==== TABLE OF CONTENTS ====
Strong-diffeomorphic-approximation-in-three-dimensions-for-p-gt-2//main.tex
Strong-diffeomorphic-approximation-in-three-dimensions-for-p-gt-2//sections/01-introduction.tex
Strong-diffeomorphic-approximation-in-three-dimensions-for-p-gt-2//sections/02-smoothing.tex
Strong-diffeomorphic-approximation-in-three-dimensions-for-p-gt-2//sections/05-high-preparation.tex
Strong-diffeomorphic-approximation-in-three-dimensions-for-p-gt-2//sections/06-quantizers.tex
Strong-diffeomorphic-approximation-in-three-dimensions-for-p-gt-2//sections/07-walls.tex
Strong-diffeomorphic-approximation-in-three-dimensions-for-p-gt-2//sections/08-mesh.tex
Strong-diffeomorphic-approximation-in-three-dimensions-for-p-gt-2//sections/09-high-assembly.tex
Strong-diffeomorphic-approximation-in-three-dimensions-for-p-gt-2//references.bib
===========================

==== BEGIN Strong-diffeomorphic-approximation-in-three-dimensions-for-p-gt-2//main.tex ====
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\hypersetup{pdftitle={Strong diffeomorphic approximation in three dimensions for p>2},pdfauthor={OpenAI}}
\title{Strong diffeomorphic approximation\\in three dimensions for $p>2$}
\author{OpenAI}
\date{September 24, 2026}
\begin{document}
\maketitle
\begin{abstract}
We resolve the three-dimensional Ball--Evans approximation problem for every
finite $p>2$. Every $W^{1,p}$ homeomorphism between arbitrary bounded domains
in $\R^3$ can be approximated strongly in $W^{1,p}$ by smooth diffeomorphisms
onto the same target.
\end{abstract}
\input{sections/01-introduction}
\tableofcontents
\input{sections/05-high-preparation}
\input{sections/06-quantizers}
\input{sections/07-walls}
\input{sections/08-mesh}
\input{sections/09-high-assembly}
\appendix
\input{sections/02-smoothing}
\begingroup
\raggedright
\bibliographystyle{plainnat}
\bibliography{references}
\endgroup
\end{document}
==== END Strong-diffeomorphic-approximation-in-three-dimensions-for-p-gt-2//main.tex ====

==== BEGIN Strong-diffeomorphic-approximation-in-three-dimensions-for-p-gt-2//sections/01-introduction.tex ====
\section{Introduction and conventions}\label{sec:introduction}

Let $\Omega,\Lambda\subset\R^3$ be domains, meaning nonempty connected
open sets.  A homeomorphism $f:\Omega\to\Lambda$ belongs to
$W^{1,p}(\Omega;\R^3)$ when its components and their weak first
derivatives are in $L^p$.  Strong $W^{1,p}$ approximation requires
convergence of both the maps and these derivatives in $L^p$.
The \emph{fixed-target Ball--Evans approximation problem} asks whether
one can choose the approximants to be smooth diffeomorphisms from
$\Omega$ onto exactly $\Lambda$.  Here a diffeomorphism and its inverse
are smooth on their open domains; this definition makes no assertion
about boundary values.

The problem originates in nonlinear elasticity, where injectivity
expresses noninterpenetration and the derivative describes local
deformation.  Ball's discussions~\cite{Ball2001,Ball2002,Ball2010}
connect the approximation question to the variational and computational
theory; the original planar approximation paper records his attribution
of the question to Evans~\cite{IwaniecKovalevOnninen2011}.
Neither ordinary convolution nor sufficiently fine vertex interpolation
automatically preserves injectivity.  We resolve the fixed-target
problem on bounded three-dimensional domains for every finite exponent
above two.

\begin{theorem}\label{main:theorem}\label{ha:approximation}
Let $\Omega,\Lambda\subset\R^3$ be bounded domains, let $2<p<\infty$,
and let $f:\Omega\to\Lambda$ be a homeomorphism in
$W^{1,p}(\Omega;\R^3)$. There exist $C^\infty$ diffeomorphisms
$f_j:\Omega\to\Lambda$, each belonging to
$W^{1,p}(\Omega;\R^3)$, such that
\begin{equation}\label{main:convergence}
\int_\Omega\bigl( |f_j-f|^p+|Df_j-Df|^p\bigr)\,dx\longrightarrow0.
\end{equation}
\end{theorem}

Neither boundary regularity, a boundary extension of $f$, inverse
Sobolev regularity, nor a nonvanishing Jacobian is assumed.  The
theorem concerns forward Sobolev convergence.  Smoothness of each
inverse does not assert Sobolev convergence of the inverses; strong
forward convergence alone also does not imply convergence of an elastic
energy with a singular determinant penalty.  These are distinct
approximation questions; see \cite{Ball2002,Pratelli2017,Hencl2025}.
The complementary range $1\le p\le2$ is treated by a different
construction in the companion article~\cite{OpenAILow2026}.
No theorem from that range is used here.  The full common smoothing
argument is included in Appendix~\ref{sec:smoothing}, so the present
range theorem has a complete local proof.

\subsection{Earlier results and the dimensional obstacle}

The planar theory provides both the principal positive precedents and
important contrasts.  Iwaniec, Kovalev, and Onninen treated the
Hilbert-space case $p=2$ using harmonic replacements in work first
circulated in June 2010~\cite{IwaniecKovalevOnninen2012}.  Their extension,
first circulated that September, covers every $1<p<\infty$ on arbitrary planar open sets
\cite[Theorem~1.1]{IwaniecKovalevOnninen2011}.  Hencl and Pratelli
proved the endpoint $p=1$ by geometric extension and grid constructions,
including locally finite piecewise affine approximation
\cite[Theorem~1.1]{HenclPratelli2018}.  Both results preserve the
target and require no inverse Sobolev hypothesis.  Thus those features
are inherited goals of the problem, not new distinctions of the
three-dimensional statement.

Geometric approximation has also been developed under stronger planar
input assumptions.  Bellido and Mora-Corral control approximation in
smaller H\"older exponents on polygonal planar domains when both
directions are H\"older
\cite{BellidoMoraCorral2011}.  Daneri and Pratelli obtain quantitative
bi-Lipschitz approximation with derivative control in both directions
for bi-Lipschitz maps~\cite{DaneriPratelli2014}; Pratelli treats
bi-$W^{1,1}$ maps~\cite{Pratelli2017}.  These results clarify the
distinction between approximating a map by maps with smooth inverses
and approximating the inverse in a specified Sobolev norm.

Piecewise affine smoothing is a separate stage.  Mora-Corral and
Pratelli established a strong planar version with control in both
directions~\cite{MoraCorralPratelli2014}.  Campbell and Soudsk\'y
obtained $C^1$ injective approximation of piecewise affine maps in
arbitrary dimension, without asserting smoothness of the inverse
\cite{CampbellSoudsky2021}.  Campbell, D'Onofrio, and V\'itek then
proved diffeomorphic approximation of locally finite piecewise affine
homeomorphisms in dimensions three and four, with uniform and
derivative-error control for both the map and its inverse
\cite[Theorem~A]{CampbellDOnofrioVitek2026}.  The remaining task for a
general Sobolev input is to obtain the required strong approximation
before this smoothing stage.  Appendix~\ref{sec:smoothing} includes
both the locally bi-Lipschitz-to-PL reduction and a PL smoothing proof
with arbitrary positive continuous value tolerances.

The dimension restriction is substantive.  Building on the $W^{1,1}$
construction of Hencl and Vejnar~\cite{HenclVejnar2016}, and repairing
a step in that construction, Campbell, Hencl, and Tengvall obtained
counterexamples in dimensions $n\ge4$ for
$1\le p<\lfloor n/2\rfloor$, with Jacobians of both signs on sets
of positive measure~\cite{CampbellHenclTengvall2018}.  Those examples
do not decide the three-dimensional problem.  Qualitative topology
does help in dimension three: the triangulation and PL-approximation
theory of Moise and Bing, in Hamilton's relative formulation,
provides the topological flexibility used below
\cite{Moise1952Approximation,Moise1952Triangulation,Bing1959,Hamilton1976}.
It supplies no strong derivative estimate.  In particular, a
topological extension chosen after a small-energy exceptional set may
have an arbitrarily large Lipschitz constant, so multiplying that
earlier energy by its later constant is not a valid approximation
argument.

\subsection{Proof strategy and organization}

We first seek a locally bi-Lipschitz homeomorphism strongly close to
$f$.  At points where $Df$ is invertible, suitably chosen small
cubes permit exact affine replacements.  The principal work concerns
the regular rank-one and rank-two data.  Their image is volume-null,
but their source Sobolev energy need not be small.  Discarding them
as an exceptional set would therefore lose the approximation theorem.
The construction retains them and recovers their gradients through
two scalar coordinates.

On each small retained patch, these coordinates have prescribed affine
first-order models.  Keeping the realized labels close to the original
scalar coordinates controls the change in their mean gradients through
a boundary integral.  Nearly
sharp upper bounds for the gradient norms then give strong gradient
closeness by uniform convexity.  This is why the construction must
control both label values and scalar parameter slopes.

\begin{enumerate}
\item \emph{Prepare exact regions and retained deficient-rank data.}
Section~\ref{sec:high-preparation} first uses the decomposability bundle
of Alberti--Marchese and the converse to Rademacher's theorem of
De Philippis--Rindler to identify transverse directions in the transported
measure~\cite{AlbertiMarchese2016,DePhilippisRindler2016}.
Thin graph bands, related to the null-set constructions of
Alberti--Cs\"ornyei--Preiss~\cite{AlbertiCsornyeiPreiss2010}, flatten
the retained data onto separated plate patches while preserving
their tangential derivatives.  These patches determine the exact
regions required of a continuous proper surjection
$F_b:\Omega\to\Lambda$, called the carrier.  It agrees with affine
or PL homeomorphisms over a prescribed open part of the target and
has controlled local Sobolev energy outside the marked exceptional
interiors that may occur when $2<p<3$.  Its only non-singleton fibres
are specified tame balls, and the proof gives their relative opening
to homeomorphisms.  The cubical cutoff uses immersions of a
two-dimensional skeleton, with the positive-codimension immersion
theorem of Hirsch~\cite{Hirsch1959} and locally flat
Schoenflies~\cite{Brown1960}.

\item \emph{Reduce the variable target to a two-dimensional complex.}
Section~\ref{sec:quantizers} constructs a Lipschitz map
$T:\Lambda\to\Lambda$ from a strongly convex potential.
Its full-dimensional input blocks lie entirely in the carrier's exact
regions; outside these blocks its image is a polyhedral two-complex.
This combines classical convex conjugacy and polyhedral diagram
geometry~\cite{Moreau1965,Aurenhammer1987}, with additional protected
paired contacts proved here.  They place the retained deficient-rank
data on rectangular faces with nearly unchanged gradients.
On a polygonal face, the two coordinates are a scaled logarithm of
the normalized radial fraction and arclength around the perimeter.

\item \emph{Realize these coordinates by source walls.}
Section~\ref{sec:walls} uses annular walls for radial levels and
disk walls for perimeter levels.  Routing removes unwanted
circle intersections without accumulating a derivative factor at
each switch or losing the length of a wall's transverse parameter
interval.  A family of reference fibre paths, called guards,
then pins the realized labels: a misplaced wall would force a
fixed positive path cost on too many small spheres.  The resulting
lower bound grows as the mesh scale tends to zero because $p>2$.
Theorem~\ref{wl:realization} is explicitly conditional on the
geometric and energy estimates proved in
Sections~\ref{sec:mesh}--\ref{sec:high-assembly}.

\item \emph{Supply calibrated meshes and parameter bounds.}
Section~\ref{sec:mesh} prepares and normalizes the source walls,
preserving the individual edge-count bounds, and constructs their
two-colour neighborhoods with simultaneous product coordinates.
Normalization and cut-and-paste belong to the normal-surface
tradition~\cite{Haken1961}.  Section~\ref{sec:calibrated-islands}
improves the estimates on selected source patches.  In aligned source
coordinates, the two scalar gradients are nearly orthogonal in rank two and
nearly parallel in rank one.  The snapping map in
Lemma~\ref{ms:snapping} removes the large parameter slopes caused
by thin intermediate configurations.  Its uniform estimate is only
in the direction from new to old coordinates, as illustrated in
Figure~\ref{ms:fig-snapping}.  Broadening then gives the sharp
scalar bounds by direct differentiation, not by differentiating an
uncontrolled ambient extension.

\item \emph{Recover strong gradients and smooth with a fixed target.}
Section~\ref{sec:high-assembly} chooses local upper bounds for
$\int G^p$, where $G$ is the wall-parameter slope density, before
choosing guards and the final mesh.  These are the capacities used
in the wall argument, not Sobolev capacities of sets.  Nearly sharp
energy and pinned mean gradients yield strong gradient approximation
by uniform convexity.  A separate compression handles marked
exceptional interiors only when $2<p<3$; for $p\ge3$ the volume
Lusin property eliminates them.  The proof ends by correcting the
reserved full-rank cubes and applying the complete smoothing theorem
of Appendix~\ref{sec:smoothing}.  The strict target maps are
self-homeomorphisms, and the final local replacements preserve their
old images; the resulting homeomorphic approximants are onto the
fixed target.
\end{enumerate}

The central quantitative distinction is between derivatives of the
two retained parameters and derivatives of a homeomorphism filling
the complementary chambers.  Only the former enter the principal
energy estimates.  The constant and error choices are recorded in
Remark~\ref{ha:scale-order}; they ensure that every uncontrolled
constant is fixed before the error required to absorb it.

\subsection{Conventions}

Unless stated otherwise, all domains, closures, local finiteness, and compactness are taken relative to the indicated open manifold. A compact set in an open domain has positive distance from its Euclidean complement. A PL map or triangulation is locally finite. A locally bi-Lipschitz homeomorphism has finite Lipschitz constants in both directions on sufficiently small neighborhoods; no global bound is implicit.

The symbol $|A|$ denotes the Frobenius norm of a matrix, and $\|A\|_{\op}$ its operator norm. We use the Frobenius norm in sharp gradient estimates. Equivalent norms give the same strong convergence in Theorem~\ref{main:theorem}. The symbols $\cL^m$ and $\cH^m$ denote Lebesgue and Hausdorff measure. Integrals on parameterized traces include their stated multiplicities. The differential along an $m$-dimensional parameter space is denoted $D_m$ when it must be distinguished from the full differential.

A \emph{fine value tolerance} on a domain $U$ is a positive continuous function on $U$. To approximate finely means to meet a prescribed such tolerance pointwise. Whenever inverse control is needed on a compact localization, it is included in the prescribed tests. Locally finite constructions permit tolerances tending to zero toward the ends of the open domains; no positive global lower bound is assumed.

Constants called \emph{absolute} depend only on dimension and on fixed universal reference shapes. A constant $C_p$ may also depend on the fixed exponent. Other dependencies are specified when the constant is introduced. A finite constant that depends on a chosen compact collection of jets or charts is fixed before an error is required to be small against it.

A homeomorphism between connected oriented domains has a constant topological orientation. If necessary, reflect the target, perform the construction for the orientation-preserving map, and undo the reflection. This convention imposes no extra hypothesis on the Sobolev differential.

\begin{lemma}[Local tolerances]\label{pre:local-tolerances}
Let $U\subset\R^n$ be open and let $a:U\to(0,\infty)$ be continuous. For every $L>0$ there is a positive $L$-Lipschitz function $d$ on $U$ such that
\[
 d(x)\le \min\{a(x),1,L\dist(x,\R^n\setminus U)\}.
\]
There are also positive smooth minorants satisfying prescribed positive continuous upper bounds on their values and first derivatives. One may additionally impose positive constant bounds on a locally finite family of compact regional tests. Finite collections of such requirements can be combined, and error budgets on a countable compact exhaustion can be made summable.
\end{lemma}
\begin{proof}
Extend $e(x)=\min\{a(x),1,L\dist(x,\R^n\setminus U)\}$ by zero outside $U$, and put
\[
 d(x)=\inf_{y\in\R^n}\{e(y)+L|x-y|\}.
\]
This function is $L$-Lipschitz and is bounded above by $e$. It is positive at an interior point $x$: on a small closed ball about $x$, $e$ has a positive minimum, and outside that ball the distance term has a positive lower bound.

For a smooth minorant take a smooth locally finite partition of unity $\{\chi_i\}$ with compact supports in $U$ and put $b=\sum_i c_i\chi_i$, with positive constants $c_i$. Given positive continuous bounds $v,w$, choose
\[
 c_i\le 2^{-i-2}\min\left\{
 \inf_{\supp\chi_i}v,
 \frac{\inf_{\supp\chi_i}w}{1+\|D\chi_i\|_\infty}
 \right\}.
\]
Then $b>0$, $b\le v$, and $|Db|\le w$. Replace a finite list of requirements by their minimum. For countably many regional tests use a locally finite compact cover with compact enlargements, refine the bounds on each intersecting support, and allocate total errors by a convergent positive series. Only finitely many regional requirements meet a given compact support.
\end{proof}

We will also use the following elementary global observation. If $H:U\to V$ is a homeomorphism and a continuous map $G:U\to V$ satisfies
\[
 |G(x)-H(x)|<\tfrac12\dist(H(x),\R^n\setminus V),
\]
then the straight homotopy stays in $V$. When $V$ is bounded, it is a proper homotopy: the preimage of a compact target set is contained, uniformly in the homotopy parameter, in the $H$-preimage of a compact set a positive distance from $\partial V$. This observation will be used together with local injectivity and degree, not as a substitute for local injectivity.
==== END Strong-diffeomorphic-approximation-in-three-dimensions-for-p-gt-2//sections/01-introduction.tex ====

==== BEGIN Strong-diffeomorphic-approximation-in-three-dimensions-for-p-gt-2//sections/02-smoothing.tex ====
\section{Qualitative topology and strong smoothing}
\label{sec:smoothing}

All PL structures in this Section are the ordinary structures on open
subsets of $\R^3$.  Triangulations are locally finite in the open domain.
In particular, neither a triangulation nor the local constants used below
need have uniform behavior at the boundary.

The qualitative PL tools belong to the three-dimensional triangulation
and approximation theory of Moise and Bing
\cite{Moise1952Approximation,Moise1952Triangulation,Bing1959}; Hamilton's
relative formulation \cite{Hamilton1976} is the precise interface used
below. Edwards--Kirby's deformation theorem supplies the supported
ambient extensions \cite{EdwardsKirby1971}. We separate these
topological existence statements from the derivative estimates:
finite local model libraries and subsequently chosen small exceptional
sets provide the latter.

\begin{theorem}[Strong smoothing of locally bi-Lipschitz maps]
\label{sm:smoothing}
Let $\Omega,\Omega'\subset\R^3$ be bounded domains, let
$1\le p<\infty$, and let $H:\Omega\to\Omega'$ be a locally
bi-Lipschitz homeomorphism in $W^{1,p}(\Omega;\R^3)$.  For every
$\eps>0$ and every continuous function $\xi:\Omega\to(0,\infty)$,
there is a $C^\infty$ diffeomorphism $g:\Omega\to\Omega'$ such that
\begin{equation}
 \label{sm:smoothing-conclusion}
 \norm{g-H}_{W^{1,p}(\Omega)}<\eps,
 \qquad |g(x)-H(x)|<\xi(x)\quad(x\in\Omega).
\end{equation}
In particular, $g$ belongs to $W^{1,p}(\Omega;\R^3)$, and the
approximations have exactly the target $\Omega'$.
\end{theorem}

The proof has two parts.  First we smooth a locally finite PL
homeomorphism with summable local derivative errors.  We then approximate
$H$ by such a PL map.  In the second part a finite collection of local
models gives derivative bounds before the measure of the exceptional
mesh cubes is made small.  The qualitative topology used to choose those
models supplies no derivative estimate.

\subsection{Relative qualitative tools}

\begin{lemma}[Fine relative PL approximation]
\label{sm:relative-pl}
Let $M,N$ be PL three-manifolds without boundary, with $N$ metrized,
and let $f:M\to N$ be a homeomorphism.  Suppose that $K\subset M$ is
closed and that $f$ is PL on an open neighborhood of $K$.  Given a
continuous function $\eta:M\to(0,\infty)$, there is a PL
homeomorphism $f_1:M\to N$ which agrees with $f$ on a neighborhood of
$K$ and satisfies
\[
 d_N(f_1(x),f(x))<\eta(x)\qquad(x\in M).
\]
Noncompact manifolds are allowed.  For manifolds with boundary the same
statement holds if $K$ contains $\partial M$ and $f$ is PL near $K$.
\end{lemma}

\begin{proof}
Choose a closed neighborhood $K^+$ of $K$ contained in the open set on
which $f$ is PL.  Transport the PL structure of $N$ to $M$ by $f$.
The identity from the original structure to the transported structure
is PL near $K^+$.  Hamilton's relative approximation Theorem
\cite[Section~3, Theorem~2.2, pp.~68--69]{Hamilton1976} applies to these
two structures.  Use the compatible metric
$d_f(x,y)=d_N(f(x),f(y))$ and the fine tolerance $\eta$.
It gives a homeomorphism $a$ that is PL between the two structures,
is the identity on $K^+$, and satisfies
$d_f(a(x),x)<\eta(x)$.  Then $f_1=f\circ a$ has all the asserted
properties.  The boundary condition in Hamilton's Theorem is precisely
the additional condition stated here; for an open Euclidean domain its
manifold boundary is empty.
\end{proof}

\begin{lemma}[Small local ambient extension]
\label{sm:local-extension}
Fix a closed Euclidean cube $B\subset\R^3$, compact sets $C,D\subset B$
with $C\subset\operatorname{int}B$, and an open neighborhood $O$ of
$C\cup D\cup\partial B$.  For every $\lambda>0$ there is a
$\delta>0$, depending only on these source sets and on $\lambda$,
with the following property.  If $e:O\to\R^3$ is an embedding,
\[
 \sup_{x\in O}|e(x)-x|<\delta,
 \qquad e=\Id\ \hbox{on }D\cup\partial B,
\]
then there is a homeomorphism $A:\R^3\to\R^3$ such that
\begin{equation}
 \label{sm:extension-conclusion}
 A=e\ \hbox{on }C,\qquad
 A=\Id\ \hbox{on }D\cup(\R^3\setminus\operatorname{int}B),
 \qquad \sup_{\R^3}|A-\Id|<\lambda.
\end{equation}
The number $\delta$ is uniform over a finite list of fixed normalized
configurations $(B,C,D,O)$.  No derivative bound on $e$ is required.
\end{lemma}

\begin{proof}
Put $E=C\cup D\cup\partial B$.  The deformation Theorem of
Edwards--Kirby \cite[Theorem~5.1]{EdwardsKirby1971}, applied at the
inclusion of $O$ in $\R^3$, deforms every sufficiently close embedding
$e$ through embeddings $e_t$, starting at $e_0=e$, so that
$e_1=\Id$ on $E$.  The deformation is unchanged outside a fixed
compact neighborhood $L$ of $E$ in $O$, and fixes the inclusion.
Its continuity at the inclusion permits us to require
\[
 |e_t(x)-x|<\lambda/2\quad(x\in L,\ 0\le t\le1),
 \qquad |e(x)-x|<\lambda/2\quad(x\in L).
\]
A sufficiently small uniform $\delta$ on $O$ meets the finitely many
compact-open conditions that specify this neighborhood of the
inclusion.  In the terminology of that Theorem a proper embedding
respects manifold boundaries; this condition is automatic for the
ambient manifold $\R^3$.

All the open images $e_t(O)$ coincide.  To see this, enclose $L$ in a
finite union of relatively compact subdomains of $O$ whose boundary collars are outside
the region where the deformation changes the map.  The embeddings
agree on that collar. Invariance of domain
\cite[Theorem~2B.3]{Hatcher2002} and degree relative to
the common boundary show that the images of the enclosed subdomain
are the same.  Outside it the maps already agree.
On this common open image define
\[
 A_0=e\circ e_1^{-1},
\]
and define $A_0$ to be the identity elsewhere.  The map is already
the identity off the compact set $e_1(L)$ in the common image, so
this is an ambient homeomorphism.  It agrees with $e$ on $E$ and has
displacement less than $\lambda$.  Since it fixes $\partial B$
pointwise, it preserves the bounded complementary component
$\operatorname{int}B$.  Restrict $A_0$ to $B$ and extend it by the
identity outside $B$ to obtain $A$.

The construction uses only the displayed source configuration and
continuity at the inclusion.  For finitely many normalized
configurations take the minimum of the resulting positive tolerances.
\end{proof}

We will use Lemma~\ref{sm:local-extension} when the input embedding is
defined by a small map $\eta$ near a new core and by the identity near
older protected sets.  The following observation specifies the required
gluing condition.  Choose open sets
$\overline{O_0}\subset U_0$, where $\eta$ is defined on $U_0$, and
an open set $V$ containing the protected sets, such that
$\eta=\Id$ on $U_0\cap V$.  If $\eta$ is sufficiently close to
the identity that $\eta(O_0)\subset U_0$, the map
\begin{equation}
 \label{sm:patched-embedding}
 e=\eta\ \hbox{on }O_0,\qquad e=\Id\ \hbox{on }V
\end{equation}
is an embedding.  Indeed, a collision $\eta(x)=y$ with
$x\in O_0$ and $y\in V$ has $y\in U_0\cap V$, whence
$\eta(y)=y$ and injectivity of $\eta$ gives $x=y$.  The open
embedding property then follows from invariance of domain.  All the
sets in this observation can be fixed in advance when the source
configurations range over a finite list.

\begin{lemma}[Fine control, properness, and the target]
\label{sm:proper-degree}
Let $h:\Omega\to\Omega'$ be an orientation-preserving homeomorphism
between bounded domains, and let $g:\Omega\to\R^3$ be a smooth
map with $\det Dg>0$.  If
\begin{equation}
 \label{sm:target-distance}
 |g(x)-h(x)|<\frac12\dist(h(x),\R^3\setminus\Omega')
 \qquad(x\in\Omega),
\end{equation}
then $g$ is a diffeomorphism of $\Omega$ onto $\Omega'$.
\end{lemma}

\begin{proof}
Write $d(y)=\dist(y,\R^3\setminus\Omega')$ and
$g_t=(1-t)h+tg$.  Condition~\eqref{sm:target-distance} puts the entire
segment $g_t(x)$ in $\Omega'$.  Since $d$ is 1-Lipschitz,
\[
 d(g_t(x))\le\frac32d(h(x)).
\]
If $K\Subset\Omega'$ and $m=\min_Kd>0$, the inverse image of $K$
under the homotopy lies in
\[
 h^{-1}\bigl(\{y\in\Omega':d(y)\ge2m/3\}\bigr)\times[0,1].
\]
The set in braces is compact because $\Omega'$ is bounded.
Thus $g_t$ is a proper homotopy.  Its endpoint $g$ has the same
degree as $h$, namely one.  A proper local diffeomorphism has
finitely many preimages of each point, and here each preimage has
positive local degree.  The degree-one identity therefore says that
every point of $\Omega'$ has exactly one preimage.  The local smooth
inverses patch to a smooth inverse on $\Omega'$.
\end{proof}

\subsection{Smoothing a locally finite PL homeomorphism}

Campbell--D'Onofrio--V\'itek proved diffeomorphic approximation of
locally finite piecewise affine homeomorphisms in dimensions three
and four, with uniform value control and derivative-error control
for both the map and its inverse
\cite[Theorem~A]{CampbellDOnofrioVitek2026}.
We give the local proof needed here, including arbitrary positive
continuous value tolerances.

\begin{proposition}[Strong PL smoothing]
\label{sm:pl-smoothing}
The conclusion of Theorem~\ref{sm:smoothing} holds if $H$ is replaced
by a locally finite PL homeomorphism
$h:\Omega\to\Omega'$ in $W^{1,p}(\Omega;\R^3)$.
\end{proposition}

\begin{proof}
An orientation-reversing Euclidean isometry in the target reduces
the proof to the orientation-preserving case.  Fix a locally finite
affine triangulation for $h$.  We prescribe summable error budgets
for its edges, vertices, and a locally finite family of compact sets
covering the remainder of the domain.  All modifications below can
also obey an arbitrary positive continuous value tolerance.

\paragraph{The open edges.}
Around each open edge choose a tube of radius $d(t)$, where
$0<t<l$ is its axial coordinate.  The tubes of distinct open edges
are disjoint, their radii are bounded by a small multiple of
$\min(t,l-t)$, and $d$ is exactly linear near both endpoints.
These choices follow from the finite geometry of each simplex star;
local finiteness makes them compatible throughout $\Omega$.
We may make $\norm{d'}_\infty$ small by decreasing the width.

In positively oriented orthogonal frames, and after translations,
the original map on this tube is
\begin{equation}
 \label{sm:edge-form}
 (t,r,\theta)\longmapsto
 \bigl(at+r b(\theta),\;r s(\theta)e_{\phi(\theta)}\bigr),
 \qquad e_\alpha=(\cos\alpha,\sin\alpha),
\end{equation}
where $a>0$, $s>0$, and the coefficients are smooth on the closed
angular sectors.  The transverse homogeneous map is injective:
otherwise two equal transverse images, taken at arbitrarily small
radius, would have their axial difference canceled by changing $t$,
contradicting injectivity of $h$ in the edge star.  Its angular
map is consequently an orientation-preserving circle
homeomorphism.  We choose a lift $\phi$ with
$\phi(\theta+2\pi)=\phi(\theta)+2\pi$.
Positive determinants on the finitely many sectors give upper and
positive lower bounds for $s$ and for the one-sided derivatives
$\phi'$.

Choose constants $c>0$ and $c'$, and for $0\le\tau\le1$ put
\[
 b_\tau=(1-\tau)b,\qquad
 s_\tau=(1-\tau)s+\tau c,\qquad
 \phi_\tau=(1-\tau)\phi+\tau(\theta+c').
\]
For fixed $\tau$, the determinant of the corresponding map in the
frame $(\partial_t,\partial_r,r^{-1}\partial_\theta)$ is
\begin{equation}
 \label{sm:edge-determinant}
 a s_\tau^2\phi_\tau'>0.
\end{equation}
On all the closed sectors and all $\tau\in[0,1]$ this has a
positive minimum.  Let $\chi$ be a smooth cutoff equal to 1 on
$(-\infty,-1]$ and to 0 on $[-1/2,\infty)$, and substitute
\[
 \tau(t,r)=\chi\!\left(\frac{\log(r/d(t))}{N}\right)
\]
in the preceding formula.  The additional derivative columns have
size at most
\begin{equation}
 \label{sm:edge-cutoff-error}
 \frac{C}{N}\bigl(1+\norm{d'}_\infty\bigr).
\end{equation}
Indeed, $|r\partial_r\tau|\le C/N$ and
$|r\partial_t\tau|\le C(r/d)|d'|/N$, while $r/d\le1$ on
the support of the modification.  The remaining factors come from
the fixed angular data.  Taking $N$ large preserves positive
determinants on every sector, including one-sided limits.
Near the axis the result is the nonsingular affine map
$(t,r,\theta)\mapsto(at,cr e_{\theta+c'})$.
Near the tube boundary it agrees with $h$.

All first derivatives are bounded by an edge-dependent constant
independent of a further common decrease of $d$.  The volume of the
tube tends to zero under that decrease.  Both its derivative error
and its value error therefore have arbitrarily small $W^{1,p}$
cost.  Choose these costs summably over the edges and call the
result $h_1$.  It is continuous and locally Lipschitz, fixes all
vertices, and has the required small total error.  Because the
tube widths are exactly linear near endpoints, $h_1-h_1(v)$ is
homogeneous of degree one on a sufficiently small ball about every
vertex $v$.

\paragraph{The faces and the punctured vertex balls.}
Away from the vertices, the closed convex hull of the nearby
almost-everywhere gradients of $h_1$ is contained in
$\GL^+(3)$ on a sufficiently small neighborhood of each point.
At a smooth point this follows by continuity.  At a nonsmooth
point the only remaining interface is one face.  If $A_-$ and
$A_+$ are its two gradient traces, continuity gives agreement on
the tangent plane, so $A_+-A_-$ has rank at most one with normal
covector to that face.  Consequently
\begin{equation}
 \label{sm:face-segment}
 \det\bigl((1-t)A_-+tA_+\bigr)
 =(1-t)\det A_-+t\det A_+>0\quad(0\le t\le1).
\end{equation}
The traces vary continuously on the two sides.  Their nearby
convex hull lies in a sufficiently small neighborhood of this
compact positive-determinant segment, proving the assertion.
Homogeneity makes this property uniform at relative scale on
a punctured ball about a vertex: cover the unit sphere by finitely
many of the corresponding neighborhoods and rescale.

Let $\rho\ge0$ be a smooth probability kernel supported in the
unit ball.  Away from vertices define
\begin{equation}
 \label{sm:variable-convolution}
 h_2(x)=\int_{\R^3}h_1(x+w(x)z)\rho(z)\,dz,
\end{equation}
where $w$ is smooth and positive there and the integration balls
lie in $\Omega$.  Its derivative is
\begin{equation}
 \label{sm:variable-convolution-derivative}
 Dh_2(x)=\int Dh_1(x+w(x)z)\rho(z)\,dz
 +\int \bigl(Dh_1(x+w(x)z)z\bigr)\otimes Dw(x)\rho(z)\,dz.
\end{equation}
The first integral is in the positive-determinant convex hull just
described.  Taking $w$ and $|Dw|$ sufficiently small locally makes
the second integral a small perturbation, and hence gives
$\det Dh_2>0$.  Formula~\eqref{sm:variable-convolution} is smooth
where $w>0$, as is also seen by writing its smooth variable kernel
in the integration variable $y=x+w(x)z$.

On a small punctured ball at $v$ choose
$w(x)=c_v|x-v|$, with $c_v>0$ sufficiently small.  This is
consistent with the relative-scale convex-hull condition and
makes $h_2-h_1(v)$ homogeneous on a smaller ball.  Set
$h_2(v)=h_1(v)$ there.  The required radius function exists by
a locally finite partition of unity with sufficiently small
positive coefficients.  Near the vertices blend this function
with $c_v|x-v|$ on disjoint annuli.  The annular cutoff terms
are bounded by a constant times $c_v$ if the competing radii
are chosen at that same small scale, so both the radius and
gradient restrictions survive.  This is also an instance of
the local minorant construction in Lemma~\ref{pre:local-tolerances}.

Here is the global error choice.  First take disjoint vertex balls
so small that their volumes, multiplied by the fixed local
gradient bounds to the power $p$, meet a summable error budget.
On these balls the gradients in
Equation~\eqref{sm:variable-convolution-derivative} are bounded independently
of a further decrease of their radii.  On a compact set outside
the balls, the local gradients are bounded and converge at every
piecewise smooth point as $w,|Dw|\to0$.  Dominated convergence
therefore gives an arbitrarily small derivative error on that
compact set.  A locally finite compact covering with summable
budgets makes the total $W^{1,p}$ error as small as prescribed.
The value error follows from local Lipschitz continuity and the
small radius.  These arguments include $p=1$.

\paragraph{The vertices.}
Center a homogeneous vertex ball at the origin and subtract
$h_1(v)$.  Write the resulting map as $F$.  Euler's identity is
$DF(x)x=F(x)$.  Since $DF(x)$ is invertible for $x\ne0$, $F(x)$
never vanishes there.  We can thus write
\begin{equation}
 \label{sm:vertex-form}
 F(r\theta)=rR(\theta)\Phi(\theta),\qquad
 R>0,\quad\theta\in S^2.
\end{equation}
Its positive determinant implies that $\Phi:S^2\to S^2$ is a
smooth orientation-preserving local diffeomorphism.  It is a
covering by compactness, and it is one-sheeted because $S^2$ is
simply connected.

Smale's Theorem gives a smooth isotopy $\Phi_\tau$ from $\Phi$
to a rotation, smooth jointly in $(\tau,\theta)$
\cite[Theorem~6]{Smale1959}; for smooth dependence on the sphere
diffeomorphism see also
\cite[Theorem~1.5]{LiWatts2011}.  Interpolate $R$ through positive
functions $R_\tau$ to a positive constant.  For fixed $\tau$ the
homogeneous map
$F_\tau(r\theta)=rR_\tau(\theta)\Phi_\tau(\theta)$
has determinant
\[
 R_\tau(\theta)^3 J_{S^2}\Phi_\tau(\theta)>0.
\]
The family has bounded first derivatives and a positive
determinant minimum, by compactness.  On an arbitrarily small
ball replace $F$ by $F_{\tau(r)}$, where $\tau$ is 0 near the
outer boundary, 1 near the center, and
$\sup_r|r\tau'(r)|$ is sufficiently small.  Such a transition
is obtained by spreading a fixed cutoff over a sufficiently
long interval of $\log r$.  Its extra derivative term is
bounded by $C|r\tau'(r)|$, so positive determinant persists.
At the center the map is a dilation followed by a rotation.
It is therefore smooth and nonsingular there.

The derivative bound for this last modification depends on the
fixed spherical isotopy but not on the support radius.  Shrinking
that radius gives arbitrarily small summable $W^{1,p}$ costs
over all vertices.  The resulting map $g$ is smooth and has
positive determinant throughout $\Omega$.

Finally, in each of the three operations impose local value
tolerances whose sum is less than
\[
 \min\!\left\{\xi(x),\frac12
 \dist(h(x),\R^3\setminus\Omega')\right\}.
\]
The preceding constructions permit these fine tolerances and a
total Sobolev error less than $\eps$.  Lemma~\ref{sm:proper-degree}
makes $g$ a diffeomorphism onto the original target.  This proves
the Proposition.
\end{proof}

It remains to approximate the locally bi-Lipschitz map by a PL
homeomorphism with strong derivative control.  The next two
subsections establish this reduction.

\subsection{Fine meshes and controlled patch insertion}

Fix a coarse, locally finite Whitney decomposition $\mathcal P$ of
$\Omega$ into closed dyadic cubes with disjoint interiors.  Choose
relatively compact enlarged neighborhoods $U_P$ that are still locally
finite.  Enlarge them a fixed finite number of times when necessary.
Since $H$ is locally bi-Lipschitz, there are constants $K_P\ge1$ such
that
\begin{equation}
 \label{sm:coarse-bilip}
 K_P^{-1}|x-y|\le |H(x)-H(y)|\le K_P|x-y|
 \qquad(x,y\in U_P).
\end{equation}
Enlarge $K_P$ also to bound $|DH|$ almost everywhere on $U_P$ in the
chosen matrix norm.  The constants may incorporate the data from finitely many neighboring
coarse cubes.  All references below to data local to $P$ allow this
finite enlargement, but never an enlargement depending on the final
fine-mesh resolution.

\begin{lemma}[Adapted dyadic meshes]
\label{sm:mesh}
There is an absolute integer $n_0$ with the following property.
Given arbitrary positive local upper bounds on the fine scale, there
is a locally finite dyadic cube decomposition $\mathcal Q$ of $\Omega$
with centers $c_Q$ and side lengths $l_Q$ for which, on writing
\begin{equation}
 \label{sm:patch-box}
 B_Q(s)=c_Q+[-s l_Q,s l_Q]^3,
\end{equation}
the following hold:
\begin{enumerate}[label=\textup{(\roman*)}]
 \item $B_Q(100)\Subset\Omega$, and cubes meeting $B_Q(50)$ have
 side lengths comparable to $l_Q$ by absolute constants.
 \item The normalized cube configurations in $B_Q(6)$ belong to a
 finite list.  There is a conforming affine triangulation of
 $\mathcal Q$ with finitely many normalized nondegenerate shapes
 on these configurations.
 \item The cubes have $n_0$ colors so that the closed boxes $B_Q(6)$
 of a single color are pairwise disjoint.
 \item Attach to $Q$ a coarse index $P(Q)$ containing $c_Q$, using
 a fixed convention on coarse boundaries.  The initial scale
 bounds can be reduced so that all interactions through any fixed
 number of neighboring patch layers are confined to a fixed
 locally finite graph of coarse indices, independent of further
 mesh refinement.  The corresponding patches lie in the
 neighborhoods on which the required bounds in
 Equation~\eqref{sm:coarse-bilip} hold.
\end{enumerate}
The upper bounds on the fine scale remain arbitrarily prescribable.
\end{lemma}

\begin{proof}
Choose a positive size function $s$ on $\Omega$ with sufficiently
small absolute Lipschitz constant, below all the prescribed local
bounds and below $10^{-3}\dist(x,\R^3\setminus\Omega)$.
Lemma~\ref{pre:local-tolerances} supplies such a minorant.  Take the
maximal dyadic cubes for which
\[
 l_Q\le\inf_Qs.
\]
Failure of the condition for the dyadic parent gives
$s(x)\le(2+C\Lip(s))l_Q$ on $Q$.  The small Lipschitz constant
then gives local comparability on $B_Q(50)$.  For example, choosing
it sufficiently small makes every dyadic ratio in this neighborhood
belong to $\{1/2,1,2\}$.  The distance bound puts $B_Q(100)$ inside
$\Omega$.  Maximality gives a decomposition, and the positive lower
bound for $s$ near every compact subset gives local finiteness.

The cube vertices have dyadic coordinates.  Bounded local scale
ratios and bounded normalized distance therefore give only finitely
many normalized configurations.  To triangulate, decompose each
cube face into its square contact facets with adjacent cubes.
Put every contact vertex and hanging subdivision point on the
perimeter of each such square.  Cone these perimeter segments from
the square center, and then cone the resulting triangulation of
each cube boundary from the cube center.  The constructions on
shared facets agree.  They give nondegenerate simplices, and their
normalized shapes range over a finite list.

Local comparability bounds the degree of the graph joining cubes
whose $B_Q(6)$ boxes meet.  A coloring with one more than that
degree proves \textup{(iii)}.  Finally choose the initial local
upper bounds so small relative to the coarse Whitney sizes that
the finitely many required patch layers stay in fixed coarse
neighborhoods.  Since the coarse enlarged cover is locally finite,
its interaction graph is locally finite.  Taking a still smaller
minorant $s$ does not change any of these conclusions.
\end{proof}

The finite lists record actual normalized geometric configurations,
including the positions of faces and vertices.  The same
dyadic-position argument applies in $B_Q(20)$, using the scale
comparability on $B_Q(50)$.  This includes the older protected
boxes needed below.  Indeed, if a box $B_R(t)$ with $t\le2$ meets
$B_Q(5)$, applying scale comparability at the larger cube gives
$l_R/l_Q\in\{1/2,1,2\}$.  Thus
$|c_R-c_Q|_\infty\le5l_Q+2l_R\le9l_Q$, and
$B_R(2)\subset B_Q(13)$.  Shrinking these boxes by one of
finitely many fixed amounts therefore leaves only finitely many
possible intersections and protection margins.

\begin{lemma}[Insertion with protected cores]
\label{sm:insertion}
Consider a patch $Q$ and a finite collection of older protected
boxes with the relative sizes and positions provided by
Lemma~\ref{sm:mesh}.  For every older box prescribe a larger
half-shrunk box and a smaller fully-shrunk box, separated by a
fixed positive normalized margin.  Let $G:\Omega\to\Omega'$ be
a homeomorphism, and let $h_Q$ be a PL embedding on a neighborhood
of $B_Q(3)$.  Assume that $h_Q=G$ on the overlaps with the
half-shrunk boxes.

For every $\lambda>0$ there is an $a>0$, depending only on
$\lambda$, the normalized source configuration, and a local
constant $K_P$ in Equation~\eqref{sm:coarse-bilip}, such that
\begin{equation}
 \label{sm:insertion-smallness}
 \sup_{B_Q(5)}|G-H|<a l_Q,
 \qquad \sup_{B_Q(3)}|h_Q-H|<a l_Q
\end{equation}
implies the following conclusion.  There is a domain
homeomorphism $A$ supported in $B_Q(5)$, with
\[
 \sup|A-\Id|<\lambda l_Q,
\]
such that $G\circ A=h_Q$ on a neighborhood of $B_Q(2)$ and
$G\circ A=G$ on the fully-shrunk older boxes.  The tolerances
are uniform over a finite list of source configurations.  They
do not depend on the derivatives of $G$ or of $h_Q$.
\end{lemma}

\begin{proof}
Scale the patch to $l_Q=1$.  First reduce $a$ using $K_P$ so
that $h_Q(B_Q(3))\subset G(\operatorname{int}B_Q(4))$.
Indeed, for $x\in B_Q(3)$ and $z\in\partial B_Q(4)$,
the distance $|H(z)-H(x)|$ is bounded below by $K_P^{-1}$.
The small errors in Equation~\eqref{sm:insertion-smallness} preserve
this separation.  Degree on $\operatorname{int}B_Q(4)$,
comparing $G$ with $H$ on the boundary, shows that $h_Q(x)$
is in its image.  The same statement holds with a small
margin around $B_Q(3)$.

The embedding
\[
 \eta=G^{-1}\circ h_Q
\]
is consequently defined there.  If $z=\eta(x)$, then
\begin{equation}
 \label{sm:inverse-value-bound}
 |z-x|\le K_P|H(z)-H(x)|
 \le K_P\bigl(|H(z)-G(z)|+|h_Q(x)-H(x)|\bigr)
 <2K_Pa.
\end{equation}
Thus its closeness to the inclusion uses no inverse-Lipschitz
bound for $G$.

Take a fixed closed neighborhood $C$ of $B_Q(2)$ inside
$B_Q(3)$, and let $D$ be the union of the fully-shrunk older
boxes intersected with $B_Q(5)$.  The half/full shrink margins
give neighborhoods on which the map $\eta$ near $C$ agrees
with the identity near $D$.  Include also an identity
neighborhood of $\partial B_Q(5)$, separated from $C$.
Choose these neighborhoods once for each normalized source
configuration.  By Equation~\eqref{sm:inverse-value-bound}, sufficiently
small $a$ makes the union of $\eta$ and these identity maps
an embedding, as in Equation~\eqref{sm:patched-embedding}.  Apply
Lemma~\ref{sm:local-extension} with $B=B_Q(5)$ and displacement
$\lambda$.  Precomposition by the resulting $A$ installs
$h_Q$ on $C$ and preserves the fully-shrunk boxes.  Rescaling
proves the Lemma.
\end{proof}

We record explicitly how tolerances in repeated insertions are chosen.
For a patch insertion supported in $B_Q(5)$,
\begin{equation}
 \label{sm:value-update}
 |G(A(x))-H(x)|
 \le |G(A(x))-H(A(x))|+K_P|A(x)-x|.
\end{equation}
After division by the local cube size, interacting patches introduce
only the bounded ratios from Lemma~\ref{sm:mesh}.  Hence, for a fixed
number of stages, all tolerances can be assigned by backward
recursion: start with the desired final value bounds; choose a small
allowed displacement at the last stage; use
Lemma~\ref{sm:insertion} to obtain its input tolerance; require the
preceding value error to be a small fraction of that tolerance; and
continue backward.  At a coarse index take the minimum over its
finitely many interacting indices at each step.  Only finitely many
steps occur.  Every tolerance remains positive, and none depends on a
PL derivative bound selected later.

\subsection{Finite PL libraries before resolution selection}

The use of finite normalized PL families with exact agreement on
earlier overlaps has a precedent in V\"ais\"al\"a's construction
\cite[Sections~2.7--2.11]{Vaisala1977}.

\begin{proposition}[Strong PL approximation of a locally bi-Lipschitz map]
\label{sm:bilip-pl}
Under the hypotheses of Theorem~\ref{sm:smoothing}, there is a
locally finite PL homeomorphism $G:\Omega\to\Omega'$ satisfying
\[
 \norm{G-H}_{W^{1,p}(\Omega)}<\eps,
 \qquad |G(x)-H(x)|<\xi(x)\quad(x\in\Omega).
\]
\end{proposition}

\begin{proof}
Fix the coarse neighborhoods, local bounds $K_P$, and finite
mesh configurations of Lemma~\ref{sm:mesh}.  We use two rounds of
$n_0$ insertion stages.  The first installs a single piecewise
affine interpolant on the cubes where $H$ is nearly affine; its
gradients there are close to $DH$.  The second installs PL models
on every cube while preserving protected neighborhoods of the
first-round cubes.  It gives
local derivative bounds everywhere, fixed before the remaining
cubes are made small in measure.  For the next steps, let
$\mathcal Q$ be any admissible mesh.  We choose the tolerances
and model libraries uniformly over all such meshes; the mesh
used for the final map will be selected only after the libraries
and their derivative bounds have been fixed.

Write $S=2n_0$ for the number of stages.  A newly installed patch
has protected radius 2.  At each subsequent stage decrease
the radius of every older protected patch by
\begin{equation}
 \label{sm:shrink-size}
 d_*=(4n_0+4)^{-1}
\end{equation}
in its own $B_Q$ coordinates.  Its final radius is greater
than $3/2$, and in particular its closed original cube $Q$
remains in the interior of its protected core.  At an
insertion we use half of the current prescribed decrease
for agreement of the new model and the old map, and the
full decrease for the sets fixed by the ambient change.
These are precisely the margins in
Lemma~\ref{sm:insertion}.

Choose all insertion and displacement tolerances by the
backward procedure following Equation~\eqref{sm:value-update}.
They can be chosen to give an arbitrarily small final
normalized value error on every patch.  If $a_{s,P}$ is
the accuracy required of a new model at stage $s$, require
the current $G-H$ bound before that stage to be less than
$a_{s,P}/8$, also on its interacting neighborhoods.  This
additional fraction is imposed during the same backward
recursion.  All these numbers are now fixed.

\paragraph{Quantized affine interpolation and good patches.}
Choose positive thresholds $\delta_P$ so small that errors
$C\delta_P$ suffice at every first-round insertion involving
that coarse neighborhood, and so that
\begin{equation}
 \label{sm:good-error-budget}
 \sum_{P\in\mathcal P}(C\delta_P)^p |U_P|<\eps^p/8.
\end{equation}
Here and below a local tolerance is reduced using finitely many
neighboring indices when necessary.  We also require
$C\delta_P<(2K_P)^{-1}$, where $C$ is the interpolation
constant for the finite mesh shapes.

We quantize the interpolation data so that their exact values on
protected overlaps, after normalization, have only finitely many
possibilities.  At each vertex $v$ of the fine triangulation, round $H(v)$
to a target dyadic grid whose spacing is a small dyadic
multiple of the smallest incident cube size.  The dyadic
factor is fixed from the coarse indices of those cubes,
small enough that all the corresponding normalized
rounding errors are at most $\delta_P$.  Interpolate
the rounded values affinely on the conforming
triangulation to obtain a continuous piecewise affine
map $L$.  We do not assert that $L$ is globally injective.

Call a cube $Q$, with $P=P(Q)$, good if there is an affine
map $A_Q$ of bi-Lipschitz constant at most $2K_P$ such that
\begin{equation}
 \label{sm:good-tests}
 \sup_{B_Q(20)}|H-A_Q|\le\delta_P l_Q,
 \qquad
 \left(\frac{1}{|Q|}\int_Q|DH-DA_Q|^p\right)^{1/p}\le\delta_P.
\end{equation}
Changing the fixed factor 20 by a larger absolute factor
would have the same effect.  The rounding and the finite
shape list imply
\begin{equation}
 \label{sm:good-interpolation}
 \begin{aligned}
 |DL-DA_Q|&\le C\delta_P
 &&\text{near }B_Q(3),\\
 \sup_{B_Q(3)}|L-H|&\le C\delta_P l_Q.
 \end{aligned}
\end{equation}
For completeness, subtract $A_Q$ at the vertices of any
such tetrahedron.  The vertex errors are at most
$C\delta_P l_Q$; the inverse matrix of its three edge
vectors has norm at most $C/l_Q$, by finite normalized
shapes.  Multiplication gives the first estimate.
The value estimate follows by affine interpolation
and Equation~\eqref{sm:good-tests}.

The first estimate in Equation~\eqref{sm:good-interpolation} makes
$L$ an embedding on a neighborhood of $B_Q(3)$.
Indeed, $L-A_Q$ is Lipschitz on a slightly larger convex
box with constant at most $C\delta_P$; integrate its
piecewise constant derivatives on segments.  Thus
\[
 |L(x)-L(y)|\ge
 \bigl((2K_P)^{-1}-C\delta_P\bigr)|x-y|>0.
\]
Invariance of domain gives the open embedding assertion.

\paragraph{The first round.}
Start with $G_0=H$.  For stages $1,\ldots,n_0$ process the
colors in order, inserting a patch only if it is good,
and taking its model to be $h_Q=L$.  On every overlap
with an older protected core this agrees with the
already installed map, because both equal the single
global map $L$.  Equation~\eqref{sm:good-interpolation}
and the chosen thresholds supply the input accuracy
for Lemma~\ref{sm:insertion}.  Supports $B_Q(5)$ of one
color are disjoint, so these insertions are simultaneous.
Their local finiteness makes the union a domain
homeomorphism.  The value invariants follow from
Equation~\eqref{sm:value-update}.  At the end of this round the
map agrees with $L$ on every good cube and on a
protected neighborhood of it.

\paragraph{Normalized data and their finiteness.}
For each $Q$ choose $m_Q\in\Z^3$ nearest to
$H(c_Q)/l_Q$ and use normalized coordinates
\begin{equation}
 \label{sm:normalization}
 z=\frac{x-c_Q}{l_Q},\qquad
 \widehat H_Q(z)=\frac{H(c_Q+l_Qz)-l_Qm_Q}{l_Q}.
\end{equation}
The value at $z=0$ is bounded by an absolute constant,
and Equation~\eqref{sm:coarse-bilip} gives a uniform local
Lipschitz bound.  The normalized vertex values of $L$
on $B_Q(4)$ lie in a bounded set on finitely many
dyadic grids.  They therefore take only finitely many
values for each coarse index.  Here it matters that
the translation $l_Qm_Q$ is included in the data:
without it one would not have finiteness of affine
maps, as opposed to finiteness of their gradients.

If $R$ interacts with $Q$, then
\begin{equation}
 \label{sm:relative-origins}
 \frac{l_Rm_R-l_Qm_Q}{l_Q}
\end{equation}
is a bounded dyadic vector with a denominator from a
finite list.  Boundedness follows by comparing
$H(c_R)$ and $H(c_Q)$ using Equation~\eqref{sm:coarse-bilip};
the denominators are fixed by the bounded dyadic
scale ratios.  Consequently the first-round exact
PL data, as viewed in any interacting normalized
patch, have only finitely many possibilities.

\paragraph{Choosing the second-round libraries.}
For stages $n_0+1,\ldots,2n_0$ we insert at every
patch of the current color.  The models are chosen
as follows, before the final fine scale is selected.
For each coarse index, the normalized maps
$\widehat H_Q$ on a fixed larger patch form a
uniformly bounded equicontinuous family.  They
admit finitely many sup-norm bins of any prescribed
positive diameter.  At stage $s$ use diameter less
than $a_{s,P}/8$.

Inductively assume that the already installed normalized PL models
have finite lists.  Let $\widehat h_R$ be an older model in its own
$R$-normalization.  In the normalized coordinates of an interacting
patch $Q$, this same map is
\begin{equation}
 \label{sm:transported-model}
 z\longmapsto
 \frac{l_R}{l_Q}\widehat h_R\!\left(
       \frac{c_Q-c_R}{l_R}+\frac{l_Q}{l_R}z\right)
       +\frac{l_Rm_R-l_Qm_Q}{l_Q}.
\end{equation}
The source similarities and protected-core positions have finitely
many possibilities by Lemma~\ref{sm:mesh} and
Equation~\eqref{sm:shrink-size}; the target translations do as well
by Equation~\eqref{sm:relative-origins}.  Only boundedly many older
cores meet $B_Q(5)$, and their coarse indices lie in a fixed finite
neighborhood.  Record each older model's library label, its
normalized source placement and target translation, and its current
protected radius.  These records determine the model on the whole
protected core and have only finitely many possibilities.  In
particular, their restrictions to the half-shrunk overlaps give a
finite list of exact PL data.  No quantization of the later models'
affine coefficients is needed.  There are consequently only finitely
many combinations of
\begin{equation}
 \label{sm:model-combination}
 \text{source configuration, normalized }H\text{-bin,
 and exact protected-model records}.
\end{equation}

For each combination which can occur, select one
representative normalized embedding $G^*$, close
to a map $H^*$ in that bin by the prescribed
current value tolerance, and realizing the
specified older data.  Take the representative
on the fixed open patch $(-4,4)^3$.  If
$\widehat K_i$ are the normalized half-shrunk
older boxes, the set
\[
 K=[-3,3]^3\cap\bigcup_i\widehat K_i
\]
is compact in that open patch.  It lies
strictly inside the currently protected
cores, so $G^*$ is PL on a neighborhood of
$K$.  Apply Lemma~\ref{sm:relative-pl} to
$G^*$ on the open patch, keeping $K$ fixed.
Obtain a PL embedding $h^*$ there with
uniform error less than $a_{s,P}/8$ on the
required compact patch.

The full protected-model records also ensure agreement on a
neighborhood of $K$ with every actual tuple having the same
combination: both its current map and $G^*$ equal the recorded
models on the protected cores, which contain the half-shrunk
boxes in their interiors.  Relative approximation makes $h^*=G^*$
on a neighborhood of $K$.  Since there are only finitely many
closed half-shrunk boxes, we can choose a neighborhood $N$ of
$[-3,3]^3$ inside $(-4,4)^3$ whose intersection with each such box
lies in this common agreement neighborhood.  Store $h^*|_N$ as the library
entry indexed by the coarse region, stage, and combination in
Equation~\eqref{sm:model-combination}.  Its entire overlap with
each half-shrunk box now has the equality required by
Lemma~\ref{sm:insertion}.

If a different actual tuple has the same
combination in Equation~\eqref{sm:model-combination}, this
single model agrees with all of its prescribed
PL overlap data.  Moreover,
\begin{equation}
 \label{sm:reuse-error}
 \norm{h^*-\widehat H_Q}_{\infty}
 \le \norm{h^*-G^*}_{\infty}
     +\norm{G^*-H^*}_{\infty}
     +\norm{H^*-\widehat H_Q}_{\infty}
 <\frac38 a_{s,P}.
\end{equation}
The three norms are on the fixed patch needed
for insertion.  The actual current map is
already within $a_{s,P}/8$ of its $H$, so
Lemma~\ref{sm:insertion} applies after undoing
the normalization.  This installs $h_Q$ while
preserving the fully-shrunk older cores.

This selection does not require compactness of
the family of possible intermediate maps $G$.
Only the $H$-family is binned; the intermediate
map is used as an existence witness for an
embedding realizing the exact finite overlap
data.  All possible normalized meshes and
inputs obeying the fixed coarse bounds may
be included when deciding which combinations
occur.  At any stage the choices for a coarse
index depend only on libraries from strictly
earlier stages at its finitely many interacting
indices, so they can be made simultaneously
over all coarse indices.  Hence the choices are independent of
the eventual mesh resolution.

Each stored map is PL on a neighborhood of a
fixed compact patch and thus has finitely many
affine pieces there.  It has a finite upper
derivative bound, unchanged by the common source and target
rescaling.  There are finitely many
stored maps for each local coarse combination,
and only finitely many stages.  After the second round every cube
retains its own installed model on a protected neighborhood,
because its protected radius remains greater than $3/2$.
Thus the final derivative on that cube comes from a stored PL
model, not from an ambient transition.  This proves by induction
the finiteness of the data needed at the next stage and gives
constants $M_P<\infty$ such that the final map satisfies
\begin{equation}
 \label{sm:model-derivative-bound}
 |DG|\le M_P\quad\hbox{almost everywhere on }Q,
 \qquad P=P(Q).
\end{equation}
The constants $M_P$ are fixed before reducing
the fine scale.  They may be arbitrarily large;
no efficient dependence on $K_P$ is asserted.

The final map is locally PL.  If a single triangulation is
desired, take common refinements of the
finitely many affine subdivisions meeting
each compact set and triangulate the resulting
polyhedral cells.  Local finiteness gives a
locally finite affine triangulation on
$\Omega$.  The map is still onto $\Omega'$
because every stage was a precomposition by
a domain homeomorphism.  On every good cube
the map still equals $L$, since its first-round
protected neighborhood was preserved throughout.

\paragraph{Selecting the resolution and paying for bad cubes.}
Only now choose the final local upper size
bounds of Lemma~\ref{sm:mesh}.  On a fixed
coarse compact region, almost every $x$ is
both a differentiability point of $H$ and a
Lebesgue point of $DH$.  The derivative at
such a point is invertible, with the upper
and lower bounds inherited from local
bi-Lipschitzness.  The affine map
\[
 A_x(y)=H(x)+DH(x)(y-x)
\]
then passes both tests in
Equation~\eqref{sm:good-tests} on every sufficiently
small cube containing $x$.  Indeed its
enlargement is contained in a ball of radius
$C l_Q$ about $x$, which gives the sup-norm
test by differentiability.  The volume of
that ball is at most a fixed multiple of
$|Q|$, which gives the derivative mean test
by the Lebesgue-point property.  Only
finitely many coarse thresholds occur near
a fixed compact set.

It follows that the measure of the union of
bad cubes with a fixed coarse index $P$
tends to zero as their local upper side
length tends to zero.  This statement is
uniform over the admissible meshes.  One
can see the uniformity by taking the union
of all bad dyadic cubes with that index
and side at most $r$.  These measurable
unions decrease as $r\downarrow0$, and
their intersection is null by the preceding
pointwise argument.  They lie in the fixed
finite-volume neighborhood $U_P$.

Choose positive numbers $b_P$ with
$\sum_Pb_P<\eps^p/8$, and reduce the
local mesh bounds until
\begin{equation}
 \label{sm:bad-error-budget}
 (M_P+K_P)^p
 \left|\bigcup_{\substack{Q\text{ bad}\\P(Q)=P}}Q\right|
 <b_P.
\end{equation}
All bounds can be imposed simultaneously
using Lemma~\ref{sm:mesh}.  The $M_P$ in
this inequality were fixed in
Equation~\eqref{sm:model-derivative-bound}; they
are unaffected by this refinement.

