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\author{Wenchang Yang}
\address{Tsinghua University, Beijing, China}

\author{Yi Wang}
\address{Tsinghua University, Beijing, China}

\subjclass[2020]{53B20, 53C42}
\keywords{local isometric immersion, smooth metric, oscillatory construction,
current linear response, Fourier localization}
\date{}


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\title[Local isometric embeddings]{Smooth Local Isometric Embeddings of Riemannian Manifolds in Optimal Dimension}
\date{}
\subjclass[2020]{53B20, 53C42, 35F50}
\keywords{local isometric embedding, smooth Riemannian metric,
characteristic waves, Fourier integral operators}

\begin{document}

\begin{abstract}
We prove that every smooth Riemannian manifold of dimension \(n\ge 5\) admits a smooth local isometric embedding into \(\mathbb{R}^{n(n+1)/2}\). This extends the classical Janet–Cartan theorem from real-analytic to arbitrary smooth metrics in these dimensions, achieving the optimal universal ambient dimension without curvature or genericity assumptions. The proof constructs characteristic waves around a flat immersion whose second fundamental form has full normal rank. Their Hermitian quadratic mean generates the required metric correction, while linear responses cancel oscillatory quadratic interactions and the remaining linear strain. Exact Fourier localization yields an iteration converging in \(C^\infty\) on a fixed neighborhood.The detailed geometric and microlocal constructions are proved in the appendices.
\end{abstract}
\maketitle
\tableofcontents
\clearpage

\section{Introduction}\label{main:sec:introduction}
The local isometric immersion problem asks whether a Riemannian metric
$g$ near a point of an $n$-manifold can be written as
\begin{equation}\label{main:eq:isometry}
 g_{ij}=\partial_iF\cdot\partial_jF
\end{equation}
for a map into a Euclidean space of prescribed dimension. The analytic
local theory of Janet (1926) and Cartan (1927) \cited{Janet26,Cartan27}
reaches
\[
 N=s_n=\frac{n(n+1)}2,
\]
the number of independent metric coefficients. For arbitrary smooth
metrics, the classical local theorem gives $\R^{s_n+n}$
\cited{Greene70}; see also \cited[\S3.7]{Gromov17} for the
perturbation and scaling argument. A free reference in that construction
has $s_n$ normal directions and an algebraic normal inverse for the
linearized metric equation \cited{Nash56}. We reach $\R^{s_n}$ for every smooth metric
in dimensions $n\geq5$. Our reference has $s_n-n$ normal directions;
the remaining $n$ stress equations form a determined first-order system.

\begin{theorem}\label{main:thm:main}
Let $n\geq5$, let $p$ belong to a smooth $n$-dimensional manifold, and let
$g$ be a smooth Riemannian metric near $p$. There exist an open
neighborhood $U$ of $p$ and a smooth embedding
\[
 F:U\longrightarrow\R^{n(n+1)/2}
\]
satisfying \eqref{main:eq:isometry} on $U$.
\end{theorem}

The same map and neighborhood work at every derivative order.
The endpoint $n=5$ gives $\R^{15}$; at $n=6$ the target is $\R^{21}$.
The theorem establishes the smooth local Schl\"afli assertion
in dimensions $n\geq5$, in the terminology of \cited[\S1.2]{Gromov17},
without an analyticity or curvature hypothesis.
The endpoint is included in the same construction: a finite-dimensional
node calculation completes the characteristic menu at $n=5$, after which
all dimensions use the same response, sampled mean and convergence proof.
No conclusion for $n=4$ is asserted.

\begin{remark}[Sharpness of the universal target dimension]
The dimension $s_n$ is necessary for a target that works for every smooth
metric germ. If $M<s_n$, the polynomial induced-metric map sends
$(k+1)$-jets of maps into $\R^M$, a space of dimension
$M\binom{n+k+1}{n}$, to $k$-jets of symmetric metrics, of dimension
$s_n\binom{n+k}{n}$. Choose $k$ with $(s_n-M)(k+1)>Mn$.
The source dimension is then strictly smaller, so the smooth image has
measure zero and misses a jet with positive definite constant term.
Its Taylor polynomial is a positive definite analytic metric on a small
neighborhood, and an isometric immersion into $\R^M$ would realize the
omitted jet, a contradiction. This is a universal lower bound; individual
metrics can have smaller embedding dimension.
\end{remark}

The reduction of the linearized equations and their characteristic
geometry were developed by Bryant--Griffiths--Yang \cited{BGY83}.
Smoothness of the characteristic variety cannot be assumed.
Han--Khuri \cited[Theorem~1.2]{HanKhuri12} prove nonsmoothness for
$n\geq5$ in a sufficiently small parameter regime under explicit
generic conditions. Our flat model also has singular characteristics,
classified in Section~\ref{main:sec:geometry}. We choose simple
carriers whose sums and nonzero differences stay in protected regular
sectors. The finite carrier menu detects every missing image direction,
including the five-dimensional node directions and those on the
multiple set: the carrier and output covectors need not coincide.
The resulting quadratic means complete the missing image directions
and give a positive symbol on the full stress space. Regular responses
cancel the oscillatory products, while this augmented mean equation
handles the full Hermitian tensor. No inverse of the determined symbol
on its singular set is used.

The analytic theory is developed in \cited{BCGGG91}.
The smooth low-dimensional results and finite conjugation methods of
Nakamura--Maeda \cited{NM89} and Goodman--Yang \cited{GY88}, and the
real-principal-type theory of Duistermaat--H\"ormander \cited{DH72},
form the background for the quantitative matrix and ambient estimates
proved here on fixed domains, with accuracy-independent losses.
The three-dimensional embedding results of Nakamura--Maeda and
Goodman--Yang have target $\R^6$ and require the Riemann curvature
tensor to be nonzero at the base point.

The argument begins with a flat immersion whose second form spans
the normal space. The kernel of its normal map is a smooth
perturbation of an explicit $n$-dimensional space of quadratic forms.
Projection onto this kernel gives a square first-order system with
simple and multiple characteristics. A finite family of three-parameter
patches on the simple locus provides the oscillations. Their signed
frequency sums and differences lie in regions admitting a linear response.

The principal correction is quadratic. A real sum of complex
characteristic waves has Hermitian self terms and oscillatory products.
A transported positive background and a mean correction adjust
the self terms; a response for the current immersion cancels the other
products. The same waves and sample weights occur throughout, so the
cancellations are identities for the metric of one map.

All waves and responses use the current immersion. A fixed finite
correction of the continuous density prepares the positive background
before sampling. High-order quadrature preserves this accuracy, and
one sampled mean iteration reaches the requested precision. Its row,
wave maps and continuum inverse remain fixed during the iteration.

The resulting increment satisfies a metric estimate of the form
\begin{equation}\label{main:eq:intro-estimate}
 \|\gamma(F+U)-g\|_{C^0}
 \leq C_Q\bigl(\delta R^{-Q}+\delta^3R^{B_3}\bigr),
\end{equation}
where $\|\gamma(F)-g\|_{C^0}\leq\delta^2$ and
$\gamma(F)=(dF)^TdF$. The constant $B_3$ is independent of the
requested accuracy $Q$. Moreover, $U$ has exact global Fourier support
below a fixed power of $R$, and its global $L^2$ norm has a similar fixed
loss. These two properties allow $Q$ to increase slowly enough to absorb
its constants while ensuring convergence in every $C^r$ norm. Thus the
same sequence converges smoothly on one neighborhood.

Section~\ref{main:sec:geometry} describes the geometric starting point.
Section~\ref{main:sec:linear} states the required linear estimates and their
uniformity. Section~\ref{main:sec:waves} constructs the waves and states the mean and
product estimates. Section~\ref{main:sec:mean-details} develops the central
normal and nonlinear mean arguments. Section~\ref{main:sec:increment}
combines these constructions and proves \eqref{main:eq:intro-estimate}.
Sections~\ref{main:sec:history}--\ref{main:sec:convergence} choose the scales and
prove Theorem~\ref{main:thm:main}. The quantitative statements in
Sections~\ref{main:sec:linear}--\ref{main:sec:mean-details} contain the
inputs used by the metric and convergence proofs, so those proofs can
be read while accepting the technical appendix results.
Table~\appref{app:tab:interfaces} gives their sources, domains,
low topologies and dependence on accuracy. The appendices prove
these inputs in separate blocks; Appendix~\appref{app:sec:ordered-scale-selection}
checks their simultaneous parameter windows.


\section{The reference geometry}\label{main:sec:geometry}
\subsection{The reduced metric equation}
For an immersion $F:W\to\R^N$, write
\[
 D\gamma_FY=(dF)^TdY+(dY)^TdF.
\]
Its exact expansion is
\begin{equation}\label{main:eq:metric-expansion}
 \gamma(F+Y)=\gamma(F)+D\gamma_FY+(dY)^TdY.
\end{equation}
Let $\mathrm{II}_F$ denote the second fundamental form, and set
$q=N-n=n(n-1)/2$. We regard it as a map
\[
 B_F:\Sym^2\R^n\longrightarrow\mathcal N_F,
 \qquad B_F(H)=\sum_{i,j}H^{ij}\mathrm{II}_{F,ij}.
\]
When $B_F$ is onto, its kernel has dimension $n$. We call this kernel
the stress space.

Write an ambient variation as
$Y=dF\,\gamma(F)^{-1}u+w$, where $u$ is a covector and $w$ is normal.
Then
\begin{equation}\label{main:eq:tangent-normal}
 D\gamma_FY=2\sym\nabla u-2\langle w,\mathrm{II}_F\rangle.
\end{equation}
Here $\nabla$ is the Levi--Civita connection of the induced metric
$\gamma(F)$, and $\sym A=(A+A^T)/2$. In particular, this connection
need not be the connection of the target metric before the iteration
has converged. The sign of the normal term follows by differentiating
$\partial_iF\cdot w=0$.

Use fixed smooth reference tensors to form the projected stress frame
$Q_i(F)=\Pi_{\ker B_F}Q_i^{\mathrm{ref}}$, with a positive Gram gap,
and write $Q:H=\sum_{i,j}Q^{ij}H_{ij}$. Define
\begin{equation}\label{main:eq:reduced-convention}
 q_F(h)=(Q_i(F):h)_{i=1}^n,\qquad
 \cP_Fu=(Q_i(F):\sym\nabla u)_{i=1}^n.
\end{equation}
Then $D\gamma_FY=h$ reduces to
\begin{equation}\label{main:eq:reduced-equation}
 \cP_Fu=\tfrac12q_F(h).
\end{equation}
The factor $1/2$ will also be used in the mean equation. Since $B_F$
is onto, its adjoint maps the normal space isomorphically onto
$(\ker B_F)^\perp$. For any $u$ and $h$, set
\begin{equation}\label{main:eq:normal-completion}
 w=(B_F^*)^{-1}\Pi_{(\ker B_F)^\perp}
                  \bigl(\sym\nabla u-\tfrac12h\bigr).
\end{equation}
For $Y=dF\,\gamma(F)^{-1}u+w$ and
$\mathsf G_F=(Q_i(F):Q_j(F))_{ij}$, the full error is
\begin{equation}\label{main:eq:completion-error}
 D\gamma_FY-h
 =2\sum_{i,j}Q_i(F)(\mathsf G_F^{-1})_{ij}
                  (\cP_Fu-\tfrac12q_F(h))_j.
\end{equation}
This follows from \eqref{main:eq:tangent-normal} and the Gram formula
for orthogonal projection. Thus a reduced solution solves the full tensor
equation; an approximate
solution has exactly the displayed stress error. We call this
algebraic step \emph{normal completion}.

\subsection{An explicit flat reference}
Let $J$ be the all-ones $n\times n$ matrix, put $c_n=(n-4)^{-2}$,
and define
\begin{equation}\label{main:eq:cube-stress}
 Q_i=e_ie_i^T-c_nJ,\qquad
 K_0=\operatorname{span}\{Q_1,\ldots,Q_n\}.
\end{equation}
With the normalization in \eqref{main:eq:reduced-convention}, the
principal symbol of the frozen reduced operator is $\ii M_0(\xi)$
under $\partial_j\mapsto\ii\xi_j$. Its real contraction pencil is
\begin{equation}\label{main:eq:pencil}
 M_0(\xi)=\diag(\xi_1,\ldots,\xi_n)
           -c_n\Bigl(\sum_i\xi_i\Bigr)J.
\end{equation}
Put $\ell_{ab}=\mathbf1-2e_a-2e_b$, $a<b$.
The $q$ rank-one forms $\ell_{ab}\ell_{ab}^T$ are independent,
annihilate every $Q_i$, and each satisfies the flat Gauss equation.
Successive circular bendings add these forms one at a time while
preserving the Euclidean metric. They give an explicit analytic
flat immersion with full normal rank and center stress space $K_0$.
Propositions~\appref{app:ch_seed:prop:seed-gram}
and~\appref{app:ch_seed:prop:seed-analytic} prove these assertions.

After one fixed dilation the actual stress
frames are uniformly close to \eqref{main:eq:cube-stress};
all later operators use these varying coefficients.

\subsection{Characteristics and product directions}
The determinant of \eqref{main:eq:pencil} is
\begin{equation}\label{main:eq:determinant}
 p_0(\xi)=\prod_i\xi_i
 -c_n\Bigl(\sum_i\xi_i\Bigr)\sum_i\prod_{j\ne i}\xi_j.
\end{equation}
The nonzero set $\{p_0=0,\ dp_0=0\}$ is the union of
\begin{equation}\label{main:eq:multiple-set}
 \{\xi_i=\xi_j=\xi_k=0\},\qquad
 \{\xi_i=\xi_j=\textstyle\sum_l\xi_l=0\},\qquad
 \R\ell_{ab},
\end{equation}
with the origin removed. The first two families have codimension three.
The additional lines have corank one and zero determinant gradient;
every other corank-one zero is regular.
Proposition~\appref{app:ch_seed:prop:seed-strata} proves this complete
classification.

At a simple characteristic covector $\xi$, the matrix $M_F(x,\xi)$
has a one-dimensional kernel, which determines a leading tangent
polarization. After normal reconstruction, the leading derivative of
its wave is a rank-one tensor of the form $\nu\otimes\xi$. Its
Hermitian square is therefore $|\nu|^2\xi\otimes\xi$. Under stress
contraction this gives the vector $|\nu|^2q_F(\xi\otimes\xi)$.
The detector inequality \eqref{main:eq:detector-gram} states how these
vectors complete the image of $M_F$ at every output direction.
Section~\ref{main:sec:mean-details} constructs the positive transported
background from this geometric input.

Quadratic products may leave a chosen characteristic patch. Their
responses must stay in the region of the chosen microlocal construction.

\begin{proposition}[Characteristic patches]\label{main:prop:patches}
For $n\geq5$, one can choose finitely many three-parameter patches of
simple characteristic covectors for the reference geometry, with
positive angular margins, such that the following properties persist
for nearby immersions on a fixed spatial chart.
\begin{enumerate}
\item For every unit output direction, the selected patches have
transverse interior incidence roots and satisfy the uniform Gram
inequality \eqref{main:eq:detector-gram} with the image of $M_F$.
\item After fixed radial labels and colors are assigned, every doubled
phase and every signed product of distinct sampled phases belongs to
the prescribed region of validity for a response operator. The same
holds along the characteristic curves used by that operator.
\item For samples separated by $h$, the product frequency $\varrho$
satisfies $c\lambda h\leq\varrho\leq C\lambda$.
\end{enumerate}
The Hermitian self terms are treated by the mean equation and are
excluded from the product assertion.
\end{proposition}

The proof is in Propositions~\appref{app:ch_seed:prop:seed-frozen-menu}--\appref{app:ch_seed:prop:seed-phase-charts}
and Theorem~\appref{app:ch_regular:thm:no-return}. Two features explain
the choices. A convex three-parameter patch controls the limiting
chords as well as the differences of separated samples. In addition,
the source region must be protected along the entire relevant
characteristic segment, in both orientations; a gap at its initial
point alone is insufficient.

The carrier construction uses the same cube in every dimension $n\geq5$.
Write $d=n-4$, $S_z=\sum_i z_i$ and
$q_0(z)=(z_i^2-d^{-2}S_z^2)_i$.
For $k\ne0$ and $0\ne s\in\ker M_0(k)$, two disjoint coordinate
pairs give transverse detecting carriers, except for a finite set of
five-dimensional node directions. At each such node an explicit
additional carrier supplies the same detection and normal-projection
gaps. Thus all dimensions use three-parameter characteristic patches,
the same signed-product separation, and the same subsequent analytic
construction. Proposition~\appref{app:ch_seed:prop:seed-frozen-menu}
proves the pair selection and the five-dimensional node calculation.

\subsection{A fixed reference and a small target}
The iteration accepts signed metric defects. We therefore initialize
it by scaling the target toward the metric of the flat reference.

\begin{proposition}[Fixed reference initialization]\label{main:prop:seed}
Choose coordinates with $g(0)=I$, and let $F_0$ be the analytic
flat reference above. One
sufficiently small fixed $\rho>0$ places
\begin{equation}\label{main:eq:normalized}
 G_{\mathrm{ref}}(y)=\rho^{-1}[F_0(\rho y)-F_0(0)],
 \qquad g^{(s)}(y)=g(s\rho y)
\end{equation}
on a fixed chart $W$, with all the required reference geometric gaps
positive. For $0\leq s\leq s_*$ and every fixed $r$,
\begin{equation}\label{main:eq:seed-small}
 \|g^{(s)}-\gamma(G_{\mathrm{ref}})\|_{C^r(W)}
                         \leq C_{r,\rho}s.
\end{equation}
The reference and its geometric bounds are independent of $s$.
\end{proposition}

\begin{proof}
Corollary~\appref{app:ch_seed:cor:seed-uniform-dilation} gives the
fixed normalized reference, with $\gamma(G_{\mathrm{ref}})=I$ and
a positive normal gap proportional to the fixed $\rho$.
Since $g^{(s)}$ is jointly smooth on the compact parameter interval
and $g^{(0)}=I$, the fundamental theorem of calculus gives
\eqref{main:eq:seed-small}, uniformly at every fixed order.
\end{proof}

After all geometric, analytic and frequency constants have been fixed,
$s>0$ can thus make the initial defect smaller than any prescribed
threshold. The positive transported background used by the mean
equation is constructed for $G_{\mathrm{ref}}$ in
Section~\ref{main:sec:mean-details}.

\section{Linear response on fixed domains}\label{main:sec:linear}
\subsection{Domains and estimates}
Fix a closed ball $D$ with positive radius, a larger physical chart
$W$, and a flat torus $\T$ containing $\overline W$ in one coordinate
cell. The limit metric equation is imposed on $D$. All measurements,
product sources, and physical output cutoffs are contained in a fixed
larger region $D_{\mathrm{lag}}\Subset W$. For the comparison source
choose cutoffs with
\begin{equation}\label{main:eq:cutoff-chain}
 D_{\mathrm{lag}}\Subset\Int\{\chi_{\mathrm{src}}=1\},
 \qquad
 \supp\chi_{\mathrm{src}}\Subset\Int\{\chi_{\mathrm{core}}=1\}
 \Subset W.
\end{equation}
The appropriate linear identities hold on a neighborhood of the
entire source support. Mean and product operators may have different
cutoffs; their supports have positive buffers inside the relevant
validity regions. This finite collection of domains is fixed before
frequency, accuracy, or sampling is chosen. The full arrangement is
specified in Setup~\appref{app:setup:domains}.

We call $D$ the \emph{core}. The larger cutoffs contain responses that
may extend beyond their source supports. Fixing them keeps the limit
neighborhood independent of accuracy.

Unqualified $H^r$ and $L^2$ norms refer to $\T$; physical $C^r$ norms
carry their domains. Tensor estimates are componentwise. Constants may
depend on the fixed dimension, chart, and geometric neighborhood.
When a bound holds at every order, $C_r$ may depend on $r$. An
\emph{outside loss} is a frequency exponent $B$ in an estimate such as
$C_rL^{r+B}$. The assertions below specify when $B$ is independent of
$r$ and of the requested accuracy.

For complex matrices, $T$ denotes transpose and $*$ conjugate
transpose. We take real parts entrywise. Thus $\Re(X^*X)$ is a real
symmetric tensor, whereas a cross term must be symmetrized when it
stands alone. A normal operator $AA^*$ acts on the output of $A$;
it is distinct from the geometric normal map $B_F$.

We use Fourier series on the fixed torus and the normalization
\[
 \widehat u(\xi)=\frac1{\operatorname{vol}(\T)}
       \int_\T e^{-\ii x\cdot\xi}u(x)\,dx,
 \qquad u(x)=\sum_{\xi\in\Lambda^*}\widehat u(\xi)e^{\ii x\cdot\xi}.
\]
Here $\Lambda^*$ is the dual lattice. The initial variables of a wave
instead use Euclidean Fourier integrals, with
$\widehat f(\xi)=(2\pi)^{-(n-1)}\int e^{-\ii y\cdot\xi}f(y)\,dy$.
This convention makes $f(y)=\int e^{\ii y\cdot\xi}\,dm(\xi)$ for
its Fourier measure; the unit atom at $0$ represents the constant $1$.

For a complex wave $Z=e^{\ii\lambda\phi}z$, the slowly varying quantity
$z$ is its demodulated amplitude. Estimates of $z$ and of
$e^{-\ii\lambda\phi}dZ$ will be carried separately. A bound for the
first does not by itself provide the second at the same scale, since
\[
 e^{-\ii\lambda\phi}dZ=\ii\lambda z\,d\phi+dz.
\]
This is why estimates below distinguish the amplitude scale $K$
from the physical frequency $\lambda$.

The main parameters and fields have the following roles:
\begin{center}
\renewcommand{\arraystretch}{1.15}
\begin{tabular}{@{}p{0.22\textwidth}p{0.72\textwidth}@{}}
$F$ & Current immersion for all waves and response operators.\\
$R,\lambda,h$ & Stage scale, carrier frequency, and separation of samples.\\
$H,K$ & Mean solve scale and Fourier scale of an amplitude.\\
$T_0$ & Derivative scale of the current immersion.\\
$\delta,b$ & Square root of the defect bound and primary size $b=\delta R^A$.\\
$P,V,S$ & Primary wave sum, mean field, and quadratic product correction.\\
$Q,r$ & Requested finite accuracy and observation derivative order.
\end{tabular}
\end{center}
Each scale is specified by a power of $R$ in
Section~\ref{main:sec:scales}. The symbols $B$ with subscripts denote
fixed loss exponents; they are not additional scales.

\subsection{The response operators}
On each protected regular sector, put
$c=|\xi|^{1-n}\operatorname{adj}M_F$ and
$h=|\xi|^{1-n}\det M_F$, so $M_Fc=hI$.
For the Hamilton phase of $h$, one matrix transport $U$ gives a
two-term graph amplitude $B$ with the exact identity
\[
 (\cP_F/i)T_B=T_UD_t+E,
\]
where $E$ has order $-1$. The weighted inverse of $T_U$ and one
characteristic integral give the sector solver.
The construction includes the source cutoff and endpoint terms;
the physical cutoff commutator vanishes on the inner plateau.
Euler's identity gives a nonzero fiber gradient also on the elliptic
region, so the same construction applies there. Equation~\eqref{main:eq:normal-completion}
completes the tensor response.

\begin{proposition}[Uniform response]\label{main:prop:response}
Fix an order $r_0$ above the construction and Sobolev shifts, and
the geometric jet order $c_{\rm jet}$. In a sufficiently small
fixed $C^{r_0+c_{\rm jet}}$ neighborhood of the reference immersion,
suppose, for some $T\geq1$,
\[
 \|F\|_{C^r(W)}\leq C_rT^{(r-r_0-c_{\rm jet})_+}
                    \quad(r\geq0).
\]
The following choices and losses are independent of accuracy.
\begin{enumerate}
\item The normal frame, metric inverse, stress frame and reduced
coefficient array $a_F$ have positive denominator gaps and satisfy
\[
 \|a_F\|_{C^r(W)}+\|c(F)\|_{C^r(W)}
 \leq C_rT^{(r-r_0)_+},
\]
where $c(F)$ denotes any coefficient in normal completion.
\item On fixed solver domains $U\Subset W$ there are a real solver
$S_F$, angular cutoff $A_\Gamma$ and complete measured residual
$\mathcal R_F$, with fixed source and measurement plateaus. For
inputs supported on the source plateau, on $U$ one has
\[
 \cP_FS_F=A_\Gamma+\mathcal R_F,\qquad
 \|\mathcal R_F\|_{L^2(W)\to H^1(U)}
 +\|\mathcal R_F\|_{H^{-1}(W)\to L^2(U)}\leq C.
\]
The literal adjoint has the same gains on the reversed domains.
The contraction cutoffs may be fixed so that
\begin{equation}\label{main:eq:response-contraction}
 K_F=\chi_2\mathcal R_F\chi_3,\qquad
 \|K_F\|_{2\to2}\leq\vartheta<1,\qquad
 T_F=(I+K_F)^{-1}.
\end{equation}
For every integer $s\geq0$, with Sobolev norms on the construction box $W$,
\begin{equation}\label{main:eq:response-tame}
 \|T_Ff\|_{H^s}+\|S_FT_Ff\|_{H^s}
 \leq C_s\bigl[\|f\|_{H^s}+T^s\|f\|_2\bigr].
\end{equation}
If $f$ is supported where $\chi_2=1$, then $z=T_Ff$ is supported
in $\operatorname{supp}\chi_2$, and on the fixed identity core
\[
 \cP_F(S_Fz)=f+(A_\Gamma-I)z.
\]
The same inverse acts at every Sobolev order.
\item The same construction has scale
$L_F=T$: its normalized phase maps $\mathcal G$ have bounded first
jets and $|D^r\mathcal G|\leq C_rT^{r-1}$ for $r\geq2$, while each
actual symbol of fixed order $p$ in this construction obeys
\[
 |D_{x,z}^{\alpha}D_\xi^\beta a_p|
 \leq C_{\alpha,\beta}\langle\xi\rangle^{p-|\beta|}
                  T^{|\alpha|+|\beta|}.
\]
Derivatives include both positions and normalized covariables.
All cutoff and endpoint factors occur in the complete residual.
\end{enumerate}
\end{proposition}

The finite construction and its complete residual are proved in
Appendix~\appref{app:sec:ch_regular_base}, in particular
Proposition~\appref{app:ch_regular_base:prop:complete-residual}
and Theorem~\appref{app:ch_regular_base:thm:sector}.
The exact correction and common graded bounds are proved in Theorems~\appref{app:ch_regular:thm:exact-correction}
and~\appref{app:ch_regular:thm:fixed-high}.
Setup~\appref{app:ch_regular:setup:base-aware-coefficients}
and Lemma~\appref{app:ch_regular:lem:finite-chain} give the total mixed
derivative requirement and protected finite-chain estimates.
Their fixed shifts are included when choosing $r_0$.

Fix the finite symbolic construction first. Higher precision estimates
additional powers of its complete residual and one remaining exact
inverse, on the same domains. The primary family below is also fixed
at every finite accuracy.

\subsection{Response to an oscillatory source}
For a tensor source $h$, apply $S_MT_M$ to
$\chi_2q_M(h)/2$ and use \eqref{main:eq:normal-completion} to define
the ambient response $\cB_Mh$. Equation~\eqref{main:eq:completion-error}
expresses its complete error as the angular discrepancy
$(A_\Gamma-I)T_M(\chi_2q_M(h)/2)$ through the bounded stress-frame inverse.
For each primary patch we choose its complex version $\cB_M^{\C}$
on the whole modulation patch before sampling.

\begin{proposition}[Preservation of amplitudes]\label{main:prop:envelopes}
Fix one primary patch, its compatible response $\cB_M^{\mathbb C}$,
source cutoff $\chi_{\mathrm{src}}$ and observation cutoff
$\chi_{\mathrm{core}}$. Their supports lie in the common buffered
chart, with the source and observation supports in the identity
region. Let $\phi_M$ be the characteristic phase with the fixed
transverse Cauchy data. Assume the profiles of
Proposition~\ref{main:prop:response} and, for fixed $\nu_0>0$,
\begin{equation}\label{main:eq:response-window}
 L_M\leq K,\qquad
 2\leq K\leq\lambda^{1-\nu_0}.
\end{equation}
Let $a$ be a complex symmetric-tensor envelope on $\T$ with
\begin{equation}\label{main:eq:response-source}
 \|a\|_{H^p}\leq C_p A K^p\quad(p\geq0),\qquad A\geq0.
\end{equation}
For the same source and response at all orders, put
\[
 h=\chi_{\mathrm{src}}e^{i\lambda\phi_M}a,\qquad
 Y=\cB_M^{\mathbb C}h,\qquad
 Y_0=e^{-i\lambda\phi_M}\chi_{\mathrm{core}}Y,\quad
 Y_1=e^{-i\lambda\phi_M}\chi_{\mathrm{core}}dY.
\]
The full ambient bounds are
\begin{align}
 \|Y_0\|_{H^s}&\leq C_s A\lambda K^s,\label{main:eq:response-displacement}\\
 \|Y_1\|_{H^s}&\leq C_s A\lambda^2 K^s.\label{main:eq:response-derivative}
\end{align}
The carrier factors $\lambda,\lambda^2$ are independent of $s,Q$.
On $D_{\mathrm{lag}}$,
\begin{equation}\label{main:eq:response-leakage}
 \|D\gamma_MY-h\|_{H^s(D_{\mathrm{lag}})}
 \leq C_{s,Q}A\lambda^{-Q}.
\end{equation}
The constants may use finitely many source profile constants;
the losses, operator, fields and
domains do not change with $s,Q$. Exact-band data use
$A=\|a\|_2$.

For each fixed $S$ and $\nu_0>0$, there is a finite jet order
$m=m(S,\nu_0)$ for which the two amplitude bounds also hold for
$0\leq s\leq S$ and $1\leq K\leq\lambda^{1-\nu_0}$:
it is enough that the map and coefficient jets through $m$ are
uniformly bounded and \eqref{main:eq:response-source} holds through $m$.
This finite-order assertion does not require the first window
in \eqref{main:eq:response-window}; it includes $K=1$.
\end{proposition}

Theorem~\appref{app:ch_response:thm:response-compatible-choice}
fixes this whole-patch response. The literal graph estimates and
finite expansion of its exact inverse prove
Proposition~\appref{app:ch_response:prop:response-graded}, including
the full tensor error \eqref{main:eq:response-leakage}.
These statements include the actual normal
completion, endpoints, cutoffs and remaining exact inverse.
Corollary~\appref{app:ch_response:cor:finite-jets} proves the
finite-order assertion by the same graph factors and the complete
residual's negative-to-positive Sobolev gain.

Choose opposite sectors by conjugation. Their principal
symbols satisfy $h_-(x,\xi)=-h_+(x,-\xi)$, so both actual components
use the same canonical time and lower integral. The factor $1/i$
then gives conjugate physical responses. Literal adjoints use upper
integrals with the input and output weights exchanged.
Proposition~\appref{app:ch_regular_base:prop:wavefront} proves the full
pairing, including all matrix components and endpoints.

\section{Characteristic waves and the mean equation}\label{main:sec:waves}
\subsection{A separated finite family}
Let $R$ be large and write
\begin{equation}\label{main:eq:basic-scales}
 \lambda=R^a,\qquad h=R^{-\beta},\qquad H=R^{1+\theta},
 \qquad \beta=(1+\epsilon)/3,
\end{equation}
where $0<\theta<\epsilon<1/8$ and $1+\theta+\beta<a<3/2$.
The three-parameter patches in Proposition~\ref{main:prop:patches} admit a
separated finite sample with
\begin{equation}\label{main:eq:sampling}
 M\leq CR^{1+\epsilon},\qquad
 \mu_\nu>0,\qquad \sum_\nu\mu_\nu\leq C.
\end{equation}
Convolve $q_{\rm quad}=2n-1$ unit-interval indicators with a fixed
nonnegative smooth function of integral one. The resulting smooth
window forms a positive partition under integer translations. In each
weighted block use an independently translated lattice of spacing
$h$. Coloring block indices modulo $q_{\rm quad}+1$ gives a fixed
finite number of colors and same-color separation $ch$.
The convolution makes each translated rule exact on polynomials
of degree below $q_{\rm quad}$. Taylor's formula therefore gives
Hermitian quadrature error $C h^{q_{\rm quad}}$ in $L^2$ on the
fixed observation support. Its required carrier jets are already
within the uniform low amplitude budget.
Fix the colors and radial labels, prepare the initial scalars as
below, and then choose the translations. One operator event controls
every dyadic input prefix; all later bands use this realization.

For the current immersion $F$ and one characteristic phase $\phi_F$,
the primary rule uses two amplitudes.
The leading amplitude $b_0$ solves $M_F(x,d\phi_F)b_0=0$ and
$l_F^T\cP_Fb_0=0$, where $l_F$ spans the left kernel of $M_F$.
Thus $\cP_Fb_0$ lies in its image. Choose $b_1$ algebraically by
$M_Fb_1=i\cP_Fb_0$, with $b_1$ perpendicular to the right kernel.
The first-order product rule gives the exact identity
\[
 \cP_F\bigl(\tau^2e^{i\phi_F/\tau}(b_0+\tau b_1)\bigr)
       =\tau^3e^{i\phi_F/\tau}\cP_Fb_1.
\]
Reconstructing the ambient field from this covector gives the
primary family, fixed before accuracy, with
$c\lambda\leq\lambda_\nu\leq C\lambda$ and phases
$\phi_\nu=\phi_{F,\nu}$. Write $L_{\rm WKB}$ for its common amplitude
scale. The actual atom has derivative amplitude
$A_\tau=e^{-i\phi_F/\tau}d\mathcal W_\tau^{\rm pr}$ satisfying
\[
 \|D_x^rD_{z,\zeta}^kA_\tau\|_\infty
       \leq C_{r,k}L_{\rm WKB}^{r}
       \quad(r\geq0,\ k\leq r_{\rm par}),\qquad r_{\rm par}=20(n+1)+3.
\]
All norms use the fixed observation supports of Section~\ref{main:sec:linear}.
The complete atomic source satisfies
\[
 \|\chi_{\rm src}e^{-i\lambda_\nu\phi_\nu}
                  D\gamma_FW_\nu(\delta_0)\|_{H^r}
       \leq C_r\lambda^{-3}L_{\rm WKB}^r.
\]
For a complex Fourier measure $m$ of variation $M_m$, supported in
$B_{C\Lambda}$, let $W_\nu(m)$ be the integral of these same atoms at
$\zeta_\nu+\lambda_\nu^{-1}\xi$ in the fixed phase buffer. If
$K\geq C\max(\Lambda,L_{\rm WKB})$, the fixed observation norms obey
\[
 \lambda_\nu\|e^{-i\lambda_\nu\phi_\nu}W_\nu(m)\|_{H^r}
 +\|e^{-i\lambda_\nu\phi_\nu}dW_\nu(m)\|_{H^r}
       \leq C_rM_mK^r.
\]
At any prescribed finite order, sufficiently many uniformly bounded
map jets give these same estimates for $K\geq1$ and $\Lambda\leq K/C$,
including the zero atom at $K=1$.
Write $W_0(m_\nu)=W_\nu(m_\nu)$. The background is the zero atom.
If $\nu_{b_0}$ denotes its transported leading normal amplitude,
its Hermitian tensor satisfies
\[
 \|\operatorname{Re}[(dW_\nu^{\rm bg})^*dW_\nu^{\rm bg}]
       -|\nu_{b_0}|^2d\phi_\nu^2\|_{C^r}
       \leq C_r\lambda^{-1}L_{\rm WKB}^{r}.
\]
Theorem~\appref{app:ch_waves:thm:wkb} proves the atom estimates;
integrating the atoms gives the measure bounds on the common supports.
The leading transported tensor and
its positive initial-density dependence are unchanged.

Appendix~\appref{app:sec:ch_waves} proves the construction and separated
sampling, using operator concentration as in \cited{MinskerBernstein}.
The physical mean and normal estimates hold for one sample realization,
so they concern the same tensor.

\subsection{One response for the finite primary strain}
Fix an order $r_*$ covering the geometric, response and finite
density-preparation constructions. The current immersion satisfies
\begin{equation}\label{main:eq:lag-hypothesis}
 \|F\|_{C^r(W)}\leq C_rT_0^{(r-r_*)_+}\qquad(r\geq0).
\end{equation}
The history argument below gives this profile. All phases and
responses belong to this same immersion.

Each primary wave is linear in a finite complex Fourier measure
\begin{equation}\label{main:eq:measures}
 m_\nu=\delta_{\{0\}}+\widehat f_{f,\nu}(\xi)\,d\xi.
\end{equation}
The atom gives the prepared positive background; the Fourier datum
adjusts its sampled mean. For a component supported in $|\xi|\leq\Lambda$,
including the zero atom, use an envelope scale $K$ in
\eqref{main:eq:response-window}, with
\begin{equation}\label{main:eq:modulation-window}
 T_0\leq K,\qquad
 \Lambda\leq K/C_1,\qquad \Lambda/\lambda_\nu\leq\epsilon_0.
\end{equation}
Here $C_1,\epsilon_0$ fix the modulation
chart. Use $K=C_{\rm env}T_0$ for the zero atom and a fixed larger
multiple of its input band for each mean datum. For $W_0(m_\nu)$ set
\begin{equation}\label{main:eq:lag-recurrence}
 H_\nu=\chi_{\mathrm{src}}D\gamma_FW_0(m_\nu),\qquad
 Z_\nu=W_0(m_\nu)-\cB_F^{\C}H_\nu.
\end{equation}
The response is fixed on the entire primary patch before sampling
and is linear in this complete measure.

\begin{lemma}[Current strain identity]\label{main:lem:telescope}
Put $E_\nu=D\gamma_F\cB_F^{\C}H_\nu-H_\nu$.
On $D_{\mathrm{lag}}$,
\begin{equation}\label{main:eq:telescope}
 D\gamma_FZ_\nu=-E_\nu.
\end{equation}
Consequently, for every finite $Q$,
\begin{equation}\label{main:eq:lag-strain}
 \|D\gamma_FZ_\nu\|_{C^0(D_{\mathrm{lag}})}
       \leq C_Q\|m_\nu\|_{\TV}\lambda^{-Q}.
\end{equation}
For the observation cutoff of Proposition~\ref{main:prop:envelopes},
\begin{equation}\label{main:eq:lag-full-envelope}
 \begin{split}
 &\lambda_\nu\|\chi_{\rm core}e^{-i\lambda_\nu\phi_{F,\nu}}Z_\nu\|_{C^r}
 +\|\chi_{\rm core}e^{-i\lambda_\nu\phi_{F,\nu}}dZ_\nu\|_{C^r}\\
 &\hspace{20mm}\leq C_r\|m_\nu\|_{\TV}
                           K^r.
 \end{split}
\end{equation}
For $Z_\nu-W_0(m_\nu)$ the right side has the additional factor
$\lambda_\nu^{-1}$.
\end{lemma}
\begin{proof}
The source equals $D\gamma_FW_0$ on $D_{\rm lag}$, giving the identity.
The fixed source cutoff and response commute with integration
against $m_\nu$. The corrected atoms of
Corollary~\appref{app:ch_response:cor:current-atoms} have arbitrary
full-strain precision on this common domain. Integrating their
bounds proves \eqref{main:eq:lag-strain}.
Proposition~\appref{app:ch_lag:prop:lag-envelopes} controls the
central phase quotient and gives both $C^r$ bounds in the same way.
\end{proof}

The fixed region $D_{\mathrm{lag}}$ contains every mean and product
measurement support used below. At fixed joint physical and carrier
orders, the normalized atomic response is $O(\lambda^{-1})$.
Corollary~\appref{app:ch_response:cor:current-atoms} proves these
bounds on the full observation support. Thus the corrected atoms
have the original phase and leading tensor, and satisfy the same
sampling and raw-input estimates as the primary family.

\subsection{The transported positive background}
The continuum family of characteristic waves has a positive normal
operator after the image channel is added. Its inverse
constructs one positive transported background and a response for
changes in the mean, on a fixed neighborhood of
Proposition~\ref{main:prop:seed}. Section~\ref{main:sec:mean-details}
states its full-output bounds and explains how the same row is
compared with one finite sample.

\begin{proposition}[Current mean]\label{main:prop:mean}
Let $e=g-\gamma(F)$, with $\|e\|_{C^0(D)}\leq\delta^2$, and put
\[
 e_R=S_RE(e|_D),\qquad b=\delta R^A,
\]
where $E,S_R$ are the fixed extension and smoothing operators of
Lemma~\ref{main:lem:smoothing}. For every finite $Q$ and the scales in
Section~\ref{main:sec:scales}, the mean construction chooses the measures
\eqref{main:eq:measures} and a real ambient field $V=V_c+V_f$ such that,
for the same $Z_\nu$ as in \eqref{main:eq:lag-recurrence},
\begin{equation}\label{main:eq:mean-equation}
 E_{\mathrm{mean}}=D\gamma_FV
   +\sum_\nu\mu_\nu\Re\bigl((dZ_\nu)^*dZ_\nu\bigr)
   -e_R/b^2
\end{equation}
satisfies $\|E_{\mathrm{mean}}\|_{C^0(D)}\leq C_QR^{-Q}$. Moreover,
\[
 \sup_\nu\|m_\nu\|_{\TV}\leq C.
\]
The initial Fourier inputs satisfy
\begin{equation}\label{main:eq:mean-input-supports}
 \supp\widehat f_{f,\nu}\subset\{|\xi|\leq CK_{\max}\},
 \qquad K_{\max}=R^{s_f},
\end{equation}
with a support constant independent of $Q$.
On the full product source support, the same waves satisfy
\begin{equation}\label{main:eq:mean-final-amplitude}
 \sup_\nu\|\chi_{\mathrm{pair}}
       e^{-i\lambda_\nu\phi_{F,\nu}}dZ_\nu\|_{C^r}
                  \leq C_rK_{\max}^r.
\end{equation}
After the fixed physical cutoff, equal to one near $D$, the fields
satisfy, for every $r\geq0$,
\begin{align}
 \|V_c\|_{C^r(\T)}+\|V_c\|_{H^r}
 &\leq C_rT_0^r,\label{main:eq:mean-coarse-size}\\
 \|V_f\|_{C^r(\T)}+\|V_f\|_{H^r}
 &\leq C_rR^{-2A}K_{\max}^{r+n/2},
       \label{main:eq:mean-fine-size}
\end{align}
Here $p=(n+1)/2$ and $1<s_f<a$.
The background preparation is fixed after $A$. The displayed field
constants are uniform once the construction threshold for $Q$ is met;
$Q$ changes the finite mean updates and that threshold.
\end{proposition}

Section~\ref{main:sec:mean-details} proves this proposition from the
prepared background and fixed canonical estimates in
Appendix~\appref{app:sec:ch_mean_pairs}. Its residual identities and
normal completion are given in the main text.

\subsection{Oscillatory quadratic products}
The normalization can be seen with a single wave. If
$dZ=e^{\ii\vartheta}A$, then
\begin{equation}\label{main:eq:single-wave-mean}
 \bigl(d(\sqrt2\Re Z)\bigr)^Td(\sqrt2\Re Z)
       =\Re(A^*A)+\Re(e^{2\ii\vartheta}A^TA).
\end{equation}
The first term is the Hermitian self tensor; the second has doubled
phase. We call the first the \emph{mean} to identify this algebraic
part of the oscillatory expansion. It is not defined by integrating
over the spatial domain. With several waves there are also sums and
differences of distinct phases. These are the products canceled by
the current response.

Define the real primary field
\begin{equation}\label{main:eq:primary}
 P=b\sqrt2\sum_\nu\sqrt{\mu_\nu}\Re Z_\nu.
\end{equation}
Its Hermitian tensor and remaining quadratic tensor are
\begin{equation}\label{main:eq:self-off}
 H_{\mathrm{self}}=b^2\sum_\nu\mu_\nu
       \Re\bigl((dZ_\nu)^*dZ_\nu\bigr),\qquad
 Q_{\mathrm{off}}=(dP)^TdP-H_{\mathrm{self}}.
\end{equation}
The first is the tensor in Proposition~\ref{main:prop:mean}. The second
contains all doubled phases and all signed products of distinct
samples, including products between different patches.

\begin{proposition}[Product correction]\label{main:prop:pairs}
Write $dZ_\nu=e^{\ii\lambda_\nu\phi_\nu}A_\nu$ on the full product
source support. Suppose the amplitudes satisfy
\[
 \|\chi_{\mathrm{pair}}A_\nu\|_{C^r}
       \leq C_{Q,r}K_{\mathrm{env}}^r
\]
and let
\[
 L_{\mathrm{pair}}=\max\{K_{\mathrm{env}},T_0\}.
\]
If, for a fixed $\sigma>0$,
\begin{equation}\label{main:eq:pair-window}
 L_{\mathrm{pair}}\leq(\lambda h)^{1-\sigma},
\end{equation}
there is a real current correction $S$, formed from the exact signed
products in $Q_{\mathrm{off}}$. After its fixed physical cutoff,
equal to one near $D$, the same field satisfies
\begin{align}
 \|D\gamma_FS+Q_{\mathrm{off}}\|_{C^0(D)}
    &\leq C_{Q,q}b^2M(\lambda h)^{-q},
        \label{main:eq:pair-error}\\
 \|S\|_{H^r}&\leq C_{Q,r}b^2M\lambda^{r+1}
       \qquad(r\geq0).
       \label{main:eq:pair-high-size}
\end{align}
The outside loss is $1$, independently of $r,q,Q$.
\end{proposition}

Apply the current response along the full characteristic segments in
Proposition~\ref{main:prop:patches}, using the complete amplitudes of
the corrected $Z_\nu$. Each integration gains
$L_{\mathrm{pair}}/\varrho$, with $\varrho\geq c\lambda h$;
the strict power in \eqref{main:eq:pair-window} gives arbitrary
finite decay. The full proof is in
Proposition~\appref{app:ch_pairs:prop:pairs-lag-sum}.

\section{The normal estimate and the nonlinear mean}\label{main:sec:mean-details}
We prove Proposition~\ref{main:prop:mean} through three arguments:
the continuum row has a right inverse, a finite density correction
prepares the background, and one recurrence adjusts its sampled mean.
The complete operator estimates and source calculations are proved in
Appendices~\appref{app:sec:ch_background}
and~\appref{app:sec:ch_mean_pairs}.

\subsection{The continuum inverse and positive background}
The transport channels $T_a$ are formed from the leading amplitudes on
the fixed characteristic patches. Their full normal operators have
order zero and homogeneous diagonal symbols
\begin{equation}\label{main:eq:transport-symbol}
 d_{0,a}(x,k)=\int_v w_a(x,v,k)
 q_F(\xi_a\otimes\xi_a)q_F(\xi_a\otimes\xi_a)^T\,dv,
\end{equation}
with Fourier normalization included in $w_a$ and positive weights
near the selected incidence roots. The real
channel $2\Re T_a$ uses both quadratures, so its real normal is
\begin{equation}\label{main:eq:real-normal}
 4\Re(T_aT_a^*).
\end{equation}
Its complexified principal symbol is
$2[d_{0,a}(x,k)+\overline{d_{0,a}(x,-k)}]$; the present $d_{0,a}$
is real symmetric, and no evenness is assumed. The incidence
change of variables and every lower-order term are proved in
Appendix~\appref{app:sec:ch_background}.

For every unit output direction $\widehat k$, the selected carriers
have interior incidence roots with
\begin{equation}\label{main:eq:detector-gram}
 \begin{gathered}
 \widehat k\cdot\partial_\xi p_F(x,\xi_a)=0,\qquad p_F=\det M_F,\\
 |M_F(x,\widehat k)^Tw|^2+
 \sum_a|w\cdot q_F(\xi_a\otimes\xi_a)|^2\geq c|w|^2
 \quad(w\in\R^n).
 \end{gathered}
\end{equation}
Here each $\xi_a$ is evaluated at its local root. On the inner
measurement region, the positive weights near these roots make
\eqref{main:eq:transport-symbol} complete the image channel.
For the positive scalars constructed below, adding a fixed exterior
channel gives a row $A_F$ on the full real output $Y=L^2(\T;\R^n)$ with
\[
 E_0=A_FA_F^*\geq cI,\qquad Q_0=E_0^{-1}.
\]
For $X^r$, the sum of the raw auxiliary $H^r$ spaces and carrier
$H^r_{z,y}$ spaces, its one inverse satisfies
\begin{equation}\label{main:eq:additive-inverse}
 \|Q_0f\|_{H^r}+\|A_F^*Q_0f\|_{X^r}
 \leq C_r\bigl[\|f\|_{H^r}
              +(1+\|F\|_{C^{r+B}(W)})\|f\|_2\bigr].
\end{equation}
The physical auxiliary realization has the same source order and
a fixed coefficient shift. These are the full-output statements of
Propositions~\appref{app:ch_background:prop:bg-frame},
\appref{app:ch_background:prop:bg-adjoint}
and~\appref{app:ch_background:prop:bg-mixed-physical}.
For the local gap, freeze the positive principal matrix at the
center of $B_r$. Its Fourier multiplier is positive; the coefficient
difference is $\sum_i x_iT_i$ with bounded order-zero $T_i$.
The complete lower-order error costs
$C\|f\|_{H^{-1}}\|f\|_2\leq Cr\|f\|_2^2$ for $f$ supported
in $B_r$. A small fixed $r$ absorbs both errors. The exterior
channel gives the full-output gap and vanishes on the mean core.

The same construction gives positive initial scalars $r_{F,a}$ and
a tangent field $V_{\rm base}=dFX_F$ on a fixed $U_{\rm fin}$ containing the mean
output support, such that
\begin{equation}\label{main:eq:positive-background}
 q_F\bigl(D\gamma_FV_{\rm base}+\mathcal H_F[r_F^2]\bigr)=0.
\end{equation}
Here $\mathcal H_F[r_F^2]$ is the continuum Hermitian tensor of the
leading amplitudes. The scalars are bounded below on the working carrier
patches, and, for one fixed $B$,
\begin{equation}\label{main:eq:positive-profile}
 \sum_a\|r_{F,a}\|_{C^r}+
 \|V_{\rm base}\|_{C^r(U_{\rm fin})}
 +\|D\gamma_FV_{\rm base}\|_{C^r(U_{\rm fin})}
 \leq C_r(1+\|F\|_{C^{r+B}(W)}).
\end{equation}
Propositions~\appref{app:ch_background:prop:bg-normalized}
and~\appref{app:ch_background:prop:bg-current} prove this by one
affine density correction at the fixed positive reference dilation.
The row $A_F$ and inverse $Q_0$ below use these corrected scalars;
their gaps and shifts are independent of accuracy.

\subsection{Preparation before sampling}
Write $A_F=(A_0,\mathcal T_0)$ for the auxiliary and carrier rows.
Proposition~\appref{app:ch_mean_pairs:prop:mean-coarse}
chooses positive initial scalars $r_{F,R}$ and a real field $U_c$
before sampling. The current zero atoms $Z_{c,\nu}$ have self tensor
$\mathcal H_c=\sum_\nu\mu_\nu\Re[(dZ_{c,\nu})^*dZ_{c,\nu}]$.
For $C_c=D\gamma_FU_c+\mathcal H_c$, it gives a full auxiliary residual
\begin{equation}\label{main:eq:prepared-residual}
 r_c=q_FC_c\text{ on }D,\qquad
 \|r_c\|_2\leq R^{-2A},\qquad \|r_c\|_{H^r}\leq C_rT_0^r.
\end{equation}
On $U_{\rm fin}$, the $C^r$ norms of $U_c$, its full strain,
$\mathcal H_c$ and $C_c$ are at most $C_rT_0^r$.
The scalar change is $O(\lambda^{-1})$ in any prescribed fixed
joint topology and has the same graded profile as the original scalar.
All required finite orders are included in $r_*$ before mean accuracy.

The proof uses the affine dependence of the leading mean on the
initial density. Write $r(v)=r_F\sqrt{1+v}$ and let
$\mathcal E_\lambda(v)$ be the measured difference between the
colored current Hermitian mean and its leading transported mean.
Linearity of each atomic wave in $r$ gives, at any fixed order,
\[
 \|\mathcal E_\lambda(v)-\mathcal E_\lambda(w)\|_{H^s}
 \leq C_s\lambda^{-1}\|v-w\|_{C^{m_s}},
 \qquad \mathcal E_\lambda(0)=O(\lambda^{-1}),
\]
with a finite input order $m_s$. Apply the original $A_F^*Q_0$
to the full residual $e_i$. The uncut density synthesis and
auxiliary field change its leading part by $-A_FA_F^*Q_0e_i=-e_i$.
Consequently the exact new residual is
\[
 e_{i+1}=\mathcal E_\lambda(v_{i+1})-\mathcal E_\lambda(v_i).
\]
For fixed $J$, choosing the finite input orders backwards gives
$e_J=O(\lambda^{-J-1})$ and $v_J=O(\lambda^{-1})$.
Take the fixed count
\begin{equation}\label{main:eq:preparation-orders}
 J=\lfloor2A/a\rfloor,\qquad
 a(J+1)>2A,\qquad \beta(2n-1)>2A.
\end{equation}
The lattice estimate then gives \eqref{main:eq:prepared-residual}.
Fixed derivative shifts and interpolation give its high profiles.

For raw auxiliary cutoff $\pi_K$ and carrier prefix $P_K$, put
\[
 \mathsf P_K=\operatorname{diag}(\pi_K,P_K),\qquad
 E_{0,K}=A_F\mathsf P_K^2A_F^*,\qquad
 \mathcal F_{0,K}=A_0\pi_K^2A_0^*.
\]
The same event compares every current prefix normal $E_K$ with
the original positive row:
\begin{equation}\label{main:eq:canonical-range}
 \sup_{K\leq H}\|E_K-E_{0,K}\|_{2\to2}
 \leq C\left[R^{-(\epsilon-\theta)/4}
                    +R^{-a+1+\theta+\beta}\right].
\end{equation}
The phase is unchanged; the prepared scalar changes each leading
normal by only $C\lambda^{-1}$, uniformly in $K$. Thus $Q_0$
still belongs to $r_F$. This is a statement on the entire real output,
including the identical exterior and auxiliary components.

