diff 61bede5..ee77827 simplicial_volume_scalar_bound_v2.txt
diff --git a/simplicial_volume_scalar_bound_v2.txt b/simplicial_volume_scalar_bound_v2.txt
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--- a/simplicial_volume_scalar_bound_v2.txt
+++ b/simplicial_volume_scalar_bound_v2.txt
@@ -1,6 +1,8 @@
\documentclass[11pt]{article}
\usepackage[margin=1in]{geometry}
\usepackage{amsmath,amssymb,amsthm,mathtools}
+\usepackage[T1]{fontenc}
+\usepackage{lmodern}
\usepackage{microtype}
\usepackage{graphicx}
\usepackage{tikz}
@@ -48,2586 +50,6713 @@
\maketitle
\begin{abstract}
-For every integer $n\ge 3$ there is a constant $a_n>0$, depending only
-on $n$, such that every closed connected oriented smooth $n$-manifold
-$M$ and every smooth Riemannian metric $g$ on $M$ satisfy
+For every integer $n\ge3$ there is a constant $a_n>0$, depending only on
+$n$, such that every closed connected oriented smooth $n$-manifold $M$
+and every smooth Riemannian metric $g$ on $M$ satisfy
\[
- \int_M (\Scal_g^-)^{n/2}\,dV_g \;\ge\; a_n\,\|M\|.
+ \int_M(\Scal_g^-)^{n/2}\,dV_g\ \ge\ a_n\,\|M\|.
\]
-Here $\Scal_g^-:=\max\{0,-\Scal_g\}$ and $\|M\|$ is the real simplicial
-volume of $M$. The proof combines the nonnegative scalar curvature
-vanishing theorem of the companion paper on rational inessentiality with
-a new bounded cohomology transfer pipeline.
+Here $\Scal_g^-=\max\{0,-\Scal_g\}$ and $\|M\|$ is the real simplicial
+volume of $M$. The left side is a local analytic quantity: it is
+dimensionless, so it does not change under a constant rescaling
+$g\mapsto\lambda^2g$, and it measures both how negative the scalar
+curvature becomes and over how much volume. The right side is a global
+topological invariant, independent of the metric. The theorem says that
+negative scalar curvature is a resource: without enough of it, measured
+in this integral sense, the manifold cannot carry a fundamental cycle of
+large simplicial volume.
+
+The proof has two inputs. Qualitatively, a closed manifold admitting a
+metric of nonnegative scalar curvature has zero real simplicial volume;
+this vanishing theorem is proved in the companion paper
+\cite{OpenAIScalarInessentiality2026} and is used here once, in
+\S\ref{sec:reduction-detector}. Quantitatively, we deform metrics by a
+prescribed-curvature graph flow, iterate it to build an entropy metric
+with a large energy coefficient, and convert the bounded-cohomology
+detector of $\|M\|$ into measured comparison degrees. Those degrees are
+bounded by a weighted integral that, after the deformations, is
+comparable to $\Vol_g(M)$. All constants depend only on $n$.
\end{abstract}
-
-\section*{Reader's guide}
-
-The theorem is a universal inequality whose two sides have very
-different flavour.
-
-\begin{itemize}
-\item The left side is analytic and local: $(\Scal_g^-)^{n/2}$
- integrated against the Riemannian volume form. It is invariant
- under a constant rescaling $g\mapsto\lambda^2g$ of the metric, and it
- records both how far below zero the scalar curvature dips and over
- how much volume it does so.
-\item The right side is topological and global: the real simplicial
- volume $\|M\|$ is the infimum of the $\ell^1$-norm of a finite real
- singular cycle representing $[M]$ in real homology. It depends only
- on the topology of $M$, not on the metric.
-\end{itemize}
-
-Loosely: negative scalar curvature is a resource that lets the
-fundamental class be killed in real homology. Admitting a metric of
-nonnegative scalar curvature forbids this at all, so the real
-simplicial volume must vanish \cite[Cor.~1.5]{OpenAIScalarInessentiality2026}.
-The point of this paper is to make that vanishing quantitative: a small
-integral of negative scalar curvature forces a small simplicial volume.
-
-\medskip
-
-\noindent\textbf{The strategy, in five steps.}
-
-\begin{enumerate}
-
-\item \emph{Reduce to $\Scal=-1$.}
- (\S\ref{sec:reduction-detector}.)
- A Yamabe minimizer in each conformal class has constant scalar
- curvature. If $\|M\|>0$, the vanishing input forces that constant
- to be negative, and a constant rescaling makes it $-1$. H\"older
- bounds the volume of the normalized metric by the integral of
- $(\Scal_g^-)^{n/2}$ of the original. So it is enough to prove
- \[
- \|M\| \le C_n\Vol_g(M) \quad \text{whenever} \quad \Scal_g\equiv -1.
- \]
-
-\item \emph{Detector.}
- (\S\ref{sec:reduction-detector}.)
- Bounded-cohomology duality gives a bounded group cocycle that pairs
- with $[M]$ to $I\ge\tfrac12\|M\|$. It becomes a closed $n$-form
- $\eta$ on the probability simplex, and an equivariant probability map
- $F\colon\widetilde M\to\ell^2(\mathcal L)$ pulls it back with
- integral $I$. So it is enough to bound $I$ by $\Vol_g(M)$.
-
-\item \emph{Graph deformation engine.}
- (\S\ref{sec:graphs}.)
- The technical core. Replace the metric by the one induced on the
- graph of a function, and change the weight in tandem so that the
- weighted volume $e^\phi dV_h$ stays fixed. The \emph{graph identity}
- writes the change of weighted scalar curvature as a prescribed
- mean-curvature square plus nonnegative errors. A solvability
- theorem for the prescribed mean-curvature equation lets us engineer
- that square to be negative. Sweeping the coordinates of $F$ one at
- a time gives an explicit reserve $\Pi_p$.
-
-\item \emph{Entropy iteration.}
- (\S\ref{sec:entropy}.)
- A bootstrap lifts the energy coefficient from $\sim 0$ to a
- dimensional positive value. Phased iterations then lift it to an
- arbitrary $E$ while keeping the accumulated volume expansion
- controlled and the residual gradient absolutely small. The output
- is an entropy datum $h,\phi,\rho,F$ with
- \[
- \wscal(h,\phi)\ge -C + 2W - n\log E + E\,e_h(F),\qquad
- e_h(F)^{1/2}\le E^{-1/100},
- \]
- where $W=\log(dV_h/\rho)\ge 0$.
-
-\item \emph{Degree.}
- (\S\ref{sec:degree}--\ref{sec:transfer}.)
- A measured-degree theorem bounds the volume of the preimage of a cube
- by an integral of $\exp(-\wscal/2)$ with derivative penalties.
- Applied with $L_T=b_*\sqrt E$ tangentially and $L_N=b_*\sqrt{qE}$
- normally, all powers of $E$ cancel. Averaging over the orthogonal
- group leaves a factor bounded in terms of $n$ alone.
-
-\end{enumerate}
-
-\noindent\textbf{On accessibility.}
-The full proof invokes bounded cohomology of groups, elliptic PDE, and
-geometric measure theory, none of which can be made elementary. What
-this version tries to do instead is to make the \emph{logic}
-transparent: every section opens by saying what it accomplishes and why
-the previous tool needed it; every lemma is motivated before being
-stated; and every parameter is explained as it appears. A reader
-without the background can follow what is proved and why each
-technical step is necessary, even when individual estimates require
-the cited tools to check.
-
-\tableofcontents
\section{Introduction}
\label{sec:introduction}
The \emph{real simplicial volume} of a closed connected oriented
$n$-manifold $M$ is the $\ell^1$-seminorm of its real fundamental class:
\[
- \|M\| = \inf \Bigl\{ \sum_j |a_j| : \sum_j a_j\sigma_j \in
- C_n^{\mathrm{sing}}(M;\R) \text{ represents } [M]_{\R} \Bigr\}.
+ \|M\|=\inf\Bigl\{\sum_j|a_j|:\ \sum_j a_j\sigma_j
+ \text{ is a finite real singular cycle representing }[M]_{\R}\Bigr\}.
\]
-It measures the cost of representing the fundamental class by singular
-simplices with real coefficients. It is a homotopy invariant which
-detects some kinds of nonexistence of positive scalar curvature
-\cite{Gromov1982} and which vanishes for many geometrically constrained
-manifolds.
-
-In 1986 Gromov proposed \cite[Conjecture 3.A, Remark~(B),
-Equation~(8$'$)]{Gromov1986} that the integral
+It is the cheapest $\ell^1$ cost of writing the fundamental class as a
+sum of singular simplices with real coefficients. It depends only on the
+topology of $M$ and not on a metric, it is a homotopy invariant, and it
+vanishes for many manifolds.
+
+In 1986 Gromov proposed \cite[Conjecture~3.A, Remark~(B),
+Equation~(8$'$)]{Gromov1986} that the integral of the $n/2$-power of the
+negative part of scalar curvature should bound $\|M\|$ above with a
+coefficient depending only on $n$. Writing
+\(\Scal_g^-:=\max\{0,-\Scal_g\}\), the quantity
\[
- \int_M (\Scal_g^-)^{n/2}\,dV_g
+ \int_M(\Scal_g^-)^{n/2}\,dV_g
\]
-of the $n/2$-power of the negative part of scalar curvature be an upper
-bound for the simplicial volume, with a coefficient depending only on
-$n$. The integral is dimensionless (invariant under
-$g\mapsto\lambda^2 g$), and it records both the size and the
-concentration of the region on which the scalar curvature is negative.
+is invariant under a constant rescaling $g\mapsto\lambda^2g$, so it is
+dimensionless; and it records both how negative the scalar curvature
+becomes and over how much volume it does so.
\begin{theorem}[Main theorem]
\label{thm:main}
-For every integer $n\ge 3$ there is a constant $a_n>0$, depending only
-on $n$, such that every closed connected oriented smooth $n$-manifold
-$M$ and every smooth Riemannian metric $g$ on $M$ satisfy
+For every integer $n\ge3$ there is a constant $a_n>0$, depending only on
+$n$, such that every closed connected oriented smooth $n$-manifold $M$
+and every smooth Riemannian metric $g$ on $M$ satisfy
\begin{equation}
- \int_M (\Scal_g^-)^{n/2}\,dV_g \;\ge\; a_n\,\|M\|.
+ \int_M(\Scal_g^-)^{n/2}\,dV_g\ \ge\ a_n\,\|M\|.
\label{eq:main}
\end{equation}
\end{theorem}
-The constant is uniform in both $M$ and $g$. A representative case is
-$g$ with $\Scal_g\ge-n(n-1)$ (the natural normalization in dimension
-$n$): then $\Scal_g^-\le n(n-1)$, so the theorem specializes to
+The constant is uniform in both $M$ and $g$. As a representative case,
+if $\Scal_g\ge-n(n-1)$, which is the natural normalization in dimension
+$n$, then $\Scal_g^-\le n(n-1)$ and the theorem reads
\[
- \Vol_g(M) \ge a_n\,[n(n-1)]^{-n/2}\,\|M\|.
+ \Vol_g(M)\ \ge\ a_n\,[n(n-1)]^{-n/2}\|M\|.
\]
-But the theorem is stronger: it allows very negative scalar curvature
-\textit{as long as it is concentrated}. A small ball with huge
-negative curvature contributes little to the integral.
+The integral form is strictly stronger than this. It permits scalar
+curvature to be arbitrarily negative, provided the region where it is
+negative has small volume: a tiny ball of huge negative curvature
+contributes almost nothing to $\int_M(\Scal_g^-)^{n/2}dV_g$.
-The theorem is also closely related to the pointwise-uniform version
-\[
- \Scal_g\ge-\kappa^2 \quad\Longrightarrow\quad
- \|M\|\le C_n\kappa^n\Vol_g(M)\quad(\kappa\ge 0).
-\]
-For a fixed dimension $n\ge 3$, the integral version of the theorem
-and the pointwise-uniform version are equivalent by a Yamabe-type
-rescaling argument; see \cite[Lemma 2.1]{MinZhengZhu2026}.
+Theorem~\ref{thm:main} is closely related to the pointwise-uniform
+assertion
+\begin{equation}
+ \Scal_g\ge-\kappa^2\quad\Longrightarrow\quad
+ \|M\|\le C_n\kappa^n\Vol_g(M),\qquad \kappa\ge0 .
+ \label{eq:pointwise-form}
+\end{equation}
+For a fixed dimension $n\ge3$ the integral statement for all metrics on
+all such manifolds and \eqref{eq:pointwise-form} for all metrics on all
+such manifolds are equivalent by a Yamabe rescaling; see
+\cite[Lemma~2.1]{MinZhengZhu2026}.
-\subsection*{Context and prior results}
+\subsection*{Prior results}
-Several cases of the conjecture were already known or partial.
+Several related statements were known before the general case.
\begin{itemize}
\item Braun and Sauer \cite{BraunSauer2021} prove a macroscopic
- counterpart when the volumes of unit balls in the universal cover
- have a uniform upper bound; their constant depends also on that
- bound.
-\item In the qualitative case $\Scal_g\ge 0$, Ma, Wang, Xie, Yu and
- Zhu \cite{MaWangXieYuZhu2026} prove vanishing of $\|M\|$ when the
- universal cover is spin. This is the companion result we build on.
+ counterpart when the volumes of unit balls in the universal cover are
+ uniformly bounded above; their constant depends also on that bound.
+\item Ma, Wang, Xie, Yu and Zhu \cite{MaWangXieYuZhu2026} prove that
+ $\|M\|=0$ when the universal cover is spin and $\Scal_g\ge0$. This is
+ the qualitative companion result used below.
\item Min, Zheng and Zhu \cite{MinZhengZhu2026} prove the integral
- estimate for every metric on every closed K\"ahler surface with
- coefficient $27/2$, and for an infinite family of non-K\"ahler
+ estimate with coefficient $27/2$ for every metric on every closed
+ K\"ahler surface, and for an infinite family of non-K\"ahler
symplectic four-manifolds.
\end{itemize}
-The present proof uses the qualitative nonnegative-scalar-curvature
-vanishing theorem of the companion manuscript
-\cite[Corollary 1.5]{OpenAIScalarInessentiality2026}:
-
-\begin{theorem}[Vanishing, \cite{OpenAIScalarInessentiality2026}]
-\label{thm:vanishing-input}
-Let $M$ be a closed connected oriented smooth manifold of positive
-dimension. If $M$ admits a smooth Riemannian metric with nonnegative
-scalar curvature, then $\|M\|=0$.
-\end{theorem}
-
-The work of this paper is the passage from this qualitative statement
-to a quantitative, dimension-uniform estimate.
+The one external input used in this paper is the qualitative vanishing
+theorem of the companion manuscript
+\cite[Corollary~1.5]{OpenAIScalarInessentiality2026}, stated precisely as
+Theorem~\ref{thm:vanishing-input} in \S\ref{sec:reduction-detector}: a
+closed connected oriented manifold of positive dimension admitting a
+metric of nonnegative scalar curvature has zero real simplicial volume.
+Everything else is proved here. The work is the passage from that
+qualitative statement to a quantitative estimate whose constant is
+uniform in the topology and in the metric.
-\subsection*{Overview of the argument}
+\subsection*{The logical spine}
-The reader's guide at the front of the paper lays out the five main
-steps. Here we summarise them in the language of the sections below.
+The proof has five stages. Each is stated as a theorem or proposition
+whose hypotheses are no stronger than what the previous stage delivers,
+and the last stage composes them.
\begin{enumerate}
-\item \emph{Yamabe reduction.}
- \S\ref{sec:reduction-detector}.
-\item \emph{The bounded-cohomology detector.}
- \S\ref{sec:reduction-detector}.
-\item \emph{Graph deformations and sweeps.}
- \S\ref{sec:graphs}.
-\item \emph{Entropy metrics and phase iteration.}
- \S\ref{sec:entropy}.
-\item \emph{Degree estimate and transfer.}
- \S\ref{sec:degree}--\ref{sec:transfer}.
+
+\item \emph{Conformal reduction to $\Scal_g=-1$}
+ (\S\ref{sec:reduction-detector}, Proposition~\ref{prop:yamabe-reduction}).
+ In every conformal class the Yamabe minimizer has constant scalar
+ curvature. If $\|M\|>0$, the vanishing theorem forces that constant to
+ be negative, and a constant rescaling normalizes it to $-1$.
+ H\"older's inequality then bounds the volume of the normalized metric
+ by $\int_M(\Scal_g^-)^{n/2}dV_g$ of the original metric. So
+ Theorem~\ref{thm:main} follows from the normalized estimate
+ \begin{equation}
+ \|M\|\le C_n\Vol_g(M)\qquad\text{whenever }\Scal_g\equiv-1,
+ \label{eq:normalized-target}
+ \end{equation}
+ which is Theorem~\ref{thm:scalar-volume}.
+
+\item \emph{A detector for $\|M\|$.}
+ (\S\ref{sec:reduction-detector}, Proposition~\ref{prop:detector}).
+ Duality between the $\ell^1$-seminorm on homology and the supremum
+ seminorm on bounded cohomology, together with the Mapping Theorem,
+ produces a bounded group cocycle $c$ of norm at most one whose value
+ $I$ on the fundamental class is at least $\|M\|/2$. The cocycle
+ becomes a closed $n$-form $\eta$ on the simplex of probability vectors,
+ and an equivariant probability map $F$ from the universal cover pulls
+ $\eta$ back with total integral $I$. So \eqref{eq:normalized-target}
+ follows once we bound $I$ by $C_n\Vol_g(M)$.
+
+\item \emph{The graph engine.}
+ (\S\ref{sec:graphs}). Replace a metric $h$ by the metric induced on the
+ graph of a function $f$, and shift the weight $\phi$ so that the
+ weighted volume $e^\phi dV_h$ does not change. The weighted graph
+ identity, Lemma~\ref{lem:graph-identity}, writes the change of weighted
+ scalar curvature as a square of a prescribed quantity $D$ plus
+ nonnegative errors. Since $D$ can be prescribed, a solvability theorem
+ for the prescribed weighted mean curvature equation
+ (Lemma~\ref{lem:graph-prescription}, localized in
+ Lemma~\ref{lem:graph-localization}) lets us make that square
+ systematically positive. Sweeping the coordinates of $F$ in order
+ yields Proposition~\ref{prop:sweep}: one pass improves the energy of $F$
+ at a cost bounded by an explicit nonnegative reserve $\Pi_p$.
+
+\item \emph{Iteration to an entropy metric.}
+ (\S\ref{sec:entropy}). A conformal bootstrap raises the energy
+ coefficient from a very small value to a fixed dimensional one
+ (Lemma~\ref{lem:entropy-bootstrap}). Phases of sweeps then raise it to
+ an arbitrary prescribed $E$ while keeping the accumulated volume
+ expansion controlled (Lemma~\ref{lem:entropy-accumulation}) and the
+ residual gradient absolutely small (Lemma~\ref{lem:entropy-phase-exit}).
+ The output is an entropy datum with
+ \[
+ \wscal(h,\phi)\ge-C+2W-n\log E+E\,e_h(F),\qquad
+ e_h(F)^{1/2}\le E^{-1/100},
+ \]
+ where $W=\log(dV_h/\rho)\ge0$ and $\rho\le dV_g$
+ (Theorem~\ref{thm:entropy}).