The stage tolerances also bound
$|G-H|/l_Q$ by fixed local constants
on each cube.  Further reduce the
local size bounds so that the actual
value error is below $\xi$ and has
$p$th integral less than the remaining
part of $\eps^p$.  For example require
it to be below a global constant
$\eps/[4(1+|\Omega|)^{1/p}]$ and below
the positive minimum of $\xi/2$ on
the relevant compact neighborhood.
These reductions only help the bad-cube
estimate.  Choose a mesh satisfying all these bounds and carry
out the two rounds with the already selected libraries, obtaining
the final map $G$.

On good cubes, Equation~\eqref{sm:good-tests} and
Equation~\eqref{sm:good-interpolation} give
\[
 \int_Q|DG-DH|^p\le(C\delta_P)^p|Q|.
\]
On bad cubes use
Equation~\eqref{sm:model-derivative-bound} and
Equation~\eqref{sm:coarse-bilip}.  Summing and
using Equation~\eqref{sm:good-error-budget} and
Equation~\eqref{sm:bad-error-budget} makes the
derivative error less than the allocated
part of $\eps^p$.

The map is locally Lipschitz,
its derivative is globally $p$-integrable
by the estimates just proved, and its
values lie in the bounded target.
Thus it belongs to $W^{1,p}(\Omega;\R^3)$
and has the required strong error.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{sm:smoothing}]
Apply Proposition~\ref{sm:bilip-pl} with
Sobolev tolerance $\eps/2$ and value
tolerance $\xi/2$, obtaining a PL
homeomorphism $h:\Omega\to\Omega'$.
Apply Proposition~\ref{sm:pl-smoothing}
to $h$ with the same tolerances.
The triangle inequality proves
Equation~\eqref{sm:smoothing-conclusion}.
Both approximations have exactly the
target $\Omega'$.
\end{proof}
==== END Strong-diffeomorphic-approximation-in-three-dimensions-for-p-gt-2//sections/02-smoothing.tex ====

==== BEGIN Strong-diffeomorphic-approximation-in-three-dimensions-for-p-gt-2//sections/05-high-preparation.tex ====
\section{Retained rank data and an exact carrier}\label{sec:high-preparation}

Our aim is to retain the source energy carried by the regular rank-one
and rank-two points while preparing most of the target for exact local
replacements.  The retained data will first be flattened onto separated
plates, with only a small change to their derivatives.  On opposite
sides of each plate we select paired target rectangles.  The carrier
of Proposition~\ref{hp:carrier} will agree with affine or PL
homeomorphisms over neighborhoods of these rectangles and over an open
set with arbitrarily small residual target volume, while agreeing with
the chosen map near the retained source data.  The order of
these two requirements explains its two rounds: endpoint neighborhoods
are fixed first; the remaining target volume is prescribed afterwards.

The carrier may collapse isolated tame balls.  Such fibres permit the
energy-controlled local replacements below and can later be opened
topologically.  For $2<p<3$, the replacements can also leave marked
source interiors whose energy is treated by a separate compression.
The final two subsections prepare that compression and the eventual
full-rank correction used in Section~\ref{sec:high-assembly}.

\subsection{Regular points and fine tolerances}

Throughout this section, $p>2$ and the topological orientation of $f$
is positive.  A reflection reduces the other
orientation to this one.  All regularity assertions below are local on
the open sets; no boundary regularity is involved.  A map has the volume
Lusin property $N$ if it maps sets of three-dimensional Lebesgue
measure zero to sets of measure zero.

We use the weighted area formula in the form recorded by
Simon~\cite[Chapter~2, Section~3, Equations~(3.4)--(3.5)]{Simon2014Draft}.
The general Lipschitz-decomposition background for Sobolev maps is
provided by Bojarski--Haj\l asz~\cite[Theorem~3(1)]{BojarskiHajlasz1993}.
The stronger ordinary differentiability and image-null assertions
needed here are proved next, using openness and two-dimensional
Sobolev estimates on spheres.

\begin{lemma}[Regular points of a Sobolev homeomorphism]\label{hp:regularity}
Let $u:G\to G'$ be a homeomorphism between open subsets of $\R^3$,
with $u\in W^{1,p}_{\mathrm{loc}}(G;\R^3)$ and $p>2$.
There is a Borel set $R_u\subset G$ of full measure such that, for
$x\in R_u$, the weak differential $A=Du(x)$ is also the Fr\'echet
differential and
\begin{equation}\label{hp:regular-point-tests}
 \lim_{r\downarrow0}\sup_{0<|z-x|\le r}
 \frac{|u(z)-u(x)-A(z-x)|}{|z-x|}=0,
 \qquad
 \lim_{r\downarrow0}\frac{1}{|B_r|}
 \int_{B_r(x)}|Du-A|^p=0.
\end{equation}
If $u$ has positive topological orientation, then $\det A>0$ at
all points of $R_u$ where $A$ is invertible.  Moreover,
\begin{equation}\label{hp:deficient-image-null}
 \cL^3\bigl(u(\{x\in R_u:\rank Du(x)<3\})\bigr)=0.
\end{equation}
If $p\ge3$, $u$ has the volume Lusin property $N$ locally, and hence
on $G$.  Finally, for every fixed coordinate direction, almost every
parallel coordinate-plane section of $G$ has image under $u$ of
three-dimensional measure zero.  This last assertion holds for every
continuous $W^{1,p}_{\mathrm{loc}}$ map, without injectivity.
\end{lemma}

\begin{proof}
We first pass from Sobolev differentiation in mean to ordinary
Fr\'echet differentiation.  At almost every $x$, with $A=Du(x)$,
\begin{equation}\label{hp:rescaled-jet-small}
 w_r(\xi):=\frac{u(x+r\xi)-u(x)}{r}-A\xi
 \longrightarrow0 \quad\hbox{in }W^{1,p}(B_2;\R^3).
\end{equation}
For completeness, the gradient assertion is the Lebesgue differentiation
Theorem applied to $Du$.  To obtain the value assertion, subtract the
affine function with gradient $A$ and apply Poincar\'e's inequality on
concentric balls.  The difference between its mean on a ball of radius
$t$ and on the concentric ball of radius $t/2$ is bounded by
\[
 Ct\left(\frac{1}{|B_t|}\int_{B_t(x)}|Du-A|^p\right)^{1/p}.
\]
Summing on dyadic radii and using the precise value $u(x)$ gives the
value assertion in \eqref{hp:rescaled-jet-small}.

Fix $0<d<1/8$.  For every $z\in B_1$, slicing the annulus
$B_{2d}(z)\setminus B_d(z)$ gives a radius $\rho\in(d,2d)$ for which
\[
 \int_{\partial B_\rho(z)}
   (|w_r|^p+|D_\tau w_r|^p)
 \le d^{-1}\|w_r\|_{W^{1,p}(B_2)}^p.
\]
The trace agrees with the continuous restriction.  Sobolev embedding
on the two-dimensional sphere, using $p>2$, therefore gives
\[
 \sup_{\partial B_\rho(z)}|w_r|
 \le C_{p,d}\|w_r\|_{W^{1,p}(B_2)}.
\]
The constant is independent of $z$.  Each coordinate of the
homeomorphism
$u_r(\xi)=(u(x+r\xi)-u(x))/r$ attains its maximum and minimum on
the boundary of $\overline B_\rho(z)$: an extremum in the interior
would contradict openness of $u_r$.  Comparing the affine function
$A\xi$ at $z$ and on this sphere yields
\[
 \sup_{z\in B_1}|w_r(z)|
 \le C\|A\|_{\op}d+C_{p,d}\|w_r\|_{W^{1,p}(B_2)}.
\]
First let $r\downarrow0$, then $d\downarrow0$.  We obtain uniform
affine approximation on $B_r(x)$, which is equivalent to the first
assertion of \eqref{hp:regular-point-tests}.  Intersecting the full
measure sets used here gives a Borel choice of $R_u$.

If $A$ is invertible, uniform approximation on a sufficiently small
sphere makes $u(x+r\xi)-u(x)$ homotopic, through maps avoiding zero,
to $rA\xi$.  The local degree of the homeomorphism is therefore
$\operatorname{sgn}\det A$, proving the orientation assertion.

The set of Fr\'echet differentiability points admits a countable
Lipschitz decomposition.  Indeed, restrict first to points where
$|Du|\le m$ and
\[
 |u(z)-u(x)-Du(x)(z-x)|\le |z-x|
 \quad(0<|z-x|<1/m),
\]
and then partition into sets of diameter less than $1/m$ in an
interior exhaustion.  On each resulting set the restriction of $u$
is Lipschitz.  Such sets cover every differentiability point after
varying $m$.  On a Lipschitz extension of each restriction, its
approximate differential agrees almost everywhere on that set with
$Du$.  The area formula, applied to the subset where $\rank Du<3$,
now proves \eqref{hp:deficient-image-null}.  Source null subsets of
these Lipschitz pieces have null images as well, so the conclusion
concerns the entire regular deficient-rank set, not merely a source
full-measure subset of it.

For the volume property $N$, it suffices to use $p=3$ locally.
For an interior cube $Q$, choose a sphere of radius comparable to
its side length, enclosing $Q$ and contained in a fixed enlargement
$C_0Q\Subset G$.  The sphere oscillation inequality and slicing give
\[
 (\operatorname{osc}_Q u)^3
 \le C\int_{C_0Q}|Du|^3.                 \tag{$*$}
\]
Here openness again bounds coordinate oscillations inside the sphere
by those on its boundary.  If $E\Subset G$ is null, choose an open
neighborhood $O\Subset G$ of $E$ with arbitrarily small
$\int_O|Du|^3$.  A Whitney decomposition of $O$ may be chosen with
$C_0Q\subset O$ and bounded overlap of these enlargements.  Summing
$(*)$ over the cubes that cover $E$ bounds the outer volume of $u(E)$
by $C\int_O|Du|^3$.  This proves the claim by exhaustion.  When
$p>3$, local $L^3$ integrability follows from H\"older's inequality.

For the last assertion, Fubini gives a continuous $W^{1,p}$
restriction to almost every coordinate-plane section.  On a compact
square in such a section, divide into squares $Q$ of side length
$a$, allowing uniformly bounded enlargements inside a slightly larger
compact square.  Planar Morrey's inequality and H\"older's inequality
give
\[
 \begin{split}
 \sum_Q\diam(u(Q))^2
 &\le C\sum_Q
 a^{2-4/p}\left(\int_{2Q}|D_\tau u|^p\right)^{2/p}\\
 &\le C\left(\sum_Q a^2\right)^{1-2/p}
       \left(\sum_Q\int_{2Q}|D_\tau u|^p\right)^{2/p}.
 \end{split}
\]
This bound is independent of $a$.  Uniform continuity gives
\[
 \max_Q\diam(u(Q))\longrightarrow0,
 \qquad \sum_Q\diam(u(Q))^3\longrightarrow0.
\]
A countable exhaustion proves the three-dimensional null-image
assertion.
\end{proof}

\begin{remark}[Fine value tolerances]\label{hp:fine-tolerance}
Lemma~\ref{pre:local-tolerances} supplies, for every positive continuous
function $a$ on an open set $G$ and every $L>0$, a positive
$L$-Lipschitz minorant $a_0\le a$ with
$a_0(x)\le L\dist(x,\R^3\setminus G)$.  We use these minorants to
combine the fine value requirements below, including distance to the
target boundary and previously fixed compact-set accuracies.
\end{remark}


\subsection{Flattening the deficient positive ranks in the target}

The regular deficient-rank image has zero target volume, but the
integral of $|Df|^p$ over its source preimage need not be small.  We
therefore organize this part of the map rather than discard it.
The next construction moves almost all of its weighted data onto
separated flat plates and preserves their tangential derivatives.

Let $R_f$ be the regular set from Lemma~\ref{hp:regularity}, and set
\[
 D=\{x\in R_f:1\le\rank Df(x)\le2\},
 \qquad
 \mu(E)=\int_{E\cap D}(1+|Df|^p)\,dx,
 \qquad
 \nu=f_\#(\mathbf1_\Omega\,\cL^3).
\]
The finite measure $\mu$ is the measure used for all losses in this
preparation.  Boundedness of $\Omega$ and $f\in W^{1,p}$ ensure
its finiteness.

The decomposability bundle $V(\nu,y)$ records the tangent directions
of rectifiable curves whose averaged measures are absolutely continuous
with respect to $\nu$.  We use the precise locality and differentiability
properties cited in the proof below to find a direction transverse to
this bundle on the retained data.

\begin{lemma}[A forbidden cone for the transported measure]
\label{hp:transported-bundle}
Let $V(\nu,y)$ be the decomposability bundle of $\nu$.  Then
\begin{equation}\label{hp:range-in-bundle}
 \operatorname{im}Df(x)\subset V(\nu,f(x))
 \quad\hbox{for almost every }x\in\Omega.
\end{equation}
Moreover, $V(\nu,y)\ne\R^3$ for $\nu$-almost every $y\in f(D)$.
Consequently, for each $0<\lambda<1$, $f(D)$ can be partitioned,
up to a $\nu$-null set, into finitely many measurable pieces with
associated unit vectors $e$ such that
\begin{equation}\label{hp:almost-normal-direction}
 |\operatorname{proj}_{V(\nu,y)}e|<\lambda/10
 \quad\hbox{on the corresponding piece}.
\end{equation}
In coordinates $(h,v)\in e^\perp\times\R$, the bundle on that piece
contains no nonzero vector in the double cone
\[
 \{(a,b):|a|\le |b|/\lambda\}.
\]
\end{lemma}

\begin{proof}
We use the decomposability bundle and its strong locality property
from Alberti--Marchese \cite[Section~2.6 and Proposition~2.9(i)]
{AlbertiMarchese2016}.  In particular, tangent directions of measurable
families of rectifiable curve measures belong to $V(\nu,\cdot)$
whenever the aggregate curve measure is absolutely continuous with
respect to $\nu$.  Weighted restrictions are allowed: a density can
be absorbed by restriction and the layer-cake formula.

Fix a coordinate direction $e_i$ and an interior rectangular box.
For almost every line in that direction, the restriction of $f$ is
absolutely continuous.  It is also injective, so the one-dimensional
area formula identifies the pushforward of
$|\partial_i f|\,dt$ with length measure on the image curve.
Integration over the transverse coordinates gives the aggregate
measure
\[
 f_\#(\mathbf1_{\text{box}}|\partial_i f|\,dx)\ll\nu.
\]
The images are rectifiable for almost every line; if necessary they
may be split into bounded-length pieces or parametrized by arclength.
At almost every parameter where $\partial_i f\ne0$, their tangent
line is $\operatorname{span}\{\partial_i f\}$.  The defining property
of the bundle therefore gives
$\partial_i f(x)\in V(\nu,f(x))$ for almost every $x$ in the box
where that derivative is nonzero.  A countable box exhaustion and the
three coordinate directions prove \eqref{hp:range-in-bundle}.

By Lemma~\ref{hp:regularity}, $f(D)$ has zero volume.  Suppose that
$A\subset f(D)$ has positive $\nu$ measure and
$V(\nu,y)=\R^3$ on $A$.  The differentiability Theorem of
Alberti--Marchese \cite[Theorem~1.1(i)]{AlbertiMarchese2016} says that
every Lipschitz function on $\R^3$ is ordinarily differentiable
$\nu\vert_A$-almost everywhere.  The converse to Rademacher's Theorem
of De Philippis--Rindler \cite[Theorem~1.14]{DePhilippisRindler2016}
then implies $\nu\vert_A\ll\cL^3$, which is impossible since
$A\subset f(D)$ is null.  This proves properness.  This invocation
uses the measure-theoretic converse to Rademacher, rather than an
uncited inference from the existence of individual curve families.

Choose a finite sufficiently fine net of directions on the unit
sphere.  Every proper linear subspace has a unit normal, so one
of these directions satisfies \eqref{hp:almost-normal-direction}.
Assign the first such direction measurably.  If $(a,b)$ belongs to
the displayed cone and is nonzero, then
\[
 \frac{|b|}{\sqrt{|a|^2+b^2}}
 \ge\frac{\lambda}{\sqrt{1+\lambda^2}}>\lambda/2,
\]
whereas any vector $w\in V(\nu,y)$ obeys
$|e\cdot w|< (\lambda/10)|w|$.  The cone is therefore forbidden.
\end{proof}

The simultaneous null-set conclusion below is a graph-parametrized form of
Alberti--Marchese's Lemma~7.5.  Its compact-convex selection mechanism
is Rainwater's lemma, as presented in
\cite[Lemma~7.1 and Corollary~7.2]{AlbertiMarchese2016}.
We give the argument because one common exceptional set for every
allowed graph is essential to the subsequent flattening.

\begin{lemma}[A simultaneous graph-null set]\label{hp:cone-null}
Fix orthogonal coordinates $(h,v)\in\R^2\times\R$ and
$0<\lambda<1$.  Suppose a finite measure $\sigma$ is concentrated on a
Borel set $A$ and is absolutely continuous with respect to $\nu$,
and that $V(\nu,y)$ contains no nonzero vector in
$\{(a,b):|a|\le |b|/\lambda\}$ for $\sigma$-almost every $y$.
There is a Borel set $A_0\subset A$ of full $\sigma$ measure such that
\begin{equation}\label{hp:graph-null}
 \bigl|\{v:(h(v),v)\in A_0\}\bigr|=0
\end{equation}
for every $1/\lambda$-Lipschitz graph $h$ on any interval.
\end{lemma}

\begin{proof}
We first work in a compact rectangle $X=H\times I$, where $H$ is a
closed horizontal box and $I$ a compact vertical interval.  Let
$\Gamma$ be the compact family of all $1/\lambda$-Lipschitz maps
$h:I\to H$, equipped with the uniform topology.  Write
\[
 \rho_h=(v\mapsto(h(v),v))_\#(\cL^1\vert_I),
 \qquad
 \mathcal C=\left\{\int_\Gamma\rho_h\,dP(h):
                         P\in\mathcal P(\Gamma)\right\}.
\]
The map $h\mapsto\rho_h$ is weakly continuous.  Thus $\mathcal C$
is convex and weakly compact in the finite measures on $X$.

Every $\rho\in\mathcal C$ is singular with respect to
$\sigma\vert_X$.  Indeed, otherwise there would be a nonzero measure
$\sigma_0\le\rho$ with $\sigma_0\ll\sigma\vert_X$.  Writing
$a=d\sigma_0/d\rho$ and $\rho=\int\rho_h\,dP(h)$ yields
\[
 \sigma_0=\int_\Gamma a\rho_h\,dP(h).
\]
Each $\rho_h$ is equivalent to length measure on its graph, with
bounded positive density.  Density restriction and layer-cake
therefore express $\sigma_0$ as a family of rectifiable curve measures.
Its aggregate is absolutely continuous with respect to $\nu$.
The tangents of the graphs must belong to $V(\nu,\cdot)$ at almost
every point charged by this family, while their directions
$(h'(v),1)$ lie in the forbidden cone.  This contradicts
$\sigma_0\ne0$.

For $0\le\phi\le1$ continuous on $X$, put
\[
 L(\phi,\rho)=\int_X\phi\,d\rho
                 +\int_X(1-\phi)\,d\sigma.
\]
Mutual singularity and regularity of finite measures imply
$\inf_\phi L(\phi,\rho)=0$ for every $\rho\in\mathcal C$.
The compact-convex minimax Theorem, with $\mathcal C$ as its compact
side \cite[Theorem~4.2]{Sion1958}, applies to this continuous affine functional
and gives
\[
 \inf_{\substack{\phi\in C(X)\\0\le\phi\le1}}
     \sup_{\rho\in\mathcal C}L(\phi,\rho)=0.
\]
Choose $\phi_n$ with
\[
 \int_X(1-\phi_n)\,d\sigma
       +\sup_{h\in\Gamma}\int_I\phi_n(h(v),v)\,dv
 \le2^{-n}.
\]
Tonelli's Theorem shows that $\phi_n\to1$ at $\sigma$-almost every
point, while, for each individual graph, $\phi_n(h(v),v)\to0$
for almost every $v$.  Hence the single Borel set
$\{y:\phi_n(y)\to1\}$ has full $\sigma\vert_X$ measure and is
null on every graph in $\Gamma$.  No intersection of graph-dependent
full-measure sets is being taken.

To pass to arbitrary graphs, use countably many rectangles with
strict interior margins and exhaust the measure by their interiors.
On the vertical interval of such a rectangle, restrict the graph
and project its horizontal component to the closed box $H$; this
preserves the Lipschitz constant and does not change graph points
inside the smaller rectangle.  A graph given on a shorter interval
extends with the same Lipschitz constant by constant extension at its
endpoints.  Taking a countable union of the resulting full-measure,
locally graph-null sets proves \eqref{hp:graph-null}.  Intersect this
set with $A$ at the end.
\end{proof}

We next convert graph-nullness into finitely many thin bands.  The
cone order, antichain decomposition and Lipschitz slabs have their
counterparts in the null-set constructions of
Alberti--Cs\"ornyei--Preiss~\cite[Sections~2.2 and~3]{AlbertiCsornyeiPreiss2010}.
The uniform near-hit estimate is the compactness mechanism used in
\cite[Lemma~4.12, Step~1]{AlbertiMarchese2016}; we retain its proof here.

\begin{lemma}[Finite thin bands]\label{hp:thin-bands}
Let $K$ be a compact subset of the interior of a rectangular chart
$H\times I\subset\R^2\times\R$, and assume that $K$ satisfies
\eqref{hp:graph-null}.  For every $\delta>0$, there are finitely many
piecewise affine functions $g_1,\dots,g_N$ on $H$, each with
Lipschitz constant at most $C\lambda$, and a common length $L>0$
such that
\[
 NL<\delta,
 \qquad
 K\Subset\bigcup_{j=1}^N
       \{(h,v):|v-g_j(h)|\le L/2\}.
\]
The values of the functions, and their bands, can be kept in a fixed
compact subinterval of the interior of $I$.  Here the containment
$\Subset$ means that the union contains an open neighborhood of $K$.
\end{lemma}

\begin{proof}
Take a cubical grid of spacing $w$ and let $P_w$ be the centers of
the grid boxes that meet $K$.  For distinct centers define
\[
 (h,v)\prec(h',v')\quad\Longleftrightarrow\quad
                         v'-v\ge\lambda|h'-h|.
\]
This is a strict partial order: transitivity follows from the
triangle inequality, and distinct comparable centers have $v'>v$.
Because vertical grid coordinates differ by integer multiples of
$w$, successive centers in a chain have vertical separation at least
$w$.

The horizontal coordinates of a chain, interpolated linearly as a
function of $v$ and extended constantly beyond its first and last
nodes, form an allowed $1/\lambda$-Lipschitz graph.  All such graphs
lie in a fixed compact graph family, after slightly enlarging $H$
and $I$ while keeping the chart margins.  For this family set
\[
 \omega(r)=\sup_{h\in\Gamma}
  \bigl|\{v\in I:\dist((h(v),v),K)\le r\}\bigr|.
\]
Then $\omega(r)\to0$ as $r\downarrow0$.  Otherwise compactness
would give uniformly convergent graphs along a sequence $r\downarrow0$;
the upper limit of the near-hit indicators is bounded by the
indicator that the limiting graph meets $K$.  Reverse Fatou and
\eqref{hp:graph-null} give a contradiction.

Around each node of a chain take a vertical interval of radius
$w/4$.  These intervals are disjoint and, for small $w$, lie inside
the fixed interval with margins.  On each such interval the graph
is within $C_\lambda w$ of $K$.  If $n_w$ is the longest chain
length, it follows that
\[
 \tfrac12 n_ww\le\omega(C_\lambda w),
 \qquad\hbox{hence}\qquad n_ww\longrightarrow0.
\]
Give every center its longest-chain rank.  Centers of the same rank
form an antichain, and two of them satisfy
$|v'-v|<\lambda|h'-h|$.  In particular, their horizontal coordinates
are distinct and their heights extend to a $\lambda$-Lipschitz
function on $H$ by the scalar Lipschitz extension formula.  Clip
the extension to an interior vertical interval containing all the
center heights.  Piecewise affine interpolation on a sufficiently
fine regular triangulation then approximates it to $o(w)$ with
Lipschitz bound $C\lambda$.

There are at most $n_w$ such functions.  A band of length $L=C_2w$
about each function contains the boxes assigned to it with a strict
margin, provided $C_2$ is fixed sufficiently large.  Thus the bands
contain a neighborhood of $K$, and
$NL\le C_2n_ww\to0$.  The initial margins and a sufficiently small
$w$ give the final assertion.
\end{proof}

\begin{proposition}[Target flattening]\label{hp:flattening}
Given $\eps>0$, $0<\lambda<1/100$, and a positive continuous fine
tolerance $a$ on $\Lambda$, one can retain a compact set $K\subset D$
with $\mu(D\setminus K)<\eps$ and construct a Lipschitz map
$S_0:\Lambda\to\Lambda$ with the following properties.
\begin{enumerate}
\item The map is the identity outside finitely many disjoint
rectangular boxes compactly contained in $\Lambda$,
$\Lip(S_0)\le C$, and $|S_0(y)-y|<a(y)$.  The constant $C$ is
absolute.  For $0<\kappa\le1$,
\[
 S_\kappa=(1-\kappa)S_0+\kappa\Id
\]
is a bi-Lipschitz homeomorphism of $\Lambda$ onto itself.
\item With $F_\kappa=S_\kappa\circ f$ for $0\le\kappa\le1$,
\begin{equation}\label{hp:flattening-derivative}
 |DF_\kappa|\le C|Df|\quad\hbox{a.e. on }\Omega,
 \qquad
 |DF_\kappa-Df|\le C\lambda|Df|
                   \quad\hbox{a.e. on }K.
\end{equation}
Moreover, $F_\kappa\to F_0$ strongly in $W^{1,p}(\Omega;\R^3)$,
and $\rank DF_0=\rank Df$ almost everywhere on $K$.
\item The compact set $F_0(K)$ is contained in finitely many compact
flat plate patches with pairwise disjoint ambient neighborhoods.
Each patch lies in a plane of slope at most $C\lambda$ in one of
the chosen coordinates.  Each patch has an assigned rank, either
one or two; the rank-one patches can in addition retain any prescribed
small directional grouping of $\operatorname{im}Df$.
\item There is a compact zero-volume set $Z\Subset\Lambda$,
contained in finitely many piecewise affine graph sheets, such that
\begin{equation}\label{hp:flattening-homeomorphic-complement}
 S_0:\Lambda\setminus S_0^{-1}(Z)\longrightarrow\Lambda\setminus Z
\end{equation}
is a locally bi-Lipschitz homeomorphism.  Its only non-singleton
fibres are the collapsed vertical intervals over points of $Z$.
Consequently $F_0$ is a homeomorphism from
$\Omega\setminus F_0^{-1}(Z)$ onto $\Lambda\setminus Z$.
\end{enumerate}
For almost every target point outside $Z$, its unique preimage under
this last homeomorphism either is a regular point of $F_0$ with
positive invertible differential and the power-mean test in
\eqref{hp:regular-point-tests}, or belongs to a fixed source null set.
When $p\ge3$, the second alternative occurs only on a target null
set.
\end{proposition}

\begin{proof}
We describe the measure selections before the geometric construction.
Since $\mu$ is finite, first discard an arbitrarily small $\mu$-mass so
that the remaining jets and the reciprocals of their nonzero singular
values are bounded.  Partition by rank and, at rank one, by a finite
angular partition of the range lines.  Lemma~\ref{hp:transported-bundle}
further partitions the data among finitely many directions $e$.
The measure $f_\#\mu$ is absolutely continuous with respect to $\nu$,
so every $\nu$-full choice in these pieces is also full for the
weight being retained.  Apply Lemma~\ref{hp:cone-null} to obtain
simultaneous graph-null data in each direction.

By inner regularity choose compact subsets of the finitely many
pieces with small total loss.  Their images are disjoint compact
sets, hence have positive separation.  In each assigned coordinate
frame take a sufficiently fine box grid inside that separation
margin and inside $\Lambda$.  Its cut planes can be chosen to have
zero $f_\#\mu$ measure: for each coordinate there are only countably
many levels with positive mass.  Discard small neighborhoods of the
finitely many cuts and take compact subsets again.  We have reduced
the problem, with arbitrarily small loss, to finitely many disjoint
boxes, with the compact data strictly inside each box and a fixed
rank and direction assignment there.  In what follows $K^*$ denotes
the compact target data in one such box.

Apply Lemma~\ref{hp:thin-bands} to $K^*$.  Sort the starts of the
length-$L$ bands pointwise, writing them as
$a_1(h)\le\cdots\le a_N(h)$.  Order statistics of finitely many
piecewise affine functions are piecewise affine after finite
subdivision, and retain their common Lipschitz bound.  Define
\begin{equation}\label{hp:separated-band-starts}
 s_1=a_1,\qquad s_j=\max\{a_j,s_{j-1}+L\}\quad(2\le j\le N).
\end{equation}
These starts remain $C\lambda$-Lipschitz, since equivalently
$s_j=\max_{i\le j}(a_i+(j-i)L)$.  Their bands have disjoint
interiors in each vertical column.

No point of an original band is lost.  To verify this, group the
successive new intervals into maximal contiguous clusters.  A cluster
starts at an unchanged start $s_i=a_i$, and every advanced interval
is attached to the top of its preceding cluster.  Since the original
starts are sorted, the portion below an advanced start belongs to
that preceding cluster; the remaining portion is covered by the
advanced interval itself.  This proves coverage by induction.
The total upward displacement is at most $NL$, so choosing $NL$
small preserves all vertical margins.

Choose a horizontal cutoff $0\le\chi\le1$, compactly supported in
the horizontal interior of the box and equal to one near the
horizontal projection of $K^*$.  Choose a nondecreasing Lipschitz
function $\eta(v)$ that is zero below a fixed level above every band
and is one near the top of the box.  Its transition is contained
strictly above all bands.  The term involving $\eta$ compensates
above the bands for the vertical length removed below it, so each
column keeps its endpoints.  For
$[t]_{[0,L]}=\max\{0,\min\{t,L\}\}$, define
\begin{equation}\label{hp:triangular-collapse}
 q_0(h,v)=v+\chi(h)\left(
       -\sum_{j=1}^N[v-s_j(h)]_{[0,L]}+NL\eta(v)\right),
 \qquad S_0(h,v)=(h,q_0(h,v)).
\end{equation}
Choose $NL$ after the cutoffs so small that
\begin{equation}\label{hp:band-cutoff-smallness}
 NL\Lip(\chi)\le\lambda,
 \qquad NL\Lip(\eta)\le1,
\end{equation}
and so that $NL$ is below the prescribed fine motion tolerance and
all unused box margins.

The correction in \eqref{hp:triangular-collapse} vanishes below all
bands, above the transition of $\eta$, and near the horizontal
boundary.  Thus $S_0$ equals the identity near the boundary of the
box.  For fixed $h$, the vertical map is nondecreasing, with slope
$1-\chi(h)$ on each band and slope
$1+\chi(h)NL\eta'(v)$ on the upper compensation interval.  Elsewhere
its slope is one.  These statements hold almost everywhere and
imply the corresponding monotonicity for the Lipschitz function.
In particular the slope lies in $[0,2]$.

The horizontal bound is independent of the number of bands.  At any
point of a piecewise affine region at most one truncated summand in
\eqref{hp:triangular-collapse} is unsaturated.  Every fully saturated
summand is the constant $L$.  The horizontal gradient of the sum
therefore has norm at most $C\lambda$.  Since the whole correction
has size at most $NL$, the cutoff term is bounded by
$NL\Lip(\chi)$.  Continuity across the finitely many pieces yields
\begin{equation}\label{hp:anisotropic-correction}
 |R(h,v)-R(\widetilde h,\widetilde v)|
 \le C\lambda|h-\widetilde h|+C|v-\widetilde v|,
 \qquad R(h,v)=q_0(h,v)-v.
\end{equation}
Thus $\Lip(S_0)\le C$ and $|S_0-\Id|\le NL$.  Each vertical
column is mapped onto itself, so the whole box is mapped onto itself.
Use this formula in each of the finitely many disjoint boxes and the
identity elsewhere.  Extension by the identity to $\R^3$ is globally
Lipschitz, as is seen by integrating along line segments across the
finitely many boxes.

For $\kappa>0$, the vertical slope of
$q_\kappa=(1-\kappa)q_0+\kappa v$ is at least $\kappa$ and at most
two.  The horizontal Lipschitz bound is still uniform.  Each column
map is therefore a bi-Lipschitz increasing homeomorphism fixing its
ends; its inverse depends Lipschitz-continuously on the horizontal
coordinate, with a bound depending on $\kappa$.  Patching with the
identity proves the asserted bi-Lipschitz property of $S_\kappa$.
No inverse bound uniform as $\kappa\downarrow0$ is asserted.

Lipschitz composition and the ACL characterization of Sobolev maps
give the first bound of \eqref{hp:flattening-derivative}.  For the
second, work in the assigned coordinates near a retained source
point and use \eqref{hp:anisotropic-correction} along almost every
source coordinate line.  At almost every such point,
\[
 |\partial_i(R\circ f)|
 \le C\lambda|\operatorname{proj}_{e^\perp}\partial_i f|
       +C|e\cdot\partial_i f|
 \le C\lambda|\partial_i f|,
\]
where Lemma~\ref{hp:transported-bundle} supplies the last inequality.
Summing the three coordinate estimates proves the claim, also on
band junctions: it does not require differentiability of $S_0$ at
the target datum.  The horizontal component of $DF_0$ is exactly
$\operatorname{proj}_{e^\perp}Df$.  The projection is injective on
$V(\nu,f(x))$, so this component has rank $\rank Df$ on the
retained data.  Conversely, Lipschitz composition cannot increase
rank here.  To see this without differentiating $S_0$ at the target
point, combine Fr\'echet differentiation of $f$ with approximate
differentiation of $F_0$: the inequality
$|F_0(x+z)-F_0(x)|\le C|Df(x)z|+o(|z|)$ implies
$|DF_0(x)v|\le C|Df(x)v|$ for every $v$, by taking approximate
limits in narrow cones about $v$.  Thus
$\ker Df(x)\subset\ker DF_0(x)$, proving equality of ranks.
Finally the exact identity
\[
 F_\kappa=(1-\kappa)F_0+\kappa f
\]
and $F_0\in W^{1,p}$ prove strong convergence as $\kappa\downarrow0$.

We next identify the exceptional set of values of $S_0$ exactly.
On a band, when $\chi(h)=1$, its image height is
\[
 b_j(h)=s_j(h)-(j-1)L.
\]
Let
\begin{equation}\label{hp:collapsed-values}
 Z_{\mathrm{box}}=
   \bigcup_{j=1}^N\{(h,b_j(h)):\chi(h)=1\},
 \qquad Z=\bigcup_{\mathrm{boxes}}Z_{\mathrm{box}}.
\end{equation}
These are compact sets inside their boxes.  Each $b_j$ is piecewise
affine with slope at most $C\lambda$, so $Z$ has zero volume.
When adjacent bands touch, their displayed image values coincide;
the resulting fibre is the entire contiguous closed interval.
These and only these intervals are the non-singleton fibres.  Indeed,
if $\chi(h)<1$, the whole column is strictly increasing.  If
$\chi(h)=1$, it is strictly increasing outside the closed union of
bands.

For a value outside $Z$ there is exactly one preimage.  Near that
preimage the vertical slope has a positive lower bound: either
$1-\chi$ is bounded below locally, or the point is at positive
distance from all closed bands.  Together with
\eqref{hp:anisotropic-correction}, this gives a local Lipschitz
inverse.  This proves the precise homeomorphic-complement statement
\eqref{hp:flattening-homeomorphic-complement}.  The selected data
lie where $\chi=1$ and in the union of bands, and hence their images
lie in $Z$.

It remains to arrange genuine separated plate patches, rather than
merely a finite union of sheets.  Refine each piecewise affine graph
into its finitely many affine pieces, including their lower-dimensional
strata.  Assign every output datum to one containing affine plane;
if sheets coincide or meet, make one measurable choice.  Use the
finite pushforward of the retained source weight under $F_0$ for
this assignment.  Inner regularity gives disjoint compact subsets
of the assignments with an arbitrarily small further loss.  These
compacts have disjoint ambient neighborhoods.  In each plane, take
a sufficiently fine rectangular grid, choose its cut lines to have
zero assigned measure, and remove thin neighborhoods of the cuts.
The remaining compact pieces lie strictly inside flat rectangles
with margins, and the rectangles and ambient neighborhoods can be
chosen disjoint using the preceding separation.  Pull back the
retained compact subsets through $F_0$ within the already compact
source data.  This gives the final compact $K$ and all the claimed
patch properties.  The initial chart separation prevents saturation
from mixing different rank or range-line assignments.  In particular,
no claim that sheet intersections or projected singular measures
have zero assigned mass is needed.  On the source subset assigned
to a given plate, the normal component of $F_0$ is constant.
Locality of weak derivatives therefore puts the range of $DF_0$
in that plate's tangent plane almost everywhere on the subset.
Together with rank preservation and
\eqref{hp:flattening-derivative}, this also preserves the prescribed
approximate line direction in the rank-one case, with an additional
$C\lambda$ angular error.  Distribute the losses in this
and the preceding selections so that their sum is less than $\eps$.

Lastly, put $U=\Omega\setminus F_0^{-1}(Z)$.  On $U$, $F_0$ is a
Sobolev homeomorphism with positive orientation onto
$\Lambda\setminus Z$.  Apply Lemma~\ref{hp:regularity} on a
countable interior exhaustion.  Its nonregular source points form
a fixed null set $E\subset U$, and the image of its regular
rank-deficient points is null.  This proves the asserted dichotomy
for almost every target point.  For $p\ge3$, the volume property
$N$ makes $F_0(E)$ null as well.  For $2<p<3$, that image is not
assumed null and is retained as the stated exceptional alternative.
\end{proof}

\subsection{Protected endpoint pairs}

The next step selects a pair of target rectangles on opposite sides of
each retained plate.  Their geometric purpose is to give exactly two
contact points for each slope in a central rectangle.  In the convex
quantizer of Section~\ref{sec:quantizers}, these paired contacts produce
a planar output face containing the eventual quantized images of the
retained data.  The carrier must
first make neighborhoods of the endpoint rectangles exact affine or
PL regions; the central data themselves will remain untouched by those
implants.

We continue to measure losses by the finite measure introduced before
Lemma~\ref{hp:transported-bundle}:
\[
 d\mu(x)=\ind_{\{1\leq\rank Df\leq2\}}(x)
             (1+|Df(x)|^p)\,dx.
\]
Only regular points, as in Lemma~\ref{hp:regularity}, enter this measure.
Every compact restriction and every finite measurable partition below may
discard a prescribed arbitrarily small amount of this finite measure.
In particular, a finite sequence of such restrictions can be made with a
prescribed total loss.  We apply Proposition~\ref{hp:flattening}, and write
$F_0=S_0f$ and $F_\kappa=S_\kappa f$ for its maps.

\begin{lemma}[Protected wells]\label{hp:wells}
Let finitely many separated compact pieces of the retained $F_0$-image
lie on flat plates, with relatively compact disjoint ambient chart
neighborhoods.  Let a positive continuous motion scale $\delta$ on
$\Lambda$ be prescribed.  After an arbitrarily small $\mu$-loss, one can
choose, on each plate, an origin $z$, a unit vector $n$ as close as
desired to its normal, a number $d>0$, and nested protected horizontal
rectangles in $n^\perp$ with the following properties.
There is a nonnegative smooth compactly supported function $b_1$ on
$\Lambda$, extended by zero to $\R^3$, such that
\begin{enumerate}[label=\textup{(\roman*)}]
\item $\Delta b_1\leq C$ with an absolute constant, and $b_1$ is as
small as prescribed relative to $\delta^2$.  In particular
$b_1=o(\dist(\cdot,\R^3\setminus\Lambda)^2)$ at the boundary.
\item For every $h$ in a protected horizontal rectangle, put $q=z+h$.
The affine function
\[
 L_h(x)=q\cdot x-\tfrac12|q|^2+d^2
\]
supports $\psi_1(x)=|x|^2/2+b_1(x)$ globally, and its contact set
consists exactly of $q+dn$ and $q-dn$.  In particular,
\begin{equation}\label{hp:paired-dual-value}
 \sup_x\{q\cdot x-\psi_1(x)\}=\tfrac12|q|^2-d^2.
\end{equation}
For a fixed compact rectangle and any fixed neighborhoods of these
two endpoint sheets, the supporting gap is uniformly positive outside
the neighborhoods.
\item The assigned central data $X=F_0(x)$ have
$|\langle X-z,n\rangle|<d/4$.  For almost every assigned $x$ with
respect to $\mu$, both points
\begin{equation}\label{hp:sampled-endpoints}
 Y_\pm(X)=X+n\bigl(\pm d-\langle X-z,n\rangle\bigr)
\end{equation}
belong to genuine homeomorphic target neighborhoods of $F_0$ and have
either a regular full-rank inverse point or an inverse point in a fixed
source null set.  The latter alternative is needed only when $2<p<3$.
\end{enumerate}
All assertions persist after shrinking the protected horizontal
rectangles.  The endpoint assignments can be restricted to compact
sets on which their inverse points and their regular or exceptional
types have fixed separated compact ranges.
\end{lemma}

Figure~\ref{hp:fig-paired-endpoints} shows the geometry of these pairs.
The endpoint rectangles will become exact target regions in the first
round of Proposition~\ref{hp:carrier}; the present lemma selects their
positions and inverse types.

\begin{figure}[htbp]
\centering
\begin{tikzpicture}[x=1.05cm,y=.9cm,>=Latex,font=\small]
  \fill[orange!8] (-2.9,1.05) rectangle (2.9,1.55);
  \fill[orange!8] (-2.9,-1.55) rectangle (2.9,-1.05);
  \draw[orange!65!black,dashed] (-2.9,1.05) rectangle (2.9,1.55);
  \draw[orange!65!black,dashed] (-2.9,-1.55) rectangle (2.9,-1.05);
  \draw[orange!80!black,very thick] (-2.5,1.3)--(2.5,1.3);
  \draw[orange!80!black,very thick] (-2.5,-1.3)--(2.5,-1.3);
  \draw[gray,densely dotted] (-3.1,0)--(3.1,0);
  \draw[blue!65!black,very thick] (-2.6,-.4)--(2.6,.4);
  \draw[gray,dashed] (1.2,-1.3)--(1.2,1.3);
  \fill (0,0) circle (1.4pt) node[below left] {$z$};
  \fill (1.2,0) circle (1.4pt) node[below right] {$q=z+h$};
  \fill[blue!65!black] (1.2,.185) circle (1.8pt)
      node[above left=2pt] {$X=F_0(x)$};
  \fill[orange!80!black] (1.2,1.3) circle (1.8pt)
      node[above=5pt] {$Y_+(X)=q+dn$};
  \fill[orange!80!black] (1.2,-1.3) circle (1.8pt)
      node[below=5pt] {$Y_-(X)=q-dn$};
  \draw[->] (-4.7,-1.5)--(-4.7,1.65) node[above] {$n$};
  \node[anchor=east] at (-3.05,1.3) {$v=d$};
  \node[anchor=east] at (-3.05,-1.3) {$v=-d$};
  \node[blue!65!black,anchor=west] at (2.8,.48) {retained plate};
  \node[anchor=west] at (3.2,0) {$v=0$};
\end{tikzpicture}
\caption{A section through a retained plate and its paired endpoint
rectangles; horizontal rectangles appear as intervals.  The chosen
normal $n$ may be slightly tilted from the plate normal, so the datum
$X=F_0(x)$ need not lie in the central plane $v=0$.  Both endpoints have
the same horizontal coordinate $h$.  The dashed boxes indicate
neighborhoods that the carrier will later make exact.  The diagram
shows $F_0$ data, before the positive parameter $\kappa$ is chosen.}
\label{hp:fig-paired-endpoints}
\end{figure}


\begin{proof}
First work on one plate.  Choose a horizontal cutoff $0\leq\theta\leq1$
which is one on a neighborhood of a compact rectangle strictly larger
than the rectangle to be protected.  Its support is compact in the
horizontal chart.  Choose a smooth nonnegative function $\beta$,
supported in $[-2,2]$, such that
\[
 u\longmapsto u^2/2+\beta(u)
\]
has minimum $1$ exactly at $u=\pm1$.  For example, first prescribe this
function to be $1+(u^2-1)^2$ for $|u|\leq3/2$; in the remaining
transition region the unperturbed function $u^2/2$ is already larger
than $1$, so the nonnegative bump can be smoothly cut off there without
creating another minimum.  In the coordinates $x=z+h+vn$, set
\begin{equation}\label{hp:well-formula}
 b_z(x)=\theta(h)d^2\beta(v/d).
\end{equation}
Choose $d$ much smaller than the horizontal cutoff margin and the
ambient chart margin.  All supports can then be made disjoint.  Since
\[
 \Delta b_z=d^2\beta(v/d)\Delta_h\theta+
                     \theta(h)\beta''(v/d),
\]
choosing $d^2\|\Delta_h\theta\|_\infty\leq1$ gives an absolute upper
bound for $\Delta b_z$.  Disjointness gives the same bound for their
sum $b_1$.  Its size is $O(d^2)$ on each chart, so the finitely many
$d$'s may also enforce every prescribed $\delta^2$ bound.  The support
is compact in $\Lambda$, proving the stated boundary decay.

For a protected $h_0$, subtract $L_{h_0}$ and write $q=z+h_0$.  Within
the chart the difference is
\[
 \psi_1(x)-L_{h_0}(x)
 =\tfrac12|h-h_0|^2+\tfrac12v^2+b_1(x)-d^2.
\]
Where $\theta=1$, the one-dimensional choice of $\beta$ makes this
nonnegative, with equality precisely when $h=h_0$ and $v=\pm d$.
Where $\theta\ne1$, the horizontal margin has been chosen so that
$|h-h_0|^2/2>d^2$.  The other bumps are nonnegative and have disjoint
supports.  The same distance estimate applies outside the chart.
This proves the global contact statement and
\eqref{hp:paired-dual-value}.  On compact $h_0$-sets the gap is
uniform away from the endpoint neighborhoods: otherwise a sequence
of almost contacts would, by quadratic growth and compactness, limit
to a third contact.

Initially the central data lie on the original flat plate.  Choose $d$
in a small positive interval $[d_-,d_+]$, and then restrict $n$ to a
sufficiently small open cap about the original normal that
$|\langle X-z,n\rangle|<d_-/4$ for all central data in the compact
patch.  All preceding supports, cutoffs and margins may be chosen
uniformly over this parameter range.  Give $n$ and $d$ smooth
probability densities in the cap and interval.  For fixed $X$, the map
in \eqref{hp:sampled-endpoints} has the form $X+r n$, where
$r=\pm d-\langle X-z,n\rangle$.  In sphere coordinates its volume
Jacobian has absolute value $r^2$: differentiation in $d$ supplies
$\pm n$, and each angular derivative has tangential part $r\,dn$.
Here $|r|\geq3d_-/4$.  Thus each endpoint law is absolutely continuous
for three-dimensional volume.

By Proposition~\ref{hp:flattening}, almost every target point off the
closed sheet-value set has exactly the stated regular or exceptional
type.  Fubini applied to the finite assignment measure therefore gives
a parameter choice for which both endpoints have an allowed type for
$\mu$-almost every assigned point.  This is an averaging argument in
three target dimensions; it uses no absolute continuity of the
horizontal assignment measure.  Push that measure to the horizontal
patch.  Split according to the two endpoint types simultaneously and
restrict each part to compact subsets of its genuine inverse charts.
The inverse endpoint maps are continuous there.  Further compact
restriction gives separated compact source ranges for the finitely
many classifications, with arbitrarily small loss.  The central
compact data remain separated from all endpoint inverse data because
their $F_0$-values lie on the central sheet, whereas the endpoints lie
in its homeomorphic complement.  A finite rectangular refinement
with smaller margins supplies the rectangles in the statement.
\end{proof}

\subsection{Implanting exact cores by cubical collapse}

The endpoint selection has located two compact sets of target values
over each retained plate.  Their inverse data are either regular full
rank or belong to a source null set, but the maps near these values are
not yet required to be affine or PL.  We now construct such exact
neighborhoods.  The regular data will permit an energy estimate with a
universal constant.  For exceptional inverse data, which occur only
when $2<p<3$, the construction will confine the uncontrolled cost to
marked source cubes.  The data are selected using $F_0$, while the
initial replacements will be made in the homeomorphism $F_\kappa$
for a sufficiently small $\kappa>0$.

The local replacement has a simple normalized goal.  Given a
homeomorphism close to the identity near the boundary of a cube, make
it exactly the identity on the inner cube, leave its outer boundary
values and its image unchanged, and control the energy in the
intervening shell.  The output is allowed to collapse isolated tame
balls.  Those fibres can subsequently be opened while preserving
previously exact target regions; the opening itself supplies no energy
estimate.

The energy estimate comes from sampling the input near a
two-dimensional cubical skeleton.  Small source charts of size $\ell$
are immersed at one common scale $s$; the factors in the derivative of
the lift cancel, and the weighted overlaps of the samples are summable.
We first construct that immersed skeleton, then carry out the
replacement and identify its fibres.

Write
\[
 B=[-1,1]^3,\qquad t(x)=\max_{1\leq j\leq3}|x_j|-1,
 \qquad B_r=\{x:t(x)<r\}.
\]
Thus the notation $B$ denotes the closed inner cube, whereas $B_r$ is
open.  Boundaries have zero volume, so this convention does not affect
any integral below.  The constants in the next two Lemmas depend on the fixed cubical
construction and, when indicated, on $p$; they do not depend on the input
homeomorphism or on the sufficiently small dyadic number $s$.

\begin{lemma}[An immersed cubical skeleton]\label{hp:immersed-skeleton}
There are positive constants $b,C_0,C_1$ and, for every sufficiently small
dyadic $s$, a locally finite conforming triangulation $\mathcal T_s$ of
$\{0<t<2s\}$ with the following properties.  If $D\in\mathcal T_s$ has
scale $\ell_D$, then
\[
 \diam D\asymp\ell_D,
 \qquad \ell_D\asymp\min\{s,t(x)\}\quad(x\in D),
\]
where the second assertion allows the uniform constants implicit in
$\asymp$.  The simplices have uniformly controlled shapes and finitely
many local configurations up to translation and rescaling.  There is a
smooth orientation-preserving local diffeomorphism $i$ on an open
neighborhood $\mathcal U$ of their two-skeleton, including a neighborhood
of the outer boundary $\{t=2s\}$, such that
\[
 i=\Id\text{ near }\{t=2s\},\qquad
 |i(x)-x|\leq C_0s.
\]
The neighborhood contains, at every simplex, a neighborhood of its
skeleton of width $b\ell_D$.  On a slightly smaller such neighborhood,
\begin{equation}\label{hp:immersion-bounds}
 |Di|\leq C_1\frac{s}{\ell_D},\qquad
 |(Di)^{-1}|\leq C_1\frac{\ell_D}{s},\qquad
 |D^2i|\leq C_1\frac{s}{\ell_D^2}.
\end{equation}
There are also uniform local inverse charts: a chart centered at a
point of the smaller neighborhood contains a source ball of radius
$b_1\ell_D$, and its image contains the corresponding target ball of
radius $b_2s$, for fixed $b_1,b_2>0$.

More precisely, after splitting into a bounded number of charts per
simplex, the formulas for $i$ have the form
\begin{equation}\label{hp:immersion-template}
 i(x_*+\ell\xi)=y_*+sI(\xi),
\end{equation}
where $I$ belongs to a fixed finite list of smooth formulas on fixed
compact chart neighborhoods, and $y_*$ belongs to a fixed rational
subdivision of the $s$-grid.  For fixed $s$ there are only finitely many
possible $y_*$ in the bounded region used by this construction.  The
map $i$ is permitted to have overlaps between different local charts.
\end{lemma}

\begin{proof}
Set $\lambda_k=2^{-k}s$, $k\geq0$.  Tile
\[
 \{\lambda_k<t<2\lambda_k\}
\]
by axis-aligned cubes with side comparable to $\lambda_k$; a fixed
dyadic subdivision can be used throughout.  This is possible with
exact alignment because $1$, $s$, and all $\lambda_k$ are dyadic.
Across consecutive layers, a face of a coarser cube is subdivided to
match its finitely many finer neighbors.  Triangulate each resulting
face by coning its subdivided boundary to its center, and cone these
triangles to the center of the cube.  Shared faces receive the same
triangulation.  The neighbor-size ratios are bounded, and all vertex
positions in a normalized cube belong to a fixed finite set.  This
proves the shape and local-configuration assertions.  The harmless
finite modifications needed near $t=2s$ can be made on fixed dyadic
levels.  In particular a thin outer band can be reserved for the
identity map.

We construct the immersion successively near vertices, edges, and
faces.  All neighborhoods in this construction are in the source;
there is no requirement that their images be disjoint.  Choose at each
vertex $v$ a representative incident scale $\ell_v$.  The ratios between
incident scales belong to a finite list.  Away from the reserved outer
band, choose a nearest point $y_v$ of the $s$-grid and prescribe
\[
 i(x)=y_v+\frac{s}{\ell_v}(x-v)
\]
on a small vertex neighborhood.  These source neighborhoods are
disjoint when their relative radii are small enough.  On the outer
band instead prescribe the identity germ.  Its normalized scale ratio
is bounded above and below, since $\ell_v\asymp s$ there.  All target
anchors, including those in the identity band, lie on a fixed rational
subdivision of the $s$-grid.  Adjacent anchors differ by at most $Cs$.

Consider an edge core outside the vertex neighborhoods.  Its two end
germs are already prescribed.  Join these germs by a regular immersed
arc in a ball of radius $Cs$ containing them.  This can be done relative
to the end intervals by the one-dimensional relative immersion
Theorem.  Choose two transverse vector fields along the arc, equal to
the prescribed transverse derivatives on the end intervals.  They
give a positive three-frame.  Relative to its positive scalar end
frames, the homotopy class of this frame path can be made trivial:
one full rotation of the two transverse vectors changes the class in
$\pi_1(GL^+(3))\cong\mathbb Z/2$.  Thus choose the rotation, if needed,
so that the entire frame path is homotopic to the path of positive
scalar frames, with its ends fixed.  Thickening the framed arc gives
an orientation-preserving local diffeomorphism on a sufficiently thin
edge neighborhood.  It agrees exactly with the old affine formulas on
the end collars.  Distinct truncated edge cores have disjoint source
neighborhoods.

For a triangular face, remove the already treated vertex and edge
neighborhoods.  Choose a rounded disk in the remaining face whose
boundary collar lies in those neighborhoods.  The prescribed local
diffeomorphism on that collar provides both a map of the collar and a
positive three-frame.  Its frame loop is null-homotopic.  Indeed, move
the loop within the treated neighborhood onto the triangle perimeter:
each edge frame path has the trivial relative class just prescribed,
and the vertex frames are positive scalars.  Concatenating these paths
therefore gives the trivial loop in $GL^+(3)$.

The target ball is contractible, so the collar map extends as a smooth
base map of the disk into a slightly larger ball of radius $Cs$.
The null-homotopy of the three-frame extends its restriction to the two
fixed tangent directions of the face.  We have consequently obtained
a formal immersion of the disk that is the differential of the
prescribed map near its boundary.  The relative Smale--Hirsch Theorem,
in source dimension two and target dimension three, gives an actual
immersed disk agreeing with the prescribed map on a smaller boundary
collar; see \cite[Theorem~5.7]{Hirsch1959}.  We use the target open ball
as the target manifold in that Theorem.  Thus the extension stays in
a ball of radius $Cs$.  In particular this is not an application of a
relative immersion extension theorem in equal dimensions.

To thicken the disk, choose a positive transverse column $\nu$ along it,
equal to the old normal derivative on the boundary collar.  This
choice extends because the set of positive transverse columns over
an oriented tangent two-frame is a contractible half-space.  In flat
source coordinates $(u,r)$ use the first-order formula
\[
 \phi(u)+r\nu(u).
\]
It is a local diffeomorphism for sufficiently small $|r|$.  On the
boundary collar, its difference from the old exact formula is
$O(r^2)$, with derivative difference $O(|r|)$.  A fixed smooth cutoff
in $u$ blends it with that old formula.  After reducing the normal
width, the differential remains positive and nonsingular, and the
formulas agree exactly on the region retained from the old
neighborhood.  Distinct face cores have disjoint source neighborhoods;
their overlaps with the old neighborhood are precisely the regions
where this matching has been imposed.  This constructs a smooth
positive local diffeomorphism on a joint neighborhood of the entire
two-skeleton.

We now justify all uniformity assertions, which are not consequences
of the qualitative immersion Theorem alone.  Normalize an incident
source configuration by its scale $\ell$ and normalize target lengths
by $s$.  There are finitely many source configurations and incident
scale ratios.  Differences between the target anchors are bounded
integer vectors on the fixed rational grid, so they also have only
finitely many possibilities.  The identity prescriptions occur only
in the outer layers, where the normalized ratios and positions have
finitely many possibilities.  First choose one edge extension for each
of these finitely many edge data, including its collar formula and
framing.  Next choose one face extension for each of the finitely many
face data determined by those choices.  Fix these choices once and
for all.  This gives a finite list of normalized smooth formulas on
compact subneighborhoods.

On each such compact subneighborhood the least singular value is
positive and all first and second derivatives are bounded.  Take the
minimum of the finitely many positive margins and the maximum of the
finitely many derivative bounds.  Shrinking once more gives a common
relative neighborhood width.  The inverse function Theorem, with
these uniform bounds and a common reduction of the chart radius,
gives source inverse-chart radii comparable to $\ell$ and image radii
comparable to $s$.  Scaling back gives
\eqref{hp:immersion-bounds} and
\eqref{hp:immersion-template}.  Finally every formula lies within
$Cs$ of its anchor, and every anchor is within $Cs$ of its source
configuration.  Hence $|i(x)-x|\leq C_0s$.
\end{proof}

\begin{lemma}[Cubical cutoff]\label{hp:cutoff}
There are constants $C>2$ and $c>0$ with the following property.
Let $1\leq p<\infty$, let $s$ be sufficiently small and dyadic, and
let $h$ be a homeomorphism on an open neighborhood of
$\overline{B_{Cs}}$, onto its image, with
$h\in W^{1,p}_{\mathrm{loc}}$.  Suppose that
\begin{equation}\label{hp:cutoff-closeness}
 |h(x)-x|\leq cs\qquad\text{whenever }|t(x)|\leq Cs.
\end{equation}
One can replace $h$ inside $B_{2s}$ by a continuous Sobolev map $F$,
keeping a neighborhood of its boundary unchanged, so that
\begin{align}
 &F=\Id\quad\text{on }B,
 \qquad F(B_{2s})=h(B_{2s}),\label{hp:cutoff-image}\\
 &\int_{0<t<2s}|DF|^p
 \leq C_p\int_{|t|\leq Cs}|Dh|^p.\label{hp:cutoff-energy}
\end{align}
The only nonsingleton fibers of $F$ are tame closed balls over a
countable set of target points.  Their diameters tend to zero at every
accumulation, and those target points can accumulate only on
$\partial B$.  These fibers can be opened simultaneously to give a
homeomorphism, altering $F$ only over arbitrarily small mutually
disjoint target neighborhoods of their values.  This last assertion
is topological; it does not assert a Sobolev energy bound for the
opened maps.  If $h$ has volume Lusin property $N$, then so does $F$.
\end{lemma}

\begin{proof}
\emph{The lift and its energy.}
Apply Lemma~\ref{hp:immersed-skeleton}.  Increase $C$ so that all
auxiliary image neighborhoods and inverse-chart neighborhoods are
contained in $\{|t|<Cs\}$.  This is possible because their centers have
$0\leq t\leq2s$ and their radii are at most a fixed multiple of $s$.
Put $\rho(x)=\min\{s,t(x)\}$ in the open shell.  This is a Lipschitz
scale function and is comparable to $\ell_D$ on every simplex $D$.

On a smaller skeleton neighborhood define $H(x)$ by the near-$x$
inverse branch of
\begin{equation}\label{hp:cutoff-lift}
 i(H(x))=h(i(x)).
\end{equation}
The right side differs from $i(x)$ by at most $cs$.  Uniform inverse
charts in Lemma~\ref{hp:immersed-skeleton} therefore make this
definition possible when $c$ is sufficiently small and give
\begin{equation}\label{hp:lift-displacement}
 |H(x)-x|\leq A c\rho(x).
\end{equation}
The branches agree on overlaps.  To see this, reduce $c$ so that any
two candidate values near a given source point lie in the same
injectivity chart; continuity of the local inverse then gives agreement
on overlapping neighborhoods.  In particular $H$ is continuous and
is locally a composition of homeomorphisms.

It is also globally injective on the smaller skeleton neighborhood.
Suppose $H(x)=H(x')$ and write $\delta=Ac$.  Then
\[
 |x-x'|\leq\delta\bigl(\rho(x)+\rho(x')\bigr).
\]
Since $\rho$ is Lipschitz, choosing $\delta<1/10$ makes $\rho(x)$ and
$\rho(x')$ comparable and gives $|x-x'|\leq3\delta\rho(x)$.
With a further fixed reduction of $c$, the points $x,x'$ and their
common lifted image lie in one injectivity chart for $i$.  Equation
\eqref{hp:cutoff-lift} and injectivity of $h$ first give
$i(x)=i(x')$, and injectivity of that chart then gives $x=x'$.
Thus $H$ is an embedding.  On the reserved outer band $i=\Id$; the
same is true at the nearby lifted points when $c$ is small.  Consequently
$H=h$ on a neighborhood of $\{t=2s\}$.

