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\title[Local isometric embeddings]{Smooth Local Isometric Embeddings of
Riemannian Manifolds in Optimal Dimension}
\author{Wenchang Yang}
\address{Tsinghua University, Beijing, China}
\author{Yi Wang}
\address{Tsinghua University, Beijing, China}
\date{}
\subjclass[2020]{53B20, 53C42, 35F50}
\keywords{local isometric embedding, smooth Riemannian metric,
characteristic waves, Fourier integral operators}

\begin{document}

%% ===== 01_intro.tex =====
\begin{abstract}
We prove that every smooth Riemannian manifold of dimension $n\geq5$
admits, near each point, a smooth isometric embedding into
$\mathbb{R}^{n(n+1)/2}$. This extends the local theorem of Janet and
Cartan from real-analytic to arbitrary smooth metrics in these
dimensions, and attains the optimal universal ambient dimension without
any curvature or genericity assumption. The proof starts from a flat
immersion whose second fundamental form has full normal rank. The
linearized metric equation then reduces to a determined first-order
system, whose characteristic variety is singular. We do not invert
this system on its singular set. Instead, the metric is corrected by the
quadratic interaction of characteristic waves: their phase-averaged
quadratic terms generate the directions missing from the image of the
reduced symbol, while the remaining oscillatory products and the
residual linear errors are removed by parametrices on regular
characteristic and elliptic patches. Exact Fourier localization of the
increments yields an iteration that converges in $C^\infty$ on a fixed
neighborhood.
\end{abstract}

\maketitle
\tableofcontents

\section{Introduction}\label{main:sec:introduction}

The local isometric embedding problem asks whether a Riemannian metric
$g$ defined near a point of an $n$-dimensional manifold can be written,
in local coordinates, as
\begin{equation}\label{main:eq:isometry}
 g_{ij}=\partial_iF\cdot\partial_jF,\qquad 1\leq i,j\leq n,
\end{equation}
for a map $F$ into a Euclidean space $\R^N$ of prescribed dimension.
Equation \eqref{main:eq:isometry} is a system of
\[
 s_n=\frac{n(n+1)}2
\]
first-order equations, one for each independent metric coefficient, so
$N=s_n$ is the smallest dimension in which one can expect solvability
for every metric. For real-analytic metrics, Janet \cite{Janet26} and
Cartan \cite{Cartan27} proved local solvability in exactly this
dimension. For general smooth metrics the classical local result, due to
Greene \cite{Greene70}, gives an isometric embedding into
$\R^{s_n+n}$; see also \cite[\S3.7]{Gromov17} for the perturbation and
scaling argument. In that construction one perturbs a \emph{free}
immersion, whose first and second derivatives span $s_n+n$ dimensions;
the linearized metric equation then has an algebraic right inverse, as
in Nash's work \cite{Nash56}, and no differential equation has to be
solved. In dimension $s_n$ such a free reference does not exist, and the
linearized problem necessarily contains a genuine system of partial
differential equations.

The main result of this paper is the following.

\begin{theorem}\label{main:thm:main}
Let $n\geq5$, let $p$ be a point of a smooth $n$-dimensional manifold,
and let $g$ be a smooth Riemannian metric defined near $p$. Then 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}

For $n=5$ the target is $\R^{15}$, and for $n=6$ it is $\R^{21}$.
In the terminology of \cite[\S1.2]{Gromov17}, Theorem~\ref{main:thm:main}
establishes the smooth local Schl\"afli assertion in all dimensions
$n\geq5$, without analyticity or curvature hypotheses. The construction,
the estimates and the convergence argument are the same in every
dimension $n\geq5$; the restriction $n\geq5$ enters through the explicit
flat reference immersion of Section~\ref{main:sec:geometry} (it uses the
constant $c_n=(n-4)^{-2}$) and through the construction of the
characteristic patches in Section~\ref{app:sec:ch_seed}. We make no
assertion for $n=4$.

The dimension $s_n$ cannot be lowered if one asks for a target that
works for every metric.

\begin{remark}[Sharpness of the universal target dimension]
\label{new:rem:sharpness}
Let $M<s_n$. We claim that some real-analytic metric germ at
$0\in\R^n$ admits no $C^\infty$ isometric immersion into $\R^M$. Fix
$k\geq0$. Composing with a translation, we may consider maps $F$ with
$F(0)=0$. The $k$-jet at $0$ of the induced metric $(dF)^TdF$ depends only
on the $(k+1)$-jet of $F$, and it does so polynomially. The space of
$(k+1)$-jets of maps into $\R^M$ vanishing at $0$ has dimension at most
$M\binom{n+k+1}{n}$, while the space of $k$-jets of symmetric matrices
has dimension $s_n\binom{n+k}{n}$. Since
\[
 \binom{n+k+1}{n}=\frac{n+k+1}{k+1}\binom{n+k}{n},
\]
the first dimension is strictly smaller than the second as soon as
$M(n+k+1)<s_n(k+1)$, that is,
\[
 (s_n-M)(k+1)>Mn,
\]
which holds for $k$ large because $s_n-M\geq1$. For such $k$ the image
of the jet map is the image of a smooth map from a space of smaller
dimension, hence has Lebesgue measure zero. It therefore misses a point
of the nonempty open set of jets whose constant term is positive
definite. The Taylor polynomial of such a jet is a positive definite
polynomial metric on a neighborhood of $0$, and an isometric immersion of
it into $\R^M$ would realize the omitted jet. This contradiction proves
the claim. The bound is universal: an individual metric may of course
embed in lower dimension.
\end{remark}

\subsection*{Background}
The reduction of the linearized isometric embedding system and the
geometry of its characteristic variety were developed by
Bryant, Griffiths and Yang \cite{BGY83}; the analytic theory is presented
in \cite{BCGGG91}. A central difficulty is that the characteristic
variety of the linearized system need not be smooth. Han and Khuri
\cite[Theorem~1.2]{HanKhuri12} proved that, for $n\geq5$ and under
explicit generic conditions in a suitable parameter regime, it is
singular. The reference immersion used here also has singular
characteristics; they are classified in Proposition~\ref{app:ch_seed:prop:seed-strata}.
Consequently one cannot hope to invert the reduced system along its whole
characteristic variety, and we never do so.

For smooth metrics in low dimensions, Nakamura and Maeda \cite{NM89} and
Goodman and Yang \cite{GY88} obtained local isometric embeddings of
three-dimensional metrics into $\R^6$ under the assumption that the
Riemann curvature tensor does not vanish at the base point. Their
methods rely on the theory of operators of real principal type of
Duistermaat and H\"ormander \cite{DH72}. The present work uses the
same circle of microlocal ideas, but only on regions of covectors where
the reduced system is elliptic or of simple real principal type. All
estimates are proved on fixed domains, with losses of derivatives that
do not depend on the accuracy required at a given step.

\subsection*{The idea of the proof}
We start from a flat immersion whose second fundamental form spans the
normal space at every point. For such an immersion the linearized
metric equation splits into an algebraic normal part and a determined
$n\times n$ first-order system for a tangential covector field. The
principal symbol of this system is explicit for our reference, and its
characteristic variety contains both simple and multiple
characteristics.

Instead of inverting the system, we use the quadratic term of the metric
expansion. A real high-frequency wave $\sqrt2\Re Z$, with
$dZ=e^{\ii\vartheta}A$, satisfies
\[
 \bigl(d(\sqrt2\Re Z)\bigr)^Td(\sqrt2\Re Z)
 =\Re(A^*A)+\Re\bigl(e^{2\ii\vartheta}A^TA\bigr).
\]
The first term is the average of the left side over the phase
$\vartheta$; we call it the \emph{mean}. If $Z$ is a characteristic
wave, so that its linearized metric change vanishes to high order, then
$A$ is approximately a rank-one matrix $\nu\otimes\xi$, with $\xi$ a
characteristic covector, and its mean is $|\nu|^2\xi\otimes\xi$. A
finite family of three-parameter patches of simple characteristic
covectors provides enough such tensors to generate, together with the
image of the reduced symbol, every direction of the reduced output
space; this is the Gram inequality \eqref{main:eq:detector-gram}. The
corresponding normal operator is coercive, and its inverse determines
how the waves must be modulated. The remaining oscillatory products,
whose phases are sums and differences of distinct phases, have
frequencies in regions where the reduced system is elliptic or of simple
real principal type; there they are cancelled by finite Neumann series
of parametrices. 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),
 \qquad\text{if}\quad\|\gamma(F)-g\|_{C^0}\leq\delta^2,
\end{equation}
where $\gamma(F)=(dF)^TdF$ is the induced metric, $R$ is a large
frequency parameter, $Q$ is an arbitrary accuracy and the exponent
$B_3$ does not depend on $Q$. Moreover, $U$ has exact Fourier support
below a fixed power of $R$, with an $L^2$ bound involving a fixed loss.
These two properties allow $Q$ to increase slowly along the iteration,
so that the constants $C_Q$ are absorbed while the increments converge
in every $C^r$ norm on one fixed neighborhood. A detailed outline is
given in Section~\ref{sec:outline}.

\subsection*{Organization of the paper}
Section~\ref{sec:outline} describes the proof step by step.
Section~\ref{app:sec:notation} fixes notation, Fourier conventions and the
domains used throughout. The geometric part comes first:
Section~\ref{main:sec:geometry} derives the reduced metric equation,
classifies the singular characteristics of the model and constructs the
flat reference immersion, and Section~\ref{app:sec:ch_seed} constructs
the characteristic patches, their phases and the Gram inequality. The
analytic tools follow: Section~\ref{app:sec:elementary-interpolation}
contains elementary smoothing and derivative estimates, and
Section~\ref{app:sec:ch_osc} the oscillatory integral estimates.
Sections~\ref{app:sec:ch_regular_base}
and~\ref{app:sec:ch_regular} construct parametrices and finite
corrections on regular patches. Section~\ref{main:sec:linear}
constructs the corrected characteristic waves, and
Section~\ref{sec:sampling} proves the sampling theorem.
Section~\ref{app:sec:ch_background} studies the continuous normal
operator and constructs the positive transported mean, and
Section~\ref{main:sec:waves} corrects the mean.
Section~\ref{app:sec:ch_pairs} cancels the oscillatory quadratic terms.
The single correction step is proved in Section~\ref{main:sec:increment},
and the iteration and the proof of Theorem~\ref{main:thm:main} are
completed in Sections~\ref{main:sec:history}--\ref{main:sec:convergence}.

\section{Outline of the proof}\label{sec:outline}

In this section we describe the structure of the argument and the
role of each ingredient. Precise statements are given in the
sections indicated. Throughout, $\gamma(F)=(dF)^TdF$ denotes the
metric induced by a map $F$, and
\[
 D\gamma_FY=(dF)^TdY+(dY)^TdF
\]
its linearization, so that
\begin{equation}\label{new:eq:outline-expansion}
 \gamma(F+Y)=\gamma(F)+D\gamma_FY+(dY)^TdY .
\end{equation}

\subsection*{Step 1: Normalization and the reference immersion}
Choose coordinates with $g(0)=I$. Let $F_0$ be an explicit analytic
flat immersion into $\R^{s_n}$ whose second fundamental form has full
normal rank (Proposition~\ref{app:ch_seed:prop:seed-analytic}); it is
built by $q=n(n-1)/2$ successive circular bendings along the vectors
$\ell_{ab}=\mathbf 1-2e_a-2e_b$. After a fixed dilation,
$G_{\rm ref}(y)=\rho^{-1}[F_0(\rho y)-F_0(0)]$ is uniformly close, in
the sense needed below, to the constant-coefficient model. The rescaled
metrics $g^{(s)}(y)=g(s\rho y)$ satisfy
$\|g^{(s)}-\gamma(G_{\rm ref})\|_{C^r}\leq C_rs$
(Proposition~\ref{main:prop:seed}). All geometric, analytic and frequency
constants are fixed first; only at the very end is $s>0$ chosen so small
that the initial defect is below the first threshold of the iteration.

\subsection*{Step 2: The reduced equation}
Let $F$ be an immersion near $G_{\rm ref}$, and write a variation as
$Y=dF\,\gamma(F)^{-1}u+w$ with $u$ a covector field and $w$ normal.
Then $D\gamma_FY=2\sym\nabla u-2\langle w,\mathrm{II}_F\rangle$
(equation~\eqref{main:eq:tangent-normal}). Since the second fundamental
form, viewed as a map $B_F:\Sym^2\R^n\to\mathcal N_F$, is onto, its kernel
$K_F$ has dimension $n$, and the normal component $w$ can absorb every
tensor orthogonal to $K_F$. The equation $D\gamma_FY=h$ therefore
reduces to a determined first-order system
\[
 \cP_Fu=\tfrac12q_F(h),\qquad \cP_Fu=(Q_i(F):\sym\nabla u)_{i=1}^n,
\]
where $Q_1(F),\dots,Q_n(F)$ is a frame of $K_F$ (equation
\eqref{main:eq:reduced-equation}); the full error of an approximate
solution is given exactly by \eqref{main:eq:completion-error}. The
principal symbol of $\cP_F$ is $\ii M_F(x,\xi)$ with
$M_F(x,\xi)_{i\cdot}=(Q_i(F)\xi)^T$. For the reference it is the pencil
$M_0(\xi)=\diag(\xi)-c_n(\sum_i\xi_i)J$, whose determinant $p_0$ has an
explicit singular set (Proposition~\ref{app:ch_seed:prop:seed-strata}):
two families of codimension-three subspaces and $q$ lines. The system is
neither elliptic nor of real principal type, so it cannot be inverted
directly.

\subsection*{Step 3: What the quadratic term can generate}
Let $\xi$ be a simple characteristic covector, $p_F(x,\xi)=0$. A wave
whose tangent part is $e^{\ii\lambda\phi}$ times a kernel vector of
$M_F(x,d\phi)$, with $p_F(x,d\phi)=0$, has linearized metric change of
lower order, and its differential is, to leading order, the rank-one
matrix $e^{\ii\lambda\phi}\,\nu\otimes d\phi$. By the identity in the
introduction, its phase average contributes $|\nu|^2\,d\phi\otimes d\phi$
to the metric, and $q_F$ maps this to $|\nu|^2q_F(d\phi\otimes d\phi)$.
The quadratic interaction of such a wave with a slow modulation of
itself has, to leading order, an output frequency tangent to the
characteristic variety at the base covector. Hence an output frequency
$k$ can be produced in this way only from base covectors $\xi_a$ with
$\widehat k\cdot\partial_\xi p_0(\xi_a)=0$; we call these the incidence
points of $\widehat k$. The key algebraic fact is
the Gram inequality \eqref{main:eq:detector-gram}:
\[
 |M_0(\widehat k)^Tw|^2+\sum_a|w\cdot q_0(\xi_a)|^2\geq c|w|^2,
 \qquad w\in\R^n,
\]
for every unit $\widehat k$, where the $\xi_a$ range over finitely many
such incidence points. Thus the quadratic tensors of characteristic waves
produce exactly the directions missing from the image of the reduced
symbol, including at the singular output covectors. This is proved in
Proposition~\ref{app:ch_seed:prop:seed-frozen-menu}.

\subsection*{Step 4: Characteristic patches and their interactions}
The waves are taken from finitely many three-parameter patches of
simple characteristic covectors (Section~\ref{app:sec:ch_seed}). Their
phases $S_F(x,\zeta)$ solve the eikonal equation by the Hamilton--Jacobi
method. Two geometric properties are essential. First, for distinct
parameters $z\ne z'$ in one patch, the direction of
$d_x\Theta(x,z)-d_x\Theta(x,z')$ stays uniformly away from the singular
set, and $|d_x\Theta(x,z)-d_x\Theta(x,z')|\geq c|z-z'|$; sums of
phases behave in the same way. Second, after multiplying the patches by
fixed, widely separated radial factors, the same holds for sums and
differences of phases from different patches. Consequently every
oscillatory product of two waves has a frequency at which the reduced
system is elliptic or of simple real principal type
(Proposition~\ref{app:ch_seed:prop:seed-actual-menu}).

\subsection*{Step 5: Parametrices on regular patches}
On a conic region where the reduced system is elliptic, or has a simple
real characteristic root $\tau=h(x,\eta)$, we construct a parametrix:
a matrix symbol inverse in the elliptic case, and a retarded oscillatory
integral with a scalar Hamilton phase and transported amplitude,
plus an algebraic complement, in the characteristic case
(Section~\ref{app:sec:ch_regular_base}). The residual gains one derivative,
and a finite Neumann series, with the microlocal flowout controlled
along the Hamilton flow, gives approximate solutions to any finite order
for sources concentrated at a large frequency $\rho$, provided their
amplitudes vary on a scale $L\leq\rho^{1-\sigma}$
(Theorem~\ref{app:ch_regular:thm:quantitative}).

\subsection*{Step 6: Corrected characteristic waves}
For each patch we solve the transport equation along the Hamilton flow,
add the algebraic second term, and then remove the remaining reduced
residual by the finite Neumann series of Step~5. This yields a family of
waves $\mathcal W^{\rm cur}_\tau(F;x,\zeta)$, linear in a prescribed
positive initial amplitude, whose linearized metric change is
$O(\lambda^{-q})$ for any prescribed $q$, and whose leading amplitude is
the transported rank-one tensor (Corollary~\ref{app:ch_response:cor:current-atoms}).
Superposing over shifted covectors $\zeta+\tau\xi$ against a finite
Fourier measure $m$ modulates the initial data without changing the
correction (Proposition~\ref{main:prop:current-waves}); this is how the
waves are adjusted to the defect.

\subsection*{Step 7: The continuous normal operator and a positive mean}
Combining the image of the reduced operator with the quadratic tensors
of the waves gives an operator $A$, and the Gram inequality yields
coercivity of $E=AA^*+1-\chi_{\rm meas}^2$
(Proposition~\ref{app:ch_background:prop:bg-frame}). Its inverse
produces a positive initial amplitude $r_F$ and a tangent field
$V_{\rm base}$ such that the transported mean of the waves with density
$r_F^2$, plus $D\gamma_FV_{\rm base}$, has zero component in $K_F$
(Proposition~\ref{app:ch_background:prop:bg-current}). This is what
allows metric defects of either sign: the positive mean of the
unmodulated waves is cancelled in advance, and the defect itself is then
matched by small modulations.

\subsection*{Step 8: Sampling}
A continuous superposition over the patch parameters cannot be used
directly, since the oscillatory products must be separated in frequency.
We therefore replace it by $M\leq CR^{1+\epsilon}$ samples at mutual
distance at least $h/4$, $h=R^{-\beta}$, with positive weights $\mu_\nu$
of total mass $O(1)$. A random construction with deletion and an operator
Bernstein inequality show that one sample approximates the continuous
normal operator in operator norm, on its full output space, provided
both operators carry the same cutoff at a fixed input frequency
$K_{\max}=R^{s_f}$ (Theorem~\ref{app:ch_waves:thm:sampling}). The
comparison is between operators with the same input cutoff; the part of
the continuous operator above $K_{\max}$ is treated separately in
Step~9.

\subsection*{Step 9: Correction of the mean}
With the sampled waves $Z_\nu$, the mean equation
\[
 D\gamma_FV+\sum_\nu\mu_\nu\Re\bigl((dZ_\nu)^*dZ_\nu\bigr)
 =e_R/b^2+O(R^{-Q})
\]
is solved for modulation measures $m_\nu$ and a field $V$, where $e_R$
is a smoothing of the defect $e=g-\gamma(F)$ at scale $R^{-1}$ and
$b=\delta R^A$ is the wave amplitude. The modulations are produced from
a source $r$ by the adjoint of the normal operator, and the
$K_F$-component of the equation becomes a quadratic equation
$g=\mathcal Er+\mathcal Q(r,r)$, with $\mathcal E=E^{\rm s}_KE_F^{-1}$.
Three operators have to be distinguished here: the continuous normal
operator $E_F$ without input cutoff, which is inverted; the continuous
operator $E_{F,K}$ with the input frequencies cut at $K=K_{\max}$; and
the sampled operator $E^{\rm s}_K$. Thus
\[
 I-\mathcal E=(E_F-E_{F,K})E_F^{-1}+(E_{F,K}-E^{\rm s}_K)E_F^{-1}.
\]
The second term is $O(R^{-\omega})$ in operator norm on all of $L^2$,
by Step~8. The first term is not small in operator norm; it is the part
of the continuous operator above the input band, and it is small only on
sources with bounds $\|r\|_{H^q}\leq C_qhR^{(q-s_0)_+}$, where it is
$O((R/K_{\max})^q)$ for every $q$. Accordingly the equation is solved by
a finite Picard iteration in which every iterate is shown to belong to
this class: the $L^2$ norms of successive differences contract by
$R^{-\omega}$, up to the frequency tail, while the higher Sobolev norms
stay bounded (Proposition~\ref{app:ch_mean_pairs:prop:mean-iteration}).
After finitely many steps the residual is $O(R^{-Q})$, and the normal
component is completed algebraically (Proposition~\ref{main:prop:mean}).

\subsection*{Step 10: Correction of the oscillatory products}
The primary field $P=b\sqrt2\sum_\nu\sqrt{\mu_\nu}\Re Z_\nu$ has
quadratic term $(dP)^TdP=H_{\rm self}+Q_{\rm off}$, where
$H_{\rm self}$ is the mean treated in Step~9 and $Q_{\rm off}$ consists of
products with phases $\theta_j\pm\theta_k$ and doubled phases. Their
frequencies $\rho_{jk}$ satisfy $c\lambda h\leq\rho_{jk}\leq C\lambda$,
while their amplitudes vary on the scale $R\leq(\lambda h)^{1-\sigma}$.
By Step~4 they lie in regular regions, and Step~5 gives a field $S$ with
$D\gamma_FS+Q_{\rm off}=O(b^2M(\lambda h)^{-q})$
(Proposition~\ref{main:prop:pairs}).

\subsection*{Step 11: One step of the iteration}
Put $U=\Pi(P+b^2V+S)$, where $\Pi$ is a Fourier multiplier cutting off
frequencies above $C\lambda^{1+\zeta}$. Substituting into
\eqref{new:eq:outline-expansion} gives an exact identity in which every
term is small: the smoothing error $e-e_R$, the linearized change of the
primary waves, the mean error, the product error, and the interaction
terms $2\sym((dP)^TdW_2)+(dW_2)^TdW_2$ with $W_2=b^2V+S$. The last terms
are of size $\delta^3R^{B_3}$ because $P=O(\delta)$ and $W_2=O(\delta^2)$
up to fixed powers of $R$. This proves \eqref{main:eq:intro-estimate},
together with the exact Fourier support and the $L^2$ bound of $U$
(Theorem~\ref{main:thm:step}).

\subsection*{Step 12: Iteration and convergence}
We use frequencies $R_j=R_0^{v^j}$ with $1<v<3/2$ and defects
$\delta_j^2=R_j^{-2k_j}$. Since
$\delta_j^3R_j^{B_3}=R_j^{-3k_j+B_3}\leq R_j^{-2v(k_j+1)}
=R_{j+1}^{-2(k_j+1)}$ when $(3-2v)k_j$ is large, the error estimate
of Step~11 reproduces the hypothesis at the next step. The increments
have Fourier support in $|\xi|\leq CR_j^{b_*}$ and $L^2$ norm
$O(R_j^{-k_j+B_0+1})$, so the Bernstein inequality makes them summable
in every $C^r$ norm. The same Fourier localization shows that the
iterates satisfy derivative bounds $\|F_j\|_{C^r}\leq C_rT_j^{(r-r_*)_+}$
with $T_j=R_j^t$, $t<1$, which are exactly the coefficient bounds
required in Steps~5--10; the accuracies $Q_j=2k_j$ are increased slowly
enough to absorb the constants $C_{Q_j}$
(Sections~\ref{main:sec:history}--\ref{main:sec:convergence}). The limit
is an isometric immersion of the dilated metric on a fixed ball;
rescaling and the inverse function theorem give
Theorem~\ref{main:thm:main}.

\subsection*{The scales}
The exponents are chosen in Proposition~\ref{app:prop:ordered-scale-selection}.
With $T_0=R^t$, $\lambda=R^a$, $h=R^{-\beta}$, $K_{\max}=R^{s_f}$ and
$\beta=(1+\epsilon)/3$, they satisfy
\[
 \tfrac23<t<1<s_f<1+\epsilon<1+\beta<a<b_*<v<\tfrac32 .
\]
The essential relations are: the coefficient scale $T_0$ and the
amplitude scale $R$ lie below the smallest product frequency $\lambda h$
(so the parametrices of Step~5 apply to the products); the input band
$K_{\max}$ lies below the wave frequency $\lambda$ (so modulations do not
destroy the characteristic structure); and the sample size
$h^{-3}=R^{1+\epsilon}$ exceeds $K_{\max}$ by a positive power (so the
sampling theorem applies on the whole input band).

\subsection*{Logical dependence}
The order of choices matters; it is listed completely in
Section~\ref{new:sec:order-of-choices}. The dimension, the reference
immersion, the characteristic patches, their radial factors, the
domains and cutoffs, and the parametrices of
Sections~\ref{app:sec:ch_regular_base}--\ref{app:sec:ch_regular} are
fixed first, using only low-order information on the immersions, together
with one fixed bound for finitely many higher derivatives. Then the
exponents and a finite derivative order $r_*$ are fixed; $r_*$ contains
every finite order used in the geometric and analytic constructions. At
each step of the iteration, a finite accuracy $Q$ then determines the
finite depths of the Neumann series and of the Picard iteration; these
depths affect only constants and frequency thresholds, never the domains,
the order $r_*$, the Fourier supports or the exponents. Finally, the
dilation parameter $s$ is chosen after all thresholds.

%% ===== 02_notation.tex =====
\section{Notation, conventions and fixed domains}\label{app:sec:notation}

The metric and normal-map conventions are introduced in
Section~\ref{main:sec:geometry}. Every reduced operator is used with the
normalization displayed there, including the factor $1/2$ in
\eqref{main:eq:reduced-equation}. In this section we collect the
conventions for norms, Fourier transforms and constants, the list of
parameters, and the fixed domains and cutoffs used in all later
constructions.

\subsection{Norms, adjoints and Fourier transforms}

\begin{notation}[Norms and Fourier transforms]\label{app:not:fourier}
For a function $f$ smooth up to the boundary of a closed set $D$ we put
\[
 \|f\|_{C^r(D)}
 =\max_{|\alpha|\leq r}\sup_{x\in D}|\partial^\alpha f(x)|,
\]
and similarly on other domains. Norms of vectors, matrices and tensors
are taken componentwise in fixed coordinates; a coefficient norm
includes every matrix entry. Fields with a fixed compact support inside
a chart are extended by zero to $\R^n$. Unless another domain is
specified, Sobolev and $L^2$ norms are taken on $\R^n$, whereas physical
$C^r$ norms always carry their domain. Integer Sobolev norms are the
$L^2$ norms of all Cartesian derivatives through the stated order; on
$\R^n$ this is equivalent to $\|f\|_{H^s}=\|\langle D\rangle^sf\|_2$,
where $D=-\ii\partial$ and $\langle\xi\rangle=(1+|\xi|^2)^{1/2}$. We
also write $\|u\|_s=\|u\|_{H^s}$ and
$H^\infty=\bigcap_{s\geq0}H^s$.

In $d$ variables we use the Fourier normalization
\[
 \widehat f(\xi)=(2\pi)^{-d}\int_{\R^d}e^{-\ii x\cdot\xi}f(x)\dd x,
 \qquad f(x)=\int_{\R^d}e^{\ii x\cdot\xi}\widehat f(\xi)\dd\xi,
\]
so that $\|f\|_2^2=(2\pi)^d\|\widehat f\|_2^2$. For the initial variables
of a wave $d=n-1$. A finite complex measure $m$ on $\R^d$ is regarded
as Fourier data: writing $dm(\xi)=\widehat f(\xi)\dd\xi$ gives
$f(y)=\int e^{\ii y\cdot\xi}\dd m(\xi)$. Its size is the total variation
$\|m\|_{\TV}$. The Dirac mass at the origin is denoted by $\delta_0$; it
represents the constant function $1$.

For complex matrices, $X^T$ denotes the transpose and $X^*$ the conjugate
transpose; real parts are taken entrywise. Thus $\Re(X^*X)$ is a real
symmetric tensor, whereas a cross term $\Re(X^*Y)$ has to be symmetrized
when it occurs alone. Complex Hilbert inner products are linear in the
first argument, and adjoints are complex Hilbert adjoints, restricted
to real fields when the operator is real. Operator norms refer to the
entire stated spaces. A normal operator $AA^*$ acts on the output space
of $A$; it should not be confused with the normal map $B_F$ of the
immersion.
\end{notation}

\subsection{Amplitudes and constants}
For a complex wave $Z=e^{\ii\lambda\phi}z$ we call $z$ its amplitude.
Estimates for $z$ and for $e^{-\ii\lambda\phi}dZ$ are stated separately.
A bound for the first does not by itself give the second at the same
scale, because
\[
 e^{-\ii\lambda\phi}dZ=\ii\lambda z\,d\phi+dz .
\]
This is why the estimates below distinguish the scale $K$ on which the
amplitude varies from the frequency $\lambda$. The leading amplitude of a
characteristic wave is a scalar transported along the Hamilton flow
times a null vector; its value at the initial time is called the
\emph{initial amplitude} and is denoted by $r$, and $r^2$ is called the
\emph{initial density}. The mean of a wave is linear in its initial
density.

The letter $M$ denotes the number of samples, while $M_{\rm an}=20(n+1)$
is a fixed order of differentiability used in the oscillatory integral
estimates. Three normal operators are distinguished in
Sections~\ref{app:sec:ch_background}--\ref{main:sec:waves}: the
continuous operator $E_F$, the continuous operator $E_{F,K}$ with input
frequencies cut at $K$, and the sampled operator $\widetilde E_K$.

Constants may depend on the dimension, the fixed charts and the fixed
geometric neighborhood. When a bound holds at every order $r$, the
constant $C_r$ may depend on $r$. In estimates of the form $C_rL^{r+B}$
we state explicitly when the exponent $B$ is independent of $r$ and of
the accuracy. At each finite order and accuracy, the constants involve
only finitely many derivative bounds of the data; they are uniform for
immersions in a fixed $C^{r_*}$ neighborhood of the reference whose
derivative bounds (see \eqref{main:eq:lag-hypothesis}) have fixed
constants. A requested accuracy determines finite iteration depths
before the waves and the sample are formed, and constants may depend on
these finite choices. Domains, the order $r_*$, Fourier supports and
exponents do not. In the single-step estimate this dependence is
displayed as $C_Q$, and the convergence argument absorbs it in one
threshold. Letters $B$ with subscripts denote fixed loss exponents.

We often use derivative bounds of the form
\begin{equation}\label{new:eq:derivative-growth}
 \|f\|_{C^r}\leq C_rT^{(r-R)_+},\qquad r\geq0,
\end{equation}
with $T\geq1$, $R\geq0$ and $x_+=\max(x,0)$: the first $R$ derivatives
are bounded uniformly, and each further derivative costs a factor $T$.
They are preserved by products, compositions, inverses and flows, with a
fixed loss in $R$ (Section~\ref{app:sec:elementary-interpolation}).

\subsection{Parameters}
The main parameters and fields have the following roles.
\begin{center}
\renewcommand{\arraystretch}{1.15}
\begin{tabular}{@{}p{0.22\textwidth}p{0.70\textwidth}@{}}
\toprule
$F$ & the immersion at the current step; all waves and correction
operators are constructed at $F$\\
$R$ & the frequency parameter of the step; also the derivative scale
of the final modulated amplitudes\\
$\lambda=R^a$ & the frequency of the characteristic waves\\
$h=R^{-\beta}$ & the separation of the samples\\
$K$, $K_{\max}=R^{s_f}$ & the amplitude scale, and the fixed bound for
the initial frequencies of the modulations\\
$T_0=R^t$ & the derivative scale of the immersion $F$\\
$\delta$, $b=\delta R^A$ & $\delta^2$ bounds the metric defect; $b$ is
the amplitude of the primary waves\\
$P$, $V$, $S$ & the primary wave sum, the mean correction field, and
the correction of the oscillatory products\\
$Q$, $r$ & the requested finite accuracy and a derivative order\\
\bottomrule
\end{tabular}
\end{center}
The exponents are fixed in Proposition~\ref{app:prop:ordered-scale-selection}.
In later sections we write $T=T_0$, and $B_{\rm cubic}=B_3$.

\begin{notation}[Scales at one step]\label{app:not:scales}
We use the scales of Section~\ref{main:sec:scales},
\[
 \lambda=R^a,\quad h=R^{-\beta},\quad
 L_F=T_0=R^t,\quad K_{\max}=R^{s_f}.
\]
The unmodulated wave, with initial data $\delta_0$, has amplitudes varying
on the scale $C_{\rm env}T_0$. Waves modulated by measures with Fourier
support in $\{|\xi|\leq CK_{\max}\}$ have amplitudes varying on the scale
$C_*K_{\max}$, and the waves obtained after the correction of the mean
have amplitudes varying on the scale $C_*R$. The two inequalities in
\eqref{main:eq:all-pair-scales} place these scales below the wave
frequency and below the smallest product frequency, respectively. All
exponents and support constants are independent of the accuracy and of
the derivative order.
\end{notation}

\subsection{Fixed domains and cutoffs}
Fix a closed ball $D$ of positive radius centered at $0$ and a larger
bounded open chart $W\subset\R^n$. The limit metric equation is imposed
on $D$. All measurements, sources of the product corrections and physical
output cutoffs are supported in a fixed compact set
$D_{\rm lag}\Subset W$. For the sources of the primary waves we 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 identities for the linearized metric hold near every measurement
support and every source of the product corrections. We write
$\chi_{\rm pair}$ for a fixed cutoff supported in $D_{\rm lag}$ and equal
to one near all product sources, in particular near $D$. The mean and the
product corrections may use different cutoffs, but all their supports
lie, with positive margins, inside the regions where the corresponding
constructions are valid. The larger cutoffs contain the corrections,
which may extend beyond the supports of their sources. All these
domains are fixed before the frequency, the accuracy and the sample are
chosen; this keeps the limit neighborhood independent of the accuracy.
The precise arrangement is as follows.

\begin{setup}[Fixed domains]\label{app:setup:domains}
Let $W\subset\R^n$ be a bounded open neighborhood of $0$. All inclusions
below have positive margins, and all choices are made before the
frequency, the sample, the accuracy and the final dilation parameter.
\begin{enumerate}
\item \emph{Primary waves.} First fix the solution operators for the
primary waves on the whole characteristic patches
(Setup~\ref{app:ch_response:thm:response-compatible-choice}), with a
fixed cutoff $\chi_{2,\rm LAG}$. The finitely many regions on which their
identities hold contain a common open set $U_{\rm LAG}$ on which
$\chi_{2,\rm LAG}=1$.
\item \emph{Measurement region and the ball $D$.} Inside these regions
choose a measurement region $V_{\rm meas}$, small enough for
Proposition~\ref{app:ch_background:prop:bg-frame}, then an open set
$U_{\rm BG}$ and the closed ball $D$, so that
\begin{equation}\label{app:eq:global-domain-chain}
 D\Subset U_{\rm BG}\Subset V_{\rm meas}\Subset\Int D_{\rm lag},
 \qquad \supp\chi_{\rm meas}\Subset V_{\rm meas},
\end{equation}
where $\chi_{\rm meas}$ equals one near $\overline{U_{\rm BG}}$.
\item \emph{Source cutoffs of the primary waves.} Choose
\begin{equation}\label{app:eq:global-cutoff-plateaus}
 \begin{split}
 D_{\rm lag}&\Subset\Int\{\chi_{\lag,\rm src}=1\},\qquad
 \supp\chi_{\lag,\rm src}\Subset
                   \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}
A larger chart in $W$ on which the primary waves are defined contains
$\overline{U_{\rm LAG}}$ and the supports of all these cutoffs.
\item \emph{Product corrections.} Every measurement support, every
source of the product corrections and every physical output support lies
in $\Int D_{\rm lag}$. The physical cutoff $\chi_{\phys}$ and the source
cutoff $\chi_{{\rm pair},\rm src}$ of the product corrections equal one
near $D$; the latter is supported in the region $X_{\rm pair}$ where the
cutoff $\chi_{2,\rm pair}$ of the regular parametrices equals one. The
amplitude cutoff $\chi_{\pair}$ satisfies
$\supp\chi_{{\rm pair},\rm src}\Subset\Int\{\chi_{\pair}=1\}$ and
$\supp\chi_{\pair}\Subset\Int D_{\rm lag}$.
\item \emph{Final cutoffs.} The cutoffs of the mean correction and the
common physical cutoff of the final increment (the sum of the primary
waves, the mean correction and the product correction) are supported
in an open set $U_{\rm fin}$ with $D\Subset U_{\rm fin}\Subset U_{\rm BG}$.
\item \emph{Cutoff for the sampling estimate.} The estimate for the
sampled operators at fixed patch parameter uses a separate cutoff
$\chi_{\rm slice}$, equal to one near $\supp\chi_{\rm meas}$ and supported
in $\Int D_{\rm lag}$. It enters only this estimate; the fields
themselves carry their own physical cutoffs.
\end{enumerate}
In Section~\ref{app:sec:ch_background}, the sets $U_{\rm core}$ and
$U_{\rm meas}$ stand for $U_{\rm BG}$ and $V_{\rm meas}$. In
Section~\ref{main:sec:linear}, $U_{\rm core}$ and the corresponding
cutoff stand for $U_{\rm LAG}$ and $\chi_{{\rm core},\rm LAG}$. The other
solution operators have their own fixed domains of validity.

The linearized metric of the primary waves, the errors of their
corrections and the identities for the linearized metric at $F$ are
estimated on $D_{\rm lag}$, which contains every measurement support and
every source of the product corrections. The smallness of the metric
defect and the smoothing of the target are imposed only on $D$.
\end{setup}

\begin{remark}\label{new:rem:domains}
The primary waves and the product corrections use different solution
operators, each on its own regular patches. The continuous operator and
its image component are constructed in Section~\ref{app:sec:ch_background}.
All samples in one characteristic patch use the same reduced solution
operator and the same reconstruction map $\mathcal V_F$, for every
initial datum, and the finite primary wave and its correction have the
same phase. For each construction the solution operator and its
residual are fixed; a requested precision only determines the number of
compositions of the residual, and the resulting polynomial in the
residual is then applied to every datum in that patch. Over the finite
list of patches we take the minimum of the margins and the maximum of
the derivative losses. Constants may depend on the finite depths, but the
depths do not change the losses.
\end{remark}

%% ===== 06_07_geometry.tex =====
\section{The reduced metric equation and a flat reference}\label{main:sec:geometry}

In this section we reduce the linearized metric equation to a determined
first-order system, introduce the constant-coefficient model, classify
its singular characteristics, and construct the flat reference immersion.

\subsection{The reduced metric equation}
For an immersion $F:W\to\R^N$ write $\gamma(F)=(dF)^TdF$ and
\[
 D\gamma_FY=(dF)^TdY+(dY)^TdF .
\]
Since $\gamma$ is quadratic in $F$, the expansion
\begin{equation}\label{main:eq:metric-expansion}
 \gamma(F+Y)=\gamma(F)+D\gamma_FY+(dY)^TdY
\end{equation}
is exact. Let $\mathcal N_F$ be the normal bundle and $\mathrm{II}_F$ the
second fundamental form, $\mathrm{II}_{F,ij}=(\partial_i\partial_jF)^{\perp}$.
Put $q=N-n$; in the case of interest $N=s_n$ and $q=n(n-1)/2$. We regard
the second fundamental form as the linear map
\[
 B_F:\Sym^2\R^n\longrightarrow\mathcal N_F,
 \qquad B_F(H)=\sum_{i,j}H^{ij}\mathrm{II}_{F,ij},
\]
where $\Sym^2\R^n$ carries the Frobenius inner product
$H:K=\sum_{i,j}H^{ij}K_{ij}$. Its adjoint is
$B_F^*w=(\langle w,\mathrm{II}_{F,ij}\rangle)_{i,j}$. We say that $F$ has
\emph{full normal rank} if $B_F$ is onto at every point. In that case
$B_F^*$ is injective and the kernel
\[
 K_F=\ker B_F=(\operatorname{im}B_F^*)^\perp
\]
has dimension $s_n-q=n$.

\begin{lem}\label{new:lem:tangent-normal}
Let $g=\gamma(F)$, let $\nabla$ be its Levi-Civita connection, and write
a variation as $Y=dF\,g^{-1}u+w$, where $u$ is a covector field and $w$ is
a normal field. Then
\begin{equation}\label{main:eq:tangent-normal}
 D\gamma_FY=2\sym\nabla u-2\langle w,\mathrm{II}_F\rangle
 =2\sym\nabla u-2B_F^*w,
\end{equation}
where $\sym A=(A+A^T)/2$.
\end{lem}

\begin{proof}
Put $X^k=g^{kl}u_l$, so that the tangential part of $Y$ is
$X^k\partial_kF$, and write $\Gamma_{jk,i}=\partial_iF\cdot\partial_j\partial_kF$
for the Christoffel symbols of the first kind of $g$. Then
\[
 \partial_iF\cdot\partial_j(X^k\partial_kF)
 =g_{ik}\partial_jX^k+X^k\Gamma_{jk,i}
 =\partial_ju_i-X^k\partial_jg_{ik}+X^k\Gamma_{jk,i}.
\]
Since $\partial_jg_{ik}=\Gamma_{ji,k}+\Gamma_{jk,i}$, the right side equals
$\partial_ju_i-X^k\Gamma_{ji,k}=\partial_ju_i-\Gamma_{ji}^lu_l=(\nabla_ju)_i$.
For the normal part, differentiating $\partial_iF\cdot w=0$ gives
$\partial_iF\cdot\partial_jw=-\partial_i\partial_jF\cdot w
=-\langle w,\mathrm{II}_{F,ij}\rangle$. Adding the $(i,j)$ and $(j,i)$
entries gives \eqref{main:eq:tangent-normal}.
\end{proof}

Note that $\nabla$ is the connection of the induced metric $\gamma(F)$,
not of the target metric; the two agree only in the limit of the
iteration.

\begin{defn}[The reduced operator]\label{new:defn:reduced}
Let $F$ have full normal rank, and let $Q_1(F),\dots,Q_n(F)$ be a frame of
$K_F$ with Gram matrix $\mathsf G_F=(Q_i(F):Q_j(F))_{i,j}$ uniformly
positive definite. (Later the frame is obtained by orthogonal projection
of fixed reference tensors, $Q_i(F)=\Pi_{K_F}Q_i^{\rm ref}$.) Define
\begin{equation}\label{main:eq:reduced-convention}
 \begin{gathered}
 q_F(h)=(Q_i(F):h)_{i=1}^n,\qquad
 \cP_Fu=(Q_i(F):\sym\nabla u)_{i=1}^n,\\
 M_F(x,\xi)_{i\cdot}=(Q_i(F)\xi)^T,\qquad
 p_F(x,\xi)=\det M_F(x,\xi).
 \end{gathered}
\end{equation}
Thus $M_F(x,\xi)$ is an $n\times n$ matrix whose $i$th row is the covector
$Q_i(F)\xi$. We also write $q_F(x,\xi)=q_F(\xi\otimes\xi)$.
\end{defn}

The operator $\cP_F$ is a first-order system with principal symbol
$\ii M_F(x,\xi)$ under $\partial_j\mapsto\ii\xi_j$; its zero-order part is
$-\bigl(\sum_{j,l}Q_i(F)^{jl}\Gamma_{jl}^ku_k\bigr)_i$. Applying $q_F$ to
\eqref{main:eq:tangent-normal} and using $Q_i(F)\in K_F\perp\operatorname{im}B_F^*$,
the normal term disappears, and the equation $D\gamma_FY=h$ implies the
\emph{reduced equation}
\begin{equation}\label{main:eq:reduced-equation}
 \cP_Fu=\tfrac12q_F(h).
\end{equation}
The factor $1/2$ is kept in the same position in the equation for the mean.
Conversely, a solution of the reduced equation can be completed to a
solution of the full equation. Since $B_F$ is onto, $B_F^*$ maps
$\mathcal N_F$ isomorphically onto $K_F^\perp$. Put
$N_F=(B_F^*)^{-1}\Pi_{K_F^\perp}$ and, for given $u$ and $h$,
\begin{equation}\label{main:eq:normal-completion}
 w=N_F\bigl(\sym\nabla u-\tfrac12h\bigr).
\end{equation}

\begin{lem}\label{new:lem:completion}
For $Y=dF\,\gamma(F)^{-1}u+w$ with $w$ given by
\eqref{main:eq:normal-completion},
\begin{equation}\label{main:eq:completion-error}
 D\gamma_FY-h
 =2\sum_{i,j}Q_i(F)(\mathsf G_F^{-1})_{ij}
                  \bigl(\cP_Fu-\tfrac12q_F(h)\bigr)_j.
\end{equation}
In particular, a solution of the reduced equation gives a solution of
$D\gamma_FY=h$, and the error of an approximate solution is the displayed
tensor in $K_F$.
\end{lem}

\begin{proof}
By \eqref{main:eq:tangent-normal} and $B_F^*N_F=\Pi_{K_F^\perp}$,
\[
 D\gamma_FY-h=2\bigl(\sym\nabla u-\tfrac12h\bigr)
 -2\Pi_{K_F^\perp}\bigl(\sym\nabla u-\tfrac12h\bigr)
 =2\Pi_{K_F}\bigl(\sym\nabla u-\tfrac12h\bigr).
\]
For any frame of $K_F$ the orthogonal projection is
$\Pi_{K_F}X=\sum_{i,j}Q_i(F)(\mathsf G_F^{-1})_{ij}(Q_j(F):X)$: the right side
lies in $K_F$, and pairing it with $Q_l(F)$ gives
$\sum_{i,j}(\mathsf G_F)_{li}(\mathsf G_F^{-1})_{ij}(Q_j:X)=Q_l:X$. Since
$Q_j(F):(\sym\nabla u-\tfrac12h)=(\cP_Fu-\tfrac12q_F(h))_j$, the formula
follows.
\end{proof}

For $h=0$ we obtain the differential lift
\begin{equation}\label{main:eq:stress-lift}
 \begin{aligned}
 \mathcal V_F(u)&=dF\,\gamma(F)^{-1}u+N_F\sym\nabla u,\\
 D\gamma_F\mathcal V_F(u)
 &=2\sum_{i,j}Q_i(F)(\mathsf G_F^{-1})_{ij}(\cP_Fu)_j .
 \end{aligned}
\end{equation}
Its linearized metric change lies in $K_F$ and vanishes exactly when
$\cP_Fu=0$. The lift $\mathcal V_F$ is used for the characteristic waves,
and the normal components of the mean and product corrections are
obtained from \eqref{main:eq:normal-completion}.

\subsection{The constant-coefficient model}

\begin{defn}[The model tensors]\label{app:ch_seed:defn:seed-cube}
Let $n\geq5$, $N=s_n=n(n+1)/2$, $q=n(n-1)/2$, $d=n-4$ and $c_n=d^{-2}$.
Let $E=\Sym^2\R^n$ carry the Frobenius pairing, and write
$S_z=\sum_iz_i$ for $z\in\R^n$, $\mathbf1=(1,\dots,1)$ and
$J=\mathbf1\mathbf1^T$. Set
\begin{equation}\label{main:eq:cube-stress}
 Q_i^0=e_ie_i^T-c_nJ,\qquad
 K_0=\operatorname{span}\{Q_1^0,\ldots,Q_n^0\},\qquad
 \ell_{ab}=\mathbf1-2e_a-2e_b\quad(a<b).
\end{equation}
With the normalization of \eqref{main:eq:reduced-convention}, the
corresponding pencil, its determinant and its quadratic map are
\begin{equation}\label{main:eq:pencil}
 \begin{gathered}
 M_0(\xi)=\diag(\xi_1,\ldots,\xi_n)-c_nS_\xi J,\qquad
 p_0(\xi)=\det M_0(\xi),\\
 q_0(\xi)=(\xi^TQ_i^0\xi)_i=M_0(\xi)\xi.
 \end{gathered}
\end{equation}
Explicitly, $q_0(\xi)_i=\xi_i^2-c_nS_\xi^2$. Note that $M_0(\xi)$ is
symmetric and linear in $\xi$, and that $r^TM_0(k)r=k\cdot q_0(r)$.
\end{defn}

By the matrix determinant lemma,
$\det(\diag(\xi)-c_nS_\xi\mathbf1\mathbf1^T)
=\det\diag(\xi)\,(1-c_nS_\xi\mathbf1^T\diag(\xi)^{-1}\mathbf1)$
when all $\xi_i\neq0$, and by continuity
\begin{equation}\label{main:eq:determinant}
 p_0(\xi)=\prod_i\xi_i
 -c_nS_\xi\sum_i\prod_{j\ne i}\xi_j .
\end{equation}

\begin{prop}[A flat form of full rank]\label{app:ch_seed:prop:seed-gram}
The $\R^q$-valued symmetric form
\[
 B_0(u,v)=\bigl((\ell_{ab}\cdot u)(\ell_{ab}\cdot v)\bigr)_{a<b}
\]
is onto, its kernel, as a map $E\to\R^q$, is $K_0$, and it satisfies the
Gauss equations of a flat metric:
$\langle B_0(X,Z),B_0(Y,W)\rangle-\langle B_0(X,W),B_0(Y,Z)\rangle=0$.
\end{prop}

\begin{proof}
As a map on $E$, $B_0(H)=(\ell_{ab}^TH\ell_{ab})_{a<b}$, so
$B_0^*$ maps the basis vector indexed by $(a,b)$ to $\ell_{ab}\ell_{ab}^T$.
The $q$ matrices $(e_a+e_b)(e_a+e_b)^T$, $a<b$, are linearly independent,
since only the one indexed by $(a,b)$ has a nonzero $(a,b)$ entry. Moreover
\begin{equation}\label{app:ch_seed:eq:seed-det}
 \ell_{ab}=-2\Bigl(I-\frac J4\Bigr)(e_a+e_b),
 \qquad \det\Bigl(I-\frac J4\Bigr)=1-\frac n4\ne0,
\end{equation}
so the matrices $\ell_{ab}\ell_{ab}^T=4L(e_a+e_b)(e_a+e_b)^TL^T$, with
$L=I-J/4$ invertible, are independent as well. Hence $B_0^*$ is
injective and $B_0$ is onto. Each $Q_i^0$ annihilates these matrices:
every coordinate of $\ell_{ab}$ is $\pm1$ and $S_{\ell_{ab}}=n-4=d$, so
\[
 Q_i^0:\ell_{ab}\ell_{ab}^T=(\ell_{ab})_i^2-c_nS_{\ell_{ab}}^2=1-c_nd^2=0 .
\]
The $Q_i^0$ are independent: if $\sum_i\alpha_iQ_i^0=\diag(\alpha)-c_n(S_\alpha)J=0$,
an off-diagonal entry gives $S_\alpha=0$ and then $\alpha=0$. Since
$\dim\ker B_0=s_n-q=n$ and $K_0\subset\ker B_0$, we get $\ker B_0=K_0$.
Finally, every component of $B_0$ is a product of two linear forms, and
for such a component the Gauss expression is
$(\ell\cdot X)(\ell\cdot Z)(\ell\cdot Y)(\ell\cdot W)
-(\ell\cdot X)(\ell\cdot W)(\ell\cdot Y)(\ell\cdot Z)=0$.
\end{proof}

\subsection{The singular characteristic set}

\begin{prop}[Singular characteristics of the model]\label{app:ch_seed:prop:seed-strata}
The singular characteristic set
$\mathfrak M=\{p_0=0,\ dp_0=0\}\setminus\{0\}$ is the union, with the
origin removed, of
\begin{equation}\label{main:eq:multiple-set}
 L_{ijl}=\{\xi_i=\xi_j=\xi_l=0\},\qquad
 L_{ijS}=\{\xi_i=\xi_j=S_\xi=0\},\qquad
 \R\ell_{ab},
\end{equation}
for distinct indices. The first two families consist of the covectors at
which $M_0$ has corank at least two; they have codimension three. Each
line $\R\ell_{ab}$ consists of covectors of corank one with
$dp_0=0$; every other zero of $p_0$ of corank one is a simple
characteristic. The lines $\R\ell_{ab}$ are also exactly the nonzero
common zeros of the components of $q_0$. We write $\mathcal L$ for the
finite family of the subspaces in \eqref{main:eq:multiple-set}, so that
$\mathfrak M=(\bigcup_{L\in\mathcal L}L)\setminus\{0\}$.
\end{prop}

\begin{proof}
\emph{The kernel of $M_0(\xi)$.} Put $\alpha=c_nS_\xi$; the kernel
equation reads
\begin{equation}\label{new:eq:model-kernel}
 \xi_iv_i=\alpha S_v\qquad(1\leq i\leq n).
\end{equation}
Let $Z=\{i:\xi_i=0\}$. If $Z=\emptyset$, then $v_i=\alpha S_v/\xi_i$ is
determined by the scalar $S_v$, so the kernel has dimension at most one.
If $Z\neq\emptyset$ and $\alpha\neq0$, an index in $Z$ gives $S_v=0$, so
$v$ is supported in $Z$ with $S_v=0$, and the kernel has dimension
$|Z|-1$. If $\alpha=0$, the kernel consists of the vectors supported in
$Z$ and has dimension $|Z|$. Thus the corank is at least two exactly on
$\{|Z|\geq3\}\cup\{|Z|\geq2,\ S_\xi=0\}$, which is the union of the spaces
$L_{ijl}$ and $L_{ijS}$; each is defined by three independent linear
equations. At such points the adjugate of $M_0$ vanishes, and since
$dp_0(\xi)[\eta]=\operatorname{tr}(\operatorname{adj}M_0(\xi)\,M_0(\eta))$,
also $dp_0(\xi)=0$.

\emph{Corank one.} If $M_0(\xi)$ has corank one and $v$ spans its kernel,
then, $M_0$ being symmetric, $\operatorname{adj}M_0(\xi)=a\,vv^T$ with
$a\neq0$. Hence
\begin{equation}\label{new:eq:dp0}
 dp_0(\xi)[\eta]=a\,v^TM_0(\eta)v=a\,\eta\cdot q_0(v),
 \qquad\text{that is}\qquad dp_0(\xi)=a\,q_0(v).
\end{equation}
So $dp_0(\xi)=0$ if and only if $q_0(v)=0$.

\emph{Common zeros of $q_0$.} Let $v\neq0$ with $q_0(v)=0$, that is,
$v_i^2=c_nS_v^2$ for all $i$. If $S_v=0$ then $v=0$; hence $S_v\neq0$ and
$v_i=\varepsilon_iS_v/d$ with $\varepsilon_i=\pm1$. Summing over $i$ gives
$\sum_i\varepsilon_i=d=n-4$, so exactly two signs are negative, and
$v\in\R\ell_{ab}$. Conversely $q_0(\ell_{ab})=0$, as computed in the proof
of Proposition~\ref{app:ch_seed:prop:seed-gram}.

\emph{Conclusion.} Let $\xi\neq0$ have corank one and $dp_0(\xi)=0$. Then
its kernel is spanned by some $\ell=\ell_{ab}$, and
\eqref{new:eq:model-kernel} gives $\xi_i\ell_i=c_nS_\xi d=S_\xi/d$,
that is, $\xi_i=(S_\xi/d)\ell_i$, since $\ell_i=\pm1$. Thus
$\xi\in\R\ell_{ab}$. Conversely, for $\xi=\ell_{ab}$ all coordinates are
nonzero, so the corank is at most one; $M_0(\ell)\ell=q_0(\ell)=0$, so it
equals one, and \eqref{new:eq:dp0} gives $dp_0(\ell)=0$.
\end{proof}

\subsection{An explicit analytic flat reference}

\begin{prop}[An explicit analytic flat reference]\label{app:ch_seed:prop:seed-analytic}
There is a real-analytic embedding $F_0:B_r\to\R^N$ of a small ball such
that
\[
 F_0(0)=0,\qquad dF_0(0)=(I,0),\qquad
 \gamma(F_0)=I,\qquad B_{F_0}(0)=B_0 ,
\]
where the normal space at $0$ is identified with the last $q$
coordinates. In particular $F_0$ has full normal rank near $0$ and
$K_{F_0}(0)=K_0$.
\end{prop}

\begin{proof}
For a unit vector $v\in\R^m$ and $Y\in\R^m$, put $t=v\cdot Y$ and define
the circular bending
\[
 \mathcal C_v(Y)=
 \Bigl(Y+\Bigl[\frac{\sin(nt)}n-t\Bigr]v,\;
             \frac{1-\cos(nt)}n\Bigr)\in\R^{m+1}.
\]
Its derivative in a direction $X$ is
\[
 d\mathcal C_v(Y)X=\bigl(X+(\cos(nt)-1)(v\cdot X)\,v,\;\sin(nt)(v\cdot X)\bigr).
\]
It maps $X\perp v$ to $(X,0)$ and $v$ to the unit vector
$(\cos(nt)v,\sin(nt))$, which is orthogonal to $(v^\perp,0)$; hence
$d\mathcal C_v(Y)$ is a linear isometry. At $Y=0$ it is the standard
inclusion $X\mapsto(X,0)$. Differentiating once more and setting $t=0$,
\[
 d^2\mathcal C_v(0)[X,Z]=\bigl(0,\;n(v\cdot X)(v\cdot Z)\bigr),
\]
that is, the second derivative at $0$ is $n\,v\otimes v$ in the new
coordinate direction.