+
+\item \emph{Degree and transfer.}
+ (\S\ref{sec:degree} and \S\ref{sec:transfer}). The measured degree
+ estimate, Theorem~\ref{thm:degree}, bounds the signed degree of a map
+ from a measured family of $d$-manifolds to a $d$-cube by an integral of
+ $\exp(-\wscal/2)$ times a penalty for each coordinate gradient. The
+ transfer of \S\ref{sec:transfer} builds such a map whose degree is the
+ detector pairing $I$ and whose tangential and normal lengths are
+ $b_*\sqrt E$ and $b_*\sqrt{qE}$. The factor $E^{n/2}$ produced by the
+ degree bound cancels the $E^{-n/2}$ from the product volume, the
+ remaining $e^{-W}$ cancels the volume expansion, and what survives is
+ an integral of $\rho$, hence $\Vol_g(M)$
+ (Theorem~\ref{thm:scalar-volume}).
\end{enumerate}
-The full proof of Theorem~\ref{thm:main} appears in \S\ref{sec:transfer},
-combining the detector construction of \S\ref{sec:reduction-detector}
-with the technical inputs of
-Sections~\ref{sec:graphs}--\ref{sec:degree}.
+The last step contains the only place where the exponent of $E$ matters:
+the tangent directions of the comparison family contribute the volume
+$L_T^m$, the normal directions contribute the reciprocal
+$L_T^{-m}L_N^{-n}$ of the degree denominator, and the entropy inequality
+contributes $E^{n/2}e^{-W}$; the three factors multiply to
+$b_*^{-n}q^{-n/2}\rho$, in which no power of $E$ appears.
+
+\subsection*{How to read this paper}
+
+The three technical engines are Sections~\ref{sec:graphs},
+\ref{sec:entropy} and \ref{sec:degree}. They are independent of each
+other in the following sense: \S\ref{sec:graphs} uses only Riemannian
+geometry and elliptic PDE; \S\ref{sec:entropy} uses only
+\S\ref{sec:graphs}; and \S\ref{sec:degree} uses only geometric measure
+theory, with no reference to simplicial volume or bounded cohomology.
+Only the final \S\ref{sec:transfer} uses all of them. A reader willing
+to accept the three engines as black boxes can read
+\S\ref{sec:reduction-detector}, \S\ref{sec:transfer} and the statements
+of Theorem~\ref{thm:entropy} and Theorem~\ref{thm:degree} for a complete
+proof of Theorem~\ref{thm:main}.
+
+Two further conventions are used throughout. Constants written
+$C_n,c_n,C$ are positive and depend only on the dimension unless a
+subscript or an explicit dependence is indicated; their values may change
+from line to line, and no effort is made to keep them sharp. Integrals
+over measured families of leaves always include the transverse measure;
+the construction is given in \S\ref{sec:reduction-detector}.
+
+\subsection*{The toolkit, in one page}
+
+Four techniques carry the proof, and the logical spine above says where
+each is used. Yamabe theory supplies, in every conformal class, a metric
+of constant scalar curvature whose sign is the Yamabe invariant; the
+vanishing theorem forces that sign to be negative, and rescaling
+normalizes the constant to $-1$. Bounded cohomology supplies a bounded
+cocycle of norm at most one, pairing with the fundamental class in an
+amount $I\ge\|M\|/2$, and a closed form on a simplex of probability
+vectors that an equivariant probability map pulls back with the same
+integral $I$. Graph deformations supply the identity
+\[
+ \wscal(h',\phi')-\wscal(h,\phi)
+ =D^2-2U(D)+|\II|^2+|d\log L|^2
+\]
+together with the fact that $D$ can be prescribed by solving the
+quasilinear equation $D=p(x,z,\nabla z)$ with $\partial_zp<0$; this is
+what makes the change of weighted scalar curvature positive on demand.
+Geometric measure theory supplies the measured degree and its bound by a
+weighted integral with gradient penalties, obtained by slicing, weighted
+perimeter minimization, coarea with the correct lapse, and varifold
+compactness with Allard regularity.
\section{Reduction and the bounded cohomology detector}
\label{sec:reduction-detector}
-We separate the conformal reduction from the quantitative part of the
-proof. We then construct a closed $n$-form on a simplex of probability
-vectors whose pairing with an equivariant map from the universal cover
-detects simplicial volume. That form remains fixed while the metric
-and the probability map are deformed.
+We first separate the conformal reduction from the quantitative part of
+the proof. We then construct a differential form on a simplex of
+probability vectors that detects simplicial volume. This form will remain fixed while
+the metric and the probability map are deformed.
\subsection{The conformal reduction}
-\begin{proposition}[Yamabe reduction]
+We use the following result, stated here in the precise form needed.
+
+\begin{theorem}[OpenAI, Corollary 1.5 of
+ \cite{OpenAIScalarInessentiality2026}]
+\label{thm:vanishing-input}
+Let \(M\) be a closed connected oriented smooth manifold of positive
+dimension. If \(M\) admits a smooth Riemannian metric with nonnegative
+scalar curvature, then its real simplicial volume is zero.
+\end{theorem}
+
+In particular, the hypothesis permits scalar curvature to vanish
+identically. Theorem~\ref{thm:vanishing-input} is the only result from
+the companion manuscript invoked below without proof.
+
+The Yamabe reformulation of the scalar curvature inequality is
+standard; \cite[Lemma 2.1]{MinZhengZhu2026} records the equivalence of
+its uniform pointwise and integral forms. We include the short
+classwise reduction that identifies the use of the vanishing input.
+
+\begin{proposition}
\label{prop:yamabe-reduction}
-Fix $n\ge 3$. Suppose a constant $C_n<\infty$ has the following
-property: every closed connected oriented $n$-manifold $M$ with a
-smooth metric $h$ of constant scalar curvature $-1$ satisfies
+Fix \(n\ge3\). Suppose that a constant \(C_n<\infty\) has the following
+property: every closed connected oriented \(n\)-manifold with a smooth
+metric \(h\) of constant scalar curvature \(-1\) satisfies
\begin{equation}
\|M\|\le C_n\Vol_h(M).
\label{eq:scalar-minus-one-target}
\end{equation}
-Then every smooth Riemannian metric $g$ on every such $M$ satisfies
+Then every smooth Riemannian metric \(g\) on every such manifold
+satisfies
\begin{equation}
- \|M\|\le C_n\int_M (\Scal_g^-)^{n/2}\,dV_g.
+ \|M\|\le C_n\int_M(\Scal_g^-)^{n/2}\,dV_g .
\label{eq:yamabe-transfer}
\end{equation}
\end{proposition}
-\noindent\emph{Idea.} The Yamabe theorem gives, in each conformal
-class, a metric of constant scalar curvature. Under $\|M\|>0$ the
-vanishing theorem forces that constant to be negative; after rescaling
-it is $-1$. H\"older's inequality then bounds the volume of such a
-constant-scalar-curvature metric by the integral in
-\eqref{eq:yamabe-transfer} for the original metric.
-
-\begin{proof}[Proof of Proposition~\ref{prop:yamabe-reduction}]
-The assertion is immediate if $\|M\|=0$, so assume $\|M\|>0$. Write
-$Y(M,[g])$ for the Yamabe constant of the conformal class of $g$:
+\begin{proof}
+The assertion is immediate when \(\|M\|=0\), so assume that
+\(\|M\|>0\). Write \(Y(M,[g])\) for the Yamabe constant of the
+conformal class of \(g\):
\[
- Y(M,[g])=\inf_{u\in C^\infty(M),\,u>0}
- \frac{\displaystyle\int_M
- \Bigl(\tfrac{4(n-1)}{n-2}|du|_g^2+\Scal_g u^2\Bigr)dV_g}
- {\displaystyle\Bigl(\int_M u^{2n/(n-2)}\,dV_g\Bigr)^{(n-2)/n}}.
+Y(M,[g])=
+\inf_{u\in C^\infty(M),\,u>0}
+\frac{\displaystyle\int_M
+ \left(\frac{4(n-1)}{n-2}|du|_g^2+\Scal_g u^2\right)dV_g}
+{\displaystyle\left(\int_Mu^{2n/(n-2)}\,dV_g\right)^{(n-2)/n}}.
\]
-The Yamabe theorem (Yamabe, Trudinger, Aubin, Schoen
-\cite{Yamabe1960,Trudinger1968,Aubin1976,Schoen1984}; see also
-\cite[Section~11]{LeeParker1987}) produces a conformal metric attaining
-this value with constant scalar curvature. Its scalar curvature is
-negative by Theorem~\ref{thm:vanishing-input}; after a constant
-rescaling it becomes a metric $h$ with $\Scal_h=-1$.
-
-Evaluating the Yamabe quotient at the constant function for this
-minimizer gives $-Y(M,[g])=\Vol_h(M)^{2/n}$. For each positive $u$,
-H\"older's inequality gives
+The Yamabe theorem, developed through the work of Yamabe, Trudinger,
+Aubin, and Schoen \cite{Yamabe1960,Trudinger1968,Aubin1976,Schoen1984},
+gives a conformal metric attaining this value and having constant
+scalar curvature; see also \cite[Section 11]{LeeParker1987}.
+Its scalar curvature is negative by
+Theorem~\ref{thm:vanishing-input}. After a constant rescaling, it becomes a
+metric \(h\) with \(\Scal_h=-1\). Evaluating the Yamabe quotient at the
+constant function for this minimizing metric gives
+\[
+ -Y(M,[g])=\Vol_h(M)^{2/n}.
+\]
+For each positive \(u\), H\"older's inequality gives
\[
- \int_M \Scal_g u^2\,dV_g \ge
- -\left(\int_M (\Scal_g^-)^{n/2}\,dV_g\right)^{2/n}
- \left(\int_M u^{2n/(n-2)}\,dV_g\right)^{(n-2)/n}.
+\int_M\Scal_g u^2\,dV_g
+\ge
+-\left(\int_M(\Scal_g^-)^{n/2}\,dV_g\right)^{2/n}
+ \left(\int_Mu^{2n/(n-2)}\,dV_g\right)^{(n-2)/n}.
\]
-The gradient term in the Yamabe quotient is nonnegative, so taking the
-infimum yields
+The gradient term in the Yamabe quotient is nonnegative. Taking the
+infimum therefore yields
\[
- -Y(M,[g]) \le \left(\int_M (\Scal_g^-)^{n/2}\,dV_g\right)^{2/n}.
+-Y(M,[g])
+\le \left(\int_M(\Scal_g^-)^{n/2}\,dV_g\right)^{2/n}.
\]
-Consequently $\Vol_h(M)\le\int_M(\Scal_g^-)^{n/2}\,dV_g$, and
-\eqref{eq:scalar-minus-one-target} implies \eqref{eq:yamabe-transfer}.
+Consequently \(\Vol_h(M)\le\int_M(\Scal_g^-)^{n/2}\,dV_g\), and
+\eqref{eq:scalar-minus-one-target} implies
+\eqref{eq:yamabe-transfer}.
\end{proof}
-So it suffices to prove
-$\|M\|\le C_n\Vol_g(M)$ uniformly whenever $\Scal_g=-1$. We work in
-this normalization from now on.
-\subsection{The bounded cohomology detector}
+For the rest of the proof we fix \(M^n\) and \(g\) with
+\(\Scal_g=-1\), and prove \eqref{eq:scalar-minus-one-target}. We may
+continue to assume \(\|M\|>0\).
-Let $\Gamma=\pi_1(M)$.
+\subsection{A bounded cocycle and its probability form}
-\medskip\noindent\textbf{Bounded cohomology duality.}
-For every integer $n\ge1$ and every closed oriented $n$-manifold $M$,
+Let \(\Gamma=\pi_1(M)\). This group is countable because \(M\) has a
+finite triangulation. We use two standard facts about bounded
+cohomology with trivial real coefficients. First, duality between the
+\(\ell^1\)-seminorm on homology and the supremum seminorm on bounded
+cohomology gives
+\begin{equation}
+\|M\|=
+\sup\left\{\langle\beta,[M]_{\R}\rangle:
+ \beta\in H_b^n(M;\R),\ \|\beta\|_\infty\le1\right\}.
+\label{eq:bounded-duality}
+\end{equation}
+See \cite[Section 1.1]{Gromov1982}.
+Second, the isometric Mapping Theorem identifies \(H_b^k(M;\R)\) with
+\(H_b^k(\Gamma;\R)\) for every \(k\ge2\), through the classifying map.
+The version used here applies to connected countable CW spaces and
+preserves the seminorms. We use the Mapping Theorem of Gromov
+\cite[Section 3.1]{Gromov1982} and its canonical group formulation in
+\cite[Theorems 8.3 and 8.6]{IvanovNotes}.
+
+We use the homogeneous model for group cochains. Thus an \(n\)-cochain
+is a bounded function on \(\Gamma^{n+1}\), invariant under simultaneous
+left translation, and its coboundary is the alternating sum obtained by
+omitting one argument. Alternation is a cochain projection of norm at
+most one and is equivariantly cochain homotopic to the identity through
+maps bounded in each degree. It therefore preserves bounded cohomology
+classes \cite[Section~4.10, Proposition~4.26]{Frigerio2016}.
+By \eqref{eq:bounded-duality}, the Mapping Theorem, and
+approximation of the seminorm by cocycle representatives, there is a
+homogeneous alternating cocycle
\[
- \|M\| = \sup\Bigl\{\langle[\alpha],[M]_{\R}\rangle :
- [\alpha]\in H_b^n(\Gamma;\R), \ \|\alpha\|_\infty\le1\Bigr\},
+ c:\Gamma^{n+1}\longrightarrow\R,\qquad \|c\|_\infty\le1,
\]
-where $[M]_{\R}\in H_n(\Gamma;\R)$ is the real fundamental class. This
-is standard (\cite{IvanovNotes, Gromov1986}). At the cocycle level
-this means: for some alternating homogeneous cocycle
-$c:\Gamma^{n+1}\to\R$ with $\|c\|_\infty\le1$, the pairing with the
-fundamental class is at least $\tfrac12\|M\|$:
+whose corresponding bounded singular cohomology class has value
\begin{equation}
- I:=\langle[c],[M]_{\R}\rangle\ge\tfrac12\|M\|.
+ I:=\langle[c],[M]_{\R}\rangle\ge\frac12\|M\|.
\label{eq:detector-value}
\end{equation}
-The factor $\tfrac12$ records the loss from choosing a class with
-value at least $\tfrac34\|M\|$, alternating, and normalizing by
-$1+\varepsilon$ with $\varepsilon\le\tfrac12$.
-
-\medskip\noindent\textbf{From cocycle to form.}
-Fix a set $\mathcal L$ with a free $\Gamma$-action (in our main
-application $\mathcal L=\Gamma\times\{1,\ldots,s\}$ for a finite $s$).
-Evaluate $c$ on $\mathcal L^{\,n+1}$ by projection to $\Gamma^{n+1}$,
-and extend multilinearly to finitely supported vectors on $\mathcal L$.
-
-Let $\Delta_{\mathcal L}$ be the probability simplex of finitely
-supported mass distributions:
+For completeness, one can choose a class in
+\eqref{eq:bounded-duality} with value at least \(3\|M\|/4\), choose a
+group representative of norm at most \(1+\varepsilon\), alternate it,
+and divide by \(1+\varepsilon\), with \(0<\varepsilon\le1/2\). Replacing
+the cocycle by its negative first, if necessary, fixes the sign.
+
+We allow a finite number of copies of every group label. More
+precisely, let \(\mathcal L=\Gamma\times\{1,\ldots,s\}\), with the free
+left \(\Gamma\)-action on the first factor, and evaluate \(c\) on
+\(\mathcal L^{n+1}\) by projection to \(\Gamma^{n+1}\). The same
+notation will be used when further finite copies of \(\mathcal L\) are
+introduced. Extend \(c\) multilinearly to finitely supported vectors
+on \(\mathcal L\).
+
+Let
\[
- \Delta_{\mathcal L}=\Bigl\{p=(p_\ell)_{\ell\in\mathcal L} :
- p_\ell\ge0,\ \sum_\ell p_\ell=1,\ p\text{ finitely supported}\Bigr\}.
+\Delta_{\mathcal L}=
+\left\{p=(p_\ell)_{\ell\in\mathcal L}:
+ p_\ell\ge0,\ \sum_\ell p_\ell=1,\ p\text{ has finite support}\right\}.
\]
-On each finite face define the $n$-form
+On each finite-dimensional face define the \(n\)-form
\begin{equation}
- \eta_p(v_1,\ldots,v_n)=n!\,c(p,v_1,\ldots,v_n).
- \label{eq:probability-form}
-\end{equation}
-These definitions agree on common faces, so $\eta$ is a well-defined
-closed $n$-form on $\Delta_{\mathcal L}$.
-
-\medskip\noindent\textbf{Root probability maps.}
-A \emph{root probability map} on a Riemannian manifold $X$ is a map
-$F=(F_\ell):X\to\ell^2(\mathcal L)$ with $F_\ell\ge0$,
-$\sum_\ell F_\ell^2=1$, locally finite support, and locally $L^2$. Its
-square $p_\ell=F_\ell^2$ takes values in $\Delta_{\mathcal L}$.
-
-\medskip\noindent\textbf{Equivariant extension.}
-Let $(\Omega_0,\mu_0)$ be a standard Borel probability space with a
-measure-preserving $\Gamma$-action. The measured family
-$\mathcal M_0=(\widetilde M\times\Omega_0)/\Gamma$ introduced above
-supports a natural equivariant extension
-$F_0$ of any $F:\widetilde M\to\ell^2(\mathcal L)$ obtained by
-$\Gamma$-invariance. When $F$ has locally finite support and finite
-energy on each compact set, the same holds for $F_0$, and
-\[
- \int_{\mathcal M_0}F_0^*\eta=\int_{\widetilde M/\Gamma}F^*\eta
- =\langle[\alpha],[M]_{\R}\rangle = I.
-\]
-Thus we may always pass to a measured equivariant family without
-changing the pairing.
-
-\medskip\noindent\textbf{The reduction.}
-Theorem~\ref{thm:main} is equivalent to the estimate
-\begin{equation}
- I \le C_n \Vol_g(M),
- \label{eq:detector-cost}
+\eta_p(v_1,\ldots,v_n)=n!\,c(p,v_1,\ldots,v_n).
+\label{eq:probability-form}
\end{equation}
-with $C_n$ depending only on $n$. The rest of the paper establishes this
-in two stages.
-
-First, the graph sweep of \S\ref{sec:graphs} and the entropy iteration
-of \S\ref{sec:entropy} produce, for every $\gamma>0$, a measured family
-$\mathcal M$, a metric $h$, a weight $\phi$, a density $\rho\le dV_g$,
-a root probability map $F$, and a coefficient $E\ge1$ satisfying
+These definitions agree on common faces.
+
+We call a map \(F=(F_\ell)\) with nonnegative coordinates and
+\(\sum_\ell F_\ell^2=1\) a \emph{root probability map}. For the maps
+used here, whose coordinate supports are locally finite, the squared
+coordinates \(p_\ell=F_\ell^2\) give the associated map to
+\(\Delta_{\mathcal L}\).
+
+\begin{proposition}
+\label{prop:detector}
+The form \(\eta\) is closed on every finite-dimensional face of
+\(\Delta_{\mathcal L}\). There is a smooth \(\Gamma\)-equivariant map
+\[
+ F:\widetilde M\longrightarrow\ell^2(\mathcal L),\qquad
+ F_\ell\ge0,\qquad \sum_\ell F_\ell^2=1,
+\]
+with finitely many label orbits and locally finite compact supports for
+its coordinates. If \(p_\ell=F_\ell^2\), then the descended form
+satisfies
\begin{equation}
- \wscal(h,\phi)\ge -C_{\mathrm{ent}}+2W-n\log E+E\,e_h(F),
- \label{eq:entropy-overview}
+ \int_M p^*\eta=I.
+ \label{eq:detector-integral}
\end{equation}
-where $W=\log(dV_h/\rho)\ge0$ and $e_h(F)^{1/2}\le E^{-1/100}$. The
-constant $C_{\mathrm{ent}}$ depends only on $n$.