The weak chain rule in a local chart gives
\[
 DH(x)=Di(H(x))^{-1}\,Dh(i(x))\,Di(x)
 \qquad\text{for almost every }x.
\]
The two scale factors in \eqref{hp:immersion-bounds} cancel, so
$|DH(x)|\leq A|Dh(i(x))|$.  Each used part of a simplex is covered by
a uniformly bounded number of injectivity charts of $i$.  Change
variables separately in those charts.  Their Jacobians satisfy
$|\det Di|\geq a(s/\ell_D)^3$, and their images lie in
$B(x_D,As)$ for any fixed $x_D\in D$, after increasing $A$.  We obtain
\begin{equation}\label{hp:lift-cell-energy}
 \int_{D\cap\operatorname{dom}H}|DH|^p
 \leq A_p\left(\frac{\ell_D}{s}\right)^3
       \int_{B(x_D,As)\cap\{|t|\leq Cs\}}|Dh(y)|^p\,dy.
\end{equation}
No global injectivity of $i$ has been used in this change of variables.

For completeness, fix a dyadic depth $\lambda\leq s$ and a target
point $y$.  Simplices of scale comparable to $\lambda$ whose balls
$B(x_D,As)$ contain $y$ lie in a cubical layer of thickness
$O(\lambda)$ within an $O(s)$-ball.  This set has volume at most
$As^2\lambda$, including near edges and corners of the cube.  Since
the simplex interiors are disjoint and have volume at least
$a\lambda^3$, there are at most $A(s/\lambda)^2$ such simplices.
Summing \eqref{hp:lift-cell-energy}, and then interchanging the
nonnegative sum and integral, gives
\begin{align}
 \sum_D\int_{D\cap\operatorname{dom}H}|DH|^p
 &\leq A_p\sum_{k\geq0}
        \left(\frac{2^{-k}s}{s}\right)^3
        \left(\frac{s}{2^{-k}s}\right)^2
        \int_{|t|\leq Cs}|Dh|^p\notag\\
 &\leq A_p\int_{|t|\leq Cs}|Dh|^p.
 \label{hp:lift-total-energy}
\end{align}
This summation is the quantitative reason for using the immersion:
the local sampling volume contributes a third power, whereas the
number of overlapping samples in one layer contributes only a second
power.

\emph{Topological hole filling.}
We next fill the parts of the simplices where the lift was not
constructed.  These fillings will supply a homeomorphism of the shell;
we will then collapse their images to remove their uncontrolled energy.  In each normalized simplex choose nested smooth convex
balls, obtained by smoothing and slightly shrinking that simplex,
\[
 V_D\Subset V_D^+\Subset U_D\Subset D.
\]
They can be chosen homothetic about a common interior point after
fixing one smooth convex model for each simplex shape.  Choose $V_D$
large enough that $D\setminus\overline{V_D}$ and a two-sided collar
of $\partial V_D$ lie in the lift neighborhood.  The three successive
margins are fixed positive multiples of $\ell_D$.  Taking $c$ small
relative to those margins gives
\[
 H(\partial V_D)\Subset V_D^+.
\]
The sphere $H(\partial V_D)$ is locally flat, since $H$ is an embedding
of a neighborhood of $\partial V_D$.  By the generalized Schoenflies
Theorem, it bounds a tame closed ball $K_D$; see
\cite[Theorem~5]{Brown1960}.  Its bounded side lies in $V_D^+$:
the complement of the convex ball $V_D^+$ is connected to infinity
and misses the sphere.  Extend the boundary homeomorphism
$H|_{\partial V_D}$ to a homeomorphism
$\overline{V_D}\to K_D$.

These extensions fit on the correct sides of the lifted boundaries.
Indeed the open shell with all closed hole interiors removed is
connected through the skeleton to the outer band.  Its lifted image
misses every $H(\partial V_D)$ by injectivity of $H$.  For a fixed
$D$, an outer-band point can be chosen outside $V_D^+$, so this
connected image lies on the exterior side of that sphere.  Different
balls $K_D$ are disjoint, since they lie inside different simplex
interiors.  Therefore filling all holes and using $H$ elsewhere gives
an injective map on the shell.

Extend it by the identity on $B$ and call the resulting map $G$.
This extension is continuous.  On the lift region its displacement
is $O(\rho)$ by \eqref{hp:lift-displacement}; within a filled hole,
both the source and image lie in the same simplex of diameter
$O(\ell_D)=O(t)$.  Thus $G(x)-x\to0$ as $x\to\partial B$ from the
shell.  Lifted points stay outside $B$ because their displacement is
less than $t(x)/2$, and filled points stay outside $B$ because
$K_D\Subset D$.  Hence there is no new failure of injectivity at
the inner boundary.

The map $G$ is consequently a continuous injection on
$\overline{B_{2s}}$, with boundary values $h$.  It is a homeomorphism
onto its compact image.  Invariance of domain and separation by the
boundary sphere show that
\[
 G(B_{2s})=h(B_{2s}):
\]
both are the bounded component enclosed by the same embedded boundary
$h(\partial B_{2s})$.  Thus $G$ glues to $h$ outside this cube.  At
this point $G$ need not have a finite Sobolev energy in the holes.

\emph{Collapsing the uncontrolled fillings.}
We eliminate that uncharged energy by a map in the output coordinates.
Choose the balls $U_D$ so that the ones in the outermost layer leave
a fixed positive multiple of $s$ to $\partial B_{2s}$.  All other
layers are farther from that boundary.  By
\eqref{hp:cutoff-closeness} and a boundary-degree argument,
$B_{2s-cs}\subset h(B_{2s})$.  Reducing $c$ therefore ensures
$U_D\Subset h(B_{2s})$ for every $D$.  The same choice leaves a
whole outer band free of these supports.

On $U_D$ construct a Lipschitz radial squeeze $Q_D$ that is constant
on $\overline{V_D^+}$, equals the identity near $\partial U_D$, and
is a homeomorphism from
$U_D\setminus\overline{V_D^+}$ onto $U_D$ with the collapsed point
removed.  Explicitly, in the uniformly controlled polar coordinates
of the smooth convex model, replace the radial variable by a
continuous function that is zero up to the radius of $V_D^+$,
increases linearly on the next radial interval, and equals the
original radius near $\partial U_D$.  The fixed relative buffer
widths give a Lipschitz constant independent of $D$ and $s$.
Extend by the identity outside $U_D$.  The supports are disjoint.
Their diameters tend to zero toward $\partial B$, so the resulting
map $Q$ is continuous there and equals the identity on $B$.
The uniform Lipschitz bound holds across the seams as well, by
decomposing any line segment into the portions inside the supports
and their complement.  The map $Q$ maps $h(B_{2s})$ onto itself.

Set $F=Q\circ G$.  On a neighborhood of every filled hole this map
is constant, because $K_D\Subset V_D^+$.  Elsewhere it is locally
the Lipschitz composition $Q\circ H$, and
\[
 |DF|\leq\Lip(Q)|DH|\quad\text{almost everywhere there}.
\]
The local Sobolev statements glue across hole boundaries because the
map is constant on two-sided collars of those boundaries.  They glue
across simplex faces because the lift formulas already agree there.
Equation~\eqref{hp:lift-total-energy} proves
\eqref{hp:cutoff-energy} on the open shell.

There is also Sobolev gluing at its limiting inner boundary.  One
direct verification uses absolute continuity on lines.  For almost
every line parallel to a coordinate axis, the restrictions in each
open component of the shell are locally absolutely continuous and
their derivatives are integrable, by Fubini and the energy bound.
An absolutely continuous function on compact subintervals with
integrable derivative and a continuous endpoint limit extends
absolutely continuously to that endpoint.  The line intersects the
cube boundary only finitely many times, except for a null family of
lines contained in boundary planes.  Since $F$ is continuous and is
the identity inside the cube, these restrictions glue to an
absolutely continuous restriction on the whole line.  This proves
the asserted Sobolev regularity and identifies the weak derivatives.
The exterior seam is unchanged on a neighborhood, so no additional
gluing issue occurs there.

\emph{Fibres and relative opening.}
The collapsed fibers have an exact description:
\[
 F^{-1}(a_D)=G^{-1}(\overline{V_D^+}),
\]
where $a_D$ is the center to which $Q_D$ collapses.  They are tame
closed balls, since $G$ is a homeomorphism and the output buffers
are tame balls.  Every other fiber is a singleton.  In any compact
subset away from $\partial B$ there are finitely many simplices;
at the remaining accumulation points their scales tend to zero.
Moreover $\diam G^{-1}(\overline{V_D^+})\leq A\ell_D$.  To verify
this last assertion outside the filled hole, use
\eqref{hp:lift-displacement}: a lifted point mapping into $V_D^+$
lies within $O(\ell_D)$ of $D$ and has comparable Whitney scale.
Inside the hole the assertion is immediate.  This proves the stated
null-size and accumulation properties.

Here is the claimed exact opening procedure.  Around each $a_D$ take
an arbitrarily small target ball $W_D$ with closure in $U_D$, mutually
disjoint from the other chosen balls.  Their radii can be reduced
further at will.  On $Q_D^{-1}(W_D)$ replace the radial collapse by
a strictly increasing radial homeomorphism onto $W_D$, agreeing
with $Q_D$ near its boundary; one can first take a small centered
ball in the polar coordinates if necessary.  Leave $Q_D$ unchanged
outside $Q_D^{-1}(W_D)$.  Composing all these changes with $G$ gives
a homeomorphism and changes $F$ only on $F^{-1}(\bigcup_D W_D)$.
The source diameters at accumulating holes tend to zero, so both
the map and its inverse remain continuous at $\partial B$.
Equivalently, one may use a boundary collar and Schoenflies to fill
each of these smaller neighborhoods.  No regularity of these
topological fillings has been used in the energy estimate.

Finally assume that $h$ has property $N$.  On the lift region,
locally $H=i^{-1}\circ h\circ i$, with smooth diffeomorphic charts
on both sides.  Such compositions inherit property $N$, and so
does their Lipschitz postcomposition by $Q$.  The filled-hole
neighborhoods map to countably many points.  On $B$ the map is the
identity, and the common inner boundary has zero volume and maps
to itself.  These countably many local descriptions prove property
$N$ for $F$.
\end{proof}

\subsection{Opening isolated ball fibres}

We record the precise relative form of the topological opening that is
used both during and after the carrier construction.  A null family
below means that only finitely many members meeting a fixed compact
localization have diameter greater than any prescribed positive
number.

\begin{lemma}[Relative opening]\label{hp:opening}
Let $F:\Omega\to\Lambda$ be a continuous proper onto map obtained by
locally finite image-preserving applications of
Lemma~\ref{hp:cutoff} in genuine homeomorphic neighborhoods.  Assume
that subsequent applications avoid earlier fibre values and their
accumulation sets.  Let $P$ be the countable set of non-singleton
fibre values.  Then its fibres are tame closed balls, form a null
family on compact localizations, and their value accumulations have
singleton fibres.  If $C\subset\Lambda$ is relatively closed and
$C\cap P=\varnothing$, there is a homeomorphism
$H:\Omega\to\Lambda$, as finely close in values to $F$ as prescribed,
such that
\begin{equation}\label{hp:relative-opening-agreement}
 H=F\quad\hbox{on }F^{-1}(C).
\end{equation}
One can require agreement on a neighborhood of every compact target
test already separated from $P$.
\end{lemma}

\begin{proof}
For a single cutoff the assertion about fibres is part of
Lemma~\ref{hp:cutoff}: its nontrivial fibres lie inside separate
Whitney tetrahedra, and their diameters tend to zero at the exact
inner seam.  The corresponding target points accumulate only at that
seam, where the fibre is a singleton.  A later operation in a genuine
homeomorphic neighborhood changes none of those fibres.  Local
finiteness of the operations therefore preserves this description and
the null-family property on every compact localization.  In
particular every $v\in P$ is isolated from the other values in $P$
and from their accumulation set.

Choose mutually disjoint small target balls $B_v$ about these values.
For each fixed $v$ their radii can be made smaller than its distance
from $C$, from the other exceptional values and accumulation seams,
and from all fixed compact tests to be preserved.  These are positive
individual clearances; a common lower bound is neither asserted nor
needed.  Enumerate the values and impose in addition radii tending to
zero, with any prescribed fine value-error bound.  Properness gives
upper semicontinuity of compact fibres: if $V$ is a neighborhood of
$F^{-1}(v)$, a sufficiently small $B_v$ satisfies
$F^{-1}(\overline B_v)\subset V$.  Otherwise points outside $V$
mapping to a sequence converging to $v$ would have a subsequence in a
compact preimage, contradicting the definition of the fibre.

The inverse of $\partial B_v$ is an embedded locally flat sphere:
$F$ is a homeomorphism on a neighborhood of this sphere.  It bounds a
tame ball containing the fibre, either directly by the radial squeeze
description in Lemma~\ref{hp:cutoff}, or by the locally flat
Schoenflies Theorem~\cite[Theorem~5]{Brown1960}.  The preceding
upper semicontinuity places this ball inside a tame neighborhood
compactly contained in $\Omega$.  Replace $F$ on the ball by a ball
homeomorphism extending its boundary parametrization.  To keep an
outer collar fixed, first use two nested target balls and do this
only on the inner one; use the old map on the annular collar.

The replacements have disjoint target images, so the resulting map
is one-to-one and onto.  At an accumulation seam its difference from
$F$ tends to zero, since a change inside $B_v$ has size at most
$\diam B_v$.  It is thus continuous everywhere.  It is a
homeomorphism by invariance of domain, and its image is precisely
$\Lambda$.  Alternatively, properness and the null-family property
also give continuity of the inverse at the singleton seams.  The
balls avoid $C$, which proves
\eqref{hp:relative-opening-agreement}.  Enlarging a compact test to
a small closed neighborhood before choosing the balls gives the last
assertion.
\end{proof}

\subsection{Exceptional boxes and regular affine cores}

We apply the cutoff in two ways, according to the endpoint inverse
classification.  For exceptional data the assignment measure may be
singular, so the first lemma selects source cubes with little lost
assignment mass and small volume and facet energy.  The second lemma
uses regular affine jets to obtain an energy bound with a universal
constant; the accuracy needed to apply it may depend on the fixed jet
bounds.

\begin{lemma}[Exceptional inverse data]\label{hp:exceptional-boxes}
Let $E$ be a compact source null set contained in finitely many genuine
homeomorphic neighborhoods of a continuous $W^{1,p}_{\rm loc}$ map
$F$, where $p>2$.  Let $\sigma$ be a finite measure supported on $E$,
and let $g\in L^1_{\rm loc}$ be nonnegative.  For any $a,b>0$, any
prescribed source and target size bounds, and any closed sets separated
from $E$, one can choose finitely many cubes $U$ with disjoint closures,
avoiding those closed sets and compact in the given neighborhoods,
such that, after a $\sigma$-loss less than $a$, the retained data lie
with positive prescribed relative margins in their interiors and
\begin{equation}\label{hp:exceptional-cube-budget}
 \int_{\bigcup U^+}g< b,
 \qquad
 \sum_U\ell_U\int_{\partial U}^{\rm ext}g\,d\cH^2<b.
\end{equation}
Here $U^+$ are sufficiently small enlarged neighborhoods, and the
superscript $\rm ext$ means that each facet is allowed a fixed small
tangential extension.  The facet positions can be chosen to be
$L^1$ Lebesgue points in their normal variable, and their $F$-images
have zero three-dimensional volume.  The relative insets and the
further margins may be prescribed arbitrarily small before the final
grid phase is selected.
\end{lemma}

\begin{proof}
Take an open neighborhood $V$ of $E$, compact in the stated charts and
disjoint from the forbidden sets, on which $\int_Vg$ is arbitrarily
small.  Such a neighborhood exists by absolute continuity of the
integral.  Choose a cubical grid with small pitch $r$; all cubes and
the small facet extensions that meet $E$ then lie in $V$.  Replace
each grid cube by a concentric inset of side $(1-\tau)r$, with
$\tau>0$ small.  Their closures are disjoint.  Average the phase over
$[0,r)^3$.  For any fixed point, the fraction of phases for which it
lies in a gap or an additional relative margin is $O(\tau)$, with
the additional margin included in this bound.  Integrating against
$\sigma$ controls the expected lost assignment mass, without any
assumption on $\sigma$.

For the facet costs, Fubini in the normal coordinate gives
\[
 \mathbb E\sum_U\ell_U
       \int_{\partial U}^{\rm ext}g\,d\cH^2
       \leq C\int_Vg.
\]
The same estimate holds for the finitely many tangentially extended
facets, with a fixed overlap constant.  Choose $\tau$ so that the
expected lost mass is much smaller than $a$, and choose $V$ so that
the expected facet cost is much smaller than $b$.  Markov's
inequality then leaves a set of phases of positive measure on which
both required bounds hold.  Almost every phase additionally has all
facet coordinates as $L^1$ Lebesgue points of the normal slices of
$g$.  Lemma~\ref{hp:regularity} gives zero image volume for these
generic facets, including their finite tangential extensions.
Intersecting with these full-measure phase conditions costs nothing.
Only finitely many cubes meet the compact set $E$.  A final compact
restriction of the retained assignment set gives strict positive
margins.  Decreasing $r$ beforehand enforces the source sizes and,
by continuity of $F$ on the compact localization, the target sizes.
\end{proof}

\begin{lemma}[Regular affine implantation]\label{hp:regular-implant}
Let $F$ be a positively oriented $W^{1,p}_{\rm loc}$ homeomorphism
on a neighborhood of a compact set of regular full-rank source
points.  Suppose its jets and inverse jets are bounded there and its
Fr\'echet and power-mean differentiation tests hold uniformly below
a fixed radius.  In sufficiently small target grid boxes one can
implant exact positive affine cores, with arbitrarily small relative
grid margins and with disjoint enlarged source comparison boxes.
For a box of target side $R$ using the affine jet $b$ at one of its
assigned inverse points, the new energy in its source support is at
most
\begin{equation}\label{hp:regular-implant-cost}
 C_p\frac{|Db|^pR^3}{\det Db}
 \leq C_p\int_{V_b}|DF|^p,
\end{equation}
where $V_b$ is the source comparison box obtained as the affine
preimage of a buffered target box.  The constant depends only
on $p$ and the fixed cubical cutoff construction, not on the condition
number of $Db$.  The choices may be made relative to previously
fixed disjoint compact sets and exact cores.
\end{lemma}

\begin{proof}
Choose a representative $x_b$ whose image belongs to the pertinent
grid box, and put $b(x)=F(x_b)+DF(x_b)(x-x_b)$.  The source preimage
under $b$ of an enlarged target box lies in a ball of radius $C_bR$
about $x_b$.  The Fr\'echet test, followed by the change of variables
$y=b(x)$, makes
\[
 \sup |F b^{-1}(y)-y|/R
 \quad\hbox{and}\quad
 R^{-3}\int|D(F b^{-1})-I|^p
\]
as small as required.  The accuracy is chosen after the finite jet
and inverse-jet bounds; this is where those bounds affect the
construction.  One may also require
\[
 \int_{V_b}|DF-Db|^p
       \leq 2^{-p-1}|Db|^p |V_b|.
\]
Consequently the normalized derivative integral is $O_p(R^3)$,
whereas $\int_{V_b}|DF|^p$ is bounded below by a fixed multiple of
$|Db|^p|V_b|$.  The volume $|V_b|$ is a fixed shape multiple of
$R^3/\det Db$.

Apply Lemma~\ref{hp:cutoff} to $F b^{-1}$ in the target-normalized
coordinates and then precompose its result with $b$.  It is the
identity on the inner normalized core, hence the resulting source
map equals $b$ on the affine-source core.  Multiplication by $Db$
and the source Jacobian give the first bound in
\eqref{hp:regular-implant-cost}; the preceding lower bound gives
the second.  An inverse condition number is not used in this final
comparison.

For disjointness, inset not only the core but every used enlarged
normalized box from its outer grid box.  Make the value-test error
smaller than the resulting target margin.  Then $F$ maps the
affine-source comparison box inside the associated outer grid box.
Since $F$ is one-to-one on these neighborhoods, source comparison
boxes corresponding to disjoint target boxes are disjoint.  Compact
separation handles different inverse charts and all previously
reserved data.  Uniform tests are obtained whenever needed by
compact restriction in the finite assignment measure before the
mesh is chosen.  This argument does not require the grid centers
themselves to be regular image points.
\end{proof}

\subsection{An exact carrier with prescribed residual volume}

We can now build the carrier.  The first round uses the endpoint
assignments to produce exact neighborhoods of both rectangles in each
pair.  Those neighborhoods and their buffers must be fixed before the
quantizer can specify how small the residual target volume should be.
After these volume thresholds are given, the second round adds exact
cores throughout the remaining target, keeping the first-round regions
and the retained source data fixed.

\begin{proposition}[Two-round exact carrier]\label{hp:carrier}
Prescribe a positive continuous source value tolerance
$a:\Omega\to(0,\infty)$ before applying Proposition~\ref{hp:flattening}.
Use there a target tolerance $a_\Lambda$ satisfying
\[
 a_\Lambda(y)\leq\tfrac14 a(f^{-1}(y)),
\]
and then fix the resulting rank data and endpoint protections of
Lemma~\ref{hp:wells}.  Fix also an enlargement factor $A>1$ for the
marked collars, before assigning their facet budgets, and let $\eps>0$.
After a total $\mu$-loss smaller than $\eps$, a first round of choices
gives $\kappa>0$, a compact retained source set $K$, a continuous
proper onto carrier $F_b^1:\Omega\to\Lambda$, an exact open target
set $O_1$, and compact endpoint rectangles strictly inside $O_1$.
These choices precede any prescribed residual-volume thresholds.
Fix a locally finite target Whitney tiling $\mathcal Q$, and choose
arbitrary numbers $a_Q>0$ for $Q\in\mathcal Q$, after $O_1$ and the
endpoint buffers have been determined.  A second round gives
$F_b:\Omega\to\Lambda$ and $O\supset O_1$ such that:
\begin{enumerate}[label=\textup{(\roman*)}]
\item $F_b$ is continuous, proper and onto, belongs to
$W^{1,p}_{\rm loc}$, and has only the countable tame ball fibres of
Lemma~\ref{hp:opening}.  It is a genuine homeomorphism on target
neighborhoods away from their values and accumulation seams.
For $p\geq3$ it satisfies volume Lusin $N$.
\item $O$ is a locally finite union of exact affine or PL core
images.  Every compact subset of such a core has a neighborhood on
which $F_b$ and its inverse are exactly those of its affine or PL
chart.  In particular no point of $O$ has a nontrivial fibre, and
\begin{equation}\label{hp:residual-volume}
 \cL^3(Q\setminus O)<a_Q\qquad(Q\in\mathcal Q).
\end{equation}
The compact endpoint rectangles retain their fixed buffers in $O_1$.
\item $F_b=F_\kappa$ on a neighborhood of $K$.  It meets the
prescribed fine value tolerance relative to $f$.  The choices of
$\kappa$ and of the retained central margins ensure that
$F_\kappa(K)$ stays in the protected central plate neighborhoods.
\item When $2<p<3$, there is a locally finite family of marked
source cubes $U$ with disjoint closures, away from $K$, such that
\begin{equation}\label{hp:small-marked-mass}
 \int_{\bigcup U}(1+|Df|^p)<\eps.
\end{equation}
For $p\geq3$ this family is empty.  Off the marked cubes the energy
has the universal buffered regional estimate described below.
\item Write $C_U(w)$ for a cube-radius collar of physical width $w$
about $\partial U$.  For the prescribed enlargement factor $A$, the
widths $w_U>0$ can be chosen after both rounds, with disjoint enlarged
marked cubes, so that
\begin{equation}\label{hp:marked-collar-budget}
 \sum_U\frac{\ell_U}{w_U}
             \int_{C_U(Aw_U)}|DF_b|^p<\eps.
\end{equation}
The same bound, multiplied by the $p$th power of the Lipschitz
constant, holds after a uniformly Lipschitz output composition.
The enlarged cubes may still have original $(1+|Df|^p)$-mass less
than $2\eps$.
\end{enumerate}

The regional estimate in \textup{(iv)} has the following meaning.
Suppose a locally finite family of regional tests $A_j$ and two
successive small open buffered enlargements $A_j^{[1]}$ and
$A_j^{[2]}$ are prescribed before the implants are made.  All implants
may be chosen so fine that
\begin{equation}\label{hp:regional-energy}
 \int_{A_j\setminus\bigcup U}|DF_b|^p
       \leq C_p\int_{A_j^{[2]}}|Df|^p
       \qquad\hbox{for every }j.
\end{equation}
Here each enlargement has positive local buffer, the supports and
comparison boxes meeting $A_j$ fit inside its first enlargement at
the corresponding round, and $C_p$ is independent of all selected
jet condition numbers, residual thresholds, and PL reference-map
constants.  In particular the tests may be reserved full-rank cut
shells or ordinary regions outside prescribed interiors.  The same
construction works for summably weighted such tests toward the ends
of the open domains.
\end{proposition}

\begin{proof}
We give the choices in order.  The initial flattening tolerance gives
\[
 |F_\kappa(x)-f(x)|
   =(1-\kappa)|S_0(f(x))-f(x)|<\tfrac14a(x)
       \qquad(0\leq\kappa\leq1).
\]
All subsequent replacements keep their old boundary values and their
old local images.  Choose their compact supports so small that the
oscillation of the map being replaced gives
\[
 |F_b^1-F_\kappa|<a/4,
 \qquad |F_b-F_b^1|<a/4.
\]
The positive minorants in Lemma~\ref{pre:local-tolerances} permit these
choices simultaneously on each locally finite family.  Thus the final
carrier satisfies $|F_b-f|<a$.  These choices control the additional
implantation motion; the first displayed bound has already reserved
the flattening motion before the rank and endpoint data were fixed.

\emph{First round: fixing the exceptional boxes.}
Classify both endpoint signs simultaneously as in
Lemma~\ref{hp:wells}.  In the range $2<p<3$, the exceptional inverse
assignments have compact ranges in a fixed source null set.  Apply
Lemma~\ref{hp:exceptional-boxes}, with $g=1+|Df|^p$, to enclose them
in finitely many marked cubes $U$, discarding only a prescribed
small assignment weight.  Make all these cubes and their enlarged
neighborhoods disjoint from $K$ and from the retained regular inverse
assignments.  Give their source integrals and extended-facet budgets
arbitrarily small shares of $\eps$.  Retain the exceptional inverse
points with strict inner margins.  In the grid phase selection impose
the generic-facet condition for both $F_0$ and the original map $f$.
Lemma~\ref{hp:regularity} supplies full-measure sets of phases for
both conditions, so their intersection retains the same budget choices.
The inset fractions can be as small as desired and are fixed at this
stage.

Inside each $U$, fix a slightly smaller cubical core $U^-$ containing
its retained inverse data with room, and leave enough space for the
cutoff support to remain compact in $U$.  The associated compact
endpoint set $L_U$ lies in $F_0(\interior U^-)$.  Choose a compact
target neighborhood $W_U$ of $L_U$ inside this open image and inside
the genuine homeomorphic target neighborhood of $F_0$.  These margins
are chosen using $F_0$, before $\kappa$ or a PL reference map is fixed.

\emph{First round: fixing the regular grids and all margins.}
First compact-restrict the regular endpoint assignments so that their
jets, inverse jets and differentiation radii for $F_0$ are uniform.
Choose small relative grid margins.  On each plate we will use the
same shifted horizontal square grid for both endpoint signs.  Choose
its pitch $R$ sufficiently small that every pertinent square has a
fixed classification for each sign, using compact separation of the
classification sets.  For an exceptional sign, require that the whole
inset endpoint rectangle lies in the interior of its previously
chosen $W_U$.  For regular signs, the uniform differentiation tests
in Lemma~\ref{hp:regular-implant} permit the required normalized value
and gradient accuracy on every buffered box.  These upper bounds on
$R$ are independent of the grid phase.

Now select that phase.  For every fixed datum, the fraction of phases
lost to the grid margins is bounded by the relative margin fraction.
Fubini therefore gives a phase losing arbitrarily little of the finite
horizontal assignment measure.  Trim compactly inside its surviving
squares, and protect smaller rectangles once more for the actual
central data.

For each square with a regular sign, choose a representative endpoint
inverse and its affine jet $b$.  Form the target three-box of horizontal
and normal side $R$, centered at the grid center at height $\pm d$.
The uniform $F_0$ tests make $F_0b^{-1}$ close to the identity on
every buffered box, including in normalized gradient power mean.
Fix these finitely many tests and their affine-source comparison
boxes.  The source boxes are disjoint because their $F_0$-images stay
strictly inside disjoint outer grid boxes.  Prior compact separation
also makes them disjoint from the marked cubes and from $K$.
All geometric tests and central margins are now fixed.

\emph{First round: choosing $\kappa$ and making the implants.}
Take $\kappa>0$ small enough that all the regular value and gradient
tests also hold for $F_\kappa b^{-1}$.  This follows from
$F_\kappa\to F_0$ in values and in $W^{1,p}$ on each fixed compact.
Require also that $F_\kappa(K)$ remains within the nested central
margins and that $F_\kappa^{-1}(W_U)\Subset\interior U^-$ for each
exceptional core.

The latter inverse requirement follows from uniform inverse convergence
on compact target sets in the homeomorphic complement of $F_0$.
Indeed, if inverse points for $F_\kappa$ failed to converge uniformly
to those for $F_0$, properness and the fine motion bound would give a
convergent subsequence limiting to a different preimage under $F_0$.
Uniqueness in the genuine inverse neighborhood excludes this.
The same compact inverse argument applies to sufficiently fine
homeomorphic approximations of the now fixed $F_\kappa$.

Apply Lemma~\ref{sm:relative-pl} to choose a PL homeomorphism
$h_{\rm ref}$ finely approximating $F_\kappa$.  Choose it so fine
that $h_{\rm ref}^{-1}F_\kappa$ satisfies the normalized identity
tolerance of Lemma~\ref{hp:cutoff} on each exceptional cube, and so
that $h_{\rm ref}^{-1}(W_U)\Subset\interior U^-$.
Inverse continuity on the compact localizations supplies the first
test, and the preceding inverse-margin argument supplies the second.
Apply the cutoff with core $U^-$ and support compact in $U$, and
postcompose by $h_{\rm ref}$.  The result equals $h_{\rm ref}$ on
$U^-$, so its exact core image contains the whole protected endpoint
rectangle with room.  Since $h_{\rm ref}$ and its inverse are locally
bi-Lipschitz, this replacement has finite local $W^{1,p}$ energy;
its possibly large constant is used only inside marked cubes.

For each regular sign, use the cutoff construction of
Lemma~\ref{hp:regular-implant} with the fixed jet and the
$F_\kappa$ tests.  The resulting map is exactly $b$
on its core, whose image contains the inset endpoint rectangle.
These supports are disjoint from the exceptional changes, and all
supports avoid a neighborhood of $K$.  The regular cost is
\[
 C_p |Db|^pR^3/\det Db
       \leq C_p\int_{V_b}|DF_0|^p
       \leq C_p\int_{V_b}|Df|^p,
\]
where the final absolute factor is supplied by
Proposition~\ref{hp:flattening}.  The comparison uses the fixed
$F_0$ jet tests; the $F_\kappa$ tests were required to be equally
accurate before applying the cutoff.  This normalization is why no
condition number enters the final energy charge.

Call the first-round map $F_b^1$, and let $O_1$ be the union of
slightly smaller exact core images.  Each is a polyhedral affine or
PL image and has an exact neighborhood with buffer.  The compact
endpoint rectangles lie strictly inside this open set.  The map is
proper, onto and continuous because the finite changes preserve
local images and agree with the old map near their outer boundaries.
Lemma~\ref{hp:cutoff} supplies its local Sobolev regularity and its
ball-fibre description.  It also preserves volume $N$ when $p\geq3$.
The fibre values and their accumulation sets have zero target
volume: the former are countable, and the latter lie on finitely
many exact affine or PL core boundaries.

\emph{Second round: choosing the sets to fill.}
The first-round choices, including $O_1$ and its endpoint buffers,
are now fixed.  Prescribe all the $a_Q$.  In the interior of each
Whitney cube $Q$, remove $O_1$ from the work region and avoid its
boundary, the first-round fibre values and their accumulations, the
images of marked-cube boundaries, and $F_b^1(K)=F_\kappa(K)$.
Except for $O_1$, all these exclusions have zero volume.  The last
one is null because the retained rank-deficient regular image under
$f$ is null and $S_\kappa$ is Lipschitz.  The first marked-cube boundaries were selected to have null images
under $f$; near them $F_b^1=F_\kappa=S_\kappa f$, and the
Lipschitz map $S_\kappa$ preserves that nullity.

On the remaining genuine homeomorphic portions,
Lemma~\ref{hp:regularity} gives the following exhaustive alternative
at almost every target point: its preimage is regular full rank, or
belongs to a fixed source null set.  Regular deficient images have
zero volume.  When $p\geq3$, volume $N$ eliminates the second
alternative.  Compactly restrict these measurable target sets,
separating their classifications, their inverse charts, and the
inside and outside of the first marked cubes.  The target volume
lost in $Q$ can be made smaller than any fixed fraction of $a_Q$.
All selected sets are compact in the remaining work region.  In
particular their source preimages are compact and all necessary
separations are positive.

\emph{Second round: implants.}
For exceptional assignments outside the old marked cubes use
Lemma~\ref{hp:exceptional-boxes} with
$g=1+|Df|^p+|DF_b^1|^p$.  This is locally integrable, and the
exceptional source set is null.  Choose new marked cubes with
summably small source and facet budgets, disjoint from all earlier
marked cubes, from the regular assignments, and from $K$.  Their
supports have images compact inside the current Whitney interior.
For exceptional assignments already inside an old marked cube,
use smaller reference-map implants there; no new marked cube is
required.  A suitable PL reference is available on the genuine
homeomorphic neighborhoods in use.  One way to obtain it globally
is to apply Lemma~\ref{hp:opening} to $F_b^1$ with a closed
neighborhood of all chosen compact tests fixed, and then use
Lemma~\ref{sm:relative-pl}.  The resulting reference approximates
$F_b^1$ arbitrarily well on these tests.  The same inverse-margin
argument as in the first round makes the chosen endpoint or volume
assignments fall within its exact core images.

For regular assignments use ordinary three-dimensional target grids
and Lemma~\ref{hp:regular-implant}, now with the jets of $F_b^1$.
Compact restriction gives the uniform tests required for a sufficiently
fine grid.  Predetermine arbitrarily small insets, then make the
value and gradient tests correspondingly fine.  Each comparison box
meeting an old marked cube is chosen wholly inside it; exterior
comparison boxes miss every marked cube.  The previously compactly
separated exceptional assignments and new marked cubes are avoided.
The regular source comparison boxes in this round are disjoint, and
their costs are bounded by $C_p\int|DF_b^1|^p$ on those boxes.

For exceptional assignments, grid phase averaging against ordinary
target volume pulled back to the null source set controls the loss
to source insets.  For regular assignments, the target grid margins
have arbitrarily small volume.  Therefore in each $Q$ the retained
assignments can be covered by exact cores with total additional
loss less than the unused share of $a_Q$.  Only finitely many compact
assignments and implants are needed in each Whitney cube.  Whitney
local finiteness and properness then make all second-round supports
locally finite in target and source.  These arguments prove
\eqref{hp:residual-volume}.  They also show that second-round
supports miss a neighborhood of every old marked-cube boundary and
of the compact retained set $K$.

Let $F_b$ be the resulting map and $O$ the union of the old and new
buffered exact core images.  The first-round endpoint cores have
been kept fixed.  Image preservation and local finiteness give
properness and surjectivity.  The local Sobolev, Lusin and fibre
properties follow as in the first round.  The additional
accumulation seams are only the new exact core boundaries, a locally
finite family of zero-volume polyhedra.  Thus
Lemma~\ref{hp:opening} applies to the combined carrier.

\emph{Universal regional charging.}
Outside the marked cubes no reference-map constant enters the
construction.  At the first round disjoint regular comparison boxes
give the energy bound by $C_p\int|Df|^p$; elsewhere
$|DF_\kappa|\leq C|Df|$.  At the second round disjoint regular
comparison boxes give the bound by $C_p\int|DF_b^1|^p$.
If a support or comparison box meets $A_j$, choose it small enough
to lie in $A_j^{[1]}$ at that round.  At the preceding round use
$A_j^{[2]}$ for this first enlargement.  Combining the two disjoint
sum estimates proves \eqref{hp:regional-energy}.  There are only two
rounds, so the number of buffer enlargements and the multiplication
of universal constants are fixed.  Local finiteness of the tests
allows all required support sizes to be chosen simultaneously from
positive local minorants.  For countably many prescribed weighted
tests, take the usual compact exhaustion and summable budgets.
This uses no positive lower bound for a buffer width at the ends
of the domains.

\emph{Marked-cube collars.}
The enlargement factor $A$ was fixed before the marked cubes were
selected.  Choose all initial and added marked-cube budgets so small
that \eqref{hp:small-marked-mass} holds and their extended-facet budgets,
including the factor $C_A$ in the slice estimate below, sum to an
arbitrarily small number.  Near the boundary of a first
marked cube the carrier still equals $F_\kappa$, whose derivative
is bounded by $C|Df|$.  Near a newly marked cube it equals the
unchanged $F_b^1$.  The modifications have positive distance from
each fixed boundary: later supports are locally finite over its
compact image and avoid it.  Thus a possibly very thin unchanged
neighborhood of each $\partial U$ remains.

Choose $w_U$ within that neighborhood.  The collar is contained in
six normal slabs of thickness $O(Aw_U)$ over the extended facets.
The $L^1$ Lebesgue property of the selected normal slices yields
\[
 \limsup_{w\downarrow0}\frac{\ell_U}{w}
       \int_{C_U(Aw)}|DF_b|^p
 \leq C_A\ell_U\int_{\partial U}^{\rm ext}g_U\,d\cH^2,
\]
where $g_U=C|Df|^p$ for old cubes and
$g_U=|DF_b^1|^p$ for added cubes.  Give each cube a further summable
error share and decrease its width to realize this estimate.  The
earlier facet budgets imply \eqref{hp:marked-collar-budget}.
Widths can simultaneously make enlarged cubes disjoint and their
additional original energy and volume arbitrarily small.  Finally,
the Sobolev chain inequality
$|D(L\circ F_b)|\leq\Lip(L)|DF_b|$ proves the assertion about
Lipschitz output compositions.  This completes all parts of the
Proposition.
\end{proof}

\subsection{Full-rank regions reserved for the final correction}

The carrier construction is now established.  The next three results
prepare a separate step in the eventual completion: reserving finitely many full-rank cubes, then making a
preliminary locally bi-Lipschitz approximant affine on those cubes.
In Section~\ref{sec:high-assembly}, the cubes and their energy tests
will be reserved before applying the carrier construction; the actual
correction will be performed only at the end.  Here the replacement must remain a homeomorphism.  We therefore return
to the normalized cube $B=[-1,1]^3$ and the shell coordinate
$t(x)=\max_j|x_j|-1$ of Lemma~\ref{hp:cutoff}, and first prove its
smooth-input version.

\begin{lemma}[Smooth cubical cutoff]\label{hp:smooth-cutoff}
Under the hypotheses of Lemma~\ref{hp:cutoff}, suppose in addition
that $h$ and its inverse are smooth on the neighborhoods in use.
For every $\varepsilon_0>0$ there is instead a locally bi-Lipschitz
homeomorphism $F$ onto the same image, equal to the identity on $B$
and equal to $h$ near and outside $\partial B_{2s}$, such that
\begin{equation}\label{hp:smooth-cutoff-energy}
 \int_{0<t<2s}|DF|^p
 \leq C_p\int_{|t|\leq Cs}|Dh|^p+\varepsilon_0.
\end{equation}
The local bi-Lipschitz constants may depend on $h,s$, and
$\varepsilon_0$.  In particular no uniform inverse Lipschitz bound
is asserted as $\varepsilon_0\downarrow0$.
\end{lemma}

\begin{proof}
Use the construction in Lemma~\ref{hp:cutoff} up to the hole-filling
stage.  The lift is now smooth on its neighborhood, and the lifted
hole boundaries are smooth embedded spheres.  The smooth
three-dimensional Schoenflies Theorem identifies their bounded sides
with smooth balls; see \cite[Section~1.1, Theorem~1.1]{Hatcher2023}.
A prescribed smooth orientation-preserving
sphere parametrization extends over a ball: after a ball
identification, isotope the resulting sphere diffeomorphism to a
rotation, and insert that isotopy in an annulus before using the
rotation in the central ball.  Smale's theorem supplies a jointly
smooth isotopy for the fixed sphere diffeomorphism
\cite[Theorem~6]{Smale1959}; for the stronger smooth dependence on
the sphere diffeomorphism, see also Li--Watts
\cite[Theorem~1.5]{LiWatts2011}.
Using collar coordinates first makes
the extension agree with the already prescribed lift on a smaller
two-sided boundary collar.  Hence each hole has a smooth, and in
particular bi-Lipschitz, filling.  It remains essential to prove that
their constants can be chosen uniformly over the infinitely many
Whitney layers.

Fix $h$ and $s$.  By \eqref{hp:immersion-template}, the normalized
lift in a local chart is
\begin{equation}\label{hp:normalized-smooth-lift}
 \xi\longmapsto
 I^{-1}\!\left(
   \frac{h(y_*+sI(\xi))-y_*}{s}
 \right),
\end{equation}
where the inverse is the specified nearby branch.  Overlaps with
neighboring charts are part of the same finite pattern data.  There
are finitely many choices of $I$ and finitely many possible target
anchors $y_*$, since $s$ is fixed and the sampling region is bounded.
The normalized hole and buffer shapes also have finitely many
choices.  Thus there are only finitely many normalized smooth collar
maps on hole boundaries, not merely a bounded family of arbitrary
boundary maps.  Choose a smooth filling once for each of these
finitely many data.  On their compact closures take the maximum of
the finitely many forward and inverse Lipschitz constants.  Call it
$L(h,s)$.  The physical source and output coordinates of a hole are
both rescaled by its scale $\ell_D$, so this constant does not change
with the depth of the Whitney layer.

The resulting pre-squeeze homeomorphism $G$ therefore has a uniform
finite bound for $|DG|$ in all filled holes, and the analogous
inverse bounds there.  Its lift pieces have such bounds as well:
there are only the finitely many normalized maps
\eqref{hp:normalized-smooth-lift}.  The agreement on collars provides
the same local bounds across all finite interfaces.

Replace the nonstrict radial squeezes by strictly increasing ones.
For a common $0<\alpha<1$, let $Q_{D,\alpha}$ be the homothety of
ratio $\alpha$ about $a_D$ on $V_D^+$, interpolate monotonically to
the identity in the surrounding fixed buffer, and keep the identity
near $\partial U_D$.  Their forward Lipschitz constants are uniformly
bounded independently of $\alpha$, and their inverse constants are
finite for every fixed $\alpha>0$.  On each filled hole,
\[
 D(Q_{D,\alpha}\circ G)=\alpha DG.
\]
Consequently
\[
 \sum_D\int_{V_D}|D(Q_{D,\alpha}\circ G)|^p
 \leq\alpha^pL(h,s)^p\sum_D|V_D|
 \leq\alpha^pL(h,s)^p|B_{2s}\setminus B|.
\]
Choose $\alpha$ so that this is below $\varepsilon_0$.  On the
remaining regions the proof of
\eqref{hp:lift-total-energy} applies without change, with the same
uniform upper bound for the Lipschitz constant of the squeezes.
This proves \eqref{hp:smooth-cutoff-energy}.

For clarity, local bi-Lipschitz regularity includes the inner seam.
For these fixed data the maps and their inverses have uniform finite
local Lipschitz bounds off $\partial B$, by the finite normalized
descriptions and the positive squeeze parameter.  They extend
continuously as the identity on $\partial B$.  On a small line
segment crossing a face, edge, or vertex of that cube, decompose the
segment into its open subintervals off the cube boundary.  Integrate
the common Lipschitz bound on those intervals and use continuity at
their endpoints; on portions inside the cube the map is the identity.
This gives the same finite Lipschitz bound across the seam.  Apply
the identical argument in the image to the inverse, which is also
the identity on $B$.  Thus the final map is locally bi-Lipschitz
throughout the open replacement domain and glues to $h$ as claimed.
\end{proof}

\begin{remark}\label{hp:cutoff-affine-scaling}
The cutoff Lemmas apply in affine output coordinates.  More
explicitly, let
\[
 b(x)=y_0+A(x-x_0),\qquad A\in GL^+(3),\qquad r>0,
\]
and normalize a map $k$ by
\[
 \widehat k(\xi)
 =\frac1r A^{-1}\bigl(k(x_0+r\xi)-y_0\bigr).
\]
If $\widehat k$ satisfies the hypotheses of either cutoff Lemma,
transforming the resulting map back makes it equal to $b$ on
$x_0+rB$ and preserves its old image and its outer boundary collar.
Writing
\[
 S=\{x_0+r\xi:0<t(\xi)<2s\},\qquad
 A^{\mathrm{band}}=\{x_0+r\xi:|t(\xi)|\leq Cs\},
\]
the smooth estimate becomes
\begin{equation}\label{hp:affine-cutoff-energy}
 \int_S|Dk_{\mathrm{new}}|^p
 \leq C_p|A|^p|A^{-1}|^p
          \int_{A^{\mathrm{band}}}|Dk|^p
       +|A|^pr^3\varepsilon_0.
\end{equation}
This follows from $D_\xi\widehat k=A^{-1}D_xk$ and the source
Jacobian $r^3$.  The last additive error can therefore be prescribed
arbitrarily in the original coordinates as well.
\end{remark}

\begin{proposition}[Postponed full-rank correction]
\label{hp:full-rank-correction}
Let $p>2$ and let $f:\Omega\to\Lambda$ be an orientation-preserving
Sobolev homeomorphism between bounded domains.  Let $R$ be the set
of its regular differentiation points at which $Df$ has rank three.
Let $P\Subset\Omega$ be any previously protected compact set
disjoint from $R$.  Given $\varepsilon>0$, one can choose finitely
many pairwise disjoint buffered source cubes, disjoint from $P$,
with centers $x_j\in R$, radii $r_j$, and positive affine jets
\[
 A_j=Df(x_j),\qquad
 b_j(x)=f(x_j)+A_j(x-x_j),
\]
such that the inner cubes $K_j=x_j+r_jB$ satisfy
\begin{equation}\label{hp:full-rank-core-tests}
 \int_{R\setminus\bigcup_jK_j}|Df|^p<\varepsilon,
 \qquad
 \sum_j\int_{K_j}|Df-A_j|^p<\varepsilon.
\end{equation}
Their buffered cutoff bands
\[
 E_j=\{x_j+r_j\xi:|t(\xi)|\leq Cs\}
\]
can have arbitrarily small total $f$-energy, even after multiplication
by all the finitely bounded affine factors
$C_p|A_j|^p|A_j^{-1}|^p$.  On those bands the normalized $f$ is
within $cs/4$ of the identity.

Write $Q_j=x_j+r_jB_{(C+1)s}$ for the buffered outer cubes.
After these choices there are numbers $v_j>0$ and $\beta>0$ such
that every smooth diffeomorphism $k:\Omega\to\Lambda$ satisfying
\[
 \sup_{Q_j}|k-f|<v_j\quad\text{for each }j,
 \qquad \sum_j\int_{E_j}|Dk|^p<\beta
\]
can be changed only inside the associated outer cubes
$x_j+r_jB_{2s}$ to a locally bi-Lipschitz homeomorphism
$\widetilde k:\Omega\to\Lambda$ with
\[
 \widetilde k=b_j\quad\text{on }K_j.
\]
The sum of the derivative-error integrals on the inner cubes and
their modified shells is less than $\varepsilon$, after starting the
selection with smaller error prescriptions if necessary.  The cubes
and tests may also be chosen to retain any prescribed positive
continuous value tolerance for the final modification.

A locally bi-Lipschitz preliminary map in $W^{1,p}(\Omega;\R^3)$
can be used in place of $k$ by first applying Theorem~\ref{sm:smoothing}, preserving the finite
value and band-energy tests with room to spare.
\end{proposition}

\begin{proof}
We spell out the order of choices.  Since $|Df|^p$ is integrable,
choose a compact set $K\subset R\setminus P$, with arbitrarily
small discarded full-rank energy, on which
\[
 |Df(x)|+|Df(x)^{-1}|\leq M
\]
for some finite $M$.  This is possible by inner regularity and
truncation of the finite values of the two matrices.  All points of
$K$ may be taken to be both Fr\'echet differentiation points and
$L^p$ differentiation points of $Df$.  The positive determinant of
their derivatives follows from the orientation and the local-degree
test at an invertible Fr\'echet differential.  Fix $M$ before making
any shell-energy choices.

Take the common dyadic $s>0$ small.  At every $x\in K$ there are
arbitrarily small radii $r$ such that the buffered outer cube
\[
 Q=x+rB_{(C+1)s}
\]
has closure in $\Omega\setminus P$, and, with $A=Df(x)$ and
$b(z)=f(x)+A(z-x)$,
\begin{align}
 &\sup_{z\in Q}|A^{-1}(f(z)-b(z))|
       <\frac{cs}{4}\,r,\label{hp:jet-value-test}\\
 &\int_Q|Df-A|^p<\eta|Q|.\label{hp:jet-energy-test}
\end{align}
Here $\eta>0$ can be prescribed after $M$ and $s$.
The first inequality follows from Fr\'echet differentiability,
using $|A^{-1}|\leq M$; the second follows from $L^p$
differentiation.  Both persist at all sufficiently small radii, so
these outer cubes form a fine Vitali family centered on $K$.
Their radii can additionally be bounded by any prescribed local
upper bounds needed for value accuracy and clearance.

Use Vitali selection and then a finite truncation to obtain disjoint
outer cubes $Q_j$ missing only an arbitrarily small amount of the
weighted measure $\mathbf1_K|Df|^p\,dx$.  The closures may be made
disjoint as well: after the finite selection, shrink the cubes by
arbitrarily small amounts, retaining the strict differentiation
tests and allowing an arbitrarily small additional weighted loss.
Start those tests with strict margins before this shrinking.  The
fixed ratio between an outer cube and its inner cube is preserved
by rescaling its $r_j$.  Set $A_j=Df(x_j)$ and use the notation in
the statement.

Each buffered band $E_j$ and each outer-minus-inner region
$Q_j\setminus K_j$ has volume at most $A s|Q_j|$, with $A$ depending
only on the fixed constant $C$.  Equation
\eqref{hp:jet-energy-test} and
$|X+Y|^p\leq2^{p-1}(|X|^p+|Y|^p)$ give
\begin{equation}\label{hp:full-rank-band-energy}
 \sum_j\int_{E_j\cup(Q_j\setminus K_j)}|Df|^p
 \leq A_p(M^ps+\eta)\sum_j|Q_j|
 \leq A_p(M^ps+\eta)|\Omega|.
\end{equation}
The same differentiation test gives
\begin{equation}\label{hp:full-rank-core-error}
 \sum_j\int_{K_j}|Df-A_j|^p\leq\eta|\Omega|.
\end{equation}
Thus the discarded full-rank energy consists of the initial compact
truncation, the finite Vitali truncation, and a part bounded by
\eqref{hp:full-rank-band-energy}.  This proves
\eqref{hp:full-rank-core-tests} after decreasing all the error
prescriptions.  It also proves the stronger band statement: every
affine cutoff multiplier is at most $C_pM^{2p}$, so choose $s$ and
then $\eta$ to make
\[
 C_pM^{2p}\,A_p(M^ps+\eta)|\Omega|
\]
as small as required.  Notice that $M$ was fixed before $s$ and
$\eta$; there is no bound being chosen after the bands to which it
must apply.  Equation~\eqref{hp:jet-value-test} is exactly the
stated normalized value test for $f$.

Now fix this finite family.  Require the preliminary smooth map
$k$ to satisfy
\begin{equation}\label{hp:preliminary-value-test}
 \sup_{Q_j}|k-f|<\frac{csr_j}{4M}
 \quad\text{for each }j.
\end{equation}
Combining this with \eqref{hp:jet-value-test} shows that
\[
 \widehat k_j(\xi)
 =r_j^{-1}A_j^{-1}\bigl(k(x_j+r_j\xi)-f(x_j)\bigr)
\]
is within $cs/2$ of $\xi$ on $|t(\xi)|\leq Cs$.
In particular it meets the closeness hypothesis of
Lemma~\ref{hp:smooth-cutoff} with a fixed positive margin.  Apply
that Lemma separately in these disjoint cubes, and return to the
original coordinates as in
Remark~\ref{hp:cutoff-affine-scaling}.  Choose the finitely many
additive errors to have arbitrarily small sum.  The new maps are
exactly $b_j$ on $K_j$, agree with $k$ on outer collars, and have
the same images as $k$ in their respective replacement cubes.
Therefore they glue to a locally bi-Lipschitz homeomorphism onto
the same fixed target $\Lambda$.

Let $S_j=\{x_j+r_j\xi:0<t(\xi)<2s\}$ denote the modified shells.
Equation~\eqref{hp:affine-cutoff-energy} yields
\[
 \sum_j\int_{S_j}|D\widetilde k|^p
 \leq C_pM^{2p}\sum_j\int_{E_j}|Dk|^p+\delta,
\]
where $\delta>0$ is freely prescribed.  Hence
\begin{equation}\label{hp:correction-shell-error}
 \sum_j\int_{S_j}|D\widetilde k-Df|^p
 \leq2^{p-1}\left(
 C_pM^{2p}\sum_j\int_{E_j}|Dk|^p+\delta
 +\sum_j\int_{S_j}|Df|^p\right).
\end{equation}
The last term was made small in
\eqref{hp:full-rank-band-energy}.  Choose a positive upper bound
for $\sum_j\int_{E_j}|Dk|^p$ after the finite affine bounds are
known, and take $\delta$ small.  Equations
\eqref{hp:full-rank-core-error} and
\eqref{hp:correction-shell-error} then give the asserted total
derivative accuracy on the corrected regions.

To retain a prescribed positive continuous value tolerance, first
make the cubes small enough that the oscillation of $f$ on each
buffered cube is much smaller than that tolerance there.  This is
allowed by the fineness of the Vitali family and continuity of $f$.
Also impose correspondingly small value errors on $k-f$.
Every modified value lies in the old image of its replacement cube.
Thus its distance from $f$ at the source point is bounded by that
cube's oscillation of $f$ plus the chosen value error of $k$.
These quantities can be made smaller than the requested tolerance.

Finally suppose the available preliminary map is only locally
bi-Lipschitz.  Apply Theorem~\ref{sm:smoothing} with sufficiently
small strong $W^{1,p}$ error and its fine value control on the
finitely many compact buffered cubes.  Strong convergence preserves
their energy upper tests after allowing a strict margin, since
\[
 \int_{E_j}|Dk|^p
 \leq2^{p-1}\int_{E_j}|Dg|^p
      +2^{p-1}\|Dk-Dg\|_{L^p(\Omega)}^p
\]
for a preliminary map $g$.  The fine value control preserves
\eqref{hp:preliminary-value-test}.  The preceding smooth correction
therefore applies.  Any change outside the selected cubes at this
preliminary smoothing stage has the arbitrarily small strong error
prescribed in Theorem~\ref{sm:smoothing}.
\end{proof}

\subsection{Compressing the marked interiors}

\begin{lemma}[Subcritical marked-cube compression]\label{hp:hide-cubes}
Suppose $2<p<3$, and let $U$ and $w_U$ be the marked cubes and
collars of Proposition~\ref{hp:carrier}.  Fix any additional error
$\varepsilon_0>0$.  Let
$k:\Omega\to\Lambda$ be a locally bi-Lipschitz homeomorphism.
Suppose on thinner collars $C_U'(w_U)$, containing a fixed
relative share of the outer half-collar, one has
\begin{equation}\label{hp:k-collar-assumption}
 \sum_U\frac{\ell_U}{w_U}
        \int_{C_U'(w_U)}|Dk|^p\leq b.
\end{equation}
There are disjoint slightly enlarged cubes $\widetilde U\supset U$,
whose boundaries lie in these outer collars, and source
self-homeomorphisms supported in their closures, such that the
resulting locally bi-Lipschitz map $\widetilde k$ agrees with $k$
outside $\bigcup\widetilde U$ and satisfies
\begin{equation}\label{hp:hidden-cost}
 \sum_U\int_{\widetilde U}|D\widetilde k|^p
                \leq C_p b+\varepsilon_0.
\end{equation}
Its value change on each $\widetilde U$ takes place inside
$k(\widetilde U)$.  Thus previously required fine compact value
accuracy is preserved by choosing the marked cubes sufficiently
fine.  In particular, if the costs in
\eqref{hp:k-collar-assumption} are bounded by a fixed multiple of
\eqref{hp:marked-collar-budget} plus errors summable after
multiplication by $\ell_U/w_U$, all marked-interior costs can be
made arbitrarily small.  No convergence to a fixed boundary trace
is assumed.
\end{lemma}

\begin{proof}
Slice the outer collar of each $U$ by concentric cube boundaries.
The coarea formula for the cube radius and
\eqref{hp:k-collar-assumption} give a slightly larger cube
$\widetilde U$ such that
\begin{equation}\label{hp:chosen-cube-trace}
 \sum_U\ell_U\int_{\partial\widetilde U}|Dk|^p\,d\cH^2
                   \leq Cb.
\end{equation}
Choose its radius also to be a Lebesgue point of the surface integral
as a function of cube radius.  Almost every radius has this property,
so it is compatible with the averaging inequality.  The boundaries
are chosen on the outer side to ensure that all of the previously
exempt interior $U$ is included.  The available disjoint collar
enlargements make the cubes $\widetilde U$ disjoint.

Write a fixed such cube as $x_0+a[-1,1]^3$, and use coordinates
$x=x_0+at\theta$, where $0\leq t\leq1$ and
$\theta\in\partial[-1,1]^3$.  The edges between facets have zero
volume and may be omitted in integration.  For $0<\alpha<1/2$
and $0<\rho<1/2$, let $\Phi$ be the radial self-homeomorphism
with radial function
\[
 \varphi(t)=
 \begin{cases}
 (1-\rho)t/\alpha,&0\leq t\leq\alpha,\\[2pt]
 1-\rho+\rho(t-\alpha)/(1-\alpha),&\alpha\leq t\leq1.
 \end{cases}
\]
It fixes the boundary and is bi-Lipschitz for these fixed positive
parameters.  On the inner cube, $k$ has a finite Lipschitz constant
$L$ on the compact cube, so
\[
 \int_{\{t<\alpha\}}|D(k\circ\Phi)|^p
                  \leq C a^3 L^p\alpha^{3-p}.
\]
Choose $\alpha$ so small that this is below an assigned summable
error.  This is exactly where $p<3$ is used.

On the outer region,
$|D\Phi|\leq C(\varphi(t)/t+\varphi'(t))\leq C/t$.
Cube polar integration therefore gives
\[
 \int_{\{\alpha<t<1\}}|D(k\circ\Phi)|^p
 \leq C_pa^3\int_\alpha^1t^{2-p}
       \int_{\partial[-1,1]^3}
          |Dk(x_0+a\varphi(t)\theta)|^p\,d\cH^2(\theta)\,dt.
\]
For fixed $\alpha$, let $\rho\downarrow0$.  The inner surface
integral is sampled from radii in $[1-\rho,1]$.  The chosen outer
radius is an $L^1$ Lebesgue point.  After the linear substitution
from $t$ to $\varphi(t)$, the weight $t^{2-p}$ is bounded by a
constant depending on the fixed $\alpha$; hence Lebesgue
differentiation applies and the limsup of the last expression is
at most
\[
 C_pa\int_{\partial\widetilde U}|Dk|^p\,d\cH^2
                   \int_\alpha^1t^{2-p}\,dt
 \leq C_p\ell_U\int_{\partial\widetilde U}|Dk|^p\,d\cH^2.
\]
The last constant may depend on $p$, since
$\int_0^1t^{2-p}\,dt=(3-p)^{-1}$.  Take $\rho$ small enough to
add only another assigned summable error.  Applying this procedure
on all cubes and using \eqref{hp:chosen-cube-trace} proves
\eqref{hp:hidden-cost}.

Each source modification fixes its boundary.  Local finiteness and
positive fixed radial slopes therefore preserve local
bi-Lipschitzness and the global homeomorphism onto the same target.
All new values on a cube belong to its old image under $k$.  If
$k$ is finely close to $f$, uniform continuity of $f$ on compact
localizations and the original fineness of the cubes make the
resulting oscillation as small as the required compact accuracy.
The proof sampled the actual ambient derivative of $k$ on a newly
chosen radius, and nowhere used convergence of derivatives on a
previously fixed boundary.
\end{proof}
==== END Strong-diffeomorphic-approximation-in-three-dimensions-for-p-gt-2//sections/05-high-preparation.tex ====

==== BEGIN Strong-diffeomorphic-approximation-in-three-dimensions-for-p-gt-2//sections/06-quantizers.tex ====
\section{Convex product quantizers and the exterior carrier}\label{sec:quantizers}

Throughout this section $p>2$.  We use the carrier
$F_b$, its exact target set $O$, the smaller exact set $O_1$, and the
protected endpoint boxes supplied by Lemma~\ref{hp:wells} and
Proposition~\ref{hp:carrier}.  We construct a Lipschitz target map $T$
whose three-dimensional input
blocks lie in $O$, where the carrier is already exact.  Outside those
blocks, $T$ takes values in a polyhedral two-complex.  The retained
deficient-rank gradients must survive on its faces.  Small central
prisms will then be removed from the input space to produce an exterior
with standard annuli and disks.  A final local replacement makes the
carrier exact near this new boundary, ready for the source-wall
construction.

The carrier $F_b$ is a proper continuous surjection in
$W^{1,p}_{\mathrm{loc}}$ and has the relative opening property of
Lemma~\ref{hp:opening}.  On the inverse of $O$ it agrees locally with
specified affine or PL homeomorphisms.  The compact retained rank data
are denoted by $K$; near $K$ the carrier equals $F_\kappa$.  All
smallness prescriptions may vary continuously toward the boundary.
A locally finite prescription therefore imposes only finitely many
accuracy requirements on each compact set.

The construction combines classical convex conjugacy and proximity
\cite{Moreau1965} with the polyhedral geometry of power diagrams
\cite{Aurenhammer1987}.  The contact exclusion, protected paired
arrays, orthogonal input products and properness estimates required
here are established below; they are not consequences of those
general constructions alone.