Order the $q$ pairs $a<b$ and put $F^{(0)}=\Id_{\R^n}$. If
$F^{(k-1)}:\R^n\supset B_r\to\R^{n+k-1}$ is defined, let $(a,b)$ be the
$k$th pair, $v=(\ell_{ab}/\sqrt n,0)\in\R^{n+k-1}$ (a unit vector, since
$|\ell_{ab}|^2=n$), and $F^{(k)}=\mathcal C_v\circ F^{(k-1)}$. All these
maps send $0$ to $0$ and have differential $(I,0)$ at $0$, and they are
isometric because each $d\mathcal C_v$ is an isometry. By the chain rule,
\[
 d^2F^{(k)}(0)[X,Z]=d\mathcal C_v(0)\bigl[d^2F^{(k-1)}(0)[X,Z]\bigr]
 +d^2\mathcal C_v(0)\bigl[dF^{(k-1)}(0)X,dF^{(k-1)}(0)Z\bigr].
\]
The first term is $d^2F^{(k-1)}(0)[X,Z]$ followed by a zero, and the
second is $(0,(\ell_{ab}\cdot X)(\ell_{ab}\cdot Z))$. By induction,
$d^2F^{(k)}(0)$ has vanishing tangential part, and its components in the
coordinates $n+1,\dots,n+k$ are $(\ell\cdot X)(\ell\cdot Z)$ for the first
$k$ pairs. For $F_0=F^{(q)}$, with values in $\R^{n+q}=\R^N$, this says
that the second fundamental form at $0$ is $B_0$. Proposition~\ref{app:ch_seed:prop:seed-gram}
shows that $B_0$ is onto with kernel $K_0$; full normal rank is an open
condition and persists near $0$. Finally, the projection of $F_0$ onto
the first $n$ coordinates has differential $I$ at $0$, so it is a local
diffeomorphism, and $F_0$ is an embedding after shrinking the ball.
\end{proof}

\begin{cor}[Rescaled flat references]
\label{app:ch_seed:cor:seed-uniform-dilation}
For all sufficiently small $\sigma>0$ define, on a fixed ball,
\[
 G_\sigma(x)=\sigma^{-1}\big(F_0(\sigma x)-F_0(0)\big).
\]
Then $\gamma(G_\sigma)=I$, the smallest singular value of $B_{G_\sigma}$ is
at least $c\sigma$, and the projected frame
$Q_i(G_\sigma)=\Pi_{K_{G_\sigma}}Q_i^0$ converges to $(Q_i^0)$ in $C^m$ for
every fixed $m\geq0$ as $\sigma\to0$. Consequently, one fixed sufficiently
small $\rho>0$ gives a reference immersion $G_{\mathrm{ref}}=G_\rho$ whose
projected frame lies in any prescribed finite-order neighborhood of the
constant frame. The metric, normal-rank and frame bounds persist on a
fixed finite-order neighborhood of $G_{\mathrm{ref}}$. All geometric
seminorms of $G_{\rm ref}$ are finite; later constants may depend on $\rho$.
\end{cor}

\begin{proof}
Let $Q_i(y)=\Pi_{\ker B_{F_0}(y)}Q_i^0$, which is smooth near $0$ with
$Q_i(0)=Q_i^0$. Since $dG_\sigma(x)=dF_0(\sigma x)$ and
$d^2G_\sigma(x)=\sigma\,d^2F_0(\sigma x)$, the tangent and normal spaces of
$G_\sigma$ at $x$ coincide with those of $F_0$ at $\sigma x$, and
\[
 B_{G_\sigma}(x)=\sigma B_{F_0}(\sigma x),\qquad
 Q_i(G_\sigma)(x)=Q_i(\sigma x).
\]
The second identity gives $\partial^\alpha[Q_i(G_\sigma)](x)
=\sigma^{|\alpha|}(\partial^\alpha Q_i)(\sigma x)$, which tends to $0$ for
$|\alpha|\geq1$, while $Q_i(\sigma x)\to Q_i^0$; this is the asserted
convergence. The first identity and the full normal rank of $F_0$ near
$0$ give the singular-value bound, and flatness gives $\gamma(G_\sigma)=I$.
The frame depends smoothly on the second jet of the immersion, so all
these bounds persist on a neighborhood in a topology with two more
derivatives than the prescribed frame topology.
\end{proof}

After this fixed dilation, the projected frames of all immersions near
$G_{\rm ref}$ are uniformly close to \eqref{main:eq:cube-stress}, and all
later operators use these variable coefficients.

\section{Characteristic patches and phases}\label{app:sec:ch_seed}

We now construct the finite family of characteristic patches which
provides the oscillations, first for the constant model and then for
immersions near the reference, together with their phases. The main
results are the Gram inequality
(Proposition~\ref{app:ch_seed:prop:seed-frozen-menu}), the
classification of all quadratic interactions as elliptic or of simple
real principal type (Propositions~\ref{app:ch_seed:prop:seed-actual-menu}
and~\ref{main:prop:patches}), and the properties of the phase
differences needed for the oscillatory integrals
(Proposition~\ref{app:ch_seed:prop:seed-phase-charts}). The construction
of phases on common domains uses the elementary estimates for flows and
inverse charts of Section~\ref{app:sec:elementary-interpolation}, which
are independent of the present section.

\subsection{Heuristics}
At a simple characteristic covector $\xi$ the matrix $M_F(x,\xi)$ has a
one-dimensional kernel, which determines the leading tangential
polarization of a wave with phase gradient $\xi$. After normal
reconstruction by \eqref{main:eq:stress-lift}, the leading part of the
differential of such a wave is a rank-one matrix $\nu\otimes\xi$, and its
Hermitian square is $|\nu|^2\xi\otimes\xi$. Applying $q_F$ gives the vector
$|\nu|^2q_F(\xi\otimes\xi)$. The waves are also slowly modulated; the
quadratic interaction between a wave with base covector $\xi_a$ and its
modulation has, to leading order, an output frequency $k$ tangent to the
characteristic variety at $\xi_a$. For the constant model these
considerations lead to the following Gram inequality, which states that
the quadratic vectors complete the image of $M_0$ at every output
direction: for each unit $\widehat k$ there are selected base covectors
$\xi_a$ with
\begin{equation}\label{main:eq:detector-gram}
 \begin{gathered}
 \widehat k\cdot\partial_\xi p_0(\xi_a)=0,\\
 |M_0(\widehat k)^Tw|^2+
 \sum_a|w\cdot q_0(\xi_a)|^2\geq c|w|^2
       \qquad(w\in\R^n).
 \end{gathered}
\end{equation}
This holds even at singular output covectors. It is proved in
Proposition~\ref{app:ch_seed:prop:seed-frozen-menu}, and the corresponding
lower bound for the continuous normal operator is proved in
Section~\ref{app:sec:ch_background}. The quadratic products of distinct
waves leave the characteristic patch; their corrections require that
their frequencies lie in regions where the reduced system is elliptic or
of simple real principal type. This is the second main property of the
construction.

\subsection{Characteristic patches for the constant model}
\label{app:ch_seed:sec:seed-frozen}

\begin{prop}[A finite family of characteristic patches]\label{app:ch_seed:prop:seed-frozen-menu}
Let $n\geq5$ and use the pencil of Definition~\ref{app:ch_seed:defn:seed-cube}.
There are finitely many smooth maps
\[
 \xi_a\colon Z_a\longrightarrow\R^n,
\]
where each $Z_a$ is a closed convex cube in $\R^3$, and a finite open
cover $(V_b)$ of the unit sphere $\mathbb S^{n-1}$, with the following
properties.
\begin{enumerate}
\item Every $\xi_a(z)$ has all coordinates nonzero, satisfies
$p_0(\xi_a(z))=0$ and $\rank M_0(\xi_a(z))=n-1$, and its norm is bounded
above and below. With $r_{a,i}=1/\xi_{a,i}$, the vector $v_a=q_0(r_a)$ is
a nonzero multiple of $dp_0(\xi_a)$. For every $L\in\mathcal L$ and all
$z,z'\in Z_a$,
\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}
The normalized directions of all nonzero differences and of all sums of
two covectors from the same patch lie at a uniform positive distance
from $\mathfrak M$.
\item For each $V_b$ there are a finite subcollection of patches and
smooth incidence points $z_a(k)$ in their interiors, $k\in V_b$, such that,
for one chosen first parameter $z_1$ on each patch,
\[
 k\cdot v_a(z_a(k))=0,\qquad
 |\partial_{z_1}(k\cdot v_a(z))|\geq c
\]
near the incidence points, and
\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 of the patches, positive radial
factors $R_g$ can be fixed so that every sum or difference
$R_g\xi_a(z)\pm R_{g'}\xi_{a'}(z')$ of covectors from different groups
has normalized direction at a uniform positive distance from
$\mathfrak M$. The factors do not depend on the later sample spacing or
accuracy.
\end{enumerate}
\end{prop}

Since $dp_0(\xi_a)$ is proportional to $v_a$, the incidence condition in
(ii) is the first line of \eqref{main:eq:detector-gram}, and
\eqref{app:ch_seed:eq:seed-frozen-detection} is its second line.

\begin{proof}
\emph{Step 1 (base covectors satisfying the Gram condition).}
Fix $k\neq0$ with $M_0(k)$ singular and a nonzero $s\in\ker M_0(k)$. Put
$\alpha=c_nS_k$ and
\[
 Q(r)=r^TM_0(k)r=k\cdot q_0(r)=\sum_lk_lr_l^2-\alpha S_r^2 .
\]
We first find two indices $i\neq j$ with $s_i+s_j\neq0$ and a smooth zero
of $Q$ in the subspace $\{r_i=r_j=0\}$ with $S_r\neq0$ and all other
coordinates nonzero.

\emph{Case A: some coordinate of $k$ vanishes.} The kernel equation
$k_ls_l=\alpha S_s$ shows that $s$ is supported on the zero coordinates of
$k$, and if $\alpha\neq0$ also $S_s=0$. If $\alpha=0$, choose $p,q$ with
$k_p>0>k_q$ (possible since $S_k=0$ and $k\neq0$). If $\alpha\neq0$, choose
$p$ with $\alpha k_p>0$ and a zero coordinate $q$ of $k$. In both cases
$s$ does not vanish identically on the remaining $n-2\geq3$ indices: if
$\alpha=0$, $s_p=s_q=0$; if $\alpha\neq0$, $s_p=0$ and $s$ has at least two
nonzero entries because $S_s=0$. Among three indices on which $s$ is not
identically zero, two satisfy $s_i+s_j\neq0$: if $s_i\neq0$ and
$s_i+s_j=s_i+s_{j'}=0$, then $s_j+s_{j'}=-2s_i\neq0$. Fix such $i,j$.
If $\alpha=0$, put $r_l=1$ for $l\notin\{i,j,p,q\}$ and choose
$r_p,r_q>0$ with $k_pr_p^2+k_qr_q^2=-\sum_{l\notin\{i,j,p,q\}}k_l$, which
is possible since $k_p>0>k_q$. Then $Q(r)=0$, $S_r>0$, and
$\partial_{r_p}Q=2k_pr_p\neq0$. If $\alpha\neq0$, choose nonzero $r_l$ for
$l\notin\{i,j,q\}$ such that $\sum_lk_lr_l^2$ has the sign of $\alpha$
(take $r_p$ large), and then choose $r_q$ with
$S_r=\pm(\sum_lk_lr_l^2/\alpha)^{1/2}$; of the two signs at least one
gives $r_q\neq0$. Since $k_q=0$, $Q(r)=0$, $S_r\neq0$, and
$\partial_{r_q}Q=-2\alpha S_r\neq0$.

\emph{Case B: all $k_i\neq0$.} The kernel equation gives
$s_i=\alpha S_s/k_i$ with $\alpha S_s\neq0$, and summing,
$\alpha\sum_ik_i^{-1}=1$. Replacing $k$ by $-k$ (which changes $M_0(k)$
to $-M_0(k)$) we may assume $\alpha>0$. Let $p$ and $q$ be the numbers of
positive and negative coordinates of $k$. The form $Q$ is indefinite:
each $\ell_{ab}$ is a null vector because $q_0(\ell_{ab})=0$, these
vectors span $\R^n$ by \eqref{app:ch_seed:eq:seed-det}, and a
semidefinite form vanishes on its null vectors' span, whereas
$M_0(k)\neq0$. To compute the inertia, consider the bordered matrix
\[
 \mathcal B=\begin{pmatrix}\diag(k)&\mathbf1\\ \mathbf1^T&\alpha^{-1}\end{pmatrix}.
\]
Its Schur complement with respect to the entry $\alpha^{-1}$ is
$\diag(k)-\alpha\mathbf1\mathbf1^T=M_0(k)$, and with respect to
$\diag(k)$ it is $\alpha^{-1}-\sum_lk_l^{-1}=0$. By the Haynsworth
inertia additivity, $\operatorname{In}(\mathcal B)=(p,q,0)+(0,0,1)
=(1,0,0)+\operatorname{In}M_0(k)$, so $\operatorname{In}M_0(k)=(p-1,q,1)$.
Indefiniteness gives $p\geq2$ and $q\geq1$. If $p\geq3$, let $i,j$ be two
positive coordinates; if $p=2$, then $q=n-2\geq3$, and let $i,j$ be two
negative coordinates. In both cases $s_i+s_j=\alpha S_s(k_i^{-1}+k_j^{-1})\neq0$.
The restriction of $Q$ to $\{r_i=r_j=0\}$ is
$\diag(k')-\alpha\mathbf1\mathbf1^T$, with $k'$ the remaining coordinates,
and the same bordered matrix argument, now with Schur complement
$\alpha^{-1}-\sum_{l\neq i,j}k_l^{-1}=k_i^{-1}+k_j^{-1}\neq0$, shows that
it is nonsingular with inertia $(p-2,q,0)$ or $(1,q-1,0)$, respectively.
It is therefore a nonsingular indefinite form in $n-2\geq3$ variables,
and its null cone is a smooth hypersurface of dimension $n-3$ away from
the origin. No open subset of this cone lies in a hyperplane
$\{\lambda=0\}$: otherwise $Q$ would vanish on an open subset of the
hyperplane, hence on the hyperplane, and $\lambda$ would divide $Q$,
forcing $\rank Q\leq2$. Hence we can choose a point of the cone with
$S_r\neq0$ and all remaining coordinates nonzero.

\emph{Excluding further degeneracies.} In both cases the zero set of $Q$
in $\{r_i=r_j=0\}$ is, near the chosen point, a smooth hypersurface of
dimension $n-3$, and $q_0(r)_i=q_0(r)_j=-c_nS_r^2\neq0$. The conditions
that two other components $q_0(r)_l,q_0(r)_m$ vanish, or that three
components $q_0(r)_l,q_0(r)_m,q_0(r)_t$ are equal, read
\[
 r_l=\varepsilon_lS_r/d,\quad r_m=\varepsilon_mS_r/d,
 \qquad\text{or}\qquad
 r_l=\varepsilon_lr_t,\quad r_m=\varepsilon_mr_t,
\]
with indices distinct and outside $\{i,j\}$ and signs $\pm1$; each is a
pair of independent linear equations, hence defines a subspace of
codimension two. (A coincidence involving $q_0(r)_i$ would force
$r_l=0$ for some $l\notin\{i,j\}$, and $q_0(r)_i$ never vanishes.) A
hypersurface of dimension $n-3$ is not contained in finitely many
subspaces of dimension $n-4$, so we may avoid all of them while keeping
the strict conditions above. Now a vector in
$L_{ijl}^\perp=\operatorname{span}\{e_i,e_j,e_l\}$ has at least $n-3\geq2$
zero components, and a vector in $L_{ijS}^\perp=\operatorname{span}\{e_i,e_j,\mathbf1\}$
has at least $n-2\geq3$ equal components. Hence $q_0(r)\notin L^\perp$ for
every codimension-three $L\in\mathcal L$.

\emph{Passing to the base covector.} At this point $r$ has exactly two
zero coordinates and $S_r\neq0$, so by the proof of
Proposition~\ref{app:ch_seed:prop:seed-strata} the kernel of $M_0(r)$ is
$\R(e_i-e_j)$. By \eqref{new:eq:dp0}, $dp_0(r)$ is a nonzero multiple of
$q_0(e_i-e_j)=e_i+e_j$, so $dp_0(r)\cdot s\neq0$. On the other hand
$dQ(r)=2M_0(k)r$ is nonzero and annihilates $s$, because
$r^TM_0(k)s=0$. Hence $dp_0(r)$ and $dQ(r)$ are independent, and
$\{p_0=0,\ Q=0\}$ is a smooth submanifold of codimension two near $r$
(note $p_0(r)=0$ since $M_0(r)$ is singular). Its tangent space
contains $e_i-e_j$: indeed $dp_0(r)(e_i-e_j)=0$, and
$dQ(r)(e_i-e_j)=2(k_ir_i-\alpha S_r)-2(k_jr_j-\alpha S_r)=0$. Moving along a
curve in this submanifold with tangent $e_i-e_j$ makes $r_i$ and $r_j$
nonzero while preserving all the open conditions obtained so far. For
such a point put $\xi_l=r_l^{-1}$. Since all $r_l\neq0$,
\eqref{main:eq:determinant} gives $p_0(r)=\prod_lr_l\,(1-c_nS_rS_\xi)=0$,
so $S_rS_\xi=d^2$, and by symmetry $p_0(\xi)=0$. The kernel of $M_0(r)$ is
spanned by $(1/r_l)_l=\xi$ and the kernel of $M_0(\xi)$ by $r$, by
\eqref{new:eq:model-kernel}; hence, by \eqref{new:eq:dp0},
$dp_0(r)\parallel q_0(\xi)$ and $dp_0(\xi)\parallel q_0(r)$. By
continuity $s\cdot q_0(\xi)\neq0$, and $q_0(r)\neq0$; thus both
covectors are simple characteristics, and
\[
 k\cdot q_0(r)=Q(r)=0,\qquad s\cdot q_0(\xi)\neq0 .
\]
The first identity is the incidence condition
$k\cdot dp_0(\xi)=0$. The map $\xi\mapsto(1/\xi_l)_l$ is a diffeomorphism
of the set of characteristic covectors with nonzero coordinates onto
itself, so it also preserves the transversality of the incidence
condition, which at the level of $r$ is $dQ(r)\neq0$ along
$\{p_0=0\}$.

\emph{Step 2 (three-parameter patches).} Let $c$ be a base covector from
Step~1 for the output covector $k_0$, put $v=q_0(c^{-1})$, which is
normal to the characteristic variety at $c$, and let $T=v^\perp$ be its
tangent space. For a codimension-three $L\in\mathcal L$ the projection
$T\to L^\perp$ is onto: its kernel $T\cap L$ has dimension
$(n-1)+(n-3)-n=n-4$, because $v\notin L^\perp$ means $L\not\subset T$. For
a line $L=\R\ell_{ab}$ the projection has rank at least $n-2\geq3$. We
choose a linear map $E_3\colon\R^3\to T$ such that every
$\pi_{L^\perp}E_3$ has rank three and $d(k_0\cdot v)E_3e_1\neq0$, where
$d(k_0\cdot v)$ denotes the differential along the characteristic variety
of $\xi\mapsto k_0\cdot q_0(\xi^{-1})$, which is nonzero by Step~1. Each of
these conditions is the nonvanishing of a polynomial in the entries of
$E_3$ (a sum of squares of $3\times3$ minors, or a linear form), and each
polynomial is not identically zero by the rank statements just proved;
hence their product is not identically zero, and a suitable $E_3$
exists.

Choose $e$ with $dp_0(c)e\neq0$. By the implicit function theorem the
characteristic variety near $c$ is parameterized as
\[
 \xi(z)=c+E_3z+u(z)e,\qquad u(0)=0,\quad Du(0)=0,\quad D\xi(0)=E_3,
\]
for $z$ near $0$ in $\R^3$ (fixing the remaining $n-4$ tangent
coordinates). Shrink to a convex cube on which
$\|\pi_{L^\perp}(D\xi(z)-E_3)\|$ is at most half the smallest singular
value $\sigma_L$ of $\pi_{L^\perp}E_3$, for every $L\in\mathcal L$. For $z,z'$
in the cube,
\[
 \pi_{L^\perp}(\xi(z)-\xi(z'))
 =\pi_{L^\perp}E_3(z-z')+\int_0^1\pi_{L^\perp}\bigl(D\xi(z'+\theta(z-z'))-E_3\bigr)(z-z')\dd\theta ,
\]
whose norm is at least $\sigma_L|z-z'|/2$. This proves
\eqref{app:ch_seed:eq:seed-frozen-chord}; the upper bound is the
Lipschitz bound of $\xi$. Since the distance of a unit vector $w$ from a
subspace $L$ equals $|\pi_{L^\perp}w|$, the two bounds give a uniform
positive distance between the normalized differences and every
$L\in\mathcal L$. On a smaller cube all sums $\xi(z)+\xi(z')$ have
directions close to that of $c$, which is a simple characteristic,
hence lie at a positive distance from $\mathfrak M$. Shrinking once more
keeps $|\partial_{z_1}(k_0\cdot v(z))|$ bounded below.

\emph{Step 3 (finite covers and radial factors).} Let $k_0$ be a unit
covector with $M_0(k_0)$ singular. By Step~1, for every unit
$s\in\ker M_0(k_0)$ there is a base covector with $s\cdot q_0(\xi)\neq0$.
By compactness of the unit sphere of $\ker M_0(k_0)$ we select finitely
many base covectors with $\sum_a|s\cdot q_0(\xi_a)|^2\geq c|s|^2$ on
$\ker M_0(k_0)$. The quadratic form
$G(w)=|M_0(k_0)^Tw|^2+\sum_a|w\cdot q_0(\xi_a)|^2$ is then positive
definite on $\R^n$: if $G(w)=0$ then $M_0(k_0)w=0$ (recall $M_0$ is
symmetric), so $w\in\ker M_0(k_0)$, and then $w=0$. If $M_0(k_0)$ is
invertible, no base covector is needed. The incidence points are
continued by the implicit function theorem as $z_1=z_1(k)$, with
$z_2=z_3=0$, for $k$ near $k_0$, and by continuity $G$ stays positive
definite on a smaller neighborhood. Compactness of $\mathbb S^{n-1}$ gives
a finite subcover, and we take the minimum of the finitely many positive
constants.

All base covectors lie in a compact set of nonzero simple
characteristics. We choose the radial factors successively, with ratios
so large that every sum or difference of covectors from different groups
has normalized direction as close as desired to $\pm$ the direction of the
covector from the group with the larger factor; a finite ratio suffices
because the norms of base covectors are bounded above and below. Each
base covector and its negative lie at a positive distance from
$\mathfrak M$, so the same holds for these directions. The factors are
fixed independently of everything chosen later. The assertions for a
single patch were proved in Step~2.
\end{proof}

\subsection{Characteristic phases near the reference immersion}
\label{app:ch_seed:sec:seed-actual}
We now continue the patches to immersions near the reference and
construct their phases on common domains.

\begin{lem}[A homogeneous characteristic phase]
\label{app:lem:common-hamilton-phase}
Let $H(t,y,\zeta)$ be real and homogeneous of degree one in $\zeta$ on
fixed physical boxes and nonzero conic boxes, with a fixed bound for its
normalized $C^3$ norm. There are common smaller boxes and a phase
$S(t,y,\zeta)$, homogeneous of degree one in $\zeta$, satisfying
\begin{equation}\label{app:ch_regular_base:eq:actual-eikonal}
 S_t=H(t,y,S_y),\qquad S(0,y,\zeta)=y\cdot\zeta .
\end{equation}
The point $S_\zeta(t,y,\zeta)$ lies in one larger box of initial
points, and $S_{y\zeta}$ is uniformly close to $I$. If, for $T\geq1$ and
$R_*\geq2$, the normalized derivatives of $H$ satisfy
$\|H\|_{C^r}\leq C_rT^{(r-R_*)_+}$, then the phase, the flow and the
inverse of the map from initial points to points satisfy bounds
$C_rT^{(r-R_*+1)_+}$. Normalized $C^{m+1}$ convergence of $H$ implies
$C^m$ convergence of these phases on the same domains. The initial time
may be any $s$ in the same short interval, and all estimates are joint
in $s$.
\end{lem}

\begin{proof}
Solve the Hamilton equations of $\tau-H(t,y,\zeta)$, on a short interval
of both signs,
\[
 \dot Y=-H_{\zeta}(t,Y,Z),\qquad \dot Z=H_{y}(t,Y,Z),\qquad (Y,Z)|_{t=0}=(u,\zeta).
\]
By the low-order bounds and homogeneity, the base point and the
normalized covector stay inside their boxes, with $|Z|\asymp|\zeta|$, and
on a common shorter interval $Y_u$ is uniformly close to $I$.
Lemma~\ref{app:lem:buffered-inverse-stability} gives the inverse
$u=U(t,y,\zeta)$ of $u\mapsto Y(t,u,\zeta)$ on common domains.

By Euler's identity $Z\cdot H_\zeta=H$; differentiating in $y$ and $\zeta$
gives $Z\cdot H_{\zeta y}=H_y$ and $Z\cdot H_{\zeta\zeta}=0$. Hence, for any
variation $\delta$ of the initial data $(u,\zeta)$,
\[
 \frac d{dt}(Z\cdot\delta Y)=H_y\,\delta Y-Z\cdot H_{\zeta y}\,\delta Y
       -(Z\cdot H_{\zeta\zeta})\,\delta Z=0,
\]
so $Z\cdot\delta Y=\zeta\cdot\delta u$, its value at $t=0$. Allowing $t$ to
vary as well, $Z\cdot dY=Z\cdot\dot Y\dd t+\zeta\cdot du=-H\dd t+\zeta\cdot du$,
that is,
\[
 Z\cdot dY+H\dd t=\zeta\cdot du .
\]
Put $S(t,y,\zeta)=U(t,y,\zeta)\cdot\zeta$. Substituting $u=U(t,y,\zeta)$,
so that $Y=y$, we get $\zeta\cdot dU=H\dd t+Z\cdot dy$, and therefore
\[
 dS=\zeta\cdot dU+U\cdot d\zeta=H\dd t+Z\cdot dy+U\cdot d\zeta .
\]
Thus $S_y=Z$, $S_t=H(t,y,S_y)$, $S_\zeta=U$ and $S(0,y,\zeta)=y\cdot\zeta$.
Scaling $\zeta$ by $c>0$ scales $Z$ and leaves $Y$ unchanged (since
$H_\zeta$ is homogeneous of degree zero and $H_y$ of degree one), so $U$ is
homogeneous of degree zero and $S$ of degree one. Finally
$S_{\zeta y}=U_y$, which is close to $I$.

The Hamilton vector field involves one derivative of $H$. The joint
estimates for flows and inverse charts in
Lemma~\ref{app:ch_regular_base:lem:buffered-interpolation} therefore give
the bound $C_rT^{(r-R_*+1)_+}$, which is preserved by multiplication by
the normalized $\zeta$. The initial time enters as one more parameter of
the same equations. Subtracting the differentiated equations for two
Hamiltonians gives the finite-order continuity.
\end{proof}

\begin{setup}[Immersions near the reference]\label{app:ch_seed:setup:seed-current}
Fix the patches, cover, incidence points and radial factors of
Proposition~\ref{app:ch_seed:prop:seed-frozen-menu}, and put
$M_{\rm an}=20(n+1)$. Let $F\colon B_1\to\R^N$ be a smooth immersion
whose second fundamental form $B_F\colon E\to\mathcal N_F$ is onto.
Extend $B_F^*$ by zero on tangent vectors and put
\begin{equation}\label{app:ch_waves:eq:ambient-normal-inverse}
 \begin{gathered}
 P_{\tan,F}=dF g^{-1}(dF)^T,\qquad
 \mathcal A_F=B_FB_F^*+P_{\tan,F},\\
 N_F=\mathcal A_F^{-1}B_F,\qquad
 \Pi_{K_F}=I-B_F^*N_F,\qquad Q_i=\Pi_{K_F}Q_i^0.
 \end{gathered}
\end{equation}
We impose two kinds of conditions on $F$. They involve a normalization
parameter $0<\sigma\leq1$, which measures the size of the second
fundamental form; the constants $C_0$, $\kappa_{\rm met}$, $\kappa$,
$\epsilon_0$ do not depend on $\sigma$.
\begin{enumerate}
\item[(L$_\sigma$)] \emph{Low-order conditions:} $\|F\|_{C^4(B_1)}\leq C_0$, the
induced metric satisfies $g=(dF)^TdF\geq\kappa_{\rm met}I$, the smallest
singular value of the normalized form $\sigma^{-1}B_F$ is at least
$\kappa>0$, and $\max_i\|Q_i-Q_i^0\|_{C^6(B_1)}<\epsilon_0$.
\item[(H)] \emph{A high-order bound:}
$\max_i\|Q_i\|_{C^{M_{\rm an}+9}(B_1)}\leq C_{\rm hi}$, where $C_{\rm hi}$
is a constant fixed once and for all in
Remark~\ref{app:ch_seed:rem:seed-map-topology}.
\end{enumerate}
On the orthogonal splitting of $\R^N$ into tangent and normal spaces,
$\mathcal A_F$ is the identity on tangent vectors and
$B_FB_F^*\geq\kappa^2\sigma^2$ on normal vectors; hence it is invertible
with $\|\mathcal A_F^{-1}\|\leq(\kappa\sigma)^{-2}$ (for $\kappa\sigma\leq1$).
The kernel $K_F$, the projection $\Pi_{K_F}$ and the frame are unchanged
if $B_F$ is replaced by $\sigma^{-1}B_F$; they are therefore smooth
functions of $(dF,\sigma^{-1}d^2F)$ with bounds depending only on the
constants of (L$_\sigma$), and not on $\sigma$. Consequently $N_F$ takes
normal values and vanishes on $K_F$, $B_F^*N_F=B_F^*(B_FB_F^*)^{-1}B_F$ is
the orthogonal projection onto $\operatorname{im}B_F^*=K_F^\perp$, and
$\Pi_{K_F}$ is the orthogonal projection onto $K_F$; on $K_F^\perp$ the
map $N_F$ inverts $B_F^*$, in agreement with the definition before
\eqref{main:eq:normal-completion}. For small $\epsilon_0$ the $Q_i$ form a
frame of $K_F$. All these are smooth functions of the second jet of $F$,
with the stated bounds for the inverses. Define $M_F$, $p_F$ and $q_F$ by
\eqref{main:eq:reduced-convention} with this frame.

We say that $F$ satisfies (L) if it satisfies (L$_\sigma$) for some
$0<\sigma\leq1$. The rescaled references $G_\sigma$ of
Corollary~\ref{app:ch_seed:cor:seed-uniform-dilation}, whose second
fundamental forms have size $\sigma$, satisfy (L$_\sigma$) with constants
independent of $\sigma$; in the proof of Theorem~\ref{main:thm:main} the
parameter is the fixed value $\sigma=\rho$ of the reference. The
conditions (L) determine every domain, box, margin and lower bound
below. Condition (H) enters only through upper bounds for finitely many
derivatives, and it is not a consequence of (L): a $C^6$ neighborhood of
the reference contains immersions whose frames have arbitrarily large
$C^{M_{\rm an}+9}$ norm, and no estimate below is claimed uniformly on
such a neighborhood without (H). The phases constructed below depend on
$F$ only through the frame $Q_i$.
\end{setup}

\begin{prop}[Characteristic phases and quadratic interactions]\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 $\epsilon_0>0$ such that, for every $F$ satisfying the low-order
conditions (L) of Setup~\ref{app:ch_seed:setup:seed-current}, each model
patch has a
jointly smooth real family of phases $\Theta_a(x,z)$ on $B_{\rm out}$ and
its fixed parameter cube, with
\[
 p_F(x,d_x\Theta_a)=0,\qquad \rank M_F(x,d_x\Theta_a)=n-1.
\]
The gradient $d_x\Theta_a$ is uniformly close to $\xi_a(z)$ in
$C_z^1C_x^0$, and is uniformly Lipschitz in $z$. 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_x\Theta_a(x,z)-d_x\Theta_a(x,z'))|\geq c|z-z'|.
\end{equation}
The normalized directions of all nonzero differences and all sums within
one patch, and of all signed sums of covectors from different groups
multiplied by the fixed radial factors, lie in one compact set $\Gamma$
at a positive distance from $\mathfrak M$. On $\Gamma$ the $(n-1)$st
singular value of $M_F$ is bounded below, and at every zero of $p_F$ in
$\Gamma$ one has $|d_\xi p_F|\geq c$. Thus every such direction is
elliptic or a simple characteristic. The domains and constants depend
only on the model data, $\kappa$, $\kappa_{\rm met}$, $C_0$ and the
$C^6$ neighborhood, not on higher norms or on the accuracy. If, in
addition, $\|F\|_{C^r}\leq C_rT^{(r-R_*)_+}$ on the fixed construction
neighborhood, with $T\geq1$ and $R_*\geq7$, then
\begin{equation}\label{app:ch_seed:eq:seed-phase-tame}
 \|\mathscr G_F\|_{C^r}
 \leq C_rT^{(r-R_*+6)_+},\qquad
 \mathscr G_F=(S_F,d\Theta_a),\quad r\geq0,
\end{equation}
where the norm is the maximum over the finite family, taken jointly in
the physical variables, the normalized covectors and the parameters, on
fixed domains.
\end{prop}

\begin{proof}
\emph{Step 1 (phases).} On each slightly enlarged model patch choose
adapted coordinates $x=(t,y)$, $\xi=(\tau,\zeta)$ with
$\partial_\tau p_0\geq c>0$. By the implicit function theorem there is a
unique homogeneous function $H_F$ with
\[
 p_F\bigl(x,(H_F(x,\zeta),\zeta)\bigr)=0
\]
near the model root, depending smoothly on the coefficients. Euler's
identity $\xi\cdot\partial_\xi p_F=n\,p_F$ (recall $p_F$ is homogeneous of
degree $n$) shows that $\zeta\neq0$ on this set of zeros, since
$\partial_\tau p_F\neq0$. Lemma~\ref{app:lem:common-hamilton-phase} gives a
phase $S_F$ on common domains, with $S_0=tH_0(\zeta)+y\cdot\zeta$ for the
constant model, and its continuity statement gives $C^5$ closeness of the
phases from $C^6$ closeness of the coefficients.

Under the derivative bounds on $F$, the projected frame is a smooth
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_rT^{(r-R_*+2)_+}.
\end{equation}
Implicit differentiation preserves this for $H_F$, and the phase lemma
uses one more derivative, giving $\|S_F\|_{C^r}\leq C_rT^{(r-R_*+3)_+}$.
None of this changes the domains or the lower bounds.

\emph{Step 2 (sums and differences).} Let $\zeta_a(z)$ be the spatial
part of the model patch and put
\[
 \Theta_a(x,z)=S_F(x,\zeta_a(z)).
\]
For the model its gradient is $\xi_a(z)$, and the $C^5$ comparison gives
$C_z^1C_x^0$ closeness of the gradients for $F$. The characteristic
equation holds exactly by construction, and a nonzero minor of order
$n-1$ of the model persists. For every $x$ and $L$, the projected map
$z\mapsto\pi_{L^\perp}d_x\Theta_a(x,z)$ is $C^1$ close to the model one on
the convex cube, so the segment argument of
Proposition~\ref{app:ch_seed:prop:seed-frozen-menu} gives
\eqref{app:ch_seed:eq:seed-actual-chord}. Together with the upper
Lipschitz bound it gives a fixed angular distance from every $L$,
including for arbitrarily short nonzero differences. The distances for
sums and for signed sums between groups persist by continuity.

Let $\Gamma_0$ be the closure of all normalized directions of these sums
and differences, including the limits of normalized difference
quotients. It is compact and disjoint from $\mathfrak M$. On $\Gamma_0$,
the $(n-1)$st singular value of $M_0$ has a positive minimum (corank at
least two occurs only on $\mathfrak M$), and by
Proposition~\ref{app:ch_seed:prop:seed-strata} $dp_0\neq0$ on
$\Gamma_0\cap\{p_0=0\}$, with a positive minimum by compactness. On a
slightly larger compact neighborhood $\Gamma$ of $\Gamma_0$, $C^1$
closeness of $M_F$ and $p_F$ to $M_0$ and $p_0$ preserves the singular
value bound and ensures $|d_\xi p_F|\geq c$ at every zero of $p_F$. This
proves the classification.

The phase estimate of Step~1 and one further derivative prove
\eqref{app:ch_seed:eq:seed-phase-tame}.
\end{proof}

For the oscillatory integrals of Sections~\ref{app:sec:ch_osc}
and~\ref{sec:sampling} we need the differences of nearby phases in a
patch, divided by the shift, and their averaged gradients. Fix a patch,
an incidence point and the corresponding base output direction
$k_b=(\omega_b,\eta_b)$ in the adapted coordinates $x=(t,y)$,
$\xi=(\tau,\eta)$ of the patch. Write $z=(s,v)$ with $s=z_1$ and
$v=(z_2,z_3)$, and let $v_b$ be the value of $v$ at the incidence point.
For $0\leq\epsilon\leq\epsilon_*$ put
\begin{equation}\label{app:ch_seed:eq:seed-exact-phase-quotient}
 \phi_{F,\epsilon}(x,z,\eta)=
 \frac{S_F(x,\zeta_a(z)+\epsilon\eta)-S_F(x,\zeta_a(z))}{\epsilon}
 =\int_0^1
 \partial_\zeta S_F(x,\zeta_a(z)+r\epsilon\eta)\cdot\eta\dd r ,
\end{equation}
\label{app:ch_seed:eq:seed-quotient-integral}%
the integral giving the value at $\epsilon=0$, and let
\[
 \Psi_{F,\epsilon}(x,x',s,v,\theta)
 =\int_0^1d_x\phi_{F,\epsilon}\bigl(x'+r(x-x'),s,v,\theta\bigr)\dd r
\]
be the averaged $x$-gradient. For the model we put
\[
 G(s,\theta)=\nabla H_0(\zeta_a(s,v_b))\cdot\theta,\qquad
 \Psi_0(s,\theta)=(G(s,\theta),\theta).
\]

\begin{prop}[Phase differences and their averaged gradients]\label{app:ch_seed:prop:seed-phase-charts}
Let $\delta_1,\delta>0$ be given. There are convex physical boxes inside
$B_{\rm out}$, convex product boxes $D_0\Subset D_1\Subset D$ in the
variables $(s,\theta)$, centered at $(s_b,\eta_b)$ and with
$\theta$-factors in a fixed annulus, a domain of $v$ around $v_b$, a
number $\epsilon_*>0$ and a bound $\epsilon_0>0$ for condition (L) of
Setup~\ref{app:ch_seed:setup:seed-current}, all depending only on the
model data and on $\delta_1,\delta$, with the following properties.
\begin{enumerate}
\item $|\partial_sG|\geq c>0$ on $D$, and $D\Psi_0$ differs on $D$ from
one constant invertible matrix $G_0$ by at most $\|G_0^{-1}\|^{-1}/4$.
\item For every $F$ satisfying (L) with this $\epsilon_0$, uniformly in
$(x,x',v)$ and $0\leq\epsilon\leq\epsilon_*$,
\begin{equation}\label{app:ch_seed:eq:seed-normal-annulus}
 \|\Psi_{F,\epsilon}-\Psi_0\|_{C^1_{s,\theta}(D)}\leq\delta_1,\qquad
 |\Psi_{F,\epsilon,y}-\theta|\leq\delta|\theta|,
 \qquad c\leq|\Psi_{F,\epsilon}|\leq C .
\end{equation}
\item If $F$ satisfies (L) and (H), then $\Psi_{F,\epsilon}$ has joint
$C^{M_{\rm an}+2}$ bounds in $(x,x',s,v,\theta,\epsilon)$, and so has
$\partial_\epsilon\Psi_{F,\epsilon}$, with constants depending only on
the data of (L) and on $C_{\rm hi}$.
\item Under the derivative bounds of
Proposition~\ref{app:ch_seed:prop:seed-actual-menu}, the maps
$\Psi_{F,\epsilon}$, their inverse charts in $(s,\theta)$ and in
$\theta$ at fixed time, and their first $\epsilon$-derivatives satisfy
\eqref{app:ch_seed:eq:seed-phase-tame} jointly in the physical, data
and $v$ variables and in $\epsilon$; we include them in $\mathscr G_F$.
\end{enumerate}
\end{prop}

Parts (i)--(iii) are exactly the hypotheses on the phase in
Setup~\ref{app:ch_osc:setup:osc-normal} below, with $M=M_{\rm an}$. The
constants $\delta_1,\delta$ are those required there and in
Proposition~\ref{app:ch_osc:prop:osc-normal}; they depend only on the
model, so the boxes, $\epsilon_*$ and $\epsilon_0$ are fixed by low-order
data alone, before any other choice.

\begin{proof}
\emph{(i)} For the model, $p_0(\tau,\zeta)=0$ near the patch is
equivalent to $\tau=H_0(\zeta)$, so $d_\xi p_0=\partial_\tau p_0\,(1,-\nabla H_0)$.
The incidence condition $k_b\cdot d_\xi p_0=0$ gives
$\omega_b=\eta_b\cdot\nabla H_0(\zeta_a(z_b))$, so $\eta_b\neq0$, and its
transversality (Proposition~\ref{app:ch_seed:prop:seed-frozen-menu}(ii))
gives $\partial_sG(s_b,\eta_b)\neq0$. For the model at $\epsilon=0$ and
$v=v_b$ we have $S_0=tH_0(\zeta)+y\cdot\zeta$, so the phase is
$tG(s,\theta)+y\cdot\theta$ and its averaged gradient is the triangular
map $\Psi_0$. Its derivative at $(s_b,\eta_b)$ is invertible, and on a
small convex box it stays within the stated distance of its value
there.

\emph{(ii)} Let $\delta_1,\delta$ be given. By
\eqref{app:ch_seed:eq:seed-exact-phase-quotient}, the map
$\Psi_{F,\epsilon}$ and its first $(s,\theta)$-derivatives involve the
derivatives of $S_F$ of order at most three on a compact set of shifted
covectors. We choose, in this order: first a domain of $v$ around $v_b$
so small that the model map at $\epsilon=0$ is within $\delta_1/3$ of
$\Psi_0$ in $C^1_{s,\theta}(D)$ for all such $v$; then $\epsilon_*$ so small
that the model maps for $0\leq\epsilon\leq\epsilon_*$ are within
$\delta_1/3$ of the map at $\epsilon=0$, and the shifted covectors stay in
the domain of the phase; finally $\epsilon_0$ so small that, by the $C^5$
comparison of the phases in
Proposition~\ref{app:ch_seed:prop:seed-actual-menu}, which uses only (L),
the maps for $F$ are within $\delta_1/3$ of the model maps. This gives
the first inequality. The $y$-component of $\Psi_0$ is $\theta$, so the
same choices, with $\delta|\theta|/3$ in place of $\delta_1/3$, give the
second inequality, and the third follows since $|\theta|$ is bounded
above and below on the annulus.

\emph{(iii)} The joint derivatives of order $M_{\rm an}+2$ of
$\Psi_{F,\epsilon}$, and of $\partial_\epsilon\Psi_{F,\epsilon}$, are
bounded by the derivatives of $S_F$ of order at most $M_{\rm an}+5$ in
$(x,\zeta)$ on the fixed compact set, multiplied by derivatives of the
fixed smooth map $\zeta_a$. The phase is obtained from the Hamiltonian
$H_F$ by Lemma~\ref{app:lem:common-hamilton-phase}, whose proof
(the induction of Lemma~\ref{app:ch_regular_base:lem:buffered-interpolation},
stopped at a finite order) bounds the $C^m$ norm of $S_F$ on the fixed
domains by the $C^{m+1}$ norm of $H_F$. In turn $H_F$ is obtained from
$p_F=\det M_F$ by the implicit function theorem with the lower bound
of (L) for $\partial_\tau p_F$, so its $C^{m+1}$ norm is bounded by the
$C^{m+1}$ norm of the frame. With $m=M_{\rm an}+5$ these orders are
covered by (H), since $M_{\rm an}+6\leq M_{\rm an}+9$.

\emph{(iv)} By \eqref{app:ch_seed:eq:seed-phase-tame} for $S_F$, the
two phase derivatives in $\Psi_{F,\epsilon}$ give the bound
$C_rT^{(r-R_*+5)_+}$. Joint inversion preserves it by
Lemma~\ref{app:ch_regular_base:lem:buffered-interpolation}, and one
$\epsilon$-derivative changes the exponent to $(r-R_*+6)_+$. The domains
were fixed in (i) and (ii) using low-order data only.
\end{proof}

\begin{rem}[Finite-order assumptions on the immersion]\label{app:ch_seed:rem:seed-map-topology}
The frame $Q_i$ depends smoothly on the second jet of $F$, so a fixed
$C^8$ neighborhood of the reference suffices for the $C^6$ frame
condition in (L). For the reference of
Corollary~\ref{app:ch_seed:cor:seed-uniform-dilation} we have
$\gamma(G_{\mathrm{ref}})=I$, so $\|dF-dG_{\mathrm{ref}}\|_{C^0}<1/2$
implies $|dF(x)v|\geq|v|/2$ and $\gamma(F)\geq I/4$. This neighborhood is
chosen together with the neighborhoods for the normal rank and the
kernel frame.

We now fix the constant $C_{\rm hi}$ of (H). Let $\sigma_1\leq1$ be so
small that the rescaled references $G_\sigma$ are defined on $B_1$ for
$0<\sigma\leq\sigma_1$. They satisfy $Q_i(G_\sigma)(x)=Q_i(\sigma x)$
(proof of Corollary~\ref{app:ch_seed:cor:seed-uniform-dilation}), where
$Q_i(y)$ is the projected frame of $F_0$; hence
$\|Q_i(G_\sigma)\|_{C^k(B_1)}\leq\|Q_i\|_{C^k(B_{\sigma_1})}$ for every
$k$, uniformly in $\sigma$, and similarly
$\|dG_\sigma\|_{C^k(B_1)}+\|\sigma^{-1}d^2G_\sigma\|_{C^k(B_1)}\leq C_k(F_0)$,
since $\sigma^{-1}d^2G_\sigma(x)=(d^2F_0)(\sigma x)$.

The frame is a smooth function of $(dF,\sigma^{-1}d^2F)$ on the set
defined by the lower bounds of (L$_\sigma$)
(Setup~\ref{app:ch_seed:setup:seed-current}), but not of the unnormalized
second jet uniformly in $\sigma$: a perturbation $Y$ of $G_\sigma$ enters
the projection through $\sigma^{-1}d^2Y$. Therefore a perturbation of
fixed size in $C^{M_{\rm an}+11}$ does not give a bound independent of
$\sigma$. We use instead the
normalized neighborhood
\begin{equation}\label{new:eq:normalized-neighborhood}
 \|d(F-G_\sigma)\|_{C^{M_{\rm an}+9}(B_1)}
 +\sigma^{-1}\|d^2(F-G_\sigma)\|_{C^{M_{\rm an}+9}(B_1)}\leq1 ,
\end{equation}
which holds in particular if $\|F-G_\sigma\|_{C^{M_{\rm an}+11}(B_1)}\leq\sigma$.
On this set the arguments $(dF,\sigma^{-1}d^2F)$ are bounded in
$C^{M_{\rm an}+9}$ by $C(F_0)+1$, and the Fa\`a di Bruno formula gives a
constant $C_{\rm hi}$, depending only on $F_0$, $n$ and the constants of
(L), such that (H) holds for every $F$ satisfying (L$_\sigma$) and
\eqref{new:eq:normalized-neighborhood} for some $0<\sigma\leq\sigma_1$.
We take $B_1$ inside the chart $W$ of Setup~\ref{app:setup:domains}, so
that bounds on $W$ imply bounds on $B_1$.

In the proof of Theorem~\ref{main:thm:main} the parameter is fixed,
$\sigma=\rho\leq\sigma_1$, before the iteration. All iterates lie in a
$C^{r_*}$ ball of radius $\varepsilon_*\leq\rho$ around
$G_{\rm ref}=G_\rho$, with $r_*\geq M_{\rm an}+11$
(Section~\ref{new:sec:order-of-choices}), so
\eqref{new:eq:normalized-neighborhood} and hence (H) hold for all of them
with this constant. Uniformity as $\sigma\to0$ is needed only for the
rescaled references $G_\sigma$ themselves (Section~\ref{app:sec:ch_background}),
for which it was shown above.
\end{rem}

\subsection{Indexing the quadratic interactions}
\label{app:sec:packet-geometry-bridge}
The characteristic patches, and therefore the sums and differences of
their covectors, are fixed before the frequencies and the sample.

\begin{defn}[Groups of patches]\label{app:defn:exhaustive-pair-groups}
Index the finite list of characteristic patches before the frequencies
and the sample are chosen. If a patch is used more than once, each
occurrence receives its own index and forms a separate group. Apply
part~(iii) of Proposition~\ref{app:ch_seed:prop:seed-frozen-menu} to this
list, so that different groups have radially separated factors. Within
one group the deletion rule of Theorem~\ref{app:ch_waves:thm:sampling}
separates distinct samples by at least $h/4$. All modulations attached
to one sample are combined into a single Fourier measure and a single
wave.
\end{defn}

\begin{rem}[Sums and differences of phases]\label{app:rem:exhaustive-signed-pairs}
For two distinct samples in the same group, the chord estimate
\eqref{app:ch_seed:eq:seed-actual-chord} keeps the direction of the
phase difference away from $\mathfrak M$ and gives it the frequency
$\lambda|z-z'|\geq c\lambda h$. The sum of two phases has frequency
comparable to $\lambda$ and the same angular separation. For different
groups the radial factors give both properties for both signs, even if
the parameters or the unscaled directions coincide. For a repeated
index, the doubled phase is separated from $\mathfrak M$, while the
Hermitian square is treated as part of the mean. These cases exhaust the
real quadratic expansion of a sum of waves. Fourier modulations are
included in the amplitudes (Section~\ref{app:sec:ch_pairs}). All group
factors are fixed before the sample is drawn.
\end{rem}

We summarize the properties of the patches used in the rest of the
paper.

\begin{proposition}[Characteristic patches]\label{main:prop:patches}
For $n\geq5$ there is a finite family of three-parameter patches of
simple characteristic covectors of the constant model with the following
properties.
\begin{enumerate}
\item For every unit output direction, the selected patches have
transversal incidence points in their interiors and satisfy the uniform
Gram inequality \eqref{main:eq:detector-gram}.
\item For immersions in a fixed neighborhood of the reference, on a fixed
chart and after fixed radial factors are assigned to the patches, every
doubled phase and every sum or difference of distinct sampled phases has
its gradient in a fixed compact conic region where the reduced system is
elliptic or of simple real principal type.
\item For samples separated by $h$, the frequency $\varrho$ of each such
product satisfies $c\lambda h\leq\varrho\leq C\lambda$.
\end{enumerate}
The Hermitian squares are treated by the equation for the mean and are
excluded from~(ii).
\end{proposition}

\begin{proof}
Part~(i) is part~(ii) of
Proposition~\ref{app:ch_seed:prop:seed-frozen-menu}. Part~(ii) is
Proposition~\ref{app:ch_seed:prop:seed-actual-menu}, with the grouping of
Definition~\ref{app:defn:exhaustive-pair-groups}.
Part~(iii) follows from Remark~\ref{app:rem:exhaustive-signed-pairs}:
the frequency of a difference is $\lambda$ times the length of a
difference of gradients, which is between $c|z-z'|$ and $C|z-z'|$ by
\eqref{app:ch_seed:eq:seed-actual-chord}, and sums and products between
groups have frequency comparable to $\lambda$.
\end{proof}

A convex three-parameter patch controls the limits of normalized
differences as well as the differences of separated samples; this is why
three parameters are used. The parametrices on the regions of~(ii) also
require control of their Hamilton trajectories. This is proved in
Theorem~\ref{app:ch_regular:thm:no-return}: each chart is inverted
separately, the successive retarded parts of the parametrix use the same
time coordinate and together occupy one fixed time interval, and the
charts are chosen so that the forward flowout of each source chart over
this interval stays in a slightly larger chart of the same regular
region.

%% ===== 04_05_preliminaries.tex =====
\section{Smoothing and derivative estimates}
\label{app:sec:elementary-interpolation}

This section contains elementary estimates which do not depend on any
phase, solution operator or frequency. Lemma~\ref{app:ch_regular_base:lem:ball-extension}
is used to smooth the metric defect (Lemma~\ref{main:lem:smoothing}).
The remaining results show that derivative bounds of the form
\eqref{new:eq:derivative-growth} are preserved, with a fixed loss, by the
algebraic operations, inverse charts and Hamilton flows used in
Section~\ref{app:sec:ch_seed} and in
Sections~\ref{app:sec:ch_regular_base}--\ref{app:sec:ch_background}.

\subsection{Smoothing on a ball}

\begin{lem}[Smoothing on a ball]
\label{app:ch_regular_base:lem:ball-extension}
Let $\Omega=B_a(c)\subset\R^n$ be a ball with $\overline\Omega\subset V$,
where $V$ is open. For each integer $m\geq1$ and all sufficiently large
$R$ there is a real linear operator
$S_{R,m}:C^\infty(\overline\Omega)\to C_c^\infty(V)$, with outputs
supported in one fixed compact subset of $V$, such that for all integers
$r\geq0$
\begin{align}
 \|S_{R,m}f\|_{C^r}+\|S_{R,m}f\|_{H^r}
 &\leq C_{r,m}R^r\|f\|_{C^0(\overline\Omega)},
 \label{app:ch_regular_base:eq:ball-extension-sobolev}\\
 \|f-S_{R,m}f\|_{C^0(\overline\Omega)}
 &\leq C_mR^{-m}\|f\|_{C^m(\overline\Omega)}.
 \label{app:ch_regular_base:eq:ball-extension-continuous}
\end{align}
The operators act componentwise on vector- and tensor-valued functions
and preserve real fields.
\end{lem}

\begin{proof}
After an affine change of variables we may assume $\Omega=B_1$; this
changes the constants and the size of $R$ only by fixed factors. Fix
$\rho\in C_c^\infty(B_{1/4})$ with $\rho\geq0$ and $\int\rho=1$. For
$R\geq2m$ choose $\chi_R\in C_c^\infty(B_{1+1/(4R)})$ with $\chi_R=1$ on
$B_1$ and $\|D^r\chi_R\|_\infty\leq C_rR^r$. Put
\begin{equation}\label{app:ch_regular_base:eq:extension-finite-moments}
 a_j=(-1)^{j-1}\binom mj\quad(1\leq j\leq m),\qquad
 \sum_{j=1}^ma_jj^k=\delta_{k0}\quad(0\leq k<m).
\end{equation}
The identity holds because the $m$th finite difference annihilates
polynomials of degree less than $m$:
$\sum_{j=0}^m(-1)^j\binom mjj^k=0$ for $0\leq k<m$, and moving the term
$j=0$ (which equals $\delta_{k0}$) to the other side gives
\eqref{app:ch_regular_base:eq:extension-finite-moments}. Define
\begin{equation}\label{app:ch_regular_base:eq:extension-radial-formula}
 S_{R,m}f(x)=\chi_R(x)\sum_{j=1}^ma_j
 \int\rho(y)f\bigl((1-j/R)x+(j/R)y\bigr)\dd y
\end{equation}
for $|x|<1+1/(2R)$, and $S_{R,m}f=0$ elsewhere.

\emph{The formula is well defined.} Let $|x|\leq1+1/(2R)$, $|y|\leq1/4$
and $1\leq j\leq m\leq R/2$. Then $1-j/R>0$ and
\[
 \bigl|(1-j/R)x+(j/R)y\bigr|
 \leq\Bigl(1-\frac jR\Bigr)\Bigl(1+\frac1{2R}\Bigr)+\frac j{4R}
 \leq1+\frac1{2R}-\frac{3j}{4R}<1 ,
\]
so every argument of $f$ lies in $B_1$. Since $\chi_R$ vanishes outside
$B_{1+1/(4R)}$, the extension by zero is smooth, and for large $R$ all
outputs are supported in a fixed compact subset of $V$.

\emph{Derivative bounds.} In the $j$th term substitute
$w=(1-j/R)x+(j/R)y$:
\[
 \int\rho(y)f\bigl((1-j/R)x+(j/R)y\bigr)\dd y
 =\Bigl(\frac Rj\Bigr)^n\int\rho\Bigl(\frac Rj\bigl(w-(1-j/R)x\bigr)\Bigr)f(w)\dd w .
\]
All $x$-derivatives now fall on $\rho$, and each produces the factor
$-(R/j)(1-j/R)=-(R/j-1)$, of absolute value at most $R$. Undoing the
substitution,
\[
 \Bigl|\partial_x^\alpha\int\rho(y)f\bigl((1-j/R)x+(j/R)y\bigr)\dd y\Bigr|
 \leq R^{|\alpha|}\|\partial^\alpha\rho\|_{L^1}\|f\|_{C^0(B_1)} .
\]
By the Leibniz rule and $\|D^r\chi_R\|_\infty\leq C_rR^r$ we obtain
$\|S_{R,m}f\|_{C^r}\leq C_{r,m}R^r\|f\|_{C^0(B_1)}$. Since the supports
lie in a fixed bounded set, the same bound holds for the $H^r$ norm.