+\end{proposition}
-Second, the degree estimate of \S\ref{sec:degree}, combined with the
-cubical transfer of \S\ref{sec:transfer}, shows that for these data
-\begin{equation}
- I\le K_n e^{C_{\mathrm{ent}}/2}\int_{\mathcal M}\rho,
- \label{eq:transfer-overview}
-\end{equation}
-with $K_n$ depending only on $n$. Since $\rho\le dV_g$ and $dV_g$
-integrates to $\Vol_g(M)$, \eqref{eq:transfer-overview} gives
-$I\le K_n e^{C_{\mathrm{ent}}/2}\Vol_g(M)$. Combined with
-$I\ge\tfrac12\|M\|$, this proves Theorem~\ref{thm:main} with
-$a_n^{-1}=2K_n e^{C_{\mathrm{ent}}/2}$.
-\section{Graph deformations and one sweep}
-\label{sec:graphs}
+\begin{proof}
+Write \(\epsilon(v)=\sum_\ell v_\ell\) for the mass of a finitely
+supported vector. Multilinear extension of the cocycle equation gives
+\[
+0=\sum_{j=0}^{n+1}(-1)^j\epsilon(w_j)
+ c(w_0,\ldots,\widehat w_j,\ldots,w_{n+1}).
+\]
+Choose \(w_0\) of mass one and let \(w_1,\ldots,w_{n+1}\) have mass
+zero. It follows that \(c(w_1,\ldots,w_{n+1})=0\). All tangent vectors
+to a probability simplex have mass zero. Differentiating
+\eqref{eq:probability-form} now shows \(d\eta=0\).
+
+To construct \(F\), choose finitely many evenly covered open sets in
+\(M\) and smooth nonnegative bumps subordinate to them whose nonzero
+sets cover \(M\). Lift each bump to one chosen sheet and take all its
+deck translates, denoting the resulting functions by \(b_\ell\).
+Their supports are compact and locally finite on \(\widetilde M\).
+Set
+\[
+ F_\ell=\frac{b_\ell}{\left(\sum_j b_j^2\right)^{1/2}}.
+\]
+The denominator is positive, and the stated properties follow.
+
+Choose a finite oriented triangulation of \(M\). On its lift, assign
+one label to every vertex equivariantly, and let \(p^{\mathrm{aff}}\)
+be the probability map affine in barycentric coordinates on each
+simplex. The homotopy
+\((1-t)p+t p^{\mathrm{aff}}\) is equivariant and has locally finite
+support. Applying Stokes' theorem to each simplex times \([0,1]\), and
+cancelling the common faces, shows that its pullback integral is
+constant. On a simplex with ordered vertex labels
+\(\ell_0,\ldots,\ell_n\), write
+\[
+p^{\mathrm{aff}}=e_{\ell_0}
++\sum_{j=1}^nt_j(e_{\ell_j}-e_{\ell_0}).
+\]
+Alternation makes the value of
+\(c(p^{\mathrm{aff}},dp^{\mathrm{aff}},\ldots,dp^{\mathrm{aff}})\)
+on the coordinate frame constant and equal to
+\(c(\ell_0,\ldots,\ell_n)\). The coordinate simplex has volume
+\(1/n!\), so the factor \(n!\) in \eqref{eq:probability-form} makes its
+integral exactly this cocycle value. Summing over the oriented
+fundamental triangulation gives \eqref{eq:detector-integral}. The
+identification with the chosen group class is the usual cochain
+realization of the Mapping Theorem: an equivariant vertex labeling on
+the universal cover evaluates the homogeneous group cochain, and two
+such labelings give cochain-homotopic maps by the simplex prism
+construction.
+\end{proof}
-The metric-changing step of the proof replaces a metric $h$ by the
-metric on the graph of a smooth function, with the weight shifted so
-that the weighted volume $e^\phi dV_h$ is preserved. The graph
-identity (Prop.~\ref{prop:graph-identity}) expresses the change of
-$\wscal$ as a tunable square plus nonnegative terms. Prescribing the
-tuned quantity then gives a controllable gain. One full sweep through
-the coordinates of a root probability map is Prop.~\ref{prop:sweep}.
-
-Throughout this section $h$ is a Riemannian metric and $\phi$ a smooth
-weight. Set $\wscal(h,\phi)=\Scal_h-2\Delta_h\phi-|d\phi|_h^2$, with
-$\diver_\phi V=e^{-\phi}\diver(e^\phi V)$ and
-$\Delta_\phi=\diver_\phi\nabla$. For $\phi=-f$, $\wscal$ is
-Perelman's modified scalar curvature of $e^{-f}dV_h$
-\cite[Section~1.3]{Perelman2002}.
+\subsection{Measured equivariant families}
-\subsection{The graph identity}
+The deformations below use auxiliary probability choices. We record
+their integration convention so that the detector value remains
+unambiguous.
-For a smooth function $f$ on $(X,h)$ set
-$L=(1+|df|_h^2)^{1/2}$, $U=\nabla f/L$, $D=-\diver_\phi U$,
-$h_f=h+df\otimes df$, $\phi_f=\phi-\log L$. Since $dV_{h_f}=L\,dV_h$,
-the weight shift makes the weighted volume invariant:
+Let \((\Omega,\mu)\) be a standard Borel probability space with a
+measure-preserving \(\Gamma\)-action. A measured equivariant family
+over \(M\) consists of \(\Gamma\)-equivariant Borel data on
+\(\widetilde M\times\Omega\), smooth in the \(\widetilde M\) variable.
+Geometrically one may regard the diagonal quotient
+\[
+ \mathcal M=(\widetilde M\times\Omega)/\Gamma
+\]
+as a space whose leaves are ordinary smooth manifolds. If
+\(\Gamma_\omega\) is the stabilizer of \(\omega\), the corresponding
+leaf is \(\widetilde M/\Gamma_\omega\); the deck action is free and
+proper in the \(\widetilde M\) factor.
+
+Related equivariant measured bundles are central to the macroscopic
+volume argument of Braun and Sauer \cite[Sections 1.2 and 4]{BraunSauer2021}.
+Here we need only Borel transverse data and use the following direct
+integration convention.
+
+Choose a finite evenly covered
+atlas \(U_a\) of \(M\), chosen lifts \(\widetilde U_a\), and a smooth
+partition of unity \(\chi_a\) subordinate to the atlas. For an
+integrable equivariant leafwise \(n\)-form \(\alpha\), set
\begin{equation}
- e^{\phi_f}dV_{h_f}=e^{\phi}dV_h.
- \label{eq:graph-volume-preservation}
+\int_{\mathcal M}\alpha
+ =
+\sum_a\int_\Omega\int_{\widetilde U_a}
+ (\chi_a\circ\pi)\,\alpha_\omega\,d\mu(\omega).
+\label{eq:measured-integral}
\end{equation}
+Equivariance and invariance of \(\mu\) show that this value is
+independent of the lifts and partition. The same convention applies
+to nonnegative densities. We write \(dV_h\) for the resulting measured
+Riemannian density when \(h=h_\omega\) is a leafwise metric.
+
+The data used in the first deformation will have bounds in a fixed
+finite lifted atlas that are uniform over \(\omega\). These bounds
+may depend on the instance: the original manifold and metric and the
+finite sequence of constructions already made. For the forms used
+here, these bounds ensure integrability over the compact base and
+justify differentiation and Fubini. Leafwise Stokes' theorem follows from
+\eqref{eq:measured-integral}: apply ordinary Stokes' theorem in each
+lifted chart to the partitioned form; the terms containing
+\(d\chi_a\) cancel on overlaps after the measure-preserving changes of
+transverse coordinate.
+
+In particular, suppose that two equivariant probability maps
+\(p^0,p^1\) have uniformly locally finite coordinate supports and
+uniform leafwise derivative bounds. Their affine homotopy has the
+same properties. Stokes' theorem on the measured product with
+\([0,1]\) gives
+\[
+\int_{\mathcal M}(p^0)^*\eta
+ =
+\int_{\mathcal M}(p^1)^*\eta.
+\]
+Thus every probability map constructed below continues to detect the
+same number \(I\). Enlarging \(\Omega\) by an independent probability
+space also preserves the integral by Fubini. We may discard invariant
+null sets whenever needed. More general measured spaces of leaves
+will enter the degree estimate in Section~\ref{sec:degree}; their chart and
+integration conventions will be specified there.
+\section{Weighted graph deformations and a sweep}
+\label{sec:graphs}
-\begin{proposition}[Graph identity]
-\label{prop:graph-identity}
-Let $f$ be smooth on $(X,h)$ with $h$ of bounded local geometry, let
-$\phi$ be a smooth weight, and let $\II_f$ be the second fundamental
-form of the graph $\{(x,f(x))\}$ in $X\times\R$ with metric
-$h\oplus dt^2$. Then
-\begin{equation}
- \wscal(h_f,\phi_f)-\wscal(h,\phi)
- =D^2-2U(D)+|\II_f|^2+|d\log L|_{h_f}^2.
- \label{eq:graph-identity}
-\end{equation}
-\end{proposition}
+This section develops a deformation that changes the metric while
+preserving a weighted volume density. Its input is a smooth function
+on a Riemannian manifold; its output is the metric induced on the
+graph of that function. We first compute the change of weighted
+scalar curvature, then solve graph equations prescribing weighted mean
+curvature to control this change. A localized version of the construction will give an
+estimate for one sweep through the coordinates of a root probability
+map.
-\noindent\emph{Consequence.} The only terms on the right that can be
-negative are those in $D$, and $D$ can be prescribed. Given any
-smooth target function $\sigma$, the equation $D=\sigma$ is a
-quasilinear second-order PDE for $f$ (a prescribed
-$\phi$-mean-curvature equation). If $\sigma$ is chosen large and
-positive, then $\wscal(h_f,\phi_f)-\wscal(h,\phi)\approx\sigma^2$ is
-large and positive. Lemma~\ref{lem:graph-prescription} below makes
-this solvable with the controlled estimates the sweep requires.
-
-\begin{proof}[Proof sketch]
-A hypersurface $\Sigma\subset(\bar h,\bar\phi)$ with unit normal $\nu$,
-mean curvature $H=\tr_{h'}\II$, and weighted mean curvature
-$D=H+\partial_\nu\bar\phi$ obeys
-\begin{equation}
- \wscal(h',\bar\phi|_\Sigma)-\wscal(\bar h,\bar\phi)
- =D^2-2P+|\II|^2,
- \label{eq:graph-general}
-\end{equation}
-with $P=|\II|^2+(\Ric_{\bar h}-\bar\nabla^2\bar\phi)(\nu,\nu)$.
-This is the general Gauss-Laplacian identity for weighted scalar
-curvature, proved by combining the Gauss equation with the
-normal decomposition of $\Delta_{\bar h}$ and $|d\bar\phi|^2$.
-Restricting to a graph in the product $X\times\R$ and computing
-$P-|\II_f|^2$ in terms of $U,D,\phi$ reduces
-\eqref{eq:graph-general} to \eqref{eq:graph-identity}. The
-identification of $P-|\II_f|^2$ is a direct calculation; the full
-computation with conventions matching ours is
-\cite[Proposition~4.1]{OpenAIScalarInessentiality2026}.
-\end{proof}
-\subsection{Prescribed graphs on complete leaves}
+For a metric \(h\) and a smooth weight \(\phi\), write
+\[
+ \wscal(h,\phi)=\Scal_h-2\Delta_h\phi-|d\phi|_h^2,
+ \qquad
+ \diver_\phi V=e^{-\phi}\diver(e^\phi V),
+ \qquad
+ \Delta_\phi=\diver_\phi\nabla .
+\]
+Thus the weighted operators use \(e^\phi dV_h\). The scalar quantity
+is Perelman's \(R^m\), with his \(f=-\phi\)
+\cite[Section~1.3]{Perelman2002}. The estimates below concern its
+behavior under particular graph deformations. For antecedents of
+the graph and vertical Jacobi identities, see
+\cite[Proposition~4.1]{OpenAIScalarInessentiality2026}; we include the
+calculation with the conventions needed here.
-The sweep requires us to solve the equation $D=\sigma$ for a smooth
-prescribed $\sigma$, with uniform control over all derivatives. This
-subsection records the necessary existence and regularity statements.
-
-\begin{definition}[Bounded geometry]
-\label{def:bounded-geometry}
-A Riemannian manifold $(X,h)$ has \emph{bounded geometry} if there is
-an $r>0$ and a covering by coordinate charts $B(p,r)$ with the
-concentric $\tfrac12r$-balls also covering, common positive buffers
-between the inner balls and chart boundaries, and uniform bounds on
-$h$, $h^{-1}$, and all their coordinate derivatives. A function is
-\emph{uniformly smooth} if all its coordinate derivatives (including
-the value) are uniformly bounded on these charts. A weight $\phi$
-needs only its positive-order derivatives to be bounded. The
-\emph{uniform H\"older space} $C^{k,\alpha}_{\mathrm{unif}}$ is the
-Banach space of functions with uniformly bounded $C^{k,\alpha}$ norms
-across all charts.
-\end{definition}
+\subsection{The graph identity}
-\begin{lemma}[Prescribed graphs on a complete manifold]
-\label{lem:graph-prescription}
-Let $(X,h)$ be complete with bounded geometry, and let $\phi$ be a
-weight with uniformly bounded positive-order derivatives. Fix
-$\epsilon>0$, $N>0$, $a=\tfrac12+\theta$ with $\theta\le1/16$, and
-$\delta>0$. Let $\alpha\ge0$ and $F:X\to[0,1]$ be uniformly smooth
-with $F$ compactly supported. Set
-$L_z=(1+|dz|_h^2)^{1/2}$, $v_z=L_z^{-1}$, $U_z=\nabla z/L_z$,
-$D_z=-\diver_\phi U_z$.
-
-\smallskip\noindent\textup{(i) \emph{Positive branch.}}
-For $H_*>0$, consider
+\begin{lemma}[Weighted graph identity]
+\label{lem:graph-identity}
+Let \((X,h)\) be a smooth Riemannian manifold, let \(\phi,f\) be
+smooth real functions, and let \(N>0\). Set
+\[
+ L=(1+N^2|df|_h^2)^{1/2},\qquad v=L^{-1},\qquad
+ U=\frac{N\nabla_h f}{L},\qquad
+ h'=h+N^2df\otimes df,\qquad \phi'=\phi-\log L.
+\]
+View \(h'\) as the induced metric on the graph
+\(\Sigma=\{(x,Nf(x)):x\in X\}\) in
+\((X\times\R,h+dz^2)\), and extend \(\phi\) constantly in the
+\(z\)-direction. Orient the graph by
+\(\nu=(-U,v)\), put
+\(\II(V,W)=\langle\bar\nabla_V\nu,W\rangle\), and let
+\[
+ H=\tr_\Sigma\II,\qquad
+ D=H+\partial_\nu\phi=-\diver_\phi U .
+\]
+Here \(D\), and its derivative \(U(D)\), are regarded as functions on
+\(X\) through the graph projection. Then
\begin{equation}
- D_z=\frac{\epsilon a(F^2+\delta^2)+(1-v_z^2)/2}{z}
- -\frac{\epsilon a+\alpha}{N^2}z.
- \label{eq:prescribed-positive}
+ \wscal(h',\phi')-\wscal(h,\phi)
+ =D^2-2U(D)+|\II|_{h'}^2+|d\log L|_{h'}^2.
+ \label{eq:graph-identity}
\end{equation}
-For every sufficiently large $H_*>0$ there is a smooth solution with
+Moreover,
\begin{equation}
- c_H:=\delta\sqrt{\frac{\epsilon a}{\epsilon a+H_*}}
- \asymp H_*^{-5/2}\le f:=z/N\le 2,
- \label{eq:prescribed-positive-bounds}
+ e^{\phi'}dV_{h'}=e^\phi dV_h,\qquad
+ N^2|df|_{h'}^2=1-v^2=|U|_h^2,\qquad
+ |d\log L|_{h'}\le L|U|_h|\II|_{h'} .
+ \label{eq:graph-basic-bounds}
\end{equation}
-and angle $v_z\ge v_0>0$ independent of $H_*$, with all derivatives
-uniformly bounded in the fixed charts.
-
-\smallskip\noindent\textup{(ii) \emph{Tangent branch.}}
-For uniformly smooth $f_0:X\to\R$ and positive constants $e,w$,
-consider
-\begin{equation}
- D_z=(\epsilon a+\alpha)\tan\bigl((f_0-ez)/w\bigr).
- \label{eq:prescribed-tangent}
-\end{equation}
-For every $\eta>0$ there is a smooth solution with
-$|(f_0-ez)/w|\le\pi/2-\eta$ and
-$|z|\le(\|f_0\|_\infty+\pi w/2)/e$, with all derivatives uniformly
-bounded.
-
-\smallskip\noindent In both cases the solution is unique among
-bounded-height solutions with uniformly bounded
-$C^{2,\alpha}_{\mathrm{unif}}$ norm and a uniform margin inside the
-prescription region. If all data are Borel in a countable chart
-enumeration and commute with an equivariant structure, the solution
-maps are Borel and equivariant, with bounds and limits uniform over
-any fixed parameter family.
\end{lemma}
-\begin{proof}[Proof sketch]
-Both cases have the form $D_z=p(x,z,v_z)$ with strict monotonicity in
-$z$:
+\begin{proof}
+Write \(\bar h=h+dz^2\), and suppress pullback by the graph
+projection. The formula for the graph mean curvature gives
+\(H=-\diver U\), while \(\partial_\nu\phi=-U(\phi)\); hence the stated
+formula for \(D\) holds. The rank-one determinant formula gives
+\(dV_{h'}=L\,dV_h\), proving the first assertion in
+\eqref{eq:graph-basic-bounds}. The inverse rank-one formula gives its
+second assertion.
+
+Let
\[
- \partial_z p=-\frac{\epsilon a(F^2+\delta^2)+(1-v^2)/2}{z^2}
- -\frac{\epsilon a+\alpha}{N^2}\le-\frac{\epsilon a}{N^2}
+ P=|\II|^2+
+ \bigl(\Ric_{\bar h}-\bar\nabla^2\phi\bigr)(\nu,\nu).
\]
-in case (i), and
+The Gauss equation and the decompositions of the Laplacian and
+gradient of \(\phi\) give
+\begin{align*}
+ \Scal_{h'}&=\Scal_{\bar h}-2\Ric_{\bar h}(\nu,\nu)
+ +H^2-|\II|^2,\\
+ \Delta_{\bar h}\phi
+ &=\Delta_\Sigma(\phi|_\Sigma)
+ +H\partial_\nu\phi+\bar\nabla^2\phi(\nu,\nu),\\
+ |d(\phi|_\Sigma)|^2&=|d\phi|_{\bar h}^2-(\partial_\nu\phi)^2.
+\end{align*}
+Since \(\wscal(\bar h,\phi)=\wscal(h,\phi)\), their combination is
+\begin{equation}
+ \wscal(h',\phi|_\Sigma)
+ =\wscal(h,\phi)+D^2-2P+|\II|^2.
+ \label{eq:graph-gauss-weighted}
+\end{equation}
+Adding \(\log v\) to the restricted weight changes the left side by
\[
- \partial_z p=-\frac{e(\epsilon a+\alpha)}{w}
- \sec^2\bigl((f_0-ez)/w\bigr)\le-c_0
+ -2\Delta_{\Sigma,\phi}\log v-|d\log v|^2
+ =-\frac{2\Delta_{\Sigma,\phi}v}{v}+|d\log v|^2 .
\]
-in case (ii), for a fixed positive $c_0$.