\subsection{Convex contacts and product cells}

At a smooth contact point, a function minus its affine support has a
local minimum, so its Laplacian is nonnegative.  We therefore force
contacts into $O$ by making the Laplacian negative outside $O$.  The
small residual volume allows this without losing the protected paired
supports or appreciably changing the potential.

\begin{lemma}[Excluding contacts by a Poisson perturbation]\label{qt:poisson}
Let $b_1$ be the protected-well perturbation of Lemma~\ref{hp:wells}.
Fix a positive $1$-Lipschitz function $\delta$ on $\Lambda$, extended by
zero outside, with $\delta\leq\dist(\cdot,\R^3\setminus\Lambda)$.
The well sizes can first be chosen sufficiently small, and the residual
volume prescriptions for $\Lambda\setminus O$ in
Proposition~\ref{hp:carrier} can subsequently be chosen sufficiently small,
so that there is $b_2\in C^\infty(\Lambda)$, zero outside $\Lambda$,
with the following properties.  For an absolute $C$,
\begin{equation}\label{qt:laplacian}
 \psi(x)=\tfrac12|x|^2+b_1(x)+b_2(x),\qquad
 \Delta\psi\leq C\text{ in }\Lambda,\qquad
 \Delta\psi<0\text{ in }\Lambda\setminus O.
\end{equation}
The function $b_2$ vanishes on neighborhoods of all protected endpoint
boxes, every prescribed affine support indexed by a protected
horizontal slope remains a global support of $\psi$, and, with
arbitrarily small $\epsilon>0$,
\begin{equation}\label{qt:potential-smallness}
 |b_1+b_2|\leq\epsilon\delta^2,\qquad
 b_1+b_2=o\bigl(\dist(x,\R^3\setminus\Lambda)^2\bigr)
 \quad(x\to\partial\Lambda).
\end{equation}
\end{lemma}

\begin{proof}
Choose closed Whitney cubes $Q$ with interior enlargements $Q^+$ and
bounded overlap of the enlargements.  For each cube the disallowed set
$E_Q=Q\setminus O$ is compact.  Its measure is smaller than a threshold
that has not yet been specified.  Smooth a neighborhood of $E_Q$ inside
$Q^+$ to obtain $0\leq\rho_Q\leq1$, equal to one near $E_Q$ and with
arbitrarily small support measure in units of $Q$.  This is possible by
outer regularity, provided the residual threshold is small enough.
In normalized coordinates put this density in a fixed three-dimensional
torus $\mathbb T$ containing $Q^+$, and solve
\[
 \Delta u_Q=-A\rho_Q+
       \frac{A}{|\mathbb T|}\int_{\mathbb T}\rho_Q,\qquad
 \frac{1}{|\mathbb T|}\int_{\mathbb T}u_Q=0.
\]
Here $A$ is fixed, larger than $3+\sup\Delta b_1+2$.
For completeness, the required smallness is already a consequence of
the periodic Green kernel: its singularities are $O(|x|^{-1})$ and those
of its first derivative are $O(|x|^{-2})$.  Both kernels belong to
$L^{q/(q-1)}$ for fixed $q>3$.  Convolution and H\"older's inequality
therefore give
\[
 \|u_Q\|_\infty+\|Du_Q\|_\infty
 \leq C_q A\|\rho_Q\|_{L^q}
 \leq C_q A|\supp\rho_Q|^{1/q}.
\]
The mean-zero Green solution is smooth because the density is smooth.
Multiply it by a cutoff equal to one on $Q$ and supported in $Q^+$,
and return to physical coordinates with the factor $\ell(Q)^2$.
The resulting $v_Q$ satisfies $\Delta v_Q\leq\varepsilon_Q$
everywhere and $\Delta v_Q\leq-A+\varepsilon_Q$ on $E_Q$;
its value and gradient are as small as desired in the corresponding
scaled units.  The errors here include the mean compensation and both
cutoff terms, which involve only $u_Q,Du_Q$.

There is a smooth cutoff $\chi$, zero on neighborhoods of the compact
protected endpoint boxes and one outside $O_1$, whose transition is
contained in $O_1$.  Its derivatives on each $Q^+$ are fixed before the
residual thresholds.  Set $b_2=\chi\sum_Qv_Q$.  The sum is locally
finite.  In
\[
 \Delta(\chi v_Q)=\chi\Delta v_Q+2D\chi\cdot Dv_Q+v_Q\Delta\chi
\]
the possibly large negative term is harmless for an upper bound, and
all other terms can be made arbitrarily small per cube.  Bounded overlap,
with the errors chosen below $1$ in total at every point, proves the
first inequality in \eqref{qt:laplacian}.  At a point outside $O$ the
cutoff is identically one locally and at least one $E_Q$ contributes
$-A$; this proves the strict negative inequality.  Make the value bounds
on each enlargement smaller also than the prescribed local multiple of
$\delta^2$, and let that multiple tend to zero with boundary distance.
This gives \eqref{qt:potential-smallness}, after the initial analogous
choice for $b_1$.

It remains to check the global supports.  For a fixed compact horizontal
endpoint box, subtract one of its prescribed affine supports from
$|x|^2/2+b_1(x)$.  Its only zeros are the two prescribed endpoint points
for that slope, by the two-well test.  Uniformly over the compact set of
slopes, it has a strictly positive minimum on every compact set outside
the endpoint neighborhoods.  At infinity the gap grows quadratically.
The added perturbation vanishes in the endpoint neighborhoods.  Choose
its absolute value elsewhere below half the preceding gap, using the
minimum over the compact slope set.  There are only finitely many protected
patches, by Lemma~\ref{hp:wells}, and their supports have been separated.
The minimum of their positive gap requirements is positive away from
the endpoint neighborhoods.  Taking the smaller of these requirements preserves all supports
simultaneously.  All these requirements determine only smaller residual
thresholds, after the endpoint neighborhoods have been fixed, as allowed
by Proposition~\ref{hp:carrier}.
\end{proof}

\begin{lemma}[Contact curvature and dual separation]\label{qt:contact}
Let $\phi$ be the lower convex envelope of $\psi$ from
Lemma~\ref{qt:poisson}, and let
\[
 F(q)=\phi^*(q)=\sup_x\{q\cdot x-\psi(x)\}.
\]
There is an absolute $C_0$ such that $\phi$ is $C^{1,1}$ and
$0\leq D^2\phi\leq C_0 I$ in the distributional sense.
With $m=1/C_0$ reduced if necessary, every actual contact maximizer
$x_i$ at slope $q_i$ satisfies
\begin{equation}\label{qt:dual-gap}
 F(q)\geq F(q_i)+x_i\cdot(q-q_i)+\tfrac m2|q-q_i|^2.
\end{equation}
For any requested fine motion tolerance the perturbations can be chosen
so that all contact maximizers at $q$ lie within that tolerance of $q$.
For $q\notin\Lambda$ the unique maximizer is $q$ and $F(q)=|q|^2/2$.
For $q\in\Lambda$ every maximizer lies in $O$; over a compact subset of
$\Lambda$ their union is compactly contained in $O$.
\end{lemma}

\begin{proof}
The function $\psi$ has quadratic growth at infinity.  A supporting
plane of its convex envelope consequently has a compact, nonempty set
of contacts, and each point of the envelope on that plane is a convex
combination of at most four contacts.  One way to see attainment is to
minimize the average of $\psi$ over probability measures of fixed
barycenter.  Quadratic growth gives tightness and a bounded second
moment; passage to the limit gives a minimizer.  Separation by a
supporting plane forces its support onto the contact set, and
Carath\'eodory's Theorem reduces it to four points.

At a contact in $\Lambda$, the smooth function $\psi$ minus its
supporting plane has a minimum.  Thus $D^2\psi$ is nonnegative there;
\eqref{qt:laplacian} bounds each eigenvalue above by $C$.  At a contact
on or outside $\partial\Lambda$, the zero extension and
\eqref{qt:potential-smallness} give the quadratic Taylor expansion of
$|x|^2/2$ with an $o(|v|^2)$ remainder.  If
$x=\sum\alpha_jx_j$ is a contact combination, use $x_j+v$ and $x_j-v$
as competing combinations to obtain
\[
 \limsup_{v\to0}
 \frac{\phi(x+v)+\phi(x-v)-2\phi(x)}{|v|^2}\leq C_0.
\]
Here $C_0$ is absolute and independent of the contact combination.
On any line this implies semiconcavity with constant $C_0$: if, after
subtracting $(C_0+\varepsilon)t^2/2$, the function dipped below one of
its chords, its chord-subtracted minimum would be interior.  The
symmetric second difference at that minimum is nonnegative, contrary
to the displayed strict negative upper bound.  Let $\varepsilon\downarrow0$.
Convexity and semiconcavity imply differentiability and the global bound
\[
 \phi(x+v)\leq\phi(x)+D\phi(x)\cdot v+\tfrac{C_0}2|v|^2.
\]
At a contact $x_i$, $D\phi(x_i)=q_i$.  Substituting
$v=(q-q_i)/C_0$ into the definition of the conjugate proves
\eqref{qt:dual-gap}.

Write $b=b_1+b_2$ and $d=|x-q|$ for a maximizing contact.  Comparison
with $x=q$ gives
\[
 \tfrac12d^2\leq |b(x)|+|b(q)|
 \leq\epsilon\{(\delta(q)+d)^2+\delta(q)^2\}.
\]
For sufficiently small $\epsilon$ this yields
$d\leq C\sqrt\epsilon\,\delta(q)$.  When $q\notin\Lambda$ it gives
$d=0$.  Choosing the positive Lipschitz minorant $\delta$ and then
$\epsilon$ proves the fine motion assertion and keeps all contacts
inside $\Lambda$ for interior $q$.  A contact in $\Lambda\setminus O$
would have nonnegative Hessian and therefore nonnegative Laplacian,
contrary to \eqref{qt:laplacian}.  Finally, maximizing contacts form a
closed graph and remain in a bounded set over bounded slopes.  For
slopes in a compact subset of $\Lambda$ their union is therefore
compact; since it is contained in $O$, its clearance from $O^c$ is
positive.
\end{proof}

\begin{proposition}[A proper product quantizer]\label{qt:quantizer}
Fix $0<c<m$ with $c<1$.  One can choose a locally finite family of
interior sites in $\Lambda$, with one contact at each ordinary site
and the two protected contacts at each paired site.  The family can
include the rectangular paired arrays constructed in
Lemma~\ref{qt:calibration}.  Index these site and contact pairs by
$i$, writing them as $(q_i,x_i)$, and define
\begin{equation}\label{qt:obstacle}
 G(q)=\sup_i\{F(q_i)+x_i\cdot(q-q_i)+\tfrac c2|q-q_i|^2\},
 \qquad T=(\partial G)^{-1},
\end{equation}
where the supremum also includes the support based at every
$q_i\notin\Lambda$.  Here $q$ is an output of $T$, and
$y\in\partial G(q)$ is an input.  Then $T:\R^3\to\R^3$ is single-valued and
$c^{-1}$-Lipschitz, equals the identity outside $\Lambda$, and restricts
to a proper onto map of $\Lambda$ to itself, as finely close to the
identity as prescribed.

Inside $\Lambda$, $G-c|q|^2/2$ is a locally finite maximum of affine
functions.  Its diagram has compact convex polyhedral cells.  Write
$p_i=x_i-cq_i$.  Over the relative interior of a face $P$, whose active
indices are $I(P)$, its exact input product is
\begin{equation}\label{qt:product}
 \partial G(q)=cq+\conv\{p_i:i\in I(P)\},\qquad
 T(cq+v)=q\quad(q\in\relint P).
\end{equation}
The two factors have orthogonal direction spaces of complementary
dimensions.  Every retained support owns a genuine three-dimensional
output cell $C_i$.  Its closed input block
\begin{equation}\label{qt:blocks}
 X_i=cC_i+p_i
\end{equation}
is compactly contained in $O$.  These closed blocks are pairwise
disjoint and have locally finite, mutually disjoint protective
neighborhoods within the exact regions of the carrier.  Outside their
union the output of $T$ lies in the two-skeleton of the diagram.
\end{proposition}

\begin{proof}
\emph{Localizing the active supports.}
For a support $L_i$ appearing in \eqref{qt:obstacle},
\eqref{qt:dual-gap} gives
\begin{equation}\label{qt:active-bound}
 L_i(q)\leq F(q)-\tfrac{m-c}{2}|q-q_i|^2.
\end{equation}
A sufficiently fine locally finite net gives $G\geq F-e$ for any
prescribed positive continuous error $e$ on $\Lambda$: for each compact
set, contacts and slopes are bounded, so the support at a nearby site
converges uniformly to $F$ there.  A Whitney refinement with compact
buffers gives these estimates simultaneously.  Also $G\leq F$.
Outside $\Lambda$ its own support is present, so $G=F=|q|^2/2$ there.
Choose $e(q)$ so small that
\[
 \tau(q)=\sqrt{2e(q)/(m-c)}<\tfrac14\dist(q,\Lambda^c),
\]
and impose any additional fine upper bounds on $\tau$ needed below.
Every active site is within $\tau(q)$ of $q$ by
\eqref{qt:active-bound}.  Exterior supports and sites outside that
compact interior neighborhood are uniformly below the supremum.
The remaining interior sites are finite locally, so the supremum is
attained and gives the asserted polyhedral diagram.  This argument
also applies on all closed cell boundaries.

\emph{The proper Lipschitz map.}
The function $G$ is $c$-strongly convex and grows quadratically.
Consequently $G(q)-y\cdot q$ has a unique minimum for every $y$.
Strong monotonicity gives
\[
 (y-y')\cdot(T(y)-T(y'))\geq c|T(y)-T(y')|^2,
\]
proving the Lipschitz bound.  At a point outside $\Lambda$, equality
$G=F$ and differentiability of $F$ imply $\partial G(q)=\{q\}$:
every supporting affine function for $G$ there also supports $F$.
Thus $T$ is the identity outside, and no interior point is sent outside.
For interior $q$, \eqref{qt:product} expresses each input as a convex
combination of
\[
 cq+p_i=x_i+c(q-q_i).
\]
The contact motion bound and the bound $|q-q_i|\leq\tau(q)$ make every
such point arbitrarily finely close to $q$; in particular it lies in
$\Lambda$.  This proves surjectivity onto $\Lambda$.  Arrange
$|T(y)-y|<\tfrac14\dist(y,\Lambda^c)$, with analogous inverse motion
control in terms of $q$.  A sequence of inputs approaching
$\partial\Lambda$ then has outputs approaching that boundary.  Since
$\Lambda$ is bounded, inverse images of compact subsets are compact,
which proves properness.

\emph{Orthogonal products and exact input blocks.}
Subtracting the common quadratic in \eqref{qt:obstacle} gives slopes
$p_i$.  Equalities between its active affine functions have normals
$p_i-p_j$.  The tangent space of a relatively open face is the common
kernel of these equalities, so their span is its entire normal space.
This proves the complementary orthogonal product assertion and the
subgradient formula.  The closure of a product over $P$ is meant here
when a face is closed; a boundary point can have further incident
products in its complete fibre.

An ordinary support at its own site is strictly above every support
from another site, by \eqref{qt:active-bound}.  The gap is uniformly
positive locally: nearby distinct sites are separated, and the
remaining sites, including the exterior family, have positive distance
from that site.  It therefore wins on an open set.  At a paired site
the two contacts have different normal components; each wins on one
open side of their common tie plane while the same other-site gap
persists.  Thus all retained supports own full cells.

Two offsets from different sites cannot coincide.  Indeed strong
monotonicity of $\partial F$, obtained by adding the two inequalities
\eqref{qt:dual-gap}, yields
\[
 (x_i-x_j)\cdot(q_i-q_j)\geq m|q_i-q_j|^2.
\]
If $p_i=p_j$, its left side is $c|q_i-q_j|^2$, a contradiction.
At one site discard identical contacts.  If two closed blocks met,
uniqueness of $T$ would give the same output $q$ and then the same
offset, another contradiction.

Choose $\tau$ smaller also than the local contact clearance in $O$
divided by $2c$, using compact neighborhoods of the slopes.
Then every $x_i+c(q-q_i)$ for $q\in C_i$ lies in $O$.  A fixed cell
cannot approach $\partial\Lambda$, since
$|q-q_i|<\dist(q,\Lambda^c)/4$ throughout it.  Boundedness makes $C_i$
and $X_i$ compact.  Over a compact input set, motion control confines
outputs and active sites to compact interior sets, so only finitely
many blocks occur.  Pairwise disjoint compact blocks in this locally
finite family admit pairwise disjoint neighborhoods in $O$, reduced
further to the prescribed exact carrier neighborhoods.  Finally, over
the interior of a full output cell the product is the singleton graph
$y=cq+p_i$, which is precisely its full input block.  This proves the
last assertion.
\end{proof}

\subsection{Calibration and strict approximation}

For the rest of the construction fix $c$ universally, for example
$c=\min\{m,1\}/2$.  It is independent of every later accuracy
parameter; constants depending only on this choice are absolute.

\begin{lemma}[Paired arrays, redistribution, and phase calibration]\label{qt:calibration}
The choices in Proposition~\ref{qt:quantizer} can be made together
with a locally bi-Lipschitz homeomorphism
$J:\Lambda\to\Lambda$, equal to the identity near every closed $X_i$,
such that $TJ$ has an absolute Lipschitz bound.  Put
\[
 \widehat F=JF_b,\qquad q_0=T\widehat F.
\]
Apart from arbitrarily small prescribed source weight, $q_0$ maps the
retained rank-one and rank-two data into the interiors of horizontal
rectangular true two-faces.
The carrier $\widehat F$ retains the exact data over the blocks and its
relative opening property, and $|Dq_0|\leq C|DF_b|$ almost everywhere.
On the retained compact rank-$k$ data, $k=1,2$,
\begin{equation}\label{qt:calibrated-gradient}
 Dq_0=a_*^{-1}\operatorname{proj}_{n^\perp}DF_\kappa,
 \qquad |Dq_0-Df|\leq C(\lambda+1-a_*)|Df|,
\end{equation}
where $c<a_*<1$ may be as close to one as required and $\lambda$ is
the preceding flattening accuracy.  The rank is exactly $k$ and
$\ker Df\subset\ker Dq_0$.  For rank one the rectangle can have any
previously prescribed small short-to-long aspect, with its first axis
arbitrarily close to the range line of $Dq_0$.  Further compact margins,
sector avoidance, and uniform jet tests may be imposed by trimming.
\end{lemma}

\begin{proof}
On a protected patch write $q=z+h+q_v n$.  At the sites $q_i=z+u_i$ of
a Cartesian lattice in $n^\perp$, with periods $d_1,d_2$, retain both
contacts $x_i^\pm=z+u_i\pm d_\perp n$.  The protected support identity is
\[
 F(z+u)=\tfrac12|z+u|^2-d_\perp^2.
\]
Exclude nonmatching sites from a narrow tube about a compact interior
part of the plane $q_v=0$.  The tube width is chosen after the error and
active-site tolerances in the preceding proof, and the periods are
chosen much smaller still.  A point in the tube has a paired site at
distance at most its width plus $\sqrt{d_1^2+d_2^2}$; local continuity
and the support formula give the required dual approximation there.
Outside use a sufficiently fine allowed net.  Thus exclusion does not
invalidate the previous approximation.

More explicitly, on the central plane the deficit of the nearest pair
is $O(d_1^2+d_2^2)$, whereas every nonmatching site has, on a smaller
protected interior, a fixed positive distance and hence a fixed
positive deficit by \eqref{qt:active-bound}.  Decrease the periods until
the former is smaller.  Dominance persists in a vertical neighborhood.
The affine parts of two matching pairs differ there by the nearest-node
quadratic comparison with coefficient $1-c>0$.  Their bisectors are
therefore exactly the Cartesian bisectors, and the two signs tie
exactly on $q_v=0$.

The subgradient formula now gives, for protected inputs
$y=z+h+vn$ with $|v|<d_\perp$,
\begin{equation}\label{qt:scalar-quantizer}
 T(y)=z+(Q_{c,1}(h_1),Q_{c,2}(h_2))_{\rm horiz}.
\end{equation}
In a period centered at node $u$ of pitch $d$, the scalar map is
$u+(h-u)/c$ for $|h-u|\leq cd/2$, and is constant, equal to the
appropriate output period endpoint, on each remaining half-gap.
At a tie all tied nodes have both signs, so their convex hull includes
the entire interval $|v|\leq d_\perp$; thus the formula also holds on
the seams.  The pertinent full output rectangles in $q_v=0$ are true
faces of the diagram, not arbitrary two-dimensional subsets of cells.

The scalar map $Q_c$ has slope $1/c$ on only a fraction $c$ of each
input period and collapses the complementary gaps.  We enlarge this
fraction to $a$ close to one, so that most retained data see the nearly
unit slope $1/a$.  For this purpose choose a Lipschitz cutoff
$a(h,v)\in[c,a_*]$, equal to $a_*$ on a
neighborhood of the protected values and equal to $c$ off a compact
subset of the separability region $|v|<d_\perp$.  In a period define
$R_a$ to map the interval of fraction $a$ linearly onto the interval of
fraction $c$, and the two complementary intervals linearly onto their
counterparts, fixing the endpoints.  Thus $Q_cR_a=Q_a$, where $Q_a$
is the same scalar map with $c$ replaced by $a$.
We must also check that the two simultaneous redistributions remain
invertible when $a$ varies with the input.  In centered coordinates
the inverse of $R_a$ is
\[
 B_a(w)=
 \begin{cases}
 (a/c)w,&|w|\leq cd/2,\\
 ad/2+\dfrac{1-a}{1-c}(w-cd/2),&cd/2\leq w\leq d/2,
 \end{cases}
\]
with the odd extension on the negative half-period.  Hence
$|\partial_a B_a|\leq d/2$, including across the fixed inverse
breakpoints.  Define simultaneously
\[
 J(h,v)=\bigl(R_{a(h,v),1}(h_1),R_{a(h,v),2}(h_2),v\bigr).
\]
For given output $(w,v)$, its inverse solves
\[
 h_j=B_{a(h,v),j}(w_j),\qquad j=1,2.
\]
Choosing the periods so that
$\sqrt{d_1^2+d_2^2}\Lip(a)<1/2$ makes this a contraction on $\R^2$,
after extending the periodic formulas and putting $a=c$ outside the
support.  It has a unique solution, Lipschitz in $w,v$.  The forward
formulas are Lipschitz for the fixed $a_*<1$; consequently $J$ is
bi-Lipschitz on this patch.  It is the identity off the patch and maps
it onto itself.  The disjoint patches assemble locally finitely.

The composite $Q_cR_a=Q_a$ has ordinary input slopes bounded
by $1/c$ and parameter derivative bounded by $C_cd$; the latter follows
either from its formula $u+(h-u)/a$ on the middle interval or from
continuity at the moving breakpoints.  The chosen period bound makes
$TJ$ uniformly Lipschitz, independently of how close $a_*$ is to one.
The support is strictly between the two endpoint heights; a full block
there would require an extreme sign, at height $\pm d_\perp$.
Thus the support avoids all closed full blocks, with a neighborhood.
The same is true of its image.  The carrier opening property transfers
under the target homeomorphism $J$, and the derivative bound follows
from the Lipschitz chain rule.

All these choices can be uniform over lattice phases.  Fix nested
horizontal boxes
$H_{\rm inner}\Subset H_{\rm array}\Subset H_{\rm endpoint}$,
the excluded tube, and the cutoff support first.  For every phase
retain all lattice nodes in $H_{\rm array}$, taking both periods below
one tenth of the nesting margins.  The nearest-node deficit is bounded
by $C(d_1^2+d_2^2)$ for every phase, whereas the nonmatching-site gap
on the inner box is fixed.  Thus a common separability slab
$|v|<d_\perp-\sigma$ contains the fixed cutoff support for all phases.
Its separation from full blocks and the inverse contraction estimate
are phase-independent.

Choose the phases after the compact data, cutoffs, and periods have
been fixed.  For any fixed datum, uniform independent translations of
the two lattice coordinates place it in their middle intervals with
probability $a_*^2$.  Fubini's Theorem for the finite source weight
therefore gives a phase losing at most the prescribed multiple of
$1-a_*^2$, with room for further losses.  Conditional on membership
in these intervals, its two normalized output coordinates are uniform
on the output rectangle: this follows directly from the linear
formula and translation-invariance of phase measure.  Consequently
any prescribed zero-area subset of a normalized rectangle, such as
its center, perimeter, or finitely many sector rays, is avoided for
almost every retained datum for almost every phase.  Neighborhoods
of those sets have losses tending to zero by the same averaging.
This argument uses no absolute continuity of the projected data
measure.  Independently shifted finer rectangular subdivisions give
the same conclusion after discarding fitting margins.

Where the retained data have $a=a_*$ and both middle intervals are
strict, differentiation of \eqref{qt:scalar-quantizer} gives
\eqref{qt:calibrated-gradient}; the preceding flattening bound and
$|n-e|\leq C\lambda$ give its error estimate.  The Lipschitz
composition annihilates every direction in $\ker Df$ at a source
point where the compared jets exist.  After the preliminary positive
singular-value bounds have been fixed, choose the errors smaller than
those bounds, so the rank cannot decrease.  It cannot increase because
of the kernel inclusion.  The first lattice axis follows the selected
rank-one range direction; its angular errors were chosen smaller than
the assigned aspect.  Finally continuity, regularity of the finite
source weight, and differentiation allow compact trimming to all
claimed interior and uniform jet margins.
\end{proof}

\begin{lemma}[Strict quantizers]\label{qt:strict}
The map $T$ has proper locally bi-Lipschitz approximations
$T_\eta:\Lambda\to\Lambda$ with the same target.  Given any positive
continuous local tolerance $\epsilon_0(y)$, they can be chosen with
\[
 |T_\eta(y)-T(y)|<\epsilon_0(y),\qquad
 |DT_\eta(y)-DT(y)|<\epsilon_0(y)\quad\text{for a.e. }y.
\]
On every product piece the derivatives of $T$ are its affine
tangential derivatives, agreeing on directions tangent to common
strata.  If $h$ is locally bi-Lipschitz, the pushforward by $Th$ of
volume restricted to each product interior is absolutely continuous
with respect to the dimensional measure on its output stratum.
\end{lemma}

\begin{proof}
For $q=T(y)$ set
\[
 T_\eta(y)=q+\eta(q)(y-q),
\]
where $\eta>0$ is smooth, bounded above by a small constant, and has a
small global Lipschitz constant.  Such functions with arbitrary local
upper bounds on their value and first derivative are obtained from a
locally finite smooth partition of unity: choose the coefficient of
each term smaller than all required bounds on its support, divided by
the local overlap count and the first derivative bound of that term.
Put $M=\sup_{\Lambda}|y-T(y)|<\infty$.  For $q_j=T(y_j)$,
monotonicity and direct subtraction give
\[
 (T_\eta(y_1)-T_\eta(y_2))\cdot(q_1-q_2)
 \geq(1-\|\eta\|_\infty-M\Lip\eta)|q_1-q_2|^2
 \geq\tfrac12|q_1-q_2|^2.
\]
Thus differences of outputs control differences of $q$.  Subtracting
again in the defining formula, and using a positive lower bound for
$\eta$ on each compact set, controls $|y_1-y_2|$ locally by the output
difference.  This proves injectivity and local inverse Lipschitz
bounds, including the case $q_1=q_2$, when the output difference is
exactly $\eta(q_1)(y_1-y_2)$.  Forward Lipschitz bounds follow from those
of $T$ and $\eta$.  Fine motion keeps the segment from $y$ to $q$
inside $\Lambda$ and makes $T_\eta$ proper, as in
Proposition~\ref{qt:quantizer}.  Invariance of domain gives an open
image; properness gives a relatively closed image.  Connectedness of
$\Lambda$ proves surjectivity.

At almost every input,
\[
 D(T_\eta-T)=\eta(T)(I-DT)
 +(y-T)\otimes\bigl(D\eta(T)DT\bigr).
\]
Properness first transfers any local input tolerance to a local output
tolerance; choose $\eta,D\eta$ below it in the preceding construction.
This proves both requested estimates.  On a product piece
\eqref{qt:product} is orthogonal projection to its face, with factor
$c^{-1}$ in tangential directions.  Its restriction to a common stratum
is the same map from either side, proving tangential agreement.
Fubini on the product, followed by the change-of-variables inequality
for a locally bi-Lipschitz $h$, proves the dimensional absolute
continuity assertion.  At a zero-dimensional output stratum the map
$Th$ is constant and its weak derivative vanishes almost everywhere
on that level set.
\end{proof}

\subsection{Face coordinates and the graph exterior}

\begin{lemma}[Subdividing the true faces]\label{qt:subdivision}
The true two-faces of the quantizer diagram have locally finite,
conforming subdivisions into convex polygonal subcells $D$ with
chosen centers $d_D$.  Protected rectangular faces may be kept as
rectangles or subdivided by finer rectangular grids.  Elsewhere the
ratio of diameter to center-boundary clearance has an absolute bound
away from arbitrarily small prescribed neighborhoods of the original
vertices.  Inside those neighborhoods it has a finite bound depending
only on the original face.  The latter constants can be confined to
regions of arbitrarily small weighted $\int|Dq_0|^p$ cost.  Additional
rays from a protected center to inserted perimeter vertices can be
chosen to avoid almost every retained datum.
\end{lemma}

\begin{proof}
In a compact convex face $P$, choose a maximal separated net of spacing
$d$ and intersect its planar Voronoi cells with $P$.  Maximality bounds
each cell's diameter by $C d$, while separation gives the intersection
of a disk of radius $c d$ about its generator with $P$.  Fix an interior
disk of $P$ once.  Interpolating the generator toward its center by a
distance proportional to $d$ gives a disk of radius $c_Pd$ in that
intersection: convexity carries the fixed interior disk into $P$, and
its interpolated center and radius both remain within the small disk
about the generator.  Thus the clearance ratio is bounded by a finite
constant depending on $P$.  Outside a prescribed neighborhood of the
vertices and for sufficiently small $d$, at most one supporting edge
line cuts this small disk.  A disk cut by one half-plane through or
outside its center contains a disk of one fixed fraction of its radius.
The clearance constant there is absolute.  Choose the center $d_D$
of each cell in the indicated inscribed disk.

Match the subdivisions of a shared original edge by overlaying the
finitely many incident subdivisions.  This inserts only collinear
perimeter vertices and does not change any cell or its clearance.
Choose the finite nets generically first, so no Voronoi vertex lies
on an original edge and each incident Voronoi segment meets that edge
transversely.  The finitely many nonzero angles and strict convexity
margins have a positive minimum.  Changes of division positions on
original edges and the corresponding changes in adjacent segments
are taken below these margins and preserve cyclic order.  They
therefore preserve convexity and the clearance bounds.  Collinear
overlay nodes may slide along a side without changing its polygon.  For a fixed interior point of a
protected rectangle, the ray from its center through that point
intersects the perimeter at one location.  A continuously distributed
perturbation of an extra perimeter vertex equals that location with
probability zero.  Fubini for the finite retained measure gives
simultaneous avoidance, even for singular data.  The original center,
corner, and sector-ray exclusions have already been arranged by
Lemma~\ref{qt:calibration}.  All choices are finite per face and locally
finite in $\Lambda$.

A Sobolev map has zero weak derivative almost everywhere on any one
of its level sets: apply the scalar statement to each coordinate,
which follows by truncating that coordinate to a shrinking interval
about the level and using the Sobolev chain rule.  In particular
$Dq_0=0$ almost everywhere on $\{q_0=v\}$ for an original vertex $v$.
On a compact localization, dominated convergence gives
\[
 \lim_{\rho\downarrow0}
 \int_{\{|q_0-v|<\rho\}}|Dq_0|^p=0.
\]
The map $q_0$ is proper and locally Sobolev, so the localizations needed
for a compact target neighborhood are compact source localizations.
Fix the original-face clearance factors first, and then choose the
vertex neighborhoods to make these integrals smaller than any given
error divided by those factors.  This also permits any previously
fixed collar weights.  Use a summable family of errors for the locally
finite vertices.  Only after these neighborhoods have been fixed take
$d$ sufficiently small.  The arbitrary original-face constants thus
multiply only the separately paid vertex contributions.
\end{proof}

\begin{lemma}[Polar coordinates and their sharp rectangular bound]\label{qt:polar}
For each subcell $D$, put
\[
 R_{D,e}(t,s)=d_D+r\bigl(\xi_e(s)-d_D\bigr),\qquad
 r=\exp(t/L_D),
\]
where $\xi_e$ is arclength on a perimeter subedge and the same arclength
label is used in all incident sectors.  Choose $L_D$ comparable to the
maximum radius of $D$; in a protected rank-one rectangle choose it
as the long-axis half-length and choose $d_D$ to be the geometric
midpoint of the rectangle.  On an open sector,
\begin{equation}\label{qt:polar-bounds}
 |D_qt|+|D_qs|\leq C_D/r,\qquad
 |\partial_tR_{D,e}|+|\partial_sR_{D,e}|\leq C r.
\end{equation}
The first constant depends only on the clearance ratio, and the second
is absolute.  For a rectangle of half-length $L$ and half-width
$\gamma L$, and its unit long-axis vector $v_1$,
\begin{equation}\label{qt:rank-one-polar}
 |D_qt(v_1)|+|D_qs(v_1)|\leq(1+\gamma)/r,
 \qquad
 \max\{|\partial_tR|,|\partial_sR|\}
 \leq\sqrt{1+\gamma^2}\,r.
\end{equation}
\end{lemma}

\begin{proof}
Translate $d_D$ to zero.  If the supporting line of $e$ is
$\nu\cdot\xi=b>0$, then $r=(\nu\cdot q)/b$,
$Dr=\nu/b$, and $Dt=L_D\nu/(br)$.  Also
\[
 \xi'(s)D s=\frac1r\left(I-\frac{q\otimes\nu}{br}\right),
\]
when restricted to the sector plane.  Since $b$ is bounded below by
the center clearance and $|q|/r$ by the maximum radius, these are the
first estimates.  Directly,
$\partial_tR=r\xi/L_D$ and $\partial_sR=r\xi'$, proving the second.
For the rectangle, on a short end $q_1=\pm rL$, so
$|D t(v_1)|=1/r$ and $|D s(v_1)|\leq\gamma/r$.
On a long side $r$ depends only on the short coordinate, so the two
values are $0$ and $1/r$.  Finally $|\xi|\leq L\sqrt{1+\gamma^2}$
and $|\xi'|=1$.  Splitting a straight side only changes the additive
origin of $s$, so none of the estimates changes.
\end{proof}

\begin{proposition}[The graph exterior and its two colours]\label{qt:exterior}
Choose a small $r_{u,D}>0$ in every subcell.  In input target
coordinates remove the full closed blocks $X_i$ and the open cores
of the interval prisms over
$d_D+r_{u,D}(D-d_D)$, with their ends attached to the full blocks.
The closure $M_y$ of the remainder is a locally finite PL
three-manifold with boundary.  The radial projection in each subcell,
identity on subedges, defines a continuous map
\[
 \pi T:M_y\longrightarrow\Gamma,
\]
where $\Gamma$ is the subedge graph.  Each component of $M_y$ is a
possibly noncompact graph handlebody.  In particular every PL embedded compact
interior two-sphere bounds a PL ball and
\begin{equation}\label{qt:exterior-homology}
 H_2(M_y;\mathbb Z)=0.
\end{equation}
For intermediate levels there are two standard families of proper
walls: $A_D(t)$ is the perimeter at common radial label $t$, times
the face-offset interval; $B_e(s)$ is the entire inverse of the
intermediate subedge point under $\pi T$.  Walls within one colour are
disjoint.  Each $A$ is an incompressible annulus and each $B$ is a disk.
An incident pair has two boundary intersection points, one on each
annular end; a nonincident pair has none.  Consequently transverse
individually standard walls with this boundary incidence have one
spanning intersection arc for each incident pair, and otherwise only
circles; every such circle is contractible in both walls.
\end{proposition}

Figure~\ref{qt:fig-product-exterior} shows one face product and the
two wall directions.  The proof also accounts for the offset polygons
and polytopes over true edges and vertices.

\begin{figure}[tb]
\centering
\begin{tikzpicture}[font=\small,>=Latex]
\begin{scope}[x=1.3cm,y=1.3cm]
 \node[font=\bfseries] at (1.15,2.97) {Output face $D$};
 \fill[gray!5] (0,0) rectangle (2.3,1.9);
 \draw[thick] (0,0) rectangle (2.3,1.9);
 \draw[blue!70!black,thick] (.46,.38) rectangle (1.84,1.52);
 \fill[white] (.92,.76) rectangle (1.38,1.14);
 \draw[densely dashed,gray] (.92,.76) rectangle (1.38,1.14);
 \draw[red!65!black,thick] (1.38,.95)--(2.3,.95);
 \fill (1.15,.95) circle (1.1pt);
 \node[below left] at (1.15,.95) {$d_D$};
 \fill[blue!70!black] (1.84,.95) circle (1.4pt);
 \node[below left,inner sep=2pt] at (1.84,.95) {$q$};
 \fill (2.3,.95) circle (1.1pt);
 \node[right] at (2.3,.95) {$\xi_e(s)$};
 \draw[->,thick] (2.46,1.25)--(2.46,1.72);
 \node[right] at (2.46,1.52) {$s$};
 \node[blue!70!black,above] at (1.15,1.54) {$t=L_D\log r$};
 \node[align=center] at (1.15,-.45)
 {$q=d_D+r(\xi_e(s)-d_D)$};
\end{scope}
\draw[->,thick] (6cm,1.65cm)--node[above] {$T$} (4.8cm,1.65cm);
\begin{scope}[shift={(7.1cm,.35cm)},
 x={(1.3cm,0cm)},y={(.42cm,.36cm)},z={(0cm,1.28cm)}]
 \node[font=\bfseries] at (1,.7,2.55) {Input face prism};
 \fill[gray!18] (0,0,0)--(2,0,0)--(2,1.4,0)--(0,1.4,0)--cycle;
 \fill[gray!12] (0,0,1.7)--(2,0,1.7)--(2,1.4,1.7)--(0,1.4,1.7)--cycle;
 \draw[gray] (0,0,0)--(2,0,0)--(2,1.4,0)--(0,1.4,0)--cycle;
 \draw[gray] (0,0,1.7)--(2,0,1.7)--(2,1.4,1.7)--(0,1.4,1.7)--cycle;
 \draw[gray] (0,0,0)--(0,0,1.7) (2,0,0)--(2,0,1.7)
  (2,1.4,0)--(2,1.4,1.7) (0,1.4,0)--(0,1.4,1.7);
 \draw[blue!70!black,thick] (.4,.28,0)--(1.6,.28,0)
  --(1.6,1.12,0)--(.4,1.12,0)--cycle;
 \draw[blue!70!black,thick] (.4,.28,1.7)--(1.6,.28,1.7)
  --(1.6,1.12,1.7)--(.4,1.12,1.7)--cycle;
 \draw[blue!70!black] (.4,.28,0)--(.4,.28,1.7)
  (1.6,.28,0)--(1.6,.28,1.7);
 \draw[blue!70!black,densely dashed] (.4,1.12,0)--(.4,1.12,1.7)
  (1.6,1.12,0)--(1.6,1.12,1.7);
 \fill[white] (.8,.56,0)--(1.2,.56,0)--(1.2,.56,1.7)
  --(.8,.56,1.7)--cycle;
 \fill[white] (1.2,.56,0)--(1.2,.84,0)--(1.2,.84,1.7)
  --(1.2,.56,1.7)--cycle;
 \draw[gray,densely dashed] (.8,.56,0)--(1.2,.56,0)
  --(1.2,.84,0)--(.8,.84,0)--cycle;
 \draw[gray,densely dashed] (.8,.56,1.7)--(1.2,.56,1.7)
  --(1.2,.84,1.7)--(.8,.84,1.7)--cycle;
 \draw[gray,densely dashed] (.8,.56,0)--(.8,.56,1.7)
  (1.2,.56,0)--(1.2,.56,1.7)
  (1.2,.84,0)--(1.2,.84,1.7);
 \fill[red!45,opacity=.3] (1.2,.7,0)--(2,.7,0)
  --(2,.7,1.7)--(1.2,.7,1.7)--cycle;
 \draw[red!65!black,thick] (1.2,.7,0)--(2,.7,0)
  --(2,.7,1.7)--(1.2,.7,1.7)--cycle;
 \draw[blue!70!black,very thick] (1.6,.7,0)--(1.6,.7,1.7);
 \fill[blue!70!black] (1.6,.7,0) circle (1.5pt);
 \fill[blue!70!black] (1.6,.7,1.7) circle (1.5pt);
 \node[left,blue!70!black] at (-.15,0,.72) {$A_D(t)$};
 \node[right,red!65!black,align=left] at (2.12,1.4,.65)
 {$B_e(s)$\\portion};
 \node[above] at (1,.7,1.92) {end on $X_+$};
 \node[below] at (1,.7,-.22) {end on $X_-$};
 \node[rotate=90,fill=white,inner sep=1pt] at (1,.64,.85) {removed core};
\end{scope}
\end{tikzpicture}
\caption{A rectangular face and its orthogonal input product,
schematically and at independent scales; $X_\pm$ denote the adjacent
full blocks.  The quantizer $T$ collapses
the normal interval and rescales tangential directions by $c^{-1}$.
After removing the central core, a radial perimeter times the interval
is the blue annulus $A_D(t)$.  The red rectangle is only the portion
of $B_e(s)$ in this prism.  Over a true edge, the complete meridian
disk also contains its offset polygon and the spoke rectangles of all
incident faces.  The blue intersection interval joins the two ends
of the annulus on the adjacent full blocks.}
\label{qt:fig-product-exterior}
\end{figure}

\begin{proof}
Over an open true face the product formula is a polygon times an
interval perpendicular to it, whose two ends are faces of the two
adjacent full blocks.  Over an open true edge it is an interval times
a convex polygon; over a true vertex it is a convex three-dimensional
offset polytope.  These products glue by their common faces because
they are subdifferentials of the same piecewise-quadratic function.
The offset dimensions are exactly complementary by
Proposition~\ref{qt:quantizer}.  Removing the central subpolygons and
the full blocks therefore leaves the usual half-space or corner
neighborhood at each boundary point; subdivisions turn all such
corners into PL half-ball charts.  This proves the manifold assertion.
The prisms are disjoint and attach along disjoint disks after the
$r_{u,D}$ have been chosen small.  Their boundaries and the full-block
boundaries give its locally finite boundary.  The radial projection
agrees on every shared subedge.  On a true edge the map $T$ is the
projection along its polygonal offset fibre, so it gives the same
value for all incident face products.  This proves continuity of
$\pi T$ through the true edges and vertices.

Cut $M_y$ at one standard meridian $B$ over an interior point of each
subedge.  We verify that every resulting chamber is a ball.  Fix a
vertex $w$ of the subdivided graph and a true face $P$ incident there.
Inside $P$, the annular portion projecting to the truncated star at
$w$ is a disk $A_{P,w}$: for each incident subcell take the radial strip
from the first cut on one side of $w$, around $w$, to the adjacent
cut.  Successive strips meet along one radial side.  Their boundary
is a single polygonal curve alternating cut spokes and inner arcs;
if $w\in\partial P$, add the one outer boundary arc through $w$.
This is a disk fan, with empty intersection with $\partial P$ in the
interior case and one boundary arc in the boundary case.
Its product with the face-offset interval is a ball.

If $w$ is interior to a true face this product already is the chamber.
If it is interior to a true edge, start with the polygon-offset prism
over the truncated edge segment.  Each incident face product attaches
along one side disk, with disjoint relative interiors of the attaching
disks.  Attaching a ball to a ball along a boundary disk yields a ball:
identify the disk with the flat disk in a half-ball chart and flatten
the two collars, then extend the resulting boundary identification
radially.  This argument is PL after subdivision.  Thus the edge
chamber is a ball.

At a true vertex start with the full convex offset polytope.  The
incident edge products attach on its facets.  The incident fan is the
normal fan of that polytope: the equalities between active affine
supports identify exactly the facet-edge incidences in question.
After the edge attachments, each $A_{P,w}$ product attaches along the
disk over its two-sided outer boundary arc and the face-offset
interval.  The two sides meet along the corresponding polytope edge.
These are boundary disks whose relative interiors are disjoint from
the other attaching disks.  The same ball-gluing argument completes
the vertex chamber.  It also shows that its frontier meets adjacent
chambers exactly in the chosen meridian disks.  This local verification
covers degeneracies of the original diagram; it uses the full active
offset polytope rather than a simplicity assumption on the vertex.

Restoring the meridian collars therefore attaches ordinary
one-handles to disjoint zero-handles.  This is a graph-handlebody
presentation, locally finite on compact sets.  It retracts onto a
one-dimensional graph, proving \eqref{qt:exterior-homology}.
For irreducibility, a compact PL embedded sphere meets only finitely many handles;
include them and their endpoints in a finite graph chunk, cutting its
remaining handles farther away from the sphere.  Each component of
this finite chunk is an ordinary compact handlebody.  To recall the
standard elementary argument, cut such a handlebody along its
meridian disks.  Put the PL sphere transverse to them and use an innermost
intersection circle and the adjacent disk in the cut ball to remove
intersection circles by disk surgery and isotopy.  Induction reduces
to a sphere in a ball, which bounds a ball by the PL Schoenflies
Theorem; restoring each removed disk collar restores a ball on the
same side of the sphere.  Equivalently this is the induction that
attaching a boundary one-handle to a union of balls preserves
irreducibility.  The bounded ball lies in the finite chunk, hence in
$M_y$.

The $A$ product is visibly an annulus.  Its core maps under $\pi T$
to the reduced perimeter loop of $D$ in $\Gamma$.  That loop traverses
successive distinct perimeter edges without cancellation, so every
nonzero power is nontrivial in the free fundamental group of the
graph.  The induced map on the annular fundamental group is therefore
injective; in particular the annulus is incompressible.
At an artificial edge, $B$ is the union of two spoke rectangles along
their common interval, a disk.  At a true edge it consists of the
convex offset polygon and one spoke rectangle on each corresponding
boundary interval.  Gluing the rectangles to this polygon successively
along boundary intervals again gives a disk.  Their descriptions
prove disjointness in each colour and the stated boundary incidence.

For transverse individually standard walls the intersection is a
compact one-manifold, with boundary precisely the prescribed wall
boundary intersections.  An incident pair therefore has exactly one
arc, joining those two endpoints, and any remaining components are
circles.  The arc joins the two annular ends and is spanning.  Every
circle bounds a disk in the $B$ wall, hence is nullhomotopic in $M_y$;
injectivity of the annular fundamental group makes it nullhomotopic
in $A$ as well.  On a disk or annulus such a simple nullhomotopic
circle bounds a disk.  This proves the last assertions.
\end{proof}

\subsection{Exact boundary data and edge matching}

The target exterior $M_y$ now has standard annuli and disks, but its
new central-prism boundaries need not lie in the exact region of
$\widehat F$.  To pull these walls back with fixed PL boundary data,
we first make the carrier exact near those boundaries.  The changes
are confined below a prescribed radial level $a_D$, leaving the
calibrated data at higher radii unchanged.  Their new local energy
need only be finite: the radial margin lets the later target map be
constant on the changed region.  The construction must also retain enough control of
fibres to preserve sampled edge intersections when it is opened.

\begin{lemma}[A carrier exact on the central tubes]\label{qt:central-carrier}
Let $p>2$.  Let $T:\Lambda\to\Lambda$ have the properties of
Proposition~\ref{qt:quantizer}, and let
$\widehat F:\Omega\to\Lambda$ have the carrier properties retained in
Lemma~\ref{qt:calibration}.  In particular, $\widehat F$ is a proper
continuous $W^{1,p}_{\mathrm{loc}}$ carrier with connected fibres and a
countable set $E_0$ of nonsingleton values, it has the relative opening
property of Lemma~\ref{hp:opening}, and its prescribed maps over
neighborhoods of the closed full blocks $X_i$ are PL homeomorphisms.
Use the locally finite face subdivision and central prisms of
Proposition~\ref{qt:exterior}.  Fix, separately from the central radii, a
positive low activation radius $a_D$ in every subcell $D$.

The radii $0<r_{u,D}\ll a_D$ can be chosen so that the resulting closed
exterior $M_y$ has the following additional properties.  There exist a
PL homeomorphism $h_*:\Omega\to\Lambda$, a proper continuous
$W^{1,p}_{\mathrm{loc}}$ map $\overline K:\Lambda\to\Lambda$, and
\[
 F_*=\overline K h_*,\qquad M_x=h_*^{-1}(M_y),\qquad q_*=TF_* ,
\]
such that:
\begin{enumerate}
\item $\overline K$ is the identity on an open neighborhood of the
full blocks and of all the removed central prisms.  Consequently
$F_*^{-1}(M_y)=M_x$, and $F_*=h_*$ on an open neighborhood of
$\partial M_x$.  These boundary data are PL and have the originally
prescribed values near the full blocks.
\item $\overline K$, and hence $F_*$, is a fine limit of actual
homeomorphisms with these exact data.  Its fibres are nonempty and
connected, and only countably many of its target values have
nonsingleton fibres.
\item All changes from $\widehat F$ occur in compactly localized,
locally finite face neighborhoods whose old and new labels lie in
$r<a_D$.  The changes meet any prescribed fine value tolerance.
Thus $F_*=\widehat F$ in the complementary high region.  On every
compact portion of $M_x$ the labels $q_*$, and the radial and perimeter
coordinates used there, have finite ambient local Sobolev budgets.
No uniform bound for these new low-region budgets is asserted.
\end{enumerate}
There is a locally finite closed family $S$ of possible limiting inner
cube seams in the changed regions.  Away from $S$, the construction
has the local fibre formula proved below.  The portions of $S$ at which
the construction is not already exact lie away from $\partial M_x$.
\end{lemma}

\begin{proof}
\emph{Fixing the target geometry and the reference homeomorphism.}
We first fix the target geometry before choosing the final PL
approximation.  Choose slightly enlarged, pairwise disjoint polyhedral
balls around the $X_i$, contained with room in the exact PL regions of
$\widehat F$, and fix smaller open protection neighborhoods there.
These protections, the central prisms, and all boxes and sampling
margins specified next depend only on the prescribed target geometry
and on the geometric constants of Lemma~\ref{hp:cutoff}; they do not
depend on a map $h_*$ or $K_0$.

Include the properness requirements at this stage.  Fix a compact
exhaustion $C_m\subset\operatorname{int}C_{m+1}$ of $\Lambda$.
Choose a positive $1$-Lipschitz function $\tau$ so small that
\[
 \tau(z)<\tfrac14\dist(z,C_m)
 \quad\text{whenever }z\notin\operatorname{int}C_{m+1}.
\]
Such a minorant exists: only finitely many of these conditions apply
near any fixed point, and the compact gaps make each applicable
bound positive.  All geometry and approximation choices below include
the requirement that the resulting target-coordinate maps $g$, and
also $K_0$, move each $y$ by less than $\tau(y)/10$.

Over a true face, the quantizer has an orthogonal product description
with an interval as normal fibre.  At each of its two ends the
attachment disk of a sufficiently small central prism lies in one of
the fixed exact neighborhoods.  Choose an elongated affine box in
the open face prism: its transverse inner rectangle contains the
central cross-section with positive margin; its longitudinal inner
interval contains the portion outside those two exact neighborhoods;
and both longitudinal end strips of its larger box lie strictly inside
the exact neighborhoods.  The larger box remains strictly between
the two endpoints of the full normal interval.  It therefore avoids
the full blocks themselves.  First decrease $r_{u,D}$, relative also
to the aspect of $D$, to put this whole transverse construction in
$r\ll a_D$; then choose the shell thickness in normalized box
coordinates.  Decrease these transverse and shell sizes also to meet
the fixed $\tau/10$ motion bound: in an affine box the lift displacement
and every hole-buffer diameter are bounded by a fixed box constant
times the normalized shell size, while the inner block will be the
identity.  This bound applies to all the later strict fills as their
images lie in those same buffers.  The two end margins absorb both
that thickness and the sampling enlargements required below.  Distinct boxes and their
sampling neighborhoods are disjoint when they belong to distinct
central prisms, and the neighborhoods form a locally finite family.
This follows by choosing them inside the open face products and
using their compact attachment margins.  The tube portions beyond
the inner longitudinal interval are already exact.

Here and below a sampling neighborhood includes a little more than
the immersion image itself.  The enlargement is needed for the
degree tests in the fibre argument.  All these neighborhoods are
chosen with compact closure in the indicated face and end regions.
Now choose a positive $1$-Lipschitz minorant $\rho\leq\tau/10$ of
all the resulting target value-closeness requirements, including their
affine normalization factors.  Open $\widehat F$ by
Lemma~\ref{hp:opening} and apply Lemma~\ref{sm:relative-pl}, relative
to the fixed enlarged protection balls, to obtain a PL homeomorphism
$h_*$ satisfying
\[
 |h_*(x)-\widehat F(x)|<\rho(\widehat F(x))/10.
\]
Define $K_0=\widehat Fh_*^{-1}$.  At $y=h_*(x)$, the Lipschitz
property of $\rho$ gives
$|K_0(y)-y|<\rho(y)/9$.  Thus all previously fixed sampling and
properness tests hold.  The map $K_0$ is proper, locally Sobolev,
exactly the identity on the protected neighborhoods, and inherits
the old relative opening property.  Its local Sobolev constants may
depend on $h_*$; only their finiteness is used below.

The motion bounds already chosen imply, for every resulting map $g$
and its sufficiently close homeomorphic approximants,
\begin{equation}\label{qt:central-proper-localization}
 g^{-1}(C_m)\subset K_0^{-1}(C_{m+1}).
\end{equation}
Indeed, $|g(y)-K_0(y)|<\tau(y)/5$, whereas
$\tau(K_0(y))>9\tau(y)/10$.  If $K_0(y)\notin C_{m+1}$, the defining
bound for $\tau$ therefore prevents $g(y)$ from lying in $C_m$.
All these requirements preceded the choice of $h_*$.

\emph{The lifted replacement.}
We describe a single normalized box.  Let $B=[-1,1]^3$ be its inner
block, and use the shell, mesh, omitted balls $V_{\sigma}$, buffer balls
$V_{\sigma}^+$, and skeleton immersion $i$ of Lemma~\ref{hp:cutoff}.
Here $\sigma$ indexes shell tetrahedra; $D$ continues to denote a
face subcell.  At depth scale $l$, the immersion has chart radii
comparable to $l$ in its domain
and to $s$ in its image.  On each injective chart $U$,
\[
 |Di|\le C s/l,\qquad |D(i|_U)^{-1}|\le C l/s.
\]
Prescribe $i=\mathrm{id}$ on a whole outer mesh band, leave that
band free of cap supports, and retain the old map there.  On all
tetrahedra whose fills are needed in the protected longitudinal end
strips, put the immersion samplings inside the exact region of
$K_0$.  Their lifts and fills are then the identity on an open
neighborhood, including the necessary collars.  This is a relative
use of the skeleton construction in Lemma~\ref{hp:cutoff}.

On the remaining usable lift region define the near-point branch
\begin{equation}\label{qt:central-lift}
 H(x)=(i|_{U_x})^{-1}\bigl(K_0(i(x))\bigr),
 \qquad i(H(x))=K_0(i(x)).
\end{equation}
The sampling requirements give
$|K_0-\mathrm{id}|\le\varepsilon_{\rm lift}s$, where
$\varepsilon_{\rm lift}>0$ is a fixed closeness constant below all
chart and buffer margins.  Hence
$|H(x)-x|\le C\varepsilon_{\rm lift}l$.
The near-point branches agree on overlaps;
their existence and agreement do not require injectivity of $K_0$.
Neither does the chain-rule estimate in the cutoff proof.  On each
tetrahedron it bounds the lift energy by
\[
 C_p(l/s)^3
       \int_{\text{enlarged image sampling}} |DK_0(y)|^p\,dy.
\]
Summing over depth scales gives the finite shell bound of that
proof.  Affine normalization introduces finite constants depending
on this fixed elongated box, which is harmless here.

For every nonidentity omitted ball, use the elementary forward
squeeze $J_{\sigma}$ which collapses $V_{\sigma}^+$ to its center $v_{\sigma}$, is a
homeomorphism off that closed ball onto the punctured output
region, and is the identity near the tetrahedron boundary.
The lifted boundary of the omitted hole lies strictly inside
$V_{\sigma}^+$.  Define the new map as $J_{\sigma}H$ where the lift is used and
as the constant $v_{\sigma}$ throughout the omitted hole.  These definitions
agree on a whole collar.  Identity-filled holes use neither a cap
nor a nonidentity fill.  Write $J_{\rm cap}$ for the disjoint union
of the squeezes.  Set the map equal to the identity on $B$ and to
$K_0$ beyond the replacement.  This defines $\overline K$.

Continuity at the inner seam follows since all displacements on a
depth-$l$ tetrahedron are $O(l)$.  More precisely, the near-branch
lift and every cap stay on the exterior side of $B$.  Indeed the
Whitney scale comparability gives $\dist(\sigma,B)\geq\alpha l$ for one
fixed $\alpha>0$.  Our choice of $\varepsilon_{\rm lift}$ includes
$C\varepsilon_{\rm lift}l<\alpha l/2$ in the lift bound.
Every lifted point is then strictly exterior to $B$.  Each
cap is supported in an exterior tetrahedron and maps its support into
itself, so cap postcomposition and the omitted-hole constants are
strictly exterior as well, at every depth scale.  The summed lift estimates, followed by the Sobolev gluing argument in
Lemma~\ref{hp:cutoff}, prove local $W^{1,p}$ regularity.  Constant
holes cost no energy.  At the outer band the immersion is the
identity and no cap acts, so the seam is unchanged on a neighborhood.
The reserved outer band and sufficiently small closeness bounds also
keep cap supports and their inverse action inside the replacement
image; hence no source outside the box is changed.  Local finiteness
now proves these assertions simultaneously for all boxes.  Their
low-region margins ensure that both old and new actual labels stay
below $a_D$.

\emph{Homeomorphic openings and connected fibres.}
Apply the old opening property to
obtain homeomorphisms $k_n\to K_0$ finely, preserving the exact
regions and end-strip samplings.  Lift $k_n$ by the same near-point
branches.  The homeomorphic cutoff argument of
Lemma~\ref{hp:cutoff} fills the omitted balls topologically, with
images inside $V_{\sigma}^+$ and the required boundary values.  Replace
each cap by a strictly increasing radial squeeze tending to $J_{\sigma}$.
The resulting maps $\bar k_n$ are homeomorphisms, agree with the
exact data, and preserve the outer replacement image.
On every omitted hole their outputs converge to the declared
constant, irrespective of the choice of topological fill.  Elsewhere
the lift formula gives convergence; at the inner seams the $O(l)$
bound gives uniform convergence by first discarding sufficiently
small depth scales.  Local finiteness gives convergence on compacts,
and diagonal choices give any prescribed fine accuracy.

By \eqref{qt:central-proper-localization}, the inverse images under
all sufficiently close $\bar k_n$ of a fixed compact lie in one
compact.  In particular $\overline K$ is proper and onto: take a
convergent subsequence of $\bar k_n^{-1}(y)$ for a given $y$.
Its fibres are connected as well.  For two points $a,b$ in a fibre
over $y$, choose closed balls $\overline B(y,\rho_n)\Subset\Lambda$,
with $\rho_n\downarrow0$, containing both $\bar k_n(a)$ and
$\bar k_n(b)$.  Their inverse images are compact connected sets
containing $a,b$ in a common compact.  A Hausdorff-convergent
subsequence has connected limit in $\overline K^{-1}(y)$ containing
$a,b$.  This proves the claim.

\emph{Local fibre identification and exact boundary data.}
The following formula gives countability of the new exceptional
values and transfers the wall tests in the next lemma to old
relative-opening tests.  Suppose $y$ is not a cap center and set
$x'=J_{\rm cap}^{-1}(y)$.  In a nonidentity lift region this inverse
is unique, and $x'$ lies outside the closed collapsed buffers
$V_\sigma^+$.  Their fixed clearance from the omitted holes
$V_\sigma$ gives a ball
$B_{\rm s}=B(x',\alpha l)$ in the usable lift region, with an
injective immersion chart on a larger neighborhood $U$.  The joint
immersion neighborhood of Lemma~\ref{hp:immersed-skeleton} supplies
these charts also across ordinary skeleton seams; the limiting inner
seams $S$ are treated separately below.  Choose
$B_{\rm t}=B(i(x'),\beta s)$ inside the enlarged image sampling
neighborhood, reducing the fixed $\beta>0$ so that its chart
pullback lies in $B(x',\alpha l/2)$.  This is possible by the
inverse derivative bound $C l/s$.

The closeness constant was chosen with
$\varepsilon_{\rm lift}<\beta/2$ and
$C\varepsilon_{\rm lift}<\alpha/2$.  On $\partial B_{\rm t}$,
closeness to the identity gives degree one for $K_0$ at $i(x')$
and excludes that value from the boundary image.  Hence the old
fibre meets $B_{\rm t}$ and misses its boundary.  Connectedness
puts the entire old fibre inside $B_{\rm t}$.  In fact every point
of this fibre lies within
$\varepsilon_{\rm lift}s$ of $i(x')$, by the same closeness test there.

Its chart pullback is therefore well inside $B_{\rm s}$.
Equation~\eqref{qt:central-lift} and the near-branch choice identify
this pullback with the local new fibre.  The relative displacement
bound excludes preimages on $\partial B_{\rm s}$.  Connectedness
of the new fibre, applied once more, excludes any other preimage.
Consequently
\begin{equation}\label{qt:central-fibre-formula}
 \overline K^{-1}(y)
   =(i|_U)^{-1}\!\left(K_0^{-1}
          \bigl(i(J_{\rm cap}^{-1}(y))\bigr)\right).
\end{equation}
All margins can be kept uniformly positive after shrinking to a
small target neighborhood $W$ of $y$.  Thus this formula holds for
\emph{every} value in $W$, with one injective chart $U$, and not
merely for its center.  The same argument crosses skeleton seams;
the outer band uses the identity immersion.

