P_2
\]
for some $K>0$, with the vectors $v_i$ as in
\eqref{cross:hessian-test}. After division by $|a_i|^2$, the first
inequality is the strict-weight condition for a function with gradient
$\nabla_i\rho$ and Hessian $P_i$. The second inequality will give the
quadratic gap. To obtain these forms by convex separation, the dual
tests are positive semidefinite forms $D$ whose diagonal blocks are
\[
d_i a_i^{\otimes2}+S_i,\qquad
d_i\ge0,\quad S_i\ge0\text{ on }a_i^\perp,
\quad \operatorname{tr} S_i=d_i|a_i|^2.
\]
This follows by separating the open convex cone consisting of positive
definite forms plus the allowed $P_1\oplus P_2$; the individual dual
cones are generated by $a_i^{\otimes2}+v_i^{\otimes2}$.
Normalize $\operatorname{tr} D=1$. To obtain a suitable $K$, it suffices by
compactness to prove $H:D>0$ on tests with $D\,d\rho=0$.
Represent such a $D$ as a covariance matrix. The normal components
measured by the two partial covectors are opposite. The diagonal-block
condition makes each normal component uncorrelated with its own
tangential component, hence with both tangential components.
Their variances coincide, giving $d_1=d_2=d>0$ and
\[
D=d(a_1,-a_2)^{\otimes2}+D_\perp,
\qquad \operatorname{tr}(D_\perp)_{ii}=d|a_i|^2.
\]
The case $d=0$ would force $D=0$. Among nonnegative tangential forms
with these two fixed traces, every extreme point has rank one:
on an image of rank $r\ge2$, a nonzero symmetric perturbation preserves
the two traces because $r(r+1)/2>2$, and small perturbations of both
signs preserve positivity. The rank-one tests are precisely those
in \eqref{cross:hessian-test}, multiplied by $d$.
The asserted positivity follows. Compactness of the normalized tests
then supplies $K$, as required.
Prescribe partial covectors $d_x\rho,d_y\rho$ and Hessians $P_i$
for smooth functions
$\theta_i$ vanishing at the respective points. In small coordinates
centered there, Taylor expansion gives
\begin{equation}\label{cross:quadratic-gap}
\theta_1(x)+\theta_2(y)
\le \widetilde\rho(x,y)-c(|x|^2+|y|^2),
\qquad \widetilde\rho=\rho+K\rho^2/2,
\end{equation}
with $c>0$. Each $\theta_i$ satisfies
\eqref{cross:strict-weight}. The positive side of $\widetilde\rho$
is the positive side of $\rho$ in this neighborhood.
\paragraph{Placing the cross and obtaining interpolation.}
Use coordinates $(s_i,z_i)$ with $s_i=\theta_i$. Fix a small
$\kappa>0$ so that $\phi(s_i)=s_i-\kappa s_i^2$ remains a strict weight.
Choose successively a small lateral radius $R$, a height
$l\ll cR^2$, and $e\ll\kappa l^2$. Take
\[
D_i=\{-l<s_i<l+4e,\ |z_i|<R\}.
\]
Use a cutoff equal to one on the inner lateral half and for
$-l+3e<s_i<l+2e$, supported in the cylinder and in
$-l+2e<s_i<l+3e$.
Let $S_i\Subset D_i$ include neighborhoods of its top and lateral
derivative supports, where respectively
\[
s_i>l+e/2\quad\text{or}\quad |z_i|>R/3,
\]
and an inner top slice near $s_i=l+5e/4$.
The two closed product sets $\overline{D_1\times S_2}$ and
$\overline{S_1\times D_2}$ lie strictly in the given side.
For a top slice, the sum of the two $s$ coordinates is positive.
For a lateral slice, it is greater than $-2l$, which is dominated
by the quadratic gap in \eqref{cross:quadratic-gap}.
Making the sets slightly smaller if necessary gives a common
$\varepsilon>0$ with $\rho>\varepsilon$ on their closures.
For an $\mathsf L_i$-solution normalized by $\|w\|_{L^2(D_i)}=1$, apply
\eqref{cross:carleman} to this cutoff times $w$. Interior estimates
bound the bottom derivative terms using the large norm and the weight
$\phi(-l+3e)$. The remaining derivative terms use
$\|w\|_{L^2(S_i)}$ and a weight at most $\phi(l+3e)$.
On a smaller inner cylinder beginning at $s_i=-e$, the weight is at
least $\phi(-e)$. Balancing the two exponentials yields
\eqref{cross:interpolation-input} with
\begin{equation}\label{cross:exponent}
\gamma_i=
\frac{\phi(-e)-\phi(-l+3e)}{
\phi(l+3e)-\phi(-l+3e)}>\frac12.
\end{equation}
Indeed at $e=0$ the quotient is $1/2+\kappa l/2$.
Values of the balancing parameter below $\tau_0$ are covered by
increasing the constant. The output cylinder may be chosen to include
a connected tube from the origin to the indicated top slice.
\paragraph{Constructing and identifying the extension.}
Lemma~\ref{cross:series} now gives a product solution on a neighborhood
of $z_0$. It agrees with the old solution there on the positive side:
from a sufficiently near positive-side point, increase $s_2$ into
the top slice. Since $\partial_{s_2}\rho>0$ locally, the segment stays
on that side and in the larger constructed cylinder. Agreement on
the slice and unique continuation along a neighborhood of the segment
prove agreement at the original point.
All choices have strict margins. Keeping their finitely many compact
sets and shrinking their output neighborhoods once proves the stated
perturbation uniformity by \eqref{cross:carleman},
Lemma~\ref{cross:series} and interior elliptic estimates.
\end{proof}
\subsection{Gluing along a compact family of surfaces}
Proposition~\ref{cross:criterion} provides a germ across one surface
point. The following formulation records the geometry needed to
combine these germs. In particular, agreement is proved on specified
overlap components, rather than inferred from the existence of a cover.
\begin{lemma}[Propagation through a compact cylinder]
\label{cross:propagation}
Suppose a region in a product patch has smooth coordinates $(z,a)$
near $B\times[a_-,a_+]$, where $B$ is a compact base, possibly with
boundary or corners. A single product solution is given on an open set $\mathcal S$ containing
neighborhoods of the top and lateral boundary.
Assume that $\mathcal S$ is upward closed in $a$ at fixed $z$. If every level $a=\text{constant}$
satisfies Proposition~\ref{cross:criterion} outside $\mathcal S$, with
the positive side pointing toward increasing $a$, the solution extends
to a neighborhood of the cylinder, agreeing on $\mathcal S$.
For fixed coordinates, compact sets and strict reference inequalities,
the extension has uniform compact-subset bounds under small smooth
perturbations, in terms of bounds on finitely many compact subsets of
the initially given region.
\end{lemma}
\begin{proof}
Starting from the top, let $a_*$ be the infimum of levels down to which
an extension agreeing with the seed has been obtained. In the uniform
version include uniform bounds in this property. At a non-seed point
of $B\times\{a_*\}$, Proposition~\ref{cross:criterion} uses compact
data strictly above that level. Take a finite cover of the level by
such output neighborhoods and seed neighborhoods. Their data are
contained above $a_*+\eta$ for one $\eta>0$, so an already reached
level supplies them with the needed bounds.
Shrink the outputs in $(z,a)$ coordinates to boxes
$B_i\times(a_*-d_i,a_*+d_i)$.
Every connected component of an overlap contains vertical segments
to a common height greater than $a_*$. Both germs equal the preceding
extension there. Unique continuation for $\mathsf L_1^x+\mathsf L_2^y$ makes them
identical on that overlap component. The same argument gives agreement
with $\mathcal S$, because a vertical segment moving upward from a
seed point stays in the seed. Boundary neighborhoods use the given
solution. A finite subcover has a positive minimum output width and
therefore passes the putative last level, or includes the endpoint
$a_-$. This proves continuation on the whole cylinder.
The same finite constructions retain all strict margins under small
perturbations. Their Carleman, series and interior estimates prove
the uniform assertion. Data bounded up to a limiting surface are
never required: each local construction uses compact sets strictly
above it.
\end{proof}
\subsection{Initialization at a fixed distance from the diagonal}
We now return to the connection Laplacians. In common coordinate
patches centered at $0$, let $(g_j,A_j)$ be smooth metrics and unitary
connection matrices obtained from interior patches of compact Dirichlet
manifolds. Use fixed local unitary frames, and let $G_j$ denote their
matrix Dirichlet Green kernels, normalized with metric volume.
Suppose that both coefficient pairs agree on an open set $W$ containing
the lower side of a smooth hypersurface through $0$, and that
$G_1(x,y)=G_2(x,y)$ for distinct $x,y\in W$. After a common linear
coordinate change, assume
\[
g_1(0)=g_2(0)=I_n,\qquad
\{x_n<q(x')\}\subset W\text{ near }0,\qquad
q(0)=0,\quad d q(0)=0,
\]
for a smooth function $q$. No regularity of the rest of $\partial W$
is assumed.
For a small spatial dilation parameter $\lambda>0$, put
\begin{equation}\label{cross:dilation}
\begin{gathered}
g_{j,\lambda}(z)=g_j(\lambda z),\qquad
A_{j,\lambda}(z)=\lambda A_j(\lambda z),\qquad
W_\lambda=\lambda^{-1}W,\\
G_{j,\lambda}(x,y)=\lambda^{n-2}G_j(\lambda x,\lambda y).
\end{gathered}
\end{equation}
On each fixed bounded patch the metrics tend smoothly to the Euclidean
metric. The corresponding dilated systems have uniformly bounded
smooth lower-order coefficients on every fixed bounded set, and
$W_\lambda$ contains every fixed compact subset of $\{z_n<0\}$ for
sufficiently small $\lambda$.
Initially define, off the diagonal,
\begin{equation}\label{cross:mixed-data}
\mathcal G_\lambda(x,y)=
\begin{cases}
G_{1,\lambda}(x,y),&y\in W_\lambda,\\
G_{2,\lambda}(x,y),&x\in W_\lambda.
\end{cases}
\end{equation}
The pieces agree on their overlap. They solve the first connection
Laplacian in $x$ and the conjugate second connection Laplacian on the
second index in $y$, in the convention at the beginning of the section.
Indeed, on the observed factor its two coefficient systems agree.
Thus \eqref{cross:mixed-data} is a product solution of the systems
already treated.
\begin{proposition}[Fixed-separation initialization]\label{cross:initial}
In this setting, let $n\ge3$ and fix $B>0$, $d>0$ and $\eta>0$.
For all sufficiently small $\lambda$, the data
\eqref{cross:mixed-data} have a product-solution continuation to a
neighborhood of
\[
\{(x,y):|x|,|y|\le B,\ |x-y|\ge d\}.
\]
Every fixed $C^m$ norm on a smaller neighborhood is bounded uniformly
in $\lambda$. It agrees with $G_{1,\lambda}$ when $y\in W_\lambda$
and with $G_{2,\lambda}$ when $x\in W_\lambda$, on the corresponding
overlaps. In particular these agreements hold on the fixed truncated
lower strips $y_n<-\eta$ and $x_n<-\eta$, respectively. The smallness threshold for $\lambda$ and
the constants may depend on $B,d,\eta,m$.
\end{proposition}
\begin{proof}
For the final comparison with the full original cross, fix at the outset
\[
\widehat B=B+\eta+4,\qquad
\widehat d=\min(d/2,1/4).
\]
We construct the continuation for these enlarged location bounds and
smaller separations, and then restrict to the stated target.
We first cross the limiting plane, then deform variable-radius tubes
to reach those separations. This follows the two-stage
construction of \cite[Section~3, ``Fixed-separation initialization'']{ScalarCompanion2026}.
We first choose the Euclidean geometric ratios, including the opening of
the region reached in the first stage. We next choose the entire tube
cylinder and a bounded positive-separation range containing it. The
first-stage constructions are then made on that range. Only after all
these fixed sets have been chosen do we decrease $\lambda$.
\paragraph{Crossing the plane and obtaining a common seed.}
We first extend slightly across the plane in one variable while keeping
the other at a comparable positive separation. The first interpolation
exponent can be made close to one; this compensates for the fixed
positive exponent obtained by propagation in the separated second
variable.
For the Euclidean equations, the following choices are invariant under
translations parallel to the plane and under a common change of spatial
scale. We make the geometric choices on a unit-scale template. They
therefore give an opening constant $c_0>0$ independent of the bounded
range on which the construction is subsequently used. On any fixed
compact range of centers and scales $0<s_{\min}\le s\le s_{\max}$,
the same strict weights and geometric fractions work for the dilated
systems once $\lambda$ is sufficiently small. The lower-order
coefficients are bounded uniformly there (and converge to zero under
the connection dilation).
Fix a point $x_0$ of height zero and a separation scale $s>0$.
In the second factor take
\[
D_2=\{s/2<|y-x_0|<2s\}
\]
and an observed ball $S_2$ about $x_0-se_n$.
In the first take a ball of radius $3\vartheta_0s$ centered at
$x_0-\vartheta_0se_n$, with $\vartheta_0>0$ fixed small enough that the two
large domains are separated. Its observed ball has radius
$(1-\alpha)\vartheta_0s$, with $\alpha>0$ to be chosen.
These observed balls lie strictly in the lower half-space. On a fixed
compact scale range their heights are bounded above by a common negative
number, since $s\ge s_{\min}>0$. Thus, after a finite covering, one
choice of small $\lambda$ places every observed ball in $W_\lambda$
with room. No neighborhood of the whole limiting plane is assumed to
belong to $W_\lambda$.
There is a two-constants estimate from $S_2$ to a slightly enlarged
closed annulus $4s/5\le|y-x_0|\le6s/5$, with an exponent
$\gamma_2>0$ independent of $\alpha$. Here is a direct way to obtain
it with room for perturbations. On a Euclidean annulus the weight
$|z-z_*|^{-1}$ is strict: its radial Hessian is
$2|z-z_*|^{-3}$ and each tangential Hessian value is
$-|z-z_*|^{-3}$. Radial cutoffs in
\eqref{cross:carleman}, followed by exponential balancing, propagate
smallness from an inner ball to an intermediate ball using the norm
on an outer ball. A finite chain of overlapping balls with enlarged
balls inside $D_2$ connects $S_2$ to the target annulus. The annulus
is connected since $n\ge3$. Multiplying the finitely many positive
exponents gives the asserted $\gamma_2$. The same weights and margins
work for sufficiently small perturbations of the Euclidean metric.
In the first factor use this radial argument centered at
$x_0-\vartheta_0se_n$. Keep the outer cutoff a fixed distance beyond radius
$\vartheta_0s$, and put the inner cutoff just inside radius
$(1-\alpha)\vartheta_0s$. Take a concentric target ball of radius
$(\vartheta_0+c)s$, with $c>0$ small; it contains the observed ball and
a ball of radius $cs$ about $x_0$.
To see the resulting exponent, let $R_{\rm in}$ be an inner cutoff
radius, chosen just inside the observed radius, let
$R_{\rm tar}=(\vartheta_0+c)s$, and let $R_{\rm out}>R_{\rm tar}$ be
a fixed starting radius for the outer derivative support. With
$\theta(R)=R^{-1}$, exponential balancing gives
\[
\gamma_1=
\frac{\theta(R_{\rm tar})-\theta(R_{\rm out})}
{\theta(R_{\rm in})-\theta(R_{\rm out})}.
\]
The inner core is bounded directly by the observed norm. As $\alpha$
and $c$ decrease, choose the inner cutoff so that both
$R_{\rm in}$ and $R_{\rm tar}$ approach $\vartheta_0s$, while
$R_{\rm out}$ stays a fixed distance beyond that radius. Thus
$\gamma_1\to1$. Choose these constants so that
$\gamma_1+\gamma_2>1$.
Lemma~\ref{cross:series} extends the kernel to a small ball about
$x_0$ times the target annulus. On these fixed separated sets the
rescaled Green kernels and their derivatives are uniformly bounded:
the same bounds hold entrywise for the smooth systems used here. To see
this, an interior inverse parametrix has order $-2$ and frequency pieces
bounded by $C_N2^{j(n-2)}(1+2^j|x-y|)^{-N}$, up to a smooth kernel.
Summing gives the local bound $C|x-y|^{2-n}$ for $n\ge3$.
The normalization in \eqref{cross:dilation} preserves this bound;
interior estimates for the uniformly elliptic dilated equations give all
fixed derivative bounds away from the pole.
The extensions agree with both lower pieces on their actual comparison
domains. For $y\in S_2$, the series equals $G_{1,\lambda}$ for all
$x$ in its first output ball. Fix any such $x$ in a truncated lower
half-space contained in $W_\lambda$. On $S_2$ both variables then lie
in $W_\lambda$, so this value also equals $G_{2,\lambda}$. Unique
continuation for the second system on the connected output annulus
proves agreement with $G_{2,\lambda}$ throughout that annulus. In the
other direction, fix $y$ in the output annulus and in $W_\lambda$.
On the first observed ball $S_1$, agreement with the second arm gives
$G_{2,\lambda}=G_{1,\lambda}$. Unique continuation for the first
system on its connected output ball gives agreement with
$G_{1,\lambda}$ there. Each comparison stays a positive distance from
the diagonal.
We make the overlap comparison precise because the two variables may
play different roles in two local constructions. Inside the output just
obtained choose a larger comparison product of the form
\[
P(x_0,s)=B(x_0,\kappa s)\times
\{(1-\zeta)s<|y-x_0|<(1+\zeta)s\},
\]
where $\kappa,\zeta>0$ are fixed geometric fractions. The first ball
is contained in the actual concentric first output ball, and the annulus
is contained in the actual second output. Keep the full original
outputs for the preceding lower-piece comparisons. For coverage retain
only
\[
P'(x_0,s)=B(x_0,\epsilon_1s)\times
\{(1-\zeta')s<|y-x_0|<(1+\zeta')s\},
\qquad 0<\epsilon_1\ll\kappa,\quad 0<\zeta'<\zeta.
\]
The transposed products are used when $y$ is the near-plane variable.