\subsection{One fixed row and one mean recurrence}
Define the source directly on the torus:
\begin{equation}\label{main:eq:actual-fine-source}
 e_f=\mathcal C(e_R/b^2)-r_c,\qquad \mathcal C=\chi_{\rm meas}q_F.
\end{equation}
On $D$ it equals $q_F(e_R/b^2-C_c)$. For
$m>\max\{r,2A/(1-t)\}$, Fourier H\"older gives
\[
 \|r_c\|_{H^r}\leq C_r\|r_c\|_2^{1-r/m}\|r_c\|_{H^m}^{r/m}
                  \leq C_rR^{-2A+r}.
\]
The smoothed target therefore gives
\begin{equation}\label{main:eq:fine-source-profile}
 \|e_f\|_{H^r}\leq C_r\epsilon_fR^r,\qquad \epsilon_f=CR^{-2A}.
\end{equation}
All coefficient and zero-atom profiles have scale $T_0<R$.

\begin{proposition}[Canonical changes]\label{main:prop:canonical}
For $T_0\leq K\leq H$ and $r\in L^2$, fix
$B_\nu=e^{-i\lambda_\nu\phi_\nu}dZ_{c,\nu}$. For raw input
$v$, let $G_{\nu,K}v$ be the demodulated derivative of its current
wave and define
\[
 A_{\nu,K}v=\mathcal C[2\sym\Re(B_\nu^*G_{\nu,K}v)],\qquad
 A_K=(\sqrt{\mu_\nu}A_{\nu,K})_\nu,\qquad
 E_K=\mathcal F_{0,K}+A_KA_K^*.
\]
The input map $B_{\nu,K}^{\rm in}$ contains the detector
$\alpha_\nu\chi_{\rm hi}$ and the dyadic half-derivative sum before
spatial cutoffs; the zero atom is uncut. For $q=Q_0r$ take
\begin{equation}\label{main:eq:old-canonical}
 f_{\nu,K}(r)=B_{\nu,K}^{\rm in}A_{\nu,K}^*q.
\end{equation}
Both real quadratures use the original raw Hilbert spaces. If
$D_{\nu,K}(r)$ is the corresponding current derivative change, then
\begin{equation}\label{main:eq:canonical-identity}
 \mathcal C\sum_\nu\mu_\nu
  2\sym\Re(B_\nu^*D_{\nu,K}(r))
                    =(E_K-\mathcal F_{0,K})Q_0r.
\end{equation}
Every update uses this fixed zero-atom row.

Write $\|D\|_{\mu,X}^2=\sum_\nu\mu_\nu\|D_\nu\|_X^2$.
Use the fixed observation cutoff in all
derivative norms below; data norms use the initial variables.
Then
\[
 \|D_K(r)\|_{\mu,H^s}\leq C_sK^{s+1/2}\|r\|_2,
 \qquad
 \|D_K(r)\|_{\mu,C^s}\leq C_sK^{s+n/2}\|r\|_2.
\]
Individually,
\[
 \|f_{\nu,K}(r)\|_{C^s}+\|D_{\nu,K}(r)\|_{C^s}
       \leq C_sK^{s+p}\|r\|_2,
\]
where $p=(n+1)/2$. The datum has Fourier support below $CK$ and
Fourier $L^1$ norm at most $CK^p\|r\|_2$. The raw auxiliary field
has, after the fixed physical cutoff,
\begin{equation}\label{main:eq:auxiliary-band-profile}
 \|V_{\rm aux,K}^{F}(r)\|_{H^s}
 +\|D\gamma_FV_{\rm aux,K}^{F}(r)\|_{H^s}
 +\|\mathcal F_{0,K}Q_0r\|_{H^s}\leq C_sK^s\|r\|_2.
\end{equation}
The last norm includes the exterior output.
\end{proposition}
\begin{proof}
Definition~\appref{app:ch_mean_pairs:defn:mean-canonical} constructs
these exact adjoints and raw maps. The comparison
\eqref{main:eq:canonical-range} bounds $A_K$.
Square the raw $H^s$ and $C^s$ bounds before summing; only the
individual estimates use $\|A_{\nu,K}\|\leq C\sqrt K$.
Proposition~\appref{app:ch_background:prop:bg-mixed-physical}
gives the full auxiliary bounds.
\end{proof}

Put $K_{\max}=H^{d_f}$, with $1/(1+\theta)<d_f<1$.
The fixed choices \eqref{main:eq:canonical-range} and
\eqref{main:eq:A-choice} give $\nu,\nu_{\rm nl}>0$ such that
\[
 \|E_K-E_{0,K}\|\leq CH^{-\nu},\qquad
 \epsilon_fK_{\max}^{p}\leq H^{-\nu_{\rm nl}}.
\]
\begin{lemma}[A finite mean with a fixed row]\label{main:lem:mean-recurrence}
For source \eqref{main:eq:fine-source-profile} and every finite $Q$,
finitely many updates of Proposition~\ref{main:prop:canonical} below
$K_{\max}$ have full residual $\|r_{\rm fin}\|_2\leq\epsilon_fH^{-Q}$.
Their source residuals have sum at most $2\epsilon_f$ in $L^2$;
the accumulated data and derivative changes have $C^s$ norm at
most $C_s\epsilon_fK_{\max}^{s+p}$. In the weighted $C^s$ norm,
the derivative changes are bounded by $C_s\epsilon_fK_{\max}^{s+n/2}$.
The data have Fourier variation
at most $C\epsilon_fK_{\max}^p$ and support below $CK_{\max}$.
The auxiliary field and full strain have $H^s$ norm at most
$C_s\epsilon_fK_{\max}^s$. These constants are independent of $Q$
once its construction threshold is met. Any prescribed finite
$C^r$ residual accuracy follows by increasing the $L^2$ precision.
\end{lemma}
\begin{proof}
Fix $0<\gamma<\min(\nu,\nu_{\rm nl})$. For precision $Q$, set
$I=\lfloor Q/\gamma\rfloor+1$ and put $d_{\rm in}=1/(1+\theta)$,
\[
 \Delta=\frac{d_f-d_{\rm in}}{I+1},\qquad
 d_l=d_{\rm in}+(l+1)\Delta,\qquad
 K_l=2^{\lceil\log_2H^{d_l}\rceil},\quad0\leq l<I.
\]
At a large threshold, $H^{d_l}\leq K_l\leq2H^{d_l}<K_{\max}$. At step
$l$, use \eqref{main:eq:old-canonical} for $r_l$ and add the
auxiliary input $\pi_{K_l}^2A_0^*Q_0r_l$. Write $D_l$ for its
derivative change, $S_l=\sum_{i<l}D_i$ and
$\mathcal N(S)=\mathcal C\sum_\nu\mu_\nu\Re(S_\nu^*S_\nu)$.
The exact full-output recurrence is
\begin{equation}\label{main:eq:full-recursion}
 r_{l+1}=(I-E_{K_l}Q_0)r_l
              -[\mathcal N(S_l+D_l)-\mathcal N(S_l)],\qquad r_0=e_f.
\end{equation}
Its linear error is
\begin{equation}\label{main:eq:linear-two-errors}
 (E_{0,K_l}-E_{K_l})Q_0r_l
           +A_F(I-\mathsf P_{K_l}^2)A_F^*Q_0r_l.
\end{equation}
Only raw inputs are cut off.

Inductively suppose $a_i:=\|r_i\|_2\leq\epsilon_fH^{-\gamma i}$
for $i\leq l$. Then $\sum_{i\leq l}a_i\leq2\epsilon_f$.
The weighted $C^0$ and $L^2$ bounds give
\[
 \|\mathcal N(S_l+D_l)-\mathcal N(S_l)\|_2
 \leq C\epsilon_fK_{\max}^{p}a_l.
\]
Leibniz and weighted Cauchy--Schwarz give, for $i\leq k$,
\[
 \left\|\mathcal C\sum_\nu\mu_\nu
        \Re(D_{i,\nu}^*D_{k,\nu})\right\|_{H^s}
 \leq C_s a_i a_k K_i^{n/2}K_k^{s+1/2}.
\]
Use the lower-band factor in weighted $C^r$ and the higher one in
weighted $H^{s-r}$; coefficient derivatives cost $T_0\leq K_i$.
The cumulative identity is
\[
 r_l=e_f-\sum_{i<l}\mathcal F_{0,K_i}Q_0r_i
   -\mathcal C\sum_\nu\mu_\nu
       [2\sym\Re(B_\nu^*S_{l,\nu})+\Re(S_{l,\nu}^*S_{l,\nu})].
\]
Since $\sum_i a_iK_i^{n/2}\leq2\epsilon_fK_{\max}^{n/2}\leq C$,
its linear and quadratic terms give
\[
 \|r_l\|_{H^s}\leq C_s\epsilon_f
   \left[R^s+\sum_{i<l}H^{-\gamma i}K_i^{s+1/2}\right].
\]

Choose one finite order $S$ with
$\Delta S>d_f/2+2\gamma+1$. Equation~\eqref{main:eq:additive-inverse}
gives
$\|A_F^*Q_0r_l\|_{X^S}\leq C_S(\|r_l\|_{H^S}+T_0^S a_l)$.
Thus the raw tail in \eqref{main:eq:linear-two-errors} is at most
\[
 C_S\epsilon_f\left[
 H^{-(l+1)\Delta S}
 +\sum_{i<l}H^{-\gamma i-(l-i)\Delta S+d_i/2}\right]
 \leq C_S\epsilon_fH^{-\gamma(l+1)-1}.
\]
For $l\geq1$ the geometric sum is bounded by a constant times its
last term, whose exponent is
$-\gamma(l-1)-\Delta S+d_{l-1}/2$; for $l=0$ the first term suffices.
The normal and nonlinear errors are at most
$C(H^{-\nu}+H^{-\nu_{\rm nl}})a_l$. At a large threshold these
and the tail give $a_{l+1}\leq\epsilon_fH^{-\gamma(l+1)}$,
closing the induction.

Consequently $\|r_I\|_2\leq\epsilon_fH^{-Q}$ and
$\sum_i a_i\leq2\epsilon_f$. The raw bounds prove all accumulated
data, variation and auxiliary estimates, with the stated uniform
constants. The high profile is at most
$C_s\epsilon_fK_{\max}^{s+1/2}$; interpolation gives each fixed
$C^r$ residual accuracy after increasing $Q$ linearly.
Choose the least such $S$; both $I$ and $S$ are $O(Q+1)$.
\end{proof}

\subsection{One normal completion}
Apply Lemma~\ref{main:lem:mean-recurrence} to $e_f$.
Sum the auxiliary fields as $V_{\rm aux}$ and let $\mathcal H$
be the final Hermitian tensor. On $U_{\rm fin}$ put
\[
 T_f=e_R/b^2-D\gamma_F(U_c+V_{\rm aux})-\mathcal H,
 \qquad \mathsf N_F=(B_F^*)^{-1},\quad
 \Pi_F^\perp=\Pi_{(\ker B_F)^\perp}.
\]
Define
\begin{equation}\label{main:eq:mean-normal-field}
 V=U_c+V_{\rm aux}-\tfrac12\mathsf N_F\Pi_F^\perp T_f.
\end{equation}
On $D$, $q_FT_f=r_I$. The metric error is
$-\Pi_{\ker B_F}T_f$, so the Gram formula proves
\eqref{main:eq:mean-equation}.

For the norms, split this same field as
\[
 \begin{aligned}
 V_c&=U_c+\tfrac12\mathsf N_F\Pi_F^\perp C_c,\\
 V_f&=V_{\rm aux}-\tfrac12\mathsf N_F\Pi_F^\perp
       (e_R/b^2-D\gamma_FV_{\rm aux}-(\mathcal H-\mathcal H_c)).
 \end{aligned}
\]
The prepared tensor profile proves \eqref{main:eq:mean-coarse-size}.
The raw auxiliary estimate and the low residual sum give
$\|V_{\rm aux}\|_{H^r}+\|D\gamma_FV_{\rm aux}\|_{H^r}
\leq C_r\epsilon_fK_{\max}^r$.
Splitting the Fourier sum at $K_{\max}$ costs $n/2$ powers.
Weighted Cauchy--Schwarz gives
$\|\mathcal H-\mathcal H_c\|_{C^r}
\leq C_r\epsilon_fK_{\max}^{r+n/2}$, using
$\epsilon_fK_{\max}^{n/2}\leq C$ for its quadratic part.
Since $R<K_{\max}$, these
prove \eqref{main:eq:mean-fine-size} on the full physical domain.
Finally,
\begin{equation}\label{main:eq:variation}
 \sum_l\|\widehat f_{\nu,K_l}\|_1
       \leq C\epsilon_fK_{\max}^p\leq C.
\end{equation}
Adding the unit atom proves the measure bound.
Equation~\eqref{main:eq:lag-full-envelope} gives the final amplitude
bound on the full product support.


\section{The metric correction}\label{main:sec:increment}
\subsection{Smoothing the defect}
We use ordinary extension, Fourier smoothing, and tame product
estimates in a form that keeps the small defect and the derivative
scale separate.

\begin{lemma}\label{main:lem:smoothing}
There is a real linear extension
$E:C^\infty(D)\to C^\infty(\T)$, supported in a fixed compact subset
of $W$, with $Ef=f$ on $D$ and
$\|Ef\|_{C^r}\leq C_r\|f\|_{C^r(D)}$ for every $r$.
Fix a smooth target tensor $g$ on $W$. Let $S_R$, $R\geq1$, be a
family of Fourier multipliers with a fixed real-even smooth cutoff
symbol, equal to one below $R$ and zero above $2R$. Suppose
$T\geq1$, $0<\delta\leq1$, and that
\[
 \|dF\|_{C^r(W)}\leq C_rT^r,\qquad
 \|g-\gamma(F)\|_{C^0(D)}\leq\delta^2.
\]
Put $e=g-\gamma(F)$. Then $e_R=S_RE(e|_D)$ satisfies
\begin{align}
 \|e_R\|_{C^r}+\|e_R\|_{H^r}&\leq C_r\delta^2R^r,
          \label{main:eq:smoothing-low}\\
 \|e-e_R\|_{C^0(D)}&\leq C\delta^2,
          \label{main:eq:smoothing-small}\\
 \|e-e_R\|_{C^0(D)}&\leq C_N(T/R)^N.
          \label{main:eq:smoothing-high}
\end{align}
The constants depend on the displayed profile constants and finitely
many seminorms of $g$. The same operators work at all orders. If
$T\leq R^t$ with $0<t<1$,
then, for every finite $Q$,
\begin{equation}\label{main:eq:smoothing-combined}
 \|e-e_R\|_{C^0(D)}\leq C_Q\delta R^{-Q}.
\end{equation}
\end{lemma}

\begin{proof}
Take the fixed extension of
Lemma~\appref{app:ch_regular_base:lem:ball-extension} and the Fourier
cutoff from Lemma~\appref{app:lem:elementary-interpolation-jets}.
Their kernel estimates give
\[
 \|S_REf\|_{C^r}+\|S_REf\|_{H^r}\leq C_rR^r\|f\|_{C^0(D)},
 \qquad
 \|(I-S_R)Ef\|_{C^0}\leq C_NR^{-N}\|f\|_{C^N(D)}.
\]
The first estimate and the second at order zero give
\eqref{main:eq:smoothing-low}--\eqref{main:eq:smoothing-small}.
Leibniz and the assumed profile imply
$\|e\|_{C^N(D)}\leq C_N(g)+C_NT^N\leq C_NT^N$,
proving \eqref{main:eq:smoothing-high}.
The geometric mean of the small and high bounds is
$C_N\delta R^{-(1-t)N/2}$. Choose $(1-t)N>2Q$.
\end{proof}

In particular, $e_R/b^2$ has $C^r$ and $H^r$ norms at most
$C_rR^{r-2A}$. The factor $\delta$ in \eqref{main:eq:smoothing-combined} comes
from combining the two bounds, not from the high tail. The defect
$e$ may have either sign.

\subsection{A single increment}
The preceding constructions give the following quantitative step.

\begin{theorem}[Finite-stage correction]\label{main:thm:step}
Fix the reference neighborhood, domains, and response constructions
above. Let $F$ be a current immersion in this neighborhood,
and assume the wave, mean, and product hypotheses of
Section~\ref{main:sec:waves}. Suppose
\[
 \|g-\gamma(F)\|_{C^0(D)}\leq\delta^2,\quad
 T_0=R^t,
\]
with $0<t<1$ and $0<\delta\leq1$, and assume \eqref{main:eq:lag-hypothesis} and
\eqref{main:eq:pair-window}.
Put
\[
 B_0=2A+1+\epsilon+a,\qquad B_3=2B_0+2a(p+1/2).
\]
For every finite $Q$ and sufficiently large $R$, a real global
increment $U$ satisfies
\begin{align}
 \|\gamma(F+U)-g\|_{C^0(D)}
 &\leq C_Q\bigl(\delta R^{-Q}+\delta^3R^{B_3}\bigr),
       \label{main:eq:step-error}\\
 \|U\|_{L^2}&\leq C_Q\delta R^{B_0},\label{main:eq:step-L2}\\
 \supp\widehat U&\subset\{|\xi|\leq CR^{b_*}\},
       \qquad b_*=a(1+\zeta)<3/2.\label{main:eq:step-band}
\end{align}
Consequently the same $U$ obeys
\begin{equation}\label{main:eq:step-Cr}
 \|U\|_{C^r(\T)}\leq C_{Q,r}\delta R^{B_0+b_*(r+n/2)}.
\end{equation}
The outside exponents and support constant are independent of $Q$.
\end{theorem}

\begin{proof}
Choose the sample for the fixed primary family and carry out
the finite mean updates at the requested accuracy. Construct $P,V,S$
as in Section~\ref{main:sec:waves}, and put
\[
 W_2=b^2V+S,\qquad U=\Pi(P+W_2),
\]
where the fields have their accepted physical cutoffs and $\Pi$ is
one real-even Fourier multiplier, equal to one below
$C_1\lambda^{1+\zeta}$ and zero above $C_2\lambda^{1+\zeta}$.
All physical cutoffs equal one near $D$.

First, \eqref{main:eq:sampling} gives
\[
 \sum_\nu\sqrt{\mu_\nu}\leq
       \bigl(M\sum_\nu\mu_\nu\bigr)^{1/2}
       \leq CR^{(1+\epsilon)/2}.
\]
The measure bound in Proposition~\ref{main:prop:mean} and
Lemma~\ref{main:lem:telescope} therefore imply
\begin{equation}\label{main:eq:primary-strain}
 \|D\gamma_FP\|_{C^0(D)}
 \leq C_Q\delta R^{A+(1+\epsilon)/2}\lambda^{-Q_p}.
\end{equation}
Choose $aQ_p>Q+A+(1+\epsilon)/2$. This gives the first term
of \eqref{main:eq:step-error}.

By \eqref{main:eq:self-off}, the remaining quadratic tensor is
exactly $Q_{\mathrm{off}}$. Proposition~\ref{main:prop:pairs},
with sufficiently large internal order, gives
\[
 \|D\gamma_FS+Q_{\mathrm{off}}\|_{C^0(D)}
                                      \leq C_Q\delta R^{-Q}.
\]
Likewise Proposition~\ref{main:prop:mean}, with accuracy greater than
$Q+2A$, gives $b^2\|E_{\mathrm{mean}}\|_{C^0(D)}
\leq C_Q\delta R^{-Q}$.

Substitution into the metric expansion yields the exact identity
\begin{align}
 \gamma(F+P+W_2)-g
 ={}&(e_R-e)+D\gamma_FP+b^2E_{\mathrm{mean}}\notag\\
 &+(D\gamma_FS+Q_{\mathrm{off}})
       +2\sym\bigl((dP)^TdW_2\bigr)+(dW_2)^TdW_2.
          \label{main:eq:metric-ledger}
\end{align}
The physical profiles of these same cut fields give
\begin{equation}\label{main:eq:joint-physical-profile}
 \|P\|_{H^r}\leq C_{Q,r}\delta R^{B_0}\lambda^r,\qquad
 \|W_2\|_{H^r}\leq C_{Q,r}\delta^2R^{B_0}\lambda^r.
\end{equation}
For $P$, use \eqref{main:eq:lag-full-envelope} and the weight sum.
For $W_2$, the pair bound costs $R^{2A+1+\epsilon}\lambda$;
the prepared field costs $R^{2A}$, and the mean update costs
$R^{s_fn/2}\leq R^{2A}$ by \eqref{main:eq:A-choice}.
All derivative slopes are at most $a$.
The same Fourier estimate gives $C^1$ bounds with the factor
$\lambda^{1+n/2}$. Since $\delta\leq1$, these
bound the last two terms in \eqref{main:eq:metric-ledger} by
$C_Q\delta^3R^{B_3}$. Lemma~\ref{main:lem:smoothing}
treats its first term.

For any integer $k>0$, the tail has $H^s$ profile
$C_{Q,k,s}\delta R^{B_0}\lambda^{s-\zeta k}$. Applying the same
Fourier bound at $s=0$ and one $s>1+n/2$ gives
\[
 \|(I-\Pi)(P+W_2)\|_{C^1}
 \leq C_{Q,k}\delta R^{B_0}\lambda^{p+1/2-\zeta k}.
\]
Thus $E_{\mathrm{cut}}=U-(P+W_2)$ has arbitrary finite $C^1$
decay with a factor $\delta$. Its exact metric contribution is
\[
 D\gamma_{F+P+W_2}E_{\mathrm{cut}}
                   +(dE_{\mathrm{cut}})^TdE_{\mathrm{cut}}.
\]
Choosing $k$ large enough absorbs the fixed polynomial size of
$F+P+W_2$ and bounds this by $C_Q\delta R^{-Q}$, proving
\eqref{main:eq:step-error}.

Equation~\eqref{main:eq:joint-physical-profile} at $r=0$
and boundedness of $\Pi$ on $L^2$
give \eqref{main:eq:step-L2}; the multiplier gives \eqref{main:eq:step-band}.
Fourier Cauchy--Schwarz and lattice counting on the fixed torus imply
\eqref{main:eq:step-Cr}. Reality follows from the real fields and the
even real multiplier.
\end{proof}

The first term in \eqref{main:eq:step-error} collects errors reduced
by finite accuracy. The second is the nonlinear interaction between
the primary and secondary fields.

\section{Histories and coefficient control}\label{main:sec:history}
Exact Fourier support gives the fixed high norm used above.

\begin{lemma}[A band-limited history]\label{main:lem:history}
Fix an integer $r_*\geq1$ and $1<b_*<v<3/2$, and put $t=b_*/v$.
For $R_0\geq2$ set $R_j=R_0^{v^j}$. Suppose real global increments
satisfy, with $k_j\geq k_0$,
\begin{equation}\label{main:eq:history-data}
 \supp\widehat U_j\subset\{|\xi|\leq CR_j^{b_*}\},\qquad
 \|U_j\|_{L^2}\leq R_j^{-k_j+B_0+1}.
\end{equation}
Put $F_j=F_{\rm init}+\sum_{i<j}U_i$, $T_j=R_j^t$ and
$b_{\rm hist}=B_0+b_*(r_*+n/2)$.
If $\mu=k_0-b_{\rm hist}-1>0$, then
\begin{equation}\label{main:eq:history-current}
 \|F_j\|_{C^r(W)}\leq C_rT_j^{(r-r_*)_+}\qquad(r\geq0),
\end{equation}
and
\begin{equation}\label{main:eq:total-small}
 \sum_j\|U_j\|_{C^{r_*}(\T)}\longrightarrow0
                      \quad\text{as }R_0\longrightarrow\infty.
\end{equation}
Both assertions are uniform for a compact smooth family of initial maps.
\end{lemma}
\begin{proof}
Fourier Cauchy--Schwarz gives
\begin{equation}\label{main:eq:history-bernstein}
 \|U_i\|_{C^{r_*}(\T)}\leq C R_i^{-k_i+b_{\rm hist}+1}.
\end{equation}
Since $v^i\geq1+i(v-1)$, their sum is at most
\[
 \frac{CR_0^{-\mu}}{1-R_0^{-\mu(v-1)}}.
\]
The periodic increment sum has uniformly bounded $C^{r_*}$ norm
and Fourier support below $CR_j^{b_*/v}=CT_j$.
Bernstein gives its higher-order profile; the fixed initial map
only changes the constants. The same sum proves the smallness assertion.
\end{proof}

\begin{proposition}[Current coefficient scale]\label{main:prop:current-scale}
\label{app:prop:fixed-high}
Enlarge the fixed order $r_0$ in Proposition~\ref{main:prop:response}
to include the fixed reserve $c_{\rm lin}+3$ of
Setup~\appref{app:ch_response:defn:response-regular} and the
sampling and physical-slice orders through $r_{\rm par}$, with
$r_0\geq D_{\rm pr}+r_{\rm par}+3$ and their fixed construction shifts.
Choose $r_*$ above the fixed response orders and the density
preparation and quadrature budgets of
Proposition~\appref{app:ch_mean_pairs:prop:mean-coarse}, including
their positive-scalar and reconstruction shifts, so subtracting
any of these shifts leaves at least $r_0$.
These orders and the resulting
geometric neighborhood precede variable mean accuracy.
If the total displacement in \eqref{main:eq:total-small} stays in
that $C^{r_*}$ neighborhood, the actual extended coefficient arrays
satisfy
\begin{equation}\label{app:eq:uniform-high-coeff}
 \|\widetilde a(F_j)\|_{C^r}\leq C_rT_j^{(r-r_0)_+}.
\end{equation}
The same bound holds for $F_j$ and the prepared positive initial
scalars, including joint data derivatives on their fixed domains.
The current response construction has scale
\[
 L_{F,j}=T_j,
\]
with the fixed outside losses of Proposition~\ref{main:prop:response}.
Since $t<1<a-\beta$, this scale lies below
$(\lambda_jh_j)^{1-\sigma}$ for some fixed $\sigma>0$ and large $R_0$.
\end{proposition}

\begin{proof}
The finite-jet bounds and fixed extension transfer the history
profile to the coefficients. Proposition~\ref{main:prop:response}
gives $L_{F,j}=T_j$. The density preparation uses only orders
within the uniform $C^{r_*}$ budget. Choose
$t<(a-\beta)(1-\sigma)$; its strict gap absorbs the fixed chart constants.
\end{proof}

This coefficient scale lies below the mean input bands and the
smallest product frequency.

\section{Choice of scales}\label{main:sec:scales}
We now explain why the estimates can be imposed simultaneously.
All exponents in this section are fixed before choosing the initial
parameter $s$ or the increasing accuracies. The detailed preparation
and mean inequalities are proved in Proposition~\appref{app:prop:ordered-scale-selection}.

\subsection{Frequency and amplitude gaps}
Use the following fixed outer exponents:
\begin{equation}\label{main:eq:fixed-outer-exponents}
\begin{gathered}
 \epsilon=\frac1{32},\quad\theta=\frac1{64},\quad
 \beta=\frac{11}{32},\quad a=\frac75,\\
 \zeta=\frac1{84},\quad b_*=\frac{17}{12},\quad
 v=\frac{35}{24},\quad t=\frac{b_*}{v}=\frac{34}{35}.
\end{gathered}
\end{equation}
Then $\beta=(1+\epsilon)/3$, $b_*=a(1+\zeta)$, and
\begin{equation}\label{main:eq:outer-window}
 1+\theta+\beta<a<b_*<v<\frac32,\qquad
 t=\frac{b_*}{v}<1.
\end{equation}
Choose the fine amplitude exponent
\begin{equation}\label{main:eq:fine-window}
 s_f=1+\theta/2.
\end{equation}
Then
\[
 1<s_f<1+\theta<a-\beta<a.
\]
Here $a-\beta-(1+\theta)=13/320$.
Thus a fine amplitude ceiling
$K_{\max}=R^{s_f}=H^{d_f}$, $d_f=s_f/(1+\theta)<1$, lies below
all product frequencies by a strict power.

The equal-gap bands of Lemma~\ref{main:lem:mean-recurrence} start
above $R$ and end below $K_{\max}$. Their common positive exponent
gap lets one finite derivative order control all omitted inputs.
Accuracy changes only this finite list within the same sampled family.

For the primary construction fix
\begin{equation}\label{main:eq:primary-fixed-scales}
 L_{\rm WKB}=T_0=R^t.
\end{equation}
The finite transport and positive background use the current
scale. Their fixed low norms cover the density preparation.
The two amplitudes are independent of accuracy; their complete
residual is included in the current source.

The common amplitude scale is
\[
 K_{\mathrm{env}}=C_*K_{\max}
\]
with one fixed $C_*$. Take $\sigma=1/1000$. Since
$t<1$ and $s_f<a-\beta$,
\begin{equation}\label{main:eq:all-pair-scales}
 C_*K_{\max},\ R^{t}\leq(\lambda h)^{1-\sigma}
\end{equation}
for large $R$. This verifies
\eqref{main:eq:pair-window}. The strict inequalities absorb
the fixed patch and radial normalization constants.

\subsection{The forcing and response size}
Set the forcing exponent
\begin{equation}\label{main:eq:A-choice}
 A=\frac{p(1+\theta)}2,\qquad
 2A-s_fp=\frac{p\theta}{2}>0.
\end{equation}
Thus the same inequality bounds the fine physical field and makes
$\epsilon_fK_{\max}^{p}$ small. The nonlinear margin may be
$\nu_{\rm nl}=p\theta/[4(1+\theta)]$ in $H$ units.
This also proves \eqref{main:eq:variation}.
Use the preparation count in \eqref{main:eq:preparation-orders}.
The quadrature margin is
$\beta(2n-1)-2A=(23n-109)/128\geq3/64$.
The normal comparison \eqref{main:eq:canonical-range} has positive
exponents because $\theta<\epsilon$ and $1+\theta+\beta<a$.

The current coefficient exponent is $t<1$ and the mean input floor
is $R$. A fixed multiple of $T_0$ is an envelope for the zero atom.
All response envelopes satisfy $K\leq\lambda^{1-\nu_0}$ with
$\nu_0=1/4$. Actual modulation supports are below their envelopes.
These are the response and source gaps used
in Proposition~\ref{main:prop:envelopes}. Proposition~\appref{app:prop:ordered-scale-selection} verifies the
individual cutoff, source, and mean inequalities without any change
of the fixed operators.

\subsection{Fixed orders and the initial defect}
Fix the density preparation count in \eqref{main:eq:preparation-orders} first.
Choose $r_*$ in Proposition~\ref{main:prop:current-scale} to include
these finite orders and their construction shifts, and put
$b_{\rm hist}=B_0+b_*(r_*+n/2)$.
Choose an integer $k_{\min}$ so that
\begin{align}
 (3-2v)k_{\min}&>B_3+2v+4,\label{main:eq:k-cubic}\\
 k_{\min}&>b_{\rm hist}+2.\label{main:eq:k-history}
\end{align}
Both are lower bounds since $3-2v>0$.
All orders and exponents are fixed before mean accuracy or target dilation.
Use
\begin{equation}\label{main:eq:Q-choice}
 Q(k)=2k.
\end{equation}
Both errors in \eqref{main:eq:step-error} are at most
$C_{2k}R^{-3k+B_3}$; \eqref{main:eq:k-cubic} controls their sum.

\section{Smooth convergence on one neighborhood}\label{main:sec:convergence}
The target family $g^{(s)}$ of Proposition~\ref{main:prop:seed}
has uniform fixed seminorms before $s$ is chosen.
Choose $\varepsilon_*>0$ so that its $C^{r_*}$ reference ball
preserves every geometric and analytic gap.

For every finite history satisfying \eqref{main:eq:history-data}
with $R_i=R_0^{v^i}$ and $k_i\geq k_{\min}$,
Lemma~\ref{main:lem:history} gives uniform map profiles.
Choose a finite nondecreasing threshold
$\mathfrak H(k)$, independent of $s$, as follows. First enlarge
$\mathfrak H(k_{\min})$ so that $R_0\geq\mathfrak H(k_{\min})$ implies
\begin{equation}\label{main:eq:displacement-budget}
 \sum_{i\geq0}C R_i^{-k_{\min}+b_{\rm hist}+1}
                                      <\varepsilon_*/2.
\end{equation}
Every such finite history then lies in the reference ball.
Proposition~\ref{main:prop:current-scale}, the scale verification
and the smoothing lemma give all step hypotheses
uniformly for these histories and the compact target family.

At each fixed $k$, Theorem~\ref{main:thm:step} uses finitely many
uniform constants at accuracy $2k$. Enlarge $\mathfrak H(k)$
monotonically so that $R_j\geq\mathfrak H(k_j)$ meets their thresholds
and bounds the metric and $L^2$ constants by $R_j$. If the current
map $M_j=G_{\rm ref}+\sum_{i<j}U_i$ has defect at most $R_j^{-2k_j}$,
its increment satisfies \eqref{main:eq:history-data} and
\begin{equation}\label{main:eq:metric-two-errors}
 \|\gamma(M_j+U_j)-g^{(s)}\|_{C^0(D)}
 \leq2C_{2k_j}R_j^{-3k_j+B_3}\leq R_j^{-2v(k_j+1)}.
\end{equation}
The last inequality follows from \eqref{main:eq:k-cubic}.

\subsection{One adaptive schedule and one target dilation}
Choose $R_0\geq\mathfrak H(k_{\min})$.
Set $k_0=k_{\min}$ and
\begin{equation}\label{main:eq:staircase}
 k_{j+1}=k_j+\mathbf1_{\{R_{j+1}\geq\mathfrak H(k_j+1)\}}.
\end{equation}
Then $R_j\geq\mathfrak H(k_j)$ and $k_j\to\infty$: an eventual
constant $K$ would be contradicted when $R_j$ exceeds the finite
$\mathfrak H(K+1)$. Put $\delta_j=R_j^{-k_j}$ and $Q_j=2k_j$.
Now choose a single $s>0$ with
\begin{equation}\label{main:eq:initial-error}
 \|g^{(s)}-\gamma(G_{\rm ref})\|_{C^0(D)}\leq\delta_0^2,
\end{equation}
using \eqref{main:eq:seed-small}. This choice follows all thresholds.

Apply this step successively.
Each increment preserves the prospective Fourier and $L^2$ budgets;
\eqref{main:eq:displacement-budget} keeps the current maps admissible.
Since $k_{j+1}\leq k_j+1$, its metric conclusion gives inductively
\begin{equation}\label{main:eq:metric-induction}
 \|g^{(s)}-\gamma(M_j)\|_{C^0(D)}\leq\delta_j^2.
\end{equation}
Thus the same construction continues on the fixed core for all $j$.

For every fixed $r$, Fourier Cauchy--Schwarz gives
\begin{equation}\label{main:eq:final-Cr}
 \|U_j\|_{C^r(\T)}
 \leq C_rR_j^{-k_j+B_0+1+b_*(r+n/2)}.
\end{equation}
Here $C_r$ is the Fourier constant, independent of $Q_j$.
As $k_j\to\infty$, this bound is eventually $C_rR_j^{-1}$,
which is summable. The same series therefore converges in every
$C^r$ to one smooth $M_\infty$ on $W$. Taking $C^1$ limits gives
$\gamma(M_\infty)=g^{(s)}$ on $\Int D$. On the fixed neighborhood
$U=s\rho\,\Int D$, set
$F(x)=s\rho M_\infty(x/(s\rho))$. Then
$\gamma(F)(x)=g^{(s)}(x/(s\rho))=g(x)$.
Positive definiteness gives $\rank dF(p)=n$.
Choose $n$ ambient coordinate components whose differential minor is
invertible at $p$. Their projection is a diffeomorphism on one smaller
neighborhood by the inverse function theorem. On that neighborhood $F$
is the graph of its other $N-n$ components, and hence an embedding.
This single restriction is independent of derivative order.

\clearpage
\newgeometry{margin=1in}
\small
\appendix
\section{Domains, Fourier conventions, and notation}\label{app:sec:notation}

The metric and normal-map conventions are those of
Section~\mainref{main:sec:geometry}. Here $T_0=T$,
$K_{\proofenv}=K_{\mathrm{env}}$ and $B_{\cubic}=B_3$ in the
main text; $B_0$ is unchanged. Every reduced operator uses its
displayed normalization, including the factor $1/2$ in
\maineqref{main:eq:reduced-equation}.
The setup below collects the common domains: the local constructions
first give buffered validity regions, and
Theorem~\ref{app:ch_response:thm:response-compatible-choice}
allows the nested choices used by the background, mean and pair steps.

\subsection*{Interfaces and verification}
Table~\ref{app:tab:interfaces} lists the results used in the main text.
Every result in the second column is proved in this paper. A local
\emph{Setup} states the hypotheses for a block of arguments; its use
in the construction is justified by the cited results. Sections~\mainref{main:sec:history}--\mainref{main:sec:convergence}
and Appendix~\ref{app:sec:ordered-scale-selection} verify the hypotheses
for the actual iterates.
The continuum row first chooses the positive scalar $r_F$ in
Proposition~\ref{app:ch_background:prop:bg-current}. The primary
family and its fixed response are linear in each prescribed scalar.
Proposition~\ref{app:ch_mean_pairs:prop:mean-coarse} then prepares
$r_{F,R}$ before sampling; the row inverse still belongs to $r_F$.
Corollary~\ref{app:ch_response:cor:current-atoms} controls this
same corrected atomic family throughout.

In the table, one fixed order $d=r_*$ dominates the geometric jets,
the normal order $M=20(n+1)$, the two primary amplitudes and all
calculus, density and reconstruction shifts. It also includes the
atomic response budget in Corollary~\ref{app:ch_response:cor:current-atoms}
and the finite density and quadrature orders chosen in
\maineqref{main:eq:preparation-orders}, after $A$ and before $Q$.
Membership in this fixed
$C^d$ neighborhood gives the stated gaps and low seminorms;
the all-order graded profiles are separate hypotheses, verified by
Lemma~\mainref{main:lem:history} and
Proposition~\mainref{main:prop:current-scale}.
All source sizes and scale windows referred to below are displayed
in the main text.

{\setlength{\tabcolsep}{3pt}
\begin{longtable}{@{}>{\raggedright\arraybackslash}p{.145\textwidth}>{\raggedright\arraybackslash}p{.20\textwidth}>{\raggedright\arraybackslash}p{.225\textwidth}>{\raggedright\arraybackslash}p{.22\textwidth}>{\raggedright\arraybackslash}p{.145\textwidth}@{}}
\caption{Interfaces used by the actual construction.}\label{app:tab:interfaces}\\
\hline
Object & Proved source & Domain and source class & Low topology and extra hypotheses & Dependence on $Q$\\
\hline
\endfirsthead
\multicolumn{5}{l}{\textsc{Table \thetable. Interfaces used by the actual construction (continued).}}\\
\hline
Object & Proved source & Domain and source class & Low topology and extra hypotheses & Dependence on $Q$\\
\hline
\endhead
\hline
\endfoot
Reference and carrier geometry
& Propositions~\ref{app:ch_seed:prop:seed-actual-menu},
\ref{app:ch_seed:prop:seed-phase-charts};
Definition~\ref{app:defn:exhaustive-pair-groups}
& Common buffered charts; detection on the whole sphere; primary and central signed directions
& Fixed $C^d$ neighborhood; positive normal, detector and chord gaps
& Geometry fixed before frequency and $Q$
\\[5pt]
Extension and finite calculus
& Lemma~\ref{app:ch_regular_base:lem:ball-extension};
Theorem~\ref{app:ch_regular_base:thm:actual-calculus}
& One extension on $H^r$ and $C^r$, $r\geq0$; localized operators on $W$
& Fixed collar and finite construction depth; fixed $C^d$ bounds
& Same extension; fixed derivative shifts
\\[5pt]
Regular inverse $S_FT_F$
& Theorems~\ref{app:ch_regular:thm:exact-correction},
\ref{app:ch_regular:thm:fixed-high},
\ref{app:ch_regular:thm:no-return}
& Full $L^2(W)$ Neumann inverse; core equation with angular error
& Gain one; fixed $C^d$ neighborhood; graded profiles
& One inverse; no extra scale power in the displayed graded bounds
\\[5pt]
Primary waves and joint sample
& Theorems~\ref{app:ch_waves:thm:wkb},
\ref{app:ch_waves:thm:sampling}
& Whole patches; $(q_{\rm quad}+1)^3$ fixed colors; full $L^2$ outputs and all prefixes; deterministic background quadrature
& $r_{\rm par}=M+3$; two amplitudes; fixed $C^d$ reserve; full strain of size $\lambda^{-3}$
& One family and sample event; fixed quadrature order before $Q$
\\[5pt]
Response $\mathcal B_M^{\mathbb C}$
& Proposition~\ref{app:ch_response:prop:response-graded};
Corollaries~\ref{app:ch_response:cor:finite-jets},
\ref{app:ch_response:cor:current-atoms}
& Source \maineqref{main:eq:response-source}; global cutoff envelopes; tensor error on $D_{\rm lag}$
& Graded profiles and \maineqref{main:eq:response-window}; fixed atomic joint jet budget
& $\lambda,\lambda^2$ carrier factors; derivative-amplitude change $\lambda^{-1}$
\\[5pt]
Continuum row and positive mean
& Propositions~\ref{app:ch_background:prop:bg-frame},
\ref{app:ch_background:prop:bg-adjoint},
\ref{app:ch_background:prop:bg-mixed-physical},
\ref{app:ch_background:prop:bg-current}
& Full $Y=L^2(\mathbb T;\mathbb R^n)$; raw $X^r$; physical fields on $U_{\rm fin}$
& Fixed positive reference dilation, then a fixed $C^d$ neighborhood
& Raw source order $r$; density source order $r+1$; fixed coefficient shifts
\\[5pt]
Prepared zero atoms and $U_c$
& Proposition~\ref{app:ch_mean_pairs:prop:mean-coarse};
Corollary~\ref{app:ch_response:cor:current-atoms}
& Fields on $U_{\rm fin}$; full auxiliary residual in \maineqref{main:eq:prepared-residual}
& Fixed positive class; finite density steps and lattice quadrature before sampling
& Preparation fixed after $A$; no extra power of $T_0$
\\[5pt]
Current waves $Z_\nu$
& Proposition~\ref{app:ch_lag:prop:lag-envelopes};
Proposition~\mainref{main:prop:mean}
& Complete finite Fourier measures; global cutoff envelopes; strain on $D_{\rm lag}$
& Same corrected atomic family; shifted support in the original chart
& No envelope offset; displacement change $\lambda^{-2}$ and derivative change $\lambda^{-1}$
\\[5pt]
Nonlinear mean
& Lemma~\mainref{main:lem:mean-recurrence};
Proposition~\mainref{main:prop:mean}
& Global residual recurrence; metric error on $D$; physical bounds \maineqref{main:eq:mean-coarse-size}--\maineqref{main:eq:mean-fine-size}
& Same current phases and sample event; canonical inverse in Proposition~\mainref{main:prop:canonical}
& $p=(n+1)/2$, $B_{\rm mean}=n/2$; nonlinear budget $p$; finite count varies
\\[5pt]
Signed product correction
& Proposition~\ref{app:ch_pairs:prop:pairs-lag-sum}
& Entire protected source product; metric error on $D$; cut global $H^r$ field
& Fixed $C^d$ neighborhood; \maineqref{main:eq:all-pair-scales}
& Source offset $0$; physical loss $1$ in \maineqref{main:eq:pair-high-size}
\\[5pt]
Stage closure
& Lemma~\mainref{main:lem:history};
Proposition~\ref{app:prop:ordered-scale-selection}
& One fixed $D$ and one global band for each increment
& Prospective $C^{r_*}$ displacement bound and ordered scale choices
& $B_0,B_3$ fixed; $Q_j=2k_j$ increases
\\
\end{longtable}}

\begin{setup}[Fixed domains]\label{app:setup:domains}
Let $D$ be a closed ball of positive radius centered at zero,
$D\Subset W$ a bounded open physical neighborhood, and $\T$ a
fixed auxiliary flat torus whose coordinate cell contains $\overline W$.
All domains below have positive fixed buffers.

First choose the whole-patch primary responses of
Theorem~\ref{app:ch_response:thm:response-compatible-choice}, with
actual contraction cutoffs $\chi_{2,\rm LAG},\chi_{3,\rm LAG}$.
Their finitely many identity cores contain a common wider core
$U_{\rm LAG}$, where $\chi_{2,\rm LAG}=1$.
Choose the background core $U_{\rm BG}$, frame ball $B_{\rm frame}$,
measurement region $V_{\rm meas}$ and source/observation cutoffs inside
these fixed validity domains, before frequency, sampling, accuracy
and the last seed parameter, so that
\begin{equation}\label{app:eq:global-domain-chain}
 D\Subset U_{\rm BG}\Subset B_{\rm frame}
 \Subset V_{\rm meas}\Subset\operatorname{Int}D_{\rm lag},
 \qquad \supp\chi_{\rm meas}\Subset V_{\rm meas},
\end{equation}
with $\chi_{\rm meas}=1$ near $\overline{B_{\rm frame}}$, and
\begin{equation}\label{app:eq:global-cutoff-plateaus}
 \begin{split}
 D_{\rm lag}&\Subset\operatorname{Int}\{\chi_{\lag,\rm src}=1\},\qquad
 \supp\chi_{\lag,\rm src}\Subset
                   \operatorname{Int}\{\chi_{{\rm core},\rm LAG}=1\},\\
 \supp\chi_{{\rm core},\rm LAG}&\Subset U_{\rm LAG}\Subset W,
 \qquad\supp\chi_{\lag,\rm src}\Subset\{\chi_{2,\rm LAG}=1\}.
 \end{split}
\end{equation}
One larger finite-primary chart in $W$ contains
$\overline{U_{\rm LAG}}$ and all these cutoffs.

Every entire measurement, protected pair-source and physical-output
support lies in $\operatorname{Int}D_{\rm lag}$.
The cutoffs $\chi_{\phys},\chi_{\pair}$ equal one near $D$.
Choose the pair cutoff inside its protected region $X_{\rm pair}$,
where $\chi_{2,\rm pair}=1$, and the mean and common final physical
cutoffs inside $U_{\rm BG}$ for the complete primary, mean and pair sum.
For the synthesis slice estimate use a separate $\chi_{\rm slice}$,
equal to one near the entire $\supp\chi_{\rm meas}$ and supported
in the larger common validity region, for example
$\operatorname{Int}D_{\rm lag}$. This cutoff belongs only to that
estimate; the actual fields use their own physical cutoffs.

The background notation $U_{\rm core},B_r,U_{\rm meas}$ means
$U_{\rm BG},B_{\rm frame},V_{\rm meas}$; primary-response notation $U_{\rm core}$
and its control cutoff means $U_{\rm LAG},\chi_{{\rm core},\rm LAG}$.
Other solver cores are their own fixed validity domains.

The primary current response and protected pair correction use
separate regular-sector operators. Appendix~\ref{app:sec:ch_background}
constructs the continuum image and transport row.
A literal sample in primary patch $g$ uses one current
$\mathcal B_F^{(g)}$ for every initial datum.
The finite primary wave and its response use the same phase.
Each construction has its own fixed domains,
angular buffers, cutoffs, real pairings and construction depths.
At each coefficient choice the actual operators and exact inverses
are the same at every precision and observation order. Take the
finite minimum of margins and maximum of constants and derivative
shifts over the patch list.

The base strain, complete response leakage and current strain identity
are estimated on $D_{\rm lag}$, covering every whole measurement and
pair source. Metric-defect smallness and target smoothing are imposed
only on $D$.
\end{setup}

\begin{notation}[Norms and Fourier transforms]\label{app:not:fourier}
For a function smooth up to the boundary of \(D\),
\[
\|f\|_{C^r(D)}
 =\max_{|\alpha|\leq r}\sup_{x\in D}|\partial^\alpha f(x)|.
\]
The same convention is used on \(W\).
Vector and tensor norms are taken componentwise in fixed coordinates.
Unless a domain is specified otherwise, global Sobolev and \(L^2\) norms
are on \(\T\).

Let \(\Lambda^\ast\) be the dual lattice of \(\T\).
We use
\[
\widehat u(\xi)
 =\frac1{\vol(\T)}
   \int_\T e^{-ix\cdot\xi}u(x)\dd x,
\qquad
u(x)=\sum_{\xi\in\Lambda^\ast}\widehat u(\xi)e^{ix\cdot\xi}.
\]
In particular,
\[
\|u\|_{L^2(\T)}^2
 =\vol(\T)\sum_{\xi\in\Lambda^\ast}|\widehat u(\xi)|^2.
\]
The integer Sobolev norm \(H^r(\T)\) is the \(L^2\) norm of all
Cartesian derivatives through order \(r\).

For Euclidean envelope variables, the convention is
\[
\widehat f(\xi)
 =(2\pi)^{-n}\int_{\R^n}e^{-ix\cdot\xi}f(x)\dd x,
\qquad
f(x)=\int_{\R^n}e^{ix\cdot\xi}\widehat f(\xi)\dd\xi.
\]
For a finite complex Fourier measure \(m\), its size is
\(\|m\|_{\TV}\).
The atom at the origin is denoted by \(\delta_0\).

On local balls, positive Sobolev spaces are restriction spaces;
negative estimates use the corresponding supported-distribution
and restriction dual pairs. Whole-space comparisons use fixed
cutoffs and the extension of Lemma~\ref{app:ch_regular_base:lem:ball-extension}.
Write $D=-i\partial$, $\|u\|_s=\|u\|_{H^s}$ and
$H^\infty=\bigcap_{s\geq0}H^s$. A coefficient norm includes every
matrix entry. Adjoints are complex Hilbert adjoints, restricted
to real fields when the operator is real.
\end{notation}

\begin{notation}[Stage scales]\label{app:not:scales}
Use the scales and strict windows of Section~\mainref{main:sec:scales}:
\[
 \lambda=R^a,\quad h=R^{-\beta},\quad H=R^{1+\theta},\quad
 L_F=T_0=R^t,\quad K_{\max}=H^{d_f}=R^{s_f}.
\]
The zero-atom and common response envelopes are respectively
$C_{\rm env}T_0<K_{\max}$ and
\[
 K_{\proofenv}=C_*K_{\max}.
\]
These constants are fixed independently of $R$, accuracy and derivative
order. Equation~\maineqref{main:eq:all-pair-scales} puts both below
the smallest protected product frequency by a strict power.
\end{notation}


\section{Interpolation and finite-jet estimates}
\label{app:sec:elementary-interpolation}
The estimates in this section are independent of every phase, solver and
frequency choice. They will be used in the geometric and analytic constructions.

\begin{lem}[One fixed extension at every nonnegative integer order]
\label{app:ch_regular_base:lem:ball-extension}
Let $\Omega=B_R(c)\subset\R^n$ and $\overline\Omega\subset V$,
with $V$ open. Use the derivative Sobolev norm
\[
 \chregularbasenorm{f}_{H^m(\Omega)}^2
 =\sum_{|\alpha|\leq m}\chregularbasenorm{D^\alpha f}_{L^2(\Omega)}^2,
 \qquad m\geq0,
\]
and the corresponding whole-space norm.
There is one real linear operator, independent of $m$,
\[
 E_\Omega\colon C^\infty(\overline\Omega)
       \longrightarrow C_c^\infty(\chregularbaseR^n),
\]
equal to $f$ on $\Omega$, supported in a fixed compact subset of $V$, with
\begin{align}
 \chregularbasenorm{E_\Omega f}_{H^m(\chregularbaseR^n)}
 &\leq C_m\chregularbasenorm{f}_{H^m(\Omega)},\label{app:ch_regular_base:eq:ball-extension-sobolev}\\
 \chregularbasenorm{E_\Omega f}_{C^m(\chregularbaseR^n)}
 &\leq C_m\chregularbasenorm{f}_{C^m(\overline\Omega)}.
 \label{app:ch_regular_base:eq:ball-extension-continuous}
\end{align}
The constants depend only on $\Omega,V$, the fixed cutoff and $m$.
The bounded extensions to $H^m$ and $C^m$ agree on their common
domains. The derivative norm is equivalent to the whole-space
restriction norm. The operator acts componentwise and complex
linearly. No negative-order extension is asserted.
\end{lem}

\begin{proof}
Choose $0<d<R/2$ with $\overline{B_{R+d}(c)}\Subset V$ and a real
$\phi\in C^\infty([0,\infty))$ equal to one on $[0,d/4]$ and zero
on $[d/2,\infty)$. For $b_k=-2^k$ and $0\leq k\leq N$ put
\[
 a_k^{(N)}=\prod_{\substack{0\leq j\leq N\\j\ne k}}
                    \frac{1+2^j}{2^j-2^k}.
\]
Lagrange interpolation at $b_0,\ldots,b_N$, evaluated at one, gives
\begin{equation}\label{app:ch_regular_base:eq:extension-finite-moments}
 \sum_{k=0}^N a_k^{(N)}b_k^\ell=1\qquad(0\leq\ell\leq N).
\end{equation}
Separating $j<k$ and $j>k$ gives
\begin{equation}\label{app:ch_regular_base:eq:extension-coefficient-bound}
 |a_k^{(N)}|\leq A^2B^{-2}2^{-k(k+1)/2},\qquad
 A=\prod_{j\geq0}(1+2^{-j})<\infty,\quad
 B=\prod_{j\geq1}(1-2^{-j})>0.
\end{equation}
For fixed $k$ the product converges. Its bound permits dominated
convergence in the moment identities, giving $a_k=\lim_Na_k^{(N)}$ with
\begin{equation}\label{app:ch_regular_base:eq:extension-infinite-moments}
 \sum_{k\geq0}a_kb_k^\ell=1,\qquad
 \sum_{k\geq0}|a_k|2^{k\ell}<\infty\qquad(\ell\geq0).
\end{equation}

Set $E_\Omega f=f$ on $\Omega$. With
$h(\omega,s)=f(c+(R-s)\omega)$, define for $t>0$
\begin{equation}\label{app:ch_regular_base:eq:extension-radial-formula}
 E_\Omega f(c+(R+t)\omega)
   =\sum_{k\geq0}a_k\phi(2^kt)h(\omega,2^kt).
\end{equation}
Terms with $2^kt\geq d/2$ are zero. The sum is finite at each
$t>0$ and supported in $\overline{B_{R+d/2}(c)}\Subset V$.