-
-\emph{Comparison.} At an interior maximum of the difference between
-a solution and a barrier, their gradients agree, so the principal
-elliptic part of $D_z$ produces a nonnegative difference of mean
-curvatures, while the strict decrease in $z$ forces a negative
-difference of prescriptions: contradiction. Upper barriers bound the
-solution from above; lower comparison is symmetric. Choosing the
-barriers $z=c_HN$ and $z=2N$ (case (i)) or
-$(f_0-ez)/w=\pm(\pi/2-\eta)$ (case (ii)) gives existence and the
-stated ranges. On a noncompact manifold a uniform chart centered at
-an extremizing sequence produces a local limit attaining the extremum,
-using the fixed $C^{2,\alpha}$ bounds of the uniqueness class.
-
-\emph{Angle.} Minimize $v_z/\xi$ for a chart cutoff $\xi$ supported
-where the height is already bounded. At an interior minimum the
-gradient of $v_z$ is controlled and the negative elliptic principal
-part of $D_z$ gives a lower bound on $v_z$.
-
-\emph{Higher regularity.} Bootstrapping from the linearized equation
-gives uniform $C^{k,\alpha}_{\mathrm{unif}}$ bounds of every order.
-\emph{Borel and equivariance.} Encoding the solution as a countable
-tuple of chart restrictions, applying the Borel injection theorem
-\cite[Corollary~15.2]{Kechris1995}, and using uniqueness for
-equivariance gives the stated conclusions.
-\end{proof}
-\subsection{Localization of the positive prescription}
-
-The positive-branch solution of Lemma~\ref{lem:graph-prescription} is
-defined on the whole complete manifold. For a probability coordinate
-$F_i$ supported on a compact set $K$, we need a graph supported near
-$K$. The following localization forces the complete solution to be
-close to a small positive constant outside a fixed annulus around
-$K$, and then cuts it off there.
-
-\begin{lemma}[Localization of the positive graph]
-\label{lem:graph-localization}
-Let $X,h,\phi$ satisfy the hypotheses of
-Lemma~\ref{lem:graph-prescription}, with $X$ connected. Let
-$h_{\mathrm{ref}}\le h$ be a second complete metric with uniform
-bounds for $h_{\mathrm{ref}},h_{\mathrm{ref}}^{-1}$ and their
-derivatives in the same charts. Suppose $F_i$ is uniformly smooth,
-$|F_i|\le1$, with compact support $K$, and $\alpha\ge0$ is uniformly
-smooth. Fix
-\[
- \epsilon>0,\quad N^2=l\epsilon,\quad
- a=\tfrac12+\theta,\quad 0<l\le1,\quad l\le\theta\le\tfrac1{16}.
-\]
-There is a radius $R_{\mathrm{loc}}$ depending only on $n$ and a
-numerical constant $c_0>0$, such that for every sufficiently large
-$H_*$ one can choose smooth functions $\beta_s\ge0$ and
-$0\le\chi\le1$ with the following properties. Put
-$\beta=\beta_s+\alpha$.
-
-\smallskip\noindent\textup{(a) Spatial control.}
-$0\le\beta\le H_*$, $N|d\beta_s|_h\le c_0(\theta\epsilon^{3/2}
-+\beta_s^{3/2})$, and $\beta_s=H_*$ on the outer region where
-$\chi=0$.
-
-\smallskip\noindent\textup{(b) Localized solution.}
-With $\tilde f=f_{\mathrm{loc}}$ the positive-branch solution of
-Lemma~\ref{lem:graph-prescription} using coefficient $\beta$, set
-$f:=\chi\tilde f$. Then $0\le f\le2$, $f\ge c_H$ on a uniform
-neighborhood of $K$, and $f\equiv0$ on a neighborhood of $K^c$ in the
-region where the outer profile has reached $H_*$.
-
-\smallskip\noindent\textup{(c) Plateau jets.}
-For every $k\ge0$ and every $M>0$ there is a $C_{k,M}$ such that, for
-every sufficiently large $H_*$, on the plateau $\{F_i=0,\beta=H_*\}$
-\[
- \|\tilde f-c_H\|_{C^{k}}\le C_{k,M}H_*^{-M}.
- \label{eq:plateau-jets}
-\]
-The constants depend only on the indicated data and parameters and
-not on $H_*$.
-
-\smallskip\noindent\textup{(d) Algebraic gain.}
-The graph $f$ satisfies $f\ge c_H$ and
+The tangential part of vertical translation is
+\[
+ T=\partial_z-v\nu=(vU,1-v^2).
+\]
+For a hypersurface variation with normal speed \(v\), the variation
+of \(H\) is
+\(-\Delta_\Sigma v-(|\II|^2+\Ric_{\bar h}(\nu,\nu))v\).
+The additional variation of \(\partial_\nu\phi\) is
+\[
+ v\bar\nabla^2\phi(\nu,\nu)
+ -\langle d(\phi|_\Sigma),dv\rangle_\Sigma .
+\]
+Including the tangential velocity \(T\), and using that vertical
+translation preserves the metric, weight, and \(D\), therefore gives
\begin{equation}
- \frac{|df|_h^2}{f}
- =O(\widetilde f\,|d\chi|_h^2/\chi+\chi|d\widetilde f|_h^2/\widetilde f)
- =O(c_H)
- \label{eq:graph-cutoff-small}
+ (\Delta_{\Sigma,\phi}+P)v=T(D).
+ \label{eq:graph-translation}
\end{equation}
-on the transition region $\{0<\chi<1\}$, and all jets of $f$ are
-controlled.
-\end{lemma}
-
-\noindent\emph{Consequences.} All of $f,\beta_s,\chi$ are Borel and
-equivariant when the data are, with uniform bounds in the family. The
-localized graph has $d\log L_z$, $D_z$, and $N|\II_z|$ all tending to
-zero on the outer transition region, so the residual contributions to
-the graph identity from that region are negligible.
-
-\begin{proof}[Proof sketch]
-\emph{Step 1: The outer profile.} Start at a fixed positive distance
-from $K$. Let $b(t)$ solve the ODE inequality
-\[
- \dot b\le\frac{c_0}{CN}\bigl(\theta\epsilon^{3/2}+b^{3/2}\bigr),
- \qquad b(0)=0.
-\]
-The total integration time from $b=0$ to $b=H_*$ is
-\[
- T=\frac{CN}{c_0}\int_0^{H_*}\frac{db}{\theta\epsilon^{3/2}+b^{3/2}}.
-\]
-Substitute $u=(b/\theta\epsilon^{3/2})^{2/3}\cdot(\theta\epsilon^{3/2})^{2/3}$;
-the integral converges as $H_*\to\infty$ to
-$C_0/(\theta\epsilon^{3/2})^{1/3}$. Hence
-\[
- T\le C\frac{N}{\theta^{1/3}\sqrt\epsilon}
- =C\frac{\sqrt l}{\theta^{1/3}}\le C,
-\]
-using $l\le\theta\le1/16$. Set $R_{\mathrm{loc}}$ slightly larger
-than this $T$. Define $\beta_s$ by composing $b$ with fixed smooth
-flattenings; this gives (a).
-
-\emph{Step 2: Plateau.} With $\beta=\beta_s+\alpha$ and $\beta_s=H_*$
-outside a fixed radius, the solution $\tilde f$ from
-Lemma~\ref{lem:graph-prescription} has $\tilde f\le2$ and $c_H\le
-\tilde f$. On the outer plateau, comparison with barriers
-$\pm\exp(-\text{const}\cdot\text{dist}/\sqrt{H_*})$ bounds
-$\tilde f-c_H$ by a superpolynomial decay in $H_*$. Coupled with the
-elliptic equation for the difference, this gives \eqref{eq:plateau-jets}.
-
-\emph{Step 3: Cutoff.} Choose $\chi$ smooth supported in the
-interior of the plateau, equal to $1$ on a neighborhood of $K$, and
-otherwise smooth with derivative bound $\sim1/R_{\mathrm{loc}}$. The
-scaling $|d\chi|^2\le C\chi$ holds because $\chi$ is bounded.
-This gives (b) and (c), and a direct computation of
-$|d(\chi\tilde f)|^2/(\chi\tilde f)$ on the transition region where
-$\chi\tilde f\asymp c_H$ gives (d).
-
-\emph{Borel and equivariance.} All constructions are scalar and
-equivariant; combined with the Borel data of the previous lemma, they
-yield the stated conclusions.
+The horizontal part of \(T\) is \(vU\), so \(T(D)=vU(D)\).
+Substitution in \eqref{eq:graph-gauss-weighted} proves
+\eqref{eq:graph-identity}. Finally, differentiating
+\(v=\langle\nu,\partial_z\rangle\) tangentially gives
+\(dv(V)=\II(V,T)\). Since \(|T|^2=1-v^2=|U|^2\), it follows that
+\(|d\log v|\le v^{-1}|U||\II|\), which is the last bound in
+\eqref{eq:graph-basic-bounds}.
\end{proof}
-\subsection{A sweep of the root probabilities}
-\label{sec:graphs-sweep}
-We now apply the localized graph to each coordinate of a root
-probability map. Two considerations drive the construction. First,
-a graph enlarges the metric for all subsequent coordinates, so ordering
-matters; we fix an ordering by finitely many colors and pass through
-the colors in order. Second, in the sweep the total change of
-$\wscal$ must be expressible through a small number of well-behaved
-"statistics" of the sweep (the quantities $m,\sigma_i,V_i$ below),
-so that later iteration can be analysed by inequalities on those
-statistics alone.
-
-Fix a smooth nonnegative nondecreasing profile
-$\Psi:[0,\infty)\to[0,\infty)$ with
+\paragraph{A one-coordinate calculation.}
+The square and differentiated terms in the graph identity suggest
+an ideal target for \(ND\), the scaled weighted mean curvature.
+Let \(F\) be one smooth input coordinate, restrict
+to a region where \(f>0\), and fix \(\epsilon,a>0\). Also let
+\(h_{\mathrm{in}}\le h\) be a comparison metric. In the later sweep,
+the incoming energy is measured in \(h_{\mathrm{in}}\), while \(h\)
+already includes the earlier graphs.
+
+Put \(A=|U|_h\). Where \(A>0\), write \(U=Ae\) with \(|e|_h=1\).
+The longitudinal unit tangent to the graph is
+\(\tau=(ve,A)\). Define
+\[
+ x=-\II(\tau,\tau),\qquad X_{\parallel}=Nx,\qquad t=A^2/f,\qquad
+ B=1-|e|_{h_{\mathrm{in}}}^2\in[0,1].
+\]
+At \(A=0\), choose any \(h\)-unit vector \(e\), put \(\tau=(e,0)\),
+and use the same definitions. With \(\tau^\flat\) denoting the dual
+covector in the graph metric \(h'\), set
+\[
+ \II^{\mathrm{rem}}=\II+x\,\tau^\flat\otimes\tau^\flat .
+\]
+The component of \(\II^{\mathrm{rem}}\) in the
+\(\tau^\flat\otimes\tau^\flat\) direction vanishes, so
\begin{equation}
- \Psi(s)=0\ (s\le8),\quad
- \Psi(s)\le Cs^2,\quad
- |\Psi'(s)|^2\le C\Psi(s),\quad
- \Psi(s)\ge cs^2\ (s\ge R_\Psi),
- \label{eq:sweep-boost-profile}
+ \begin{gathered}
+ NU(f)=LA^2,\qquad
+ |\II|_{h'}^2=x^2+|\II^{\mathrm{rem}}|_{h'}^2,\qquad
+ \tr_{h'}\II^{\mathrm{rem}}=H+x,\\
+ x=e(A)\quad\text{where }A>0.
+ \end{gathered}
+ \label{eq:graph-longitudinal}
\end{equation}
-with $R_\Psi$ numerical. A prototype is the square of a function
-that is zero up to $8$, has bounded derivative, and equals $s-8$ from
-$s=R_\Psi$ on. We take either $\psi=0$ or $\psi=\kappa\Psi$ for a
-small constant $\kappa>0$.
-
-\subsubsection{Sweep notation}
+For the last identity, use \(\nu=(-Ae,v)\) and \(A^2+v^2=1\) to
+compute
+\[
+ \II(\tau,\tau)=-v^2e(A)+Av\,e(v)=-e(A).
+\]
+The first identity in \eqref{eq:graph-longitudinal} follows directly
+from the definitions of \(U,L,A\).
-Fix pass parameters
-\begin{equation}
- \epsilon>0,\quad N^2=l\epsilon,\quad a=\tfrac12+\theta,\quad
- \lambda=\tfrac12-2\theta,\quad 0<l\le1,\quad l\le\theta\le\tfrac1{16}.
- \label{eq:sweep-parameters}
-\end{equation}
-
-Start with $(h_0,\phi_0,F)$ where $F$ is a root probability map with
-$F_i\ge0$, $\sum_i F_i^2=1$, and locally finite support. Suppose
-nonnegative smooth bumps $f_i$ form a locally finite family ordered
-in finitely many colors with disjoint supports within each color,
-such that $F_i\ne0$ implies $f_i>0$. Attach nonnegative coefficients
-$\beta_i$. At this stage these are candidates; no graph equation is
-yet imposed.
-
-Apply in color order the graph of height $N$ times the sum of the
-bumps in each color. On the support of $f_i$, let $h_{i-1},\phi_{i-1}$
-be the pair before its color, and $h_i,\phi_i$ the pair afterward.
-Write $j<i$ for colors strictly before $i$, $L_i,U_i,D_i,\II_i,H_i$
-for the individual graph quantities of $f_i$, and $h_+,\phi_+$ for
-the final pair.
-
-The graph identities give:
-\begin{equation}
- \begin{gathered}
- h_+=h_0+N^2\sum_i df_i\otimes df_i,\qquad
- W_p:=\sum_i\log L_i=\log(dV_{h_+}/dV_{h_0})\ge0,\\
- \phi_+=\phi_0-W_p,\qquad
- e^{\phi_+}dV_{h_+}=e^{\phi_0}dV_{h_0},\qquad
- r:=\sum_i f_i^2>0,\qquad Y_i=f_i/\sqrt r .
- \end{gathered}
- \label{eq:sweep-output}
-\end{equation}
-The sums are pointwise finite, with at most one $f_i$ nonzero per
-color, and $r>0$ because $F_i\ne0\Rightarrow f_i>0$.
-
-For each index with $f_i>0$, use the longitudinal notation of the
-one-coordinate calculation:
-\begin{equation}
- A_i=|U_i|_{h_{i-1}},\quad U_i=A_i e_i,\quad
- x_i=-\II_i(\tau_i,\tau_i),\quad X_i=N x_i,\quad
- t_i=A_i^2/f_i,
- \label{eq:sweep-longitudinal-notation}
-\end{equation}
-with $\tau_i=(L_i^{-1}e_i,A_i)$ the longitudinal unit tangent, and
-the "residual" second fundamental form
-$\II_i^{\mathrm{rem}}:=\II_i+x_i\tau_i^\flat\otimes\tau_i^\flat$.
-Then
+Define the ideal target for \(ND\) by
\begin{equation}
- |\II_i|^2=x_i^2+|\II_i^{\mathrm{rem}}|^2,\quad
- \tr\II_i^{\mathrm{rem}}=H_i+x_i .
- \label{eq:sweep-longitudinal}
+ Z=\epsilon a(F^2/f-f)+A^2/(2f).
+ \label{eq:graph-ideal-target}
\end{equation}
-The comparison errors from the sweep metrics are
+Since \(NU(A^2)=2A^2X_{\parallel}\), with both sides zero at \(A=0\),
+differentiation gives
+\begin{align}
+ -2NU(Z)
+ ={}&-4\epsilon a(F/f)NAe(F)
+ +2\epsilon a(F^2/f^2+1)LA^2 \notag\\
+ &-2X_{\parallel}t+Lt^2.
+ \label{eq:graph-ideal-derivative}
+\end{align}
+The first term couples the prescribed \(F^2/f\) to the incoming
+energy. The \(A^2/(2f)\) term completes the longitudinal square:
\begin{equation}
- B_i=1-|e_i|_{h_0}^2,\qquad
- C_i=1-\frac{N^2|df_i|_{h_+}^2}{A_i^2}\ \ (A_i>0),
- \label{eq:sweep-comparison-errors}
+ \begin{gathered}
+ X_{\parallel}^2-2X_{\parallel}t+Lt^2
+ =(X_{\parallel}-t)^2+(L-1)t^2,\\
+ \begin{aligned}
+ &\epsilon N^2|dF|_{h_{\mathrm{in}}}^2
+ -4\epsilon a(F/f)NAe(F)\\
+ &\quad=\epsilon\left|N\,dF
+ -2a(F/f)A\,h_{\mathrm{in}}(e,\cdot)\right|_{h_{\mathrm{in}}}^2
+ -4\epsilon a^2(F/f)^2A^2(1-B)\\
+ &\quad\ge-4\epsilon a^2(F/f)^2A^2(1-B).
+ \end{aligned}
+ \end{gathered}
+ \label{eq:graph-ideal-completions}
\end{equation}
-and $0\le B_i,C_i\le1$ by metric monotonicity.
+Consequently
+\begin{align}
+ &Z^2-2NU(Z)+N^2|\II|_{h'}^2+N^2|d\log L|_{h'}^2
+ +\epsilon N^2|dF|_{h_{\mathrm{in}}}^2 \notag\\
+ \ge{}&Z^2+(X_{\parallel}-t)^2+N^2|\II^{\mathrm{rem}}|_{h'}^2
+ +N^2|d\log L|_{h'}^2 \notag\\
+ &+\epsilon A^2\bigl[(2a-4a^2)F^2/f^2+2a\bigr]
+ +4\epsilon a^2(F^2/f^2)A^2B \notag\\
+ &+(L-1)\bigl[2\epsilon a(F^2/f^2+1)A^2+t^2\bigr].
+ \label{eq:graph-ideal-bound}
+\end{align}
+At \(A=0\), \(t=0\) and
+\((X_{\parallel}-t)^2+N^2|\II^{\mathrm{rem}}|^2=N^2|\II|^2\); every other
+direction-dependent term vanishes. Thus the inequality is independent
+of the arbitrary direction at such points.
-\subsubsection{Sweep statistics}
+This calculation is algebraic and assumes no equation for \(D\).
+The complete equation below realizes \(ND=Z-\beta f\) up to a positive
+floor, and localization introduces controlled cutoff errors. A sweep
+will also have to account for normalization of the output map and the
+interaction of the ordered graph metrics.
-Set the probability weights $\nu_i=f_i^2/r$, and write
-$\E_\nu b_i=\sum_i\nu_i b_i$. Define
-\begin{equation}
- m=\sum_i A_i^2,\quad u_i=\frac{A_i^2}{\epsilon\nu_i},\quad
- \bar u=\E_\nu u_i=\frac m\epsilon,\quad
- \sigma_i=\frac{F_i^2}{\nu_i},\quad
- M_0=a(1-r)-\frac{\bar u}{2},
- \label{eq:sweep-stats-1}
-\end{equation}
+\subsection{Prescribed graphs on complete leaves}
+
+We specify the uniformity required for the analytic construction.
+A manifold has \emph{bounded geometry} here if it has coordinate
+charts of a common positive size whose smaller concentric domains
+cover the manifold, with a common positive buffer to each chart
+boundary, and with uniform bounds on the metric, its inverse, and
+all their coordinate derivatives. These bounds, as well as the
+chart size, may depend on the instance. A function is uniformly
+smooth if all its coordinate derivatives, including its value, are
+uniformly bounded on these charts. For a weight \(\phi\), only
+positive-order derivatives are required to be uniformly bounded.
+The \emph{uniform Hölder space} below means the Banach space
+obtained by taking the supremum of the corresponding Hölder norms in
+these charts, after a fixed shrinkage within the buffers.