Take a countable cover by these charts.  Outside cap centers,
\eqref{qt:central-fibre-formula} shows that a nonsingleton value
must arise from a chart inverse of an old exceptional value.  There
are countably many such values.  Equivalently, a local diffeomorphism
has discrete, hence countable, point preimages in this second-countable
domain.  Identity-filled regions have open identity behavior and
thus singleton fibres by connectedness.  A point of an inner seam
is also isolated in its fibre: on the inner side the map is the
identity, and nearby exterior outputs stay strictly exterior even
after the cap operation.  Its entire fibre is therefore singleton.
Finally, in an unchanged open region a singleton old fibre gives
an isolated point in the new fibre, so a nonsingleton new fibre
meeting that region must have an old exceptional value.  These
cases exhaust the construction and prove countability.

The identity neighborhood of the removed thick cores has singleton
fibres, again because each such point is isolated in its connected
fibre.  It follows that $\overline K^{-1}(M_y)=M_y$, including the
boundary, and gives the asserted exact boundary data after composing
with $h_*$.  The pullbacks of the affine inner cube boundaries form
the stated locally finite family $S$; the end strips make its
nonexact portions disjoint from a neighborhood of $\partial M_x$.
Finally, $T$ is locally Lipschitz, and on $M_y$ every central radius
is positive.  On compact portions only finitely many product and
sector charts occur.  Their radial and perimeter functions have
finite Lipschitz constants there, and near the boundary $F_*$ is
PL.  Sobolev composition, with local Lipschitz extensions of these
coordinate functions, supplies the claimed finite ambient budgets.
\end{proof}

The later wall estimates count intersections along the carrier's mesh
edge paths.  Opening the carrier must preserve these very hits, or the
counts would no longer describe the pulled-back walls.  The next lemma
does this for the sampled wall families, using their whole-wall
avoidance of nonsingleton values.

\begin{lemma}[Exact edge matching when opening the central carrier]
\label{qt:edge-matching}
Use the carrier and exterior of Lemma~\ref{qt:central-carrier}.
Let $\mathcal E$ be a locally finite family of mesh edges in $M_x$,
and let $\mathcal W$ be a locally finite family of sampled standard
walls of Proposition~\ref{qt:exterior}.  Assume:
\begin{enumerate}
\item the entire union $W=\bigcup\mathcal W$ is relatively closed
in $M_y$ and avoids all nonsingleton values of $\overline K$;
\item the actual hit set
$H=(\bigcup\mathcal E)\cap F_*^{-1}(W)$ is finite on compact
localizations, and no hit outside the exact regions lies on $S$.
\end{enumerate}
Then there are homeomorphisms $f_n:\Omega\to\Lambda$ converging
finely to $F_*$, equal to $F_*$ on a neighborhood of $\partial M_x$
and on the other prescribed exact regions, for which
\[
 f_n^{-1}(W_0)\cap e=F_*^{-1}(W_0)\cap e
 \quad\text{for every }W_0\in\mathcal W, e\in\mathcal E.
\]
In particular each restriction maps $M_x$ onto $M_y$ with the exact
prescribed boundary map.  The hypotheses concern sampled edge
tests; they make no relative-opening assertion for arbitrary new
closed tests accumulating through the inner lift scales.
\end{lemma}

\begin{proof}
Work first in the target-domain coordinates given by $h_*$.
Consider a hit $x$ outside an exact region, and write
$y=\overline K(h_*(x))$.  Because $W$ avoids all new nonsingleton
values, $y$ is not a cap center.  The seam-avoidance assumption
permits a target neighborhood $V$ in which the chart formula
\eqref{qt:central-fibre-formula} holds with fixed margins.  For each
wall meeting $y$, take a compact patch $P$ of that wall in $V$,
containing a relative neighborhood of $y$.  The formula shows that
\begin{equation}\label{qt:central-old-wall-test}
 C_P=i\bigl(J_{\rm cap}^{-1}(P)\bigr)
             \quad\text{is compact and satisfies}\quad C_P\cap E_0=\varnothing.
\end{equation}
Indeed, a nonsingleton old fibre at any value of $C_P$ would give
a nonsingleton new fibre at the corresponding point of $P$, which
is excluded by the hypothesis on the \emph{whole wall}.

If the hit lies in an unchanged open region, the same conclusion
uses a direct patch $P$.  Its old fibre at $y$ is singleton:
otherwise the old connected fibre would have an isolated point
where the new fibre is singleton.  Properness then places the old
inverse image of a sufficiently small neighborhood of $y$ entirely
inside the unchanged region.  To see this, a contrary sequence of
values tending to $y$ and preimages outside that region has a
convergent preimage subsequence, contradicting the singleton fibre.
For every value on the smaller wall patch the old fibre consequently
agrees with the new one and is singleton.  Hits in exact open
regions require no additional tests.

There are only finitely many hits on each compact localization.
Thus their neighborhoods can be chosen locally finitely, avoiding
$S$ wherever required.  Only finitely many depth scales occur in
each compact collection of these charts.  Choose the old patch
images in the compactly localized sampling neighborhoods used in
the preceding Lemma.  The resulting family of compact sets $C_P$
is locally finite in $\Lambda$; in particular its union is relatively
closed.  Adjoin the closed protected exact data, slightly inside
their exact neighborhoods.  All these old target tests avoid $E_0$.
Apply Lemma~\ref{hp:opening} to obtain arbitrarily close old
homeomorphisms $k_n$ agreeing with $K_0$ over every one of these
tests and retaining the exact strips.  Here agreement over $C_P$
means the exact old evaluations on $K_0^{-1}(C_P)$; since these
fibres are singleton and $k_n$ is bijective, it also means equality
of the inverse evaluations.

Lift and fill these homeomorphisms as before.  For $y\in P$ the
fixed branch and \eqref{qt:central-old-wall-test} give exactly the
same inverse image of $J_{\rm cap}^{-1}(y)$ under the new lift as
under the carrier lift.  Choose the strict radial cap openings so
that their inverse maps agree with $J_{\rm cap}^{-1}$ on all tested
patches.  This is possible by changing a collapse only over an
arbitrarily small ball about its center: each center has positive
distance from the relatively closed union of the tests relevant
to it, although no uniform distance is required.  The ordinary
piecewise radial interpolation is strictly increasing inside that
ball and unchanged outside it.  Null mesh sizes at the limiting
seams and the reserved outer bands retain the convergence and
support conclusions of the preceding proof.  The resulting
homeomorphism therefore has exactly the carrier inverse over every
tested wall patch; global uniqueness follows from its bijectivity.

Shrink the source hit neighborhoods so their old images, and then
their sufficiently close new images, meet walls only in the chosen
patches.  On the rest of the edges, fine closeness prevents new
hits.  Explicitly, after removing these neighborhoods, the edge
remainder on each compact localization has compact image disjoint
from the relatively closed wall union, hence a positive gap.
An exhaustion and a positive minorant impose all these local gap
conditions simultaneously.  This proves the claimed equality for
every wall and edge, and composition with $h_*$ returns to source
coordinates.  The exact boundary neighborhood is retained throughout,
so the homeomorphisms preserve $M_x$ and its prescribed boundary
map as asserted.
\end{proof}

\begin{remark}
The edge hypotheses can be obtained by the later ACL and coarea
sampling step: choose edges for which the relevant label paths are
absolutely continuous, arrange that their intersections with the
nonexact inner seams have parameter measure zero, and avoid the
resulting null sets of scalar labels when sampling levels.  Finite
variation gives finitely many hits at almost every sampled level.
Countably many exceptional values exclude only countably many
radial or intermediate meridian levels; a graph vertex is not an
intermediate meridian value.  These sampling facts are hypotheses
of the preceding Lemma, rather than an assertion that arbitrary
edges or arbitrary sampled levels satisfy them.
\end{remark}
==== END Strong-diffeomorphic-approximation-in-three-dimensions-for-p-gt-2//sections/06-quantizers.tex ====

==== BEGIN Strong-diffeomorphic-approximation-in-three-dimensions-for-p-gt-2//sections/07-walls.tex ====
\section{Wall routing, guard barriers, and realization}\label{sec:walls}\label{wl:section}

Fix $2<p<\infty$.  We use the quantizer and carrier of
Proposition~\ref{qt:quantizer} and Lemma~\ref{qt:central-carrier}, the exterior
of Proposition~\ref{qt:exterior}, and the notation
\[
 M_y\subset\Lambda,\qquad M_x=h_*^{-1}(M_y),\qquad
 F_*=\overline K h_*,\qquad q_*=TF_*,\qquad q_0=T\widehat F.
\]
In particular $h_*$ is fixed, $q_*=Th_*$ near $\partial M_x$, and the
prescribed map on every full block is fixed.  Both exteriors are
irreducible and have vanishing absolute second homology.  The two
standard wall families are the annuli $A(D,t)$ and meridian disks
$B(e,s)$ of Proposition~\ref{qt:exterior}.  Their radial and perimeter
coordinates are
\[
 t=L_D\log r,\qquad
 q=d_D+r\bigl(\xi_e(s)-d_D\bigr),\qquad r\geq r_{u,D}>0.
\]
All families and coordinate decompositions below are locally finite.
A compact wall is understood as a proper wall in the manifold with
boundary $M_x$ or $M_y$.

This section separates the topological conclusion from its quantitative
inputs.  The pre-stack geometry is supplied in
Proposition~\ref{ms:prestacks}; the capacities used to choose guards are
fixed in Lemma~\ref{ha:capacities}, and
Proposition~\ref{ha:initial-walls} constructs the initial walls with
those bounds.  In particular, a large
Lipschitz constant for a subsequently chosen extension is never used as
an input to the guard construction.

\subsection{Sampling transverse labels}

Fix $0<\zeta<1$.  Let $j$ index a sample in one radial or perimeter
feature.  Write
$[0,w_j]$ for its root-parameter interval and
\[
 p_j:[0,w_j]\longrightarrow I_j
\]
for its increasing affine identification with a label interval about
the sample.  Within one feature the intervals $I_j$ are ordered and
disjoint, and $|I_j|\asymp c_*w_j$, where $c_*>0$ is fixed and small.
The number $w_j$ measures transverse parameter length, not physical
collar thickness.  Routing will retain a finite union
$E_j\subset(0,w_j)$ of closed parameter intervals with
\begin{equation}\label{wl:occupied-length}
 |E_j|\geq(1-\zeta)w_j.
\end{equation}
The same affine $p_j$ is used on every component of $E_j$.
We first construct stairs for arbitrary such sets.  Before the retained
intervals are chosen, we use the whole-interval case $E_j=[0,w_j]$
to test labels.  After realization, the stairs will be constant off
$p_j(E_j)$, so their nonzero derivatives occur only where a wall
parameter has been prescribed.

\begin{lemma}[Stairs and sampling]\label{wl:stairs}
On a feature interval of length $\ell$, one can choose $N$ equal
weights $w_j=w$ with $Nw/\ell=1+O(\zeta)$, and construct an
endpoint-fixing nondecreasing stair $S$, constant outside the selected
$p_j(E_j)$, such that its total increase over each $I_j$ is $\ell/N$.
Its slope with respect to the occupied root parameter is at most
$1+O(\zeta)$, and its Lipschitz constant in the target label is
$O(c_*^{-1})$.  The stair can be made arbitrarily close to the identity
on each compact feature.

The choices allow finitely many forbidden intervals of arbitrarily
small prescribed total length.  For a fixed finite list of integrable
edge-count tests, the weighted sample counts have expected density at
most $1+O(\zeta+c_*)$ against label measure.  They satisfy the
corresponding upper bounds with arbitrarily small additive errors and
probability tending to one as the slots are refined.
\end{lemma}

\begin{proof}
Remove the forbidden intervals and divide each remaining component
into slots.  Choose the numbers of slots approximately proportional to
component lengths.  Taking $w$ sufficiently small makes both rounding
errors and the difference of each slot length from $w$ as small as
required.  This also works for finitely many features using one common
$w$; no exact divisibility is required.  Sample in each slot outside a
small endpoint margin wide enough to contain $I_j$.  The total omitted
relative length and the rounding loss can be included in $\zeta$.

Give $S$ increase $\ell/N$ at constant speed over the occupied
root length $|E_j|$, and no increase on the complementary pieces.
Its speed is
\[
 \frac{\ell}{N|E_j|}\leq
 \frac{\ell}{Nw(1-\zeta)}=1+O(\zeta).
\]
Since $p_j'=|I_j|/w\asymp c_*$, the target-label slope is
$O(c_*^{-1})$.  Start the first flat portion at the left endpoint and
use all $N$ increments, so the final value is the right endpoint.
The ordered slots show that the displacement is bounded by their
largest length, the rounding error, and the total forbidden length.
These quantities can all be prescribed small.  The same conclusion
holds when $E_j=[0,w_j]$, a version used before selecting stable
occupied intervals.

For a nonnegative integrable count function $n$, sampling uniformly in
the allowed middle of a slot $J_j$ gives
\[
 \mathbb E\bigl[w_j n(\tau_j)\bigr]
 =\frac{w_j}{|J_j^{\rm allow}|}
       \int_{J_j^{\rm allow}}n(t)\,dt.
\]
The density is at most $1+O(\zeta+c_*)$.  For bounded $n$, independence
and $\max_jw_j\to0$ make the variance of the weighted sum tend to
zero.  For general $n\in L^1$, truncate it, apply this argument to the
bounded part, and use the first moment of the integrable tail.  A
finite list of tests can be imposed simultaneously.  On an exhaustion,
feature weights and additive errors can be chosen locally, with
summable failure allowances.
\end{proof}

\subsection{The two-colour pre-stack input}
\label{wl:prestack-subsection}

Before routing, a pre-stack is a product neighborhood of a sampled source
wall, equipped with its transverse scalar parameter.  Opposite colours may
still meet in circles as well as in the prescribed spanning arcs.  The
following definition records the simultaneous product structure and the
pointwise slope control used in routing and in the guard estimates.

\begin{definition}[Pre-stack contract]\label{wl:prestack}
Let $M_x,M_y$ be the source and target exteriors and let
$h_*:M_x\to M_y$ be the fixed reference homeomorphism.  A pre-stack
system consists of a locally finite, countable, two-colour family of
compact collars
\[
 \mathcal E_j:\Sigma_j\times[0,w_j]\longrightarrow C_j\subset M_x,
 \qquad u_j(\mathcal E_j(z,\lambda))=\lambda,\qquad w_j>0,
\]
whose central sampled wall is the level $u_j=w_j/2$, subject to the
following conditions.
\begin{enumerate}
\item Each $\mathcal E_j$ is a bi-Lipschitz homeomorphism onto its image.
For colour $A$, $\Sigma_j$ is an annulus whose core is essential in
$M_x$; for colour $B$, $\Sigma_j$ is a disk.
Each level is properly embedded, and
$C_j\cap\partial M_x=\mathcal E_j(\partial\Sigma_j\times[0,w_j])$.
The collars of a single colour are disjoint, including their boundaries.
The family of closed collars is locally finite.

\item There is a specified affine increasing correspondence from
$[0,w_j]$ to a true target label interval $I_j$.
On $\partial M_x$ every level has exactly the boundary trace of its
specified standard target wall, pulled back by $h_*$.
The increasing-$u_j$ coorientation agrees with that target label
coorientation at every boundary component; for an annulus this includes
both ends.  The target incidence says that an incident opposite-colour
pair has two boundary intersection points, one on each annulus end,
and that a nonincident pair has none.

\item For each opposite-colour pair $j,k$, the intersection
$C_j\cap C_k$ is a finite disjoint union of bi-Lipschitz product boxes
\[
 Q\times[0,w_j]\times[0,w_k],
 \qquad u_j=\lambda,\quad u_k=\mu,
\]
where $Q$ is a circle or a compact interval.  The interval endpoints
are exactly the parts of that box on $\partial M_x$.
These are full parameter rectangles: no box terminates at an interior
value of either parameter.  The level sheets meet transversely in
these product coordinates.  There are no further intersection pieces.
In particular, for an incident pair every level pair has the specified
single spanning arc, together with possible circles; a nonincident pair
has only possible circles.

\item In each collar the subdivision by all its opposite-colour boxes
is jointly bi-Lipschitz trivial over $u_j$, preserving every concurrent
opposite parameter.  Precisely, for some reference level there is a
bi-Lipschitz product parametrization that carries every intersection
box, every complementary piece, and every boundary piece at the
reference level to its counterpart at each level; on an opposite box
it preserves $u_k$.  Its restriction to a boundary piece remains in
$\partial M_x$.  No uniform quantitative bound for this
trivialization is required.

\item The ambient PL structure and the physical metric are compatible
in the following explicit sense.  Every locally finite arrangement
of pieces obtained by finitely many constant or affine parameter cuts
on each compact set admits common ambient coordinate changes that are
locally bi-Lipschitz in the physical metric and make the pieces PL
simultaneously.  These changes preserve the manifold boundary and
the specified incidences.  Separate unrelated PL charts for individual
pieces do not suffice for this requirement.

\item Every $u_j$ is Lipschitz on its closed collar in the physical
metric.  Any subsequent estimate using a density $G$ assumes, as an
additional quantitative input, a nonnegative locally measurable $G$
dominating the point-to-base upper local slopes
\[
 \operatorname{lip}_{C_j}u_j(x)
   =\limsup_{\substack{y\to x\\y\in C_j,\ y\ne x}}
         \frac{|u_j(y)-u_j(x)|}{|y-x|}.
\]
Use the adjacent maxima at internal interfaces and include both
parameter bounds at an essential intersection box.  On exceptional
collar-boundary or mesh-interface strata one may enlarge $G$ to the
actual local Lipschitz bounds of the fixed compact collars.  These
strata, in the common working coordinates or their bi-Lipschitz
images, have zero volume; the enlargement therefore changes no
volume capacity.  This convention does not require a two-point
Lipschitz bound across gaps of a nonconvex union of pieces.  The
contract alone asserts neither an integral estimate for $G$ nor
independence of that estimate from later choices of guards.
\end{enumerate}
\end{definition}

The full parameter rectangles and the simultaneous trivializations
let the pieces cut out by one colour be tracked as the other root
parameter varies, while preserving the concurrent opposite parameter.
The common ambient coordinates
serve a different purpose: they permit the eventual endpoint walls to
be cut and filled in one PL structure.  Constructions of these data,
including the sufficient test in Lemma~\ref{wl:pattern}, are given in
Sections~\ref{sec:mesh} and~\ref{sec:calibrated-islands}.

\subsection{Diagonal routing and conservation of periods}
\label{wl:routing-subsection}

The switching construction is related to regular exchange in
normal-surface theory~\cite{Haken1961}.  Here the additional requirement
is to preserve transverse parameter length and exact boundary labels,
including along a possibly infinite rooted assembly.
We work in the full source exterior $M_x$, including any boundary-collar
continuations used in preparing the input walls.  The constructions in
this subsection
assume the pre-stack contract of Definition~\ref{wl:prestack}.  They are
qualitative until explicit parameter-slope bounds are imposed.  We also use
the previously established irreducibility and vanishing
$H_2(M_x;\mathbb Z)=0$ of the exterior.

Fix one level in each of two intersecting opposite-colour collars.  A circle
of intersection bounds a disk on the disk wall.  It cannot be essential on
the annular wall, since an annular core is noncontractible in $M_x$.
Consequently every intersection circle is trivial on both walls.  A pair
with no prescribed boundary incidence has no intersection arc.  An incident
pair has exactly two boundary intersection points, one on each end of the
annulus, and hence exactly one proper intersection arc.  That arc spans the
annulus.  Any additional intersections are circles.

In a circle box write its coordinates as
\[
                (\theta,u,v)\in S^1\times[0,a]\times[0,b],
                \qquad a=w_j,\quad b=w_k.
\]
Reverse $u$ or $v$ in this box if necessary.  We arrange that the rim $v=0$
of the strip on a $j$-layer faces the exterior of its bounded inner disk,
and that the rim $u=0$ on a $k$-layer has the same property.  Call these two
sides OUT and the sides $v=b$ and $u=a$ IN.  The pattern trivializations
ensure that these choices hold throughout the box.  Introduce
\begin{equation}
\label{wl:diagonal-coordinate}
                              d=a-u+v.
\end{equation}
For $d\in(0,a+b)\setminus\{a,b\}$ the segment of this level in the parameter
rectangle has one endpoint on OUT and the other on IN.  Its product with
$S^1$ is an annulus.  On OUT the two sides give the $d$-intervals $[0,a]$
and $[a,a+b]$, while on IN they give $[0,b]$ and $[b,a+b]$.
Thus these annuli match the combined OUT parameter length to the combined
IN parameter length bijectively, by partial affine maps of slope $1$ or
$-1$.  The corner values $0,a,b,a+b$ are excluded.  No diagonal replacement
is made in an arc box: both original parameters are retained there.
Figure~\ref{wl:routing-figure} shows this parameter matching.

\begin{figure}[htbp]
\centering
\begin{tikzpicture}[x=1.1cm,y=1.1cm,>=Stealth]
 \fill[blue!4] (0,0) rectangle (4,3);
 \draw[thick] (0,0) rectangle (4,3);
 \draw[blue!65!black,thick,->] (3,0)--(4,1);
 \draw[blue!65!black,thick,->] (1.6,0)--(4,2.4);
 \draw[blue!65!black,thick,->] (0,.8)--(2.2,3);
 \draw[blue!65!black,thick,->] (0,2)--(1,3);
 \node[below] at (2,0) {OUT: $v=0$};
 \node[left,align=right] at (0,1.5) {OUT\\$u=0$};
 \node[above] at (2,3) {IN: $v=b$};
 \node[right,align=left] at (4,1.5) {IN\\$u=a$};
 \draw[->] (0,-.85)--(4,-.85) node[right] {$u$};
 \node[below] at (0,-.85) {$0$};
 \node[below] at (4,-.85) {$a=w_j$};
 \draw[->] (-1.15,0)--(-1.15,3) node[above] {$v$};
 \node[left] at (-1.15,0) {$0$};
 \node[left] at (-1.15,3) {$b=w_k$};
 \node[fill=white,inner sep=3pt] at (2.05,1.45) {$d=a-u+v$};
\end{tikzpicture}
\caption{The circle-box parameter rectangle.  Each regular $d$-segment
joins one OUT side to one IN side, with partial parameter transport of
absolute slope one.  Its product with the circle is the replacement
annulus.  Corner values are discarded; essential arc boxes retain both
parameters and are not routed.}
\label{wl:routing-figure}
\end{figure}


\begin{proposition}[Routing and good parameters]
\label{wl:routing}
For every pre-stack root $j$, outside a null subset of $(0,w_j)$, the diagonal
rule constructs a connected, oriented, properly embedded surface
$L_{j,\lambda}$ whose boundary, with coorientation, is the prescribed boundary
of the original $j$-level at parameter $\lambda$.  Different constructed
surfaces are disjoint except for the prescribed essential arc intersections
between opposite-colour roots.  Each such surface is obtained by attaching
punctured disks and annuli along a rooted tree; it may have ends at infinity.
The following stronger local finiteness holds for each constructed surface:
for every compact $K\subset M_x$, only finitely many of its reached states
have an original parent collar meeting $K$.
\end{proposition}

\begin{proof}
Cut every original level along the open strips belonging to circle boxes.
The circles are trivial and disjoint on that level.  Their bounded disks
are therefore nested or disjoint.  There is one root piece containing the
actual boundary of the wall, and, for an annulus, both actual boundary
components belong to this same piece.  The root is an annulus or a disk with
finitely many open disks removed.  Every other piece is a punctured disk.
Each nonroot piece has exactly one IN rim, namely its rim toward its parent
in the nesting, and any remaining rims are OUT.  The root has no IN rim.
All essential arc boxes lie on root pieces: a spanning arc or a proper arc
with endpoints on the actual boundary cannot enter a bounded disk through
one of its disjoint circle boundaries.  There are finitely many pieces in
any one collar, and they are tracked over its whole parameter interval by
the pattern trivialization.

A state is a pair $(P,t)$ consisting of such a tracked piece and its local
parameter.  Starting at the root state $(R_j,\lambda)$, follow every OUT rim
through its diagonal annulus to the matched IN rim, and continue at that
state.  Every nonroot state has exactly one incoming predecessor wherever
the matching is defined: its unique IN rim has precisely one OUT partner.
Root states have none.  This proves that no state reached from a root can
repeat.  Indeed a repetition along an ancestral path would give a cycle of
predecessors, which could never terminate at the root.  If two distinct
ancestral paths reached the same state, following its unique predecessors
backwards would force their paths, root indices, and root parameters to
coincide.  This also rules out duplication between different root-parameter
starts.  The states reached from one start consequently form a tree, with
finite branching at each state.  Cycles or infinite backward orbits that
are not reachable from any root play no role.

There are countably many tracked pieces and countably many finite transition
itineraries.  On its domain each itinerary maps the starting parameter to
a local parameter by $t=\lambda+c$ or $t=-\lambda+c$.  A corner or breakpoint
condition therefore excludes at most one starting value for that itinerary.
Discard the union of these countable sets.  All transitions then occur along
nondegenerate segments and are local partial isometries of parameter
intervals.

Here is the estimate that controls possibly infinite trees.  For a tracked
piece $P$ belonging to collar $C_k$, let $N_P(j,\lambda)$ count the states
of the tree started at $(j,\lambda)$ that use $P$.  The image parameter sets
in $(0,w_k)$ of all valid itineraries from all roots to $P$ are pairwise
disjoint.  Otherwise one state would have two backward ancestries.  Each
itinerary preserves one-dimensional Lebesgue measure.  Decomposing its
valid domain into its measurable partial intervals and using countable
additivity gives
\begin{equation}
\label{wl:visit-bound}
             \sum_j\int_0^{w_j} N_P(j,\lambda)\,d\lambda\leq w_k.
\end{equation}
The valid domains are measurable: finite itineraries impose affine interval
conditions, and the excluded parameter set is countable.

For a compact $K$, let $\mathcal P(K)$ be all tracked pieces whose original
parent collars meet $K$.  This set is finite, by local finiteness of the
compact collars and the finite number of pieces in each collar.  Summing
\eqref{wl:visit-bound} yields
\[
 \sum_j\int_0^{w_j}
      \sum_{P\in\mathcal P(K)}N_P(j,\lambda)\,d\lambda
       \leq\sum_{P\in\mathcal P(K)}w_{k(P)}<\infty.
\]
It follows that for almost every parameter of every root, only finitely
many reached states have parent collars meeting $K$.  Intersect these
full-measure sets over a countable compact exhaustion of $M_x$.  Call the
remaining parameters good, also requiring avoidance of all corner values.

For a good parameter, glue all reached pieces to the associated diagonal
annuli.  The gluing is locally finite in the ambient manifold: a band meeting
a compact set lies in a circle box, hence in the original collar of its
parent state, and those states are finite in number by goodness.  Each
piece or band is compact and includes its attaching rims.  Thus the assembly
is locally closed and properly embedded.  Each rim has exactly two local
half-surface pieces, giving an ordinary surface chart there.  The pre-stack
charts make the same assertion at corners and at the actual manifold
boundary.  Connectedness follows from the tree construction.  Only the root
piece meets the actual boundary, so the boundary of the assembly is exactly
the original root boundary.

Orient the root according to its prescribed coorientation in the oriented
three-manifold.  At each tree edge orient the attached band and then its
child piece so that the common boundary orientations cancel.  There is no
orientation consistency obstruction because no state is attached by two
ancestral paths.  All child pieces are planar and all bands are annuli.
A finite tree assembly, capped at its frontier rims by disks, is therefore
a disk or annulus of the original root type: removing an inner disk and
attaching a punctured disk followed by disks at its remaining holes simply
replaces that inner disk by a disk.

Finally, two assemblies cannot meet in one original piece at the same
parameter or in one diagonal annulus, by uniqueness of backwards ancestry
and the bijective diagonal matching.  Their only possible remaining
intersections are in arc boxes.  Those boxes belong to root pieces on both
sides and are exactly the retained essential intersections.  A single
assembly contains only one root, so these intersections cause no
self-intersection.
\end{proof}

\begin{remark}
\label{wl:routing-no-width-arithmetic}
The argument uses ordinary Lebesgue length in each original parameter
interval and partial isometries between those intervals.  The widths $w_j$
need not be equal or commensurable.  Goodness controls states whose
\emph{entire original collars} meet a compact set; controlling only the
routed fragments would not suffice for the cap argument below.
\end{remark}

The routed surface has the right boundary, but it need not have been
obtained by a compact isotopy.  To compare its crossings with those of
the standard wall, we truncate the routing tree and cap its frontier.
This reduces every compact path test to a compact homology calculation.

\begin{proposition}[Conservation of oriented periods]
\label{wl:period}
Let $L=L_{j,\lambda}$ be a good routed surface.  Let $W_{j,\lambda}\subset M_x$
be the pullback by $h_*$ of the standard cut at the exact affine label
corresponding to $\lambda\in(0,w_j)$.  Give both surfaces the coorientation of
increasing standard parameter.  For every compact path $\gamma$ with endpoints
on $\partial M_x\setminus\partial L$, their relative oriented intersection
numbers agree:
\begin{equation}
\label{wl:period-identity}
                  I(\gamma,L)=I(\gamma,W_{j,\lambda}).
\end{equation}
Here intersection numbers may be defined after general-position perturbation
relative to the endpoints.  If the original path has finitely many hits on
$L$, its signed local crossings give the same number and
\[
                       |I(\gamma,L)|\leq\#(\gamma\cap L),
\]
where hits are counted with their occurrences along the path.
\end{proposition}

\begin{proof}
Truncate the routing tree at a finite depth $n$.  At every frontier OUT rim,
cap with the original inside disk bounded by that rim in its original wall
at the corresponding local parameter.  Orient each cap to cancel the
frontier boundary.  The result is a finite compact oriented two-chain $A_n$.
The caps need not be disjoint from one another or from the truncated assembly;
only their boundary identities are used.  Every cap is contained in the
original parent collar of its frontier state.

Let $K$ be a compact neighborhood of the path, or of any prescribed compact
homotopy of paths relative to their endpoints.  Goodness says that only
finitely many reached states have original collars meeting $K$.  Their
depths have a finite maximum.  If $n$ exceeds that maximum, every omitted
piece and every new cap lies outside $K$.  Taking a slightly larger compact
neighborhood first shows that $A_n$ agrees with $L$ in a neighborhood of the
path or homotopy.  This is the reason for using whole parent collars in the
definition of goodness.

The actual boundary of $A_n$ is the root boundary.  The pre-stack boundary
contract, including agreement of coorientations at both annular ends, gives
\[
                     \partial A_n=\partial W_{j,\lambda}
\]
as oriented one-chains.  A common subdivision and boundary parametrization
makes these chains literally identical.  Hence
$Z_n=A_n-W_{j,\lambda}$ is an absolute compact two-cycle.  Since
$H_2(M_x;\mathbb Z)=0$, it bounds a finite compact three-chain $B_n$.

We spell out why the endpoints on the manifold boundary cause no extra
intersection term.  First take the path in the interior except at its
endpoints, using a boundary collar and keeping the endpoints fixed.  Neither
endpoint lies on the support of $Z_n$ at the boundary, since the two surface
boundaries agree and the endpoints avoid that common boundary.  Push the
cycle and its bounding chain into the interior by a sufficiently small
collar push.  During this push the cycle avoids neighborhoods of the two
endpoints, so its intersection with the path is unchanged.  The pushed
three-chain is disjoint from the path endpoints.  The boundary formula for
oriented intersections, after general position, therefore gives
\[
                  I(\gamma,Z_n)
                    =I(\gamma,\partial B_n)=0.
\]
The same formula applies to the pushed chains; the notation suppresses that
harmless push.  Since $A_n=L$ near the path, this proves
\eqref{wl:period-identity}.  For a homotopy relative to the endpoints, choose
one $n$ for its entire compact image.  The same finite-chain argument proves
homotopy invariance.  Equivalently, one can push that homotopy into the
interior away from the fixed endpoint neighborhoods before applying the
intersection formula.

If the unperturbed path has finitely many hits, choose disjoint small
parameter intervals about those hits, each mapping into a two-sided surface
chart.  Before and after its isolated hit the path lies on a definite local
side.  Its local algebraic crossing is $1$, $-1$, or $0$, according to these
two sides and the coorientation.  A sufficiently small general-position
perturbation has exactly that total algebraic crossing in the chart.  The
same reasoning works in a proper half-space chart at a boundary hit; push
its path segment slightly inward and keep its ends fixed.  Outside these
intervals there are positive gaps on compact remainders, so no new crossings
are needed.  Summing proves the asserted agreement and the bound by the
number of hits.  No isotopy relative to any later guard system has been
used.
\end{proof}


\subsection{Graph potentials and guards}

On $M_y$ put $\beta=\pi T\in\Gamma$, with subedge lengths inherited
from Euclidean arclength.  Define $\alpha$ in a metric star having one
branch for each subcell $D$, with branch coordinate $b=-t\geq0$ and
common node $b=0$.  Thus all $r=1$ points have the same $\alpha$.
The coordinates are continuous and locally Lipschitz in the product
charts.  If $q=T(y)$, then
\begin{equation}\label{wl:graph-physical}
 |q-\beta|\leq Cb.
\end{equation}
At a fixed compatible $\beta$ on one branch, the physical point varies
with $b$ at speed at most $C$; at fixed $b$ in one subcell it varies by
at most the Euclidean displacement of $\beta$.  These follow directly
from the polar formula and $L_D\asymp\max|\xi_e-d_D|$.

For a layer rooted at $A(D,t_L)$ write $\alpha_L$ for its standard
star label.  For a layer rooted at $B(e,s_L)$ write $\beta_L$ for its
standard perimeter label.  The following potentials, each
$1$-Lipschitz in the appropriate graph metric, detect distance from
the entire physical support of the root layer:
\begin{equation}\label{wl:potentials}
\begin{array}{c|c|c}
 \hbox{root}&\hbox{star potential}&\hbox{perimeter potential}\\ \hline
 A(D,t_L)&\dist(\alpha,\alpha_L)&\dist_{\R^3}(\beta,\partial D)\\
 B(e,s_L)&\dist(\alpha,\mathcal S_e)&|\beta-\beta_L|.
\end{array}
\end{equation}
Here $\mathcal S_e$ is the subtree consisting of the common node and
all full branches corresponding to subcells incident to $e$.

To verify detection, suppose both displayed distances are small.  In
the $A$ case, if the guard is on the same branch, compare at a nearby
perimeter point of $\partial D$ and then at the same $b$.  On another
branch, small star distance implies that both branch heights are
small, so \eqref{wl:graph-physical} applies.  In the $B$ case, on a
compatible branch compare using the same $b$ and $\beta_L$; on an
incompatible branch small distance to $\mathcal S_e$ makes the guard
height small.  Thus the physical distance to the standard support is
at most a fixed multiple of the sum of the two potential values.
Both potentials vanish on that support.  Moreover, when the potential
tests the colour opposite to the root, it vanishes identically along
every feature having an essential intersection with the root.

\begin{definition}[Guards and prior capacities]\label{wl:guards}
Choose a locally finite compact cover of $M_x$ by smaller buffered
coordinate regions $K_a$, with larger regions $U_a$ carrying fixed
bi-Lipschitz ball or half-ball charts.  All smaller regions have
positive margins in the larger charts.  Fix finite numbers $C_a$ as
upper capacities for
\begin{equation}\label{wl:prior-capacity}
 \int_{U_a}G^p\leq C_a.
\end{equation}
These numbers, the coordinate bounds and the prescribed continuous
label accuracies are fixed before guards, sample weights and the
final resolution are chosen.

A guard system is obtained from a sufficiently fine homeomorphic
approximation $h_g$ to $F_*$, agreeing with $h_*$ on the exact boundary
data and respecting the exteriors.  Choose a locally finite source
net of sufficiently small local spacing.  For every net point $g$,
choose two paths in the constant $Th_g$ fibre from $g$ to boundary
anchors whose $h_*$-images are extremal points of the target fibre
on distinct full blocks, and let $\kappa_g$ be their union.  Thus
$\kappa_g$ is compact and connected, and each of its points has a
path within $\kappa_g$ to a boundary anchor.  Write
$q_g=Th_g(g)$ and require $Th_g$ to be sufficiently close to $q_*$.
The pre-stacks must be disjoint from all $\kappa_g$.
\end{definition}

The guards have a locally uniform positive reach, independent of the
net spacing and the approximating $h_g$.  Indeed a target fibre in
the exterior has paths to at least two extremal endpoints on distinct
full blocks, also over true edges and vertices.  Those endpoints have
positive mutual separation on compact label tests.  Their source
positions use the fixed boundary map $h_*^{-1}$, so a guard through
any one point has uniformly positive diameter.  Properness of
$Th_g$ makes the family of guards locally finite: if a guard meets a
fixed compact set, its label belongs to a compact label set, and its
net point belongs to the compact inverse image of that set.

For later use, the order in Definition~\ref{wl:guards} can be
implemented without a relative isotopy fixing guards.  Reserve tiny
forbidden parameter neighborhoods of the guard labels in each
feature before sampling.  Their total length can be arbitrarily
small.  Properness bounds the number of relevant guards for each
compact feature using the prescribed net; hence the accuracy of
$h_g$ can be chosen before the actual gaps are placed.  Carrier
levels outside slightly enlarged gaps have positive buffers from the
guards.  Make the mesh and boundary motions smaller than these
buffers.  The preparation in Proposition~\ref{wl:normalization} ends with
wall pieces lying only in tetrahedra incident to original possible
edge hits.
Controlled placement, thickening and cell-sized snapping then stay
within the buffers.  This controls the final wall supports; no
normalization isotopy relative to the guards is required.
This argument is useful only when
\eqref{wl:prior-capacity} remains valid at all such smaller
resolutions; that is the independent capacity assertion of
Lemma~\ref{ha:capacities}, with the geometry of
Proposition~\ref{ms:prestacks}.  Proposition~\ref{ha:initial-walls}
carries out the preparation with these choices.

\begin{lemma}[Crossing lower bound]\label{wl:crossing-bound}
Assume the pre-stack hypotheses of Definition~\ref{wl:prestack} and
its slope majorant $G$, and let the pre-stacks avoid the guards.
Suppose a guard label has physical distance at least $\delta>0$ from
the standard support of a good routed layer $L$.  On a fixed compact
separation test, sufficiently accurate feature stairs give a
constant $c_\delta>0$ such that every compact rectifiable path $\sigma$
joining $L$ to that guard satisfies
\begin{equation}\label{wl:path-cost}
 \int_\sigma G\,d\mathcal H^1\geq c_\delta.
\end{equation}
The constant depends on the fixed geometry and separation test, not
on the sample weights or the number of switches in an itinerary.
\end{lemma}

\begin{proof}
We may assume $\int_\sigma G\,d\mathcal H^1<\infty$; otherwise
the conclusion is immediate.  The coarea bound below will make the
weighted crossing sums absolutely integrable.

Use the whole-interval stairs of Lemma~\ref{wl:stairs}, before
restricting to stable occupied intervals.  They act on the star and
on $\Gamma$, fixing vertices and each edge setwise; on the star use
$b=-t$ with its orientation reversed.  Choose one of the potentials
$U$ in \eqref{wl:potentials} giving a separation comparable to
$\delta$.  For each root $j$ of the tested colour set
\begin{equation}\label{wl:crossing-coefficient}
 c_j(\lambda)=\frac{d}{d\lambda}
                 U\bigl(S(p_j(\lambda))\bigr).
\end{equation}
These coefficients are uniformly bounded.  In the opposite-colour
case they vanish almost everywhere on all essentially incident
features.  The potential at the boundary anchor of $L$ is as close
to zero as required, whereas the potential at the guard anchor stays
quantitatively positive.

Extend $\sigma$ along $L$ and along the guard to a path $\gamma$
from a boundary anchor $a_L$ of $L$ to a boundary anchor $a_g$ of
the guard.  For almost every tested root parameter with nonzero
coefficient, the added portions avoid the corresponding routed
layer: the guard avoids all pre-stacks, distinct routed layers are
disjoint except at essential root boxes, and coefficients on those
excluded essential features vanish.  The layer containing the
chosen starting point itself is a single null parameter value.
Let $\ell$ denote the tested graph coordinate, either $\alpha$ or
$\beta$.  The weighted crossing identity is
\begin{equation}\label{wl:weighted-period}
\begin{aligned}
 \sum_j\int_0^{w_j}c_j(\lambda)I(\gamma,L_{j,\lambda})\,d\lambda
 &=\sum_j\int_0^{w_j}c_j(\lambda)I(\gamma,W_{j,\lambda})\,d\lambda\\
 &=U\bigl(S(\ell(h_*(a_g)))\bigr)
       -U\bigl(S(\ell(h_*(a_L)))\bigr).
\end{aligned}
\end{equation}
Here the sum is over roots of the tested colour.  The first equality
is Proposition~\ref{wl:period}.  For the second, choose a fixed
piecewise regular representative of $\gamma$, homotopic relative
to its endpoints.  It meets only finitely many standard features;
its graph-coordinate
paths are piecewise Lipschitz.  Oriented one-dimensional level
counting and the Fundamental Theorem of Calculus on every graph
edge give the displayed difference of $U\circ S$ at the two anchors.
There is no error accumulated along a cycle or a long path.

On the other hand, one-dimensional coarea bounds the integrated
number of hits on $\sigma$.  Within an unchanged pre-stack piece it
uses $u_j$; within a transition rectangle it uses
$d=w_j-u+v$.  The upper local slope of $d$ is at most the sum of the
two original slopes.  Partial transition itineraries have derivative
$+1$ or $-1$, and Proposition~\ref{wl:routing} gives a unique reachable
ancestry for every piece-value and every diagonal-segment value.
Thus coarea pays for all reached root layers there without an
itinerary multiplicity.  Same-colour disjointness bounds overlap;
at an essential box count both parameters.  Apply coarea on the
closed subsets of path portions and partition the affine itinerary
rules countably if necessary.  Interface slopes use the convention in
Definition~\ref{wl:prestack}.  More explicitly, the fixed parameter
function is Lipschitz on each closed collar.  On the closed subset
of times when a rectifiable path lies in that collar, its scalar
trace has a Lipschitz extension; at almost every density time its
absolute derivative is bounded by the point-to-base upper slope
times the path speed.  The one-dimensional coarea inequality on
that subset therefore uses exactly this $G$.  Hence
\[
 \sum_j\int_0^{w_j}
       \#\bigl(\sigma\cap L_{j,\lambda}\bigr)\,d\lambda
       \leq C\int_\sigma G\,d\mathcal H^1.
\]
The right side is finite by assumption.  Hence almost every count
is finite and the weighted signed sum in \eqref{wl:weighted-period} is absolutely
integrable.  The appended portions contribute zero for almost every
parameter with nonzero coefficient, so the displayed coarea bound
controls its absolute value.  The bounded coefficients and the
positive potential difference prove \eqref{wl:path-cost}.

There is no infinite list of conflicting stair accuracies hidden in
this argument.  The perimeter projection is proper.  Near a compact
guard perimeter-label test only finitely many subcells and subedges
occur, and they use finite lists of accuracy requirements.  For a remote $A$
feature use distance to its whole $\partial D$; for a remote $B$
feature use distance to its whole closed subedge.  These supports are
invariant under their stairs, and only stair accuracy near the guard
label is needed.  The proper map $\pi q_*$ sends a locally finite
compact source cover to locally finite compact perimeter-label sets.
Choose locally finite, relatively compact enlargements of these sets
and accuracies smaller than their margins.  This supplies consistent
local prescriptions on all features.
\end{proof}

\begin{lemma}[Spherical path estimate]\label{wl:sphere}
Let $p>2$, and let $\Sigma_R$ be a sphere of radius $R$, either whole
or intersected with a closed half-space containing its center.  There
is a constant $C_p$ such that, for any two points $z_1,z_2\in\Sigma_R$
and nonnegative measurable $g$ with finite surface $L^p$ norm, some
rectifiable path in $\Sigma_R$ joining them has cost at most
\begin{equation}\label{wl:sphere-upper}
 C_p R^{1-2/p}\left(\int_{\Sigma_R}g^p\,d\mathcal H^2\right)^{1/p}.
\end{equation}
Fixed bi-Lipschitz chart changes alter only the constant.
\end{lemma}

\begin{proof}
Scale first to $R=1$.  Choose a fixed cap strictly above the equator
parallel to the cutting plane.  From an endpoint on or above the
equator, shortest arcs to cap points stay in the upper hemisphere.
For an endpoint below the equator, restrict cap points to those with
nonnegative horizontal scalar product with its radius direction.
This retains a fixed fraction of the cap.  The initial vertical
velocity of the shortest arc is then nonnegative.  While its height
is negative, the geodesic equation gives nonnegative second
vertical derivative; once it becomes positive it cannot return
negative before its positive endpoint on an arc shorter than $\pi$.
The arc therefore stays above the height of its starting point and
inside the prescribed half-space.

Average the cost of these arcs over the permitted cap fraction.
Spherical polar integration from the endpoint gives a kernel bounded
by $C/\sin\theta$ against surface measure.  Its conjugate power is
integrable precisely because $p'=p/(p-1)<2$; the endpoint and
antipodal singularities both satisfy
$\int_0^\epsilon\theta^{1-p'}\,d\theta<\infty$.
Hölder's inequality bounds the averaged cost by $C_p\|g\|_{L^p}$.
Do this at both endpoints.  Join the resulting two cap points by a
shortest arc in the cap's containing upper hemisphere; averaging
also this middle cost has the same bound.  Some choice of the two
cap points has total cost bounded by the sum of the averages.
Scaling surface measure and length restores the factor
$R^{1-2/p}$.  The same proof bounds the costs after a fixed chart
change, using its length and surface-measure constants.
\end{proof}

\begin{proposition}[Guard barrier]\label{wl:barrier}
Fix the data of Definition~\ref{wl:guards}, the capacities $C_a$ and
a positive continuous local layer-support accuracy.  There are
sufficiently fine net spacings and sufficiently small approximation
and stair errors, depending only on these prior data, with the
following property.  If a pre-stack system satisfies
\eqref{wl:prior-capacity} and avoids the resulting guards, then every
point of every good routed layer is within the prescribed local
accuracy, under $q_*$, of that layer's standard physical support.
\end{proposition}

\begin{proof}
It suffices to impose countably many locally finite compact tests
with positive accuracy constants.  Suppose one fails: at a point
$x$ of a good layer, its physical-support discrepancy exceeds
$\delta$.  The layer is connected to its actual boundary, where its
label is exact.  Uniform continuity of $q_*$ on a buffered chart
therefore gives a path stretch of the layer of diameter at least
$a>0$ on which the discrepancy exceeds $\delta/2$.  The number $a$
can be chosen from the fixed smaller-chart margins, the modulus of
continuity and $\delta$, independently of the layer and sampling.
This also applies near the manifold boundary, using half-ball
charts; a discrepant point cannot end its path in the exact boundary
without leaving this discrepant neighborhood.

For small $h$, pack at least $c a/h$ disjoint chart balls of radius
$2h$ centered along this stretch.  A maximal separated selection on a
connected set of diameter $a$ gives this elementary packing bound.
Place a guard net point within $h/2$ of each center.  Its label is
still separated by a fixed fraction of $\delta$ from the layer
support, by uniform continuity and the chosen approximation
accuracy.  The layer stretch and the guard each meet every chart
sphere of radius between $h$ and $2h$ about the center: each has a
point inside that radius and a connected continuation outside it.
For the guard this uses the prior uniform reach; for the layer it
uses the diameter of the selected stretch and smaller-chart
margins.

Every spherical path between such two hits has $G$-cost at least a
fixed $c_\delta>0$ by Lemma~\ref{wl:crossing-bound}.  Apply
Lemma~\ref{wl:sphere} to the whole or truncated sphere.  For almost
every radius $R\in[h,2h]$,
\[
 \int_{\Sigma_R}G^p\,d\mathcal H^2\geq cR^{2-p}.
\]
Integrating in radius gives at least $c h^{3-p}$ in each chart ball.
The disjoint balls therefore force
\begin{equation}\label{wl:barrier-divergence}
 \int_{U_a}G^p\geq c a h^{2-p}.
\end{equation}
Since $p>2$, choose $h$ using the previously fixed $C_a$ to contradict
\eqref{wl:prior-capacity}.  All constants used before choosing $h$
come from the fixed test, not the pre-stack resolution or weights.
Spheres whose centers are near the boundary are exactly the
half-space intersections in Lemma~\ref{wl:sphere}.  Paths are allowed
in the whole manifold chart, including through guards; no estimate
on a punctured chart is used.  A locally finite choice of spacings,
with positive continuous minorants for the requested errors, proves
all the compact tests simultaneously.
\end{proof}

\subsection{Stable occupied stacks and their realization}

Choose the layer-support errors of Proposition~\ref{wl:barrier} also
small enough toward infinity to have two consequences.  A layer
with a fixed compact standard support is confined to a compact
source set; and on any fixed compact source set only finitely many
root indices can occur.  To make these choices explicitly, take a
compact exhaustion of the locally finite label complex, enlarge
each member by a compact neighborhood, and use properness of $q_*$
to pull these neighborhoods back.  Require the local error outside
the corresponding inverse image to be less than its positive
separation from the smaller label compact.  Conversely, on a compact
source test choose the error small enough to keep any meeting
standard support in a fixed compact enlargement of its $q_*$ image.
Only finitely many compact features, and finitely many samples on
each feature, occur there.

\begin{lemma}[Stable intervals]\label{wl:stable}
Under these confinement choices, a good routed layer is a compact
disk or annulus of its root type.  For every root $j$ and every
$\zeta>0$ one can choose finitely many separated closed intervals
$E_j\subset(0,w_j)$ satisfying \eqref{wl:occupied-length}, such that
their routed layers form compact, locally finite, bi-Lipschitz
product stacks.  These products preserve all concurrent essential
parameters and the exact boundary data.  Different stacks are
disjoint except for the genuine essential-pair products.
\end{lemma}

\begin{proof}
A good tree visits only finitely many states whose entire initial
collars meet a given compact set.  Confinement therefore makes the
whole reached tree finite.  It has no frontier, and attaching its
finite tree of punctured disk pieces merely fills the root's inner
disk holes by disks.  Thus its completed surface has the original
disk or annulus topology and is compact.

All corner and breakpoint values have been discarded.  Finitely
many strict affine itinerary inequalities then remain valid on an
open interval about the starting parameter.  The same finite tree
and switching sides persist throughout this interval.  The good
parameters have full measure, so a finite collection of separated
closed intervals inside these stability neighborhoods captures at
least $(1-\zeta)w_j$ of the root interval.  Locally finite confinement
persists by continuity from the dense good parameters.

For product triviality use the pre-stack trivializations under the
affine parameter transports on each reached piece.  In a circle
box, parametrize each nonvanishing diagonal segment between its
fixed pair of sides by fractional affine position.  Its product
with the circle gives a band bundle.  Match the boundary-circle
parametrizations of this band to the adjacent piece
trivializations: relative to a reference parameter, lift the
orientation-preserving circle reparametrizations continuously to
monotone maps of $\R$ commuting with integral translation, and
interpolate their lifts across the band.  On a compact stability
interval both the maps and their inverses have finite Lipschitz
bounds, as do the segment fractions.  Gluing the finitely many
bundles gives a bi-Lipschitz product.  Essential boxes lie on the
root piece and their other parameter is preserved by the original
trivialization, so the modifications do not alter them.  The
compatible changed-coordinate PL hypothesis in
Definition~\ref{wl:prestack} applies to all endpoint arrangements.
\end{proof}

\begin{theorem}[Conditional wall realization]\label{wl:realization}
Fix the quantizer, carrier, exterior and boundary data above.  Fix
$0<\zeta<1$, $c_*>0$, the requested locally positive label and value
accuracies, and locally finite buffered charts with finite
capacities $C_a$.  Choose guards and feature-stair accuracies in the
order of Definition~\ref{wl:guards} and
Proposition~\ref{wl:barrier}.  Suppose the resulting sampled
pre-stacks satisfy Definition~\ref{wl:prestack}, avoid these guards,
and admit a slope majorant $G$ obeying the previously fixed
capacities \eqref{wl:prior-capacity}.  The sample intervals are
ordered, have the affine parametrizations $p_j$, and satisfy the
slot, gap and accuracy choices of Lemma~\ref{wl:stairs}.

Then there are finite unions of occupied closed intervals $E_j$
satisfying \eqref{wl:occupied-length} and a locally bi-Lipschitz
homeomorphism
\[
 h:M_x\longrightarrow M_y,\qquad h|_{\partial M_x}=h_*|_{\partial M_x},
\]
which sends every occupied routed layer at parameter $\lambda$ to
the standard layer at label $p_j(\lambda)$, with the same affine
parametrization on all occupied components.  It extends by the
prescribed $h_*$ on the removed cores and tubes.  Moreover
$q_h=Th$ is as finely close to $q_*$ on $M_x$ as prescribed.

The Theorem uses only the preassigned capacities for its choice of
guards.  Existence of a system meeting its hypotheses at that
resolution is the separate content of
Proposition~\ref{ms:prestacks}, Lemma~\ref{ha:capacities}, and
Proposition~\ref{ha:initial-walls}.
\end{theorem}

\begin{proof}
Lemma~\ref{wl:stable} supplies the occupied stacks.  We first construct
$h$ with their exact parameter correspondence and then prove the
asserted label accuracy.

Cut at both endpoints of every occupied $B$ interval, thus at
multiple gates on every subedge.  The corresponding target
components are balls by the graph-handlebody description in
Proposition~\ref{qt:exterior}; this includes the intermediate
chambers along a subedge.  A target ball boundary consists of a
punctured-sphere patch of $\partial M_y$ and its distinct gate disks.
At the source the corresponding fixed boundary patch, capped with
the new gates, is an embedded sphere.  Trim a small product collar
of the manifold boundary and carry this sphere into the trimmed
copy.  It bounds a ball by irreducibility.  The ball lies on the
inward side: the discarded collar escapes to the original manifold
boundary without meeting the sphere, so it cannot be contained in
the ball.  Pulling back from the trimmed copy gives the required
ball bounded by the original patch and gates.

No additional gate lies inside this ball, since its boundary lies
outside the prescribed patch and it is disjoint from the ball's
gates.  Along every gate, the two prescribed patch-balls occupy its
two local sides, as already seen near its rim.

The patch-ball interiors are disjoint.  Perform the following
comparison in the preceding trimmed construction.  For any two
patch-balls, choose a point just inside each of their distinct
boundary patches, lying outside the other ball.  If the capped
spheres share gates, their opposite local sides permit arbitrarily
small compatible collar pushes there that make the spheres disjoint
while preserving these two points.  If their interiors overlapped,
the pushes could also retain a point of that overlap.  The resulting
disjoint spheres would then bound nested balls, contradicting one of
the two distinguished points.  Hence the original interiors cannot
overlap.

There is no leftover component.  Indeed each patch-ball minus its
gates is a relatively
open and closed connected region off the gate system.  A compact
transverse path from an unassigned component to an outer boundary
patch would have finitely many gate crossings by local finiteness;
it can neither have a first entry through a gate, whose two sides
are already assigned, nor reach the endpoint without such an
entry.  Thus the source and target $B$-ball decompositions
correspond exactly.

On each gate, the endpoint $A$ incidences are disjoint proper arcs
with prescribed endpoint pairs.  They correspond relative to the
circle boundary by ordinary cutting of a disk along proper arcs.
Cutting an endpoint annulus along all its spanning gate arcs leaves
disks.  Their boundary slots on the two annulus ends identify the
pieces, even if the spanning arcs twist.  There are at least two
distinct cuts on every sampled subedge: each occupied interval has
two endpoints and each feature is sampled.  Thus successive arcs
leave actual disk pieces with slots on both ends; with more than
two arcs the slot not containing another endpoint uniquely
identifies adjacent pieces.  Along a gate interior there is one
adjacent piece on each local side, in the two distinct gate balls.

These $A$ disks lie in the matching $B$ balls, as their end segments
already do.  Their boundary loops correspond on the ball spheres.
A disjoint family of proper tame disks in a ball cuts it into balls,
indexed by the sphere patches cut along those loops.  Each such
patch reaches an open patch of the manifold boundary: on a gate,
every complementary region of disjoint proper arcs reaches an open
rim interval.  Consequently the endpoint chamber incidence agrees
on the source and target.  The collar structure makes membership
in the interior of an occupied interval constant on an endpoint
cell interior, and its boundary patch determines the corresponding
active or inactive side.

All of these disk and ball recognitions can be made in the common
changed polyhedral coordinates required by
Definition~\ref{wl:prestack}.  Their finite PL models are
bi-Lipschitz equivalent to ordinary disks and balls.  Tame
Schoenflies and the relative PL structure are the qualitative
topological inputs; see \cite[Theorem~5]{Brown1960} and
\cite[Theorem~2.2]{Hamilton1976}.  No uniform constant for these
models is claimed or required.

It remains to map the occupied interiors, not just their endpoint
walls.  A region where both colours are active is an arc times the
two occupied parameter intervals.  Its two boundary end patches
already carry the prescribed correspondence $h_*$.  Extend this as
a product, interpolating the interval coordinate along the arc
between its two increasing endpoint correspondences, and preserve
the two root parameters exactly.  In a $B$ stack, removing the
interiors of these opposite-colour strips from each disk fibre
leaves disk fibres.  In an $A$ stack the analogous removal of
spanning strips leaves disk fibres as well.  Their incidence was
identified above, and Lemma~\ref{wl:stable} gives products over the
remaining single active parameter that preserve all concurrent
box parameters.

The boundaries of each such disk fibre are already assigned, either
on a both-active product or on $\partial M_x$.  In disk-product
coordinates extend them by coning.  More explicitly, for a jointly
bi-Lipschitz family of circle maps $g_\lambda$, the map
\[
 (r\theta,\lambda)\longmapsto
             (r g_\lambda(\theta),\lambda)
\]
is jointly bi-Lipschitz, as is its inverse; the factor $r$ controls
the parameter variation at the disk center.  The boundary circle
maps have these bounds because their finitely many pieces and the
product charts do.  This defines the single-active portions
compatibly with the both-active ones.

These active products occupy the appropriate endpoint cells
completely: their fibres reach the true boundary, their side walls
are precisely the prescribed cuts, and the collar/product
structure makes them open in the relevant cell interiors and
closed up to their indicated boundary.  Every remaining open cell
is therefore a neither-active ball.  Its boundary map is already
assigned and extends by coning in bi-Lipschitz ball coordinates.
The resulting finite-on-compacts maps glue to a locally
bi-Lipschitz map in both directions; use the common polyhedral
coordinates at their interfaces.  Source and target local
finiteness gives a global homeomorphism.  On the cores and tubes
use $h_*$, matching the boundary values.

Finally remake the stairs with their increments allocated only on
$E_j$; Lemma~\ref{wl:stairs} still bounds their root-parameter speeds.
We now pin the point label $q_h(x)$, rather than the support of an
individual routed layer.  Fix a value $q_x=q_h(x)$ and a guard
$\kappa_g$.  Take any point
$z$ on the constant-$q_x$ fibre and any point $y\in\kappa_g$.
The target product-fibre description gives a piecewise linear path
from $h(z)$ to a boundary anchor; pulling it back by $h$ gives
a rectifiable path $\nu_z$ from $z$ to a boundary anchor $a_z$.
For almost every occupied parameter, $\nu_z$ misses its routed
layer: by exact realization, a hit would require $q_x$ to have that
single standard label.  Choose a path $\nu_y\subset\kappa_g$
from $y$ to a boundary anchor $a_g$.  This path misses every
pre-stack, hence every routed layer.  For any rectifiable connector
$\sigma$ from $z$ to $y$, form the boundary-to-boundary path
$\gamma=\nu_z^{-1}*\sigma*\nu_y$.

Use the point-distance potentials for the star and perimeter
coordinates of $q_x$.  The comparison
\eqref{wl:graph-physical} shows that, if the guard label is separated
from $q_x$, one of these potentials separates the two labels.
Its value at $q_x$ is zero before applying the stairs.  Sufficiently
small stair displacement keeps its value at the first transformed
anchor small and its value at the guard anchor bounded away from
zero.  Apply the period identity to $\gamma$ and weight its crossings by
the occupied-stair coefficients as in
Lemma~\ref{wl:crossing-bound}.  The two appended fibre paths contribute
zero for almost every parameter with nonzero coefficient.  Thus the
potential difference at $a_z,a_g$ is bounded by the weighted unsigned
crossings on $\sigma$, and hence by $C\int_\sigma G$.
The coefficient bound changes only by $1+O(\zeta)$.
The same invariant-support tests for remote subedges and vertices
make these prescriptions locally consistent.

If $q_h(x)$ differs too far from $q_*(x)$, follow a path in its
constant-$q_x$ fibre toward an exact boundary anchor.  Since $q_h$
is constant along this path and
$q_*$ agrees with it at the anchor, continuity of $q_*$ produces a
fixed-diameter stretch with a definite discrepancy.  Pack small
balls along this stretch and use nearby guards.  The connector
bound just proved applies to every spherical path between the fibre
and a guard.  The sphere argument of Proposition~\ref{wl:barrier}
therefore gives \eqref{wl:barrier-divergence}, contradicting the same
previously scheduled capacity.  The initial guard spacings accommodate
both the earlier layer-support tests and these point-label tests.
This proves the requested accuracy of $q_h$; the argument estimates
labels after $T$, not $Dh$.
\end{proof}

\subsection{Target stairs, parameter costs, and strict approximation}

Fix the homeomorphism $h$ and occupied intervals supplied by
Theorem~\ref{wl:realization}.  Thus $h$ sends every occupied root
parameter $\lambda\in E_j$ to its standard label $p_j(\lambda)$,
agrees with the prescribed map on full blocks, and respects the
central tubes.  We now collapse the unused target label intervals.
This makes the derivative of the resulting singular composition
depend on the prescribed scalar parameters.