For $p=2,\infty$ and $|\beta|+\ell\leq m$, the substitution $s=2^kt$
in a fixed finite angular atlas gives
\begin{equation}\label{app:ch_regular_base:eq:extension-collar-l2}
 \left\|a_k2^{k\ell}
 [D_\omega^\beta\partial_s^\ell(\phi h)](\omega,2^kt)\right\|_{L^p}
 \leq C_m|a_k|2^{k(\ell-1/p)}\|f\|_{W^{m,p}(\Omega)},
 \qquad 1/\infty=0.
\end{equation}
The same summable bound proves both norm estimates and uniform
convergence of every derivative at $t=0$. Its boundary value is
$(-1)^\ell D_\omega^\beta\partial_s^\ell h(\omega,0)$ by
\eqref{app:ch_regular_base:eq:extension-infinite-moments}.
These are exactly the interior jets of $h(\omega,-t)$, so the
joined function is smooth, with no boundary terms in weak derivatives.
For $n=1$, use the two radial directions without angular variables.

Contraction by $x\mapsto c+(1-\epsilon)(x-c)$, followed by
mollification on the resulting larger ball, proves smooth density
in $H^m$, $C^m$ and their intersections. The bounds extend this
same $E_\Omega$ consistently. Restriction and $E_\Omega$ give the two quotient-norm
inequalities; Plancherel compares the whole-space derivative and
Bessel-potential norms. The coefficients are real.
\end{proof}

\begin{lem}[Fourier interpolation at a prescribed scale]
\label{app:lem:sharp-fourier-embedding}
On $\R^n$, the fixed auxiliary torus, or a fixed ball with the
preceding extension, let $r\geq0$ and $m>r+n/2$ be integers.
For $K\geq1$,
\begin{equation}\label{app:eq:sharp-fourier-embedding}
 \|u\|_{C^r}\leq C_{r,m}\left[
 K^{r+n/2}\|u\|_2+K^{r+n/2-m}\|u\|_{H^m}\right].
\end{equation}
Thus a Sobolev profile $\|u\|_{H^s}\leq C_sAK^s$ gives
$\|u\|_{C^r}\leq C_rAK^{r+n/2}$, using only $s=0,m$.
\end{lem}
\begin{proof}
Fourier inversion bounds each order-$r$ derivative by the
frequency integral, or sum, of $|\xi|^r|\widehat u(\xi)|$.
Cauchy--Schwarz on $|\xi|\leq K$ gives the first term. On
$|\xi|>K$, the squared weight $|\xi|^{2r-2m}$ has integral
or lattice sum at most $C_{r,m}K^{2r+n-2m}$, giving the second.
Lower derivatives obey the same bound since $K\geq1$.
The fixed extension is bounded at orders zero and $m$.
\end{proof}

\begin{lem}[Elementary interpolation, derivative products and finite jets]
\label{app:lem:elementary-interpolation-jets}
On a fixed auxiliary torus, or a fixed chart with a bounded extension,
let $0\leq R<j<k$ be integers. Then
\[
 \lVert u\rVert_{C^j}\leq C_{R,j,k}
 \lVert u\rVert_{C^R}^{1-(j-R)/(k-R)}
 \lVert u\rVert_{C^k}^{(j-R)/(k-R)}.
\]
The endpoint and bounded low orders have their direct norm bounds.

Let \(f_i\) be finitely many primitive arrays, with possibly different
nonnegative integer baselines \(b_i\), and assume that their
\(C^{b_i}\) norms are bounded.
For an integer $W\geq0$, suppose that a product of primitive
derivatives, allowing repeated primitives, has orders \(j_h\) satisfying
\[
 \sum_h(j_h-b_{i(h)})_+\leq W.
\]
Then
\begin{equation}\label{app:ch_regular_base:eq:joint-primitive-product}
 \prod_h\chregularbasenorm{D^{j_h}f_{i(h)}}_{L^\infty}
 \leq C_W\left[
 1+\sum_i\chregularbasenorm{f_i}_{C^{b_i+W}}
 \right].
\end{equation}
Matrix factors remain in their original order.
The constant may depend on the number of factors and their fixed low
bounds.

Let $\mathcal F(J^cF)$ be a fixed smooth finite-jet expression, including
finitely many matrix inversions whose singular-value gaps stay positive.
If its low jet ranges in a fixed bounded set with those gaps, then
\[
 \lVert\mathcal F(J^cF)\rVert_{C^r}
       \leq C_r(1+\lVert F\rVert_{C^{r+c}}).
\]
Here the finite shift may be enlarged once for finitely many such operations.
These are algebraic jet estimates; dependence of Hamilton flows, retarded
operators and their inverses is proved separately below.
\end{lem}
\begin{proof}
Use the fixed extension on a chart. Let $a=\|u\|_{C^R}$ and
$b=\|u\|_{C^k}$, so $a\leq b$. Choose one real even $\psi\in C_c^\infty$ equal to one near zero.
Its inverse Fourier kernel has mass one and all positive moments
zero, by differentiation of $\psi$ at zero. Periodizing its
scale-$t$ dilate gives $P_t$, for $0<t\leq t_0$, with
\[
 \|P_tu\|_{C^j}\leq Ct^{R-j}a,\qquad
 \|(I-P_t)u\|_{C^j}\leq Ct^{k-j}b.
\]
The first bound follows by moving derivatives to the kernel; the second
follows from Taylor's integral remainder and the moment conditions.
If $a=0$, then $u=0$. Otherwise choose
$t=(a/b)^{1/(k-R)}$ when it is at most $t_0$. This gives
\begin{equation}\label{app:eq:elementary-interpolation}
 \|u\|_{C^j}\leq C_{R,j,k}a^{1-\theta}b^\theta,
 \qquad\theta=\frac{j-R}{k-R}.
\end{equation}
When that choice exceeds $t_0$, the ratio $b/a$ is bounded and the
direct bound $\|u\|_{C^j}\leq b$ proves the same estimate.
The endpoint and bounded low orders follow directly.

For $W>0$ set $H_i=1+\|f_i\|_{C^{b_i+W}}$ and
$\theta_h=(j_h-b_{i(h)})_+/W$. Interpolation at the indicated
baselines gives
\[
 \prod_h\|D^{j_h}f_{i(h)}\|_\infty
 \leq C_W\prod_hH_{i(h)}^{\theta_h}
 \leq C_W\left(1+\sum_iH_i\right),
 \qquad\sum_h\theta_h\leq1,
\]
by weighted arithmetic-geometric mean. For $W=0$ use the baseline
bounds. This proves \eqref{app:ch_regular_base:eq:joint-primitive-product}
for ordered matrix products as well.

For the finite-jet assertion put $U=J^cF$. Its low range and
inverse gaps bound every derivative of the finite-dimensional
coefficient function. Each order-$r$ chain-rule term is a bounded
coefficient times $\prod_iD^{\alpha_i}U$, where $|\alpha_i|\geq1$
and $\sum_i|\alpha_i|=r$. The product estimate with baseline zero
therefore gives $C_r(1+\|U\|_{C^r})$.
Fixed-gap inversions are smooth coefficient functions; finitely
many jet operations enlarge the shift only by a fixed amount.
\end{proof}



\begin{cor}[A fixed profile under finite derivative loss]
\label{app:cor:fixed-profile-transfer}
Suppose $T\geq1$ and
$\|F\|_{C^j}\leq A_jT^{(j-R)_+}$ for every $j\geq0$.
Let a fixed finite family of arrays on the same buffered charts satisfy
$\|\mathcal F_i(F)\|_{C^r}\leq C_r(1+\|F\|_{C^{r+c_i}})$,
with fixed shifts $c_i$. If $R\geq\max_i c_i+\nu$, then
\begin{equation}\label{app:eq:fixed-profile-transfer}
 \|\mathcal F_i(F)\|_{C^r}\leq C_rT^{(r-\nu)_+},\qquad r\geq0.
\end{equation}
Any fixed finite list of parameter derivatives can be included among
the arrays. For normalized symbols the radial order stays separate.
This applies to fixed constructions, not to an accuracy-dependent
number of compositions.
\end{cor}
\begin{proof}
Substitution gives $C_rT^{(r+c_i-R)_+}$, which has the stated bound.
\end{proof}

\begin{lem}[Common buffered inverse domains]
\label{app:lem:buffered-inverse-stability}
Let $D_0\Subset D_1\Subset D$ be bounded convex boxes in $\R^d$
and $F_0$ smooth near $\overline D$. Suppose a constant invertible
matrix $G$ satisfies
$\sup_D\|G^{-1}(DF_0-G)\|\leq1/4$.
For all $F$ in one sufficiently small $C^1$ neighborhood of $F_0$,
$F$ is injective on $D$, with a uniform lower Lipschitz bound, and
one open set $V$ satisfies
$F(D_0)\Subset V\Subset F(D_1)$ with common positive buffers.
The assertion is uniform for parameter families in this same neighborhood.
\end{lem}
\begin{proof}
Choose the $C^1$ neighborhood so
$\sup_D\|I-G^{-1}DF\|\leq1/2$. The segment identity gives
$|G^{-1}(F(u)-F(v))|\geq|u-v|/2$.
Choose one radius $r>0$ so every $\overline B_r(u_0)$,
$u_0\in\overline D_0$, lies in $D_1$.
A fixed small neighborhood $V$ of $F_0(\overline D_0)$ and a small
$C^0$ neighborhood of $F_0$ ensure that every $v\in\overline V$
has such a $u_0$ with $|G^{-1}(v-F(u_0))|<r/2$.
Then
\[
 u\longmapsto u-G^{-1}(F(u)-v)
\]
is a $1/2$ contraction of $\overline B_r(u_0)$ into itself.
Its fixed point gives $\overline V\subset F(D_1)$.
Reducing the $C^0$ neighborhood also gives
$F(\overline D_0)\Subset V$. All buffers and constants use only
these fixed low choices.
\end{proof}

\begin{lem}[Uniform interpolation and joint derivative budgets]
\label{app:ch_regular_base:lem:buffered-interpolation}
Use finitely many fixed working balls with larger buffered
coordinate neighborhoods, including every physical variable,
normalized covariable and parameter. The interpolation estimate of
Lemma~\ref{app:lem:elementary-interpolation-jets} applies jointly
in these coordinates. Right-hand norms may be on the larger domains.
For an entire working ball, use its single bounded extension at
every order; maxima over the fixed finite family obey the same bound.

Use \eqref{app:ch_regular_base:eq:joint-primitive-product} for
complete ordered products on these domains.

These estimates are uniform on normalized dyadic shells.
For a degree-one phase and an order-\(p\) symbol, the normalized
arrays are
\[
 S_\mu(x,\omega)=\mu^{-1}S(x,\mu\omega),
 \qquad a_\mu(x,\omega)=\mu^{-p}a(x,\mu\omega),
 \qquad \mu\geq1,
\]
on fixed buffered normalized annuli and angular charts.
Both physical variables are included for a two-position amplitude.
The estimates control coefficient products; explicit frequency
powers, shell ratios, and integration measures remain separate.

Fixed-gap matrix inverses, existing implicit and inverse charts, and
fixed-time flows preserve these joint derivative budgets.
All estimates use the same fixed working domains and inverse gaps.
\end{lem}

\begin{proof}
Apply \eqref{app:ch_regular_base:eq:joint-primitive-product} to the complete
primitive product before substituting coefficient estimates:
if $\|f_i\|_{C^m}\leq C_m(1+\|a\|_{C^{m+c_i}})$, its result uses
the maximum index $b_i+W+c_i$, without multiplying high coefficient
norms. The corresponding datum interpolation is
\[
 \|v\|_{p+k}\leq C_{p,s}\|v\|_p^{1-k/s}\|v\|_{p+s}^{k/s}
 \quad(0\leq k\leq s),
 \qquad \|f\|_{C^j}\leq C_{j,L}\|f\|_{H^{j+L}}\quad(L>d/2).
\]
These follow respectively from Fourier H\"older and Fourier
Cauchy--Schwarz; fixed extensions give the local versions.

Fixed-gap matrix inverses are finite-jet expressions from
Lemma~\ref{app:lem:elementary-interpolation-jets}. For compositions
with bounded first jets, the derivative partitions satisfy
\[
 (q-1)+\sum_{i=1}^q(\alpha_i-1)=r-1,
 \qquad\alpha_i\geq1,\quad\sum_i\alpha_i=r.
\]
This is the baseline-one budget for a composition.

For an existing inverse $X=F^{-1}$, differentiate $F\circ X=\operatorname{Id}$
on the same fixed working balls. The first-order inverse gap
bounds $DX$. Inductively, all lower derivatives of $X$ are bounded
by the corresponding $C^j$ norms of $F$ on the fixed larger range
charts. Every term other than $DF(X)D^rX$ has $q\geq2$ and
$\alpha_i<r$. Substitution of the inductive bounds leaves products
of $F$ derivatives with the displayed excess at most $r-1$.
Equation~\eqref{app:ch_regular_base:eq:joint-primitive-product}
therefore bounds them by $C_r(1+\|F\|_{C^r})$; the inverse gap
proves the same bound for $X$. Implicit maps use the inverse of
$(z,w)\mapsto(z,F(z,w))$, including all external parameters.

For the fixed-time flow $\partial_t\Psi=V(\Psi)$, the top
variational term is $DV(\Psi)D^r\Psi$. The other partitions have
the same budget and involve only lower flow derivatives. Induction
bounds their forcing by $C_r(1+\|V\|_{C^r})$, and Gronwall gives
the same estimate for the flow on the fixed interval. Time
derivatives follow from the equation. Variable initial sections
are compositions with this flow and obey the same joint budget.

For $U'=-CU$, $U(0)=I$, apply the flow estimate to $-CU$ and
$(U^{-1})'=U^{-1}C$, including time and the constant tube parameters
as coordinates. Gronwall keeps both matrix ranges bounded, giving
\[
 \|U\|_{C^r}+\|U^{-1}\|_{C^r}
 \leq C_r(1+\|C\|_{C^r}).
\]
Variable initial values multiply this fundamental matrix. Their
composition with the fixed flow obeys the same joint product budget.

For normalized symbols and phases the exact identities are
\[
 \partial_x^\alpha\partial_\omega^\gamma a_\mu
 =\mu^{|\gamma|-p}(\partial_x^\alpha\partial_\eta^\gamma a)(x,\mu\omega),
 \qquad
 \partial_x^\alpha\partial_\omega^\gamma S_\mu
 =\mu^{|\gamma|-1}(\partial_x^\alpha\partial_\eta^\gamma S)(x,\mu\omega).
\]
All interpolation domains are consequently fixed. Normalize two
noncomparable frequencies separately and leave their ratios, powers
and shell volumes outside the primitive norms. The fixed cutoff pullbacks of
Setup~\ref{app:ch_regular_base:setup:finite-calculus}
obey the same composition budget.

\end{proof}

\section{Oscillatory estimates and canonical retarded integration}\label{app:sec:ch_osc}
This section proves three elementary operator estimates used later:
exact normal comparison, localization of oscillatory sources, and
canonical retarded integration. The phase hypotheses below
are verified for the carrier phases in
Proposition~\ref{app:ch_seed:prop:seed-phase-charts}; each result
otherwise uses only its displayed phase and amplitude bounds.
\subsection{Exact normal comparison}
\label{app:ch_osc:sec:osc-normal}


\begin{notation}\label{app:ch_osc:defn:osc-fourier}
Use the Fourier convention of Notation~\ref{app:not:fourier} and
$\|f\|_{H^s}=\|\langle D\rangle^sf\|_2$ on $\R^n$, where
$\langle\xi\rangle=(1+|\xi|^2)^{1/2}$.
Compactly supported chart outputs are extended by zero.
Complex Hilbert inner products are linear in the first argument;
operator norms are on the entire stated spaces. Derivative order
means total multi-index order, and matrix dimensions are fixed.
\end{notation}

\begin{lem}[Two-position amplitudes on fixed supports]
\label{app:ch_osc:lem:osc-amplitude}
Let $d\geq1$ and fix compact physical supports inside bounded boxes
of $\R^d$. Suppose $b(x,y,\xi)$ is smooth, has these supports in
$x,y$, and
\[
 \sup_\xi|D_{x,y}^\alpha b(x,y,\xi)|\leq A,
 \qquad |\alpha|\leq2(d+1).
\]
Then
\[
 Bf(x)=(2\pi)^{-d}\iint e^{i(x-y)\cdot\xi}b(x,y,\xi)f(y)\,dy\,d\xi
 \qquad\hbox{satisfies}\qquad \|B\|\leq C A.
\]
The frequency support may be unbounded; $C$ depends on the fixed
physical boxes and $d$. Fixed matrix amplitudes obey the same bound.

Let $T:L^2(D)\to L^2(\R^d)$ have a smooth integral kernel $t(x,u)$
on fixed compact supports. For $c(x,x',u)$ with the same physical
supports and the preceding derivative bounds in $x,x'$, uniformly
in $u$, the full kernel
\[
 \mathcal N(c,g)(x,x')=\int t(x,u)c(x,x',u)
                                  \overline{t(x',u)}g(u)\,du
\]
has the bound
\begin{equation}\label{app:ch_osc:eq:two-position-normal-bound}
 \|\mathcal N(c,g)\|\leq CA\|T\|^2\|g\|_\infty.
\end{equation}
\end{lem}
\begin{proof}
Expand the two physical variables on one fixed larger torus, with
outer cutoffs removing the periodic copies. Integration by parts
with $(1-\Delta_x-\Delta_y)^{d+1}$ gives
\[
 \sup_\xi|b_{pq}(\xi)|+\sup_u|c_{pq}(u)|
 \leq CA\langle(p,q)\rangle^{-2(d+1)}.
\]
With the outer cutoffs included in the bounded modulations $M_p,M_q$,
the exact operators are
\[
 B=\sum_{p,q}M_pb_{pq}(D)M_q,\qquad
 \mathcal N(c,g)=\sum_{p,q}M_pT[c_{pq}g]T^*M_q.
\]
The coefficients are summable in $2d$ dimensions. Plancherel and
the bound on $T$ prove both assertions by absolute convergence in
operator norm. Compact frequency regularization, followed by its
distributional limit, justifies the unbounded-frequency formula.
Apply the argument entrywise for fixed matrices.
\end{proof}

\begin{setup}[Uniform normal hypotheses]\label{app:ch_osc:setup:osc-normal}
Fix $n\geq2$, $r\geq1$, $M=20(n+1)$, bounded convex open sets $X,D\subset\choscR^n$, compact sets $X_0\Subset X$, $D_0\Subset D$, and $\epsilon_0>0$. Let the real phase $\phi_\epsilon(x,u)$ and the complex $r$-vector amplitude $a_\epsilon(x,u)$ be smooth for $0\leq\epsilon\leq\epsilon_0$ on neighborhoods of $\overline X\times\overline D$, with smooth extensions across the parameter endpoints. Assume
\[
 \choscsupp a_\epsilon\subset X_0\times D_0,
 \qquad |D_{x,u}^\alpha\phi_\epsilon|+
 |D_{x,u}^\alpha a_\epsilon|+
 |\partial_\epsilon D_{x,u}^\alpha\phi_\epsilon|+
 |\partial_\epsilon D_{x,u}^\alpha a_\epsilon|\leq C_\alpha
 \quad(|\alpha|\leq M).
\]
Define the exact averaged gradient
\[
 \Psi_\epsilon(x,x',u)=\int_0^1
 d_x\phi_\epsilon(x'+t(x-x'),u)\choscdd t.
\]
Assume each $\Psi_\epsilon(x,x',\cdot)$ is injective on $D$,
with a common buffered inverse chart containing the images of $D_0$,
and $|\det d_u\Psi_\epsilon|\geq c_*>0$ throughout these boxes.
\end{setup}

\begin{prop}[Exact masked normal comparison]\label{app:ch_osc:prop:osc-normal}
In Setup~\ref{app:ch_osc:setup:osc-normal}, define for $K\geq1$
\[
 T_\epsilon^Kf(x)=\left(\frac K{2\pi}\right)^{n/2}
 \int_D e^{iK\phi_\epsilon(x,u)}a_\epsilon(x,u)f(u)\choscdd u.
\]
This maps the entire $L^2(D;\choscC)$ into
$L^2(\choscR^n;\choscC^r)$, and
\begin{equation}\label{app:ch_osc:eq:osc-normal-basic}
 \|T_\epsilon^K\|\leq C.
\end{equation}
Let $0<h\leq1$ and let $m_h$ be real and smooth on $D$, with
$0\leq m_h\leq1$ and $\|Dm_h\|_\infty\leq C/h$.
For $E_\epsilon^K=T_\epsilon^Km_h(T_\epsilon^K)^*$,
\begin{equation}\label{app:ch_osc:eq:osc-normal-mask}
 \|E_\epsilon^K-E_0^K\|\leq C\epsilon/h,\qquad K\geq1.
\end{equation}
The constant uses only the fixed sets, buffers, dimensions,
Jacobian gap and finite phase and amplitude bounds. The comparison
uses the complete kernels and only the displayed first mask derivative.
\end{prop}
\begin{proof}
\noindent\textbf{Step 1. The mask-free norm.}
Write $\Phi_\epsilon=\phi_\epsilon(x,u)-\phi_\epsilon(x',u)$ and
$A_\epsilon=a_\epsilon(x,u)a_\epsilon(x',u)^*$.
Convexity gives $\Phi_\epsilon=(x-x')\cdot\Psi_\epsilon$.
Let $u_\epsilon(x,x',v)$ be the common inverse. Its fixed buffer
and Jacobian gap give uniform derivative bounds by implicit
differentiation. The single change $v=\Psi_\epsilon(x,x',u)$ gives
the exact full kernel
\begin{equation}\label{app:ch_osc:eq:osc-exact-normal-kernel}
 E_\epsilon^K(x,x')=\left(\frac K{2\pi}\right)^n
 \int e^{iK(x-x')\cdot v}b_\epsilon(x,x',v)\choscdd v,
\end{equation}
where
\[
 b_\epsilon=
 \frac{A_\epsilon(x,x',u_\epsilon)m_h(u_\epsilon)}
 {|\det D_u\Psi_\epsilon(x,x',u_\epsilon)|}.
\]
The inverse buffer makes its zero extension smooth, jointly with
all physical and parameter variables. For mask $1$
its needed derivatives are uniformly bounded. Set $\xi=Kv$ and apply
Lemma~\ref{app:ch_osc:lem:osc-amplitude} to $b_\epsilon(x,x',\xi/K)$.
Its bounds are uniform for $K\geq1$, proving
$\|T_\epsilon^K\|^2=\|T_\epsilon^K(T_\epsilon^K)^*\|\leq C$.

\medskip\noindent\textbf{Step 2. Differentiation in the original input variable.}
Set
\[
 w_\epsilon=(D_u\Psi_\epsilon)^{-1}\partial_\epsilon\Psi_\epsilon.
\]
Then $\partial_\epsilon\Phi_\epsilon
=w_\epsilon\cdot D_u\Phi_\epsilon$ exactly.
Choose fixed scalar $\chi_x,\chi_u$ equal to one on the original
supports, inside $X,D$ and the common inverse buffer.
Step 1 bounds the scalar channel
$T_{\epsilon,\chi}^K$ with amplitude $\chi_x\chi_u$.
Let $\mathcal N_\epsilon(c,g)$ be the second operator in
Lemma~\ref{app:ch_osc:lem:osc-amplitude} for this channel.
Differentiation and integration by parts in the original $u$ give
\begin{gather*}
 \partial_\epsilon E_\epsilon^K
 =\mathcal N_\epsilon(c_0,m_h)
       +\sum_j\mathcal N_\epsilon(c_j,\partial_{u_j}m_h),\\
 c_0=\partial_\epsilon A_\epsilon-\operatorname{div}_u(w_\epsilon A_\epsilon),
 \quad c_j=-w_{\epsilon,j}A_\epsilon.
\end{gather*}
The compact input support removes every boundary term. These full
amplitudes have uniform spatial derivatives through $2(n+1)$:
computing them uses at most $2(n+1)+3$ phase derivatives and one
$\epsilon$ derivative, within the fixed $M$ budget. Equation~\eqref{app:ch_osc:eq:two-position-normal-bound} gives
$\|\partial_\epsilon E_\epsilon^K\|\leq C(1+\|Dm_h\|_\infty)
\leq C/h$. At each fixed $K$ the compact kernel is differentiable
in Hilbert--Schmidt norm. Integrating from $0$ to $\epsilon$ proves
\eqref{app:ch_osc:eq:osc-normal-mask}.
\end{proof}

\begin{cor}[Both real quadratures]\label{app:ch_osc:cor:osc-real}
Regard $L^2(D;\choscC)$ as a real Hilbert space with inner product $\choscRea\int f\overline g$, and set
\[
 C_\epsilon^Kf=2\choscRea(T_\epsilon^Kf)\in L^2(\choscR^n;\choscR^r).
\]
For every real output $r_0$,
\begin{equation}\label{app:ch_osc:eq:osc-factor-four}
 (C_\epsilon^K)^*_{\choscR}r_0=2(T_\epsilon^K)^*r_0,
 \qquad C_\epsilon^Km_h(C_\epsilon^K)^*_{\choscR}r_0
 =4\choscRea(E_\epsilon^Kr_0).
\end{equation}
The bounds of Proposition~\ref{app:ch_osc:prop:osc-normal} hold for these real operators with fixed numerical factors. The input still contains both complex quadratures.
\end{cor}
\begin{proof}
For complex $f$ and real $r_0$,
\[
 \langle2\choscRea(Tf),r_0\rangle_{\choscR}
 =2\choscRea\langle Tf,r_0\rangle
 =2\choscRea\langle f,T^*r_0\rangle.
\]
This proves the adjoint formula. Since the mask is real, applying $C$ to $2m_hT^*r_0$ proves the normal identity and its factor $4$. Taking norms proves the assertions. There is no restriction to real input and no extra conjugate cross operator.
\end{proof}

\begin{prop}[A full-space single-channel estimate]\label{app:ch_osc:prop:osc-channel}
Assume additionally that $X=J_{\rm time}\times Y$ and $D=I_{\rm carrier}\times D_\theta$, where $J_{\rm time},I_{\rm carrier}$ are bounded open intervals and $Y,D_\theta\subset\choscR^{n-1}$ are bounded convex open sets. Write $x=(t,y)$ and $u=(z,\theta)$. Suppose, uniformly in $t,z,\epsilon$, the maps
\[
 \theta\longmapsto\int_0^1
 d_y\phi_\epsilon(t,y'+s(y-y'),z,\theta)\choscdd s
\]
satisfy the buffered inverse, Jacobian and derivative assumptions of Setup~\ref{app:ch_osc:setup:osc-normal} in dimension $n-1$. Define
\[
 L_{\epsilon,z}^Kf(t,y)=\left(\frac K{2\pi}\right)^{n/2}
 \int e^{iK\phi_\epsilon(t,y,z,\theta)}
 a_\epsilon(t,y,z,\theta)f(\theta)\choscdd\theta.
\]
Then, on the entire indicated input and output spaces,
\[
 \|L_{\epsilon,z}^K\|_{L^2(D_\theta;\choscC)\to
 L^2(\choscR^n;\choscC^r)}^2\leq CK.
\]
The realification $2\choscRea L_{\epsilon,z}^K$ has the same squared bound with a factor $4$.
\end{prop}
\begin{proof}
Put $d=n-1$. For each fixed $t,z,\epsilon$, the slice
\[
 A_{t,z,\epsilon}^Kf(y)=\left(\frac K{2\pi}\right)^{d/2}
 \int e^{iK\phi_\epsilon(t,y,z,\theta)}
 a_\epsilon(t,y,z,\theta)f(\theta)\choscdd\theta
\]
has a uniformly bounded $L^2$ norm by the mask-free proof of Proposition~\ref{app:ch_osc:prop:osc-normal}, now in dimension $d$. At time $t$, the operator $L_{\epsilon,z}^K$ is $\sqrt{K/(2\pi)}$ times this slice. Integrating the squared slice estimate over the fixed bounded interval $J_{\rm time}$ proves the claim. The inequality $\|2\choscRea v\|\leq2\|v\|$ gives the real estimate.
\end{proof}

For the actual carrier families,
Proposition~\ref{app:ch_seed:prop:seed-phase-charts} verifies these
hypotheses for the exact finite-difference phase
\eqref{app:ch_seed:eq:seed-exact-phase-quotient}, including its
smooth value at zero from \eqref{app:ch_seed:eq:seed-quotient-integral}.


\subsection{Exact phase localization and Sobolev tails}
\label{app:ch_osc:sec:osc-tails}


\begin{prop}[A radial shell and negative Sobolev gain]\label{app:ch_osc:prop:osc-source-shell}
Fix $n\geq1$, a bounded box $X\subset\choscR^n$, a compact $K\Subset X$, a finite complex measure $\mu$ on a parameter set $D$, $0<c<C$, $b\geq0$, and $0<\sigma<1$. For $\rho\geq2$ and $L\geq1$, let $\phi(x,u)$ be a smooth real phase and $a(x,u)$ a smooth complex amplitude of fixed finite dimension, supported in $K$ in $x$. Assume uniformly in $u\in D$
\[
 c\leq|d_x\phi|\leq C,\qquad
 |D_x^r\phi|\leq C_rL^{r-1}\ (r\geq2),\qquad
 |D_x^ra|\leq C_rAL^{r+b}\ (r\geq0),
\]
where $A\geq0$. Set
\[
 h_\rho(x)=\int_D e^{i\rho\phi(x,u)}a(x,u)\,d\mu(u).
\]
Choose one fixed rescaled smooth radial multiplier $\Pi_\rho$ equal to $1$ on $c\rho/2\leq|\xi|\leq2C\rho$, with support in $c\rho/4\leq|\xi|\leq4C\rho$. For integers $m\geq0$ and $q>m+n/2$,
\begin{equation}\label{app:ch_osc:eq:osc-source-tail}
 \|(I-\Pi_\rho)h_\rho\|_{H^m}
 \leq C_{m,q}A\|\mu\|_{\rm TV}L^b
 \rho^{m+n/2}(L/\rho)^q.
\end{equation}
If $L\leq\rho^{1-\sigma}$, then for every integer $M\geq0$,
\begin{equation}\label{app:ch_osc:eq:osc-source-negative}
 \|h_\rho\|_{H^{-M}}
 \leq C_{M,\sigma}A\|\mu\|_{\rm TV}L^b\rho^{-M}.
\end{equation}
The source, cutoff family and derivative exponent $b$ are fixed for all requested observation and tail orders.
More generally, the tail estimate uses only the upper phase bounds
and the gap
$|\rho d_x\phi-\xi|\geq c_1(\rho+|\xi|)$ on the entire complement
of a fixed bounded rescaled Fourier mask. If that mask is supported
in a fixed annulus, the negative-norm conclusion follows under
the same scale gap.
For $d\mu=F\,du$ on a set of measure $V$,
$\|\mu\|_{\rm TV}\leq V^{1/2}\|F\|_2$.
\end{prop}
\begin{proof}
For fixed $u$, the Fourier kernel is
\[
 \widehat h_u(\xi)=(2\pi)^{-n}\int_K
 e^{i[\rho\phi(x,u)-x\cdot\xi]}a(x,u)\choscdd x.
\]
On the entire support of $1-\Pi_\rho$, either $|\xi|<c\rho/2$ or $|\xi|>2C\rho$, including its transition regions. Thus
\[
 |\rho d_x\phi-\xi|\geq c_1(\rho+|\xi|).
\]
Integrate by parts with
\[
 \mathcal V_\xi=\frac{(\rho d_x\phi-\xi)\cdot\partial_x}
 {i|\rho d_x\phi-\xi|^2}.
\]
Reciprocal differentiation and the phase bounds show that its coefficients have derivatives of order $j$ bounded by $C_jL^j/(\rho+|\xi|)$. An amplitude derivative costs one factor $L$. Induction on the number $q$ of transposed integrations gives
\[
 |(1-\pi_\rho(\xi))\widehat h_u(\xi)|
 \leq C_qAL^{q+b}(\rho+|\xi|)^{-q}.
\]
All boundary terms vanish because $K$ is a common compact support.
Integrate against $|\mu|$.
For $q>m+n/2$, rescaling $\xi=\rho\omega$ gives
\[
 \int_{\choscR^n}\langle\xi\rangle^{2m}(\rho+|\xi|)^{-2q}\choscdd\xi
 \leq C_{m,q}\rho^{2m+n-2q}.
\]
Taking square roots proves \eqref{app:ch_osc:eq:osc-source-tail}.

The amplitude bound and fixed output support give
$\|h_\rho\|_2\leq CA\|\mu\|_{\rm TV}L^b$.
The annular split and the tail estimate at $m=0$ therefore give
\[
 \|h_\rho\|_{H^{-M}}
 \leq C_qA\|\mu\|_{\rm TV}L^b
       [\rho^{-M}+\rho^{n/2}(L/\rho)^q].
\]
Take $q>n/2$ with $\sigma q\geq M+n/2$ to prove
\eqref{app:ch_osc:eq:osc-source-negative}.
Only the complementary gap was used for the tail, and annular support
for the negative norm.
\end{proof}

\begin{rem}\label{app:ch_osc:rem:osc-difference-frequency}
The true frequency in Proposition~\ref{app:ch_osc:prop:osc-source-shell} may be $\rho=\lambda|z-z'|$ for a normalized same-carrier difference phase. The estimate applies when the normalized phase satisfies the displayed gradient and graded derivative hypotheses. The phase is used exactly, rather than expanded in the source parameter.
\end{rem}

\subsection{Canonical retarded integration}
\label{app:ch_osc:sec:osc-retarded}
The literal integral gives all integer Sobolev bounds.

\begin{defn}[The fixed retarded operator]\label{app:ch_osc:setup:osc-retarded}
Fix real smooth cutoffs $\chi_{\rm in},\chi_{\rm out}$ compactly
supported in one rectangle $(t_-,t_+)\times Y\subset\R\times\R^d$,
$n=d+1$, with a fixed positive buffer after $t_-$ for
$\supp\chi_{\rm in}$. Put
\[
 Jf(t,y)=\chi_{\rm out}(t,y)\int_{t_-}^t
                  \chi_{\rm in}(s,y)f(s,y)\,ds.
\]
\end{defn}

\begin{prop}[The fixed retarded bounds]
\label{app:ch_osc:prop:osc-retarded-tube}
The literal adjoint is
\begin{equation}\label{app:ch_osc:eq:retarded-literal-adjoint}
 J^*v(s,y)=\chi_{\rm in}(s,y)
                 \int_s^{t_+}\chi_{\rm out}(t,y)v(t,y)\,dt.
\end{equation}
Both operators are bounded on $H^m(\R^n)$ for every integer $m$;
on buffered rectangles the negative bounds use the reversed
supported/restriction pairs.
For real $\lambda$
and transverse $\zeta$,
\begin{equation}\label{app:ch_osc:eq:osc-retarded-commute}
 e^{-i\lambda\zeta\cdot y}J[e^{i\lambda\zeta\cdot y}f]=Jf.
\end{equation}
\end{prop}
\begin{proof}
Fubini gives the adjoint. Cauchy--Schwarz on the fixed time
interval gives both $L^2$ bounds. For $j\geq1$,
\[
 \partial_t^j\int_{t_-}^t g(s,y)\,ds
                  =\partial_t^{j-1}g(t,y),
\]
and transverse derivatives pass under the integral. Leibniz' rule
gives the $H^m$ bounds for $m\geq0$, with the same argument for
the upper integral $J^*$. Their literal adjoint relation gives all
negative indices on the reversed supported/restriction pairs.
The transverse multiplier commutes with both weights and the integral.
\end{proof}


\section{A fixed flat reference and its carrier geometry}\label{app:sec:ch_seed}
The flat reference, finite carrier menu and common Hamilton charts
are constructed here. Setup~\ref{app:ch_seed:setup:seed-current}
specifies the current-map neighborhood; its metric, normal and
stress gaps hold near the reference by
Corollary~\ref{app:ch_seed:cor:seed-uniform-dilation}.
Every later use of this geometry refers to these proved constructions.
\subsection{An explicit flat cube and its characteristics}
\label{app:ch_seed:sec:seed-cube}

\begin{defn}[Cube notation]\label{app:ch_seed:defn:seed-cube}
Let $n\geq5$, $N=n(n+1)/2$, $q=n(n-1)/2$, $d=n-4$ and
$c_n=d^{-2}$. Let $E=\chseedSym^2\chseedR^n$ have its Frobenius
pairing, and write $S_z=\sum_i z_i$ and $J=\mathbf1\mathbf1^T$.
Set
\[
 Q_i^0=e_ie_i^T-c_nJ,\qquad
 K_0=\chseedspanof\{Q_i^0\}_{i=1}^n,\qquad
 \ell_{ab}=\mathbf1-2e_a-2e_b\quad(a<b).
\]
The model pencil and its quadratic map are
\[
 M_0(\xi)=\chseeddiag(\xi_i)-c_nS_\xi J,\qquad
 p_0(\xi)=\det M_0(\xi),\qquad
 q_0(\xi)=(\xi^TQ_i^0\xi)_i=M_0(\xi)\xi.
\]
For an immersion $F$, put $\gamma(F)=(dF)^TdF$ and regard its
second form as $B_F:E\to N_F$. Its stress space is $\ker B_F$.
Full normal rank means that $B_F$ is onto; then $B_F^*$ is
injective.
\end{defn}

\begin{prop}[A flat cube form]\label{app:ch_seed:prop:seed-gram}
The form
\[
 B_0(u,v)=\big((\ell_{ab}\cdot u)(\ell_{ab}\cdot v)\big)_{a<b}
\]
is onto, has kernel $K_0$ and satisfies the flat Gauss equations.
\end{prop}
\begin{proof}
The matrices $(e_a+e_b)(e_a+e_b)^T$, $a<b$, are independent by
their distinct off-diagonal entries. Since
\[
 \ell_{ab}=-2(I-J/4)(e_a+e_b),\qquad \det(I-J/4)=1-n/4\ne0,
\]
the $q$ matrices $\ell_{ab}\ell_{ab}^T$ are independent.
Every $Q_i^0$ annihilates them: each coordinate of $\ell_{ab}$
has square one and $S_{\ell_{ab}}=d$. The $n$ matrices $Q_i^0$
are independent, so dimension gives $\ker B_0=K_0$.
Each rank-one component satisfies the Gauss equations separately.

The rank-one determinant formula gives
\begin{equation}\label{app:ch_seed:eq:seed-det}
 p_0(\xi)=\prod_i\xi_i-c_nS_\xi\sum_i\prod_{j\ne i}\xi_j.
\end{equation}
\end{proof}

\begin{prop}[The complete bad set]\label{app:ch_seed:prop:seed-strata}
The nonzero bad set $\mathfrak M=\{p_0=0,\ dp_0=0\}\setminus\{0\}$
is the union of
\[
 L_{ijl}=\{\xi_i=\xi_j=\xi_l=0\},\qquad
 L_{ijS}=\{\xi_i=\xi_j=S_\xi=0\},
\]
and the lines $\chseedR\ell_{ab}$, with the origin removed.
These lines are also precisely the nonzero common zeros of $q_0$.
Let $\mathcal L$ consist of these codimension-three spaces and node
lines, so $\mathfrak M=(\bigcup_{L\in\mathcal L}L)\setminus\{0\}$.
\end{prop}
\begin{proof}
If $S_\xi=0$, the matrix is diagonal. Otherwise its kernel equation is
\[
 \xi_i v_i=c_nS_\xi\sum_jv_j.
\]
With no zero coordinates this has at most one free scalar.
A zero coordinate forces $\sum_jv_j=0$ and confines the kernel
to the zero positions. Thus corank at least two is equivalent to
membership in the displayed spaces, and makes the adjugate zero.

At corank one, $\operatorname{adj}M_0=a\,vv^T$ with $a\ne0$,
so $dp_0(\xi)=a\,q_0(v)$. A nonzero common zero of the quadrics
satisfies $v_i=\pm S_v/d$ and $S_v\ne0$. Summing gives sign sum
$d=n-4$, hence exactly two negative signs: $v\in\R\ell_{ab}$.
The kernel equation then gives $\xi=(S_\xi/S_v)v$.
Conversely each node line has corank one and zero determinant
gradient. This proves both assertions.
\end{proof}

\subsection{The actual strain and an analytic flat reference}
\label{app:ch_seed:sec:seed-geometry}
Successive circular bendings realize the flat form and fix the later geometry.

For an immersion $F$ and variation $V$, put
$u_i=\langle\partial_iF,V\rangle$. The proved identity
\maineqref{main:eq:tangent-normal} gives
$\tfrac12Q_a:D\gamma_FV=Q_a:\operatorname{sym}\nabla^{\gamma(F)}u$
for every stress $Q_a$. Thus the reduced principal row is
$(Q_a\xi)^T$ and its zero-order term is
$-\sum_{ij}(Q_a)_{ij}\Gamma_{ij}^ku_k$.

\begin{prop}[An explicit analytic flat reference]\label{app:ch_seed:prop:seed-analytic}
There is a real analytic local embedding $F_0:B_r\to\chseedR^N$
such that
\[
 F_0(0)=0,\qquad dF_0(0)=(I,0),\qquad
 \gamma(F_0)=I,\qquad B_{F_0}(0)=B_0.
\]
Its center stress space is $K_0$.
\end{prop}
\begin{proof}
For a unit vector $v\in\chseedR^m$ and $t=v\cdot Y$, define
\[
 \mathcal C_v(Y)=
 \left(Y+\left[\frac{\sin(nt)}n-t\right]v,
             \frac{1-\cos(nt)}n\right)\in\chseedR^{m+1}.
\]
Its derivative is an isometry: it fixes $v^\perp$ and sends
$v$ to the unit vector $(\cos(nt)v,\sin(nt))$.
At zero its first derivative is the standard inclusion and its
second derivative is $n\,v\otimes v$ in the new normal direction.

Start with the identity on $\chseedR^n$. In a fixed order of the
$q$ pairs $a<b$, compose with $\mathcal C_v$ using
$v=(\ell_{ab}/\sqrt n,0)\in\chseedR^m$.
All maps have center zero and standard first derivative there.
Each composition leaves the preceding center second forms
unchanged and adds exactly $\ell_{ab}\otimes\ell_{ab}$.
The resulting analytic map has the stated metric and second form.
Its projection to the first $n$ coordinates is a local
diffeomorphism, so it is an embedding after shrinking.
Proposition~\ref{app:ch_seed:prop:seed-gram} gives full normal
rank at the center, which persists nearby.
\end{proof}

\begin{cor}[One fixed normalized flat reference]
\label{app:ch_seed:cor:seed-uniform-dilation}
For any prescribed finite kernel topology and sufficiently small
neighborhood of the constant cube, one can choose a fixed $\rho>0$
so that
\[
 G_{\mathrm{ref}}(y)=
 \rho^{-1}\big(F_0(\rho y)-F_0(0)\big)
\]
is defined on a fixed ball, has $\gamma(G_{\mathrm{ref}})=I$,
and has an actual stress frame in that neighborhood.
Its normal gap is at least $c\rho>0$. After $\rho$ is fixed,
all its geometric seminorms are finite, and the required metric,
normal and kernel gaps persist on one fixed finite-topology
neighborhood of $G_{\mathrm{ref}}$.
\end{cor}
\begin{proof}
Full normal rank gives a smooth stress frame with $Q_i(0)=Q_i^0$.
The dilated frame is $Q_i(\rho y)$. Its zeroth difference from
$Q_i^0$ tends to zero; every positive-order derivative of order $j$
has the factor $\rho^j$. Thus its difference tends to zero in any
one prescribed finite topology.

Moreover $dG_{\mathrm{ref}}=dF_0(\rho y)$ and
$d^2G_{\mathrm{ref}}=\rho d^2F_0(\rho y)$. The normal spaces agree,
so $B_{G_{\mathrm{ref}}}=\rho B_{F_0}(\rho y)$ and its least
singular value is at least $c\rho$ on a smaller fixed ball.
The metric remains exactly $I$. Smooth dependence of the normal
projector and kernel frame gives the stated neighborhood, using
two additional map derivatives for the chosen kernel topology.
All later constants may depend on this fixed $\rho$; the actual
stress frame need not be constant away from zero.
\end{proof}

\subsection{Frozen three-parameter carrier geometry}
\label{app:ch_seed:sec:seed-frozen}


\begin{prop}[A finite three-parameter menu]\label{app:ch_seed:prop:seed-frozen-menu}
Let $n\geq5$ and use the cube pencil of
Definition~\ref{app:ch_seed:defn:seed-cube}.
There are finitely many simple-characteristic patches
\[
 \xi_a\colon Z_a\longrightarrow\chseedR^n,
\]
where $Z_a$ is a fixed closed convex three-dimensional parameter cube,
and a finite open cover $V_b$ of $\mathbb S^{n-1}$, with the following
uniform properties.
\begin{enumerate}
\item Every carrier has all coordinates nonzero, $p_0(\xi_a)=0$,
$\chseedrank M_0(\xi_a)=n-1$, and norm bounded above and below.
With $r_{a,i}=1/\xi_{a,i}$, the vector $v_a=q_0(r_a)$ is a
nonzero characteristic normal. For every $L\in\mathcal L$ and
$z,z'\in Z_a$,
\begin{equation}\label{app:ch_seed:eq:seed-frozen-chord}
 |\pi_{L^\perp}(\xi_a(z)-\xi_a(z'))|\geq c|z-z'|,
 \qquad |\xi_a(z)-\xi_a(z')|\leq C|z-z'|.
\end{equation}
Every nonzero same-patch difference and every same-patch sum has
a normalized direction uniformly separated from $\mathfrak M$.
\item For each $V_b$ there is a finite selected subcollection and
smooth incidence points $z_a(k)$ in the patch interiors such that
\[
 k\cdot v_a(z_a(k))=0,\qquad
 |\partial_{z_1}(k\cdot v_a(z))|\geq c
\]
near the selected roots, for one chosen first parameter on each patch.
Moreover,
\begin{equation}\label{app:ch_seed:eq:seed-frozen-detection}
 |M_0(k)^Ts|^2+\sum_a|s\cdot q_0(\xi_a(z_a(k)))|^2
 \geq c|s|^2,\qquad s\in\R^n,\quad k\in V_b.
\end{equation}
\item For any fixed finite grouping into color groups, positive radial
labels $R_g$ can be fixed so that every cross-group sum or difference
$R_g\xi_a(z)\pm R_{g'}\xi_{a'}(z')$ has normalized direction
uniformly separated from $\mathfrak M$.
Labels are independent of later sampling spacing and accuracy.
Within one patch and group,
\eqref{app:ch_seed:eq:seed-frozen-chord} controls every nonzero
difference and the sum bound controls every sum.
\end{enumerate}
This proposition concerns frozen geometry, not a current-field
parametrix or an analytic frame lower bound.
\end{prop}
\begin{proof}
\noindent\textbf{Step 1. Paired detecting carriers.}
For all-nonzero $r$ and $\xi_i=r_i^{-1}$, the determinant formula
gives $p_0(r)=p_0(\xi)=0$ exactly when $S_rS_\xi=d^2$.
In that case $M_0(\xi)$ and $M_0(r)$ have respective kernels
$\R r$ and $\R\xi$.
Fix $k\ne0$ and $0\ne s\in\ker M_0(k)$.
We seek disjoint coordinate pairs $I,J$ such that, with
\[
 K_I=\sum_{i\in I}k_i,\quad S_I=\sum_{i\in I}s_i,
 \qquad K_J=\sum_{i\in J}k_i,\quad S_J=\sum_{i\in J}s_i,
\]
one has
\begin{equation}\label{app:ch_seed:eq:seed-pair-criterion}
 \begin{gathered}
 (S_I,S_J)\ne(0,0),\qquad
 \bigl[K_I=K_J=0\quad\hbox{or}\quad K_IK_J(K_I+K_J)\ne0\bigr].
 \end{gathered}
\end{equation}
The kernel equation is
\begin{equation}\label{app:ch_seed:eq:seed-kernel-pair}
 k_is_i=d^{-2}S_kS_s.
\end{equation}
If $k$ has a zero coordinate, this confines $s$ to its zero
coordinates; if also $S_k\ne0$, it gives $S_s=0$.

First let $n\geq6$.
With at least four zero coordinates, choose a pair of zeros whose
$s$-sum is nonzero, and another disjoint pair of zeros.
Such a first pair exists because vanishing of all pair sums on at
least three positions forces every component to vanish.
With two or three zero coordinates, at least three nonzero entries
of $k$ remain. Two have nonzero sum; pair them with different zero
coordinates, including one where $s_i\ne0$.
With one zero coordinate, at least five nonzero entries contain
three of one sign. Pair two of these together and the third with
the zero coordinate.
These choices satisfy \eqref{app:ch_seed:eq:seed-pair-criterion}.
If $k$ has no zero coordinate, \eqref{app:ch_seed:eq:seed-kernel-pair}
implies $S_k\ne0$ and $s_i=C/k_i$ with $C\ne0$.
Four entries of one sign give the required disjoint pairs.
The remaining case has $n=6$ with three entries of each sign.
Over all positive pairs $I$ and negative pairs $J$,
\[
 \sum_{I,J}(K_I+K_J)=6S_k\ne0.
\]
Some summand is nonzero, and the corresponding same-sign pair sums
of both $k$ and $s$ are nonzero. This proves the criterion for $n\geq6$.

Now let $n=5$, so $d=1$, and let $z$ be the number of zero
coordinates of $k$.
For $z=4$, use two pairs of zero coordinates, choosing the first
with nonzero $s$-sum.
For $z=3$ and $S_k=0$, the two nonzero entries of $k$ form one
pair with $K_I=0$; choose as the other pair two zeros with nonzero
$s$-sum.
For $z=3$ and $S_k\ne0$, pair the two nonzero entries separately
with two zeros, choosing at least one zero where $s_i\ne0$.
The two $K$-sums are nonzero and their sum is $S_k\ne0$.
For $z=2$, choose two of the three nonzero entries of $k$ whose
sum is nonzero, and pair them separately with the two zeros.
For $z=1$, \eqref{app:ch_seed:eq:seed-kernel-pair} and $s\ne0$
force $S_k=0$. Choose two nonzero entries with nonzero sum as
one pair. Pair the unique zero with one of the two remaining
nonzero entries. The two $K$-sums are nonzero; their sum is the
negative of the last unused entry and is therefore nonzero.
The pair containing the zero has nonzero $s$-sum.

It remains to consider $z=0$ in dimension five. Again
$S_k\ne0$, $s_i=C/k_i$, and
\begin{equation}\label{app:ch_seed:eq:seed-five-reciprocal}
 S_k\sum_i\frac1{k_i}=1.
\end{equation}
If four entries have one sign, choose two disjoint pairs of that
sign. Otherwise, after replacing $k$ by $-k$ if necessary and
permuting coordinates, write
$k=(a,b,c,u,v)$ with $a,b,c>0$ and $u,v<0$.
Take $J=\{4,5\}$ and any pair $I\subset\{1,2,3\}$.
Unless one of these choices satisfies $K_I+K_J\ne0$, all three
positive pair sums equal $-(u+v)$; hence
$a=b=c=A>0$ and $u+v=-2A$.
Then $S_k=A$, and \eqref{app:ch_seed:eq:seed-five-reciprocal} gives
\[
 1=A\left(\frac3A+\frac{u+v}{uv}\right)
   =3-\frac{2A^2}{uv}.
\]
Thus $uv=A^2$, and $u=v=-A$.
Consequently the pair criterion holds in dimension five except
possibly when $k$ belongs to one of the node lines $\R\ell_{ab}$.

Whenever the pair criterion holds, choose $\alpha,\beta$ with
$K_I\alpha+K_J\beta=0$ and
$\alpha\beta(\alpha-\beta)\ne0$.
Use $(\alpha,\beta)=(1,2)$ when $K_I=K_J=0$, and
$(K_J,-K_I)$ otherwise.
Put $A=t\alpha$, $B=t\beta$.
On $I$ let $r$ have entries $-\sqrt{1+A},\sqrt{1+A}$;
on $J$ use $-\sqrt{1+B},\sqrt{1+B}$; elsewhere put $r_i=1$.
For all sufficiently small $t$, these entries are nonzero and,
with $\xi_i=r_i^{-1}$,
\[
 S_r=S_\xi=d,\qquad
 k\cdot q_0(r)=0,\qquad
 s\cdot q_0(\xi)
 =-\frac{t[\alpha S_I+\beta S_J
                  +t\alpha\beta(S_I+S_J)]}
              {(1+t\alpha)(1+t\beta)}.
\]
The numerator is not identically zero: if both coefficients
vanished, then $S_I+S_J=0$ and
$(\alpha-\beta)S_I=0$, contradicting the criterion.
Choose small nonzero $t$ with nonzero pairing.
Then $\xi$ is regular and simple, and $q_0(r)$ has entries
$A,A,B,B$ on the two pairs and zero elsewhere.
It has four nonzero coordinates and no $n-2$ equal coordinates.
It therefore lies in none of the spaces $L_{ijl}^\perp$ or
$L_{ijS}^\perp$.

\medskip\noindent\textbf{Step 2. The five-dimensional node carriers.}
In dimension five consider the node direction
$\ell=(1,1,1,-1,-1)$; all other node lines follow by permutation.
Its kernel is $\ker M_0(\ell)=\R\ell$.
Set
\begin{equation}\label{app:ch_seed:eq:seed-five-node-carrier}
 \begin{aligned}
 r&=\left(1,1,-\tfrac12,
              \tfrac{-1+\sqrt3}{2},\tfrac{-1-\sqrt3}{2}\right),\\
 \xi&=\left(1,1,-2,1+\sqrt3,1-\sqrt3\right)=r^{-1}.
 \end{aligned}
\end{equation}
Here $S_r=1/2$, $S_\xi=2$, so both vectors are all-nonzero
simple characteristics. Direct calculation gives
\[
 \begin{aligned}
 q_0(r)&=\left(\tfrac34,\tfrac34,0,
          \tfrac34-\tfrac{\sqrt3}{2},
          \tfrac34+\tfrac{\sqrt3}{2}\right),\\
 q_0(\xi)&=(-3,-3,0,2\sqrt3,-2\sqrt3),\\
 \ell\cdot q_0(r)&=0,\qquad \ell\cdot q_0(\xi)=-6.
 \end{aligned}
\]
Thus this carrier detects every nonzero stress in the kernel at
every $k\in\R\ell\setminus\{0\}$ and has the required incidence.
Its normal $q_0(r)$ has four nonzero coordinates and no three
equal coordinates, and therefore again avoids every
$L_{ijl}^\perp$ and $L_{ijS}^\perp$.
Both characteristic normals are nonzero, so both characteristic
points are regular.