+
+The use of complete prescribed graphs has precedents in
+\cite[Theorem~5.2]{OpenAIScalarInessentiality2026}. The
+following statement records the precise uniform and measurable
+conclusions used here and in the hypersurface argument. Its positive
+branch realizes the designed prescription with a positive floor and
+will be localized for the sweep. Its tangent branch supplies the value
+approximation and the intrinsic gradient bound used in
+Section~\ref{sec:degree}.
+
+\begin{lemma}[Prescribed graphs on a complete manifold]
+\label{lem:graph-prescription}
+Let \((X,h)\) be a complete smooth Riemannian manifold without
+boundary, with bounded geometry as above, and suppose the
+positive-order derivatives of \(\phi\) are uniformly bounded. For
+a graph of height \(z\), set
+\[
+ L_z=(1+|dz|_h^2)^{1/2},\quad v_z=L_z^{-1},\quad
+ U_z=\frac{\nabla_hz}{L_z},\quad
+ D_z=-\diver_\phi U_z .
+\]
+Consider either of the following equations \(D_z=p(x,z,v_z)\).
+
+\begin{enumerate}
+\renewcommand{\labelenumi}{\textup{(\roman{enumi})}}
+\item Let \(N,\epsilon,a>0\), \(0<\delta\le1\), and \(H_*\ge0\).
+ Suppose \(F_i,\beta\) are uniformly smooth,
+ \(|F_i|\le1\), and \(0\le\beta\le H_*\). On the region \(z>0\),
+ prescribe
+ \begin{equation}
+ p(x,z,v)=
+ \frac{\epsilon a(F_i(x)^2+\delta^2)+(1-v^2)/2}{z}
+ -(\epsilon a+\beta(x))\frac{z}{N^2}.
+ \label{eq:graph-prescription-positive}
+ \end{equation}
+\item Let \(e,w>0\) be constants. Suppose \(B,f_0\) are uniformly
+ smooth and \(B\ge B_{\min}>0\). On the region
+ \(|(f_0(x)-ez)/w|<\pi/2\), prescribe
+ \begin{equation}
+ p(x,z,v)=B(x)\tan\bigl((f_0(x)-ez)/w\bigr).
+ \label{eq:graph-prescription-tangent}
+ \end{equation}
+\end{enumerate}
+
+In either case there is a smooth solution of bounded height with
+uniform bounds on all its derivatives, and it lies uniformly inside
+the stated open prescription region. Its angle satisfies
+\(v_z\ge v_0>0\). In Case \textup{(i)} it satisfies
\begin{equation}
- V_i=a(\sigma_i-1)+\lambda(u_i-\bar u),\quad
- \Var_\nu u:=\E_\nu(u_i-\bar u)^2 .
- \label{eq:sweep-stats-2}
+ c_{\min}:=\delta\sqrt{\frac{\epsilon a}{\epsilon a+H_*}}
+ \ \le\ f:=z/N\ \le 2.
+ \label{eq:graph-height-positive}
\end{equation}
+In Case \textup{(ii)} there is \(\eta>0\) such that
+\[
+ \left|\frac{f_0-ez}{w}\right|\le\frac{\pi}{2}-\eta;
+ \qquad
+ |ez-f_0|<\frac{\pi w}{2},\quad
+ |z|\le\frac{\|f_0\|_\infty+\pi w/2}{e}.
+\]
+The constants \(v_0,\eta\) and the derivative bounds may depend on
+all the indicated parameters and common bounds of the data.
+The solution is unique among bounded-height solutions whose
+\(C^{2,\alpha}\) norms in all buffered charts are uniformly bounded
+for some \(0<\alpha<1\), and whose values have a uniform margin inside
+the prescription region. Each such solution may have its own
+exponent, bound, and margin.
+
+The graph metric \(h_z=h+dz\otimes dz\) is again complete and has
+bounded geometry, and \(\phi_z=\phi-\log L_z\) has uniform
+positive-order derivatives. In particular,
+\[
+ e^{\phi_z}dV_{h_z}=e^\phi dV_h,\qquad
+ |dz|_{h_z}^2=1-v_z^2<1.
+\]
+For the family assertion, let \(\mathcal T\) be a standard Borel
+parameter space indexing complete problems of a fixed dimension.
+Assume a common chart radius and buffers, common bounds for the
+metric, its inverse, and all their chart derivatives, and common
+bounds for the positive-order derivatives of \(\phi\). Require
+common bounds, including order zero, for all chart derivatives of
+\(F_i,\beta\) in Case \textup{(i)} or \(B,f_0\) in Case
+\textup{(ii)}. In Case \textup{(i)}, let \(N,\epsilon,a,\delta\)
+range in fixed compact subsets of their permitted positive ranges,
+and require a common finite upper bound on \(H_*\). In Case
+\textup{(ii)}, let \(e,w\) range in fixed compact subsets of
+\((0,\infty)\), and require \(B\ge B_{\min}\) for one common
+\(B_{\min}>0\). Suppose also that the data and chart overlaps are
+Borel and that the charts accessible along each leaf admit a
+countable Borel enumeration. Then the preceding estimates for the
+solutions and their graph metrics and weights hold with constants common to the family,
+including a common positive height margin in Case \textup{(i)}
+and a common margin from the poles in Case \textup{(ii)}. The
+solutions are Borel in the chart representations and commute with
+any equivariant identifications of the complete problems.
+\end{lemma}
-Finally, define two nonnegative quantities
+\begin{proof}
+We give existence and the estimates together, since the estimates
+also give uniqueness and measurable dependence.
+
+\emph{Height barriers and comparison.}
+In Case \textup{(i)}, replace \(F_i,\beta\) by \(tF_i,t\beta\),
+\(0\le t\le1\). At \(t=0\) the constant \(f=\delta\) is a
+solution. For every \(t\), the constant graphs \(f=c_{\min}\) and
+\(f=2\) are a lower and an upper barrier, respectively. Indeed their
+weighted mean curvature is zero, and at angle \(v=1\) the right side
+of \eqref{eq:graph-prescription-positive} is nonnegative at
+\(f=c_{\min}\) and strictly negative at \(f=2\). In Case
+\textup{(ii)}, replace \(f_0\) by \(tf_0\), starting with \(z=0\).
+For \(\tau=\pi/2-\eta\), the graphs
+\[
+ z_-=\frac{tf_0-w\tau}{e},\qquad
+ z_+=\frac{tf_0+w\tau}{e}
+\]
+have arguments \(+\tau\) and \(-\tau\), respectively. Their
+weighted mean curvatures have a common bound independent of
+\(\eta\), because their derivatives do not involve \(\tau\).
+Choose \(\eta>0\) so small that \(B_{\min}\tan\tau\) exceeds this
+bound. These are then lower and upper barriers for every \(t\).
+They give the asserted height and pole margins.
+
+Both comparisons use strict decrease of \(p\) in its height
+argument, with the other arguments held fixed:
\begin{align}
- T_0={}&\sum_i A_i^2\bigl[C_i+4a^2\sigma_iB_i
- +(L_i-1)\{2a(\sigma_i+r)+u_i\}\bigr],
- \label{eq:sweep-T}\\
- \mathcal X={}&\sum_i\bigl[(X_i-t_i)^2+N^2|\II_i^{\mathrm{rem}}|^2
- +N^2|d\log L_i|_{h_i}^2+\beta_i^2f_i^2\bigr].
- \label{eq:sweep-X}
+ p_z&=-\frac{\epsilon a(F_i^2+\delta^2)+(1-v^2)/2}{z^2}
+ -\frac{\epsilon a+\beta}{N^2}
+ \le-\frac{\epsilon a}{N^2}
+ &&\text{in Case \textup{(i)}}, \notag\\
+ p_z&=-\frac{eB}{w}\sec^2\bigl((f_0-ez)/w\bigr)
+ \le-\frac{eB_{\min}}w
+ &&\text{in Case \textup{(ii)}}.
+ \label{eq:graph-height-monotonicity}
\end{align}
-Here $T_0$ records the gains from the metric comparisons and from
-each graph's stretching; its full coefficient will yield the sharp
-quadratic term $m^2/n$ in \S\ref{sec:entropy}. The quantity
-$\mathcal X$ retains the derivative squares and the applied
-coefficients; together with the mean and variance squares below, it
-controls the graph curvature.
-
-\subsubsection{The sweep proposition}
-
-\begin{proposition}[One sweep]
-\label{prop:sweep}
-Fix $(h_0,\phi_0,F)$ a root probability map with $F_i\ge0$,
-$\sum_i F_i^2=1$, and locally finite support. Fix pass parameters
-\eqref{eq:sweep-parameters}, with $\epsilon\ge\epsilon_*$ in the
-boosted case (below), where $\epsilon_*>0$ is the dimensional
-threshold of Lemma~\ref{lem:entropy-bootstrap}.
-
-For any $\delta_0,\eta_0>0$ there is a candidate sweep
-$(h_0,\phi_0,F;f_i,\beta_i,\chi_i)$ with all bumps $f_i\le2$,
-$\supp f_i$ in a uniform neighborhood of $\supp F_i$, and the
-following properties. The output
-$(h_+,\phi_+,Y)$ from \eqref{eq:sweep-output} is again a root
-probability map with the same label orbits, equivariant and Borel
-when the data are, with uniform bounds in the fixed charts.
-
-For each $i$, the coefficient has the form
+At an interior positive maximum of the difference between a
+solution and an upper barrier, their gradients, hence their angles,
+agree. The negative elliptic principal part of \(D_z\) gives a
+nonnegative difference of mean curvatures there, whereas strict
+height decrease gives a negative difference of prescriptions.
+This is impossible. The lower comparison and uniqueness follow in
+the same way. On a noncompact manifold the maximum can be replaced
+by a supremum: take uniform charts centered at an extremizing
+sequence, subtract the value of \(\phi\) at each center, and pass to
+a local limit. For any individual solution in the stated uniqueness
+class, its uniform \(C^{2,\alpha}\) bound and the coefficient bounds
+give such a limit, which attains the extremum at the center of an
+interior chart. This
+justifies the same comparison without compactness.
+
+\emph{The angle bound.}
+For a solution, \eqref{eq:graph-translation} and the chain rule give
\begin{equation}
- \beta_i=\beta_{s,i}+\epsilon\psi(R_i),\qquad R_i=\sum_{j<i}f_j^2 ,
- \label{eq:sweep-beta}
+ (\Delta_{\Sigma,\phi}+P)v_z
+ =v_z\langle p_x,U_z\rangle
+ +p_z(1-v_z^2)+p_vT(v_z).
+ \label{eq:graph-angle-equation}
\end{equation}
-where $\beta_{s,i}\ge0$ vanishes near $\supp F_i$ and satisfies the
-spatial cutoff conditions of Lemma~\ref{lem:graph-localization} in
-$h_{i-1}$. Writing
+Here \(p_x\) differentiates with \(z,v\) fixed, and we used
+\(T(z)=1-v_z^2\). The ambient curvature and weight bounds imply
+\(P\ge-C\), since the remaining term \(|\II|^2\) is nonnegative.
+At an infimum of \(v_z\), taken by the same recentering procedure,
+\(\Delta_{\Sigma,\phi}v_z\ge0\) and \(T(v_z)=0\). Consequently, if
+\(v_z\le1/2\) there,
\begin{equation}
- ND_i^0=\epsilon a(F_i^2/f_i-f_i)+A_i^2/(2f_i)
- \label{eq:sweep-ND0}
+ (1-v_z^2)|p_z|\le v_z(C+|p_x|).
+ \label{eq:graph-angle-infimum}
\end{equation}
-on $\{f_i>0\}$, extended by zero off the bump, one has
+The height margins just proved bound \(p_x\) uniformly, and
+\eqref{eq:graph-height-monotonicity} bounds \(|p_z|\) below by a
+positive constant. Since \(1-v_z^2\ge3/4\) in this case,
+\eqref{eq:graph-angle-infimum} gives a uniform positive lower bound
+for \(v_z\). In using recentered limits at this stage, a solution
+is first in an individual uniform \(C^{2,\alpha}\) class. Ordinary
+local bootstrapping gives its individual higher regularity before
+the common estimates are deduced, so the differentiations and
+limits in this argument are justified.
+
+\emph{Higher estimates.}
+In a uniform chart let
+\(\omega=e^\phi\sqrt{\det h}\), and divide \(\omega\) by its value
+at the chart center. The resulting density has a common upper
+bound and a positive lower bound, and all its derivatives are
+bounded. The equation is of the form
\begin{equation}
- ND_i=ND_i^0-\beta_i f_i+\mathrm{err}_i,\qquad
- \sum_i|\mathrm{err}_i|^2\le\eta_0 .
- \label{eq:sweep-prescription-error}
+ -\partial_j A^j(x,\partial z)=F(x,z,\partial z).
+ \label{eq:graph-divergence-equation}
\end{equation}
-\end{proposition}
+For example \(A^j=\omega h^{jk}z_k/L_z\) after the normalization.
+On the established height and gradient ranges the flux and the
+right side are smooth with uniformly bounded derivatives, and
+\(\partial A^j/\partial z_k\) is uniformly elliptic. Differentiating
+\eqref{eq:graph-divergence-equation} gives linear divergence
+equations for \(\partial_kz\). Their leading coefficients are
+bounded and uniformly elliptic. Gradient dependence of \(F\)
+produces a bounded first-order coefficient; the remaining terms
+are a bounded function plus the divergence of a bounded vector
+field. The term \(F_z\partial_kz\) is included in the bounded
+source, since the gradient is already bounded.
+
+The interior De Giorgi--Nash estimate for these equations, with
+bounded lower-order coefficients and source data in \(L^{s/2}\)
+and \(L^s\) for \(s>\dim X\), gives a common Hölder bound for
+\(\partial z\); see \cite[Theorem~8.24]{GilbargTrudinger}.
+The coefficients in the nondivergence form of the original
+equation are now Hölder continuous. Interior Schauder estimates
+give a uniform \(C^{2,\alpha}\) bound for some \(\alpha>0\), after
+shrinking within the chart buffers. Differentiating and applying
+Schauder estimates again yields uniform bounds for every
+derivative. All these estimates are local with common buffers, so
+their constants are uniform throughout \(X\).
+
+\emph{Continuation.}
+Work in uniform \(C^{2,\alpha}\) spaces on the open set of functions
+with a uniform height margin. The equation \(D_z-p(x,z,v_z)=0\)
+defines a smooth map to uniform \(C^{0,\alpha}\). At a solution
+its linearization has negative uniformly elliptic principal part,
+uniform Hölder coefficients, and zeroth-order coefficient
+\(-p_z\) bounded below by a positive constant. This linear
+operator is invertible on the uniform spaces. To see
+surjectivity, exhaust a noncompact component by smooth relatively
+compact domains and solve the linear Dirichlet problem with zero
+boundary data on each domain. The maximum principle bounds the
+supremum of the solutions by the supremum of the right side
+divided by the positive zeroth-order lower bound. Interior
+Schauder estimates then give the same \(C^{2,\alpha}\) bound in
+every buffered chart once it lies in the exhaustion. A diagonal
+local limit solves the equation on the component with the uniform
+bound. On a compact component the corresponding closed-manifold
+elliptic theory gives the same conclusion. Uniqueness follows
+from the maximum principle, using recentered limits on noncompact
+components. The estimate also makes the inverse bounded.
+
+The inverse function theorem gives openness of the set of
+homotopy parameters with a solution. The a priori height, angle,
+and higher bounds are common over the homotopy. For a sequence
+of parameters converging to a limit, diagonal smooth convergence
+on a countable chart cover therefore gives a solution at that
+limit with the same bounds and height margin. This gives
+closedness. Each homotopy starts from the constant solution
+specified above, so existence follows. Comparison gives the
+claimed uniqueness.
+
+The graph estimates now follow from
+\eqref{eq:graph-basic-bounds}. In particular \(h_z\ge h\), and
+the gradient bound also gives a uniform upper comparison in the
+given charts. Thus \(h_z\) is complete and has uniform buffered
+charts with bounds on all derivatives. The corresponding
+derivative bounds on \(\log L_z\) give those for \(\phi_z\).
+This also permits successive applications of the lemma.
+
+\emph{Borel dependence.}
+For completeness, fix a starting plaque and enumerate the charts
+accessible along its leaf by words in the Borel overlap maps.
+Represent a solution by its functions in this countable family
+of charts, in a countable product of smooth function spaces.
+Compatibility on overlaps, the equation, the height margin, and
+the common derivative bounds are Borel conditions: each can be
+tested on countably many compact subcharts and dense sets, where
+the relevant functions and their derivatives are continuous.
+The chart data and changes of coordinates are Borel by
+hypothesis. The resulting Borel relation between data and local
+representatives has a singleton fiber over each set of data,
+by existence and uniqueness. The Borel injection theorem
+\cite[Corollary~15.2]{Kechris1995},
+applied to its projection to the data, makes the inverse
+projection Borel. This proves measurable dependence. The
+coding refers to a full prescribed problem on the same complete
+leaf. It applies independently of the transverse mass of the
+chart family. Finally, applying an equivariant identification
+to a solution produces a solution of the identified problem, so
+uniqueness proves equivariance.
+\end{proof}
+
+\subsection{Localization of the positive prescription}
-The estimate \eqref{eq:sweep-prescription-error} says that the
-prescribed mean curvature of the graph realizes $ND_i^0-\beta_if_i$
-up to arbitrarily small residuals. In the concrete construction
-below, the $\mathrm{err}_i$ arise from the smoothing of the cutoff and
-from the positive floor $\varpi=\epsilon a\delta^2/f$ that
-Lemma~\ref{lem:graph-prescription}(i) carries; the estimate records
-that both are under control.
-\subsubsection{The one-sweep inequality}
-
-We now prove the analytic inequality controlling one sweep. The
-quantity of interest, which measures the total "improvement" of the
-probability-map functional at fixed weighted volume, is
+The positive solution in Lemma~\ref{lem:graph-prescription} is
+defined on the complete manifold and need not have compact support.
+For a probability coordinate we need a graph supported near that
+coordinate. We obtain one by forcing the complete solution close
+to a small constant on an outer region, and cutting it off there.
+The following lemma states the quantitative features of this
+procedure that enter the sweep.
+
+\begin{lemma}[Localization of the positive graph]
+\label{lem:graph-localization}
+Let \(X,h,\phi\) satisfy the hypotheses of
+Lemma~\ref{lem:graph-prescription}, with \(X\) connected, and let
+\(h_{\mathrm{ref}}\le h\) be a complete second metric. Require the
+coefficients of \(h_{\mathrm{ref}}\), its inverse, and all their
+coordinate derivatives to have uniform bounds in the same buffered
+charts used for \(h\).
+Suppose \(F_i\) is uniformly smooth, \(|F_i|\le1\), with compact
+support \(K\), and let \(\alpha\ge0\) be a uniformly smooth
+function. Fix
+\[
+ \epsilon>0,\qquad N^2=l\epsilon,\qquad
+ a=\tfrac12+\theta,\qquad 0<l\le1,\qquad l\le\theta\le\tfrac1{16}.
+\]
+There is a radius \(R_{\mathrm{loc}}\), depending only on the
+dimension and a fixed sufficiently small numerical constant
+\(c_0>0\), with the following property. For every sufficiently
+large \(H_*\), one can choose smooth functions
+\(\beta_s\ge0\) and \(0\le\chi\le1\) such that, with
+\(\beta=\beta_s+\alpha\),
\begin{equation}
- Q:=\frac{rN^2}{\epsilon}
- \bigl[\wscal(h_+,\phi_+)-\wscal(h_0,\phi_0)
- +\epsilon e_{h_0}(F)-\epsilon e_{h_+}(Y)\bigr].