\emph{Approximation.} Let $x\in B_1$, so that $\chi_R(x)=1$. For
$y\in B_{1/4}$ the segment from $x$ to $y$ lies in $B_1$, and Taylor's
formula at $x$ gives
\[
 f\bigl(x+(j/R)(y-x)\bigr)
 =\sum_{k=0}^{m-1}\frac{(j/R)^k}{k!}D^kf(x)[(y-x)^{k}]
  +E_j(x,y),
\]
where $|E_j(x,y)|\leq C_m(j/R)^m\|f\|_{C^m(B_1)}$.
Multiplying by $a_j\rho(y)$, integrating in $y$ and summing over $j$, the
term of order $k$ is multiplied by $\sum_ja_jj^k=\delta_{k0}$. Hence
\[
 S_{R,m}f(x)-f(x)=\sum_{j=1}^ma_j\int\rho(y)E_j(x,y)\dd y,
\]
whose absolute value is at most $C_mR^{-m}\|f\|_{C^m(B_1)}$.
This proves \eqref{app:ch_regular_base:eq:ball-extension-continuous}.
The operator is real and acts componentwise by construction.
\end{proof}

\subsection{Derivative bounds}
The following elementary inequality is used repeatedly: if $R\geq0$ and
$j_1,\dots,j_k\geq0$, then
\begin{equation}\label{new:eq:plus-subadditive}
 \sum_{i=1}^k(j_i-R)_+\leq\Bigl(\sum_{i=1}^kj_i-R\Bigr)_+ .
\end{equation}
Indeed, if all $j_i\leq R$ the left side vanishes; otherwise let
$I=\{i:j_i>R\}\neq\emptyset$. Then the left side equals
$\sum_{i\in I}j_i-|I|R\leq\sum_ij_i-R$.

\begin{lem}[Derivative products]
\label{app:lem:elementary-interpolation-jets}
Let $T\geq1$ and $R\geq0$, and suppose finitely many arrays $f_i$ on
fixed charts satisfy $\|f_i\|_{C^r}\leq C_rT^{(r-R)_+}$ for all
$r\geq0$. Then
\begin{equation}\label{app:ch_regular_base:eq:joint-primitive-product}
 \prod_i\|D^{j_i}f_i\|_\infty
 \leq C_{j_1,\ldots,j_k}T^{(\sum_i j_i-R)_+}.
\end{equation}
Let $\Phi$ be a smooth function, all of whose derivatives are bounded on
a fixed bounded set containing the ranges of the $f_i$. Then
$\Phi(f_1,\dots,f_k)$ satisfies the same bounds
$\|\Phi(f)\|_{C^r}\leq C_rT^{(r-R)_+}$. This applies in particular to
matrix inversion on a set of matrices with a fixed lower bound for
their smallest singular value. All coordinate and parameter derivatives
are counted jointly.
\end{lem}

\begin{proof}
The product estimate follows from the hypothesis and
\eqref{new:eq:plus-subadditive}. For the second assertion, the
Fa\`a di Bruno formula expresses $D^r[\Phi(f)]$, $r\geq1$, as a finite
sum of terms
\[
 (D^q\Phi)(f)\bigl[D^{\alpha_1}f,\dots,D^{\alpha_q}f\bigr],\qquad
 1\leq q\leq r,\quad \alpha_i\geq1,\quad \alpha_1+\dots+\alpha_q=r,
\]
with universal coefficients. The factor $(D^q\Phi)(f)$ is bounded by
assumption, and the product of the remaining factors is bounded by
$CT^{(r-R)_+}$ by \eqref{app:ch_regular_base:eq:joint-primitive-product}.
\end{proof}

\begin{cor}[Derivative estimates for finite jets]
\label{app:cor:fixed-profile-transfer}
Let $T\geq1$ and suppose $\|F\|_{C^j}\leq A_jT^{(j-R)_+}$ for all $j\geq0$.
Let $\mathcal F_i(J^{c_i}F)$ be a fixed finite family of smooth
functions of the $c_i$-jets of $F$, and assume that the values of these
jets lie in fixed bounded sets on which the $\mathcal F_i$ have bounded
derivatives of all orders (this includes fixed lower bounds for
determinants to be inverted). If $\nu\geq0$ and $R\geq\max_ic_i+\nu$, then
\begin{equation}\label{app:eq:fixed-profile-transfer}
 \|\mathcal F_i(J^{c_i}F)\|_{C^r}
       \leq C_rT^{(r-\nu)_+},\qquad r\geq0.
\end{equation}
Additional parameter derivatives of fixed order are included in $c_i$.
\end{cor}

\begin{proof}
A derivative of order $r$ of $J^{c_i}F$ is a derivative of $F$ of order
at most $r+c_i$, so
$\|J^{c_i}F\|_{C^r}\leq CT^{(r+c_i-R)_+}\leq CT^{(r-\nu)_+}$, since
$R-c_i\geq\nu$. Lemma~\ref{app:lem:elementary-interpolation-jets}, applied
with $\nu$ in place of $R$, gives the claim.
\end{proof}

\begin{lem}[Common inverse domains]
\label{app:lem:buffered-inverse-stability}
Let $D_0\Subset D_1\Subset D$ be bounded convex boxes in $\R^d$ and let
$F_0$ be smooth near $\overline D$. Suppose a constant invertible matrix
$G$ satisfies $\sup_D\|G^{-1}(DF_0-G)\|\leq1/4$. Then for all $F$ in a
sufficiently small $C^1$ neighborhood of $F_0$, the map $F$ is injective
on $D$, with $|G^{-1}(F(u)-F(v))|\geq|u-v|/2$, and there is one open set
$V$ with $F(D_0)\Subset V\Subset F(D_1)$, with positive distances between
these sets that do not depend on $F$. The same holds uniformly for
parameter families of maps in this neighborhood.
\end{lem}

\begin{proof}
Choose the $C^1$ neighborhood so small that
$\sup_D\|I-G^{-1}DF\|\leq1/2$. For $u,v\in D$, convexity gives
\[
 G^{-1}(F(u)-F(v))=(u-v)-\int_0^1\bigl(I-G^{-1}DF(v+\theta(u-v))\bigr)(u-v)\dd\theta,
\]
and the integral has norm at most $|u-v|/2$; this proves the lower
Lipschitz bound and injectivity.

Choose $r>0$ such that $\overline B_r(u_0)\subset D_1$ for every
$u_0\in\overline D_0$, and let $V$ be the open
$(r/(4\|G^{-1}\|))$-neighborhood of $F_0(\overline D_0)$. If
$\|F-F_0\|_{C^0}<r/(4\|G^{-1}\|)$, then every $v\in\overline V$ has a point
$u_0\in\overline D_0$ with $|G^{-1}(v-F(u_0))|<r/2$. Consider
\[
 \Phi(u)=u-G^{-1}(F(u)-v),\qquad u\in\overline B_r(u_0).
\]
Since $D\Phi=I-G^{-1}DF$ has norm at most $1/2$, $\Phi$ is a
$1/2$-contraction, and
\[
 |\Phi(u)-u_0|\leq|\Phi(u)-\Phi(u_0)|+|\Phi(u_0)-u_0|
 <\tfrac12|u-u_0|+\tfrac r2\leq r .
\]
Thus $\Phi$ maps $\overline B_r(u_0)$ into itself, and its fixed point
$u$ satisfies $F(u)=v$. Hence $\overline V\subset F(D_1)$. Finally,
$F(\overline D_0)$ lies within distance $r/(4\|G^{-1}\|)$ of
$F_0(\overline D_0)$; after halving the $C^0$ neighborhood once more we
obtain $F(\overline D_0)\Subset V$ with a fixed margin. All constants
depend only on $r$, $G$ and the size of the neighborhood.
\end{proof}

\begin{lem}[Composition, inverse charts and flows]
\label{app:ch_regular_base:lem:buffered-interpolation}
Work on fixed coordinate domains, which may include physical variables
and external parameters, and let $T\geq1$ and $R\geq1$. Bounds of the form
$\|f\|_{C^r}\leq C_rT^{(r-R)_+}$ are preserved, with the same index $R$:
\begin{enumerate}
\item by composition of maps satisfying such bounds;
\item by inverse and implicit function charts, whenever the charts
exist on the given domains and the relevant first derivative has a
uniformly bounded inverse;
\item by the flow of a vector field $V(t,x)$ satisfying such bounds,
on a fixed time interval on which the trajectories stay in the working
domains; the initial section may satisfy the same bounds;
\item by the solutions $U$ and $U^{-1}$ of $U'=-CU$, $U(0)=I$, when the
matrix $C$ satisfies such bounds.
\end{enumerate}
The constants depend only on the constants in the hypotheses and on the
fixed lower bounds.
\end{lem}

\begin{proof}
\emph{The basic inequality.} Let $q\geq1$ and
$\alpha_1,\dots,\alpha_q\geq1$ with $\sum_i\alpha_i=r$. We claim
\begin{equation}\label{app:eq:graded-composition-budget}
 (q-R)_++\sum_{i=1}^q(\alpha_i-R)_+\leq(r-R)_+ .
\end{equation}
If $q>R$, then $(\alpha_i-R)_+\leq\alpha_i-1$ for every $i$ (because
$\alpha_i\geq1$ and $R\geq1$), so the left side is at most
$(q-R)+\sum_i(\alpha_i-1)=r-R$. If $q\leq R$, the first term vanishes and
the claim is \eqref{new:eq:plus-subadditive}.

\emph{Composition.} By the Fa\`a di Bruno formula, $D^r(f\circ g)$ is a
sum of terms $(D^qf)(g)[D^{\alpha_1}g,\dots,D^{\alpha_q}g]$ with
$\sum\alpha_i=r$, $\alpha_i\geq1$. The first factor is bounded by
$CT^{(q-R)_+}$ and the others by $CT^{(\alpha_i-R)_+}$, so
\eqref{app:eq:graded-composition-budget} gives the bound $CT^{(r-R)_+}$.

\emph{Inverse charts.} Let $X=F^{-1}$. The first derivative
$DX=DF(X)^{-1}$ is bounded by the hypothesis on the inverse. For $r\geq2$,
differentiating $F\circ X=\Id$ $r$ times gives
\[
 DF(X)\,D^rX=-\sum(D^qF)(X)\bigl[D^{\alpha_1}X,\dots,D^{\alpha_q}X\bigr],
\]
where the sum runs over $q\geq2$ and $\alpha_i\geq1$ with
$\sum\alpha_i=r$; in particular every $\alpha_i<r$. Assume inductively
that $\|D^\alpha X\|_\infty\leq C_\alpha T^{(\alpha-R)_+}$ for
$\alpha<r$. Since $(D^qF)(X)$ is bounded by $CT^{(q-R)_+}$,
\eqref{app:eq:graded-composition-budget} bounds the right side by
$CT^{(r-R)_+}$, and multiplication by $DF(X)^{-1}$ completes the
induction. Implicit functions are obtained by inverting
$(z,w)\mapsto(z,F(z,w))$, with the parameters included in $z$.
Existence of the charts on common domains is a separate low-order
matter, treated by Lemma~\ref{app:lem:buffered-inverse-stability}.

\emph{Flows.} Let $\partial_t\Psi=V(t,\Psi)$. Differentiating $r$ times
in the initial point gives
\[
 \partial_tD^r\Psi=D_xV(t,\Psi)\,D^r\Psi
 +\sum_{q\geq2}(D_x^qV)(t,\Psi)\bigl[D^{\alpha_1}\Psi,\dots,D^{\alpha_q}\Psi\bigr],
\]
where again $\alpha_i<r$. By induction the sum is bounded by
$CT^{(r-R)_+}$, and the coefficient $D_xV$ is bounded; Gronwall's
inequality on the fixed time interval gives
$\|D^r\Psi\|\leq C_rT^{(r-R)_+}$. Joint estimates in time and parameters
follow by adjoining $t$ and the parameters as coordinates with
$\dot t=1$ and vanishing derivative. A derivative in the elapsed time is
replaced by the equation, which costs no further derivative of $\Psi$;
a variable initial section is handled by composition.

\emph{Linear matrix equations.} Gronwall's inequality bounds $U$ and
$U^{-1}$, which solves $(U^{-1})'=U^{-1}C$. The flow argument applied to
these linear equations on the resulting bounded ranges gives the
derivative bounds.
\end{proof}

\begin{rem}[Homogeneous symbols]\label{new:rem:homogeneous-normalization}
Symbols and phases which are homogeneous in a covector variable are
estimated through their restrictions to a fixed annulus. For a symbol
$a$ of order $p$ and a phase $S$ of degree one put
$a_\mu(x,\omega)=\mu^{-p}a(x,\mu\omega)$ and
$S_\mu(x,\omega)=\mu^{-1}S(x,\mu\omega)$ for $\omega$ in a fixed annulus.
Then
\[
 \partial_x^\alpha\partial_\omega^\gamma a_\mu
 =\mu^{|\gamma|-p}(\partial_x^\alpha\partial_\eta^\gamma a)(x,\mu\omega),
\]
and similarly for $S$ with $p=1$. Thus the usual symbol estimates for
$a$ are equivalent to uniform bounds for the normalized functions
$a_\mu$, to which Lemmas~\ref{app:lem:elementary-interpolation-jets}
and~\ref{app:ch_regular_base:lem:buffered-interpolation} apply. Amplitudes
depending on two positions are treated in the same way, and different
frequency shells are normalized separately; their radial powers and
volume factors are kept outside the derivative bounds.
\end{rem}

\section{Oscillatory integral estimates}\label{app:sec:ch_osc}

This section contains $L^2$ bounds for oscillatory integral operators
whose amplitudes have finitely many derivatives, an exact formula for
their normal operators, a comparison estimate, and an estimate for the
Fourier tails of oscillatory functions. The phase hypotheses below are
those established for the phases of our construction in
Proposition~\ref{app:ch_seed:prop:seed-phase-charts}; otherwise each
result uses only the displayed bounds. The results are applied to the
continuous normal operator (Section~\ref{app:sec:ch_background}), the
sampled operators (Section~\ref{sec:sampling}), and the parametrices
(Sections~\ref{app:sec:ch_regular_base} and~\ref{app:sec:ch_regular}).

\subsection{Normal operators}
\label{app:ch_osc:sec:osc-normal}

\begin{notation}\label{app:ch_osc:defn:osc-fourier}
We use the Fourier convention of Notation~\ref{app:not:fourier} and the
norm $\|f\|_{H^s}=\|\langle D\rangle^sf\|_2$ on $\R^n$. Outputs with
compact support in a chart are extended by zero. Derivative orders are
total orders of multi-indices, and matrix sizes are fixed.
\end{notation}

\begin{lem}[Multiplication of kernels by smooth amplitudes]
\label{app:ch_osc:lem:osc-amplitude}
Fix compact sets in $\R^d$, $d\geq1$, and put $\ell=\lfloor d/2\rfloor+1$.
\begin{enumerate}
\item Let $U$ be a measure space and let
$T\colon L^2(U)\to L^2(\R^d)$ be a bounded integral operator with kernel
$K(x,u)$. For a function $b(x,u)$ supported, in $x$, in the fixed
compact set, let $T[b]$ be the operator with kernel $b(x,u)K(x,u)$. Then
\begin{equation}\label{app:ch_osc:eq:amplitude-row}
 \|T[b]\|\leq C\|T\|\sup_u\|b(\cdot,u)\|_{H^\ell}.
\end{equation}
\item Let $b(x,y,\xi)$ be supported, in $(x,y)$, in a fixed compact
set, and assume $\sup_\xi\|b(\cdot,\cdot,\xi)\|_{C^{2\ell}}\leq A$. Then
the operator
\[
 Bf(x)=(2\pi)^{-d}\iint e^{\ii(x-y)\cdot\xi}b(x,y,\xi)f(y)\dd y\dd\xi
\]
satisfies $\|B\|_{L^2\to L^2}\leq CA$.
\end{enumerate}
No derivatives in $u$ or $\xi$ are needed, and the frequency support may
be unbounded. The constants depend only on $d$ and the compact sets; the
same bounds hold for matrix-valued amplitudes of fixed size.
\end{lem}

\begin{proof}
(i) For $a\in\R^d$ define $G(a,x)=T\bigl(b(a,\cdot)f\bigr)(x)$, so that
$T[b]f(x)=G(x,x)$. Since $\ell>d/2$, the Sobolev embedding in the
variable $a$ gives, for every fixed $x$,
\[
 |G(x,x)|^2\leq\sup_a|G(a,x)|^2
 \leq C\sum_{|\alpha|\leq\ell}\int|\partial_a^\alpha G(a,x)|^2\dd a,
\]
where we use the continuous representative in $a$; the support of $b$ in
its first variable is a fixed compact set, so the integral is over a
fixed bounded region. Integrating in $x$ and using Fubini's theorem,
\[
\begin{split}
 \|T[b]f\|_2^2
 &\leq C\sum_{|\alpha|\leq\ell}\int
       \bigl\|T\bigl(\partial_a^\alpha b(a,\cdot)f\bigr)\bigr\|_2^2\dd a
 \leq C\|T\|^2\sum_{|\alpha|\leq\ell}\int\!\!\int_U
       |\partial_a^\alpha b(a,u)|^2|f(u)|^2\dd u\dd a\\
 &= C\|T\|^2\int_U\|b(\cdot,u)\|_{H^\ell}^2|f(u)|^2\dd u
 \leq C\|T\|^2\sup_u\|b(\cdot,u)\|_{H^\ell}^2\|f\|_2^2 .
\end{split}
\]

(ii) For $a\in\R^d$ let $T_a$ be the operator
$T_ag(y)=(2\pi)^{-d}\int e^{\ii y\cdot\xi}b(a,y,\xi)^*g(\xi)\dd\xi$ from
$L^2(\R^d_\xi)$ to $L^2(\R^d_y)$, and put
$G(a,x)=(\mathcal F^{-1}T_a^*f)(x)$, where $\mathcal F^{-1}$ is the inverse
Fourier transform $g\mapsto\int e^{\ii x\cdot\xi}g(\xi)\dd\xi$. Writing out
$T_a^*f(\xi)=(2\pi)^{-d}\int e^{-\ii y\cdot\xi}b(a,y,\xi)f(y)\dd y$, we get
$G(a,x)=(2\pi)^{-d}\iint e^{\ii(x-y)\cdot\xi}b(a,y,\xi)f(y)\dd y\dd\xi$, and
hence $Bf(x)=G(x,x)$. As in~(i),
\[
 \|Bf\|_2^2\leq C\sum_{|\alpha|\leq\ell}\int
 \|\mathcal F^{-1}\partial_a^\alpha T_a^*f\|_2^2\dd a
 \leq C\sum_{|\alpha|\leq\ell}\int\|\partial_a^\alpha T_a\|^2\dd a\;\|f\|_2^2 .
\]
For each fixed $a$ and $\alpha$, the operator $\partial_a^\alpha T_a$ is,
up to the constant $(2\pi)^{-d}$ and the unitary Fourier transform, the
multiplication of the kernel of the inverse Fourier transform by the
amplitude $\partial_a^\alpha b(a,y,\xi)^*$. Part~(i), applied in the
variable $y$, bounds its norm by
$C\sup_\xi\|\partial_a^\alpha b(a,\cdot,\xi)\|_{H^\ell}\leq CA$, because
$|\alpha|\leq\ell$ derivatives in $a$ and at most $\ell$ in $y$ use at most
$2\ell$ derivatives. Since $b$ has compact support in $a$, this proves
$\|B\|\leq CA$. For unbounded frequency supports one first inserts a
frequency cutoff, which multiplies $T_a^*f$, and passes to the limit by
dominated convergence in $H^\ell_a(L^2_\xi)$; the bound is unchanged.
The same argument works if $\mathcal F^{-1}$ is replaced by any bounded
operator $L^2_\xi\to L^2_x$, at the cost of its norm.
\end{proof}

\begin{setup}[Phases with conic structure]\label{app:ch_osc:setup:osc-normal}
Write $x=(t,y)\in\R\times\R^{n-1}$ and $z=(s,v)$, where $v$ ranges over a
fixed compact set of remaining parameters. On fixed physical and
covector boxes let $S(x,\zeta)$ and $\zeta(z)$ be smooth and real, and put
\[
 \Phi_\tau(x,z,\eta)=\frac{S(x,\zeta(z)+\tau\eta)-S(x,\zeta(z))}{\tau}
 =\int_0^1 d_\zeta S(x,\zeta(z)+r\tau\eta)\cdot\eta\dd r,
 \qquad \phi_\epsilon=\Phi_\epsilon ,
\]
with the integral giving the value at $\tau=0$. The physical boxes are
convex. Fix convex product boxes $D_0\Subset D_1\Subset D$ in the
variables $(s,\theta)$, whose $\theta$-factors lie in one annulus
around $(s_b,\eta_b)$, $\eta_b\neq0$, and a smooth map
$\Psi_0(s,\theta)=(G(s,\theta),\theta)$ with $|\partial_sG|\geq c>0$. The
boxes are chosen so that $D\Psi_0$ stays close to one constant invertible
matrix, as in Lemma~\ref{app:lem:buffered-inverse-stability}. We assume
that the averaged $x$-gradients
\[
 \Psi_\epsilon^{\rm ann}(x,x',s,v,\theta)
 =\int_0^1d_x\phi_\epsilon\bigl(x'+r(x-x'),s,v,\theta\bigr)\dd r
\]
are uniformly close to $\Psi_0$ in $C^1(s,\theta)$ on $D$, and have fixed
joint $C^{M}$ bounds in the physical, data, $v$ and $\epsilon$ variables,
where $M=M_{\rm an}=20(n+1)$ and $0\leq\epsilon\leq\epsilon_0$. The
closeness is measured relative to the fixed domains and their margins.
For the phases of our construction these hypotheses are parts
(i)--(iii) of Proposition~\ref{app:ch_seed:prop:seed-phase-charts}; the
closeness uses only the low-order conditions (L) of
Setup~\ref{app:ch_seed:setup:seed-current}, and the $C^M$ bounds use the
high-order bound (H).

The amplitudes $a_\tau(x,z,\eta)$ have fixed compact physical and
parameter supports, with $(s,|\eta_b|\widehat\eta)$ in a fixed compact
subset of $D_0$. For $|\eta|\geq1/2$ and $\tau|\eta|$ sufficiently small
we assume
\[
 |D_{x,z}^\alpha D_\eta^\beta a_\tau|
       \leq A|\eta|^{-|\beta|},\qquad |\alpha|+|\beta|\leq M.
\]
Finally, $p_\Lambda$ is a real multiplier whose square is smooth,
supported in $\{1/2\leq|\eta|\leq C\Lambda\}$, with
$|D_\eta^\beta(p_\Lambda^2)|\leq C_\beta|\eta|^{-|\beta|}$; its lower
cutoff does not depend on $\Lambda$.
\end{setup}

\begin{prop}[Normal operators and fixed-parameter estimates]
\label{app:ch_osc:prop:osc-normal}\label{app:ch_osc:prop:osc-channel}
Assume Setup~\ref{app:ch_osc:setup:osc-normal}. The averaged full and
transverse gradients have inverse charts on common smaller domains, with
fixed joint $C^M$ bounds. For $\Lambda\geq1$ and $\tau\Lambda$
sufficiently small define
\[
 T_\tau f(x)=(2\pi)^{-n/2}\int e^{\ii\Phi_\tau(x,z,\eta)}
 |\eta|^{1/2}p_\Lambda(\eta)a_\tau(x,z,\eta)f(z,\eta)\dd z\dd\eta,
\]
and let $T_{\tau,z}$ denote the operator obtained by fixing $z$, acting on
functions of $\eta$. On the entire complex input and output spaces,
\begin{equation}\label{app:ch_osc:eq:osc-normal-basic}
 \|T_\tau\|\leq CA,\qquad
 \|T_{\tau,z}\|^2\leq CA^2\Lambda,\qquad
 \|T_{\tau,z}\|_{\rm HS}^2\leq CA^2\Lambda^n.
\end{equation}
Without the factor $|\eta|^{1/2}$ the norm of $T_{\tau,z}$ is at most
$CA$. The first bound is also uniform when $v$ is fixed, and at $\tau=0$
it holds without an upper frequency cutoff. On an unchanged phase, an
amplitude difference of size $d$ (in the norms above) changes $T_\tau$ by
at most $Cd$ in norm and $T_{\tau,z}$ by at most $C\sqrt\Lambda d$.

The normal operator $T_\tau T_\tau^*$ has an exact kernel of the form
\eqref{app:ch_osc:eq:osc-exact-normal-kernel}, with an amplitude $b_\tau$
of order zero, and it depends continuously on the data in operator norm:
if two sets of data (phases and amplitudes) satisfy the hypotheses with
the same constants and differ by at most $d$ in the joint $C^{2\ell+2}$
norms of the normalized phases $\Psi^{\rm ann}_\epsilon$ and of the
normalized amplitudes $a_\tau(x,z,|\eta|\omega)$, $\omega$ in the unit
sphere, with $\ell=\lfloor n/2\rfloor+1$, then the two normal operators
differ by at most $Cd$ in norm, where $C$ depends only on the constants
in the hypotheses. No such continuity is asserted for
$T_\tau$ itself when the phase changes: a phase difference of size $d$
is multiplied by the unbounded covector $\eta$ in $e^{\ii\Phi_\tau}$, and
$T_\tau$ is in general not continuous in operator norm with respect to
the phase. Only the normal operators and the uniform bounds are used
below.

Assume in addition that
$|D_{x,z}^\alpha D_\eta^\beta\partial_\tau a_\tau|\leq A|\eta|^{1-|\beta|}$
for $|\alpha|+|\beta|\leq M-1$. Then for every fixed smooth real weight
$m(s,v)$,
\begin{equation}\label{app:ch_osc:eq:osc-normal-mask}
 \|T_\tau mT_\tau^*-T_0mT_0^*\|
        \leq CA^2\tau\Lambda,
\end{equation}
where $C$ depends on finitely many derivatives of $m$.
\end{prop}

\begin{proof}
\emph{Step 1 (inverse charts).}
Lemma~\ref{app:lem:buffered-inverse-stability} gives inverse charts of
$(s,\theta)\mapsto\Psi_\epsilon^{\rm ann}$ on common domains. When $t'=t$,
the $y$-component of $\Psi^{\rm ann}_\epsilon$ is the averaged transverse
gradient, which is uniformly $C^1$ close to $\theta$; the same lemma
gives its inverse charts at fixed time. Implicit differentiation
(Lemma~\ref{app:ch_regular_base:lem:buffered-interpolation}) gives the
joint $C^M$ bounds of the inverses. We choose the closeness so that
\begin{equation}\label{new:eq:transverse-closeness}
 |\Psi^{\rm ann}_{\epsilon,y}-\theta|\leq\delta|\theta|,
\end{equation}
with $\delta$ small compared with the margins of the data boxes.

\emph{Step 2 (scaling).} Let $\Psi_\tau$ be the averaged $x$-gradient of
$\Phi_\tau$. Substituting $\eta=\rho\theta$ in the integral formula for
$\Phi_\tau$ gives $\Phi_\tau(x,z,\rho\theta)=\rho\,\Phi_{\tau\rho}(x,z,\theta)$,
hence
\begin{equation}\label{new:eq:normal-scaling}
 \Psi_\tau(x,x',s,v,\rho\theta)
       =\rho\,\Psi^{\rm ann}_{\tau\rho}(x,x',s,v,\theta).
\end{equation}
Suppose $\Psi_\tau(s,\eta)=\Psi_\tau(s',\eta')$ at fixed $(x,x',v)$. By
\eqref{new:eq:transverse-closeness} and \eqref{new:eq:normal-scaling}
the $y$-components satisfy $|\Psi_{\tau,y}-\eta|\leq\delta|\eta|$, so
$|\eta-\eta'|\leq\delta(|\eta|+|\eta'|)$, and $|\eta|$, $|\eta'|$ are
comparable. Hence one common $\rho>0$ puts both $\eta/\rho$ and
$\eta'/\rho$ in the annular chart, with the same parameter $\tau\rho$,
and injectivity of the annular map gives $(s,\eta)=(s',\eta')$. Thus
$(s,\eta)\mapsto\Psi_\tau$ is injective on the whole cone; the same
argument applies at fixed time. The support margins and a small
threshold for $\tau\Lambda$ keep $(s,\eta/\rho)$ in $D_0$ for the choice
$\rho=|k_y|/|\eta_b|$ made below, which depends only on the output point
$k$.

For fixed $\rho$, differentiating \eqref{new:eq:normal-scaling} in
$(s,\theta)$ shows that
$\det D_{(s,\eta)}\Psi_\tau=\rho\det D_{(s,\theta)}\Psi^{\rm ann}_{\tau\rho}$,
and the second determinant is bounded above and below. Since
$\rho\asymp|\eta|$,
\begin{equation}\label{app:ch_background:eq:bg-jacobian}
 J_\tau=|\det D_{(s,\eta)}\Psi_\tau|\asymp|\eta|,
 \qquad |\det D_\eta\Psi_{\tau,y}|\asymp1 .
\end{equation}
By the same scaling, each derivative of the inverse map in the covector
variable lowers its order of homogeneity by one, while physical
derivatives preserve it, uniformly on the cone.

\emph{Step 3 (the kernel of the normal operator).} The kernel of
$T_\tau T_\tau^*$ is
\[
 (2\pi)^{-n}\int e^{\ii(\Phi_\tau(x,z,\eta)-\Phi_\tau(x',z,\eta))}
 |\eta|p_\Lambda^2(\eta)a_\tau(x,z,\eta)a_\tau(x',z,\eta)^*\dd z\dd\eta .
\]
By the fundamental theorem of calculus along the segment from $x'$ to
$x$, the phase equals $(x-x')\cdot\Psi_\tau(x,x',z,\eta)$ exactly.
Changing variables $(s,\eta)\mapsto k=\Psi_\tau(x,x',s,v,\eta)$, which is
legitimate by Step~2, we obtain the kernel
\begin{equation}\label{app:ch_osc:eq:osc-exact-normal-kernel}
 (2\pi)^{-n}\int e^{\ii(x-x')\cdot k}b_\tau(x,x',k)\dd k,
 \qquad b_\tau=\int
 \left.\frac{|\eta|\,p_\Lambda^2\,a_\tau(x,z,\eta)a_\tau(x',z,\eta)^*}
 {J_\tau}\right|_{(s,\eta)=\Psi_\tau^{-1}(k)}\dd v.
\end{equation}
The factor $|\eta|$ cancels the Jacobian, so $b_\tau$ is bounded by
$CA^2$. Its $(x,x')$-derivatives are bounded by $CA^2$ as well: a
physical derivative falling on $a_\tau$ costs $A$, one falling on the
inverse point $(s,\eta)$ produces a derivative of $\eta$ of order
$|\eta|$, which is compensated by the decay $|\eta|^{-1}$ of the
$\eta$-derivative of $a_\tau$, $p_\Lambda^2$ or $|\eta|/J_\tau$. The
amplitude vanishes near the boundary of the inverse charts, so its
extension by zero is smooth.

\emph{Step 4 (the full bound).} Part~(ii) of
Lemma~\ref{app:ch_osc:lem:osc-amplitude}, with $d=n$ and $2\ell\leq M$,
gives $\|T_\tau T_\tau^*\|\leq CA^2$, that is, $\|T_\tau\|\leq CA$. At
$\tau=0$ the amplitude bounds do not use the upper cutoff, and the
estimate without it follows by inserting a frequency cutoff and letting
it tend to one.

\emph{Step 4$'$ (continuity of the normal operator).} Let primed
quantities refer to a second set of data. Both normal operators have
kernels of the form \eqref{app:ch_osc:eq:osc-exact-normal-kernel} with the
same exponential $e^{\ii(x-x')\cdot k}$, so by
Lemma~\ref{app:ch_osc:lem:osc-amplitude}(ii) it suffices to show
$\sup_k\|b_\tau-b_\tau'\|_{C^{2\ell}_{x,x'}}\leq Cd$. By
\eqref{new:eq:normal-scaling}, the inverse point $(s,\eta)$ of $k$ is
$(s,\rho\theta)$, where $(s,\theta)$ is the inverse of the normalized map
$\Psi^{\rm ann}_{\tau\rho}$ at $k/\rho$ and $\rho=|k_y|/|\eta_b|$ depends
only on $k$. Implicit differentiation, with the common lower bound for
the Jacobian, shows that the normalized inverses of the two data sets
differ by $O(d)$ in $C^{2\ell+1}$. Hence $\eta-\eta'=O(d|\eta|)$ at the
two inverse points, together with $2\ell$ physical derivatives. Each
factor of the integrand of $b_\tau$, namely $a_\tau$, $p_\Lambda^2$ and
$|\eta|/J_\tau$, has $\eta$-derivative $O(|\eta|^{-1})$, so it changes by
$O(d)$ when $\eta$ is replaced by $\eta'$, and by $O(d)$ when the data
are replaced at a fixed point. This proves the continuity assertion. The
argument concerns the kernel of $T_\tau T_\tau^*$ after the change of
variables $k=\Psi_\tau$; no comparison of the operators $T_\tau$ with
different phases is involved.

\emph{Step 5 (fixed parameters).} Fix $z$ and $t$. The operator
$f\mapsto T_{\tau,z}f(t,\cdot)$ acts from $L^2(d\eta)$ to $L^2(dy)$. Its
normal operator has the kernel
$(2\pi)^{-n}\int e^{\ii(y-y')\cdot\Psi_{\tau,y}}|\eta|p_\Lambda^2a_\tau a_\tau^*\dd\eta$,
and the change of variables $\eta\mapsto\kappa=\Psi_{\tau,y}$, whose
Jacobian is comparable to one by \eqref{app:ch_background:eq:bg-jacobian},
brings it to the form of Lemma~\ref{app:ch_osc:lem:osc-amplitude}(ii) in
dimension $n-1$, with amplitude bounded by $CA^2|\eta|\leq CA^2\Lambda$.
Hence $\|T_{\tau,z}f(t,\cdot)\|_{L^2_y}^2\leq CA^2\Lambda\|f\|_2^2$, and
integration over the fixed $t$-interval gives
$\|T_{\tau,z}\|^2\leq CA^2\Lambda$. Without the factor $|\eta|^{1/2}$ the
same argument gives $CA^2$. The Hilbert--Schmidt norm is the $L^2$ norm
of the kernel:
\[
 \|T_{\tau,z}\|_{\rm HS}^2
 \leq CA^2\int_{\{|\eta|\leq C\Lambda\}}|\eta|\dd\eta
 \leq CA^2\Lambda^n ,
\]
since $\eta\in\R^{n-1}$. Amplitude differences are estimated by the same
argument applied to the difference divided by $d>0$.

\emph{Step 6 (comparison).} Insert the weight $m(s,v)$ in the numerator
of \eqref{app:ch_osc:eq:osc-exact-normal-kernel}, and call the resulting
amplitude $b_\tau^m$. At fixed $k$ the inverse point is obtained from the
annular inverse at parameter $\tau\rho$, with $\rho\asymp|\eta|$ independent
of $\tau$. Hence $\partial_\tau s=O(|\eta|)$ and
$\partial_\tau\eta=O(|\eta|^2)$, together with their physical
derivatives. Consequently
\[
 \bigl|\partial_\tau\bigl[a_\tau(x,z,\eta(\tau))\bigr]\bigr|
 \leq|\partial_\tau a_\tau|+|\partial_sa_\tau||\partial_\tau s|
 +|\partial_\eta a_\tau||\partial_\tau\eta|
 \leq CA|\eta|,
\]
and the same bound holds for $p_\Lambda^2$, the weight $m$ and
$|\eta|/J_\tau$, each of which loses one power of $|\eta|$ under an
$\eta$-derivative. On the support, $|k|\asymp|\eta|\leq C\Lambda$. The
mean value theorem in $\tau$ therefore gives
\[
 \sup_k|D_{x,x'}^\alpha(b_\tau^m-b_0^m)|
       \leq CA^2\tau\Lambda,\qquad |\alpha|\leq2(n+1),
\]
and Lemma~\ref{app:ch_osc:lem:osc-amplitude}(ii), applied to the
difference of amplitudes, proves \eqref{app:ch_osc:eq:osc-normal-mask}.
\end{proof}

\begin{cor}[Real and imaginary parts of the input]\label{app:ch_osc:cor:osc-real}
Regard the complex input space as a real Hilbert space with inner
product $\Re\langle\cdot,\cdot\rangle$, and put $C=2\Re T$. For real
outputs $r$,
\begin{equation}\label{app:ch_osc:eq:osc-factor-four}
 C^*r=2T^*r,\qquad CmC^*r=4\Re(TmT^*r).
\end{equation}
Thus the preceding bounds hold for the real operators, with fixed
numerical factors, when both the real and the imaginary parts of the
input are used.
\end{cor}

\begin{proof}
For real $r$ and complex $f$,
$\langle2\Re Tf,r\rangle=2\Re\langle Tf,r\rangle=\Re\langle f,2T^*r\rangle$,
which is the first identity. Applying $C=2\Re T$ to $2mT^*r$ gives the
second.
\end{proof}

\subsection{Fourier localization of oscillatory functions}
\label{app:ch_osc:sec:osc-tails}

\begin{prop}[Fourier tails of oscillatory functions]\label{app:ch_osc:prop:osc-source-shell}
Fix $n\geq1$, a bounded box $X\subset\R^n$, a compact set $K\Subset X$ and
$0<c<C$. For $\rho\geq2$ and $L\geq1$ let $\phi$ be a smooth real phase and
$a$ a smooth complex amplitude of fixed finite dimension, supported in
$K$, with
\[
 c\leq|d_x\phi|\leq C,\qquad
 |D_x^r\phi|\leq C_rL^{r-1}\ (r\geq2),\qquad
 |D_x^ra|\leq C_rAL^r\ (r\geq0),
\]
where $A\geq0$. Put $h_\rho=e^{\ii\rho\phi}a$. Let $\Pi_\rho=\pi(D/\rho)$,
where $\pi$ is a fixed smooth radial function equal to $1$ on
$\{c/2\leq|\xi|\leq2C\}$ and supported in $\{c/4\leq|\xi|\leq4C\}$. Then
for all 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\rho^{m+n/2}(L/\rho)^q.
\end{equation}
More generally, the same estimate holds for any fixed bounded rescaled
Fourier cutoff, provided that on the support of $1-\Pi_\rho$ one has
$|\rho d_x\phi(x)-\xi|\geq c_1(\rho+|\xi|)$ for all $x\in K$.
\end{prop}

\begin{proof}
The Fourier transform is
\[
 \widehat h_\rho(\xi)=(2\pi)^{-n}\int_K
 e^{\ii[\rho\phi(x)-x\cdot\xi]}a(x)\dd x.
\]
On the support of $1-\pi(\xi/\rho)$ either $|\xi|<c\rho/2$ or
$|\xi|>2C\rho$. In the first case
$|\rho d\phi-\xi|\geq c\rho/2$ and $\rho+|\xi|\leq(1+c/2)\rho$. In the
second case $|\rho d\phi-\xi|\geq|\xi|-C\rho\geq|\xi|/2$, and
$|\xi|/2>C\rho\geq c\rho$. In both cases, with
$c_1=\min\{c/(2+c),1/4,c/2\}$,
\[
 |\rho d_x\phi(x)-\xi|\geq c_1(\rho+|\xi|)\qquad(x\in K).
\]
Let $w=\rho d_x\phi-\xi$ and
\[
 \mathcal V_\xi=\frac{w\cdot\partial_x}{\ii|w|^2},
 \qquad\mathcal V_\xi e^{\ii[\rho\phi-x\cdot\xi]}=e^{\ii[\rho\phi-x\cdot\xi]} .
\]
Since $a$ vanishes near $\partial K$, integration by parts $q$ times gives
\[
 \widehat h_\rho(\xi)=(2\pi)^{-n}\int e^{\ii[\rho\phi-x\cdot\xi]}
 (\mathcal V_\xi^T)^qa\dd x,\qquad
 \mathcal V_\xi^Tf=-\partial_x\cdot\Bigl(\frac{w}{\ii|w|^2}f\Bigr).
\]
The coefficient $w/|w|^2$ is bounded by $C/(\rho+|\xi|)$. Its $j$th
derivative is a sum of terms involving derivatives of $w$, that is
$\rho D^{k+1}\phi$, divided by powers of $|w|$; since
$\rho|D^{k+1}\phi|\leq C\rho L^{k}\leq CL^k(\rho+|\xi|)$, an induction on
$j$ shows that the $j$th derivative is bounded by
$C_jL^j(\rho+|\xi|)^{-1}$. Each derivative of $a$ costs a factor $L$. By
induction on $q$, $(\mathcal V_\xi^T)^qa$ is a sum of terms bounded by
$C_qAL^q(\rho+|\xi|)^{-q}$, so
\[
 \bigl|(1-\pi(\xi/\rho))\widehat h_\rho(\xi)\bigr|
 \leq C_qAL^q(\rho+|\xi|)^{-q}.
\]
For $q>m+n/2$ the substitution $\xi=\rho\omega$ gives
\[
 \int_{\R^n}\langle\xi\rangle^{2m}(\rho+|\xi|)^{-2q}\dd\xi
 \leq C\rho^{2m+n-2q}\int_{\R^n}(1+|\omega|)^{2m-2q}\dd\omega
 \leq C_{m,q}\rho^{2m+n-2q}.
\]
By Plancherel,
$\|(I-\Pi_\rho)h_\rho\|_{H^m}^2\leq C_{m,q}A^2L^{2q}\rho^{2m+n-2q}$, which
is \eqref{app:ch_osc:eq:osc-source-tail}. For a general mask the proof
used only the lower bound for $|\rho d\phi-\xi|$ on its complement.
\end{proof}

\begin{rem}\label{app:ch_osc:rem:osc-difference-frequency}
Proposition~\ref{app:ch_osc:prop:osc-source-shell} is applied with
$\rho=\lambda|z-z'|$ to normalized phase differences
$\lambda[\Theta(x,z)-\Theta(x,z')]/\rho$ of two waves in the same
characteristic patch. The phase difference is used exactly, not expanded
in the patch parameter; the hypotheses of the proposition are verified
for it in Proposition~\ref{app:ch_seed:prop:seed-actual-menu}.
\end{rem}

%% ===== 08_09_parametrix.tex =====
\section{A parametrix on regular patches}\label{app:sec:ch_regular_base}

In this section we construct a parametrix for a real first-order
$r\times r$ system on a conic region of covectors where the system is
elliptic, or has one simple real characteristic root. The construction
is used in Section~\ref{app:sec:ch_regular} to build finite corrections,
which in turn correct the characteristic waves
(Section~\ref{main:sec:linear}) and the oscillatory quadratic terms
(Section~\ref{app:sec:ch_pairs}).

\subsection{The system and the main result}\label{app:ch_regular_base:sec:introduction}
Consider the real 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}
whose coefficient array $a=(A_1,\ldots,A_n,B)$ of real $r\times r$
matrices is smooth near a point $x_0\in\R^n$. Put
\begin{equation}\label{app:ch_regular_base:eq:initial-symbols}
 p_a=\det H_a .
\end{equation}
A real zero $\xi\neq0$ of $p_a(x,\cdot)$ is a simple characteristic if
$\partial_\xi p_a(x,\xi)\ne0$. On a characteristic patch we use one
retarded oscillatory integral with a scalar Hamilton phase and two matrix
amplitudes, together with an algebraic inverse on the complementary
directions; on an elliptic patch we invert the principal symbol.

\begin{thm}[A regular patch]\label{app:ch_regular_base:thm:sector}
Fix a smooth reference coefficient array $a_0$ for
\eqref{app:ch_regular_base:eq:initial-system} and nested closed conic sets
$\Gamma_{\rm in}\Subset\Int\Gamma_{\rm out}$. Assume that on a fixed
spatial neighborhood of $x_0$ one of the following holds:
\begin{enumerate}
\item $H_{a_0}(x,\xi)$ is invertible for $\xi\in\Gamma_{\rm out}\setminus0$;
\item in a coordinate system $x=(t,y)$, $\xi=(\tau,\eta)$, the
equation $p_{a_0}=0$ has on $\Gamma_{\rm out}$ exactly one root
$\tau=h_0(x,\eta)$, which is simple, with $|\eta|\asymp|\xi|$ and
$|\partial_\tau p_{a_0}|$ bounded below on unit covectors.
\end{enumerate}
No condition is imposed outside $\Gamma_{\rm out}$. Then, after one
spatial shrinking, there are fixed balls $U\Subset W$, a neighborhood of
$a_0$ in a fixed $C^k$ norm, and a scalar pseudodifferential cutoff
$A_\Gamma$, equal to the identity microlocally on $\Gamma_{\rm in}$ at
high frequency and with conic support in $\Gamma_{\rm out}$, such that
the following holds for every smooth real $a$ in that neighborhood. For
$h\in C_c^\infty(W,\C^r)$ there are complex linear operators
$S_{a,\Gamma}$, $R_{a,\Gamma}$ with smooth outputs, satisfying
\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}
and
\begin{equation}\label{app:ch_regular_base:eq:sector-theorem-solution}
 \|S_{a,\Gamma}h\|_{L^2(W)}
 +\|R_{a,\Gamma}h\|_{H^1(U)}\leq C\|h\|_{L^2(W)}.
\end{equation}
For every integer $s\geq0$ the operators
$S_{a,\Gamma}:H^s(W)\to H^s(W)$ and
$R_{a,\Gamma}:H^{\max(s-1,0)}(W)\to H^s(U)$ have norms bounded by
polynomials in finitely many coefficient seminorms. The residual
$R_{a,\Gamma}$ is a finite sum of pseudodifferential operators of order
$-1$ and of operators containing one retarded oscillatory integral with
an amplitude of order $-1$; it includes all source and low-frequency
terms. The domains, the coefficient neighborhood, the derivative losses
and the orders of the operators are fixed before $s$ and before any
later accuracy. All statements include fixed input and output cutoffs.
\end{thm}

The proof is given at the end of the section, after the necessary
estimates.

\subsection{Pseudodifferential estimates}

\begin{setup}[Symbols on fixed boxes]
\label{app:ch_regular_base:setup:finite-calculus}
All evaluations of coefficients and all Hamilton trajectories lie in
fixed relatively compact subdomains of $W$. Local symbols and amplitudes
are multiplied by their cutoffs before being extended by zero, so their
seminorms are controlled by the coefficient norms on $W$. Every
homogeneous symbol is multiplied by one fixed cutoff vanishing near
$\xi=0$, and the complementary low-frequency part is kept in the
formulas. We write
\[
 T_bf(x)=(2\pi)^{-n}\iint e^{\ii(x-z)\cdot\xi}
                      b(x,z,\xi)f(z)\dd z\dd\xi,
\]
and $\operatorname{Op}(b)=T_b$ when $b$ does not depend on $z$. For an
amplitude of order $p$ we use the seminorms
\[
 |b|_{p,j}=\max_{|\alpha|+|\beta|+|\gamma|\leq j}
 \sup\langle\xi\rangle^{-p+|\gamma|}
 |\partial_x^\alpha\partial_z^\beta\partial_\xi^\gamma b|.
\]
\end{setup}

\begin{lem}[Pseudodifferential bounds]
\label{app:ch_regular_base:thm:actual-calculus}
An amplitude of order $0$ defines a bounded operator on $L^2$, and an
amplitude of order $-1$ defines an operator which, together with its
adjoint, maps $L^2$ to $H^1$. For each integer $s\geq0$ these operators
map $H^{\max(s+p,0)}$ to $H^s$, where $p\in\{0,-1\}$ is the order. The
bounds involve finitely many symbol seminorms, a fixed number of them
beyond $s$, and depend polynomially on finitely many coefficient
seminorms.
\end{lem}

\begin{proof}
Lemma~\ref{app:ch_osc:lem:osc-amplitude}(ii) gives
$\|T_b\|_{2\to2}\leq C|b|_{0,2(n+1)}$ (only $x$- and $z$-derivatives are
used). For order $-1$, an output derivative gives
$D_{x_j}T_b=T_{\xi_jb+D_{x_j}b}$, which has order zero, so $T_b:L^2\to H^1$;
the adjoint of $T_b$ is $T_{b^\dagger}$ with $b^\dagger(x,z,\xi)=b(z,x,\xi)^*$,
an amplitude of the same order. Integration by parts in $z$ gives the
exact identity
\[
 D_x^\alpha T_bf
   =\sum_{\beta\leq\alpha}\binom{\alpha}{\beta}
       T_{D_{x+z}^\beta b}D^{\alpha-\beta}f,
 \qquad D_{x+z}=D_x+D_z,
\]
from which the $H^s$ bounds follow by the $L^2$ estimate; for order $-1$
we first take one output derivative and then argue at order $s-1$. The
symbols are fixed algebraic expressions in the coefficients, which gives
the polynomial dependence.
\end{proof}

\subsection{Retarded oscillatory integrals}

\begin{lem}[Bounds for retarded kernels]
\label{app:ch_regular_base:lem:retarded-kernel}
Let $d=n-1$ and let $\Psi(t,s,y,\eta)$ be homogeneous of degree one in
$\eta$, with $\Psi(s,s,y,\eta)=y\cdot\eta$ and $\Psi_{y\eta}$ uniformly
close to $I$ on fixed boxes with interior margins. For a spatial
amplitude $b(t,s,y,\eta)$ of order $p\in\{0,-1\}$ put
\[
 U_b(t,s)f(y)=(2\pi)^{-d}\iint
 e^{\ii(\Psi(t,s,y,\eta)-z\cdot\eta)}b(t,s,y,\eta)f(z)\dd z\dd\eta,
\]
and
\[
 \mathcal E_bf(t)=\int_{t_-}^tU_b(t,s)f(s)\dd s .
\]
Assume that $b$ has fixed physical and conic margins, contains the fixed
low-frequency cutoff, and vanishes for $s$ near the initial time $t_-$.
Let $B=\beta(D_t,D_y)$ be a fixed Fourier multiplier of order zero
supported in $\{|\tau|\leq C|\eta|\}$. Then
$\mathcal E_bB:H^{\max(s+p,0)}\to H^s$ for every integer $s\geq0$, with
bounds given by finite products of seminorms of the phase and the
amplitude.
\end{lem}

\begin{proof}
\emph{Fixed times.} For fixed $t,s$, the operator $U_b(t,s)U_b(t,s)^*$
has the kernel
\[
 (2\pi)^{-d}\int e^{\ii(\Psi(t,s,y,\eta)-\Psi(t,s,y',\eta))}
 b(t,s,y,\eta)b(t,s,y',\eta)^*\dd\eta .
\]
The phase difference equals $(y-y')\cdot\widetilde\Psi$, where
$\widetilde\Psi=\int_0^1\Psi_y(t,s,y'+\theta(y-y'),\eta)\dd\theta$ is
homogeneous of degree one in $\eta$ and, since $\Psi_{y\eta}$ is close to
$I$, defines a homogeneous change of variables $\eta\mapsto\widetilde\Psi$
with Jacobian close to one. After this change of variables the kernel has
the form of Lemma~\ref{app:ch_osc:lem:osc-amplitude}(ii) with an amplitude
of order zero, so $\|U_b(t,s)\|_{L^2\to L^2}\leq C$ uniformly.

\emph{A fixed-time representation.} Put $x=(t,y)$, $\xi=(\tau,\eta)$ and
$r=t-s$. Since $b$ vanishes near $s=t_-$, we may extend it by zero to
$s<t_-$, and on the fixed output support we obtain
\begin{equation}\label{app:ch_regular_base:eq:fixed-time-representation}
\begin{gathered}
 \mathcal E_bB=\int_0^\Delta\mathcal U_{r,b}\dd r,
 \qquad \Theta_r(t,y,\tau,\eta)=(t-r)\tau+\Psi(t,t-r,y,\eta),\\
 \mathcal U_{r,b}f(x)=(2\pi)^{-n}\iint
 e^{\ii[\Theta_r(x,\xi)-z\cdot\xi]}
 b(t,t-r,y,\eta)\beta(\xi)f(z)\dd z\dd\xi ,
\end{gathered}
\end{equation}
with a fixed $\Delta>0$. Indeed, $\mathcal U_{r,b}f(t,\cdot)$ is
$U_b(t,t-r)$ applied to $(Bf)(t-r,\cdot)$, whose time Fourier variable
produces the factor $e^{\ii(t-r)\tau}$. By time translation and the
fixed-time bound, each $\mathcal U_{r,b}$ of order zero is bounded on
$L^2$ uniformly in $r$. For amplitudes depending on both positions we
use the last assertion of Lemma~\ref{app:ch_osc:lem:osc-amplitude}, with
outer operator $\mathcal U_{r,*}\mathcal F^{-1}$ and fixed larger spatial
and Fourier cutoffs equal to one on the support of the amplitude.

\emph{Derivatives.} On the support of $\beta$ we have
$|\tau|\leq C|\eta|$, so the amplitude has full spacetime order $p$. An
output derivative of the phase factor produces an order-zero coefficient
times $\xi$, and integration by parts in $z$ transfers this factor to a
derivative of the input. For $p=-1$ one output derivative first makes
the amplitude of order zero. The full-amplitude bound then gives the
$H^s$ estimates uniformly in $r$, and integration over $[0,\Delta]$
completes the proof.
\end{proof}

\subsection{A characteristic patch}

\begin{lem}[A characteristic parametrix]
\label{app:ch_regular_base:lem:characteristic-kernel}
On a regular characteristic patch write $x=(t,y)$, $\xi=(\tau,\eta)$,
with $|\eta|\asymp|\xi|$ and $|\partial_\tau p_a|$ bounded below. Choose a
fixed Fourier cutoff $\beta(\tau,\eta)$ supported in this cone, including
the low-frequency cutoff, and a spatial source cutoff $\sigma(x,\eta)$
whose forward Hamilton flowout over the time interval stays in the
patch. Put
\[
 B=\beta(D_t,D_y),\qquad \alpha=\sigma\beta,
 \qquad A=\operatorname{Op}(\alpha).
\]
There is a fixed operator
\[
 S=\chi_{\rm out}\Bigl(\mathcal E_WB+\frac1{\ii}\operatorname{Op}(b_{-1}\alpha)\Bigr),
\]
where $W=W_0+W_1$ is an amplitude with terms of orders $0$ and $-1$ and
$b_{-1}$ is a symbol of order $-1$, such that $P_aS=A+R$ on the inner
physical box, with
\begin{equation}\label{app:ch_regular_base:eq:energy-complete-residual}
 R=\operatorname{Op}(A_tW_1(t,t)\beta)
       +\mathcal E_{P_aW_1}B
       +\frac1{\ii}\operatorname{Op}\bigl(P_x(b_{-1}\alpha)\bigr).
\end{equation}
Here $A_t$ is the coefficient of $\partial_t$, $W_1(t,t)$ denotes the
amplitude at $s=t$, and $P_x$ applies $P_a$ to the physical variable of a
symbol. The construction uses a Hamilton phase and scalar transport. If
the coefficients satisfy $\|a\|_{C^q}\leq C_qT^{(q-R_*)_+}$, then the
phase gradients, flow maps and amplitudes satisfy the same normalized
mixed derivative bounds with $R_*$ replaced by $R_*-c$, for a fixed loss
$c$.
\end{lem}

\begin{proof}
Write $P=P_a$, $H=H_a$, and let $\tau=h(x,\eta)$ be the simple root.

\emph{Null vectors.} Since the root is simple,
$\partial_\tau p=\operatorname{tr}(\operatorname{adj}H\cdot A_t)\neq0$ at
$\tau=h$, so $\operatorname{adj}H\neq0$ and $H(x,h,\eta)$ has corank one.
Using a fixed nonvanishing minor, choose smooth real vectors
$k(x,\eta)$, $l(x,\eta)$, homogeneous of degree zero, with
\[
 H(x,h,\eta)k=0,\qquad l^TH(x,h,\eta)=0,
 \qquad l^TA_tk=1 .
\]
The last normalization is possible because
$\operatorname{adj}H(x,h,\eta)=c\,kl^T$ with $c\neq0$, so
$\partial_\tau p=c\,l^TA_tk\neq0$. Since $H$ is linear in $\tau$,
\begin{equation}\label{new:eq:left-null}
 l^TH(x,\tau,\eta)=(\tau-h)\,l^TA_t .
\end{equation}

\emph{The algebraic inverse.} For $\tau$ near $h$, $H(x,\tau,\eta)$ maps
the hyperplane $\ker(l^TA_t)$ into $\ker l^T$, by \eqref{new:eq:left-null}.
At $\tau=h$ this map is an isomorphism: its kernel is
$\R k\cap\ker(l^TA_t)=0$, and both spaces have dimension $r-1$. Hence it
remains an isomorphism on a fixed conic neighborhood of the root, and we
choose $\beta$ supported in that neighborhood. Put
\begin{equation}\label{app:ch_regular_base:eq:complement-inverse}
 \Pi=A_tkl^T,\qquad
 b_{-1}(x,\tau,\eta)=\bigl(H(x,\tau,\eta)|_{\ker(l^TA_t)\to\ker l^T}\bigr)^{-1}(I-\Pi).
\end{equation}
Since $\Pi^2=\Pi$ and $l^T(I-\Pi)=0$, the matrix $I-\Pi$ takes values in
$\ker l^T$, and by construction
\begin{equation}\label{app:ch_regular_base:eq:complement-symbol}
 H(x,\tau,\eta)\,b_{-1}(x,\tau,\eta)=I-\Pi .
\end{equation}
The restricted matrix is homogeneous of degree one, so $b_{-1}$ is
homogeneous of degree $-1$. Its product with $\alpha$ is extended by zero
outside the fixed conic neighborhood and is a spacetime symbol of order
$-1$.