\medskip\noindent\textbf{Step 3. Transversality and three-dimensional patches.}
The preceding constructions give, for every relevant $k,s$,
all-nonzero reciprocal characteristics $r,\xi$ with
$k\cdot q_0(r)=0$, $s\cdot q_0(\xi)\ne0$ and a normal
$q_0(r)$ outside every codimension-three bad-normal space.
Incidence is transverse on the characteristic hypersurface.
Indeed, in reciprocal coordinates its defining function is
$E_k(r)=k\cdot q_0(r)=r^TM_0(k)r$, with gradient $2M_0(k)r$.
This gradient annihilates $s$, whereas the characteristic normal
$q_0(\xi)$ does not.
The incidence gradient cannot vanish: if $M_0(k)r=0$, the
coordinate equations and $r_i\ne0$ give
$k=d^{-2}S_kS_r\xi$, with a nonzero scalar factor.
Then $s\in\ker M_0(\xi)=\R r$, contradicting
$s\cdot q_0(\xi)\ne0$, since
$r\cdot q_0(\xi)=\xi^TM_0(r)\xi=0$.
Thus the two hypersurfaces meet transversely.
Reciprocal inversion is a local diffeomorphism, so this proves
the claimed transversality in the $\xi$ coordinates as well.

At a detecting carrier $c$ for $k_0$, put
$v=q_0(c^{-1})$ and $T=v^\perp$.
For a codimension-three $L$, the projection $T\to L^\perp$
is onto since $v\notin L^\perp$.
For a node line $L$, that projection has rank at least
$n-2\geq3$.
Consequently each Gram determinant
\[
 \det\bigl[\pi_{L^\perp}e_i\cdot\pi_{L^\perp}e_j
                                                  \bigr]_{i,j=1}^3
\]
is a nonzero polynomial on $T^3$.
Their finite product with $d(k_0\cdot v)(e_1)$ is also a
nonzero polynomial, by incidence transversality.
Choose a triple at which this product does not vanish.
Its span $E_3$ projects injectively to every $L^\perp$ and has
the required incidence derivative.

For a transverse direction $e$ with $dp_0(c)e\ne0$, the implicit
function theorem parameterizes a characteristic patch as
\[
 \xi(z)=c+E_3z+u(z)e,\qquad D\xi(0)=E_3.
\]
Here $E_3$ also denotes the injective linear map with the chosen
three columns; $u(0)=0$ and $Du(0)=0$.
Shrink a convex parameter cube so every
$\pi_{L^\perp}D\xi(z)$ stays within half the least singular
value of $\pi_{L^\perp}E_3$ in operator norm.
For $z,z'$ in the cube, subtract the linear map
$\pi_{L^\perp}E_3$ inside the segment identity; the resulting
error is at most half its lower singular bound times $|z-z'|$.
This proves \eqref{app:ch_seed:eq:seed-frozen-chord}.
Combining its lower and upper bounds separates every normalized
nonzero difference from every $L$.
On a smaller cube all sums have directions close to the fixed
nonzero regular carrier $c$, hence have a fixed gap from
$\mathfrak M$.
Rename the first parameter so that its incidence derivative is
nonzero, and shrink to keep a positive lower bound.

\medskip\noindent\textbf{Step 4. Finite covers and radial labels.}
At each $k_0$, the preceding detecting linear forms have no common
nonzero nullvector on $\ker M_0(k_0)$.
Compactness of its unit sphere selects finitely many carriers
with a positive detector gap there; since $M_0(k_0)$ is symmetric,
the Gram form in \eqref{app:ch_seed:eq:seed-frozen-detection}
is positive on $\R^n$.
For invertible $M_0(k_0)$ no detector is needed.
The implicit function theorem continues the transverse incidence
roots as $z_1=z_1(k)$, with $z_2=z_3=0$, in a neighborhood
of $k_0$.
Continuity preserves the full Gram gap on a smaller neighborhood.
Compactness of $\mathbb S^{n-1}$ supplies a finite subcover;
take the minimum of its finitely many positive margins.

All carriers lie in a compact nonzero regular set.
Choose radial labels successively with ratios sufficiently large
that a cross-group sum or difference has normalized direction as
close as desired to the carrier of the larger group, up to sign.
A finite ratio suffices because carrier norms have common positive
upper and lower bounds.
Each carrier and its negative have a positive gap from
$\mathfrak M$, so the cross-group directions have one as well.
The labels are fixed independently of later sampling or accuracy.
The same-patch assertions were already proved by the chord and
sum estimates.
\end{proof}

\subsection{Actual Hamilton geometry and buffered phase charts}
\label{app:ch_seed:sec:seed-actual}
We deform the frozen menu to actual current immersions and verify the common inverse-chart hypotheses for exact full-dispersion phases.

\begin{lem}[A common homogeneous Hamilton phase]
\label{app:lem:common-hamilton-phase}
Let $h_a$ be a real degree-one principal symbol on fixed physical
and nonzero conic boxes, in a sufficiently small fixed $C^3$
neighborhood of $h_0$, with $|\partial_{\xi_1}h_a|\geq c>0$.
Assume its normalized order-$m$ seminorm is bounded by
$C_m(1+\|a\|_{C^{m+c_0}})$, with fixed $c_0$.
There are common buffered physical and canonical boxes and a
homogeneous phase, for $\eta=(e,\eta')$, satisfying
\begin{equation}\label{app:ch_regular_base:eq:actual-eikonal}
 h_a(x,S_x(x,\eta))=e,\qquad S(0,x',\eta)=x'\cdot\eta'.
\end{equation}
Its coordinates $S_\eta=(t,u')$ are Hamilton time and the initial
section point. The entire canonical preimage of the inner physical
box lies in a fixed larger source rectangle.
The matrix $S_{x\eta}$ has a uniform inverse gap, and
\begin{equation}\label{app:ch_regular_base:eq:averaged-gradient}
 \Xi(x,\widetilde x,\eta)
 =\int_0^1S_x(\widetilde x+\tau(x-\widetilde x),\eta)\,d\tau
\end{equation}
is injective in $\eta$ on the buffered convex chart and its conic
extension, with a common buffered inverse domain.
All joint phase, flow and inverse derivatives through order $m$
have a one-high bound at coefficient order $m+c$, for fixed $c$.
The construction is continuous from the normalized $C^{m+1}$ symbol
topology to $C^m$ phases, at each fixed $m$.
All phase assertions concern the homogeneous principal symbol;
low-frequency and lower-order operator terms are separate.
\end{lem}
\begin{proof}
The map $\xi\mapsto(h_a(x,\xi),\xi')$ has Jacobian
$h_{a,\xi_1}$. Euler's identity excludes zero from its image on
a nonzero transverse patch. Its homogeneous inverse therefore has
$c|\eta|\leq|\xi|\leq C|\eta|$ on common buffered convex conic
boxes, also when $e\ne0$ and $\eta'=0$.
The transverse gap gives the unique homogeneous root
$h_a(x,g(x,e,\eta'),\eta')=e$, with
$g_e=(h_{a,\xi_1})^{-1}$. Use the Hamilton field
$X_h=(h_{a,\xi},-h_{a,x})$ starting at
$(0,u',g(0,u',e,\eta'),\eta')$.
Its low bounds and homogeneity give, for $\omega=\xi/|\xi|$,
\[
 |x(t)-x(0)|+|\omega(t)-\omega(0)|\leq C|t|,\qquad
 e^{-C|t|}\leq|\xi(t)|/|\xi(0)|\leq e^{C|t|}.
\]
Choose one short interval for both signs inside the spatial and
angular buffers; a first-exit argument gives the entire flow there.
At $t=0$, the differential of $(t,u')\mapsto x$ has columns
$h_{a,\xi},e_2,\ldots,e_n$ and determinant of magnitude
$|h_{a,\xi_1}|\geq c$.
After one refinement it stays close to a central invertible matrix.
Lemma~\ref{app:lem:buffered-inverse-stability} gives the common
inverse projection and preimage buffer, uniformly in $(e,\eta')$.

For a variation of initial data at fixed time, Hamilton's equations
and degree-one homogeneity give
\[
 \frac d{dt}(\xi\cdot\delta x)
 =(-h_x+\xi\cdot h_{\xi x})\delta x
                +(\xi\cdot h_{\xi\xi})\delta\xi=0.
\]
Initially $\xi\cdot\delta x=\eta'\cdot\delta u'$.
Since $\xi\cdot\dot x=h_a=e$, allowing time to vary gives
$\xi\cdot dx=e\,dt+\eta'\cdot du'$. Therefore
\[
 S(x,\eta)=e\,t(x,\eta)+u'(x,\eta)\cdot\eta',\qquad
 dS=\xi\cdot dx+t\,de+u'\cdot d\eta'.
\]
This proves the eikonal and canonical-coordinate identities.
Positive covector scaling leaves the base flow unchanged, so $S$
is homogeneous. The joint implicit, flow and inverse estimates of
Lemma~\ref{app:ch_regular_base:lem:buffered-interpolation} use symbol
orders at most $m+1$, and give the claimed fixed-shift bound.
Subtracting the differentiated equations also proves the asserted
finite-order continuity on the same domains.

On the section, $S_{x'\eta'}=I$, $S_{x'e}=0$ and
$S_{x_1e}=(h_{a,\xi_1})^{-1}$. Shrink the boxes and strip once
so $S_{x\eta}$ and hence $\Xi_\eta$ stay close to a central
invertible matrix. Lemma~\ref{app:lem:buffered-inverse-stability}
gives the common inverse domain. Since $\Xi_\eta$ is degree zero,
the same matrix gap holds on the convex conic extension; its
segment identity proves injectivity there. Inverse differentiation
and the joint estimates give the one-high bounds.
\end{proof}

\begin{setup}[A finite current neighborhood]\label{app:ch_seed:setup:seed-current}
Fix the patches, cover, roots and labels of
Proposition~\ref{app:ch_seed:prop:seed-frozen-menu}, and put
$M_{\rm an}=20(n+1)$. Let $F\colon B_1\to\chseedR^N$ be a real
smooth immersion with $\|F\|_{C^4}\leq C_0$, induced metric
$g=(dF)^TdF\geq\kappa_{\rm met}I$, and actual second form
$B_F\colon E\to N_F$ onto with least normal singular value at
least $\kappa>0$. Use throughout the projected cube frame
\[
 \begin{aligned}
 P^\perp&=I-dF g^{-1}(dF)^T,
 &\mathcal A_F&=B_FB_F^*+(I-P^\perp),\\
 \Pi_{K_F}&=I-B_F^*\mathcal A_F^{-1}B_F,
 &Q_i&=\Pi_{K_F}Q_i^0,
 \end{aligned}
\]
and assume $\max_i\|Q_i-Q_i^0\|_{C^6(B_1)}<\epsilon_0$.
The two gaps are separate hypotheses; they bound
$\|g^{-1}\|\leq\kappa_{\rm met}^{-1}$ and
$\|\mathcal A_F^{-1}\|\leq\max(1,\kappa^{-2})$.
The small low neighborhood makes $Q_i$ a frame of $K_F=\ker B_F$.
Define
\[
 M_F(x,\xi)_{i\cdot}=(Q_i(x)\xi)^T,
 \qquad p_F(x,\xi)=\det M_F(x,\xi),
 \qquad q_F(x,\xi)=(Q_i(x):\xi\xi^T)_i.
\]
\end{setup}

\begin{prop}[Actual carrier phases, signed pairs and detection]\label{app:ch_seed:prop:seed-actual-menu}
There are fixed balls $B_{\rm in}\Subset B_{\rm out}\Subset B_1$ and a fixed choice of $\epsilon_0>0$ such that every $F$ in Setup~\ref{app:ch_seed:setup:seed-current} has the following properties on the same $B_{\rm out}$.
\begin{enumerate}
\item For every frozen patch there is a jointly smooth real family $\Theta_a(x,z)$ on its fixed convex parameter cube, with
\[
 p_F(x,d_x\Theta_a)=0,\qquad \chseedrank M_F(x,d_x\Theta_a)=n-1.
\]
Its gradient is uniformly close to $\xi_a(z)$ in $C_z^1C_x^0$, and has a uniform upper parameter Lipschitz bound. For all $L\in\mathcal L$, all $x$ and all $z,z'$ in one patch,
\begin{equation}\label{app:ch_seed:eq:seed-actual-chord}
 |\pi_{L^\perp}(d\Theta_a(x,z)-d\Theta_a(x,z'))|\geq c|z-z'|.
\end{equation}
All nonzero same-patch differences, same-patch sums, and the cross-group signed pairs with the fixed radial labels have normalized directions in one compact set separated from $\mathfrak M$. On those directions the $(n-1)$st singular value of $M_F$ has a positive lower bound. At any actual determinant zero there, $|d_\xi p_F|\geq c$. Thus the directions are elliptic or simple real principal type.
\item On each member of a fixed smaller cover $V_b$ of $\mathbb S^{n-1}$, the selected patches have jointly smooth interior roots $z_a(F;x,k)$ satisfying
\[
 k\cdot d_\xi p_F(x,d\Theta_a(x,z_a))=0,
 \quad
 |\partial_{z_1}[k\cdot d_\xi p_F(x,d\Theta_a(x,z))]|\geq c.
\]
The same roots satisfy
\begin{equation}\label{app:ch_seed:eq:seed-actual-detection}
 |M_F(x,k)^Ts|^2+
 \sum_a|s\cdot q_F(x,d\Theta_a(x,z_a(F;x,k)))|^2
 \geq c|s|^2,\qquad s\in\R^n.
\end{equation}
\end{enumerate}
All low domains and gaps depend only on the frozen finite data,
$\kappa,\kappa_{\rm met},C_0$ and the fixed $C^6$ neighborhood,
not on higher norms or accuracy. The same zero-energy phase $S_F$
used below, carrier gradients and incidence roots obey
one estimate on their stated fixed buffered domains:
\begin{equation}\label{app:ch_seed:eq:seed-phase-tame}
 \|\mathscr G_F\|_{C^r}
 \leq C_r(1+\|F\|_{C^{r+6}}),\qquad
 \mathscr G_F=(S_F,d\Theta_a,z_a),\quad r\geq0.
\end{equation}
The norm is the maximum over the finite family, jointly in physical
variables, normalized covectors and its parameters. The right-hand
norm is on the fixed construction neighborhood.
\end{prop}
\begin{proof}
\noindent\textbf{Step 1.} We use the zero-energy part of the common phase construction.
On each enlarged frozen patch choose adapted physical coordinates
$x=(t,y)$, $\xi=(\tau,\zeta)$ with a fixed
$\partial_\tau p_0>c>0$ gap. Put $h_F=p_F|\xi|^{1-n}$.
Lemma~\ref{app:lem:common-hamilton-phase} applies directly to
$h_F$. On its zero set the Hamilton field is
$|\xi|^{1-n}X_{p_F}$; finite minima give common buffers for the menu.
Restrict the full-energy phase to $e=0$ and call it $S_F$. It satisfies
\[
 \partial_tS_F=H_F(t,y,d_yS_F),\qquad
 p_F(x,(H_F(x,\zeta),\zeta))=0,\qquad
 S_F(0,y,\zeta)=y\cdot\zeta.
\]
The root is the unique homogeneous continuation of $H_0$, and
$S_0=tH_0(\zeta)+y\cdot\zeta$. Thus the primary characteristic
phase is exactly the zero-energy restriction of the current
full-energy phase, with the same section data and sign convention.

The continuity in Lemma~\ref{app:lem:common-hamilton-phase}
gives $C^5$ closeness of these phases from the $C^6$ coefficient
closeness. The displayed
projector is a fixed-gap rational function of the second jet of $F$,
so Lemma~\ref{app:lem:elementary-interpolation-jets} gives
\begin{equation}\label{app:ch_seed:eq:seed-frame-tame}
 \max_i\|Q_i\|_{C^r}\leq C_r(1+\|F\|_{C^{r+2}}).
\end{equation}
That lemma uses symbol derivatives through $r+1$, so
$\|S_F\|_{C^r}\leq C_r(1+\|F\|_{C^{r+3}})$.
All choices use the fixed low bounds, without high-norm smallness.

\medskip\noindent\textbf{Step 2.} We keep the complete signed-pair geometry.
Let $\zeta_a(z)$ be the spatial projection of the frozen patch and set
\[
 \Theta_a(x,z)=S_F(x,\zeta_a(z)).
\]
The frozen gradient is $\xi_a(z)$. The $C^5$ comparison gives $C_z^1C_x^0$ closeness of the actual gradients to it. The characteristic equation is exact, and a frozen nonzero rank-$(n-1)$ minor persists.

For every $x,L$, the actual projected parameter map is $C^1$ close
to the frozen one on its convex cube. The same segment estimate
therefore preserves the lower chord bound
\eqref{app:ch_seed:eq:seed-actual-chord}. Together with the upper
derivative bound it gives a fixed angular gap, including arbitrarily
short nonzero chords. The compact gaps for sums and fixed-label
cross-group signed pairs persist by low continuity.

Take the closure of all normalized pair directions, including limits of normalized derivative chords. It is compact and avoids $\mathfrak M$. The $(n-1)$st singular value of $M_0$ has a positive minimum on it. Proposition~\ref{app:ch_seed:prop:seed-strata} makes the determinant gradient nonzero on its reference zero subset, and compactness gives a positive minimum there. On a slightly larger compact angular set, low $C^1$ closeness of $p_F$ and $M_F$ preserves the singular-value bound and ensures that every actual determinant zero keeps $|d_\xi p_F|\geq c$. This proves the claimed elliptic-or-simple classification.

\medskip\noindent\textbf{Step 3.} We continue incidence roots and stress detection.
The actual $d_\xi p_F(x,d\Theta_a)$ is $C_z^1$ close to a
nonzero scalar multiple of $v_a$. At incidence, differentiating
that scalar contributes zero because $k\cdot v_a=0$.
The implicit function theorem therefore continues the transverse
frozen roots to actual interior roots, uniformly for all $x$ on
the common ball and $k$ in a fixed smaller $V_b$.
The phase bound and joint implicit estimate use coefficient order
at most $r+4$, proving \eqref{app:ch_seed:eq:seed-phase-tame}.
The full-matrix gap \eqref{app:ch_seed:eq:seed-frozen-detection}
persists by continuity of $M_F$, $q_F$ and the roots, giving
\eqref{app:ch_seed:eq:seed-actual-detection}. Take a finite minimum
over the cover.

\end{proof}

\begin{prop}[Exact full-dispersion buffered charts]\label{app:ch_seed:prop:seed-phase-charts}
Under Setup~\ref{app:ch_seed:setup:seed-current}, fix an incidence patch and a base direction $k_b=(\omega_b,\eta_b)$ in its adapted coordinates. There are fixed compact convex spatial-covector and data charts, nested common convex physical boxes inside $B_{\rm out}$, and a fixed $\epsilon_*>0$, with the following property. With spectator parameters $z_2,z_3$ in a fixed sufficiently small box, write
\[
 D_{\rm chart}=I_{z_1}\times B_\eta,
 \qquad D_0\Subset D_{\rm chart},
\]
where $B_\eta\subset\chseedR^{n-1}$ is convex and contains $\eta_b$. For $0\leq\epsilon\leq\epsilon_*$ define
\begin{equation}\label{app:ch_seed:eq:seed-exact-phase-quotient}
 \phi_{F,\epsilon}(x,z_1,\eta)=
 \frac{S_F(x,\zeta_a(z)+\epsilon\eta)-S_F(x,\zeta_a(z))}{\epsilon},
\end{equation}
with its integral-derivative value at zero. Put
\[
 \Psi_{F,\epsilon}(x,x',u)=\int_0^1
 d_x\phi_{F,\epsilon}(x'+t(x-x'),u)\chseeddd t,
 \qquad u=(z_1,\eta).
\]
The normalized data charts stay in a fixed annulus and satisfy
\begin{equation}\label{app:ch_seed:eq:seed-normal-annulus}
 |\Psi_{F,\epsilon,y}-\eta|\leq\tfrac12|\eta|,
 \qquad c\leq|\Psi_{F,\epsilon}|\leq C
\end{equation}
on the entire indicated physical and data boxes.
For every $x,x'$ in those boxes, this map is globally one-to-one on $D_{\rm chart}$ and has a uniform positive absolute Jacobian determinant. It has an inverse on one fixed open $V_{\rm chart}$, with all images $\Psi_{F,\epsilon}(D_0)$ in a fixed compact subset of $V_{\rm chart}$. The inverse and its first $\epsilon$ derivative satisfy the same bound
\eqref{app:ch_seed:eq:seed-phase-tame} at every order $r$, jointly
on these fixed charts.

Uniformly in $t,z_1,\epsilon$, the transverse maps
\[
 \eta\longmapsto\int_0^1d_y\phi_{F,\epsilon}
 (t,y'+s(y-y'),z_1,\eta)\chseeddd s
\]
have analogous common buffered inverse charts and the same bounds on
fixed convex $\eta$ boxes. Include these maps, their inverses and
first $\epsilon$ derivatives in the same finite family $\mathscr G_F$;
all estimates are joint in physical, data, spectator and $\epsilon$
variables. These are the phase and chart hypotheses of
Setup~\ref{app:ch_osc:setup:osc-normal} and
Proposition~\ref{app:ch_osc:prop:osc-channel}.
\end{prop}
\begin{proof}
At the frozen reference,
$d_\xi p_0=p_{0,\tau}(1,-\nabla H_0)$. Incidence gives
$\omega_b=\eta_b\cdot\nabla H_0(\zeta_a(z))$, so $\eta_b\ne0$;
its transversality gives
$\partial_{z_1}[\eta_b\cdot\nabla H_0(\zeta_a(z))]\ne0$.
On sufficiently small convex data and spectator boxes, put
\[
 G(z_1,\eta)=\nabla H_0(\zeta_a(z))\cdot\eta,
 \qquad \Psi_0=(G,\eta).
\]
Then $\phi_0=tG+y\cdot\eta$ and
$\det D\Psi_0=\partial_{z_1}G$ has a fixed nonzero sign.
Shrink once so $D\Psi_0$ stays close to its central matrix.

The exact identity
\begin{equation}\label{app:ch_seed:eq:seed-quotient-integral}
 \phi_{F,\epsilon}(x,z_1,\eta)=\int_0^1
 \partial_\zeta S_F(x,\zeta_a(z)+s\epsilon\eta)\cdot\eta\,ds
\end{equation}
includes $\epsilon=0$ smoothly. The $C^5$ phase comparison in
Proposition~\ref{app:ch_seed:prop:seed-actual-menu} makes
$\Psi_{F,\epsilon}$ uniformly $C^1$ close to $\Psi_0$; these
derivatives and their first $\epsilon$ derivatives use phase jets
through order four. First choose the low kernel neighborhood,
then $\epsilon_*$ small. The transverse averaged-gradient map is
likewise $C^1$ close to $\eta\mapsto\eta$.
Since $\eta$ stays in a fixed annulus, the same comparison proves
\eqref{app:ch_seed:eq:seed-normal-annulus} on the full boxes.
Lemma~\ref{app:lem:buffered-inverse-stability}, applied to both
families with all physical and spectator parameters, gives their
global injectivity, Jacobian gaps and common buffered inverse domains.

The integral quotient and its gradient use two phase derivatives.
One $\epsilon$ derivative uses a third. The joint inverse estimate
therefore uses $S_F$ through order $r+3$, hence $F$ through $r+6$.
This proves the stated bound, on domains fixed by low data alone.
\end{proof}

\begin{rem}[Finite map topology]\label{app:ch_seed:rem:seed-map-topology}
Equation~\eqref{app:ch_seed:eq:seed-frame-tame} makes a fixed $C^8$
map neighborhood sufficient for the low $C^6$ kernel condition.
The fixed order $C^{M_{\rm an}+11}$ controls phase jets through
$M_{\rm an}+5$, carrier gradients and roots through $M_{\rm an}+3$,
and chart inverse jets through $M_{\rm an}-2$ with one $\epsilon$
derivative. Corollary~\ref{app:cor:fixed-profile-transfer} applies
to this same finite geometric family, without changing its domains.
For the reference of
Corollary~\ref{app:ch_seed:cor:seed-uniform-dilation},
$\gamma(G_{\mathrm{ref}})=I$, so
$\|dF-dG_{\mathrm{ref}}\|_{C^0}<1/2$ gives
$|dF(x)v|\geq|v|/2$ and $\gamma(F)\geq I/4$.
Choose this neighborhood together with the normal and kernel
neighborhoods. The positive $\rho$ stays fixed thereafter.
\end{rem}


\subsection{Signed carrier groups}
\label{app:sec:packet-geometry-bridge}
The actual menu fixes the central signed directions before frequencies and sampling.

\begin{defn}[Exhaustive literal carrier groups]\label{app:defn:exhaustive-pair-groups}
Freeze the finite list of full carrier-patch specifications before frequencies and sampling. An unmerged duplicate detector entry remains distinct. A group consists of that identifier and a block color modulo $q_{\rm col}=q_{\rm quad}+1$ in Theorem~\ref{app:ch_waves:thm:sampling}; each chart has $q_{\rm col}^3$ colors. The quadrature order is fixed by \maineqref{main:eq:preparation-orders} before the radial labels. Apply the finite-group choice in Proposition~\ref{app:ch_seed:prop:seed-frozen-menu} to this entire list. Different specifications or colors have radially separated labels, fixed before frequencies and sampling.

Within one block the translated lattice has spacing $h$.
Different blocks of one color have centers at least $(q_{\rm quad}+1)h$
apart in some coordinate and support width $(q_{\rm quad}+1/2)h$.
Their sample distance is therefore at least $h/2$ in every realization.
All band updates at a literal carrier enter its one
final Fourier measure and complex wave; they are not new sample indices.
\end{defn}

\begin{rem}[Exhaustion of all signed terms]\label{app:rem:exhaustive-signed-pairs}
For two distinct sample indices in the same full group, the strong chord gives the protected minus phase with true beat $\lambda|z-z'|\geq c\lambda h$, and the same-patch sum bound gives the plus phase with beat comparable to $\lambda$. For indices in different full groups, prior radial domination gives both signs with beat comparable to $\lambda$, even for equal parameter values or equal unscaled carrier directions. For a repeated single index $j=j$, the doubled plus phase is protected; its entire Hermitian minus tensor is the self contribution assigned to the mean. These cases exhaust the ordered real quadratic expansion. There is no distinct-index zero-beat copy of a literal carrier. These are central phase directions; all modulation enters the complete amplitudes in Appendix~\ref{app:sec:ch_pairs}. The finite group choice is available before the sample event, so this convention introduces neither a new random event nor an additional separation claim.
\end{rem}



\section{The regular-sector construction}\label{app:sec:ch_regular_base}
\subsection{The selected sectors and their estimates}\label{app:ch_regular_base:sec:introduction}
Consider the real matrix system
\begin{equation}\label{app:ch_regular_base:eq:initial-system}
 P_a=\sum_{j=1}^n A_j(x)\partial_{x_j}+B(x),
 \qquad H_a(x,\xi)=\sum_{j=1}^n A_j(x)\xi_j,
\end{equation}
where the $r\times r$ coefficient array $a=(A_1,\ldots,A_n,B)$
is smooth near $x_0\in\R^n$. Set $L_a=P_a/i$ and
\begin{equation}\label{app:ch_regular_base:eq:initial-symbols}
 p_a=\det H_a,\qquad q_a=p_a|\xi|^{1-r},\qquad \xi\ne0.
\end{equation}
A characteristic is simple when $\partial_\xi p_a\ne0$.
We construct a finite inverse on selected regular sectors, keeping
both the angular complement and characteristic integration.
The latter can turn a symbolic error into a nonpseudodifferential
residual. Finite conjugation and composition have the classical
background \cited{NM89,GY88,DH72}; the actual-domain estimates used
here are proved below.

\begin{thm}[Regular-sector approximate inverse]\label{app:ch_regular_base:thm:sector}
\label{app:ch_regular_base:prop:adjoint-gain}
\label{app:ch_regular_base:thm:finite-gap}
For the system \eqref{app:ch_regular_base:eq:initial-system}, fix
a smooth reference $a_0$ and write $H_0=H_{a_0}$, $p_0=\det H_0$.
Let \(\Gamma_{\mathrm{in}},\Gamma_{\mathrm{out}}\subset\chregularbasesphere\) be fixed antipodally symmetric compact sets, with
\(\Gamma_{\mathrm{in}}\subset\operatorname{int}\Gamma_{\mathrm{out}}\).
Assume, on a fixed spatial neighborhood of \(x_0\), that
\[
 p_0(x,\omega)=0,\quad\omega\in\Gamma_{\mathrm{out}}
 \quad\Longrightarrow\quad
 \partial_\xi p_0(x,\omega)\neq0.
\]
There is no hypothesis on characteristic directions outside
\(\Gamma_{\mathrm{out}}\).

After one spatial shrinking, there exist fixed balls \(U\Subset W\), an integer
\(k_0\), a finite \(C^{k_0}\) neighborhood of \(a_0\), a real even order-zero angular/source cutoff \(A_\Gamma\), and one fixed nonnegative integer
\(\beta\), with the following properties.
The operator \(A_\Gamma\) is the identity microlocally on
\(\Gamma_{\mathrm{in}}\) at high frequency and has angular microsupport in
\(\Gamma_{\mathrm{out}}\).

For each smooth real \(a\) in this neighborhood, one real linear operator
\[
 S_{a,\Gamma}\colon C_c^\infty(W,\chregularbaseR^r)\longrightarrow C^\infty(W,\chregularbaseR^r)
\]
and one complete residual \(R_{a,\Gamma}\) satisfy
\begin{equation}\label{app:ch_regular_base:eq:sector-theorem-identity}
 P_aS_{a,\Gamma}h=A_\Gamma h+R_{a,\Gamma}h
 \quad\text{on }U.
\end{equation}
For every integer \(s\geq0\),
\begin{equation}\label{app:ch_regular_base:eq:sector-theorem-solution}
 \chregularbasenorm{S_{a,\Gamma}h}_{H^s(W)}
 \leq C_s\left[
 \chregularbasenorm{h}_{H^s(W)}
 +(1+\chregularbasenorm{a}_{H^{s+\beta}(W)})
       \chregularbasenorm{h}_{L^2(W)}\right].
\end{equation}
For fixed real $\chi_{\rm in}\in C_c^\infty(W)$ and
$\chi_{\rm meas}\in C_c^\infty(U)$, define the measured pair by
zero extension:
\begin{equation}\label{app:ch_regular_base:eq:measured-adjoint}
 \mathcal R_a=\chi_{\rm meas}R_{a,\Gamma}\chi_{\rm in},\qquad
 \mathcal R_a^*=\chi_{\rm in}R_{a,\Gamma}^*\chi_{\rm meas}.
\end{equation}
For $T=R_{a,\Gamma},\mathcal R_a$ use the domain pair $W\to U$;
for $T=\mathcal R_a^*$ use $U\to W$.
On these input and output domains, every integer $s\geq0$ satisfies
\begin{equation}\label{app:ch_regular_base:eq:sector-theorem-residual}
 \|Th\|_{H^s}
 \leq C_s\left[
 \|h\|_{H^{\max(s-1,0)}}
 +(1+\|a\|_{H^{s+\beta}(W)})\|h\|_{L^2}\right].
\end{equation}
Both measured operators also map $H^t$ to $H^{t+1}$ for every
fixed negative integer $t$, on the corresponding supported/restriction
pairs. In particular, one fixed finite low coefficient neighborhood controls
\begin{equation}\label{app:ch_regular_base:eq:finite-gap-theorem}
 \mathcal R_a^{(*)}:L^2\longrightarrow H^1,\qquad
 \mathcal R_a^{(*)}:H^{-1}\longrightarrow L^2.
\end{equation}

The construction has one fixed finite accuracy. Its domains, cutoffs, low coefficient neighborhood, and derivative losses are independent of \(s\).
The residual includes characteristic integration, finite conjugation, inverse-coordinate errors, all source and spatial-cutoff terms, and the low-frequency term.
\end{thm}

\subsection{Adjugate amplitudes and domains}
\label{app:ch_regular_base:sec:preliminaries}
\label{app:ch_regular_base:sec:finite-construction}







\begin{prop}[Adjugates and regular charts]\label{app:ch_regular_base:prop:normalization}
The protected unit set in Theorem~\ref{app:ch_regular_base:thm:sector}
has a finite cover by buffered conic patches. On each patch a fixed
unit vector $v$ satisfies, in one common coefficient neighborhood,
\[
 v\cdot\partial_\xi q_a\geq c>0
\]
and, for $\xi\ne0$, the adjugate amplitudes are
\begin{equation}\label{app:ch_regular_base:eq:schur-principal}
 c_a=|\xi|^{1-r}\operatorname{adj}H_a,
 \qquad h_a=|\xi|^{1-r}\det H_a=q_a.
\end{equation}
The matrix $c_a$ has order zero, and
\begin{equation}\label{app:ch_regular_base:eq:normalization-exact}
 H_ac_a=h_aI.
\end{equation}
These symbols have fixed-shift one-high coefficient bounds.
All low and angular cutoffs will be in the graph amplitudes.
Opposite sectors use the exact conjugate factors chosen below.
\end{prop}
\begin{proof}
Euler's identity $\xi\cdot\partial_\xi q_a=q_a$ excludes a zero
fiber gradient wherever $q_a\ne0$; the theorem's hypothesis
excludes it on $q_a=0$. After one spatial shrinking, compactness
of the protected unit set gives a uniform gradient gap for $q_0$.
A finite cover and fixed rotations give common transverse gaps
in one low coefficient neighborhood, including nonzero-energy
patches, each inside one larger phase chart.
The adjugate identity gives $H_ac_a=h_aI$; the polynomial adjugate
and normalized power of $|\xi|$ have the stated bounds by
Lemma~\ref{app:lem:elementary-interpolation-jets}.
No inverse of $c_a$ is used.
\end{proof}

\subsection{Finite calculus on common charts}

\begin{setup}[Actual short section-energy charts]\label{app:ch_regular_base:setup:finite-calculus}
Use the symbols of Proposition~\ref{app:ch_regular_base:prop:normalization}
and the common buffered physical and canonical charts of
Lemma~\ref{app:lem:common-hamilton-phase}, in their fixed finite
coefficient neighborhood. The ordinary pseudodifferential formulas
also apply independently. Write $x=(x_1,x')$, $\eta=(e,\eta')$
and $y=(t,z)$.
On those boxes fix real physical cutoffs \(\chi_s,\chi_t,\chi_e\) and source and target
conic Fourier multipliers \(\Pi_s,\Pi_t\).
The source cutoff \(\chi_s\) equals one on the inner source patch and
the canonical preimage of the inner target patch.
The target cutoff \(\chi_t\) equals one near \(\chregularbasesupp\chi_e\).
The microsupports of \(\Pi_s,\Pi_t\) lie strictly inside their corresponding
conic boxes.
All transitions have positive fixed buffers.

The graph operators use this actual type-I phase
$S(x,\eta)-y\cdot\eta$.
Their inverse-coordinate factors are the weighted adjoints \(V\)
constructed below from the full matrix transport and its inverse.
A fixed low-frequency cutoff is included in every homogeneous construction;
its complementary terms are kept.

All physical, proper-support, angular and frequency cutoffs are
fixed smooth profiles on the common boxes, rescaled with their
indicated frequencies. Their phase and inverse-coordinate pullbacks,
including $\chi_s(S_\eta)$, obey the joint composition estimate of
Lemma~\ref{app:ch_regular_base:lem:buffered-interpolation}.
On dyadic shells $\lambda$ and $\mu$, each external $\eta$
derivative gains $\lambda^{-1}$, and each derivative in the other
covariable gains $\mu^{-1}$. These actual profiles and pullbacks
have fixed support buffers at every order.

For a full symbol $a$ of order $m$ use
\[
 |a|_{m,j}=\max_{|\alpha|+|\beta|+|\gamma|\leq j}
 \sup\langle\eta\rangle^{-m+|\gamma|}
       |\partial_x^\alpha\partial_z^\beta\partial_\eta^\gamma a|,
\]
with absent variables omitted, and the degree-one seminorm for the phase.

Two-position pseudodifferential amplitudes keep their physical
and proper-support masks. Their diagonal terms and exact
order-lower remainders are used only on the stated plateaus.
Physical multipliers and finite products stay in their literal order.
\end{setup}



\begin{lem}[Fixed ordered products]
\label{app:ch_regular_base:lem:ordered-products}
Use the fixed whole-space representatives of the buffered Sobolev
domain pairs. Let $H_s\geq1$ be nondecreasing. Suppose finitely many
operators $T_j$ and their literal adjoints have uniformly bounded
$L^2$ norms and, for fixed nonnegative integers $d_j$, satisfy
\[
 \|T_jf\|_{H^s}
 \leq C_s\bigl[\|f\|_{H^{\max(s-d_j,0)}}+H_s\|f\|_2\bigr],
 \qquad s\geq0,
\]
for integers $s\geq0$, with the same estimate for $T_j^*$ on its reversed domain pair.
Every fixed compatible product has this bound with gain
$d=\sum_jd_j$, as does its literal adjoint. Fixed sums have the
minimum gain. The high quantity remains $H_s$.

\end{lem}
\begin{proof}
For two factors put $t=\max(s-d_1,0)$. Applying the first estimate
at order $s$ and the second at orders $t$ and zero gives
\[
 \|T_1T_2f\|_{H^s}
 \leq C_s\bigl[\|f\|_{H^{\max(s-d_1-d_2,0)}}
                  +(H_t+H_s)\|f\|_2\bigr].
\]
Here the high term uses only the uniform $L^2$ bound of $T_2$.
Induction and reverse order prove the product and adjoint assertions.

\end{proof}

\begin{lem}[Exact cutoff errors on an inner plateau]
\label{app:ch_regular_base:lem:inner-cutoffs}
Let $T_A$ have phase $S(x,\eta)-y\cdot\eta$ and a full order-$p$
amplitude $A(x,y,\eta)$ with fixed compact physical supports.
If a scalar cutoff $b$ vanishes near every $S_\eta(x,\eta)$ on the
inner output and covector supports, then
\begin{equation}\label{app:ch_regular_base:eq:cutoff-graph-amplitude}
 \chi_eT_Ab=T_{r_b},\qquad
 r_b=-i\sum_j\partial_{\eta_j}\left[
 \chi_e A\int_0^1(\partial_jb)
       (S_\eta+t(y-S_\eta))\,dt\right].
\end{equation}
This full amplitude has order $p-1$.
For a compactly supported $b$, a fixed source cutoff equal to one on
$\operatorname{supp}b$ may first be included in $A$.

The identity keeps the actual cutoffs inside the displayed amplitude.
Its normalized derivatives have one-high bounds with
fixed shifts whenever the original primitives do. Literal adjunction
reverses this exact kernel.
\end{lem}
\begin{proof}
Expand $b(y)$ to first order at $S_\eta$ and use
$(y_j-S_{\eta_j})e^{i(S-y\cdot\eta)}=i\partial_{\eta_j}e^{i(S-y\cdot\eta)}$.
One integration by parts proves the identity; derivatives of
$S_\eta$ in $\eta$ have order $-1$.
The joint derivative budgets of
Lemma~\ref{app:ch_regular_base:lem:buffered-interpolation} prove the
fixed-shift bounds, including every position and mask.
\end{proof}

\begin{thm}[Finite calculus on an inner physical plateau]
\label{app:ch_regular_base:thm:actual-calculus}
Fix the boxes and low coefficient neighborhood of
Setup~\ref{app:ch_regular_base:setup:finite-calculus}.
After one reduction of these boxes and that finite neighborhood, the
following assertions hold.

\begin{enumerate}
\item The phase and boxes are those of
Lemma~\ref{app:lem:common-hamilton-phase}, applied to $h_a$.
Its eikonal, canonical preimage, mixed inverse and averaged-gradient
gaps hold on the full energy chart, with fixed-shift one-high bounds.

\item Let \(T_p\) have phase \(S(x,\eta)-y\cdot\eta\) and a full
matrix amplitude \(A_p(x,y,\eta)\) of order $p=-d$, $d\in\{0,1\}$, with fixed
compact physical and conic supports. Source-independent amplitudes
are also allowed.
Declare a finite low phase and amplitude bound \(M_{\mathrm{low}}\)
large enough for the low \(L^2\) estimates.
For \(A=T_p\), and separately for its literal Hilbert adjoint \(A=T_p^*\),
\begin{equation}\label{app:ch_regular_base:eq:fio-input}
 \chregularbasenorm{Av}_s
 \leq C_s\bigl[\chregularbasenorm{v}_{\max(s-d,0)}
                                      +H_s\chregularbasenorm{v}_0\bigr],
 \qquad s\geq0.
\end{equation}
Here one may take
\[
 H_s=1+|A_p|_{s+\mu}+|S|_{s+\mu},
\]
where the bars are the symbol and homogeneous phase seminorms on the
fixed buffered boxes and \(\mu\) is independent of \(s\).
The constants may depend on \(M_{\mathrm{low}}\), but not on higher
seminorms.
For the amplitudes constructed below,
\begin{equation}\label{app:ch_regular_base:eq:actual-one-high}
 H_s\leq C_s\bigl(1+\chregularbasenorm{a}_{H^{s+\beta}(W)}\bigr)
\end{equation}
with fixed \(\beta\).
An order \(-1\) error and its literal adjoint gain one derivative
in this form. At every fixed integer index the corresponding bounds
hold on the buffered supported/restriction pairs, with a finite
coefficient bound.

\item Every complete two-position pseudodifferential amplitude is
its diagonal principal term, with a fixed source cutoff, plus an exact
one-order-lower full amplitude. The exact graph normal becomes such an
amplitude by an averaged-gradient change. Right multiplication of a
source-independent graph by a fixed scalar source cutoff has the
exact formula below; right Fourier multiplication is exact amplitude
multiplication. These formulas handle the actual localizations.
The pure type-I normal is computed before source multiplication;
the source-cut normal includes its exact source-mismatch term.
Principal terms, complete remainders and literal adjoints have the
indicated orders and one-high bounds, using only index $s+c$ at
output index $s$, with fixed $c$. All angular, low-frequency and
separated-support terms are included.

The transported matrix and its inverse define the order-zero
weighted adjoint \(V\) in
\eqref{app:ch_regular_base:eq:actual-matrix-inverse}. It has the
strong tame estimate, and
\begin{equation}\label{app:ch_regular_base:eq:actual-right-inverse}
 \Pi_t\chi_e(TV-\chregularbaseId)\chi_e\Pi_t
\end{equation}
has order at most \(-1\), including the full source-cutoff error.

\item For $L_a=P_a/i$, one full matrix transport constructs the
preceding $T$, an order-zero graph $\widetilde T$ and an order-$-1$
graph error $E$ such that
\begin{equation}\label{app:ch_regular_base:eq:actual-inner-conjugation}
 \chi_e(L_a\widetilde T-TD_{y_1})\Pi_s
 =\chi_e E\Pi_s+R_{\mathrm{cut}},
\end{equation}
where \(R_{\mathrm{cut}}\) is a finite sum of full cutoff factors
with at least one order-\(-1\) graph or diagonal factor.
The physical transition term is
$[L_a,\chi_t]\widetilde T_0\chi_s$; its inner evaluation is exactly
zero since $L_a$ is differential and $\chi_t=1$ near $\supp\chi_e$.

The complete products after characteristic integration are specified
in Proposition~\ref{app:ch_regular_base:prop:complete-residual}.
\end{enumerate}

The operator calculus in items~(2)--(3) also holds for any real
degree-one type-I phase on these buffered boxes with the stated
mixed gap and injective averaged-gradient chart. Its bounds use
the phase and amplitude seminorms in $H_s$; the weighted inverse
uses the stated matrix inverse gap. The coefficient version
\eqref{app:ch_regular_base:eq:actual-one-high} applies when the
primitives have those coefficient bounds. Steps~1--4 below use
only these assumptions.

The regular construction uses fixed finite depths. Its buffers,
low topology and derivative shifts are fixed before $s$ and correction
precision. Larger integration-by-parts orders only estimate the same operators.
\end{thm}

\begin{proof}
\noindent\textbf{Step 1. The exact normal and the low operator bound.}
Before source multiplication write
\[
 T_0v(x)=(2\pi)^{-n}\iint e^{i(S(x,\eta)-y\cdot\eta)}
                           A_0(x,\eta)v(y)\chregularbasedd y\chregularbasedd\eta.
\]
Its literal normal is
\begin{equation}\label{app:ch_regular_base:eq:actual-normal-kernel}
 (T_0T_0^*)(x,\widetilde x)
 =(2\pi)^{-n}\int e^{i(S(x,\eta)-S(\widetilde x,\eta))}
 A_0(x,\eta)A_0(\widetilde x,\eta)^*\chregularbasedd\eta.
\end{equation}
The exact identity $S(x,\eta)-S(\widetilde x,\eta)
=(x-\widetilde x)\cdot\Xi(x,\widetilde x,\eta)$ and the change
$\xi=\Xi$ give the full pseudodifferential amplitude
\begin{equation}\label{app:ch_regular_base:eq:actual-normal-amplitude}
 b(x,\widetilde x,\xi)
 =\frac{A_0(x,\eta(x,\widetilde x,\xi))
             A_0(\widetilde x,\eta(x,\widetilde x,\xi))^*}
 { |\det\Xi_\eta(x,\widetilde x,\eta(x,\widetilde x,\xi))| }.
\end{equation}
The same identity holds for a cross-product, with the second
amplitude replaced by another amplitude of the same phase. The compact
amplitude support lies inside its inverse chart, so zero extension
is smooth; the low-frequency pieces are smooth kernels.

Lemma~\ref{app:ch_osc:lem:osc-amplitude} bounds this order-zero full
amplitude on $L^2$ from finitely many derivatives. The declared low
bound and inverse gap control them, hence $\|T_0\|_{0\to0}\leq C$.
The literal adjoint has the same $L^2$ norm.

For a full amplitude compactly supported in $y$, expand that variable
in a Fourier series on one fixed larger box, with a fixed cutoff
inside the box. Thus $T_A=\sum_kT_{a_k}M_{\beta e^{ik\cdot y}}$ exactly.
Integration in $y$ gives $|a_k|_{0,\mu_0}\leq
C_N\langle k\rangle^{-N}|A|_{0,\mu_0+N}$.
The source multipliers are uniformly bounded on $L^2$; choose
one $N>n+1$. The series and its literal adjoint give the
same low bound from a fixed total derivative reserve. This Fourier
series is used only for the norm estimate; the full amplitudes and
their original supports remain the operators used below.

For an order-$-1$ amplitude, one output derivative of the graph or
its literal adjoint has order zero. These same bounds therefore give
$L^2\to H^1$ in both directions, with a fixed finite derivative reserve.

\medskip
\noindent\textbf{Step 2. Exact diagonal and source formulas.}
For a full order-$q$ amplitude $c(x,y,\xi)$ on the fixed physical
boxes, choose $\beta\in C_c^\infty$ equal to one near both its
physical supports. Taylor's formula in $y$, followed by one
integration by parts in $\xi$, gives the exact identity
\begin{equation}\label{app:ch_regular_base:eq:actual-pdo-remainder}
 \begin{split}
 P_c&=\operatorname{Op}(c(x,x,\xi))\beta+P_r,\\
 r(x,y,\xi)&=-i\sum_j\partial_{\xi_j}\left[
 \beta(y)\int_0^1\partial_{y_j}c(x,x+t(y-x),\xi)\,dt\right].
 \end{split}
\end{equation}
Indeed, $\beta(y)c(x,y,\xi)=c(x,y,\xi)$ and
$(y_j-x_j)e^{i(x-y)\cdot\xi}=i\partial_{\xi_j}e^{i(x-y)\cdot\xi}$.
All segments lie in one fixed larger convex box. The remainder
has compact physical supports, order $q-1$ and seminorm bounded by
$C_s|c|_{q,s+2}$. Compact frequency regularization proves the
identity distributionally. Its literal adjoint reverses the full
kernel and has the same bounds. Choose $\beta=1$ near every
actual physical input cutoff.
This applies directly to the normal amplitude in
\eqref{app:ch_regular_base:eq:actual-normal-amplitude}.

For fixed $m\in C_c^\infty$, choose $\Pi_{\rm in}=1$ on
$\supp\chi_g$ inside the larger phase chart. Fixed angular and
low-frequency cutoffs extend $S$ to a real smooth symbol, agreeing
on both buffered supports, with the same one-high bounds; extend $A_p$
by zero. Fourier inversion gives the absolutely convergent formula
\begin{equation}\label{app:ch_regular_base:eq:fixed-source-multiplication}
 \begin{split}
 (T_{A_p}m)\Pi_{\rm in}&=T_b\Pi_{\rm in},\\
 b(x,\eta)&=\int\widehat m(v)
 e^{i[S(x,\eta+v)-S(x,\eta)]}A_p(x,\eta+v)\,dv\\
 &=A_p(x,\eta)m(S_\eta(x,\eta))+r_m(x,\eta).
 \end{split}
\end{equation}
For $|v|\leq\delta\langle\eta\rangle$, choose $\delta$ so the
whole covector segment stays in the buffered chart. Apply the
fundamental theorem of calculus to
\[
 e^{i[S(x,\eta+tv)-S(x,\eta)+(1-t)S_\eta(x,\eta)\cdot v]}
                         A_p(x,\eta+tv),\qquad 0\leq t\leq1.
\]
Its differentiated amplitude is $v\cdot A_{p,\eta}$ plus
$i[S_\eta(x,\eta+tv)-S_\eta(x,\eta)]\cdot v\,A_p$.
Both lower the order by one, with respective factors $|v|$ and
$|v|^2$. External covector derivatives gain their radial powers; joint
product differentiation uses only $s+c$ primitive derivatives.
The Schwartz moments of $\widehat m$ give the full order-$p-1$
bound with one fixed shift.
On $|v|\geq c\langle\eta\rangle$, the same moments absorb every
frequency power, including all cutoff transitions and bounded
frequencies. This exact complement has arbitrary negative order.
For phase $x\cdot\xi$, the same formula makes each fixed
physical/Fourier commutator order $-1$. Thus the added $\beta$
can be moved past a fixed Fourier multiplier and removed against
its actual physical input cutoff. Every commutator is kept.

Finally, substitute the coefficient estimates into these exact
amplitudes. Lemma~\ref{app:lem:common-hamilton-phase} bounds the phases in the normal
and source formulas; the symbols have the stipulated coefficient
bounds; and the full matrix
transport and its inverse in Steps 4--5 give
$|A_p|_{p,m}\leq C_m(1+\|a\|_{C^{m+c_A}})$ for these fixed constructions.
The joint budget above therefore gives
$C_s(1+\|a\|_{C^{s+\widetilde\beta}})$ for every kept
coefficient and complete remainder. The Schwartz moments depend only on the fixed source cutoff. Fixed extensions
and Sobolev embedding replace this by
$C_s(1+\|a\|_{H^{s+\beta}(W)})$, with fixed $\beta$.

\medskip
\noindent\textbf{Step 3. Strong forward and literal adjoint estimates.}
Let $\mu_0$ include every fixed low derivative cost in Step 1. First
consider order zero. Take as primitives
the normalized amplitude, $S_{x\eta}$, its inverse $B$, and all
cutoff profiles; choose $\mu$ to cover $\mu_0$ and their fixed
jet shifts. Their $C^{\mu_0}$ norms are uniformly bounded on
the normalized boxes.

For the forward graph, Euler's identity $S_x=S_{x\eta}\eta$
trades each phase derivative for a datum derivative. For its
literal adjoint the exact identity is
\begin{equation}\label{app:ch_regular_base:eq:actual-adjoint-trade}
 i\eta_i e^{i(y\cdot\eta-S(x,\eta))}
 =-\sum_jB_{ij}\partial_{x_j}e^{i(y\cdot\eta-S(x,\eta))}.
\end{equation}
Integrate by parts in $x$; compact support removes the boundary
terms. Thus, for either $A=T_p$ or $A=T_p^*$,
\begin{equation}\label{app:ch_regular_base:eq:graph-derivative-transfer}
 D^\alpha A=\sum_{|\beta|\leq s}A_{\alpha\beta}D^\beta,
 \qquad |\alpha|=s,
\end{equation}
where the full amplitudes of $A_{\alpha\beta}$ have order zero and
primitive derivative excess at most $s-|\beta|$. Derivatives in the
second physical position enter this same count. This induction uses
only the displayed phase-gradient identity and the low $L^2$ bound
for the complete modified amplitudes. Including a low
operator seminorm costs only $\mu_0$ additional derivatives. If
$r_h$ are the resulting primitive orders and $k=|\beta|$, then
$\sum_h r_h\leq s-k+\mu_0$, hence
$\sum_h(r_h-\mu_0)_+\leq s-k$.
Interpolation from the bounded $C^{\mu_0}$ baseline to
$C^{s+\mu_0}$ therefore gives total high exponent at most
$(s-k)/s$. Step 1 yields
\[
 \|A_{\alpha\beta}\|_{L^2\to L^2}
 \leq C_sH_s^{1-|\beta|/s},\qquad
 H_s=1+|A_p|_{s+\mu}+|S|_{s+\mu}.
\]
For $k=|\beta|$, Fourier interpolation and Young's inequality give
\[
 H_s^{1-k/s}\|v\|_k
 \leq(H_s\|v\|_0)^{1-k/s}\|v\|_s^{k/s}
 \leq H_s\|v\|_0+\|v\|_s.
\]
Summing proves \eqref{app:ch_regular_base:eq:fio-input} for $d=0$; $s=0$
is Step 1. Fixed low-frequency pieces have the same estimate directly.

The coefficient bounds at the end of Step 2 give
\eqref{app:ch_regular_base:eq:actual-one-high}, after enlarging
$\beta$ by the fixed $\mu$.

For order $-1$ and $s\geq1$, apply the order-zero estimate at $s-1$
to each first output derivative of the graph or its literal adjoint.
Together with the low $L^2$ bound, this proves
\eqref{app:ch_regular_base:eq:fio-input} with gain one. Increasing
$\mu$ once includes that first derivative.
The opposite positive estimate and supported/restriction duality
give gain one at every fixed negative integer index. The full cutoff
amplitudes in Lemma~\ref{app:ch_regular_base:lem:inner-cutoffs} have
these same order-$-1$ bounds and literal reversed domain pairs.