The activation radii and full-cell layer widths are fixed target
data; the occupied stair intervals are supplied by $h$.  Denote the
radial and perimeter stairs by $S_D$ and $S_e$.  For a fixed activation
radius $a$ with $r_{u,D}\ll a\ll1$, use a nondecreasing Lipschitz
cutoff $H_a$ with an absolute Lipschitz bound,
\[
 H_a(u)=0\ (u\leq a),\qquad H_a(u)=u\ (u\geq3a),\qquad
 0\leq H_a(u)\leq u,
\]
with the choices made locally feature by feature.  On the output
two-complex define
\begin{align}
 P_0\bigl(d_D+r(\xi_e(s)-d_D)\bigr)
   &=d_D+\rho_D(r)\bigl(\xi_e(S_e(s))-d_D\bigr),
       \label{wl:target-map}\\
 \rho_D(r)&=H_a\bigl(\exp(S_D(L_D\log r)/L_D)\bigr)
       \quad(r\geq r_{u,D}),\label{wl:radial-stair}
\end{align}
with $\rho_D=0$ farther inward.  The shared subedge stairs make these
formulas agree across sectors and subcells.  By making the stair
accuracy small relative to $L_D$, one has
\begin{equation}\label{wl:target-speeds}
 \rho_D(r)\leq Cr,\qquad |\rho_D'(r)|\leq C/c_*.
\end{equation}
The derivative of the output with respect to one occupied root
parameter is $O(r)$.  Above the central transition, its output
log-radius or perimeter parameter has speed at most $1+O(\zeta)$.
The first bound cancels the polar $1/r$ appearing in weak coarea
estimates when the true and final labels are pinned.  No ratio of
subedge count to subedge length enters these speeds.

Extend $P_0$ to a full convex output cell using a fixed interior point
$z_i$.  Write a point as $z_i+b(y-z_i)$ with $y$ on the cell boundary.
Let $\lambda(b)$ be zero away from a thin boundary layer and rise to
one at $b=1$.  On the boundary, let $\mathcal P_\lambda$ interpolate
the radial factor between $r$ and $\rho_D(r)$ and the perimeter
factor between $s$ and $S_e(s)$, separately and linearly.  Set
\begin{equation}\label{wl:conical-extension}
 P_0\bigl(z_i+b(y-z_i)\bigr)
   =z_i+b\bigl(\mathcal P_{\lambda(b)}(y)-z_i\bigr).
\end{equation}
Here $P_0$ denotes the final singular map; $\mathcal P_0$ is the
identity boundary map.  This notation avoids identifying those two
uses of zero.  The interpolation preserves each boundary sector and
subcell.

Tangential Lipschitz bounds in \eqref{wl:conical-extension} depend only
on the fixed cell geometry, the subcell aspect bounds and $c_*$.  The
speed in $\lambda$ tends uniformly to zero with the radial cutoff
scale and stair displacement.  In fixing these target data, first choose the boundary-layer
thickness using the fixed tangential bounds and absolute continuity
of the relevant integrals; then make the factor displacements small
compared with that thickness.  Thus the extra full-cell energy of
$P_0Th$ can be made summably small.  This uses that $h=h_*$ on the
full input blocks and that $T$ is the prescribed homothety there.
On full blocks outside these thin boundary layers the composition
remains $q_0$.

\begin{proposition}[Parameter cost after realization]\label{wl:parameter-cost}
On occupied pre-stack pieces, a final active root parameter has
local differential $\pm du_j$.  On a routed circle box it has local
differential
\begin{equation}\label{wl:diagonal-gradient}
 \pm d(w_j-u_j+u_k)=\pm(-du_j+du_k),
\end{equation}
with the local coordinate reversals understood.  At such a switched
point only one root scalar is active.  At an essential box both
original root scalars persist independently.  Consequently the
exterior derivative costs of $P_0Th$ depend on these parameter
differentials and the speeds in \eqref{wl:target-speeds}, with no
factor involving $Dh$ or the number of switches.
\end{proposition}

\begin{proof}
Every partial itinerary is a composition of translations and
reflections of the root parameter.  Thus its derivative is $+1$
or $-1$, irrespective of its length.  A diagonal segment has one
reachable ancestry, so there is only one active scalar on the
corresponding circle annulus.  Essential boxes were never switched
and retain both root parameters.  Under the realization, each
occupied interval corresponds by the original affine map $p_j$ to
its target label.  The subsequent stair has bounded forward speed
against that root parameter by Lemma~\ref{wl:stairs}.  In inactive
label directions the stair is constant.  The chain rule in the
piecewise product coordinates therefore yields the asserted local
formulas and the $O(r)$ output factors; interfaces have measure
zero.  This accounts for the derivative entirely in terms of the
original parameter functions.  The qualitative disk and ball
extension supplies no additional energy constant.
\end{proof}

\begin{proposition}[Strict target compositions]\label{wl:strict}
For a fixed realization $h$ as above, the singular composition $P_0Th$
admits locally bi-Lipschitz homeomorphic approximations
$P^{\varepsilon}T_\eta h$ with arbitrarily small prescribed local
$W^{1,p}$ differences and value errors.  Prescriptions may be
summable on a compact exhaustion.  Only local finiteness of the
uncontrolled strict derivatives is required in regions whose energy
is subsequently removed by a source collapse.
\end{proposition}

\begin{proof}
Replace the radial and perimeter factors by
\[
 (1-\varepsilon_D)\rho_D(r)+\varepsilon_Dr,
 \qquad (1-\varepsilon_e)S_e(s)+\varepsilon_es,
\]
with positive, sufficiently small constants on each feature.  Each
factor is strictly increasing with a positive lower slope on a
compact feature.  The interpolating boundary maps in
\eqref{wl:conical-extension} are then homeomorphisms at every radius.
Polar coordinates, including their centers, and the conical cell
coordinates give both local Lipschitz directions.  Hence
$P^{\varepsilon}$ is a locally bi-Lipschitz self-homeomorphism of
$\Lambda$.  Piecewise Lipschitz derivative bounds for the differences
of the factors imply
\[
 P^{\varepsilon}Th\longrightarrow P_0Th
       \quad\hbox{in }W^{1,p}_{\rm loc}
\]
as the feature parameters decrease, with arbitrary compact-local
error prescriptions.

Now fix $P^{\varepsilon}$ and $h$, and use the strict quantizer
$T_\eta$ from Lemma~\ref{qt:strict}.  The composition convergence
requires checking the tangential derivative at a collapsed stratum,
not merely invoking uniform convergence.  Use stairs and cutoffs
that are piecewise smooth, with finitely many pieces on each compact.
On the relative interior of a true face or edge, the pushforward of
source volume by $Th$ is absolutely continuous with respect to the
corresponding dimensional measure: this follows from the product
fibres of $T$ and local bi-Lipschitzness of $h$.  Sector cuts and stair
knots therefore have null inverse image within these strata.

At a generic face point, the derivatives of the full-cell conical
extension on face-tangent vectors converge to the boundary
transformation's derivatives.  At a generic edge point approached
through an incident face sector, an edge-tangent vector keeps $r$
fixed and the perimeter derivative tends to the common subedge
derivative at $r=1$.  In any incident full-cell cone the conical
radial coordinate has derivative zero on that edge-tangent vector;
thus the same compatibility holds on approach through the full
cell.  On a true-vertex level set $D(Th)=0$ almost everywhere.
Consequently the tangential limit supplied by $DT_\eta$ gives
\[
 D(P^{\varepsilon}T_\eta h)
      \longrightarrow D(P^{\varepsilon}Th)
       \quad\hbox{almost everywhere locally}.
\]
The fixed piecewise bounds give an integrable local majorant after
$\eta$ and its gradient are restricted as in
Lemma~\ref{qt:strict}.  Bounded convergence proves the local
power estimates.  Properness and local finiteness allow a diagonal
choice meeting summable compact prescriptions.  Before passage to
the limit, $T_\eta h$ is itself locally bi-Lipschitz, so the ordinary
piecewise chain rule applies.
\end{proof}
==== END Strong-diffeomorphic-approximation-in-three-dimensions-for-p-gt-2//sections/07-walls.tex ====

==== BEGIN Strong-diffeomorphic-approximation-in-three-dimensions-for-p-gt-2//sections/08-mesh.tex ====
\section{Meshes and wall neighborhoods}
\label{sec:mesh}

This section constructs wall neighborhoods satisfying the geometric
hypotheses of Definition~\ref{wl:prestack}, with fixed-factor bounds
for their scalar parameter slopes.  Section~\ref{sec:calibrated-islands}
will improve those bounds on the calibrated source patches.
The analytic selection of the edge tests and the
guard-independent integral capacities is made in
Lemma~\ref{ha:ambient-tests} and Lemma~\ref{ha:capacities};
Proposition~\ref{ha:initial-walls} transfers their counts to the actual
starting walls.  We separate those
selections from the geometric estimates: all statements below concerning
an edge count are conditional on that count, and their constants do not
depend on the final sample weights or on the mesh resolution.

Two different estimates are needed.  Away from the calibrated source
patches, a fixed multiple of the weighted edge counts suffices to bound
the scalar parameter slopes.  On those finitely many patches, the
later gradient argument requires nearly sharp bounds: a fixed
multiplicative loss would remain in the approximation error.  We first
construct the ordinary collars.  Section~\ref{sec:calibrated-islands}
then arranges the calibrated collars around nearly planar disks.
That construction controls thin
intermediate collars by a change of coordinates and broadens selected
unchanged portions to obtain the sharp scalar bounds.  All these
estimates concern the collar parameters; the derivative of the eventual
ambient wall realization is not estimated here.

We work in the polyhedral exterior pulled back by the fixed map $h_*$.
Its working coordinates are locally bi-Lipschitz and piecewise smooth
in physical coordinates.  A compact set meets finitely many working
charts and finitely many smooth pieces.  A \emph{fixed local constant}
may depend on these charts, their incidence numbers, and the already
chosen feature lengths, but not on subsequent collar thinning, guard
gaps, sample weights, or mesh refinement.  In calibrated islands it may
also depend on a fixed compact set of basis matrices and on the slot
margin specified below.  A constant called universal has none of these
additional dependencies.

\subsection{Compatible fine meshes and lattice insertions}

The boundary square complex has the following form.  Squares on exposed
full blocks use normalized $(t,s)$ coordinates, and squares on the
vertical sides of the inner tubes use normalized interval-height and
$s$ coordinates.  Across a seam the varying coordinate agrees affinely,
possibly after reversal.  If that coordinate carries an intermediate
label, both sides carry the same feature with the same normalized
correspondence; otherwise it is unused on both sides.  Thus the tested
boundary levels are axis lines before the modifications below.

\begin{lemma}[Fine meshes and prescribed lattice cores]
\label{ms:meshes}
Fix the working polyhedral structure and a boundary product collar.
There are arbitrarily fine, locally finite triangulations with the
following properties.
\begin{enumerate}
\item Their boundary meshes have bounded shapes and bounded incidence
inside each normalized square.  Their volume shapes and incidences
have fixed local bounds.
\item Any prescribed positive local upper bound for mesh size can be
met.  In finitely many inset coarse-simplex cores one may require a
common constant background step $H$ and exact Kuhn meshes of fixed
bases $A$ on prescribed smaller patches.
\item Transition shape constants depend on the fixed bases and coarse
charts, but are independent of $H$ and of the final guard requirements.
Most of each smooth coarse-chart interior can instead be meshed by
tetrahedra with universally bounded physical shapes.  All remaining
transition and chart-error regions may be confined to predetermined
sets with arbitrarily small integrals of any fixed locally integrable
weights.
\end{enumerate}
These statements remain valid after pushing the boundary to a cut at
depth $d_0(x)>0$, with chart constants independent of the smallness of
$d_0$, provided its absolute slope in square coordinates is bounded.
\end{lemma}

\begin{proof}
Give the vertices of the coarse triangulation a global order.  On each
coarse tetrahedron use the corresponding ordered Kuhn-simplex
coordinates.  The dyadic subdivisions restrict to the same subdivisions
on common faces; on a right-triangle face they are the subdivisions by
side parallels.  Choose a positive mesh-size function $\delta$ with
arbitrarily small absolute Lipschitz constant, subordinate to all
required upper bounds.  Refine until the simplex scale is at most a
fixed small multiple of $\delta$ throughout that simplex.  Positivity
and the small slope imply that neighboring stopping levels, including
levels throughout a vertex star, differ by a bounded amount.

Make the subdivision conforming by overlaying the nested subdivisions
on a common face, splitting their edges, triangulating the resulting
face pieces, and coning the interior pieces.  There are only finitely
many normalized configurations when the level difference is bounded.
Thus this operation has a finite shape list and bounded local
incidence.  It need not alter a uniform Kuhn region whose star has no
hanging data.  To reserve a constant core, choose its preferred dyadic
size much smaller than the allowed slope times its distance from the
coarse facets.  The infimum of these preferred values plus the allowed
slope times path distance is a Lipschitz minorant.  The unit-complex
metric separates each fixed star interior from remote simplices, so
this construction is positive locally and permits the stated constant
cores.  Ordinary positive piecewise-linear minorants give the same
construction on a locally finite complex.

The lattice insertion uses the classical empty-sphere construction
of Delaunay~\cite{Delaunay1934}.  Its buffer, jitter and determinant
estimates are supplied here.  We describe the insertion of a second lattice,
since the absence of a
cut-dependent sliver constant is useful later.  Suppose first that the
two lattice bases have orthogonal columns in one fixed positive
metric.  In mesh units, retain pure portions of the two lattices
separated by a buffer much wider than the covering radius.  Fill the
intermediate region with a separated finite-density net of free sites,
allowing a small fixed jitter of each site.  The net has a bounded
covering radius and bounded local candidate counts.  Choose the
buffer width so that a Delaunay cell can contain fixed sites from at
most one pure lattice.

Place the free sites successively, avoiding small neighborhoods of the
affine spans of at most three previously placed or fixed nearby sites.
There are boundedly many conditions at every step.  Their excluded
volumes can be made smaller than the allowed jitter volume.  The
affinely independent prefixes of fixed lattice points have fixed
positive discrete determinant gaps.  Induction therefore gives a
positive mesh-unit lower bound for the volume of every nondegenerate
candidate tetrahedron.  Affinely dependent prefixes cannot occur in a
nondegenerate tetrahedron.  Resolve nonsimplicial Delaunay cells by a
pulling order that agrees, on each deep pure lattice portion, with the
lattice-coordinate order producing the Kuhn tetrahedra.  The covering
radius bounds Delaunay locality.  Far enough inside a pure portion the
cells are precisely its metric-orthogonal boxes, so the construction
attaches to the unchanged Kuhn meshes.  It can be performed with the
outer lattice extended indefinitely, avoiding an artificial outer
boundary issue.

For an arbitrary inserted basis $A=U\Sigma V^T$, first couple the
identity lattice with $U\Sigma$ in the Euclidean metric.  Then couple
$U\Sigma$ with $A$ in the metric
$(U\Sigma)^{-T}(U\Sigma)^{-1}$.  Both pairs have orthogonal columns in
their respective metrics.  A pure intermediate band, with the same
Kuhn order on both sides, joins the two constructions.  All constants
depend on the fixed bases and buffer ratios, not on $H$.

On a smooth coarse-chart piece, take finitely many inset patches on
which the chart derivative is close to an invertible matrix $B$.
Insert the basis $B^{-1}$ there.  Physical tetrahedra on the smaller
patches then have universal shapes.  The matrices and their inverses
have fixed bounds on each coarse compact.  Having fixed these bounds,
make the patch margins, uncovered sets, and neighborhoods of coarse
facets and nonsmooth chart loci small in the chosen integrable
weights.  Absolute continuity allows this, since the latter loci are
locally finite null sets and the transition constants just constructed
do not involve the shrinking physical margin.  Take $H$ smaller
afterward.  Summable local prescriptions handle the noncompact
exterior.

Finally, in collar coordinates $(x,d)$ send $d=0$ to $d=d_0(x)$ by a
monotone piecewise-linear depth change which is the identity for
$d\ge C d_0(x)$.  Its depth slopes can be bounded above and below by
fixed positive constants, and its cross slopes by a fixed multiple of
$|\nabla d_0|$.  It preserves $x$.  Transporting the coarse collar
coordinates by this map proves the last assertion.
\end{proof}

\subsection{Boundary warping, straight traces, and joint collars}

Choose the cut depth $d_0$ within the region of exact $h_*$ behavior,
with several multiples of $d_0$ still in that region.  It can be
arbitrarily small and have a small bounded absolute slope.  Throughout
the boundary construction impose $\delta\ll d_0$.

\begin{lemma}[Boundary trace preparation]
\label{ms:boundary-collar}
The boundary mesh and the central wall traces can be arranged so that:
\begin{enumerate}
\item each central trace in a boundary triangle is a straight normal
arc with at most one hit on an individual edge;
\item near each selected sample, a closed label interval has a
piecewise-linear ribbon continuation matching the normal-disk prism
construction, and distinct intervals of one colour are disjoint;
\item the two-colour pattern throughout the outer collar is jointly
locally bi-Lipschitz equivalent to the cut pattern times depth,
preserving both parameters;
\item the inverse label slopes and the edge-coarea bounds on the
modified collar have fixed local bounds independent of $d_0$, of the
excluded belt widths, and of the final mesh and sample scales.
\end{enumerate}
In normalized square coordinates, if a feature has physical label
length $\ell$, an occupied interval of label length comparable to
$c_*w_j$ gives on each crossed boundary edge a fractional width at
least
\begin{equation}
 \frac{c\varepsilon c_*w_j}{\ell H_0}.
 \label{ms:boundary-width}
\end{equation}
Here $H_0$ is the local dyadic square scale, $\varepsilon>0$ is fixed
before final sampling, and $c>0$ is independent of the subsequent
choices.
\end{lemma}

\begin{proof}
\emph{Compression and inverse response.}
Write $z=(z_1,z_2)$ for the original normalized square coordinates;
their axis levels carry the prescribed labels.  The warped spatial
position will be denoted by $x$.  Thus all exclusions of label belts
below are made in the $z$ coordinates, before the warp.
When $H_0=2^{-k}$ and $H_0/2\le\delta(x)\le H_0$, set
$\chi(x)=2(1-\delta(x)/H_0)$.  Let $\phi_k$ be an increasing
piecewise-linear map of the unit interval, symmetric under reversal
and fixing its endpoints.  On the interior of each dyadic bin of
length $H_0$, except for narrow endpoint belts, it has slope
$\varepsilon$ and image in an $O(\varepsilon H_0)$ window about the
bin midpoint.  In the belts it expands to join the fixed bin
endpoints.  Its minimum slope is $\varepsilon$, regardless of belt
width.  Choose the relative belt lengths summably small over $k$.
After a compact feature's finitely many relevant scales have been
specified, exclude all their belts from sample points and their
closed label intervals.

For $z$ in a square define $x$ by
\begin{equation}
 x_i=(1-\chi(x))\phi_k(z_i)+\chi(x)\phi_{k+1}(z_i),
 \qquad i=1,2.
 \label{ms:warp}
\end{equation}
Since $|\phi_k-\phi_{k+1}|\le C H_0$ and
$\Lip\chi\le 2\Lip\delta/H_0$, the right side has Lipschitz
constant at most $C\Lip\delta$ in $x$.  Choose this less than $1/4$.
The Contraction Theorem gives a unique $x$ for each $z$.  Conversely,
at a fixed $x$, each coordinate equation is inverted by a strictly
increasing scalar map of minimum slope $\varepsilon$.  This proves
that the map is onto and one-to-one and that its inverse has Lipschitz
constant at most $C/\varepsilon$, independently of belt widths.
At dyadic scale changes the formulas agree.  Reversal symmetry and
the common seam coordinate make them compatible on the square
complex.

The same construction with the right side interpolated with $z_i$
turns the warp on.  At every stage the minimum scalar response is
at least $\varepsilon$.  A level $z_i=\text{constant}$ is a graph in
the other coordinate with slope $O(\Lip\delta)$.  In an allowed bin
configuration its normal response to $z_i$ is
$\varepsilon(1+O(\Lip\delta))$: differentiate the scalar contraction
with the running coordinate held fixed.  This assertion also follows
by difference quotients at the finitely many piecewise-linear breaks.
For partial turn-on the response has the same positive lower bound.
The needed other inverse ordinate exists by the scalar monotonicity
already proved.

Use normalized depth $d/d_0(x)$ as an independent product coordinate
for these operations.  Turning on the warp moves points by $O(\delta)$.
Converting back to physical depth adds at most
$C(\delta/d_0)(1+|\nabla d_0|)$ to the relevant inverse response
bounds.  It therefore causes no inverse power of the collar thickness
in the final estimate.

\emph{Mesh placement and straight traces.}
The possible positions of allowed traces near one triangle star lie
within $C(\varepsilon+\Lip\delta)H_0$ of a bounded list of
references: blend the two relevant bin midpoints using $\chi$ at a
fixed nearby point.  There are boundedly many such references per
star, and they stay a fixed multiple of $H_0$ from the square's axis
ends.  Perturb each mesh vertex by a small shape-preserving fraction
of its local size, freely in a square or along a macro seam, and keep
actual square corners fixed.  We may require, with a fixed $c>0$,
that vertices stay $cH_0$ from relevant axis references, that
non-seam edges have both coordinate components of magnitude at
least $cH_0$, and that all edges avoid the $cH_0$ balls about relevant
pair-reference points.  Here lists from both endpoint stars and all
incident triangles are included.

For completeness these requirements can be imposed simultaneously.
Give the allowed vertex perturbations independent uniform
distributions on fixed small balls or seam intervals.  Each bad event
involves only boundedly many such variables.  Its probability tends
uniformly to zero as $c\downarrow0$, as follows.  Normalize the local
mesh scale to one; the perturbation balls and intervals then have
fixed positive sizes.  With a free endpoint, condition on the other
endpoint.  Except when that endpoint lies within $\sqrt c$ of the
forbidden pair-reference point, the edge--point distance test excludes
a strip of width $O(\sqrt c)$ in the free endpoint's ball.  The
exception itself has probability $O(\sqrt c)$ if the other endpoint
is movable; a fixed corner already has a fixed positive clearance.
For endpoints on two different macro sides near a corner, write them
as $(a,0)$ and $(0,b)$ and the pair-reference point as $(r,s)$.
The line determinant is $ab-as-br$.  Outside
$|b-s|<\sqrt c$, a distance at most $c$ excludes only
$O(\sqrt c)$ of the allowed $a$ interval; the omitted $b$ interval
has probability $O(\sqrt c)$ as well.  The axis-reference and
edge-component tests give ordinary interval exclusions of vanishing
length.  These bounds are uniform in the normalized configurations.
Fixed corner
coordinates are already separated from the references; a seam edge
is automatically separated from pair references.  Take the boundary
mesh fine enough that no edge joins two actual square corners.
There is then no exception involving two immovable endpoints.
Each variable belongs
to boundedly many tests, even at a high-valence actual corner,
because that corner is fixed and every other variable belongs to
bounded stars in at most two squares.  The classical Lov\'asz local lemma of Erd\H{o}s and Lov\'asz
\cite{ErdosLovasz1975}, in the formulation recalled in
\cite[Theorem~1.1]{MoserTardos2010},
therefore applies after making the single-event probability smaller
than $1/(e(D+1))$, where $D$ bounds the dependency degree.  Apply it on
finite exhaustions and pass to a convergent subnet in the compact
product of closed perturbation sets; impose twice the desired slack
before passing to the limit.  This gives all requirements on the
locally finite complex.  Now fix $c$ and take $\varepsilon$ and
$\Lip\delta$ much smaller than its associated tolerances.

The jitter extends to the collar: interpolate the vertex movement
piecewise linearly on each square and taper it over depth
$O(\delta)$, keeping it constant on an initial product band.
Its slopes are bounded, its size is a small fraction of the mesh,
and it respects seams.  Working triangulation coordinates are pulled
back by this change.

In a fully warped triangle every allowed trace is consequently a
single graph arc.  Its endpoints are uniformly away from triangle
vertices, and its running-coordinate span is at least $c'H_0$.
Opposite-colour crossings are transverse and stay away from edges.
At a moving endpoint the running speed is at most $C(c)$ times the
normal speed, by edge obliqueness; a macro edge parallel to the arc
cannot be an endpoint edge.  The same pair of endpoint edges persists
through each allowed bin interval.  Replace the arc by its chord at
fixed running coordinate.  If its endpoints are $(a_i(z),b_i(z))$,
the normal response of the chord at a fixed running coordinate is a
convex combination of
\[
 b_i'(z)-m(z)a_i'(z),
\]
where $m(z)$ is the chord slope.  The endpoint cone and the smallness
of $m$ give the lower bound $c'\varepsilon$.  Chords of disjoint
full cuts of a convex triangle have the same order.  Convex
interpolation from the original arc to its chord therefore preserves
order and the response bound, and agrees across triangle edges.

\emph{Ribbons and continuation through the cut.}
We must also match the closed parameter intervals to affine disk
prisms.  On either side of a sample, take the ordered stair
triangulation of the chord times a half-interval, moving one endpoint
per stair triangle between the specified end placements.  At fixed
abstract arc fraction, its parameter velocity is the relevant
endpoint secant velocity.  It lies in the same response cone.
Choose the label interval much shorter than the local mesh scale;
then the arc-fraction tangential derivative has positive running
component at least $c'H_0$, while its normal slope remains small.
Interpolate between the varying chord and this stair ribbon in both
coordinates at fixed oriented arc fraction.  The same determinant
calculation, namely positive running speed times positive normal
response with a smaller cross term, gives a family of ordered graph
arcs.  The edge trajectories are monotone between the same
endpoints, so each half-ribbon fills exactly the shell between its
end arcs.  Equivalently its positive Jacobians and embedded boundary
give degree one.  Hence intervals stay disjoint, opposite crossings
stay transverse, and the central sample chord is unchanged.  Equal
subdivision of the arcs, used later, causes no loss: tangential
corrections involve differences of endpoint displacements.

Perform these stages successively across the outer cut collar, starting
with the identity at the true boundary $d=0$ and reaching the prepared
chord-and-ribbon pattern at the cut $d=d_0(x)$.  Thus the true boundary
traces remain the prescribed $h_*$ traces at every label.
Inside the cut mirror
the central homotopy back to the unwarped central curves, keeping a
straight-chord product germ at the cut.  Stretch the remaining depth
coordinate back to the full exterior, with fixed two-sided slope
bounds and identity past several multiples of $d_0$, and pull back
the starting carrier walls there.  This changes only the exact
$h_*$ region and preserves every individual wall's topology.
The normal boundary arcs have the required per-wall per-edge count.
On the modified collar, inverse level maps on allowed strips have
the fixed bounds just proved, so one-dimensional coarea on edges
has fixed local constants.  The stretched exact data have the same
property.  These constants may involve feature lengths,
$1/\varepsilon$, and $1/c_*$, but not $d_0$, belt widths, guards, or
resolution.  Formula \eqref{ms:boundary-width} follows by converting
label speed $c'\varepsilon/\ell$ to a fractional edge displacement
on an edge of scale $H_0$.

\emph{The simultaneous product collar.}
At this point each family has a continuation through the collar.  We
now construct a common product identification that preserves both
parameters.  At each normalized depth every used trace
in a whole square is a graph
$x_i=F^i_{z_i}(x_{\bar i})$ with small cross-slope, strict order,
and locally two-sided Lipschitz dependence on each occupied label
interval.  Adjacent triangle pieces have strictly increasing
running-coordinate spans.  Fill gaps between the occupied intervals
by linear interpolation at fixed running coordinate, fixing the
axis endpoints.  The gaps have positive local response for the
fixed data by strict separation and compactness.  All these
definitions agree at macro seams.  Solve the two graph equations
jointly by contraction.  The resulting depth-dependent
homeomorphisms of the square complex, compared with the map at the
cut, identify both families with the cut pattern times depth and
preserve both parameters.  Their local bi-Lipschitz constants need
not be uniform, which is sufficient for this assertion.
\end{proof}

\subsection{Generic volume edges and count estimates}

\begin{lemma}[Generic edges and conditional count charging]
\label{ms:edge-tests}
Away from the constrained boundary collar, the meshes above admit
shape-preserving ambient jitter for which their edges are Sobolev ACL
edges with the usual chain rule.  If $W_e$ is the sum of sample weights
over central wall hits of both colours on an edge, samples can then be
chosen so that, on every unmodified edge,
\begin{equation}
 W_e\le (1+C(\zeta+c_*))\operatorname{Var}_e(q_*)+\epsilon_e,
 \label{ms:weighted-count}
\end{equation}
with separate-colour versions and arbitrarily small prescribed
positive errors $\epsilon_e$.  Here variation means the appropriate
star or perimeter-graph variation; in a smooth sector it can instead
be tested by the individual scalar directional derivatives.  On the
modified boundary collar the corresponding coarea bounds use the fixed
local inverse-label constants of Lemma~\ref{ms:boundary-collar}.
The expected ambient charges for powers of the edge-variation means have
bounded overlap and fixed local density factors independent of mesh
resolution.
\end{lemma}

\begin{proof}
Interpolate small random physical vertex vectors by the working
simplex barycentric coordinates and multiply by a cutoff vanishing
at the constrained collar.  Choose the local amplitudes to make the
ambient displacement have a small local Lipschitz constant.  In
poorly shaped charts use smaller fixed local fractions; in a regular
physical bulk use a universal fraction; in a calibrated patch use a
prescribed arbitrarily small fraction.  Include chart derivatives
and incidence constants when fixing these fractions.  Taper only in
the exact region, where fixed label Lipschitz bounds are available,
and take the mesh much finer than taper depth.  The resulting
ambient maps are locally bi-Lipschitz; fine proper displacement and
degree give global homeomorphisms.

At a fixed edge parameter outside the cutoff region, at least one
endpoint's barycentric coefficient is bounded below.  The perturbed
point therefore has a density at most $C h^{-3}$ in a neighborhood
of scale $h$, with $C$ depending only on the fixed jitter fraction
and local charts.  Its speed in normalized edge time is at most
$C h$.  Fubini applied first to smooth approximations of the
Sobolev labels and then to their gradients gives ACL, the chain
rule, and the expected integral estimates.  The neighborhoods lie
in bounded mesh stars, so multiplying by adjacent tetrahedron
volumes and summing gives bounded overlap.  At inner limiting lift
seams the edge-time set is almost surely null; ACL makes its label
length zero.  This suffices for generic sample avoidance without
requiring finitely many seam hits.

One-dimensional coarea gives integrable hit-count functions on the
finite feature tests associated with each compact edge.  The
stratified slot sampling, performed after the edges are fixed,
gives \eqref{ms:weighted-count}, its partial-colour versions, and any
finite collection of directional tests with the stated errors.
Generic samples avoid vertices, nongenuine chart breaks, tangencies,
and corners, and have finitely many hits on each compact edge.
Near the normalization cut use the PL general-position counts and
the straight-chord germs from Lemma~\ref{ms:boundary-collar}; elsewhere
isolated crossings can be cleaned off a smaller cut collar.

Away from modifications, the graph speeds are those of $q_*$; on a
high-radius sector they are bounded by the appropriate aspect
factor times $|Dq_0|/r$.  On collar modifications the fixed inverse
level bounds from Lemma~\ref{ms:boundary-collar} replace them.  Local
irregular-chart factors charge only the preselected error regions
and slightly enlarged buffers.  For simultaneous finite aggregate
tests use Markov's inequality with room; compact-local upper tests
can be assigned summable failure probabilities.  This Lemma is a
count estimate, not yet an estimate for the final derivative.
\end{proof}

\subsection{Preparing and normalizing the source walls}
\label{wl:normalization-subsection}

The edge estimates are used on actual locally flat source walls.
Proposition~\ref{ha:initial-walls} supplies those systems by a common
opening of the carrier, with the prescribed boundary collars and finite
isolated edge hits.  Counts alone do not supply such surfaces.  The two
results below prepare one colour at a time: first make its walls PL
without adding hits, then replace them by normal disks while retaining
every individual wall--edge upper bound.  Neither replacement is
required to preserve the other colour.

All PL assertions below concern the abstract triangulation.  They remain
applicable when its realization in the physical exterior is given by locally
bi-Lipschitz charts.  For the standard exterior these charts are obtained from
sector boxes with coordinates $(t,s,\text{interval height})$, from edge products
with the edge coordinate and the offset-polygon coordinates, and from the
vertex offset polytopes.  The logarithmic radial coordinate is used only outside
the removed inner polygon.  Common boundary cells have affine identifications
after normalizing the interval coordinates.  Subdivision therefore gives a
compatible locally finite triangulation.

\begin{lemma}[Relative PL preparation with edge counts]\label{wl:relative-pl}
Let $M$ be a three-dimensional PL manifold, possibly with boundary, with a
locally finite triangulation.  Physical coordinates on $M$ may be obtained
from its PL coordinates by locally bi-Lipschitz homeomorphisms.  Let
$(S_j)$ be a locally finite disjoint family of compact, properly embedded,
locally flat surfaces.  Suppose that each $S_j$ avoids the mesh vertices
and has only finitely many, isolated intersections with mesh edges.
On a neighborhood of $\partial M$ suppose the surfaces already have
prescribed PL product collars, with transverse edge crossings and straight
traces in boundary triangles.  The protected collars may be made smaller
before the construction.

Then the family can be replaced by a family of PL surfaces of the same
individual topological types, retaining the prescribed smaller boundary
collars, such that the number of intersections of each individual surface
with each individual edge does not increase.  Every retained interior
edge intersection can be required to have a flat transverse disk germ.
The changes can be confined to any prescribed neighborhood of the original
family and can be arbitrarily small in a prescribed positive continuous
position tolerance.  No bound on the Lipschitz constants of these
preparatory changes is asserted.
\end{lemma}

\begin{proof}
We first explain the preparation at an isolated interior edge hit $x$.
Choose a surface chart in which the relevant part of $S_j$ is a proper
unknotted disk, with no other sheet present, and choose the support to
avoid all other mesh edges and the prescribed boundary collar.  In the
original PL coordinates the edge is straight near $x$.  Take a short
polygonal cylinder about it, with both caps disjoint from the surface.
The cap positions are chosen first, and then the radius is decreased.
Isolation of the hit and continuity of the inverse surface
parametrization ensure that every intersection with the cylinder side
lies in a small subdisk of the surface chart.  Make the surface PL and
transverse on a neighborhood of this side, leaving a neighborhood of the
axis unchanged.  This local preparation follows by approximating the
flattening chart on the compact annular region by a PL chart and
extending the small embedding change across its collar.  Its support
has positive distance from the axis and from the fixed part of the
surface.

There are now finitely many intersection loops on the cylinder side.
An innermost loop that is null-homotopic in the side annulus bounds a
disk on that side whose interior misses the surface.  Replace the
corresponding surface subdisk by a small pushoff of that side disk,
matching the unchanged surface along a collar of the loop.  The surface
remains a locally flat disk, the operation creates no axis intersections,
and it removes that loop without adding side intersections.  Continue
until all remaining loops are essential in the side annulus.  Every
remaining loop has linking number $1$ or $-1$ with the extended straight
axis, so its bounded surface subdisk contains the unique possible axis
hit.  These subdisks are nested.  Replace the outermost one by an
unknotted disk in the cylinder with the same rim, meeting the axis once
transversely and flat near that crossing.  Such a disk is obtained by
isotoping its essential rim to a cross-section in an outer annular
collar and adjoining the flat cross-section.  If no side loop remains,
the surface misses the cylinder: its caps are disjoint from the surface
and the boundary of the surface chart is outside the cylinder.
Thus the final number of hits in this support is zero or one.  The old
and new proper locally flat disks agree near their outer rim and split
the enclosing chart ball into balls; the relative disk-extension consequence of generalized
Schoenflies \cite[Theorem~5, p.~76]{Brown1960} therefore extends the disk replacement to an
ambient homeomorphism fixed at the enclosing ball boundary.  All these
supports may be chosen disjoint and locally finite.  In particular the
construction preserves surface type and same-colour disjointness.

Retain smaller PL balls at the surviving crossings, together with a
smaller prescribed boundary collar.  Outside these protected sets,
every surface has a positive gap from the mesh edges on each compact
set.  It remains to approximate the surfaces while preserving these
gaps and the protected PL germs.  Here are the relative triangulation
details.  In a compact neighborhood of a finite subsystem, cut along
the collared locally flat surfaces.  Triangulate the two copies of
each cut surface compatibly, extending the triangulations already
prescribed at their rims and in the crossing balls.  Hamilton's existence construction, started from the prescribed
boundary collars~\cite[Theorem~2.1 and Section~3, p.~69]{Hamilton1976},
extends these triangulations over the cut manifolds; gluing the copies back gives a PL structure in
which all the surfaces are PL.  This structure agrees with the original
one on smaller protected neighborhoods.  Equivalently, to arrange the
boundary agreement before extending, the uniqueness of the PL structure
on a surface supplies an isotopy between the two boundary
triangulations, and a collar extends that isotopy to the interior.

The relative three-dimensional PL approximation Theorem
\cite[Section~3, Theorem~2.2]{Hamilton1976} now gives a
PL homeomorphism from this auxiliary PL structure to the original
structure, arbitrarily close to the identity and equal to the identity
on the smaller protected neighborhoods.  Choose its position tolerance
below the gaps to all unprotected edge segments.  Its images of the
surfaces are PL; they retain exactly the retained edge hits and create
no others.  PL general position relative to the protected germs is
then imposed with the same gap restriction.  Only the surface sets,
and not a pre-existing parametrizing homeomorphism, have been
prescribed to be PL.

For the locally finite family, choose pairwise disjoint small regular
neighborhoods of its members, with locally finite closures.  The
preceding compact construction is performed in these neighborhoods,
retaining their frontiers and the protected boundary data.  The
constructions agree with the identity near the frontiers and therefore
glue.  Local finiteness supplies continuity in both directions and
preserves all the asserted local restrictions.  The same argument
applies in the abstract PL coordinates when the physical charts are
bi-Lipschitz.
\end{proof}

\begin{remark}
The edge-count conclusion uses the flat crossing germs and the positive
gaps on the remaining edge segments.  Uniform approximation by itself
does not control the number of intersections with an edge.  The
preparation is applied to one colour at a time; it imposes no relative
condition on the other colour.  The preceding proof uses the classical
relative triangulation, PL approximation, and locally flat
Schoenflies Theorems in dimension three only in their qualitative
forms.
\end{remark}

The following argument follows Haken's normalization method
\cite[Chapter~I, \S5]{Haken1961}, using disk and bigon moves that reduce
edge weight.  Its needed conclusion keeps a separate
bound for every wall and every mesh edge, so we give the relative
argument rather than substitute a total-weight normalization theorem.

\begin{proposition}[Normalization with individual edge bounds]
\label{wl:normalization}
Let $M$ be an orientable irreducible PL three-manifold with a locally finite
triangulation.  Let $(S_j)$ be a locally finite family of pairwise disjoint,
compact, properly embedded PL surfaces in general position.  Each $S_j$ is a
disk or an annulus; in the annulus case its core is noncontractible in $M$.
Assume that the prescribed boundary traces are straight normal arcs in boundary
triangles and that
\[
                  \#(S_j\cap e)\leq 1
       \quad\hbox{for every boundary edge $e$ and every $j$}.
\]
Then there is a locally finite disjoint family $(S'_j)$ with the same individual
topologies, coorientations, and exact boundary traces, consisting of normal
triangles and quadrilaterals, such that
\[
                  \#(S'_j\cap e)\leq\#(S_j\cap e)
                    \quad\hbox{for every $j$ and every edge $e$}.
\]
For a finite family the replacements can be obtained by an ambient isotopy
relative to the boundary.  Consequently any initially prescribed upper bounds
on the individual edge counts are retained.  This assertion concerns one
colour only; it imposes no requirement to preserve the other colour.
\end{proposition}

\begin{proof}
Every surface under consideration is incompressible in the following form:
a simple closed curve in its interior that is null-homotopic in $M$ bounds a
disk in that surface.  This is immediate for a disk.  On an annulus a simple
closed curve not bounding a disk represents a core, contradicting the core
hypothesis if it is null-homotopic in $M$.

First suppose that the family is finite.  Among its positions obtained by
ambient isotopies relative to the boundary and satisfying the initial
individual edge bounds, choose one minimizing the total number of edge hits;
subject to this, minimize the total number of circular components of
intersection with faces.  These are nonnegative integers and the class of
admissible positions is nonempty.  All moves used below have final positions
within every individual edge bound.

Suppose that a face contains an intersection circle.  Choose an innermost
circle in the entire family.  Its face disk $D$ has interior disjoint from all
the surfaces.  Incompressibility supplies a disk $E$ on the surface containing
the circle.  The union $D\cup E$ is an embedded sphere in the interior of $M$,
after rounding its corner, and hence bounds a ball.  No other surface is in
this ball: it is disjoint from the sphere and has boundary on $\partial M$.
The part of the same surface outside $E$ is also outside, since it is connected
and reaches its actual boundary.  Thus $E$ can be moved across the ball to a
small pushoff of $D$, without moving any other surface.  The new disk has no
edge hits.  Face-interior collars make its seam transverse without creating
new face circles.  The move either decreases edge weight or, with that weight
unchanged, decreases the number of face circles, a contradiction.

If an intersection arc in a face returns to the same edge, choose a returning
arc cutting off an innermost face bigon.  Its interior is disjoint from the
surface family.  The edge side of the bigon has no further hit: any such hit
would continue into the face on the bigon side and yield an arc there.  The
edge is not a boundary edge, since the returning arc belongs to one wall and
would give two hits of that wall on the edge.  The usual bigon isotopy removes
the two hits and is supported away from every other edge and from
$\partial M$.  One precise description is to move the edge segment across
the empty bigon and slightly past its surface side, and then apply the inverse
ambient isotopy to the surfaces.  This again decreases edge weight.  Hence
all face intersections are normal arcs.

Consider now the pieces of the surfaces in a tetrahedron.  They have nonempty
polygonal boundary; a closed piece would be a whole connected component of
an original surface, whereas every such component has actual boundary.
If a piece is not a disk, the collection of pieces has a compression disk
whose interior misses the entire collection.  To justify this assertion, cut
the tetrahedron along bicollars of all pieces.  If some inclusion of a surface
copy into an adjacent cut region fails to be injective on fundamental groups,
the Loop Theorem produces the asserted disk.  Theorem~2.A.5 of \cite[p.~13]{Stallings1971} allows a surface
contained in the manifold boundary; apply it to the interior of that
surface copy.  Round the polygonal corners and push the
disk interior off the boundary of the region.  Extend its boundary back
to the original piece through that piece's product collar; the collars
are disjoint, so this does not meet any other piece.
If all those inclusions were
injective, reconstruct the tetrahedron as a graph of spaces.  Its vertex
spaces are the connected cut regions, and its edge spaces are the
bicollars of the connected surface pieces; an edge may have both ends
at the same cut region.  The two attaching maps for every edge space
induce the assumed injections.  The graph-of-spaces normal form
therefore injects each piece's fundamental group into that of the
tetrahedron.  A connected compact surface
with boundary other than a disk has nontrivial fundamental group, whereas the
tetrahedron has trivial fundamental group.  This is impossible.

Let $D$ be the resulting compression disk and let $\gamma=\partial D$ lie in
the interior of a piece $P$.  Incompressibility of the whole wall supplies a
disk $E$ in that wall with boundary $\gamma$.  Since $\gamma$ is essential in
$P$, the disk $E$ is not contained in $P$.  Choose a path in $E$ from
$\gamma$ to a point outside $P$.  Its first exit from $P$ crosses a
component $\beta$ of $\partial P$.  The connected curve $\beta$ is
disjoint from $\gamma=\partial E$ and meets $\interior E$, so it lies
entirely in $\interior E$.  Thus $E$ contains a whole boundary polygon
of $P$.
This polygon is in the interior of the wall and thus meets only interior
edges of $M$.  In particular $E$ has at least one edge hit.  The sphere
$D\cup E$ bounds a ball.  As above, all the other walls and the part of this
wall outside $E$ lie outside that ball because they reach $\partial M$.
Replacing $E$ by a pushoff of $D$ removes its edge hits and introduces none,
contrary to minimality.  All tetrahedron pieces are consequently disks.

Disjoint proper disks cut a tetrahedron into balls, with a dual tree.
Traversing an edge of the tetrahedron gives a walk on this tree.  If one disk
is met twice, that walk contains an immediate backtrack.  There is therefore
an edge segment with no intervening surface hit whose endpoints lie on the
same disk.  Join its endpoints by an arc in that disk, interior except at its
ends.  In the intervening complementary ball the two arcs lie on the boundary
sphere and form a simple closed curve.  A boundary disk pushed into the ball
gives an empty bigon, with interior disjoint from all surfaces and from the
other tetrahedron faces and edges.  Its edge side cannot be a boundary edge
by the per-wall boundary-edge hypothesis.  The bigon move again decreases
weight without increasing any individual edge count.  Thus every disk meets
each tetrahedron edge at most once.

A simple normal loop on the tetrahedron boundary meeting each edge at most
once separates the four vertices into two nonempty sets.  It crosses exactly
the edges joining the two sets.  A $1+3$ partition gives a triangle and a
$2+2$ partition gives a quadrilateral.  Straightening the face arcs and
replacing the filling disks by compatible standard normal disks preserves
the surface family: proper tame disks in a ball with the same boundary
pattern are carried to one another by a boundary-relative ball
homeomorphism.  The construction is an ambient isotopy relative to the
boundary, and establishes the finite assertion.

For a locally infinite family, exhaust its index set by finite subfamilies
and perform the finite construction.  A fixed wall initially meets only
finitely many edges, because it is compact and the triangulation is locally
finite.  Its normalized disks can occur only in tetrahedra incident to these
edges, since no new edge hit is permitted.  There are finitely many such
tetrahedra, and only finitely many normal combinatorial possibilities under
the fixed hit bounds.  Include the wall labels, coorientations, matching data
across faces, and order of hits along edges among these finite data.  A
successive subsequence extraction stabilizes all these data wall by wall.
For a compact subset of $M$, only finitely many walls can appear in this
extraction: the finitely many relevant tetrahedron stars are compact, and
any candidate wall originally met an edge in those stars.  Initial local
finiteness applies.  Thus the stabilized data realize a locally finite,
simultaneously disjoint normal family.  Each whole wall has stabilized on its
finite support, so its topology and prescribed boundary placements are
retained.  Ordered placements on interior edges and standard fillings realize
the data; boundary-edge placements are the originally prescribed ones.
No convergence of an infinite ambient isotopy is needed.
\end{proof}

\subsection{Normal disks and affine shell prisms}

\begin{lemma}[Normal templates]
\label{ms:normal-templates}
In a normalized tetrahedron fix an endpoint margin $\gamma>0$.
For every normal disk type use its triangle, or the two triangles
of a quadrilateral on one chosen combinatorial diagonal.  We call these
pieces the main triangles and, in the quadrilateral case, call their
shared diagonal the main crease.  Require all crossing-edge fractions
to lie in $[\gamma,1-\gamma]$.
Compatible within-colour edge orders give disjoint templates.
Their separations are at least $c(\gamma)$ times the minimum edge
gap.  Between two ordered placements of the same disk type the
ordered affine stair prisms give a PL homeomorphism onto the entire
shell.  If its scalar parameter has range of length at most $w$,
then
\begin{equation}
 |\nabla u|\le C(\gamma)
       \max_{e\text{ crossed}}\frac{w}{\Delta_e},
 \label{ms:normal-slope}
\end{equation}
where $\Delta_e$ is the fractional width on the crossed edge.
Physical and other working coordinates add their usual fixed shape
and scale factors.
\end{lemma}

\begin{proof}
Write the vertex partition as $I\mid J$.  For a quadrilateral both
groups have two vertices.  In barycentric probability coordinates
write
\[
 \lambda=((1-u)a,ub),\qquad a_0+a_1=b_0+b_1=1,
 \qquad r_{ij}=\frac{t_{ij}}{1-t_{ij}}.
\]
Choose diagonal $00,11$.  On the triangle $00,01,11$ the disk is
the graph
\begin{equation}
 \frac{u}{1-u}
   =\frac{r_{01}a_0+r_{11}a_1}{(r_{01}/r_{00})b_0+b_1},
 \qquad r_{11}a_1b_0\le r_{00}a_0b_1,
 \label{ms:quad-graph}
\end{equation}
with the symmetric formula on the other half.  The two agree on the
diagonal.  The displayed expression is monotone in each of its edge
positions.  In particular its derivative in $r_{01}$ has the sign
of $a_0b_1-(r_{11}/r_{00})a_1b_0$, nonnegative by the side
condition; the other direct derivatives are nonnegative.  The
second half gives the remaining derivatives.  A common multiplier
of the $r_{ij}$ multiplies the whole odds graph by that multiplier.
Compactness of the allowed fractions therefore gives a lower
response $c(\gamma)$ to a common edge increment and bounded graph
slopes.  Ordered same-type disks have at least the asserted
separation.  Their edge orders agree on all crossed edges, since
disjoint boundary cuts of the tetrahedron sphere are nested between
the two vertex groups.  The triangular case follows from the
affine separating plane, or the same formula with one group a
single vertex.

Different compatible types include a triangular disk.  By the
face-arc order every crossing vertex of the other disk lies on the
specified side of that triangle plane.  So does its convex hull,
and hence its chosen disk.  An edge gap gives the same quantitative
separation by the triangle's uniformly positive separating
altitudes.  Edges not crossed by the triangle already have vertex
clearance.  This proves the simultaneous disjointness assertion.

Triangulate a reference disk consistently, and use the standard
ordered triangulation of each triangle times an interval.  Map its
bottom and top vertices to the two placements.  A stair tetrahedron
contains three mixed-level vertices and one duplicate vertex moving
along its crossing edge.  Its mixed triangle still has a strictly
separating plane.  For example the first quadrilateral half has
plane coefficients proportional to
\[
 (-r_{01},-r_{11},r_{01}/r_{00},1)
\]
on $(I_0,I_1,J_0,J_1)$, with strict separating signs for all allowed
mixed placements.  Consequently the oriented altitude of the
duplicate-edge motion has the correct sign and is at least
$c(\gamma)$ times that edge displacement.  The other half and
triangular types are identical in this respect.  All stair
tetrahedra have positive orientation and volumes bounded below by
that displacement times a fixed normalized area factor.

The common order on neighboring triangle prisms makes the boundary
ribbons agree.  On a tetrahedron face each ribbon lies between
disjoint full chord cuts with monotone endpoint tracks.  Thus the
prism boundary is embedded and bounds precisely the desired shell.
Positive nondegenerate simplex signs give local openness in the
interior: an interior preimage is isolated locally, and a nearby
generic value in an incident simplex image has positive local
degree.  The embedded sphere boundary has degree one.  Summing
the positive local degrees proves that the entire prism map is a
homeomorphism.  This degree argument also excludes possible
branching at internal simplex faces and edges.

On each stair tetrahedron, inversion of its affine matrix and the
altitude lower bound give \eqref{ms:normal-slope}.  One may use two
half-prisms with the central disk exactly prescribed.  If all
placements are close to one template, the horizontal layer
tangents are close to that template's main triangle: differentiating
at fixed layer parameter compares mixed vertices at that same
parameter, so only displacement differences enter the tangential
columns.
\end{proof}

\subsection{Simultaneous product collars}

Normal prisms give product coordinates for each individual wall.
Definition~\ref{wl:prestack} also requires their intersections to vary
as products preserving both scalar parameters.  In a candidate collar
$\mathcal E_j$, write $f_k=u_k\circ\mathcal E_j$ for an
opposite-colour parameter on its domain.  The following criterion
obtains the simultaneous products from an intrinsic direction in which
$f_k$ increases strictly.  The proof of Lemma~\ref{ms:weak-templates}
will supply these directions at smooth crossings, main creases, and
tetrahedron faces.

\begin{lemma}[A sufficient pattern-triviality test]\label{wl:pattern}
Let $\Sigma$ be a compact smooth disk or annulus, let $J$ be a compact
interval, and let $E:\Sigma\times J\to C$ be bi-Lipschitz.
For finitely many indices $k$, suppose that a Lipschitz function
$f_k(z,\lambda)$ is defined on an open neighborhood of its compact
enlarged strip
\[
 K_k=\{(z,\lambda):a_k-\delta_k\leq f_k(z,\lambda)
                         \leq b_k+\delta_k\},
 \qquad a_k<b_k,\quad\delta_k>0.
\]
The sets $K_k$ are assumed disjoint, and all relevant strip points
belong to these compact sets, with a margin inside their domains of
definition.  Each strip is the full inverse image of its indicated
label interval in its isolated neighborhood: apart from
$\partial\Sigma$, there is no additional lateral frontier inside
the enlarged label interval.  At every point of $K_k$ suppose there is a smooth vector
field $V$ on a surface neighborhood, tangent to $\partial\Sigma$ at
boundary points, and a neighborhood in $\Sigma\times J$ on which
\[
 D_z f_k(z,\lambda)V(z)\geq c>0
 \quad\hbox{for almost every }(z,\lambda).
\]
Equivalently one may impose the integrated strict increase
along the local $V$-flow for every nearby $\lambda$.
The constant and direction may depend on the neighborhood.

Then the simultaneous subdivision of $C$ by the strips
$a_k\leq f_k\leq b_k$ is jointly bi-Lipschitz trivial over $J$,
preserving $f_k$ on each strip and preserving the manifold boundary.
For each fixed $\lambda$, every connected strip is also a
bi-Lipschitz product over its $f_k$ parameter.  Thus, when its level
fibres are the prescribed arcs or circles, it has the full product-box
form required in Definition~\ref{wl:prestack}.
\end{lemma}

\begin{proof}
Fix $\lambda_0\in J$.  Compactness and a partition of unity combine
the finitely many admissible local directions into a smooth field
$V_k$ near the $k$th enlarged strip at $\lambda_0$.  After decreasing
the parameter neighborhood, the same field works for every
$\lambda$ in that neighborhood, with a uniform positive lower bound
$c_k$.  Indeed differentiation in the direction
$\sum_i\psi_iV_i$ is $\sum_i\psi_iD_zf_kV_i$; there is no
derivative of the coefficients in this identity.  All fields are
tangent to the boundary, and the supports belonging to different
$k$ can be kept disjoint.  Extend them smoothly, using slightly
larger neighborhoods, and write $\Phi_k^t$ for their flows.

The derivative lower bound implies
\[
 c_k(t_2-t_1)\leq
 f_k(\Phi_k^{t_2}z,\lambda)-f_k(\Phi_k^{t_1}z,\lambda)
 \leq M_k(t_2-t_1)
\]
whenever the indicated flow segment stays in the enlarged strip,
for a finite $M_k$.  To justify this for every flow line, use a
smooth flow box.  The inequality holds on almost every parallel
line for almost every parameter by the Lipschitz chain rule and
Fubini, hence on all parallel lines and at every parameter by
continuity.  It applies to the boundary lines by continuity from the
interior.

For $z$ in a smaller enlarged strip and $\lambda$ close to
$\lambda_0$, strict monotonicity gives a unique small number
$t_k(z,\lambda)$ satisfying
\[
 f_k(\Phi_k^{t_k(z,\lambda)}z,\lambda)
       =f_k(z,\lambda_0).
\]
The available strip margin ensures existence in both time directions.
The lower slope bound and joint Lipschitz continuity give
$|t_k(z,\lambda)|\leq C|\lambda-\lambda_0|$ and joint
Lipschitz continuity of $t_k$ in $(z,\lambda)$, by subtracting the
two defining equations and applying the lower slope bound to the
time difference.

Choose a Lipschitz cutoff $\chi_k$, equal to one on
$[a_k-\delta_k/2,b_k+\delta_k/2]$ and zero outside a slightly
larger interval strictly inside
$(a_k-\delta_k,b_k+\delta_k)$, and set
\[
 P_\lambda(z)=
 \Phi_k^{\chi_k(f_k(z,\lambda_0))t_k(z,\lambda)}z
\]
on its support, extending by the identity elsewhere.  The supports
for distinct $k$ are disjoint.  We verify the required injectivity,
since a small displacement by itself would not suffice.  In flow-box
coordinates $(y,s)$, write $F_\lambda(y,s)=f_k(z,\lambda)$.
The uncut reparametrization $h_\lambda$ is determined by
$F_\lambda(y,h_\lambda(y,s))=F_{\lambda_0}(y,s)$.
For $s_2>s_1$ it satisfies
\[
 h_\lambda(y,s_2)-h_\lambda(y,s_1)
       \geq (c_k/M_k)(s_2-s_1).
\]
The cut reparametrization is
$s+\chi_k(F_{\lambda_0}(y,s))(h_\lambda(y,s)-s)$.
Its derivative in $s$, where defined, is at least
\[
 \min\{1,c_k/M_k\}
 -\Lip(\chi_k\circ F_{\lambda_0})
                      \|h_\lambda-s\|_\infty>0
\]
after making $|\lambda-\lambda_0|$ small.  It has a finite upper
Lipschitz bound as well.  Thus it is strictly increasing on every
flow line, with a uniform lower slope, and is jointly Lipschitz in
the transverse coordinates and in $\lambda$.  Solving this scalar
monotone equation gives a jointly Lipschitz inverse.  The maps agree
with the identity off their supports.  Globally on each flow orbit
they are increasing reparametrizations with bounded displacement,
or increasing degree-one maps on a periodic orbit, and hence are
onto.  Consequently $P_\lambda$ is a homeomorphism of $\Sigma$,
and $(z,\lambda)\mapsto(P_\lambda(z),\lambda)$ is bi-Lipschitz
on the compact local parameter product.  Tangency of the flows
preserves $\partial\Sigma$.

On the actual strip the cutoff is one, so
$f_k(P_\lambda z,\lambda)=f_k(z,\lambda_0)$.
Its image is the entire corresponding strip.  Indeed, by smallness
of the displacement and the extra half-margin, the inverse of a
point in the target strip lies where the cutoff is one; the same
identity then identifies its value.  Complementary components and
boundary components are carried to their counterparts by the
resulting ambient surface homeomorphism.

Cover $J$ by finitely many such parameter neighborhoods and
subdivide it into successive closed intervals subordinate to this
cover.  Transport from the first endpoint across one interval at a
time, composing the transports already constructed.  At a subdivision
point the next transport starts with the identity.  The resulting
transport is continuous, preserves all the specified strip values,
and is jointly bi-Lipschitz, together with its inverse, because it
has only finitely many bi-Lipschitz pieces in the parameter interval.
Composing with $E$ gives the asserted simultaneous trivialization.

Finally fix $\lambda$.  A field with the preceding strictly positive
lower slope can be chosen on the entire compact enlarged strip.
Every one of its flow lines starting at $f_k=a_k$ reaches
$f_k=b_k$ before it can leave the enlarged strip, and does so in
time at most $(b_k-a_k)/c_k$.  Every point of the strip reaches
$f_k=a_k$ uniquely by the reverse flow.  The first hitting time
of a specified intermediate value is jointly Lipschitz, by the same
monotone-equation estimate.  Flow boxes show that $f_k=a_k$ is a
compact Lipschitz one-manifold, with its boundary on
$\partial\Sigma$.  The first hitting maps therefore give the
claimed bi-Lipschitz product of each of its arc or circle components
with $[a_k,b_k]$.  Combining this product with the preceding
parameter transport preserves both parameters.
\end{proof}

\begin{remark}
The test is qualitative: its constants may depend on the fixed
compact collar and its intersection pattern.  A global locally
bi-Lipschitz change of coordinates preserving the parameters transfers
the conclusion directly.  At a crease of a triangulated sheet, a
direction along the crease can be used when the required positive
one-sided slope holds on both incident sheet pieces; integration in
the corresponding intrinsic flow boxes gives precisely the hypothesis
used above.  The test supplies pattern triviality, not the common
ambient changed-PL condition or the later quantitative density bound;
those remain separate parts of Definition~\ref{wl:prestack}.
\end{remark}

\begin{lemma}[Uniform weak templates]
\label{ms:weak-templates}
After the separate-colour normalizations, all normal disks can be
thickened into pre-stacks satisfying Definition~\ref{wl:prestack}, with the
boundary data of Lemma~\ref{ms:boundary-collar}.  On a physical tetrahedron
$\sigma$ of size $h_\sigma$ with universally bounded shape, away
from special boundary widths, the scalar-slope majorant satisfies
\begin{equation}
 G|_\sigma\le C h_\sigma^{-1}
                    \max_{e\subset\sigma} W_e.
 \label{ms:weak-G}
\end{equation}
Fixed local chart factors and the fixed boundary terms described
below apply in the preselected error regions.  No quantitative
general-position bound depending on the number of actual sheets is
required.
\end{lemma}

\begin{proof}
On each interior edge put each colour's central hits near its own
mid-edge reference, with the two reference clusters separated.
Preserve the within-colour order and give each hit two-sided
fractional widths at least $c w_j/W_e$, using its prescribed
coorientation.  The total widths fit for a fixed small $c$.
At boundary edges keep the actual prescribed widths.  The warp,
obliqueness, and pair-reference avoidance give bounded reference
lists there, separated between colours and away from vertices.
All actual central positions lie in small windows about these
references.  Formula \eqref{ms:boundary-width} and
\eqref{ms:normal-slope} show that a boundary edge contributes only a
fixed local term, of scale $C\ell/(\varepsilon c_*)$ in square
coordinates, instead of an inverse mesh or belt scale.  Mixed
interior and boundary widths are allowed by the maximum in
\eqref{ms:normal-slope}.

We now arrange opposite-colour transversality uniformly over the
finite reference tables.  In one tetrahedron move the second
system by a smooth diffeomorphism equal to the identity near the
boundary.  Ordinary relative general position for the finite lists
of triangles and main creases makes smooth pieces transverse,
makes every crease meet opponent triangles transversely along the
crease, and keeps opponent creases disjoint.  Near the tetrahedron
boundary the face arcs already cross transversely and no pair meets
an edge, so the motion can indeed be supported away from it.
Approximate this diffeomorphism by a PL homeomorphism whose
simplex derivatives are uniformly close to its derivatives, still
fixed in a smaller boundary collar.  Fine affine interpolation of
a smooth diffeomorphism supplies such an approximation; taking the
derivative error smaller than its inverse-derivative margin keeps
it locally invertible, and its fixed boundary and degree give a
homeomorphism.  Take the error also smaller than all the finite
transversality margins.