- \label{eq:sweep-Q}
+ 0\le\beta\le H_*,
+ \qquad
+ N|d\beta_s|_h
+ \le c_0\bigl(\theta\epsilon^{3/2}+\beta_s^{3/2}\bigr).
+ \label{eq:graph-spatial-cutoff}
\end{equation}
-
-\smallskip\noindent\emph{Step 1: raw estimate.} Apply
-Proposition~\ref{prop:graph-identity} to the $i$-th coordinate, using
-the prescribed data of \eqref{eq:sweep-prescription-error} and the
-metric comparison \(B_i,C_i\in[0,1]\). After summing over $i$:
+The function \(\beta_s\) vanishes on a neighborhood of \(K\).
+The function \(\chi\) is one there, is supported in
+\(\{\dist_{h_{\mathrm{ref}}}(x,K)<R_{\mathrm{loc}}\}\), and changes
+from one to zero only within a region where \(F_i=0\) and
+\(\beta=H_*\). This region and the support of \(d\chi\) have fixed
+positive buffers in \(h_{\mathrm{ref}}\), independent of \(H_*\).
+The derivatives of \(\chi\) are bounded independently of \(H_*\),
+\(\chi\) is flat on its zero set, and \(|d\chi|_h^2\le C\chi\).
+
+Set \(\delta=H_*^{-2}\), and let \(\widetilde f=z/N\) be the
+complete solution of \eqref{eq:graph-prescription-positive} for
+this \(\beta,\delta\). Define \(f=\chi\widetilde f\) and form the
+graph of height \(Nf\). Denote its quantities by \(L,U,D,\II\),
+put \(A=|U|_h\), and set, where \(f>0\),
\begin{equation}
- Q\ge\frac r\epsilon\sum_i\bigl[Z_i^2-2NU_i(Z_i)
- +N^2|\II_i|_{h_i}^2+N^2|d\log L_i|_{h_i}^2\bigr]
- +\frac r\epsilon\sum_i\epsilon N^2|dF_i|_{h_{i-1}}^2
- -\frac r\epsilon\delta_0,
- \label{eq:sweep-raw}
+ ND^0=\epsilon a(F_i^2/f-f)+A^2/(2f).
+ \label{eq:graph-ideal-prescription}
\end{equation}
-where \(Z_i=ND_i^0\) is the ideal target from \eqref{eq:sweep-ND0}.
-
-\smallskip\noindent\emph{Step 2: ideal bound.} Using the longitudinal
-calculation \eqref{eq:sweep-longitudinal} and completing the squares
-in \(Z_i\) and in the longitudinal energy, one obtains
+The expressions \(A^2/f\) and \(ND^0\) are extended by zero
+where \(f=0\). The bump \(f\) is smooth,
+\(0\le f\le2\), is positive on \(K\), and is flat on its zero set.
+On the region \(\chi=1\) it satisfies
\begin{equation}
- \begin{split}
- &Z_i^2-2NU_i(Z_i)+N^2|\II_i|_{h_i}^2+N^2|d\log L_i|_{h_i}^2
- +\epsilon N^2|dF_i|_{h_{i-1}}^2\\
- &\quad\ge Z_i^2+(X_i-t_i)^2+N^2|\II_i^{\mathrm{rem}}|^2
- +N^2|d\log L_i|_{h_i}^2\\
- &\quad+\epsilon A_i^2\bigl[(2a-4a^2)F_i^2/f_i^2+2a\bigr]
- +4\epsilon a^2\sigma_iB_iA_i^2\\
- &\quad+(L_i-1)\bigl[2a(\sigma_i+r)+u_i\bigr]A_i^2 .
- \end{split}
- \label{eq:sweep-ideal-bound}
-\end{equation}
-The completion uses \(NU(A_i^2)=2A_i^2X_i\) and the energy estimate
-$\epsilon|dF|^2-4\epsilon a(F/f)A\langle dF,\cdot\rangle
-\ge-4\epsilon a^2(F/f)^2A^2(1-B)$ applied pointwise.
-
-\smallskip\noindent\emph{Step 3: summing.}
-Multiplying \eqref{eq:sweep-ideal-bound} by \(r/\epsilon\), summing
-over $i$, and using \(rF_i^2/f_i^2=\sigma_i\), gives
-\begin{align}
- Q\ge{}&T_0+\frac r\epsilon\mathcal X
- +\epsilon\E_\nu g_i^2
- +\sum_i A_i^2\bigl[(2a-4a^2)\sigma_i+2ar-1\bigr] \notag\\
- &+\frac r\epsilon\Bigl(\sum_i\mathcal B_i-\delta_0\Bigr),
- \label{eq:sweep-expanded}
-\end{align}
-where \(g_i=a(\sigma_i-r)+u_i/2\), and $\mathcal B_i$ is the boost
-cost (defined below).
-
-\smallskip\noindent\emph{Step 4: the boost.} Suppose the pass uses
-$\psi=\kappa\Psi$ for a small \(\kappa>0\) (the non-boost case
-\(\psi=0\) gives $\mathcal B_i=0$). The boost cost, expressed in
-the units of $Q$, is bounded by
+ ND=ND^0-\beta f+\epsilon a\delta^2/f .
+ \label{eq:graph-core-prescription}
+\end{equation}
+On the region where the cutoff changes, the following quantities
+tend uniformly to zero as \(H_*\to\infty\), with the preceding
+data and parameters fixed:
\begin{equation}
- \sup_{\text{sweep}}\Bigl|\frac r\epsilon\sum_i\mathcal B_i\Bigr|
- \le C(\epsilon+\epsilon^2)\kappa r^2\indic_{\{r\ge8\}}.
- \label{eq:sweep-boost-cost}
+ \begin{gathered}
+ N^2\bigl|\wscal(h+N^2df\otimes df,\phi-\log L)
+ -\wscal(h,\phi)\bigr|,\quad
+ A^2/f,\quad |ND^0|,\quad \beta f,\\
+ |ND|,\quad N|\II|,\quad N|d\log L|_{h+N^2df\otimes df}.
+ \end{gathered}
+ \label{eq:graph-cutoff-small}
\end{equation}
-Two positive terms of \eqref{eq:sweep-expanded} cover it.
-
-\emph{(a) Mean square.} For \(r\ge8\), \(a(1-r)\le-7r/16\) and
-$\bar u\ge0$, so $-M_0\ge7r/16$, giving $M_0^2\ge49r^2/256$ and hence
-$\epsilon^2M_0^2/r\ge c\epsilon^2r$.
+In particular, on all of \(X\) the function
+\(\mathrm{err}:=ND-(ND^0-\beta f)\) tends uniformly to zero.
+All data retain the corresponding uniform bounds for each fixed
+choice of \(H_*\), with the positive-order convention for weights.
+For the family assertion, let a standard Borel parameter space index
+such problems of a fixed dimension. Require common sizes and buffers
+for these simultaneous charts, and common bounds there for
+\(h,h^{-1},h_{\mathrm{ref}},h_{\mathrm{ref}}^{-1}\) and all their
+coordinate derivatives. Require common bounds for all derivatives,
+including values, of \(F_i,\alpha\), and for the positive-order
+derivatives of \(\phi\). Let \((\epsilon,l,\theta)\) range in a fixed
+compact subset of the permitted parameter set. Require the data,
+including \(h_{\mathrm{ref}}\), to
+be Borel in the charts and preserved by the equivariant identifications
+under consideration. Require also the Borel chart overlaps and
+countable Borel enumerations of accessible charts in
+Lemma~\ref{lem:graph-prescription}. Then the choices are Borel and
+commute with those identifications; their bounds for each fixed
+\(H_*\), and the limits above as \(H_*\to\infty\), are uniform over
+the family. When \(K\) is empty one may take \(f=0\) and omit the
+operation.
+\end{lemma}
-\emph{(b) Boost square.} Since $f_i\le2$, the increments $f_i^2$ in
-$R_i=\sum_{j<i}f_j^2$ are at most $4$. For large $r$, the rows with
-\(R_i\ge r/2\) carry total increment at least \(r/4\). Each such row
-contributes (using $\Psi\ge cs^2$ for $s\ge R_\Psi$) at least
-$c\epsilon^2\kappa^2 f_i^2 R_i^2$, so
-\begin{equation}
- \sum_i f_i^2\psi(R_i)^2\ge c\epsilon^2\kappa^2r^5
- \quad(r\ge C_*).
- \label{eq:sweep-boost-square}
-\end{equation}
-Choosing \(\kappa\) sufficiently small (uniformly in $r$) lets (a) and
-(b) absorb \eqref{eq:sweep-boost-cost} in the two regimes
-$8\le r\le\kappa^{-1/2}$ and $r\ge\kappa^{-1/2}$, respectively, and
-leaves a fixed fraction of $\sum_i\beta_i^2f_i^2$ free for $\mathcal X$.
-
-\smallskip\noindent\emph{Step 5: conclusion.} Combining
-\eqref{eq:sweep-expanded} with the boost absorption gives the
-\emph{one-sweep inequality}
+\begin{proof}
+We first construct the outer coefficient while keeping its
+transition width bounded. Starting at a fixed positive distance
+from \(K\), a scalar profile \(b\) can increase from zero to
+\(H_*\) at speed at most
+\[
+ \frac{c_0}{C N}\bigl(\theta\epsilon^{3/2}+b^{3/2}\bigr),
+\]
+where the fixed factor \(C\) leaves room for the estimates below.
+Choose the speed comparable to this expression during the
+increase, with smooth flattening at either end. The width needed
+is at most
\begin{equation}
- Q\ge\Pi_p-4\theta^2m-\frac r\epsilon\delta_0,
- \label{eq:sweep-inequality}
+ C N\int_0^\infty\frac{dt}
+ {\theta\epsilon^{3/2}+t^{3/2}}
+ =C\frac{N}{\theta^{1/3}\sqrt\epsilon}
+ =C\frac{\sqrt l}{\theta^{1/3}}\le C .
+ \label{eq:graph-cutoff-width}
\end{equation}
-where the \emph{positive reserve} is
+For example one can use the primitive of the reciprocal of a smooth
+positive function comparable to
+\(\theta\epsilon^{3/2}+t^{3/2}\) as the new profile coordinate,
+and compose with fixed smooth flattenings. The last inequality
+uses \(l\le\theta\le1/16\).
+
+One may use this profile in a smooth approximation of
+\(\dist_{h_{\mathrm{ref}}}(x,K)\) with gradient at most \(2\)
+and a uniformly small error. Here is a construction with the
+needed uniformity. Shrinking the controlled charts gives normal
+charts at a fixed positive radius, by the uniform geodesic
+equation estimates. Average the distance function in such a
+chart centered at \(x\) against a fixed smooth radial kernel in
+\(T_xX\), at a sufficiently small radius. Written as an integral
+against a kernel in \((x,y)\), this is a smooth function of \(x\).
+The error is at most the averaging radius. To bound the first
+derivative, move the center and parallel transport the averaging
+vectors. At a sufficiently small radius the displacement of
+the exponential images is at most twice the displacement of the
+center; the distance function is one-Lipschitz. The derivative
+bound follows. Higher derivatives have uniform bounds from the
+smooth local kernels: in differentiating the normalized kernel
+one may subtract the distance at the center, leaving a uniformly
+bounded oscillation on its support. These derivative bounds hold in
+the common buffered charts, since the \(h_{\mathrm{ref}}\) exponential
+maps and kernels have uniformly bounded derivatives there. The inverse metric
+inequality \(h^{-1}\le h_{\mathrm{ref}}^{-1}\) then bounds the
+gradient of the composed profile in \(h\).
+
+Choose this raw profile \(b\) to vanish near \(K\), to equal
+\(H_*\) past the width in \eqref{eq:graph-cutoff-width}, and to
+leave fixed outer buffers. To accommodate \(\alpha\), take a
+smooth nondecreasing saturation
+\(\sigma_{H_*}:[0,\infty)\to[0,H_*]\) that equals the identity
+up to \(H_*/3\), equals \(H_*\) from \(H_*\) on, and has a
+derivative bounded by a numerical constant. Put
+\[
+ \beta=\sigma_{H_*}(b+\alpha),\qquad
+ \beta_s=\sigma_{H_*}(b+\alpha)-\alpha .
+\]
+Take \(H_*\) much larger than the value and derivative bounds
+of the already fixed \(\alpha\). Then \(\beta_s\ge0\), and it
+equals \(b\) near \(K\). Wherever the saturation differs from
+the identity, both \(\beta_s\) and the scale of \(b\) are
+comparable to \(H_*\). The formula
+\[
+ d\beta_s=\sigma_{H_*}'(b+\alpha)\,db
+ +\bigl(\sigma_{H_*}'(b+\alpha)-1\bigr)d\alpha
+\]
+therefore proves \eqref{eq:graph-spatial-cutoff}: choose the
+initial fixed speed factor large enough to handle the first
+term, and then choose \(H_*\) large enough for the fixed
+\(d\alpha\) term to be a small fraction of
+\(H_*^{3/2}/N\). Where saturation is the identity this is just
+the raw profile bound. Where \(b=H_*\), the total \(\beta\)
+equals \(H_*\). A cutoff supported beyond fixed additional
+buffers can consequently be chosen once at a radius
+\(R_{\mathrm{loc}}\) as in the statement. Take it to be the
+square of a smooth nonnegative cutoff that is flat at its outer
+end. This gives \(|d\chi|^2\le C\chi\) and the other asserted
+cutoff properties. If some outer regions are empty, the
+corresponding assertions are vacuous.
+
+We next prove that the complete solution can be cut on this
+plateau. All constants in the rest of the proof may use the
+fixed data and parameters, but not \(H_*\). Write
+\[
+ c_H=H_*^{-2}\sqrt{\frac{\epsilon a}{\epsilon a+H_*}}
+ \asymp H_*^{-5/2}.
+\]
+By \eqref{eq:graph-height-positive},
+\(c_H\le\widetilde f\le2\). Cover the cutoff region by inner chart
+boxes, each with a fixed succession of larger buffered boxes whose
+closures lie in the plateau \(F_i=0,\ \beta=H_*\). Their sizes may
+use the current \(h\)-chart bounds inside the fixed
+\(h_{\mathrm{ref}}\) buffers; all these bounds were fixed before
+\(H_*\), so the boxes and their buffers have sizes independent of
+\(H_*\). Our target is that, for every integer \(j\ge0\) and every
+\(M>0\), there is \(C_{j,M}\) such that, for all sufficiently large
+\(H_*\),
\begin{equation}
- \Pi_p:=T_0+c_0\Bigl(\frac r\epsilon\mathcal X
- +\epsilon(\bar u^2+\Var_\nu u)
- +c_*\epsilon r^6\indic_{\{r\ge C_*\}}\Bigr)
- \label{eq:sweep-positive-reserve}
-\end{equation}
-with a small universal $c_0>0$ and $c_*>0$ a small multiple of
-$\kappa^2$. All terms in $\Pi_p$ are nonnegative.
-
-The inequality \eqref{eq:sweep-inequality} is what we will iterate in
-\S\ref{sec:entropy}. It roughly says: as long as the sweep has
-either many active coordinates or a substantial mismatch between
-$F_i^2$ and $r f_i^2$, the functional $Q$ is bounded below by a
-positive fraction of the "size" of the sweep. If neither is true, we
-need no improvement. This dichotomy is made precise in
-\S\ref{sec:entropy}.
-\section{Entropy metrics with controlled labels}
-\label{sec:entropy}
-
-This section iterates the graph sweep of \S\ref{sec:graphs} to build up
-a large \emph{energy coefficient} \(E\) while controlling the volume
-expansion. The output is Theorem~\ref{thm:entropy}: an entropy datum
-$(h,\phi,\rho,F)$ with respect to the input metric $g$ (which
-satisfies $\Scal_g=-1$), for which the inequality
-\[
- \wscal(h,\phi)\ge-C+2W-n\log E+E\,e_h(F),
-\]
-holds with $W=\log(dV_h/\rho)\ge0$ and $\rho\le dV_g$. Two extra
-properties are essential for \S\ref{sec:transfer}: the \emph{absolute
-gradient smallness} $e_h(F)^{1/2}\le E^{-1/100}$, which makes the label
-overlaps bounded by a fixed graph, and the linear paying term
-$E\,e_h(F)$, which absorbs the coordinate-gradient penalties in the
-degree estimate.
-
-\subsection{The entropy theorem}
-
-\begin{theorem}[Entropy metrics]
-\label{thm:entropy}
-Suppose $\Scal_g=-1$ on a closed oriented $M$. For every prescribed
-elementary function $\mathcal B$ and every finite $E_{\min}$, there
-exist:
-\begin{itemize}
-\item a standard Borel probability $\Gamma$-space $(\Omega,\mu)$
- extending an initial space $(\Omega_0,\mu_0)$ (with an equivariant
- measure-preserving extension $\Omega\to\Omega_0$);
-\item a countable label set $\mathcal L$ with finitely many free
- $\Gamma$-orbits;
-\item equivariant leafwise data on $\widetilde M\times\Omega$: a
- metric \(h\), a function $\phi$, a positive density $\rho$, and a
- root probability map $F$,
-\end{itemize}
-with the following properties. On an invariant conull set of
-transverse parameters,
+ \|\widetilde f-c_H\|_{C^j}
+ \le C_{j,M}H_*^{-M},
+ \label{eq:graph-plateau-jets}
+\end{equation}
+on the inner boxes, with norms in these fixed coordinates.
+The constants use only the indicated data and parameters fixed
+before \(H_*\). The family assertion of
+Lemma~\ref{lem:graph-prescription} is applied separately for each
+fixed \(H_*\); the estimates that follow establish this
+superpolynomial decay uniformly as \(H_*\) grows.
+
+There is first a global angle bound
\begin{equation}
- \rho\le dV_g,\qquad
- W:=\log(dV_h/\rho)\ge0,
- \label{eq:entropy-density}
+ v\ge\frac{c_1}{1+\sqrt{H_*}}
+ \label{eq:graph-global-plateau-angle}
\end{equation}
+for the uncut graph. In fact, at a recentered infimum with
+\(v\le1/2\), the formulas for \(p_x,p_z\) and
+\eqref{eq:graph-spatial-cutoff} give
+\[
+ |p_z|\ge c_2(1+\beta+\widetilde f^{-2}),\qquad
+ |p_x|\le C\bigl(\widetilde f^{-1}
+ +\widetilde f(1+\beta^{3/2})\bigr).
+\]
+For the first bound use \(1-v^2\ge3/4\) in \(p_z\); for the
+second use the fixed bounds on \(F_i,\alpha\) and their
+derivatives. Since \(\widetilde f\le2\) and \(\beta\le H_*\),
+the ratio of the right side of the second estimate, plus a
+constant, to the right side of the first is at most
+\(C(1+\sqrt{H_*})\). Equation
+\eqref{eq:graph-angle-infimum} proves
+\eqref{eq:graph-global-plateau-angle}. This recentering is done
+for each fixed \(H_*\), using the individual regularity supplied
+by Lemma~\ref{lem:graph-prescription}.
+
+On each of the outer plateau boxes the equation is
\begin{equation}
- \wscal(h,\phi)\ge-C+2\bigl(W-\tfrac n2\log E\bigr)+E\,e_h(F),
- \qquad e_h(F)^{1/2}\le E^{-1/100},
- \label{eq:entropy-contract}
+ ND=\frac{\epsilon a\delta^2}{\widetilde f}
+ +\frac{1-v^2}{2\widetilde f}
+ -(\epsilon a+H_*)\widetilde f,
+ \qquad
+ (\epsilon a+H_*)c_H^2=\epsilon a\delta^2 .