If a point belongs to two retained products, their scales are comparable
to each other and to $|x-y|$, with fixed constants. Choose $\epsilon_1$
small enough, relative to $\kappa$ and $\zeta-\zeta'$, that a downward
translation of size at most a fixed multiple of $\epsilon_1$ times
either scale stays inside each larger comparison product whenever it
acts on a retained near-plane variable. This follows directly from
the triangle inequality for the ball and annular distances. On the
chosen compact scale range take $\eta_0>0$ much smaller than the
remaining margins times $s_{\min}$, and with $\eta_0\le\eta$.
For products of the same orientation, lower only their common near-plane
variable by
\[
h=\max(x_n,0)+2\eta_0
\]
(or the analogous expression in $y$). The other variable remains
fixed. For products of opposite orientations, both $x$ and $y$ lie
in retained near-plane balls; lower both by
\[
h=\max(x_n,y_n,0)+2\eta_0.
\]
The scale comparability and the preceding choices keep the entire path
in the intersection of the two larger comparison products. Its
endpoint has the near-plane variable below $-\eta_0$ in the first case,
and both variables below $-\eta_0$ in the second. For small enough
$\lambda$ these endpoint neighborhoods lie in the original lower cross,
where the preceding comparisons identify both germs. Unique
continuation for the elliptic sum on the component of the larger
intersection containing the path identifies the germs at the starting
point. The path need not remain in the retained products. This proves
agreement on every retained overlap without a connectedness assumption
on that overlap.
Apply the same construction with the variable roles and their tensor
factors interchanged, and then return the values to the original tensor
order. The resulting germs solve the same two equations as the first
orientation. The overlap comparison therefore applies to them and
produces, for some fixed $c_0>0$, a single-valued continuation in
\begin{equation}\label{cross:coarse-region}
\min(x_n,y_n)<c_0|x-y|,
\end{equation}
on every fixed compact range of bounded locations and positive
separations. The estimates are uniform in $\lambda$ on that range.
For coverage near the plane, take $x_0=(x',0)$ and
$s=|y-x_0|$ when the first variable has the smaller nonnegative height;
then $x$ is in the retained first ball and $y$ in the retained annulus
if $c_0$ is small; for instance
$c_0/(1-c_0)<\epsilon_1$ guarantees the required ball inclusion.
Points already below the plane are covered by the
same construction near it or by the given cross farther below it.
The preceding overlap argument makes these choices compatible.
\paragraph{Sweeping tubes inward from the common seed.}
The region \eqref{cross:coarse-region} is now the seed for the second
stage. At fixed center $y$ and direction $\omega$, we move $x$ along the
ray from $y$: large radii lie in the seed, and decreasing the radius
will reach the target pairs. Choose $M>\widehat B+3$ and introduce
\begin{equation}\label{cross:tubes}
t=M+y_n+e_0|y'|^2,\qquad r_a(y)=a t^{1+\sigma},\qquad
x=y+r_a(y)\omega,\quad |\omega|=1,
\end{equation}
where $\sigma,e_0>0$ will be small. The base is
\[
y_n\ge-\widehat B-1,\qquad t\le T_0,\qquad \omega\in\mathbb S^{n-1},
\]
and $a$ decreases through a fixed interval $[a_{\min},a_{\max}]$.
The term $e_0|y'|^2$ makes this base compact. The parameter $a$ decreases
while $(y,\omega)$ remains fixed, as required by the compact-cylinder
propagation lemma.
Put $m=M-\widehat B-1>2$. Reserve the bounds
$0<\sigma\le1$, $e_0\le(1+\widehat B^2)^{-1}$, and
$e_0T_0\le1/4$. Once $\sigma$ is chosen below, take
\[
0<a_{\min}<
\frac{\widehat d}{2(M+\widehat B+1)^{1+\sigma}},
\qquad
a_{\max}>\max\left(a_{\min},\frac{2}{c_0m^\sigma}\right),
\]
and then choose
\[
T_0>2(M+\widehat B+1),\qquad
c_0a_{\min}T_0^\sigma>2.
\]
For every target center, $t\le M+\widehat B+1$; hence its target
separations exceed $r_{a_{\min}}(y)$. On the initial level,
$c_0r_{a_{\max}}>2t>y_n$, so this level lies in
\eqref{cross:coarse-region}. The bottom base boundary has $y_n<0$,
and on the face $t=T_0$ we have $c_0r_a>2t>y_n$ for all allowed $a$.
Thus the initial level and both base boundary faces are supplied by
the seed. The power $1+\sigma>1$ is what permits this last inequality
at the top face.
This known region is upward closed in $a$: since
$\min(x_n,y_n)=y_n+r\min(\omega_n,0)$, its defining
inequality is $y_n<(c_0-\min(\omega_n,0))r$, whose radius coefficient
is positive.
It remains to check the strict extension criterion outside that region.
At the Euclidean metrics use
\[
\rho=\tfrac12(|x-y|^2-r_a(y)^2),\qquad b=\nabla r_a.
\]
Away from $\omega+b=0$, multiply the vectors in
\eqref{cross:hessian-test} by the common positive factor $r_a$ and
write their real and imaginary parts as
\begin{equation}\label{cross:null-vectors}
\begin{aligned}
U_1&=\omega+iv,& |v|=1,\quad v\perp\omega,\\
U_2&=\frac{\omega+b}{|\omega+b|^2}+iw,
&w\perp(\omega+b),\quad |w|=|\omega+b|^{-1}.
\end{aligned}
\end{equation}
The identity $\omega\cdot(U_1-U_2)=b\cdot U_2$ cancels the normal
part of the distance Hessian. Consequently the test is
\begin{equation}\label{cross:tube-test}
\big|\pi_{\omega^\perp}(U_1-U_2)\big|^2
-(r_a\partial^2r_a)(U_2,\overline{U_2}).
\end{equation}
For bounded $b$, outside fixed small neighborhoods of
$b=-\omega$ and $b=-2\omega$, the first term is at least $c|b|^2$.
To see this, a zero forces the real part of $U_2$ to be parallel to
$\omega$, and equality of the projected imaginary lengths gives
$|\omega+b|=1$. Thus $b=0$ or $b=-2\omega$.
Near $b=0$, the real projected difference controls
$|\pi_{\omega^\perp}b|$ to first order; also
\[
|\pi_{\omega^\perp}w|
=1-\omega\cdot b+O(|b|^2),
\]
so the imaginary projected difference controls $|\omega\cdot b|$
to first order. This proves the quadratic lower bound near zero;
compactness proves it on the remaining range.
Outside \eqref{cross:coarse-region}, $y_n\ge c_0r_a$, so
$r_a/t\le1/c_0$ and
\[
b=(1+\sigma)(r_a/t)\nabla t
\]
has $|b|\le4/c_0$ under the reserved bounds. This fixes a compact
range for $b$ before $\sigma$, the $a$-interval, $T_0$, and $e_0$
receive their final values. When $\nabla t$ is close to $e_n$, the two excluded
neighborhoods already lie in \eqref{cross:coarse-region}. Indeed,
\[
\frac{x_n}{r_a}
\le \frac{(1+\sigma)|\nabla t|}{|b|}+\omega_n,
\]
which is arbitrarily small near $b=-\omega$ and negative near
$b=-2\omega$ as $\sigma$ and the gradient error tend to zero.
On the complement $|U_2|$ is bounded and differentiation of
\eqref{cross:tubes} gives
\begin{equation}\label{cross:tube-error}
r_a|\partial^2r_a|
\le C(\sigma+T_0e_0)|b|^2.
\end{equation}
Here the choices can be made in a definite order. First choose a
neighborhood radius $\delta_0>0$ for $b=-\omega$ and $b=-2\omega$,
in terms of $c_0$, and upper bounds on $\sigma$ and
$|\nabla t-e_n|$ so that their closures satisfy $x_n/r_a<c_0/2$.
On the complementary bounded $b$-range let $c>0$ be the constant in the
quadratic lower bound. There $|U_2|^2\le2/\delta_0^2$.
Direct differentiation sharpens \eqref{cross:tube-error} to
\[
\frac{r_a|\partial^2r_a|}{|b|^2}
\le\sigma+2e_0T_0.
\]
Choose $\sigma>0$ small enough that its contribution to this bound is
less than $c\delta_0^2/8$. Choose the $a$-interval and $T_0$ as above,
and finally take $e_0>0$ sufficiently small that all reserved bounds hold,
$|\nabla t-e_n|\le2\sqrt{e_0T_0}$ has the required smallness, and
$\sigma+2e_0T_0<c\delta_0^2/4$. The second term of
\eqref{cross:tube-test} is then at most $(c/2)|b|^2$ in absolute value.
The whole test is bounded below by
\[
\frac c2|b|^2
\ge\frac c2(1+\sigma)^2a_{\min}^2m^{2\sigma}>0
\]
where continuation is needed. Also the two partial gradients of
$\rho$ have lengths at least $r_a$ and $\delta_0r_a$, respectively.
This supplies fixed strict margins before any metric perturbation.
All these sets are bounded and stay at separation at least
$a_{\min}(M-\widehat B-1)^{1+\sigma}>0$. Smooth convergence of the metrics
therefore preserves the strict tests for sufficiently small $\lambda$.
At this point the tube parameters and its compact range have all been
fixed. Apply the first-stage construction on a slightly larger bounded
range of locations and separations containing this cylinder, and then
choose $\lambda$ small enough for both its data and the strict tube
tests. Lemma~\ref{cross:propagation} continues the kernel through the
cylinder, with uniform estimates and agreement on the seed. A target pair has
$a=|x-y|/t^{1+\sigma}\ge a_{\min}$; if $a>a_{\max}$ it is already in
the coarse region. Hence every target pair is covered.
It remains to identify this extension with both pieces of the full
cross \eqref{cross:mixed-data}, including parts of $W_\lambda$ not in
a truncated half-space. For $|y|\le B$ put
\[
D_y=B(0,\widehat B)\setminus\overline{B(y,\widehat d)},\qquad
A=B(-(B+\eta+2)e_n,1/4).
\]
The excluded closed ball lies strictly inside $B(0,\widehat B)$,
so $D_y$ is connected; it contains the fixed lower ball $A$, which
is disjoint from every excluded ball and lies below $x_n=-\eta$.
For fixed $y\in W_\lambda$ in the target range, the constructed kernel
and $G_{1,\lambda}$ solve the same first-variable system on $D_y$.
On $A$ the constructed kernel equals $G_{2,\lambda}$, which equals
$G_{1,\lambda}$ because both variables are in $W_\lambda$.
Unique continuation on $D_y$ proves agreement on the target overlap.
Interchanging the variables proves agreement with the other piece.
The inequalities defining $\widehat B$ and $\widehat d$ leave open
room around the original target, so this also gives the claimed
neighborhood and its compact-subset estimates.
\end{proof}
The constants in Proposition~\ref{cross:initial} may deteriorate as
$d\downarrow0$. Its role is to initialize the sphere construction at fixed positive
separations. Lemma~\ref{cross:propagation} will subsequently give
existence at each positive radius of a shrinking family. The separate
estimate on that family's transfer density, rather than an estimate
obtained by iterating this continuation, will control the shrinking
limit.
==== END Determination-of-a-metric-and-a-unitary-connection-from-one-boundary-patch//03-product.tex ====
==== BEGIN Determination-of-a-metric-and-a-unitary-connection-from-one-boundary-patch//04-spheres.tex ====
\section{Shrinking spheres and harmonic transfer}
\label{sec:spheres}\label{sph:section}
We now work with the contact data of Proposition~\ref{bdy:contact}.
The two systems agree on the lower side of a smooth hypersurface
through the origin, and their Green matrices agree when both arguments
are on that side. Our aim is to represent a map between their local
harmonic sections by integration over small spheres. Product
continuation constructs this map at each positive radius. A separate
estimate in the next section will control its density as the radii
collapse.
The sphere geometry and radius profiles below are adapted from
\cite[Section~4]{ScalarCompanion2026}. We give their proofs, including
the strict Hessian calculation, and formulate the resulting transfer
for the present matrix equations. The coordinates need not be harmonic.
\subsection{A concave depth and the radius profiles}
Apply the spatial dilation~\eqref{bdy:dilation-formulas}. In this section we
suppress the dilation subscript on $g_j,A_j,G_j,F_j$ and on the paired
functions, while retaining $\lambda$ for the dilation parameter.
On every fixed bounded coordinate region, $g_j\to I$ smoothly and
$A_j\to0$ smoothly as $\lambda\downarrow0$. The harmonic frames and
their inverses have uniform smooth bounds there. All bounded regions
and positive separation scales used below are fixed before the final
upper bound on $\lambda$ is chosen.
Put
\[
Y=\{y=(y',y_n):|y'|\le1,\ -1\le y_n\le1\},
\qquad t_0(y)=y_n+|y'|^2.
\]
For a sufficiently large fixed $L$, define
\begin{equation}\label{sph:normalization}
\gamma=e^{-2Lt_0}g_1,\qquad a=\gamma^{-1},\qquad
Q=\frac{n g_2^{-1}}{\operatorname{tr}(\gamma g_2^{-1})},
\qquad h=Q-a.
\end{equation}
These tensors are defined on a fixed neighborhood of $Y$, also used
for the first-variable points. Specify the scalar multipliers in the
harmonic-frame convention~\eqref{bdy:harmonic-normalization} by
\begin{equation}\label{sph:frame-normalization}
\mathcal L_jv=-c_jF_j^{-1}L_j(F_jv),\qquad
c_1=e^{2Lt_0},\qquad
c_2=\frac{n}{\operatorname{tr}(\gamma g_2^{-1})}>0.
\end{equation}
Their second-derivative coefficient matrices are $a$ and $Q$,
respectively, and their zeroth-order terms vanish because the columns
of $F_j$ are harmonic. These changes leave the solution spaces
unchanged after the frame identification. For product continuation
we use the positive-principal-symbol convention: $-\mathcal L_j$ in
harmonic frames, or equivalently $c_jL_j$ in unitary frames. The
corresponding principal metrics are $\gamma$ and $Q^{-1}$.
The normalization ensures
\begin{equation}\label{sph:trace-free}
\operatorname{tr}(\gamma h)=0,\qquad h\longrightarrow0
\quad\hbox{in every fixed smooth norm as }\lambda\downarrow0.
\end{equation}
Define a depth function by
\begin{equation}\label{sph:depth}
t(y)=\int_{1/8}^{t_0(y)}e^{-2Ls}\,ds.
\end{equation}
It is negative near the origin and has nonvanishing differential.
Its useful feature is strict concavity for the reference metric:
\begin{equation}\label{sph:depth-hessian}
\operatorname{Hess}_{\gamma}t
=e^{-2Lt_0}\bigl(
\operatorname{Hess}_{g_1}t_0
-L|dt_0|_{g_1}^2g_1\bigr)\le-c\gamma.
\end{equation}
Indeed the conformal connection formula cancels the two
$dt_0\otimes dt_0$ terms. The norm of $dt_0$ is bounded below on the
fixed region, whereas $\operatorname{Hess}_{g_1}t_0$ is uniformly bounded
for small $\lambda$. Choosing $L$ first proves the last inequality.
All constants describing this reference geometry are henceforth fixed.
For an integer $p_*\ge2$, an amplitude $R_*>0$, and $0<\delta\le1$, put
\begin{equation}\label{sph:radius}
\ell_\delta(t)=\delta\log(1+e^{t/\delta}),\qquad
r_\delta(y)=R_*\ell_\delta(t(y))^{p_*},\qquad
f_*=\log r_\delta,\qquad \beta=|df_*|_\gamma.
\end{equation}
We use the $\gamma$-geodesic ball with center $y$ and radius
$r_\delta(y)$. These balls shrink to a point on $\{t\le0\}$, including
a neighborhood of the contact point. At positive depths bounded away
from zero, their radii retain a positive lower bound. This will let us
place cutoff derivatives where the transfer is already controlled.
\begin{lemma}[Uniform profile bounds]\label{sph:profiles}
For the geometry just fixed and all sufficiently small $\lambda$, the
profiles~\eqref{sph:radius} satisfy
\begin{equation}\label{sph:profile-bounds}
\beta\ge cp_*,\qquad
\operatorname{Hess}_\gamma f_*\le-c\beta\gamma,
\qquad |\partial^j f_*|\le C_j\beta^j\quad(j\ge1),
\end{equation}
uniformly in $0<\delta\le1$. The constants $c,C_j$ are independent of
$p_*\ge2$. For fixed $p_*,R_*$, the functions $r_\delta^{1/p_*}$ have
a uniform Lipschitz bound. The radii increase strictly with $\delta$,
converge to zero where $t\le0$, and obey
\begin{equation}\label{sph:amplitude-smallness}
\sup_{0<\delta\le1,\ y\in Y}r_\delta\beta^2
\longrightarrow0\quad\hbox{as }R_*\downarrow0
\end{equation}
for each fixed $p_*$.
\end{lemma}
\begin{proof}
Write $a_\delta=(\log\ell_\delta)'$. With $s=t/\delta$,
\[
a_\delta(t)=\delta^{-1}A(s),\qquad
A(s)=\frac{e^s}{(1+e^s)\log(1+e^s)}.
\]
The function $A$ is positive and decreasing: setting $u=e^s$ gives
\[
A'(s)=\frac{u\bigl(\log(1+u)-u\bigr)}
{(1+u)^2\log(1+u)^2}<0.
\]
At the negative end $A$ tends to one, and at the positive end it is
comparable to $(1+s)^{-1}$. Differentiating its convergent expressions
at the two ends gives $|A^{(j)}|\le C_jA^{j+1}$; positivity gives the
same bounds on the remaining compact interval. Consequently, on the
fixed range of $t$,
\[
a_\delta\ge c>0,\qquad a_\delta'\le0,\qquad
|\partial_t^j\log\ell_\delta|\le C_j a_\delta^j,
\qquad a_\delta\le\ell_\delta^{-1}.