There are finitely many combinatorial reference types, while their
positions range in compact sets with the fixed endpoint and
opposite-colour slack.  Each of the preceding choices works on an
open neighborhood of its data.  A finite subcover therefore gives
a finite list of PL motions, a common allowed position and tangent
perturbation, and common finite complexity and bi-Lipschitz bounds.
No general position within one colour's candidate list is needed:
the same ambient motion preserves all its disjointness relations.
Make the actual clusters narrow enough for these uniform
tolerances.  The normal prisms then give the desired broad layers.

Here is the required joint, rather than merely separate,
transversality verification.  At an interior smooth crossing use
an intrinsic direction of one sheet transverse to the other.  At
a main crease crossing use the crease direction on that sheet.
Conversely, in the opponent tangent plane there is a direction
crossing both adjacent facets with the same strict sign: the
crease-normal-plane normals of the two facets are not opposite,
and the opponent plane projects surjectively along the crease.
At a tetrahedron-face crossing use the boundary-arc direction,
which crosses the opponent arc with the same sign from either
tetrahedron.  Pair crossings avoid all endpoints of these intrinsic
edges.  These strict directional cones persist for the actual
prism simplices and the fine PL approximation of the smooth
motion.  Orient them using increasing stack parameter and retain
small extension margins.  More precisely, extend each
edge-displacement formula to a slightly larger parameter interval,
with the same orders and shared prism triangulations.  Compactness
of each wall and the strict separation and directional inequalities
permit a positive extension retaining these properties.  Main
creases and tetrahedron faces are glued internal sides of the
enlarged collar.  Since the wall is proper, its only remaining sides
are its parameter ends and its product side on the manifold
boundary.  Restriction to a smaller enlarged interval therefore
gives the open domain, compact margin, and absence of any additional
lateral frontier required in Lemma~\ref{wl:pattern}.

The sufficient test in Lemma~\ref{wl:pattern}
therefore supplies the entire pattern trivialization, preserving
the opposite parameter.  The joint boundary collar from
Lemma~\ref{ms:boundary-collar} extends it to the true boundary.

Finally apply \eqref{ms:normal-slope} to the proportional widths.
The finite template motions multiply the bounds by fixed factors,
giving \eqref{ms:weak-G}.  Parameters and endpoint arrangements are
PL in the working coordinates, followed by the common locally
bi-Lipschitz collar changes.  Upper local slopes at interfaces may
be bounded by the adjacent maxima.  This is precisely the
regularity required in Definition~\ref{wl:prestack}; it makes no claim of a
sharp derivative constant.
\end{proof}

\section{Calibrated islands and sharp parameter estimates}
\label{sec:calibrated-islands}

We improve the ordinary pre-stacks of Section~\ref{sec:mesh} on the
finitely many aligned source patches used to recover the deficient-rank
gradients.  The product structures and boundary data of
Definition~\ref{wl:prestack} are preserved; the two scalar gradient
bounds become nearly sharp.

\subsection{Calibrated slots in protected lattice islands}

Only finitely many protected lattice patches require sharp geometry.
Work in their aligned affine units, before the tiny ambient
edge-generic jitter.  First-derivative chart errors and the jitter
fractions will be prescribed as small as needed.  A common background
step $H$ is available from Lemma~\ref{ms:meshes}.  Keep all sharp operations
inside the pure lattice, away from the boundary collar.  Reserve nested
positive patch margins: the placements become fully clustered while
still in the aligned lattice, and all later fixed-number mesh buffers
fit inside these margins.  On the enlarged islands assume that all
eligible nearby labels belong to the specified smooth $(D,e)$ sectors
of $q_*=q_0$.  Use a common weight $W=w_j$ for every feature relevant
there and in the required finite-on-compacts buffer.  The ratio $W/H$
is chosen arbitrarily small only after the edges and the preceding
geometry have been fixed.  The order of these choices is collected in
the proof of Proposition~\ref{ms:prestacks}.

The two ideal scalar-covector modes in aligned coordinates are
\begin{equation}
 (e_1^T,e_2^T)
 \quad\text{and}\quad
 (b_1e_1^T,b_2e_1^T),\qquad |b_1|+|b_2|=1.
 \label{ms:modes}
\end{equation}
Call them independent and parallel, respectively.  They are the
normal forms of the two scalar derivatives in the nonsingular source
frames chosen in Subsection~\ref{ha:patches}; here they are inputs to
the geometric construction.  Signs of the coefficients affect label
coorientations but not the following geometric orders.  An inter-edge
for a family increases its indicated
axis by $H$; a level edge has constant indicated coordinate.  An edge
is called \emph{eligible} when it lies in the sharp placement region
and its upper counts $N_{e,i}$ for the relevant families satisfy
\begin{equation}
 N_{e,i}\le (B_i+\delta')H/W\quad\text{on inter-edges},
 \qquad
 N_{e,i}\le\delta'H/W\quad\text{on level edges},
 \label{ms:regular-counts}
\end{equation}
where $B_i=1$ in independent mode and $B_i=|b_i|$ in parallel mode.
A Kuhn tetrahedron is called \emph{regular} when all its required
edges are eligible.
The inter-edge excess may use a different small tolerance, as long as
it is much smaller than the slot loss below.  The level tolerance
$\delta'$ is free to be made smaller after constants depending on
that slot loss have been fixed.

\begin{lemma}[Slots, slab slopes, and gaps]
\label{ms:slots}
Fix a small slot-loss tolerance $\eta>0$.  Choose the tolerances in
\eqref{ms:regular-counts} sufficiently small relative to $\eta$.
Central hits on eligible inter-edges can be placed in consecutive
slots of axial spacing at least $(1-C\eta)W$, starting at a common
offset of order $\eta H$ from the lower endpoint.  Fractions stay
in a fixed interval $[\gamma(\eta),1-\gamma(\eta)]$.

On each regular tetrahedron all but $C\delta'H/W$ disks of a family
are slabs separating its lower and upper vertex layers.  The main
triangles of these slabs have slopes $O_\eta(\delta')$ from the
indicated coordinate planes.  Successive slabs are separated along
that axis by at least $(1-C\eta)W$, with a smaller fixed choice of
the slot slack.  In parallel mode this also holds between the two
families, placed in consecutive separate bands.  These statements
allow arbitrarily small generic perturbations of the slot positions.
\end{lemma}

\begin{proof}
In independent mode allocate slots separately for the appropriate
colour on each inter-edge.  In parallel mode use one list, all
first-colour slots before all second-colour slots.  The sum of the
two count bounds is at most $(1+2\delta')H/W$.  Shortening the slot
spacing by a sufficiently large multiple of $\eta W$ leaves room
for both endpoint margins.  Preserve the normal-surface order of
each family.  An edge failing its eligibility test, or lying outside
the sharp placement region, instead uses arbitrary margin-spaced
placements with same-colour gap at least
$cH/(1+\#\text{hits})$; in the outer buffer use the clustered
placements of Lemma~\ref{ms:weak-templates}.  A tetrahedron is used for
the sharp conclusion only when every required edge uses the slots.
Use the same combinatorial quadrilateral diagonal for a fixed cut
type, also across the two colours.

A Kuhn tetrahedron has two nonempty vertex layers in each indicated
coordinate, at heights separated by $H$.  A normal disk other than
the cut between those layers hits a level edge.  Hence the total
number of non-slabs is at most the sum of the level-edge counts,
which is $C\delta'H/W$.  Every slab hits every inter-edge once.
Slabs of a fixed family are nested, so their order toward the upper
layer is the same on every such edge, even when their positive
label coorientations differ.  Only non-slabs can change their ranks
in the complete edge lists.  Thus the ranks differ by
$O(\delta'H/W)$, and the corresponding fractional positions differ
by $O(\delta')$.  In parallel mode the length of the preceding
first-colour band varies between edges only by the same non-slab
count.  This proves the same estimate for the second band.

Apply \eqref{ms:quad-graph}, or its triangular analogue, to these
almost-identical fractional positions.  Its graph and the two main
triangles are $O_\eta(\delta')$ perturbations of the constant-height
template.  This gives the claimed slopes.  To verify the sharp gap,
move all fractions of a lower slab by an increment
$\Delta\asymp W/H$ no larger than the least edge-slot gap to the
next slab.  Its odds multipliers are
\[
 1+\frac{\Delta}{t_*(1-t_*)}
       +O_\eta\bigl(\Delta(\delta'+\Delta)\bigr),
\]
where the old fractions differ from $t_*$ by $O(\delta')$.
Monotonicity and the common-multiplier property in
Lemma~\ref{ms:normal-templates} give at least
$(1-O_\eta(\delta'+\Delta))\Delta$ response in group height $u$ at
fixed probability coordinates $(a,b)$.

This last comparison must be converted to a fixed physical
transverse foot.  It suffices to consider points with possible
separation $O(W)$.  Since $u$ is bounded off $0$ and $1$, their
probability coordinates differ by $O_\eta(W/H)$.  The near-horizontal
slab graph has $(a,b)$ slopes $O_\eta(\delta')$, so this changes
$u$ by only $O_\eta(\delta'W/H)$.  The group height component is
exactly the indicated coordinate divided by $H$.  This proves the
same local gap on a common transverse foot.  Choose $\delta'$,
$W/H$, and the further generic displacements small enough compared
with the previously reserved slot slack.  No lower bound on the
number of surviving slabs was used.
\end{proof}

\begin{lemma}[A neighboring majorant]
\label{ms:majorant}
On a finite enlarged lattice region containing the equal-weight
island and its clustered buffer, give original tetrahedra their
star-neighbor graph.  Its degree is bounded by a fixed $D$.
Let $N_e$ be the total initial wall-hit upper count of both colours on
$e$; the counts after separate-colour normalization are bounded by
these numbers.  Put
\begin{equation}
 m_\sigma=1+\frac WH\max_{e\subset\sigma}N_e,
 \qquad
 M_\sigma=\sum_\tau
       \theta^{\operatorname{dist}_{\rm star}(\sigma,\tau)}m_\tau,
 \qquad
 s_\sigma=\frac W{M_\sigma},
 \label{ms:majorant-def}
\end{equation}
where $0<\theta<1/(2D)$ is fixed.  Then $M\ge m$,
neighboring values of $M$ differ by at most a factor $\theta^{-1}$,
and for $p\ge1$
\begin{equation}
 \sum_\sigma H^3M_\sigma^p
       \le C_{p,D,\theta}\sum_\sigma H^3m_\sigma^p.
 \label{ms:majorant-power}
\end{equation}
If $m_\sigma\le B+e_\sigma$ with fixed $B$ and a small
$\sum H^3e_\sigma^p$, the portion of $M$ above a sufficiently large
fixed threshold, including any fixed number of neighboring layers,
has correspondingly small $p$-cost.
\end{lemma}

\begin{proof}
For neighbors $\sigma,\rho$, graph distances to any $\tau$ differ
by at most one, giving
$\theta M_\rho\le M_\sigma\le\theta^{-1}M_\rho$.
Both row sums and column sums of the symmetric kernel
$\theta^{\operatorname{dist}_{\rm star}}$ are at most
$C_\theta=\sum_{k\ge0}D^k\theta^k$.  Jensen's inequality with
these row sums, followed by summation in $\sigma$, proves
\eqref{ms:majorant-power}.  The same estimate applies separately
to the convolution of $e$.  The baseline convolution is at most
$BC_\theta$.  On $\{M>2BC_\theta+1\}$ the error convolution
dominates $M$ up to a fixed factor.  The asserted small cost follows.
Neighbor comparability and bounded degree extend it to a fixed
number of graph layers.  All statements remain valid with the
common lattice-volume weight $H^3$.
\end{proof}

\begin{lemma}[Auxiliary mesh]
\label{ms:auxiliary}
On an inset union of the original lattice tetrahedra, with its outer
boundary deep in the clustered buffer, there is a conforming
auxiliary triangulation $\mathcal G$ whose cells in an original
tetrahedron have sizes comparable to $s_\sigma$.  Its shapes and
star degrees have fixed bounds.  In particular all auxiliary sizes
are at most $CW$.
\end{lemma}

\begin{proof}
Refine each original tetrahedron dyadically to scale comparable to
$s_\sigma$.  By Lemma~\ref{ms:majorant} the dyadic level difference on
neighboring original tetrahedra is bounded by a constant depending
only on $\theta$.  Overlay the nested face subdivisions and cone
as in Lemma~\ref{ms:meshes}.  The resulting finite normalized
configuration list gives the asserted bounds.  Since $M\ge1$,
$s_\sigma\le W$.
\end{proof}

\subsection{A master system and its translated regular portions}

We now combine the sharp slot placements with the ordinary clustered
collars.  The first common construction may have extremely thin strips:
its purpose is to establish the simultaneous pattern of both colours.
On regular portions we replace these strips by small axial translations
of their supporting triangles.  We then protect smaller translated
portions while changing coordinates elsewhere to control all parameter
slopes.  The final broadening takes place inside those protected
portions, where the axial formulas have remained unchanged.

\begin{lemma}[Master pre-stacks]
\label{ms:master}
The slot placements and their weak clustered continuations admit
compact master pre-stacks satisfying Definition~\ref{wl:prestack}.  Before the
uniformly controlled template adjustments in fully clustered tetrahedra,
the main central disks are the normal triangles and quadrilaterals.  On
nonclustered diagrams their widths may be microscopic.  In the
equal-weight fully clustered belt they may be chosen comparable to
$s_\sigma$, with all neighboring tetrahedra included in the
equal-weight cost region.  Each main sheet triangle may be
subdivided by one common dyadic factor comparable to $H/W$, without
increasing its baseline tangential error or its inverse-parameter
estimate by that factor.
\end{lemma}

\begin{proof}
First make all nonclustered central diagrams qualitatively generic
by arbitrarily small allowed variations of edge positions.  Two
meeting main planes are transverse generically.  Containment of a
main crease in an opponent plane imposes an equality between its
endpoint fractions and the intersections of that plane with the
crossed edges, including the fourth relation if it is part of a
quadrilateral.  It is therefore avoided generically.  Two opponent
creases are disjoint generically: an interior point on one crease
prescribes, by barycentric projection onto the two skew edges of
the other, a possible endpoint pair; these pairs form a
one-dimensional null set in the two-dimensional endpoint space.
Face crossings and edge separation have the same direct generic
description.  Only finitely many strict conditions occur locally.
In fully clustered tetrahedra use instead the robust finite-list
motion from Lemma~\ref{ms:weak-templates}, equal to the identity near the
tetrahedron faces.  No such motion is needed in nonclustered
tetrahedra.

Assign two endpoint displacements along each crossed edge,
strictly ordered according to the disk's coorientation.  In a
non-robust diagram make them as small as necessary for all its
central transversality margins.  This is purely qualitative and
may require microscopic widths.  In an equal-weight fully
clustered belt use instead $c\min_{\sigma\supset e}s_\sigma$ as
the aligned displacement scale, with a fixed small $c$.  The
neighbor-ratio bound makes this comparable to the local
$s_\sigma$.  Farther out return to the ordinary weak proportional
widths.  Leave several original tetrahedron layers in the
quantitative clustered regime between every possibly microscopic
width and the eventual boundary of the snapping patch.

All central edge gaps in this region are at least $c s_\sigma$.
This follows either from the $W$-slot gaps, or from
$H/(1+N_e)\ge W/(1+N_eW/H)\ge s_\sigma$, up to the fixed cluster
factor.  Edges incident to a non-robust tetrahedron may retain
microscopic widths even if another incident tetrahedron is fully
clustered.  The robust construction allows such mixed widths.
At a mixed face the common boundary-arc direction has the same
strict crossing sign from both sides, so sufficiently small
baseline tangent errors preserve the joint pattern test.
Likewise the last stars on which $s_\sigma$ widths are used stay
inside the equal-weight budget region; their transition to
ordinary widths occurs only in the already clustered belt.
Nested positive patch margins make all these requirements
compatible after $H$ is made small.

Subdivide every main triangle by the same dyadic barycentric
factor comparable to $H/W$.  At a new vertex $P$ interpolate the
original endpoint displacement vectors affinely on its main
triangle, obtaining $d^-(P)$ and $d^+(P)$.  Use a globally
consistent ordering for the stair prisms on the fine triangles
and the two half-intervals.  On a stair simplex, choose its
duplicate vertex as origin for the tangential columns.  At fixed
layer parameter each other vertex can be compared with that
duplicate at the same layer.  The tangential correction is then
a difference of the interpolated displacements; their common
translation cancels.  Since these displacements are affine on a
main triangle, an original $O(\epsilon H)$ bound produces a
relative $O(\epsilon)$ tangential correction on every fine
triangle, independent of the subdivision factor.

The vertical column uses one fine-vertex displacement per
half-parameter width.  That vector is a convex combination of
the consistently oriented crossing-edge displacements.  Its
component against the main separating normal has the correct
sign and is at least a fixed multiple of the same convex
combination of their magnitudes.  In particular a microscopic
width does not lose its relative normal margin.  The determinant
and inverse parameter estimates of Lemma~\ref{ms:normal-templates}
therefore persist.  The endpoint sheets and their face ribbons
bound embedded shells; the same positive-degree argument fills
them.  On the subdivided cut-boundary ribbons the endpoint
secant velocities and their convex combinations obey the
response cone of Lemma~\ref{ms:boundary-collar}; again tangential
columns contain only displacement differences.  Thus the fine
subdivision is compatible also with the prescribed boundary
intervals and the ordinary weak costs elsewhere.  Strict
opponent directional cones survive, so Lemma~\ref{wl:pattern} applies
to the entire master system.
\end{proof}

\begin{lemma}[Thin axial translations]
\label{ms:translations}
Fix a low-leakage threshold $M_\sigma\le C_0$.  In regular
tetrahedra satisfying this bound, at a sufficiently large fixed
$W$-multiple of
clearance from tetrahedron faces, main slab creases, non-slab
disks, nonregular tetrahedra, and placement transitions, the master
may be changed so that its slab strips are exact axial
translations of their supporting main triangles, of half-width
$a_0=c_0W$, where $c_0>0$ is fixed sufficiently small.  The
modified system remains a valid system of master pre-stacks.  Deep in such
a portion its layer coordinate is exactly the affine conversion
of $l_P/a_0\in[-1,1]$ to $[0,W]$, where $l_P$ is signed axial
height above the supporting plane.
\end{lemma}

\begin{proof}
On these tetrahedra $W/C_0\le s_\sigma\le W$, so a displacement
of size $a_0=c_0W$ is bounded by $c_0C_0s_\sigma$.
At each fine vertex in the safe portion replace the two endpoint
displacements by $\pm a_0e_i$, with signs determined by the local
sheet coorientation.  Start the change only with the stated
clearance, so every affected triangle and possible opponent has
regular near-horizontal slab geometry.  Main triangle shapes
have fixed bounds from the slot margins; the fine sides are
comparable to $W$.  Even an abrupt vertex switch therefore adds
at most $C(\eta)c_0$ to the tangential derivative.  The old
nonclustered displacements here can be taken microscopic in
advance.  Convex combinations of old and new vertex motions
retain their strict normal components, including at unswitched
vertices.  Take $c_0$ small relative to the slot margins and the
near-axis transversality bounds.

We verify embedding through the switch.  On each affected main
triangle, projection perpendicular to its indicated axis is a
uniformly small Lipschitz perturbation of the central triangle's
projection in its abstract sheet coordinates.  It is one-to-one
and covers the common interior footprints with margin, because
the motion is small and the switch stays away from the triangle
boundary.  At a fixed projected foot, height increases strictly
with the oriented layer parameter.  Indeed the tangential
compensation needed to fix that foot has only the small central
height slope, whereas the vertical column has a definite normal
component relative to its magnitude.  This remains true even
where that magnitude is microscopic.  It holds on affine pieces
and then everywhere by continuity and integration.  All altered
images have such interior feet, while outside the change region
the original embedded geometry is unchanged.

The axial gaps of Lemma~\ref{ms:slots} keep same-axis slabs disjoint.
The full exclusion buffers keep the motion away from non-slabs,
main creases, and nonregular diagrams.  Opponents are either
disjoint in parallel mode or retain independent near-axis
normals.  The same arguments hold throughout the continuous
interpolation of the vertex motions, proving that the pattern
trivialities are preserved.  When all vertices of a fine prism
use the same axial translation, its affine stair maps interpolate
that translation exactly at fixed layer parameter.  Hence its
coordinate is precisely the stated axial-height formula.
\end{proof}
\subsection{Finite-type snapping with protected buffers}

The change of coordinates used below is directed from the new configuration
to the old one.  Its forward Lipschitz constant is uniform; its inverse
Lipschitz constant need only be finite for the particular configuration.
This distinction permits the old affine pieces to have arbitrarily small
altitudes.



We use the scales of Lemma~\ref{ms:majorant} and the auxiliary mesh of
Lemma~\ref{ms:auxiliary}.  Thus, on an original tetrahedron $\sigma$,
\[
 s_\sigma=\frac{W}{M_\sigma},\qquad M_\sigma\geq 1,
 \qquad 0<s_\sigma\leq W,
\]
and the scales on incident original tetrahedra are comparable by a fixed
factor.  Every auxiliary simplex $D$ of positive dimension has diameter
$h_D$ comparable to the relevant $s_\sigma$, has a uniformly controlled
shape after rescaling by $h_D$, and belongs to a uniformly bounded star.
All constants in this subsection may depend on these fixed comparisons.

\begin{lemma}[Bounded local complexity of the master data]
\label{ms:bounded-complexity}
Consider an equal-weight patch with $W\leq H$.  Suppose that the following
properties hold in every original tetrahedron relevant to the canonical
region.
\begin{enumerate}
\item The central normal disks of each one of the two families are
pairwise separated by at least $c s_\sigma$.  Each disk consists of a
bounded number of main triangles of diameter comparable to $H$, with
uniformly controlled shapes.  The constants are those of the fixed
endpoint margins and normal templates of
Lemma~\ref{ms:normal-templates}.
\item Each main triangle is subdivided in barycentric coordinates by a
common dyadic factor comparable to $H/W$.  Its fine triangles therefore
have diameter comparable to $W$ and uniformly controlled shapes.
The two half-prisms over a fine triangle are triangulated by the fixed
ordered stair rule.  All their vertex displacements have magnitude at
most $A s_\sigma$, for a fixed $A$.
\item Any subsequent template adjustment in an original tetrahedron is
a PL homeomorphism of that tetrahedron with Lipschitz constant and
inverse Lipschitz constant at most $L$.  It is affine on a partition
with a uniformly bounded number of convex polyhedral pieces, whose
numbers of faces are uniformly bounded.  The identity is allowed.
\end{enumerate}
Then there is a constant $N$, independent of the number of disks and
of the small angles between the two families, with the following
property.  On each auxiliary simplex, the master collar domains and
all their affine parameter pieces admit a common straight simplicial
subdivision having at most $N$ simplices.  These subdivisions can be
chosen consistently on the whole canonical subcomplex.  Each collar
domain in that subcomplex is a union of closed simplices, and its
parameter is affine on every relevant simplex, with values in $[0,W]$.
The simplices of this subdivision are nondegenerate for the fixed
configuration, but no uniform shape bound for them is asserted.
\end{lemma}

\begin{proof}
First fix an auxiliary top-dimensional simplex $D$ inside an original
tetrahedron $\sigma$, and let $\Psi$ be its template adjustment.  The
set $\Psi^{-1}(D)$ has diameter at most $L h_D$, hence at most
$C s_\sigma$.  If an adjusted prism piece meets $D$, a point of its
unadjusted prism lies in $\Psi^{-1}(D)$.  The affine prism construction
puts every such point within $A s_\sigma$ of the corresponding central
fine triangle: write the point as a convex combination of prism
vertices and omit their displacement vectors.  Consequently its
central disk meets a ball of radius $C s_\sigma$.

Choose one central-disk point in this ball for each relevant disk of
one family.  Points chosen from different disks have mutual distances
at least $c s_\sigma$.  Disjoint balls of radius $c s_\sigma/3$ about
these points fit in a ball of radius $(C+c)s_\sigma$.  Comparing their
Euclidean volumes bounds the number of relevant disks by a constant
depending only on $C/c$.  Apply this argument to both families.
There are only a bounded number of original tetrahedra to consider
when $D$ is a lower-dimensional auxiliary simplex, since the original
lattice stars are bounded and their scales are comparable.

For any one of these disks, a relevant fine triangle meets a ball of
radius $C s_\sigma\leq C W$.  Every fine triangle has diameter at most
$C_1W$ and area at least $c_1W^2$.  The interiors of the fine triangles
in one main triangle are disjoint.  The relevant fine triangles lie
in a planar ball of radius $(C+C_1)W$, so area comparison bounds their
number uniformly.  The number of main triangles is bounded as well.
Each fine triangle produces only a fixed number of stair tetrahedra.
Cutting one such tetrahedron by the fixed finite partition of $\Psi$
produces a bounded number of affine polyhedral pieces with bounded
numbers of faces.  We have therefore bounded the entire list of
affine collar pieces that can meet $D$.

For completeness, the asserted central-disk separation is precisely
the quantitative consequence of the normal-template hypotheses
needed here.  For equal disk types, the common-direction graph-order
estimate bounds the separation below by a fixed multiple of the
minimum crossed-edge gap.  For different compatible disk types, one
disk is triangular.  Every vertex of the other disk lies on the
prescribed side of its separating plane.  On a shared crossed edge
its distance to that plane is at least a fixed multiple of the edge
gap; on an uncut edge the endpoint margin gives an even larger
bound.  Convexity gives the same separation for the whole other
disk.  The gaps used here are at least a fixed multiple of
$s_\sigma$: slot gaps are comparable to $W\geq s_\sigma$, and, if an
edge has $n$ hits, the alternative margin-spaced placement has gap
at least a fixed multiple of $H/(1+n)$.  Since $W\leq H$ and
$M_\sigma\geq 1+nW/H$, one has
\[
 \frac{H}{1+n}\geq\frac{W}{1+nW/H}
 \geq\frac{W}{M_\sigma}=s_\sigma.
\]
The fixed cluster factors only alter the separation constant.
Equivalently, after adjustment the separation is retained up to the
factor $L$; the preceding packing argument used the unadjusted
configuration so that no smallness of the adjustment relative to
$s_\sigma$ was needed.

We now construct the common subdivision.  In a top-dimensional
auxiliary simplex take all supporting planes of its relevant affine
polyhedral pieces.  A bounded number of planes cuts the simplex into
a bounded number of convex polyhedra.  Every collar piece is a union
of these polyhedra, and its parameter has a single affine formula on
each of them.  On every auxiliary face, overlay the restrictions of
the subdivisions from all incident auxiliary simplices.  There are
only a bounded number of incident simplices and a bounded number of
planes from each, so all these overlays have bounded complexity.
On auxiliary edges use all induced cut points, and use the same
points from every incident face.

Triangulate each polygonal face consistently, respecting its already
subdivided boundary, by joining an interior point to its boundary
segments.  Apply the same rule to shared interior faces of the
convex polyhedra, including any subdivisions forced at their
boundaries by the auxiliary-face overlays.  Finally, join an
interior point of each full-dimensional convex polyhedron to its
triangulated boundary.  Lower-dimensional pieces are treated first,
so all choices on shared pieces agree.  Degenerate intersections
are treated in their actual dimension; an interior point is chosen
in the relative interior of each positive-dimensional piece.
Thus all resulting simplices are nondegenerate.  The bounded
numbers of planes, incidences, boundary segments, and stars give a
uniform bound for the number of resulting simplices in each
auxiliary simplex.  Taking the largest of the finitely many
dimension-dependent bounds gives $N$.
\end{proof}

\begin{lemma}[Canonical snapping and protected buffers]
\label{ms:snapping}
Let $\mathcal G$ be the finite auxiliary triangulation in the patch.
Let $\mathcal C$ and $\mathcal P$ be disjoint subcomplexes, respectively
the canonical and protected subcomplexes.  On $\mathcal C$ suppose
that a common subdivision as in
Lemma~\ref{ms:bounded-complexity} has been fixed, with at most $N$
simplices in each auxiliary simplex.  In particular, every master
collar domain is a subcomplex of that subdivision and every relevant
master parameter $u_j$ is affine there with values in $[0,W]$.

On each remaining positive-dimensional auxiliary simplex
$D\notin\mathcal C$, suppose the old master parameters already have
upper local slope at most $B W/h_D$ on their closed parameter parts.
Here the upper local slope of a function on a closed set $A$ at
$x\in A$ means
\[
 \limsup_{\substack{y\to x\\y\in A,\ y\ne x}}
 \frac{|u(y)-u(x)|}{|y-x|},
\]
with value zero at an isolated point; bounds on different incident
cells may be combined by taking their maximum.  The master
parameters are locally Lipschitz for the fixed configuration.

There is a cell-preserving PL homeomorphism $J$ of the patch, from
new coordinates to old coordinates, such that:
\begin{enumerate}
\item $J$ is the identity on $\mathcal P$;
\item $\Lip(J|_D)\leq C$ on every auxiliary simplex $D$, with $C$
depending only on $N$, the auxiliary shape bounds, and the dimension;
\item on each positive-dimensional canonical cell $D$, the transported parameter
$u_j\circ J$ has a $C W/h_D$-Lipschitz extension to all of $D$;
\item on every other positive-dimensional cell, the transported parameters have upper
local slope at most $C B W/h_D$ on their parameter parts.
\end{enumerate}
Consequently the transported upper slope density is at most
$C(1+B)W/s_\sigma=C(1+B)M_\sigma$, with incident maxima at
interfaces.  The constants do not depend on the smallest altitude
of an old subdivision simplex or on the smallest angle between
opposite master sheets.
If the outer boundary of the patch belongs to $\mathcal P$, then
$J$ extends by the identity outside the patch.  The inverse of $J$
is locally Lipschitz for the fixed configuration, without a uniform
bound being required.
\end{lemma}

\begin{proof}
We first specify the finite representatives, then give the extension
estimate used both in the canonical construction and in the buffers.

\emph{Finite marked representatives.}
For each dimension $d\leq 3$, fix a standard $d$-simplex $\Delta_d$.
The type of a subdivision records its abstract simplicial complex,
the ordering of the vertices of its coarse simplex, and, for every
subdivision simplex, the smallest coarse face containing it.
Thus all coarse corners and faces are marked.  A bound on the number
of subdivision simplices bounds the number of vertices.  There are
only finitely many such abstract marked types.

Retain only the types that have a straight nondegenerate realization
as a subdivision of a simplex with these markings.  For each retained
type choose one such realization on $\Delta_d$, once and for all.
Existence follows by taking any realization of that type and applying
the affine map of its coarse simplex to $\Delta_d$.  There is no
claim that these representatives are optimally shaped.  Each chosen
representative has finitely many nondegenerate simplices, and the
list of representatives is finite.  Hence the inverses of all its
simplex edge matrices have a finite common bound.  This fixed bound
is the only representative-shape bound needed below.

For a canonical auxiliary simplex $D$, let
\[
 R_D:D\longrightarrow\Delta_d
\]
be the simplicial map carrying its old subdivision to its chosen
marked representative.  This is a PL homeomorphism and preserves
every marked coarse face.  For positive-dimensional $D$, its inverse has the bound
\[
 \Lip(R_D^{-1})\leq C h_D.
\]
Indeed, on a representative $d$-simplex with vertices
$a_0,\ldots,a_d$, whose corresponding old vertices are
$v_0,\ldots,v_d$, the derivative of $R_D^{-1}$, in orthonormal
coordinates on the affine spans, is
\[
 [v_1-v_0\ \cdots\ v_d-v_0]
 [a_1-a_0\ \cdots\ a_d-a_0]^{-1}.
\]
The first matrix has norm at most $C_d h_D$, since its vertices lie
in $D$.  The second matrix belongs to the finite representative
list.  This proves the bound on each affine piece, and hence on
the convex simplex $\Delta_d$ by subdividing a line segment at the
finitely many faces it meets.  No inverse edge matrix of an old
subdivision simplex appears in this estimate.

\emph{An explicit coning estimate.}
Let $D_0,E_0$ be convex simplices, with interior centers $c,c'$,
respectively.  Suppose
\[
 B(c,r)\subset D_0\subset B(c,R),\qquad
 B(c',r')\subset E_0\subset B(c',R').
\]
Let $g:\partial D_0\to\partial E_0$ be a PL homeomorphism that is
$L_0$-Lipschitz for the ambient Euclidean distances.  Every
$x\ne c$ has a unique expression
$x=c+t(z-c)$, with $z\in\partial D_0$ and $0<t\leq 1$.
Define
\[
 C_g(c)=c',\qquad
 C_g\bigl(c+t(z-c)\bigr)=c'+t\bigl(g(z)-c'\bigr).
\]
This is a homeomorphism, whose inverse is the same construction
with $g^{-1}$, and it is PL after coning a boundary triangulation
on which $g$ is affine.

To estimate it, write
$x=c+t(z-c)$ and $y=c+s(w-c)$, with $t\geq s$.
The radial gauge $p(x-c)=t$ is the maximum of the normalized
linear forms defining the facets of $D_0$.  Their gradients have
norm at most $1/r$, so $|t-s|\leq |x-y|/r$.  Moreover,
\[
 t|z-w|\leq |x-y|+(t-s)|w-c|
 \leq (1+R/r)|x-y|.
\]
It follows that
\[
 \Lip(C_g)\leq L_0(1+R/r)+R'/r.                 \tag{*}
\]
The case $y=c$ follows directly from the same estimate or by
continuity.  If $g^{-1}$ has Lipschitz constant $L_0'$, the inverse
cone similarly satisfies
\[
 \Lip(C_g^{-1})\leq L_0'(1+R'/r')+R/r'.
\]
In particular, the forward estimate in $(*)$ requires no bound on
$\Lip(g^{-1})$.

We will also use the fact that compatible Lipschitz bounds on the
facets give an ambient Euclidean Lipschitz bound on the whole
boundary, with a constant depending only on the coarse simplex
shape.  Here is a direct verification.  Let $a$ be its diameter and
$h$ its smallest altitude.  If $x,y$ belong to a common facet,
use the segment joining them.  Otherwise choose facets $F_i,F_j$
containing $x,y$, with $i\ne j$, and choose $k\ne i,j$.
With vertices $v_i$ and barycentric coordinates $\lambda_i$, put
\[
 z_x=x+\lambda_j(x)(v_k-v_j),\qquad
 z_y=y+\lambda_i(y)(v_k-v_i).
\]
Both points lie in $F_i\cap F_j$.  Since
$\lambda_j(y)=\lambda_i(x)=0$ and each barycentric coordinate has
Lipschitz constant at most $1/h$,
\[
 |x-z_x|+|y-z_y|\leq 2(a/h)|x-y|.
\]
The three boundary segments from $x$ to $z_x$, then to $z_y$,
then to $y$ have total length at most
$(1+4a/h)|x-y|$.  Sum the facetwise Lipschitz estimates along this
path.  This proves the assertion in dimensions at least two;
for a one-dimensional simplex its two boundary points are handled
directly.  Applying the same argument to the inverse boundary map
gives the corresponding two-sided assertion when every facet is
mapped to its prescribed facet.

\emph{Canonical face corrections.}
We construct $J_D$ on all canonical simplices by induction on
dimension, in the form
\[
 J_D=R_D^{-1}\circ K_D,
\]
where $K_D:D\to\Delta_d$ is PL and, for $d\geq 1$, satisfies
\[
 \Lip(K_D)\leq C/h_D,
 \qquad \Lip(K_D^{-1})\leq C h_D.
\]
These are uniform two-sided estimates in normalized cell units.
At vertices the map $J_D$ is the identity.  For an edge, prescribe
the marked endpoint values and use the affine map $K_D$ to its
standard interval.

Suppose now that the maps have been constructed on all proper
faces of a canonical simplex $D$.  On a facet $F$ prescribe
\[
 K_D|_F
   =R_D|_F\circ J_F
   =\bigl(R_D|_F\circ R_F^{-1}\bigr)\circ K_F.       \tag{**}
\]
The parenthesized map is a simplicial comparison between the
chosen representative of the face type and the face subdivision
induced by the chosen representative of the type of $D$.
Both are fixed realizations of the same marked old face
combinatorics.  There are only finitely many possible comparisons,
including the finitely many vertex-identification choices, and
each comparison is bi-Lipschitz.  Their two-sided constants are
therefore uniformly bounded.  Uniform shape control gives
$h_F\asymp h_D$, so the induction hypothesis in $(**)$ gives the
required normalized two-sided bounds on every facet.

The facet prescriptions agree on their intersections: on any
smaller face they equal $R_D\circ J$ with the already constructed
map of that face.  They therefore form a PL boundary homeomorphism
onto $\partial\Delta_d$, mapping each facet to its marked facet.
The preceding boundary estimate supplies uniform two-sided bounds
on the whole boundary.  Cone from the barycenter of $D$ to the
barycenter of $\Delta_d$.  The coning estimates give the asserted
bounds for $K_D$; only the dimension and normalized shape constants
enter.  Formula $(**)$ also proves
$J_D|_F=J_F$, as required for gluing.

The induction has at most three levels.  Thus the constants depend
only on the fixed representative list and the coarse shape bounds,
not on the shapes of any old subdivision simplices.  Combining
the estimates for $R_D^{-1}$ and $K_D$ gives
\[
 \Lip(J_D)\leq (C h_D)(C/h_D)\leq C.
\]

\emph{Parameter estimate on canonical cells.}
Fix a positive-dimensional canonical cell $D$ and a parameter $u_j$
on its old closed domain in $D$.  On the
representative triangulation its pullback $u_j\circ R_D^{-1}$
is affine on the relevant subcomplex.  Give every representative
vertex outside that subcomplex the value zero, retain the
prescribed values at all vertices in the subcomplex, and extend
affinely over every representative simplex.  This defines a
continuous PL function $\widetilde u_{j,D}$ on all of $\Delta_d$.
It agrees with $u_j\circ R_D^{-1}$ on its domain and all its vertex
values lie in $[0,W]$.

The fixed inverse edge-matrix bounds for the representative
simplices imply
$\Lip(\widetilde u_{j,D})\leq C W$ on the convex standard simplex.
Consequently
\[
 \widetilde u_{j,D}\circ K_D
\]
is a $C W/h_D$-Lipschitz extension to $D$ of the transported
parameter $u_j\circ J_D$.  This argument applies separately to
each label.  It does not require the numerical vertex labels to
belong to a finite set; only their common range $[0,W]$ is used.

\emph{Protected cells and buffer cells.}
Set $J$ equal to the identity on the protected subcomplex.
There is no conflict with the canonical construction because
$\mathcal C\cap\mathcal P=\varnothing$.  Extend over all remaining
auxiliary vertices by the identity, and then over the remaining
positive-dimensional simplices by increasing dimension.  On a remaining
simplex $D$, its proper faces already carry compatible
cell-preserving PL homeomorphisms with uniformly bounded forward
Lipschitz constants.  They give a boundary homeomorphism of $D$,
again with a uniform forward constant by the boundary estimate.
Cone this map from the barycenter of $D$ to itself.  The forward
bound in $(*)$ gives $\Lip(J_D)\leq C$, regardless of the inverse
Lipschitz constants on its boundary.  Since the dimension is
bounded, iterating this step preserves a uniform forward bound.
No regular subdivision of a protected trace is needed for this
step: its map is the identity, and only the forward bound of the
prescribed boundary map enters the cone estimate.

Let $A_j$ be an old parameter domain and put
$A_j^{\rm new}=J^{-1}(A_j)$.  The map is cell-preserving, so on a
buffer cell $D$ it maps $A_j^{\rm new}\cap D$ into $A_j\cap D$.
The elementary composition inequality for upper local slopes,
together with $\Lip(J_D)\leq C$, yields
\[
 \operatorname{lip}_{A_j^{\rm new}\cap D}(u_j\circ J)(x)
 \leq C\,\operatorname{lip}_{A_j\cap D}u_j(J(x))
 \leq C B W/h_D.
\]
On a protected cell the same statement holds with $J$ the identity.
At an interface, every sequence approaching a point lies, after
passing to a subsequence, in one of its finitely many incident
closed auxiliary cells.  Taking the maximum of their bounds gives
the asserted upper local slope on the whole parameter domain.
Scale comparability changes $W/h_D$ to at most $C W/s_\sigma$.

All the cell maps agree on common faces and are homeomorphisms
preserving each cell.  They therefore glue to a PL homeomorphism
of the finite patch.  Each affine piece is nondegenerate for the
fixed input, so the inverse is locally Lipschitz for that input;
its constant may deteriorate with old sliver degeneracy.  If the
outer boundary is protected, the identity extension agrees there
and gives the asserted ambient change of coordinates.  Pulling
the collar system back by $J$ preserves its intersections, labels,
and parameter trivializations exactly.  A previously given
locally bi-Lipschitz trivialization remains such after composition
with $J^{-1}$, since only a finite, input-dependent inverse bound
is needed for that qualitative statement.
\end{proof}

Figure~\ref{ms:fig-snapping} summarizes the directions of the maps
in the preceding proof and distinguishes their uniform forward
bounds from the input-dependent inverse bound.

\begin{figure}[htbp]
\centering
\begin{tikzpicture}[
  box/.style={draw,rounded corners,align=center,minimum width=3.1cm,
              minimum height=1.1cm,font=\small},
  bounded/.style={-{Stealth[length=2mm]},thick,blue!65!black},
  qualitative/.style={-{Stealth[length=2mm]},dashed,gray!75!black},
  every edge quotes/.style={font=\small}]
\node[box] (new) {new cell $D$\\parameter $u\circ J_D$};
\node[box,right=1.2cm of new] (rep) {fixed representative\\$\Delta_d$};
\node[box,right=1.2cm of rep] (old) {old cell $D$\\parameter $u$};
\draw[bounded] (new) -- node[above,font=\small] {$K_D$} (rep);
\draw[bounded] (rep) -- node[above,font=\small] {$R_D^{-1}$} (old);
\draw[qualitative] (old.south) to[bend left=28]
 node[below,font=\small,align=center] {$J_D^{-1}$: finite inverse bound\\for fixed data only} (new.south);
\node[below=2.45cm of rep,font=\small,align=center]
 {solid: uniform upper Lipschitz bound\qquad dashed: qualitative bound};
\end{tikzpicture}
\caption{The direction of canonical snapping, in normalized cell units.
Both solid arrows have uniform upper Lipschitz bounds; $K_D$ also has
a uniform inverse bound.  No uniform bound is required for
$R_D$ or $J_D^{-1}$.  The new parameter is the old parameter composed
with $J_D=R_D^{-1}K_D$.  The diagram records map directions and bounds,
not the shapes of the triangulations.}
\label{ms:fig-snapping}
\end{figure}

The disjoint subcomplexes required in
Lemma~\ref{ms:snapping} are selected using the already quantitative
belts of the master construction.  Put in the canonical complex
every closed auxiliary cell on which a microscopic master width
can cause an uncontrolled parameter slope, together with all its
faces.  Select the protected cells strictly inside the fully
translated regular region, and at the outer edge of the snap patch
strictly inside the quantitative clustered belt.  Leave several
auxiliary cell layers of the same quantitative regime between
these protected cells and the uncontrolled cells.  Equivalently,
trim the initial protected regions by a fixed number of closed
auxiliary stars before selecting their closed cell subcomplexes.
Since every auxiliary cell has diameter at most $C W$, this only
enlarges the prescribed geometric buffers by a fixed multiple of
$W$.  The two closed subcomplexes are then disjoint, and every
remaining buffer cell has the master slope bound used in the
Lemma.  At the outer transition the canonical complex is kept
inside the region where displacements are at most $C s_\sigma$;
any return to larger ordinary weak widths takes place in the
quantitative cells outside it.  Thus the bounded-complexity
hypothesis is used exactly where it is available.

The separation from protected cells is essential to the stated
proof.  On canonical faces, the uniformly controlled correction
is obtained by comparing two finite representatives as in $(**)$.
The identity is prescribed only on the disjoint protected
subcomplex; the intervening cells use the forward coning estimate
and the old quantitative parameter bound.  No uniform inverse
estimate is imposed on those buffer maps.

Figure~\ref{ms:fig-protected-buffers} shows how the quantitative buffer
separates the canonical construction from an unchanged translated
region.  Broadening will take place farther inside that protected
region.

\begin{figure}[htbp]
\centering
\begin{tikzpicture}[x=1cm,y=1cm,font=\small]
\filldraw[fill=blue!10,draw=blue!65!black] (0,0) rectangle (4,1.25);
\filldraw[fill=gray!12,draw=gray!65!black] (4,0) rectangle (6.7,1.25);
\filldraw[fill=green!8,draw=green!45!black] (6.7,0) rectangle (12.5,1.25);
\node[align=center,text width=3.7cm] at (2,.625)
  {canonical cells $\mathcal C$\\possibly thin strips};
\node[align=center,text width=2.4cm] at (5.35,.625)
  {buffer cells\\bounded slopes};
\node[align=center,text width=5.5cm] at (9.6,.625)
  {protected translated cells $\mathcal P$\\axial strip formulas unchanged};
\draw[decorate,decoration={brace,mirror,amplitude=4pt}]
  (0,-.12) -- (6.65,-.12)
  node[midway,below=6pt,align=center] {snapping may change coordinates};
\draw[decorate,decoration={brace,mirror,amplitude=4pt}]
  (6.75,-.12) -- (12.5,-.12)
  node[midway,below=6pt] {$J=\Id$};
\draw[very thick,green!45!black] (8,1.48) -- (11.5,1.48);
\node[above,align=center] at (9.75,1.48)
  {support of the later broadening};
\end{tikzpicture}
\caption{A schematic local passage from canonical cells to an interior
protected region.  The buffer already has quantitative parameter bounds,
so only the forward Lipschitz bound for its snapping map is needed.
Snapping fixes the protected region; broadening is supported strictly
inside it.  The bands depict adjacency, not metric widths or the global
topology of the subcomplexes.  Protected cells at the outer patch boundary
instead lie in the quantitative clustered belt.}
\label{ms:fig-protected-buffers}
\end{figure}

\subsection{Broadening the protected strips}

\begin{lemma}[Simultaneous broadening by an ambient flow]
\label{ms:broadening}
On the deep protected translation regions of
Lemma~\ref{ms:translations}, after snapping, the half-width can be
increased from $a_0=c_0W$ to
\begin{equation}
 a_1=(1-C\eta)W/2,
 \label{ms:wide-halfwidth}
\end{equation}
with fixed slack below half the regular slab gap.  It equals $a_1$
outside larger excluded-set buffers and equals $a_0$ near the edge
of the protected region.  The change is induced by a common
ambient locally bi-Lipschitz homeomorphism preserving every layer
parameter.  The resulting pre-slope density is bounded by
$CM_\sigma$ throughout the sharp construction.  On a region of
constant width $a_1$,
\begin{equation}
 \nabla u_j=\epsilon_j e_i+O(\eta),\qquad
 |\nabla u_j|\le1+C\eta,
 \qquad \epsilon_j\in\{-1,1\},
 \label{ms:sharp-scalar}
\end{equation}
where the error is in aligned coordinates and may be made smaller
than any prescribed multiple of $\eta$ except for the fixed
slot-loss contribution in the principal component.
\end{lemma}

\begin{proof}
Choose a common smooth cutoff supported well inside the unchanged
translation subcomplexes, equal to one beyond larger excluded-set
buffers.  Mollified distance cutoffs give derivative
$O(1/(NW))$ when the buffer width is $NW$.  First take the fixed
integer $N$ sufficiently large, depending on the slot margins and
the previously fixed constants.  Let $a(x)$ vary between $a_0$
and $a_1$ using this cutoff, so $|\nabla a|\le C/N$.

On a supporting main triangle $P$, let $l_P(x)$ be signed axial
height above its plane, with sign chosen according to increasing
layer parameter.  Use the strip $|l_P|\le a(x)$ and the layer
parameter given by affine conversion of $l_P/a$:
\begin{equation}
 u_P(x)=\frac W2\left(1+\frac{l_P(x)}{a(x)}\right).
 \label{ms:wide-parameter}
\end{equation}
The support of the change is far enough from the main triangle
boundary, from all unswitched fine triangles, and from all
non-slab, crease, face, and nonregular exclusions that every
supporting-plane formula required within several multiples of
$W$ is valid.  Same-family gaps from Lemma~\ref{ms:slots}, including
the opposite parallel band, stay larger than the two half-widths
with fixed slack.  The slice slopes are small because the
supporting planes are near horizontal and $|\nabla a|$ is small.
In independent mode the only possible concurrent strips have
near-independent coordinate normals.  These facts remain true
on a small open enlargement of each interval
$l_P/a\in[-1,1]$.

We verify that this simultaneous change preserves the parameter
topology.  Interpolate smoothly in time from $a_0$ to $a(x)$,
writing $a_\tau(x)$, $0\le\tau\le1$, with the same spatial
cutoff.  On each relevant plane neighborhood impose
\begin{equation}
 D_x(l_P/a_\tau)X_\tau
       +\partial_\tau(l_P/a_\tau)=0.
 \label{ms:material}
\end{equation}
Locally the list of possible constraints contains one member or
an independent pair.  Their gradients have the same independence
as the near-axis normals, since
\[
 D_x(l_P/a_\tau)
   =a_\tau^{-1}
       \left(\nabla l_P-(l_P/a_\tau)\nabla a_\tau\right).
\]
Thus the linear equations have a smooth local solution.  For
example use the right inverse of the one-row or two-row gradient
matrix.  The right sides are linear in $\partial_\tau a_\tau$,
and this choice has size $O(W)$ on the necessary open strip
neighborhoods.  Take a finite space-time cover such that every
neighborhood's list includes all constraints potentially present
there.  A smooth partition of unity preserves the linear
equations on their required sets.  Include the complementary
region with zero field, and arrange zero wherever
$\partial_\tau a_\tau=0$; the local choices are linear in that
quantity.  The field is compactly supported in the finite patch.

Its flow exists on the whole time interval.  Increase the buffer
factor if necessary so its $O(W)$ motion stays within all the
supporting-plane and endpoint extension margins.  Equation
\eqref{ms:material} makes $l_P/a_\tau$ constant along the flow.
The backward flow gives the converse, so all strip boundaries and
all layer parameters are transported exactly.  In particular this
is a common ambient change of both systems.  It leaves the
nonflat seams unchanged and preserves the joint product
trivializations and the PL-in-changed-coordinates property.
No uniform derivative bound for the flow or its inverse is needed.

The quantitative bound is obtained directly from
\eqref{ms:wide-parameter}, not by differentiating that flow:
\[
 \nabla u_P
   =\frac W{2a}\left(\nabla l_P-(l_P/a)\nabla a\right).
\]
Since $a\ge c_0W$, $|l_P|\le a$, and the plane slopes and
$|\nabla a|$ are bounded, this is bounded by a fixed constant.
Broadening occurs only on low-leakage tetrahedra, where
$s_\sigma$ is comparable to $W$.  Elsewhere the snapped bound is
$CM_\sigma$.  Thus this bound holds everywhere.  When $a=a_1$,
$\nabla a=0$, $\nabla l_P=\epsilon_je_i+O_\eta(\delta')$, and
$W/(2a_1)=1+O(\eta)$.  Taking the remaining tolerances as in
Lemma~\ref{ms:slots} proves \eqref{ms:sharp-scalar}.
\end{proof}

\begin{proposition}[Sharp-island geometric contract]
\label{ms:sharp-contract}
Fix $\eta$, the compact chart and basis bounds, and the finite
template choices above.  The calibrated islands admit pre-stacks
with all the qualitative properties of Definition~\ref{wl:prestack} and
with the following quantitative properties.
\begin{enumerate}
\item Throughout an enlarged equal-weight island the pre-slope
density is at most $CM_\sigma$ in aligned units.  This constant
is independent of microscopic master widths, central crossing
angles, final guard gaps, $H$, $W$, and the final level-count
tolerance $\delta'$.
\item In deep regular low-leakage portions, the scalar gradients
obey \eqref{ms:sharp-scalar}.  At concurrent independent strips
the two-coordinate squared Frobenius cost is at most $2+C\eta$.
At an occupied diagonal transition after routing, the active
coordinate has gradient $\pm\nabla u_j\pm\nabla u_k$, whose
squared norm has the same bound.  In parallel mode the opponent
bands are disjoint in these portions and the squared norm is at
most $1+C\eta$.  Occupied-stair speeds multiply these bounds by
at most $(1+C\zeta)^2$.
\item In a regular tetrahedron the excluded $CNW$ neighborhoods
of its faces, all its main slab creases, and its non-slab disks
occupy relative volume at most
\begin{equation}
 C_N\left(\frac WH+\delta'\right).
 \label{ms:dirty-volume}
\end{equation}
The canonical protection and broadening margins only change the
fixed factor $C_N$.  Nonregular tetrahedra, high-leakage
tetrahedra, their required fixed-neighbor layers, and patch
boundary margins are charged by their full $M^p$-cost.
\end{enumerate}
The norm statements concern the smooth compatible label sectors
after the combined label barrier, away from central radial
activation cutoffs.  They are geometric scalar-pair estimates;
their physical conversion and the saturation argument are made
in Lemma~\ref{ha:saturation}.
\end{proposition}

\begin{proof}
The master construction, the translation switch, the canonical
snapping, and the common broadening flow preserve the entire
two-colour pattern and its parameters, as proved above.  In the
canonical region all displacements are at most $Cs_\sigma$.
In particular the larger translation displacement $a_0$ is used
only when $s_\sigma$ is comparable to $W$.  Thus
Lemma~\ref{ms:bounded-complexity} applies there, and
Lemma~\ref{ms:snapping} gives the first assertion in the formerly
uncontrolled region.  Its buffer cells use only a uniform upper
Lipschitz bound for the map from new to old coordinates and an
already quantitative master bound.  Ordinary weak widths may
be larger outside the canonical region, but their transition is
fully clustered and has the same $CM_\sigma$ bound in the
equal-weight belt, or the ordinary edge bound outside it.
Finally Lemma~\ref{ms:broadening} supplies the remaining transition and
deep-strip estimates.  The finite representatives, slot margins,
and chart factors were all fixed before the small count errors,
so no new dependence on those errors has entered.

In independent mode the two indicated axes are orthogonal.
Equation \eqref{ms:sharp-scalar} gives squared Frobenius norm at
most $2+C\eta$ when both coordinates contribute.  A routed
diagonal has parameter $d=w_j-u_j+u_k$, with possible
coorientation changes.  Hence its gradient is the asserted
signed sum of two near-orthogonal covectors, with the same bound.
Only one output coordinate is active on that transition.  In
parallel mode the two bands are separated by the axial gap, so
they have no such overlap in the protected region.  The
occupied-interval stair conversion has speed at most
$1+C\zeta$, giving the stated factor.  These statements use the
combined label pinning to identify the intended smooth sector;
they assert no estimate across an unrelated sector or an active
central cutoff.

For the volume estimate, there are $O(H/W)$ slab disks, each
with at most one main crease of length $O(H)$.  Their
$CNW$-tubes cost at most $C_NH^2W$.  There are at most
$C\delta'H/W$ non-slabs, each of area $O(H^2)$; thickening
their finitely many main triangles by $CNW$ costs at most
$C_N\delta'H^3$ (the smaller edge and vertex terms are absorbed
when $W/H$ is small).  The tetrahedron-face buffers cost
$C_NH^2W$.  Divide by volume comparable to $H^3$ to obtain
\eqref{ms:dirty-volume}.  Auxiliary cells have size at most
$CW$, so trimming protected subcomplexes and reserving the
flow neighborhoods only enlarge this fixed buffer factor.
For nonregular and high-leakage sets use
Lemma~\ref{ms:majorant}; the appropriate analytic error tests in
Lemma~\ref{ha:sharp-counts} make their full costs small.  No small
volume assertion for those uncontrolled tetrahedra is being
used in place of that $M^p$ payment.
\end{proof}

\subsection{The pre-stack contract and the order of constants}

\begin{proposition}[Geometric realization of pre-stacks]
\label{ms:prestacks}
Suppose the central starting systems have the boundary data of
Lemma~\ref{ms:boundary-collar} and the wall-by-wall edge upper counts
used in the separate-colour normalization of
Proposition~\ref{wl:normalization}.  Suppose also that the weighted edge tests
have been selected as in Lemma~\ref{ms:edge-tests}.  The constructions
of Sections~\ref{sec:mesh} and~\ref{sec:calibrated-islands} give the
pre-stacks required in
Definition~\ref{wl:prestack}: same-colour compact disjoint collars; opposite
arc or circle products with the full parameter rectangles;
individual pattern trivializations preserving the opposite
parameter; the exact boundary correspondence; and locally
bi-Lipschitz changed coordinates in which the relevant endpoint
arrangements and parameters are PL.

Their pre-parameter slopes admit a density $G$ satisfying the
ordinary bound \eqref{ms:weak-G}, the fixed boundary and chart
bounds of Lemma~\ref{ms:boundary-collar}, and the island bound
$G\le CM_\sigma$ of Proposition~\ref{ms:sharp-contract}.  All constants in
these inequalities may be fixed before choosing final guard
gaps, mesh resolutions, and sample weights.  Consequently any
crude compact-local integral capacities deduced from these
inequalities and the conditional edge-variation bounds are
independent of those later choices.  The actual simultaneous
choice of the capacities, meshes, and samples is supplied by
Lemma~\ref{ha:capacities} and Proposition~\ref{ha:initial-walls}.
\end{proposition}

\begin{proof}
Normalize one colour at a time, keeping its specified boundary
traces and without increasing any individual wall--edge count.  For this
conclusion only the resulting normal data
are needed; an ambient normalization isotopy need not fix the
other colour.  The weak templates give the claimed collar and
pattern properties away from the calibrated islands.  On the
islands the master construction has the same properties and
the subsequent common ambient changes preserve them exactly.
All attachments to the outer collar use its joint trivialization.
In particular no step merely chooses independent individual
product charts and assumes they preserve the other coordinate.
Compactness of each individual sampled wall and local
finiteness of the family give the required qualitative local
bi-Lipschitz statements for each fixed construction.

We make the constant order explicit.  First fix coarse working
charts, compact basis bounds, feature lengths and aspect bounds,
the slot loss $\eta$, and the weak boundary-jitter slack.  Fix
the resulting finite weak template list, its angle tolerances,
and the boundary compression $\varepsilon$.  These fix the
normal-template shapes, the auxiliary neighbor-ratio bounds,
the cardinality of the canonical arrangements, all finite
representatives in Lemma~\ref{ms:snapping}, and the translation and
broadening buffer factors.  On a given compact their constants
are finite, though they need not be small.  Next choose the
level-edge tolerance and generic perturbation fractions small
against those fixed bounds; in calibrated patches fix also the
chart and ambient-jitter error fractions.  Their possibly large
density constants are then known.  The analytic mean errors
are chosen smaller afterward.  None of these choices involves
the eventual guard gaps.

Fix the local edge-variation and density bounds, including any
chart weights, before the guard nets are set.  They give finite
candidate capacities for $\int G^p$ by
\eqref{ms:weak-G}, \eqref{ms:majorant-power}, and the fixed
boundary terms.  Having chosen the guards and their forbidden
label neighborhoods, take $d_0$, the mesh, and every displacement
fine enough for the associated spatial buffers.  Boundary warp
motion has size $O(\delta)$ with $\delta\ll d_0$; its inverse
label constants do not deteriorate under this thinning.  The
interior meshes keep their shape lists under refinement and
their fixed jitter density factors.  Sample last, taking all
feature slots sufficiently fine and, in the finitely many
equal-weight islands, $W/H$ as small as necessary.  Arbitrarily
small allowed feature components do not obstruct this final
refinement.  Whole-interval and occupied-interval stair errors
charge absolute gap lengths and rounding errors, so they may
also honor the already prescribed guard and label accuracies.

Microscopic master widths introduce no additional capacity
constant.  They are removed precisely on the canonical cells
by a map with uniformly bounded upper Lipschitz constant, while
the direct representative-label estimate is $CW/s_\sigma$.
Subdivision uses displacement differences and introduces no
$H/W$ loss.  The final broadening estimate comes directly
from \eqref{ms:wide-parameter}.  The qualitative inverse bounds
of the common coordinate changes, the pattern trivializations,
and the eventual product or ball extensions in
Theorem~\ref{wl:realization} are never substituted into these estimates.

For path estimates take upper slopes relative to the closed
parameter domains, as defined in Lemma~\ref{ms:snapping}, with
adjacent-cell maxima.  The fixed-input parameters are locally
Lipschitz on their compact collars.  On exceptional
lower-dimensional interface strata one may enlarge $G$ further
to any necessary actual local collar Lipschitz bound, with
locally finite prescriptions.  Such strata have zero ambient
volume.  This gives the pointwise control needed for
one-dimensional coarea on restricted path portions without
asserting a uniform two-point Lipschitz bound across arbitrary
gaps between parameter pieces.  It changes none of the stated
integral capacities.
\end{proof}
==== END Strong-diffeomorphic-approximation-in-three-dimensions-for-p-gt-2//sections/08-mesh.tex ====

==== BEGIN Strong-diffeomorphic-approximation-in-three-dimensions-for-p-gt-2//sections/09-high-assembly.tex ====
\section{Assembly for exponents greater than two}
\label{ha:section}\label{sec:high-assembly}

We now choose the data in the carrier, quantizer, wall, and mesh
constructions simultaneously.  Throughout this section, put
\[
 \Lambda=\Omega',\qquad E=\int_\Omega |Df|^p,\qquad p>2.
\]
Matrix norms without a subscript are Frobenius norms.  All local
finiteness statements refer to the open source or target, rather than
to their closures.

Our objective is to prove Theorem~\ref{main:theorem} by constructing
an approximant with any prescribed positive error in
Equation~\eqref{main:convergence}.  The construction first gives a
locally bi-Lipschitz homeomorphism.  The last use of
Theorem~\ref{sm:smoothing} gives the smooth map.
We keep separate the constants allowed to multiply an
initial energy tail and those allowed only in later absolute-error
prescriptions.  This distinction is essential: the constants of an
individual topological normalization or an individual chart need not
be bounded in terms of $p$.