+ \label{eq:graph-plateau-equation}
\end{equation}
+There is an exponential value estimate on an inner box:
\begin{equation}
- E>\mathcal B(q),
- \label{eq:entropy-E-lower}
-\end{equation}
-where $C=C(n,\mathcal B)<\infty$, and $q$ is a label-complexity
-parameter defined just below. All output objects are Borel
-transversely and smooth along leaves. In the same lifted charts, the
-coefficients of \(h\), their derivatives of every order, and the
-coefficients of \(h^{-1}\) have uniform bounds; $\phi$, the density
-ratio $\rho/dV_g$, its reciprocal, and their derivatives of every
-order have uniform bounds. The coordinates of $F$ have the same
-derivative and uniform local-finiteness properties as the input. The
-supports of all these objects are contained in equivariant compact
-sets, uniformly in the transverse parameter.
-
-There is, in addition, a fixed $\Gamma$-invariant graph $\mathcal G$
-on $\mathcal L$, independent of the transverse parameter, of finite
-maximum degree $\Delta$, containing an edge between every pair of
-distinct labels that are simultaneously nonzero at any point of any
-leaf. We may take
-\[
- q\ge\max\{n+1,\ \#(\Gamma\backslash\mathcal L),\ \Delta+1\},
+ 0\le\widetilde f-c_H\le C\exp(-c_3\sqrt{H_*}).
+ \label{eq:graph-plateau-value}
+\end{equation}
+To prove it, let \(d_{\mathrm{face}}\) be ordinary coordinate
+distance to a face of the outer box and put
+\[
+ w_1=\sum_{\mathrm{faces}}3
+ \exp(-\gamma\sqrt{H_*}\,d_{\mathrm{face}}).
\]
-and a Borel equivariant coloring
-$\chi:\mathcal L\times\Omega\to\{1,\ldots,q\}$ such that labels at
-graph distance one or two have different colors.
-\end{theorem}
+On the boundary, \(c_H+w_1\ge3\ge\widetilde f\).
+Its first and second coordinate derivatives are bounded by
+\(C\gamma\sqrt{H_*}w_1\) and \(C\gamma^2H_*w_1\),
+respectively. Direct differentiation of the weighted graph
+divergence therefore bounds its mean curvature from below by
+\[
+ -C(\gamma^2H_*+\gamma\sqrt{H_*})w_1 .
+\]
+For this graph, the height terms in its prescription satisfy
+\[
+ \frac{\epsilon a\delta^2}{N(c_H+w_1)}
+ -\frac{(\epsilon a+H_*)(c_H+w_1)}N
+ =-\frac{\epsilon a+H_*}{N}w_1
+ \left(1+\frac{c_H}{c_H+w_1}\right).
+\]
+The remaining angle term is at most
+\(C\gamma^2H_*w_1\), because
+\((1-v^2)/(c_H+w_1)\le
+ N^2|dw_1|^2/(c_H+w_1)\le C\gamma^2H_*w_1\).
+Choosing \(\gamma>0\) sufficiently small and then \(H_*\)
+sufficiently large makes \(c_H+w_1\) an upper barrier.
+Comparison with this barrier and the lower constant \(c_H\)
+proves \eqref{eq:graph-plateau-value} on any inner box a fixed
+distance from the faces.
+
+On a smaller box the angle is bounded below independently of
+\(H_*\). To see this, choose a smooth chart cutoff
+\(0\le\xi\le1\), supported in a region where
+\eqref{eq:graph-plateau-value} holds and equal to one on the
+smaller box. Minimize \(v/\xi\) over \(\{\xi>0\}\).
+The global lower bound \eqref{eq:graph-global-plateau-angle}
+makes a minimum with value below \(1/2\), if one exists,
+interior, and at that minimum
+\[
+ v<\xi/2,\qquad \xi\ge c/\sqrt{H_*},\qquad
+ \nabla_\Sigma v=(v/\xi)\nabla_\Sigma\xi,\qquad
+ \Delta_{\Sigma,\phi}v\ge(v/\xi)\Delta_{\Sigma,\phi}\xi.
+\]
+The formula for the restriction of an ambient function to the
+graph gives
+\(\Delta_{\Sigma,\phi}\xi\ge-C(1+|\II|)\). The positive
+\(|\II|^2\) term in \(P\), and completion of its square against
+\(C|\II|/\xi\), show that the left side of
+\eqref{eq:graph-angle-equation} is at least \(-CH_*v\).
+On this plateau \(p_x=0\). Since \(v<1/2\), the \(p_z\) term
+on its right side is at most \(-c/\widetilde f^2\).
+Also \(p_v=-v/(N\widetilde f)\) and
+\(|T(v)|\le Cv/\xi\), so that right side is at most
+\[
+ -\frac{c}{\widetilde f^2}
+ +\frac{C\sqrt{H_*}\,v}{\widetilde f}.
+\]
+Combining the two bounds and multiplying by \(\widetilde f^2\)
+would give
+\[
+ c\le CH_*\widetilde f^2+
+ C\sqrt{H_*}\,\widetilde f .
+\]
+By \eqref{eq:graph-plateau-value},
+\(\widetilde f\le2c_H\) for large \(H_*\), and the last right
+side tends to zero. This contradiction proves \(v\ge1/2\) on
+the smaller box.
+
+It remains to control derivatives on the cutoff region without
+using any growth rate for the global elliptic constants.
+In a still smaller box, rescale coordinates \(x=x_0+c_Hy\) and
+write \(\widetilde f=c_Hg\). The weighted graph curvature in
+the rescaled chart is \(c_HD\), and its right side becomes
+\begin{align*}
+ c_Hp(x,Nc_Hg,v)
+ &=\frac{\epsilon a\delta^2+(1-v^2)/2}{Ng}
+ -\frac{c_H^2(\epsilon a+H_*)g}{N}\\
+ &=\frac{\epsilon a\delta^2+(1-v^2)/2}{Ng}
+ -\frac{\epsilon a\delta^2g}{N}.
+\end{align*}
+Here \(1\le g\le2\) for large \(H_*\), and the angle bound gives
+a uniform gradient bound. The rescaled metric and normalized
+weight have uniformly bounded derivatives; the displayed
+right side and all its derivatives are uniformly bounded on
+these ranges. The local higher estimates in the proof of
+Lemma~\ref{lem:graph-prescription}, applied on unit boxes in
+these coordinates, consequently give uniform bounds for every
+derivative of \(g\). Scaling back gives bounds of the form
+\(C_jc_H^{1-j}\) for the \(j\)-th derivatives of
+\(\widetilde f\). Interior interpolation between these polynomial
+bounds and the exponential bound \eqref{eq:graph-plateau-value}, on
+further fixed buffered boxes, proves \eqref{eq:graph-plateau-jets}.
+Indeed the usual finite-difference interpolation of a
+\(j\)-th derivative between the supremum norm and a higher
+derivative norm gives an exponential factor times a power of
+\(c_H^{-1}\), which is smaller than every power of \(H_*^{-1}\).
+
+We can now cut off. On the transition region,
+\(\widetilde f\asymp c_H\), its positive-order derivatives are
+smaller than every power of \(H_*^{-1}\), and the derivatives
+of \(\chi\) are fixed. On \(\{\chi>0\}\),
+\[
+ \frac{|d(\chi\widetilde f)|^2}{\chi\widetilde f}
+ \le
+ 2\widetilde f\,\frac{|d\chi|^2}{\chi}
+ +2\chi\,\frac{|d\widetilde f|^2}{\widetilde f}
+ =O(c_H).
+\]
+It follows that \(A^2/f\le N^2|df|^2/f\) tends to zero.
+All fixed derivatives of \(f\) are \(O(c_H)\) there.
+The graph formulas therefore give \(ND,N|\II|\), and
+\(N|d\log L|\) tending to zero. They also control
+\(\nabla D\), so \eqref{eq:graph-identity} gives the first
+limit in \eqref{eq:graph-cutoff-small}. Finally \(F_i=0\) in
+this region, \(ND^0=-\epsilon af+A^2/(2f)\), and
+\(\beta f\le H_*\,O(c_H)\to0\). This proves every limit in
+\eqref{eq:graph-cutoff-small}. To verify the extensions at the zero
+set, write \(\chi=\zeta^2\), where \(\zeta\) is smooth and flat there.
+On \(\{\zeta>0\}\),
+\[
+ \frac{|df|_h^2}{f}
+ =4\widetilde f\,|d\zeta|_h^2
+ +4\zeta\langle d\zeta,d\widetilde f\rangle_h
+ +\zeta^2\frac{|d\widetilde f|_h^2}{\widetilde f}.
+\]
+The right side extends smoothly and flatly across \(\{\zeta=0\}\),
+since \(\widetilde f>0\). The same is therefore true of
+\[
+ \frac{A^2}{f}
+ =\frac{N^2}{1+N^2|df|_h^2}\,\frac{|df|_h^2}{f}.
+\]
+Moreover, \(F_i\) vanishes on a neighborhood of the zero set of
+\(f\). Thus \(ND^0\) extends smoothly by zero there, and all jets
+of \(f=\zeta^2\widetilde f\) vanish on its zero set.
-The proof of Theorem~\ref{thm:entropy} is the content of the rest of
-this section.
+On the uncut region the last term
+\(\varpi=\epsilon a\delta^2/f\) in
+\eqref{eq:graph-core-prescription} has
+\begin{equation}
+ \|\varpi\|_\infty
+ \le\delta\sqrt{\epsilon a(\epsilon a+H_*)}\longrightarrow0,
+ \qquad
+ -2NU(\varpi)
+ =\frac{2\epsilon a\delta^2 L A^2}{f^2}\ge0 .
+ \label{eq:graph-floor-derivative}
+\end{equation}
+Since \(ND^0-\beta f\ge-(\epsilon a+\beta)f\), its contribution
+inside the curvature square satisfies
+\begin{equation}
+ (ND^0-\beta f+\varpi)^2
+ \ge (ND^0-\beta f)^2
+ -2\epsilon a\delta^2(\epsilon a+H_*).
+ \label{eq:graph-floor-square}
+\end{equation}
+The last loss tends to zero. These estimates on the core and
+\eqref{eq:graph-cutoff-small} on the transition prove the
+claimed bound for \(\mathrm{err}\). The curvature estimates
+use the derivative sign in \eqref{eq:graph-floor-derivative}
+and the full transition estimates, as well as that bound on
+the value of \(\mathrm{err}\).
+
+Every construction has uniform bounds for fixed \(H_*\).
+For Borel data, the distance to the open nonzero set of \(F_i\)
+is Borel: using the enumerated charts it is the infimum of
+lengths of a countable dense collection of chart paths ending
+where \(F_i\ne0\). The normalized smoothing kernels and all
+subsequent scalar operations are Borel and equivariant.
+Lemma~\ref{lem:graph-prescription} supplies these same
+properties for the complete solution. This proves the final
+assertions.
+\end{proof}
-\subsection{The bootstrap}
+\subsection{A sweep of the root probabilities}
-The first stage produces an initial datum with a small but positive
-energy coefficient.
+We now apply the localized graph to each coordinate of a root
+probability map. The order of the operations matters: a graph
+enlarges the metric available to all subsequent coordinates. We
+use finitely many colors so that graphs in the same color have
+disjoint supports and can be performed simultaneously.
+For a root probability map, its energy is
+\[
+ e_h(F)=|dF|_{h,\HS}^2=\sum_i|dF_i|_h^2.
+\]
-\begin{lemma}[Conformal bootstrap]
-\label{lem:entropy-bootstrap}
-There are dimensional $\epsilon_*>0$ and $C_{\mathrm{boot}}<4$ with
-the following property. Starting from any root probability map $F$
-and a metric $h_0$ with $\Scal_{h_0}\ge-C_0+\epsilon e_{h_0}(F)$ for
-some $C_0\in[2,4]$ and $0<\epsilon\le\epsilon_b$, finitely many
-sweeps without boost, each followed by a conformal change, produce a
-metric \(h_*\), root probability map \(F_*\), and
-$\epsilon_0\in[\epsilon_*,2\epsilon_*]$ with
+Fix a smooth nonnegative nondecreasing function
+\(\Psi:[0,\infty)\to[0,\infty)\) such that
\begin{equation}
- \rho:=dV_{h_*}\le dV_g,\qquad
- \Scal_{h_*}\ge-C_{\mathrm{boot}}+\epsilon_0e_{h_*}(F_*).
- \label{eq:entropy-bootstrap-output}
+ \Psi(s)=0\quad(s\le8),\qquad
+ \Psi(s)\le Cs^2,\qquad |\Psi'(s)|^2\le C\Psi(s),\qquad
+ \Psi(s)\ge cs^2\quad(s\ge R_\Psi),
+ \label{eq:sweep-boost-profile}
\end{equation}
-The metric and map have the uniformity and support properties of
-Theorem~\ref{thm:entropy}, and the number of sweeps may depend on the
-input.
-\end{lemma}
+where \(R_\Psi\) and the constants are numerical. Such a
+function can be obtained as the square of a smooth
+nondecreasing function that is zero up to \(8\), has bounded
+derivative, and equals \(s-8\) from a fixed larger value on.
+We use either \(\psi=0\), or \(\psi=\kappa\Psi\) for a small
+constant \(\kappa>0\). In the latter case,
+\[
+ \psi\le C\kappa s^2,\qquad
+ |\psi'|^2\le C\kappa\psi,\qquad
+ \psi(s)\ge c\kappa s^2\quad(s\ge R_\Psi).
+\]
-\begin{proof}[Proof sketch]
-Take $\phi_0=0$, no boost, and $l=\theta$ fixed small. Apply a sweep
-to produce $(h_+,\phi_+)$, then reset by the conformal change
-\(k=e^{-2W_p/(n-1)}h_+\). Since $\phi_+=-W_p$ after a non-boost
-sweep, the scalar conformal formula and the definition of $\wscal$
-give
+\paragraph{Notation for an ordered sweep.}
+We introduce the pointwise quantities for any candidate sequence of
+graphs before asserting that the sequence can be constructed. The
+pass parameters are
\begin{equation}
- \Scal_k=e^{2W_p/(n-1)}
- \Bigl(\wscal(h_+,\phi_+)+\frac{|dW_p|_{h_+}^2}{n-1}\Bigr),
- \label{eq:entropy-bootstrap-reset}
+ \epsilon>0,\qquad N^2=l\epsilon,\qquad
+ a=\tfrac12+\theta,\qquad \lambda=\tfrac12-2\theta,\qquad
+ 0<l\le1,\quad l\le\theta\le\tfrac1{16},
+ \label{eq:sweep-parameters}
\end{equation}
-and the volume changes as
+with \(N>0\). On one leaf, start with \((h_0,\phi_0,F)\), where
+\(F_i\ge0\) and \(\sum_iF_i^2=1\). Suppose nonnegative smooth
+bumps \(f_i\) form a locally finite family ordered in finitely many
+colors, with disjoint supports within each color, and that
+\(F_i\ne0\) implies \(f_i>0\). Attach also nonnegative functions
+\(\beta_i\). At this stage these are candidate data, with no graph
+equation imposed.
+
+Apply in color order the graph of height \(N\) times the sum of the
+bumps in that color. On the support of \(f_i\), write
+\(h_{i-1},\phi_{i-1}\) for the pair before its color and
+\(h_i,\phi_i\) for the pair afterward; write \(j<i\) for an earlier
+color. Let \(L_i,U_i,D_i,\II_i,H_i\) be the individual graph
+quantities for \(f_i\), and let \(h_+,\phi_+\) be the final pair.
+The subscript \(p\) denotes the current pass in \(W_p\) and in the
+reserve \(\Pi_p\) below.
+The graph identities give the output and its normalization:
\begin{equation}
- dV_k=e^{-nW_p/(n-1)}dV_{h_+}
- =e^{-W_p/(n-1)}dV_{h_0}\le dV_{h_0}.
- \label{eq:entropy-bootstrap-density}
-\end{equation}
-The claim is that \(C_b,\epsilon_b>0\) exist so that, whenever
-\(2\le C_0\le4\), \(0<\epsilon\le\epsilon_b\), and
-$\Scal_{h_0}\ge-C_0+\epsilon e_{h_0}(F)$, one can choose the sweep
-errors to achieve
-\begin{equation}
- \Scal_k\ge-(C_0+C_bl\epsilon)+(1+l)\epsilon e_k(Y).
- \label{eq:entropy-bootstrap-step}
-\end{equation}
-Indeed, on the region where \eqref{eq:sweep-inequality} has a
-positive reserve $\Pi_p\ge cm$, the inequality itself provides the
-gain. On the complementary region $m<m_0$, $|1-r|<1/4$ (where
-$\Pi_p$ may be small), the mean square $M_0^2$ and the off-diagonal
-terms cover the loss. The full absorption is the content of the
-original derivation; the bootstrap constant $C_b$ is dimensionally
-bounded and $\epsilon_b$ can be chosen small in terms of the
-absorption constants. This proves \eqref{eq:entropy-bootstrap-step}.
-
-Iterating \eqref{eq:entropy-bootstrap-step}: starting with
-$\epsilon\le\epsilon_b$, after $k$ sweeps,
-$\epsilon_k=(1+l)^k\epsilon$ and
-$C_k=C_0+C_b\sum_{p<k}l\epsilon_p$. Since $l$ is fixed and the
-coefficient grows geometrically, finitely many sweeps (depending on
-the input) reach $\epsilon_0\in[\epsilon_*,2\epsilon_*]$ and
-$\Scal_{h_*}\ge-C_{\mathrm{boot}}+\epsilon_0e_{h_*}(F_*)$ with
-$C_{\mathrm{boot}}<4$.
-\end{proof}
-\subsection{The entropy increment}
-
-The increment lemma is the key analytic step of \S\ref{sec:entropy}.
-One sweep with boost converts the one-sweep inequality
-\eqref{eq:sweep-inequality} into a growth of the entropy functional
-$\mathcal F:=\wscal(h,\phi)-\epsilon e_h(F)$ at controlled volume
-growth. The precise form of the increment is
+ \begin{gathered}
+ h_+=h_0+N^2\sum_i df_i\otimes df_i,\qquad
+ W_p:=\sum_i\log L_i=\log(dV_{h_+}/dV_{h_0})\ge0,\\
+ \phi_+=\phi_0-W_p,\qquad
+ e^{\phi_+}dV_{h_+}=e^{\phi_0}dV_{h_0},\qquad
+ r:=\sum_i f_i^2>0,\qquad Y_i=f_i/\sqrt r .
+ \end{gathered}
+ \label{eq:sweep-output}
+\end{equation}
+The sums are pointwise finite, with at most one nonzero bump per
+color. The implication \(F_i\ne0\Rightarrow f_i>0\) ensures
+\(r>0\). In all the quantities below, indices are restricted to
+\(f_i>0\).
+
+For each such index, use the longitudinal notation of the
+one-coordinate calculation, with \(h=h_{i-1}\),
+\(h'=h_i\), and \(h_{\mathrm{in}}=h_0\):
\[
- \mathcal F(\text{output})\ge
- \mathcal F(\text{input})+2W_p-n\log(1+l)-C_nl^2+P_p,
+ A_i=|U_i|_{h_{i-1}},\qquad U_i=A_i e_i,\qquad
+ x_i=-\II_i(\tau_i,\tau_i),\qquad X_i=Nx_i,\qquad
+ t_i=A_i^2/f_i.
\]
-where the surplus $P_p$ is positive whenever the sweep has either
-many active coordinates or a mismatch between $F_i^2$ and $rf_i^2$.