\]
Since $|dt|_\gamma$ is bounded above and below,
$\beta=p_*a_\delta|dt|_\gamma$. The identity
\[
\operatorname{Hess}_\gamma f_*
=p_*a_\delta\operatorname{Hess}_\gamma t
+p_*a_\delta'\,dt\otimes dt
\]
and~\eqref{sph:depth-hessian} prove the first two bounds
in~\eqref{sph:profile-bounds}. The chain rule gives the derivative
bounds; their constants can be independent of $p_*$ because
$p_*a_\delta^j\le p_*^j a_\delta^j$ for $j\ge1$.
The estimate $0<\ell_\delta'<1$ proves the Lipschitz assertion.
Moreover,
\[
\partial_\delta\ell_\delta
=\log(1+e^s)-\frac{s e^s}{1+e^s}>0.
\]
The displayed expression tends to zero at both ends and its derivative
in $s$ is $-s e^s/(1+e^s)^2$, so it is strictly positive at finite $s$.
The limit of $\ell_\delta$ is $\max(t,0)$. Finally,
\[
r_\delta\beta^2\le C R_*p_*^2\ell_\delta^{p_*-2},
\]
and $\ell_\delta$ is uniformly bounded on the fixed depth interval.
This proves~\eqref{sph:amplitude-smallness}.
\end{proof}
\subsection{The core, cutoff, and order of choices}\label{sph:core-cutoff}
Choose $t_b>0$ so small that $3t_b<t(1/4)$, where $t(1/4)$ denotes
the value of the increasing function in~\eqref{sph:depth} at
$t_0=1/4$. Define
\begin{equation}\label{sph:cutoff}
\begin{gathered}
K=\{y\in Y:y_n\ge-1/2,\ t(y)\le2t_b\},\\
\chi\in C_c^\infty(Y^\circ),\qquad \chi=1\text{ near }K,\\
\operatorname{supp}(d\chi)\subset
\{y_n<-1/4\}\cup\{t>t_b\}.
\end{gathered}
\end{equation}
Such a cutoff exists. Indeed, on $K$ one has $t_0<1/4$, and thus
$|y'|^2<3/4$ and $y_n<1/4$; hence $K\Subset Y^\circ$.
Multiply a cutoff changing from zero to one between $y_n=-3/4$
and $y_n=-1/2$ by a cutoff of $t$ changing from one to zero between
$2t_b$ and $3t_b$, moving its endpoints slightly to leave room around
$K$. The inequality $3t_b<t(1/4)$ keeps the support away from the
lateral and upper faces of $Y$. On the second cutoff region,
\begin{equation}\label{sph:positive-depth-radius}
r_\delta\ge R_*t_b^{p_*}>0.
\end{equation}
Every point of $K$ can be joined to the known lower region by the
vertical segment with the same $y'$ and initial height $-1/2$.
That segment stays in $K$; both $t$ and $r_\delta$ are nondecreasing
along it. For sufficiently small $\lambda$, its initial point and a
neighborhood of it lie in the known side.
We will use the following temporary metric comparison:
\begin{equation}\label{sph:smallness}
H_8(y):=\sum_{|\alpha|\le8}|\partial^\alpha h(y)|
\le\varepsilon r_\delta(y)\qquad(y\in Y).
\end{equation}
Coordinate norms in this definition are uniformly equivalent to the
reference-metric norms. This condition will later be improved and
then removed by a continuity argument.
\begin{remark}[Parameter choices]\label{sph:choices}
The reference geometry, $t_b$, $K$, and $\chi$ are fixed first.
Choose $p_*$ sufficiently large and then $\varepsilon>0$ sufficiently
small. The lower bounds on $p_*$ and upper bounds on $\varepsilon$
used below depend only on the fixed geometry and coefficient bounds,
before $R_*$ and the final dilation are chosen.
Choose $R_*$ so that every ball lies in a normal neighborhood with
fixed coordinate room and
\begin{equation}\label{sph:radius-smallness}
r_\eta|d\log r_\eta|_\gamma^2\le\varepsilon
\qquad(0<\eta\le1,\ y\in Y).
\end{equation}
The fixed-separation scales determined by these choices are positive,
although they may be small. Only then reduce $\lambda$ to meet the
fixed-separation initialization and the coefficient bounds. The
parameter $\delta$ remains free subject to~\eqref{sph:smallness}.
In particular the upper bound on $\lambda$ does not depend on
$\delta$.
\end{remark}
\subsection{The strict Hessian test for a moving sphere}
The difficulty in approaching the diagonal is that the radius varies
with the center. The leading terms from its gradient cancel in the
product Hessian test. The strict concavity of its logarithm provides
the positive remainder.
\begin{lemma}[Strict sphere geometry]\label{sph:strict}
Let $r=r_\delta$ satisfy~\eqref{sph:profile-bounds},
\eqref{sph:smallness}, and~\eqref{sph:radius-smallness}.
For $p_*$ sufficiently large and $\varepsilon$ sufficiently small,
the hypersurface
\[
\rho(x,y)=\tfrac12\bigl(d_\gamma(x,y)^2-r(y)^2\bigr)=0
\]
satisfies the strict criterion of Proposition~\ref{cross:criterion}
for principal metrics $\gamma$ in the first factor and $Q^{-1}$ in
the second. Both partial gradients are nonzero, and the positive
side is the exterior of the moving ball.
\end{lemma}
\begin{proof}
Put $b=\nabla^\gamma r$, so that $|b|=r\beta$. Parallel transport
along the radial $\gamma$-geodesic identifies the endpoint tangent
spaces; let $\omega$ be its unit direction from $y$ to $x$.
Multiply the normalized vectors in Proposition~\ref{cross:criterion}
by the common factor $r$. Their real and imaginary parts can be
written as
\[
U_1=\omega+i v,\qquad v\perp\omega,\quad |v|=1,
\qquad U_2=q+i w.
\]
All lengths in the following calculation use $\gamma$. Set
$s=\omega+b$. In a $\gamma$-orthonormal frame at $y$, the exact
conditions on the second vector are
\begin{equation}\label{sph:test-vectors}
q=\frac{Qs}{s^TQs},\qquad s\cdot w=0,\qquad
w^TQ^{-1}w=\frac1{s^TQs}.
\end{equation}
Here the matrix of $a$ is the identity. Therefore
\begin{equation}\label{sph:test-vector-bounds}
s\cdot U_2=1,\qquad
q=\frac{s}{|s|^2}+O(|h|),\qquad
|w|=|s|^{-1}+O(|h|).
\end{equation}
The assumptions make $b$ and $h$ small. Both partial gradients of
$\rho$ are consequently nonzero, and $|U_2|$ is bounded above and below.
With the product $\gamma$-connection, the endpoint Hessian of half
the squared distance has value
\begin{equation}\label{sph:distance-hessian}
|U_1-U_2|^2+O(r^2)(|U_1|^2+|U_2|^2).
\end{equation}
To obtain the error, parametrize the radial geodesic on $[0,1]$.
In a parallel frame its Jacobi field with prescribed endpoint values
differs from the affine interpolant by $O(r^2)$ in $C^1$, because the
curvature is bounded and the velocity has size $r$. The second
variation formula gives~\eqref{sph:distance-hessian}.
Replacing the second connection by that of $Q^{-1}$ contributes only
$O(r^2)$: its difference from the $\gamma$-connection is
$O(|h|+|\partial h|)=O(\varepsilon r)$, while $|d\rho|=O(r)$.
The relation $s\cdot U_2=1$ implies
$\omega\cdot(U_1-U_2)=b\cdot U_2$. Also
\[
\operatorname{Hess}_\gamma(r^2/2)
=2\,dr\otimes dr+r^2\operatorname{Hess}_\gamma f_*.
\]
Thus the Hessian sum in the criterion, multiplied by $r^2$, equals
\begin{equation}\label{sph:test-expression}
|\pi_{\omega^\perp}(U_1-U_2)|^2-|b\cdot U_2|^2
-r^2\operatorname{Hess}_\gamma f_*(U_2,\overline{U_2})
+O(r^2).
\end{equation}
For a real bilinear form, evaluation on a complex vector and its
conjugate means the sum of the evaluations on its real and imaginary
parts, as in the criterion.
We record the cancellation quantitatively. The real projected
difference is
$-\pi_{\omega^\perp}b+O(|b|^2+|h|)$. The imaginary part obeys
\[
|\pi_{\omega^\perp}w|
=1-\omega\cdot b+O(|b|^2+|h|).
\]
Comparing this length with $|v|=1$ and squaring gives
\[
|\pi_{\omega^\perp}(U_1-U_2)|^2
\ge |b|^2-C\bigl(|b|^3+|b||h|+|h|^2\bigr).
\]
The real and imaginary directions of $U_2$ are almost orthonormal;
hence
\[
|b\cdot U_2|^2\le
\bigl(1+C(|b|+|h|)\bigr)|b|^2.
\]
Using $|b|=r\beta$, $|h|\le\varepsilon r$, and
\eqref{sph:profile-bounds}, expression~\eqref{sph:test-expression}
is bounded below by
\begin{equation}\label{sph:test-lower-bound}
c r^2\beta-
C\bigl(r^2+r^3\beta^3+\varepsilon r^2\beta+
\varepsilon^2r^2\bigr).
\end{equation}
Divide the error terms by $r^2\beta$. Their ratios are bounded by
$C/\beta$, $Cr\beta^2$, $C\varepsilon$, and
$C\varepsilon^2/\beta$, respectively. Increasing $p_*$ controls the
first, and then decreasing $\varepsilon$ controls the others by
\eqref{sph:radius-smallness}. This proves strict positivity.
\end{proof}
\begin{proposition}[Continuation to the sphere family]
\label{sph:continuation}
Make the choices of Remark~\ref{sph:choices}. For all sufficiently
small $\lambda$ and every $0<\delta\le1$ satisfying
\eqref{sph:smallness}, the mixed Green matrix extends to a neighborhood
of
\[
\{(x,y):y\in Y,\ d_\gamma(x,y)=r_\delta(y)\}.
\]
It solves the first system in $x$ and the conjugate second system in
the second tensor index. On the lower center region $y_n<-1/4$ it
agrees with $G_1$, and on the prescribed positive-depth region
$t>t_b$ it agrees with the fixed-separation continuation of
Proposition~\ref{cross:initial}. The extension is smooth for each
positive $\delta$.
\end{proposition}
\begin{proof}
Deform $\eta$ from $1$ down to $\delta$, using $(y,\omega)$ in the
unit sphere bundle of $\gamma$ as base coordinates and
\[
x=\exp_y^\gamma(r_\eta(y)\omega).
\]
The base over the closed cylinder $Y$ is compact. The radius has
strictly positive derivative in $\eta$, so these variables give
cylinder coordinates on every fixed interval $\delta\le\eta\le1$.
The positive side of a level is the direction of increasing $\eta$.
Since $r_\eta\ge r_\delta$, condition~\eqref{sph:smallness} holds
at every intermediate radius. Lemma~\ref{sph:strict} applies there.
At $\eta=1$ the separations have a positive minimum on $Y$.
The same is true throughout $t>t_b$ by
\eqref{sph:positive-depth-radius}. Proposition~\ref{cross:initial}
therefore supplies the top and positive-depth data, with open room and
uniform bounds at any fixed order. In the lower strip $y_n<-1/4$,
use the original kernel $G_1$; this strip lies in the dilated known
side for small $\lambda$. The initialization agrees with this kernel
on their overlap.
These two base regions cover every lateral face of $Y$: its bottom
face is in the lower strip, and on a remaining lateral or upper face
outside that strip one has $t_0\ge3/4$. Both regions are independent
of $\eta$ and hence upward closed. Together with a neighborhood of
the top level they supply exactly the seed required by
Lemma~\ref{cross:propagation}. That lemma and the strict sphere test
extend the kernel through the whole cylinder, preserving agreement
on the seed. Normal-neighborhood room and positive radii on each
fixed interval ensure that all surfaces remain off the diagonal.
\end{proof}
The proposition gives existence at every admissible positive radius.
It does not give a bound for the kernel as $\delta\downarrow0$.
We next pass from the kernel to its action on harmonic sections;
this action has a sphere density that can be estimated separately.
\subsection{The transfer and its matrix density}
Fix $\delta>0$ satisfying~\eqref{sph:smallness}, write $r=r_\delta$,
and denote the continued kernel by $\mathcal G(x,y)$.
For a first-system harmonic section $u$ defined near the closed
$\gamma$-ball centered at $y$, set
\begin{equation}\label{sph:flux}
S(y;u)=\int_{\partial B_\gamma(y,r(y))}
\left[\mathcal G(x,y)^*\nabla^{A_1}_{\nu_1}u(x)
-\bigl(\nabla^{A_1}_{\nu_1,x}\mathcal G(x,y)\bigr)^*u(x)\right]
\,dS_1(x).
\end{equation}
Here $\nu_1$ and $dS_1$ are the outward unit normal and hypersurface
density for the actual metric $g_1$. The derivative of the kernel
acts on its first fiber index. Consequently the displayed integrand
is a section of the second fiber, with the adjoints in the indicated
order. For a first harmonic-frame function $v$, define
\begin{equation}\label{sph:transfer-definition}
T_yv=F_2(y)^{-1}S(y;F_1v).
\end{equation}
Let $S_y$ denote the unit sphere of $(T_y\mathbb R^n,\gamma_y)$ and
$d\sigma_y$ its rotation-invariant probability measure.
\begin{proposition}[Matrix harmonic transfer]\label{sph:transfer}
Under the assumptions of Proposition~\ref{sph:continuation}, there
is a smooth matrix function $\psi=\psi_\delta$ on the unit sphere
bundle over $Y^\circ$ such that
\begin{equation}\label{sph:density-representation}
T_yv=\int_{S_y}\psi(y,\omega)
v\bigl(\exp_y^\gamma(r(y)\omega)\bigr)\,d\sigma_y(\omega).
\end{equation}
For every $y_0\in Y^\circ$ and every first harmonic-frame function
$v$ defined near $\overline{B_\gamma(y_0,r(y_0))}$, the function
$y\mapsto T_yv$ is second harmonic-frame harmonic on a neighborhood
of $y_0$. For each pair $(v_1,v_2)$ of first and second harmonic-frame
functions defined on the fixed coordinate neighborhoods and agreeing
on the known side,
\begin{equation}\label{sph:reproduction}
\int_{S_y}\psi(y,\omega)\,d\sigma_y(\omega)=I,
\qquad T_yv_1=v_2(y).
\end{equation}
\end{proposition}
\begin{proof}
The Dirichlet problem on each ball is invertible. In the unitary
frame this follows from the positive connection energy: a zero-boundary
null solution is parallel and hence zero. Invertibility is preserved
by the harmonic-frame change and by positive scalar multiplication of
the equation. Its Dirichlet solution operator determines the boundary
covariant normal derivative from the boundary value. Transpose this
boundary operator onto the smooth matrix coefficient in the first term
of~\eqref{sph:flux}, then multiply by the frame factors and pull the
surface density back under $\omega\mapsto\exp_y^\gamma(r(y)\omega)$.
This gives the smooth matrix density in
\eqref{sph:density-representation}. Smooth dependence of the ball
Dirichlet problem on its center and positive radius gives smooth
dependence of $\psi$.
To prove the local harmonicity assertion, fix $y_0$ and $u=F_1v$.
The kernel is defined on an open neighborhood of the compact sphere
over $y_0$. Choose a fixed slightly larger surrounding surface inside
that neighborhood and inside the domain of $u$. After restricting
the center neighborhood, this surface surrounds all the moving
spheres and the intervening collars remain in the common domain.
Green's identity on each collar replaces~\eqref{sph:flux} by its
integral over the fixed surface. Both first-variable equations are
homogeneous in the collar. Applying the second-variable operator
under the fixed integral now proves that $S(y;u)$ is second-system
harmonic: the columns of $\mathcal G(x,y)^*$ solve that system.
Equation~\eqref{sph:transfer-definition} proves the assertion in
harmonic frames. This argument applies to arbitrary local harmonic
inputs; they need not extend throughout the coordinate region.
If $y_n<-1/4$, the kernel is $G_1$ and the Green representation formula
gives $S(y;u)=u(y)$. On that region $F_1=F_2$, so $T_yv=v(y)$.
For a fixed pair $(v_1,v_2)$, both $T_yv_1$ and $v_2(y)$ solve the
second harmonic-frame equation on connected $Y^\circ$ and agree in
the lower strip. Unique continuation proves their equality.
Finally constant columns are harmonic in both harmonic frames and
agree there, so the same argument yields the first identity
in~\eqref{sph:reproduction}.
\end{proof}
\begin{lemma}[Bounds where the cutoff varies]\label{sph:cutoff-bounds}
After the choices in Remark~\ref{sph:choices}, for every fixed integer
$m$ the angular $H^m$ norms of $\psi_\delta$ and its first base
derivatives on $\operatorname{supp}(d\chi)$ are bounded independently
of admissible $\delta$ and sufficiently small $\lambda$.
\end{lemma}
\begin{proof}
On the lower cutoff region, $T_y$ is evaluation at the center in the
harmonic frame. Normalize its equation to principal matrix
$a=\gamma^{-1}$, use a smooth $\gamma$-orthonormal frame to pull it
back under $z\mapsto\exp_y^\gamma(rz)$, and multiply by $r^2$.
With center and radius regarded as independent parameters,
the resulting Dirichlet systems extend smoothly to $r=0$, are uniformly
elliptic, and have the Euclidean Laplacian as limit; their lower-order
coefficients vanish at $r=0$. The Dirichlet inverses are uniformly
bounded, either by their limiting inverse and perturbation or by the
positive energy before the change of frame. Elliptic boundary
regularity, applied after differentiation in the parameters, shows
that the evaluation density at the center has uniform angular smooth
bounds and smooth center--radius dependence. The center remains a
fixed positive distance from the unit sphere in these coordinates.
Substituting the varying radius costs only
$|dr|_\gamma=r\beta$, which is uniformly bounded by
\eqref{sph:radius-smallness} and the lower bound for $\beta$.