\emph{The phase.} On a common short time interval,
Lemma~\ref{app:lem:common-hamilton-phase}, with initial time $s$, gives
\[
 \Psi_t=h(t,y,\Psi_y),\qquad \Psi(s,s,y,\eta)=y\cdot\eta,
 \qquad \Psi_{y\eta}\ \text{uniformly close to }I .
\]
Below, $k$ and $l$ are evaluated at $(t,y,\Psi_y(t,s,y,\eta))$. We choose
$\sigma$ equal to one on the inner source region, with its complete
flowout inside the larger chart, and vanishing near the initial time.
Since $|\Psi_y|\asymp|\eta|$, nested low-frequency cutoffs leave every
homogeneous formula unchanged near the transported support of $\sigma$.

\emph{Transport.} Let the row vector $a(t,s,y,\eta)$ solve
\[
 (\partial_t-h_\eta\cdot\partial_y)a+l^T(Pk)\,a=0,
 \qquad a(s,s,y,\eta)=l(s,y,\eta)^T\sigma(s,y,\eta),
\]
where $h_\eta$ is evaluated at $(t,y,\Psi_y)$ and $Pk$ denotes $P$ applied
to the function $(t,y)\mapsto k(t,y,\Psi_y)$. Put $W_0=ka$. Differentiating
$H(x,h(x,\eta),\eta)k(x,\eta)=0$ in $\eta_j$ and multiplying by $l^T$ gives
$h_{\eta_j}+l^TA_{y_j}k=0$, where $A_{y_j}$ is the coefficient of
$\partial_{y_j}$. Therefore
\[
 l^TPW_0=l^TA_tk\,\partial_ta+\sum_jl^TA_{y_j}k\,\partial_{y_j}a+l^T(Pk)a
 =(\partial_t-h_\eta\cdot\partial_y)a+l^T(Pk)a=0 .
\]
Moreover, since $\Psi_t=h(t,y,\Psi_y)$,
\[
 P(e^{\ii\Psi}W_0)=e^{\ii\Psi}\bigl(\ii H(x,\Psi_t,\Psi_y)W_0+PW_0\bigr)
 =e^{\ii\Psi}PW_0 .
\]
Put $W_1=\ii\,b_{-1}(t,y,\Psi_t,\Psi_y)PW_0$. By
\eqref{app:ch_regular_base:eq:complement-symbol} and $l^TPW_0=0$ (which
gives $\Pi PW_0=0$), we get $HW_1=\ii(I-\Pi)PW_0=\ii PW_0$, and hence
\[
 P(e^{\ii\Psi}W_1)=e^{\ii\Psi}\bigl(\ii HW_1+PW_1\bigr)=e^{\ii\Psi}(-PW_0+PW_1).
\]
Adding, we obtain the exact cancellation
\begin{equation}\label{app:ch_regular_base:eq:two-term-null-identity}
 P\{e^{\ii\Psi}(W_0+W_1)\}=e^{\ii\Psi}PW_1 .
\end{equation}
No pseudodifferential operator acts on the phase. The transported
support stays in the patch, and all symbols are extended by zero only
outside a fixed larger margin.

\emph{The residual.} For $W=W_0+W_1$, differentiation of the retarded
integral gives
\[
 P\mathcal E_Wg(t)=A_tU_W(t,t)g(t)+\int_{t_-}^tPU_W(t,s)g(s)\dd s .
\]
By \eqref{app:ch_regular_base:eq:two-term-null-identity},
$PU_W(t,s)=U_{PW_1}(t,s)$, and since $\Psi(t,t,y,\eta)=y\cdot\eta$, the
operator $U_W(t,t)$ is the spatial pseudodifferential operator
$\operatorname{Op}_y(W(t,t))$. Because $B$ is a Fourier multiplier and
$W(t,t)$ does not depend on $\tau$,
$\operatorname{Op}_y(A_tW(t,t))B=\operatorname{Op}(A_tW(t,t)\beta)$
exactly. Moreover $A_tW_0(t,t)=A_tk\,l^T\sigma=\Pi\sigma$, so
\[
 P\mathcal E_WB=\operatorname{Op}(\Pi\alpha)
 +\operatorname{Op}(A_tW_1(t,t)\beta)+\mathcal E_{PW_1}B .
\]
For the algebraic term, $P\operatorname{Op}(c)=\operatorname{Op}(\ii Hc+P_xc)$,
so by \eqref{app:ch_regular_base:eq:complement-symbol}
\[
 \frac1{\ii}P\operatorname{Op}(b_{-1}\alpha)
 =\operatorname{Op}((I-\Pi)\alpha)+\frac1{\ii}\operatorname{Op}\bigl(P_x(b_{-1}\alpha)\bigr).
\]
Adding these identities gives $PS=A+R$ with $R$ as in
\eqref{app:ch_regular_base:eq:energy-complete-residual}, on the set
where $\chi_{\rm out}=1$; there the commutator of $P$ with $\chi_{\rm out}$
vanishes. Both pseudodifferential terms in $R$ have spacetime order $-1$,
because $\beta$ restricts to $|\tau|\leq C|\eta|$, and the retarded term
gains one derivative by Lemma~\ref{app:ch_regular_base:lem:retarded-kernel}.
These are complete formulas, including low frequencies, and a common
input cutoff can be placed on the right of all terms.

\emph{Derivative bounds.} Every division has a fixed lower bound.
Lemma~\ref{app:ch_regular_base:lem:buffered-interpolation}, applied to the
Hamilton flow, its inverse charts and the scalar transport equation,
gives the stated derivative bounds, and they pass to
$W_0$, $W_1$, $PW_1$ and $b_{-1}$ through the explicit formulas and the
fixed cutoffs. One fixed loss in $R_*$ covers all these factors,
independently of later compositions.
\end{proof}

\subsection{Elliptic patches and the proof of the main result}
\label{app:ch_regular_base:sec:energy-construction}

On an elliptic patch choose a scalar source symbol $\alpha$, equal to one
on the inner region and containing the fixed low-frequency cutoff, and
put $S=\chi_{\rm out}\operatorname{Op}(H_a^{-1}\alpha)/\ii$. As above,
$PS=\operatorname{Op}(\alpha)+R$ where $\chi_{\rm out}=1$, with the
residual $R=\operatorname{Op}(P_x(H_a^{-1}\alpha))/\ii$ of order $-1$.

\begin{proof}[Proof of Theorem~\ref{app:ch_regular_base:thm:sector}]
In case (i) use the elliptic construction just described, and in case
(ii) the parametrix of Lemma~\ref{app:ch_regular_base:lem:characteristic-kernel};
in both cases $A_\Gamma=\operatorname{Op}(\alpha)$ with $\alpha$ equal to
one on $\Gamma_{\rm in}$ at high frequency and supported in
$\Gamma_{\rm out}$. All hypotheses are open conditions on a fixed
finite jet of the coefficients, which gives the coefficient
neighborhood. This proves \eqref{app:ch_regular_base:eq:sector-theorem-identity},
with the residual of the stated form. The bounds
\eqref{app:ch_regular_base:eq:sector-theorem-solution} and the $H^s$
bounds follow from Lemmas~\ref{app:ch_regular_base:thm:actual-calculus}
and~\ref{app:ch_regular_base:lem:retarded-kernel}, whose constants are
polynomial in finitely many coefficient seminorms. All boxes, cutoffs,
phases and amplitudes are fixed before any later accuracy.
\end{proof}

\section{Finite correction on regular patches}\label{app:sec:ch_regular}

We now cover a compact regular set of covectors by finitely many
characteristic and elliptic charts and show that a finite Neumann series
of the parametrices of Section~\ref{app:sec:ch_regular_base} solves the
system to any prescribed finite order for oscillatory sources.

\subsection{The choice of charts}
\label{app:ch_regular:sec:introduction}
\label{app:ch_regular:sec:preliminaries}

\begin{notation}\label{app:ch_regular:notation:conventions}
Compactly supported functions are extended by zero, and all cutoffs
have supports strictly inside the stated domains. Conic sectors are
compact subsets of the unit sphere together with their conic extensions;
the source cover is taken symmetric under $\xi\mapsto-\xi$. We write
$\mathfrak cf=\overline f$ for complex conjugation. For a size $A_\rho$, a
\emph{tail of arbitrary order} means a term bounded by
$C_{m,Q}A_\rho\rho^{-Q}$ in each fixed $H^m$, for every $Q$.
\end{notation}

\begin{setup}[The local solution operators]
\label{app:ch_regular:setup:analytic}
We consider the system of Section~\ref{app:sec:ch_regular_base} on a
compact set of nonzero covectors which are elliptic or simple
characteristics. The finite cover by charts and the source regions are
chosen in Theorem~\ref{app:ch_regular:thm:no-return}. On each selected
chart we use Theorem~\ref{app:ch_regular_base:thm:sector}, with an input
cutoff equal to one near $\overline U$. The residual has the terms
\eqref{app:ch_regular_base:eq:energy-complete-residual}, or their elliptic
counterparts, and satisfies $R_j:L^2(W)\to H^1(U)$; order-zero factors are
bounded on $L^2$, and every fixed Sobolev norm has a polynomial cost in
finitely many coefficient seminorms. Output cutoff derivatives vanish
on $U$, and all other cutoff terms are included. One finite order
$k_{\rm lin}$ of coefficient derivatives controls the $L^2$ and
$L^2\to H^1$ constants, and one fixed loss $c_{\rm lin}$ covers the higher
symbol, flow and Fourier estimates. These choices are fixed before
every later accuracy.
\end{setup}

\subsection{Correction on each chart}
\label{app:ch_regular:sec:extension}

\begin{thm}[A common choice of charts]
\label{app:ch_regular:thm:no-return}
Let $V_0'$ be a compact conic set of regular covectors as in
Setup~\ref{app:ch_regular:setup:analytic}. After one common shrinking of
the physical domains there are scalar source operators $A_j^0$ with
\[
 \sum_j(A_j^0+\mathfrak cA_j^0\mathfrak c)=A_\Gamma,
\]
where $A_\Gamma$ is the identity microlocally near $U\times V_0'$. Each
index $j$ selects one member of a pair of characteristic or elliptic
charts symmetric under $\xi\mapsto-\xi$. There are larger source cutoffs
$\widetilde A_j$ and fixed complex linear operators with
\[
 P_aS_j=\widetilde A_j+R_j\quad\text{on }U,
\]
satisfying the bounds of Setup~\ref{app:ch_regular:setup:analytic}. On
each characteristic chart put $\ell_j=\tau-h_j(x,\eta)$. The forward
Hamilton flowout of $\ell_j$, over the entire fixed time interval, of
every point in the conic support of $A_j^0$ stays in a comparable
annulus and in the region where $\widetilde A_j$ is the identity. The
physical source region contains $\overline U$.

Having fixed these data, choose $0\leq\chi_2\leq1$ in $C_c^\infty(U)$,
equal to one on a fixed set $U_{\rm core}$, and put
\[
 K_j=\chi_2R_j,\qquad
 \mathcal T_{j,N}=\sum_{0\leq\nu<N}(-K_j)^\nu.
\]
For $f=\chi_2h$ and $z_{j,N}=\mathcal T_{j,N}f$ we have the exact identity
\begin{equation}\label{app:ch_regular:eq:exact-physical}
 P_aS_jz_{j,N}=h+(\widetilde A_j-I)z_{j,N}
             +(-1)^{N-1}K_j^Nf\quad\text{on }U_{\rm core}.
\end{equation}
The set of normalized sums and differences of phase gradients of
Proposition~\ref{app:ch_seed:prop:seed-actual-menu} admits one such
choice on a fixed neighborhood of the reference immersion. All flowouts
lie in one fixed $W_0\Subset W$, where the operators use the coefficients
of the immersion $F$. These choices precede the accuracy; only the
finite depth $N$ changes with it.
\end{thm}

\begin{proof}
At a simple characteristic Euler's identity gives
$\xi\cdot\partial_\xi p=(\deg p)\,p=0$, so in coordinates with
$\partial_\tau p\neq0$ we have $\eta\neq0$, and the root is of the form
required in Theorem~\ref{app:ch_regular_base:thm:sector}(ii). Cover $V_0'$
by finitely many root charts and elliptic charts, symmetric under
$\xi\mapsto-\xi$, and select one member of each symmetric pair. By
compactness there are nested conic sets
$\Gamma_j^0\Subset\Gamma_j^1\Subset\Gamma_j^2$ inside larger charts, with
uniform margins. Choose $A_j^0$ with conic support in $\Gamma_j^0$ forming
the stated partition, and $\widetilde A_j$ equal to one on $\Gamma_j^1$
and supported in $\Gamma_j^2$, with fixed low-frequency and physical
cutoffs.

On a characteristic chart, the Hamilton equations of $\ell_j$,
restricted to the unit sphere in $\xi$, have bounded angular velocity,
and $|\xi|$ changes by a bounded logarithmic rate. We choose the time
interval so short that a trajectory starting in $\Gamma_j^0$ stays in
$\Gamma_j^1$ and in a comparable annulus. The same shrinking puts the
transported support from $\Gamma_j^2$ and all its physical supports in a
fixed $W_0\Subset W$ inside the larger chart. Elliptic factors involve
only the identity relation.

Apply Theorem~\ref{app:ch_regular_base:thm:sector} on each selected
chart, with source symbol $\widetilde A_j$ (equal to $\sigma_j\beta_j$ on
characteristic charts); this defines $S_j$ and $R_j$ with all cutoff
errors. Then choose $\chi_2$ inside the common source region. On $U$,
\[
 P_aS_jz_{j,N}=\widetilde A_jz_{j,N}+R_jz_{j,N}
 =z_{j,N}+(\widetilde A_j-I)z_{j,N}+R_jz_{j,N},
\]
and on $U_{\rm core}$, where $\chi_2=1$, we have $R_jz_{j,N}=K_jz_{j,N}$.
The telescoping identity
\[
 (I+K_j)z_{j,N}=\sum_{0\leq\nu<N}(-K_j)^\nu f-\sum_{1\leq\nu\leq N}(-K_j)^\nu f
 =f+(-1)^{N-1}K_j^Nf
\]
and $f=h$ on $U_{\rm core}$ give \eqref{app:ch_regular:eq:exact-physical}.
The source $f$ is supported in $\supp\chi_2$, and the fixed Sobolev bounds
compose with polynomial costs; we take the maximum of the finitely many
derivative losses once. The set of normalized sums and differences is
compact and lies at a positive distance from the singular set by
Proposition~\ref{app:ch_seed:prop:seed-actual-menu}, so the same
construction applies to all of them. The construction preserves the
derivative bounds of the coefficients, with a fixed loss.
\end{proof}

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

\begin{setup}[Coefficient estimates]
\label{app:ch_regular:setup:base-aware-coefficients}
We use the fixed chart operators of Theorem~\ref{app:ch_regular:thm:no-return}
and assume
\begin{equation}\label{app:ch_regular:eq:initial-coefficient-profile}
 \|a\|_{C^r(W)}\leq D_rT^{(r-R_*)_+},\qquad r\geq0,
 \quad R_*>c_{\rm lin}+2,
\end{equation}
and put $L=T\geq1$. By the derivative bounds of
Lemma~\ref{app:ch_regular_base:lem:characteristic-kernel}, the normalized
flow maps and their inverses have bounded derivatives of orders zero and
one, and the phase function $\mathcal G$ of each retarded factor
satisfies
\begin{equation}\label{app:ch_regular:eq:high-phase}
 |D^r\mathcal G|\leq C_rL^{r-1}\quad(r\geq2).
\end{equation}
Every symbol or amplitude of order $p$ satisfies
\begin{equation}\label{app:ch_regular:eq:high-amplitude}
 |D_{x,z}^{\alpha}D_\xi^\beta b_p|
 \leq C_{\alpha,\beta}\langle\xi\rangle^{p-|\beta|}
                     L^{|\alpha|+|\beta|},
\end{equation}
including both positions and all cutoffs; the covariable is $\eta$ for
spatial symbols, and spacetime bounds are used only on their cones. At
each fixed Sobolev order the operator norms of the factors are at most
$C_sL^{B_s}$ for some finite $B_s$. Since $R_*>c_{\rm lin}+2$, the $L^2$ and
$L^2\to H^1$ bounds do not depend on $L$. For $\rho\geq2$ and a fixed
$0<\sigma<1$ we assume
\begin{equation}\label{app:ch_regular:eq:slow-versus-frequency}
 1\leq L\leq\rho^{1-\sigma}.
\end{equation}
The index $R_*$, the orders of the construction and all factors are fixed
before the accuracy.
\end{setup}

\begin{thm}[Approximate solutions to any finite order]\label{app:ch_regular:thm:quantitative}
Assume Setup~\ref{app:ch_regular:setup:base-aware-coefficients}. Choose a
compact conic set $V_0\Subset V_0'$ and a fixed smooth rescaled Fourier
cutoff $\Pi_{0,\rho}$, equal to one on $\rho V_0\cap\{|\xi|\asymp\rho\}$
and supported in $\rho V_0'$. Let $h_\rho$ be real and smooth, supported
where $\chi_2=1$, with
\begin{align}
 \|h_\rho\|_2&\leq CH_\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\text{for all fixed }m,Q.
 \label{app:ch_regular:eq:source-tail}
\end{align}
For each finite $N\geq1$ define
\[
\begin{gathered}
 \mathcal D_{j,N}=S_j\mathcal T_{j,N}\chi_2A_j^0,\qquad
 \mathcal D_{a,N}=\sum_j
       (\mathcal D_{j,N}+\mathfrak c\mathcal D_{j,N}\mathfrak c),\\
 u_{\rho,N}=\mathcal D_{a,N}h_\rho .
\end{gathered}
\]
Then $\mathcal D_{a,N}$ is complex linear and preserves real fields. For
all fixed $m,Q$ and every integer $N\geq m+Q$,
\begin{equation}\label{app:ch_regular:eq:pde-arbitrary}
 \|P_au_{\rho,N}-h_\rho\|_{H^m(U_{\rm core})}
       \leq C_{N,m,Q}H_\rho\rho^{-Q},
\end{equation}
and for every fixed $s$
\begin{equation}\label{app:ch_regular:eq:solution-profile}
 \|u_{\rho,N}\|_{H^s}\leq C_{N,s}H_\rho\rho^s .
\end{equation}
A higher accuracy changes only the finite depth $N$; the derivative
losses, physical domains, conic source cutoffs and orders of the kernels
are fixed. The constants and the frequency threshold may depend on $N$,
$m$, $Q$ and on the constants $D_r$ in
\eqref{app:ch_regular:eq:initial-coefficient-profile}, but not on
$\rho$ or $L$ within \eqref{app:ch_regular:eq:slow-versus-frequency}.
\end{thm}

\begin{proof}
\emph{Step 1 (localization of one factor).} Let $f$ satisfy
$\|f\|_{H^s}\leq C_sA\rho^s$ for all $s$, and suppose that, up to a tail
of arbitrary order, its Fourier transform is concentrated in a compact
conic subset, with a margin, of one source chart, within a comparable
annulus $\{|\xi|\asymp\rho\}$. We show that the image of $f$ under each
parametrix factor or residual factor is, up to a tail of arbitrary order,
concentrated in any prescribed open conic neighborhood of the forward
Hamilton flowout of this set (including the set itself), within a
comparable annulus. The constants depend on the chosen neighborhoods
but not on $\rho$.

For a retarded factor we use the representation
\eqref{app:ch_regular_base:eq:fixed-time-representation}. At a fixed
elapsed time $r$, the critical points of the phase
$\Theta_r(x,\xi)-z\cdot\xi$ relate the input point and the output
covector by
\[
 (s,z)=(t-r,\Psi_\eta),\qquad
 (\tau_{\rm out},\eta_{\rm out})=(\tau+\Psi_t+\Psi_s,\Psi_y).
\]
The Hamilton--Jacobi identities $\Psi_t=h(t,y,\Psi_y)$ and
$-\Psi_s=h(s,z,\eta)$ show that this relation is the forward Hamilton
flow of $\ell=\tau-h$ for the time $r$, at every value of $\tau$.
Pseudodifferential terms and fixed Fourier cutoffs have the identity
relation, corresponding to $r=0$.

Compose a fixed-$r$ factor with an input cutoff concentrated near the
given conic set and an output cutoff vanishing near its flowout (these
cutoffs are used only in this proof), and rescale comparable covectors by
$\rho$. Away from the flowout, the combined phase, as a function of the
integration variables, has a gradient bounded below by $c\rho$, uniformly
in $0\leq r\leq\Delta$. Each integration by parts, at fixed $r$, gains
$\rho^{-1}$ and costs at most a factor $L$ by
\eqref{app:ch_regular:eq:high-phase} and
\eqref{app:ch_regular:eq:high-amplitude}. Hence for all fixed $m,q$ the
resulting operator is bounded in the relevant Sobolev norms by
\begin{equation}\label{app:ch_regular:eq:evolution-off-relation}
 C_{m,q}\rho^{m+C}(L/\rho)^q .
\end{equation}
The integrations include all amplitude positions and the source masks.
Frequencies in non-comparable annuli are separated from the output by a
frequency gap in a physical derivative of the phase; their dyadic sum
converges after increasing $q$. Fixed physical supports, Schur's test
and the bounded $r$-interval give the Sobolev bounds. Since
$L/\rho\leq\rho^{-\sigma}$ by \eqref{app:ch_regular:eq:slow-versus-frequency},
increasing $q$ gives a tail of arbitrary order. By the choice of charts,
the short flow keeps the covectors in a comparable annulus.

\emph{Step 2 (finite powers).} Put $X=\supp\chi_2$, $f_j=\chi_2A_j^0h_\rho$
and $v_{j,d}=K_j^df_j$. We show that, for $d\leq N+1$, $v_{j,d}$ is
concentrated, up to a tail of arbitrary order, in a fixed conic region
where $\widetilde A_j$ is the identity, and that
\begin{equation}\label{app:ch_regular:eq:finite-power-gain}
 \|v_{j,d}\|_{H^s}\leq C_{d,s}H_\rho\rho^{s-d}
       \qquad(d,s\geq0).
\end{equation}
On a characteristic chart we use its time coordinate $t$, which ranges
over a fixed interval $[t_-,t_+]$, and the Hamilton flow $\Phi^r$ of
$\ell_j=\tau-h_j$; along this flow $\dot t=1$. Let $\Gamma_j^0$ be the
conic support of $A_j^0$ and put
\[
 \Lambda_j=\{\Phi^r(p):p\in\Gamma_j^0,\ 0\leq r\leq t_+-t(p)\}.
\]
By the choice of charts in Theorem~\ref{app:ch_regular:thm:no-return},
$\Lambda_j$ lies in the open conic region where $\widetilde A_j$ is the
identity, and $|\xi|$ varies on $\Lambda_j$ only within a fixed factor of
its initial value.

\emph{(a) The total elapsed time.} By Step~1, the relation of each factor
of $K_j$, and of $S_j$, is contained in
$\{(\Phi^r(p),p):0\leq r\leq t_+-t(p)\}$, the identity relation
corresponding to $r=0$. Since $\Phi^{r_2}\circ\Phi^{r_1}=\Phi^{r_1+r_2}$
and the time coordinate increases by $r$ along $\Phi^r$, the composition
of any number of such relations is contained in the same set: the total
elapsed time is $t_{\rm out}-t_{\rm in}\leq t_+-t_-$, independently of the
number of factors. Hence $\Lambda_j$ is invariant under the relation of
every factor. In particular the frequency band reached by the iterates
is the fixed band of $\Lambda_j$, determined by the bounded logarithmic
rate of change of $|\xi|$ over the time $t_+-t_-$; it does not widen
with $d$. On an elliptic chart all relations are the identity and
$\Lambda_j=\Gamma_j^0$.

\emph{(b) Auxiliary sets depending on the depth.} The relations are
closed, and their restrictions to the unit cosphere bundle are compact.
Hence for every open conic set $O\supset\Lambda_j$ there is an open conic
set $O'$ with $\Lambda_j\subset O'$ whose closure is mapped by every
relation into $O$. Starting from an open conic set $O_{N+1}\supset\Lambda_j$,
over a neighborhood of $X$, on which $\widetilde A_j$ is the identity, we
choose successively $O_N,\dots,O_0$ with $\Lambda_j\subset O_k$,
$\overline{O_k}\subset O_{k+1}$, and with the closure of $O_k$ mapped by
every relation into $O_{k+1}$. These sets, and the cutoffs of Step~1
equal to one near $\overline{O_k}$ and vanishing outside $O_{k+1}$,
depend on $N$ but not on $\rho$. They are used only in this proof and
affect only the constants and the frequency threshold.

\emph{(c) Induction.} The source $f_j$ is concentrated in $O_0$ up to a
tail of arbitrary order, by the conic support of $A_j^0$ and
\eqref{app:ch_regular:eq:source-tail}. If $v_{j,d}$ is concentrated in
$O_d$, Step~1 shows that $v_{j,d+1}=K_jv_{j,d}$ and $S_jv_{j,d}$ are
concentrated in $O_{d+1}$, up to tails; at each step we increase the
order of the tail to absorb the polynomial Sobolev costs of the
subsequent factors. Elliptic factors have only the identity relation,
and low-frequency pieces are separated by the annular gap. Thus, for
$d\leq N+1$, all these functions are concentrated in $O_{N+1}$, where
$\widetilde A_j$ is the identity. This proves the localization,
including all source and cutoff terms.

Fix $j$ and write $v_d=v_{j,d}$, so that $\|v_0\|_2\leq CH_\rho$. For
$d\geq1$, since $v_d$ is concentrated in an annulus of radius comparable
to $\rho$ up to a tail, and $K_j:L^2\to H^1$ with a bound independent of
$L$,
\[
 \|v_d\|_2\leq C\rho^{-1}\|v_d\|_{H^1}+O(H_\rho\rho^{-\infty})
       \leq C\rho^{-1}\|v_{d-1}\|_2+O(H_\rho\rho^{-\infty}).
\]
Induction gives $\|v_d\|_2\leq C_dH_\rho\rho^{-d}$. In the other
direction, frequency localization gives
$\|v_d\|_{H^s}\leq C_{s}\rho^s\|v_d\|_2+O(H_\rho\rho^{-\infty})$, which
proves \eqref{app:ch_regular:eq:finite-power-gain}.

\emph{Step 3 (the finite correction).} Put $z_{j,N}=\mathcal T_{j,N}f_j$.
By the localization just proved,
\[
 (\widetilde A_j-I)K_j^\nu f_j
       =O_{\nu,m,Q}(H_\rho\rho^{-Q})\quad\text{in }H^m .
\]
Each $S_j$ is bounded on $L^2$, and by Step~1 every $S_jK_j^\nu f_j$ is
concentrated in a comparable annulus up to a tail; the same frequency
localization and the finite sum give \eqref{app:ch_regular:eq:solution-profile}.

For real $h_\rho$ we have $\mathfrak ch_\rho=h_\rho$, so
$\mathfrak c\mathcal D_{j,N}\mathfrak ch_\rho=\overline{\mathcal D_{j,N}h_\rho}$
and $u_{\rho,N}=2\Re\sum_jS_jz_{j,N}$. Since $P_a$ is real, applying
\eqref{app:ch_regular:eq:exact-physical} with $f=f_j$ (on
$U_{\rm core}$, $f_j=A_j^0h_\rho$) and taking real parts gives, on
$U_{\rm core}$,
\[
 P_au_{\rho,N}-h_\rho
 =(A_\Gamma-I)h_\rho
 +2\Re\sum_j\left[(\widetilde A_j-I)z_{j,N}
                    +(-1)^{N-1}K_j^Nf_j\right],
\]
where we used $\sum_j2\Re(A_j^0h_\rho)=\sum_j(A_j^0+\mathfrak cA_j^0\mathfrak c)h_\rho=A_\Gamma h_\rho$.
The source tail \eqref{app:ch_regular:eq:source-tail} and the choice of
$A_\Gamma$ make the first term a tail of arbitrary order, and the
localization makes the second one such a tail. The last term is bounded
in $H^m$ by $C_{N,m}H_\rho\rho^{m-N}$ by
\eqref{app:ch_regular:eq:finite-power-gain}, which is at most
$C_{N,m}H_\rho\rho^{-Q}$ when $N\geq m+Q$.
\end{proof}

%% ===== 10_11_waves_sampling.tex =====
\section{Corrected characteristic waves}\label{main:sec:linear}\label{app:sec:ch_waves}\label{app:sec:ch_response}\label{app:sec:ch_lag}

In this section we construct, for an immersion $F$ near the reference,
families of complex waves whose linearized metric change vanishes to any
prescribed order on a fixed domain, whose leading amplitude is an
explicit rank-one tensor, and which depend linearly on Fourier data. The
construction has three stages: a WKB approximation with a Hamilton
phase and a transported amplitude; a correction of the remaining
residual by a finite Neumann series of the parametrix of
Section~\ref{app:sec:ch_regular_base}; and a superposition over shifted
covectors against a finite Fourier measure. The main result is
Proposition~\ref{main:prop:current-waves} at the end of the section.

\subsection{Normal reconstruction and conventions}
\label{app:ch_waves:sec:introduction}\label{app:ch_waves:sec:preliminaries}

\begin{notation}\label{app:ch_waves:notation:conventions}
We use the dimensions, the tensors $Q_i^0$ and the Frobenius pairing of
Definition~\ref{app:ch_seed:defn:seed-cube}; all orthogonal complements
and projections of tensors refer to this pairing. Spatial and parameter
supports have fixed larger margins. Constants may depend on these
supports, on the fixed neighborhood of the reference, on lower bounds for
inverses and on finitely many constants in the derivative bounds, but not
on frequencies or the number of samples unless this is indicated. A
statement that a datum has Fourier support in a ball refers to the
datum before any cutoff in the initial variables; such a cutoff is kept
in the amplitude of the wave, and if it is applied to the datum, its
Fourier tail is estimated separately. In particular, $\kappa(y)P_Kf(y)$ is
not regarded as band limited.
\end{notation}

\begin{defn}[Normal reconstruction]\label{app:ch_waves:defn:geometry}
Let $G$ be an immersion as in Setup~\ref{app:ch_seed:setup:seed-current},
with induced metric $g$, normal map $N_G$ and projection $\Pi_{K_G}$ as in
\eqref{app:ch_waves:eq:ambient-normal-inverse}, and projected frame
$Q_i=\Pi_{K_G}Q_i^0$. We use the objects of
Definition~\ref{new:defn:reduced} and \eqref{main:eq:stress-lift} for this
frame, and write them out for later reference:
\begin{equation}\label{app:ch_waves:eq:reconstruction}
\mathcal V_G(u)
=\sum_{i,j}g^{ij}u_i\partial_jG+
N_G(\sym\nabla^g u),
\end{equation}
\begin{equation}\label{app:ch_waves:eq:reduced-symbol}
\begin{gathered}
P_Gu=\cP_Gu=(Q_i:\sym\nabla^g u)_{i=1}^n,\\
M_G(x,\xi)_{i\bullet}=(Q_i(x)\xi)^T,\qquad
p_G(x,\xi)=\det M_G(x,\xi).
\end{gathered}
\end{equation}
The operator $P_G$ includes the Christoffel terms of $g$. For a covector
$\xi$ and a vector $b$ we write $\xi\sym b=(\xi b^T+b\xi^T)/2$; note that
$Q_i:(\xi\sym b)=\xi^TQ_ib=(M_G(x,\xi)b)_i$.
\end{defn}

The patches and samples are indexed as in
Definition~\ref{app:defn:exhaustive-pair-groups}, and their quadratic
interactions are listed in Remark~\ref{app:rem:exhaustive-signed-pairs}.

\begin{setup}[Oscillatory estimates for phase differences]\label{app:ch_waves:setup:oscillatory}
We use Setup~\ref{app:ch_osc:setup:osc-normal} with the full and
fixed-time charts of Proposition~\ref{app:ch_seed:prop:seed-phase-charts},
$x=(t,y)$, $z=(s,v)$ and $v\in\R^2$. Their lower bounds, larger supports
and margins are fixed with the order $M=M_{\rm an}=20(n+1)$, and the
immersions satisfy (L) and (H) of
Setup~\ref{app:ch_seed:setup:seed-current}, so that
Proposition~\ref{app:ch_osc:prop:osc-normal} applies uniformly to the
scalar and matrix amplitudes below, on the entire complex input and
output spaces. Both the real and the imaginary parts of the input are
used; by Corollary~\ref{app:ch_osc:cor:osc-real}, on real outputs,
\begin{equation}\label{app:ch_waves:eq:real-normal}
 (2\Re T)(2\Re T)^*=4\Re(TT^*).
\end{equation}
\end{setup}

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

\begin{thm}[Transport amplitudes and normal reconstruction]\label{app:ch_waves:thm:leading}
\label{app:ch_waves:thm:mixed-tame}
Consider one of the three-parameter characteristic patches, with its
adapted coordinates $x=(t,y)$, its map $z\mapsto\zeta(z)$ of base
covectors and the larger charts for shifted covectors of
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 lower bounds for the metric, for the
singular values of the second fundamental form, for the minors defining
the null vectors and for $\partial_\tau p_G$, and use full and fixed-time
inverse charts on the larger nested convex source domains. One
sufficiently small finite $C^m$ neighborhood of the reference gives all
these bounds; the boxes and lower bounds do not depend on frequencies or
on the accuracy.

For a real initial amplitude $r_z(y)$ with fixed positive lower bound and
fixed $C^m$ upper bound, there are a real phase $S_G$ and an amplitude
$b_G=\alpha_Gk_G$ satisfying
\[
 p_G(x,d_xS_G)=0,\qquad S_G(0,y,\zeta)=y\cdot\zeta,
 \qquad \alpha_G|_{t=0}=r_z,\qquad l_G^TP_Gb_G=0,
\]
where $k_G$, $l_G$ are real nonzero right and left null vectors of
$M_G(x,d_xS_G)$ and $\alpha_G$ has a positive lower bound. The
reconstruction \eqref{app:ch_waves:eq:reconstruction} satisfies
\begin{equation}\label{app:ch_waves:eq:exact-half-strain}
 \tfrac12D\gamma_G\mathcal V_G(u)
                 =\Pi_{K_G}(\sym\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(\xi\sym b),\qquad
 A_0(b)=-\nu_b\,\xi^T .
\end{equation}
Then, for $\tau>0$,
\begin{equation}\label{app:ch_waves:eq:exact-W-amplitude}
 \mathcal V_G(\tau^2e^{\ii S_G/\tau}b)
 =e^{\ii S_G/\tau}
       (\ii\tau\nu_b+\tau^2\mathcal V_G(b)),
\end{equation}
and $e^{-\ii S_G/\tau}d\,\mathcal V_G(\tau^2e^{\ii S_G/\tau}b)=A_0(b)+O(\tau)$.
For $b=b_G$ we have $|\nu_b|,|\xi|\geq c>0$ and
$A_0(b)^*A_0(b)=|\nu_b|^2\xi\xi^T$; this transported tensor is linear in
the initial density $r_z^2$.

Suppose moreover that $T\geq1$ and
$\|G\|_{C^r}+\|r_z\|_{C^r}\leq C_rT^{(r-R_*)_+}$ for all $r$. There is one
fixed loss $D$ such that, if $R_*>D$, then for all $r\geq0$
\begin{equation}\label{app:ch_waves:eq:joint-tame}
\begin{split}
&\|S_G\|_{C^r(x,\zeta)}+\|b_G\|_{C^r(x,z,\zeta)}
 +\|\nu_b\|_{C^r(x,z,\zeta)}\\
&\qquad+\|\mathcal V_G(b)\|_{C^r(x,z,\zeta)}
 +\|A_0(b)\|_{C^r(x,z,\zeta)}
 \leq C_rT^{(r-R_*+D)_+},
\end{split}
\end{equation}
as joint norms in the physical variables and the normalized covectors,
on the same larger domains.
\end{thm}

\begin{proof}
\emph{Phase and transport.} The phase is given by
Proposition~\ref{app:ch_seed:prop:seed-actual-menu} and
Lemma~\ref{app:lem:common-hamilton-phase}; the inverse metric, the normal
map and the null vectors are smooth functions of a fixed jet of $G$
under the stated lower bounds. We use the real null vectors and the
scalar transport equation of
Lemma~\ref{app:ch_regular_base:lem:characteristic-kernel}, with initial
time $0$ and initial datum $r_z$. As computed there, the transport
equation is equivalent to $l_G^TP_G(\alpha_Gk_G)=0$, and since it is a
linear first-order equation along the projected Hamilton flow, its
solution is
\[
 \alpha_G=a_{\rm flow}\,r_z(y_{\rm foot}),
\]
where $y_{\rm foot}$ is the initial point of the trajectory through $x$
and $a_{\rm flow}>0$ is the exponential of the integral of $-l^T(Pk)$
along it. Since $a_{\rm flow}$ does not depend on $r_z$, $\alpha_G$ is
positive and linear in $r_z$, so $\alpha_G^2$ is linear in $r_z^2$.

\emph{Reconstruction.} By \eqref{main:eq:tangent-normal} with the normal
field $w=N_G(\sym\nabla^gu)$,
\[
 D\gamma_G\mathcal V_G(u)=2\sym\nabla^gu-2B_G^*N_G(\sym\nabla^gu)
 =2\Pi_{K_G}(\sym\nabla^gu),
\]
which is \eqref{app:ch_waves:eq:exact-half-strain}. For
$u=\tau^2e^{\ii S_G/\tau}b$ the tangential part of $\mathcal V_G(u)$ is
$\tau^2e^{\ii S_G/\tau}dG\,g^{-1}b$, and
\[
 \sym\nabla^g\bigl(e^{\ii S_G/\tau}b\bigr)
 =e^{\ii S_G/\tau}\Bigl(\sym\nabla^gb+\frac{\ii}{\tau}\,\xi\sym b\Bigr).
\]
Multiplying by $\tau^2$ and applying $N_G$ gives
\eqref{app:ch_waves:eq:exact-W-amplitude}. Differentiating it,
\[
 e^{-\ii S_G/\tau}d\bigl(e^{\ii S_G/\tau}(\ii\tau\nu_b+\tau^2\mathcal V_G(b))\bigr)
 =-\nu_b\xi^T+\tau\bigl(\ii\,d\nu_b+\ii\mathcal V_G(b)\xi^T\bigr)+\tau^2d\mathcal V_G(b),
\]
whose leading term is $A_0(b)$. For real $\nu_b$,
$A_0(b)^*A_0(b)=\xi\nu_b^T\nu_b\xi^T=|\nu_b|^2\xi\xi^T$.

\emph{Lower bounds.} For $b=b_G$ the null relation gives
$q_G(\xi\sym b_G)=M_G(x,\xi)b_G=0$, so $\xi\sym b_G\in K_G^\perp$ and
$B_G^*\nu_b=\Pi_{K_G^\perp}(\xi\sym b_G)=\xi\sym b_G$. Moreover
\[
 |\xi\sym b|^2=\tfrac14|\xi b^T+b\xi^T|^2
 =\tfrac12\bigl(|\xi|^2|b|^2+(\xi\cdot b)^2\bigr)\geq\tfrac12|\xi|^2|b|^2 .
\]
Hence $|\nu_b|\geq\|B_G^*\|^{-1}|\xi\sym b_G|\geq c$, using the lower
bounds for $|\xi|$, $\alpha_G$ and $|k_G|$.

\emph{Derivative bounds.} The phase, the algebraic formulas and the
transport amplitudes satisfy the derivative bounds with a fixed loss, by
Lemmas~\ref{app:lem:elementary-interpolation-jets},
\ref{app:ch_regular_base:lem:buffered-interpolation} and
Proposition~\ref{app:ch_seed:prop:seed-actual-menu}; in
$\alpha_G=a_{\rm flow}r_z(y_{\rm foot})$, joint composition and
multiplication are covered by
Lemma~\ref{app:ch_regular_base:lem:buffered-interpolation}. The
reconstruction uses finitely many further derivatives, so one common
loss $D$ proves \eqref{app:ch_waves:eq:joint-tame}.
\end{proof}

\subsection{A two-term WKB approximation}
\label{app:ch_waves:sec:wkb}
One algebraic correction reduces the reduced residual by one more power
of $\tau$. With the phase and leading transport of
Theorem~\ref{app:ch_waves:thm:leading}, put $b_0=b_G$ and, using the
symbol \eqref{app:ch_regular_base:eq:complement-inverse} of the system
$P_G$,
\begin{equation}\label{app:ch_waves:eq:primary-algebraic-correction}
 b_1=\ii\, b_{-1,G}(x,dS_G)P_Gb_0 .
\end{equation}
For $\lambda=\tau^{-1}$ define, on the fixed compact set of shifted
covectors $\zeta'$,
\begin{equation}\label{app:ch_waves:eq:WKB-u}
 v_\tau(x,z,\zeta')
 =\tau^2e^{\ii S_G(x,\zeta')/\tau}(b_0+\tau b_1).
\end{equation}
It is linear in the initial amplitude.

\begin{lem}\label{new:lem:two-term-residual}
\begin{equation}\label{app:ch_waves:eq:WKB-residual}
 P_Gv_\tau=\tau^3e^{\ii S_G/\tau}P_Gb_1 .
\end{equation}
\end{lem}

\begin{proof}
Apply \eqref{app:ch_regular_base:eq:two-term-null-identity} with
$\Psi=S_G/\tau$, $W_0=\tau^2b_0$ and $W_1=\tau^3b_1$. Its hypotheses hold:
$\Psi$ solves $\Psi_t=h(t,y,\Psi_y)$ because $S_G$ does and $h$ is
homogeneous of degree one; $l^TP_GW_0=\tau^2l^TP_Gb_0=0$; and, since
$b_{-1}$ is homogeneous of degree $-1$,
\[
 \ii\,b_{-1}(x,d\Psi)P_GW_0=\ii\,\tau\,b_{-1}(x,dS_G)\,\tau^2P_Gb_0=\tau^3b_1=W_1 .
\]
Hence $P_G(e^{\ii\Psi}(W_0+W_1))=e^{\ii\Psi}P_GW_1=\tau^3e^{\ii S_G/\tau}P_Gb_1$.
\end{proof}

The correction constructed in the next subsections removes this
residual; the joint amplitude bounds, including those of $v_\tau$, are
proved together in Corollary~\ref{app:ch_response:cor:current-atoms}.

\subsection{The solution operator}\label{app:ch_response:sec:response-inputs}

\begin{setup}[A common solution operator]
\label{app:ch_response:thm:response-compatible-choice}
\label{app:ch_response:defn:response-regular}
We use the immersion $G$ and the reduced system of
Definition~\ref{app:ch_waves:defn:geometry}. On a compact characteristic
patch, the phase of the waves and the characteristic parametrix use the
same simple root $\tau=h_G(t,y,\eta)$, so that
\begin{equation}\label{app:ch_response:eq:response-exact-phase}
 \partial_t\phi_G=h_G(t,y,\partial_y\phi_G).
\end{equation}
On one chart containing the whole compact set of shifted covectors,
Lemma~\ref{app:ch_regular_base:lem:characteristic-kernel} gives
$P_GS_+=A_++R_+$ on the inner region, with
\begin{equation}\label{app:ch_response:eq:response-actual-solver-formula}
 S_+=\chi_{\rm out}\Bigl(\mathcal E_WB_+
            +\frac1{\ii}\operatorname{Op}(b_{-1}\alpha_+)\Bigr),
 \qquad B_+=\beta_+(D).
\end{equation}
The conic cutoff $\alpha_+$ and $\beta_+$ are chosen equal to one near
all these phase gradients throughout $\supp\chi_2$. By compactness, the
finitely many patches have common source and time margins, common lower
bounds and one common ball on which the identities hold. The larger
physical and conic domains are convex and contain the source and
observation boxes, all short flowouts and all initial points. For finite
powers of the residual we use an observation domain containing
$\supp\chi_2$ and the core cutoff. All factors are fixed independently of
the patch parameters and of the accuracy. Put
\[
 \mathcal K=\chi_2R_+,\qquad
 \mathcal T_N=\sum_{\nu=0}^{N-1}(-\mathcal K)^\nu .
\]
The source cutoff of the primary waves is $\chi_{\rm src}=\chi_{\lag,\rm src}$,
supported inside the primary chart and in $\Int\{\chi_2=1\}$; the source
and observation supports are those of Setup~\ref{app:setup:domains}, and
the region where the identities hold is $U_{\rm LAG}$.

The additional quantitative hypothesis is
\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}
where $a_G$ consists of the principal and Christoffel coefficients of
the reduced system. The metric, normal and Gram lower bounds are
positive, and the reconstruction, including one derivative of
$\mathcal V_G$, is a smooth function of a fixed jet of $G$. The parametrix
has the fixed factors of Setup~\ref{app:ch_regular:setup:analytic}. The
waves are complex; real fields are obtained by taking real parts. We
choose $R_*$ above the fixed geometric and analytic losses, so that every
finite composition has polynomial costs at each physical or parameter
order.
\end{setup}

Lemma~\ref{main:lem:history} and Proposition~\ref{app:prop:fixed-high}
verify \eqref{app:ch_response:eq:response-actual-G-profile} along the
iteration.

\subsection{Oscillatory inputs depending on parameters}
\label{app:ch_response:sec:response-mixed}

\begin{lem}[Oscillatory inputs depending on parameters]
\label{app:ch_response:lem:energy-carrier-jets}
Let $\mathcal A$ be one of the retarded or pseudodifferential factors of
the parametrix of Lemma~\ref{app:ch_regular_base:lem:characteristic-kernel},
of order $p\in\{0,-1\}$. Let $\phi(x;c)$ solve
\eqref{app:ch_response:eq:response-exact-phase} on the source flowout,
with phase gradients in the interior of the region where the source
cutoff equals one, depending smoothly on parameters $c$. Let
$T\leq\lambda^{1-\nu}$ for a fixed $\nu>0$. There is a fixed loss $b$ such
that the coefficient and phase bounds with index $R+b$ imply the
following. If $a_c$ is supported in the fixed source chart and
\[
 \|\partial_c^\alpha a_c\|_{H^s}
 \leq C_{s,\alpha}A T^{(s+|\alpha|-R)_+},
 \qquad |\alpha|\leq k_0,
\]
then, on the fixed observation support,
\[
 \bigl\|\partial_c^\alpha\bigl(e^{-\ii\lambda\phi}\mathcal A(e^{\ii\lambda\phi}a_c)\bigr)\bigr\|_{H^s}
 \leq C_{s,\alpha}A\lambda^pT^{(s+|\alpha|-R)_+},\qquad|\alpha|\leq k_0 .
\]
If the amplitude of $\mathcal A$ vanishes near the phase gradients
(a fixed nonstationary gap), the left side is $O(A\lambda^{-Q})$ at each
fixed joint order, for every $Q$. The loss $b$ does not depend on $s$,
$k_0$ or the length of a finite composition, and finite compositions
satisfy the same bounds with the product of the factors $\lambda^p$.
\end{lem}

\begin{proof}
For bounded $\lambda$ the claim follows from the ordinary estimates, so
let $\lambda$ be large. Include the fixed input cutoff in $a_c$ and put
$u_c=e^{\ii\lambda\phi}a_c$.

\emph{Frequency localization.} Choose a fixed spacetime annular
multiplier $P_\lambda=\pi(D/\lambda)$ with $\pi=1$ near all phase
gradients. By Sobolev embedding the hypothesis gives
$|D_x^r\partial_c^\alpha a_c|\leq C_{r}AT^{\ell}T^r$, so
Proposition~\ref{app:ch_osc:prop:osc-source-shell}, with $\rho=\lambda$,
$L=T\leq\lambda^{1-\nu}$ and amplitude constant
$AT^\ell\lambda^{|\alpha|}\leq A\lambda^{\ell+k_0}$ (the factor
$\lambda^{|\alpha|}$ comes from parameter derivatives of the phase factor),
gives $(I-P_\lambda)u_c=O(A\lambda^{-Q})$ for every $Q$, at every fixed
joint order. The powers of $\lambda$ produced by derivatives and by the
ordinary bounds for $\mathcal A$ are absorbed by increasing $Q$; this
uses higher physical derivatives of $a_c$, but does not change $R$ or the
order $k_0$. In \eqref{app:ch_regular_base:eq:fixed-time-representation}
the multiplier $P_\lambda$ enters the amplitude as the factor
$\pi(\xi/\lambda)$, so all frequency integrals may be restricted to
$|\xi|\asymp\lambda$; the multiplier $\beta_+$ remains in the amplitude.

\emph{Stationary structure.} Fix $r$, write $\xi=\lambda\theta$ and
remove the factor $e^{\ii\lambda\phi(x)}$. The amplitude of order $p$
contributes a factor $\lambda^p$, and the phase becomes $\lambda f$,
\[
 f(x,c;z,\theta)=\Theta_r(x,\theta)-z\cdot\theta+\phi(z;c)-\phi(x;c),
\]
with $\Theta_0=x\cdot\theta$ for a pseudodifferential factor. The
critical points in $w=(z,\theta)$ are given by $\theta=d\phi(z;c)$ and
$z=\partial_\theta\Theta_r(x,\theta)$, that is, $z$ is the initial point
of the Hamilton trajectory of time $r$ ending at $x$, with initial
covector $d\phi(z)$. Since $\phi$ solves the eikonal equation, the
Hamilton flow preserves the graph of $d\phi$, and the projection of the
flow is close to the identity on the short interval; hence there is
exactly one critical point $w_0(x,c)$, and by the Hamilton--Jacobi
identities of Lemma~\ref{app:lem:common-hamilton-phase} the critical value
is $f(w_0)=0$. The Hessian in $w$ is
\[
 \begin{pmatrix}\phi_{zz}(z;c)&-I\\-I&(\Theta_r)_{\theta\theta}(x,\theta)\end{pmatrix},
\]
with $\|\phi_{zz}\|\leq C$ and
$(\Theta_r)_{\theta\theta}=\diag(0,\Psi_{\eta\eta})=O(r)$. On a fixed
short interval every segment average of this Hessian has the same form
and is uniformly invertible, and the amplitude support and the critical
point lie in one fixed convex product domain.

\emph{Derivatives in $x$ and $c$.} Let $\partial_e$ be a derivative in an
output variable or a parameter. Since the critical value vanishes
identically, $\partial_ef(w_0)=0$, and the identities
\[
\begin{gathered}
 \nabla_wf=\Bigl(\int_0^1f_{ww}(w_0+\theta(w-w_0))\dd\theta\Bigr)(w-w_0),\\
 \partial_ef=\Bigl(\int_0^1\nabla_w\partial_ef(w_0+\theta(w-w_0))\dd\theta\Bigr)
 \cdot(w-w_0)
\end{gathered}
\]
give $\partial_ef=A_e\cdot\nabla_wf$, with
\[
 A_e=\Bigl(\int_0^1f_{ww}(w_0+\theta(w-w_0))\dd\theta\Bigr)^{-T}
 \int_0^1\nabla_w\partial_ef(w_0+\theta(w-w_0))\dd\theta .
\]
Hence $\partial_e(e^{\ii\lambda f})=A_e\cdot\nabla_we^{\ii\lambda f}$, and
one integration by parts in $w=(z,\theta)$ gives
\begin{equation}\label{new:eq:carrier-ibp}
 \partial_e\!\int\! e^{\ii\lambda f}\alpha\,a_c\dd w
 =\int e^{\ii\lambda f}\bigl[\partial_e\alpha\,a_c+\alpha\,\partial_ea_c
 -\nabla_w\cdot(A_e\alpha)\,a_c-\alpha A_e\cdot\nabla_wa_c\bigr]\dd w ,
\end{equation}
where $\alpha$ is the amplitude of the factor, multiplied by the input
cutoff, and $\partial_ea_c=0$ unless $e$ is a parameter. No power of
$\lambda$ is produced: a derivative in $\theta$ is $\lambda\partial_\xi$,
which preserves the order of the amplitude in the annulus
$|\xi|\asymp\lambda$, and $a_c$ depends on $w$ only through $z$. The new
integrand has the same phase.

We now count derivatives. Let $\kappa$ be the fixed number of derivatives
of the coefficients of $P$ needed to write down $f_{ww}$,
$\nabla_w\partial_ef$ and the amplitude $\alpha$; it consists of the
losses of the Hamilton--Jacobi phases and their inverse charts
(Lemma~\ref{app:lem:common-hamilton-phase} and
Proposition~\ref{app:ch_seed:prop:seed-actual-menu}), of the null vectors
and of the transport equation, plus the two derivatives in $f_{ww}$ and
$\nabla_w\partial_ef$. With $\ell=\lfloor n/2\rfloor+1$ we take
\[
 b=\kappa+2\ell .
\]
If the coefficients satisfy bounds with index $R+b$, then, by
Lemmas~\ref{app:lem:elementary-interpolation-jets}
and~\ref{app:ch_regular_base:lem:buffered-interpolation} (the averaged
Hessian has a uniformly bounded inverse), every factor $\beta$ among
$A_e$ and $\alpha$ satisfies
\begin{equation}\label{new:eq:carrier-coefficient}
 \|\partial^\delta\partial^\gamma\beta\|_\infty
 \leq C_{\gamma}T^{(|\gamma|-R)_+},\qquad |\delta|\leq2\ell,
\end{equation}
where $\gamma$ counts the derivatives produced by the integrations by
parts and $\delta$ the derivatives used below for the $L^2$ bound. After
$q$ derivatives $\partial_{e_1}\cdots\partial_{e_q}$, repeated use of
\eqref{new:eq:carrier-ibp} writes the derivative of the output as a
finite sum of terms
\[
 \int e^{\ii\lambda f}\beta\;\partial_z^\gamma\partial_c^{\alpha_1}a_c\dd w,
 \qquad j=|\gamma|+|\alpha_1|\leq q,
\]
in which $\beta$ is a product of derivatives of the factors $A_{e_i}$ and
$\alpha$ of total order $q-j$: each of the $q$ derivatives falls either
on the input or on one of these factors. By
\eqref{new:eq:carrier-coefficient} and \eqref{new:eq:plus-subadditive},
the product $\beta$ satisfies \eqref{new:eq:carrier-coefficient} with
$|\gamma|$ replaced by $q-j$. Restoring the phase factor, the term is
$e^{-\ii\lambda\phi}\mathcal A[\beta](e^{\ii\lambda\phi}\partial_z^\gamma\partial_c^{\alpha_1}a_c)$,
where $\mathcal A[\beta]$ is the factor with its amplitude multiplied by
$\beta$. The fixed-time bound in the proof of
Lemma~\ref{app:ch_regular_base:lem:retarded-kernel}, in the form for
amplitudes depending on both positions
(Lemma~\ref{app:ch_osc:lem:osc-amplitude}), uses at most $2\ell$
derivatives of $\beta$ in the physical variables, so
$\|\mathcal A[\beta]\|_{L^2\to L^2}\leq C\lambda^pT^{(q-j-R)_+}$. The
hypothesis gives
$\|\partial_z^\gamma\partial_c^{\alpha_1}a_c\|_2\leq CAT^{(j-R)_+}$ when
$|\alpha_1|\leq k_0$. By \eqref{new:eq:plus-subadditive},
\[
 C_qA\lambda^pT^{(q-j-R)_+}T^{(j-R)_+}
       \leq C_qA\lambda^pT^{(q-R)_+}.
\]
Integration over the fixed $r$-interval proves the assertion for a
retarded factor, including time derivatives.

The loss $b$ is thus consumed entirely by the coefficient factors $A_e$
and $\alpha$, each of which is differentiated at most $2\ell$ times
beyond its share $\gamma$. The input enters only through its derivatives
of total order $j$, at the same index $R$ as in the hypothesis. This is
why $b$ does not depend on $s$, on $k_0$, or on the number of factors in
a composition.