\medskip
\noindent\textbf{Step 4. A weighted adjoint with identity principal amplitude.}
Let $U$ be the invertible full transport matrix constructed in Step 5.
Choose one real conic cutoff $\chi_g(\eta)$, including the fixed
low-frequency cutoff, equal to one on the smaller source and target
frequency regions and supported inside the common larger phase chart.
Choose $\chi_s=1$ near all canonical preimages of the inner output
for every frequency in $\supp\chi_g$. The amplitude of $T_0$ is $\chi_gU$.
Put $j_S=|\det S_{x\eta}|$ and define
\begin{equation}\label{app:ch_regular_base:eq:actual-matrix-inverse}
 \begin{split}
 V&=\chi_sV_0\chi_t,\\
 V_0f(y)&=(2\pi)^{-n}\iint e^{i(y\cdot\eta-S(x,\eta))}
        \chi_g(\eta)j_S(x,\eta)U(x,\eta)^{-1}f(x)\,dx\,d\eta.
 \end{split}
\end{equation}
It is the literal adjoint graph whose forward amplitude is
$\chi_gj_SU^{-*}$, where $U^{-*}=(U^{-1})^*$.
The inverse and determinant gaps and Step 3 give its order-zero
strong estimates, with a fixed coefficient shift.

The exact cross-normal formula
\eqref{app:ch_regular_base:eq:actual-normal-amplitude} gives $T_0V_0$
the diagonal amplitude
\[
 b^\vee(x,x,\xi)=\chi_g(\eta(x,x,\xi))^2I.
\]
It equals $I$ on the inner frequency plateau. Step 2 gives the
complete order-$-1$ reduction error. Its source multiplier is
one on the inner input. Move a fixed physical multiplier past
the diagonal principal operator by the source formula of Step 2
with phase $x\cdot\xi$; its commutator has order $-1$.
Right multiplication by $\Pi_t$ then multiplies the symbol
exactly, so the principal defect vanishes. The low-frequency
term is smooth. This proves the paired gain-one bounds for
the pure localized defect, with the same fixed shift.

For source-cut factors $T=T_t\chi_s$, $W=W_t\chi_s$ of the same
phase, the exact identity is
\begin{equation}\label{app:ch_regular_base:eq:source-cut-normal}
 TW^*=T_tW_t^*-T_t(1-\chi_s^2)W_t^*.
\end{equation}
It includes the literal normal when $W=T$ and the weighted product
$TV$ for the preceding choice. Formula~\eqref{app:ch_regular_base:eq:fixed-source-multiplication}
gives $T_t\chi_s^2$ the principal amplitude
$A_t(x,\eta)\chi_s^2(S_\eta(x,\eta))$ and an exact order-$-1$
remainder. On the inner output the first multiplier is one.
Here $\Pi_{\rm in}V_0=V_0$ exactly, since its canonical Fourier
support lies in $\supp\chi_g$ inside the common larger phase chart. The remaining cross-normal
is therefore a full order-$-1$ amplitude by Step 1.
Consequently $TV=I-E_0$ on the stated inner localizations, with
all source, physical, angular and low-frequency terms included.

\medskip
\noindent\textbf{Step 5. Two graph amplitudes and the physical plateau.}
Put $\Xi=S_x$. With the symbol derivatives evaluated at $(x,\Xi)$,
define the real matrix and vector field
\begin{equation}\label{app:ch_regular_base:eq:actual-effective-transport}
 \begin{split}
 C_{\rm tr}&=\sum_jA_j\partial_{x_j}c_a+Bc_a
          +\frac12\sum_{j,l}h_{a,\xi_j\xi_l}S_{x_jx_l}I,\\
 X&=\sum_jh_{a,\xi_j}\partial_{x_j}.
 \end{split}
\end{equation}
Here $\partial_{x_j}c_a$ is taken at fixed $\xi$.
The eikonal gives $XS_{\eta_1}=1$ and
$XS_{\eta_j}=0$ for $j\geq2$, so $X=\partial_t$, with zero
canonical time on the section.

Solve the full matrix equations
\begin{equation}\label{app:ch_regular_base:eq:actual-transport-solution}
 U'=-C_{\rm tr}U,\quad U(0)=I,\qquad
 (U^{-1})'=U^{-1}C_{\rm tr}.
\end{equation}
Fixed-time Gronwall and the matrix transport estimate of
Lemma~\ref{app:ch_regular_base:lem:buffered-interpolation} give
\[
 |U|_{0,s}+|U^{-1}|_{0,s}
 \leq C_s(1+\|a\|_{C^{s+c}})
\]
with a fixed shift and uniform low inverse bounds. The matrix $U$ is real.
Define, with every derivative of $c_a$ evaluated at $(x,\Xi)$,
\begin{equation}\label{app:ch_regular_base:eq:direct-graph-amplitudes}
 \begin{split}
 b_0&=c_a(x,\Xi)U,\\
 b_{-1}&=-i\sum_j c_{a,\xi_j}\partial_{x_j}U
       -\frac i2\sum_{j,l}c_{a,\xi_j\xi_l}S_{x_jx_l}U.
 \end{split}
\end{equation}
These have orders $0$ and $-1$. Put
$\widetilde T_0=T_{\chi_g(b_0+b_{-1})}$ and $T_0=T_{\chi_gU}$.
The following identity is exact:
\begin{equation}\label{app:ch_regular_base:eq:direct-graph-identity}
 L_a\widetilde T_0-T_0D_t=T_{\chi_g r_{-1}},\qquad
 r_{-1}=\sum_jA_jD_{x_j}b_{-1}-iBb_{-1}.
\end{equation}
Indeed, $T_UD_t=T_{eU}$ by integration in canonical time.
Differentiating $H_ac_a=h_aI$ in $\xi$ gives
\[
 H_ac_{a,\xi_j}=h_{a,\xi_j}I-A_jc_a,\qquad
 H_ac_{a,\xi_j\xi_l}=h_{a,\xi_j\xi_l}I
                         -A_jc_{a,\xi_l}-A_lc_{a,\xi_j}.
\]
The product rule in $L_aT_{b_0+b_{-1}}$ now gives the order-one
term $eU$ and the order-zero term $-i(X+C_{\rm tr})U=0$; the remaining
term is precisely $r_{-1}$. Matrix factors stay in the displayed
order. Since $\chi_g$ depends only on $\eta$, it preserves the identity.
The explicit formulas and joint product bounds give $r_{-1}$ and
$\widetilde T_0$ fixed-shift one-high bounds. No inverse of $c_a$ is needed.

For $\widetilde T=\chi_t\widetilde T_0\chi_s$ and
$T=\chi_tT_0\chi_s$, put $E=T_{\chi_g r_{-1}}$. Then
\begin{align}
 L_a\widetilde T-TD_{y_1}
 =\chi_tE\chi_s+[L_a,\chi_t]\widetilde T_0\chi_s
                       +\chi_tT_0[D_{y_1},\chi_s].
 \label{app:ch_regular_base:eq:actual-global-cutoff}
\end{align}
The target cutoff is a function of physical $x$, and
$[L_a,\chi_t]=-i\sum_jA_j\partial_{x_j}\chi_t$.
Thus its inner evaluation is exactly zero.
The last term has the cutoff form
\eqref{app:ch_regular_base:eq:cutoff-graph-amplitude} after a fixed
source localization equal to one on its support.
For angular localizations, right Fourier multiplication is exact;
moving a fixed physical multiplier uses Step 2 with phase
$x\cdot\xi$. Its order-$-1$ error and the inner-cutoff errors
remain in their original order. Thus
\eqref{app:ch_regular_base:eq:actual-inner-conjugation} follows on
the stated plateau, as a fixed finite chain of full graph and
diagonal factors.

\end{proof}





A negative-order error before characteristic integration does not
identify the complete residual. Proposition~\ref{app:ch_regular_base:prop:complete-residual}
includes the ordered integrated products and their physical
transition terms.

\subsection{Characteristic integration and the complete residual}\label{app:ch_regular_base:sec:integration}

\begin{notation}[Construction domains and characteristic integral]
\label{app:ch_regular_base:setup:construction}
\label{app:ch_regular_base:defn:characteristic-integral}
Use the finite regular cover of
Proposition~\ref{app:ch_regular_base:prop:normalization} and the
common buffered charts of
Theorem~\ref{app:ch_regular_base:thm:actual-calculus}.
Take $W$ containing all construction supports. After one fixed
inner shrinking, choose all canonical rectangles with their complete
physical flow images in the prescribed $W_0\Subset W$, before
the final core and accuracy. All graph, source and low-frequency
cutoff pieces stay in their literal order.
Extend $a-a_0$ by the single componentwise operator of
Lemma~\ref{app:ch_regular_base:lem:ball-extension}, and add a fixed
extension of $a_0$. This agrees with $a$ on the construction region
and preserves every nonnegative Sobolev and $C^m$ order.
Negative estimates use the supported/restriction dual pairs.

On each canonical rectangle use $J$ from
Definition~\ref{app:ch_osc:setup:osc-retarded}, with lower endpoint
$-T$. Choose both $\chi_{\rm in}$ and $\chi_{\rm out}$ equal to
one near $\supp\chi_s$, inside one larger fixed rectangle. The
input still vanishes near the lower endpoint. Put
\begin{equation}\label{app:ch_regular_base:eq:characteristic-integral}
 \mathcal J=iJ,\qquad D_t=-i\partial_t.
\end{equation}
These choices are fixed at every observation order.
\end{notation}

\begin{prop}[The complete finite construction]
\label{app:ch_regular_base:prop:finite-family}
\label{app:ch_regular_base:prop:complete-residual}
\label{app:ch_regular_base:prop:wavefront}
Use the preceding buffered cover and the actual factors
$\widetilde T_j,T_j,V_j$ of
Theorem~\ref{app:ch_regular_base:thm:actual-calculus}, with the cutoffs
in \eqref{app:ch_regular_base:eq:actual-matrix-inverse} and
\eqref{app:ch_regular_base:eq:actual-global-cutoff}. Write
\begin{equation}\label{app:ch_regular_base:eq:finite-conjugation}
 L_a\widetilde T_j-T_jD_{t_j}=E_j+\mathcal C_j+\mathcal S_j,
 \qquad T_jV_j=I-E_{0,j}.
\end{equation}
Here $E_j,\mathcal C_j,\mathcal S_j$ are respectively the graph term,
target commutator and source commutator of
\eqref{app:ch_regular_base:eq:actual-global-cutoff}.
The target commutator vanishes on the inner measurement plateau.
The graph and localized identity errors have measured order $-1$,
including every localization difference. All factors and their literal
adjoints have the preceding strong estimates at their respective
orders, with one fixed common coefficient shift.

Choose a fixed Fourier angular partition with fixed physical
multipliers, $A_\Gamma=A_{\mathrm{low}}+\sum_jA_j$, inside the
matched target frequency plateaus. Each localized sector copy
is accompanied by its exact conjugate. With $\mathfrak c$
denoting conjugation on each physical or canonical space, set
\[
 \widetilde T_-=\mathfrak c\widetilde T_+\mathfrak c,\qquad
 A_-=\mathfrak cA_+\mathfrak c,
\]
and choose the entire source-cut factors by
\begin{align}
 T_-&=\mathfrak c_xT_+\mathfrak c_{\rm can},
 \label{app:ch_regular_base:eq:signed-graph-factor}\\
 V_-&=\mathfrak c_{\rm can}V_+\mathfrak c_x,
 \label{app:ch_regular_base:eq:signed-coordinate-inverse}\\
 \mathcal J_-&=-\mathfrak c_{\rm can}\mathcal J_+
                         \mathfrak c_{\rm can}=\mathcal J_+.
 \label{app:ch_regular_base:eq:signed-canonical-integral}
\end{align}
All partner cutoffs have the same physical and canonical roles.
Put
\begin{equation}\label{app:ch_regular_base:eq:QN-definition}
 Q_N=\sum_j\widetilde T_j\mathcal J_jV_jA_j.
\end{equation}
All factors include their fixed buffered representatives.
The complete physical residual is
\begin{equation}\label{app:ch_regular_base:eq:scalar-residual-definition}
 \mathcal E_a=A_\Gamma-L_aQ_N.
\end{equation}
With $\mathcal F_j=E_j+\mathcal C_j+\mathcal S_j$, including
all exact source and target localization differences, one has
\begin{equation}\label{app:ch_regular_base:eq:complete-residual}
 \mathcal E_a=E_{\mathrm{part}}
       +\sum_j[E_{0,j}-\mathcal F_j\mathcal J_jV_j]A_j.
\end{equation}
Here $E_{\mathrm{part}}=A_{\mathrm{low}}$, including any further
low-frequency partition adjustments. Each $\mathcal F_j$ is a finite
sum of full matrix graph and diagonal products, with measured order
at most $-1$ before its integral and the preceding strong bounds.
Its source commutators and angular differences are estimated on the
inner measurement and actual source supports. The physical transition
vanishes there; no global gain at that transition is asserted.
\end{prop}
\begin{proof}
Choose $c_-(x,\xi)=c_+(x,-\xi)$. The principal adjugate has
parity $(-1)^{r-1}$, so $h_-=(-1)^{r-1}q_a=-h_+(x,-\xi)$
and its type-I phase is
\begin{equation}\label{app:ch_regular_base:eq:signed-type-one-phase}
 S_-(x,e,\eta')=-S_+(x,-e,-\eta').
\end{equation}
At paired covectors $S_{-,\eta_-}=S_{+,\eta_+}$; hence the
canonical base point and time are unchanged. The weighted amplitude
$j_SU^{-1}$ and the source-cut cross-normal conjugate with all
their cutoffs. Thus the physical source gaps and separations persist
with the same nested canonical localizers on both full branches
of the antipodal regular buffer.

Since $\mathfrak cL_a\mathfrak c=-L_a$ and $\mathcal J=iJ$
with $J$ real, the paired factors give $Q_-=-\mathfrak cQ_+\mathfrak c$.
Thus
\begin{equation}\label{app:ch_regular_base:eq:signed-physical-normalization}
 \mathfrak cQ_N\mathfrak c=-Q_N,\qquad
 Q_N/i\ \hbox{is real},\qquad
 \mathfrak c\mathcal E_a\mathfrak c=\mathcal E_a.
\end{equation}
The nesting gives $\chi_{\rm in}V=V$, $T\chi_{\rm out}=T$ and
$T[D_t,\chi_{\rm out}]=0$ as exact operator identities.
Direct differentiation therefore yields
\begin{equation}\label{app:ch_regular_base:eq:DJ-identity}
 T D_t\mathcal JV=TV.
\end{equation}
The endpoint term vanishes by support, as does its literal adjoint.
The exact graph identity and source-cut inverse now give
\begin{equation}\label{app:ch_regular_base:eq:local-exact-product}
 I-L_a\widetilde T_j\mathcal J_jV_j
                       =E_{0,j}-\mathcal F_j\mathcal J_jV_j.
\end{equation}
Formula~\eqref{app:ch_regular_base:eq:cutoff-graph-amplitude}
handles the source commutator with its cutoffs in their displayed
order. Substitution in
\eqref{app:ch_regular_base:eq:QN-definition} gives
\eqref{app:ch_regular_base:eq:complete-residual}.
These formulas use nested inner localizers. Fixed physical/Fourier
commutators use Step 2 of
Theorem~\ref{app:ch_regular_base:thm:actual-calculus}; graph cutoff
errors use \eqref{app:ch_regular_base:eq:cutoff-graph-amplitude}.
Each exact order-$-1$ term stays before the next nonlocal factor. Additional copies of a sector index distinguish different
localized inputs. Hence every complete component is an exact finite
product of the selected graph, weighted adjoint, diagonal and buffered
integral factors, with at least one order-$-1$ factor. Derivatives of
an amplitude do not change its phase. Lemma~\ref{app:ch_regular_base:lem:ordered-products}
therefore gives the paired gain-one bounds for these actual products
and their literal reverse products.
Literal adjunction reverses every full product and gives the upper
integral \eqref{app:ch_regular_base:eq:reverse-integral}, with its
input and output weights exchanged and no time reflection.
The bounded-frequency terms and their conjugates are smooth directly.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{app:ch_regular_base:thm:sector}]
Use the fixed finite construction of
Proposition~\ref{app:ch_regular_base:prop:complete-residual}, with
its complete conjugate pairs. For real $h$, define
\begin{equation}\label{app:ch_regular_base:eq:real-sector-normalization}
 S_{a,\Gamma}h=Q_Nh/i,
\end{equation}
with a fixed output cutoff equal to one near $\overline U$ and
supported in $W$. Since $P_a=iL_a$ is real,
\[
 P_aS_{a,\Gamma}h=A_\Gamma h-\mathcal E_ah
 \quad\text{on }U.
\]
Thus the theorem's plus-residual convention is
\begin{equation}\label{app:ch_regular_base:eq:two-residual-conventions}
 R_{a,\Gamma}=-\mathcal E_a
 \quad\text{on real inputs and their complexifications}.
\end{equation}

Proposition~\ref{app:ch_osc:prop:osc-retarded-tube} bounds
$\mathcal J_j$ and its literal adjoint
\begin{equation}\label{app:ch_regular_base:eq:reverse-integral}
 (\mathcal J_j^*v)(\sigma,z)
 =-i\chi_{\mathrm{in},j}(\sigma,z)
       \int_\sigma^T\chi_{\mathrm{out},j}(t,z)v(t,z)\,dt
\end{equation}
on every integer supported/restriction pair.
Theorem~\ref{app:ch_regular_base:thm:actual-calculus} and
Lemma~\ref{app:ch_regular_base:lem:ordered-products} apply in both
factor orders to the finite factorization \eqref{app:ch_regular_base:eq:QN-definition},
proving \eqref{app:ch_regular_base:eq:sector-theorem-solution}.
Every complete residual term and its literal adjoint has a gain-one
factor on its measured supports. The same lemma therefore proves
\eqref{app:ch_regular_base:eq:sector-theorem-residual}, including
the low-frequency, integrated and measured boundary terms, with
one maximum fixed shift $\beta$. In particular,
\begin{equation}\label{app:ch_regular_base:eq:adjoint-integrated-term}
 (\mathcal F_j\mathcal J_jV_jA_j)^*
       =A_j^*V_j^*\mathcal J_j^*\mathcal F_j^*.
\end{equation}
The actual source and measurement cutoffs and the literal upper
integral stay in this reverse product. For $T=\mathcal R_a$ or
$\mathcal R_a^*$ and $t=-k-1$, supported/restriction duality gives
\begin{equation}\label{app:ch_regular_base:eq:negative-duality}
 \|T\|_{H^{-k-1}\to H^{-k}}
       =\|T^*\|_{H^k\to H^{k+1}},\qquad k\geq0,
\end{equation}
with equivalent norms under the fixed extensions. Each fixed $k$
uses finitely many coefficient seminorms. All domains and low
choices are fixed before $s$.
\end{proof}

\begin{rem}[The kept angular complement]\label{app:ch_regular_base:rem:angular-complement}
For an arbitrary source \(h\), the identity on \(U\) is
\[
 P_aS_{a,\Gamma}h
 =h-(\chregularbaseId-A_\Gamma)h+R_{a,\Gamma}h.
\]
The term \((\chregularbaseId-A_\Gamma)h\) is not a smoothing error under the hypotheses of
Theorem~\ref{app:ch_regular_base:thm:sector}.
In particular, inversion of a localized residual alone does not remove this term.
Repeated residual corrections also require separate control of their angular support.
No accuracy-independent angular tail estimate is asserted.
\end{rem}



\section{The fixed-domain inverse}\label{app:sec:ch_regular}
\subsection{The fixed inverse and its inputs}\label{app:ch_regular:sec:introduction}

\subsubsection*{The complete residual}

Apply Appendix~\ref{app:sec:ch_regular_base} to the real first-order system
\begin{equation}\label{app:ch_regular:eq:system}
 P_a=\sum_{\nu=1}^n A_\nu(x)\partial_{x_\nu}+B(x).
\end{equation}
The coefficient array is denoted by \(a\).
The matrices have size \(r\times r\).
Its principal matrix, determinant, and homogeneous normalization are
\begin{equation}\label{app:ch_regular:eq:principal-symbol}
 H_a(x,\xi)=\sum_{\nu=1}^n A_\nu(x)\xi_\nu,\qquad
 p_a(x,\xi)=\det H_a(x,\xi),\qquad
 q_a(x,\xi)=p_a(x,\xi)|\xi|^{1-r}.
\end{equation}
We work on angular sectors where every characteristic is simple real principal
type and \(H_a\) is invertible away from the characteristic set.

Setup~\ref{app:ch_regular:setup:analytic} specifies the proved operator
bounds used here. The localized identity is
\begin{equation}\label{app:ch_regular:eq:localized-parametrix}
 P_aS_{a,\Gamma}=A_\Gamma+R_{a,\Gamma}.
\end{equation}
Here \(R_{a,\Gamma}\) is the complete residual.
It includes the direct graph and coordinate-inverse errors,
the canonical source commutators and the low-frequency partition.
The physical cutoff commutators vanish on the measurement core.
The correction below uses this entire operator. Quantitative localization
also requires the actual coefficient profile and protected source
product stated below; Proposition~\mainref{main:prop:current-scale}
and Theorem~\ref{app:ch_regular:thm:no-return} verify them in the iteration.



\subsection{The fixed regular family}\label{app:ch_regular:sec:preliminaries}

\begin{notation}\label{app:ch_regular:notation:conventions}
Sobolev norms are taken on the fixed construction box, on \(W\), or on the
indicated core.
Compactly supported functions are extended by zero when this is required
by the displayed operator.
All cutoff multiplications used with a zero extension have support strictly
inside the relevant domain.
Matrix-valued norms are taken componentwise.

An angular sector is a compact subset of the unit sphere together with its
positive conic lift.
All sectors are antipodally paired.
The notation \(E\Subset F\) for angular sets means containment with a
positive angular gap.
Fourier transforms use Notation~\ref{app:not:fourier}.
For a family with size \(H_\rho\), an arbitrary-accuracy tail means a bound
\(C_{m,Q}H_\rho\rho^{-Q}\) in each fixed \(H^m\).
Its input precision may be increased before propagation through a fixed
finite operator chain.
\end{notation}

\begin{setup}[The proved regular-sector construction]\label{app:ch_regular:setup:analytic}
Fix a smooth reference coefficient array \(a_0\) for \eqref{app:ch_regular:eq:system}.
Fix compact antipodally symmetric sectors
\[
 \Gamma_{\mathrm{in}}\Subset\Gamma_{\mathrm{out}}
\]
such that, on the construction neighborhood, every zero of
\(p_0=\det H_0\) in \(\Gamma_{\mathrm{out}}\) has nonzero fiber gradient.
Assume that \(H_0\) is invertible at every other point of this sector.
Shrink one finite coefficient neighborhood once so that the corresponding
simple-characteristic, elliptic, chart-projection, and normalization gaps
remain uniformly positive.

Theorems~\ref{app:ch_regular_base:thm:actual-calculus}
and~\ref{app:ch_regular_base:thm:sector}, with
Proposition~\ref{app:ch_regular_base:prop:finite-family}, give the following
construction on this coefficient neighborhood.
\begin{enumerate}
\item
There are fixed domains \(U\Subset W\), a real angular source cutoff
\(A_\Gamma\), a real localized parametrix \(S_{a,\Gamma}\), and a complete
residual \(R_{a,\Gamma}\) satisfying
\eqref{app:ch_regular:eq:localized-parametrix} on \(U\).
The cutoff equals the identity microlocally on a fixed neighborhood of
the protected inner sector at sufficiently high frequencies.
Any fixed low-frequency modification belongs to the complete residual.
Before choosing $\chi_2,\chi_3$, fix a real source cutoff equal to
one near $\overline U$ and a larger measurement plateau containing
$\overline U$. Include that source cutoff on the right of
$S_{a,\Gamma},A_\Gamma,R_{a,\Gamma}$, and use the buffered restriction
for the low maps below. The identity on $U$ is unchanged, and every
later $\chi_3$ lies in this source plateau.

\item
The complete residual has uniform one-gain bounds
\begin{equation}\label{app:ch_regular:eq:residual-negative-positive}
 R_{a,\Gamma}:L^2(W)\longrightarrow H^1(U),\qquad
 R_{a,\Gamma}:H^{-1}(W)\longrightarrow L^2(U).
\end{equation}
Its literal adjoint has the corresponding bounds on the reversed
supported/restriction domains. At every fixed order both obey the
additive one-gain estimates of the cited theorem.
After fixed physical cutoffs,
\eqref{app:ch_regular_base:eq:sector-theorem-residual} gives, for one
fixed nonnegative integer $b$ and every integer $s\geq1$,
\begin{equation}\label{app:ch_regular:eq:one-gain-tame}
 \|Kv\|_{H^s}
 \leq C_s\left[
 \|v\|_{H^{s-1}}
 +(1+\|a\|_{H^{s+b}})\|v\|_{L^2}
 \right].
\end{equation}
The constants are uniform in the declared low neighborhood.
The actual solver obeys
\eqref{app:ch_regular_base:eq:sector-theorem-solution}:
\begin{equation}\label{app:ch_regular:eq:parametrix-tame}
 \|S_{a,\Gamma}v\|_{H^s}
 \leq C_s\left[
 \|v\|_{H^s}
 +(1+\|a\|_{H^{s+b_S}})\|v\|_{L^2}
 \right],\qquad s\geq0,
\end{equation}
with one fixed shift $b_S$. The fixed low class includes the
$s=0$ coefficient norm, so $\|S_{a,\Gamma}\|_{2\to2}\leq C$.
All selected cutoffs are included in these estimates.

\item
The complete residual is $R_{a,\Gamma}=-\mathcal E_a$, with
$\mathcal E_a$ exactly as in
\eqref{app:ch_regular_base:eq:complete-residual}; in particular
$R_{\rm low}=-A_{\rm low}$. Every measured component is a fixed
product of full graph, weighted-coordinate-inverse, diagonal and
buffered integral factors, with one order-$-1$ factor. All physical
cutoffs, localized sector copies, coordinate errors, source
commutators and angular discrepancies are included. The outer
physical commutator vanishes on the core; the differentiated integral
cutoff vanishes by \eqref{app:ch_regular_base:eq:DJ-identity}.

\item
For a positive representative each canonical integration chart has coordinates
\((t,y;\tau,\eta)\).
Its integral is retarded on one fixed interval
\([t_-,t_+]\).
Writing the actual two cutoffs in that chart as \(\alpha,\beta\), its
localized form is
\begin{equation}\label{app:ch_regular:eq:retarded}
 Jf(t,y)=i\alpha(t,y)\int_{t_-}^{t}\beta(s,y)f(s,y)\,ds.
\end{equation}
The input factor \(\beta f\) vanishes on one fixed interval immediately
after \(t_-\).
Both weights are one near the canonical source support, as in
\eqref{app:ch_regular_base:eq:DJ-identity}. The source-cut commutator
has the fixed preimage gap on the final physical core.
The full signed identities
\eqref{app:ch_regular_base:eq:signed-graph-factor}--%
\eqref{app:ch_regular_base:eq:signed-physical-normalization} give
\begin{equation}\label{app:ch_regular:eq:regular-signed-Ji}
 J_-=-\mathfrak cJ_+\mathfrak c=J_+,
\end{equation}
where $\mathfrak c$ is conjugation. Both partners have the same
lower endpoint and input buffer. Literal adjunction instead
exchanges the weights and reverses the limits, as in
\eqref{app:ch_regular_base:eq:reverse-integral}, with the literal
Hilbert densities on each supported/restriction pair.

\item
The two graph factors $T_j,\widetilde T_j$, inverse graphs, matrix
transport, weighted adjoints and exact Taylor-integral
remainders use one fixed finite set of operations.
Their low inverses, first flow derivatives, and graph Jacobians are
uniformly controlled by one finite norm \(C^{k_{\rm lin}}\).
There is one fixed extra coefficient-derivative count \(c_{\rm lin}\) for their
tame high derivative estimates, covering the preceding Sobolev
shifts and the fixed low symbol seminorms for full two-position
$L^2$ bounds.
No operation is repeated to an order depending on Fourier-tail accuracy.
\end{enumerate}
The diagonal and characteristic relations, with quantified off-relation
errors, follow from the full kernels in
Lemma~\ref{app:ch_regular:lem:finite-chain}.
\end{setup}

The additional coefficient profile for quantitative localization is
Setup~\ref{app:ch_regular:setup:base-aware-coefficients}.


\subsection{The exact small-support inverse}\label{app:ch_regular:sec:inverse}

The first result identifies what remains after the complete residual has been
inverted.
All domains and all cutoffs in this statement are fixed before an accuracy
is requested.

\begin{thm}[Exact small-support correction]\label{app:ch_regular:thm:exact-correction}
Use the fixed real operators of
Setup~\ref{app:ch_regular:setup:analytic} on \(U\Subset W\).
There are real cutoffs \(\chi_2,\chi_3\in C_c^\infty(U)\), with
\[
 0\leq\chi_2,\chi_3\leq1,\qquad
 \chi_3=1\ \hbox{on }\chregularsupp\chi_2,\qquad
 \chi_2=1\ \hbox{on }U_{\mathrm{core}},
\]
such that the zero extension to \(W\) of
\[
 K_a=\chi_2R_{a,\Gamma}\chi_3
\]
satisfies
\[
 \|K_a\|_{L^2(W)\to L^2(W)}\leq\theta<1
\]
uniformly on the fixed sufficiently small finite coefficient neighborhood.
Consequently
\[
 T_a=(I+K_a)^{-1}
\]
exists on the entire real space \(L^2(W)\).
It preserves smooth input functions for each fixed smooth \(a\).
There is a fixed integer \(\beta_*\), independent of requested
accuracy, such that
\begin{equation}\label{app:ch_regular:eq:inverse-tame}
 \|T_af\|_{H^s(W)}
 \leq D_s\left[
 \|f\|_{H^s(W)}
 +(1+\|a\|_{H^{s+\beta_*}(W)})\|f\|_{L^2(W)}
 \right],
 \qquad s\geq0.
\end{equation}
For every smooth real \(h\) supported in \(U\), set
\[
 z=T_a(\chi_2h),\qquad u=S_{a,\Gamma}z.
\]
Then \(\chregularsupp z\subset\chregularsupp\chi_2\), and the exact physical identity on
\(U_{\mathrm{core}}\) is
\begin{equation}\label{app:ch_regular:eq:exact-physical}
 P_au=h+(A_\Gamma-I)z.
\end{equation}
The same \(S_{a,\Gamma},R_{a,\Gamma},K_a,T_a\), cutoffs, and core apply to all
such sources.
\end{thm}

\begin{proof}[Proof of Theorem~\ref{app:ch_regular:thm:exact-correction}]
Choose $p>2$ with $H^1(U)\hookrightarrow L^p(U)$. Equation~
\eqref{app:ch_regular:eq:residual-negative-positive} bounds
$R_{a,\Gamma}:L^2(W)\to L^p(U)$ uniformly before the final supports
are chosen. For the stated cutoffs, H\"older gives
\[
 \|\chi_2R_{a,\Gamma}\chi_3f\|_2
 \leq |\chregularsupp\chi_2|^{1/2-1/p}\|R_{a,\Gamma}\chi_3f\|_{L^p(U)}
 \leq C|\chregularsupp\chi_2|^{1/2-1/p}\|f\|_2.
\]
Fix the output support so this operator bound is at most $\theta<1$.
Their derivatives affect subsequent constants. The real operator
\begin{equation}\label{app:ch_regular:eq:neumann-series}
 T_a=\sum_{j=0}^{\infty}(-K_a)^j
\end{equation}
converges on the entire real $L^2(W)$, with norm at most
$(1-\theta)^{-1}$.

Fix smooth $a$ and $f\in H^s$, $s\geq1$.
Starting with $z=T_af\in L^2$, the identity $z=f-K_az$ and
\eqref{app:ch_regular:eq:one-gain-tame} at orders $1,\ldots,s$
give $z\in H^s$ and
\[
 \|z\|_{H^s}\leq\|f\|_{H^s}+C_s\|z\|_{H^{s-1}}
       +C_s(1+\|a\|_{H^{s+b}})\|z\|_2.
\]
Induction from $\|z\|_2\leq(1-\theta)^{-1}\|f\|_2$ proves
\eqref{app:ch_regular:eq:inverse-tame} with $\beta_*=b$, since
the preceding coefficient norms are bounded by the one at order $s+b$.
The same operator preserves smooth input. No parameter-tame
dependence is asserted.

Finally $z=T_a(\chi_2h)$ satisfies
$z=\chi_2h-K_az$, so $\chregularsupp z\subset\chregularsupp\chi_2$ and
$\chi_3z=z$. On $U_{\mathrm{core}}$ this becomes $z+R_{a,\Gamma}z=h$.
Applying the localized identity to this compactly supported $z$ yields
\[
 P_aS_{a,\Gamma}z=A_\Gamma z+R_{a,\Gamma}z
                 =h+(A_\Gamma-I)z,
\]
which proves \eqref{app:ch_regular:eq:exact-physical}.
\end{proof}

\subsection{Common graded profiles}\label{app:ch_regular:sec:profiles}

\begin{thm}[Common graded bounds]\label{app:ch_regular:thm:fixed-high}\label{app:ch_response:prop:response-mixed}
Use the fixed finite construction of
Setup~\ref{app:ch_regular:setup:analytic}. Suppose its actual coefficient
array satisfies
\begin{equation}\label{app:ch_regular:eq:initial-coefficient-profile}
 \|a\|_{C^r(W)}\leq D_rT^{(r-R_*)_+},\qquad r\geq0,\quad T\geq1,
\end{equation}
with one fixed $R_*>c_{\rm lin}+2$ above the construction orders.
Set $L=T$. Every normalized phase map $\mathcal G=\Phi_x$ or $\Phi_\xi$
has bounded zeroth and first jets and
\begin{equation}\label{app:ch_regular:eq:high-phase}
 |D^r\mathcal G|\leq C_rL^{r-1},\qquad r\geq2.
\end{equation}
Derivatives are joint in physical and normalized covariable coordinates.
Every actual full symbol of fixed order $p$, including the finite
remainders, satisfies, with $r=|\alpha|+|\beta|$,
\begin{equation}\label{app:ch_regular:eq:high-amplitude}
 |D_{x,z}^{\alpha}D_\xi^\beta a_p(x,z,\xi)|
 \leq C_{\alpha,\beta}\langle\xi\rangle^{p-|\beta|}L^r.
\end{equation}
All domains, operators, cutoffs and derivative reserves are fixed before accuracy.
\end{thm}
\begin{proof}
Theorem~\ref{app:ch_regular_base:thm:actual-calculus} and
Proposition~\ref{app:ch_regular_base:prop:finite-family} bound the
normalized phase maps and full symbols by
$C_r(1+\|a\|_{C^{r+c_{\rm lin}}})$.
This includes both amplitude positions, inverse charts, matrix transport,
cutoff jets and the exact inner-cutoff factors.
Corollary~\ref{app:cor:fixed-profile-transfer} gives directly
\begin{equation}\label{app:ch_regular:eq:fixed-high-derivative}
 1+\|a\|_{C^{r+c_{\rm lin}}}\leq C_rT^{(r-1)_+}.
\end{equation}
The phase and normalized symbol bounds follow; radial rescaling gives
\eqref{app:ch_regular:eq:high-amplitude}.
\end{proof}

\subsection{Protection by characteristic phase graphs}\label{app:ch_regular:sec:extension}

\begin{defn}[Protected source product]\label{app:ch_regular:defn:protected-saturation}
Fix $X\Subset W_0\Subset W$, with $X$ compact and every complete
physical flow tube of the selected factors contained in $W_0$.
A compact annular covector set $E_0$ over $X$ is protected if its characteristic
part lies in a conic set $\mathscr P\subset\{q_a=0\}$ invariant under
Hamilton segments wholly in $W_0$, in both orientations.
The normalized restriction of $\mathscr P$ over $X$, together with the
elliptic points of $E_0$, must be compactly inside $\Gamma_{\rm in}$,
with a positive gap from the nonprincipal set.
Uniform protection means uniform compact buffers on the fixed low
coefficient neighborhood.
\end{defn}

Proposition~\ref{app:ch_seed:prop:seed-actual-menu} puts all normalized
signed pair directions, including limiting difference chords and
antipodes, in one compact set $\chregularcC_{\rm quad}$ separated
from the frozen nonprincipal set $\mathfrak M$.

\begin{thm}[Protection of the complete source product]\label{app:ch_regular:thm:no-return}
Use that current immersion and pair family in a sufficiently small
fixed $C^{k_*}$ neighborhood of the cube, with $k_*$ above the regular
construction orders. There are fixed balls $X\Subset W_0\Subset W$,
with $X$ closed,
a real smooth coefficient extension $\widetilde a$ equal to $a_F$ on
$W_0$, and symmetric angular sets
\[
 \Omega_0\Subset\Omega_0'\Subset\Gamma_{\rm in}\Subset\Gamma_{\rm out}
\]
with these properties. Every actual normalized pair covector on $X$
lies inside $\Omega_0$; its magnitude is comparable to its true beat
$\lambda|z-z'|$, or to $\lambda$ for sums and cross-group pairs.
The entire product $X\times\Omega_0'$ is uniformly protected for
$\widetilde a$. All actual flow tubes lie in $W_0$ and the operator
agrees with $P_F$ on the core. The extension stays in the fixed low
class and preserves every stated coefficient profile.
\end{thm}
\begin{proof}
Let $a_0$ have the frozen cube principal matrices and a fixed smooth
lower-order completion. Choose small fixed balls and the affine extension
\begin{equation}\label{app:ch_regular:eq:coefficient-extension}
 \widetilde a=a_0+\mathcal E(a_F-a_0),\qquad
 \|\widetilde a\|_{C^r}\leq C_r(1+\|a_F\|_{C^r(W_0)}),
\end{equation}
using Lemma~\ref{app:ch_regular_base:lem:ball-extension}.
Shrink the current neighborhood by its fixed low operator norm.
Choose $\Omega_0\Subset\Omega_0'$ around $\chregularcC_{\rm quad}$;
the signed chord and sum bounds give the true beats.

Apply Lemma~\ref{app:lem:common-hamilton-phase} to $q_{\widetilde a}$
on a finite cover of its regular directions. Write these comparison
phases as $\Phi_\alpha(x,e,\eta')$. Choose compact annular parameter
sets $V_\alpha$ so that every characteristic point of
$X\times\Omega_0'$ has the form
\[
 (x,\Phi_{\alpha,x}(x,0,\eta')),\qquad \eta'\in V_\alpha,
\]
and these graphs for all $x\in\overline{W_0}$ lie in fixed compact
regular angular sets $D_\alpha$, with norms bounded above and below.
This is possible with uniform buffers: the frozen phase gradient
is independent of $x$, and the actual maps and their inverses are
uniformly close in the fixed low topology. Include the antipodes.
Choose $\Gamma_{\rm in},\Gamma_{\rm out}$ around these images and
$\Omega_0'$. Choose the actual finite regular factors with their
complete canonical rectangles having physical flow images inside
$W_0$; their source, target and both integral weights use the inner
buffers. These are fixed shrinkages in the same phase construction.

For fixed $\eta'$, put $\psi=\Phi_\alpha(\cdot,0,\eta')$.
Differentiating $q_{\widetilde a}(x,d\psi)=0$ gives
\[
 \psi_{xx}q_{\widetilde a,\xi}=-q_{\widetilde a,x}.
\]
Hamilton's equations and uniqueness therefore keep each segment
wholly in $W_0$ on this same graph, in either orientation.
Let $\mathscr P$ be the positive conic lift of the union of these
graphs over the compact $V_\alpha$. Its restriction over $X$ has
compact normalized image inside $\Gamma_{\rm in}$. Every complete
residual relation is diagonal or one of the actual characteristic
segments, so each finite chain preserves $\mathscr P$.
This proves protection of the whole source product, including
elliptic identity images, with no condition on exterior trajectories.
\end{proof}

\subsection{Finite powers and arbitrary accuracy}\label{app:ch_regular:sec:quantitative}

\begin{setup}[Actual coefficients and total mixed grades]
\label{app:ch_regular:setup:base-aware-coefficients}
Use Setup~\ref{app:ch_regular:setup:analytic} and the actual profile
\eqref{app:ch_regular:eq:initial-coefficient-profile} with $L=T=T_0$.
The operators and cutoffs are those of
Theorem~\ref{app:ch_regular:thm:exact-correction}.
For a compact annular frequency set $V_0'$, require uniform protection
of the whole $X\times V_0'$, where $X=\operatorname{supp}\chi_2$,
in Definition~\ref{app:ch_regular:defn:protected-saturation}.
The entire invariant set over $X$ lies in the selected buffered charts. For $\rho\geq2$
and fixed $0<\sigma<1$, assume
\begin{equation}\label{app:ch_regular:eq:slow-versus-frequency}
 1\leq L\leq\rho^{1-\sigma}.
\end{equation}
\end{setup}

The product protection is Definition~\ref{app:ch_regular:defn:protected-saturation};
Theorem~\ref{app:ch_regular:thm:fixed-high} gives the total mixed
phase and full amplitude grades from the stated coefficient bounds.

\begin{thm}[Arbitrary-accuracy correction]\label{app:ch_regular:thm:quantitative}
Use Setup~\ref{app:ch_regular:setup:base-aware-coefficients} and its
fixed operators and cutoffs. Choose a compact annular set
$V_0\Subset V_0'$ and one fixed smooth rescaled Fourier cutoff
$\Pi_{0,\rho}$, equal to one on the actual input frequency set
in $\rho V_0$ and supported in $\rho V_0'$.
Let \(h_\rho\) be real and smooth, supported where \(\chi_2=1\), and satisfy
\begin{align}
 \|h_\rho\|_{L^2}
 &\leq C H_\rho,\label{app:ch_regular:eq:source-ltwo}\\
 \|(I-\Pi_{0,\rho})h_\rho\|_{H^m}
 &\leq C_{m,Q}H_\rho\rho^{-Q}
 \quad\hbox{for every fixed }m,Q.\label{app:ch_regular:eq:source-tail}
\end{align}
The same exact inverse gives
\[
 z_\rho=T_ah_\rho,\qquad u_\rho=S_{a,\Gamma}z_\rho,
\]
with, for every fixed $m,Q$,
\begin{equation}\label{app:ch_regular:eq:pde-arbitrary}
 \|P_au_\rho-h_\rho\|_{H^m(U_{\mathrm{core}})}
 \leq C_{m,Q}H_\rho\rho^{-Q}.
\end{equation}
For every fixed $s$,
\begin{equation}\label{app:ch_regular:eq:solution-profile}
 \|u_\rho\|_{H^s}\leq C_sH_\rho\rho^s.
\end{equation}
The operator, its finite symbolic order, its physical domains, and its
angular source cutoff are unchanged when \(Q\) changes.
The conclusions keep genuine outgoing branches and make no assertion near
excluded nonprincipal characteristic directions.
\end{thm}

\subsubsection{Finite-chain localization}

\begin{lem}[Base-aware localization through the actual finite chain]
\label{app:ch_regular:lem:finite-chain}
Use Setup~\ref{app:ch_regular:setup:base-aware-coefficients}. For each fixed finite chain of graph, inverse-graph, diagonal, retarded and complete residual factors, one can choose buffered proof localizers in the factors' own canonical coordinates. Across an off-relation gap their full composed kernels satisfy
\begin{equation}\label{app:ch_regular:eq:chain-tail}
 \|(I-M_{\rm out})A M_{\rm in}\|_{L^2\to H^m}
 \leq C_{m,q}\rho^{m+C_A}L^b(L/\rho)^q.
\end{equation}
Here $C_A,b$ are fixed independently of $q$ and requested accuracy. For each fixed number of complete residual powers the final physical mask lies in an annulus $[c_j\rho,C_j\rho]$, $0<c_j<C_j<\infty$, with angular support strictly inside $\Gamma_{\rm in}$. Intermediate canonical masks need not lie in that physical cone. The discarded output has arbitrary prescribed $H^m$ decay. All actual operators, cutoffs and their finite symbolic order remain the same.
\end{lem}
\begin{proof}
The joint grades \eqref{app:ch_regular:eq:high-phase}--%
\eqref{app:ch_regular:eq:high-amplitude} hold at $L=T_0$ for both
amplitude positions, all cutoff jets and fixed Taylor parameters.
The buffered proof masks have the same grades. A fixed product
adds only one fixed amplitude offset $b$.

\noindent\textbf{A common kernel estimate.}
On comparable normalized annuli, a localized composed kernel has
phase $\rho F$ and compact base support. Off its full stationary
relation, compactness gives $|\nabla F|\geq\delta>0$.
The field $\nabla F\cdot\nabla/(i\rho|\nabla F|^2)$ then has
transpose cost $CL/\rho$ at each application, since
$|D^r\nabla F|\leq C_rL^r$. After $q$ integrations,
the fixed symbol orders, Fourier measures and $m$ output derivatives
give $C_{m,q}\rho^{m+C_A}L^b(L/\rho)^q$.
Schur's inequality proves \eqref{app:ch_regular:eq:chain-tail}.
For a half-space kernel, use a tangential field in a fixed collar
where the boundary tangential gradient has a gap, and a full-gradient
field in the interior. This gives the same count without differentiating
the boundary jump. We verify the relations and gaps for every factor.

\medskip\noindent\textbf{Graph factors.}
Write the actual kernel as
\[
 Af(u)=\int e^{i[\Phi(u,\xi)-z\cdot\xi]}
                  a_A(u,z,\xi)f(z)\,dz\,d\xi.
\]
Use the right input symbol $m_{\rm in}(y,\eta/\rho)$ and left
output symbol $m_{\rm out}(x,\zeta/\rho)$. The former has compact
base and annular support; the latter is one near the complete
canonical image, with base support compact on the actual output.
After frequency normalization their exact joint phase is
\begin{equation}\label{app:ch_regular:eq:joint-kernel-phase}
 F=\phi(u,\xi)-z\cdot\xi+(z-y)\cdot\eta+(x-u)\cdot\zeta,
 \qquad\Phi(u,\rho\xi)=\rho\phi(u,\xi).
\end{equation}
Stationarity in $(u,z,\xi,\eta,\zeta)$ means
\begin{equation}\label{app:ch_regular:eq:joint-stationary}
 \phi_u=\zeta,\qquad\eta=\xi,\qquad\phi_\xi=z,
 \qquad z=y,\qquad x=u.
\end{equation}
These equations describe precisely that full image. Thus
$1-m_{\rm out}$, including its identity contribution, has the
required joint gap; the input-base equation is essential.
For the literal reverse phase $z\cdot\xi-\Phi(u,\xi)$ the equations
are $\eta=\phi_u$, $\zeta=\xi$, $z=\phi_\xi$, $u=y$, $x=z$,
giving the inverse full graph relation. The weighted coordinate
inverse has this phase and its actual two-position amplitude.
Diagonal factors use $\phi(u,\xi)=u\cdot\xi$; the two-term
$\widetilde T$ and every negative-order graph use the same argument.

Outside comparable annuli, forward graphs have the larger-frequency
gap $\eta-\xi$ in $z$, or $\Phi_u-\zeta$ in $u$; reverse graphs
use $\eta-\Phi_u$ or $\xi-\zeta$. Homogeneity and the inverse
mixed gap give $|\Phi_u|\asymp|\xi|$. Each integration costs $L$
and one inverse larger frequency. Taking $q$ above a fixed
integrability threshold sums all dyadic shells with only a fixed
enlargement of $C_A$. This includes bounded frequencies and all
source shoulders, with both amplitude positions unchanged.

\medskip\noindent\textbf{Retarded and signed factors.}
The actual lower input buffer removes the term $H(t_--s)$ from
the primitive. Its remaining kernel is
\[
 i\alpha(t,y)H(t-s)\beta(s,y)\delta(y-y').
\]
With input $(t_i,y_i)$ and output $(t_o,y_o)$ its localized phase is
\[
 (t_o-t)\kappa+(y_o-y)\cdot\zeta
                 +(s-t_i)\tau+(y-y_i)\cdot\eta.
\]
Put $d=t-s\geq0$. At $d=0$, tangential stationarity gives the
full diagonal: $t_o=t_i$, $y_o=y_i$ and
$(\kappa,\zeta)=(\tau,\eta)$. In $d>0$ the additional equation
gives $\kappa=\tau=0$, $t_o\geq t_i$, $y_o=y_i$ and $\zeta=\eta$.
This is the oriented characteristic translation; its transverse
covector is nonzero by the input annulus. A chart with
$|\tau|\geq c|(\tau,\eta)|$ has only the diagonal branch.
Away from buffered neighborhoods of these images, compactness
gives the boundary tangential gap and the interior full gap used
above. For noncomparable frequencies the tangential derivatives
$\tau-\kappa$ and $\eta-\zeta$ give the larger-frequency gap.
The same dyadic count proves the full tail and nonzero output annulus,
without division by $\tau$ or an omitted endpoint value.
Literal $J^*$ exchanges the weights and reverses the translation;
its upper input buffer removes the fixed upper-endpoint term.

The exact signed identities
\eqref{app:ch_regular_base:eq:signed-graph-factor}--%
\eqref{app:ch_regular_base:eq:signed-physical-normalization} have
$A_-=c_A\mathfrak c_{\rm out}A_+\mathfrak c_{\rm in}$, $|c_A|=1$.
Use conjugate proof masks. Conjugation preserves supported integer
Sobolev norms and radial annuli, reflects covectors and cancels
between adjacent factors. Hence it proves the same bounds for
every negative chain. The complete residual $R=-\mathcal E$,
including its source-cut branches, consists of these actual ordered
factors by Lemma~\ref{app:ch_regular_base:lem:inner-cutoffs} and
Theorem~\ref{app:ch_regular_base:thm:sector}; every cutoff and
negative-order factor is included.

\medskip\noindent\textbf{The protected finite chain.}
Partition the entire source product $X\times\rho V_0'$.
At each factor choose a buffered full image in its own canonical
coordinates. Since $L/\rho\leq\rho^{-\sigma}$, a sufficiently
large $q$ gives any prescribed tail in
\eqref{app:ch_regular:eq:chain-tail}; increase its accuracy by the
polynomial Sobolev costs of the remaining finitely many factors.
No datum is differentiated.

Every complete residual relation is diagonal or a zero-characteristic
segment wholly in $W_0$. In each chart $h_j=\pm q_a$ by
\eqref{app:ch_regular_base:eq:schur-principal}, so either orientation
preserves $\mathscr P$. The actual $\chi_2$ puts every next source
base in $X$, where $\chi_3=1$. Thus final physical covectors lie
strictly inside $\Gamma_{\rm in}$; intermediate canonical masks
need not. The compact buffer permits finitely many nested proof
masks at each fixed depth. Homogeneity and the nonzero transverse
retarded covector give positive annular endpoints. Induction proves
the assertion, including original Fourier tails and localizer leakage
at a higher internal precision. No actual operator, cutoff or order
depends on that precision.
\end{proof}

\subsubsection{One estimate for the remaining inverse}

\begin{lem}[The remaining exact inverse]
\label{app:ch_regular:lem:finite-propagation-inverse}
Use the fixed complete $K_a$ and $T_a=(I+K_a)^{-1}$ above.
Suppose the actual graded profile gives
\[
 1+\|a\|_{H^{r+b}}\leq C_rL^r,\qquad 1\leq L\leq\rho,
\]
where $b$ covers their fixed tame shifts.
If $\|h\|_{H^{-N}}\leq C_NA\rho^{-N}$, then for fixed integers
$N,s\geq0$,
\begin{equation}\label{app:ch_regular:eq:inverse-remainder}
 \|T_aK_a^{N+s}h\|_{H^s}
 \leq C_{N,s}A L^s(L/\rho)^N.
\end{equation}
For bounded coefficient jets through the finitely many proof orders,
take $L=1$.
All products use the original supported/restriction domains.
\end{lem}
\begin{proof}
Both $K_a$ and $K_a^*=\chi_3R_{a,\Gamma}^*\chi_2$ have uniform
$L^2$ bounds and the same additive one-gain estimate on their
fixed reversed support pairs. Induction gives, for $B=K_a,K_a^*$,
\[
 \|B^jv\|_{H^j}\leq C_j(1+\|a\|_{H^{j+b}})\|v\|_2.
\]
At each step the new coefficient term multiplies only
$\|B^{j-1}v\|_2$; the other term uses the preceding Sobolev order.
Dualize the entire $(K_a^*)^N$ to obtain
\[
 \|K_a^Nh\|_2
 \leq C_N(1+\|a\|_{H^{N+b}})\|h\|_{H^{-N}}
 \leq C_NA(L/\rho)^N.
\]
The same induction at order $s$ gives
\[
 \|K_a^sv\|_{H^s}\leq C_sL^s\|v\|_2.
\]
Apply it to $v=K_a^Nh$. The tame estimate for $T_a$, using its
uniform $L^2$ bound in the high term, gives
\eqref{app:ch_regular:eq:inverse-remainder}.
\end{proof}

For every integer $d\geq1$, the same $L^2$ inverse satisfies
\begin{equation}\label{app:ch_regular:eq:finite-neumann-identity}
 T_ah=\sum_{j=0}^{d-1}(-K_a)^jh+(-1)^dT_aK_a^dh.
\end{equation}
If the original source obeys $\|h\|_{H^{-N}}\leq C_NA\rho^{-N}$
for all $N$ and $L\leq\rho^{1-\sigma}$, any operator $B_\rho$
with fixed polynomial bounds
$\|B_\rho v\|_{H^s}\leq C_s\rho^{c_s}\|v\|_{H^{t_s}}$,
with fixed nonnegative integers $t_s$, satisfies
\begin{equation}\label{app:ch_regular:eq:finite-observation}
 B_\rho T_ah=\sum_{j<d}(-1)^jB_\rho K_a^jh
                   +O_{s,Q}(A\rho^{-Q})\quad\hbox{in }H^s
\end{equation}
for one sufficiently large proof depth $d=d(s,Q)$.
Indeed, the lemma with $d=N+t_s$ bounds the last term by
$C A\rho^{c_s+t_s-\sigma N}$. The operators and domains are unchanged.

\begin{proof}[Proof of Theorem~\ref{app:ch_regular:thm:quantitative}]
The original annular source and its complementary tail give
\[
 \|h_\rho\|_{H^s}\leq C_sH_\rho\rho^s,\qquad
 \|h_\rho\|_{H^{-N}}\leq C_NH_\rho\rho^{-N}.
\]
The actual profile, with its reserve covering the fixed inverse
and solver shifts, bounds their coefficient terms by $C_sL^s$.
Their tame estimates prove \eqref{app:ch_regular:eq:solution-profile}
directly, since $L\leq\rho$.