This evaluation density is the density constructed in
Proposition~\ref{sph:transfer}. Indeed both represent the same
functional on traces of functions harmonic near the closed ball.
These traces are dense in smooth boundary data. For a prescribed
smooth trace $\varphi(\omega)$, solve on the ball of radius
$r(1+\eta)$ with boundary value $\varphi$ transported along the radial
rays. On pulling this problem back to the closed unit ball, its
solution $z_\eta$ converges in every $C^k$ norm to the solution $z_0$
at radius $r$, by smooth dependence of the Dirichlet inverse and
boundary regularity. Its trace on the original sphere is
$z_\eta(\omega/(1+\eta))$, which converges in $C^k(S_y)$ to
$z_0(\omega)=\varphi(\omega)$. The corresponding physical solutions
are harmonic near the original closed ball. Thus the two smooth
density matrices agree.
On the other cutoff region $t>t_b$, the radii have the positive lower
bound~\eqref{sph:positive-depth-radius}; all their required derivatives
are uniformly bounded in $\delta$. Proposition~\ref{cross:initial}
gives uniform kernel bounds there. The smooth Dirichlet construction
of the density therefore gives the same bounds for $\psi_\delta$ and
its first base derivatives. Compactness of the cutoff derivative
support completes the proof.
\end{proof}
We have constructed a matrix density that reproduces the matched
harmonic pairs and has controlled data wherever the cutoff changes.
The remaining estimate is a uniform bound on $\chi\psi_\delta$ over
the shrinking part of the sphere family.
==== END Determination-of-a-metric-and-a-unitary-connection-from-one-boundary-patch//04-spheres.tex ====
==== BEGIN Determination-of-a-metric-and-a-unitary-connection-from-one-boundary-patch//05-angular.tex ====
\section{Uniform estimates for the matrix transfer density}
\label{ang:section}\label{sec:angular}
The previous section constructs a smooth matrix density on every sphere
of positive radius. We prove that its $L^2$ norm stays bounded when the
spheres shrink, provided the two normalized principal matrices differ by
$O(r)$. The main analytic input is a scalar coercivity argument from
\citet[Section~5]{ScalarCompanion2026}, whose proof we reproduce below.
We then compare the matrix equation with that scalar operator. The
comparison is of lower differential order; it requires bounded matrix
drifts, but no smallness assumption on their difference.
\subsection{Moving traces and the matrix equation}
Use the reference metric and normalized inverse matrices from
\eqref{sph:normalization}. The harmonic-frame convention
\eqref{bdy:harmonic-normalization}, with the multipliers $c_j$ in
\eqref{sph:frame-normalization}, gives equations whose second-derivative
coefficient matrices are positive definite:
\begin{equation}\label{ang:frame-equations}
\mathcal L_1=a^{ij}\partial_i\partial_j+B_1^i\partial_i,
\qquad
\mathcal L_2=Q^{ij}\partial_i\partial_j+B_2^i\partial_i.
\end{equation}
The $B_j^i$ are complex $2\times2$ matrices acting on columns from the
left. There is no zeroth-order term because each column of the defining
frame $F_j$ is harmonic. All coefficients have uniform smooth bounds
on the fixed coordinate neighborhood for sufficiently small spatial
dilations. Suppress the dilation and shrinking parameters for now, and
write
\[
r=r_\delta,\qquad f_*=\log r,\qquad
\beta=|df_*|_\gamma,\qquad f_i=\nabla_i f_*.
\]
The profile bounds supplied by Lemma~\ref{sph:profiles} have the form
\begin{equation}\label{ang:profile-bounds}
\beta\ge\beta_0,\qquad
\operatorname{Hess}_\gamma f_*\le-c_0\beta\gamma,\qquad
|\nabla^j f_*|\le C_j\beta^j\quad(1\le j\le8).
\end{equation}
The lower threshold $\beta_0$ can be increased by increasing $p_*$.
Throughout this section we impose
\begin{equation}\label{ang:smallness}
\sum_{|\alpha|\le8}|\partial_y^\alpha h|\le\varepsilon r,
\qquad r\beta^2\le\varepsilon.
\end{equation}
The first is the bootstrap assumption of Section~\ref{sec:spheres};
the second is ensured by the choice of radius amplitude.
Let $S_\gamma Y$ be the tangent unit-sphere bundle with fiber probability
measure $d\sigma_y$. Horizontal differentiation $\nabla$ uses parallel
transport for $\gamma$, including any base tensor indices, and acts
componentwise on the fixed harmonic-frame components. The measure
$d\sigma_y$ is parallel. Our Hilbert norms use
$dV_\gamma(y)d\sigma_y(\omega)$ and the standard Hermitian norm, or the
Frobenius norm for matrices.
There are two different transpose operations in the argument. If
$\mathcal C$ is a sphere operator on columns, its \emph{bilinear row
transpose} $\mathcal C^t$ is defined by
\begin{equation}\label{ang:row-transpose}
\int_{S_y}(\mathcal C^t\psi)V\,d\sigma_y
=\int_{S_y}\psi(\mathcal C V)\,d\sigma_y.
\end{equation}
Here the identity is applied to each row of $\psi$; it involves no
complex conjugation. In particular, a matrix multiplication $M$ on
columns becomes right multiplication $\psi\mapsto\psi M$, whereas a
matrix multiplying the output of the transfer map acts on $\psi$ from
the left. A star will denote the Hermitian sphere adjoint of a scalar
operator, and a dagger its full Hilbert-space adjoint, including the
base integration. Real scalar sphere operators have the same
bilinear transpose and Hermitian adjoint.
For a column $v$ solving $\mathcal L_1v=0$ near a closed moving ball, set
\[
V(y,\omega)=v\bigl(\exp_y^\gamma(r(y)\omega)\bigr).
\]
Solving the ball Dirichlet problem expresses the boundary derivatives of
$v$ in terms of its trace and therefore defines sphere operators
$\mathcal C_i$ satisfying $\nabla_iV=\mathcal C_iV$.
For scalar functions write $\Delta_g=\operatorname{div}_g\nabla$.
Let $\mathcal C_{i,0}$ be the corresponding moving trace operators for
$c_1\Delta_{g_1}$, whose second-derivative coefficient matrix is
$a=\gamma^{-1}$. This scalar comparison equation need not have zero
coordinate drift.
On each tangent unit ball, let $\mathsf B$ be the radial boundary
derivative of Euclidean harmonic extension and $\mathsf A_i$ its ambient
derivatives in a $\gamma$-orthonormal frame. On spherical harmonics
$\mathcal H_k$ of degree $k$, $\mathsf B=k$ and
$\mathsf A_i:\mathcal H_k\to\mathcal H_{k-1}$. The operators are parallel
with their tensor indices, and
\begin{equation}\label{ang:elementary-identities}
[\mathsf B,\mathsf A_i^*]=\mathsf A_i^*,
\qquad \sum_i\mathsf A_i^*\mathsf A_i^*=0.
\end{equation}
The second identity is the adjoint of the vanishing Laplacian of a
harmonic extension. Define
\[
\|w(y,\cdot)\|_s=\|(1+\mathsf B)^s w(y,\cdot)\|_{L^2(S_y)}.
\]
These norms are equivalent to the usual sphere Sobolev norms, uniformly
in $y$. They also apply componentwise to tensors and matrices.
Write $\Psi^m$ for classical angular pseudodifferential operators of
order $m$, scalar or matrix valued as specified. The notation
$c(y)\Psi^m$ means that division by $c(y)>0$ leaves uniformly bounded
symbol and operator seminorms in the finitely many orders being used.
An allowance $\beta^j$ for $j$ base derivatives means the corresponding
seminorms are bounded by $C_j\beta^j$.
\begin{lemma}[Trace expansions]\label{ang:trace}
Under \eqref{ang:profile-bounds}--\eqref{ang:smallness},
\begin{equation}\label{ang:C-expansion}
\mathcal C_{i,0}^*
=r^{-1}\mathsf A_i^*+f_i\mathsf B+E_i+F_i,
\qquad E_i\in r\Psi^1,\quad F_i\in\Psi^0.
\end{equation}
The normalized remainders $r^{-1}E_i,F_i$ have the allowance $C_j\beta^j$
through six base derivatives. Moreover, on rows,
\begin{equation}\label{ang:trace-difference}
K_i:=\mathcal C_i^t-\mathcal C_{i,0}^*
\in\Psi^0,\qquad \nabla K_i\in\beta\Psi^0.
\end{equation}
All constants depend on fixed smooth background bounds, not on the
lower threshold $\beta_0$ once $\beta_0\ge1$.
\end{lemma}
\begin{proof}
First regard the radius $s$ as an independent parameter. Pull the first
scalar equation back by $z\mapsto\exp_y^\gamma(sz)$ and multiply by
$s^2$. Its principal coefficients are $\delta^{ij}+O(s^2)$ and its
first-order coefficients are $O(s)$. The same statement holds for the
system, with scalar principal coefficients times $I$. Both Dirichlet
operators extend smoothly to $s=0$, where they equal the Euclidean
Laplacian. On each fixed Sobolev scale their Dirichlet inverses are
uniformly bounded and smooth for sufficiently small $s$, by elliptic
Dirichlet estimates and invertibility at zero.
For these smooth strongly elliptic Dirichlet families on the unit ball,
the boundary radial derivative operator is classical of order one. Its
principal symbol depends only on the principal coefficients and the
differentiating vector. The boundary symbol construction, obtained by
choosing the decaying normal root and solving the successive transport
equations, is smooth in $(y,s)$; see, for example,
\citet[Section~3, Proposition~3.1]{JoshiLionheart2005}.
Their factorization is for the Hodge Laplacian; its construction applies
to the present Dirichlet systems because their
principal symbol is scalar times $I$, while the transport equations
allow matrix coefficients. The true Dirichlet inverse corrects the
parametrix by a smoothing operator. Differentiating this inverse on
arbitrarily high fixed Sobolev scales bounds every required smooth
kernel seminorm of the correction. Thus these assertions hold in the
full symbol topology, including smoothing remainders and parameter
derivatives.
At $s=0$ the radial operator is $\mathsf B$, and the first $s$-derivative
of its principal symbol vanishes because the principal coefficients
have no linear term in $s$. Taylor expansion in the symbol topology
therefore gives, for the scalar equation,
\begin{equation}\label{ang:radial-expansion}
\mathsf B+s\mathcal B_1(y)+s^2\mathcal B_2(y,s),
\qquad \mathcal B_1\in\Psi^0,\quad\mathcal B_2\in\Psi^1.
\end{equation}
The system and scalar radial operators have identical principal
symbols for every $s$ and coincide at zero. Their difference is
consequently $s$ times a smooth family in $\Psi^0$.
Differentiating the center of the exponential map while transporting
$\omega$ parallel gives a Jacobi field with initial derivative zero.
In the unit-ball coordinates its velocity is
\[
s^{-1}(e_i+O(s^2));
\]
putting $s=r(y)$ adds $f_i\omega$. Tangential derivatives act directly
on boundary data and are identical for the scalar and matrix equations.
For the radial component use \eqref{ang:radial-expansion}. The leading
terms are $r^{-1}\mathsf A_i+f_i\mathsf B$; the order-one errors have
size $O(r)$ and the order-zero errors are bounded, since
$r\beta\le\varepsilon/\beta$. This proves \eqref{ang:C-expansion}
after sphere adjunction. The radial difference $r\Psi^0$, multiplied
by the center velocity $O(r^{-1}+\beta)$, proves
\eqref{ang:trace-difference}. Base derivatives cost at most one factor
$\beta$ each by the profile bounds. Sphere transposition and
commutation with the smooth angular connection fields preserve these
orders and bounds.
\end{proof}
Write the second system in the reference connection as
$\mathcal L_2=Q^{ij}\nabla_i\nabla_j+b_2^i\nabla_i$; equivalently,
$b_2^k=B_2^k+Q^{ij}\Gamma_{ij}^k(\gamma)I$ in coordinates.
Set $D_i=\nabla_i+\mathcal C_i^t$. The matrix density of
Proposition~\ref{sph:transfer} satisfies
\begin{equation}\label{ang:P-equation}
P\psi=0,\qquad
P=r\left[Q^{ij}D_iD_j+b_2^iD_i\right].
\end{equation}
Products are covariant products, and $b_2^i$ multiplies $D_i\psi$ from
the left. In particular, $Q$ is not differentiated by a base adjoint.
Indeed, differentiation of $T_yv=\int\psi V\,d\sigma_y$ gives
$\nabla_i(T_yv)=\int(D_i\psi)V\,d\sigma_y$ by
\eqref{ang:row-transpose}. Differentiate once more and apply the second
system equation, which holds for every local first-system harmonic
$v$ by Proposition~\ref{sph:transfer}. The resulting expression pairs
to zero with all such traces. These traces are dense in smooth boundary
data: solve the Dirichlet problems on slightly enlarged balls with
transported smooth boundary values and let their radii decrease to the
given radius. Smooth parameter dependence gives convergence in every
fixed boundary smooth norm. This proves \eqref{ang:P-equation}.
Write the scalar comparison equation on the second side as
$c_2\Delta_{g_2}=Q^{ij}\nabla_i\nabla_j+b_{2,0}^i\nabla_i$.
Define the scalar operator,
acting entrywise on matrices,
\begin{equation}\label{ang:P0}
P_0=r\left[Q^{ij}D_{i,0}D_{j,0}+b_{2,0}^iD_{i,0}\right],
\qquad D_{i,0}=\nabla_i+\mathcal C_{i,0}^*.
\end{equation}
The next subsections prove coercivity for $P_0$. The proof uses
only its trace expansion and coefficient bounds; it assumes no scalar
transfer density or scalar Green-kernel identity.
\subsection{The model operator and its principal perturbations}
We separate a degree-raising model operator from the remaining
principal and lower-order terms of the moving trace equation. We continue under the hypotheses of
Lemma~\ref{ang:trace}, including \eqref{ang:smallness}.
Define
\begin{equation}\label{ang:T-definition}
D_i^+=\nabla_i+f_i\mathsf B,\qquad D_i^-=-\nabla_i+f_i\mathsf B,
\qquad
T=\sum_i \mathsf A_i^*D_i^+,\quad T^\flat=\sum_i \mathsf A_iD_i^-.
\end{equation}
The symbol $T$ here denotes a differential operator on sphere functions;
the transfer functional continues to be denoted $T_y$.
Expand \eqref{ang:P0} by
\eqref{ang:C-expansion}. The identities
\eqref{ang:elementary-identities} give
\[
D_i^+(r^{-1}\mathsf A_j^*)=r^{-1}\mathsf A_j^*D_i^+,
\qquad \sum_i\mathsf A_i^*\mathsf A_i^*=0.
\]
Thus, in ordinary base coordinates and an orthonormal sphere
trivialization,
\begin{equation}\label{ang:P-decomposition}
P_0=2T+\mathcal U+\mathcal J,
\end{equation}
where $\mathcal J$ is a uniformly bounded family in $\Psi^1$, with no base
derivatives, and
\begin{equation}\label{ang:U-bounds}
\mathcal U=\sum_{|\alpha|\le2}\mathcal U_\alpha(y)\partial_y^\alpha,
\qquad \mathcal U_\alpha\in\varepsilon\beta^{-|\alpha|}
\Psi^{2-|\alpha|}.
\end{equation}
Up to five derivatives of these coefficients have the additional
allowance $C_j\beta^j$. The second-base-derivative term is precisely
\begin{equation}\label{ang:real-principal-part}
rQ^{ij}\partial_{y_i}\partial_{y_j};
\end{equation}
its coefficients are real scalars independent of the sphere variable.
We detail these bounds. The pure angular second derivative contracted
with $a$ vanishes. The remaining contraction is
$(h^{ij}/r)\mathsf A_i^*\mathsf A_j^*$, and belongs to the zero-base-derivative term of
\eqref{ang:U-bounds}. The mixed correction is
$2h^{ij}\mathsf A_i^*D_j^+$ and has the stated small bounds. The $D^+D^+$ term
has coefficient $r$; its base orders two, one and zero have coefficients
$O(r)$, $O(r\beta)$ and $O(r\beta^2)$, respectively. The condition
$r\beta^2\le\varepsilon$ gives \eqref{ang:U-bounds} for all three.
Terms containing $E_i$ obey the same bounds. Terms containing $F_i$
also do so, except for their cross terms with $r^{-1}\mathsf A_j^*$, which are
bounded angular order-one terms and belong to $\mathcal J$. The drift term with
$\mathsf A_i^*$ belongs to $\mathcal J$ as well. Differentiating any of these expressions
gives the stated coefficient bounds by
\eqref{ang:profile-bounds} and \eqref{ang:smallness}.
A horizontal connection written in this trivialization adds a bounded
angular first derivative, which preserves the same estimates. The bounds
for $\mathcal J$ use the fixed background and remain uniform as $\beta_0$ is
increased.
The decomposition identifies the model operator $2T$ and the
second-order perturbation $\mathcal U$. To carry the model estimate over to $P_0$
we shall need more than an ordinary first-derivative estimate: the error forms
also contain mixed base and angular derivatives. For a sphere function
$\varphi$, put
\begin{equation}\label{ang:N-definition}
\mathcal N(\varphi)^2
=\int_Y\left(
\|\nabla\varphi\|_0^2+\beta^2\|\varphi\|_1^2
+\beta^{-1}\|\nabla\varphi\|_{1/2}^2
+\beta\|\varphi\|_{3/2}^2\right)dV_\gamma.
\end{equation}
The last two terms of $\mathcal N$ place half an angular derivative
on $\nabla\varphi$ and three halves on $\varphi$. They will bound the
remaining error forms left by the comparison with $P_0$.
\subsection{A comparison of raising and lowering operators}
The next estimate supplies these two derivative strengths for the model
operator. Spherical harmonics are trace-free symmetric tensors, and
comparing their raising and lowering operators is familiar from energy
identities in tensor tomography; see, for example, \cite{PSU15,GPSU16}.