\emph{Nonstationary case and compositions.} Fixed low-frequency pieces
contribute only tails. If the amplitude vanishes near the critical set,
$|\nabla_wf|$ is bounded below on its support, and $J$ integrations by
parts give $C_{q,J}A\lambda^{C_q}(T/\lambda)^J$ at joint order $q$;
choosing $J>(Q+C_q)/\nu$ gives $O(A\lambda^{-Q})$. For a finite
composition $\mathcal A_m\cdots\mathcal A_1$, the output
$a_c'=e^{-\ii\lambda\phi}\mathcal A_1(e^{\ii\lambda\phi}a_c)$, multiplied
by the fixed cutoff in front of the next factor, satisfies by the
estimate just proved the hypothesis of the lemma with $A$ replaced by
$CA\lambda^{p_1}$, with the same index $R$ and the same $k_0$. The
coefficient index $R+b$ required for the next factor is the same as for
the first one. Hence the estimate iterates, with the product of the
factors $\lambda^{p_i}$ and constants depending on the number of factors,
but without any increase of the required derivative order.
\end{proof}

\subsection{The corrected fields}

\begin{cor}[The corrected fields]
\label{app:ch_response:cor:current-atoms}
Use the two-term approximation \eqref{app:ch_waves:eq:WKB-u}, with a real
initial amplitude, on the whole compact set of shifted covectors, and assume
\[
 \|a_G\|_{C^r}+\|G\|_{C^r}+\|r_z\|_{C^r}
 \leq D_rT^{(r-R_*)_+},\qquad r\geq0,
 \qquad 1\leq T\leq\lambda^{1-\nu},\quad\nu>0.
\]
Recall $M_{\rm an}=20(n+1)$, and put $r_{\rm par}=M_{\rm an}+3$ and
$\ell=\lfloor n/2\rfloor+1$. Before the sample and the later accuracy of
the mean are chosen, fix
\[
 R_{\rm env}\geq M_{\rm an}+\ell+2+r_{\rm par},
 \qquad k_0=r_{\rm par}.
\]
There is a fixed $B_{\rm at}$ such that $R_*\geq R_{\rm env}+B_{\rm at}$
covers the losses in the coefficients, in $P_Gb_1$, in the
reconstruction and in the transfer to shifted covectors, including the
finite orders required by the iteration for the mean; the two
derivatives of the reconstruction and the Sobolev embedding loss are
included in $R_{\rm env}$. After $R_{\rm env}$ is fixed, prescribe finitely
many derivative orders for the linearized metric and an accuracy, and
choose a sufficiently large finite $N$. Suppressing $N$ from the
notation, define
\begin{equation}\label{app:ch_response:eq:current-reduced-atom}
 \begin{gathered}
 u_\tau=S_+\mathcal T_N
                 (\chi_{\rm src}P_Gv_\tau),\qquad
 \mathcal Y_\tau=\mathcal V_G(u_\tau),\\
 \mathcal W_\tau^{\rm cur}=\mathcal V_G(v_\tau-u_\tau),\qquad
 A_\tau^{\rm cur}=e^{-\ii S_G/\tau}d\mathcal W_\tau^{\rm cur}.
 \end{gathered}
\end{equation}
Write $c=(z,\zeta')$. On the full observation support, for
$k\leq r_{\rm par}$ and uniformly over the fixed radial factors,
\begin{align}
 &\lambda\|D_x^rD_c^k(e^{-\ii S_G/\tau}\mathcal Y_\tau)\|_\infty
 +\|D_x^rD_c^k(e^{-\ii S_G/\tau}d\mathcal Y_\tau)\|_\infty
       \leq C_{r,k}\lambda^{-1}T^{(r-M_{\rm an})_+},
 \label{app:ch_response:eq:current-atomic-amplitudes}\\
 &\lambda\|D_x^rD_c^k(e^{-\ii S_G/\tau}\mathcal W_\tau^{\rm cur})\|_\infty
 +\|D_x^rD_c^kA_\tau^{\rm cur}\|_\infty
       \leq C_{r,k}T^{(r-M_{\rm an})_+},\notag\\
 &\|D_x^rD_c^k(A_\tau^{\rm cur}-A_0)\|_\infty
       \leq C_{r,k}\lambda^{-1}T^{(r-M_{\rm an})_+},
 \label{app:ch_response:eq:current-atomic-reserve}
\end{align}
where $A_0=A_0(b_0)$ is the leading transported tensor. More precisely,
for every $r\geq0$ and $k\leq k_0$ the bounds for $A_\tau^{\rm cur}$,
$A_0$ and their difference hold with $T^{(r+k+\ell+2-R_{\rm env})_+}$ in
place of $T^{(r-M_{\rm an})_+}$, the difference carrying the factor $\lambda^{-1}$.
The linearized metric $D\gamma_G\mathcal W^{\rm cur}_\tau$ is bounded by
the prescribed accuracy on $D_{\rm lag}$, where the source cutoff equals
one and the reduced identity holds. These are the joint bounds assumed
in Theorem~\ref{app:ch_waves:thm:sampling}, which therefore applies to
this family. The construction is linear in the initial amplitude and is
fixed before the sample. Constants may depend on $N$; the lower bounds
at the finitely many orders used in the sampling theorem are uniform
above a frequency threshold depending on $N$, because the corrections
carry a factor $\lambda^{-1}$.
\end{cor}

\begin{proof}
\emph{Step 1 (the reduced residual).} By \eqref{app:ch_waves:eq:exact-half-strain},
$D\gamma_G\mathcal V_G(v)=2\Pi_{K_G}(\sym\nabla^gv)$, whose components in
the frame are determined by $P_Gv$ through
\eqref{main:eq:stress-lift}. Thus the normal completion has no source,
and it suffices to correct the reduced residual $P_Gv_\tau$. The finite
array $b_0,b_1,P_Gb_1$, together with $\nu_{b_j}$, $\mathcal V_G(b_j)$ and
their first physical derivatives, $j=0,1$, is given by the transport
equation and fixed algebraic formulas, and satisfies the joint
derivative bounds with index $R_{\rm env}$ after one fixed increase of
$B_{\rm at}$. Put $h_\tau=\chi_{\rm src}P_Gv_\tau$. By
\eqref{app:ch_waves:eq:WKB-residual}, $h_\tau=e^{\ii S_G/\tau}\tau^3\chi_{\rm src}P_Gb_1$,
an oscillatory function with amplitude of size $\lambda^{-3}$ and
derivative bounds with index $R_{\rm env}$.

\emph{Step 2 (the correction $u_\tau$).}
Lemma~\ref{app:ch_response:lem:energy-carrier-jets} gives the same bounds
for every $\mathcal K^jh_\tau$, $j<N$, and then for
$u_\tau=S_+\mathcal T_Nh_\tau$; each new residual starts inside the
original source region. Hence
\[
 \|\partial_c^\alpha(e^{-\ii S_G/\tau}
              \chi_{\rm core}u_\tau)\|_{H^s}
 \leq C_{N,s,\alpha}\lambda^{-3}
             T^{(s+|\alpha|-R_{\rm env})_+}.
\]

\emph{Step 3 (amplitude bounds).} The lift $\mathcal V_G$ and the
differential use at most two derivatives, and Sobolev embedding uses
$\ell$ further orders. After $r$ physical and $k$ parameter derivatives
the exponent is at most $(r+k+2+\ell-R_{\rm env})_+\leq(r-M_{\rm an})_+$, because
$k\leq r_{\rm par}$ and $R_{\rm env}\geq M+\ell+2+r_{\rm par}$. Each derivative
of the phase factor costs at most $\lambda$, and at most two such
factors occur, which proves
\eqref{app:ch_response:eq:current-atomic-amplitudes}. For $v_\tau$
substitute $b=b_0+\tau b_1$ in \eqref{app:ch_waves:eq:exact-W-amplitude}
and differentiate once: the leading term of the demodulated derivative
is $A_0(b_0)$, and every other term carries a factor $\tau$; the
demodulated field itself carries a factor $\tau$. The same bounds for the
finite array and the subtraction of $\mathcal Y_\tau$ prove
\eqref{app:ch_response:eq:current-atomic-reserve}.

\emph{Step 4 (the linearized metric).} On the region where the reduced
identity holds,
\[
 P_G(v_\tau-u_\tau)=(1-\chi_{\rm src})P_Gv_\tau
      -(A_+-I)\mathcal T_Nh_\tau
      -(-1)^{N-1}\mathcal K^Nh_\tau ,
\]
by the telescoping identity of Theorem~\ref{app:ch_regular:thm:no-return}.
The first term vanishes on $D_{\rm lag}$. Each term of the residual
\eqref{app:ch_regular_base:eq:energy-complete-residual} has one factor of
order $-1$, so the preceding lemma, after restoring the phase factor,
gives
\[
 \|\partial_c^\alpha\mathcal K^Nh_\tau\|_{H^s}
       \leq C_{N,s,\alpha}\lambda^{s+|\alpha|-N-3}.
\]
The conic complement $A_+-I$ vanishes near the phase gradients, so the
nonstationary case of the lemma makes the second term $O(\lambda^{-Q})$.
By \eqref{main:eq:stress-lift} and the Gram formula, with $T\leq\lambda$,
increasing $N$ gives the prescribed accuracy for the linearized metric.
The index $R_{\rm env}$ and all factors are independent of $N$.
\end{proof}

\subsection{Finite Fourier measures}\label{app:ch_lag:sec:lag-inputs}
The same correction applies to every Fourier modulation of the initial
scalar. We integrate the corrected fields over the fixed set of shifted
covectors; the source and observation cutoffs are those of
Setup~\ref{app:setup:domains}.

\begin{prop}[Bounds for Fourier measures]
\label{app:ch_lag:prop:lag-envelopes}
\label{app:ch_lag:prop:lag-telescope}
Apply Corollary~\ref{app:ch_response:cor:current-atoms} at the immersion
$F$, with coefficient scale $T\geq2$ and phase $S_F(x,\zeta)$, and with
its finite depth chosen for a prescribed derivative order $r_{\rm obs}$
and accuracy $Q$. Let $\lambda=\tau^{-1}\geq2$, $T\leq\lambda^{1-\nu}$ for a
fixed $\nu>0$, and let $m$ be a finite complex measure supported in
$B_\Lambda\subset\R^{n-1}$. Suppose $K\geq\max(T,C_{\rm mod}\Lambda,2)$ and
that $\zeta+\tau B_\Lambda$ lies in the fixed set of shifted covectors.
Put $\|m\|=\|m\|_{\TV}$, $\phi_F=S_F(\cdot,\zeta)$ and
\begin{equation}\label{app:ch_lag:eq:lag-recurrence}
 Z(m)=\int\mathcal W_\tau^{\rm cur}
                  (F;x,\zeta+\tau\xi)\dd m(\xi).
\end{equation}
Then for every $s\geq0$
\begin{equation}\label{app:ch_lag:eq:lag-all-envelopes}
 \lambda\|\chi_{\rm core}e^{-\ii\lambda\phi_F}Z\|_{C^s}
 +\|\chi_{\rm core}e^{-\ii\lambda\phi_F}dZ\|_{C^s}
 \leq C_s\|m\|K^s,
\end{equation}
and on $D_{\rm lag}$, for $0\leq r\leq r_{\rm obs}$,
\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 map $m\mapsto Z(m)$ is complex linear. The depth is chosen before $m$
and the sample; the geometry, the derivative losses and the domains do
not depend on it. The fixed radial factors affect only the constants.
\end{prop}

\begin{proof}
Since the shifted covectors range over a compact set, we may
differentiate under the integral. Demodulating by the central phase,
\[
 e^{-\ii\lambda\phi_F}\mathcal W^{\rm cur}_\tau(F;x,\zeta+\tau\xi)
 =e^{\ii\Psi(x,\xi)}\,e^{-\ii S_F(x,\zeta+\tau\xi)/\tau}
 \mathcal W^{\rm cur}_\tau(F;x,\zeta+\tau\xi),
\]
where
\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\dd u.
\end{equation}
By the joint phase bounds of Proposition~\ref{app:ch_seed:prop:seed-actual-menu},
$|D_x^j\Psi|\leq C_j\Lambda T^{j-1}$ for $j\geq1$ and $|\xi|\leq\Lambda$. By
the Leibniz and chain rules, $D_x^s(e^{\ii\Psi}w)$ is a sum of terms
$e^{\ii\Psi}\,D^{j_1}\Psi\cdots D^{j_k}\Psi\,D^{i}w$ with
$j_1+\dots+j_k+i=s$ and $j_\iota\geq1$. The amplitude bounds of
Corollary~\ref{app:ch_response:cor:current-atoms} give
$\|D^iw\|_\infty\leq C_iT^i$ for both demodulated amplitudes (the field
with the factor $\lambda$, and its differential). Since
$\sum_\iota(j_\iota-1)=s-i-k$, such a term is bounded by
\[
 C\Lambda^kT^{s-i-k}\,\|D^iw\|_\infty
 \leq C\Lambda^kT^{s-k}\leq C_sK^s ,
\]
using $\Lambda\leq K$ and $T\leq K$. Integrating against
$|m|$ proves \eqref{app:ch_lag:eq:lag-all-envelopes}. The linearized
metric of each field is bounded by the prescribed accuracy on $D_{\rm lag}$,
uniformly over the shifted covectors and including all source terms,
in every $C^r$ norm with $r\leq r_{\rm obs}$; integration proves
\eqref{app:ch_lag:eq:lag-current-strain}.
\end{proof}

\subsection{The waves at one step}
We now state the result in the form used at one step of the iteration.
Fix a finite order $r_*$ containing all geometric, analytic and density
derivative orders used below. The immersion $F$ lies in a fixed
$C^{r_*}$ neighborhood of $G_{\rm ref}$ and satisfies
\begin{equation}\label{main:eq:lag-hypothesis}
 \|F\|_{C^r(W)}\leq C_rT_0^{(r-r_*)_+},\qquad T_0\geq2 .
\end{equation}
The estimates of Section~\ref{main:sec:history} verify both conditions for
every iterate. All constructions use the same immersion $F$, at which the
metric is linearized.

On a whole characteristic patch we use $x=(t,y)$ and $\zeta\in\R^{n-1}$,
and the Hamilton phase with
\[
 p_F(x,d_x\phi_F)=0,\qquad
 \phi_F(0,y,\zeta)=y\cdot\zeta .
\]
For each prescribed integer $q\geq1$,
Corollary~\ref{app:ch_response:cor:current-atoms} gives a family
$\mathcal W_\tau^{\rm cur}(F;x,\zeta)$: a transported characteristic wave
corrected by a finite Neumann series applied to its reduced residual.
The correction preserves the phase and the leading transported tensor.
The family depends linearly on a prescribed real initial amplitude, with
complex extension for Fourier data; its depth may depend on $q$ but not
on the frequency or the base covector. The family is chosen on the
entire compact set of shifted covectors before the sample is drawn. For a
sample $\nu$ with $\lambda_\nu=\tau_\nu^{-1}$ and central phase
$\phi_\nu=\phi_F(\cdot,\zeta_\nu)$ we define the complex linear map
\begin{equation}\label{main:eq:lag-recurrence}
 Z_\nu(m)=\int\mathcal W_{\tau_\nu}^{\rm cur}
          (F;x,\zeta_\nu+\tau_\nu\xi)\dd m(\xi).
\end{equation}
The Dirac mass $m=\delta_0$ gives the unmodulated wave; a finite
Fourier measure changes its initial data without changing the
correction operator.

\begin{proposition}[Corrected characteristic waves]\label{main:prop:current-waves}
Fix $q\geq1$. Assume \eqref{main:eq:lag-hypothesis}, the fixed domains of
Setup~\ref{app:setup:domains}, and use the positive initial amplitude $r_F$
of Proposition~\ref{app:ch_background:prop:bg-current}, whose derivative
losses are included in $r_*$. The family has the original phase and
leading transported tensor, and $\|d\phi_\nu\|_{C^r}\leq C_rT_0^r$ on the
full observation support. Let $m$ have finite total variation and support
in $B_\Lambda$, and choose a fixed $\nu_0>0$ with
\begin{equation}\label{main:eq:response-window}
 C_{\rm env}\max(T_0,\Lambda)\leq K
       \leq\lambda^{1-\nu_0},\qquad
 c\lambda\leq\lambda_\nu\leq C\lambda .
\end{equation}
Then, for large $\lambda$, the shifted covectors stay in the domain of the
phase, and on the full observation support
\begin{equation}\label{main:eq:lag-full-envelope}
 \lambda_\nu\|\chi_{\rm core}e^{-\ii\lambda_\nu\phi_\nu}Z_\nu(m)\|_{C^r}
 +\|\chi_{\rm core}e^{-\ii\lambda_\nu\phi_\nu}dZ_\nu(m)\|_{C^r}
 \leq C_r\|m\|_{\rm TV}K^r .
\end{equation}
The depth of the correction can be chosen so that
\begin{equation}\label{main:eq:lag-strain}
 \|D\gamma_FZ_\nu(m)\|_{C^0(D_{\rm lag})}
       \leq C_q\|m\|_{\rm TV}\lambda^{-q}.
\end{equation}
The domains, the order $r_*$ and the frequency exponents do not depend
on $q$.
\end{proposition}

\begin{proof}
Apply Proposition~\ref{app:ch_lag:prop:lag-envelopes} with $T=T_0$,
$\tau=\tau_\nu$, $\zeta=\zeta_\nu$ and accuracy $Q$ with $Q\geq q+1$. Its
hypotheses hold: by \eqref{main:eq:response-window},
$T_0\leq K\leq\lambda^{1-\nu_0}\leq C\lambda_\nu^{1-\nu_0}$, so
$T\leq\lambda_\nu^{1-\nu}$ for any fixed $\nu<\nu_0$ and large $\lambda$;
$K\geq C_{\rm env}\Lambda\geq C_{\rm mod}\Lambda$ after enlarging
$C_{\rm env}$; and $\tau_\nu\Lambda\leq C\lambda^{-1}K\leq C\lambda^{-\nu_0}$,
so the shifted covectors $\zeta_\nu+\tau_\nu B_\Lambda$ stay in the fixed
domain of the phase for large $\lambda$. The coefficient bounds required
in Corollary~\ref{app:ch_response:cor:current-atoms} follow from
\eqref{main:eq:lag-hypothesis} by Proposition~\ref{app:prop:fixed-high}
and the bounds for $r_F$ in
Proposition~\ref{app:ch_background:prop:bg-current}. Then
\eqref{app:ch_lag:eq:lag-all-envelopes} is
\eqref{main:eq:lag-full-envelope} (with $\lambda$ replaced by the
comparable $\lambda_\nu$), and \eqref{app:ch_lag:eq:lag-current-strain}
with $r=0$ gives \eqref{main:eq:lag-strain}. The bound for $d\phi_\nu$
is part of \eqref{app:ch_seed:eq:seed-phase-tame}: no derivative loss is
hidden in passing from a single shifted covector to finite measures,
because the phase quotient \eqref{app:ch_lag:eq:lag-exact-phase} varies
on the scale $\max(T_0,|\xi|)\leq K$.
\end{proof}

\section{Separated sampling}\label{sec:sampling}

A continuous superposition of waves over the patch parameters is
replaced by a finite sum over sample points which are mutually separated,
so that the products of distinct waves oscillate at frequencies at least
comparable to $\lambda h$. In this section we show that such a sample
reproduces the continuous normal operator in operator norm, uniformly on
the whole output space, and the continuous Hermitian mean in every
Sobolev norm.

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

\begin{construction}[The sampling procedure]\label{new:constr:sampling}
Let finitely many patches with parameters $z=(s,v)$ in buffered
three-dimensional charts be given, together with fixed smooth nonnegative
compactly supported weights $\psi_a(z)$. In each chart and for
$0<h\leq h_0$ consider the cubes $C_q=h(q+[0,1)^3)$, $q\in\Z^3$. Take all
cubes meeting $\supp\psi_a$ together with their neighbors, and draw
independent uniformly distributed points $U_q\in C_q$; different charts
use independent draws. Discard a point if another original point of the
same chart has $\ell^\infty$ distance less than $h/4$ from it. For
$z\in C_q$ put
\[
 p_q(z)=\prod_{q'\ne q}
 \left(1-\frac{|C_{q'}\cap(z+[-h/4,h/4]^3)|}{h^3}\right);
\]
only the $26$ neighboring cubes contribute. To each surviving point
$U_q$ of a cube meeting $\supp\psi_a$ assign the weight
\[
 \mu_q=h^3\psi_a(U_q)/p_q(U_q),
\]
and omit zero weights. Number the resulting samples by $j$, with patch
index $a(j)$ and parameter $Z_j$. The fixed $h_0$ keeps all points inside
the chart margins.
\end{construction}

\begin{lem}[Separation and unbiasedness]\label{new:lem:sampling-basic}
For every realization there are at most $Ch^{-3}$ samples, the weights
are at most $Ch^3$, the total mass is bounded, and distinct samples in
the same chart are separated by at least $h/4$. Moreover $p_q\geq7/8$,
and for every integrable family $\Phi$ with values in a Banach space,
\[
 \mathbb E\Bigl[\mathbf1_{\{U_q\ \text{survives}\}}
      \frac{h^3\psi_a(U_q)}{p_q(U_q)}\Phi(U_q)\Bigr]
       =\int_{C_q}\psi_a(z)\Phi(z)\dd z .
\]
\end{lem}

\begin{proof}
Conditioned on $U_q=z$, the point survives if and only if no other
point $U_{q'}$ lies in $z+(-h/4,h/4)^3$. These events are independent,
and the probability that $U_{q'}$ avoids the box is
$1-|C_{q'}\cap(z+[-h/4,h/4]^3)|/h^3$ (the boundary has measure zero).
Hence the conditional survival probability is exactly $p_q(z)$. The box
has volume $(h/2)^3=h^3/8$, so the excluded volumes in the other cubes sum
to at most $h^3/8$, and $p_q(z)\geq1-\sum_{q'}x_{q'}\geq7/8$, where
$x_{q'}$ are the factors subtracted from one. Since $U_q$ has density
$h^{-3}$ on $C_q$,
\[
 \mathbb E\Bigl[\mathbf1_{\rm surv}\frac{h^3\psi_a(U_q)}{p_q(U_q)}\Phi(U_q)\Bigr]
 =\int_{C_q}h^{-3}p_q(z)\frac{h^3\psi_a(z)}{p_q(z)}\Phi(z)\dd z
 =\int_{C_q}\psi_a\Phi\dd z .
\]
If two surviving points were at distance less than $h/4$, both would
have been discarded, which gives the separation. There is at most one
sample per cube, and there are $O(h^{-3})$ cubes; the weights are at most
$\tfrac87h^3\|\psi_a\|_\infty$, and the total mass is at most a constant.
No derivative of $p_q$ is used.
\end{proof}

We use the following form of the operator Bernstein inequality with
intrinsic dimension; see \cite{MinskerBernstein} and
\cite[Theorem~7.3.1]{Tropp15}.

\begin{lem}[Operator Bernstein inequality]\label{new:lem:operator-bernstein}
Let $Y_1,\dots,Y_m$ be independent random self-adjoint operators on a
finite-dimensional Hilbert space with $\mathbb EY_q=0$ and
$\|Y_q\|\leq\mathsf R$ almost surely. Let $V$ be a positive semidefinite
operator with $\sum_q\mathbb EY_q^2\leq V$, and put
$d=\operatorname{tr}V/\|V\|$. Then for $u\geq\|V\|^{1/2}+\mathsf R/3$,
\[
 \mathbb P\Bigl(\Bigl\|\sum_qY_q\Bigr\|\geq u\Bigr)
 \leq8d\exp\Bigl(-\frac{u^2/2}{\|V\|+\mathsf Ru/3}\Bigr).
\]
\end{lem}

\begin{thm}[Separated sampling]\label{app:ch_waves:thm:sampling}
Use finitely many buffered three-parameter charts with weights $\psi_a$
as in Construction~\ref{new:constr:sampling}, and a smooth family of
ambient waves $\mathcal W_\tau(x,z,\zeta')$ with the phase $S_G$ and the
leading tensor $A_0=A_0(b_G)$ of Theorem~\ref{app:ch_waves:thm:leading}.
Put $A_\tau=e^{-\ii S_G/\tau}d\mathcal W_\tau$, and assume,
for one scale $L\geq1$, every physical order $r$ and every joint order
$k\leq M_{\rm an}+3$ in the patch parameters and shifted covectors,
\begin{equation}\label{app:ch_waves:eq:sampling-amplitudes}
 \|D_x^rD_{z,\zeta'}^kA_\tau\|_\infty
 +\tau^{-1}\|D_x^rD_{z,\zeta'}^k(A_\tau-A_0)\|_\infty
 \leq C_{r,k}L^r .
\end{equation}
Assume also uniform joint $C^{M_{\rm an}+2}$ bounds for the amplitudes and the
inverse charts, with an $O(\tau)$ difference of amplitudes in that norm,
including the $\epsilon$-derivatives of the leading homotopy. No
hypothesis on the linearized metric of the waves is needed. All
physical, parameter and conic cutoffs lie inside the larger charts, and
frequencies below one are omitted.

Fix the band $\Lambda\geq2$ before sampling, with $\Lambda/\lambda$
inside the phase margin. Each indexed patch $a$ has a fixed positive
radial factor $R_a$; put $\tau_a=(R_a\lambda)^{-1}$. Fix a smooth radial
cutoff $0\leq\vartheta\leq1$, equal to one on $B_1$ and zero outside $B_2$,
and put
\begin{equation}\label{app:ch_waves:eq:single-raw-cutoff}
 P_K(\eta)=\vartheta(\eta/K),\qquad K\geq1.
\end{equation}
The background uses the unshifted wave $\mathcal W_{\tau_a}(x,z,\zeta(z))$.
With fixed measurement and frame cutoffs $\chi,\chi_{\rm in}$ define, at
each fixed $z$,
\begin{align}
 T_{a,z}F(x)
 &=\int e^{\ii\Phi_{\tau_a}(x,z,\eta)}|\eta|^{1/2}
       \chi_{\rm hi}P_\Lambda(\eta)a_{\tau_a}(x,z,\eta)F(\eta)\dd\eta,
 \label{app:ch_waves:eq:actual-T}\\
 \Phi_\tau
 &=\frac{S_G(x,\zeta(z)+\tau\eta)-S_G(x,\zeta(z))}{\tau},
 \label{app:ch_waves:eq:phase-quotient}\\
 a_\tau
 &=\chi\chi_{\rm in}
 \bigl(Q_i:A_\tau(x,z,\zeta(z))^*
       A_\tau(x,z,\zeta(z)+\tau\eta)\bigr)_{i=1}^n,
 \label{app:ch_waves:eq:cross-amplitude}
\end{align}
the quotient being extended by its integral formula at $\tau=0$. The
input space is the complex $L^2(d\eta)$ and the output space is
$\mathcal H=L^2(\R^n;\C^n)$. The cutoffs belong to these operators; the
ambient fields are not cut. The same random sample may serve finitely
many further operators of this type with the same lower bounds, or scalar
operators with amplitude $1$ and any fixed physical cutoff in the common
larger phase chart. Define
\begin{equation}\label{app:ch_waves:eq:sampling-prefix-normal}
 E_{\rm sample}=\sum_j\mu_jT_{a(j),Z_j}T_{a(j),Z_j}^*,\qquad
 E_{\rm cont}=\sum_a\int\psi_aT_{a,z}T_{a,z}^*\dd z.
\end{equation}
Then the following hold.
\begin{enumerate}
\item The operators satisfy, on the whole spaces,
\begin{equation}\label{app:ch_waves:eq:sampling-operator-bounds}
\|T_{a,z}\|^2\leq C\Lambda,\quad
\|T_{a,z}\|_{\mathrm{HS}}^2\leq C\Lambda^n,\quad
\|E_{\rm cont}\|\leq C,\quad
\operatorname{tr}E_{\rm cont}\leq C\Lambda^n.
\end{equation}
\item For fixed $\epsilon>0$ and $h^{-3}\asymp\Lambda^{1+\epsilon}$, with
probability tending to one as $\Lambda\to\infty$,
\begin{equation}\label{app:ch_waves:eq:sampling-all-prefixes}
 \|E_{\rm sample}-E_{\rm cont}\|\leq\Lambda^{-\epsilon/4}.
\end{equation}
The same holds for the real operators $2\Re T$, whose normal operators are
$4\Re E$, with fixed constants.
\item Put, with a fixed compact spatial cutoff,
\[
 H_{a,z}=\Re\bigl[
 A_{\tau_a}(x,z,\zeta(z))^*A_{\tau_a}(x,z,\zeta(z))\bigr],\qquad
 H_{\rm cont}=\sum_a\int\psi_aH_{a,z}\dd z .
\]
If $\sup_{a,z}\|H_{a,z}\|_{H^s}\leq B_s$ for all integers $s\geq0$, the
event in (ii) can be chosen so that, simultaneously for all $s$,
\begin{equation}\label{app:ch_waves:eq:background-quadrature}
 \Bigl\|\sum_j\mu_jH_{a(j),Z_j}-H_{\rm cont}\Bigr\|_{H^s}
       \leq C2^{s/2}hB_s ,
\end{equation}
at the cost of an additional failure probability at most $Ch$.
Moreover, with $H_{\mathrm{transport},a,z}=|\nu_{b_0}|^2\,dS(x,\zeta(z))^{\otimes2}$,
\[
 \Bigl\|H_{\rm cont}-\sum_a\int\psi_a
 H_{\mathrm{transport},a,z}\dd z\Bigr\|_{L^2}\leq C\lambda^{-1}.
\]
\item Let $E_{\mathrm{transport,band}}$ be the continuous normal operator
with the same weights and cutoffs, but with $\Phi_{\tau_a}$ replaced by
$\Phi_0$ and $A_{\tau_a}$ by $A_0(b_0)$. Then
\begin{equation}\label{app:ch_waves:eq:continuous-normal-comparison}
 \|E_{\rm cont}-E_{\mathrm{transport,band}}\|
       \leq C\lambda^{-1}\Lambda .
\end{equation}
\end{enumerate}
In particular one separated sample approximates the transported normal
operator on the whole output space within $C\lambda^{-1}\Lambda+\Lambda^{-\epsilon/4}$,
so that a lower bound for the latter on a given class of outputs passes
to the former. The wave family and the band are fixed before the sample;
separation, count, weight and mass bounds hold for every realization.
\end{thm}

\begin{proof}
\emph{Operator bounds.} The joint amplitude bounds give the hypotheses
of Setup~\ref{app:ch_waves:setup:oscillatory}, and
Proposition~\ref{app:ch_osc:prop:osc-normal}, with
$p_\Lambda^2=\chi_{\rm hi}^2P_\Lambda^2$, gives the bounds for
$T_{a,z}$. Since $E_{\rm cont}=\sum_aT_a\psi_aT_a^*$ with the full
operators $T_a$, the full bound gives $\|E_{\rm cont}\|\leq C$, and
$\operatorname{tr}E_{\rm cont}=\sum_a\int\psi_a\|T_{a,z}\|_{\rm HS}^2\dd z\leq C\Lambda^n$.
For several operator families we use the block-diagonal normal operator
on the finite direct sum of the output spaces.

\emph{Decomposition into independent classes.} Let $X_q$ be the positive
operator contributed by the cube $q$, that is
$\mu_qT_{a,U_q}T_{a,U_q}^*$ if $U_q$ survives and $0$ otherwise, and put
$Y_q=X_q-\mathbb EX_q$. By Lemma~\ref{new:lem:sampling-basic},
$\sum_q\mathbb EX_q=E_{\rm cont}$. Each $X_q$ depends only on the draws in
the cubes $q+\{-1,0,1\}^3$. Partition the indices according to
$q\bmod3\in(\Z/3)^3$; for two distinct indices in the same class the sets
$q+\{-1,0,1\}^3$ are disjoint, so within each of the $27$ classes the
operators $X_q$ are independent. Fix a class $\mathcal I$ and put
$\mathsf R=C\Lambda h^3$. Then $0\leq X_q\leq\mathsf RI$, so
$\|Y_q\|\leq\mathsf R$, and $X_q^2\leq\mathsf RX_q$ gives
\[
 \sum_{q\in\mathcal I}\mathbb EY_q^2\leq\sum_{q\in\mathcal I}\mathbb EX_q^2
 \leq\mathsf R\sum_{q\in\mathcal I}\mathbb EX_q\leq\mathsf RE_{\rm cont}.
\]

\emph{Concentration.} Let $P_\ell\uparrow I$ be finite-rank orthogonal
projections, and let $E_1$ be a rank-one projection with range in that
of $P_\ell$. Apply Lemma~\ref{new:lem:operator-bernstein} to the compressions
$P_\ell Y_qP_\ell$, with
$V_\ell=\mathsf RP_\ell E_{\rm cont}P_\ell+\mathsf RE_1$. The variance
condition holds, since $(P_\ell Y_qP_\ell)^2\leq P_\ell Y_q^2P_\ell$ for
self-adjoint $Y_q$. The term $\mathsf RE_1$ makes $V_\ell\neq0$, so that
the intrinsic dimension is defined even when the variance of the
compressed sum vanishes. Then $\mathsf R\leq\|V_\ell\|\leq C\mathsf R$ and
$\operatorname{tr}V_\ell\leq\mathsf R(C\Lambda^n+1)$, so
$d\leq C\Lambda^n$. Take $u=\Lambda^{-\epsilon/4}/27$. Since
$h^{-3}\asymp\Lambda^{1+\epsilon}$, $\mathsf R\asymp\Lambda^{-\epsilon}$, and
for large $\Lambda$ we have $u\geq(C\mathsf R)^{1/2}+\mathsf R/3$ and
\[
 \frac{u^2/2}{C\mathsf R+\mathsf Ru/3}\geq c\Lambda^{-\epsilon/2}\Lambda^{\epsilon}
 =c\Lambda^{\epsilon/2}.
\]
Hence
$\mathbb P(\|P_\ell\sum_{q\in\mathcal I}Y_qP_\ell\|\geq u)\leq C\Lambda^n\exp(-c\Lambda^{\epsilon/2})$,
uniformly in $\ell$. The sum $\sum_{q\in\mathcal I}Y_q$ is a finite sum of
trace class operators, hence compact, so its compressions converge to it
in norm; by Fatou's lemma the same bound holds for
$\|\sum_{q\in\mathcal I}Y_q\|$. A union bound over the $27$ classes and the
triangle inequality give \eqref{app:ch_waves:eq:sampling-all-prefixes}
with failure probability at most $C\Lambda^n\exp(-c\Lambda^{\epsilon/2})$.
The assertion for the real operators follows from
\eqref{app:ch_waves:eq:real-normal}.

\emph{Background quadrature.} The tensor contributed by one cube has
$H^s$ norm at most $Ch^3B_s$. Centering these vectors and summing within
one class, independence and the Hilbert space identity
$\mathbb E\|\sum_qZ_q\|^2=\sum_q\mathbb E\|Z_q\|^2$ for independent
centered $Z_q$ give a bound $Ch^{-3}(Ch^3B_s)^2=Ch^3B_s^2$, and the
triangle inequality in $L^2(\mathbb P;H^s)$ over the $27$ classes gives
\[
 \mathbb E\Bigl\|\sum_j\mu_jH_{a(j),Z_j}-H_{\rm cont}\Bigr\|_{H^s}^2
       \leq Ch^3B_s^2 .
\]
By Chebyshev's inequality, the probability that the norm exceeds
$C2^{s/2}hB_s$ is at most $C2^{-s}h$. Summing over $s\geq0$ and
intersecting with the event of~(ii) proves
\eqref{app:ch_waves:eq:background-quadrature}. By
\eqref{app:ch_waves:eq:sampling-amplitudes},
$|A_\tau-A_0|\leq C\tau\leq C\lambda^{-1}$, and
$A_0^*A_0=|\nu_{b_0}|^2\xi\xi^T$ by Theorem~\ref{app:ch_waves:thm:leading};
this gives the comparison with the transported tensor.

\emph{Comparison of normal operators.} First replace the amplitudes by
the leading ones on the unchanged phase. Their difference is
$O(\tau_a)$ in the relevant norm, so by
Proposition~\ref{app:ch_osc:prop:osc-normal} the normal operator changes
by at most $C\tau_a$. The leading amplitudes along the homotopy in
$\tau$ have $\tau$-derivatives of order $|\eta|$, so
\eqref{app:ch_osc:eq:osc-normal-mask}, applied with each weight $\psi_a$
and summed over the finitely many charts, bounds the change of phase by
$C\tau_a\Lambda$. Since $\tau_a\asymp\lambda^{-1}$, this proves
\eqref{app:ch_waves:eq:continuous-normal-comparison}.
\end{proof}

\begin{rem}\label{app:ch_waves:rem:reference-status}
For a fixed band the sampled normal operators are compact on the whole
output space. Their approximation in norm therefore transfers a lower
bound only on a class of outputs for which such a bound has already been
established, as in Section~\ref{app:sec:ch_background}.
\end{rem}

%% ===== 12_background.tex =====
\section{The continuous normal operator and a positive mean}\label{app:sec:ch_background}

In this section we combine the image of the reduced operator with the
quadratic tensors of characteristic waves into one operator $A$, and
show that $E=AA^*+1-\chi_{\rm meas}^2$ is coercive; this is the analytic
form of the Gram inequality \eqref{main:eq:detector-gram}. We then
estimate the inverse of $E$ in Sobolev spaces, and use it to construct a
positive initial amplitude $r_F$ and a tangent field $V_{\rm base}$ such
that the transported mean of the unmodulated waves with density
$r_F^2$, together with $D\gamma_FV_{\rm base}$, has zero component in
$K_F$. This is what permits metric defects of either sign. Finally we
identify the continuous operator with the leading part of the normal
operator of the waves, which is approximated by the sample in
Section~\ref{sec:sampling}.

In summary, the input of this section is an immersion $G$ in a fixed
$C^{r_*}$ neighborhood of the reference, with derivative bounds of the
form \eqref{new:eq:derivative-growth}; the output consists of the
coercive operator $E_G$ (Proposition~\ref{app:ch_background:prop:bg-frame}),
Sobolev bounds for its inverse with a fixed loss
(Proposition~\ref{app:ch_background:prop:bg-inverse}), and the positive
initial amplitude $r_G$ with the base field $V_{\rm base}$
(Proposition~\ref{app:ch_background:prop:bg-current}). In the proof of
Theorem~\ref{main:thm:main}, $E_G^{-1}$ produces the modulations of the
waves in Section~\ref{main:sec:waves}, and $r_G$, $V_{\rm base}$ cancel
the positive mean of the unmodulated waves.

We use the reference immersion and phase charts of
Section~\ref{app:sec:ch_seed}, the transported fields of
Section~\ref{main:sec:linear} and the domains of
Setup~\ref{app:setup:domains}. All derivative orders used here are
included in $r_*$.

\subsection{Geometry and the associated operators}\label{app:ch_background:sec:bg-inputs}

\begin{defn}[Second fundamental form and reduced operator]\label{app:ch_background:defn:bg-geometry}
For an immersion $G$ near the reference we use the normal map $N_G$ and
the projected frame $Q_{G,i}=\Pi_{K_G}Q_i^0$ of
\eqref{app:ch_waves:eq:ambient-normal-inverse}, and the reduced operator
$P_G$ of \eqref{app:ch_waves:eq:reduced-symbol}. Put
\[
 g_G=\gamma(G),\qquad
 \mathscr G_G=(Q_{G,i}:Q_{G,j})_{i,j=1}^n,\qquad
 q_G(H)=(Q_{G,i}:H)_{i=1}^n,
\]
and $q_G(x,\xi)=q_G(\xi\otimes\xi)$. The neighborhood of
Setup~\ref{app:ch_seed:setup:seed-current} gives a lower bound for the
Gram matrix $\mathscr G_G$. These are smooth functions of the second jet
of $G$, so by Corollary~\ref{app:cor:fixed-profile-transfer} they satisfy
the derivative bounds of $G$ with a loss of two. Replacing $B_G$ by
$B_G/\sigma$ does not change the kernel projection; hence uniform
normalized lower bounds and fixed-order bounds for $dG$, $B_G/\sigma$ and
the metric connection give uniform bounds for the frame and the reduced
operator. The identities \eqref{main:eq:tangent-normal} and
\eqref{app:ch_waves:eq:exact-half-strain} hold pointwise on $W$.
\end{defn}

\begin{defn}[Normalized reference and characteristic phases]\label{app:ch_background:defn:bg-carriers}
We use the flat family $G_\sigma$ of
Corollary~\ref{app:ch_seed:cor:seed-uniform-dilation}, whose projected
frames tend to the constant frame of
Definition~\ref{app:ch_seed:defn:seed-cube}, and the finite cover of the
sphere, the characteristic patches with parameters
$z=(s,v)\in\R\times\R^2$, and the phase charts of
Propositions~\ref{app:ch_seed:prop:seed-frozen-menu}--\ref{app:ch_seed:prop:seed-phase-charts}.
These constructions hold on a fixed $C^{r_*}$ neighborhood, and for the
normalized family they hold on common boxes with normalized lower bounds
down to $\sigma=0$.

For each such immersion $G$ we use the phase $S_G$, the transported
amplitude $b_G$, the normal field $\nu_G$ and the tensor $A_0=A_0(b_G)$ of
Theorem~\ref{app:ch_waves:thm:leading}, with the projected frame and a
prescribed positive initial amplitude; their lower bounds and derivative
bounds are those of that theorem. In the adapted coordinates $x=(t,y)$
put
\[
 \xi_{G,a}=d_xS_G(x,\zeta_a(z)),\quad
 Y_{G,a}=d_\zeta S_G(x,\zeta_a(z)),\quad
 \Phi_{G,a}(x,z,\eta)=Y_{G,a}(x,z)\cdot\eta .
\]
Thus $\Phi_{G,a}$ is the phase $\Phi_\tau$ of
Setup~\ref{app:ch_osc:setup:osc-normal} at $\tau=0$, and
Proposition~\ref{app:ch_seed:prop:seed-phase-charts} gives its normalized
full and fixed-time inverse charts, together with those of the exact
difference quotients. The parameter neighborhoods are taken large enough
to contain every shifted covector in
Definition~\ref{app:ch_background:setup:bg-wkb-input}.
\end{defn}

\subsection{\texorpdfstring{The limit $\sigma\to0$}{The limit sigma to 0}}
The normalized family has a regular limit. Let $\widehat S$ and
$\widehat b$ be the phase and the transported amplitude for $F_0$ with
initial amplitude $1$, and use the initial amplitude $\sigma$ for $G_\sigma$.

\begin{lem}\label{new:lem:scaling}
For $x$ in the fixed ball,
\begin{equation}\label{app:ch_background:eq:bg-Nsigma}
\begin{gathered}
 B_{G_\sigma}(x)=\sigma B_{F_0}(\sigma x),\qquad
 N_{G_\sigma}(x)=\sigma^{-1}N_{F_0}(\sigma x),\\
 S_\sigma(x,\zeta)=\sigma^{-1}\widehat S(\sigma x,\zeta),\qquad
 b_\sigma(x,\zeta)=\sigma\widehat b(\sigma x,\zeta),\\
 \xi_\sigma(x,\zeta)=\widehat\xi(\sigma x,\zeta),\qquad
 \nu_\sigma(x,\zeta)=\widehat\nu(\sigma x,\zeta).
\end{gathered}
\end{equation}
Moreover
\[
 S_\sigma(x,\zeta)=\int_0^1
 x\cdot d_x\widehat S(\theta\sigma x,\zeta)\dd\theta ,
\]
so the phases, and with them the normalized second fundamental form,
its inverse, the transported amplitudes and the normal fields, extend
smoothly to $\sigma=0$ in all fixed mixed derivatives. At $\sigma=0$ the
tensors $Q_i^0$, $\xi_0$ and $\nu_0$ do not depend on $x$, $Y_0$ is linear
in $x$, and $\nu_0\neq0$.
\end{lem}

\begin{proof}
The first identity was proved in
Corollary~\ref{app:ch_seed:cor:seed-uniform-dilation}. The normal spaces of
$G_\sigma$ at $x$ and of $F_0$ at $\sigma x$ coincide, so $K_{G_\sigma}(x)=K_{F_0}(\sigma x)$,
and since $N_G$ vanishes on $K_G$ and inverts $B_G^*=\sigma B_{F_0}^*$ on
$K_G^\perp$, we get $N_{G_\sigma}(x)=\sigma^{-1}N_{F_0}(\sigma x)$. Since
$Q_i(G_\sigma)(x)=Q_i(\sigma x)$, the principal symbols satisfy
$M_{G_\sigma}(x,\xi)=M_{F_0}(\sigma x,\xi)$, and both metrics are the
Euclidean one, so there are no Christoffel terms. The function
$\sigma^{-1}\widehat S(\sigma x,\zeta)$ has gradient
$(d_x\widehat S)(\sigma x,\zeta)$, hence solves the eikonal equation for
$G_\sigma$, and at $t=0$ it equals $\sigma^{-1}(\sigma y\cdot\zeta)=y\cdot\zeta$;
by uniqueness for the Hamilton--Jacobi problem it is $S_\sigma$, and
$\xi_\sigma(x)=\widehat\xi(\sigma x)$. The null vectors of $G_\sigma$ at $x$ are
those of $F_0$ at $\sigma x$, and
\[
 P_{G_\sigma}\bigl[\sigma\widehat b(\sigma x)\bigr]
 =\bigl(Q_i(\sigma x):\sym\partial_x[\sigma\widehat b(\sigma x)]\bigr)_i
 =\sigma^2(P_{F_0}\widehat b)(\sigma x),
\]
so $\sigma\widehat b(\sigma x)$ solves the transport equation for $G_\sigma$
with initial amplitude $\sigma$; by uniqueness it equals $b_\sigma$. Then
$\nu_\sigma=N_{G_\sigma}(\xi_\sigma\sym b_\sigma)
=\sigma^{-1}N_{F_0}(\sigma x)\bigl(\widehat\xi\sym\sigma\widehat b\bigr)(\sigma x)=\widehat\nu(\sigma x)$.

Since $\widehat S(0,\zeta)=0$, the fundamental theorem of calculus along
the segment from $0$ to $\sigma x$ gives the integral formula, which is
smooth in $\sigma$ including $\sigma=0$, where
$S_0(x,\zeta)=x\cdot d_x\widehat S(0,\zeta)$ is linear in $x$. The
remaining quantities are of the form $f(\sigma x)$ and are smooth in
$(\sigma,x)$. Finally, $\xi_0\neq0$ and the null vector $k_0\neq0$ give
$\xi_0\sym k_0\neq0$ (see the proof of
Theorem~\ref{app:ch_waves:thm:leading}); this tensor lies in $K_0^\perp$,
and its preimage under the injective map $B_{F_0}(0)^*$ is a nonzero
multiple of $\nu_0$. Compact parameter supports give a uniform positive
lower bound for $|\nu|$.
\end{proof}

Only these normalized algebraic coefficients are used at $\sigma=0$;
$G_0$ itself is not an immersion of full normal rank. All derivative
losses are fixed before the observation order and the accuracy.

\subsection{Coercivity of the continuous normal operator}\label{app:ch_background:sec:bg-frame}

\begin{defn}[The continuous operator]\label{app:ch_background:defn:bg-row}
We first use the normalized family of coefficients, including its limit
at $\sigma=0$, and then a small fixed $C^{r_*}$ neighborhood of a chosen
positive $G_{\rm ref}=G_\sigma$. Fix domains
$U_{\rm core}\Subset U_{\rm meas}\Subset W$ and a cutoff
$0\leq\chi_{\rm meas}\leq1$ equal to one near $\overline{U_{\rm core}}$ and
supported in $U_{\rm meas}$; the measurement region is chosen small
below, and these cutoffs are fixed before $\sigma$ and the neighborhood of
$G_\sigma$. For every patch choose a smooth function
$\rho_a(z,\widehat\eta)\geq0$, positive on fixed neighborhoods of the
model incidence points and of the corresponding directions $\widehat\eta$,
and compactly supported in the input charts. Let $\chi_{\rm hi}(|\eta|)$
vanish for $|\eta|\leq1$ and equal one for $|\eta|\geq2$.

With a fixed Fourier normalization, define the complex operators
\begin{equation}\label{app:ch_background:eq:bg-T}
 T_af(x)=\int e^{\ii\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)\dd z\dd\eta ,
\end{equation}
acting on the complex space $L^2(z,\eta)$, with values in
$L^2(\R^n;\C^n)$, and put $C_a=2\Re T_a$. The operator
\[
 A=\Bigl(
 \chi_{\rm meas}P_G\langle D\rangle^{-1},\ (C_a)_a\Bigr)
\]
acts on the direct sum of $L^2(\R^n;\R^n)$ (the image component) and the
real Hilbert spaces underlying the inputs of the $T_a$, with values in
$L^2(\R^n;\R^n)$. Put
\[
 E=AA^*+1-\chi_{\rm meas}^2 .
\]
\end{defn}

The image component corresponds to tangential corrections, whose
linearized metric is $2\sym\nabla u$, and the operators $C_a$ correspond
to modulations of the waves: by Theorem~\ref{app:ch_waves:thm:leading},
changing the initial density of a wave by a real modulation changes its
transported mean by $|\nu_a|^2\xi_a\otimes\xi_a$ times the modulation
evaluated at the initial point $Y_{G,a}$, and $q_G$ of this tensor is the
amplitude in \eqref{app:ch_background:eq:bg-T}.

\begin{prop}[Coercivity]\label{app:ch_background:prop:bg-frame}
Fix a finite order $k_{\rm gap}$ containing the derivatives used in the
operator norm estimates of Proposition~\ref{app:ch_osc:prop:osc-normal}.
One sufficiently small measurement region and fixed cutoffs work for
every $0\leq\sigma\leq\sigma_0$ in the normalized family. All the operators
above are bounded, and
\begin{equation}\label{app:ch_background:eq:bg-global-bound}
 E\geq c_0I ;
\end{equation}
in particular $E$ has a bounded self-adjoint inverse on $L^2$. The same
conclusions hold on a small finite $C^{r_*}$ neighborhood of each fixed
positive $G_\sigma$, and after multiplying the initial amplitude by
$\sqrt{1+v_a(z,y)}$ with $\|v_a\|_{C^{k_{\rm gap}}}$ sufficiently small.
\end{prop}

\begin{proof}
\emph{Boundedness.} Proposition~\ref{app:ch_osc:prop:osc-normal}, at
$\tau=0$ with $p_\infty=\chi_{\rm hi}$, applies to $T_a$; with $J_a$ as in
\eqref{app:ch_background:eq:bg-jacobian}, the change of variables
$k=\Psi_a$ shows that the kernel of $T_aT_a^*$ is
$\int e^{\ii(x-x')\cdot k}b_{T,a}(x,x',k)\dd k$, where, at the inverse
point $(s,\eta)=u_a(x,x',v,k)$,
\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
 \frac{\rho_a^2\chi_{\rm hi}^2|\eta|}{J_a}
 |\nu_a(x,z)|^2|\nu_a(x',z)|^2\\
 &\qquad\cdot q_G(x,\xi_a(x,z))
              q_G(x',\xi_a(x',z))^T\dd v .
\end{split}
\end{equation}
This is an amplitude of order zero. Hence $\|T_a\|\leq C$, uniformly on
the coefficient neighborhood, and by the continuity statement of
Proposition~\ref{app:ch_osc:prop:osc-normal} the normal operator
$T_aT_a^*$ depends continuously in operator norm on finitely many
normalized derivatives of the data. By
Corollary~\ref{app:ch_osc:cor:osc-real}, $C_aC_a^*=4\Re(T_aT_a^*)$ on real
outputs, so the same holds for $C_aC_a^*$. The image component is a
product of coefficient multiplications, the Fourier multipliers
$D_j\langle D\rangle^{-1}$ and $\langle D\rangle^{-1}$, and the
measurement cutoff; it is bounded and depends continuously on the
coefficients, and so does its normal operator. Since
\[
 E=\chi_{\rm meas}P_G\langle D\rangle^{-2}P_G^*\chi_{\rm meas}
 +\sum_aC_aC_a^*+1-\chi_{\rm meas}^2 ,
\]
the operator $A$ is bounded uniformly on the neighborhood, and $E$
depends continuously on the coefficients in operator norm. We do not
claim, and do not use, continuity of $A$ itself in operator norm: the
phases $\Phi_{G,a}$ enter $T_a$ through $e^{\ii\Phi_{G,a}}$ with unbounded
$\eta$. All later perturbation arguments are applied to $E$.

\emph{The limit $\sigma=0$.} By Lemma~\ref{new:lem:scaling} the
coefficients are constant in $x$ at $\sigma=0$, and the phases are linear
in $x$. The reduced operator is then $P_0=M_0(D)$ (up to the factor
$\ii$), and the kernels of $T_aT_a^*$, apart from the measurement
cutoffs, depend only on $x-x'$. Hence
\[
 AA^*=\chi_{\rm meas}\,e_0(D)\,\chi_{\rm meas},
\]
with a matrix-valued multiplier $e_0(k)\geq0$: the image component
contributes $M_0(k)M_0(k)^T\langle k\rangle^{-2}$, and each wave
component contributes the (realified) amplitude
\eqref{app:ch_background:eq:bg-full-complex-amplitude}, which is a
positive semidefinite matrix; realification adds the nonnegative
contributions at $k$ and $-k$. For $|k|\geq2$ the multiplier is
homogeneous of degree zero up to the factor $|k|^2/\langle k\rangle^2$.

Suppose $s\in\R^n$ and $e_0(k)s\cdot s=0$ for some large $|k|$. Then
$M_0(k)^Ts=0$, and the wave contributions give
\[
 \int\frac{\rho_a^2|\eta|}{J_a}|\nu_a|^4\,
 \bigl|s\cdot q_0(\xi_a(z))\bigr|^2\dd v=0
\]
at the inverse points of $k$. At an incidence point of $\widehat k$ the
map $\eta\mapsto D_\zeta d_xS_0\cdot\eta$ is onto the tangent hyperplane of
the characteristic variety, which contains $\widehat k$; hence there is a
unique nonzero $\eta$ with $d_x\Phi_a=\widehat k$, and the weight
$\rho_a$ is positive there. Therefore $s\cdot q_0(\xi_a(z_a(\widehat k)))=0$
for every selected patch, and the Gram inequality
\eqref{app:ch_seed:eq:seed-frozen-detection} gives $s=0$. Since $e_0$ is
continuous and asymptotically homogeneous, compactness of the sphere
gives $e_0(k)\geq c$ for $|k|\geq R_0$, with fixed $R_0$; we used
$\chi_{\rm hi}=1$ there and $|k|^2/\langle k\rangle^2\geq1/2$.

For $u=\chi_{\rm meas}f$ we obtain, by Plancherel,
\[
 \|A^*f\|_2^2=\langle e_0(D)u,u\rangle
 \geq c\|u\|_2^2-C\int_{|k|\leq R_0}|\widehat u(k)|^2\dd k
 \geq c\|u\|_2^2-C\|u\|_1^2 ,
\]
using $\|\widehat u\|_\infty\leq C\|u\|_1$. By the Cauchy--Schwarz
inequality $\|u\|_1^2\leq|\supp\chi_{\rm meas}|\,\|u\|_2^2$, hence
\[
 \|A^*f\|_2^2\geq\bigl(c-C|\supp\chi_{\rm meas}|\bigr)\|\chi_{\rm meas}f\|_2^2
 \geq\tfrac c2\|\chi_{\rm meas}f\|_2^2
\]
if the measurement support is small enough. Since
$\|\chi_{\rm meas}f\|_2^2+\langle(1-\chi_{\rm meas}^2)f,f\rangle=\|f\|_2^2$,
\[
 \langle Ef,f\rangle=\|A^*f\|_2^2+\langle(1-\chi_{\rm meas}^2)f,f\rangle
 \geq\min\bigl(\tfrac c2,1\bigr)\|f\|_2^2 .
\]

\emph{Perturbation.} With these cutoffs fixed, the continuity of $E$ in
operator norm, proved above, gives $\sigma_0>0$ such that
\eqref{app:ch_background:eq:bg-global-bound} holds for
$0\leq\sigma\leq\sigma_0$ with $c_0$ halved: by
Lemma~\ref{new:lem:scaling} the normalized data of $G_\sigma$ converge in
every fixed $C^m$ norm as $\sigma\to0$. Then a finite-order neighborhood
of each fixed positive $G_\sigma$ gives the same bound with a common
constant. A bounded self-adjoint operator with $E\geq c_0$ is
invertible, with $\|E^{-1}\|\leq c_0^{-1}$.

\emph{Change of the initial amplitude.} The transported amplitude is linear
in the initial amplitude, and $|\nu_a|^2$ is quadratic in it. Multiplying the
initial amplitude by $\sqrt{1+v_a}$ therefore multiplies the amplitude of
the kernel of $T_aT_a^*$ by
$[1+v_a(z,Y_a(x,z))][1+v_a(z,Y_a(x',z))]$, which changes the normal
operator by at most $C\max_a\|v_a\|_{C^{k_{\rm gap}}}$ in operator norm.
Small $v$ therefore preserves coercivity with a smaller constant.
\end{proof}

\subsection{The inverse and its realization}\label{app:ch_background:sec:bg-inverse}
\label{app:ch_background:sec:bg-adjoint}
We realize every image input by a tangential field:
\begin{equation}\label{app:ch_background:eq:bg-image-realization}
 \mathcal J_G h=\tfrac12dG\,g_G^{-1}
       \langle D\rangle^{-1}h .
\end{equation}
By \eqref{main:eq:tangent-normal} with $u=\tfrac12\langle D\rangle^{-1}h$
and $w=0$,
\[
 D\gamma_G\mathcal J_Gh=\sym\nabla^{g_G}(\langle D\rangle^{-1}h),\qquad
 q_GD\gamma_G\mathcal J_Gh=P_G\langle D\rangle^{-1}h .
\]
The single derivative is compensated by $\langle D\rangle^{-1}$, so the
field and its linearized metric have order at most zero.