Apply Lemma~\ref{app:ch_regular:lem:finite-chain} to each fixed
complete $K_a^jh_\rho$. Every following residual starts over
$X=\operatorname{supp}\chi_2$, where $\chi_3=1$. Its diagonal or
characteristic relation stays in the protected set $\mathscr P$. The resulting physical mask is supported strictly inside
$\Gamma_{\rm in}$, and its complementary tail has arbitrary precision.
The actual source plateau therefore gives
\[
 (A_\Gamma-I)K_a^jh_\rho
       =O_{j,m,Q}(H_\rho\rho^{-Q})\quad\hbox{in }H^m,
\]
including the fixed low-frequency modification. Intermediate proof
masks stay in their own canonical coordinates; every original cutoff
and complete error is included.

Apply \eqref{app:ch_regular:eq:finite-observation} with
$B_\rho=A_\Gamma-I$. The preceding finite-chain tails give
\begin{equation}\label{app:ch_regular:eq:resolved-angular-tail}
 \|(A_\Gamma-I)T_ah_\rho\|_{H^m}
               \leq C_{m,Q}H_\rho\rho^{-Q}.
\end{equation}
The exact identity \eqref{app:ch_regular:eq:exact-physical} proves
\eqref{app:ch_regular:eq:pde-arbitrary} on the same physical core.
\end{proof}

\section{Primary waves and simultaneous sampling}\label{app:sec:ch_waves}
\subsection{The primary family and its inputs}\label{app:ch_waves:sec:introduction}
We construct one finite primary family and separated finite-band
samples on the entire output space. The same
samples approximate the Hermitian background, while individual
corrections have no sample-count loss. Throughout, the
stress space is $\ker B_G$ and tangent covectors are raised by the actual
metric in the ambient reconstruction. The finite transport and sampling are completed in this section.
Appendix~\ref{app:sec:ch_response} then constructs the response used
to correct their full strain for the current immersion.

\subsection{Geometric and oscillatory estimates}\label{app:ch_waves:sec:preliminaries}

\begin{notation}\label{app:ch_waves:notation:conventions}
Use the dimensions, frozen stresses $Q_i^0$ and Frobenius pairing
of Definition~\ref{app:ch_seed:defn:seed-cube}. All tensor
complements and projections use that pairing. The common Fourier
and adjoint conventions are in Notation~\ref{app:not:fourier}.
Spatial and carrier supports have fixed larger buffers. Constants
may depend on those supports, the fixed low class, inverse gaps
and finitely many declared profile constants, but not on frequencies
or sample counts unless displayed.
An exact Fourier-band hypothesis refers to the canonical datum before any physical initial-variable cutoff. Such a cutoff stays in the wave amplitude; if it is applied to the datum, its Fourier leakage is estimated explicitly. In particular, $\kappa(y)P_Kf(y)$ is not declared band limited.
\end{notation}

\begin{defn}[Actual second-form reconstruction]\label{app:ch_waves:defn:geometry}
Let $G$ be an actual smooth immersion into $\chwavesR^N$, and put $g=\gamma(G)$. Write
\[
\mathrm{II}_{ij}=\Pi_{\mathrm{Nor}_G}\partial_{ij}G.
\]
Define
\[
B_G\colon\chwavesSym^2\chwavesR^n\longrightarrow\mathrm{Nor}_G,
\qquad B_G(T)=\sum_{i,j}T_{ij}\mathrm{II}_{ij}.
\]
Assume that $B_G$ is surjective. Its stress space and its orthogonal projection are
\[
K_G=\ker B_G,\qquad
\Pi_{K_G}=\chwavesId-B_G^*(B_GB_G^*)^{-1}B_G.
\]
In particular, $K_G$ is not $\ker B_G^*$. The inverse of
$B_G^*\colon\mathrm{Nor}_G\to K_G^\perp$ is
\[
N_G=(B_G^*)^{-1}
=(B_GB_G^*)^{-1}B_G\big|_{K_G^\perp}.
\]
For formulas on the fixed ambient space, extend $B_G^*$ by zero
on tangent vectors and put
\begin{equation}\label{app:ch_waves:eq:ambient-normal-inverse}
 \begin{gathered}
 P_{\tan,G}=dG\,g^{-1}(dG)^T,\qquad
 \mathsf A_G=B_GB_G^*+P_{\tan,G},\qquad
 \mathsf N_G=\mathsf A_G^{-1}B_G,\\
 \Pi_{K_G}=I-B_G^*\mathsf N_G,\qquad
 \mathsf N_G\big|_{K_G^\perp}=N_G.
 \end{gathered}
\end{equation}
The ambient matrix $\mathsf A_G$ is invertible on the orthogonal
normal and tangent summands. These are rational fixed-jet formulas.
For a tangent covector $u=(u_i)$, set
\begin{equation}\label{app:ch_waves:eq:reconstruction}
\mathcal V_G(u)
=\sum_{i,j}g^{ij}u_i\partial_jG+
N_G\Pi_{K_G^\perp}(\chwavessym\nabla^g u).
\end{equation}
Thus the tangential term is literally $dG\,g^{-1}u$.

For a real frame $Q_1,\ldots,Q_n$ of $K_G$, put
\begin{equation}\label{app:ch_waves:eq:reduced-symbol}
P_Gu=(Q_i:\chwavessym\nabla^g u)_{i=1}^n,\qquad
M_G(x,\xi)_{i\bullet}=(Q_i(x)\xi)^T,\qquad
p_G(x,\xi)=\det M_G(x,\xi).
\end{equation}
The operator $P_G$ includes the actual Christoffel terms. We write
$\xi\chwavessym b=(\xi b^T+b\xi^T)/2$.
\end{defn}



Full carrier groups
and all literal sample pairs use Definition~\ref{app:defn:exhaustive-pair-groups}
and Remark~\ref{app:rem:exhaustive-signed-pairs}.


\begin{setup}[The common oscillatory interface]\label{app:ch_waves:setup:oscillatory}
Use the full and slice inverse charts of
Proposition~\ref{app:ch_seed:prop:seed-phase-charts}, with
$x=(t,y)$, $z=(s,v)$, $v\in\chwavesR^2$, and normalized annular
$\theta\in\chwavesR^{n-1}$. They satisfy
Setup~\ref{app:ch_osc:setup:osc-normal} and
Proposition~\ref{app:ch_osc:prop:osc-channel} with $u=(s,\theta)$
and fixed spectator $v$. Use their common larger supports and
buffers at $M=20(n+1)$; the bounds include first parameter
derivatives and the literal full-dispersion phases.

For the normalized full operator $T_\epsilon^K$ and fixed-carrier
slice $L_{\epsilon,z}^K$ of those results, the entire-space bounds are
\[
 \|T_\epsilon^K\|\leq C\|a_\epsilon\|_{C^M},\qquad
 \|L_{\epsilon,z}^K\|\leq C\sqrt K\|a_\epsilon\|_{C^M}.
\]
The input is the entire complex chart $L^2$ space and the output
the entire physical $L^2$ space. On a fixed phase, an amplitude
difference of $C^M$ norm $\delta$ therefore costs $C\delta$ and
$C\sqrt K\delta$, respectively: for $\delta>0$, apply the cited mask-free proof
to that amplitude divided by $\delta$. Fixed matrix components
and spectator integration obey the same bounds.

For $0<h\leq1$ and real $0\leq m_h(s,v)\leq1$ with
$\|Dm_h\|_\infty\leq C/h$,
Proposition~\ref{app:ch_osc:prop:osc-normal} gives
\begin{equation}\label{app:ch_waves:eq:abstract-normal-two}
 \|T_\epsilon^Km_h(T_\epsilon^K)^*
       -T_0^Km_h(T_0^K)^*\|
 \leq C\epsilon/h,\qquad K\geq1.
\end{equation}
Both complex quadratures are used. By
Corollary~\ref{app:ch_osc:cor:osc-real}, the real channel $2\chwavesRea T$
has, on real outputs, the exact normal
\begin{equation}\label{app:ch_waves:eq:real-normal}
 (2\chwavesRea T)(2\chwavesRea T)^*=4\chwavesRea(TT^*).
\end{equation}
\end{setup}

\Needspace{8\baselineskip}
\subsection{Hamilton phases and transport amplitudes}\label{app:ch_waves:sec:actual}

\begin{thm}[Phases, transport and ambient reconstruction]\label{app:ch_waves:thm:leading}
\label{app:ch_waves:thm:mixed-tame}
Use an actual three-parameter primary patch, its adapted
coordinates $x=(t,y)$, carrier map $\zeta(z)$ and larger full
modulation charts from
Propositions~\ref{app:ch_seed:prop:seed-actual-menu} and
\ref{app:ch_seed:prop:seed-phase-charts}. Use the projected frame
$Q_i(G)=\Pi_{K_G}Q_i^0$. Fix the immersion, normal-rank, null-minor
and root gaps and the convex buffered full and slice inverse charts
on the entire larger source boxes. One sufficiently small finite
$C^m$ neighborhood fixes all required low bounds; the boxes and
gaps are independent of frequencies and accuracy.

For a real positive initial scalar $r_z(y)$ with fixed positive
lower and $C^m$ upper bounds, there are a real phase and amplitude
$b_G=\alpha_Gk_G$ satisfying
\[
 p_G(x,d_xS_G)=0,\qquad S_G(0,y,\zeta)=y\cdot\zeta,
 \qquad \alpha_G|_{t=0}=r_z,\qquad l_G^TP_Gb_G=0.
\]
Here $k_G,l_G$ are real nonzero right and left nullvectors of
$M_G(x,dS_G)$, and $\alpha_G$ has a positive lower bound.
The reconstruction \eqref{app:ch_waves:eq:reconstruction} obeys
\begin{equation}\label{app:ch_waves:eq:exact-half-strain}
 \tfrac12D\gamma_G\mathcal V_G(u)
                 =\Pi_{K_G}(\chwavessym\nabla^gu).
\end{equation}
For any amplitude $b$, put $\xi=d_xS_G$ and
\begin{equation}\label{app:ch_waves:eq:arbitrary-b-amplitudes}
 \nu_b=N_G\Pi_{K_G^\perp}(\xi\chwavessym b),\qquad
 A_0(b)=-\nu_b\otimes\xi.
\end{equation}
The exact product identity is
\begin{equation}\label{app:ch_waves:eq:exact-W-amplitude}
 \mathcal V_G(\tau^2e^{\chwavesii S_G/\tau}b)
 =e^{\chwavesii S_G/\tau}
       (\chwavesii\tau\nu_b+\tau^2\mathcal V_G(b)).
\end{equation}
For $b=b_G$, $|\nu_b|,|\xi|\geq c>0$ and
$A_0(b)^*A_0(b)=|\nu_b|^2\xi\xi^T$.
This transported tensor is linear in the initial density $r_z^2$.

One fixed shift $D$ gives, for every integer $r\geq0$,
\begin{align}
&\|S_G\|_{C^r(x,\zeta)}+\|b_G\|_{C^r(x,z,\zeta)}
 +\|\nu_b\|_{C^r(x,z,\zeta)}\notag\\
&\hspace{15mm}+\|\mathcal V_G(b)\|_{C^r(x,z,\zeta)}
 +\|A_0(b)\|_{C^r(x,z,\zeta)}
 \leq C_r\left(1+\|G\|_{C^{r+D}(B_1)}
                   +\|r_z\|_{C^{r+D}(y,z)}\right).
 \label{app:ch_waves:eq:joint-tame}
\end{align}
These are joint physical and normalized carrier norms on the same
larger domains at every order.
\end{thm}
\begin{proof}
The stated propositions give the phases, full modulation charts
and fixed inverse gaps. The metric, normal inverse and nullvectors
are smooth fixed-jet formulas on these gaps.
Choose real nullvectors with the reference orientation. Since
$\operatorname{adj}M_G$ is a nonzero multiple of $k_Gl_G^T$,
\[
 \partial_{\xi_t}p_G
   =(\text{nonzero scalar})\,l_G^TC_tk_G\ne0,
\]
where $C_t$ is the time coefficient of $P_G$. Thus the equation
for $\alpha_G$ is the noncharacteristic scalar transport
\[
 \sum_a(l_G^TC_ak_G)\partial_a\alpha_G
                 +(l_G^TP_Gk_G)\alpha_G=0.
\]
Its characteristic formula is
$\alpha_G=a_{\rm flow}\,r_z(y_{\rm foot})$, where the positive
flow exponential $a_{\rm flow}$ is independent of $r_z$.
This proves positivity and the density assertion.

The Gauss formula \maineqref{main:eq:tangent-normal} with
$\nu=N_G\Pi_{K_G^\perp}(\sym\nabla^gu)$ proves
\eqref{app:ch_waves:eq:exact-half-strain}; the product rule gives
\eqref{app:ch_waves:eq:exact-W-amplitude}.
For $b_G$, the null relation puts $\xi\chwavessym b_G$ in
$K_G^\perp$, and
\[
 |\xi\chwavessym b_G|^2\geq\tfrac12|\xi|^2|b_G|^2,
 \qquad B_G^*\nu_b=\xi\chwavessym b_G.
\]
The fixed gaps give the asserted lower bounds.

Equation~\eqref{app:ch_seed:eq:seed-phase-tame} gives the joint
one-high phase bound. The jet and characteristic formulas, with
Lemma~\ref{app:ch_regular_base:lem:buffered-interpolation}, give
one high norm of $G$ or $r_z$ for $b_G$; other factors use their
fixed low bounds. Reconstruction uses only a fixed number of derivatives,
so one fixed reserve $D$ proves \eqref{app:ch_waves:eq:joint-tame}.
\end{proof}

One algebraic correction reduces the strain by one further power.
The same current response will correct the complete residual.

\subsection{A two-term primary wave}
\label{app:ch_waves:sec:wkb}

\begin{thm}[One finite primary family]
\label{app:ch_waves:thm:wkb}
Use Theorem~\ref{app:ch_waves:thm:leading}. Set
$M=20(n+1)$ and $r_{\rm par}=M+3$. Assume
\[
 \|a_G\|_{C^r}+\|G\|_{C^r}+\|r_z\|_{C^r}
       \leq D_rL^{(r-R_*)_+},\qquad r\geq0,\quad L\geq1.
\]
Choose $R_*$ above the construction orders and
$D_{\rm pr}+r_{\rm par}+2$, where $D_{\rm pr}$
is one fixed one-high derivative reserve for the algebraic operations,
complete strain and differentiated reconstruction below. It is independent of accuracy. All estimates hold on the
larger primary boxes containing every later source and measurement
support. Put $b_0=b_G$ and
\begin{equation}\label{app:ch_waves:eq:primary-algebraic-correction}
 \mathscr R=(M_G(x,dS_G)|_{k_G^\perp\to l_G^\perp})^{-1}\Pi_{l_G^\perp},
 \qquad b_1=i\mathscr R P_Gb_0.
\end{equation}
For sufficiently large $\lambda=\tau^{-1}$, define
\begin{align}
 v_\tau(x,z,\zeta')
 &=\tau^2e^{iS_G(x,\zeta')/\tau}(b_0+\tau b_1),
 \label{app:ch_waves:eq:WKB-u}\\
 \mathcal W_\tau^{\rm pr}&=\mathcal V_G(v_\tau),\qquad
 W_{\tau,z}^{\rm pr}(\mu)
 =\int\mathcal W_\tau^{\rm pr}(x,z,\zeta(z)+\tau\eta)\,d\mu(\eta).
 \label{app:ch_waves:eq:WKB-W}
\end{align}
Here $\mu$ is any finite complex Fourier measure whose shifted
carriers lie in the original chart; the background uses $\mu=\delta_0$.
The fields are separately linear in the initial scalar and measure.
Ordinary data mean $d\mu=\widehat f\,d\eta$; on $B_{CK}$ their
variation is at most $CK^{(n-1)/2}\|f\|_2$.

Put $A_\tau^{\rm pr}=e^{-iS_G/\tau}d\mathcal W_\tau^{\rm pr}$.
For every physical order $r$ and joint carrier/patch parameter order
$k\leq r_{\rm par}$,
\begin{align}
 \|D_x^rD_{z,\zeta'}^k A_\tau^{\rm pr}\|_\infty
       &\leq C_{r,k}L^r,\notag\\
 \|D_x^rD_{z,\zeta'}^k(A_\tau^{\rm pr}-A_0(b_0))\|_\infty
       &\leq C_{r,k}\tau L^r.
 \label{app:ch_waves:eq:WKB-amplitude-difference}
\end{align}
The displacement amplitude has the first bound times $\tau$.
The bounds are uniform over the fixed positive radial labels.
The strain amplitude $\mathscr E_\tau=e^{-iS_G/\tau}
D\gamma_G\mathcal W_\tau^{\rm pr}$ satisfies
\begin{equation}\label{app:ch_waves:eq:WKB-strain}
 \|D_x^rD_{z,\zeta'}^k\mathscr E_\tau\|_\infty
       \leq C_{r,k}\lambda^{-3}L^r.
\end{equation}
The current response corrects this full atomic strain.

The phase maps and frame satisfy
\begin{equation}\label{app:ch_waves:eq:graded-phase}
 \|D^r(S_{G,x},S_{G,\zeta})\|\leq C_rL^{r-1},\qquad r\geq1,
 \qquad \|Q_i(G)\|_{C^r}\leq C_rL^r.
\end{equation}
The real Hermitian background satisfies
\begin{equation}\label{app:ch_waves:eq:WKB-background}
 \left\|\operatorname{Re}[(dW_{\tau,z}^{\rm pr,bg})^*
                    dW_{\tau,z}^{\rm pr,bg}]
       -|\nu_{b_0}|^2dS_G^2\right\|_{C^r}
                         \leq C_r\tau L^r.
\end{equation}
Its derivative stays nonzero, and its leading transported mean is
linear in the initial density $r_z^2$.

Fix smooth compact physical and input cutoffs $\chi,\chi_{\rm in}$.
For $K\geq1$ and $\epsilon=\tau K\leq\epsilon_0$, the actual
Hermitian channel, with input half-derivative and unitary dilation
$\eta=K\theta$, is, up to one fixed Fourier scalar,
\begin{align}
 T_{\tau,v}^{K,\rm pr}F(x)
 &=\left(\frac K{2\pi}\right)^{n/2}
   \int e^{iK\phi_\epsilon(x,s,v,\theta)}
    a_{\tau,\epsilon}^{\rm pr}(x,s,v,\theta)F(s,\theta)\,ds\,d\theta,
 \label{app:ch_waves:eq:actual-T}\\
 \phi_\epsilon
 &=\frac{S_G(x,\zeta(z)+\epsilon\theta)-S_G(x,\zeta(z))}{\epsilon},
 \label{app:ch_waves:eq:phase-quotient}\\
 a_{\tau,\epsilon}^{\rm pr}
 &=\chi\chi_{\rm in}|\theta|^{1/2}
 \bigl(Q_i:A_\tau^{\rm pr}(x,z,\zeta(z))^*
       A_\tau^{\rm pr}(x,z,\zeta(z)+\epsilon\theta)\bigr)_{i=1}^n.
 \label{app:ch_waves:eq:cross-amplitude}
\end{align}
The quotient has its smooth value at zero. The input is the entire
complex $L^2(s,\theta)$ and the output the entire $L^2(x;\mathbb C^n)$.
Measurement and initial cutoffs occur in this channel, not in the
ambient field. For $0<h\leq1$ and real masks $0\leq m_h\leq1$,
$\|Dm_h\|_\infty\leq C/h$, put
$E_{\tau,v}^{K,\rm pr}=T_{\tau,v}^{K,\rm pr}m_h(T_{\tau,v}^{K,\rm pr})^*$.
Let $E_{0,v}^K$ set $\tau=\epsilon=0$, so its amplitude is $A_0(b_0)$.
The whole-space channels are bounded, and their normals satisfy
\begin{equation}\label{app:ch_waves:eq:WKB-normal-two}
 \|E_{\tau,v}^{K,\rm pr}-E_{0,v}^K\|
 \leq C\tau K/h,\qquad K\geq1.
\end{equation}
The squared whole-space slice norm is at most $CK$. Spectator
integration and realification have the same conclusions. All fields
and constants use only finitely many profile constants at each order.
\end{thm}

\begin{proof}
Write $P=P_G$. The leading transport gives $M_G(x,dS_G)b_0=0$ and
$l_G^TPb_0=0$, so the algebraic correction satisfies
\[
 iM_G(x,dS_G)b_1+Pb_0=0.
\]
The finite list consists of $b_0,b_1,Pb_1$, together with
$\nu_{b_j},\mathcal V_G(b_j)$ and their first physical derivatives,
$j=0,1$. It uses the fixed-gap jet formulas and the already
constructed leading transport. Lemma~\ref{app:ch_regular_base:lem:buffered-interpolation}
gives one common one-high reserve $D_{\rm pr}$.
Corollary~\ref{app:cor:fixed-profile-transfer}, after at most
$r_{\rm par}$ parameter derivatives, gives
\begin{equation}\label{app:ch_waves:eq:WKB-grading}
 \|D_x^rD_{z,\zeta'}^kc\|_\infty\leq C_{r,k}L^r,
 \qquad k\leq r_{\rm par},
\end{equation}
for every member $c$. The same argument gives the phase and frame
bounds and uniform joint low norms through $M+2$.

Put $\operatorname{Gram}_G=(Q_i:Q_j)$. The two algebraic cancellations and exact
half-strain formula give
\begin{equation}\label{app:ch_waves:eq:WKB-residual}
 Pv_\tau=\tau^3e^{iS_G/\tau}Pb_1,
 \qquad
 \mathscr E_\tau
 =2\tau^3\sum_{i,j}Q_i(\operatorname{Gram}_G^{-1})_{ij}(Pb_1)_j.
\end{equation}
The fixed frame and inverse Gram have the same grades, proving
\eqref{app:ch_waves:eq:WKB-strain}.

Substitute $b=b_0+\tau b_1$ in
\eqref{app:ch_waves:eq:exact-W-amplitude} and differentiate once.
The leading derivative amplitude is $A_0(b_0)$; every other term
has a factor $\tau$. This proves the amplitude bounds.
Their product estimate, with
$A_0(b_0)^*A_0(b_0)=|\nu_{b_0}|^2dS_G^2$, proves
\eqref{app:ch_waves:eq:WKB-background} and nonvanishing.

The exact differentiated fields give the displayed channel.
Its phase and the full and slice inverse charts obey
Setup~\ref{app:ch_waves:setup:oscillatory} on the larger supports.
The phase and amplitude bounds through $M$, and inverse-chart
bounds through $M-2$, are uniform in $L$, including first
$\epsilon$ derivatives; the primary reserve covers the two extra
phase derivatives in the quotient. On this same phase, replacing
$A_\tau^{\rm pr}$ by $A_0(b_0)$ changes the $C^M$ amplitude by $O(\tau)$, hence its
channel and normal by $C\tau$. The remaining homotopy from
$\epsilon=\tau K$ to zero costs $C\epsilon/h$ at every $K\geq1$.
Since $\tau\leq\tau K/h$, this proves the normal comparison. The same setup gives
the slice and spectator bounds and the real normal
$4\operatorname{Re}E_{\tau,v}^{K,\rm pr}$ on real outputs.
\end{proof}

\subsection{Separated finite-band sampling}\label{app:ch_waves:sec:sampling}

\begin{thm}[Separated lattice sampling]\label{app:ch_waves:thm:sampling}
Use finitely many buffered three-parameter charts $z=(s,v)$ and
the finite primary family of Theorem~\ref{app:ch_waves:thm:wkb},
with fixed smooth nonnegative compact weights $\psi(z)$ and one
amplitude scale $L$ for all radial labels. Its fixed low topology
and parameter orders give the uniform $C^{M+2}$ amplitude and
inverse-chart bounds used below, including the $\epsilon$ derivatives
of the leading homotopy. All spatial, angular and annular measurement and input
cutoffs are fixed inside those larger charts; the finite collection
covers the selected angular support. Frequencies below one are omitted.
Put $q_{\rm quad}=2n-1$ and $q_{\rm col}=q_{\rm quad}+1$; use
$q_{\rm col}^3$ radial colors per chart, fixed before frequency and accuracy.

For $\Lambda\geq2$, $\lambda\geq1$, with $\Lambda/\lambda$ below the phase buffer, fix finitely many positive radial labels $R_c$ bounded above and below and set $\tau_c=(R_c\lambda)^{-1}$. Use one smooth dyadic square partition of $1\leq|\eta|\leq C\Lambda$, with scales $K\geq1$, and the same angular cutoffs for every color. Define actual fields by \eqref{app:ch_waves:eq:WKB-u}--\eqref{app:ch_waves:eq:WKB-W}, with $\tau=\tau_c$, keeping the entire phase, and define backgrounds directly at $\eta=0$.

Let $T_{c,z,K}$ be their actual background-modulation Hermitian contraction with $Q_i$, or any selected smooth tensor contraction satisfying the same fixed low amplitude bounds, with input half-derivative $|\eta|^{1/2}$ and the fixed localizations. Choose the fixed Fourier scalar so that dilation $\eta=K\theta$ gives prefactor $(K/(2\pi))^{n/2}$ and phase $K\phi_{\tau_cK}$. Let
\[
T_{c,z}=(T_{c,z,K})_K
\]
be the row on the Hilbert direct sum of entire annular input spaces. Its output is the entire
\[
\chwavescH=L^2(\chwavesR^n;\chwavesC^n),
\]
with compact measurement. Arbitrary row data correspond to finite sums of actual modulated ambient fields.

Choose $\kappa\geq0$ smooth of integral one, supported in
$(-1/4,1/4)$, and put
$w=\kappa*\mathbf1_{[-1/2,1/2]}^{*q_{\rm quad}}$.
For $0<h\leq h_0\leq1$, define
\[
 c_q=hq,\qquad \rho_q(z)=\prod_{\ell=1}^3w(z_\ell/h-q_\ell),
 \qquad N_s=h^{-3}.
\]
The fixed $h_0$ respects the enlarged block supports and chart buffers.
For each block meeting $\operatorname{supp}\psi$, independently
draw $U_q$ uniformly in $[0,h)^3$ and, for $m\in\mathbb Z^3$, set
\[
 Z_{q,m}=c_q+hm+U_q,\qquad
 \mu_{q,m}=h^3(\psi\rho_q)(Z_{q,m}).
\]
Omit zero weights; each block has at most $C_{q_{\rm quad}}$ nodes.
Color them by $q\bmod q_{\rm col}$ and assign each color its fixed
radial label. Distinct nodes
of one color are separated by $ch$. There are at most $CN_s$
samples, their weights are at most $C/N_s$, and their total mass
is bounded. Number the literal samples by $j$ and set
\[
 m_c=\sum_{q:\,q\bmod q_{\rm col}=c}\rho_q,
\]
and let $\mathscr D_\Lambda$ be the fixed dyadic scale set.
For each prefix define, with finite chart sums understood,
\begin{equation}\label{app:ch_waves:eq:sampling-prefix-normal}
\begin{aligned}
 T_{c,z}^{\leq K}&=(T_{c,z,\kappa})_{\kappa\in\mathscr D_\Lambda,\,\kappa\leq K},\\
 E_{\rm sample}^{\leq K}
   &=\sum_j\mu_jT_{c(j),Z_j}^{\leq K}(T_{c(j),Z_j}^{\leq K})^*,\\
 E_h^{\leq K}
   &=\sum_c\int\psi m_cT_{c,z}^{\leq K}(T_{c,z}^{\leq K})^*\,dz.
\end{aligned}
\end{equation}
Write $E_{\rm sample},E_h$ for the largest prefix. Each normal is
exactly the sum of its annular normals, since its input is the same
Hilbert direct sum restricted to that prefix. The actual wave maps,
radial labels, colors, weights and samples are unchanged.

The following whole-space bounds hold:
\begin{align}
\|T_{c,z}\|^2&\leq C\Lambda,&
\|T_{c,z}\|_{\chwavesHS}^2&\leq C\Lambda^n,\notag\\
\|E_h\|&\leq C,&
\chwavestr E_h&\leq C\Lambda^n.
\label{app:ch_waves:eq:sampling-operator-bounds}
\end{align}
For fixed $\epsilon>0$, set $h=\Lambda^{-(1+\epsilon)/3}$.
With probability tending to one,
\begin{equation}\label{app:ch_waves:eq:sampling-all-prefixes}
 \max_{K\in\mathscr D_\Lambda}
 \|E_{\rm sample}^{\leq K}-E_h^{\leq K}\|
                \leq\Lambda^{-\epsilon/4}.
\end{equation}
The ceiling and
complete dyadic partition are fixed before sampling; later choices of prefixes do not change the event
or its constants. Both input quadratures remain physical real data;
$2\chwavesRea T$ has normal $4\chwavesRea E$, with the same conclusions
up to fixed constants.

Define
\[
H_{c,z}=\chwavesRea[(dW_{\tau_c,z}^{\mathrm{pr,bg}})^*
                  dW_{\tau_c,z}^{\mathrm{pr,bg}}],
\qquad
H_{\mathrm{transport},z}=|\nu_{b_0}|^2dS(x,\zeta(z))^2.
\]
Localize tensor norms by a fixed compact spatial cutoff and put
\[
 H_{\rm col}=\sum_c\int\psi m_cH_{c,z}\,dz.
\]
Since $q_{\rm quad}<r_{\rm par}$, the stated low amplitude budget
already covers its carrier derivatives. Every realization satisfies
\begin{equation}\label{app:ch_waves:eq:background-quadrature}
 \left\|\sum_j\mu_jH_{c(j),Z_j}-H_{\rm col}\right\|_{L^2}
       \leq Ch^{q_{\rm quad}}.
\end{equation}
The primary amplitude comparison also gives
$\|H_{\rm col}-\int\psi H_{\mathrm{transport},z}\,dz\|_{L^2}
\leq C\lambda^{-1}$.
For each integer $r\geq0$ it also satisfies
\begin{equation}\label{app:ch_waves:eq:background-quadrature-rough}
\left\|\sum_j\mu_jH_{c(j),Z_j}
-H_{\rm col}\right\|_{C^r}
\leq C_rL^r.
\end{equation}
The primary family and common band ceiling are fixed before
sampling. Only the all-prefix normal estimate requires a
probability event; the background estimate and same-color spacing
hold for every realization.

Let $E_{\mathrm{transport,band}}$ be the continuous uncolored normal on the same bands and cutoffs, with $\phi_{\tau_cK}$ replaced by $\phi_0$ and $A_{\tau_c}^{\rm pr}$ by $A_0(b_0)$. Then
\begin{equation}\label{app:ch_waves:eq:colored-normal-comparison}
\|E_h-E_{\mathrm{transport,band}}\|
\leq C\lambda^{-1}\Lambda/h.
\end{equation}
Thus one separated sample set approximates that normal on the entire output space within the right side plus $\Lambda^{-\epsilon/4}$. A separately established lower bound on a specified output class
transfers by this perturbation estimate.

The conclusions apply to any same-phase ambient family satisfying
\eqref{app:ch_waves:eq:WKB-amplitude-difference}, the fixed full
and slice seminorms, and the stated background jets, with the same
leading tensor. In particular the proof uses no formula for its
linear strain.

Finally,
\begin{equation}\label{app:ch_waves:eq:physical-P}
\mathcal P=\sqrt2\sum_j\sqrt{\mu_j}\chwavesRea W_j^{\mathrm{bg}}
\end{equation}
is an actual field in $\chwavesR^N$, whose same-carrier Hermitian metric contribution is $\sum_j\mu_jH_j$. Doubled same-carrier oscillations and all cross-carrier products
remain as unremoved metric terms.
\end{thm}

\begin{proof}
\noindent\textbf{Operator bounds and separated blocks.}
Theorem~\ref{app:ch_waves:thm:wkb} and
Setup~\ref{app:ch_waves:setup:oscillatory} give, on entire chart spaces,
\[
 \|T_{c,z,K}\|^2\leq CK,
 \qquad\|T_{c,z,K}\|_{\chwavesHS}^2\leq CK^n.
\]
The second inequality is the squared-kernel integral after dilation:
the prefactor is $(K/(2\pi))^{n/2}$ and the amplitude and its physical
and normalized-frequency supports are uniformly bounded. Since
$T_{c,z}T_{c,z}^*=\sum_KT_{c,z,K}T_{c,z,K}^*$, geometric dyadic sums
give the first two bounds in \eqref{app:ch_waves:eq:sampling-operator-bounds}.
First discard the masks by positivity. By
\eqref{app:ch_seed:eq:seed-normal-annulus} and the exact change
\eqref{app:ch_osc:eq:osc-exact-normal-kernel}, each unmasked band
normal has a full two-position amplitude supported in
$cK\leq|\xi|\leq CK$, with uniform physical derivatives through
$2(n+1)$; the fixed smooth weight $\psi$ is part of this amplitude.
At any $\xi$ only a uniformly bounded number of bands contribute.
Lemma~\ref{app:ch_osc:lem:osc-amplitude} bounds their sum, giving
$\|E_h\|\leq C$ after the fixed spectator integral and color sum.
Tonelli gives
\[
 \chwavestr E_h=\sum_c\int\psi m_c\|T_{c,z}\|_{\chwavesHS}^2
                   \chwavesdd z\leq C\Lambda^n.
\]

Convolution with $\kappa$ makes $w$ smooth and nonnegative.
Fubini, the unit-box periodization and the unit masses give
$\sum_{k\in\mathbb Z}w(u-k)=1$. The $q_{\rm quad}$ box factors
give Fourier zeros of order at least $q_{\rm quad}$ at every nonzero
$2\pi k$. The support length is at most $q_{\rm quad}+1/2$.
There are $O(h^{-3})$ blocks, each with a fixed number of nodes
at spacing $h$ and total weight $Ch^3$. Distinct blocks of one
color have gap at least $h/2$. This proves separation, count
and mass bounds. Uniform lattice translation has mean
Lebesgue measure, so \eqref{app:ch_waves:eq:sampling-prefix-normal}
is its exact expectation, with $0\leq m_c\leq1$ and $\sum_cm_c=1$.

\medskip\noindent\textbf{Concentration on the entire output.}
Use the independent positive blocks
$X_q=\sum_m\mu_{q,m}T_{q,m}T_{q,m}^*$.
Their total weight is at most $Ch^3$. Put
$Y_q=X_q-\mathbb EX_q$ and $R=C\Lambda h^3$. Then
\[
 \begin{gathered}
 0\leq X_q\leq RI,\qquad \|Y_q\|\leq R,\qquad
 V=\sum_q\mathbb EY_q^2\leq RE_h,\\
 \|V\|\leq v:=CR,\qquad
 \operatorname{tr}V\leq CR\Lambda^n.
 \end{gathered}
\]
These bounds hold for every prefix with the same upper values.

Equations~(3.6)--(3.9) of \cited{MinskerBernstein}, with the positive
variance upper bound $v$, give
\[
 \mathbb P\!\left(\left\|\sum_qY_q\right\|>t\right)
 \leq\frac{14\operatorname{tr}V}{v}
       \exp\!\left(-\frac{t^2}{2(v+Rt/3)}\right),
 \qquad t^2\geq v+Rt/3.
\]
To justify this bound on the whole output, apply its finite-dimensional
form to $P_\ell Y_qP_\ell$ for finite-rank orthogonal
$P_\ell\uparrow I$. Their variance is at most $P_\ell VP_\ell$,
so the same $v$ and trace bound apply. Each $X_q$ and its expectation
are trace class, so $Y_q$ is compact;
the finite centered sum therefore converges in norm under these
compressions, and Fatou's lemma passes the bound to the full sum.
This also covers zero variance.

For $t=\Lambda^{-\epsilon/4}$ and $h^{-3}=\Lambda^{1+\epsilon}$,
the threshold holds at large $\Lambda$. The bound and a union over
the fixed prefixes give failure probability at most
\[
 C\Lambda^{n+1}\exp[-c\Lambda^{\epsilon/2}],
\]
proving \eqref{app:ch_waves:eq:sampling-all-prefixes} for the
unchanged channels and samples. Equation~\eqref{app:ch_waves:eq:real-normal}
gives the real-channel assertion.

\medskip\noindent\textbf{Background quadrature.}
For $W=w\otimes w\otimes w$, every nonzero integer lattice
Fourier coefficient of $pW$ vanishes when $\deg p<q_{\rm quad}$.
Indeed it differentiates the product $\prod_\ell\widehat w$
fewer than $q_{\rm quad}$ times; a nonzero coordinate keeps its
zero. Poisson summation therefore gives, for every shift $u$,
\[
 \sum_{m\in\mathbb Z^3}W(m+u)p(m+u)=\int_{\mathbb R^3}W(z)p(z)\,dz.
\]
The Hermitian phase cancels. Taylor-expand the $L^2$-valued
function $\psi H_{c,z}$ at $c_q$ through degree $q_{\rm quad}-1$.
The displayed identity cancels its polynomial part exactly in
block $q$. The remainder is $C h^{q_{\rm quad}}$ on that block;
its sampled and integral masses are $Ch^3$. Summing $O(h^{-3})$
blocks proves \eqref{app:ch_waves:eq:background-quadrature}, uniformly
in all shifts. Bounded total mass and the amplitude profiles give
the rough bound. Replacing each actual Hermitian tensor by its
leading one costs $C\lambda^{-1}$ in the prescribed fixed norm.

The carrier masks satisfy $\sum_cm_c=1$ and
$\|Dm_c\|_\infty\leq C/h$. Sum
\eqref{app:ch_waves:eq:WKB-normal-two} on all dyadic bands. Since
\[
 \sum_{K\leq C\Lambda}K/h\leq C\Lambda/h,
\]
this proves
\eqref{app:ch_waves:eq:colored-normal-comparison}. At transport order
phase, amplitude and input cutoffs are independent of the radial label,
so the masks sum to the uncolored normal. The triangle inequality
transfers any already known lower bound on its specified output class.

Finally, for a complex derivative matrix $D$,
\[
 2(\chwavesRea D)^T(\chwavesRea D)
     =\chwavesRea(D^*D)+\chwavesRea(D^TD).
\]
Applied to \eqref{app:ch_waves:eq:physical-P}, this identifies its
same-carrier Hermitian term as $\sum_j\mu_jH_j$. The second term is
the doubled oscillation, and every distinct pair gives a cross term;
these metric contributions remain separate.
\end{proof}

\begin{rem}\label{app:ch_waves:rem:reference-status}

At fixed bandwidth the sampled normals are compact on the entire output space. Their norm approximation therefore transfers coercivity only on an output class for which a lower bound has already been established.
\end{rem}

\subsection{Raw inputs and primary fields}\label{app:ch_waves:sec:individual}

\begin{thm}[Raw primary fields]\label{app:ch_waves:thm:individual}
Choose a finite family from Theorem~\ref{app:ch_waves:thm:wkb} on
its larger buffered boxes, and a compact inner box $X$.
Write $W_j,S_j,\zeta_j,\tau_j$ for its maps, phases, carriers and
inverse frequencies. Fix a band ceiling $\Lambda$ whose shifted
covectors stay in the original charts. The exact representation is
\begin{align}
e^{-iS_j(x,\zeta_j)/\tau_j}dW_j(f)
 &=\int e^{i\phi_j(x,\eta)}a_j(x,\eta)\widehat f(\eta)\,d\eta,\notag\\
 \phi_j(x,\eta)
 &=\frac{S_j(x,\zeta_j+\tau_j\eta)-S_j(x,\zeta_j)}{\tau_j}.
 \label{app:ch_waves:eq:individual-representation}
\end{align}
Equations~\eqref{app:ch_waves:eq:WKB-amplitude-difference}
and~\eqref{app:ch_waves:eq:graded-phase} give
\begin{equation}\label{app:ch_waves:eq:individual-grading}
 |D_x^ra_j|\leq C_rL^r,\qquad
 |D_x^r\partial_\zeta S_j|\leq C_rL^{r-1}\quad(r\geq1).
\end{equation}
The amplitudes and the required low slice seminorms are uniformly
bounded. Constants are independent of sample count and weights.

For $1\leq K\leq\Lambda$, let $X_K$ be the real Hilbert space underlying
the finite direct sum of complex initial Fourier $L^2$ spaces on dyadic
annuli up to $CK$, with fixed bounded-overlap cutoffs $\beta_\ell$. Put
\[
 \widehat{\mathcal B_Kv}(\eta)
 =\sum_\ell|\eta|^{1/2}\beta_\ell(\eta)v_\ell(\eta).
\]
This is the Fourier input before any spatial initial cutoff, and
for every integer $m\geq0$ it satisfies
\begin{equation}\label{app:ch_waves:eq:B-bounds}
 \|\mathcal B_Kv\|_{C^m}\leq C_mK^{m+n/2}\|v\|_{X_K},
 \qquad\|\widehat{\mathcal B_Kv}\|_1\leq CK^{n/2}\|v\|_{X_K}.
\end{equation}
The same bounds hold after fixed detector and high-pass masks,
with uniform rescaled seminorms, are included in each $\beta_\ell$.
These masks act on the annular datum; the separate zero atom is
unchanged. The corresponding raw map may depend on $j$.
Choose a fixed smooth $\chi_{\rm slice}$ inside the larger wave
validity region, equal to one near $X$ and the measurement support.
Then
\begin{equation}\label{app:ch_waves:eq:individual-raw-slice}
 \mathcal G_j^Kv=\chi_{\rm slice}e^{-iS_j/\tau_j}
                         dW_j(\mathcal B_Kv),\qquad
 \|\mathcal G_j^Kv\|_{H^s(\chwavesR^n;\chwavesC^{N\times n})}
 \leq C_sK^{s+1/2}\|v\|_{X_K},\qquad K\geq L.
\end{equation}
At order zero the bound holds for all $K\geq1$.
This cutoff is used only in the norm; the actual fields are unchanged.
At any prescribed finite order the raw slice bound holds for
$K\geq1$ under bounded coefficient and initial-data jets through
the fixed shifted orders. For every $f$ with $\operatorname{supp}\widehat f
\subset B_{CK}$, the physical bound is
\begin{equation}\label{app:ch_waves:eq:raw-physical-bound}
 \|e^{-iS_j/\tau_j}dW_j(f)\|_{C^m(\operatorname{supp}\chi_{\rm slice})}
 \leq C_m(K+L)^m\|\widehat f\|_1.
\end{equation}
These are the same smooth integral fields for arbitrary $v\in X_K$.
\end{thm}

\begin{proof}
On an initial annulus of radius $\kappa$, dimension $n-1$, Fourier
Cauchy--Schwarz with the input half-derivative gives
$\|D^m\mathcal B_\kappa v\|_\infty
 \leq C_m\kappa^{m+n/2}\|v\|_2$.
Cauchy--Schwarz over the annuli and their geometric squared-power
sum prove \eqref{app:ch_waves:eq:B-bounds}.

Use the componentwise slice estimate of
Setup~\ref{app:ch_waves:setup:oscillatory} on the entire larger
support. On an initial annulus $\kappa\leq CK$, a term with $q$
phase factors after $s$ physical derivatives has prefactor $\kappa^q$.
Its primitive excess through the additional $M$ slice derivatives
is at most $s-q+M+D_{\rm pr}$. The fixed reserve
$D_{\rm pr}+r_{\rm par}+2$ therefore gives the $C^M$ bound
$C_s\kappa^qL^{s-q}\leq C_sK^s$. The squared slice bound
is $C_sK^{2s}\kappa$; Cauchy--Schwarz and
$\sum_{\kappa\leq CK}\kappa\leq CK$ prove
\eqref{app:ch_waves:eq:individual-raw-slice}.
At order zero only uniform low slice seminorms enter; bounded jets
at prescribed finite orders allow $L=1$.

The quotient formula \eqref{app:ch_waves:eq:phase-quotient} gives
$|D_x^r\phi_j|\leq C_rKL^{r-1}$ for $r\geq1$. A term with $q$
phase derivatives in an order-$m$ product costs
$K^qL^{m-q}\leq(K+L)^m$. Integrating against $|\widehat f|$
proves \eqref{app:ch_waves:eq:raw-physical-bound}.
Compactly Fourier-supported $L^2$ data have all polynomial
moments in $L^1$, proving smoothness at every order.
\end{proof}

\section{Ambient response to modulated waves}\label{app:sec:ch_response}
\subsection{The fixed response and its inputs}\label{app:ch_response:sec:response-inputs}
We select the regular construction of
Setup~\ref{app:ch_regular:setup:analytic} on an entire primary patch,
using the full graphs and literal weighted coordinate inverse of
Appendix~\ref{app:sec:ch_regular_base}.

\begin{thm}[A fixed real response on an entire primary patch]\label{app:ch_response:thm:response-compatible-choice}
Use one compact pointed primary patch of
Theorem~\ref{app:ch_waves:thm:leading}, with section $t=0$,
phase $\phi_G$, a fixed compact neighborhood $Z$ of every allowed
shifted carrier, and the larger convex modulation chart. Write $\mathsf h_G=p_G|\xi|^{1-n}$.
The simple-root branch gives
\[
 \partial_t\phi_G=H_G(t,y,\partial_y\phi_G),\qquad
 \phi_G(0,y,\zeta)=y\cdot\zeta.
\]
Choose symmetric regular sectors
$\Gamma_{\rm in}\Subset\Gamma_{\rm out}$ inside the common
energy chart of this branch and its antipode, containing the entire
positive family and its antipode with compact buffers on the
larger physical box. All root and projection gaps are uniform on
the fixed low neighborhood.

The real regular family of Setup~\ref{app:ch_regular:setup:analytic}
can use just these two conjugate charts. The positive chart has
section-energy phase $\Phi_G(x,e,\eta')$ satisfying
\[
\begin{gathered}
 \mathsf h_G(x,d_x\Phi_G(x,e,\eta'))=e,\qquad
 \Phi_G(0,y,e,\eta')=y\cdot\eta',\\
 \Phi_G(x,0,\zeta)=\phi_G(x,\zeta)\qquad(\zeta\in Z).
\end{gathered}
\]
Both canonical projections have uniform inverse gaps. The literal
coordinate inverse is the weighted adjoint of
\eqref{app:ch_regular_base:eq:actual-matrix-inverse}, with its
original source and target cutoffs; the central inverse carrier
is $(0,\zeta)$. For either primary sign, the opposite source
piece and $A_\pm-I$ have a fixed normalized source gap, including
all shoulders. The full low-frequency part and source-cut errors
remain in the selected residual.

This family obeys Theorem~\ref{app:ch_regular:thm:exact-correction}:
its fixed complete residual $R_G$ and cutoffs give
$\mathcal T=(I+\chi_2R_G\chi_3)^{-1}$ on the entire $L^2$ space.
The signed partners and literal adjoints use the full identities in
Setup~\ref{app:ch_regular:setup:analytic}, including both integral
weights and endpoint buffers.

For finitely many primary patches, choose common finite-low bounds,
buffer minima and one response ball $U_{\rm resp}$ inside all patch
cores. All source and observation plateaus are fixed there before
frequencies, samples and requested accuracies. Every Fourier atom
or measure datum in a patch uses that patch's same selected map.
\end{thm}
\begin{proof}
On the compact primary family,
$\partial_{\xi_t}\mathsf h_G=|\xi|^{1-n}\partial_{\xi_t}p_G$
has a uniform nonzero gap. Its transverse covector is also bounded
away from zero: if $\xi'=0$, Euler's identity would give
$0=\mathsf h_G=\xi_t\partial_{\xi_t}\mathsf h_G$.
Lemma~\ref{app:lem:common-hamilton-phase} therefore gives the unique
homogeneous root and phase
\begin{align}
 \mathsf h_G(x,h_G(x,e,\eta'),\eta')&=e,
 \qquad |e|\leq\epsilon_e|\eta'|,
 \label{app:ch_response:eq:response-energy-root}\\
 \partial_t\Phi_G&=h_G(t,y,e,\partial_y\Phi_G),
 \qquad\Phi_G(0,y,e,\eta')=y\cdot\eta',
 \label{app:ch_response:eq:response-energy-Cauchy}
\end{align}
with common buffered projection gaps and joint one-high bounds.
At $e=0$ this is the primary Cauchy problem; uniqueness proves
phase equality on the whole shifted-carrier compact.

Shrink the physical strip and coefficient neighborhood once. Choose
a smooth angular $a_+$, independent of the source base, equal to one
near that positive family and supported in its common energy chart,
with support disjoint from its antipode. Put
\begin{equation}\label{app:ch_response:eq:response-whole-source-partition}
 a_-(\omega)=a_+(-\omega),\qquad a_\Gamma=a_++a_-.
\end{equation}
The same $T,\widetilde T,V,J$ handles the whole positive support,
including $e\ne0$; its exact partner handles the negative support.
Use the common high-frequency cutoff and a right source plateau
near $\operatorname{supp}\chi_3$, leaving the low piece in the
complete residual. Both inactive differences have the stated
full source gap.

Choose the literal coordinate inverse's source plateau to contain
the full canonical preimage of the inner target. At $(e,\eta')=(0,\zeta)$,
phase equality and Euler's identity give
\[
 d_x\Phi_G=d_x\phi_G,\qquad
 \Phi_G=\zeta\cdot\Phi_{G,\eta'}=z'\cdot\zeta,
 \qquad z=\Phi_{G,\eta}.
\]
The mixed inverse gap makes this the unique central inverse carrier.
Proposition~\ref{app:ch_regular_base:prop:finite-family} applied
to this phase and partition gives the actual weighted inverse,
its source-cut error, signed factors and complete residual, with
all original cutoffs and integral buffers. Theorem~
\ref{app:ch_regular:thm:exact-correction} gives $\mathcal T$.
Finite buffer minima and a ball inside every patch core give the
common response choices.
\end{proof}

The operator and all carrier-compatible geometric choices are now fixed.
Theorem~\ref{app:ch_response:thm:response-literal-transfer} below proves
the quantitative packet estimates for this selection from its literal
kernels and the separate profile hypotheses.


\begin{setup}[The proved response and its profile]\label{app:ch_response:defn:response-regular}
Use the immersion and reduced system of
Definition~\ref{app:ch_waves:defn:geometry}, and the per-patch real
response just fixed on a buffered chart $X$. The metric, normal and
Gram gaps are positive. Write
\[
 \mathcal K=\chi_2R_G\chi_3,\qquad
 \mathcal T=(I+\mathcal K)^{-1},\qquad V_G=\mathcal V_G.
\]
Fix a source cutoff $\chi\in C_c^\infty(X)$ inside
$\operatorname{Int}\{\chi_2=1\}$ and the primary phase chart.
Source and observation supports are those of
Setup~\ref{app:setup:domains}; the primary identity region is
$U_{\rm LAG}$.

The additional quantitative hypothesis is the actual profile
\begin{equation}\label{app:ch_response:eq:response-actual-G-profile}
 \|a_G\|_{C^v}+\|G\|_{C^v}\leq D_vT^{(v-R_*)_+},
 \qquad v\geq0,\quad T\geq1.
\end{equation}
Here $a_G$ contains the principal and connection coefficients of
$P_G$; it does not determine $G$. Choose $R_*>c_{\rm lin}+2$,
also above the fixed geometric jet reserves, before observation
and tail orders. The reconstruction coefficients, including one
derivative of $V_G$, are smooth functions of a fixed jet
$j^{q_{\rm geo}}G$ with the one-high bounds of
Lemma~\ref{app:lem:elementary-interpolation-jets}.

Theorem~\ref{app:ch_regular_base:thm:actual-calculus} and
Proposition~\ref{app:ch_regular_base:prop:finite-family} give the
selected full graphs, weighted coordinate inverses, diagonal and
source factors, and exact inner-cutoff products uniform $L^2$
bounds and fixed-shift Sobolev estimates. Supported-adjoint duality
gives negative integer orders; the signed integrals and their
literal adjoints preserve these orders with the original weights
and endpoint buffers. The inverse and solver obey
\eqref{app:ch_regular:eq:inverse-tame} and
\eqref{app:ch_regular:eq:parametrix-tame}. Together with the profile,
these give polynomial costs at every fixed order used below.
Ambient differentiation is counted separately.

With the original cutoffs, the complex-linear extension of the real
solver in \eqref{app:ch_regular_base:eq:signed-physical-normalization} is
\begin{equation}\label{app:ch_response:eq:response-actual-solver-formula}
 S_{G,\Gamma}^{\chresponseC}
 =\frac{\chi_{\rm out}}{i}
       \sum_{\pm}\widetilde T_\pm\mathcal J_\pm V_\pm A_\pm.
\end{equation}
Only the transport graph $T$, with invertible matrix amplitude
$U$, is inverted by $V$. The complete antipodal sum is real;
all low and cutoff errors belong to $R_G$.
\end{setup}

Lemma~\mainref{main:lem:history} and
Proposition~\ref{app:prop:fixed-high} verify this profile for the
iteration.




\subsection{Carrier transfer through the fixed factor family}
\label{app:ch_response:sec:response-mixed}

The actual profiles in
Setup~\ref{app:ch_response:defn:response-regular} give the common
scale $L_*=T$ by Theorem~\ref{app:ch_regular:thm:fixed-high}.

\begin{thm}[Carrier transfer through the fixed factor family]
\label{app:ch_response:thm:response-literal-transfer}
Use that setup and its whole-patch choice. On every active chart,
with $\zeta_c=(0,\zeta)$, the section-energy phase satisfies
\begin{equation}\label{app:ch_response:eq:response-exact-phase}
 S(x,\zeta_c)=\phi_G(x,\zeta).
\end{equation}
For fixed $0<\nu<1$, suppose
\[
 L_*\leq K,\qquad 2\leq K\leq\lambda^{1-\nu}.
\]
Let $A$ be $T$, $\widetilde T$, $V$, a full graph error from this
active chart, or any literal adjoint, with order at most zero and
its actual cutoffs. The forward carriers are
$E_{\rm in}=e^{i\lambda z\cdot\zeta_c}$ and
$E_{\rm out}=e^{i\lambda\phi_G(x)}$; adjoint-phase graphs use the
reverse pair. If $\|f\|_{H^p}\leq C_pA_0K^p$ for $p\geq0$, then
\begin{equation}\label{app:ch_response:eq:fixed-factor-transfer}
 \|E_{\rm out}^{-1}AE_{\rm in}f\|_{H^s}
                         \leq C_sA_0K^s.
\end{equation}
The same profile holds through every fixed finite sum or chain with
matching carriers, including physical multipliers, the buffered
signed integrals and their literal adjoints. It holds for the complete
residual and solver chains after the source cutoff. Every inactive
source component and $A_+-I$ applied to the original supported carrier
input has norm $C_{s,Q}A_0\lambda^{-Q}$ in physical and demodulated
$H^s$. All statements use full supports and literal Hilbert measures;
constants may use finitely many profile constants, while factors,
cutoffs and their orders are fixed.