Two kinds of estimates enter the proof.  Ordinary meshes give the
\emph{weak estimate}: an energy bound in terms of the carrier energy.
Around the retained rank-one and rank-two data, calibrated meshes give
the \emph{sharp estimate}: an energy bound close to that of the affine
comparison.  Label pinning controls the mean gradient there, so a
quantitative uniform-convexity argument yields strong gradient accuracy;
this is the classical energy-to-strong-convergence principle associated
with Clarkson~\cite{Clarkson1936}.  The precise pointwise estimate used
here is proved in Lemma~\ref{ha:saturation}.  The complete order of choices
is collected in Remark~\ref{ha:scale-order}.

The estimates have four spatial roles.  The sharp estimate recovers
the derivative on small source patches containing almost all the
retained rank-one and rank-two energy.  Reserved full-rank cubes
will instead receive their affine comparison maps at the end.
When $2<p<3$, marked interiors are also set aside, for a later
source compression.  On the remaining region, and on the shells
and collars needed to make these replacements, the weak estimate
must give small energy.  These buffered regions can overlap;
their separate bounds will be imposed simultaneously.  We first
reserve the full-rank cubes and the carrier budgets, then choose
the sharp patches and realize the wall parameters that recover
their gradients.

\subsection{Initial parameters and carrier budgets}
\label{ha:initial}

Choose a small aspect $\gamma>0$ for the protected rank-one rectangles;
use squares for rank two.  Choose a small slot-loss parameter $\eta>0$,
and then $\zeta,c_*>0$ sufficiently small compared with $\eta$.
Here $\zeta$ controls occupied root length and sampling density, and
$c_*$ is the ratio of a true target interval length to its root weight.
The ordinary weak constant, denoted $C_{\mathrm w}$, may depend on
\[
                       p,\gamma,\eta,\zeta,c_* ,
\]
but it is independent of all later jet truncations, chart bounds,
mesh resolutions, and guard spacings.  The physical bulk meshes and
clustered templates of Proposition~\ref{ms:prestacks} supply precisely
this kind of constant.  Protected rectangle aspects are included in
$C_{\mathrm w}$.  We will prove a leading sharp error bound
\begin{equation}\label{ha:leading-error}
 C_p(\eta+\gamma)(1+E)+\text{freely small errors},
\end{equation}
where $C_p$ is independent of $\gamma,\eta$ and of the finite jet
bounds.  Thus $\gamma$ and $\eta$ can be fixed at the outset to make
the first term as small as required.

In what follows, a freely small error is prescribed only after every
factor that will multiply it has been fixed.  On noncompact parts we
use a locally finite compact cover and positive prescriptions with
sum as small as required.  A finite number of additional local weights
can be included in these prescriptions by dividing the allowed error
by their maximum.  Equivalently, for a countable collection of local
costs, prescribe the $j$th weighted cost to be less than
$\delta 2^{-j}$, after its weights are known.  Positive tolerance minorants and simultaneous locally summable
choices are supplied by Lemma~\ref{pre:local-tolerances}.  This
convention never replaces the initial choice of tails against
$C_{\mathrm w}$.

By Lemma~\ref{hp:regularity}, we may choose compact sets of regular
rank-one and rank-two points covering those ranks except for an
arbitrarily small tail in the measure
\[
                         d\mu=(1+|Df|^p)\,dx.
\]
First make this tail small compared with the already fixed weak
constant.  On the selected sets the positive singular values can be
bounded above and bounded away from zero.  Separately choose a compact
set of full-rank points with finite jet and inverse-jet bounds,
leaving an equally affordable tail in $\int |Df|^p$ on that rank.
The full-rank points and the deficient compact data are separated.

Use Proposition~\ref{hp:full-rank-correction} to reserve finitely many
full-rank patches.  Write $B_i$ for their inner cubes and $b_i$ for
the positive affine comparison jets.  The enlarged support and
comparison cubes are disjoint across distinct patches.
In normalized cube coordinates,
let $t_i=0$ on $\partial B_i$ and let $s>0$ be the cut-band parameter.
The finite jet factors are fixed before $s$ is made small.  For any
fixed large $C'$, all bands
\begin{equation}\label{ha:full-bands}
                         |t_i|\le C's
\end{equation}
can consequently have arbitrarily small total $|Df|^p$ integral,
even with the jet factors and $C_{\mathrm w}$ included.  The constant
$C'$ includes the finitely many buffer enlargements used below.
The power-mean and normalized value tests at the full-rank points
are then imposed on sufficiently small outer cubes.  Vitali selection
and finite truncation give coverage by the inner cubes, and even by
$\{t_i<-3s\}$, up to the required full-rank tail.  We retain
\begin{equation}\label{ha:affine-full-tests}
 \sum_i\int_{B_i}|Df-Db_i|^p\ \text{as small as required},
 \qquad
 |b_i^{-1}f-\mathrm{Id}|\ll cs
 \quad\text{in normalized cube units}.
\end{equation}
The cube sizes also make the eventual value cost of these corrections
arbitrarily small.  Subsequent value approximations will preserve
the second test with room.

Before fixing the flattening and endpoint data, choose a positive
continuous source tolerance $a$ small enough for the desired value
error and the finite normalized full-rank tests, reserving room for
all later changes.  In Proposition~\ref{hp:flattening} use a target
tolerance $a_\Lambda(y)\le a(f^{-1}(y))/4$.  Then
$|F_\kappa-f|<a/4$ for every $0\le\kappa\le1$.  After this share
has been reserved, the two carrier rounds use the remaining
allowance in $a$, as in Proposition~\ref{hp:carrier}.
When $2<p<3$, fix the marked-collar enlargement factor large enough
for the fixed two-sided buffers used below, before assigning the
carrier's facet budgets; only the collar widths are chosen after
both rounds.

Apply the flattening and two-round carrier construction, using
Proposition~\ref{hp:flattening} and Proposition~\ref{hp:carrier}, and then
the quantizer calibration of Lemma~\ref{qt:calibration}.
Choose the line directions and the flattening tolerance $\lambda$
so accurately that, on the ultimately retained rank-one data, the
range direction of $Dq_0$ differs from the long rectangle axis by
$o(\gamma^2)$.  The derivative of the flattening has relative error
$C\lambda$ there.  The central fraction of the quantizer uses a common
scalar times orthogonal projection onto the protected plane; its
scalar $1/a_*$ can be taken arbitrarily close to one.  Trimming to
these fractions loses arbitrarily little $\mu$-mass.  Kernel inclusion
follows from Lipschitz composition on the retained data, and the
positive singular-value bounds preserve the rank when the errors are
small enough.

Denote by $K_c$ the common compact data surviving the carrier rounds,
before the later phase losses.  In both rounds the implant supports
\emph{and their regular comparison boxes} avoid $K_c$ with room.
Indeed the regular assignments are first restricted to compact sets
separated from the central data; only finitely many operations meet
a neighborhood of $K_c$.  The two rounds can therefore use common
positive separations.  In particular,
\begin{equation}\label{ha:q0-weak}
 F_b=F_\kappa\quad\hbox{near }K_c,
 \qquad q_0=TJF_b,
 \qquad |Dq_0|\le C|DF_b|\quad\hbox{a.e.}
\end{equation}
All supports and comparisons are chosen sufficiently fine relative
to the reserved full-rank patches and their bands.  The regional
charging conclusion of Proposition~\ref{hp:carrier} has only the two
preassigned buffer enlargements, so this is a requirement on those
comparisons before they are fixed.

Here are the resulting regional budgets, stated explicitly for their
later uses.  Put
\[
                     Z=\Omega\setminus\bigcup_i\{t_i\le-s\}.
\]
Outside $K_c$ and the marked cubes, the $|DF_b|^p$ integral on $Z$ is
small against $C_{\mathrm w}$.  Each regular implant contributing
there charges a disjoint comparison away from $K_c$.  Exterior
comparisons in the second round also avoid the first-round marked
cubes.  Both charging enlargements stay in $\{t_i\ge-3s\}$ for every
$i$.  The unchanged derivative is bounded by $C|Df|$, while exceptional
implant costs are confined to marked interiors.  The rank-zero set
contributes no $|Df|^p$ cost.  The same assertion holds on the enlarged
full-rank shells outside marked interiors, with the finite correction
jet factors included, because their charging enlargements lie in the
larger bands \eqref{ha:full-bands}.

If $2<p<3$, use the marked cubes $U$ of
Proposition~\ref{hp:carrier}.  Their enlarged union has arbitrarily
small $\int(1+|Df|^p)$.  After both carrier rounds choose collar widths
$d_U>0$, with fixed-factor two-sided buffers and mutual separation,
so that
\begin{equation}\label{ha:collar-budget}
 \sum_U\frac{\ell_U}{d_U}
             \int_{\mathcal C_U}|DF_b|^p
                  \quad\hbox{is as small as required}.
\end{equation}
Here $\ell_U$ is the side scale, and $\mathcal C_U$ is an enlarged
collar of width comparable to $d_U$ about $\partial U$.
The prescribed smallness includes $C_{\mathrm w}$ and the full-rank
jet factors wherever those tests overlap.  The unchanged-base slice
estimate in Proposition~\ref{hp:carrier} allows the widths to be
chosen after the rounds, however small the unchanged neighborhoods
have become.  The marked cubes, their enlargements and collars avoid
the deficient protected data.  Their sizes are also fine enough to
make the oscillation of $f$ on each enlarged cube meet all required
value tests.  There are no marked cubes for $p\ge3$.  The weights
$\ell_U/d_U$ are now fixed for all later absolute-error prescriptions.

\subsection{Polar localization and the active derivative estimate}
\label{ha:active-localization}

The central-tube modification will change $q_0$ to a map $q_*$
with only finite local Sobolev bounds in its low-radius regions.
We arrange that the final nonstrict map is constant there.
Thus the weak derivative estimate will use $q_*=q_0$ wherever
an output parameter is active; the larger low-region bounds will
enter only the crude capacities needed to pin the labels.

We next choose the protected arrays, phases and subcells from
Lemma~\ref{qt:calibration} and the polar setup of
Lemma~\ref{qt:polar}.  Arrays can be arbitrarily fine for the value
accuracy and for the quantizer's maximum-activity and clearance tests.
Phase selection retains the protected $\mu$-weight in rectangular
interiors, up to arbitrarily small loss, and avoids atoms at the
centers and the rays to corners.  Additional subedge divisions along
straight protected sides can also avoid ray levels of positive
$\mu$-mass: before overlaying the polygon meshes, take strict interior
cuts in the individual clipped diagrams and slide these finitely
localized cuts generically in their available intervals.

All full-cell and lower-face geometries, and their compact-local
aspect bounds, are now fixed.  For unprotected polygons, use the
original-vertex localization in Lemma~\ref{qt:subdivision} before
fixing the fine subcells.  In detail, the worst relevant aspect factor
on each true polygon is finite at this stage.  Moreover $Dq_0=0$ a.e.
on each vertex level set.  Thus the integral of $|Dq_0|^p$, multiplied
by that fixed aspect factor and any already assigned local weights,
tends to zero on the preimages of shrinking vertex neighborhoods.
Properness makes these tests locally finite.  Choose the neighborhoods
with summable costs, and then fit the fine subcells.  Everywhere else
the aspect factors in the ordinary weak estimate are absolute or
are the already fixed protected rectangle factor.

Choose first the conical boundary-layer widths in the full output
cells, as in \eqref{wl:conical-extension}.  Their weighted excess
costs can be made summably small before choosing the central activation
radii.  To see the order, in one boundary polygon the two-factor
interpolation has a Lipschitz bound depending on its fixed aspect and
$c_*$: its radial factor satisfies $\rho/r\le C$ and
$|\rho'|\le C/c_*$, while the inverse angular derivative contributes
an aspect factor times $1/r$.  The $r$ cancels.  Once the layer
thickness is fixed, its normal derivative is controlled by requiring
the factor motions to be sufficiently small.  The parametrizations
of the full input blocks are already fixed exact locally bi-Lipschitz
data, so these are ordinary absolute-continuity choices on known
maps.

Now take the activation radii $a_D$ sufficiently small for those
motion tests, and retain compact data of both positive deficient
ranks where $r>5a_D$, strictly inside smooth sectors.  Center and ray
avoidance makes the additional loss arbitrarily small; no absolute
continuity of the projected protected measure is being assumed.
Choose $r_{u,D}\ll a_D$ and use
Lemma~\ref{qt:central-carrier} to obtain
$M_y,h_*,M_x,F_*,q_*$.  The exterior changes from $q_0$ are confined
to deep low regions within individual subcells, which we may require
to satisfy
\begin{equation}\label{ha:deep-changes}
                         r(q_*)<a_D/100.
\end{equation}
The relative radius is set equal to one on subedges, giving a
continuous function on the two-complex.  All openings, central changes
and labels retain the preliminary fine value tests.

Write $K^\sharp\subset K_c$ for the final protected compact set.
It lies in the interior of $M_x$; the slab heights, tube clearances
and strict sector insets give room to require $q_*=q_0$ on its
neighborhood.  Lemma~\ref{qt:calibration} gives, a.e. there,
\begin{equation}\label{ha:relative-calibration}
                   |Dq_0-Df|\le C(\lambda+1-a_*)|Df|.
\end{equation}
All losses from $K_c$ made so far are small against $C_{\mathrm w}$.

The wall barriers will require arbitrarily fine local closeness of
$q_h=Th$ to $q_*$.  Prescribe that closeness so that the relevant radii
are comparable, including on all mesh-neighborhood enlargements.
The radii are positive and locally bounded away from zero on $M_x$.
Whenever $q_h$ lies on an edge, or in the possible active face range
$r(q_h)\ge a_D/2$, the comparison neighborhood is contained in an
open high-budget region where $q_*=q_0$.  Such neighborhoods exist by
\eqref{ha:deep-changes}.  Prescribe sufficiently small stair shifts,
also relative to $L_D$ in the logarithmic coordinate, that the
nonstrict map is constant below this active range.

Let $G$ be the upper physical pre-parameter slope supplied by the
mesh and wall constructions, allowing a fixed factor for sums of
two slopes.  On the active region, Lemma~\ref{wl:stairs},
\eqref{wl:target-speeds}, and Proposition~\ref{wl:parameter-cost} give
\begin{equation}\label{ha:active-bound}
                      |D(P_0q_h)|\le Cr(q_h)G
                      \quad\hbox{a.e.}
\end{equation}
Indeed a nonzero stair derivative occurs in an occupied stack.  Its
root parameter has one pre-parameter derivative or a sum/difference
of two derivatives on a switched diagonal.  Every intervening
OUT-to-IN identification has absolute slope one, so there is no
factor depending on itinerary length.  The occupied stair has root
speed at most $1+O(\zeta)$, and the forward polar map costs $O(r)$,
including the activation ramp.  On edge interiors this is the
perimeter estimate with $r=1$; on vertex levels the derivative
vanishes.  The formulas suffice a.e. at knots and piece interfaces:
$h$ is locally bi-Lipschitz, the face and edge product projections
are absolutely continuous for their respective dimensional measures,
and derivatives vanish on constant level sets.  In full blocks,
away from the already paid conical layers, $P_0q_h=q_0$.  On the
remaining central tube interiors outside $M_x$ the nonstrict map is
constant by the exact correspondence.  Thus only the prescribed
high-budget regions require the weak estimate
\eqref{ha:active-bound}.

\subsection{Calibrated source patches}
\label{ha:patches}

Use the fixed cuboidal and coarse chart structures on $M_x$.
Their walls and nonsmooth piece loci are locally finite ambient-null
sets.  Trim $K^\sharp$ away from them with room, losing arbitrarily
little $\mu$-mass.  At a retained Lebesgue gradient point $z$, within
a single smooth $(D,e)$ sector, put
\begin{equation}\label{ha:Y0}
              Y_0=(t(q_0),s(q_0)),\qquad B=D_xY_0(z).
\end{equation}
We choose a nonsingular spatial basis $V$ as follows.  If $B$ has
rank two, let $k$ be a unit vector in its kernel and take
\[
                    V=[B^\dagger,k/\|B\|_{\mathrm{op}}].
\]
The columns of $B^\dagger$ are orthogonal to $k$, and the singular
values of $V$ show that
\begin{equation}\label{ha:rank-two-basis}
             BV=[I_2,0],\qquad
             \|V^{-1}\|_{\mathrm{op}}=\|B\|_{\mathrm{op}}.
\end{equation}
If $B$ has rank one, write $B=a v^T$, where $|v|=1$, put
$\alpha=|a_1|+|a_2|$, and choose an orthogonal $Q$ with $Qe_1=v$.
Then $V=Q/\alpha$ gives
\begin{equation}\label{ha:rank-one-basis}
 BV=\begin{pmatrix}b_1&0&0\\b_2&0&0\end{pmatrix},
 \qquad |b_1|+|b_2|=1.
\end{equation}
For the rank-one rectangle,
\begin{equation}\label{ha:b-small}
                        \min(|b_1|,|b_2|)\le C\gamma.
\end{equation}
On a long-side sector the long-axis direction has only perimeter
speed, of order $1/r$.  On an end sector it has logarithmic speed
of order $1/r$ and perimeter speed at most $C\gamma/r$.  The
$o(\gamma^2)$ direction error, combined with inverse sector cost
$O(1/(\gamma r))$, preserves these estimates.  For rank two the
rectangles are squares.  In both cases Lemma~\ref{qt:polar} and
\eqref{ha:relative-calibration} give the absolute bound
\begin{equation}\label{ha:V-cancellation}
                r(z)\|V^{-1}\|_{\mathrm{op}}\le C|Df(z)|,
                \qquad r(z)=r(q_0(z)).
\end{equation}
This is the cancellation needed when the normalized sharp estimate
is returned to physical coordinates: the target reconstruction
costs $O(r)$, while the source change of coordinates costs
$\|V^{-1}\|_{\mathrm{op}}$.  Their product is controlled by the
original gradient, without a factor from the finite jet bounds.
On the retained compact set, $V$ and $V^{-1}$ have finite bounds:
use the positive rank, the singular-value truncations, the fixed
sector insets, and the finite chart bounds.

In a smooth coarse chart $x=g(\xi)$ use aligned coordinates
\[
 \xi=\xi_z+A y,\qquad A=Dg(\xi_z)^{-1}V.
\]
Choose small outer $y$-cubes centered at $y=0$, corresponding to $z$,
with nested inner cubes.
On their enlargements require: a single smooth sector with small
label oscillation; chart derivatives close to their center values;
and arbitrarily small power means of
\[
                         DY_0-B,\qquad Df-Df(z).
\]
All required eccentricities are bounded on the retained compact, so
these are a Vitali differentiation family.  Select disjoint outer
cubes and then a finite subfamily.  Choose the nested ratios close
enough to one that the inner cubes still capture the required weight,
including a further inset for residual tests.  We call them the
\emph{saturation cubes}.  Their physical images are separated from the
boundary of $M_x$, the marked cubes and the full-rank correction
supports.

There is no circular dependence between their small tests and their
geometric constants.  First fix the finite chart, sector, basis,
slot-shape and template bounds on the prospective compact data.
The finite triangulation lists in Lemma~\ref{ms:snapping}
depend on bounded cardinalities and fixed slot margins, not on
microscopic noncluster angles or master widths.  The transition mesh
bounds in Lemma~\ref{ms:meshes} depend on the fixed bases, not on the
relative thickness of patch buffers.  Next fix the level-edge
count tolerance, then smaller chart-error and jitter fractions,
and then still smaller power-mean data errors, allowing the inverse
jitter-density factors.  Finally choose patch radii realizing these
tests.  The mesh step $H$ and then $W/H$ can be arbitrarily smaller.

Thin patch margins, including equal-weight, clustering and Delaunay
exit belts, are chosen after the fixed local derivative and density
factors, so their volume and needed Sobolev integrals are small.
They can be resolved without deterioration of the shape factors.
Keep every equal-weight convolution cost inside enlarged pure lattice
regions where the affine tests remain valid.  Microscopic master
widths occur only inside the snapping construction; leave several
layers of widths comparable to $s_\sigma$ in the clustered
equal-weight belt before returning to ordinary widths.  Sharp and
mixed-position tetrahedra use the noncluster generic convention;
there is no robust-template motion of an opponent system in the
sharp bulk.  The wholly clustered convention resumes past the
buffer.  Outside-chart and unequal-orientation transitions use the
fixed coarse-shape weak bound on their paid regions.

\subsection{Ambient edge tests and weak costs}
\label{ha:weak-costs}

Outside saturation interiors use the physical-mode overrides of
Lemma~\ref{ms:meshes} on smooth coarse chart interiors.  Reserve small
neighborhoods of coarse facets, chart-piece seams, patch and Delaunay
transitions, and nonphysical mesh regions.  Their integrals of
$1+|Dq_*|^p$, with every assigned compact-local factor and residual
weight, can be made summably small.  Indeed these factors are fixed
before the neighborhood widths, and finitely many inset physical
patches cover a coarse compact except for arbitrarily small weighted
error.  Enlarge the error sets slightly to include neighboring mesh
stars.  The actual cut-boundary adjustment and exact-region jitter
taper can be made equally thin, later if necessary.

The last assertion uses the width-independent bound in
Lemma~\ref{ms:boundary-collar}.  The horizontal warp and chord inverse
response have fixed bounds, and motions depending on $d/d_0$ have
size comparable to the boundary mesh scale, which is much smaller
than $d_0$.  The slopes of $d_0$ are controlled.  Thus the label
slope, including its conversion to a root parameter, has no inverse
power of $d_0$; a fixed factor $1/c_*$ is allowed.  Equal subdivision
of the boundary chords changes only displacement differences in
tangential derivatives and adds no subdivision factor.

The exterior residual test lies in $Z$, outside slightly shrunken
saturation interiors, and enters a marked interior only through
its enlarged collar.  The other two tests are the buffered
full-rank correction shells and the marked collars with weight
$\ell_U/d_U$.  Mesh stars and jitter displacements will fit inside
the enlargements already reserved for these tests.

\begin{lemma}[Ambient edge tests and weak costs]\label{ha:ambient-tests}
With the preceding geometry and local weights fixed, the ordinary
weak costs are controlled by the carrier budgets with constant
$C_{\mathrm w}$, plus freely small absolute errors.  The control is
uniform over meshes sufficiently fine for prescribed buffer and
guard requirements.  It applies simultaneously to the exterior
residual, the full-rank correction shells, and the marked collars
weighted by $\ell_U/d_U$.
\end{lemma}

\begin{proof}
For a mesh edge or edge portion $l$, let $W_l$ be the sum of the
weights of its initial central hits.  On whole edges these sums
are upper bounds for the normalized systems: cleaning, opening and
normalization do not increase any individual wall--edge count.
Edge portions are used only to estimate the initial counts by
coarea; their contributions are added before normalization.
On an edge portion outside the modified
boundary collar, conditional on a suitable ACL mesh,
Lemma~\ref{wl:stairs} gives
\begin{equation}\label{ha:weighted-coarea}
 W_l\le(1+C(\zeta+c_*))\operatorname{Var}_l(q_*)+\varepsilon_l.
\end{equation}
Here the variation is the sum of star variation for the radial family
and graph arclength variation for the perimeter family; the two may
also be tested separately.  The additive errors $\varepsilon_l$ may
be as small as required relative to the actual edge and its weights.
Each feature has only finitely many relevant edge tests on a compact
localization, by properness.

The coefficient close to one is the weighted sampling density of
Lemma~\ref{wl:stairs}, not the width of $I_j$.  Its stratified
sampling argument, applied to the integrable crossing-count tests
in Lemma~\ref{ms:edge-tests}, gives the inequality with arbitrarily
high probability after the edges have been chosen.  It also permits
a common weight on finitely many features and all almost-sure null
avoidances.

In a high-budget region, the two graph speeds satisfy
\begin{equation}\label{ha:graph-speed}
                 |D(\hbox{graph labels})|
                    \le \frac{C_{\mathrm{asp}}}{r}|Dq_0|
\end{equation}
along the relevant ACL edge paths.  The bound is a stratumwise length
bound; an inverse ambient dihedral angle is unnecessary.  On a radial
spoke the perimeter-vertex coordinate is constant, and
$q=d_D+r(\xi-d_D)$ gives
$|dt|\le (L_D/|\xi-d_D|)|dq|/r$.  On a subedge the perimeter
projection is the identity and the star coordinate is constant.
At vertices or graph nodes the corresponding coordinate speed
vanishes a.e. on its level set.  Restrictions at density and
differentiability times have the ambient ACL derivatives, so these
bounds combine along the path.  Local Lipschitz graph coordinates,
possibly with larger fixed seam constants, ensure the required
absolute continuity.  The estimate itself uses just the sector
aspect factors.

On modified cut-collar portions replace \eqref{ha:graph-speed} by
the fixed local label-slope bound of
Lemma~\ref{ms:boundary-collar}.  The moving graphs and chords have
inverse-response bounds on the allowed bins, also across the
triangle pieces.  Restricted one-dimensional coarea on the disjoint
allowed parameter strips gives the corresponding weighted-count
bound in terms of these prepared labels, with the stated fixed
factors.  The central prepared trace family is fixed before sampling;
its later interval-ribbon adaptation keeps the sampled central
chords unchanged.  We split an edge into its unchanged and modified
collar portions when applying these bounds.  The depth-stretched
portions use the adjoining exact-region bounds.

For an ordinary all-clustered tetrahedron $\sigma$ of physical scale
$h_\sigma$, Proposition~\ref{ms:prestacks} and
\eqref{ha:active-bound} give the active slope majorant
\begin{equation}\label{ha:weak-tet-slope}
                 C r_\sigma h_\sigma^{-1}
                     \max_{l\subset\sigma}W_l.
\end{equation}
The radii on the required stars are comparable to $r_\sigma$.
There is no fixed baseline in \eqref{ha:weak-tet-slope}: a
contributing pre-parameter has width proportional to its root
weight, and zero crossings require no disk.  Apply
\eqref{ha:weighted-coarea}, \eqref{ha:graph-speed} and H\"older's
inequality on an edge $x_l(u)$, $0\le u\le1$, parametrized at speed
$O(h_\sigma)$.  The $p$th power of
\eqref{ha:weak-tet-slope}, times $|\sigma|$, is at most an arbitrarily
small additive error plus
\begin{equation}\label{ha:edge-power}
 C h_\sigma^3\sum_{l\subset\sigma}
       \int_0^1 C_{\mathrm{asp}}(x_l(u))^p
                    |Dq_0(x_l(u))|^p\,du.
\end{equation}
Only tetrahedra in deterministic high-budget buffers require this
estimate.  Costs which become inactive after pinning the labels are
not included in the weak objective.

Lemma~\ref{ms:edge-tests} supplies the ambient expectation estimate:
on an unconstrained jittered edge, the point at fixed normalized
edge time has density at most $C h_\sigma^{-3}$ in a neighboring
star, and the normalized edge speed is at most $C h_\sigma$.
The lemma also supplies ACL and the chain rule.  Thus for any
nonnegative ambient integrable weight $F$,
\begin{equation}\label{ha:jitter-expectation}
 \mathbb E\left[h_\sigma^3\int_0^1F(x_l(u))\,du\right]
       \le C\int_{\operatorname{star}^+(\sigma)}F(x)\,dx.
\end{equation}
Here the enlarged stars have bounded overlap.  Jitter radii are
chosen to stay within those stars.  The taper into nonrandom data is
wholly in exact regions, where fixed local Lipschitz bounds replace
the expectation estimate.  In the physical bulk, shapes, displacement
fractions and overlap are universal.  Keep any additional mesh layers
needed to reach that bulk inside the paid transition regions.
Summing \eqref{ha:jitter-expectation} therefore charges
\eqref{ha:edge-power} to the prescribed ambient budgets with only
$C_{\mathrm w}$ in the ordinary bulk.  In the exceptional zones it
charges $1+|Dq_*|^p$ with the already fixed local factors.  Near the
original polygon vertices it charges the affordable weighted
$|Dq_0|^p$ selected before subdivision.

We now charge the deterministic envelopes specified above.  The
part of the exterior-residual envelope in $K_c$ is small in
$|Df|^p$ by weighted capture and \eqref{ha:q0-weak}; on its complement
use the regional $DF_b$ budget and \eqref{ha:collar-budget}.  The
full-rank shells and marked-collar envelopes lie in their preassigned
enlargements and avoid the sharp construction.  Mesh stars and
jitter displacements are finer than all these buffers.  The sharp
transition and equal-weight belts use the margin and leakage estimate
proved below.  Add the already small full-cell conical-layer costs;
elsewhere in full blocks the map is $q_0$.  The objectives are a
bounded number of aggregate nonnegative costs, and their local
weighted summands can be combined into one objective.  Their
expectations have the asserted smallness.  Lemma~\ref{ha:capacities}
selects the mesh and sample data simultaneously with the crude guard
capacities; Proposition~\ref{ha:initial-walls} realizes the actual
wall systems with these same bounds.
\end{proof}

\subsection{Sharp counts, convolution and discarded regions}
\label{ha:sharp-errors}

On an enlarged calibrated patch let $H$ be the common aligned Kuhn
step.  The ambient volume jitter is included in its chart $g'$ from
$y$ to $x$.  The small first-chart errors and jitter fraction ensure
that $Dg'$ is as close to $V$ as prescribed, in relative matrix norm.
All relevant features use a common root weight $W$, with $W/H$
arbitrarily small.

\begin{lemma}[Sharp counts and discarded costs]\label{ha:sharp-counts}
The meshes and the subsequent stratified samples can be chosen so
that the sharp-template count conditions hold outside tetrahedra
of arbitrarily small aggregate pre-derivative $p$-cost.  More
precisely, for the two family counts on an edge $l$,
\begin{equation}\label{ha:sharp-edge}
 \frac{N_{l,i}W}{H}
 \le (1+C(\zeta+c_*))
          \left|(BV)_i\frac{\Delta y_l}{H}\right|+e_l,
 \qquad e_l\ge0,
\end{equation}
and the aggregate $\sum_lH^3e_l^p$ can be arbitrarily small after all
fixed template, transition and physical conversion factors.
The dirty portions left by Proposition~\ref{ms:sharp-contract} have
arbitrarily small aggregate evaluated pre-derivative $p$-cost in
aligned coordinates.
\end{lemma}

\begin{proof}
Use the separate coordinate variation tests in the single smooth
sector.  Along an edge, subtract the affine comparison $BVy$.
The resulting variation is bounded by the integral of
$|DY_0-B|$ times the bounded chart speed, plus the chart/jitter
derivative error times $\|B\|$.  Normalize by $H$ and use
H\"older's inequality.  The expectation estimate
\eqref{ha:jitter-expectation} bounds the sum of the resulting
$p$-errors by the power-mean data errors on the enlarged patch.
The inverse jitter-density factor may be large, but was fixed before
those tests.  The small chart/jitter terms are deterministic.
Conditional sampling gives the multiplicative factor in
\eqref{ha:sharp-edge} and arbitrarily small additive errors, which
are included in $e_l$.  There are only finitely many sharp patches,
and all matrix bounds are finite.  We may therefore impose an
aggregate error with any prescribed finite patchwise weights;
there is no requirement of simultaneous probabilistic control of a
relative mean on every individual small cube.

Fix first a count threshold much smaller than $\eta$, and also a
level-edge threshold $\delta'>0$, which may be arbitrarily small
after the $\eta$-dependent template constants.  Discard tetrahedra
where a relevant $e_l$ exceeds these thresholds.  On an inter-edge,
\eqref{ha:sharp-edge} bounds each independent count by
$(1+o(\eta))H/W$.  In parallel mode the sum of the two ideal counts
is $H/W$, by \eqref{ha:rank-one-basis}, so the combined excess is
also $o(\eta)H/W$.  On a level edge the ideal variation vanishes:
its count is at most $\delta'H/W$, with no multiplicative bias.
These distinct tolerances have distinct uses.  Inter-edge excess
controls fitting the lists into slots of spacing $(1-C\eta)W$;
level-edge errors control non-slabs and discrepancies in slab ranks.
Start the lists at common offsets of order $\eta H$ from the lower
axis end and perturb only within the unused slack.  Across a
regular tetrahedron the rank offsets of corresponding slabs differ
by at most $O(\delta'H/W)$, including the first band before the
second band in parallel mode.  These are the hypotheses of
Proposition~\ref{ms:sharp-contract}; no lower bound for the number
of surviving slabs is needed.

We give the cost calculation for discarded tetrahedra.  In the
finite enlarged equal-weight region, put
\[
 m_\sigma=1+\max_{l\subset\sigma}N_lW/H,
 \qquad
 M_\sigma=\sum_{\tau}\theta^{d(\sigma,\tau)}m_\tau,
\]
where $N_l=N_{l,1}+N_{l,2}$ is the total two-colour upper count,
the star graph has bounded degree, and $\theta$ is smaller
than a fixed reciprocal of that degree.  The initial weighted
counts remain upper bounds for all normalized counts, so they can
be used in these majorants.  Equation~\eqref{ha:sharp-edge} implies
$m_\sigma\le m_0+C e_\sigma$, where $m_0$ is an absolute baseline
and $e_\sigma$ is the largest of its finitely many edge errors.
The kernel has uniformly bounded row and column sums.  Thus its
convolution is bounded on every $\ell^p$, and
\begin{equation}\label{ha:convolution-error}
 M_\sigma\le M_0+L_\sigma,
 \qquad
 \sum_\sigma H^3L_\sigma^p
       \le C\sum_\sigma H^3e_\sigma^p.
\end{equation}
Adjacent $M$ values also have bounded ratios.

For a fixed threshold $\tau>0$, the number-weighted volume of
$\{e_\sigma>\tau\}$ is at most
$\tau^{-p}\sum H^3e_\sigma^p$.  Its $M^p$ cost is small as well:
pay the baseline $M_0^p$ by that volume and the remainder by
\eqref{ha:convolution-error}.  On $\{M_\sigma>2M_0\}$, the full
$M^p$ cost is bounded by a constant times the $L^p$ cost.  Fixed
neighbor layers of these sets have the same conclusions, using
bounded degree and adjacent ratios.  In thin patch margins within
the convolution region, pay the baseline by volume and the leakage
by \eqref{ha:convolution-error}.  Mixed-width tetrahedra in the last
quantitative belt can use both this majorant and the ordinary edge
bound; they still lie in the paid margin.

On low-leakage regular tetrahedra, fix the buffer multiple $N$
sufficiently large after the slot margins.  It accommodates the
flat translations, the auxiliary-cell trimming for snapping, and
the slow broadening to strip half-width
$a_1=(1-O(\eta))W/2$.  Proposition~\ref{ms:sharp-contract} gives
relative dirty volume at most
\begin{equation}\label{ha:dirty-fraction}
                           C_N(W/H+\delta').
\end{equation}
The pre-slope there is at most $CM_\sigma$, and $M_\sigma$ is
bounded on these tetrahedra.  First make $\delta'$ sufficiently
small, then $W/H$ sufficiently small.  The remaining bad-tetrahedron
and high-leakage costs are small by
\eqref{ha:convolution-error}, and the patch-boundary buffers are
paid margins.  Altogether the integral of the pre-slope to the
$p$th power on the dirty portions is arbitrarily small.  The
opposite-band gap and slope errors away from these portions are
$o(\eta)$ by the same level tolerance, negligible generic motions,
and $W/H$.  All constants used here were fixed before the final
aggregate error was prescribed.
\end{proof}

\subsection{Capacities before guards and simultaneous choices}
\label{ha:guard-schedule}

Here a capacity means a finite upper bound for a local integral
$\int G^p$, not a Sobolev capacity of a set.  The preceding estimates
are uniform over sufficiently fine mesh
resolutions.  They therefore allow the crude $G$ capacities to be
fixed before the guard nets, even though the final meshes must be
finer than the guard buffers.

There are two tasks.  We first select edges and samples for which
the count-based slope bounds fit the chosen capacities and energy
budgets.  We then construct the actual initial walls and pre-stacks,
retaining those very counts.  The second task is carried out in
Proposition~\ref{ha:initial-walls}; until then, the selected bounds
are upper budgets for the eventual slope density $G$.

\begin{lemma}[Prior capacities and simultaneous choice]\label{ha:capacities}
For each member of a locally finite family of buffered compact
barrier charts one can fix a finite capacity $A_j$, independently of
guard spacing and final mesh resolution.  After fixing these
capacities, one can choose the guards, mesh and samples so that
substitution of the selected edge counts into the slope bounds of
Proposition~\ref{ms:prestacks} gives local integral upper budgets
at most $A_j$.  The same data meet the weak aggregate bounds of
Lemma~\ref{ha:ambient-tests} and the sharp count and discarded-cost
bounds of Lemma~\ref{ha:sharp-counts}.
\end{lemma}

\begin{proof}
\emph{Uniform preliminary budgets.}
For the crude bound on a compact chart, include the upper bound for
every pre-slope, including ones that will later be inactive.
In nonsharp tetrahedra,
Proposition~\ref{ms:prestacks} bounds the slope by the weighted edge
variations with the fixed local chart, aspect and inverse-radius
factors, together with the allowed fixed width terms and the
cut-collar bounds.  Equations~\eqref{ha:weighted-coarea} and
\eqref{ha:jitter-expectation} give a finite expectation majorant
using $1+|Dq_*|^p$ on an enlarged compact.  The local factors are
bounded there.  In equal-weight regions the aligned pre-slope is
at most $CM_\sigma$, and the bounded convolution estimate charges
the whole enlarged patch to its $m$ budget.  Physical conversion
has only fixed compact factors.  Sharp overrides occur in a finite
interior compact.  Thus, for the $j$th barrier chart, there is a
finite number $B_j$ bounding the expected crude upper budget, uniformly
over all sufficiently fine resolutions and all permitted guard
gaps.

This expected cost is an upper-budget construction.  First require
the weighted sample counts to be bounded by their variations plus
the prescribed errors, conditional on the edges.  Take the $p$th
power of those variation upper bounds and then the ambient jitter
expectation.  No high moment of the unweighted crossing count is
being assumed.  The additive count errors can be chosen to obey
both the small aggregate objectives and the crude capacities.
The boundary-collar constants are independent of the excluded belt
widths and of $d_0$, and the mesh constants are independent of guard
spacing.  A guard-induced allowed component may be very short;
only the eventual sample weight, not these constants, must then
be smaller.

Choose capacities, for example,
\[
                            A_j=2^{j+6}(1+B_j).
\]
The probability that the corresponding upper budget exceeds $A_j$
is at most $2^{-j-6}$.  The small objectives are finitely many
aggregate nonnegative random variables.  Their expected values
can be made smaller than their desired bounds by any prescribed
factor, using the initial affordable tails and the subsequent
absolute errors already described.  Fix these with enough room
that their Markov failure probabilities sum to less than $1/8$.
The sharp affine-error objective is treated in the same way.
The sum of the crude-capacity failure probabilities is also less
than $1/8$.  Thus a positive-probability set of edge jitters
satisfies all these upper-budget inequalities.

\emph{Guards, meshes, and samples.}
Now use these fixed capacities in Proposition~\ref{wl:barrier} to choose
the guard spacings and their buffers.  The desired accuracies include
those used in Subsection~\ref{ha:active-localization}, all local value tests,
and the following sharper requirement on every saturation cube:
\begin{equation}\label{ha:sharp-label-pinning}
                  Y=(S_D(t(q_h)),S_e(s(q_h))).
\end{equation}
Require this pair to be uniformly as close to $Y_0$ as prescribed
relative to the aligned cube side length.  Both labels stay in the
same smooth sector, and the activation
function $H_{a_D}$ is the identity at their resulting radii.
These are physical label requirements after the finite patches
have been selected.  They can be imposed by a sufficiently fine
combined label accuracy and sufficiently close stairs.

Properness bounds the number of guard-label gap exclusions relevant
to a compact feature.  Choose the preliminary guard realization
accuracies relative to these prescribed gap budgets, which may be
smaller than the relative tolerance $\zeta$ to ensure fine stair
motion.  The dyadic sampling belts in
Lemma~\ref{ms:boundary-collar} can meet these prescriptions by
using summably small fractions across scales and requiring every
scale actually used on a compact feature to lie sufficiently far
down the tail.  Only finitely many such scales are used there
once a mesh has been selected.

Choose $d_0$ and the meshes small enough for the guard, high-label
and regional buffers.  The size functions have constant cores and
a common sharp-interior step $H$, as in Lemma~\ref{ms:meshes}.
The push into the cut boundary preserves the already fixed local
bi-Lipschitz constants.  Apply the allowed boundary and volume
jitter.  The uniform upper-budget bounds just proved apply to this
mesh, so choose a jitter satisfying them and all almost-sure ACL
conditions.  Finally choose the central sampling weights after
these edges.  In the finite equal-weight regions take $W/H$ as
small as required.  Split allowed components sufficiently finely
and place intervals $I_j$ of width comparable to $c_*w_j$, with
small sampling margins.  Lemma~\ref{wl:stairs} makes all required
weighted count inequalities hold with simultaneous positive
conditional probability.  Countably many local sampling tests
use summable conditional failure probabilities.  Hence at least
one joint choice of edges and samples satisfies everything.
The displacement of whole-interval and occupied-interval stairs
is charged only to absolute gaps, rounding and slot size; keeping
$c_*>0$ fixed introduces no residual motion error.

In the same conditional selection impose the almost-sure tests
needed to open the carrier: avoid labels of nonsingleton fibre
values, labels attained on nonexact seams along edges, and mesh
vertices; require finite original-carrier hit sets, including in
the exact collar.  These tests use only coarea and null avoidance,
and add no numerical energy constraint.  Their applicability is
verified in the next proof.  Intersecting them with the
positive-probability set just obtained completes the selection.
\end{proof}

\subsection{From sampled counts to the actual wall systems}
\label{ha:wall-transfer}

The sampled estimates refer to the original carrier away from the
boundary adjustment and to the prepared trace family inside its
exact collar.  We now obtain actual walls with these counts.  A
single opening is used for both colours; its equality with the
carrier in the exact collar then allows the boundary preparation
to realize the second set of tests.

\begin{proposition}[Initial walls with the selected budgets]
\label{ha:initial-walls}
The guards, mesh, samples and count bounds selected in
Lemma~\ref{ha:capacities} admit pre-stacks satisfying
Definition~\ref{wl:prestack} and avoiding the selected guards.
Their actual slope majorant $G$ satisfies
\[
                         \int_{U_j}G^p\le A_j
\]
on every buffered barrier chart $U_j$, together with the weak and
sharp estimates of Lemmas~\ref{ha:ambient-tests} and
\ref{ha:sharp-counts}.  The mesh and samples are unchanged.
Consequently the hypotheses of Theorem~\ref{wl:realization} hold.
\end{proposition}

\begin{proof}
\emph{A common opening and the initial walls.}
We now pass from the sampled carrier levels to the initial systems
required by Proposition~\ref{ms:prestacks}; this step precedes routing.
Let $\mathcal W$ be the entire two-colour family of sampled standard
walls in $M_y$.  Each wall is compact and properly embedded, and
there are finitely many samples per feature.  The standard feature
family is locally finite.  Hence $\mathcal W$ is locally finite and
its union is relatively closed in $M_y$.  The nonsingleton values
of the central carrier in Lemma~\ref{qt:central-carrier} are
countable.  Each such value forbids at most finitely many radial
or intermediate meridian levels, and a graph vertex is not an
intermediate meridian level.  These countably many labels are among
the almost-sure sample avoidances in Lemma~\ref{ha:capacities}.
Thus the whole wall union,
not merely its edge-hit values, avoids every nonsingleton value.

For the opening use the \emph{original} carrier $F_*$ and the
actual final mesh edges, before the boundary preparation has
changed the walls.  Away from the exact collar, the already chosen
ACL and coarea tests give finite hit sets on compact localizations.
The nonexact seam set $S$ of Lemma~\ref{qt:central-carrier} has
edge-parameter measure zero by volume jitter.  The relevant label
paths are absolutely continuous, so the images of these parameter
sets have one-dimensional measure zero.  These scalar label sets
and the mesh-vertex labels were also excluded in the same sample
selection.  This gives no hit on $S$ outside the exact regions.
In the exact collar the unmodified carrier labels along the final
edges are locally Lipschitz.  Their additional coarea tests therefore
give finite hits almost surely; these tests require no numerical
energy budget.  The finite feature lists permit all of them in
the selection of Lemma~\ref{ha:capacities}.  A
compact localization meets finitely many edges and sampled walls,
so its total actual hit set is finite.

Lemma~\ref{qt:edge-matching} therefore supplies \emph{one common}
opening homeomorphism $f_{\mathrm o}:\Omega\to\Lambda$ for the
entire two-colour wall union, with exactly its original carrier
hits on every mesh edge.  Choose it sufficiently finely close to
$F_*$ for the prescribed guard-gap buffers.  It equals $F_*=h_*$
on an exact neighborhood containing all supports of the boundary
preparation, as well as on the other prescribed exact data, and
maps $M_x$ onto $M_y$ with the fixed boundary correspondence.
Pull back both standard families by this same $f_{\mathrm o}$.
They are compact properly embedded locally flat systems of the
required individual types, with noncontractible annular cores.

Now apply the common joint boundary adjustment of
Lemma~\ref{ms:boundary-collar}: the warp and chord preparation,
its mirrored continuation inside the cut, and the depth stretch.
This operation is supported entirely in the exact neighborhood
where $f_{\mathrm o}=F_*=h_*$.  Therefore the resulting collar
walls and their actual edge counts are precisely the prepared
data whose weighted coarea bounds were used above.  Outside that
neighborhood their edge counts are the original carrier counts,
preserved by the opening.  The interval-ribbon adaptation keeps
the central sample chords fixed.  Consequently all previously
selected weighted and directional upper counts apply to these
actual initial systems.  No sampling estimate on a
sample-dependent pullback of an edge is being used.

Apply Lemma~\ref{wl:relative-pl} separately to each colour,
retaining the smaller prescribed PL boundary product collars.
Its hypotheses hold by the finite isolated edge hits, vertex
avoidance and the exact prepared boundary data.  Choose its fine
position tolerances within the guard buffers.  It gives the
required PL systems and flat transverse interior crossing germs,
without increasing any individual wall--edge count.  Apply
Proposition~\ref{wl:normalization} separately to these two systems,
retaining their boundary traces, topologies and coorientations,
and again without increasing an individual edge count.  Neither
application requires a relative isotopy fixing the other colour.
Proposition~\ref{ms:prestacks} now supplies the pre-stacks.
No individual edge count has increased.  The same count-based
slope upper budgets selected in Lemma~\ref{ha:capacities} therefore
bound the actual integrals $\int_{U_j}G^p$ by $A_j$ and give the
stated weak and sharp estimates.  Their guard avoidance follows from the
reserved label gaps and the chosen finer mesh: each final normal
disk lies in a tetrahedron incident to an original possible hit,
and placements, thickening and snapping stay in its prescribed
mesh buffers.  The preliminary opening and PL changes were chosen
within the same positive buffers.  Thus this execution establishes
the pre-stack hypotheses of Theorem~\ref{wl:realization} without
changing the previously fixed capacities or the guard order.

\emph{Slopes on lower-dimensional interfaces.}
For the rectifiable-path use of the barrier, define $G$ on mesh
interfaces and exceptional collar-boundary strata to include the
larger adjacent actual local collar slopes.  These strata have
ambient volume zero and locally finite prescriptions, so this
changes none of the volume budgets.  The pre-parameters are
Lipschitz on their compact product collars; restricted path coarea
therefore uses these pointwise upper slopes.  On open pieces the
volume bounds are exactly those already proved.  In particular,
no estimate for an inverse distance across a gap at a manipulated
boundary is needed in the integral capacity.
\end{proof}

\begin{remark}\label{ha:scale-order}
The order established above can be recorded without a forward
parameter dependency.  Fix $p$ and the desired power error; choose
$\gamma,\eta$, then $\zeta,c_*$ and the ordinary weak constant.
Choose affordable rank tails and finite jet truncations, then the
full-rank shell tests.  Construct the carrier and fix its regional
and weighted marked-collar budgets.  Fix the polar data, aspect
localizations, full-cell layer widths, activation radii and exact
central carrier.  Fix compact sharp-template and basis bounds,
then level tolerances, chart/jitter fractions and smaller
power-mean tests; select the finite sharp patches.  Choose the paid
transition regions and the crude capacities.  Only then choose
guard accuracies and spacings.  Finally take the boundary widths
and meshes fine enough for all buffers, select a permissible
jitter, and sample with sufficiently small weights.  The expectation
bounds are uniform in these last resolutions, which is why the
capacities may precede the guards.  Strict parameters, marked-cube
collapses and the final smoothing are chosen afterward.
\end{remark}

\subsection{Saturation and physical gradient accuracy}
\label{ha:saturation-subsection}

Apply the wall construction to the pre-stacks of
Proposition~\ref{ha:initial-walls}, with the capacities and samples
selected in Lemma~\ref{ha:capacities}.  Theorem~\ref{wl:realization} gives the
locally bi-Lipschitz $h$, the actual occupied intervals and the
label pinning.  We estimate $P_0Th$, not the derivative of the
qualitative product and ball extension used to construct $h$.

\begin{lemma}[Saturation and physical gradient accuracy]\label{ha:saturation}
On the union of the saturation interiors,
\begin{equation}\label{ha:physical-saturation}
 \int |D(P_0Th)-Df|^p
       \le C_p(\eta+\gamma)(1+E)
                         +\text{freely small errors}.
\end{equation}
The constant in the displayed leading term is independent of the
compact jet and chart bounds.
\end{lemma}

\begin{proof}
In a saturation cube $Q$ in aligned coordinates put
\[
                  \widetilde Y=Y\circ g',\qquad D_0=BV.
\]
Proposition~\ref{ms:sharp-contract} and
Proposition~\ref{wl:parameter-cost} give, off the dirty portions,
\begin{equation}\label{ha:Frobenius}
 |D_y\widetilde Y|\le
 \begin{cases}
   \sqrt2+C\eta,&\rank D_0=2,\\
   1+C\eta,&\rank D_0=1.
 \end{cases}
\end{equation}
At an essential both-active crossing, the two scalar gradients
are separately near-unit orthogonal axis gradients.  On a switched
diagonal only one output scalar is active and its derivative is
their sum or difference, with the same squared-norm bound.  In
parallel mode the opponent strips in this deep flat region are
disjoint.  Strip width $a_1$, flat slopes and occupied-root speeds
contribute $O(\eta)$ after the subsequent smaller choices.  Thus
arbitrary partial switches do not increase the upper bound in
\eqref{ha:Frobenius}.  By Lemma~\ref{ha:sharp-counts}, the evaluated
$p$-integrals on dirty portions are arbitrarily small in aggregate,
with any fixed physical conversion weights.

The mean gradient has the required affine value:
\begin{equation}\label{ha:mean-gradient}
                \left|\frac{1}{|Q|}\int_Q D_y\widetilde Y-D_0\right|
                      \quad\hbox{is arbitrarily small}.
\end{equation}
For $Y_0\circ g'$ this follows from the enlarged power-mean tests,
the bounded Jacobian factors and the chart/jitter error.  For the
difference, integration on the boundary of the cube gives
\[
 \left|\frac{1}{|Q|}\int_Q D_y\bigl((Y-Y_0)\circ g'\bigr)\right|
       \le \frac{C}{\operatorname{side}Q}
                       \|(Y-Y_0)\circ g'\|_{L^\infty(Q)}.
\]
This is valid for the continuous Sobolev functions at issue by
the trace formula; the right side is small by
\eqref{ha:sharp-label-pinning}.  The cube remains compactly in one
smooth sector.

We use the elementary uniform-convexity inequality
\begin{equation}\label{ha:convexity}
 c_p|Z-D_0|^p\le |Z|^p-|D_0|^p
             -p|D_0|^{p-2}D_0:(Z-D_0),\qquad p>2.
\end{equation}
One proof integrates the Hessian of $|X|^p$ along the segment
$D_0+t(Z-D_0)$.  Its quadratic form is at least
$p|X|^{p-2}|Z-D_0|^2$.  After scaling $|Z-D_0|$ to one, an interval
of fixed positive length in $0\le t\le3/4$ is a fixed positive
distance from the possible zero of the projection of that segment
onto $Z-D_0$.  The integrated Hessian is therefore bounded below
by a positive constant depending only on $p$.  This proves
\eqref{ha:convexity} uniformly in both matrices.

In rank two, $|D_0|=\sqrt2$.  In rank one,
\eqref{ha:rank-one-basis} and \eqref{ha:b-small} imply
$1-C\gamma\le|D_0|\le1$.  Integrate
\eqref{ha:convexity} with $Z=D_y\widetilde Y$.
Equation~\eqref{ha:Frobenius} bounds its energy on the clean set;
the dirty energy is already small.  The linear term is controlled
by \eqref{ha:mean-gradient}, since $|D_0|\le\sqrt2$.  We obtain
\begin{equation}\label{ha:normalized-saturation}
 \int_Q|D_y\widetilde Y-D_0|^p
      \le C_p(\eta+\gamma)|Q|
                   +\text{mean and dirty errors}.
\end{equation}

It remains to verify that the physical conversion does not multiply
the leading error by a large jet condition number.  On this cube
$P_0q_h=\mathcal R(Y)$, with the smooth sector reconstruction
\[
       \mathcal R(t,s)=d_D+\exp(t/L_D)(\xi_e(s)-d_D).
\]
There is no central activation transition.  Since $s$ is arclength
and $L_D$ controls the polygon diameter, its two derivative columns
are bounded by $Cr$; this bound is independent of the thin
rectangle aspect.  Require the chart error to satisfy
$\|(Dg'-V)V^{-1}\|_{\mathrm{op}}\ll1$.  Then
\begin{equation}\label{ha:physical-multiplier}
 \|D\mathcal R(\widetilde Y)\|_{\mathrm{op}}
       \|(Dg')^{-1}\|_{\mathrm{op}}
       \le Cr(z)\|V^{-1}\|_{\mathrm{op}}
       \le C|Df(z)|,
\end{equation}
and $|\det Dg'|=(1+o(1))|\det V|$ relatively.  All these
requirements are possible after the finite basis bounds.

Split the physical derivative into the term containing
$D_y\widetilde Y-D_0$ and the comparison term
\[
             D\mathcal R(\widetilde Y)D_0(Dg')^{-1}.
\]
The latter differs as little as prescribed from $Dq_0(z)$, by the
label, patch and chart tests.  At the central data the identity
$D\mathcal R(Y_0(z))B=Dq_0(z)$ holds.  Now use
\eqref{ha:relative-calibration} and the affine-gradient tests on
$Df$ to compare further with $Df(x)$.  The mean, dirty and comparison
errors may use every finite conversion factor because they were
chosen after those factors.  For the leading term, however,
\eqref{ha:physical-multiplier} and
\eqref{ha:normalized-saturation} give exactly
\[
          C_p(\eta+\gamma)
                      |Df(z)|^p|\det V|\,|Q|.
\]
The selected outer patches are disjoint, their affine-gradient
tests are accurate, and their chart Jacobians are relatively close
to $|\det V|$.  Hence
\[
                  \sum_Q|Df(z_Q)|^p|\det V_Q|\,|Q|
                         \le C(E+1).
\]
Summing proves \eqref{ha:physical-saturation}.  The flattening angle
and central-fraction losses in \eqref{ha:relative-calibration} were
freely chosen small relative to this same finite energy.
\end{proof}

\subsection{Strict maps, marked interiors and completion}
\label{ha:completion}

\begin{proof}[Proof of Theorem~\ref{ha:approximation}]
Carry out the preceding choices with sufficient room in each target
error.  Let $u=P_0Th$ be the nonstrict map.  On saturation interiors
it satisfies Lemma~\ref{ha:saturation}.  On the exterior residual,
full-rank shells and weighted marked collars, the derivative costs
are arbitrarily small by Lemma~\ref{ha:ambient-tests} and the
simultaneous choice and transfer in Lemma~\ref{ha:capacities} and
Proposition~\ref{ha:initial-walls}.  The $|Df|^p$
integral on the residual is also small: the positive deficient ranks
are captured by the saturation interiors up to the affordable
losses, the full-rank part is captured by its inner cubes up to the
chosen tail, and the rank-zero derivative vanishes.  The corresponding
$|Df|^p$ costs on the correction shells and enlarged marked cubes
were fixed small at the initial stage.

Use Proposition~\ref{wl:strict} and Lemma~\ref{qt:strict} to
replace $u$ by
\[
                            k=P^\epsilon T_\delta h.
\]
The symbols $\epsilon$ and $\delta$ here denote the strict target
parameters, not the slot loss.  Choose them locally with summably
small derivative and value errors on every prescribed test, including
the finite full-rank tests and the marked-collar weights.  The stated
tangential compatibility on faces and edges ensures this derivative
convergence; at vertex levels the nonstrict derivative is zero.
The map $k$ is a locally bi-Lipschitz homeomorphism of $\Omega$ onto
$\Lambda$.  We impose no global energy bound on its still exempt
marked interiors at this stage.

If $2<p<3$, apply Lemma~\ref{hp:hide-cubes}, using
\eqref{ha:collar-budget} and its transferred weighted volume tests
for $k$.  Choose the rescaled boundary in a slightly outer collar
so that the entire exempt cube is enclosed.  We recall why this
step can make its energy small despite an arbitrarily large fixed
interior Lipschitz constant.  In cube polar coordinates, the source
self-homeomorphism sends $[0,\alpha]$ linearly to $[0,1-\rho]$ and
$[\alpha,1]$ to $[1-\rho,1]$, fixing the boundary.  The inner energy
is bounded by a fixed compact constant times $\alpha^{3-p}$ and
can be made arbitrarily small after $k$ has been chosen.  For fixed
$\alpha$, take $\rho\downarrow0$ at the chosen radial Lebesgue
boundary.  The outer energy is bounded by a constant times
\[
 \ell_U\int_{\text{chosen facets}}|Dk|^p
                     \int_\alpha^1 t^{2-p}\,dt,
 \qquad \int_\alpha^1 t^{2-p}\,dt\le \frac{1}{3-p}.
\]
Slice selection bounds the first factor by the weighted collar
volume test.  The factor depending on $p<3$ is fixed before those
budgets are prescribed.  Use summably small choices for the marked
cubes, also including any finite full-rank shell factors.  The changes
avoid all saturation patches.  Their value error is small because
$f$ has the prescribed small oscillation on each enlarged cube and
$k$ already meets the fine value test there.  Denote the resulting
locally bi-Lipschitz homeomorphism by $k_1$.  If $p\ge3$, put
$k_1=k$.

The map $k_1$ now belongs to global $W^{1,p}$.  The residual,
saturation, shell and marked-interior derivative costs have summable
bounds.  The only possibly unpaid full-rank interiors lie in a
finite compact subset of $\Omega$, where local bi-Lipschitzness
gives finite $p$-energy.  Finally its values lie in the bounded
$\Lambda$, and $\Omega$ has finite measure.  This proves actual
membership, rather than merely local membership, before invoking
the smoothing Theorem.

Apply Theorem~\ref{sm:smoothing} to $k_1$, keeping room in the
energy and value tests, in particular the normalized full-rank
sup tests.  Perform the finitely many smooth cubical jet corrections
of Proposition~\ref{hp:full-rank-correction}.  The map equals $b_i$
on the exact inner cubes, where the error is
\eqref{ha:affine-full-tests}.  Its changed shells have arbitrarily
small cost by the buffered budgets including the fixed jet factors.
Their supports avoid the saturation patches.  On the complement,
the sharp estimate remains in force and the residual small-energy
estimates for both maps control the gradient difference.  Smooth
once more by Theorem~\ref{sm:smoothing}, with an arbitrarily small
additional $W^{1,p}$ error.

All operations have the required fixed target.  The strict target
maps are self-homeomorphisms of $\Lambda$; the wall realization
maps onto $\Lambda$; the marked-cube operations are source
self-homeomorphisms fixed at their boundaries; the cubical
corrections preserve their previous images; and the smoothing
Theorem preserves the source and target.  Local finiteness and the
fine properness tests in the preceding constructions apply on the
open domains throughout.
Thus surjectivity holds for every individual approximant, rather
than only for a limiting image.  Its everywhere invertible smooth
differential supplies local smooth inverses, which bijectivity
assembles into a smooth inverse on $\Lambda$.

We may also make the value power error arbitrarily small.  One
explicit way to pass from the fine local tests to this assertion is
to choose a compact subset of $\Omega$ whose complement has
arbitrarily small measure.  On the compact impose arbitrarily small
uniform value error through the preliminary constructions,
corrections and smoothing.  On the complement use the fixed bound
on the diameter of $\Lambda$.  Full-rank correction supports and
marked cubes were chosen with the additional small oscillation
requirements needed for their localized motions.  Thus none of the
last operations obstructs this value estimate.

Combining the estimates gives a final derivative error bounded by
\[
                    C_p(\eta+\gamma)(1+E)+o(1),
\]
where $o(1)$ denotes the later freely prescribed errors, after all
of their multipliers are fixed.  First choose $\eta,\gamma$ so that
the displayed leading term is smaller than the desired error share;
then choose the tails against $C_{\mathrm w}$ and the successive
absolute errors in Remark~\ref{ha:scale-order}.  The value error
and the two smoothing errors receive the remaining shares.  This
gives the prescribed bound $\eps$.  Apply the construction with
$\eps=2^{-j}$ and undo the initial reflection if needed.
The resulting sequence satisfies Equation~\eqref{main:convergence},
with each approximant in global $W^{1,p}$ and onto the same fixed target.
\end{proof}
==== END Strong-diffeomorphic-approximation-in-three-dimensions-for-p-gt-2//sections/09-high-assembly.tex ====

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@misc{OpenAILow2026,
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  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Strong-diffeomorphic-approximation-in-three-dimensions-for-1-le-p-le-2-September-24-2026/main.pdf}{OAI:Strong-diffeomorphic-approximation-in-three-dimensions-for-1-le-p-le-2-September-24-2026}},
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==== END Strong-diffeomorphic-approximation-in-three-dimensions-for-p-gt-2//references.bib ====