The following elementary inequality is used repeatedly: for $L\geq1$,
$r_0\geq0$ and integers $0\leq i\leq s$,
\begin{equation}\label{app:ch_background:eq:bg-grade-endpoints}
 L^{(s-i-r_0)_+}\|u\|_{H^i}
 \leq C_s\bigl[\|u\|_{H^s}+L^{(s-r_0)_+}\|u\|_2\bigr].
\end{equation}
If $s-i-r_0\leq0$ this is $\|u\|_{H^i}\leq\|u\|_{H^s}$. Otherwise split
$u$ in frequency at $|\xi|=L$: the high part satisfies
$\|u_{\rm hi}\|_{H^i}\leq CL^{i-s}\|u\|_{H^s}$ and
$L^{s-i-r_0}L^{i-s}\leq1$, while the low part satisfies
$\|u_{\rm lo}\|_{H^i}\leq CL^i\|u\|_2$ and
$L^{s-i-r_0}L^i=L^{s-r_0}$.

\begin{prop}[Inverse estimates and realization]
\label{app:ch_background:prop:bg-inverse}
\label{app:ch_background:prop:bg-adjoint}
\label{app:ch_background:prop:bg-normalized}
\label{app:ch_background:prop:bg-mixed-physical}
Work in the coefficient neighborhood of
Proposition~\ref{app:ch_background:prop:bg-frame}, including its positive
initial amplitudes $r_a$, and write
\[
 A=A_{G,r}=(A_{\rm im},T),\qquad
 \mathcal F=A_{\rm im}A_{\rm im}^*+1-\chi_{\rm meas}^2,\qquad E=\mathcal F+TT^*,
\]
where $A_{\rm im}$ is the image component and $T$ collects the real wave
components. Let $X^s$ be the direct sum of $H^s$ for the image input and
of $H^s_{z,y}$ for the wave inputs, the latter measured in the initial
variables after inverse Fourier transform in $\eta$. With the cutoff $P_K$
of \eqref{app:ch_waves:eq:single-raw-cutoff} define
\[
 E_K=\mathcal F+TP_K^2T^*,\qquad
 V_{\rm aux}(e)=\mathcal J_G A_{\rm im}^*E^{-1}e .
\]
Suppose that, for some $L\geq1$,
\[
 \|G\|_{C^m}+\sum_a\|r_a\|_{C^m}
       \leq C_mL^{(m-R_*)_+},\qquad m\geq0 .
\]
There is a fixed $B$ such that, whenever $R_*\geq r_0+B$ with $r_0\geq0$,
for every integer $s\geq0$,
\begin{equation}\label{app:ch_background:eq:bg-r-inverse}
\begin{split}
 &\|E^{-1}e\|_{H^s}+\|A^*E^{-1}e\|_{X^s}
  +\|\mathcal FE^{-1}e\|_{H^s}+\|V_{\rm aux}(e)\|_{H^s(W)}\\
 &\qquad\leq C_s\bigl[\|e\|_{H^s}+L^{(s-r_0)_+}\|e\|_2\bigr].
\end{split}
\end{equation}
The constants depend on the constants in the hypothesis but not on $L$
or $K$. The bound for $V_{\rm aux}$ includes its linearized metric and
holds on $\R^n$ after a fixed physical cutoff. Before that cutoff,
$q_GD\gamma_GV_{\rm aux}(e)=P_G\langle D\rangle^{-1}A_{\rm im}^*E^{-1}e$, and on
$U_{\rm core}$, where $\chi_{\rm meas}=1$, this equals
$A_{\rm im}A_{\rm im}^*E^{-1}e=\mathcal FE^{-1}e$. The inverse preserves
$H^\infty$, and $AA^*E^{-1}e=e$ on $U_{\rm core}$. For every integer
$N\geq0$,
\begin{equation}\label{app:ch_background:eq:bg-carrier-tail}
 \|(E-E_K)E^{-1}e\|_2
 \leq C_NK^{-N}\bigl[\|e\|_{H^N}+L^{(N-r_0)_+}\|e\|_2\bigr].
\end{equation}
For the normalized family $G_\sigma$ with initial amplitude $\sigma$, the lower
bound and all constants are uniform as $\sigma\to0$, and the right side of
\eqref{app:ch_background:eq:bg-r-inverse} becomes $C_s\|e\|_{H^s}$.
\end{prop}

\begin{proof}
\emph{Derivative losses.} The algebraic formulas, fixed-time flows and
inverse charts preserve the derivative bounds by
Lemma~\ref{app:ch_regular_base:lem:buffered-interpolation} and
\eqref{app:ch_regular_base:eq:joint-primitive-product}. We choose $B$ to
include their fixed losses and all fixed amplitude tests below. Then an
$s$th derivative of an operator applied to a source, with $i$ of the
derivatives falling on the source, carries the coefficient factor
$L^{(s-i-r_0)_+}$, and \eqref{app:ch_background:eq:bg-grade-endpoints}
converts all such terms into the right side of
\eqref{app:ch_background:eq:bg-r-inverse}.

\emph{The inverse.} Let $U_zf(x)=f(x+z)$ and $E_z=U_zEU_{-z}$. Conjugation
by translations translates the coefficients; Fourier multipliers commute
with translations. In the amplitude of the wave component, a derivative
in $z$ acts as $\partial_x+\partial_{x'}$, and the image component and
$1-\chi_{\rm meas}^2$ are finite products of smooth coefficients and fixed
multipliers. Taylor's formula and the amplitude bounds show that
$z\mapsto E_z$ is smooth in operator norm, with
\[
 \|\partial_z^\alpha E_z\|_{2\to2}
          \leq C_\alpha L^{(|\alpha|-r_0)_+}.
\]
Since $E_z\geq c_0$ for all $z$, the resolvent identity shows that
$z\mapsto E_z^{-1}$ is smooth in operator norm, and differentiating
$E_zE_z^{-1}=I$ gives
\[
 \partial_z^\alpha E_z^{-1}
 =\sum(-1)^m\frac{\alpha!}{\alpha_1!\cdots\alpha_m!}\,
 E_z^{-1}(\partial^{\alpha_1}E_z)E_z^{-1}\cdots(\partial^{\alpha_m}E_z)E_z^{-1},
\]
summed over ordered decompositions $\alpha=\alpha_1+\dots+\alpha_m$ with
$\alpha_i\neq0$. By \eqref{new:eq:plus-subadditive},
\[
 \|\partial_z^\alpha E_z^{-1}\|_{2\to2}
          \leq C_\alpha L^{(|\alpha|-r_0)_+}.
\]
For smooth $e$, the identity $U_z(E^{-1}e)=E_z^{-1}U_ze$ shows that
$z\mapsto U_z(E^{-1}e)$ is smooth with values in $L^2$, and its derivatives
at $z=0$ are the weak derivatives of $E^{-1}e$:
\[
 \partial^\alpha(E^{-1}e)=\sum_{\beta\leq\alpha}\binom\alpha\beta
 (\partial_z^\beta E_z^{-1})|_{z=0}\,\partial^{\alpha-\beta}e .
\]
Hence
$\|\partial^\alpha E^{-1}e\|_2\leq C\sum_\beta L^{(|\beta|-r_0)_+}\|e\|_{H^{|\alpha|-|\beta|}}$,
and \eqref{app:ch_background:eq:bg-grade-endpoints} gives
\eqref{app:ch_background:eq:bg-r-inverse} for $E^{-1}e$. By density the
bound extends to all $e\in H^s$; in particular $E^{-1}$ preserves
$H^\infty$.

\emph{The adjoint.} The image adjoint has order zero. For a phase
$\Phi=Y(x,z)\cdot\eta$ with $B=D_yY$ invertible (rows indexed by the
physical $y$ variables), the exact identities
\begin{equation}\label{app:ch_background:eq:bg-derivative-trade}
 d_y\Phi=B\eta,\qquad \eta=B^{-1}d_y\Phi,\qquad
 \partial_{z_\ell}\Phi=V_\ell\cdot d_y\Phi,\quad
 V_\ell=B^{-T}\partial_{z_\ell}Y,
\end{equation}
hold. Here $Y=d_\zeta S_G$, and the common charts make $B$ uniformly
close to $I$. Consequently
$\eta_je^{-\ii\Phi}=\ii(B^{-1}\partial_y)_je^{-\ii\Phi}$ and
$\partial_{z_\ell}e^{-\ii\Phi}=V_\ell\cdot\partial_ye^{-\ii\Phi}$, so every
factor $\eta$ and every parameter derivative of the phase factor in
$T_a^*v$ can be transposed to a $y$-derivative falling on $v$ and on the
amplitude. The compact measurement support removes boundary terms. After
$s$ such operations, a term with $i$ derivatives on $v$ has coefficient
factor $L^{(s-i-r_0)_+}$. Its amplitude still has the factor
$|\eta|^{1/2}$, and its normal operator has order zero because of the
degree-one Jacobian \eqref{app:ch_background:eq:bg-jacobian}. The
amplitude bounds, Plancherel and
\eqref{app:ch_background:eq:bg-grade-endpoints} give
\[
 \|A^*v\|_{X^s}
 \leq C_s\bigl[\|v\|_{H^s}+L^{(s-r_0)_+}\|v\|_2\bigr],
\]
and we insert $v=E^{-1}e$. The transported amplitude equals the initial
scalar $r(z,Y_G)$ times a flow factor, so these estimates include
variable initial amplitudes.

\emph{The remaining bounds.} The image component $A_{\rm im}$ has order zero, and
\eqref{app:ch_background:eq:bg-image-realization} gives the bounds for
$V_{\rm aux}$ and its linearized metric; a fixed output cutoff adds only
a lower-order term. The identity $AA^*E^{-1}=I-(1-\chi_{\rm meas}^2)E^{-1}$
gives $AA^*E^{-1}e=e$ on $U_{\rm core}$. Every wave adjoint contains the
factor $\chi_{\rm hi}$, and $1-P_K^2$ vanishes for $|\eta|\leq K$, so
\[
 \|(I-P_K^2)T^*E^{-1}e\|_2\leq K^{-N}\||\eta|^NT^*E^{-1}e\|_2
 \leq C_NK^{-N}\bigl[\|e\|_{H^N}+L^{(N-r_0)_+}\|e\|_2\bigr],
\]
by the adjoint estimate. Composing with the bounded operator $T$ gives
\eqref{app:ch_background:eq:bg-carrier-tail}.

\emph{The normalized family.} For $G_\sigma$, Lemma~\ref{new:lem:scaling}
bounds every fixed derivative of the normalized amplitudes and of the
coefficients in \eqref{app:ch_background:eq:bg-derivative-trade}
uniformly in $\sigma$, and Proposition~\ref{app:ch_background:prop:bg-frame}
gives a common lower bound. The proof above therefore has uniform
constants, and since all derivatives are bounded uniformly, the factor
$L^{(s-r_0)_+}$ can be replaced by a constant.
\end{proof}

\subsection{The positive transported mean}
\label{app:ch_background:sec:bg-positive}
\label{app:ch_background:sec:bg-current}
The transported mean is affine in the initial density. One correction of
the density and one tangential field cancel its component in $K_G$.

\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
Definition~\ref{app:ch_background:defn:bg-carriers}. Choose smooth
weights $\psi_a\geq0$, compactly supported in the parameter charts and
equal to one on a neighborhood of the projection
$\pi_z(\supp\rho_a)$. For $\sigma>0$ and an immersion $G$ near $G_\sigma$,
let $A=A_{G,\sigma}$ and $E=AA^*+1-\chi_{\rm meas}^2$ be the operators with
constant initial amplitude $\sigma$. For the leading normal field
$\nu_{G,\sigma,a}$ with initial amplitude $\sigma$, define the transported mean
\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
 \bigl[1+v_a(z,Y_{G,a})\bigr]\xi_{G,a}\otimes\xi_{G,a}\dd z .
\end{equation}
Write $H_\sigma=H_{G_\sigma}[\sigma^2]$, and let $H_0$ be its constant
limit. Fix $X_0(x)=-H_0x/2$ and $U_{\rm fin}\Subset U_{\rm core}$, and
define
\begin{align}
 R_0&=D\gamma_G(dG\,X_0)+H_G[\sigma^2],\notag\\
 e&=-\chi_{\rm meas}q_G(R_0),\qquad h=A^*E^{-1}e,\notag\\
 v_a&=2\Re\mathcal F_\eta^{-1}
 \bigl[\rho_a\chi_{\rm hi}(|\eta|)|\eta|^{1/2}h_a\bigr],
 \label{app:ch_background:eq:bg-density-input}
\end{align}
where $h=(h_{\rm im},(h_a)_a)$, the source is extended smoothly by zero
to $\R^n$, and the inverse Fourier transform in $\eta$ is taken without a
cutoff in the initial variables. Put
\[
 r_{G,a}=\sigma\sqrt{1+v_a},\qquad
 V_{\rm base}=dG\,X_0+\mathcal J_Gh_{\rm im}.
\]
Then, for all sufficiently small $\sigma>0$, and on one sufficiently small
fixed $C^{r_*}$ neighborhood of $G_\sigma$, these fields are real and
smooth, $1+v_a\geq1/2$, the scalar $r_{G,a}$ is positive on the supports
of the cutoffs $\rho_a$, 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 normal operator for this positive scalar is coercive, and its
inverse satisfies the estimates of
Proposition~\ref{app:ch_background:prop:bg-inverse}. For one fixed $B$,
if $\|G\|_{C^m(W)}\leq C_mL^{(m-R_*)_+}$ with $L\geq1$, $r_0\geq0$ and
$R_*\geq r_0+B$, then
\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_sL^{(s-r_0)_+},
\end{equation}
where the norm of the scalar is taken in all variables $(z,y)$ on the
fixed patches. These choices precede the accuracy and the final
dilation, and fixed-order bounds are uniform on smooth compact families.
Along $G=G_\sigma$,
\begin{equation}\label{app:ch_background:eq:bg-positive-profile}
 \sum_a\|v_a\|_{C^s}
 +\|V_{\rm base}-dG\,X_0\|_{C^s(W)}\leq C_s\sigma,
\end{equation}
the identity holds on $U_{\rm core}$, and the coercivity and the
fixed-order inverse bounds for the corrected scalar are uniform as
$\sigma\to0$.
\end{prop}

\begin{proof}
\emph{The identity.} The density $v_a$ has parameter support in
$\{\psi_a=1\}$, so $\psi_av_a=v_a$, and the transport evaluates it at the
initial point $Y_{G,a}$. Writing out the inverse Fourier transform,
\[
 v_a(z,Y_{G,a}(x,z))=2\Re\int e^{\ii\Phi_{G,a}(x,z,\eta)}
 \rho_a\chi_{\rm hi}|\eta|^{1/2}h_a(z,\eta)\dd\eta .
\]
Multiplying by $\chi_{\rm meas}\psi_a|\nu_{G,\sigma,a}|^2q_G(\xi_{G,a}\otimes\xi_{G,a})$,
integrating in $z$ and using $\psi_a\rho_a=\rho_a$, we get, on the whole
measurement support,
\begin{equation}\label{app:ch_background:eq:bg-density-row}
 \chi_{\rm meas}q_G\big(H_G[\sigma^2(1+v)]-H_G[\sigma^2]\big)
                  =\sum_aC_ah_a .
\end{equation}
The same identity holds for any fixed positive initial amplitude and the
corresponding operators. On $U_{\rm core}$ we have $\chi_{\rm meas}=1$,
so, by \eqref{app:ch_background:eq:bg-image-realization},
\[
 q_G\bigl(D\gamma_GV_{\rm base}+H_G[r_G^2]\bigr)
 =q_GR_0+P_G\langle D\rangle^{-1}h_{\rm im}+\sum_aC_ah_a
 =q_GR_0+Ah .
\]
Since $Ah=AA^*E^{-1}e=e=-q_GR_0$ on $U_{\rm core}$, the right side
vanishes.

\emph{Size at the reference.} By Lemma~\ref{new:lem:scaling},
$H_\sigma(x)=\widehat H(\sigma x)$ with $\widehat H$ smooth, so $H_0=\widehat H(0)$
is constant and $H_\sigma-H_0=O_s(\sigma)$ in every $C^s$. At $G_\sigma$ the
metric is $I$ and the connection is flat, so by
\eqref{main:eq:tangent-normal}
\[
 D\gamma_G(dG\,X_0)=2\sym dX_0=-H_0 ,
\]
and $R_0=H_\sigma-H_0=O_s(\sigma)$. The density formula is a multiplier of
order one half in $y$ with fixed smooth masks in $z$, so, before any
spatial cutoff,
\[
 \|v\|_{H^s_{z,y}}\leq C_s\|h\|_{X^{s+1}}.
\]
The uniform inverse and adjoint bounds for the normalized family,
together with Sobolev embedding, give $h=O_s(\sigma)$, $v=O_s(\sigma)$ and
$V_{\rm base}-dG\,X_0=O_s(\sigma)$. At the single output order $k_{\rm gap}$
these estimates give $\|v\|_{C^{k_{\rm gap}}}\leq C\|e\|_{H^m}$ for one fixed
$m$, with a constant depending only on the lower bound and finitely many
normalized seminorms; these input orders are included in $r_*$. We choose
$\sigma$ so small that this bound is below the threshold of
Proposition~\ref{app:ch_background:prop:bg-frame} with a fixed margin;
then the corrected operator is coercive, $1+v_a\geq1/2$, and $v=O_s(\sigma)$
gives uniform bounds for the corrected coefficients at every fixed order.

\emph{Nearby immersions.} Now fix this positive $\sigma$. On a small
$C^{r_*}$ neighborhood of $G_\sigma$, the finite-jet and flow formulas
preserve these bounds and keep $\|e\|_{H^m}$ small, so
$\|v\|_{C^{k_{\rm gap}}}$ stays small and the supports are unchanged.

\emph{Derivative bounds.} Let $c$ include the losses of the density
multiplier and of Sobolev embedding. After increasing $B$ once, the
finite-jet and flow formulas give $\|e\|_{H^s}\leq C_sL^{(s-r_0-c)_+}$.
Proposition~\ref{app:ch_background:prop:bg-inverse}, applied with
$r_0+c$ in place of $r_0$ to the operator with constant initial amplitude
$\sigma$, the density estimate and Sobolev embedding give
$\|v\|_{C^s}+\|\mathcal J_Gh_{\rm im}\|_{C^s}\leq C_sL^{(s-r_0)_+}$, also
for the linearized metric of the field. The positive square root and
$dG\,X_0$ satisfy the same bounds by
Lemma~\ref{app:lem:elementary-interpolation-jets}. This proves
\eqref{app:ch_background:eq:bg-current-profile}. Continuity of the normal
operator gives the lower bound for the corrected scalar, and
Proposition~\ref{app:ch_background:prop:bg-inverse} applies to it.
\end{proof}

\begin{defn}[The operators with the corrected scalar]\label{app:ch_background:defn:bg-mixed}
We use the preceding operators with the positive scalar $r_G$ of
Proposition~\ref{app:ch_background:prop:bg-current}, and write
\[
 \begin{gathered}
 A_G=(A_{{\rm im},G},T_G),\quad
 \mathcal F_G=A_{{\rm im},G}A_{{\rm im},G}^*+1-\chi_{\rm meas}^2,\quad E_G=\mathcal F_G+T_GT_G^*,\\
 E_{G,K}=\mathcal F_G+T_GP_K^2T_G^*,
 \end{gathered}
\]
and $V_{\rm aux}^G$ for the field $V_{\rm aux}$ of these operators. The
inverse estimates and the tail estimate of
Proposition~\ref{app:ch_background:prop:bg-inverse} apply to them.
\end{defn}

\subsection{Comparison with the sampled waves}
\label{app:ch_background:sec:bg-prior-wave-inputs}
\label{app:ch_background:sec:bg-matching}
The fixed weights identify the continuous operator with the leading part
of the normal operator of the waves.

\begin{defn}[The associated family of waves]
\label{app:ch_background:setup:bg-wkb-input}
Use the positive scalar $r_{G,a}$ of
Proposition~\ref{app:ch_background:prop:bg-current}, the measure
$\psi_a\dd z$ and the cutoff $\rho_a$. In
Theorem~\ref{app:ch_waves:thm:sampling} take the phase $S_G$, the frame
$Q_{G,i}$, the measurement cutoff $\chi_{\rm meas}$ and the cutoff
$\rho_a$. The wave family has leading tensor $A_0(b_G)$ and satisfies
the hypotheses of the sampling theorem by
Corollary~\ref{app:ch_response:cor:current-atoms}. We use the fixed radial
factors, $\tau_a=(R_a\lambda)^{-1}$, and
\begin{equation}\label{app:ch_background:eq:bg-WKB-conditions}
 \Lambda/\lambda<\text{the fixed margin of the phase charts}.
\end{equation}
Every shifted covector stays in its common chart. We write
$H_G=H_G[r_G^2]$ for the positive transported mean.
\end{defn}

At $\tau=0$, the amplitude in \eqref{app:ch_waves:eq:actual-T} is
$\chi_{\rm meas}\rho_a|\nu_{G,a}|^2q_G(x,\xi_{G,a})$, because
$A_0^*A_0=|\nu|^2\xi\xi^T$. Since $\psi_a\rho_a^2=\rho_a^2$, summing the
weighted real normal operators gives exactly $T_GP_\Lambda^2T_G^*$, the
factor $4\Re$ coming from real adjunction
(Corollary~\ref{app:ch_osc:cor:osc-real}). The image part $\mathcal F_G$ is
added unchanged.

%% ===== 13_mean.tex =====
\section{Correction of the mean}\label{main:sec:waves}\label{app:sec:ch_mean_pairs}

In this section we construct the part of the metric correction which
is not oscillatory. The primary field of the metric correction is a
weighted sum of real parts of complex characteristic waves $Z_\nu$.
If $dZ=e^{\ii\vartheta}A$ with $\vartheta$ real, then
\[
 \frac1{2\pi}\int_0^{2\pi}
 \bigl(d\sqrt2\Re(e^{\ii\theta}Z)\bigr)^T
 d\sqrt2\Re(e^{\ii\theta}Z)\,d\theta=\Re(A^*A),
\]
since the remaining term $\Re(e^{2\ii(\vartheta+\theta)}A^TA)$ has
mean zero in $\theta$. We call the Hermitian tensor $\Re(A^*A)$ the
\emph{mean} of the wave; it is the average over the phase, not an
average over the spatial domain. The quadratic tensor of the primary
field is the weighted sum of these means plus oscillatory products;
the latter are treated in Section~\ref{app:sec:ch_pairs}. Here we
choose the initial data of the waves and a real ambient field $V$ so
that the linearized metric $D\gamma_FV$ plus the weighted sum of the
means equals the rescaled smoothed defect up to an error $O(R^{-Q})$.
The result is Proposition~\ref{main:prop:mean}, which is stated and
proved in Section~\ref{new:sec:mean-proof}.

\subsection{Scales and the sample}\label{new:sec:mean-scales}
Let $R$ be large and put
\begin{equation}\label{main:eq:basic-scales}
 \lambda=R^a,\qquad h=R^{-\beta},\qquad K_{\max}=R^{s_f},
 \qquad\beta=(1+\epsilon)/3,
\end{equation}
where $0<\epsilon<1/8$, $1<s_f<1+\epsilon$ and $1+\beta<a<3/2$.
Here $\lambda$ is the frequency of the waves, $h$ is the separation of
the sample, and $K_{\max}$ is a fixed upper bound for the Fourier
support of the initial data. The condition $s_f>1$ places $K_{\max}$
above the derivative scale $R$ of the sources constructed below,
while $s_f<1+\epsilon$ places it below the number $h^{-3}=R^{1+\epsilon}$
of samples, which is needed for the concentration of the sampled
normal operator. Section~\ref{main:sec:scales} verifies all required
inequalities between these exponents at once.

\medskip\noindent\emph{The positive initial amplitude and the sample.}
We use the positive initial amplitude $r_F$ and the tangent field
$V_{\rm base}$ of Proposition~\ref{app:ch_background:prop:bg-current},
for which the leading transported mean of the waves, together with
$D\gamma_FV_{\rm base}$, has zero projection onto $K_F$; this is what
permits target defects of either sign. We sample the unmodulated waves,
that is, the waves with initial data the Dirac mass $\delta_0$ and this
initial amplitude, by Construction~\ref{new:constr:sampling} on the
three-parameter characteristic patches. This produces $M$ samples with
weights $\mu_\nu$ such that
\begin{equation}\label{main:eq:sampling}
 M\leq CR^{1+\epsilon},\qquad
 \mu_\nu>0,\qquad\sum_\nu\mu_\nu\leq C,
\end{equation}
separated by at least $h/4$ within each patch. One realization in the
event of Theorem~\ref{app:ch_waves:thm:sampling} approximates the
continuous normal operator with the input cutoff at $K_{\max}$ on the
whole output space, and the continuous mean with error $O(h)$ in every
Sobolev norm.

\subsection{The fixed operators and scales}
\label{app:ch_mean_pairs:sec:mean-appendix-inputs}
\label{app:ch_mean_pairs:sec:mean-current}

In the rest of this section the symbol $Q$ denotes the requested
accuracy, $Q_i(F)$ the projected frame, and $\mathcal Q$ the quadratic
operator defined below. Three normal operators occur: the continuous
operator $E_F$ of Definition~\ref{app:ch_background:defn:bg-mixed} at
$G=F$, without input cutoff, which is inverted; the continuous operator
$E_{F,K}$ with the input cutoff $P_K$; and the sampled operator
$\widetilde E_K$ defined in \eqref{new:eq:sampled-normal} below, which
carries the same input cutoff.

\begin{setup}[Operators and scales]
\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}
We use Definitions~\ref{app:ch_background:defn:bg-geometry}
and~\ref{app:ch_background:defn:bg-mixed} at the immersion $F$, on the
domains
\[
 D\Subset U_{\rm fin}\Subset U_{\rm core}
 \Subset U_{\rm meas}\Subset W
\]
of Setup~\ref{app:setup:domains}, with the same auxiliary spaces.
Write $\mathcal C=\chi_{\rm meas}q_F$. For the positive initial amplitude
$r_F$ of Proposition~\ref{app:ch_background:prop:bg-current} (with
$G=F$), set
\[
 \begin{gathered}
 A_F=(A_{{\rm im},F},T_F),\qquad
 \mathcal F_F=A_{{\rm im},F}A_{{\rm im},F}^*+1-\chi_{\rm meas}^2,\qquad
 E_F=\mathcal F_F+T_FT_F^*,\\
 E_{F,K}=\mathcal F_F+T_FP_K^2T_F^*,
 \end{gathered}
\]
where $P_K$ is the initial-frequency cutoff
\eqref{app:ch_waves:eq:single-raw-cutoff}. Let $H_F=H_F[r_F^2]$ be the
positive transported mean and $V_{\rm base}$ the base field of
Proposition~\ref{app:ch_background:prop:bg-current}, and let
$V_{\rm aux}^F$ be the auxiliary field of
Proposition~\ref{app:ch_background:prop:bg-mixed-physical}, together
with the bounds for its linearized metric $D\gamma_FV_{\rm aux}^F$ and
its fixed physical cutoff.

Let $m_0=\lfloor n/2\rfloor+2$, $s_0=2m_0$ and $A=m_0+\frac12$, as in
\eqref{main:eq:A-choice}, and assume
\begin{equation}\label{app:ch_mean_pairs:eq:mean-stage-window}
\begin{gathered}
 0<\epsilon<\tfrac18,\quad\beta=(1+\epsilon)/3,\quad
 1<s_f<1+\epsilon,\quad 1+\beta<a<\tfrac32,\\
 \lambda=R^a,\quad h=R^{-\beta},\qquad T=R^t,\quad2/3<t<1.
\end{gathered}
\end{equation}
Before the sample is drawn we fix the input frequency bound
\[
 K=K_{\max}=R^{s_f};
\]
all wave data below have Fourier support in this range. The immersion
satisfies $\|F\|_{C^r}\leq C_rT^{(r-r_*)_+}$, and its positive initial
scalar, its phases and its corrected waves satisfy derivative bounds
of the same form, with fixed shifts of the index $r_*$. We
choose $r_*$ so large that
\eqref{app:ch_background:eq:bg-current-profile} gives $r_F$ the
derivative bounds required below. The data in the patch variables are
fixed before the spatial source cutoffs and the cutoffs in the initial
variables are applied, and $1-\chi_{\rm meas}^2=0$ on a neighborhood of
$\overline{U_{\rm fin}}$, in particular on $D$.

The order of choices is the following. The patch labels and the
coefficient neighborhood are fixed first. The low derivative orders
and the indices in the derivative bounds depend only on $n$ and on the
fixed margins between exponents. The integer $r_*$ is chosen above all
required orders before the sample, the accuracy $Q$ and the target
dilation are chosen.
\end{setup}

\subsection{The sampled operators and their corrections}

The $\nu$-th sample has patch label $a(\nu)$, patch parameter $z_\nu$,
base covector $\zeta_\nu=\zeta_{a(\nu)}(z_\nu)$, weight $\mu_\nu$ and
frequency $\lambda_\nu=\tau_\nu^{-1}$, $\tau_\nu=(R_{a(\nu)}\lambda)^{-1}$.
We write $\mathcal L_\nu$ for the complex-linear map $m\mapsto Z_\nu(m)$
of \eqref{main:eq:lag-recurrence}.

\begin{defn}[The sampled corrections]
\label{app:ch_mean_pairs:defn:mean-wave-input}
\label{app:ch_mean_pairs:defn:mean-lag-input}
\label{app:ch_mean_pairs:defn:mean-canonical}
The number of terms of the finite correction in
Corollary~\ref{app:ch_response:cor:current-atoms} is fixed before
sampling, and the positive initial amplitude $r_F$ is used in the
resulting maps $\mathcal L_\nu$ of Section~\ref{app:sec:ch_lag}. In
this section constants may depend on this number and on the finite
number of iteration steps; the input frequency bound, $r_*$,
$R_{\rm env}$ and the exponents do not. Set
\[
 \vartheta_\nu=S_F(x,\zeta_\nu)/\tau_\nu=\lambda_\nu\phi_\nu,\qquad
 B_\nu=e^{-\ii\vartheta_\nu}d\mathcal L_\nu(\delta_0),
\]
so that $B_\nu$ is the demodulated derivative of the unmodulated wave.
A function $f$ of the initial variables is identified with the
measure $dm=\widehat f\,d\eta$. On the real Hilbert space underlying
$L^2(d\eta)$ define $\mathcal B_K$ by
$\widehat{\mathcal B_Kv}=|\eta|^{1/2}P_Kv$. These compactly supported
Fourier densities are integrable by the Cauchy--Schwarz inequality, so
Proposition~\ref{app:ch_lag:prop:lag-envelopes} defines the
corresponding waves for every $L^2$ input. Write
$\rho_\nu=\rho_{a(\nu)}(z_\nu,\widehat\eta)$ and, with the high-pass
multiplier $\chi_{\rm hi}$ of
Definition~\ref{app:ch_background:defn:bg-row}, put
\[
 B_{\nu,K}^{\rm in}=\rho_\nu\chi_{\rm hi}\mathcal B_K,\qquad
 G_{\nu,K}v=e^{-\ii\vartheta_\nu}d\mathcal L_\nu(B_{\nu,K}^{\rm in}v),
 \qquad
 A_{\nu,K}v=\mathcal C[2\sym\Re(B_\nu^*G_{\nu,K}v)].
\]
Thus $A_{\nu,K}$ is the linearization at $\delta_0$ of the reduced
mean of the $\nu$-th wave. For any family $V=(V_\nu)_\nu$ of derivative
amplitudes, let $\widetilde A_K[V]$ be the operator from the direct sum of the
input spaces to the reduced output space given by
\[
 \widetilde A_K[V](v_\nu)_\nu=\sum_\nu\sqrt{\mu_\nu}\,
      \mathcal C[2\sym\Re(V_\nu^*G_{\nu,K}v_\nu)],
\]
and put
\begin{equation}\label{new:eq:sampled-normal}
 \widetilde A_K=\widetilde A_K[(B_\nu)_\nu],\qquad
 \widetilde E_K=\mathcal F_F+\widetilde A_K\widetilde A_K^*
 =\mathcal F_F+\sum_\nu\mu_\nu A_{\nu,K}A_{\nu,K}^*.
\end{equation}
The operator $\widetilde A_K[V]$ is an aggregated operator: it acts on the
whole family of inputs at once, and its norm is controlled by the
sampling theorem, whereas each individual operator $G_{\nu,K}$ has norm
of order $K^{1/2}$ (Proposition~\ref{app:ch_osc:prop:osc-normal}). This
distinction is used in the proof of
Proposition~\ref{app:ch_mean_pairs:prop:canonical-estimates}.
For a real source $r$, define the initial data and the demodulated
derivatives
\begin{equation}\label{app:ch_mean_pairs:eq:mean-current-canonical}
 f_\nu[r]=B_{\nu,K}^{\rm in}A_{\nu,K}^*E_F^{-1}r,\qquad
 D_\nu r=e^{-\ii\vartheta_\nu}d\mathcal L_\nu(f_\nu[r])
 =G_{\nu,K}A_{\nu,K}^*E_F^{-1}r.
\end{equation}
If the initial data of the $\nu$-th wave are changed from $\delta_0$ to
$\delta_0+\widehat f_\nu[r]\,d\eta$, the weighted mean changes by
$\sum_\nu\mu_\nu$ times the exact tensor increment
\begin{equation}\label{app:ch_mean_pairs:eq:mean-exact-self-change}
 2\sym\Re(B_\nu^*D_\nu r)+\Re((D_\nu r)^*D_\nu r).
\end{equation}
Together with the auxiliary field $V_{\rm aux}^F(r)$, the linear part
of this change equals $\widetilde E_KE_F^{-1}r$ on $U_{\rm core}$ after application of
$q_F$; see Lemma~\ref{new:lem:mean-source-identity}. Both real
quadratic forms are taken with respect to the stated Hilbert space
inner products. No inverse of a sample weight occurs. The spatial
cutoffs in the initial variables are applied after these data are
formed, and observation cutoffs are used in norms only. We use the
domains of Setup~\ref{app:setup:domains}; the set $D_{\rm lag}$
contains the measurement supports, the product source supports and
the physical output supports. The maps $\mathcal L_\nu$, the
unmodulated waves, the adjoints and the operator $\widetilde A_K$ are the same
throughout the iteration below.
\end{defn}

\subsection{The equation for the source}\label{new:sec:mean-source-equation}

We now derive the equation that the source $r$ must satisfy. Put
\begin{equation}\label{new:eq:mean-operators}
 \mathcal E=\widetilde E_KE_F^{-1},\qquad
 \mathcal Q(u,v)=\mathcal C\sum_\nu\mu_\nu
                  \sym\Re((D_\nu u)^*D_\nu v),
\end{equation}
and
\begin{equation}\label{new:eq:mean-source}
 r_0=\mathcal C\Bigl[H_F-\sum_\nu\mu_\nu\Re(B_\nu^*B_\nu)\Bigr],\qquad
 g=r_0+\mathcal C(e_R/b^2),
\end{equation}
where $e_R=S_{R,m}(e|_D)$ and $b=\delta R^A$ are the smoothed defect
and the amplitude of Proposition~\ref{main:prop:mean}. The operator $\mathcal Q$ is real
bilinear and symmetric, because $D_\nu$ is real linear and
$\Re(X^*Y)^T=\Re(Y^*X)$ for complex matrices, so that
$\sym\Re(X^*Y)=\frac12[\Re(X^*Y)+\Re(Y^*X)]$ is symmetric in $X,Y$.
In particular
\begin{equation}\label{new:eq:Q-difference}
 \mathcal Q(u,u)-\mathcal Q(v,v)=\mathcal Q(u-v,u+v).
\end{equation}

\begin{lem}[The reduced mean]\label{new:lem:mean-source-identity}
Let $r\in H^\infty(\R^n;\R^n)$ be real, let the $\nu$-th wave have the
initial data $m_\nu=\delta_0+\widehat f_\nu[r]\,d\eta$, and put
\[
 \mathcal H=\sum_\nu\mu_\nu\Re\bigl((dZ_\nu)^*dZ_\nu\bigr),\qquad
 Z_\nu=\mathcal L_\nu(m_\nu),\qquad
 V_0=V_{\rm base}+V_{\rm aux}^F(r).
\]
Then, on $U_{\rm fin}$,
\begin{equation}\label{new:eq:reduced-mean-identity}
 q_F\bigl(D\gamma_FV_0+\mathcal H\bigr)
 =\mathcal Er+\mathcal Q(r,r)-r_0.
\end{equation}
\end{lem}
\begin{proof}
By linearity of $\mathcal L_\nu$ and
\eqref{app:ch_mean_pairs:eq:mean-current-canonical},
$e^{-\ii\vartheta_\nu}dZ_\nu=B_\nu+D_\nu r$. Since $\vartheta_\nu$ is
real, the phase factor cancels in $(dZ_\nu)^*dZ_\nu$, and
\[
 \Re\bigl((dZ_\nu)^*dZ_\nu\bigr)
 =\Re(B_\nu^*B_\nu)+2\sym\Re(B_\nu^*D_\nu r)
  +\Re((D_\nu r)^*D_\nu r),
\]
which is \eqref{app:ch_mean_pairs:eq:mean-exact-self-change}. On
$U_{\rm fin}$ we have $\chi_{\rm meas}=1$, hence $q_F=\mathcal C$
there. By the definitions of $A_{\nu,K}$ and $D_\nu$,
\[
 \mathcal C\sum_\nu\mu_\nu2\sym\Re(B_\nu^*D_\nu r)
 =\sum_\nu\mu_\nu A_{\nu,K}A_{\nu,K}^*E_F^{-1}r=\widetilde A_K\widetilde A_K^*E_F^{-1}r,
\]
and $\mathcal C\sum_\nu\mu_\nu\Re((D_\nu r)^*D_\nu r)=\mathcal Q(r,r)$.
By Proposition~\ref{app:ch_background:prop:bg-inverse}, applied to the
operators of Definition~\ref{app:ch_background:defn:bg-mixed},
$q_FD\gamma_FV_{\rm aux}^F(r)=\mathcal F_FE_F^{-1}r$ on $U_{\rm core}$ (here, and in
the lemma, $V_{\rm aux}^F(r)$ is the field before its fixed physical
cutoff; this cutoff equals one near $D$, so nothing changes on $D$). By
\eqref{app:ch_background:eq:bg-current-identity},
$q_FD\gamma_FV_{\rm base}=-q_FH_F$ on $U_{\rm fin}$. Adding these
identities gives, on $U_{\rm fin}$,
\[
 q_F(D\gamma_FV_0+\mathcal H)
 =-q_FH_F+q_F\sum_\nu\mu_\nu\Re(B_\nu^*B_\nu)
  +(\mathcal F_F+\widetilde A_K\widetilde A_K^*)E_F^{-1}r+\mathcal Q(r,r),
\]
which is \eqref{new:eq:reduced-mean-identity}.
\end{proof}

By \eqref{new:eq:reduced-mean-identity}, the projection of the mean
error onto $K_F$ vanishes exactly when the source solves the quadratic
equation
\begin{equation}\label{new:eq:fixed-point}
 g=\mathcal Er+\mathcal Q(r,r).
\end{equation}
The component of the error orthogonal to $K_F$ is then removed
algebraically by a normal field; see Section~\ref{new:sec:mean-normal}.
We do not solve \eqref{new:eq:fixed-point} exactly; a finite number of
steps of a Picard iteration suffices. The reasons are the following.
\begin{enumerate}
\item The operator $\mathcal E$ is close to the identity on the sources
that occur, but not on all of $L^2$. Indeed $E_FE_F^{-1}=I$, so
\[
 I-\mathcal E=(E_F-E_{F,K})E_F^{-1}+(E_{F,K}-\widetilde E_K)E_F^{-1}.
\]
The second term has norm $O(R^{-\omega})$ on $L^2$ for a fixed
$\omega>0$, by the concentration of the sampled normal operator and the
comparison of phases in Theorem~\ref{app:ch_waves:thm:sampling}; both
operators carry the same input cutoff $P_K$. The first term is the part
of the continuous operator above the input frequency $K$. It is not
small in operator norm, but by \eqref{app:ch_background:eq:bg-carrier-tail}
it is $O(h(R/K)^q)$ in $L^2$, for every $q$, on sources $u$ with
$\|u\|_{H^q}\leq C_qhR^{(q-s_0)_+}$.
\item The operator $\mathcal Q$ is quadratic and small: it satisfies
$\|\mathcal Q(u,v)\|_2\leq C\|u\|_2\|v\|_{H^{s_0/2}}$, and the source
$g$ has size $O(h)$.
\item Consequently the map $r\mapsto g+(I-\mathcal E)r-\mathcal Q(r,r)$,
whose fixed points are the solutions of \eqref{new:eq:fixed-point},
contracts differences of iterates in $L^2$ by a factor $CR^{-\omega}$, up
to the frequency tail, as long as the iterates stay in the class of (i).
Each step gains $R^{-\omega}$, and $J$ steps give an $L^2$ error
$O(h[R^{-\omega J}+(R/K)^q])$.
\item High Sobolev norms are not contracted, but they remain bounded:
every iterate satisfies $\|r^{(j)}\|_{H^s}\leq C_{j,s}hR^{(s-s_0)_+}$
for all $s$, with constants depending on $j$. Since only finitely many
steps are taken, this suffices. An interpolation inequality between
$L^2$ and $H^{s_0}$, $s_0>n$, then turns the $L^2$ smallness into
uniform smallness.
\end{enumerate}
The bounds in (i) and (ii), and their higher-order versions, are
proved in Proposition~\ref{app:ch_mean_pairs:prop:canonical-estimates};
the iteration is carried out in
Proposition~\ref{app:ch_mean_pairs:prop:mean-iteration}.

\subsection{Estimates for the sampled corrections}

We first prove two elementary inequalities.

\begin{lem}\label{new:lem:exponent-inequality}
Let $s_0\geq0$, and let $s,i,j\geq0$ be real numbers with $i+j\leq s$
and $j\leq s/2$. Then
\[
 (s-i-j-s_0)_++(i-s_0)_++(j-s_0/2)_+\leq(s-s_0)_+.
\]
\end{lem}
\begin{proof}
Put $x=s-j-s_0$. We first show
$(x-i)_++(i-s_0)_+\leq x_+$. If $i\leq s_0$, the left side is
$(x-i)_+\leq x_+$, since $i\geq0$. If $i>s_0$ and $x\geq i$, the left
side is $x-i+i-s_0=x-s_0\leq x_+$. If $i>s_0$ and $x<i$, the left side
is $i-s_0$, and $i-s_0\leq s-j-s_0=x$ because $i+j\leq s$.

It remains to show $(s-j-s_0)_++(j-s_0/2)_+\leq(s-s_0)_+$. If
$j\leq s_0/2$, this follows from $(s-j-s_0)_+\leq(s-s_0)_+$. If
$j>s_0/2$, then $s\geq2j>s_0$. When $s-j-s_0\geq0$ the left side is
$s-\frac32s_0\leq s-s_0$; when $s-j-s_0<0$ it is
$j-\frac{s_0}2\leq\frac{s-s_0}2\leq s-s_0$.
\end{proof}

\begin{lem}\label{new:lem:interpolation}
Let $s_0>n$. There is a constant $C$ such that, for all
$f\in H^{s_0}(\R^n)$,
\[
 \|f\|_{C^0}\leq C\|f\|_2^{1/2}\|f\|_{H^{s_0}}^{1/2}.
\]
\end{lem}
\begin{proof}
We use $\|f\|_{H^{s_0}}=\|\langle D\rangle^{s_0}f\|_2$; integer Sobolev
norms are equivalent. We may assume $f\neq0$. Put $X=\|f\|_2$ and
$Y=\|f\|_{H^{s_0}}$, so that $X\leq Y$. With the Fourier normalization
of Section~\ref{app:sec:notation}, $f(x)=\int e^{\ii x\cdot\xi}
\widehat f(\xi)\,d\xi$ and $\|\widehat f\|_2=(2\pi)^{-n/2}\|f\|_2$; we
show below that $\widehat f\in L^1$, so $f$ has a continuous
representative with $\|f\|_{C^0}\leq\|\widehat f\|_{L^1}$. For
$\Theta>0$ we split the frequencies. By the Cauchy--Schwarz inequality,
\[
 \int_{|\xi|\leq\Theta}|\widehat f|\,d\xi
 \leq|B_\Theta|^{1/2}\|\widehat f\|_2\leq C\Theta^{n/2}X,
\]
and, since $2s_0>n$,
\[
 \int_{|\xi|>\Theta}|\widehat f|\,d\xi
 \leq\Bigl(\int_{|\xi|>\Theta}|\xi|^{-2s_0}\,d\xi\Bigr)^{1/2}
 \Bigl(\int|\xi|^{2s_0}|\widehat f|^2\,d\xi\Bigr)^{1/2}
 \leq C\Theta^{n/2-s_0}Y.
\]
Choose $\Theta=(Y/X)^{1/n}$. The first bound becomes $CX^{1/2}Y^{1/2}$,
and the second becomes
\[
 C(Y/X)^{1/2-s_0/n}Y=CX^{1/2}Y^{1/2}(X/Y)^{s_0/n-1}
 \leq CX^{1/2}Y^{1/2},
\]
since $s_0/n>1$ and $X\leq Y$.
\end{proof}

\begin{prop}[Estimates for the sampled corrections]
\label{app:ch_mean_pairs:prop:canonical-estimates}
Put $\ell=m_0-1=\lfloor n/2\rfloor+1$. There is a fixed $B_{\rm can}$
such that, if
\[
 r_*\geq s_0+B_{\rm can},\qquad
 R_{\rm env}\geq s_0+B_{\rm can}+\ell+2+r_{\rm par},
\]
then the following bounds hold for every realization of the sample in
the event of Theorem~\ref{app:ch_waves:thm:sampling}.
\begin{enumerate}
\item For all smooth sources $u,v$,
\begin{equation}\label{app:ch_mean_pairs:eq:mean-quadratic-low}
 \|\mathcal Q(u,v)\|_2\leq C\|u\|_2\|v\|_{H^{s_0/2}}.
\end{equation}
\item For all integers $s\geq0$,
\begin{equation}\label{app:ch_mean_pairs:eq:block-normal}
 \|\mathcal Eu\|_{H^s}
 \leq C_s[\|u\|_{H^s}+T^{(s-s_0)_+}\|u\|_2].
\end{equation}
\item If $u,v$ satisfy $\|u\|_{H^s}+\|v\|_{H^s}\leq C_shR^{(s-s_0)_+}$
for all $s$, then
\begin{equation}\label{app:ch_mean_pairs:eq:block-bilinear}
 \|\mathcal Q(u,v)\|_{H^s}
 \leq C_sh^2R^{(s-s_0)_+},\qquad s\geq0,
\end{equation}
where $C_s$ depends on the constants in the hypothesis.
\item For all integers $s\geq0$,
\begin{equation}\label{app:ch_mean_pairs:eq:block-fields}
 \begin{split}
 &\sup_\nu\|D_\nu u\|_{H^s}+\|V_{\rm aux}^F(u)\|_{H^s}
                +\|D\gamma_FV_{\rm aux}^F(u)\|_{H^s}\\
 &\qquad\leq C_s[\|u\|_{H^{s+1}}
              +T^{(s+1-s_0)_+}\|u\|_2].
 \end{split}
\end{equation}
\item The measures $\widehat f_\nu[u]\,d\eta$ have support in
$\{|\eta|\leq CK\}$ and total variation at most $C\|u\|_{H^{m_0}}$.
\end{enumerate}
The bounds hold on the fixed supports and are uniform in $K$. No
regularity with respect to the sample index $\nu$ is used.
\end{prop}

\begin{proof}
\medskip\noindent\emph{Step 1 (Operator bounds).}
We use the scalar auxiliary operator of
Theorem~\ref{app:ch_waves:thm:sampling}, with a cutoff $\chi_*$ equal to
one near every measurement support and near the supports of the
fixed-parameter operators. Its sampled operator $\mathbb T_K$ is
bounded on the event of that theorem. By positivity this bounds every
operator with a smaller cutoff. For an amplitude
$b(x,\nu,\eta)$ supported where $\chi_*=1$, let $\mathbb T_K[b]$ be the
operator whose kernel is multiplied by $b$.
Lemma~\ref{app:ch_osc:lem:osc-amplitude} gives
\begin{equation}\label{new:eq:amplitude-test}
 \|\mathbb T_K[b]\|\leq C\sup_{\nu,\eta}\|b(\cdot,\nu,\eta)\|_{H^\ell}.
\end{equation}
Only physical derivatives of $b$ occur; no derivatives in the initial
frequency or in the sample index are needed. In particular
$\|\widetilde A_K\|=\|\widetilde A_K^*\|\leq C$, and $\|E_F^{-1}\|\leq C$ by
Proposition~\ref{app:ch_background:prop:bg-frame}.

\medskip\noindent\emph{Step 2 (Moment estimates for the adjoint).}
The exact phase of the $\nu$-th operator is $\Phi_\nu=Y_\nu\cdot\eta$,
where
\[
 Y_\nu=\int_0^1d_\zeta S_F(x,\zeta_\nu+t\tau_\nu\eta)\,dt,
\]
and its $y$-derivative is uniformly close to the identity. Hence the
identities \eqref{app:ch_background:eq:bg-derivative-trade} hold, with a
matrix $\mathsf B_\nu$, uniformly invertible, such that
$d_y\Phi_\nu=\mathsf B_\nu\eta$. Consequently
\[
 \eta_ke^{-\ii\Phi_\nu}
 =\ii\sum_l(\mathsf B_\nu^{-1})_{kl}\,\partial_{y_l}e^{-\ii\Phi_\nu}.
\]
The adjoint $A_{\nu,K}^*w$ is an integral of $e^{\mp\ii\Phi_\nu}$ against
an amplitude times $\mathcal C^*w=\chi_{\rm meas}q_F^*w$, which has
compact support in the measurement region. Multiplying by $\eta_k$ and
integrating by parts in $y$ therefore produces no boundary terms and
moves one derivative onto $\mathsf B_\nu^{-1}$, onto the amplitude, or
onto $w$. Thus one factor of the input frequency is exchanged for one
physical derivative of the source or of the coefficients.

For an integer $q\geq0$ we have
$|\eta|^{2q}=\sum_{|\alpha|=q}\frac{q!}{\alpha!}\eta^{2\alpha}$, so
\[
 \||\eta|^qf\|_2\leq C_q\sum_{|\alpha|=q}\|\eta^\alpha f\|_2,
\]
and it suffices to treat the monomials $\eta^\alpha$. After $q$
integrations by parts, a term with $i$ derivatives on $w$ carries
$q-i$ derivatives on the coefficients. It is an operator of the same
form as $\widetilde A_K^*$ with a modified amplitude, applied to a derivative of
order $i$ of $w$. We bound it by \eqref{new:eq:amplitude-test}, which
costs $\ell$ further derivatives of the amplitude. The coefficients
satisfy derivative bounds of the form $C_mT^{(m-r_*+B)_+}$ with fixed
shifts $B$; including these shifts, the amplitude test and the shifts
of the inverse and of the corrected waves in $B_{\rm can}$, the
condition $r_*\geq s_0+B_{\rm can}$ gives the coefficient factor
$T^{(q-i-s_0)_+}$. With $w=E_F^{-1}u$ and the inverse estimate
\eqref{app:ch_background:eq:bg-r-inverse} (with $L=T$ and $r_0=s_0$),
$\|E_F^{-1}u\|_{H^i}\leq C_i[\|u\|_{H^i}+T^{(i-s_0)_+}\|u\|_2]$. Since
$(q-i-s_0)_++(i-s_0)_+\leq(q-s_0)_+$, we obtain
\begin{equation}\label{app:ch_mean_pairs:eq:raw-adjoint-moments}
 \||\eta|^q\widetilde A_K^*E_F^{-1}u\|_2
 \leq C_q\sum_{i=0}^qT^{(q-i-s_0)_+}\|u\|_{H^i}.
\end{equation}
For an individual adjoint $A_{\nu,K}^*$ we remove the weight
$|\eta|^{1/2}$ and use the bound for the fixed-parameter operator
without this weight in Proposition~\ref{app:ch_osc:prop:osc-normal}.
The same argument shows that the right side of
\eqref{app:ch_mean_pairs:eq:raw-adjoint-moments} also bounds
$\sup_\nu\||\eta|^{q-1/2}A_{\nu,K}^*E_F^{-1}u\|_2$; we call this the
individual moment of order $q$.

\medskip\noindent\emph{Step 3 (Proof of \eqref{app:ch_mean_pairs:eq:block-normal}).}
A derivative of order $s$ of the forward operator $\widetilde A_K$ produces at
most $s$ factors of the input frequency, from derivatives of
$e^{\ii\Phi_\nu}$; a term with $k$ such factors carries a coefficient
factor $T^{(s-k-s_0)_+}$. By \eqref{new:eq:amplitude-test},
\[
 \|\widetilde A_K\widetilde A_K^*E_F^{-1}u\|_{H^s}
 \leq C_s\sum_{k=0}^sT^{(s-k-s_0)_+}\||\eta|^k\widetilde A_K^*E_F^{-1}u\|_2.
\]
The products of the differentiated coefficients obey
\eqref{app:ch_regular_base:eq:joint-primitive-product}, so these
operations use the same $B_{\rm can}$. Inserting
\eqref{app:ch_mean_pairs:eq:raw-adjoint-moments} and using
$(s-k-s_0)_++(k-i-s_0)_+\leq(s-i-s_0)_+$ for $0\leq i\leq k\leq s$, we
obtain terms $C_sT^{(s-i-s_0)_+}\|u\|_{H^i}$, $0\leq i\leq s$. By
\eqref{app:ch_background:eq:bg-grade-endpoints} with $L=T$ and
$r_0=s_0$, each of them is at most
$C_s[\|u\|_{H^s}+T^{(s-s_0)_+}\|u\|_2]$. The term
$\mathcal F_FE_F^{-1}u$ satisfies the same bound by
\eqref{app:ch_background:eq:bg-r-inverse}. This proves
\eqref{app:ch_mean_pairs:eq:block-normal}.

\medskip\noindent\emph{Step 4 (Proof of \eqref{app:ch_mean_pairs:eq:block-fields} and of (v)).}
By \eqref{app:ch_mean_pairs:eq:mean-current-canonical},
$D_\nu u=G_{\nu,K}A_{\nu,K}^*E_F^{-1}u$, and $G_{\nu,K}$ is a fixed-parameter
operator with input weight $|\eta|^{1/2}$. By the bound for the
unweighted fixed-parameter operator in
Proposition~\ref{app:ch_osc:prop:osc-normal}, a term of an order-$s$
derivative of $D_\nu u$ with $k$ frequency factors is bounded by
$C_sT^{(s-k-s_0)_+}\||\eta|^{k+1/2}A_{\nu,K}^*E_F^{-1}u\|_2$, that is, by
the individual moment of order $k+1$. Step 2 and the endpoint
inequality \eqref{app:ch_background:eq:bg-grade-endpoints} at order
$s+1$ give the bound for $D_\nu u$ in
\eqref{app:ch_mean_pairs:eq:block-fields}. The auxiliary field and its
linearized metric obey the same bound by
Proposition~\ref{app:ch_background:prop:bg-mixed-physical}.

For the measures, $\widehat f_\nu[u]=\rho_\nu\chi_{\rm hi}|\eta|^{1/2}
P_KA_{\nu,K}^*E_F^{-1}u$ vanishes for $|\eta|\leq1$ and for $|\eta|\geq2K$,
which gives the support assertion. Since the initial variables lie in
$\R^{n-1}$ and $2(m_0-1)=2\lfloor n/2\rfloor+2>n-1$, the
Cauchy--Schwarz inequality gives
\[
 \int|\widehat f_\nu[u]|\,d\eta
 \leq C\Bigl(\int_{|\eta|\geq1}|\eta|^{2(1-m_0)}d\eta\Bigr)^{1/2}
 \||\eta|^{m_0-1/2}A_{\nu,K}^*E_F^{-1}u\|_2
 \leq C\|u\|_{H^{m_0}},
\]
by the individual moment of order $m_0$ and $m_0<s_0$.

\medskip\noindent\emph{Step 5 (Proof of \eqref{app:ch_mean_pairs:eq:mean-quadratic-low}).}
By the definitions and the symmetry of $\mathcal Q$,
\[
\begin{split}
 \widetilde A_K[(D_\nu v)_\nu]\widetilde A_K^*E_F^{-1}u
 &=\sum_\nu\mu_\nu\mathcal C[2\sym\Re((D_\nu v)^*G_{\nu,K}A_{\nu,K}^*E_F^{-1}u)]\\
 &=2\mathcal Q(v,u)=2\mathcal Q(u,v).
\end{split}
\]
The operator $\widetilde A_K[(D_\nu v)_\nu]$ is obtained from the
aggregated operator of Step~1 by multiplying its amplitude by
$(D_\nu v)^*$ and by coefficient factors with uniform $C^\ell$ bounds;
the input $\widetilde A_K^*E_F^{-1}u$ is the aggregated adjoint, which is
bounded on $L^2$, and no individual operator $G_{\nu,K}$ is estimated
separately. By \eqref{new:eq:amplitude-test}
and \eqref{app:ch_mean_pairs:eq:block-fields} with $s=\ell$,
\[
 \|\widetilde A_K[(D_\nu v)_\nu]\|\leq C\sup_\nu\|D_\nu v\|_{H^\ell}
 \leq C\|v\|_{H^{m_0}},
\]
since $\ell+1=m_0<s_0$. Together with $\|\widetilde A_K^*\|\leq C$ and
$\|E_F^{-1}\|\leq C$ this proves
\eqref{app:ch_mean_pairs:eq:mean-quadratic-low}, because $m_0=s_0/2$.