These conclusions hold jointly on any fixed compact smooth parameter
box $c$, with $\zeta_c=(0,\zeta(c))$ in the full shifted-covector chart.
The frequency and all actual
operators are fixed when differentiating $c$. If
$\|\partial_c^\alpha f_c\|_{H^p}\leq C_{p,\alpha}A_0K^{p+|\alpha|}$,
the left side of \eqref{app:ch_response:eq:fixed-factor-transfer},
after $\partial_c^\alpha$, is at most
$C_{s,\alpha}A_0K^{s+|\alpha|}$. Parameter derivatives of the
complete complementary errors have the same arbitrary precision.
\end{thm}

\begin{proof}
Put $L=L_*$. Phase equality follows from the common Cauchy problem.
The full symbols have the bounds
\eqref{app:ch_regular:eq:high-amplitude}. Each order-at-most-zero
factor has uniform $L^2$ norm by the cited calculus; its required
low seminorms lie in the fixed $C^{R_*}$ bound, independently of
$K,\lambda$.
We first estimate each near-carrier graph on the whole envelope
Sobolev space and then bound its complete complement.

\noindent\textbf{Derivative transfer in both graph directions.}
Set $\eta=\lambda\zeta_c+\delta$ and
$\Delta S=S(x,\lambda\zeta_c+\delta)-S(x,\lambda\zeta_c)$.
In the proof select $|\delta|\leq c\lambda$ inside the larger
convex carrier chart. For one small fixed $c>0$,
\[
 B(x,\delta)=\int_0^1S_{x\xi}(x,\lambda\zeta_c+t\delta)\,dt,
 \qquad D_x\Delta S=B\delta
\]
has a uniform inverse: $B$ is uniformly close to
$S_{x\xi}(x,\lambda\zeta_c)$ on every physical support.
All other supports belong to the complete complement below.
Homogeneity gives the parameter identity
\[
 \partial_c\Delta S=C_c\delta,\qquad
 C_c=(\partial_c\zeta_c)^T
       \int_0^1S_{\xi\xi}(x,\zeta_c+t\delta/\lambda)\,dt.
\]
These coefficients have normalized order zero and the joint grades of $B$.

The exact demodulated forward phase is $\Delta S-z\cdot\delta$.
An output derivative produces $B\delta$; integration in $z$
transfers $\delta$ to an input derivative, or differentiates the
full amplitude. For the reverse phase $z\cdot\delta-\Delta S$,
use $\delta=B^{-1}D_x\Delta S$ and integrate in $x$.
The parameter term $C_c\delta$ transfers in the same way;
$\partial_ca(x,z,\lambda\zeta_c+\delta)$ contributes only
normalized covector derivatives of order zero.
The actual compact weights remove boundary terms. The induction
\eqref{app:ch_regular_base:eq:graph-derivative-transfer} now applies
with $B$ in place of $S_{x\xi}$. Each modified amplitude has
primitive derivative excess $s-|\beta|$, including the second
physical position. Its low $L^2$ seminorm costs the fixed
$c_{\rm lin}$ extra orders. The actual profile and
$R_*>c_{\rm lin}+2$ bound it by $C_sL^{s-|\beta|}$.
Covector derivatives are normalized on $\eta\asymp\lambda$;
the selector costs $\lambda^{-1}$ per derivative, so every
modified amplitude still has order zero.
The actual $L^2$ calculus, in both directions, gives
\begin{equation}\label{app:ch_response:eq:direct-envelope-profile}
 \|A_\lambda f\|_{H^s}
 \leq C_s\sum_{r=0}^s L^{s-r}\|f\|_{H^r}.
\end{equation}
This includes the weighted amplitude $j_SU^{-1}$ of the literal
coordinate inverse. Since $L\leq K$, it proves the stated profile
for the near-carrier graph, without projecting the datum.
Counting physical and parameter derivatives together gives the joint
assertion: each derivative lands on the source or adds one graded
amplitude derivative. With finitely many bounded jets, take $L=1$,
$K\geq1$ at the prescribed joint orders.

For the complete complement put $K_{\rm far}=c\lambda^{1-\nu/2}$
and split the source envelope there. On the lower part, the full
source-gradient gap outside the preceding carrier neighborhood is
$c(\lambda+|\eta|)$: use $z$ for a forward graph and $x$ for its
adjoint phase. Source integration and dyadic summation, as in
Lemma~\ref{app:ch_regular:lem:finite-chain}, give the bound
\[
 C_{s,q}A_0\lambda^{c_s}K_{\rm far}^{n/2}(L/\lambda)^q,
 \qquad L/\lambda\leq\lambda^{-\nu}.
\]
Here $c_s$ is fixed before $q$. Every radial or angular shoulder
and bounded-frequency piece has this same gap. The upper source has
\[
 \|f_{>K_{\rm far}}\|_{H^t}
 \leq C_{t,p}A_0K^pK_{\rm far}^{t-p}
 \leq C_{t,p}A_0\lambda^{t-\nu p/2}.
\]
Increasing $q,p$ pays every fixed tame, carrier and demodulation
cost, giving any prescribed precision in $\lambda$ for the full
complement. Proposition~\ref{app:ch_osc:prop:osc-source-shell}
gives the same bounds for the pure Fourier inactive source factors
and $A_+-I$. A fixed cutoff transfers torus profiles to the physical
chart by Leibniz's rule, so this argument also covers torus data.

Physical multipliers preserve the profile since their jets have
grade $L^r\leq K^r$. The zero-time canonical carrier commutes
exactly with the signed integrals. Their direct Sobolev estimates
in Proposition~\ref{app:ch_osc:prop:osc-retarded-tube} preserve
the same profile, including the literal adjoints with exchanged
weights and upper input buffers. Thus every near-carrier factor
preserves the profile, and every complement has arbitrary
$\lambda$ precision. Propagate each complement through the
remaining fixed factors using their polynomial Sobolev costs,
choosing its precision higher beforehand. This proves the assertion
for fixed chains without successive envelope projections.
The complete signed identities preserve all original cutoff orders. A fixed number of parameter derivatives in the original
complementary expressions adds only a fixed power of $\lambda$;
increasing the internal precision gives their stated joint bounds.

For the actual solver and residual, first apply their literal source
operators. Every input is first multiplied by $\chi_3$ or is supported in
$\operatorname{supp}\chi_2$, where the right physical source cutoff
is one. Thus $A_+$ is the identity up to the preceding $\lambda$-small error
and every other source term has the same precision. In the positive
residual keep the exact ordered terms $(TV-I)A_+$ and
$(L_G\widetilde T-TD_t)JVA_+$ from the differential identity and
the buffered $D_tJ$ identity. Estimate $TV$ and $I$ separately for
carrier transfer; their difference is never set to zero. Its gain-one
bound is the independent complete-normal estimate of Appendix~\ref{app:sec:ch_regular_base}.
The remaining measured error is a full order-$-1$ graph or the exact
source-cut graph product. Thus the graph and integral estimates apply
to these complete chains, with every original cutoff in its original
order.
\end{proof}

\subsection{Full ambient strain reconstruction}\label{app:ch_response:sec:response-ambient}
\begin{prop}[Exact full-tensor identity]\label{app:ch_response:prop:response-ambient}
Let $\operatorname{Gram}_G=(Q_i:Q_j)$ and define, for a symmetric tensor $h$ supported where $\chi_2=1$,
\begin{equation}\label{app:ch_response:eq:response-full-map}
 e_i=\tfrac12Q_i:h,\qquad
 z=\mathcal T(\chi_2e),\qquad u=S_{G,\Gamma}z,\qquad
 \mathcal B_Gh=V_G(u)-\tfrac12\mathsf N_Gh.
\end{equation}
This is one real linear map, extended complex linearly when needed. On $U_{\rm core}$,
\begin{equation}\label{app:ch_response:eq:response-full-identity}
 D\gamma_G(\mathcal B_Gh)-h
 =2\sum_{i,j}Q_i(\operatorname{Gram}_G^{-1})_{ij}(A_\Gamma-I)z_j.
\end{equation}
All complete residual components remain inside this same $z$.
\end{prop}
\begin{proof}
The map is exactly the normal completion in
\maineqref{main:eq:normal-completion}. Substitute
$P_Gu=e+(A_\Gamma-I)z$ from
\eqref{app:ch_regular:eq:exact-physical} into
\maineqref{main:eq:completion-error} to obtain the stated identity.
The real solver and normal inverse give real linearity; complex
linear extension preserves the formula.
\end{proof}

For a reduced source $e$ satisfying
Theorem~\ref{app:ch_regular:thm:quantitative},
\eqref{app:ch_response:eq:response-full-identity} converts its angular
error into the full tensor error. This uses that theorem's actual coefficient profile and protection
of the entire product $\chresponsesupp\chi_2\times V_0'$.
The fixed frame multipliers cost only a polynomial at each observation
order, absorbed by a higher internal accuracy.

\subsection{Graded sources and the full ambient response}
\label{app:ch_response:sec:response-graded}

\begin{prop}[Displacement and derivative of the same graded response]
\label{app:ch_response:prop:response-band-ambient}
\label{app:ch_response:prop:response-graded}
Keep the same fixed real ambient map, source cutoff and buffered
observation cutoff $\chi_{\rm core}$. Let $\lambda,K\geq2$ obey
\begin{equation}\label{app:ch_response:eq:response-windows}
 L_*\leq K\leq\lambda^{1-\nu},\qquad 0<\nu<1.
\end{equation}
Let $a$ be a smooth complex symmetric-tensor envelope on the fixed
auxiliary torus or on
$\mathbb R^n$, with
\begin{equation}\label{app:ch_response:eq:response-graded-source}
 \|a\|_{H^p}\leq C_pAK^{p},\qquad p\geq0,
\end{equation}
where $A\geq0$. Put
$h=\chi e^{i\lambda\phi_G}a$, $Y=\mathcal B_G^{\mathbb C}h$, and
\begin{equation}\label{app:ch_response:eq:response-two-envelopes}
 U=e^{-i\lambda\phi_G}\chi_{\rm core}Y,
 \qquad D=e^{-i\lambda\phi_G}\chi_{\rm core}dY.
\end{equation}
The explicit bounds are
\begin{align}
 \|U\|_{H^s}&\leq C_sA\lambda K^{s},
 \label{app:ch_response:eq:response-graded-displacement}\\
 \|D\|_{H^s}&\leq C_sA\lambda^2K^{s},
 \label{app:ch_response:eq:response-graded-derivative}
\end{align}
for every fixed $s$. On each fixed compact inner identity core,
\begin{equation}\label{app:ch_response:eq:response-graded-strain}
 \|D\gamma_GY-h\|_{H^s}\leq C_{s,Q}A\lambda^{-Q}.
\end{equation}
All orders use the same field and map. In particular exact-band data
$\operatorname{supp}\widehat a\subset B_K$ satisfy these conclusions
with $A=\|a\|_2$. Physical derivatives keep their
carrier factors; $D$ is the demodulated physical derivative.
For the parameter families of
Theorem~\ref{app:ch_response:thm:response-literal-transfer}, assume
$\|\partial_c^\alpha a_c\|_{H^p}\leq C_{p,\alpha}AK^{p+|\alpha|}$.
The same envelope bounds hold after $\partial_c^\alpha$, with an
additional $K^{|\alpha|}$ on the right.
\end{prop}
\begin{proof}
The profile in Setup~\ref{app:ch_response:defn:response-regular},
with $R_*>q_{\rm geo}$, gives by
Corollary~\ref{app:cor:fixed-profile-transfer}
\begin{equation}\label{app:ch_response:eq:response-reconstruction-grading}
 \|c(G)\|_{C^r}+\|d\phi_G\|_{C^r}\leq C_rL_*^r.
\end{equation}
Leibniz's rule and $L_*\leq K$ give $f=(Q_i:a/2)_i$ the profile
$\|f\|_{H^p}\leq C_pAK^{p}$.
Thus $z=\mathcal T(\chi e^{i\lambda\phi_G}f)$ and
$u=S_{G,\Gamma}z$.

Theorem~\ref{app:ch_response:thm:response-literal-transfer} applies
directly to this graded $f$. For $h_0=\chi e^{i\lambda\phi_G}f$,
every fixed $\mathcal K^jh_0$ and $S_{G,\Gamma}\mathcal K^jh_0$
has demodulated profile $C_{j,s}AK^s$. Append the angular source
factor to each finite chain. Its input is supported in
$\operatorname{supp}\chi_2$, inside the original right source
plateau, so
\[
 (A_\Gamma-I)\mathcal K^jh_0
       =O_{j,s,Q}(A\lambda^{-Q})\quad\hbox{in }H^s.
\]
Only for the original source, the same full source gap and high-envelope
split give $\|h_0\|_{H^{-N}}\leq C_NA\lambda^{-N}$.
Apply \eqref{app:ch_regular:eq:finite-observation} with
$\rho=\lambda$ and the operators $e^{-i\lambda\phi_G}S_{G,\Gamma}$
and $A_\Gamma-I$, whose Sobolev costs are fixed polynomials.
Since $L_*/\lambda\leq\lambda^{-\nu}$, the finite-chain estimates prove
\begin{equation}\label{app:ch_response:eq:response-graded-reduced}
 \|e^{-i\lambda\phi_G}u\|_{H^s}\leq C_sAK^s,\qquad
 \|(A_\Gamma-I)z\|_{H^s}\leq C_{s,Q}A\lambda^{-Q}.
\end{equation}
The fixed Gram and frame multipliers cost only a polynomial at
each observation order. Increase the internal precision by that
fixed cost and apply \eqref{app:ch_response:eq:response-full-identity}
to prove the strain estimate.

Write $u=e^{i\lambda\phi_G}g$. The literal reconstruction gives
$du=e^{i\lambda\phi_G}(i\lambda d\phi_G\,g+dg)$.
By \eqref{app:ch_response:eq:response-reconstruction-grading} and
$K\leq\lambda$, $V_G(u)$ costs at most $\lambda$ and
$dV_G(u)$ at most $\lambda^2$ beyond the reduced profile.
Every additional derivative acts on an envelope or slow coefficient
and costs $K$. The term $-\mathsf N_Gh/2$ obeys the same bounds,
also after the fixed observation cutoff.
For parameter derivatives apply the joint factor bounds to the same
finite expansion. In its last term the fixed $\mathcal K,\mathcal T$
are unchanged; derivatives hit the original source or final carrier. At total order $k$ their additional cost is
at most $\lambda^k$, absorbed by a larger finite depth. Reconstruction
then gives the stated joint bounds. Real data give real responses by
the unchanged conjugate pairing.
\end{proof}


\begin{cor}[A fixed finite jet budget]\label{app:ch_response:cor:finite-jets}
Fix a nonnegative integer $S$ and $0<\nu<1$. There is a finite
integer $m=m(S,\nu)$, depending also on the fixed construction,
with the following property. Suppose
$\|a_G\|_{C^m}+\|G\|_{C^m}$ is uniformly bounded and the joint
source bounds of the preceding proposition hold through total order
$m$ (with no parameter derivatives for a single source). If
\[
 \lambda\geq2,\qquad 1\leq K\leq\lambda^{1-\nu},
\]
then the same response and envelopes in
\eqref{app:ch_response:eq:response-two-envelopes} satisfy
\[
 \|\partial_c^\alpha U\|_{H^s}\leq C_{s,\alpha}A\lambda K^{s+|\alpha|},\qquad
 \|\partial_c^\alpha D\|_{H^s}\leq C_{s,\alpha}A\lambda^2K^{s+|\alpha|},
 \qquad s+|\alpha|\leq S.
\]
The budget is fixed before frequency and changes none of the
operators, fields or domains. It is independent of any later
strain accuracy.
\end{cor}
\begin{proof}
The preceding proof through joint order $s_1=S+2$ uses only finitely
many source and coefficient derivatives. At those orders use
\eqref{app:ch_response:eq:direct-envelope-profile} with $L=1$,
$K\geq1$, and the same complete source split. In
\eqref{app:ch_regular:eq:inverse-remainder}, take $N>s_1+2$;
apply the bounded fixed-order solver and demodulate. The last term is at most
$C_{N,s_1}A\lambda^{s_1-N}$ in $H^{s_1}$.
The finite sum has the proved envelope profiles, and the same
reconstruction adds at most two derivatives and carrier factors.
Choose the internal complement precision once to absorb these
fixed costs, and let $m$ be the largest derivative order used.
Only $S,\nu$ and the fixed construction enter this choice.
\end{proof}


\begin{cor}[The corrected atomic family]\label{app:ch_response:cor:current-atoms}
Use the two-term primary on its whole compact shifted-carrier patch.
Choose the fixed history reserve to include
$m(M+2+\ell+r_{\rm par},\nu)$, the primary shifts and the initial-data
orders, where $\ell=\lfloor n/2\rfloor+1$ and $r_{\rm par}=M+3$.
For the same source and observation cutoffs, put
\[
 \mathcal Y_\tau=\mathcal B_G^{\mathbb C}
       (\chi_{\rm src}D\gamma_G\mathcal W_\tau^{\rm pr}),
 \qquad \mathcal W_\tau^{\rm cur}=\mathcal W_\tau^{\rm pr}-\mathcal Y_\tau.
\]
Write $A_\tau^{\rm cur}=e^{-iS_G/\tau}d\mathcal W_\tau^{\rm cur}$.
On the observation support, through parameter orders $k\leq r_{\rm par}$
and uniformly over the fixed radial labels,
\begin{equation}\label{app:ch_response:eq:current-atomic-amplitudes}
 \lambda\|D_x^rD_c^k(e^{-iS_G/\tau}\mathcal Y_\tau)\|_\infty
 +\|D_x^rD_c^k(e^{-iS_G/\tau}d\mathcal Y_\tau)\|_\infty
 \leq C_{r,k}\lambda^{-1}T^{(r-M)_+}.
\end{equation}
Thus the corrected family has the original phase and the mixed
amplitude bounds \eqref{app:ch_waves:eq:WKB-amplitude-difference},
with the same leading tensor. Its full strain has arbitrary precision on the inner
identity core. The raw slice and sampling conclusions of
Theorems~\ref{app:ch_waves:thm:individual} and
\ref{app:ch_waves:thm:sampling} apply to this same corrected family.
All choices precede sampling and accuracy.

There is also one fixed shift $D_{\rm at}$. If an initial scalar has
$\|r\|_{C^s}\leq C_sT^{s+B}$ with $B\geq0$ and the prescribed
bounded low jets, then
\begin{equation}\label{app:ch_response:eq:current-atomic-offset}
 \|D_x^sD_c^k(A_\tau^{\rm cur}(r)-A_0(r))\|_\infty
 \leq C_{s,k}\lambda^{-1}T^{s+k+B+D_{\rm at}}.
\end{equation}
The shift is independent of $s,k,B$ and accuracy.

The atomic maps and $A_0$ are linear in their initial scalar $r$.
At every prescribed joint order $s$, one finite jet budget $d_s$
gives, for $\mathcal E_\tau(r)=
\Re[(A_\tau^{\rm cur}(r))^*A_\tau^{\rm cur}(r)]-A_0(r)^*A_0(r)$,
\begin{equation}\label{app:ch_response:eq:current-density-lipschitz}
 \|\mathcal E_\tau(r)-\mathcal E_\tau(\widetilde r)\|_{C^s}
 \leq C_s\lambda^{-1}
 (\|r\|_{C^{d_s}}+\|\widetilde r\|_{C^{d_s}})
 \|r-\widetilde r\|_{C^{d_s}}.
\end{equation}
The same coefficient budget is imposed on $G$; $d_s$ is independent
of frequencies and any later accuracy.
\end{cor}
\begin{proof}
The complete atomic strain is $\lambda^{-3}$ times the graded
amplitude in \eqref{app:ch_waves:eq:WKB-residual}.
For each $k\leq r_{\rm par}$, the finite-jet corollary at $K=1$
gives both normalized response envelopes in $C^{M+2}$ with size
$C\lambda^{-1}$. The joint graded response at $K=T$ gives their
$C^j$ norm at most $C_{j,k}\lambda^{-1}T^{j+k+\ell}$.
For $r>M+2$, choose $j\geq r+(M+k+\ell)(r-M-1)$.
Interpolation from $C^{M+1}$ in $x$ has exponent
$(j+k+\ell)(r-M-1)/(j-M-1)\leq r-M$.
The low estimate covers the remaining orders.
The full response identity proves the strain assertion.

The corrected field is still an integral on the exact original phase;
linearity of the fixed response permits integration of its atoms.
Its full and slice amplitudes on the shifted-carrier charts satisfy
the hypotheses used in the two cited proofs;
the reserve $M$ pays the additional slice seminorms after physical
differentiation. Their operator bounds, normal homotopy and independent lattice sampling
apply without changing cutoffs or data; $q_{\rm quad}<r_{\rm par}$
also covers the background quadrature. No derivative of the actual
inverse with respect to a parameter is used.

For the offset profile, the fixed one-high primary bounds give the
complete source size $\lambda^{-3}T^{B+D_{\rm pr}}$ at scale $K=T$,
including joint derivatives. Apply the graded response and the fixed
physical Sobolev embedding, with $D_{\rm at}=D_{\rm pr}+\ell$.
This proves \eqref{app:ch_response:eq:current-atomic-offset}.
Linearity follows from the scalar transport, algebraic correction
and fixed response. The finite-jet estimates applied to an initial
scalar difference give
$\|A_\tau^{\rm cur}(r)-A_0(r)\|_{C^s}
\leq C_s\lambda^{-1}\|r\|_{C^{d_s}}$.
Expand the two Hermitian products and use bilinearity to obtain
\eqref{app:ch_response:eq:current-density-lipschitz}.
\end{proof}
\section{Transported positive backgrounds and current rows}\label{app:sec:ch_background}
\subsection{The common geometry and row}\label{app:ch_background:sec:bg-inputs}
We use the actual reference and carrier charts of
Appendix~\ref{app:sec:ch_seed} and the primary maps of
Appendix~\ref{app:sec:ch_waves} on the
domains of Setup~\ref{app:setup:domains}. The choices below fix
their common finite-low neighborhood and define the continuum row.
We prove its full-output lower bound, one inverse and positive
density correction, then compare that same row with the sampled
waves. Appendix~\ref{app:sec:ordered-scale-selection} verifies the
later frequency and history windows.

\begin{defn}[Second form and reduced strain]\label{app:ch_background:defn:bg-geometry}
Use the actual geometry and ambient formulas of
Definition~\ref{app:ch_waves:defn:geometry}, with
$g_G=\gamma(G)$ and $\mathcal N_G=\mathrm{Nor}_G$.
For the fixed cube tensors of Appendix~\ref{app:sec:ch_seed}, put
\[
 Q_{G,i}=\Pi_{K_G}Q_i^0,\qquad
 \mathscr G_G=(Q_{G,i}:Q_{G,j})_{i,j=1}^n.
\]
Proposition~\ref{app:ch_seed:prop:seed-actual-menu} gives this frame
and its positive Gram gap on one fixed $C^2$ neighborhood.
Use $P_G$ of \eqref{app:ch_waves:eq:reduced-symbol} with this frame,
and write
\[
 q_G(H)=(Q_{G,i}:H)_{i=1}^n,\qquad
 q_G(x,\xi)=q_G(\xi\otimes\xi).
\]
The rational second-jet formulas and
Lemma~\ref{app:lem:elementary-interpolation-jets} give
\begin{equation}\label{app:ch_background:eq:bg-frame-tame}
 \|\Pi_{K_G}\|_{C^r}+\sum_i\|Q_{G,i}\|_{C^r}
 +\|\mathscr G_G^{-1}\|_{C^r}+\|\operatorname{coeff}(P_G)\|_{C^r}
 \leq C_r(1+\|G\|_{C^{r+2}}).
\end{equation}
Replacing $B_G$ by $B_G/\sigma$ leaves its kernel projection
unchanged. Uniform normalized gaps and fixed-order bounds for
$dG$, $B_G/\sigma$ and the metric connection therefore give
uniform frame and reduced-operator bounds.
The tangent, normal and reconstruction identities are
\maineqref{main:eq:tangent-normal} and
\eqref{app:ch_waves:eq:exact-half-strain}, pointwise on $W$.
\end{defn}

The actual flat reference and its fixed neighborhood are given by
Proposition~\ref{app:ch_seed:prop:seed-analytic},
Corollary~\ref{app:ch_seed:cor:seed-uniform-dilation} and
Proposition~\ref{app:ch_seed:prop:seed-actual-menu}.


\begin{defn}[Cube geometry and Hamilton carriers]\label{app:ch_background:defn:bg-carriers}
Use the analytic flat $F_0$ of
Proposition~\ref{app:ch_seed:prop:seed-analytic}, with surjective
actual second form and projected frame
$Q_{F_0,i}=\Pi_{K_{F_0}}Q_i^0$, $Q_{F_0,i}(0)=Q_i^0$.
Keep these fixed reference tensors and put
\[
 G_\sigma(x)=\frac{F_0(\sigma x)-F_0(0)}{\sigma}.
\]
For small positive $\sigma$, its metric is $I$, its normal gap
is positive and its actual frame is $C^6$ close to
\[
 Q_i^0=e_ie_i^T-c_n\mathbf J,
 \qquad M_0(\xi)=\operatorname{diag}(\xi_i)
       -c_n\Big(\sum_i\xi_i\Big)\mathbf J,
\]
where $c_n=(n-4)^{-2}$ and $\mathbf J$ is the all-ones matrix.
Use the finite full-sphere cover, simple-characteristic
patches $z=(s,v)\in\chbackgroundR\times\chbackgroundR^2$ and
buffered phase charts of
Propositions~\ref{app:ch_seed:prop:seed-frozen-menu}--\ref{app:ch_seed:prop:seed-phase-charts}.
These constructions hold on one fixed finite $C^d$ neighborhood.
For the normalized family use common boxes and normalized gaps
down to $\sigma=0$.

On each patch use adapted coordinates $x=(t,y)$, transverse covector $\zeta\in\chbackgroundR^{n-1}$, and an actual Hamilton phase satisfying
\[
 p_G(x,d_xS_G(x,\zeta))=0,
 \qquad S_G(0,y,\zeta)=y\cdot\zeta.
\]
The carrier map is $\zeta_a(z)$. Write
\[
 \xi_{G,a}=d_xS_G(x,\zeta_a(z)),\quad
 Y_{G,a}=d_\zeta S_G(x,\zeta_a(z)),\quad
 \Phi_{G,a}(x,z,\eta)=Y_{G,a}(x,z)\cdot\eta.
\]
The matrix $M_G(x,\xi_{G,a})$ has rank $n-1$. Smooth real right and left nullvectors $k_G,l_G$ are selected by fixed nonzero minors. A positive real scalar transport with prescribed positive initial scalar gives $b_G=\alpha_G k_G$ and satisfies $l_G^TP_Gb_G=0$. Its normal field and transport derivative are
\[
 \nu_G=\mathsf N_G(\xi_G\mathbin{\chbackgroundSym}b_G),
 \qquad A_0=-\nu_G\otimes\xi_G.
\]
Here $\xi\mathbin{\chbackgroundSym}b=(\xi b^T+b\xi^T)/2$. Both $\nu_G$ and $\xi_G$ have positive lower bounds on the selected compact supports. Their mixed spatial, carrier and transverse-covector derivatives satisfy
\begin{equation}\label{app:ch_background:eq:bg-mixed-input}
 \chbackgroundnorm{D^m(S_G,A_0,\nu_G)}
 \leq C_m(1+\chbackgroundnorm G_{C^{m+D}})
\end{equation}
with one fixed $D$, under the fixed finite low coefficient bounds. For variable initial scalar $r$, the right side also includes one $\chbackgroundnorm r_{C^{m+D}}$; this follows from its initial value evaluated at $Y_G$ times the scalar flow exponential.

Use the interior incidence roots $z_a(G;x,\widehat k)$ of
Proposition~\ref{app:ch_seed:prop:seed-actual-menu}. They satisfy
\[
 \widehat k\cdot d_\xi p_G(x,\xi_{G,a})=0,
 \quad |\partial_s(\widehat k\cdot d_\xi p_G(x,\xi_{G,a}))|\geq c,
\]
and the augmented Gram bound
\[
 |M_G(x,\widehat k)^Tw|^2+
 \sum_a|w\cdot q_G(x,\xi_{G,a})|^2\geq c|w|^2
 \qquad(w\in\chbackgroundR^n)
\]
on each neighborhood of the full-sphere cover.

For fixed spectator $v$, the averaged-gradient map
\begin{equation}\label{app:ch_background:eq:bg-Psi}
 \Psi_G(x,x',s,v,\eta)=
 \int_0^1d_x\Phi_G(x'+t(x-x'),s,v,\eta)\chbackgrounddd t
\end{equation}
is injective on one fixed buffered annular chart in $(s,\eta)$ and has a common inverse output chart. Its determinant is bounded away from zero on that chart. The corresponding averaged $y$-gradient charts are also buffered and injective. The full-dispersion phases
\[
 \phi_\epsilon(x,z,\theta)=
 \frac{S_G(x,\zeta_a(z)+\epsilon\theta)-S_G(x,\zeta_a(z))}{\epsilon}
 =\int_0^1d_\zeta S_G(x,\zeta_a(z)+t\epsilon\theta)\cdot\theta\chbackgrounddd t
\]
have the same buffered charts for $0\leq\epsilon\leq\epsilon_0$;
at zero they give the transport phase. Proposition~\ref{app:ch_seed:prop:seed-phase-charts}
gives the finite-low inverse-chart bounds, and
Theorem~\ref{app:ch_waves:thm:leading} gives
\eqref{app:ch_background:eq:bg-mixed-input}. Use their common
parameter neighborhoods, containing every shifted covector in
Definition~\ref{app:ch_background:setup:bg-wkb-input}.
\end{defn}


Use the full-amplitude bounds and diagonal reduction of
Theorem~\ref{app:ch_regular_base:thm:actual-calculus}, with its
finite-low derivative requirements included in the fixed neighborhood.
The transport normal becomes a PDO amplitude under the exact conic
change \eqref{app:ch_background:eq:bg-Psi}; fixed-jet substitution
and \eqref{app:ch_regular_base:eq:joint-primitive-product} give its
one-high estimates, also for literal adjoints.
For $C=2\Re T$, the raw inner product is $\Re\int f\overline h$
on both quadratures, and
\begin{equation}\label{app:ch_background:eq:bg-realification}
 C^*w=2T^*w,\qquad CC^*w=4\Re(TT^*w)\quad(w\text{ real}).
\end{equation}
All derivative shifts precede observation and accuracy.

\subsection{The full continuum normal and its low-frequency completion}\label{app:ch_background:sec:bg-frame}
We prove a full-output lower bound from the global incidence Gram estimate.

\begin{defn}[Continuum row]\label{app:ch_background:defn:bg-row}
Fix one sufficiently rescaled reference $G_{\rm ref}=G_\sigma$ and a small fixed $C^d$ neighborhood of it. Fix domains
\[
 U_{\rm core}\Subset B_r\Subset U_{\rm meas}\Subset W
\]
and a flat auxiliary torus containing $W$. The radius $r$ will be chosen sufficiently small. All extensions and cutoffs are fixed before $G$ varies. Take $\chi_{\rm meas}=1$ throughout $B_r$, with support strictly inside $U_{\rm meas}$, and $\chi_{\rm src}=1$ near $\overline {B_r}$, with support in $W$. For every carrier patch choose a smooth nonnegative $\rho_a(z,\widehat\eta)$, positive near the selected incidence roots and their modulation directions, compactly supported in the buffered input charts. Let $\chi_{\rm hi}=0$ for $|\eta|\leq1$ and $\chi_{\rm hi}=1$ for $|\eta|\geq2$.

With one fixed Fourier normalization, define the complex channel by
\begin{equation}\label{app:ch_background:eq:bg-T}
 T_af(x)=\int e^{i\Phi_{G,a}(x,z,\eta)}
 \chi_{\rm meas}\rho_a\chi_{\rm hi}(|\eta|)|\eta|^{1/2}
 |\nu_a(x,z)|^2q_G(x,\xi_a(x,z)) f(z,\eta)\chbackgrounddd z\chbackgrounddd\eta.
\end{equation}
The output is zero outside the chart, and the input is the full complex $L^2(z,\eta)$. Set $C_a=2\Re T_a$. The physical row is the direct-sum row
\[
 A_{\rm phys}=\left(
 \chi_{\rm meas}P_G\langle D\rangle^{-1}\chi_{\rm src},\ (C_a)_a\right).
\]
Take a real $\psi$ with $\psi=1$ on $U_{\rm core}$ and $\chbackgroundsupp\psi\Subset B_r$. Append $A_{\rm out}=1-\psi$ and write
\[
 A_{\rm aug}=(A_{\rm phys},A_{\rm out}),
 \qquad E_{\rm aug}=A_{\rm aug}A_{\rm aug}^*.
\]
The exterior input is auxiliary data on the torus. It is not a Euclidean ambient coordinate.
\end{defn}

\begin{prop}[Continuum lower bound and auxiliary inverse]\label{app:ch_background:prop:bg-frame}
For one sufficiently small fixed $r$ and one sufficiently small finite $C^d$ neighborhood of $G_{\rm ref}$, every row in Definition~\ref{app:ch_background:defn:bg-row} is bounded on its full $L^2$ spaces. Uniformly for real $f$ supported in $B_r$,
\begin{equation}\label{app:ch_background:eq:bg-local-bound}
 \chbackgroundnorm{A_{\rm phys}^*f}_2^2\geq c\chbackgroundnorm f_2^2.
\end{equation}
On the entire real auxiliary torus,
\begin{equation}\label{app:ch_background:eq:bg-global-bound}
 E_{\rm aug}\geq c_0I.
\end{equation}
It has one bounded self-adjoint $L^2$ inverse. The exterior channel vanishes on $U_{\rm core}$.
The same choices work after multiplying each prescribed initial
scalar by $\sqrt{1+v_a(z,y)}$, with
$1/2\leq1+v_a\leq3/2$ and $\|v_a\|_{C^{d_0}}\leq1$, where
$d_0$ includes the complete normal proof's fixed low derivative reserves.
\end{prop}

\begin{proof}
Fix $\sigma>0$ and one common finite neighborhood preserving the
stated gaps, chart buffers and amplitude bounds. In this proposition
the geometric constants may depend on $\sigma$. All choices
include the stated fixed-low class of initial multipliers.

\noindent\textbf{Step 1. The exact transport normal.}
The transport derivative has
$Q_{G,i}:A_0^*A_0=|\nu_a|^2Q_{G,i}:(\xi_a\otimes\xi_a)$,
which gives the amplitude in \eqref{app:ch_background:eq:bg-T}.
For fixed spectator $v$,
$\Phi_a(x,s,v,\eta)-\Phi_a(x',s,v,\eta)
=(x-x')\cdot\Psi_a(x,x',s,v,\eta)$.
The annular inverse extends homogeneously to the cone over a smaller
compact support. The transverse output block is close to $\eta$,
so equal outputs have input-magnitude ratio close to one. Reduce
the fixed angular support and perturbation size to fit the annular
buffer. A common normalization then puts both inputs in that
buffered chart, where injectivity holds. Homogeneity and the inverse identity give
\begin{equation}\label{app:ch_background:eq:bg-jacobian}
 J_a=|\det D_{(s,\eta)}\Psi_a|,
 \qquad c|\eta|\leq J_a\leq C|\eta|.
\end{equation}
Changing exactly to $k=\Psi_a$ in $T_aT_a^*$ gives its kernel
\[
 K_{T_aT_a^*}(x,x')=\int e^{i(x-x')\cdot k}
                         b_{T,a}(x,x',k)\chbackgrounddd k,
\]
where, at $(s,\eta)=u_a(x,x',v,k)$,
\begin{equation}\label{app:ch_background:eq:bg-full-complex-amplitude}
\begin{split}
 b_{T,a}(x,x',k)
 ={}&\chi_{\rm meas}(x)\chi_{\rm meas}(x')\int_v
 \frac{\rho_a(s,v,\widehat\eta)^2\chi_{\rm hi}(|\eta|)^2|\eta|}{J_a}\\
 &\quad\cdot|\nu_a(x,z)|^2|\nu_a(x',z)|^2
 q_G(x,\xi_a(x,z))q_G(x',\xi_a(x',z))^T\chbackgrounddd v.
\end{split}
\end{equation}
Thus the normalized PDO amplitude in
Theorem~\ref{app:ch_regular_base:thm:actual-calculus} is
$(2\pi)^n b_{T,a}$. The input support lies strictly
inside the inverse chart, so zero extension is smooth. The two half
weights cancel the degree-one Jacobian. Differentiating the inverse
identity gives order-zero full-amplitude estimates, with one high
geometric norm; each $k$ derivative lowers order. The finite-low
full-space bound therefore controls $T_a$ and $C_a$.

For $k\ne0$, define its homogeneous diagonal by
\begin{equation}\label{app:ch_background:eq:bg-symbol}
 d_{0,a}(x,k)=\left.b_{T,a}(x,x,k)\right|_{\chi_{\rm hi}=1}.
\end{equation}
Since $|\eta|\asymp|k|$, the part containing
$\chi_{\rm hi}^2-1$ is a bounded-frequency smoothing term.
The identity $C_aC_a^*=4\Re(T_aT_a^*)$ and full-amplitude
reduction in Theorem~\ref{app:ch_regular_base:thm:actual-calculus}
give the real principal matrix
\begin{equation}\label{app:ch_background:eq:bg-real-symbol}
 b_a(x,k)=2(2\pi)^n[d_{0,a}(x,k)+d_{0,a}(x,-k)]
                 \geq2(2\pi)^n d_{0,a}(x,k).
\end{equation}
No evenness of the input mask is required. There are an actual
order-minus-one reduction error and the bounded-frequency
error. Their literal adjoints have the same gains and fixed shifts.

For the image normal, the fixed-source formula at phase $x\cdot\xi$
moves $\chi_{\rm src}^2$ past $\langle D\rangle^{-1}$ with an
order-minus-two error. The next inverse multiplier and the two
first-order factors $P_G,P_G^*$ leave an order-minus-one error.
The leading full amplitude is
$M_G(x,\xi)M_G(x',\xi)^*\langle\xi\rangle^{-2}$ times the fixed
cutoffs. The same direct diagonal reduction therefore applies.

\noindent\textbf{Step 2. Principal positivity.}
Differentiating $p_G(x,d_xS_G)=0$ in $\zeta$ shows that
$D_\zeta d_xS_G$ maps onto the characteristic tangent hyperplane;
its transverse block has rank $n-1$.
At each incidence root there is therefore a unique nonzero
$\eta_a$ with $d_x\Phi_a=\widehat k$. The incidence derivative
bounds $J_a/|\eta_a|$ above and below. Choose $\rho_a$ positive
near these root/modulation pairs.

The image normal has principal matrix $M_GM_G^T$ up to a fixed
positive normalization. If it and all $d_{0,a}$ annihilate $w$,
then $M_G^Tw=0$. The homogeneous positive weights in
\eqref{app:ch_background:eq:bg-symbol} and continuity also give
$w\cdot q_G(x,\xi_{G,a})=0$ at every selected incidence root.
The augmented Gram bound forces $w=0$. Compactness and continuity
therefore give a uniform positive gap on the entire sphere.
For the allowed initial multiplier, scalar transport multiplies
$\nu_a$ by $\sqrt{1+v_a(z,Y_{G,a})}$. Every detector symbol
weight is therefore multiplied by $(1+v_a)^2\geq1/4$, and its
finite-low full-amplitude bounds remain uniform.
The gap persists under the chosen finite-low perturbations.

\noindent\textbf{Step 3. Low frequencies and small support.}
Fix a plateau $\chi=1$ near the eventual $B_r$, inside the positive
principal-symbol region. Full-amplitude reduction gives
\[
 \chi A_{\rm phys}A_{\rm phys}^*=\chi\operatorname{Op}(e_0)+R_0,\qquad
 R_0:H^{-1}\longrightarrow L^2.
\]
Choose a fixed smooth low-frequency extension with
$e_0(0,k)\geq cI$. Near $B_r$, Taylor's formula gives
\[
 \operatorname{Op}(e_0)-e_0(0,D)=\sum_i x_iT_i.
\]
The order-zero $T_i$ have symbols
$\int_0^1\partial_{x_i}e_0(tx,k)\,dt$, extended with fixed
cutoffs, and are uniformly bounded. For $f$ supported in $B_r$,
positivity of the frozen Fourier multiplier gives
\[
\begin{split}
 \|A_{\rm phys}^*f\|_2^2
 &\geq c\|f\|_2^2-\sum_i\|x_if\|_2\|T_if\|_2
                 -C\|f\|_{H^{-1}}\|f\|_2\\
 &\geq(c-Cr)\|f\|_2^2.
\end{split}
\]
Here $\|f\|_{H^{-1}}\leq Cr\|f\|_2$ follows from Sobolev
duality and H\"older on the small support. Choose $r$ inside the
fixed plateaus to absorb both errors. This proves
\eqref{app:ch_background:eq:bg-local-bound}.

\noindent\textbf{Step 4. Exterior completion.}
Apply the local estimate to $\psi f$. Global row boundedness gives
\[
 \|f\|_2\leq\|\psi f\|_2+\|(1-\psi)f\|_2
 \leq C\|A_{\rm phys}^*f\|_2+C'\|(1-\psi)f\|_2.
\]
Squaring proves \eqref{app:ch_background:eq:bg-global-bound}.
This bounded self-adjoint operator has closed range and zero
kernel, so its range is also dense. Its inverse is bounded and
self-adjoint.
The exterior row vanishes on $U_{\rm core}$.
\end{proof}

\subsection{One inverse and its initial-variable adjoint}\label{app:ch_background:sec:bg-inverse}
\label{app:ch_background:sec:bg-adjoint}
The $L^2$ inverse, derivative commutators and one derivative trade
give the raw canonical data and physical density correction.

\begin{prop}[The inverse and initial-variable adjoint]
\label{app:ch_background:prop:bg-inverse}
\label{app:ch_background:prop:bg-adjoint}
Use the common finite-low class of
Proposition~\ref{app:ch_background:prop:bg-frame}, including its
positive initial scalars $r_a$, and write
\[
 A=A_{G,r},\qquad E=AA^*,\qquad
 \mathcal C_m=1+\|G\|_{C^m}+\sum_a\|r_a\|_{C^m}.
\]
Let $X^s$ be the direct sum of auxiliary $H^s$ inputs and carrier
initial-variable $H^s_{z,y}$ inputs before their physical cutoffs.
For one fixed $B$, the same inverse satisfies
\begin{equation}\label{app:ch_background:eq:bg-r-inverse}
 \|E^{-1}e\|_{H^s}+\|A^*E^{-1}e\|_{X^s}
 \leq C_s[\|e\|_{H^s}+\mathcal C_{s+B}\|e\|_2],
 \qquad s\geq0.
\end{equation}
It preserves $H^\infty$, and $A^*E^{-1}$ is a right inverse,
whose exterior component is zero on $U_{\rm core}$.
For the real carrier adjoint $C_a^*$ set
\[
 D_af=2\Re\{\chi_y(z,y)\mathcal F_\eta^{-1}
 [\rho_a(z,\widehat\eta)\chi_{\rm hi}(|\eta|)|\eta|^{1/2}f(z,\eta)](y)\},
\]
where the fixed $\chi_y$ equals one at the core's footpoints.
After enlarging $B$ once,
\begin{equation}\label{app:ch_background:eq:bg-density-adjoint}
 \sum_a\|D_aC_a^*E^{-1}e\|_{H^s_{z,y}}
 \leq C_s[\|e\|_{H^{s+1}}+\mathcal C_{s+B}\|e\|_2].
\end{equation}
This includes all carrier and initial coordinates and also holds
before $\chi_y$.
\end{prop}
\begin{proof}
The image, transport and exterior normals make $E$ an order-zero
PDO, with the full-amplitude one-high bounds.
For $|\alpha|=s\geq1$, these bounds give
\begin{equation}\label{app:ch_background:eq:bg-commutator}
 \|[D^\alpha,E]v\|_2
 \leq C_s[\|v\|_{H^{s-1}}+\mathcal C_{s+B}\|v\|_2].
\end{equation}
In the commutator, at least one derivative hits the full amplitude;
interpolation from its fixed first-derivative baseline, jointly
with $v$ at $H^{s-1}$, gives the displayed additive bound.

Let $z=E^{-1}e$. At order one, difference quotients justify
\eqref{app:ch_background:eq:bg-commutator}: the PDO translation
commutator is uniformly bounded on $L^2$ by the fundamental
theorem of calculus. Thus $E^{-1}$
preserves $H^1$; self-adjunction extends the same inverse to $H^{-1}$.
If $z\in H^{s-1}$, the distributional identity
\[
 ED^\alpha z=D^\alpha e-[D^\alpha,E]z
\]
has right side in $L^2$. Uniqueness in $H^{-1}$ gives
$D^\alpha z=E^{-1}(D^\alpha e-[D^\alpha,E]z)\in L^2$.
Integer induction proves the first bound in
\eqref{app:ch_background:eq:bg-r-inverse}, with the same $B$.

Image and exterior adjoints are order zero. For a carrier
put $\mathsf A=D_\zeta d_xS_G$ and
$L=(\mathsf A^T\mathsf A)^{-1}\mathsf A^T$. The transverse gap gives
\begin{equation}\label{app:ch_background:eq:bg-derivative-trade}
 d_x\Phi=\mathsf A\eta,\quad \eta=Ld_x\Phi,\qquad
 \partial_{z_\ell}\Phi=V_\ell\cdot d_x\Phi,\quad
 V_\ell=(\partial_{z_\ell}d_\zeta S_G)L.
\end{equation}
Thus every initial or carrier phase derivative of $T_a^*v$
can be transposed to an $x$ derivative. Compact measurement
support removes boundary terms. After $s$ derivatives, a term
with $k$ derivatives on $v$ has coefficient excess $s-k$ plus
one fixed reserve. Its amplitude still has order $|\eta|^{1/2}$;
its paired normal has order zero by the degree-one conic Jacobian.
The full-amplitude bound, Plancherel and interpolation give
\[
 \mathcal C_{s-k+c}\|v\|_{H^k}
 \leq C_s[\|v\|_{H^s}+\mathcal C_{s+c}\|v\|_2].
\]
Sum these terms and insert $v=E^{-1}e$, using the inverse at the
high order and only its $L^2$ bound in the coefficient term.
This proves the raw estimate. Scalar transport is $r(z,Y_G)$
times its flow factor, so the same calculation has one high $G$
or $r$ norm when the initial scalar varies.

Finally $D_a$ is a fixed half-order multiplier in $y$, followed
by real part and $\chi_y$. Its carrier derivatives act on the
fixed mask, so its $H^s$ norm is bounded by the raw $H^{s+1}$
norm, also before $\chi_y$. This proves
\eqref{app:ch_background:eq:bg-density-adjoint}.
The identity $AA^*E^{-1}=I$ proves the right-inverse assertion;
its exterior component is $(1-\psi)E^{-1}e$.
\end{proof}

\begin{prop}[Uniform normalization as the second form shrinks]\label{app:ch_background:prop:bg-normalized}
For $0<\sigma\leq\sigma_0$, take $G_\sigma$ from Definition~\ref{app:ch_background:defn:bg-carriers} and use initial scalar $\sigma$ for the null transports. All domains, cutoffs and carrier charts in Definition~\ref{app:ch_background:defn:bg-row} can be chosen independently of $\sigma$. The fields
\[
 \sigma^{-1}B_{G_\sigma},\quad \sigma\mathsf N_{G_\sigma},\quad
 \sigma^{-1}b_{\sigma,a},\quad S_{G_\sigma},\quad\nu_{\sigma,a},\quad A_{0,\sigma}
\]
extend smoothly through $\sigma=0$ in every fixed mixed space/carrier derivative. The normal fields $\nu_{\sigma,a}$ keep a uniform positive lower bound. For fixed smooth compact nonnegative carrier weights $w_a$, put
\[
 H_\sigma(x)=\sum_a\int w_a(z)|\nu_{\sigma,a}(x,z)|^2
                \xi_{\sigma,a}(x,z)\otimes\xi_{\sigma,a}(x,z)\chbackgrounddd z.
\]
Its limit $H_0$ is constant and positive semidefinite, and
\begin{equation}\label{app:ch_background:eq:bg-Hsigma}
 \chbackgroundnorm{H_\sigma-H_0}_{C^s}\leq C_s\sigma.
\end{equation}
The normalized augmented normal satisfies, on the entire auxiliary $L^2$,
\begin{equation}\label{app:ch_background:eq:bg-uniform-normalized}
 E_\sigma\geq cI,
 \qquad\|E_\sigma^{-1}f\|_{H^s}+\|A_\sigma^*E_\sigma^{-1}f\|_{X^s}
 \leq C_s\|f\|_{H^s},
\end{equation}
with $c,C_s$ independent of $\sigma$. The density estimate
\eqref{app:ch_background:eq:bg-density-adjoint} is uniform as well,
with right side $C_s\|f\|_{H^{s+1}}$. The algebraic limit at zero
is used for these estimates only; $G_0$ is not asserted to have full normal rank.
\end{prop}

\begin{proof}
On one small box for $F_0$, construct the phase $\widehat S$
and null transport $\widehat b$ with initial scalar $1$.
Choose a fixed normalized box with $\sigma x$ in that domain.
Corollary~\ref{app:ch_seed:cor:seed-uniform-dilation} gives its flat
metric and the scaled second form and stress frame. The ambient
formula and common Hamilton Cauchy data then give
\begin{equation}\label{app:ch_background:eq:bg-Nsigma}
\begin{gathered}
 \mathsf N_{G_\sigma}(x)=\sigma^{-1}\mathsf N_{F_0}(\sigma x),\qquad
 S_\sigma(x,\zeta)=\sigma^{-1}\widehat S(\sigma x,\zeta),\\
 b_\sigma(x,\zeta)=\sigma\widehat b(\sigma x,\zeta),\qquad
 \xi_\sigma(x,\zeta)=\widehat\xi(\sigma x,\zeta),\qquad
 \nu_\sigma(x,\zeta)=\widehat\nu(\sigma x,\zeta).
\end{gathered}
\end{equation}
Indeed the displayed phase has initial value $y\cdot\zeta$,
and
$P_{G_\sigma}[\sigma\widehat b(\sigma x)]
 =\sigma^2(P_{F_0}\widehat b)(\sigma x)$.
Thus it has the same leading scalar transport and initial scalar
$\sigma$; uniqueness proves the identities.

Since $\widehat S(0,\zeta)=0$,
\[
 S_\sigma(x,\zeta)=\int_0^1
 x\cdot d_x\widehat S(t\sigma x,\zeta)\,dt.
\]
Every normalized field therefore extends smoothly in all mixed
variables through zero. There the nonzero tensor
$\xi_0\mathbin{\chbackgroundSym}k_0$ lies in $K_0^\perp$;
its inverse image under $B_{F_0}(0)^*$ is nonzero. Compact detector
supports give a common positive lower bound for $\nu$.
For the fixed carrier weights, the same identities give
$H_\sigma(x)=\widehat H(\sigma x)$ and $H_0=\widehat H(0)$.
The fundamental theorem of calculus proves
\eqref{app:ch_background:eq:bg-Hsigma} at every fixed order.

The normalized image and detector symbols have the frozen positive
joint gap at zero. Their full amplitudes, chart inverses and
literal adjoints have uniform
fixed-order bounds. Apply the proofs of
Propositions~\ref{app:ch_background:prop:bg-frame}
and~\ref{app:ch_background:prop:bg-adjoint} to this compact
normalized family, including its fixed-low initial multiplier
class: one common small ball absorbs the local errors,
and the same exterior row gives the full $L^2$ gap. The inverse,
raw and density estimates have uniform coefficients at every
fixed order. The physical domains stay fixed, with zero exterior
row on $U_{\rm core}$. No norm continuity of individual graph
operators is used.
\end{proof}

\subsection{The positive transported mean}
\label{app:ch_background:sec:bg-positive}
\label{app:ch_background:sec:bg-current}
The mean is affine in its initial density. One density correction
and tangent field cancel its reduced strain, at the normalized
reference and throughout its fixed neighborhood.

\begin{prop}[A positive mean at the reference and nearby immersions]
\label{app:ch_background:prop:bg-positive}
\label{app:ch_background:prop:bg-current}
Use the normalized family of
Proposition~\ref{app:ch_background:prop:bg-normalized}.
Choose smooth nonnegative cutoffs $\chi_a$ compactly supported in
the carrier charts and equal to one near every detector projection.
Fix the external carrier measure $\psi_a\,dz=\chi_a^2\,dz$ and
put $\alpha_a=\rho_a$, so $\rho_a=\chi_a\alpha_a$.
Let $H_0$ be the corresponding constant limiting mean and fix
$X_0(x)=-H_0x/2$, a source cutoff $\theta\in C_c^\infty(B_r)$
equal to one on $U_{\rm core}$, and $U_{\rm fin}\Subset U_{\rm core}$.