Here the negative Hessian of the spatial weight supplies the positive
term that absorbs the bounded curvature. The particular weighted estimate below is adapted from
\citet[Section~5, weighted raising estimate]{ScalarCompanion2026}.
We include the degree calculation because it is the source of the
additional half angular derivative used in the perturbation argument.
\begin{lemma}[Weighted raising estimate]\label{ang:raising}
Let $n\ge3$, and suppose \eqref{ang:profile-bounds} holds on a fixed
base region with uniformly bounded reference geometry. There are
$\beta_0$ and $c>0$, depending only on that geometry and the constants in
\eqref{ang:profile-bounds}, such that every smooth, compactly
base-supported function with zero fiber mean satisfies
\begin{equation}\label{ang:raising-estimate}
\|T\varphi\|^2-\|T^\flat\varphi\|^2
\ge c\,\mathcal N(\varphi)^2.
\end{equation}
\end{lemma}
\begin{proof}
Both operators change angular degree by exactly one, so it suffices to
prove the estimate at one input degree $k\ge1$. Identify that component
with a harmonic homogeneous polynomial $v(z)$ of degree $k$. Use the
Fischer inner product, in which the monomials $z^\alpha$ are orthogonal
with squared norm $\alpha!$. Polynomial differentiation $a_i=\partial_{z_i}$
is adjoint to multiplication $M_i=z_i$. On harmonic polynomials of degree
$k$ the Fischer squared norm is the normalized squared sphere norm
multiplied by
\[
N_k=n(n+2)\cdots(n+2k-2),\qquad N_0=1.
\]
For completeness, integration against a standard real Gaussian identifies
the Fischer norm on trace-free homogeneous polynomials with their
Gaussian $L^2$ norm;
integrating radially then gives the factor $N_k$. These classical
identities are also recorded in \cite[Equation~(2.1) and
Theorem~2.1]{Render08}.
Set $d=k+n/2-1$ and use $D_i^\pm=\pm\nabla_i+k f_i$ at this fixed degree.
For harmonic polynomials $q_i$ of degree $k$, the Laplacian of
$\sum_iM_iq_i$ is $2\sum_i a_iq_i$. Since
\[
\Delta_z(|z|^2s)=4d\,s\qquad(s\in\mathcal H_{k-1}),
\]
its harmonic projection is
$\sum_iM_iq_i-|z|^2\sum_i a_iq_i/(2d)$. Orthogonality of this projection
and its complement shows that its squared norm is
\[
\sum_i\|q_i\|^2+\sum_{i,j}(a_jq_i,a_iq_j)
-d^{-1}\Bigl\|\sum_i a_iq_i\Bigr\|^2.
\]
Changing from Fischer adjoints to sphere adjoints gives
\[
\mathsf A_i^*|_{\mathcal H_k}
=\frac{N_{k+1}}{N_k}\Pi_{k+1}M_i,
\]
where $\Pi_{k+1}$ is harmonic projection in homogeneous degree $k+1$.
Consequently $N_k$ times the left side of
\eqref{ang:raising-estimate} is
\begin{equation}\label{ang:Fock-difference}
\begin{split}
(2d+2)\left[\|D^+v\|^2
+\sum_{i,j}(a_jD_i^+v,a_iD_j^+v)
-\frac1d\Bigl\|\sum_i a_iD_i^+v\Bigr\|^2\right]
-2d\Bigl\|\sum_i a_iD_i^-v\Bigr\|^2.
\end{split}
\end{equation}
All norms in this calculation include integration over the base and use
the polynomial inner product in the fibers. The factors are
$N_{k+1}/N_k=2d+2$ and $N_k/N_{k-1}=2d$.
Write $C_{ij}=2k(\operatorname{Hess}_\gamma f_*)_{ij}$ and
\[
C[av]=\int_Y\sum_{i,j}C_{ij}\langle a_iv,a_jv\rangle\,dV_\gamma.
\]
Covariant integrations by parts give
\begin{align}
\sum_{i,j}(a_jD_i^+v,a_iD_j^+v)
&=\Bigl\|\sum_i a_iD_i^-v\Bigr\|^2-C[av]
+O(k^2)\|v\|^2,\label{ang:ibp-one}\\
\Bigl\|\sum_i a_iD_i^+v\Bigr\|^2
&\le k\|D^-v\|^2-C[av]+O(k^2)\|v\|^2,\label{ang:ibp-two}\\
\|D^-v\|^2
&=\|D^+v\|^2+\int_Y\tr C\,|v|^2\,dV_\gamma.
\label{ang:ibp-three}
\end{align}
Here and below a form denoted $O(k^2)\|v\|^2$ has absolute value at most
a fixed constant times that quantity. To verify the first two formulas,
move a $D_j^+$ across the pairing and commute
$D_j^-D_i^+$ to $D_i^+D_j^--C_{ij}$. The reordered term in
\eqref{ang:ibp-two} is at most $k\|D^-v\|^2$, because
$\sum_i|a_iw|^2=k|w|^2$ for $w\in\mathcal H_k$.
The extra commutator is curvature: it acts as a sum of rotations of norm
$O(k)$ at degree $k$, and the two contractions supply one more factor $k$.
This proves the claimed error bounds. Expanding the two squares proves
\eqref{ang:ibp-three}. For complex functions all the displayed forms
are understood through their real parts.
Substituting \eqref{ang:ibp-one} into
\eqref{ang:Fock-difference} rewrites that expression as
\begin{equation}\label{ang:after-first-ibp}
\begin{split}
&(2+2/d)\left[d\|D^+v\|^2
-\Bigl\|\sum_i a_iD_i^+v\Bigr\|^2-dC[av]\right]\\
&\qquad+2\Bigl\|\sum_i a_iD_i^-v\Bigr\|^2
+O(k^3)\|v\|^2.
\end{split}
\end{equation}
We must control the remaining negative divergence square while retaining
both an ordinary derivative estimate and an additional half-order
angular derivative estimate. The latter requires a coefficient of
size $k/\beta$ in front of $|D^+v|^2$, including when $k$ is much larger
than $\beta$. Choose a small constant $\vartheta>0$.
On the fraction $1-\vartheta$ of its negative divergence square use
\eqref{ang:ibp-two}--\eqref{ang:ibp-three}. On the remaining
fraction use the pointwise convex combination
\begin{equation}\label{ang:convex-bound}
\begin{split}
\Bigl|\sum_i a_iD_i^+v\Bigr|^2
&\le(1-\zeta)(k+n-1)|D^+v|^2\\
&\quad+\zeta\left(2\Bigl|\sum_i a_iD_i^-v\Bigr|^2
+8k^3\beta^2|v|^2\right),
\qquad \zeta=c_1/\beta.
\end{split}
\end{equation}
The fraction $1-\vartheta$ in the integrated estimates is constant;
the $y$-dependent $\zeta$ is used only in this pointwise inequality.
The first bound follows from the adjoint multiplication operators, since
the contraction $(q_i)\mapsto\sum_i a_iq_i$ has squared norm at most
$k+n-1$ on polynomials of degree $k$. The second follows from
$D_i^++D_i^-=2k f_i$ and
$|\sum_i f_i a_i v|^2\le k\beta^2|v|^2$.
Combining these bounds in \eqref{ang:after-first-ibp}, the coefficient
retained for $|D^+v|^2$ is
\begin{equation}\label{ang:positive-degree-factor}
(2+2/d)\bigl[d-k-\vartheta(n-1)
+\vartheta\zeta(k+n-1)\bigr].
\end{equation}
Since $d-k=(n-2)/2$, a sufficiently small fixed $\vartheta$ makes this
at least a fixed positive constant plus a positive multiple of
$c_1k/\beta$. The coefficient of
$|\sum_i a_iD_i^-v|^2$ is nonnegative when $\beta_0$ is sufficiently
large. The Hessian contribution is exactly
\begin{equation}\label{ang:hessian-contribution}
-(2+2/d)\left[(d-1+\vartheta)C[av]
+(1-\vartheta)k\int_Y\tr C\,|v|^2\,dV_\gamma\right].
\end{equation}
For $k\ge1$ and $n\ge3$, both coefficients inside this expression are
positive. The Hessian hypothesis bounds it below by
$c k^3\int\beta|v|^2$. This absorbs the curvature error $O(k^3)\|v\|^2$
by increasing $\beta_0$, and absorbs the last term of
\eqref{ang:convex-bound} by choosing $c_1$ sufficiently small.
We have therefore retained positive multiples of
\[
\|D^+v\|^2,\qquad
\int_Y\frac{k}{\beta}|D^+v|^2\,dV_\gamma,
\qquad k^3\int_Y\beta|v|^2\,dV_\gamma.
\]
The first controls both $\|\nabla v\|^2$ and
$k^2\int\beta^2|v|^2$, since its cross term is
$-k\int\Delta_\gamma f_*\,|v|^2\ge0$. For the weighted derivative use the
pointwise inequality
\[
\frac{k}{\beta}|\nabla v|^2
\le2\frac{k}{\beta}|D^+v|^2+2k^3\beta|v|^2.
\]
No differentiation of $\beta^{-1}$ is required. Dividing by $N_k$, and
then summing the orthogonal degree components, proves
\eqref{ang:raising-estimate}; the weights $(1+k)^s$ are comparable
to $k^s$ because the zero degree is excluded.
\end{proof}
\subsection{Keeping the principal perturbations}
The raising estimate controls the model operator. The scalar operator
\eqref{ang:P0} also contains second base derivatives and a
small second-order angular perturbation. We retain these terms in a
comparison of squared norms.
\begin{proposition}[Scalar coercivity]
\label{ang:scalar-coercivity}
Fix uniform smooth bounds for the reference geometry and scalar
comparison equations, and the constants in \eqref{ang:profile-bounds}.
Suppose $P_0$ is the scalar operator
\eqref{ang:P0}, with the scalar trace operators of
Lemma~\ref{ang:trace}. There exist $\beta_0$ sufficiently large,
$\varepsilon_0>0$ and $C<\infty$, depending only on these fixed bounds, such that
\eqref{ang:smallness} with $\varepsilon\le\varepsilon_0$ implies
\begin{equation}\label{ang:coercive-estimate}
\mathcal N(\varphi)\le C\|P_0\varphi\|
\end{equation}
for every smooth, compactly base-supported function $\varphi$ with
zero mean on each sphere. These constants are independent of the
shrinking parameter. Only the eight indicated derivatives of the small
tensor $h$ are required.
\end{proposition}
\begin{proof}
Let a dagger denote the full adjoint for $dV_\gamma d\sigma$.
Covariant integration and the degree shift give
\begin{equation}\label{ang:Z-definition}
Z:=T^\flat-T^\dagger=\sum_i f_i\mathsf A_i.
\end{equation}
A direct expansion yields
\begin{equation}\label{ang:norm-comparison}
\begin{split}
&\|(2T+\mathcal U)\varphi\|^2-
\|(2T^\flat+\mathcal U^\dagger)\varphi\|^2\\
&\qquad=4\bigl(\|T\varphi\|^2-\|T^\flat\varphi\|^2\bigr)
+\langle\mathcal E\varphi,\varphi\rangle,
\end{split}
\end{equation}
where
\begin{equation}\label{ang:error-operator}
\mathcal E=2[T^\dagger,\mathcal U]+2[\mathcal U^\dagger,T]+[\mathcal U^\dagger,\mathcal U]
-2\bigl(\mathcal U Z+(\mathcal U Z)^\dagger\bigr).
\end{equation}
We will prove
\begin{equation}\label{ang:error-form-bound}
|\langle\mathcal E\varphi,\varphi\rangle|
\le C\varepsilon\mathcal N(\varphi)^2.
\end{equation}
Here are the complete order and weight counts needed for this claim.
An operator has count $(m,s)$ if its coefficient at $\partial_y^\alpha$
is angular order $m-|\alpha|$, of size $O(\beta^{s-|\alpha|})$,
with the derivative allowances already specified.
Adjoints preserve the count, and products add the counts. When a base
derivative hits a coefficient it consumes one differential order and
adds a factor $\beta$, so it preserves the weight index. A commutator
of scalar angular operators loses one angular order: the products of
their principal symbols cancel. Hence a scalar commutator loses one in
total order while preserving the sum of the weight indices.
The counts of $T,T^\dagger,\mathcal U,Z$ are, respectively,
\[
(2,1),\quad(2,1),\quad(2,0),\quad(1,1),
\]
and every occurrence of $\mathcal U$ carries the factor $\varepsilon$. Thus
\eqref{ang:error-operator} has count at most $(3,1)$ and a factor
$O(\varepsilon)$; the quadratic $\mathcal U$ term is smaller.
There is a further cancellation that the total count alone does not
express: no third base derivative survives. To check it explicitly,
use an orthonormal sphere trivialization and write the base measure as
$\mu(y)\,dy$. The sphere probability measure is then fixed. Write
\[
\mathcal U=c^{ij}\partial_i\partial_j+\mathcal B^i\partial_i+\mathcal C,
\qquad c^{ij}=rQ^{ij}=c^{ji}\in\mathbb R.
\]
The coefficients $\mathcal B^i$ and $\mathcal C$ are angular
operators. Replacing a horizontal derivative by a coordinate derivative
plus its angular connection term changes only base orders at most one,
so these are all the second-base coefficients. If a star denotes the
sphere adjoint, direct integration by parts gives
\begin{align*}
\mathcal U^\dagger-\mathcal U
={}&\left(2\mu^{-1}\partial_j(\mu c^{ij})
-\mathcal B^i-(\mathcal B^i)^*\right)\partial_i\\
&+\mu^{-1}\partial_i\partial_j(\mu c^{ij})
-\mu^{-1}\partial_i\bigl(\mu(\mathcal B^i)^*\bigr)
+\mathcal C^*-\mathcal C.
\end{align*}
In particular it has base order at most one.
In $[T^\dagger,\mathcal U]$ and
its partner, the possible third-base-derivative coefficient is a commutator of an angular coefficient
with $rQ^{ij}$, which is zero by
\eqref{ang:real-principal-part}. For $[\mathcal U^\dagger,\mathcal U]$, first write
it as $[\mathcal U^\dagger-\mathcal U,\mathcal U]$. The operator $\mathcal U^\dagger-\mathcal U$ has no second base
derivative, since the coefficient in
\eqref{ang:real-principal-part} is real. Taking adjoints against
the smooth base density only produces lower base orders. Its remaining
possible third-base-derivative coefficients commute for the same reason.
Finally, $\mathcal U Z$ already has at most two base derivatives.
It follows that $\mathcal E$ is a sum of terms of precisely the following
three permitted types:
\begin{equation}\label{ang:error-types}
\varepsilon\beta^{-1}\Psi^1\partial_y^2,
\qquad \varepsilon\Psi^2\partial_y,
\qquad \varepsilon\beta\Psi^3.
\end{equation}
This argument also covers lower orders, since $\beta\ge1$.
For the first type in \eqref{ang:error-types}, integrate once in the
base. The principal form is bounded by
$C\varepsilon\int\beta^{-1}\|\partial_y\varphi\|_{1/2}^2$.
A derivative of its coefficient costs one factor $\beta$ and leaves an
$\varepsilon\Psi^1\partial_y$ term. This includes differentiation of the
weight itself: $|d\beta|\le C\beta^2$ implies
$|d(\beta^{-1})|\le C$. The second type has the form bound
\[
C\varepsilon\int_Y
\|\partial_y\varphi\|_{1/2}\|\varphi\|_{3/2}\,dV_\gamma
\le C\varepsilon\int_Y\left(
\beta^{-1}\|\partial_y\varphi\|_{1/2}^2
+\beta\|\varphi\|_{3/2}^2\right)dV_\gamma.
\]
The third type is bounded by
$C\varepsilon\int\beta\|\varphi\|_{3/2}^2$.
The lower-order term from differentiating a coefficient satisfies the
same bound. Replacing coordinate derivatives by horizontal derivatives
adds a bounded angular first derivative, again controlled by these norms.
A fixed finite partition of the base region has the same harmless
lower-order errors. This proves \eqref{ang:error-form-bound}.
The constants in the preceding estimates are uniform for all
$\beta\ge\beta_0$ once a fixed lower threshold is met. Choose
$\varepsilon_0$ small relative to the constant in
Lemma~\ref{ang:raising}. Dropping the nonnegative second squared
norm in \eqref{ang:norm-comparison} gives
$\mathcal N(\varphi)\le C\|(2T+\mathcal U)\varphi\|$.
Finally,
\[
\|\mathcal J\varphi\|\le C\|\varphi\|_{L^2(Y;H^1(S_y))}
\le C\beta_0^{-1}\mathcal N(\varphi),
\]
so a sufficiently large $\beta_0$ absorbs $\mathcal J$ and proves
\eqref{ang:coercive-estimate}.
All these operations use angular symbol estimates pointwise in the base,
and ordinary base differential operators of degree at most two. Formal
adjoints, the commutator expansion and the one integration by parts use
at most the five base coefficient derivatives specified in
\eqref{ang:U-bounds}. Forming these coefficients uses at most one base
derivative of a trace operator; six trace derivatives therefore suffice.
Since its weight dependence is through $r$ and $f_i$, profile derivatives
through order seven suffice, within the eight allowed in
\eqref{ang:profile-bounds}. Differentiating a factor such as $h/r$ five
times uses only five derivatives of $h$, with the remaining factors
bounded by the permitted powers of $\beta$. Higher angular symbol
seminorms depend on the fixed smooth first-metric family: the tensor $h$
is evaluated at the center, so angular differentiation introduces no
further center derivatives of it. The scalar drifts have fixed smooth
background bounds. Thus the eight smallness derivatives in
\eqref{ang:smallness} suffice in every fixed dimension; no smallness of
all derivatives of $h$ is used.
\end{proof}
\subsection{The matrix comparison and the density bound}
The scalar calculation is now complete. We use it entrywise before
introducing any matrix coefficients into a squared-norm comparison.
This order matters: matrix principal symbols need not commute, whereas
the preceding commutator argument is scalar.
\begin{proposition}[System coercivity]\label{ang:system-coercivity}
Fix uniform smooth bounds for the harmonic-frame systems and the
reference geometry, and the constants in \eqref{ang:profile-bounds}.