\medskip\noindent\emph{Step 6 (Proof of \eqref{app:ch_mean_pairs:eq:block-bilinear}).}
Fix $s$ and a multi-index $\sigma$ with $|\sigma|=s$. By the Leibniz rule,
$\partial^\sigma\mathcal Q(u,v)$ is a finite sum of terms
\begin{equation}\label{new:eq:Q-derivative-term}
 (\partial^{\sigma_1}\mathcal C)\sum_\nu\mu_\nu
 \sym\Re\bigl((\partial^{\sigma_2}D_\nu u)^*\,\partial^{\sigma_3}D_\nu v\bigr),
 \qquad c=|\sigma_1|,\ r=|\sigma_2|,\ t=|\sigma_3|,\quad c+r+t=s .
\end{equation}
Since $\mathcal Q$ is symmetric and the hypothesis is symmetric in $u$
and $v$, we may assume $t\leq r$, hence $t\leq s/2$.

The two factors are treated differently. The factor with $r$ derivatives
cannot be estimated sample by sample: by Step~4, $\sup_\nu\|D_\nu u\|_{H^r}$
is controlled only through the individual moment of order $r+1$, that is,
with one more derivative of $u$, and the resulting bound
$h^2R^{(s+1-s_0)_+}$ is not sufficient, since $hR\geq1$. We therefore keep
this factor inside the aggregated operator of Step~1 and place the other
factor in its amplitude.

\emph{The factor with $r$ derivatives.} Write $w_\nu=A_{\nu,K}^*E_F^{-1}u$,
so that $D_\nu u=G_{\nu,K}w_\nu$. Each physical derivative of the phase
factor $e^{\ii\Phi_\nu}$ in $G_{\nu,K}$ produces one factor of the input
frequency $\eta$, multiplied by a coefficient with bounded physical
derivatives, because $\Phi_\nu=Y_\nu\cdot\eta$ with $Y_\nu$ as in Step~2,
so that $d_x\Phi_\nu=(D_xY_\nu)^T\eta$. Hence
\[
 \partial^{\sigma_2}D_\nu u=\sum_{|\kappa|\leq r}
 G_{\nu,K}[\gamma_{\nu,\kappa}]\bigl(\eta^\kappa w_\nu\bigr),
\]
where $G_{\nu,K}[\gamma]$ is the operator $G_{\nu,K}$ with its amplitude
multiplied by $\gamma$, and the coefficient $\gamma_{\nu,\kappa}$ is a
product of derivatives of the phase, of the amplitude of the wave and of
the cutoffs, of total order $r-|\kappa|$. Including the shifts in
$B_{\rm can}$, its first $\ell$ physical derivatives are bounded by
$CT^{(r-|\kappa|-s_0)_+}$, uniformly in $\nu$ and $\eta$.

\emph{The aggregated form.} Substituting this into
\eqref{new:eq:Q-derivative-term} and recalling the definition of
$\widetilde A_K[V]$ and $\widetilde A_K^*E_F^{-1}u=(\sqrt{\mu_\nu}\,w_\nu)_\nu$, the
term with a given $\kappa$, $k=|\kappa|$, equals
\[
 \tfrac12\,\widetilde A_K[b_\kappa]\bigl(\eta^\kappa\widetilde A_K^*E_F^{-1}u\bigr),
\]
where $\widetilde A_K[b_\kappa]$ denotes the operator
$\widetilde A_K[(\partial^{\sigma_3}D_\nu v)_\nu]$ in which $\mathcal C$ is
replaced by $\partial^{\sigma_1}\mathcal C$ and $G_{\nu,K}$ by
$G_{\nu,K}[\gamma_{\nu,\kappa}]$. It differs from the aggregated operator
of Step~1 only by an amplitude factor $b_\kappa(x,\nu,\eta)$, built from
the coefficients of $\partial^{\sigma_1}\mathcal C$, from
$\gamma_{\nu,\kappa}$ and from $(\partial^{\sigma_3}D_\nu v)^*$. Thus the
factor with $t$ derivatives, together with the differentiated
coefficients, has become part of the amplitude of the aggregated
operator. By \eqref{new:eq:amplitude-test}, which uses $\ell$ physical
derivatives of the amplitude, and by Step~4 at order $t+\ell$,
\[
 \|\widetilde A_K[b_\kappa]\|
 \leq CT^{(c+r-k-s_0)_+}\sup_\nu\|D_\nu v\|_{H^{t+\ell}}
 \leq CT^{(c+r-k-s_0)_+}\bigl[\|v\|_{H^{t+m_0}}+T^{(t+m_0-s_0)_+}\|v\|_2\bigr],
\]
since $\ell+1=m_0$. By the hypothesis on $v$, $T<R$ and $s_0=2m_0$, the
bracket is at most $C_thR^{(t+m_0-s_0)_+}=C_thR^{(t-s_0/2)_+}$. The input
is controlled by the aggregated moment estimate
\eqref{app:ch_mean_pairs:eq:raw-adjoint-moments} and the hypothesis on
$u$:
\[
 \|\eta^\kappa\widetilde A_K^*E_F^{-1}u\|_2
 \leq C_k\sum_{i\leq k}T^{(k-i-s_0)_+}\|u\|_{H^i}
 \leq C_kh\sum_{i\leq k}T^{(k-i-s_0)_+}R^{(i-s_0)_+}.
\]

\emph{The exponents.} Using $T<R$ and \eqref{new:eq:plus-subadditive} to
combine $(c+r-k-s_0)_++(k-i-s_0)_+\leq(c+r-i-s_0)_+$, each term is bounded
by $C_sh^2R^{\varepsilon_i}$ with
\[
 \varepsilon_i=(s-i-t-s_0)_++(i-s_0)_++(t-s_0/2)_+ ,
\]
because $c+r=s-t$. Here $i\leq k\leq r$, so $i+t\leq s$, and $t\leq s/2$.
Lemma~\ref{new:lem:exponent-inequality}, with $j=t$, gives
$\varepsilon_i\leq(s-s_0)_+$. Summing the finitely many terms proves
\eqref{app:ch_mean_pairs:eq:block-bilinear}.

The index $s_0/2$ in the last summand of $\varepsilon_i$ is what makes
this distribution of derivatives possible: the factor carried in the
amplitude loses $m_0=s_0/2$ derivatives (one for the individual operator
and $\ell$ for the amplitude test), and this loss is compensated by
$t\leq s/2$. Only source norms of order at most $\max(s,s_0)$ occur, so
the source regularity required in the iteration is fixed.
\end{proof}

\subsection{A finite iteration}

\begin{prop}[Finite correction of the mean]
\label{app:ch_mean_pairs:prop:mean-iteration}
We work under Setup~\ref{app:ch_mean_pairs:defn:mean-geometry-input}
and Definition~\ref{app:ch_mean_pairs:defn:mean-wave-input}, with the
positive initial amplitude $r_F$, the base field $V_{\rm base}$, the
continuous operator $A_F$ and the inverse $E_F^{-1}$, and we sample the
unmodulated waves with the input frequency bound $K$. Let
$\mathcal E$, $\mathcal Q$, $r_0$ and $g$ be as in
\eqref{new:eq:mean-operators}--\eqref{new:eq:mean-source}. For every
finite $Q$ and all sufficiently large $R$, the finite iteration
\begin{equation}\label{app:ch_mean_pairs:eq:single-iteration}
 r^{(0)}=0,\qquad
 r^{(j+1)}=g+(I-\mathcal E)r^{(j)}-\mathcal Q(r^{(j)},r^{(j)}),
\end{equation}
stopped after a suitable number $J$ of steps, gives a real source
$r=r^{(J)}$ whose residual satisfies
\begin{equation}\label{main:eq:full-recursion}
 r_{\rm rem}=g-\mathcal Er-\mathcal Q(r,r),\qquad
 \|r_{\rm rem}\|_{C^0}\leq R^{-Q}.
\end{equation}
The measures $\widehat f_\nu[r]\,d\eta$ have support in
$\{|\eta|\leq CK\}$ and total variation at most $C$. On the fixed
supports, for every $s\geq0$,
\begin{equation}\label{app:ch_mean_pairs:eq:single-mean-profiles}
 \sup_\nu\|D_\nu r\|_{C^s}
 +\|V_{\rm aux}^F(r)\|_{H^s}
 +\|D\gamma_FV_{\rm aux}^F(r)\|_{H^s}
 \leq C_{s,Q}R^s.
\end{equation}
The input frequency bound, $r_*$ and $R_{\rm env}$ do not depend on
$Q$. The family of waves is fixed before the sample is drawn, and the
sample does not depend on the number $J$ of Picard steps; $J$, the
constants and the threshold for $R$ depend on $Q$.
\end{prop}

\begin{proof}
\medskip\noindent\emph{Step 1 (The sample).}
We apply Theorem~\ref{app:ch_waves:thm:sampling} with $\Lambda=K=R^{s_f}$.
Then $h^{-3}=R^{1+\epsilon}=\Lambda^{1+\epsilon_{\rm samp}}$ with
$\epsilon_{\rm samp}=(1+\epsilon)/s_f-1>0$, and
$\Lambda^{-\epsilon_{\rm samp}/4}=R^{-(1+\epsilon-s_f)/4}$. We fix a
realization of the sample in the event of that theorem; this event is
nonempty for large $R$, since its probability tends to one. The
concentration bound \eqref{app:ch_waves:eq:sampling-all-prefixes}, the
comparison of phases \eqref{app:ch_waves:eq:continuous-normal-comparison},
and the identification of the continuous operator at $\tau=0$ with
$T_FP_K^2T_F^*$ (end of
Section~\ref{app:sec:ch_background}) give
\begin{equation}\label{app:ch_mean_pairs:eq:mean-stage-normal-errors}
 \|\widetilde E_K-E_{F,K}\|
 \leq C[R^{-(1+\epsilon-s_f)/4}+R^{-a+s_f}]
 \leq CR^{-\omega},\qquad \omega=\frac{1+\epsilon-s_f}4,
\end{equation}
where $\omega$ is the exponent of \eqref{main:eq:preparation-orders};
the second inequality uses $a-s_f>\omega$, verified in
Section~\ref{main:sec:scales}. The sample has at most $CR^{1+\epsilon}$
points, bounded total mass, weights at most $CR^{-1-\epsilon}$ and
separation $ch$ within each patch. The orders $s_0+B_{\rm can}$ and the
shifts of the primary waves and of the background are included in the
fixed order $r_*$.

\medskip\noindent\emph{Step 2 (Bounds for the source).}
We take $R_{\rm env}\geq s_0+B_{\rm can}+\ell+2+r_{\rm par}$ in
Corollary~\ref{app:ch_response:cor:current-atoms}. The quadrature
estimate \eqref{app:ch_waves:eq:background-quadrature}, valid on the
same event at all orders, and the comparison of the corrected and the
leading amplitudes \eqref{app:ch_response:eq:current-atomic-reserve},
together with $\lambda^{-1}\leq h$, give
\begin{equation}\label{app:ch_mean_pairs:eq:comparison-input}
 \|r_0\|_{H^s}\leq C_shT^{(s-s_0)_+},\qquad
 \sup_\nu\|B_\nu\|_{C^s}+\|\mathcal C\|_{C^s}
       \leq C_sT^{(s-s_0)_+}.
\end{equation}
By \eqref{main:eq:smoothing-low} and $b^2=\delta^2R^{2A}$, the
rescaled defect satisfies $\|e_R/b^2\|_{H^s}\leq C_sR^{s-2A}$, and
$2A=s_0+1$. By the Leibniz rule and
\eqref{app:ch_mean_pairs:eq:comparison-input}, the zero extension of
$\mathcal C(e_R/b^2)$ satisfies
\[
 \|\mathcal C(e_R/b^2)\|_{H^s(\R^n)}
 \leq C_s\sum_{i\leq s}T^{(s-i-s_0)_+}R^{i-s_0-1}
 \leq C_sR^{s-s_0-1}.
\]
Since $T<R$ and $R^{s-s_0-1}\leq R^{-1}R^{(s-s_0)_+}\leq hR^{(s-s_0)_+}$
(because $\beta<1$), the complete source obeys
\begin{equation}\label{new:eq:source-bound}
 \|g\|_{H^s}\leq C_shR^{(s-s_0)_+}\qquad(s=0,1,\ldots).
\end{equation}

\medskip\noindent\emph{Step 3 (Bounds for the iterates).}
Fix a finite $J$. We claim that
\begin{equation}\label{new:eq:iterate-bound}
 \|r^{(j)}\|_{H^s}\leq C_{J,s}hR^{(s-s_0)_+},\qquad
 0\leq j\leq J+1,\ s\geq0.
\end{equation}
This holds for $j=0$. If it holds for $j$, then
\eqref{app:ch_mean_pairs:eq:block-normal} and $T<R$ give
$\|\mathcal Er^{(j)}\|_{H^s}\leq C_shR^{(s-s_0)_+}$, and
\eqref{app:ch_mean_pairs:eq:block-bilinear} gives
$\|\mathcal Q(r^{(j)},r^{(j)})\|_{H^s}\leq C_sh^2R^{(s-s_0)_+}$.
With \eqref{new:eq:source-bound} and
\eqref{app:ch_mean_pairs:eq:single-iteration} this gives the claim for
$j+1$. Each of the finitely many steps changes only the constants. All
iterates are real and belong to $H^\infty$.

\medskip\noindent\emph{Step 4 (The difference equation).}
Put $d_j=r^{(j+1)}-r^{(j)}$. Since $r^{(0)}=0$ and $\mathcal Q(0,0)=0$,
$d_0=r^{(1)}=g$. Subtracting two consecutive steps of
\eqref{app:ch_mean_pairs:eq:single-iteration} and using
\eqref{new:eq:Q-difference},
\begin{equation}\label{new:eq:difference-equation}
\begin{split}
 d_{j+1}&=(I-\mathcal E)d_j-\mathcal Q(r^{(j+1)},r^{(j+1)})
          +\mathcal Q(r^{(j)},r^{(j)})\\
        &=(I-\mathcal E)d_j-\mathcal Q(d_j,r^{(j+1)}+r^{(j)}).
\end{split}
\end{equation}
For the linear term we use $E_FE_F^{-1}=I$:
\[
 I-\mathcal E=(E_F-E_{F,K})E_F^{-1}+(E_{F,K}-\widetilde E_K)E_F^{-1}.
\]
By \eqref{app:ch_mean_pairs:eq:mean-stage-normal-errors} and the
frequency tail estimate \eqref{app:ch_background:eq:bg-carrier-tail}
with $L=T$ and $r_0=s_0$, for every integer $q\geq0$,
\[
\begin{split}
 \|(I-\mathcal E)d_j\|_2
 &\leq CR^{-\omega}\|d_j\|_2
 +C_qK^{-q}[\|d_j\|_{H^q}+T^{(q-s_0)_+}\|d_j\|_2]\\
 &\leq CR^{-\omega}\|d_j\|_2+C_{J,q}h(R/K)^q,
\end{split}
\]
where in the second line we used \eqref{new:eq:iterate-bound} for
$r^{(j+1)}$ and $r^{(j)}$ and $T<R$. For the quadratic term,
\eqref{app:ch_mean_pairs:eq:mean-quadratic-low} and
\eqref{new:eq:iterate-bound} at order $s_0/2<s_0$ give
\[
 \|\mathcal Q(d_j,r^{(j+1)}+r^{(j)})\|_2
 \leq C\|d_j\|_2\|r^{(j+1)}+r^{(j)}\|_{H^{s_0/2}}
 \leq C_Jh\|d_j\|_2.
\]
Hence, for $j<J$,
\[
 \|d_{j+1}\|_2
 \leq[CR^{-\omega}+C_Jh]\|d_j\|_2+C_{J,q}h(R/K)^q.
\]
Since $\beta>\omega$, we have $h\leq R^{-\omega}$, and the bracket is at
most $C_JR^{-\omega}\leq1$ for large $R$. Iterating from
$\|d_0\|_2=\|g\|_2\leq Ch$ and absorbing the finite powers of $C_J$
into $C_{J,q}$ gives
\begin{equation}\label{new:eq:difference-bound}
 \|d_J\|_2\leq C_{J,q}h[R^{-\omega J}+(R/K)^q].
\end{equation}

\medskip\noindent\emph{Step 5 (The residual and the choice of $J$ and $q$).}
By \eqref{app:ch_mean_pairs:eq:single-iteration},
$d_J=r^{(J+1)}-r^{(J)}=g-\mathcal Er^{(J)}-\mathcal Q(r^{(J)},r^{(J)})$,
so the residual is exactly $r_{\rm rem}=d_J$. By
\eqref{new:eq:iterate-bound}, $\|d_J\|_{H^{s_0}}\leq C_Jh$. Since
$s_0>n$, Lemma~\ref{new:lem:interpolation} and
\eqref{new:eq:difference-bound} give
\[
 \|r_{\rm rem}\|_{C^0}
 \leq C\|d_J\|_2^{1/2}\|d_J\|_{H^{s_0}}^{1/2}
 \leq C_{J,q}h[R^{-\omega J}+(R/K)^q]^{1/2}.
\]
Now $(R/K)^q=R^{-q(s_f-1)}$. We choose the number of steps $J$ with
$\omega J>2Q$. The integer $q$ is not part of the construction; it is
only the order of the tail estimate, and we choose it with
$q(s_f-1)>2Q$. Then the right side is at most
$C_{J,q}R^{-Q-\kappa}$ for some $\kappa>0$, and after increasing the
threshold for $R$ it is at most $R^{-Q}$. This proves
\eqref{main:eq:full-recursion}.

\medskip\noindent\emph{Step 6 (The fields and the measures).}
By \eqref{app:ch_mean_pairs:eq:block-fields} at order $s+\ell$, the
Sobolev embedding $H^{s+\ell}\hookrightarrow C^s$ and
\eqref{new:eq:iterate-bound},
\[
 \sup_\nu\|D_\nu r\|_{C^s}
 \leq C_s[\|r\|_{H^{s+m_0}}+T^{(s+m_0-s_0)_+}\|r\|_2]
 \leq C_{J,s}hR^{(s+m_0-s_0)_+}\leq C_{s,Q}R^s,
\]
since $\ell+1=m_0$ and $s_0=2m_0$, so that $(s+m_0-s_0)_+=(s-m_0)_+\leq s$.
The auxiliary field and its linearized metric obey the same bounds in
$H^s$. By Proposition~\ref{app:ch_mean_pairs:prop:canonical-estimates}(v)
the total variation of the data is at most
$C\|r\|_{H^{m_0}}\leq C_Jh$, hence bounded, and the support is still
determined by the cutoff $P_K$. This proves
\eqref{app:ch_mean_pairs:eq:single-mean-profiles}.
\end{proof}

\subsection{The normal component}\label{new:sec:mean-normal}
Let $r$ be the source of
Proposition~\ref{app:ch_mean_pairs:prop:mean-iteration}. Put
$m_\nu=\delta_0+\widehat f_\nu[r]\,d\eta$, let
$\mathcal H=\sum_\nu\mu_\nu\Re((dZ_\nu)^*dZ_\nu)$ be the Hermitian
tensor of the waves $Z_\nu=\mathcal L_\nu(m_\nu)$, and write
$V_0=V_{\rm base}+V_{\rm aux}^F(r)$. On $U_{\rm fin}$ set
\[
 T_f=e_R/b^2-D\gamma_FV_0-\mathcal H,
\]
and define
\begin{equation}\label{main:eq:mean-normal-field}
 V=V_0-\tfrac12N_F T_f,
\end{equation}
where $N_F$ is the normal map of \eqref{main:eq:normal-completion}.

\begin{lem}[The error of the mean]\label{new:lem:mean-exact}
On $U_{\rm fin}$, in particular on $D$,
\begin{equation}\label{new:eq:qT-identity}
 q_FT_f=r_{\rm rem},
\end{equation}
and the tensor $E_{\rm mean}$ of \eqref{main:eq:mean-equation} satisfies
\begin{equation}\label{new:eq:mean-error-formula}
 E_{\rm mean}=-\Pi_{K_F}T_f
 =-\sum_{i,j}Q_i(F)(\mathsf G_F^{-1})_{ij}(r_{\rm rem})_j.
\end{equation}
\end{lem}
\begin{proof}
On $U_{\rm fin}$ we have $q_F=\mathcal C$. By
Lemma~\ref{new:lem:mean-source-identity} and the definition of $g$,
\[
 q_FT_f=q_F(e_R/b^2)-\bigl[\mathcal Er+\mathcal Q(r,r)-r_0\bigr]
 =g-\mathcal Er-\mathcal Q(r,r)=r_{\rm rem}.
\]
Next, by the definitions of $E_{\rm mean}$ and $T_f$,
\[
 E_{\rm mean}=D\gamma_FV_0-\tfrac12D\gamma_F(N_FT_f)+\mathcal H-e_R/b^2
 =-T_f+D\gamma_F\bigl(-\tfrac12N_FT_f\bigr).
\]
The field $w=-\frac12N_FT_f$ is normal, and by
\eqref{main:eq:tangent-normal} with $u=0$ its linearized metric is the
algebraic expression $D\gamma_Fw=-2\langle w,\mathrm{II}_F\rangle=-2B_F^*w$.
Since $N_F=(B_F^*)^{-1}\Pi_{K_F^\perp}$, where $B_F^*$ maps the normal
space isomorphically onto $K_F^\perp=\operatorname{im}B_F^*$,
\[
 D\gamma_F\bigl(-\tfrac12N_FT_f\bigr)
 =B_F^*(B_F^*)^{-1}\Pi_{K_F^\perp}T_f=\Pi_{K_F^\perp}T_f.
\]
Hence $E_{\rm mean}=-T_f+\Pi_{K_F^\perp}T_f=-\Pi_{K_F}T_f$. (This is
also \eqref{main:eq:completion-error} with $u=0$ and $h=T_f$.)
Finally, the Gram formula for the orthogonal projection onto the span
of the frame $Q_1(F),\ldots,Q_n(F)$ reads
\[
 \Pi_{K_F}H=\sum_{i,j}Q_i(F)(\mathsf G_F^{-1})_{ij}(Q_j(F):H),
\]
since the right side lies in $K_F$ and its pairing with each $Q_k(F)$
equals
\[
 \sum_{i,j}(\mathsf G_F)_{ki}(\mathsf G_F^{-1})_{ij}(Q_j(F):H)=Q_k(F):H .
\] With $H=T_f$ and $(Q_j(F):T_f)_j=q_FT_f=r_{\rm rem}$, this
proves \eqref{new:eq:mean-error-formula}.
\end{proof}

\subsection{Correction of the mean}\label{new:sec:mean-proof}

\begin{proposition}[Correction of the mean]\label{main:prop:mean}
There is a positive initial amplitude $r_F$, depending only on $F$, for
which Proposition~\ref{main:prop:current-waves} applies and the
following conclusions hold. Let $e=g-\gamma(F)$, with
$\|e\|_{C^0(D)}\leq\delta^2$ and $0<\delta\leq1$. Fix a finite
smoothing order $m\geq1$ and put
\[
 e_R=S_{R,m}(e|_D),\qquad b=\delta R^A,
\]
where $S_{R,m}$ is the operator of Lemma~\ref{main:lem:smoothing}.
Given a finite $Q$, at the scales of Section~\ref{main:sec:scales} and
for sufficiently large $R$, there are a family of measures $m_\nu$ and a
real ambient field $V$ with the following properties, where $Z_\nu$ is
given by \eqref{main:eq:lag-recurrence} with $m=m_\nu$.
\begin{enumerate}
\item The tensor
\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 $\|E_{\mathrm{mean}}\|_{C^0(D)}\leq C_QR^{-Q}$.
\item The measures satisfy $\sup_\nu\|m_\nu\|_{\TV}\leq C$ and
\begin{equation}\label{main:eq:mean-input-supports}
 \supp m_\nu\subset\{|\xi|\leq CK_{\max}\},
 \qquad K_{\max}=R^{s_f},
\end{equation}
with a support constant independent of every accuracy choice.
\item On the whole product source support, the waves satisfy
\begin{equation}\label{main:eq:mean-final-amplitude}
 \sup_\nu\|\chi_{\mathrm{pair}}
       e^{-\ii\lambda_\nu\phi_\nu}dZ_\nu\|_{C^r}
                  \leq C_rR^r.
\end{equation}
\item After the fixed physical cutoff, equal to one near $D$, the field
$V$ satisfies, for every $r\geq0$,
\begin{equation}\label{main:eq:mean-size}
 \|V\|_{H^r}\leq C_rR^r.
\end{equation}
\end{enumerate}
\end{proposition}

The source order $s_0$ and the coefficient index $r_*$ are fixed before
the accuracy. The number of iteration steps may depend on $Q$; the
input frequency bound and the exponents in the derivative bounds do
not. The constants $C_r$ may depend on $m$ and $Q$.

\begin{proof}[Proof of Proposition~\ref{main:prop:mean}]
\medskip\noindent\emph{Step 1 (The initial amplitude and the waves).}
Let $r_F$ be the positive initial amplitude of
Proposition~\ref{app:ch_background:prop:bg-current} with $G=F$; it is
determined by $F$ and the fixed choices of
Section~\ref{app:sec:ch_background}. Its derivative bounds
\eqref{app:ch_background:eq:bg-current-profile} are of the form
required in Proposition~\ref{main:prop:current-waves}, whose joint
coefficient shifts are included in $r_*$. The measures below have
Fourier support in $\{|\xi|\leq CK_{\max}\}$, and $C_*K_{\max}\leq
\lambda^{1-\nu_0}$ by \eqref{main:eq:all-pair-scales}, so
Proposition~\ref{main:prop:current-waves} applies to the waves $Z_\nu$.

\medskip\noindent\emph{Step 2 (Choice of the data and of $V$).}
We work under Setup~\ref{app:ch_mean_pairs:defn:mean-geometry-input};
its conditions \eqref{app:ch_mean_pairs:eq:mean-stage-window} are those
of \eqref{main:eq:basic-scales} together with $T=T_0=R^t$. We apply
Proposition~\ref{app:ch_mean_pairs:prop:mean-iteration} with accuracy
$Q$, and obtain the sample with \eqref{main:eq:sampling} and the
source $r$. We put
\[
 m_\nu=\delta_0+\widehat f_\nu[r]\,d\eta,
\]
and define $V$ by \eqref{main:eq:mean-normal-field}. It is real, since
$r$, $V_{\rm base}$, $V_{\rm aux}^F$, $e_R$ and $\mathcal H$ are real.

\medskip\noindent\emph{Step 3 (The error).}
By Lemma~\ref{new:lem:mean-exact}, on $D$,
\[
 E_{\rm mean}=-\sum_{i,j}Q_i(F)(\mathsf G_F^{-1})_{ij}(r_{\rm rem})_j.
\]
The frame $Q_i(F)$ is bounded on $D$, and $\mathsf G_F^{-1}$ is bounded
by the uniform Gram gap of
Definition~\ref{app:ch_background:defn:bg-geometry}. Hence, by
\eqref{main:eq:full-recursion},
\[
 \|E_{\rm mean}\|_{C^0(D)}\leq C\|r_{\rm rem}\|_{C^0}\leq CR^{-Q}.
\]
Thus the mean equation holds up to $O(R^{-Q})$: the iteration solves
its projection onto $K_F$ up to the residual $r_{\rm rem}$, and the
normal field $-\frac12N_FT_f$ removes the component orthogonal to $K_F$
exactly.

\medskip\noindent\emph{Step 4 (The measures).}
Since $\|\delta_0\|_{\TV}=1$, Proposition~\ref{app:ch_mean_pairs:prop:mean-iteration}
gives $\|m_\nu\|_{\TV}\leq1+C\leq C$. The Dirac mass is supported at
the origin and $\widehat f_\nu[r]$ vanishes for $|\eta|\geq2K$; this
proves \eqref{main:eq:mean-input-supports}, with a support constant
fixed by the cutoff $P_K$.

\medskip\noindent\emph{Step 5 (The amplitudes).}
We have $e^{-\ii\lambda_\nu\phi_\nu}dZ_\nu=B_\nu+D_\nu r$. The
amplitude $B_\nu$ of the unmodulated wave has derivative scale $T<R$
by \eqref{app:ch_mean_pairs:eq:comparison-input} (equivalently, by
\eqref{main:eq:lag-full-envelope} with $m=\delta_0$ and $K=C_{\rm env}T_0$),
and \eqref{app:ch_mean_pairs:eq:single-mean-profiles} bounds $D_\nu r$
in $C^s$ by $C_{s,Q}R^s$ on the fixed supports, which contain
$\supp\chi_{\rm pair}$. This proves
\eqref{main:eq:mean-final-amplitude}. We remark that this bound does not
follow from \eqref{main:eq:lag-full-envelope} with the Fourier support
of $m_\nu$, which would give the larger scale $K_{\max}=R^{s_f}$; it
uses the fact that the data $f_\nu[r]$ are produced by the adjoint from
a source with bounds $\|r\|_{H^s}\leq C_shR^{(s-s_0)_+}$.

\medskip\noindent\emph{Step 6 (The size of $V$).}
By Step 5, the Leibniz rule and $\sum_\nu\mu_\nu\leq C$,
\[
 \|\mathcal H\|_{C^s}
 \leq\sum_\nu\mu_\nu\sum_{i\leq s}C_s\|B_\nu+D_\nu r\|_{C^i}
 \|B_\nu+D_\nu r\|_{C^{s-i}}\leq C_{s,Q}R^s
\]
on the fixed supports. By
\eqref{app:ch_background:eq:bg-current-profile}, $V_{\rm base}$ and
$D\gamma_FV_{\rm base}$ have $C^s(U_{\rm fin})$ norms at most
$C_sT^s\leq C_sR^s$; by
\eqref{app:ch_mean_pairs:eq:single-mean-profiles}, $V_{\rm aux}^F(r)$
and $D\gamma_FV_{\rm aux}^F(r)$ have $H^s$ norms at most $C_{s,Q}R^s$;
and $\|e_R/b^2\|_{H^s}\leq C_sR^{s-2A}$. Hence $T_f$ is bounded by
$C_{s,Q}R^s$ in $H^s(U_{\rm fin})$, after the fixed physical cutoff.
The coefficients of the normal map $N_F$ are smooth second-jet
expressions in $F$ with derivative bounds $C_sT^s\leq C_sR^s$
on $U_{\rm fin}$. The Leibniz rule in $H^s$ for a product with a
smooth coefficient $c$, $\|cu\|_{H^s}\leq C_s\sum_{i\leq s}
\|c\|_{C^{s-i}}\|u\|_{H^i}$, therefore gives
$\|N_FT_f\|_{H^s}\leq C_{s,Q}R^s$ after the fixed physical cutoff, which
is supported in $U_{\rm fin}$. Together with the bounds for $V_0$ this
proves \eqref{main:eq:mean-size}.

\medskip\noindent\emph{Step 7 (Order of choices).}
The input frequency bound $K_{\max}$, $r_*$, $R_{\rm env}$ and the
exponents do not depend on $Q$. The accuracy $Q$ enters only through the
number $J$ of iteration steps in
Proposition~\ref{app:ch_mean_pairs:prop:mean-iteration}, the constants
and the threshold for $R$; the depth of the wave correction is fixed by
the accuracy of Theorem~\ref{main:thm:step} before the sample is drawn
(Section~\ref{new:sec:order-of-choices}).
\end{proof}

%% ===== 14_16_products_iteration.tex =====
\section{Correction of the oscillatory quadratic terms}\label{app:sec:ch_pairs}

In this section we construct a correction, linearized at the immersion
$F$, which cancels all oscillatory terms in the quadratic part
$(dP)^TdP$ of the metric of $F+P$. The Hermitian part of $(dP)^TdP$ is
not treated here; it is the tensor corrected by
Proposition~\ref{main:prop:mean}. The input is the family of sampled
waves of Proposition~\ref{main:prop:mean}, whose amplitudes vary on the
scale $R$, and the output is a real field $S$, constructed at $F$, whose
linearized metric cancels all products of distinct phases and all
doubled phases up to $O(b^2M(\lambda h)^{-q})$
(Proposition~\ref{main:prop:pairs}). The construction applies the finite
corrections of Theorem~\ref{app:ch_regular:thm:quantitative} to each
product separately; this is possible because every product has its
frequency in a regular region (Proposition~\ref{main:prop:patches}) and
at least $c\lambda h$, which exceeds the amplitude scale $R$ by a fixed
power.

\subsection{The average over the phase}\label{new:sec:phase-average}
The normalization of the primary field is already visible for a single
wave. Let $Z$ be a complex ambient field with
$dZ=e^{\ii\vartheta}A$, where $\vartheta$ is a real phase and $A$ is a
complex $N\times n$ matrix. 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}
Indeed, $d(\Re Z)=\Re(dZ)=\tfrac12(e^{\ii\vartheta}A+e^{-\ii\vartheta}\overline A)$, so
\[
 \bigl(d(\sqrt2\Re Z)\bigr)^Td(\sqrt2\Re Z)
 =\tfrac12\bigl(e^{2\ii\vartheta}A^TA+e^{-2\ii\vartheta}\,\overline A{}^T\overline A
   +A^T\overline A+\overline A{}^TA\bigr).
\]
Since $\overline A{}^TA=A^*A$ and $A^T\overline A=\overline{A^*A}$,
the last two terms add up to $2\Re(A^*A)$, while the first two are
complex conjugate to each other and add up to
$2\Re(e^{2\ii\vartheta}A^TA)$. This proves
\eqref{main:eq:single-wave-mean}.

The first term on the right of \eqref{main:eq:single-wave-mean} is the
mean of the wave in the sense of Section~\ref{main:sec:waves}; the second
oscillates with the doubled phase $2\vartheta$. For a sum of several
waves the quadratic tensor contains, in addition to the means and the
doubled phases, the sums and differences of distinct phases. All these
oscillatory terms are cancelled by the correction constructed in this
section.

Let $Z_\nu$ be the waves and $\mu_\nu$ the weights of
Proposition~\ref{main:prop:mean}, and let $b=\delta R^A$. 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 the 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 tensor $H_{\mathrm{self}}$ is the one appearing in the mean
equation \eqref{main:eq:mean-equation}. The tensor $Q_{\mathrm{off}}$
contains all doubled phases and all signed products of distinct
samples, including the products between different patches;
Lemma~\ref{app:ch_pairs:lem:pairs-ledger} below gives its exact
expansion.

\subsection{The quadratic sources}
\label{app:ch_pairs:sec:pairs-inputs}
\label{app:ch_pairs:sec:pairs-data}
The geometric and linear results used below have been proved in
Sections~\ref{app:sec:ch_seed} and~\ref{app:sec:ch_regular}.

\begin{setup}[Geometry and amplitude bounds]
\label{app:ch_pairs:setup:pairs-current}
\label{app:ch_pairs:setup:pairs-lag}
We work with the geometry of
Definition~\ref{app:ch_background:defn:bg-geometry}, the domains of
Setup~\ref{app:setup:domains}, and the finite regular corrections
of Theorem~\ref{app:ch_regular:thm:quantitative} at the immersion $F$.
\begin{enumerate}
\item \emph{Coefficients.} Assume
\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 index $r_*$ exceeds the derivative shifts of the
coefficients in Theorem~\ref{app:ch_regular:thm:quantitative} and in the
normal reconstruction. By Corollary~\ref{app:cor:fixed-profile-transfer},
every reconstruction coefficient $c_F$ (the inverse metric
$\gamma(F)^{-1}$, the normal map $N_F$, the frame $Q_i(F)$ and
$\mathsf G_F^{-1}$), multiplied by any fixed cutoff, satisfies
\begin{equation}\label{app:ch_pairs:eq:pairs-lag-geometric-profile}
 \|D^rc_F\|_\infty\leq C_rT^r.
\end{equation}
\item \emph{Phases and frequencies.} We use the Hamilton phases $S_F$
and the groups of patches of
Definition~\ref{app:defn:exhaustive-pair-groups}. The samples are
indexed by $j=1,\dots,M$, with base covectors $\zeta_j$ and
frequencies $\tau_j^{-1}$ satisfying
$c\lambda\leq\tau_j^{-1}\leq C\lambda$, and two distinct samples in
the same group have patch parameters at distance at least $ch$. For an
ordered pair $(j,k)$ and a sign $\pm$ we write
$\theta_j\pm\theta_k=\rho_{jk}\varphi$, where the frequency
$\rho=\rho_{jk}$ is chosen as follows:
\[
 \begin{array}{c|c}
 \text{distinct same-group differences}&\rho=\lambda|z-z'|\\
 \text{sums, doubled phases, and either sign between groups}&\rho=\lambda.
 \end{array}
\]
Here $z,z'$ are the patch parameters of the two samples. The distinct
radial labels separate different groups even when their unscaled
directions agree. The equal-index differences $\theta_j-\theta_j=0$
form the Hermitian diagonal, which is excluded. All Fourier modulations
at the same patch parameter are combined in one field. Thus every
oscillatory quadratic term has
\[
 2\leq c\lambda h\leq\rho_{jk}\leq C\lambda.
\]
\item \emph{Phase bounds.} The joint phase bounds of
Proposition~\ref{app:ch_seed:prop:seed-actual-menu}, with the fixed
geometric shifts included in $r_*\geq8$, give
\[
 |D_x^rS_F|+|D_x^rD_\zeta S_F|+|D_x^rD_zS_F|\leq C_rT^{r-1}
 \qquad(r\geq1).
\]
The segment formula on each convex patch gives the same bound for the
divided differences of the phase in the patch parameter.
Proposition~\ref{app:ch_seed:prop:seed-actual-menu} and
Theorem~\ref{app:ch_regular:thm:no-return} place every normalized
covector $d\varphi$ of a sum or a difference in the fixed buffered
regular region. All source terms use the chart partition and the larger
sets where the cutoffs of that theorem equal one.
\item \emph{Cutoffs.} The source cutoff is
$\chi_{\rm src}=\chi_{{\rm pair},\rm src}$ of
Setup~\ref{app:setup:domains}; it is supported where $\chi_2=1$ and
equals one near $D$. It lies compactly inside $D_{\rm lag}$ and is
distinct from the source cutoff of the primary waves in
Section~\ref{app:sec:ch_lag}. All measurement and output supports lie in
the common charts of validity.
\item \emph{Amplitudes.} Let $W_j$ be the complete complex fields,
with
\begin{equation}\label{app:ch_pairs:eq:pairs-lag-demod}
 dW_j=e^{\ii\theta_j}A_j,\qquad
 \theta_j=S_F(x,\zeta_j)/\tau_j.
\end{equation}
After a fixed cutoff equal to one on every quadratic 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}
\end{enumerate}
For the final measures of Proposition~\ref{main:prop:mean},
\eqref{main:eq:mean-final-amplitude} proves this amplitude bound with
$K_{\rm env}=C_*R$ on the full source support; along the iteration the
coefficient bounds are verified in Proposition~\ref{app:prop:fixed-high}.
\end{setup}

For ambient derivative matrices we write
\[
 \mathsf B(A,B)=\tfrac12(A^TB+B^TA).
\]
This is a symmetric bilinear form with values in symmetric matrices,
and $\overline{\mathsf B(A,B)}=\mathsf B(\overline A,\overline B)$.

\subsection{Quadratic expansion and its correction}
\label{app:ch_pairs:sec:pairs-sum}
The correction below is linear in the coefficients of the quadratic
expansion. The Hermitian diagonal term is handled by the correction of
the mean.

\begin{lem}[The quadratic expansion]\label{app:ch_pairs:lem:pairs-ledger}
Let $b>0$, $\mu_j\geq0$, $dW_j=e^{\ii\theta_j}A_j$ with real phases
$\theta_j$, and set
\[
 P=b\sqrt2\sum_{j=1}^M\sqrt{\mu_j}\Re W_j,
 \qquad H_{\rm self}=b^2\sum_j\mu_j\Re(A_j^*A_j).
\]
For ordered pairs $(j,k)$ put
\[
 q_{jk}^+=\Re\bigl(e^{\ii(\theta_j+\theta_k)}\mathsf B(A_j,A_k)\bigr),
 \qquad
 q_{jk}^-=\Re\bigl(e^{\ii(\theta_j-\theta_k)}\mathsf B(A_j,\overline{A_k})\bigr).
\]
Then
\begin{equation}\label{app:ch_pairs:eq:pairs-exact-offself}
 Q_{\rm off}:=(dP)^TdP-H_{\rm self}
 =b^2\Bigl[\sum_{j,k}\sqrt{\mu_j\mu_k}\,q_{jk}^+
       +\sum_{j\ne k}\sqrt{\mu_j\mu_k}\,q_{jk}^-\Bigr].
\end{equation}
The first sum includes the doubled phases $j=k$. The omitted diagonal
of the differences is the complete Hermitian tensor, including every
Fourier modulation and every correction contained in the fields $W_j$.
\end{lem}

\begin{proof}
Let $D,E$ be complex $N\times n$ matrices. Since
$\Re D=\tfrac12(D+\overline D)$ and $\mathsf B$ is bilinear,
\[
 4\mathsf B(\Re D,\Re E)
 =\mathsf B(D,E)+\mathsf B(\overline D,\overline E)
  +\mathsf B(D,\overline E)+\mathsf B(\overline D,E).
\]
The first two terms are complex conjugate, and so are the last two.
Hence
\begin{equation}\label{new:eq:real-part-B}
 2\mathsf B(\Re D,\Re E)
 =\Re\mathsf B(D,E)+\Re\mathsf B(D,\overline E).
\end{equation}
Moreover,
\[
 \mathsf B(D,\overline D)
 =\tfrac12\bigl(D^T\overline D+\overline D{}^TD\bigr)
 =\tfrac12\bigl(\overline{D^*D}+D^*D\bigr)=\Re(D^*D).
\]
Now put $D_j=dW_j=e^{\ii\theta_j}A_j$. Then
$dP=b\sqrt2\sum_j\sqrt{\mu_j}\Re D_j$, and since the double sum is
symmetric in $(j,k)$,
\[
 (dP)^TdP=2b^2\sum_{j,k}\sqrt{\mu_j\mu_k}\,(\Re D_j)^T\Re D_k
 =2b^2\sum_{j,k}\sqrt{\mu_j\mu_k}\,\mathsf B(\Re D_j,\Re D_k).
\]
By \eqref{new:eq:real-part-B} and the reality of the phases,
\[
 2\mathsf B(\Re D_j,\Re D_k)
 =\Re\bigl(e^{\ii(\theta_j+\theta_k)}\mathsf B(A_j,A_k)\bigr)
 +\Re\bigl(e^{\ii(\theta_j-\theta_k)}\mathsf B(A_j,\overline{A_k})\bigr)
 =q_{jk}^++q_{jk}^-.
\]
For $j=k$ the second term is
$q_{jj}^-=\mathsf B(A_j,\overline{A_j})=\Re(A_j^*A_j)$. Summing over
$j$ with the weights $b^2\mu_j$ gives exactly $H_{\rm self}$, and
subtracting it proves \eqref{app:ch_pairs:eq:pairs-exact-offself}.
\end{proof}

With $W_j=Z_\nu$ and $\vartheta=\theta_j$, the case $M=1$, $b\sqrt{\mu_1}=1$
of the lemma is \eqref{main:eq:single-wave-mean}, since
$\mathsf B(A,A)=A^TA$.

\begin{prop}[Correction of all oscillatory quadratic terms]
\label{app:ch_pairs:prop:pairs-lag-sum}
We work under the assumptions of Setup~\ref{app:ch_pairs:setup:pairs-lag}
and assume $\sum_j\mu_j\leq C_\mu$. Let $N_F$ and $V_F=\mathcal V_F$ be
the reconstruction maps of Definition~\ref{app:ch_waves:defn:geometry}.
Given finite $s$ and $Q$, choose one depth $N=N(s,Q)$ in the real
operator $\mathcal D_{F,N}$, that is, the operator $\mathcal D_{a,N}$ of
Theorem~\ref{app:ch_regular:thm:quantitative} for the coefficients of
$\cP_F$, with all its fixed cutoffs. Let $Q_{\rm off}$ be the exact
tensor \eqref{app:ch_pairs:eq:pairs-exact-offself}, put
\[
 q=\chi_{\rm src}Q_{\rm off},\qquad
 h_i=-\tfrac12Q_i(F):q,
 \qquad u=\mathcal D_{F,N}h,
\]
and define the real field
\begin{equation}\label{app:ch_pairs:eq:pairs-weighted-U}
 S_{\rm lag}=V_F(u)+\tfrac12N_Fq.
\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^r(W)}
 &\leq C_{r,N}b^2\mathcal W_\mu \lambda^{r+1}
       \qquad(r\geq0).
 \label{app:ch_pairs:eq:pairs-lag-sum-size}
\end{align}
For a fixed physical cutoff $\chi_{\rm out}$, equal to one near $D$ and
supported in the observation chart, the same size bound holds for
$\chi_{\rm out}S_{\rm lag}$ on $\R^n$.
\end{prop}

The depth $N$ may depend on $s$ and $Q$; the additional power of
$\lambda$ in \eqref{app:ch_pairs:eq:pairs-lag-sum-size} and all
derivative shifts are fixed. All doubled phases and all signed terms
of distinct samples are included, and the complete Hermitian diagonal
tensor is excluded. The number of samples enters only through
$\mathcal W_\mu\leq MC_\mu$, which follows from the Cauchy--Schwarz
inequality $(\sum_j\sqrt{\mu_j})^2\leq M\sum_j\mu_j$.

\begin{proof}
\medskip\noindent\emph{Step 1 (One signed source).}
Fix an ordered pair $(j,k)$ and a sign, and apply the construction in
the statement to the single source $q=\chi_{\rm src}q_{jk}^{\pm}$ in
place of $\chi_{\rm src}Q_{\rm off}$. Denote the resulting reduced and
ambient fields by $u_{jk}^{\pm}$ and $U_{jk}^{\pm}$. Write the complex
product defining $q_{jk}^{\pm}$ as $e^{\ii\rho\varphi}a$, with
$\rho=\rho_{jk}$ as in Setup~\ref{app:ch_pairs:setup:pairs-lag},
$\rho\varphi=\theta_j\pm\theta_k$, and $a=\mathsf B(A_j,A_k)$ or
$a=\mathsf B(A_j,\overline{A_k})$. The reduced source is
\[
 h_{jk,i}^{\pm}=-\tfrac12Q_i(F):\chi_{\rm src}q_{jk}^{\pm}
 =\Re\bigl(e^{\ii\rho\varphi}a_{jk,i}\bigr),\qquad
 a_{jk,i}=-\tfrac12\chi_{\rm src}\,Q_i(F):a,
\]
because $Q_i(F)$ and $\chi_{\rm src}$ are real. By the Leibniz rule,
\eqref{app:ch_pairs:eq:pairs-lag-geometric-profile} and
\eqref{app:ch_pairs:eq:pairs-lag-envelope},
\[
 \|a_{jk}\|_{C^s}+\|a_{jk}\|_{H^s}
 \leq C_s\sum_{s_1+s_2+s_3=s}T^{s_1}K_{\rm env}^{s_2+s_3}
 \leq C_sL_{\rm pair}^s,
\]
where the $H^s$ bound follows from the $C^s$ bound and the fixed
compact support. Only the total derivative order enters.

\medskip\noindent\emph{Step 2 (Fourier localization).}
By Setup~\ref{app:ch_pairs:setup:pairs-lag}(iii), the normalized phase
satisfies $|D^r\varphi|\leq C_rL_{\rm pair}^{r-1}$ for $r\geq2$, and
$d\varphi$ takes values in a fixed compact set of regular covectors,
in particular $c\leq|d\varphi|\leq C$. Choose a fixed compact annular
Fourier mask containing a neighborhood of this compact set and of its
antipodal set, and let $\Pi_{0,\rho}$ be the corresponding rescaled
multiplier of Theorem~\ref{app:ch_regular:thm:quantitative}. On the
complement of the rescaled mask,
$|\rho\,d\varphi-\xi|\geq c(\rho+|\xi|)$. The general form of
Proposition~\ref{app:ch_osc:prop:osc-source-shell}, applied with
amplitude constant $A=1$ and scale $L=L_{\rm pair}$, gives for every
$m\geq0$ and every integer $k>m+n/2$
\[
 \|(I-\Pi_{0,\rho})(e^{\ii\rho\varphi}a_{jk})\|_{H^m}
 \leq C_{m,k}\rho^{m+n/2}(L_{\rm pair}/\rho)^k
 \leq C_{m,k}\rho^{m+n/2-\sigma k},
\]
where the last step uses $L_{\rm pair}\leq\rho^{1-\sigma}$. Choosing
$k\geq(m+n/2+Q')/\sigma$ gives the decay $\rho^{-Q'}$ for any requested
$Q'$. Conjugation gives the same estimate for the term with the
opposite phase. Hence $h_{jk}^{\pm}$ satisfies the hypotheses
\eqref{app:ch_regular:eq:source-ltwo}--\eqref{app:ch_regular:eq:source-tail}
of Theorem~\ref{app:ch_regular:thm:quantitative} with $H_\rho$ equal to
a fixed constant. The scale condition
\eqref{app:ch_regular:eq:slow-versus-frequency} holds because
$1\leq T\leq L_{\rm pair}\leq\rho^{1-\sigma}$, and the sources are
supported where $\chi_2=1$. The chart source sets and complete flowouts
are those of Setup~\ref{app:ch_pairs:setup:pairs-lag}.

\medskip\noindent\emph{Step 3 (Approximate solution and normal
reconstruction).}
By \eqref{main:eq:completion-error}, applied to the tensor $-q$, the
field $U=V_F(u)+\tfrac12N_Fq=dF\,\gamma(F)^{-1}u+N_F(\sym\nabla u+\tfrac12q)$
satisfies the exact identity
\begin{equation}\label{new:eq:pair-completion}
 D\gamma_FU+q
 =2\sum_{i,l}Q_i(F)(\mathsf G_F^{-1})_{il}\,(\cP_Fu-h)_l,
 \qquad h_l=-\tfrac12Q_l(F):q.
\end{equation}
Theorem~\ref{app:ch_regular:thm:quantitative}, with a depth
$N\geq s+Q'$, gives
$\|\cP_Fu_{jk}^{\pm}-h_{jk}^{\pm}\|_{H^s(U_{\rm core})}\leq C_{N,s,Q'}\rho^{-Q'}$.
Multiplication by the coefficients in \eqref{new:eq:pair-completion},
whose $C^s$ norms are at most $C_sT^s\leq C_s\rho^s$, costs at most a
factor $C_s\rho^s$ in $H^s(D)$. Taking $Q'=Q+s$ and using $D\subset
U_{\rm core}$ and $\chi_{\rm src}=1$ near $D$, we obtain
\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}
One such depth works for the compact sets of regular covectors and the
finite list of charts; it is independent of the frequency $\rho_{jk}$.

\medskip\noindent\emph{Step 4 (The exact weighted sum).}
The operator $\mathcal D_{F,N}$, the frame and the cutoffs are the same
for all pairs. By linearity, $S_{\rm lag}$ is the sum of the fields
$U_{jk}^{\pm}$ with the coefficients of
\eqref{app:ch_pairs:eq:pairs-exact-offself}, and
$D\gamma_FS_{\rm lag}+Q_{\rm off}$ is, on $D$, the same combination of
the left-hand sides of \eqref{app:ch_pairs:eq:pairs-lag-one-error}.
The total absolute coefficient is
\[
 b^2\Bigl[\sum_{j,k}\sqrt{\mu_j\mu_k}+\sum_{j\ne k}\sqrt{\mu_j\mu_k}\Bigr]
 \leq2b^2\Bigl(\sum_j\sqrt{\mu_j}\Bigr)^2=2b^2\mathcal W_\mu
 \leq2b^2MC_\mu .
\]
Since $\rho_{jk}\geq c\lambda h$ for every oscillatory term,
\eqref{app:ch_pairs:eq:pairs-lag-one-error} gives
\eqref{app:ch_pairs:eq:pairs-lag-sum-error}, with $c^{-Q}$ absorbed in
the constant.

\medskip\noindent\emph{Step 5 (Size bounds).}
Put $\mathfrak A=b^2\mathcal W_\mu$. Each derivative of
$e^{\ii\rho\varphi}$ costs at most $C(\rho+L_{\rm pair})\leq C\lambda$,
and each derivative of $a_{jk}$ costs at most $CL_{\rm pair}\leq C\lambda$.
The Leibniz rule and the coefficient sum of Step 4 therefore give
\[
 \|q\|_{H^s}+\|h\|_{H^s}\leq C_s\mathfrak A\lambda^s .
\]
For each signed source, \eqref{app:ch_regular:eq:solution-profile}
gives $\|u_{jk}^{\pm}\|_{H^r}\leq C_{r,N}\rho_{jk}^r$. Since
$\rho_{jk}\leq C\lambda$, the same coefficient sum gives
$\|u\|_{H^r}\leq C_{r,N}\mathfrak A\lambda^r$. The field $V_F(u)$
contains $u$ and one derivative of $u$, multiplied by coefficients
whose derivatives of order $i$ are bounded by $C_iT^i\leq C_i\lambda^i$;
the Leibniz rule gives
$\|V_F(u)\|_{H^r(W)}\leq C_{r,N}\mathfrak A\lambda^{r+1}$. Similarly
$\|N_Fq\|_{H^r(W)}\leq C_r\mathfrak A\lambda^r$. This proves
\eqref{app:ch_pairs:eq:pairs-lag-sum-size}; multiplication by the fixed
cutoff $\chi_{\rm out}$ changes only the constants. The operators are
real and uniform on the fixed family of charts.
\end{proof}

\subsection{The product correction for the sampled waves}
\label{new:sec:product-correction}
We now apply Proposition~\ref{app:ch_pairs:prop:pairs-lag-sum} to the
waves constructed in Proposition~\ref{main:prop:mean}.

\begin{prop}[Product correction]\label{main:prop:pairs}
Let $Z_\nu$ and $\mu_\nu$ be the waves and weights of
Proposition~\ref{main:prop:mean}, and let $P$, $Q_{\rm off}$ be given by
\eqref{main:eq:primary} and \eqref{main:eq:self-off}. Assume that, for a
fixed $\sigma>0$ and a fixed amplitude constant $C_*$,
\begin{equation}\label{main:eq:pair-window}
 C_*R\leq(\lambda h)^{1-\sigma}.
\end{equation}
Then, for every finite $q\geq1$, there is a real correction $S$,
constructed at $F$ from the exact signed products in $Q_{\mathrm{off}}$,
such that, after its fixed physical cutoff equal to one near $D$,
\begin{align}
 \|D\gamma_FS+Q_{\mathrm{off}}\|_{C^0(D)}
    &\leq C_qb^2M(\lambda h)^{-q},
        \label{main:eq:pair-error}\\
 \|S\|_{H^r}&\leq C_rb^2M\lambda^{r+1}
       \qquad(r\geq0).
       \label{main:eq:pair-high-size}
\end{align}
The finite depth of the construction depends on $q$; the same field
satisfies the size bound at every $r$, with the same additional power
of $\lambda$.
\end{prop}

The mechanism is the following. The correction solves the linearized
equation along the full characteristic segments of
Proposition~\ref{main:prop:patches}, using the complete amplitudes of
the corrected waves $Z_\nu$, whose derivative scale is $R>T_0$. Each
step of the finite Neumann series in
Theorem~\ref{app:ch_regular:thm:quantitative} gains a factor
$C_*R/\varrho$, with $\varrho\geq c\lambda h$, and the strict power in
\eqref{main:eq:pair-window} turns this into arbitrary finite decay.

\begin{proof}[Proof of Proposition~\ref{main:prop:pairs}]
We apply Lemma~\ref{app:ch_pairs:lem:pairs-ledger} and
Proposition~\ref{app:ch_pairs:prop:pairs-lag-sum} with the samples
$j=\nu$, the fields $W_\nu=Z_\nu$, the phases
$\theta_\nu=\lambda_\nu\phi_\nu=S_F(x,\zeta_\nu)/\tau_\nu$, and the
amplitudes $A_\nu=e^{-\ii\theta_\nu}dZ_\nu$. Since $\theta_\nu$ is real,
$(dZ_\nu)^*dZ_\nu=A_\nu^*A_\nu$, so the fields $P$, $H_{\rm self}$ and
$Q_{\rm off}$ of \eqref{main:eq:primary}--\eqref{main:eq:self-off} are
those of Lemma~\ref{app:ch_pairs:lem:pairs-ledger}. By
\eqref{main:eq:sampling}, $\mu_\nu>0$ and $\sum_\nu\mu_\nu\leq C=:C_\mu$.