For a positive $\sigma$ and an immersion $G$ near $G_\sigma$, use
initial scalar $\sigma$ and its uncorrected continuum row
$A=A_{G,\sigma}$, $E=AA^*$.
For the leading normal $\nu_{G,\sigma,a}$ with initial scalar
$\sigma$, define the transported mean with this fixed measure:
\begin{equation}\label{app:ch_background:eq:bg-affine-mean}
 H_G[\sigma^2(1+v)]
 =\sum_a\int\psi_a|\nu_{G,\sigma,a}|^2
 [1+v_a(z,Y_{G,a})]\xi_{G,a}\otimes\xi_{G,a}\,dz.
\end{equation}
Define, with the same source and raw-input cutoffs,
\begin{align}
 R_0&=D\gamma_G(dGX_0)+H_G[\sigma^2],\notag\\
 e&=-\tfrac12\theta q_G(R_0),\qquad h=A^*E^{-1}e,\notag\\
 u_{\rm corr}&=\langle D\rangle^{-1}\chi_{\rm src}h_{\rm im},\notag\\
 X_G&=X_0+g_G^{-1}u_{\rm corr},
 \label{app:ch_background:eq:bg-current-raising}\\
 v_a&=4\Re\mathcal F_\eta^{-1}
 [\alpha_a\chi_{\rm hi}(|\eta|)|\eta|^{1/2}h_a].
 \label{app:ch_background:eq:bg-density-four}
\end{align}
The source is smoothly zero-extended to the torus; the density
synthesis is uncut in the initial spatial variables. Put
\[
 r_{G,a}=\sigma\sqrt{1+v_a},\qquad V_{\rm base}=dGX_G.
\]
For all sufficiently small $\sigma>0$, and then on one sufficiently
small fixed finite $C^d$ neighborhood of $G_\sigma$, these fields
are real smooth, $1+v_a\geq1/2$, the scalar $r_{G,a}$ is positive
on detector supports, and
\begin{equation}\label{app:ch_background:eq:bg-current-identity}
 q_G\bigl(D\gamma_GV_{\rm base}+H_G[r_G^2]\bigr)=0
                   \quad\text{on }U_{\rm fin}.
\end{equation}
The corrected positive row has a full $L^2$ gap and one
all-Sobolev inverse. For one fixed $B$,
\begin{equation}\label{app:ch_background:eq:bg-current-profile}
 \sum_a\|r_{G,a}\|_{C^s}+\|V_{\rm base}\|_{C^s(U_{\rm fin})}
 +\|D\gamma_GV_{\rm base}\|_{C^s(U_{\rm fin})}
 \leq C_s(1+\|G\|_{C^{s+B}(W)}).
\end{equation}
The scalar norm includes all carrier and initial variables $(z,y)$
on the fixed buffered patches;
$B$ includes the one derivative in the tangent strain.
These choices precede accuracy and any later target dilation;
fixed-order bounds are uniform on smooth compact families.

Along $G=G_\sigma$, the construction satisfies
\begin{equation}\label{app:ch_background:eq:bg-positive-profile}
 \sum_a\|v_a\|_{C^s}+\|X_G-X_0\|_{C^s(W)}\leq C_s\sigma,
\end{equation}
the identity holds on $U_{\rm core}$, and the corrected normal
gap and fixed-order inverse bounds are uniform as $\sigma\to0$.
\end{prop}
\begin{proof}
The density $v_a$ has carrier support inside $\{\chi_a=1\}$,
so $\psi_av_a=v_a$. Scalar transport evaluates it at $Y_{G,a}$.
The factor $4$ in the uncut synthesis and $C_a=2\Re T_a$ give,
on the entire measurement support,
\begin{equation}\label{app:ch_background:eq:bg-density-twice}
 \chi_{\rm meas}q_G\big(H_G[\sigma^2(1+v)]-H_G[\sigma^2]\big)
                  =2\sum_aC_ah_a.
\end{equation}
The same identity holds for any fixed positive initial scalar and
its row. On $U_{\rm core}$,
$\theta=\chi_{\rm meas}=\psi=1$, so covector raising gives
\[
 \begin{aligned}
 q_G(D\gamma_GV_{\rm base}+H_G[r_G^2])
 &=q_GR_0+2P_Gu_{\rm corr}+2\sum_aC_ah_a\\
 &=q_GR_0+2e=0.
 \end{aligned}
\]

At $G_\sigma$ the metric is $I$, so
$D\gamma_G(dGX_0)=-H_0$ and $R_0=H_\sigma-H_0=O_s(\sigma)$.
The uncorrected row is the normalized row. Its uniform inverse, raw-adjoint and density
bounds give $h=O_s(\sigma)$, $v=O_s(\sigma)$ and
$X_G-X_0=O_s(\sigma)$.
Choose $\sigma$ so that $v$ lies in the fixed-low multiplier
class of Proposition~\ref{app:ch_background:prop:bg-frame}. Its
normalized proof gives a uniform corrected gap. Since $v=O_s(\sigma)$,
the same inverse proof has uniform coefficients at every fixed order.

Now fix this positive $\sigma$. Choose the neighborhood order
above $d_0$ plus the density and inverse shifts, keeping the
preselected support. A small neighborhood in that finite topology
preserves the uncorrected gap and the needed low smallness of $e,v$. The fixed-jet and transport formulas give
$\|R_0\|_{C^s}+\|e\|_{H^s}\leq C_s(1+\|G\|_{C^{s+B_1}})$.
Apply the inverse, adjoint, density and physical realization
estimates; every high coefficient multiplies a bounded low source
norm. Covector raising, the positive square root and the one
tangent-strain derivative enlarge this fixed shift once, proving
the profile. The same fixed-low multiplier class gives the corrected gap and
inverse throughout this neighborhood.
\end{proof}

\subsection{The row and its auxiliary fields}\label{app:ch_background:sec:bg-mixed}
The continuum inverse uses one positive row.
We truncate every raw input before its physical realization.

\begin{defn}[One row and its input cuts]\label{app:ch_background:defn:bg-mixed}
For the positive base of Proposition~\ref{app:ch_background:prop:bg-current},
write
\[
 A_G=(A_{0,G},T_G),\qquad E_G=A_GA_G^*,\qquad Q_G=E_G^{-1},
\]
where $A_{0,G}$ contains the image and exterior channels, and $T_G$ the real carrier channels. Every geometric
coefficient in this row belongs to $G,r_G$.

Let $\pi_K$ be one real even Fourier contraction on each auxiliary
torus input, equal to one below $K$ and zero above $2K$.
On the carrier input let $P_K^2=\sum_{\kappa\leq K}\beta_\kappa^2$
be the fixed dyadic square prefix, from an annular partition
that sums to one on $\operatorname{supp}\chi_{\rm hi}$. Set
\[
 \mathsf P_K=\operatorname{diag}(\pi_K,P_K),\qquad
 \mathcal F_{G,K}=A_{0,G}\pi_K^2A_{0,G}^*,\qquad
 E_{G,K}=A_G\mathsf P_K^2A_G^*.
\]
All squares act on raw inputs. The spatial source and footpoint
cutoffs remain in the original operators after these multipliers.
The raw norm $X^s$ is the direct sum of auxiliary $H^s$ norms and
carrier initial-variable $H^s_{z,y}$ norms, before the footpoint
cutoff. Each carrier component of $A_G^*$ contains $\chi_{\rm hi}$,
and $P_K^2=1$ on that support below $cK$. Hence
\[
 \|(I-\mathsf P_K^2)v\|_{X^0}\leq C_sK^{-s}\|v\|_{X^s}
 \qquad(v\in\operatorname{Ran}A_G^*\cap X^s).
\]
\end{defn}

Proposition~\ref{app:ch_background:prop:bg-adjoint}, composed with
$\pi_K^2$ and the same strong order-zero auxiliary row, gives
\begin{equation}\label{app:ch_background:eq:bg-mixed-raw}
\begin{split}
 &\|Q_Ge\|_{H^s}+\|A_G^*Q_Ge\|_{X^s}
       +\|\mathcal F_{G,K}Q_Ge\|_{H^s}\\
 &\qquad\leq C_s[\|e\|_{H^s}+\mathcal C_{s+B}(G,r_G)\|e\|_2].
\end{split}
\end{equation}
Here $\mathcal C_m(G,r_G)=1+\|G\|_{C^m}+\sum_a\|r_{G,a}\|_{C^m}$.
The fixed loss and constants are uniform in $K$; at each prescribed
low order the coefficient norm is bounded. The inverse and raw
cutoff order are unchanged.

\begin{prop}[Physical realization of truncated data]\label{app:ch_background:prop:bg-mixed-physical}
For $h=A_{0,G}^*Q_Ge$, put
\begin{equation}\label{app:ch_background:eq:bg-Vaux}
\begin{gathered}
 u_K=\langle D\rangle^{-1}\chi_{\rm src}\pi_K^2h_{\rm im},\\
 V_{\rm aux,K}^{G}(e)=\tfrac12dG\,g_G^{-1}u_K.
\end{gathered}
\end{equation}
On the physical core, $q_GD\gamma_GV_{\rm aux,K}^{G}=P_Gu_K$;
the exterior output is zero. For one fixed shift $B_{\rm phys}$,
uniformly in $K$,
\begin{equation}\label{app:ch_background:eq:bg-Vaux-tame}
 \|V_{\rm aux,K}^{G}(e)\|_{H^s(W)}
 \leq C_s[\|e\|_{H^s}+\mathcal C_{s+B_{\rm phys}}(G,r_G)\|e\|_2].
\end{equation}
The same ordered physical factors, started after the raw Fourier
cut, also give
\begin{equation}\label{app:ch_background:eq:bg-Vaux-raw-band}
 \|V_{\rm aux,K}^{G}(e)\|_{H^s(W)}
 +\|\mathcal F_{G,K}Q_Ge\|_{H^s}
 \leq C_s[K^s+\mathcal C_{s+B_{\rm phys}}(G,r_G)]\|e\|_2.
\end{equation}
In the field estimates here and below one may also include
$\|D\gamma_GV_{\rm aux,K}^{G}(e)\|_{H^s(W)}$ on the left.
A fixed physical cutoff gives the same field and strain bounds
on the torus. In particular its $C^2$ norm is at most $C\|e\|_{H^{s_0}}$
at one fixed $s_0$ under the prescribed fixed-low bounds.

Suppose
\[
 \mathcal C_m(G,r_G)\leq C_mT^{(m-R_*)_+},\qquad T\geq2,
\]
with one fixed $R_*$ above every displayed shift. Then
\begin{equation}\label{app:ch_background:eq:bg-graded-raw}
 \|A_G^*Q_Ge\|_{X^s}+\|\mathcal F_{G,K}Q_Ge\|_{H^s}
 \leq C_s[\|e\|_{H^s}+T^s\|e\|_2],
\end{equation}
and, if $T\leq K$,
\begin{equation}\label{app:ch_background:eq:bg-Vaux-band}
 \|V_{\rm aux,K}^{G}(e)\|_{H^s(W)}
 +\|\mathcal F_{G,K}Q_Ge\|_{H^s}
 \leq C_sK^s\|e\|_2.
\end{equation}
At any fixed finite set of orders where the coefficient norms
are bounded, this last conclusion holds for every $K\geq2$.
\end{prop}
\begin{proof}
Equation~\eqref{app:ch_background:eq:bg-mixed-raw} controls $h$.
The tangent identity gives
$D\gamma_GV_{\rm aux,K}^{G}=\chbackgroundSym\nabla^{g_G}u_K$.
Its first derivative is paired with $\langle D\rangle^{-1}$,
so the field and full strain in
\eqref{app:ch_background:eq:bg-Vaux} have order at most zero.
Their one-high bounds prove
\eqref{app:ch_background:eq:bg-Vaux-tame}; a final fixed cutoff
adds only a lower-order term to the strain. Starting instead with
$\|\pi_K^2h\|_{H^s}\leq C_sK^s\|e\|_2$ proves
\eqref{app:ch_background:eq:bg-Vaux-raw-band}, including the
complete auxiliary row and its exterior output.

Corollary~\ref{app:cor:fixed-profile-transfer} gives
$\mathcal C_{s+c}\leq C_sT^s$ for every displayed fixed shift $c$.
Apply this to the
one-high shifts in \eqref{app:ch_background:eq:bg-mixed-raw}
and \eqref{app:ch_background:eq:bg-Vaux-raw-band}. This proves
both graded bounds; bounded coefficient norms give the finite-order
conclusion directly.
\end{proof}

\subsection{The same primary row and separated samples}
\label{app:ch_background:sec:bg-prior-wave-inputs}
\label{app:ch_background:sec:bg-matching}
The fixed external weights identify the positive continuum row
with the leading normal of its actual primary waves.

\begin{defn}[Primary realization of the positive row]
\label{app:ch_background:setup:bg-wkb-input}
Use the positive scalar $r_{G,a}$ of
Proposition~\ref{app:ch_background:prop:bg-current}, its fixed
measure $\psi_a\,dz=\chi_a^2\,dz$ and detector mask
$\alpha_a=\rho_a$, so $\chi_a\alpha_a=\rho_a$. The primary atom
$\mathcal W_{\tau,a,z}^{\rm pr}(x,\zeta')$ is
\eqref{app:ch_waves:eq:WKB-W}, with the common Hamilton Cauchy
data and one algebraic correction. Use its proved mixed grades
at the same scale $L$ as $G,r_G$, including the fixed parameter
order $r_{\rm par}=M+3$.

For $\zeta=\zeta_a(z)$ define the background-modulation tensor
and real channel on both quadratures by
\[
\begin{split}
 \mathcal K_{\tau,a,z}(x,\eta)
 &=\chbackgroundSym[(d\mathcal W_\tau(x,\zeta))^*
                          d\mathcal W_\tau(x,\zeta+\tau\eta)],\\
 \mathscr U_{\tau,a,z,K}f
 &=2\Re\int\chi_{\rm meas}\alpha_a\chi_{\rm hi}|\eta|^{1/2}
                    q_G(\mathcal K_{\tau,a,z})\beta_Kf(\eta)\,d\eta.
\end{split}
\]
Its exact phase is
$[S_G(x,\zeta+\tau\eta)-S_G(x,\zeta)]/\tau$; at zero,
$\mathcal K_{0,a,z}=e^{i\Phi_G}|\nu_G|^2\xi_G\otimes\xi_G$.
Use the fixed dyadic square partition and radial labels of
Theorem~\ref{app:ch_waves:thm:sampling}, with
$\tau_c=(R_c\lambda)^{-1}$ and
\begin{equation}\label{app:ch_background:eq:bg-WKB-conditions}
 \Lambda/\lambda<\text{the fixed phase margin}.
\end{equation}
Every shifted covector stays in its common buffered chart.
Write $H_G=H_G[r_G^2]$ for this same positive transported mean.
\end{defn}

The square weights satisfy $\rho_a^2=\psi_a\alpha_a^2$.
Thus the transport normal of these channels is exactly
$T_GP_K^2T_G^*$; real adjunction gives $CmC^*=4\Re(TmT^*)$.
Summing the fixed radial colors removes their masks at transport
order. The auxiliary normal is appended with the same raw cut.
The fixed-low full-amplitude bounds also show that an initial-scalar
change of size $\eta$ in those seminorms changes every continuum
prefix normal by $C\eta$, uniformly in its ceiling.

\section{Current correction for Fourier measures}\label{app:sec:ch_lag}
\subsection{One current patch and its two envelopes}\label{app:ch_lag:sec:lag-inputs}
The finite primary and its complete response use the same immersion.
We apply the response once to the finite strain of each Fourier
measure, on the common source and observation supports.

\begin{setup}[The current phase and full response]
\label{app:ch_lag:defn:lag-geometry}
\label{app:ch_lag:defn:lag-response}
Use the actual geometry of Definition~\ref{app:ch_background:defn:bg-geometry}
at $F$, in its fixed finite-low neighborhood. On one buffered
simple patch let $\phi_F(x)=S_F(x,\zeta)$ be the Hamilton phase
with the fixed transverse Cauchy data. Use the finite primary of
Theorem~\ref{app:ch_waves:thm:wkb}, including its positive initial
scalar, and the compatible complex-linear response
$\mathcal B_F^{\mathbb C}$ of
Theorem~\ref{app:ch_response:thm:response-compatible-choice}.
This is the fixed full ambient map
\eqref{app:ch_response:eq:response-full-map}, including its
literal inverse and normal reconstruction.

Use the wider response core $U_{\rm LAG}$ of
Setup~\ref{app:setup:domains}, with
$D\Subset\operatorname{Int}D_{\rm lag}\Subset U_{\rm LAG}$.
The entire $\operatorname{supp}\chi_{\rm meas}$ and every eventual
mean, protected pair-source and physical-output support lie
compactly inside $\operatorname{Int}D_{\rm lag}$.
Fix real source and observation cutoffs with
\begin{equation}\label{app:ch_lag:eq:lag-buffer}
 \operatorname{supp}\chi_{\rm src}\Subset\{\chi_{\rm core}=1\},
 \quad\operatorname{supp}\chi_{\rm core}\Subset U_{\rm LAG},
 \quad\operatorname{supp}\chi_{\rm src}\subset\{\chi_2=1\}.
\end{equation}
Require $D_{\rm lag}\Subset\operatorname{Int}\{\chi_{\rm src}=1\}$
and the full observation support inside the primary's validity
chart. These choices precede sampling, frequency and accuracy.
All envelopes have fixed extensions to the auxiliary torus.
\end{setup}

\begin{setup}[Source scales and finite measures]\label{app:ch_lag:defn:lag-scales}
For a fixed $r_*$ above the finite construction shifts and the joint
budget of Corollary~\ref{app:ch_response:cor:current-atoms}, suppose
\begin{equation}\label{app:ch_lag:eq:lag-history}
 \|F\|_{C^r}\leq C_rT^{(r-r_*)_+},\qquad T\geq2.
\end{equation}
The phase, finite primary and response have their proved grades
at this same scale. Let $\lambda=\tau^{-1}\geq2$ and choose
$K\geq2$ with fixed $\nu>0$ and
\begin{equation}\label{app:ch_lag:eq:lag-scale-windows}
 T\leq K,\qquad T\leq\lambda^{1-\nu}.
\end{equation}
Let $m$ be a finite complex measure in the $(n-1)$ initial Fourier
variables, with $M=\|m\|_{\rm TV}$, and
\[
 \operatorname{supp}m\subset B_\Lambda,\qquad
 2\leq\Lambda\leq K/C_{\rm mod},\qquad
 \Lambda/\lambda<\text{the fixed phase margin}.
\]
The allowed modulation chart includes this entire support and any
atom at zero. No relation between $T$ and $\Lambda$ is required.
For fixed radial labels $\lambda_j=c_jR^a$, strict power gaps
absorb the positive constants $c_j$ uniformly.
\end{setup}

\subsection{Finite measures and one current response}\label{app:ch_lag:sec:lag-base}
\label{app:ch_lag:sec:lag-induction}
\begin{defn}[The finite primary and its strain correction]\label{app:ch_lag:defn:lag-recurrence}
Put
\begin{equation}\label{app:ch_lag:eq:lag-W0-definition}
 W_0(m)=\int\mathcal W_\tau^{\rm pr}(F;x,\zeta+\tau\xi)\,dm(\xi),
\end{equation}
and define
\begin{equation}\label{app:ch_lag:eq:lag-recurrence}
 H=\chi_{\rm src}D\gamma_FW_0,\qquad
 Y=\mathcal B_F^{\mathbb C}H,\qquad Z=W_0-Y.
\end{equation}
Every map is complex linear in the complete measure. The finite
primary, response, inverse and cutoffs precede the datum and accuracy.
\end{defn}

\begin{prop}[Finite-measure envelopes and current strain]\label{app:ch_lag:prop:lag-envelopes}
\label{app:ch_lag:prop:lag-telescope}
Under the preceding setups, the same fields satisfy
\begin{align}
 \lambda\|\chi_{\rm core}e^{-i\lambda\phi_F}Z\|_{C^s}
 +\|\chi_{\rm core}e^{-i\lambda\phi_F}dZ\|_{C^s}
 &\leq C_sMK^s,
 \label{app:ch_lag:eq:lag-all-envelopes}\\
 \lambda\|\chi_{\rm core}e^{-i\lambda\phi_F}Y\|_{C^s}
 +\|\chi_{\rm core}e^{-i\lambda\phi_F}dY\|_{C^s}
 &\leq C_sM\lambda^{-1}K^s.
 \label{app:ch_lag:eq:lag-Y-envelope}
\end{align}
On $D_{\rm lag}$, for every finite $r,Q$,
\begin{equation}\label{app:ch_lag:eq:lag-current-strain}
 \|D\gamma_FZ\|_{C^r(D_{\rm lag})}\leq C_{r,Q}M\lambda^{-Q}.
\end{equation}
The fields and domains are independent of these proof orders.

\end{prop}
\begin{proof}
The full source cutoff and the response are fixed and linear.
Compact shifted support permits integration in every fixed Sobolev
space; continuity of the same response therefore gives
\[
 (Y,Z)=\int(\mathcal Y_\tau,\mathcal W_\tau^{\rm cur})
                (F;x,\zeta+\tau\xi)\,dm(\xi).
\]
These are the original complete fields, including all source leakage.
The central demodulation phase is
\begin{equation}\label{app:ch_lag:eq:lag-exact-phase}
 \Psi(x,\xi)=\frac{S_F(x,\zeta+\tau\xi)-S_F(x,\zeta)}{\tau}
 =\int_0^1d_\zeta S_F(x,\zeta+u\tau\xi)\cdot\xi\,du.
\end{equation}
Its joint grades give $|D_x^r\Psi|\leq C_r\Lambda T^{r-1}$.
A derivative monomial of order $s$ in $e^{i\Psi}$, with $k$
phase factors, costs $C_s\Lambda^kT^{s-k}\leq C_sK^s$.
Multiply the atomic envelopes of
Corollary~\ref{app:ch_response:cor:current-atoms} by this phase
and integrate against $|m|$ to prove both envelope bounds.
The same atomic theorem gives full strain $C_{r,Q}\lambda^{-Q}$
on the common inner core, uniformly throughout the shifted compact
support. Its finite higher precision includes the physical $C^r$
derivatives. Integration proves
\eqref{app:ch_lag:eq:lag-current-strain} on $D_{\rm lag}$.
\end{proof}

\section{The sampled current mean}\label{app:sec:ch_mean_pairs}
\subsection{Preparation and the original positive row}
\label{app:ch_mean_pairs:sec:mean-appendix-inputs}
\label{app:ch_mean_pairs:sec:mean-current}

\begin{setup}[One row and the stage window]
\label{app:ch_mean_pairs:defn:mean-geometry-input}
\label{app:ch_mean_pairs:defn:mean-continuum-input}
\label{app:ch_mean_pairs:defn:mean-inverse}
\label{app:ch_mean_pairs:defn:mean-stage}
Use Definitions~\ref{app:ch_background:defn:bg-geometry}
and~\ref{app:ch_background:defn:bg-mixed} at the current $F$ on
$D\Subset U_{\rm fin}\Subset U_{\rm core}\Subset B_r
\Subset U_{\rm meas}\Subset W$ and the common auxiliary torus.
Write $\mathcal C=\chi_{\rm meas}q_F$ and
$\ell=\lfloor n/2\rfloor+1$. For the original positive scalar $r_F$, set
\[
 A_F=(A_0,\mathcal T_0),\quad E_0=A_FA_F^*,\quad Q_0=Q_F,\quad
 \mathcal F_{0,K}=A_0\pi_K^2A_0^*,\quad E_{0,K}=A_F\mathsf P_K^2A_F^*.
\]
Use $A=(n+1)(1+\theta)/4$ from \maineqref{main:eq:A-choice} and
\begin{equation}\label{app:ch_mean_pairs:eq:mean-stage-window}
\begin{gathered}
 0<\theta<\epsilon<\tfrac18,\quad\beta=(1+\epsilon)/3,\quad
 1+\theta+\beta<a<\tfrac32,\\
 \lambda=R^a,\quad H=R^{1+\theta},\quad h=R^{-\beta},\qquad
 T=R^t,\quad0<t<1.
\end{gathered}
\end{equation}
The current history profile is
$\|F\|_{C^r}\leq C_rT^{(r-r_*)_+}$; its positive base, phases
and finite primary have the same derivative scale. Choose $r_*$
above the fixed shifts in
Proposition~\ref{app:ch_background:prop:bg-mixed-physical}, which gives
\begin{equation}\label{app:ch_mean_pairs:eq:mean-input-raw-one-high}
 \|A_F^*Q_0e\|_{X^s}+\|\mathcal F_{0,K}Q_0e\|_{H^s}
 \leq C_s[\|e\|_{H^s}+T^s\|e\|_2],
\end{equation}
and, for $T\leq K$,
\begin{equation}\label{app:ch_mean_pairs:eq:mean-input-physical-one-high}
 \|V_{\rm aux,K}^{F}(e)\|_{H^s(W)}
 +\|\mathcal F_{0,K}Q_0e\|_{H^s}
 \leq C_sK^s\|e\|_2.
\end{equation}
The latter includes the complete strain and a fixed physical cutoff.
All raw inputs precede physical source and footpoint cutoffs.
The exterior output is included on the torus and vanishes on $D$.

The fixed order $q_{\rm quad}=2n-1$ precedes the finite colors,
radial labels and positive-row neighborhood. The preparation orders
below depend only on $n$ and the fixed power margins; put their
budget into $r_*$ before sampling, fine accuracy and target dilation.
\end{setup}

\begin{prop}[A prepared positive background]\label{app:ch_mean_pairs:prop:mean-coarse}
Use the positive scalar $r_F$, base field and row $A_F=(A_0,\mathcal T_0)$
of Appendix~\ref{app:sec:ch_background}, its inverse $Q_0$, the
current atomic family, and Setup~\ref{app:ch_mean_pairs:defn:mean-stage}.
Put $J=\lfloor2A/a\rfloor$.
Then $a(J+1)>2A$ and $\beta q_{\rm quad}>2A$.
Use the convolution weights, their fixed radial colors and spacing
$h$ of Theorem~\ref{app:ch_waves:thm:sampling}. The weights and
colors precede the following density preparation and the random shifts.

There are positive initial scalars $r_{F,R}$ and a real field $U_c$,
chosen before sampling, such that one sample event gives the current
zero-atom fields $Z_{c,j}=\mathcal W_{\tau_j,z_j}^{\rm cur}(r_{F,R})$ and
\[
 H_c=\sum_j\mu_j\Re[(dZ_{c,j})^*dZ_{c,j}],\qquad
 C_c=D\gamma_FU_c+H_c,
\]
with a full auxiliary residual satisfying
\begin{equation}\label{app:ch_mean_pairs:eq:mean-prepared-residual}
 r_c=q_FC_c\text{ on }D,\qquad
 \|r_c\|_2\leq R^{-2A},\qquad
 \|r_c\|_{H^s}\leq C_sT^s.
\end{equation}
The initial scalar stays in the fixed positive class of the original
row. For every fixed reserve $R_0$, the history order can be chosen
so that
\begin{equation}\label{app:ch_mean_pairs:eq:mean-prepared-scalar}
 \|r_{F,R}-r_F\|_{C^{R_0}}\leq C\lambda^{-1},\qquad
 \|r_{F,R}\|_{C^s}\leq C_sT^{(s-R_0)_+}.
\end{equation}
All carrier and initial variables are included. The field and full
self tensor obey
\begin{equation}\label{app:ch_mean_pairs:eq:mean-coarse-field-profile}
 \|U_c\|_{C^s}+\|H_c\|_{C^s}+\|C_c\|_{C^s}\leq C_sT^s
                       \quad\text{on }U_{\rm fin},
\end{equation}
also after a fixed physical cutoff.
The same event has at most $CR^{1+\epsilon}$ nodes, weights at most
$CR^{-1-\epsilon}$, bounded mass and same-color separation $ch$.
For every prefix $K\leq H$, its current normal satisfies
\begin{equation}\label{app:ch_mean_pairs:eq:mean-stage-normal-errors}
 \|E_K-E_{0,K}\|
 \leq C[R^{-(\epsilon-\theta)/4}
                 +R^{-a+1+\theta+\beta}].
\end{equation}
The inverse $Q_0$ still belongs to $r_F$. All preparation orders,
profiles and domains precede the accuracy of the later mean solve.
\end{prop}
\begin{proof}
Write $r(v)=r_F\sqrt{1+v}$ and let $\mathscr H_\lambda(v)$ be
the colored continuum Hermitian mean of its current zero atoms.
Put
\[
 \mathcal E_\lambda(v)=\mathcal C\bigl[
   \mathscr H_\lambda(v)-H_{\rm transport}[r(v)^2]\bigr].
\]
For each fixed $s$, the current-atom estimates and linearity in the
initial scalar give, on a bounded finite-jet positive class,
\begin{equation}\label{app:ch_mean_pairs:eq:mean-density-lipschitz}
 \|\mathcal E_\lambda(v)-\mathcal E_\lambda(w)\|_{H^s}
 \leq C_s\lambda^{-1}\|v-w\|_{C^{m_s}_{z,y}},\qquad
 \|\mathcal E_\lambda(0)\|_{H^s}\leq C_s\lambda^{-1}.
\end{equation}
Here $m_s$ is finite. Indeed both the finite primary and its fixed
response are linear in $r$. Their derivative amplitude is its
transport amplitude plus a $C^s$ error of size $\lambda^{-1}$
with a finite initial-data shift. Subtract the two Hermitian squares
and apply the positive square root. The masks are only integrated;
$0\leq m_c\leq1$ and $\sum_cm_c=1$ make these constants uniform in $h$.

Start $v_0=0$, $U_0=V_{\rm base}$ and $e_0=\mathcal E_\lambda(0)$.
For $i<J$, put $h_i=A_F^*Q_0e_i$ and use the uncut density synthesis
of \eqref{app:ch_background:eq:bg-density-four}:
\[
 \begin{split}
 \delta v_{i,a}&=-2\Re\mathcal F^{-1}
       [\alpha_a\chi_{\rm hi}|\eta|^{1/2}h_{i,a}],\\
 v_{i+1}&=v_i+\delta v_i,\qquad
 U_{i+1}=U_i-\tfrac12dF\,g_F^{-1}
                   \langle D\rangle^{-1}\chi_{\rm src}h_{i,\rm im}.
 \end{split}
\]
Define the full residual, including the exterior component, by
\[
 e_i=\mathcal C[\mathscr H_\lambda(v_i)-H_{\rm transport}[r_F^2]]
                         -\sum_{k<i}A_0A_0^*Q_0e_k.
\]
Equation~\eqref{app:ch_background:eq:bg-density-twice} gives the
full measured density change $-\mathcal T_0\mathcal T_0^*Q_0e_i$;
the auxiliary change is
$-A_0A_0^*Q_0e_i$. Since $A_FA_F^*Q_0=I$ on the whole torus,
\begin{equation}\label{app:ch_mean_pairs:eq:mean-density-recursion}
 e_{i+1}=\mathcal E_\lambda(v_{i+1})-\mathcal E_\lambda(v_i).
\end{equation}
No source is cut again between these steps.

The inverse and density estimates have fixed shifts. Start above
the initial-scalar orders of Corollary~\ref{app:ch_response:cor:current-atoms},
then choose their finite input orders and
$m_s$ backwards through the fixed $J$ steps, then put this finite
budget into $r_*$. Induction gives $e_i=O(\lambda^{-i-1})$ and
$v_i=O(\lambda^{-1})$ at all prescribed low orders. Thus $1+v_i\geq1/2$
at a large threshold, and $r_{F,R}=r(v_J)$ is defined.
For the high profiles, a scalar bound $C_sT^{s+b}$ gives a
primary strain of size $\lambda^{-3}T^{s+b+D_{\rm pr}}$.
The graded response at $K=T$ and fixed Sobolev embedding give
current envelopes $C_sT^{s+b+D_{\rm pr}+\ell}$. One-high products,
the density inverse and the auxiliary field add only fixed shifts.
After $J$ steps all fields therefore have bounds $C_sT^{s+B_J}$
with $B_J$ independent of $s$. Include a fixed low order above
$R_0+B_J+1$, and interpolate with a sufficiently high order; this
proves the scalar profile in
\eqref{app:ch_mean_pairs:eq:mean-prepared-scalar}. The same argument
gives grade $T^s$ for $U_J$ and its full strain. These are finite
choices independent of later accuracy.

Now fix this family and sample. The deterministic lattice estimate
at order $q_{\rm quad}<r_{\rm par}$,
gives $\|H_c-\mathscr H_\lambda(v_J)\|_2\leq Ch^{q_{\rm quad}}$.
Set $U_c=U_J$ and
\[
 r_c=e_J+\mathcal C[H_c-\mathscr H_\lambda(v_J)].
\]
The strict choices of $J,q_{\rm quad}$ give its $L^2$ bound. On $D$, the
original positive-base identity and vanishing exterior row give
$r_c=q_FC_c$. Bounded mass and the current envelopes prove all
field and global residual profiles.

The sampling theorem and current-atom theorem compare every current
prefix with the leading row at $r_{F,R}$ by the right side of
\eqref{app:ch_mean_pairs:eq:mean-stage-normal-errors}. The phase is
unchanged, and the scalar difference is $O(\lambda^{-1})$ in the
fixed full-amplitude seminorms. Thus replacing that leading row by
the original $E_{0,K}$ adds only $C\lambda^{-1}$, uniformly in $K$,
by the exact normal formula of Appendix~\ref{app:sec:ch_background}.
This is absorbed by the displayed bound. The random event is chosen
only after the complete prepared family has been fixed.
\end{proof}

\subsection{One fixed current row}\label{app:ch_mean_pairs:sec:mean-fixed-row}
Fix the prepared scalar and the sample event above. All subsequent
wave maps use this scalar, while $A_F,Q_0$ keep the original $r_F$.

\begin{defn}[The fixed current measure maps]
\label{app:ch_mean_pairs:defn:mean-wave-input}
\label{app:ch_mean_pairs:defn:mean-lag-input}
At each literal sample let $\mathcal L_j(m)$ be the current map
$Z(m)$ of Definition~\ref{app:ch_lag:defn:lag-recurrence}, with
initial scalar $r_{F,R}$ and
\[
 \tau_j=\lambda_j^{-1},\qquad
 \vartheta_j=S_F(x,\zeta_j)/\tau_j.
\]
Function data mean $dm=\widehat f\,d\xi$; the zero atom evaluates
the original integrand at $\xi=0$. Use the full domains and cutoffs of
Setup~\ref{app:ch_lag:defn:lag-response}. The region $D_{\rm lag}$
contains every measurement, extension, pair-source and physical-output
support. Corollary~\ref{app:ch_response:cor:current-atoms} proves the
joint low bounds for the raw slice spaces, with both quadratures.
These wave maps and responses stay fixed in every update.
\end{defn}

\begin{defn}[One current canonical row]\label{app:ch_mean_pairs:defn:mean-canonical}
Use the graded response window and the current zero-atom
background $B_j=e^{-i\vartheta_j}d\mathcal L_j(\delta_0)$.
Use the raw space $X_K$ of Theorem~\ref{app:ch_waves:thm:individual}
with both real quadratures, and set
$B_{j,K}^{\rm in}=\alpha_j\chi_{\rm hi}\mathcal B_K$ in the initial
Fourier variables. These are the detector and high-pass masks of
the positive row; the zero atom above is uncut. Define
\[
 G_{j,K}v=e^{-i\vartheta_j}d\mathcal L_j(B_{j,K}^{\rm in}v),
 \qquad A_{j,K}v=\mathcal C[2\sym\Re(B_j^*G_{j,K}v)],
\]
and put
\[
 A_K=(\sqrt{\mu_j}A_{j,K})_j,\qquad
 E_K=\mathcal F_{0,K}+A_KA_K^*.
\]
The slice cutoff, equal to one on every measurement support, is
understood only in norm estimates. Spatial initial cutoffs remain
inside the physical wave map, after the Fourier input. Every mean step uses this same background and canonical row.

The current-atom theorem and the raw slice bound give
\begin{equation}\label{app:ch_mean_pairs:eq:mean-current-raw}
 \|G_{j,K}v\|_{H^s}\leq C_sK^{s+1/2}\|v\|_2.
\end{equation}
Proposition~\ref{app:ch_mean_pairs:prop:mean-coarse} compares all
sampled prefixes with $E_{0,K}$ on the entire output. Hence $A_K$
is bounded. Only the continuum normal is inverted.

For $q=Q_0r$, define the canonical data by
\begin{equation}\label{app:ch_mean_pairs:eq:mean-current-canonical}
 f_{j,K}[q]=B_{j,K}^{\rm in}A_{j,K}^*q,
 \qquad D_{j,K}[q]=e^{-i\vartheta_j}d\mathcal L_j(f_{j,K}[q]).
\end{equation}
Adjoints use the real Hilbert products; the row weight and physical
normalization cancel, with no inverse sample weight. The complete
linear response at $B$ is exactly $E_Kq$.
The individual raw bound $\|A_{j,K}\|\leq C\sqrt K$,
\eqref{app:ch_waves:eq:B-bounds} and the current version of
\eqref{app:ch_waves:eq:raw-physical-bound} give
\begin{equation}\label{app:ch_mean_pairs:eq:mean-canonical-bounds}
 \|f_{j,K}[Q_0r]\|_{C^m_{\rm initial}}
 +\|D_{j,K}[Q_0r]\|_{C^m}
 \leq C_mK^{m+p}\|r\|_2,\qquad p=(n+1)/2.
\end{equation}
The initial Fourier support is below $CK$, and its Fourier $L^1$
norm is at most $CK^p\|r\|_2$. Apply the raw Sobolev and physical
bounds before summing, then use boundedness of $A_K$:
\begin{align}
 \sum_j\mu_j\|D_{j,K}[Q_0r]\|_{H^s}^2
 &\leq C_sK^{2s+1}\|A_K^*Q_0r\|_2^2
 \leq C_sK^{2s+1}\|r\|_2^2,
 \label{app:ch_mean_pairs:eq:mean-canonical-weighted-low}\\
 \sum_j\mu_j\|D_{j,K}[Q_0r]\|_{C^s}^2
 &\leq C_sK^{2s+n}\|A_K^*Q_0r\|_2^2
 \leq C_sK^{2s+n}\|r\|_2^2.
 \label{app:ch_mean_pairs:eq:mean-canonical-weighted-C}
\end{align}
These bounds hold when $T\leq K$. For a complete derivative change
$D$, the exact self-tensor increment from $B$ is
\begin{equation}\label{app:ch_mean_pairs:eq:mean-exact-self-change}
 2\sym\Re(B_j^*D_j)+\Re(D_j^*D_j).
\end{equation}
Equal phases cancel before differentiation.
\end{defn}

For the mean solve the fixed normal comparison is
\begin{equation}\label{app:ch_mean_pairs:eq:mean-normal-comparison}
 \|E_K-E_{0,K}\|_{2\to2}\leq CH^{-\nu},\qquad\nu>0.
\end{equation}
Proposition~\ref{app:ch_mean_pairs:prop:mean-coarse} proves it for
all bands at once, before any canonical update.

\section{Signed quadratic products}\label{app:sec:ch_pairs}
\subsection{The complete current sources and one solver}
\label{app:ch_pairs:sec:pairs-inputs}
\label{app:ch_pairs:sec:pairs-data}
The geometric and linear inputs below have been proved in
Appendices~\ref{app:sec:ch_seed} and \ref{app:sec:ch_regular}.
Proposition~\mainref{main:prop:mean} gives the complete current
derivative envelopes used here.

\begin{setup}[Current geometry and derivative envelopes]
\label{app:ch_pairs:setup:pairs-current}
\label{app:ch_pairs:setup:pairs-lag}
Use the actual geometry of
Definition~\ref{app:ch_background:defn:bg-geometry}, the domains of
Setup~\ref{app:setup:domains}, and the fixed regular solver of
Theorems~\ref{app:ch_regular:thm:exact-correction}
and~\ref{app:ch_regular:thm:quantitative} at $F$. Suppose
\begin{equation}\label{app:ch_pairs:eq:pairs-lag-map-profile}
 \|F\|_{C^r}\leq D_rT^{(r-r_*)_+},\qquad L_F=T,
\end{equation}
where the fixed $r_*$ exceeds the coefficient shifts in that solver
and in ambient reconstruction. Corollary~\ref{app:cor:fixed-profile-transfer}
gives, for every reconstruction coefficient and fixed cutoff $c_F$,
\begin{equation}\label{app:ch_pairs:eq:pairs-lag-geometric-profile}
 \|D^rc_F\|_\infty\leq C_rT^r.
\end{equation}

Use the exact current Hamilton phases and literal groups of
Definition~\ref{app:defn:exhaustive-pair-groups}, with
$\tau_j^{-1}\asymp\lambda$. The same-group spacing is $ch$.
The protected signed classes and their true frequencies are
\[
 \begin{array}{c|c}
 \text{distinct same-group differences}&\rho=\lambda|z-z'|\\
 \text{sums, doubled carriers, and either sign between groups}&\rho=\lambda.
 \end{array}
\]
Distinct radial labels protect different groups even when their
unscaled directions agree. Only the equal-index Hermitian difference
is a self term. All band updates at one literal carrier are combined
in one field. Thus every off-self term has
$2\leq c\lambda h\leq\rho_{jk}\leq C\lambda$.

The mixed phase grades of Theorem~\ref{app:ch_waves:thm:wkb} give
$|D_x^rS|+|D_x^rD_\zeta S|+|D_x^rD_zS|\leq C_rT^{r-1}$ for $r\geq1$.
The segment formula on each convex carrier patch gives the same
bound for a divided difference. Proposition~\ref{app:ch_seed:prop:seed-actual-menu}
and Theorem~\ref{app:ch_regular:thm:no-return} place every normalized
pair covector in the fixed buffered regular region. The whole source
product is protected there, and every complete characteristic segment
stays in $W_0$ in either orientation. Use their original source cutoff
$\chi_{\rm src}$, supported where $\chi_2=1$ and equal to one near $D$;
all measurement and output supports lie in the common validity charts.

Let $W_j^{\rm lag}$ be the complete complex fields, with
\begin{equation}\label{app:ch_pairs:eq:pairs-lag-demod}
 dW_j^{\rm lag}=e^{i\theta_j}A_j,\qquad
 \theta_j=S_F(x,\zeta_j)/\tau_j.
\end{equation}
After one fixed envelope cutoff equal to one on every pair source,
assume
\begin{equation}\label{app:ch_pairs:eq:pairs-lag-envelope}
 \|A_j\|_{C^s}\leq C_sK_{\rm env}^s,\qquad
 L_{\rm pair}=\max(K_{\rm env},T)\leq\rho_{jk}^{1-\sigma},
 \quad K_{\rm env}\geq2,\quad\sigma>0.
\end{equation}
For a fixed mean construction, these fields and coefficients serve
every product accuracy and observation order.
Equation~\maineqref{main:eq:mean-final-amplitude}
proves this envelope for the final measures on the whole protected
source support; Proposition~\ref{app:prop:fixed-high} verifies the
separate history bound in the iteration.
\end{setup}

\begin{defn}[The one complete regular solve]\label{app:ch_pairs:setup:pairs-solver}
The preceding real operators and literal source cutoffs give
\[
 \mathcal C_F=\chi_2\mathcal R_F\chi_3,\quad
 \|\mathcal C_F\|_{2\to2}\leq\kappa_0<1,\quad
 \mathcal T_F=(I+\mathcal C_F)^{-1},\quad
 u=\mathcal S_F\mathcal T_Fh.
\]
All outgoing branches, endpoints, cutoff and low-frequency terms
are included. The quantitative theorem applies to a source with its
full complementary tail; the actual inverse and physical cutoffs do
not depend on the beat or proof accuracy.

Write $N_F=\mathsf N_F$, $V_F=\mathcal V_F$ for the reconstruction
of \eqref{app:ch_waves:eq:reconstruction}. For a real metric source
$q$, put $h_i=-\tfrac12Q_i(F):q$ and define
\begin{equation}\label{app:ch_pairs:eq:pairs-ambient-U}
 U=V_F(u)+\tfrac12N_F\Pi_{K_F^\perp}q.
\end{equation}
Equation~\maineqref{main:eq:completion-error}, with target $-q$,
converts $P_Fu-h$ to the full error $D\gamma_FU+q$ on $D$ through
the actual Gram inverse.
\end{defn}

For ambient derivative matrices use
$\mathsf B(A,B)=(A^TB+B^TA)/2$.

\subsection{Exact signed expansion and weighted correction}
\label{app:ch_pairs:sec:pairs-sum}
We sum the actual pair corrections with the literal physical coefficients and leave the complete Hermitian self contribution for the current mean.

\begin{lem}[The exact signed expansion]\label{app:ch_pairs:lem:pairs-ledger}
Let $b>0$, $\mu_j\geq0$, $dW_j=e^{i\theta_j}A_j$, and set
\[
 P=b\sqrt2\sum_{j=1}^M\sqrt{\mu_j}\chpairsRea W_j,
 \qquad H_{\rm self}=b^2\sum_j\mu_j\chpairsRea(A_j^*A_j).
\]
For the ordered signed pairs put
\[
 q_{jk}^+=\chpairsRea\big(e^{i(\theta_j+\theta_k)}\mathsf B(A_j,A_k)\big),
 \qquad
 q_{jk}^-=\chpairsRea\big(e^{i(\theta_j-\theta_k)}\mathsf B(A_j,\overline{A_k})\big).
\]
Then
\begin{equation}\label{app:ch_pairs:eq:pairs-exact-offself}
 Q_{\rm off}:=(dP)^TdP-H_{\rm self}
 =b^2\left[\sum_{j,k}\sqrt{\mu_j\mu_k}\,q_{jk}^+
       +\sum_{j\ne k}\sqrt{\mu_j\mu_k}\,q_{jk}^-\right].
\end{equation}
The first sum includes doubled carriers. The omitted difference
diagonal is the complete Hermitian self tensor, with every modulation
and current correction.
\end{lem}
\begin{proof}
For complex matrices $D,E$,
\[
 2\mathsf B(\Re D,\Re E)
 =\Re\mathsf B(D,E)+\Re\mathsf B(D,\overline E),
 \qquad\mathsf B(D,\overline D)=\Re(D^*D).
\]
Expand $dP$ using the first identity and subtract its difference
diagonal using the second.
\end{proof}

\begin{prop}[One current solve for the complete off-self tensor]
\label{app:ch_pairs:prop:pairs-lag-sum}
Use Setup~\ref{app:ch_pairs:setup:pairs-lag},
Definition~\ref{app:ch_pairs:setup:pairs-solver}, and
$\sum_j\mu_j\leq C_\mu$. Use the exact tensor $Q_{\rm off}$ of
\eqref{app:ch_pairs:eq:pairs-exact-offself}. Put
\[
 q=\chi_{\rm src}Q_{\rm off},\qquad
 h_i=-\tfrac12Q_i(F):q,
 \qquad z=\mathcal T_Fh,\quad u=\mathcal S_Fz,
\]
and define one real current field
\begin{equation}\label{app:ch_pairs:eq:pairs-weighted-U}
 S_{\rm lag}=V_F(u)+\tfrac12N_F\Pi_{K_F^\perp}q.
\end{equation}
With $\mathcal W_\mu=(\sum_j\sqrt{\mu_j})^2\leq MC_\mu$,
\begin{align}
 \|D\gamma_FS_{\rm lag}+Q_{\rm off}\|_{H^s(D)}
 &\leq C_{s,Q}b^2\mathcal W_\mu (\lambda h)^{-Q},
 \label{app:ch_pairs:eq:pairs-lag-sum-error}\\
 \|S_{\rm lag}\|_{H^s(W)}
 &\leq C_sb^2\mathcal W_\mu \lambda^{s+1}.
 \label{app:ch_pairs:eq:pairs-lag-sum-size}
\end{align}
For a fixed physical cutoff $\chi_{\rm out}=1$ near $D$ and
supported in the observation chart, the same size bound holds
for $\chi_{\rm out}S_{\rm lag}$ on the torus. The literal field and operators serve all $s,Q$. All doubled and
distinct signed terms are included; the complete Hermitian self
tensor is excluded. The sample count enters only through
$\mathcal W_\mu\leq MC_\mu$.
\end{prop}
\begin{proof}
\noindent\textbf{Step 1. One signed source at its true scale.}
Let $U_{jk}^{\pm}$ and $u_{jk}^{\pm}$ be its ambient and reduced
fields at $q=\chi_{\rm src}q_{jk}^{\pm}$.
Leibniz, including the source cutoff and current reduced
multiplier, bounds each product amplitude directly by
\[
 \|a_{jk}\|_{C^s}+\|a_{jk}\|_{H^s}
 \leq C_sL_{\rm pair}^s.
\]
The geometric coefficients have the derivative scale
\eqref{app:ch_pairs:eq:pairs-lag-geometric-profile}; only the
total derivative order enters.

Write the product as $e^{i\rho\varphi}a_{jk}$ at its true
$\rho=\rho_{jk}$. The normalized phase has
$|D^r\varphi|\leq C_rL_{\rm pair}^{r-1}$ and a fixed nonzero
gradient. Its compact protected covector set gives
$|\rho d\varphi-\xi|\geq c(\rho+|\xi|)$ on the entire
complementary mask. Proposition~\ref{app:ch_osc:prop:osc-source-shell}
therefore bounds the off-mask source in $H^s$ by
\[
 C_{s,k}\rho^{s+n/2}
                       (L_{\rm pair}/\rho)^k.
\]
Since $L_{\rm pair}\leq\rho^{1-\sigma}$, a finite $k$ gives
any requested decay. The source plateau and protected complete segments
are those of Definition~\ref{app:ch_pairs:setup:pairs-solver}.
The fixed solve and \maineqref{main:eq:completion-error} give
\begin{equation}\label{app:ch_pairs:eq:pairs-lag-one-error}
 \|D\gamma_FU_{jk}^{\pm}+q_{jk}^{\pm}\|_{H^s(D)}
 \leq C_{s,Q}\rho_{jk}^{-Q}.
\end{equation}
A higher finite proof accuracy pays the fixed frame and Gram
derivative costs, without changing the field.

\noindent\textbf{Step 2. The exact weighted sum.}
The solver, inverse, frame and cutoffs are fixed across all pair
scales. By Cauchy--Schwarz, the total absolute coefficient in
\eqref{app:ch_pairs:eq:pairs-exact-offself} is at most
$2b^2\mathcal W_\mu\leq2b^2MC_\mu$. By linearity,
\eqref{app:ch_pairs:eq:pairs-lag-one-error} and
$\rho_{jk}\geq c\lambda h$ give the full tensor error on $D$.
For the size, put $\mathfrak A=b^2\mathcal W_\mu$.
Leibniz and the same coefficient sum give
\[
 \|q\|_{H^s}+\|h\|_{H^s}\leq C_s\mathfrak A\lambda^s.
\]
Corollary~\ref{app:cor:fixed-profile-transfer} absorbs the fixed
shifts in the strong
tame estimates for $\mathcal S_F\mathcal T_F$; since $L_F\leq\lambda$,
$\|u\|_{H^s}\leq C_s\mathfrak A\lambda^s$.
The geometric profile and one derivative in $V_F$ prove the
ambient size bound, also after physical cutoff. The operators
are real and uniform on the fixed chart menu.
\end{proof}

\section{Verification of the fixed scales}
\label{app:sec:ordered-scale-selection}
We check the choices of Section~\mainref{main:sec:scales} on the
fixed domains, with the same operators and sample family.

\begin{prop}[The fixed scale windows]\label{app:prop:ordered-scale-selection}
The stated choices satisfy all primary, current-response, mean and
product hypotheses. Their exponents and finite low topologies are
independent of stage accuracy and the final target dilation.
\end{prop}
\begin{proof}
\noindent\textbf{1. Preparation and one sample.}
The fixed quadrature order is $q_{\rm quad}=2n-1<r_{\rm par}$.
Its color list depends only on $n$ and changes only constants.
Choose $A$ by \maineqref{main:eq:A-choice} and
$J=\lfloor2A/a\rfloor$. Then
\[
 a(J+1)>2A,\qquad \beta(2n-1)-2A\geq3/64.
\]
The radial labels, common geometric neighborhood and finite backward
input orders then precede sampling and mean accuracy. The existing
joint amplitude budget covers the quadrature. The prepared scalars
stay in the original positive class. Every translated lattice has
the required background error; the full-prefix normal event has
positive probability at large $R$.
In $R$ units, the two normal-error exponents are
\[
 (\epsilon-\theta)/4=1/256,\qquad
 a-1-\theta-\beta=13/320.
\]
The $O(\lambda^{-1})$ change from the original leading row is smaller.
The inverse and all auxiliary input cuts are unchanged.

\noindent\textbf{2. Modulation and product bands.}
Use a fixed multiple of $T_0$ for the zero atom and a fixed larger
multiple of the actual input band for each mean datum. With
$\sigma_{\rm pair}=1/1000$ and $\nu=1/4$, direct calculation gives
\[
\begin{aligned}
 1-t&=1/35,\\
 (1-\nu)a-s_f&=27/640,\qquad
 (1-\sigma_{\rm pair})(a-\beta)-s_f=7581/160000.
\end{aligned}
\]
These verify $T_0\leq K\leq\lambda_j^{1-\nu}$ and the product
window at every true beat $c\lambda h\leq\rho\leq C\lambda$.
The input support lies below $K/C_1$, and its ratio to $\lambda_j$
tends to zero. The primary and coefficient scale is $T_0$; all
physical slopes are at most $a<b_*$. Strict gaps absorb the fixed
support and radial constants.

\noindent\textbf{3. The sampled mean.}
For $p=(n+1)/2$ and $A=p(1+\theta)/2$,
\[
 \epsilon_fK_{\max}^{p}
 =CR^{-p\theta/2}\leq H^{-\nu_{\rm nl}},\qquad
 \nu_{\rm nl}=\frac{p\theta}{4(1+\theta)}>0.
\]
Choose a contraction exponent below this and the comparison
exponents in Step 1. The prepared residual has $L^2$ size
$R^{-2A}$ and global $H^s$ profile $C_sR^{ts}$.
Interpolation with $t<1$ gives the source bound
$C_s\epsilon_fR^s$ in \maineqref{main:eq:fine-source-profile}.
Its slope $1/(1+\theta)$ and the coefficient slope
$t/(1+\theta)$ lie below the first input band. At every finite
update count the equal-gap bands have a positive gap, so one finite
derivative order controls every omitted input. The low residual
sum proves the variation and physical field bounds.

\noindent\textbf{4. Increasing precision.}
Choose the fixed high history order after the preparation orders,
then $k_{\min}$ by
\maineqref{main:eq:k-cubic}--\maineqref{main:eq:k-history};
$3-2v=1/12>0$. Set $Q(k)=2k$.
Take $aQ_{\rm primary}>Q(k)+A+(1+\epsilon)/2$, mean accuracy
above $Q(k)+2A+2$, and smoothing order $(1-t)N>2Q(k)$.
The product and physical-cutoff gaps
$\sigma_{\rm pair}(a-\beta)>0$ and $b_*-a>0$
give every finite precision for the same primary family and responses.
Only the finite mean updates vary with accuracy.
\end{proof}


\section*{Acknowledgements}
Generative AI was used in developing the proofs.

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\end{thebibliography}
\end{document}