There are $\beta_0$ sufficiently large, $\varepsilon_0>0$, and $C$,
depending only on these bounds, such that
\eqref{ang:smallness} with $\varepsilon\le\varepsilon_0$ implies
\[
\mathcal N(\varphi)\le C\|P\varphi\|_{L^2}
\]
for every smooth compactly base-supported matrix function $\varphi$
with zero fiber mean. The constants are independent of the shrinking
parameter and of sufficiently small spatial dilations.
\end{proposition}
\begin{proof}
Put $q^i=b_2^i-b_{2,0}^iI$ and use
$D_i=D_{i,0}+K_i$ from \eqref{ang:trace-difference}.
Let $L_M$ denote left multiplication by a matrix $M$. Expansion, without
interchanging any factors, gives the identity
\begin{equation}\label{ang:comparison-identity}
\begin{split}
P-P_0=r\bigl[&Q^{ij}(D_{i,0}K_j+K_iD_{j,0}+K_iK_j)\\
&+L_{q^i}D_{i,0}+L_{b_2^i}K_i\bigr].
\end{split}
\end{equation}
Thus $D_{i,0}K_j$ denotes composition and includes differentiation of
$K_j$. There is no derivative of $q^i$ or $b_2^i$ in this formula.
Left multiplication commutes with the row action of a sphere operator
when its matrix depends only on the center, but no such commutation is
needed in \eqref{ang:comparison-identity}.
The trace expansion gives
$\mathcal C_{i,0}^*\in(r^{-1}+\beta+1)\Psi^1$.
Together with $K_i\in\Psi^0$, $\nabla K_i\in\beta\Psi^0$, and bounded
drifts, \eqref{ang:comparison-identity} consequently yields
\begin{equation}\label{ang:comparison-bound}
\|(P-P_0)\varphi\|_{L^2}
\le C\left(\|\varphi\|_{L^2(Y;H^1(S_y))}
+\|r\nabla\varphi\|_{L^2}\right).
\end{equation}
To see the weights explicitly, every term with a derivative of
$\varphi$ in the base is $r\Psi^0\nabla\varphi$.
Products of $K_i$ with the leading $r^{-1}\mathsf A_j^*$ leave a bounded
$\Psi^1$ term after multiplication by $r$; products with
$f_j\mathsf B$ have coefficient $O(r\beta)$.
Differentiating $K_j$ gives $r\beta\Psi^0$. The remaining products
have smaller orders and bounded coefficients. Since $r\beta\le
\varepsilon/\beta$, all constants in \eqref{ang:comparison-bound}
are uniform as the lower threshold $\beta_0$ increases.
Apply Proposition~\ref{ang:scalar-coercivity} to the four entries of
$\varphi$. By the definition of $\mathcal N$,
\[
\|\varphi\|_{L^2(Y;H^1)}\le\beta_0^{-1}\mathcal N(\varphi),
\qquad
\|r\nabla\varphi\|_{L^2}
\le\varepsilon\beta_0^{-2}\mathcal N(\varphi).
\]
It follows that
\[
\mathcal N(\varphi)
\le C_0\|P\varphi\|_{L^2}
+C_0C(\beta_0^{-1}+\varepsilon\beta_0^{-2})
\mathcal N(\varphi).
\]
Increase the fixed lower threshold so that the last coefficient is at
most $1/2$. This proves the assertion. In the parameter choices for
the construction, this increase is made when $p_*$ is chosen, before
$\varepsilon$ and the radius amplitude are fixed.
\end{proof}
We record two bounds needed to use coercivity on the actual density.
First, $P_0 1$ is uniformly bounded: the raising part in
\eqref{ang:P-decomposition} annihilates constants, the zeroth base
coefficient of $\mathcal U$ is bounded in $\Psi^2$, and $\mathcal J$ is bounded in
$\Psi^1$. Equation~\eqref{ang:comparison-bound} then shows that
$PI$ is uniformly bounded in $L^2$.
Second, for any fixed smooth base cutoff $\chi$,
\begin{equation}\label{ang:cutoff-bound}
\|[P,\chi]w\|_{L^2}
\le C_\chi\left(
\|w\|_{L^2(\mathcal O_\chi;H^1(S_y))}
+\|r\nabla w\|_{L^2(S_\gamma\mathcal O_\chi)}\right),
\end{equation}
where $\mathcal O_\chi$ is any fixed small neighborhood of the supports
of its derivatives. Indeed,
$[2T,\chi]=2\sum_i\mathsf A_i^*(\nabla_i\chi)$,
$\mathcal U$ has second base coefficient $rQ^{ij}$ and first base coefficients
bounded in $\Psi^1$, and $\mathcal J$ commutes with $\chi$.
The additional first base coefficients of $P-P_0$ are $r\Psi^0$,
by \eqref{ang:comparison-identity}. These observations give exactly
\eqref{ang:cutoff-bound}.
\begin{proposition}[Uniform transfer density]\label{ang:density}
Choose the fixed geometry as in Section~\ref{sec:spheres}, then choose
$p_*$ sufficiently large, $\varepsilon>0$ sufficiently small, and the
radius amplitude $R_*>0$ sufficiently small that
$r_\delta\beta_\delta^2\le\varepsilon$ for $0<\delta\le1$.
For all sufficiently small spatial dilations there is a constant $C$,
independent of the dilation and $\delta$, such that
\begin{equation}\label{ang:density-bound}
\|\chi\psi_\delta\|_{L^2(S_\gamma Y)}\le C
\end{equation}
whenever $\sum_{|\alpha|\le8}|\partial^\alpha h|
\le\varepsilon r_\delta$ on $Y$.
Here $\chi$ is the fixed cutoff from \eqref{sph:cutoff} and
$\psi_\delta$ is the left-acting matrix density of
Proposition~\ref{sph:transfer}.
\end{proposition}
\begin{proof}
The reproduction identity $\int\psi_\delta\,d\sigma_y=I$ makes
$\varphi=\chi(\psi_\delta-I)$ mean zero on each sphere. It is smooth
and compactly supported in the base. Equation~\eqref{ang:P-equation}
gives
\[
P\varphi=-\chi PI+[P,\chi](\psi_\delta-I).
\]
The first term has the uniform bound just established. For the second,
\eqref{ang:cutoff-bound} requires only one angular derivative and
$r\nabla\psi_\delta$ on the cutoff derivative region.
The uniform bounds of Lemma~\ref{sph:cutoff-bounds} supply these:
in the lower region the density is the normalized ball evaluation
kernel, and in the other region the radii have a fixed positive lower
bound. System coercivity bounds $\mathcal N(\varphi)$, hence
$\|\varphi\|_{L^2}$, uniformly. Adding the fixed matrix function
$\chi I$ proves \eqref{ang:density-bound}.
\end{proof}
The estimate controls the density of the actual transfer functional.
In the next section its first and second Taylor moments will turn this
bound into quantitative information about the difference of the two
harmonic-frame equations.
==== END Determination-of-a-metric-and-a-unitary-connection-from-one-boundary-patch//05-angular.tex ====
==== BEGIN Determination-of-a-metric-and-a-unitary-connection-from-one-boundary-patch//06-matrix.tex ====
\section{Matrix moments and extension across the contact point}
\label{sec:matrix}
\label{mat:section}
The transfer density constructed in Section~\ref{sec:spheres} has a
uniform $L^2$ bound whenever the two normalized principal tensors differ
by at most a small multiple of the sphere radius. We now improve that
comparison from order $r$ to order $r^{3/2}$. This strict improvement will
allow the radii to tend to zero while keeping one fixed dilation of the
original coordinates.
The scalar argument uses affine and quadratic moments of the transfer
density; see \cite[Section~6]{ScalarCompanion2026}. In a harmonic frame,
the first moments are matrices and need not vanish. We approximate
them by a matrix-valued displacement independent of the shrinking
parameter. Its component complementary
to the real scalar matrices satisfies a first-order system with
conformal Killing principal part. The
finite-type property of that system provides the additional power of
the radius.
\subsection{A finite family of harmonic jets}
We use the coordinates and harmonic frames of
Proposition~\ref{bdy:contact}. Before dilation, their matrix coordinate
functions satisfy
\[
\frac12\operatorname{Re}\operatorname{tr}X_j^l(x)=x^l,
\qquad X_j^l(0)=0,
\qquad \partial_iX_j^l(0)=\delta_i^l I,
\qquad 1\le l\le n.
\]
There are also finitely many paired harmonic columns whose value and
gradient vanish at the origin and whose first-side Hessians form a
basis of the vector-valued trace-free symmetric Hessians there. All
these pairs agree on the known side of the contact hypersurface.
For the dilation parameter $\lambda$, replace the coordinate matrices
by
\begin{equation}
\widehat X_j^l(y)=\lambda^{-1}X_j^l(\lambda y),
\label{mat:renormalized-coordinates}
\end{equation}
and replace each of the additional columns $u_j$ by
$\lambda^{-2}u_j(\lambda y)$. Let
\[
(v_{1,\nu},v_{2,\nu}),\qquad 1\le\nu\le N,
\]
be the resulting list of column pairs, including all columns of the
$\widehat X_j^l$. The list is finite and fixed independently of the
shrinking parameter $\delta$. Taylor's formula at the origin shows
that these functions have uniform bounds of every fixed smooth order
on the fixed dilation range. To obtain the second-side bounds, note
that each paired difference, expressed in the common coordinates,
vanishes on an open set accumulating at the origin. Every derivative
therefore vanishes at the origin by continuity. Each additional
column pair thus has zero value and gradient on both sides, as
required for its quadratic rescaling; the coordinate pairs have the
common zero value required for linear rescaling. Taylor's formula
then gives uniform bounds of every fixed order on both sides.
Recall the harmonic-frame equations, written with positive definite
second-derivative coefficients:
\begin{equation}
\mathcal L_1=a^{ij}\partial_i\partial_j+B_1^i\partial_i,
\qquad
\mathcal L_2=Q^{ij}\partial_i\partial_j+B_2^i\partial_i,
\qquad Q=a+h.
\label{mat:operators}
\end{equation}
The tensors $a,Q$ are real and scalar in the fiber index; the $B_j^i$
are complex $2\times2$ matrices. All coefficients have uniform smooth
bounds for small $\lambda$. There is no zeroth-order term because the
columns of the frames are harmonic. In ordinary coordinates,
$B_j^i\longrightarrow0$ smoothly under dilation, whereas
$a\longrightarrow e^{2Lt_0}I_n$.
\begin{lemma}[Uniform spanning by the selected jets]
\label{mat:spanning}
For all sufficiently small $\lambda$ and every $y\in Y$, the vectors
\[
\big(\partial_i v_{1,\nu}(y),
\partial_i\partial_j v_{1,\nu}(y)\big),
\qquad 1\le\nu\le N,
\]
span, over $\mathbb C$, the space of pairs $(p,H)$ with
$p_i\in\mathbb C^2$ and
$H_{ij}=H_{ji}\in\mathbb C^2$ satisfying
\begin{equation}
a^{ij}H_{ij}+B_1^ip_i=0.
\label{mat:jet-constraint}
\end{equation}
The coefficients realizing prescribed such data may be chosen by a
smooth right inverse, with uniform bounds to every fixed order.
\end{lemma}
\begin{proof}
As $\lambda\to0$, the coordinate columns converge smoothly to
$y^l e_\alpha$, $1\le l\le n$, $1\le\alpha\le2$, where
$e_\alpha$ is the standard basis of $\mathbb C^2$. The additional
columns converge to the trace-free quadratic polynomials specified by
their Hessians at the origin. At any point $y$, the latter give
arbitrary trace-free Hessians, and the affine columns correct their
gradients to any prescribed values. Thus the limiting jets span all
$(p,H)$ with $\sum_iH_{ii}=0$, uniformly for $y\in Y$.
The spaces in \eqref{mat:jet-constraint} form a smooth vector bundle of
constant rank: contraction with the positive definite tensor $a$ is
onto $\mathbb C^2$. They can be trivialized by the gradient and the
trace-free part of the Hessian, the remaining trace being determined
by \eqref{mat:jet-constraint}. In this trivialization the selected jets
give a smooth matrix of constant full row rank at $\lambda=0$.
Compactness of $Y$ preserves this rank, with a positive lower bound on
its nonzero singular values, for small $\lambda$. If this matrix is
$T$, the right inverse $T^*(TT^*)^{-1}$ has the asserted smoothness and
uniform bounds. Each selected column satisfies
\eqref{mat:jet-constraint}, so its range is precisely the stated space.
\end{proof}
\subsection{Displacement and weighted remainders}
Throughout the next three subsections we fix $0<\delta\le1$ and assume
the provisional comparison
\begin{equation}
H_8(y):=\sum_{|\alpha|\le8}|\partial^\alpha h(y)|
\le\varepsilon r_\delta(y),\qquad y\in Y.
\label{mat:provisional}
\end{equation}
Write $r=r_\delta$. Since $r_{\delta'}\ge r_\delta$ when
$\delta'\ge\delta$, the bound \eqref{mat:provisional} also holds with
every intermediate radius used to continue from $\delta'=1$ to
$\delta'=\delta$. Proposition~\ref{sph:transfer} and
Proposition~\ref{ang:density} give
\begin{equation}
\begin{split}
v_{2,\nu}(y)
&=\int_{S_y}\psi(y,\omega)\,
v_{1,\nu}\big(\exp_y^\gamma(r(y)\omega)\big)
\,d\sigma_y(\omega),\\
\int_{S_y}\psi(y,\omega)\,d\sigma_y(\omega)&=I,
\qquad \|\chi\psi\|_{L^2(Y\times S_y)}\le C.
\end{split}
\label{mat:transfer-input}
\end{equation}
The constant is independent of $\delta$ and of sufficiently small
$\lambda$. The measure in the last norm is
$dV_\gamma(y)d\sigma_y(\omega)$; on the fixed coordinate range it is
uniformly equivalent to the corresponding Euclidean base measure.
Set $x(y,\omega)=\exp_y^\gamma(r(y)\omega)$. The first Taylor
coefficients of the transfer formula involve the matrix moments
\[
M^i(y)=\int_{S_y}\psi(y,\omega)
\big(x^i(y,\omega)-y^i\big)\,d\sigma_y(\omega).
\]
These moments depend on $\delta$ through the density and the sphere.
We need a displacement that remains fixed as the radii shrink. The
paired coordinate matrices provide such a choice, and the moment
estimate below will show that it differs from $M$ only by a
second-order error.
Define matrices $D^i(y)$, $1\le i\le n$, by the linear system
\begin{equation}
\widehat X_2^l-\widehat X_1^l
=\sum_iD^i\partial_i\widehat X_1^l,
\qquad 1\le l\le n.
\label{mat:displacement}
\end{equation}
The map on the right sends the $n$ matrices $D^i$ to $n$ matrices by
right multiplication with the indicated coefficients. It converges
uniformly, with derivatives, to the identity map because
$\partial_i\widehat X_1^l\to\delta_i^l I$. Hence
\eqref{mat:displacement} defines smooth $D^i$ with uniform smooth
bounds. In particular, $D$ is independent of $\delta$. For each
selected pair define
\begin{equation}
R_\nu=v_{2,\nu}-v_{1,\nu}-D^i\partial_i v_{1,\nu}.
\label{mat:remainder}
\end{equation}
These remainders also have uniform smooth bounds and are independent
of $\delta$.
\begin{lemma}[Weighted moment bounds]
\label{mat:moments}
Under \eqref{mat:provisional},
\begin{equation}
\|\chi D/r\|_{L^2(Y)}\le C,
\qquad
\|\chi R_\nu/r^2\|_{L^2(Y)}\le C,
\qquad 1\le\nu\le N.
\label{mat:weighted-moments}
\end{equation}
The constants are uniform in $\delta$ and small $\lambda$.
\end{lemma}
\begin{proof}
Writing $\rho(y)=\|\psi(y,\cdot)\|_{L^2(S_y)}$, uniform normal
coordinates and Cauchy--Schwarz give $|M(y)|\le Cr(y)\rho(y)$.
Taylor expansion in the fixed chart, followed by
\eqref{mat:transfer-input}, gives
\begin{equation}
v_{2,\nu}-v_{1,\nu}
=M^i\partial_i v_{1,\nu}+E_\nu,
\qquad |E_\nu(y)|\le Cr(y)^2\rho(y).
\label{mat:taylor-transfer}
\end{equation}
This calculation uses only the uniform second derivatives of the
selected columns and $|x-y|\le Cr$; the coordinate displacement is a
scalar, so no matrix factors have been interchanged.
Apply the same calculation to the coordinate matrices. Subtracting
\eqref{mat:displacement} shows that the uniformly invertible map
defining $D$ sends $D-M$ to an error bounded by $Cr^2\rho$. Therefore
\[
|D-M|\le Cr^2\rho,
\qquad |D|\le Cr\rho,
\qquad |R_\nu|\le Cr^2\rho.
\]
Multiplication by $\chi$ and integration prove
\eqref{mat:weighted-moments}.
\end{proof}
The next estimate turns these weighted $L^2$ bounds into bounds for
enough derivatives to compare the equations. No derivative of the
shrinking radius is taken in this interpolation step.
\begin{lemma}[Pointwise interpolation]
\label{mat:interpolation}
Choose
\[
d_*=10+\frac n2,\qquad \eta=\frac1{2d_*},\qquad
p_*>2d_*,\qquad m\ge\lceil10+3d_*\rceil.
\]
There is a fixed $\rho_0>0$ such that, if
$y\in K$ and $r(y)<\rho_0$, then
\begin{equation}
\sum_{|\alpha|\le10}|\partial^\alpha D(y)|\le Cr(y)^{1/2},
\qquad
\sum_{|\alpha|\le10}|\partial^\alpha R_\nu(y)|
\le Cr(y)^{3/2}.
\label{mat:pointwise}
\end{equation}
The threshold and constants are independent of $\delta$ and small
$\lambda$.