We verify the hypotheses of Setup~\ref{app:ch_pairs:setup:pairs-lag}.
The coefficient bound \eqref{app:ch_pairs:eq:pairs-lag-map-profile} is
\eqref{main:eq:lag-hypothesis} with $T=T_0=R^t$. The phases, groups and
frequencies are those of the sampling in Proposition~\ref{main:prop:mean};
by Proposition~\ref{main:prop:patches}(iii), every oscillatory term has
frequency $c\lambda h\leq\varrho\leq C\lambda$, and $c\lambda h\geq2$ for
large $R$ because $a>\beta$. For the amplitudes,
\eqref{main:eq:mean-final-amplitude} gives
$\|\chi_{\rm pair}A_\nu\|_{C^r}\leq C_rR^r$, where $\chi_{\rm pair}$
equals one on every quadratic source (Setup~\ref{app:setup:domains}).
Thus \eqref{app:ch_pairs:eq:pairs-lag-envelope} holds with
$K_{\rm env}=C_0R$ for any fixed $C_0\geq2$. Since $t<1$, we have
$T_0=R^t\leq C_0R$, so $L_{\rm pair}=\max(K_{\rm env},T_0)=C_0R$. For
every oscillatory term, \eqref{main:eq:pair-window} gives
\[
 \rho_{jk}^{1-\sigma}\geq(c\lambda h)^{1-\sigma}
 =c^{1-\sigma}(\lambda h)^{1-\sigma}\geq c^{1-\sigma}C_*R
 \geq C_0R=L_{\rm pair},
\]
provided the amplitude constant satisfies $C_*\geq c^{\sigma-1}C_0$;
this is how the constant $C_*$ in \eqref{main:eq:pair-window} is
understood. Hence all hypotheses of
Setup~\ref{app:ch_pairs:setup:pairs-lag} hold.

Apply Proposition~\ref{app:ch_pairs:prop:pairs-lag-sum} with
$s=\lfloor n/2\rfloor+1$ and $Q=q$, and put
$S=\chi_{\rm out}S_{\rm lag}$. Since $\chi_{\rm out}=1$ near $D$ and
$D\gamma_F$ is a first-order differential operator,
$D\gamma_FS=D\gamma_FS_{\rm lag}$ on $D$. Since $s>n/2$ and $D$ is a
ball, the Sobolev embedding $H^s(D)\subset C^0(D)$ and
\eqref{app:ch_pairs:eq:pairs-lag-sum-error} give
\[
 \|D\gamma_FS+Q_{\rm off}\|_{C^0(D)}
 \leq C\|D\gamma_FS_{\rm lag}+Q_{\rm off}\|_{H^s(D)}
 \leq C_qb^2\mathcal W_\mu(\lambda h)^{-q}
 \leq C_qb^2M(\lambda h)^{-q},
\]
which is \eqref{main:eq:pair-error}. The bound
\eqref{main:eq:pair-high-size} follows from the cut form of
\eqref{app:ch_pairs:eq:pairs-lag-sum-size} and $\mathcal W_\mu\leq MC_\mu$.
The depth $N=N(s,q)$ depends only on $q$, so the constants $C_r$ may
depend on $q$.
\end{proof}


\section{The metric correction at one step}\label{main:sec:increment}

In this section we combine the primary waves, the correction of the mean
and the correction of the products into one increment. The input is an
immersion $F$ with derivative bounds \eqref{main:eq:lag-hypothesis}, a
metric defect of size $\delta^2$ on $D$, a frequency $R$ and an accuracy
$Q$. The output is a real increment $U$ with exact Fourier support in
$\{|\xi|\leq CR^{b_*}\}$, an $L^2$ bound with a fixed loss, and new
defect $O(\delta R^{-Q}+\delta^3R^{B_3})$
(Theorem~\ref{main:thm:step}). The exponents $b_*$, $B_0$, $B_3$ and the
support constant do not depend on $Q$; this is what makes the iteration
of Sections~\ref{main:sec:history}--\ref{main:sec:convergence} possible.

\subsection{Smoothing the defect}
The smoothing operator acts on the defect on $D$ and gives two separate
bounds for the approximation error: one by the size of the defect, and
one by its derivatives.

\begin{lemma}\label{main:lem:smoothing}
For each integer $m\geq1$, let $S_{R,m}$ be the real smoothing
operator of Lemma~\ref{app:ch_regular_base:lem:ball-extension} for the
ball $\Omega=D$, with output supported in a fixed compact subset of
$W$. Fix a smooth target tensor $g$ on $W$. Suppose that
$T\geq1$, $0<\delta\leq1$, and
\[
 \|dF\|_{C^r(W)}\leq C_rT^r,\qquad
 \|g-\gamma(F)\|_{C^0(D)}\leq\delta^2.
\]
Put $e=g-\gamma(F)$. For all sufficiently large $R$, the tensor
$e_R=S_{R,m}(e|_D)$ is smooth and satisfies, for every integer $r\geq0$,
\begin{align}
 \|e_R\|_{C^r}+\|e_R\|_{H^r}&\leq C_{r,m}\delta^2R^r,
          \label{main:eq:smoothing-low}\\
 \|e-e_R\|_{C^0(D)}&\leq C_m\delta^2,
          \label{main:eq:smoothing-small}\\
 \|e-e_R\|_{C^0(D)}&\leq C_m(T/R)^m.
          \label{main:eq:smoothing-high}
\end{align}
The constants depend on the constants $C_r$ above and on finitely many
seminorms of $g$. If $T\leq R^t$ with $0<t<1$, then for every finite
$Q$ the choice $(1-t)m>2Q$ gives
\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}
By \eqref{app:ch_regular_base:eq:ball-extension-sobolev} and
$\|e\|_{C^0(D)}\leq\delta^2$,
\[
 \|e_R\|_{C^r}+\|e_R\|_{H^r}\leq C_{r,m}R^r\|e\|_{C^0(D)}
 \leq C_{r,m}\delta^2R^r,
\]
which is \eqref{main:eq:smoothing-low}. Taking $r=0$ and using the
triangle inequality,
$\|e-e_R\|_{C^0(D)}\leq\|e\|_{C^0(D)}+\|e_R\|_{C^0}\leq(1+C_{0,m})\delta^2$,
which is \eqref{main:eq:smoothing-small}. For
\eqref{main:eq:smoothing-high}, the Leibniz rule applied to
$\gamma(F)=(dF)^TdF$ gives
\[
 \|D^m\gamma(F)\|_{C^0(D)}
 \leq\sum_{i=0}^m\binom mi\|D^idF\|_{C^0}\|D^{m-i}dF\|_{C^0}
 \leq C_mT^m,
\]
and hence $\|e\|_{C^m(D)}\leq\|g\|_{C^m(D)}+C_mT^m\leq C_mT^m$, since
$T\geq1$. Now \eqref{app:ch_regular_base:eq:ball-extension-continuous}
gives $\|e-e_R\|_{C^0(D)}\leq C_mR^{-m}T^m$.

Finally, if $T\leq R^t$, then the minimum of the two bounds
\eqref{main:eq:smoothing-small} and \eqref{main:eq:smoothing-high} is at
most their geometric mean:
\[
 \|e-e_R\|_{C^0(D)}
 \leq\bigl(C_m\delta^2\cdot C_m(T/R)^m\bigr)^{1/2}
 \leq C_m\delta R^{-(1-t)m/2}\leq C_m\delta R^{-Q},
\]
because $(1-t)m/2>Q$ and $R\geq1$.
\end{proof}

In particular, since $b^2=\delta^2R^{2A}$, the normalized tensor $e_R/b^2$
has $C^r$ and $H^r$ norms at most $C_{r,m}R^{r-2A}$. The factor $\delta$
in \eqref{main:eq:smoothing-combined} comes from combining the two
bounds, not from the high-order bound alone. The defect $e$ may have
either sign.

\subsection{Two Fourier inequalities}
We use two elementary consequences of the Fourier inversion formula and
the Cauchy--Schwarz inequality.

\begin{lem}\label{new:lem:bernstein}
Let $m_0=\lfloor n/2\rfloor+2$.
\begin{enumerate}
\item For every $f\in H^{m_0}(\R^n)$,
 $\|f\|_{C^1(\R^n)}\leq C\|f\|_{H^{m_0}}$.
\item \textup{(Bernstein inequality.)} If $f\in L^2(\R^n)$ and
 $\supp\widehat f\subset\{|\xi|\leq\Lambda\}$ with $\Lambda\geq1$, then
 for every integer $r\geq0$
 \[
  \|f\|_{C^r(\R^n)}\leq C_r\Lambda^{r+n/2}\|f\|_{L^2}.
 \]
\end{enumerate}
\end{lem}

\begin{proof}
For a multi-index $\alpha$, the inversion formula gives
$|\partial^\alpha f(x)|\leq\int|\xi|^{|\alpha|}|\widehat f(\xi)|\,d\xi$.
For (i) and $|\alpha|\leq1$, the Cauchy--Schwarz inequality bounds this
by
\[
 \Bigl(\int\langle\xi\rangle^{2(|\alpha|-m_0)}d\xi\Bigr)^{1/2}
 \Bigl(\int\langle\xi\rangle^{2m_0}|\widehat f|^2d\xi\Bigr)^{1/2}
 \leq C\|f\|_{H^{m_0}},
\]
since $2(m_0-1)=2\lfloor n/2\rfloor+2>n$. For (ii), with $|\alpha|\leq r$,
\[
 |\partial^\alpha f(x)|\leq\Lambda^{|\alpha|}
 \bigl|\{|\xi|\leq\Lambda\}\bigr|^{1/2}\|\widehat f\|_{L^2}
 =C\Lambda^{|\alpha|+n/2}\|f\|_{L^2},
\]
by the Plancherel identity $\|f\|_2^2=(2\pi)^n\|\widehat f\|_2^2$.
\end{proof}

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

\begin{theorem}[Correction at one step]\label{main:thm:step}
Fix the reference neighborhood, the domains, and the correction
operators constructed above. Let $F$ be an immersion in this
neighborhood, and assume the hypotheses of
Propositions~\ref{main:prop:current-waves}, \ref{main:prop:mean}
and~\ref{main:prop:pairs}. Suppose that
\[
 \|g-\gamma(F)\|_{C^0(D)}\leq\delta^2,\qquad
 T_0=R^t,
\]
with $0<t<1$ and $0<\delta\leq1$, and that \eqref{main:eq:lag-hypothesis}
and \eqref{main:eq:pair-window} hold. Put $m_0=\lfloor n/2\rfloor+2$ and
\[
 B_0=2A+1+\epsilon+a,\qquad B_3=2B_0+2am_0.
\]
For every finite $Q$ and all sufficiently large $R$, there is one real
global increment $U=U_Q$ satisfying
\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,
\begin{equation}\label{main:eq:step-Cr}
 \|U\|_{C^r(\R^n)}\leq C_{r,Q}\delta R^{B_0+b_*(r+n/2)}.
\end{equation}
The frequency exponents and the support constant $C$ in
\eqref{main:eq:step-band} are independent of $Q$.
\end{theorem}

The exponent $B_0$ is the size at order zero of the quadratic
correction: by \eqref{main:eq:pair-high-size}, \eqref{main:eq:sampling}
and $b=\delta R^A$,
\[
 b^2M\lambda\leq C\delta^2R^{2A}\cdot R^{1+\epsilon}\cdot R^a
 =C\delta^2R^{B_0}.
\]
The exponent $B_3$ arises from the product of the $C^1$ norms of the
primary field and of the quadratic correction, each estimated through
Lemma~\ref{new:lem:bernstein}(i) by $\delta^{\ell}R^{B_0}\lambda^{m_0}$
with $\ell=1,2$; see Step~5 below. The first term in
\eqref{main:eq:step-error} comes from the finite depths of the
constructions and from Fourier tails; the second is the nonlinear
interaction between the primary field and the quadratic correction.

\begin{proof}
\medskip\noindent\emph{Step 1 (Choice of the finite accuracies).}
Fix $Q$. Choose integers $m$, $Q_p$, $Q_m$, $q$ and $k$ with
\begin{equation}\label{new:eq:step-accuracies}
\begin{gathered}
 (1-t)m>2Q,\qquad aQ_p>Q+A+\tfrac{1+\epsilon}2,\qquad
 Q_m>Q+2A,\\
 (a-\beta)q>Q+2A+1+\epsilon,\qquad a\zeta k>Q+B_3 .
\end{gathered}
\end{equation}
These depend only on $Q$ and the fixed exponents. We use $m$ for the
smoothing $e_R=S_{R,m}(e|_D)$, the depth $Q_p$ in
Proposition~\ref{main:prop:current-waves} (the wave families are
chosen before sampling), the accuracy $Q_m$ in
Proposition~\ref{main:prop:mean}, the accuracy $q$ in
Proposition~\ref{main:prop:pairs}, and $k$ in Step~6. Then we choose the
sample on the fixed input band, the measures $m_\nu$ and the field $V$
of Proposition~\ref{main:prop:mean}, the primary field $P$ of
\eqref{main:eq:primary} and the field $S$ of
Proposition~\ref{main:prop:pairs}. All these fields carry their fixed
physical cutoffs, which equal one near $D$. Put
\[
 W_2=b^2V+S,\qquad U=\Pi(P+W_2),
\]
where $\Pi$ is the Fourier multiplier with symbol
$\pi_0(\xi/\lambda^{1+\zeta})$, and $\pi_0$ is a fixed smooth real even
function equal to one for $|\xi|\leq C_1$ and to zero for $|\xi|\geq
C_2$. Note that $\lambda^{1+\zeta}=R^{a(1+\zeta)}=R^{b_*}$. Since the
symbol is real and even, $\Pi$ maps real fields to real fields. Finally
$R$ is taken large enough for all propositions at these accuracies.

\medskip\noindent\emph{Step 2 (The exact identity).}
Put $Y=P+W_2$. Since the physical cutoffs equal one near $D$ and the
metric involves only first derivatives, we may compute on $D$ with the
uncut fields. By \eqref{main:eq:metric-expansion},
\[
 \gamma(F+Y)=\gamma(F)+D\gamma_FP+b^2D\gamma_FV+D\gamma_FS+(dY)^TdY,
\]
and
\[
 (dY)^TdY=(dP)^TdP+2\sym\bigl((dP)^TdW_2\bigr)+(dW_2)^TdW_2 .
\]
By \eqref{main:eq:self-off}, $(dP)^TdP=H_{\rm self}+Q_{\rm off}$, and by
\eqref{main:eq:mean-equation},
\[
 b^2E_{\rm mean}=b^2D\gamma_FV+H_{\rm self}-e_R .
\]
Substituting these identities and $\gamma(F)-g=-e$, we obtain on $D$ 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}

\medskip\noindent\emph{Step 3 (The linear terms).}
We bound the first four terms of \eqref{main:eq:metric-ledger} by
$C_Q\delta R^{-Q}$. The first is bounded by
Lemma~\ref{main:lem:smoothing}, applied with $T=T_0=R^t$; its
hypothesis $\|dF\|_{C^r(W)}\leq C_rT_0^r$ follows from
\eqref{main:eq:lag-hypothesis}, since
$(r+1-r_*)_+\leq r$ for $r_*\geq1$.

For the second term, \eqref{main:eq:sampling} and the Cauchy--Schwarz
inequality give
\[
 \sum_\nu\sqrt{\mu_\nu}\leq
       \Bigl(M\sum_\nu\mu_\nu\Bigr)^{1/2}
       \leq CR^{(1+\epsilon)/2}.
\]
Since $D\gamma_F$ is real, $D\gamma_FP=b\sqrt2\sum_\nu\sqrt{\mu_\nu}
\Re(D\gamma_FZ_\nu)$. The measure bound $\|m_\nu\|_{\TV}\leq C$ of
Proposition~\ref{main:prop:mean} and \eqref{main:eq:lag-strain} with
depth $Q_p$ (recall $D\subset D_{\rm lag}$ and $\lambda=R^a$) give
\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}
 =C_Q\delta R^{A+(1+\epsilon)/2-aQ_p}\leq C_Q\delta R^{-Q},
\end{equation}
by the choice of $Q_p$ in \eqref{new:eq:step-accuracies}.

For the third term, Proposition~\ref{main:prop:mean} at accuracy $Q_m$
gives
\[
 b^2\|E_{\mathrm{mean}}\|_{C^0(D)}\leq C_Q\delta^2R^{2A-Q_m}
 \leq C_Q\delta R^{-Q},
\]
since $\delta^2\leq\delta$ and $Q_m>Q+2A$.

For the fourth term, Proposition~\ref{main:prop:pairs} at accuracy $q$,
together with $M\leq CR^{1+\epsilon}$ and $\lambda h=R^{a-\beta}$, gives
\[
 \|D\gamma_FS+Q_{\mathrm{off}}\|_{C^0(D)}
 \leq C_Q\delta^2R^{2A}R^{1+\epsilon}R^{-(a-\beta)q}
 \leq C_Q\delta R^{-Q},
\]
since $(a-\beta)q>Q+2A+1+\epsilon$.

\medskip\noindent\emph{Step 4 (Sobolev bounds for $P$ and $W_2$).}
We show that
\begin{equation}\label{main:eq:joint-physical-profile}
 \|P\|_{H^r}\leq C_r\delta R^{B_0}\lambda^r,\qquad
 \|W_2\|_{H^r}\leq C_r\delta^2R^{B_0}\lambda^r .
\end{equation}
For $P$, put $z_\nu=\chi_{\rm core}e^{-\ii\lambda_\nu\phi_\nu}Z_\nu$.
By \eqref{main:eq:lag-full-envelope}, with an amplitude scale
$K\leq\lambda^{1-\nu_0}\leq\lambda$, we have
$\|z_\nu\|_{C^r}\leq C_r\lambda^{-1}K^r\leq C_r\lambda^{r-1}$. Since
$\|d\phi_\nu\|_{C^r}\leq C_rT_0^r$ and $T_0\leq\lambda$, every term in
the expansion of a derivative of order $j$ of $e^{\ii\lambda_\nu\phi_\nu}$
is a product of factors $\lambda_\nu D^{i}\phi_\nu$ with total order $j$,
each bounded by $C\lambda T_0^{i-1}\leq C\lambda^i$. Hence
$\|\chi_{\rm core}Z_\nu\|_{C^r}\leq C_r\lambda^{r-1}$. The physical
cutoff of $P$ is supported in $D_{\rm lag}$, where $\chi_{\rm core}=1$ by
\eqref{main:eq:cutoff-chain}. Using the weight sum of Step~3 and the
fixed support,
\[
 \|P\|_{H^r}\leq C_rb\sum_\nu\sqrt{\mu_\nu}\,\lambda^{r-1}
 \leq C_r\delta R^{A+(1+\epsilon)/2}\lambda^{r-1}
 \leq C_r\delta R^{B_0}\lambda^r .
\]
For $W_2$, \eqref{main:eq:mean-size} and $R\leq\lambda$ give
$b^2\|V\|_{H^r}\leq C_r\delta^2R^{2A}R^r\leq C_r\delta^2R^{B_0}\lambda^r$,
and \eqref{main:eq:pair-high-size} gives
\[
 \|S\|_{H^r}\leq C_r\delta^2R^{2A}R^{1+\epsilon}\lambda^{r+1}
 =C_r\delta^2R^{2A+1+\epsilon+a}\lambda^r=C_r\delta^2R^{B_0}\lambda^r .
\]
In all cases each derivative costs at most a factor $\lambda=R^a$.

\medskip\noindent\emph{Step 5 (The quadratic terms).}
By Lemma~\ref{new:lem:bernstein}(i) and
\eqref{main:eq:joint-physical-profile} with $r=m_0$,
\[
 \|P\|_{C^1}\leq C\delta R^{B_0+am_0},\qquad
 \|W_2\|_{C^1}\leq C\delta^2R^{B_0+am_0}.
\]
Therefore, using $\delta\leq1$,
\[
 \bigl\|2\sym\bigl((dP)^TdW_2\bigr)\bigr\|_{C^0(D)}
 +\|(dW_2)^TdW_2\|_{C^0(D)}
 \leq C\bigl(\delta^3+\delta^4\bigr)R^{2B_0+2am_0}
 \leq C_Q\delta^3R^{B_3}.
\]
Together with Step~3, the identity \eqref{main:eq:metric-ledger} gives
\begin{equation}\label{new:eq:uncut-error}
 \|\gamma(F+P+W_2)-g\|_{C^0(D)}\leq C_Q\bigl(\delta R^{-Q}+\delta^3R^{B_3}\bigr).
\end{equation}

\medskip\noindent\emph{Step 6 (The Fourier truncation).}
Since $1-\pi_0(\xi/\lambda^{1+\zeta})$ vanishes for
$|\xi|\leq C_1\lambda^{1+\zeta}$, we have
$\langle\xi\rangle^{m_0}\leq(C_1\lambda^{1+\zeta})^{-k}\langle\xi\rangle^{m_0+k}$
on its support. By Lemma~\ref{new:lem:bernstein}(i),
\eqref{main:eq:joint-physical-profile} and $\delta\leq1$,
\[
\begin{split}
 \|(I-\Pi)(P+W_2)\|_{C^1}
 &\leq C\|(I-\Pi)(P+W_2)\|_{H^{m_0}}
 \leq C_k\lambda^{-(1+\zeta)k}\|P+W_2\|_{H^{m_0+k}}\\
 &\leq C_k\delta R^{B_0}\lambda^{m_0-\zeta k}.
\end{split}
\]
Thus $E_{\mathrm{cut}}=U-(P+W_2)=-(I-\Pi)(P+W_2)$ is small in $C^1$,
with a factor $\delta$. Put $G=F+P+W_2$. By \eqref{main:eq:metric-expansion},
\[
 \gamma(F+U)-\gamma(G)
 =D\gamma_{G}E_{\mathrm{cut}}+(dE_{\mathrm{cut}})^TdE_{\mathrm{cut}},
 \qquad D\gamma_GE=(dG)^TdE+(dE)^TdG .
\]
By Step~5, $\|G\|_{C^1(D)}\leq C+C\delta R^{B_0+am_0}\leq C_QR^{B_0+am_0}$.
Hence
\[
\begin{split}
 \|\gamma(F+U)-\gamma(G)\|_{C^0(D)}
 &\leq C_Q R^{B_0+am_0}\cdot\delta R^{B_0+am_0-a\zeta k}
 +C_Q\delta^2R^{2B_0+2am_0-2a\zeta k}\\
 &\leq C_Q\delta R^{B_3-a\zeta k}\leq C_Q\delta R^{-Q},
\end{split}
\]
since $a\zeta k>Q+B_3$. Combined with \eqref{new:eq:uncut-error}, this
proves \eqref{main:eq:step-error}.

\medskip\noindent\emph{Step 7 (The $L^2$, band and $C^r$ bounds).}
The multiplier $\Pi$ has norm at most $\sup|\pi_0|=1$ on $L^2$ (we
take $0\leq\pi_0\leq1$), so \eqref{main:eq:joint-physical-profile} at
$r=0$ gives $\|U\|_{L^2}\leq\|P\|_{L^2}+\|W_2\|_{L^2}\leq C_Q\delta
R^{B_0}$, which is \eqref{main:eq:step-L2}. The symbol of $\Pi$ vanishes
for $|\xi|\geq C_2\lambda^{1+\zeta}=C_2R^{b_*}$, which gives
\eqref{main:eq:step-band}; and $b_*<3/2$ by
\eqref{main:eq:fixed-outer-exponents}. Lemma~\ref{new:lem:bernstein}(ii)
with $\Lambda=C_2R^{b_*}$ gives
\[
 \|U\|_{C^r(\R^n)}\leq C_rR^{b_*(r+n/2)}\|U\|_{L^2}
 \leq C_{r,Q}\delta R^{B_0+b_*(r+n/2)},
\]
which is \eqref{main:eq:step-Cr}. The field $U$ is real because $P$, $V$
and $S$ are real and $\Pi$ preserves real fields.
\end{proof}


\section{Uniform estimates along the iteration}\label{main:sec:history}
The exact Fourier support of the increments gives the high-order bounds
on the iterates which are required in the hypotheses of
Theorem~\ref{main:thm:step}. We prove these bounds
(Lemma~\ref{main:lem:history}), transfer them to the coefficients of all
constructions (Proposition~\ref{app:prop:fixed-high}), and conclude that
the single-step theorem applies at every iterate with uniform constants
(Proposition~\ref{main:prop:current-scale}). The parameters are fixed in
Section~\ref{main:sec:scales}, and the iteration is carried out in
Section~\ref{main:sec:convergence}.

\begin{lemma}[Estimates for sums of Fourier-localized increments]\label{main:lem:history}
Fix an integer $r_*\geq1$ and numbers $1<b_*<v<3/2$, and put $t=b_*/v$.
For $R_0\geq2$ set $R_j=R_0^{v^j}$. Suppose that real global increments
$U_j$ satisfy, with integers $k_j\geq k_0$,
\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 $F_j=F_{\rm init}+\sum_{i<j}U_i$ on $W$, $T_j=R_j^t$ and
$b_\sharp=B_0+b_*(r_*+n/2)$. If $\mu=k_0-b_\sharp-1>0$, 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_*}(\R^n)}\longrightarrow0
                      \quad\text{as }R_0\longrightarrow\infty.
\end{equation}
Both assertions are uniform for a compact smooth family of initial maps
$F_{\rm init}$.
\end{lemma}

\begin{proof}
By the Bernstein inequality, Lemma~\ref{new:lem:bernstein}(ii), with
$\Lambda=CR_i^{b_*}$, and by \eqref{main:eq:history-data},
\begin{equation}\label{main:eq:history-bernstein}
 \|U_i\|_{C^{r_*}(\R^n)}\leq CR_i^{b_*(r_*+n/2)}R_i^{-k_i+B_0+1}
 =CR_i^{-k_i+b_\sharp+1}.
\end{equation}
The constant $C$ depends only on $n$, $r_*$ and the support constant.
Since $k_i\geq k_0$, the exponent is at most $-\mu$, so
$\|U_i\|_{C^{r_*}}\leq CR_0^{-\mu v^i}$. Bernoulli's inequality
$v^i\geq1+i(v-1)$ and $R_0>1$ give
$R_0^{-\mu v^i}\leq R_0^{-\mu}\bigl(R_0^{-\mu(v-1)}\bigr)^i$, and hence
\begin{equation}\label{new:eq:geometric-sum}
 \sum_{i\geq0}\|U_i\|_{C^{r_*}(\R^n)}
 \leq\frac{CR_0^{-\mu}}{1-R_0^{-\mu(v-1)}} .
\end{equation}
The right-hand side tends to zero as $R_0\to\infty$, which proves
\eqref{main:eq:total-small}.

For $r\leq r_*$, \eqref{new:eq:geometric-sum} gives
$\|F_j\|_{C^r(W)}\leq\|F_{\rm init}\|_{C^{r_*}(W)}+C$, uniformly in $j$.
Let $r>r_*$, and put $\Psi_j=\sum_{i<j}U_i$. For $i<j$ we have
$R_i^{b_*}\leq R_{j-1}^{b_*}=R_j^{b_*/v}=T_j$, so
$\supp\widehat\Psi_j\subset\{|\xi|\leq CT_j\}$. Fix
$\chi\in C_c^\infty(\R^n)$ equal to one on $\{|\xi|\leq C\}$, put
\[
 \kappa(x)=(2\pi)^{-n}\int e^{\ii x\cdot\xi}\chi(\xi)\dd\xi,
 \qquad \kappa_j(x)=T_j^n\kappa(T_jx).
\]
With the normalization of Notation~\ref{app:not:fourier} the convolution
theorem reads $\widehat{f*g}=(2\pi)^n\widehat f\,\widehat g$, and
$\widehat\kappa_j(\xi)=(2\pi)^{-n}\chi(\xi/T_j)$; hence
$\widehat{\kappa_j*\Psi_j}=\chi(\cdot/T_j)\widehat\Psi_j=\widehat\Psi_j$,
that is, $\Psi_j=\kappa_j*\Psi_j$. For
$|\alpha|=r$ write $\alpha=\beta+\gamma$ with $|\beta|=r_*$; then
$\partial^\alpha\Psi_j=(\partial^\gamma\kappa_j)*\partial^\beta\Psi_j$
and $\|\partial^\gamma\kappa_j\|_{L^1}=T_j^{r-r_*}\|\partial^\gamma\kappa\|_{L^1}$.
Young's inequality gives
\[
 \|\partial^\alpha\Psi_j\|_{L^\infty}
 \leq C_rT_j^{r-r_*}\|\Psi_j\|_{C^{r_*}}\leq C_rT_j^{r-r_*}.
\]
Since $F_{\rm init}$ is smooth on $\overline W$ and $T_j\geq1$, also
$\|F_{\rm init}\|_{C^r(W)}\leq C_r\leq C_rT_j^{r-r_*}$. This proves
\eqref{main:eq:history-current}. The initial map is used only on $W$,
and all constants depend on finitely many seminorms of $F_{\rm init}$,
which gives the uniformity.
\end{proof}

The derivative bounds \eqref{main:eq:history-current} for the iterates
imply the coefficient bounds used in
Sections~\ref{app:sec:ch_regular}--\ref{app:sec:ch_pairs}, uniformly along
such sequences.

\begin{prop}[Coefficient estimates along the iteration]
\label{app:prop:fixed-high}
Choose a fixed order $\bar r_0$ above the orders used in the constructions
and the source order $s_0$, and choose $r_*$ above $\bar r_0$ by the fixed
shifts of the coefficients and of the reconstruction. For the
sequences of Lemma~\ref{main:lem:history}, the reduced coefficients
$a(F_j)$ on $W$ and the positive initial amplitudes satisfy
\begin{equation}\label{app:eq:uniform-high-coeff}
 \|a(F_j)\|_{C^r(W)}
 +\sum_a\|r_{F_j,a}\|_{C^r_{z,y}}
 \leq C_rT_j^{(r-\bar r_0)_+}.
\end{equation}
Consequently the estimates for phases, symbols and waves apply with
$L=T_j$, and these derivative bounds give the bounds for the source and
for the coefficients used in
Proposition~\ref{app:ch_mean_pairs:prop:mean-iteration} for all
fixed-parameter operators. The shifts are fixed before the accuracy;
the finite number of correction steps for the waves and the number of
iteration steps enter only the constants and the thresholds.
\end{prop}
\begin{proof}
The inverse metric $g_F^{-1}$, the normal map $N_F$ and the projected
frame $Q_i=\Pi_{K_F}Q_i^0$ are smooth functions of the second jet of
$F$ on a set where the rank and Gram gaps are bounded below. By
Lemma~\ref{app:lem:elementary-interpolation-jets}, such functions
transfer the derivative bounds \eqref{main:eq:history-current} of
$F_j$, with a fixed shift of the index. The same holds for the
positive initial amplitudes by
\eqref{app:ch_background:eq:bg-current-profile}. This proves
\eqref{app:eq:uniform-high-coeff}. The kernel estimates of
Setup~\ref{app:ch_regular:setup:base-aware-coefficients} and
Corollary~\ref{app:ch_response:cor:current-atoms} then give the
conclusions for the phases and for the corrected waves with
initial data $\delta_0$. No regularity with respect to the sample
index is required.
\end{proof}

\begin{proposition}[Uniformity of the correction estimates]
\label{main:prop:current-scale}
Choose $r_*$ larger than the fixed derivative orders of
Propositions~\ref{main:prop:current-waves}, \ref{main:prop:mean}
and~\ref{main:prop:pairs}, including the source order of
Proposition~\ref{main:prop:mean}. Consider a sequence satisfying the
hypotheses of Lemma~\ref{main:lem:history} whose total displacement stays
in the corresponding $C^{r_*}$ neighborhood. Then these propositions
apply successively at $F_j$, with $T_0=T_j$ and the scales of
Section~\ref{main:sec:scales}. At each fixed accuracy their constants are
uniform over all such sequences. The same $r_*$ works for every accuracy.
\end{proposition}

\begin{proof}
Equation \eqref{main:eq:history-current} is
\eqref{main:eq:lag-hypothesis} with $T_0=T_j$, and
\eqref{main:eq:total-small} keeps $F_j$ in the fixed $C^{r_*}$
neighborhood. This neighborhood preserves the low-order conditions (L)
of Setup~\ref{app:ch_seed:setup:seed-current} and all lower bounds, and,
since $r_*\geq M_{\rm an}+11$, also the high-order bound (H), by
Remark~\ref{app:ch_seed:rem:seed-map-topology}.
Proposition~\ref{app:prop:fixed-high} transfers
\eqref{main:eq:history-current} to the coefficient bounds used by the
three propositions. These constants depend only on the constants in
\eqref{main:eq:history-current}, which are uniform by
Lemma~\ref{main:lem:history}.
\end{proof}

Since $t<1$, the coefficient scale $T_j=R_j^t$ lies below the input
frequency bound $K_{\max}=R_j^{s_f}$ of the mean and below the smallest
product frequency $c\lambda h$. The next section verifies both strict
gaps.


\section{Choice of parameters}\label{main:sec:scales}
\label{app:sec:ordered-scale-selection}
The estimates require only an open set of exponents. The following
choice verifies all of them, before the accuracy or the target dilation
is chosen.

\begin{proposition}[Compatible scales]\label{app:prop:ordered-scale-selection}
Choose
\begin{equation}\label{main:eq:fixed-outer-exponents}
 0<\epsilon<\tfrac18,\quad \beta=\frac{1+\epsilon}{3},\quad
 1<s_f<1+\epsilon,\quad 1+\beta<a<b_*<v<\tfrac32.
\end{equation}
Set
\begin{equation}\label{main:eq:outer-window}
 \frac23<t=\frac{b_*}{v}<1,\qquad \zeta=\frac{b_*}{a}-1>0,
\end{equation}
and
\begin{equation}\label{main:eq:A-choice}
 m_0=\lfloor n/2\rfloor+2,\qquad s_0=2m_0,\qquad A=m_0+\tfrac12.
\end{equation}
Use $\lambda=R^a$, $h=R^{-\beta}$, $K_{\max}=R^{s_f}$ and
\begin{equation}\label{main:eq:primary-fixed-scales}
 T_0=R^t.
\end{equation}
There are fixed $\sigma,\nu_0>0$ such that the conditions
\eqref{main:eq:response-window} of
Proposition~\ref{main:prop:current-waves},
\eqref{main:eq:pair-window} of Proposition~\ref{main:prop:pairs}, and the
conditions of Proposition~\ref{main:prop:mean} hold for all large $R$.
The fixed derivative orders are chosen before the accuracy and the target
dilation.
\end{proposition}

\begin{proof}
\medskip\noindent\emph{Step 1 (The ordering).}
Since $\epsilon<1/8$, we have $\beta=(1+\epsilon)/3<(9/8)/3=3/8$, so
$1+\beta<11/8<3/2$ and the interval $(1+\beta,3/2)$ is nonempty; hence
$a,b_*,v$ can be chosen. Next, $\epsilon<\beta$ is equivalent to
$3\epsilon<1+\epsilon$, that is $\epsilon<1/2$; thus
$s_f<1+\epsilon<1+\beta<a$. Since $b_*>a>1$ and $v<3/2$, we get
$t=b_*/v>2/3$, and $t<1$ because $b_*<v$; also $\zeta>0$ because
$b_*>a$, and $b_*=a(1+\zeta)$. Altogether
\[
 2/3<t<1<s_f<a<b_*<v<3/2,\qquad a-\beta>1 .
\]

\medskip\noindent\emph{Step 2 (The strict gaps).}
Since $a-\beta>1$ and $s_f<a$, we may choose $\sigma,\nu_0>0$ so small
that $(1-\sigma)(a-\beta)>1$ and $s_f<(1-\nu_0)a$. Then
$(\lambda h)^{1-\sigma}=R^{(1-\sigma)(a-\beta)}$ and
$\lambda^{1-\nu_0}=R^{(1-\nu_0)a}$ are strictly larger powers of $R$ than
$R$ and $K_{\max}=R^{s_f}$, respectively, so every fixed constant is
absorbed for large $R$:
\begin{equation}\label{main:eq:all-pair-scales}
 C_*R\leq(\lambda h)^{1-\sigma},\qquad
 C_*K_{\max}\leq\lambda^{1-\nu_0}.
\end{equation}
The first inequality is \eqref{main:eq:pair-window}: the amplitude scale
$R$ of the mean lies below the product frequencies. The second puts the
input frequency bound below the wave frequency, as required in
\eqref{main:eq:response-window} with $\Lambda=CK_{\max}$ and
$K=C_*K_{\max}$; there also $C_{\rm env}T_0\leq K$ because $t<1<s_f$.
Moreover $c\lambda h=cR^{a-\beta}\geq2$ for large $R$.

\medskip\noindent\emph{Step 3 (The exponents of the mean).}
At the fixed input band the concentration exponent is
\begin{equation}\label{main:eq:preparation-orders}
 \omega=(1+\epsilon-s_f)/4>0,
\end{equation}
which is positive because $s_f<1+\epsilon$. The phase error is smaller:
by Step~1,
\[
 a-s_f>(1+\beta)-(1+\epsilon)=\beta-\epsilon=\frac{1-2\epsilon}3
 >\frac{1-1/4}{3}=\frac14,
\]
while $\omega<\epsilon/4<1/32$ because $s_f>1$. Hence $\omega<a-s_f$.
The source order satisfies $s_0=2m_0\geq2\bigl(\tfrac{n-1}2+2\bigr)=n+3>n$,
and
\[
 2A-s_0=2m_0+1-2m_0=1>\beta .
\]
By the remark after Lemma~\ref{main:lem:smoothing}, the normalized target
satisfies $\|e_R/b^2\|_{H^{s_0}}\leq CR^{s_0-2A}=CR^{-1}\leq CR^{-\beta}$,
and the background error of the sampling is $O(h)=O(R^{-\beta})$. Thus
the background and the target together have size $O(R^{-\beta})$ in the
fixed Sobolev norm $H^{s_0}$. The normal error and the quadratic estimate
give a small $L^2$ factor for successive differences of the iteration in
Proposition~\ref{main:prop:mean}; the tail of the initial-frequency
cutoff is controlled by the same derivative bounds at higher orders. The
amplitude scale $R$ and the coefficient scale $T_0$ lie below the fixed
input band $K_{\max}=R^{s_f}$, because $t<1<s_f$; this permits arbitrary
finite precision for this sample.
\end{proof}

\subsection{The accuracy as a function of the defect}
We fix $r_*$ after these derivative orders and after all geometric and
analytic shifts. With $b_\sharp=B_0+b_*(r_*+n/2)$, choose an integer
$k_{\min}$ such that
\begin{align}
 (3-2v)k_{\min}&>B_3+2v+4,\label{main:eq:k-cubic}\\
 k_{\min}&>b_\sharp+2.\label{main:eq:k-history}
\end{align}
These are two lower bounds for $k_{\min}$, and they are compatible
because $3-2v>0$, by $v<3/2$. We take the accuracy
\begin{equation}\label{main:eq:Q-choice}
 Q(k)=2k.
\end{equation}
At defect size $\delta=R^{-k}$, both terms in \eqref{main:eq:step-error}
are bounded by $C_{2k}R^{-3k+B_3}$:
\[
 \delta R^{-Q(k)}=R^{-3k}\leq R^{-3k+B_3},\qquad
 \delta^3R^{B_3}=R^{-3k+B_3}.
\]
For $k\geq k_{\min}$ and $R\geq1$, \eqref{main:eq:k-cubic} gives
\begin{equation}\label{new:eq:cubic-gain}
 R^{-3k+B_3}=R^{-2v(k+1)}\,R^{-(3-2v)k+B_3+2v}
 \leq R^{-2v(k+1)}\,R^{-4}.
\end{equation}
The strict gaps proved above permit all the internal accuracy choices of
Theorem~\ref{main:thm:step} without changing the domains, the derivative
orders or the frequency exponents.


\section{Initialization and convergence}\label{main:sec:convergence}

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

\begin{proposition}[Fixed reference initialization]\label{main:prop:seed}
Choose coordinates centered at $p$ with $g(0)=I$, and let $F_0$ be the
analytic flat reference of Section~\ref{main:sec:geometry}. There is a
fixed $\rho>0$ such that the maps
\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}
are defined on the fixed chart $W$, and $G_{\rm ref}$ has uniform rank
and normal-rank bounds there. 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 $s$. The same
reference admits the positive transported background and the inverse of
the continuous normal operator used in Section~\ref{app:sec:ch_background},
on a fixed smaller core containing the ball $D$ and the support of the
output of the mean.
\end{proposition}

\begin{proof}
By Corollary~\ref{app:ch_seed:cor:seed-uniform-dilation}, for every
sufficiently small $\rho>0$ the map $G_{\rm ref}=G_\rho$ lies in the
prescribed finite-order neighborhood of the constant frame, satisfies
$\gamma(G_{\mathrm{ref}})=I$, and its least normal singular value is at
least $c\rho$.

Since $g$ is smooth near $0$ and $W$ is bounded, $g^{(s)}$ is defined on
$W$ for $0\leq s\leq s_*$ if $s_*$ is small. For a multi-index $\alpha$
with $|\alpha|\geq1$,
$\partial_y^\alpha g^{(s)}(y)=(s\rho)^{|\alpha|}(\partial^\alpha g)(s\rho y)$,
so $|\partial_y^\alpha g^{(s)}|\leq(s\rho)^{|\alpha|}\|g\|_{C^{|\alpha|}}
\leq C_{r,\rho}s$ for $s\leq s_*\leq1$. For $\alpha=0$, the fundamental
theorem of calculus and $g(0)=I$ give
$|g^{(s)}(y)-I|\leq s\rho|y|\,\|dg\|_{C^0}\leq C_\rho s$ on $W$. Since
$\gamma(G_{\rm ref})=I$, this proves \eqref{main:eq:seed-small},
uniformly at every fixed order.

For the background, Proposition~\ref{app:ch_background:prop:bg-normalized}
gives bounds and a coercivity gap which are uniform for the normalized
family $G_\sigma$ as $\sigma\to0$, with fixed cutoffs chosen before
$\rho$. Proposition~\ref{app:ch_background:prop:bg-current} constructs
the density change $v_a$, of size $O(\sigma)$ at every fixed order by
\eqref{app:ch_background:eq:bg-positive-profile}; for small $\sigma$ we
have $1+v_a\geq1/2$, so the initial amplitude $\sigma\sqrt{1+v_a}$ is
positive and the corresponding normal operator is coercive with a
bounded inverse. Reducing $\rho=\sigma$ once gives both properties for
$G_{\rm ref}$ and for a fixed $C^{r_*}$ neighborhood of it. We also take
$\rho\leq\sigma_1$, so that the constant $C_{\rm hi}$ of
Remark~\ref{app:ch_seed:rem:seed-map-topology} applies to the $C^{r_*}$
ball of radius $\rho$ around $G_{\rm ref}$, by the normalized
neighborhood \eqref{new:eq:normalized-neighborhood} with $\sigma=\rho$. The later target parameter $s$
does not enter these choices.
\end{proof}

After all geometric, analytic and frequency constants have been fixed,
the parameter $s>0$ can therefore make the initial defect smaller than
any prescribed threshold.

\subsection{The order of choices}\label{new:sec:order-of-choices}
All constants of the proof are chosen in the following order; each
item depends only on the preceding ones.
\begin{enumerate}
\item The dimension $n\geq5$, the model, the characteristic patches, the
cover of the sphere, the radial factors and the groups of patches
(Proposition~\ref{app:ch_seed:prop:seed-frozen-menu},
Definition~\ref{app:defn:exhaustive-pair-groups}).
\item The low-order data (L) of Setup~\ref{app:ch_seed:setup:seed-current},
the constant $C_{\rm hi}$ of (H)
(Remark~\ref{app:ch_seed:rem:seed-map-topology}), the phase boxes
(Proposition~\ref{app:ch_seed:prop:seed-phase-charts}), the charts and
parametrices on regular patches
(Theorems~\ref{app:ch_regular_base:thm:sector}
and~\ref{app:ch_regular:thm:no-return}), the solution operators of the
waves (Setup~\ref{app:ch_response:thm:response-compatible-choice}), the
measurement region and cutoffs
(Proposition~\ref{app:ch_background:prop:bg-frame}), and the domains
$D\Subset\cdots\Subset W$ of Setup~\ref{app:setup:domains}.
\item The dilation parameter $\rho$ of the reference
(Proposition~\ref{main:prop:seed}).
\item The exponents of Proposition~\ref{app:prop:ordered-scale-selection},
and $m_0$, $s_0$, $A$, $B_0$, $B_3$.
\item The order $r_*$: it is at least $M_{\rm an}+11$ and exceeds all
fixed derivative orders and losses of the constructions
($R_{\rm env}+B_{\rm at}$, $s_0+B_{\rm can}$, $\bar r_0$ and the shifts of
Proposition~\ref{app:prop:fixed-high}, $k_{\rm gap}$ and the orders of
Section~\ref{app:sec:ch_background}). None of these depends on $r_*$.
\item The radius $\varepsilon_*\leq\rho$ of a $C^{r_*}(W)$ ball around
$G_{\rm ref}$ on which (L), (H) and all geometric and analytic gaps hold.
\item The integer $k_{\min}$ of \eqref{main:eq:k-cubic}--\eqref{main:eq:k-history};
it depends on $r_*$ through $b_\sharp$.
\item The threshold function $\mathfrak H$ below. Only here do the
accuracies enter: at accuracy $Q=2k$ they determine the smoothing order,
the depths of the Neumann series for the waves and for the products,
the number of Picard steps, the constants $C_{2k}$ and the threshold
for $R$, and nothing else.
\item The initial frequency $R_0\geq\mathfrak H(k_{\min})$ and the schedule
$(R_j,k_j)$.
\item The target dilation $s$.
\end{enumerate}
Within one step of the iteration, at $F=M_j$, the accuracy $Q_j=2k_j$
first fixes the finite depths; then the wave families are formed, a
realization of the sample is chosen in the event of
Theorem~\ref{app:ch_waves:thm:sampling}, and the Picard iteration and the
corrections are computed. The domains, $r_*$, the Fourier supports and
the exponents are the same at every step, so that all iterates are
defined on the fixed chart $W$ and the equation is imposed on the fixed
ball $D$.

\subsection{Admissible sequences and thresholds}
The target family $g^{(s)}$, $0<s\leq s_*$, has uniformly bounded
seminorms of every fixed order. Let $\varepsilon_*\leq\rho$ be as in
item~(vi) above, and fix $r_*$, $k_{\min}$ and the exponents of
Section~\ref{main:sec:scales}. Every map in the $C^{r_*}(W)$ ball of radius
$\varepsilon_*$ around $G_{\rm ref}$ satisfies (H), by
Remark~\ref{app:ch_seed:rem:seed-map-topology}, since
$r_*\geq M_{\rm an}+11$ and $\varepsilon_*\leq\rho$, so that \eqref{new:eq:normalized-neighborhood} holds with $\sigma=\rho$.

We call a finite sequence of real increments $U_0,\dots,U_{j-1}$
\emph{admissible} if it satisfies \eqref{main:eq:history-data} with
$R_i=R_0^{v^i}$ and integers $k_i\geq k_{\min}$. By
\eqref{main:eq:k-history}, $\mu=k_{\min}-b_\sharp-1>1$, so
Lemma~\ref{main:lem:history} applies to the maps
$M_j=G_{\rm ref}+\sum_{i<j}U_i$, with constants independent of the
sequence.

We now choose a finite nondecreasing threshold function
$k\mapsto\mathfrak H(k)$, defined for integers $k\geq k_{\min}$ and
independent of $s$.
\begin{enumerate}
\item First, $\mathfrak H(k_{\min})$ is chosen so large that
 $R_0\geq\mathfrak H(k_{\min})$ implies $T_0=R_0^t\geq2$ and
 \begin{equation}\label{main:eq:displacement-budget}
  \sum_{i\geq0}C R_i^{-k_{\min}+b_\sharp+1}
                                       <\varepsilon_*/2,
 \end{equation}
 where $C$ is the constant of \eqref{main:eq:history-bernstein}; this is
 possible by \eqref{new:eq:geometric-sum}. Then, for every admissible
 sequence, $\|M_j-G_{\rm ref}\|_{C^{r_*}(W)}<\varepsilon_*/2$, so all
 maps $M_j$ lie in the chosen $C^{r_*}$ neighborhood.
 Proposition~\ref{main:prop:current-scale},
 Proposition~\ref{app:prop:ordered-scale-selection} and
 Lemma~\ref{main:lem:smoothing} then give all hypotheses of
 Theorem~\ref{main:thm:step} at $F=M_j$, $R=R_j$, $T_0=R_j^t$, uniformly
 for admissible sequences and for the compact target family.
\item For each $k\geq k_{\min}$, Theorem~\ref{main:thm:step} at accuracy
 $Q=2k$ involves finitely many constants, uniform in the sense of (i).
 Let $C_{2k}$ denote the largest of the constants in
 \eqref{main:eq:step-error} and \eqref{main:eq:step-L2}. We enlarge
 $\mathfrak H(k)$ so that $R\geq\mathfrak H(k)$ implies that $R$ exceeds
 the threshold for "sufficiently large $R$" in
 Theorem~\ref{main:thm:step} at accuracy $2k$, and
 \[
  C_{2k}\leq R,\qquad 2C_{2k}\leq R^4 .
 \]
 Finally we replace $\mathfrak H(k)$ by
 $\max_{k_{\min}\leq k'\leq k}\mathfrak H(k')$, which makes it
 nondecreasing.
\end{enumerate}

\begin{lem}[One step of the iteration]\label{new:lem:one-step}
Let $U_0,\dots,U_{j-1}$ be admissible, let $k_j\geq k_{\min}$ and
$R_j\geq\mathfrak H(k_j)$, and suppose that
$\|g^{(s)}-\gamma(M_j)\|_{C^0(D)}\leq R_j^{-2k_j}$. Let $U_j$ be the
increment of Theorem~\ref{main:thm:step} at $F=M_j$, $R=R_j$,
$\delta=R_j^{-k_j}$ and $Q=2k_j$. Then $U_0,\dots,U_j$ is admissible,
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}
\end{lem}

\begin{proof}
By the choice of $\mathfrak H$, Theorem~\ref{main:thm:step} applies.
Its band condition \eqref{main:eq:step-band} is the first condition in
\eqref{main:eq:history-data}, with a support constant independent of
$Q$, and \eqref{main:eq:step-L2} gives
$\|U_j\|_{L^2}\leq C_{2k_j}R_j^{-k_j+B_0}\leq R_j^{-k_j+B_0+1}$ since
$C_{2k_j}\leq R_j$. Thus the extended sequence is admissible. By
\eqref{main:eq:step-error} and the computation after
\eqref{main:eq:Q-choice}, the defect of $M_j+U_j$ is at most
$2C_{2k_j}R_j^{-3k_j+B_3}$. By \eqref{new:eq:cubic-gain}, which uses
\eqref{main:eq:k-cubic}, and by $2C_{2k_j}\leq R_j^4$, this is at most
$R_j^{-2v(k_j+1)}$.
\end{proof}

\subsection{One schedule and one target dilation}
Choose $R_0\geq\mathfrak H(k_{\min})$. Set $k_0=k_{\min}$ and define
recursively
\begin{equation}\label{main:eq:staircase}
 k_{j+1}=k_j+\mathbf1_{\{R_{j+1}\geq\mathfrak H(k_j+1)\}}.
\end{equation}
Thus the accuracy index increases by one exactly when the next scale
reaches the threshold of the next index. We claim that
\[
 R_j\geq\mathfrak H(k_j)\quad\text{for all }j,\qquad
 k_{j+1}\leq k_j+1,\qquad k_j\to\infty .
\]
The first assertion holds for $j=0$. If $k_{j+1}=k_j+1$, then
$R_{j+1}\geq\mathfrak H(k_j+1)=\mathfrak H(k_{j+1})$ by
\eqref{main:eq:staircase}; if $k_{j+1}=k_j$, then
$R_{j+1}\geq R_j\geq\mathfrak H(k_j)=\mathfrak H(k_{j+1})$. The second
assertion is clear. For the third, the sequence $k_j$ is nondecreasing.
If it were bounded, it would be equal to some $K$ for all $j\geq j_0$.
Since $R_j\to\infty$ and $\mathfrak H(K+1)$ is finite, there is
$j\geq j_0$ with $R_{j+1}\geq\mathfrak H(K+1)$, and then $k_{j+1}=K+1$,
a contradiction.

Put $\delta_j=R_j^{-k_j}$ and $Q_j=2k_j$. The schedule $(R_j,k_j)$ is
determined by $R_0$ and $\mathfrak H$ alone. Now choose a single
$s\in(0,s_*]$ with
\begin{equation}\label{main:eq:initial-error}
 \|g^{(s)}-\gamma(G_{\rm ref})\|_{C^0(D)}\leq\delta_0^2,
\end{equation}
which is possible by \eqref{main:eq:seed-small} with $r=0$, for instance
$s=\min(s_*,\delta_0^2/C_{0,\rho})$. This choice is made after all
thresholds; since $\mathfrak H$ is independent of $s$, it does not affect
them.

\subsection{Convergence}
Let $M_0=G_{\rm ref}$ and, inductively, let $U_j$ be the increment of
Lemma~\ref{new:lem:one-step} at $M_j$, and $M_{j+1}=M_j+U_j$. We prove
by induction that $U_0,\dots,U_{j-1}$ is admissible and
\begin{equation}\label{main:eq:metric-induction}
 \|g^{(s)}-\gamma(M_j)\|_{C^0(D)}\leq\delta_j^2.
\end{equation}
For $j=0$ this is \eqref{main:eq:initial-error}. If it holds for $j$,
then Lemma~\ref{new:lem:one-step} applies, because $R_j\geq\mathfrak
H(k_j)$ and $\delta_j^2=R_j^{-2k_j}$. It shows that $U_0,\dots,U_j$ is
admissible, and, since $R_j^v=R_{j+1}$, $k_{j+1}\leq k_j+1$ and
$R_{j+1}\geq1$,
\[
 \|g^{(s)}-\gamma(M_{j+1})\|_{C^0(D)}
 \leq R_j^{-2v(k_j+1)}=R_{j+1}^{-2(k_j+1)}
 \leq R_{j+1}^{-2k_{j+1}}=\delta_{j+1}^2 .
\]
Thus the construction continues on $D$ for all $j$.

For every fixed $r$, the Bernstein inequality,
Lemma~\ref{new:lem:bernstein}(ii), and \eqref{main:eq:history-data} give
\begin{equation}\label{main:eq:final-Cr}
 \|U_j\|_{C^r(\R^n)}
 \leq C_rR_j^{-k_j+B_0+1+b_*(r+n/2)}.
\end{equation}
Here $C_r$ is the Fourier constant, independent of $Q_j$, because the
constants of Theorem~\ref{main:thm:step} have been absorbed by $R_j$.
Since $k_j\to\infty$, there is $j_r$ with
$k_j\geq B_0+2+b_*(r+n/2)$ for $j\geq j_r$, and then
$\|U_j\|_{C^r}\leq C_rR_j^{-1}=C_rR_0^{-v^j}$, which is summable. The
series $G_{\rm ref}+\sum_jU_j$ therefore converges in $C^r(W)$ for every
$r$, to one smooth map $M_\infty$ on $W$. Since $M_j\to M_\infty$ in
$C^1(D)$, we have $\gamma(M_j)\to\gamma(M_\infty)$ uniformly on $D$, and
\eqref{main:eq:metric-induction} with $\delta_j\leq R_j^{-1}\to0$ gives
$\gamma(M_\infty)=g^{(s)}$ on $D$.

\begin{proof}[Proof of Theorem~\ref{main:thm:main}]
In the coordinates of Proposition~\ref{main:prop:seed}, $p=0$. Let
$U=s\rho\,\Int D$ and define
\[
 F(x)=s\rho\,M_\infty\bigl(x/(s\rho)\bigr),\qquad x\in U .
\]
Then $dF(x)=dM_\infty(x/(s\rho))$, so
\[
 \gamma(F)(x)=\gamma(M_\infty)\bigl(x/(s\rho)\bigr)
 =g^{(s)}\bigl(x/(s\rho)\bigr)=g(x),
\]
which is \eqref{main:eq:isometry} on $U$, and $F$ is smooth. Since
$(dF)^TdF=g$ is positive definite, $dF(x)$ is injective, so
$\rank dF(0)=n$. Choose $n$ of the $N=n(n+1)/2$ ambient coordinates whose
$n\times n$ minor of $dF(0)$ is invertible, and let $\pi$ be the
projection onto them. By the inverse function theorem, $\pi\circ F$ is a
diffeomorphism from a smaller neighborhood $U'\subset U$ of $0$ onto an
open set $\Omega$. On $U'$ the map $F$ is injective, and
$F(U')=\{(z,\phi(z)):z\in\Omega\}$ is the graph of the smooth map
$\phi$ given by the other $N-n$ components of $F\circ(\pi\circ F)^{-1}$,
in the corresponding ordering of the coordinates. The inverse of
$F:U'\to F(U')$ is $(\pi\circ F)^{-1}\circ\pi$, which is continuous.
Hence $F|_{U'}$ is a smooth embedding into $\R^{n(n+1)/2}$ satisfying
\eqref{main:eq:isometry}. This single restriction does not depend on any
derivative order, and the same map $F$ is the $C^r$ limit, for every
$r$, of the rescaled iterates.
\end{proof}

%% ===== 99_back.tex =====
\section*{Acknowledgements}
Generative AI was used in developing the proofs.

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\end{document}