\end{lemma}
\begin{proof}
By Lemma~\ref{sph:profiles}, $r^{1/p_*}$ has a uniform Lipschitz
constant. Set $s=r(y)^\eta$. Since
$\eta>1/p_*$, for $r(y)$ sufficiently small every $z\in B(y,s)$
satisfies
\[
\big|r(z)^{1/p_*}-r(y)^{1/p_*}\big|
\le C r(y)^\eta\le\tfrac12 r(y)^{1/p_*}.
\]
Consequently $c r(y)\le r(z)\le C r(y)$ on this ball, with constants
depending only on the fixed $p_*$. Because $\chi=1$ on a neighborhood
of the compact set $K$, the ball lies in that neighborhood after a
further fixed reduction of $\rho_0$. The weighted bounds yield
\[
\|D\|_{L^2(B(y,s))}\le Cr(y),\qquad
\|R_\nu\|_{L^2(B(y,s))}\le Cr(y)^2.
\]
For any smooth vector- or matrix-valued function $U$ with a bounded
$C^m$ norm, Taylor expansion to degree $m-1$ and equivalence of norms
on the finite-dimensional polynomial space give
\begin{equation}
|\partial^\alpha U(y)|
\le C s^{-|\alpha|-n/2}\|U\|_{L^2(B(y,s))}
+C_m s^{m-|\alpha|},\qquad |\alpha|<m.
\label{mat:taylor-interpolation}
\end{equation}
Indeed the Taylor remainder has $L^2$ norm at most
$C_m s^{m+n/2}$, and, after rescaling the ball to unit radius, each
coefficient of its Taylor polynomial is bounded by that polynomial's
$L^2$ norm. This proves the displayed inequality componentwise.
For $|\alpha|\le10$, the first term of
\eqref{mat:taylor-interpolation} applied to $D$ is at most
$Cr(y)^{1-\eta d_*}=Cr(y)^{1/2}$; for $R_\nu$ it is at most
$Cr(y)^{2-\eta d_*}=Cr(y)^{3/2}$. The second term is at most
$C_m r(y)^{\eta(m-10)}\le C_m r(y)^{3/2}$, taking
$\rho_0\le1$. This proves \eqref{mat:pointwise}.
\end{proof}
\subsection{The equation for the displacement}
The weighted moments give a preliminary $r^{1/2}$ bound for $D$.
To improve it, we compare the two harmonic equations on the selected
columns. This comparison separates the scalar tensor $h$ from the
matrix part of $D$.
For a selected first-side column write $w=v_{1,\nu}$ and $R=R_\nu$.
Using $v_{2,\nu}=w+D^k\partial_kw+R$, expand
$\mathcal L_2v_{2,\nu}=0$ and use
$\mathcal L_1w=0$. In the third derivatives use
\[
a^{ij}\partial_i\partial_j\partial_kw
=-(\partial_ka^{ij})\partial_i\partial_jw
-(\partial_kB_1^i)\partial_iw
-B_1^i\partial_i\partial_kw.
\]
The resulting identity is
\begin{align}
&\big[h^{ij}I+a^{li}\partial_lD^j+a^{lj}\partial_lD^i
+\mathcal B^{ij}(D)\big]
\partial_i\partial_jw
+\mathcal Y^i\partial_iw\notag\\
&\hspace{15mm}=-\mathcal L_2R
-2h^{ij}(\partial_iD^k)\partial_j\partial_kw
-D^kh^{ij}\partial_i\partial_j\partial_kw,
\label{mat:comparison-identity}\\
&\mathcal B^{ij}(D)
=-D^k\partial_ka^{ij}
+\tfrac12\big(B_2^iD^j+B_2^jD^i
-D^jB_1^i-D^iB_1^j\big),
\label{mat:B-definition}\\
&\mathcal Y^k=B_2^k-B_1^k
+Q^{ij}\partial_i\partial_jD^k
+B_2^i\partial_iD^k-D^l\partial_lB_1^k.
\label{mat:Y-definition}
\end{align}
In particular, $\mathcal Y$ is independent of the selected column.
The order of the factors in \eqref{mat:B-definition} records the
noncommutativity of the matrix drifts. The drift of $\mathcal L_2$ acting on
$D^k\partial_kw$ places $B_2^i$ on the left of $D^k$, while
differentiating the first-side equation places $B_1^i$ on its right.
Let $\Pi$ be the projection of symmetric contravariant spatial
tensors onto their $\gamma$-trace-free part, acting entrywise on
matrices:
\[
(\Pi T)^{ij}=T^{ij}
-\frac1n\big(\gamma_{kl}T^{kl}\big)a^{ij}.
\]
Recall that $\gamma_{ij}h^{ij}=0$.
\begin{lemma}[Projected displacement equation]
\label{mat:projected-equation}
There is a smooth matrix-valued symmetric tensor $\mathcal E$ such
that
\begin{equation}
h^{ij}I+
\Pi\big(a^{li}\partial_lD^j+a^{lj}\partial_lD^i
+\mathcal B^{ij}(D)\big)
=\mathcal E^{ij}.
\label{mat:projected}
\end{equation}
For $y\in K$ with $r(y)<\rho_0$,
\begin{equation}
\sum_{|\alpha|\le8}|\partial^\alpha\mathcal E(y)|
\le Cr(y)^{3/2}.
\label{mat:E-bound}
\end{equation}
\end{lemma}
\begin{proof}
Denote the right-hand side of
\eqref{mat:comparison-identity} for the column $v_{1,\nu}$ by
$Z_\nu$. Its derivatives through order eight are bounded by
$Cr^{3/2}$ on the stated set. For $\mathcal L_2R_\nu$ this uses
derivatives of $R_\nu$ through order ten in
\eqref{mat:pointwise}. Every derivative of either remaining product
contains a derivative of $h$ of order at most eight and a derivative
of $D$ of order at most nine or eight, respectively. Thus
\eqref{mat:provisional} and \eqref{mat:pointwise} give the factor
$r\,r^{1/2}$. The other factors are fixed smooth derivatives of the
selected columns.
At each point, Lemma~\ref{mat:spanning} permits linear combinations of
the identities \eqref{mat:comparison-identity} for which the combined
gradient is zero and the combined Hessian is any prescribed
$a$-trace-free symmetric covariant tensor with values in
$\mathbb C^2$. Such Hessians satisfy
\eqref{mat:jet-constraint} with zero gradient. They annihilate the
$\mathcal Y$ term and test precisely the $\gamma$-trace-free part of
the coefficient of the Hessian.
More explicitly, choose a smooth uniformly bounded basis $H_s$ of
scalar symmetric Hessians with $a^{ij}(H_s)_{ij}=0$, and use the right
inverse in Lemma~\ref{mat:spanning} to realize $H_se_\alpha$, for
$\alpha=1,2$, with zero gradient. The resulting combinations of
$Z_\nu$ give the two columns of the contraction of the projected
coefficient with $H_s$. Inverting this spatial duality defines
$\mathcal E$ and proves \eqref{mat:projected}. The coefficients
realizing these prescribed jets are smooth functions of $y$ and give
zero combined gradient identically. We form these combinations
before differentiating, so the $\mathcal Y$ term is identically zero.
The resulting formula expresses $\mathcal E$ using only the $Z_\nu$
and coefficients with uniformly bounded derivatives. Differentiating
that formula through order eight proves \eqref{mat:E-bound}, without
introducing derivatives of $\mathcal Y$.
\end{proof}
Define the real-linear projection on matrices
\[
q(Z)=Z-\tfrac12\operatorname{Re}\operatorname{tr}(Z)I,
\qquad
\mathcal V_0=\{Z\in\mathbb C^{2\times2}:
\operatorname{Re}\operatorname{tr}Z=0\}.
\]
Write $D^i=d^iI+W^i$, where $d^i\in\mathbb R$ and
$W^i\in\mathcal V_0$. Taking half the real trace in
\eqref{mat:displacement}, and using the exact trace coordinate
identity for both sides, gives
\begin{equation}
d^l=-\frac12\operatorname{Re}\operatorname{tr}
\sum_i W^i\partial_i\widehat X_1^l.
\label{mat:scalar-direction}
\end{equation}
Thus $d$ is a zero-order real-linear expression in $W$ with uniformly
smooth coefficients.
Apply $q$ in the matrix index to \eqref{mat:projected}. The term
$h^{ij}I$ vanishes, as do all derivative terms involving $d^iI$.
Substituting \eqref{mat:scalar-direction} in the remaining
zero-order terms gives
\begin{equation}
\Pi\big(a^{li}\partial_lW^j+a^{lj}\partial_lW^i\big)
+\mathcal B_0^{ij}(W)=\mathcal E_0^{ij},
\qquad \mathcal E_0=q(\mathcal E).
\label{mat:CK-system}
\end{equation}
Here $\mathcal B_0$ is a real-linear zeroth-order operator with
uniformly smooth coefficients, and $\mathcal E_0$ satisfies
\eqref{mat:E-bound}. There are no hidden first derivatives in
$\mathcal B_0$: differentiation of \eqref{mat:scalar-direction}
occurs only in the terms proportional to $I$, which have already
been removed by $q$.
\subsection{Finite type and the improved estimate}
We prove the finite-type statement needed for
\eqref{mat:CK-system}. In Euclidean space, conformal Killing vector
fields have degree at most two; see, for example,
\cite[Section~3]{Eastwood2005}. The calculation below establishes the
corresponding injectivity on third derivatives and includes the
uniformity needed for the variable-coefficient equation.
\begin{lemma}[Prolongation of the conformal Killing principal part]
\label{mat:finite-type}
Let $n\ge3$, let $a=\gamma^{-1}$ be a smooth positive definite tensor
on a coordinate domain, and let $V$ be a finite-dimensional real vector
space. Suppose that a $V$-valued vector field $W$ satisfies
\[
\Pi\big(a^{li}\partial_lW^j+a^{lj}\partial_lW^i\big)
+C^{ij}(W)=E^{ij},
\]
where $C$ is a smooth real-linear zeroth-order operator. Every third
derivative of $W$ is then a smooth linear combination of derivatives
of $W$ through order two and derivatives of $E$ through order two.
The coefficient bounds are uniform when $a$ is uniformly elliptic
and the indicated background coefficients range in bounded smooth
sets.
\end{lemma}
\begin{proof}
Differentiate the equation twice. At a point, its highest-order part
is a linear map from the tensor of third derivatives of $W$ to the
twice-differentiated trace-free symmetric gradient. We first show
that this map is injective. A constant linear change of coordinates
sets $a^{ij}=\delta^{ij}$ at the point, and each real component of
$V$ can be considered separately.
A tensor in the kernel defines a homogeneous cubic vector field $Z$
whose trace-free symmetric gradient vanishes: that gradient is
homogeneous quadratic and all of its second derivatives are zero.
Hence
\begin{equation}
\partial_iZ_j+\partial_jZ_i=2\delta_{ij}\sigma,
\qquad \sigma=\frac1n\operatorname{div}Z.
\label{mat:CK-cubic}
\end{equation}
Differentiating and combining the three permutations of the indices
gives
\[
\partial_i\partial_jZ_k
=\delta_{jk}\partial_i\sigma
+\delta_{ik}\partial_j\sigma
-\delta_{ij}\partial_k\sigma.
\]
Taking divergence in $k$ yields
\[
(n-2)\partial_i\partial_j\sigma=-\delta_{ij}\Delta\sigma.
\]
Its trace gives $(2n-2)\Delta\sigma=0$; since $n\ge3$, the Hessian of
$\sigma$ is zero. The preceding formula therefore gives
$\partial_i\partial_j\partial_lZ_k=0$, so the homogeneous cubic $Z$
vanishes. This proves injectivity.
Let $T_a$ denote this injective map on third-derivative tensors, using
fixed Euclidean inner products on the finite-dimensional tensor
spaces. It has the smooth left inverse
$(T_a^*T_a)^{-1}T_a^*$. Uniform ellipticity and compactness of bounded
coefficient ranges give uniform bounds for this inverse. All other
terms in the twice-differentiated equation contain at most two
derivatives of $W$ or of $E$. Applying the left inverse proves the
claim, including the smooth coefficient bounds after further fixed
numbers of differentiations.
\end{proof}
\begin{proposition}[Strict improvement of the principal tensors]
\label{mat:improvement}
With the fixed choices in Lemma~\ref{mat:interpolation}, there are
constants $C$ and $\rho_0>0$, uniform for sufficiently small
$\lambda$, such that \eqref{mat:provisional} implies
\begin{equation}
H_8(y)\le C r_\delta(y)^{3/2}
\qquad\text{if }y\in K\text{ and }r_\delta(y)<\rho_0.
\label{mat:improved-h}
\end{equation}
On the same set, the derivatives of $D$ through order nine are bounded
by $Cr_\delta^{3/2}$.
\end{proposition}
\begin{proof}
Apply Lemma~\ref{mat:finite-type} to
\eqref{mat:CK-system} with $V=\mathcal V_0$. For each multi-index
$\alpha$ of order three, it gives an identity of the form
\begin{equation}
\partial^\alpha W
=\sum_{|\zeta|\le2}C_{\alpha\zeta}\partial^\zeta W
+\sum_{|\zeta|\le2}F_{\alpha\zeta}
\partial^\zeta\mathcal E_0,
\label{mat:prolongation}
\end{equation}
with uniformly smooth coefficient maps.
Fix $y\in K$ with $r(y)<\rho_0$. By the core geometry of
Section~\ref{sph:core-cutoff}, the vertical segment
\[
z(s)=(y',s),\qquad -\tfrac12\le s\le y_n,
\]
stays in $K$, and $r(z(s))\le r(y)$. At its initial point, $D$ and
all its derivatives vanish: the paired coordinate matrices agree on
an open neighborhood in the known lower region. The same holds for
$W$.
Let $\mathcal W(s)$ consist of all coordinate derivatives of $W$
through order two at $z(s)$. Differentiating this vector with respect
to $s$ either gives another one of its components or a third
derivative, to which \eqref{mat:prolongation} applies. Thus
\[
\mathcal W'(s)=A(s)\mathcal W(s)+F(s),
\qquad \mathcal W(-\tfrac12)=0,
\qquad |A(s)|\le C,
\qquad |F(s)|\le Cr(y)^{3/2}.
\]
The bound for $F$ follows from \eqref{mat:E-bound}, since the entire
segment has radius below $\rho_0$. Its length is uniformly bounded,
so the elementary integral form of Gronwall's inequality gives
\[
\sum_{|\alpha|\le2}|\partial^\alpha W(y)|
\le Cr(y)^{3/2}.
\]
This argument holds at every point of the stated set. Differentiating
\eqref{mat:prolongation} $k-3$ times, for $3\le k\le9$, expresses
the derivatives of $W$ of order $k$ in terms of its derivatives
through order $k-1$ and derivatives of $\mathcal E_0$ through order
$k-1$. Induction and \eqref{mat:E-bound} therefore give the same
bound through order nine. Notice that the last step uses precisely
eight derivatives of $\mathcal E_0$.
Equation~\eqref{mat:scalar-direction} now gives the same estimates for
$d$ and hence for $D$. Finally, differentiate
\eqref{mat:projected} through order eight. Its right-hand side uses
derivatives of $D$ through order nine and of $\mathcal E$ through
order eight, all bounded by $Cr^{3/2}$. This proves
\eqref{mat:improved-h}.
\end{proof}
\subsection{One dilation for all shrinking radii}
We have proved the strict improvement assuming the provisional bound.
We now remove that assumption for all positive shrinking parameters.
The order of choices matters: the threshold for the improvement must
be fixed before the final choice of the dilation.
\begin{proposition}[Local equality of the normalized equations]
\label{mat:local}
In the contact coordinates and harmonic frames of
Proposition~\ref{bdy:contact}, there is a neighborhood of the origin
on which the two metrics are smoothly conformal and the
harmonic-frame equations, multiplied by the
positive scalars making their principal tensors equal, coincide.
All the finitely many selected pairs of harmonic-frame columns agree
on that neighborhood.
\end{proposition}
\begin{proof}
First fix the reference geometry and uniform smooth bounds for all
sufficiently small dilations. Choose $p_*$ large enough for the sphere
geometry, the angular estimate, and
Lemma~\ref{mat:interpolation}. Next choose $\varepsilon$ sufficiently
small and $R_*$ sufficiently small to satisfy those results, including
$r\beta^2\le\varepsilon$. As explained in
Remark~\ref{sph:choices}, every fixed-separation range needed for the
continuation and cutoff bounds is then fixed before $\lambda$.
Proposition~\ref{ang:density} and all estimates of this section have
constants uniform for sufficiently small $\lambda$.
Choose $0<\rho_*<\rho_0$, with $\rho_*\le1$, so small that
$C\rho_*^{1/2}\le\varepsilon/2$ in
\eqref{mat:improved-h}. Thus, wherever that estimate applies and
$r_\delta<\rho_*$, it improves
\eqref{mat:provisional} to
$H_8\le\varepsilon r_\delta/2$.
As $\lambda\to0$, $h\to0$ in $C^8(Y)$. The numbers
\[
\rho_*,\qquad \min_Y r_1,\qquad R_*(2t_b)^{p_*}
\]
are positive and fixed independently of $\lambda$. We can therefore
choose one sufficiently small $\lambda$ such that
\begin{equation}
\sup_Y H_8\le\frac{\varepsilon}{2}
\min\big\{\rho_*,\min_Yr_1,R_*(2t_b)^{p_*}\big\}.
\label{mat:final-dilation}
\end{equation}
We also make the known lower region contain $\{y_n<-1/4\}$ on the
fixed ranges, as in the sphere construction. Fix this $\lambda$ for
the rest of the proof.
Let
\[
\mathcal I=\{\delta\in(0,1]:
H_8(y)\le\varepsilon r_\delta(y)
\text{ for every }y\in Y\}.
\]
It contains $1$ by \eqref{mat:final-dilation}. For any
$\delta\in\mathcal I$, the preceding estimates improve the inequality
to $H_8\le\varepsilon r_\delta/2$ throughout $Y$. Indeed,
Proposition~\ref{mat:improvement} handles the points in $K$ with
$r_\delta<\rho_*$. Equation~\eqref{mat:final-dilation} handles all
points with $r_\delta\ge\rho_*$. At a point outside $K$, either
$y_n<-1/2$, where equality of the coefficients is already known, or
$t>2t_b$. In the latter case
$\ell_\delta(t)\ge t$, so
$r_\delta\ge R_*(2t_b)^{p_*}$ and
\eqref{mat:final-dilation} again applies.
The set $\mathcal I$ is relatively closed in $(0,1]$ by continuity
and compactness of $Y$. It is also relatively open: at any positive
$\delta$, the radius has a positive minimum on $Y$, so the strict
factor $1/2$ just proved persists for nearby parameter values in the
weaker defining inequality. Since $(0,1]$ is connected,
$\mathcal I=(0,1]$.
On the fixed open set $K^\circ\cap\{t<0\}$, which contains the
origin, $r_\delta\to0$ as $\delta\to0$. The coefficients $h,D,R_\nu$
are independent of $\delta$. The provisional bound and
\eqref{mat:pointwise} therefore imply
\[
h=0,\qquad D=0,\qquad R_\nu=0
\]
there. In particular all selected pairs agree. Subtracting their
equations now gives
\[
(B_2^i-B_1^i)\partial_i v_{1,\nu}=0
\qquad(1\le\nu\le N).
\]
The gradient projection of the space
\eqref{mat:jet-constraint} is all of $(\mathbb C^2)^n$: for any
gradient one can choose the trace of the Hessian to satisfy that
constraint. Lemma~\ref{mat:spanning} therefore implies $B_1=B_2$.
Together with $Q=a$, this proves equality of the normalized
harmonic-frame operators. The definition of $Q$ shows that the two
inverse metrics are positive scalar multiples, so the metrics are
smoothly conformal. Undoing the fixed dilation proves the stated
conclusion in the original contact coordinates.
\end{proof}
The remaining conformal factor and the metric carried by the harmonic
frames are addressed next. The local conclusion just proved supplies
the equality of equations needed to recover both the Riemannian metric
and a unitary connection identification.
==== END Determination-of-a-metric-and-a-unitary-connection-from-one-boundary-patch//06-matrix.tex ====
==== BEGIN Determination-of-a-metric-and-a-unitary-connection-from-one-boundary-patch//07-global.tex ====
\section{Exact identification and the measured boundary}
\label{glob:section}\label{sec:global}
Proposition~\ref{mat:local} identifies the metric up to conformal
scale and identifies the normalized equations in harmonic frames
near a contact point. We first show why, for a connection
Laplacian without an additional potential, this is already an
exact local isometry with a unitary connection identification.
We then return to the maximal match of Section~\ref{bdy:section}
and prove that it extends over the whole manifold and fixes every
part of the measured boundary.
\subsection{Removing the conformal factor}
\begin{lemma}[Exact local identification]\label{glob:conformal}
In the contact coordinates and unitary frames of
Proposition~\ref{bdy:contact}, suppose that on a connected
neighborhood \(\Omega\) of zero the metrics satisfy
\(g_2=e^{2\omega}g_1\), and that the equations in the harmonic
frames \(F_j\), after scalar normalization to the same principal
part, have equal first-order coefficients. Then
\[
g_2=g_1,\qquad J_{\mathrm{loc}}=F_1F_2^{-1}\in U(2),\qquad
d_{A_1}(J_{\mathrm{loc}}u)=J_{\mathrm{loc}}d_{A_2}u
\quad\text{on }\Omega.
\]
Moreover this identification agrees with the old match on the
known side and matches all corresponding source evaluations
throughout \(\Omega\).
\end{lemma}
\begin{proof}
Let
\[
C_j=F_j^{-1}(dF_j+A_jF_j).
\]
These are the connection matrices in the harmonic frames; they
need not be skew-Hermitian in the standard matrix inner product.
The negative of the actual connection Laplacian, expressed in
such a frame, is
\[
\Delta_{g_j,C_j}
=g_j^{il}\bigl((\partial_i+C_{j,i})(\partial_l+C_{j,l})
-\Gamma^k_{il}(g_j)(\partial_k+C_{j,k})\bigr).
\]
Its zeroth-order term is zero, because the columns of \(F_j\)
are harmonic. The common-principal-part hypothesis therefore
says
\begin{equation}\label{glob:equal-frame-operators}
e^{2\omega}\Delta_{g_2,C_2}=\Delta_{g_1,C_1}.
\end{equation}
Indeed their second-order coefficients agree after the indicated
conformal normalization, their first-order coefficients agree
by assumption, and both zeroth-order coefficients vanish.
Put \(\phi=(n-2)\omega/2\). The conformal change of the
Christoffel symbols gives
\[
e^{2\omega}\Delta_{g_2,C_2}
=\Delta_{g_1,C_2}
+2\langle d\phi,\partial+C_2\rangle_{g_1}.
\]
Comparison of first-order terms with
\eqref{glob:equal-frame-operators} yields
\(C_2=C_1-d\phi\,I\). Substitution, with matrix multiplication
in its displayed order, gives
\begin{align*}
\Delta_{g_1,C_1-d\phi I}
+2\langle d\phi,\partial+C_1-d\phi I\rangle_{g_1}
&=\Delta_{g_1,C_1}
-(\Delta_{g_1}\phi+|d\phi|_{g_1}^2)I.
\end{align*}
Here \(\Delta_{g_1}=\operatorname{div}_{g_1}\nabla_{g_1}\).
The zeroth-order terms in
\eqref{glob:equal-frame-operators} consequently imply
\[
\Delta_{g_1}\phi+|d\phi|_{g_1}^2=0,
\qquad \Delta_{g_1}(e^\phi)=0.
\]
On the known open side the metrics already agree, so
\(e^\phi=1\). Unique continuation on \(\Omega\) gives
\(e^\phi=1\) everywhere. Since \(n\ge3\) and \(\omega\) is
real, \(\omega=0\), and hence \(C_1=C_2\).
The last equality is equivalent to
\(d_{A_1}(F_1F_2^{-1}u)=F_1F_2^{-1}d_{A_2}u\).
Although the harmonic frames are not unitary,
\(J_{\mathrm{loc}}=F_1F_2^{-1}\) is. To see this, parallel
transport the intertwining identity along any path starting in
the known side. Both connection transports are unitary, and
\(J_{\mathrm{loc}}\) is the old unitary identification at the
initial point. It remains unitary along the path; connectedness
of \(\Omega\) proves the assertion everywhere.
For an arbitrary source \(f\in C_c^\infty(U_0;\mathbb C^2)\),
both \(w_{1,f}\) and \(J_{\mathrm{loc}}w_{2,f}\), in the common
coordinate domain, are solutions of the first homogeneous
connection equation. The charts avoid \(\overline{U_0}\).
Their difference vanishes on the known side, so unique
continuation makes it zero on \(\Omega\). Thus all evaluation
identities hold, not only those for the finite family used in
the local argument. On an overlap with the old match, the two
proposed first-side images of a point, together with their invertible fiber maps,
would impose an invertible linear relation between the two
first-side evaluation maps. Joint surjectivity in
Lemma~\ref{bdy:evaluations} forces the image points to coincide.
Surjectivity at the second-side point then forces the fiber maps
to coincide as well. Thus the local identification agrees with
the old match on every overlap.
\end{proof}
\subsection{The match fills both interiors}
The source-evaluation argument below follows the embedding architecture
of \cite[Section~6.1]{LLS20}. The maximal matching, boundary completion,
and recovery on the whole measured patch adapt
\cite[Section~7]{ScalarCompanion2026}; we give the details for the
unitary bundle identification as well as the base map.
\begin{proposition}[Global interior identification]\label{glob:interior}
The maximal match of Lemma~\ref{bdy:matches} is defined on
\(N_2^\circ\), and its base map is an onto isometry
\(\Phi:N_2^\circ\to N_1^\circ\).
\end{proposition}
\begin{proof}
If its domain had an interior frontier,
Proposition~\ref{bdy:contact} and the dilation in
Section~\ref{bdy:dilation} would supply the local data used in
Sections~\ref{sph:section}--\ref{mat:section}.
Proposition~\ref{mat:local} gives conformally equal metrics and
equal normalized harmonic-frame equations on a neighborhood of
the contact point in the original charts. Apply
Lemma~\ref{glob:conformal}.
Coordinate identity and \(J_{\mathrm{loc}}\) extend the old match
to a neighborhood containing the frontier point. The concluding
assertion of Lemma~\ref{glob:conformal} ensures agreement on every
overlap. This contradicts maximality. The
domain, already nonempty and open, has no relative frontier in
the connected interior, and is therefore all of \(N_2^\circ\).
The image is open because \(\Phi\) is a local isometry. To see
that it is relatively closed, suppose \(\Phi(p_k)\to q\in
N_1^\circ\). Compactness gives a subsequence with \(p_k\to p\in
N_2\) and \(J(p_k)\to J_0\in U(2)\). If \(p\in\partial N_2\),
continuity of the evaluations and \eqref{bdy:matching} would give
\(I_1(q)=J_0I_2(p)=0\), contrary to
Lemma~\ref{bdy:evaluations}. Thus \(p\) is interior, and
continuity gives \(\Phi(p)=q\). Connectedness of \(N_1^\circ\)
makes the image all of that interior. Injectivity was proved in
Lemma~\ref{bdy:matches}; the inverse is smooth because the map
is a bijective local diffeomorphism.
\end{proof}
\subsection{Boundary extension and the original gauge}
\begin{proposition}[Completion and the measured patch]\label{glob:boundary}
The interior maps \((\Phi,J)\) extend smoothly to the extended
manifolds, with \(\Phi:N_2\to N_1\) a diffeomorphism and \(J\)
a unitary bundle map intertwining the connections. They preserve
the original copies of \(M\) and satisfy
\[
\Phi|_\Gamma=\operatorname{Id},\qquad J|_\Gamma=I.
\]
\end{proposition}
\begin{proof}
An onto interior isometry preserves lengths of curves in both
directions and therefore preserves intrinsic distance. The
metric completion of the interior of a compact smooth
Riemannian manifold with boundary is the manifold with its length
metric. In particular, a curve touching the boundary can be
pushed slightly into an inward collar with arbitrarily small
change of length; uniform local metric comparisons identify
the resulting completion with the original compact topology.
Thus \(\Phi\) and its inverse extend to mutually inverse
homeomorphisms of \(N_2,N_1\), mapping boundary to boundary.
The extensions are smooth. In a sufficiently thin uniform
collar, the distance \(s_j\) to \(\partial N_j\) is smooth and
its gradient generates inward unit normal flow. The isometry
preserves distance to boundary, so
\(s_1\circ\Phi=s_2\), and in the interior it intertwines these
gradient fields. On a fixed small level \(s_2=s_0>0\), the map
is already smooth. For \(0\le s\le s_0\), express it using
that smooth level map and the smooth normal flows on the two
sides. This gives a smooth extension to \(s=0\); the inverse
has the same construction. Parallel transport for \(A_1,A_2\)
along these normal curves similarly extends \(J\) smoothly,
retaining unitarity and the connection intertwining identity.
The maps are identity on the open cap, hence on its closure by
continuity. With closure taken in \(N_j\), the original interiors satisfy
\[
M_j^\circ=N_j^\circ\setminus\overline E.
\]
Bijectivity and identity on \(\overline E\) preserve these
complements, and hence their closures, the original copies of
\(M\). The restricted diffeomorphism fixes the attachment patch
\(P=\{b>0\}\), and \(J=I\) there.
We must recover the boundary identity on every part of
\(\Gamma\), which may have more than one component. For
\(f\in C_c^\infty(\Gamma;\mathbb C^2)\), let \(u_j\) now denote
the harmonic solutions on the original manifolds with boundary
value \(f\). On the second manifold define
\(\widetilde u_1=J^{-1}(u_1\circ\Phi)\). The metric and
connection identities make \(\widetilde u_1\) second-harmonic.
On \(P\), it has the same boundary value as \(u_2\). It also
has the same covariant normal derivative there. Indeed equality
of the measured forms, tested against arbitrary functions
supported in \(P\), gives the equality of smooth densities
\[
(\nabla^{A_1}_{\nu_1}u_1)\,dS_{g_1}
=(\nabla^{A_2}_{\nu_2}u_2)\,dS_{g_2}
\quad\text{on }P.
\]
The densities agree by Lemma~\ref{bdy:jets}, the isometry
transports the outward normal, and the bundle map equals \(I\)
on \(P\). Thus \(\widetilde u_1-u_2\) has zero Cauchy data
there. Boundary and interior unique continuation give
\(\widetilde u_1=u_2\) throughout \(M\).
Taking boundary traces yields
\begin{equation}\label{glob:boundary-tests}
J(p)f(p)=f(\Phi(p))
\quad\bigl(p\in\partial M,\ f\in C_c^\infty(\Gamma;\mathbb C^2)\bigr).
\end{equation}
If \(p\in\Gamma\) and \(\Phi(p)\ne p\), choose \(f\) nonzero
at \(p\), supported in a small neighborhood of \(p\) avoiding
\(\Phi(p)\). Equation~\eqref{glob:boundary-tests} contradicts
invertibility of \(J(p)\). Hence \(\Phi(p)=p\) on \(\Gamma\).
Arbitrary vector values of \(f(p)\) then give \(J(p)=I\) there.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:main}]
The boundary-identity gauges \(V_j\) of
Section~\ref{bdy:section} put the connections in normal gauge.
Lemma~\ref{bdy:jets} and Proposition~\ref{bdy:cap} produce common
exterior Green data. Lemmas~\ref{bdy:evaluations} and
\ref{bdy:matches} define the maximal match, and
Proposition~\ref{bdy:contact} supplies the local contact data
wherever an interior frontier is present. Proposition~\ref{mat:local}, followed by Lemma~\ref{glob:conformal}, removes
that frontier. Propositions~\ref{glob:interior} and
\ref{glob:boundary} give a smooth global metric and connection
identification fixing the entire measured patch.
To distinguish the original connections in this final step, write
\(A_j^{\mathrm{orig}}\) for them, so the connections used in the
proof were
\[
A_j=V_j^{-1}A_j^{\mathrm{orig}}V_j+V_j^{-1}dV_j.
\]
In the original trivializations the bundle map from second to
first fiber is
\[
U=(V_1\circ\Phi)J V_2^{-1}.
\]
It is smooth and unitary, and equals \(I\) on \(\Gamma\), since
both original gauges equal \(I\) on the entire boundary.
The intertwining identity is
\(d_{\Phi^*A_1^{\mathrm{orig}}}(Uu)=U d_{A_2^{\mathrm{orig}}}u\).
Expanding it gives
\[
A_2^{\mathrm{orig}}
=U^{-1}(\Phi^*A_1^{\mathrm{orig}})U+U^{-1}dU,
\qquad g_2=\Phi^*g_1.
\]
These are the asserted identities. The argument uses densities
and a single nonempty cap; it requires neither orientability nor
connectedness of the boundary or of \(\Gamma\).
\end{proof}
==== END Determination-of-a-metric-and-a-unitary-connection-from-one-boundary-patch//07-global.tex ====
==== BEGIN Determination-of-a-metric-and-a-unitary-connection-from-one-boundary-patch//paper.tex ====
\documentclass[11pt,a4paper]{article}
\input{preamble}
\title{Determination of a metric and a unitary connection\\from one boundary patch}
\author{OpenAI}
\date{October 5, 2026}
\begin{document}
\maketitle
\begin{abstract}
We prove that zero-frequency boundary measurements on any nonempty open
boundary patch determine both a smooth Riemannian metric and a smooth
unitary connection on the trivial Hermitian rank-two bundle over a compact
connected smooth manifold of dimension at least three with smooth boundary.
Both inputs and observations are restricted to the same patch. The metric
and connection are determined up to a diffeomorphism and a unitary gauge
that restrict to the identity on the measured patch.
\end{abstract}
{\small\tableofcontents}
\input{sections/01-introduction}
\input{sections/02-boundary}
\input{sections/03-product}
\input{sections/04-spheres}
\input{sections/05-angular}
\input{sections/06-matrix}
\input{sections/07-global}
\bibliographystyle{plainnat}
\bibliography{references}
\end{document}
==== END Determination-of-a-metric-and-a-unitary-connection-from-one-boundary-patch//paper.tex ====
==== BEGIN Determination-of-a-metric-and-a-unitary-connection-from-one-boundary-patch//preamble.tex ====
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==== END Determination-of-a-metric-and-a-unitary-connection-from-one-boundary-patch//preamble.tex ====
==== BEGIN Determination-of-a-metric-and-a-unitary-connection-from-one-boundary-patch//references.bib ====
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publisher = {Sociedade Brasileira de Matem{\'a}tica},
address = {Rio de Janeiro},
year = {1980},
pages = {65--73},
note = {Reprinted in Computational \& Applied Mathematics 25 (2006), nos. 2--3, 133--138}
}
@article{GPSU16,
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note = {\href{https://arxiv.org/abs/1502.04720v2}{arXiv:1502.04720v2}}
}
@book{HormanderIV2009,
author = {H{\"o}rmander, Lars},
title = {{The Analysis of Linear Partial Differential Operators IV: Fourier Integral Operators}},
series = {Classics in Mathematics},
publisher = {Springer},
address = {Berlin, Heidelberg},
year = {2009},
doi = {10.1007/978-3-642-00136-9},
url = {https://link.springer.com/book/10.1007/978-3-642-00136-9},
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author = {{OpenAI}},
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howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Smooth-Anisotropic-Uniqueness-in-the-Calderon-Problem-from-One-Boundary-Patch-September-24-2026/paper.pdf}{OAI:Smooth-Anisotropic-Uniqueness-in-the-Calderon-Problem-from-One-Boundary-Patch-September-24-2026}},
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==== END Determination-of-a-metric-and-a-unitary-connection-from-one-boundary-patch//references.bib ====