-
-\begin{lemma}[Entropy increment]
-\label{lem:entropy-increment}
-There are dimensional constants $m_0>0$, $D_*\ge1$, $l_0>0$,
-$\gamma_n>0$, and $C_n<\infty$ with the following property. For
-$\epsilon\ge\epsilon_*$ from Lemma~\ref{lem:entropy-bootstrap}, take a
-boosted sweep with
+Here \(\tau_i=(L_i^{-1}e_i,A_i)\) and \(x_i=e_i(A_i)\) when
+\(A_i>0\). At \(A_i=0\) use any unit direction as in
+\eqref{eq:graph-longitudinal}. In the graph metric \(h_i\), define
+\begin{equation}
+ \begin{gathered}
+ \II_i^{\mathrm{rem}}
+ :=\II_i+x_i\,\tau_i^\flat\otimes\tau_i^\flat,\\
+ |\II_i|^2=x_i^2+|\II_i^{\mathrm{rem}}|^2,\qquad
+ \tr\II_i^{\mathrm{rem}}=H_i+x_i .
+ \end{gathered}
+ \label{eq:sweep-longitudinal}
+\end{equation}
+In particular, at \(A_i=0\) the combination
+\((X_i-t_i)^2+N^2|\II_i^{\mathrm{rem}}|^2=N^2|\II_i|^2\)
+is independent of the chosen direction. The losses from comparing
+the current graph with the input and final metrics are
\[
- 0<l\le l_0,\qquad \theta=D_*l\le\tfrac1{16},\qquad N^2=l\epsilon.
+ B_i=1-|e_i|_{h_0}^2,\qquad
+ C_i=
+ \begin{cases}
+ 1-N^2|df_i|_{h_+}^2/A_i^2,&A_i>0,\\
+ 0,&A_i=0.
+ \end{cases}
\]
-The errors $(\delta_0,\eta_0)$ in Proposition~\ref{prop:sweep} can be
-chosen so that
+Metric monotonicity and the inverse graph formula give
+\(0\le B_i,C_i\le1\).
+
+The normalization gives probability weights \(\nu_i=f_i^2/r\).
+Write \(\E_\nu b_i=\sum_i\nu_i b_i\). The mean and centered
+quantities used in the estimate are
\begin{equation}
-\begin{split}
- &\wscal(h_+,\phi_+)-(1+l)\epsilon e_{h_+}(Y)\\
- &\quad\ge \wscal(h_0,\phi_0)-\epsilon e_{h_0}(F)
- +2W_p-n\log(1+l)-C_nl^2+P_p,
-\end{split}
-\label{eq:entropy-increment}
+ \begin{gathered}
+ m=\sum_iA_i^2,\qquad
+ u_i=\frac{A_i^2}{\epsilon\nu_i},\qquad
+ \bar u=\E_\nu u_i=\frac m\epsilon,\\
+ \sigma_i=\frac{F_i^2}{\nu_i},\qquad \E_\nu\sigma_i=1,\\
+ M_0=a(1-r)-\bar u/2,\qquad
+ V_i=a(\sigma_i-1)+\lambda(u_i-\bar u),\qquad
+ \Var_\nu u=\E_\nu(u_i-\bar u)^2 .
+ \end{gathered}
+ \label{eq:sweep-statistics}
\end{equation}
-where the nonnegative surplus is
-\begin{equation}
- P_p=\gamma_n\frac{\epsilon}{rN^2}\Pi_p\,
- \indic_{\{m\ge m_0\}\cup\{|1-r|\ge1/4\}}.
- \label{eq:entropy-surplus-definition}
-\end{equation}
-Here $\Pi_p\ge0$ is the one-sweep positive reserve
-\eqref{eq:sweep-positive-reserve}; on the indicated region, $P_p$
-retains a fixed fraction of every positive term in the sweep
-inequality, and on the complementary region it is zero.
-\end{lemma}
+The identity for \(\E_\nu\sigma_i\) uses
+\(F_i\ne0\Rightarrow f_i>0\); the individual ratios \(\sigma_i\)
+need not be at most one.
-\begin{lemma}[Surplus bounds]
-\label{lem:entropy-surplus}
-There are dimensional $m_0>0$ and $C_*<\infty$ such that the surplus
-$P_p$ of \eqref{eq:entropy-surplus-definition} satisfies
+Finally, define two nonnegative quantities:
\begin{align}
- m&\le C_n\epsilon\bigl(1+P_p/\epsilon\bigr)^{3/5},
- \label{eq:entropy-m-surplus}\\
- W_p&\le C_n m\bigl(1+\log(1+P_p/m)\bigr),
- \label{eq:entropy-log-surplus}
+ T_0={}&\sum_i A_i^2\left[
+ C_i+4a^2\sigma_iB_i+(L_i-1)\{2a(\sigma_i+r)+u_i\}\right],
+ \label{eq:sweep-T}\\
+ \mathcal X={}&\sum_i\left[
+ (X_i-t_i)^2+N^2|\II_i^{\mathrm{rem}}|^2
+ +N^2|d\log L_i|_{h_i}^2+\beta_i^2f_i^2\right].
+ \label{eq:sweep-X}
\end{align}
-when $m>0$, and $W_p=0$ when $m=0$.
-\end{lemma}
-
-\begin{proof}[Proof sketch of Lemma~\ref{lem:entropy-increment}]
-Use the one-sweep inequality \eqref{eq:sweep-inequality} and the
-conformal reset $k=e^{-2W_p/(n-1)}h_+$. Substitute the definition
-\eqref{eq:sweep-Q} of $Q$; separate the $-4\theta^2m$ term; use the
-boost absorption already in $\Pi_p$; and account for the conformal
-reset through the scalar formula \eqref{eq:entropy-bootstrap-reset}.
-The volume reset contributes the $+2W_p$ in the target inequality,
-and the entropy-coefficient scaling $\epsilon\mapsto(1+l)\epsilon$
-matches the $(1+l)$ coefficient on the output side. Combined with
-$n\log(1+l)=nl-\tfrac n2l^2+O(l^3)$, the target
-\eqref{eq:entropy-increment} emerges with the $O_n(l^2)$ loss
-identified as $C_nl^2$.
-
-The key content is the identification of the surplus. On the
-region $\{m\ge m_0\}\cup\{|1-r|\ge1/4\}$, the one-sweep inequality
-provides a positive reserve $\Pi_p$; the surplus $P_p$ is exactly a
-fixed fraction of $\Pi_p$ scaled by $\epsilon/(rN^2)$, adjusted so
-that the sum of all individually absorbed terms matches the increment.
-On the complementary region $m<m_0$, $|1-r|<1/4$, no surplus is
-produced but the loss is bounded by $C_nl^2$ alone, which is
-absorbed in the target.
-\end{proof}
-
-\begin{proof}[Proof sketch of Lemma~\ref{lem:entropy-surplus}]
-Fix a small dimensional $m_0>0$ (with $m_0\ll\epsilon_*$ and
-$m_0<1/4$).
+The quantity \(T_0\) records the gains from the metric comparisons
+and from each graph's stretching. Its full coefficient will yield
+the sharp quadratic term \(m^2/n\) in Section~\ref{sec:entropy}.
+The quantity \(\mathcal X\) retains the derivative squares and the
+applied coefficients. Together with the mean and variance squares
+below, it controls graph curvature; those moment squares also control
+the normalization.
-\emph{Coarse coercivity.} Claim:
-\begin{equation}
- T_0+\epsilon M_0^2\ge c_n(m+rW_p)
- \quad\text{on }\{m\ge m_0\}\cup\{m<m_0,\ |1-r|\ge1/4\}.
- \label{eq:entropy-coarse-coercivity}
-\end{equation}
-For $m\ge2n$, the $C_i$ terms of $T_0$ give
-$T_0\ge m-m_f\ge m-n\ge m/2$. For $m_0\le m\le2n$:
-if every $A_i^2$ is at most $\eta=\eta(n,m_0)>0$, then
-$2W_p\le(1+C\eta)m$, and $m-m_f\ge c_nm$ by the trace identity;
-if some $A_i^2>\eta$, the self term of that index in $T_0$ is at
-least $c_n(r+u_i)$, which together with the alternative
-control of $M_0$ and $\bar u$ bounds the remaining terms. The
-volume contribution $rW_p$ is handled by $A_i\ge1/4$ indices
-(which contribute to $r(L_i-1)$ in $T_0$) and by the bound $|M_0|
-\ge c(r+\bar u)$ for $r>2$; for $r\le2$ the $m$-bound absorbs it.
-
-\emph{Small-$m$ region: $m<m_0$, $|1-r|<1/4$.}
-Here $T_0\ge c_nm^2$. Expanding the exponential in $W_p$:
-\[
- \sum_iA_i^2C_i-m_f\ge c_nm^2-C_n\sum_iA_i^4,
-\]
-and the self terms $\sum_iA_i^2(L_i-1)2ar$ are at least
-$c_n\sum_iA_i^4$ (by $r>3/4$ and $L_i-1\ge A_i^2/2$). Adding these
-two estimates gives $T_0\ge c_nm^2$, which controls the budget
-whenever $m\ge Kl$ for a fixed large dimensional $K$.
-
-\emph{Critical regime.} For $m\le Kl$ under $|1-r|<1/4$, expand the
-RHS of the target budget:
-\[
- lm+2lrW_p-nr\log(1+l)=2lm-nl^2+O_n(l^3)+O_n(l^2)|r-1|.
-\]
-The quadratic target $m^2/n$ is covered by
-$\frac{m^2}{n}-2lm+nl^2=(\frac m{\sqrt n}-\sqrt nl)^2\ge0$; the
-residual is bounded by the boost square whenever $r$ is close to $1$
-and $m$ is small.
-
-\emph{Conclusion.} Combining the three cases above, on the coarse
-region $P_p\ge c_n\epsilon\bar u^{5/3}$ (from \eqref{eq:entropy-m-surplus}
-when $\bar u$ is large), giving \eqref{eq:entropy-m-surplus}. The
-$\log$-surplus \eqref{eq:entropy-log-surplus} is an immediate
-consequence of $W_p\le C_nm$ on the small-$m$ region and of the
-self-term bound for $A_i>1/2$.
-\end{proof}
-\subsection{Accumulation and the step schedule}
-\label{sec:entropy-accumulation}
-
-The increment \eqref{eq:entropy-increment} admits iteration until the
-energy coefficient reaches $E$. Two constraints shape the step
-schedule: the losses $\sum l(\epsilon_p)^2$ must be summable (so that
-the scalar constant stays bounded independent of the input), and the
-number of sweeps up to $E$ must grow at most cubically in
-$\log(1+E)$ (a constraint that the graph structure and support radii
-can keep pace with).
-
-The step is
+\begin{proposition}[One sweep]
+\label{prop:sweep}
+There is a numerical \(c>0\) with the following property.
+For a sweep without boost, set \(\psi=0\). For a boosted sweep,
+given \(\epsilon_*>0\), there are constants
+\(\kappa>0\), \(c_*>0\), and \(C_*<\infty\), depending only on
+\(\epsilon_*\), for which one may use \(\psi=\kappa\Psi\) whenever
+\(\epsilon\ge\epsilon_*\).
+
+Let \(h_0,\phi_0,F\) be a smooth equivariant Borel family on
+\(\widetilde M\times\Omega\), with bounds uniform in the lifted
+charts for the instance. Suppose \(h_0\) is complete, its
+geometry and the positive-order derivatives of \(\phi_0\) have
+the uniform bounds above, and
+\[
+ F=(F_i)_{i\in\mathcal L},\qquad F_i\ge0,\qquad
+ \sum_iF_i^2=1
+\]
+has finitely many free label orbits and uniformly locally finite
+compact coordinate supports. For each orbit representative,
+suppose all its transverse supports lie in a fixed compact
+subset of \(\widetilde M\). Let \(h_{\mathrm{ref}}\le h_0\)
+be a complete equivariant Borel leafwise metric whose coefficients,
+inverse, and all coordinate derivatives have uniform bounds in the
+same lifted buffered charts as \(h_0\). Suppose a Borel
+equivariant coloring
+\(\chi:\mathcal L\times\Omega\to\{1,\ldots,k\}\), \(k<\infty\),
+has been chosen such that, for each retained \(\omega\) and distinct
+labels \(i,j\) with \(\chi(i,\omega)=\chi(j,\omega)\), the
+\(R_{\mathrm{loc}}\)-neighborhoods in \(h_{\mathrm{ref}}(\omega)\)
+of \(\supp F_i(\cdot,\omega)\) and \(\supp F_j(\cdot,\omega)\)
+are disjoint. Fix the pass parameters in
+\eqref{eq:sweep-parameters}, with \(\epsilon\ge\epsilon_*\) in
+the boosted case.
+
+For any \(\delta_0,\eta_0>0\), there is a candidate sweep as above
+with smooth nonnegative bumps \(f_i\le2\), supported in the
+indicated enlargements, positive on the supports of \(F_i\), and
+flat on their zero sets. Its output is given by
+\eqref{eq:sweep-output}. The lower bound for \(r\) is uniform
+for the instance. The bumps, coefficients, all intermediate and
+final graph metrics and weights, their smooth graph quantities, and
+\(Y\) are equivariant and Borel and retain the corresponding uniform
+chart bounds, with the positive-order convention for weights.
+The map \(Y\) is again a
+root probability map with the same label orbits.
+
+For each \(i\), the coefficient furnished by
+Lemma~\ref{lem:graph-localization} has the form
\begin{equation}
- l(\epsilon)=\frac{l_*}{(1+\log(1+\epsilon))^2},
- \label{eq:entropy-step-schedule}
+ \beta_i=\beta_{s,i}+\epsilon\psi(R_i),\qquad
+ R_i=\sum_{j<i}f_j^2 ,
+ \label{eq:sweep-beta}
\end{equation}
-where the dimensional $l_*>0$ is chosen small (below $l_0$ from
-Lemma~\ref{lem:entropy-increment}).
-
-\begin{lemma}[Accumulation]
-\label{lem:entropy-accumulation}
-Suppose at parameter $\epsilon_{\mathrm{in}}\ge\epsilon_*$ that
-\begin{equation}
- \wscal(h,\phi)\ge-C+2W-n\log\epsilon_{\mathrm{in}}
- +\epsilon_{\mathrm{in}}e_h(F)+P_{\mathrm{in}},
- \label{eq:entropy-phase-input}
-\end{equation}
-with $W\ge0$ and $P_{\mathrm{in}}\ge0$. Successive sweeps of
-Lemma~\ref{lem:entropy-increment} with
-$\epsilon_{\mathrm{new}}=(1+l(\epsilon))\epsilon$ preserve this form:
-at the final parameter $E$
+Here \(\beta_{s,i}\) vanishes near \(\supp F_i\), and the coefficients
+satisfy \eqref{eq:graph-spatial-cutoff} in \(h_{i-1}\).
+Writing
+\[
+ ND_i^0=\epsilon a(F_i^2/f_i-f_i)+A_i^2/(2f_i)
+\]
+on \(\{f_i>0\}\), extended by zero off the bump, one has
\begin{equation}
- \wscal(h',\phi')\ge-C+2W_{\mathrm{new}}-n\log E
- +E e_{h'}(F')+P,
- \label{eq:entropy-phase-output}
+ ND_i=ND_i^0-\beta_i f_i+\mathrm{err}_i,\qquad
+ \sum_i|\mathrm{err}_i|^2\le\eta_0 .
+ \label{eq:sweep-prescription-error}
\end{equation}
-with $P=P_{\mathrm{in}}+\sum_p P_p$, $W_{\mathrm{new}}=W+\sum_pW_p$,
-and the scalar constant increased by $C_n\sum_p l(\epsilon_p)^2$.
-Along the entire increasing parameter schedule,
+Define the nonnegative reserve
\begin{equation}
- \sum_p l(\epsilon_p)^2\le C_n,
- \qquad
- \#\{p:\epsilon_p<E\}\le C_n(1+\log^3(1+E)).
- \label{eq:entropy-summability}
+ \begin{split}
+ \Pi_p:={}&
+ T_0+c\epsilon\bigl(M_0^2+\E_\nu V_i^2+\theta\Var_\nu u\bigr)
+ \\
+ &+c\frac r\epsilon\mathcal X
+ +c_*\epsilon r^6\indic_{\{r\ge C_*\}},
+ \end{split}
+ \label{eq:sweep-positive-reserve}
\end{equation}
-If the initial volume logarithm is zero, then for sufficiently large
-dimensional $E$,
+where the final term is omitted in a sweep without boost. Then,
+pointwise,
+\begin{align}
+ Q:=\frac{rN^2}{\epsilon}\bigl[
+ &\wscal(h_+,\phi_+)-\wscal(h_0,\phi_0)
+ +\epsilon e_{h_0}(F)-\epsilon e_{h_+}(Y)\bigr]
+ \notag\\
+ &\ge\Pi_p-4\theta^2m-\frac r\epsilon\delta_0 .
+ \label{eq:sweep-inequality}
+\end{align}
+The numerical constant \(c\) is independent of the instance, the
+number of colors, and the pass parameters. The errors
+\(\delta_0,\eta_0\) can be prescribed independently before the pass.
+\end{proposition}
+
+\begin{proof}
+Process the colors in their prescribed order. At a label in
+the current color, the sum \(R_i\) of earlier bumps and all its
+derivatives have common bounds for that color, so the function
+\(\alpha_i=\epsilon\psi(R_i)\) is fixed and uniformly smooth.
+Apply Lemma~\ref{lem:graph-localization} with this
+\(\alpha_i\), using a common sufficiently large \(H_*\) for
+the color. Uniqueness of the complete solutions and the
+equivariant cutoff construction make all choices equivariant.
+Their Borel dependence follows from the same lemma. Bumps in
+the same color are separated, so their sum gives exactly the
+individual graph at each point of their supports. The graph
+metric and weight bounds hold in the same lifted charts as the input
+bounds. The reference metric is unchanged during the pass, so the
+simultaneous chart control required by
+Lemma~\ref{lem:graph-localization} persists for the next color.
+
+At any point some \(F_i\) is nonzero, and its cutoff is one
+there. The lower barrier for that color gives
+\(f_i\ge c_{\min}>0\). Taking the minimum over the finitely
+many colors gives the stated instance-uniform lower bound for
+\(r\); also \(r\le4k\). Thus \(Y\) is smooth with uniform
+bounds and \(\sum_iY_i^2=1\). The graph determinant and
+density identities prove \eqref{eq:sweep-output}.
+By making the error in each color small enough that its square
+is at most \(\eta_0/k\), the last conclusion of
+Lemma~\ref{lem:graph-localization} gives
+\eqref{eq:sweep-prescription-error}.
+
+We prove the curvature estimate in the units before
+multiplication by \(r/\epsilon\). Write
+\[
+ G=N^2\bigl[\wscal(h_+,\phi_+)-\wscal(h_0,\phi_0)
+ +\epsilon e_{h_0}(F)-\epsilon e_{h_+}(Y)\bigr],
+ \qquad Z_i=ND_i^0,
+\]
+so \(Q=(r/\epsilon)G\). For each \(i\), the one-coordinate calculation
+applies with \(h=h_{i-1}\), \(h'=h_i\), and
+\(h_{\mathrm{in}}=h_0\). We first account for the difference between
+the ideal target \(Z_i\) for \(ND_i\) and the realized prescription.
+
+For the moment omit the floor in
+\eqref{eq:graph-core-prescription}. Replacing \(Z_i\) by
+\(Z_i-\beta_i f_i\) in the first two terms of
+\eqref{eq:graph-identity}, after multiplying by \(N^2\),
+adds exactly
+\begin{align*}
+ &(Z_i-\beta_i f_i)^2-2NU_i(Z_i-\beta_i f_i)
+ -Z_i^2+2NU_i(Z_i)\\
+ &\hspace{1cm}=\beta_i^2f_i^2
+ +2\epsilon a\beta_i(f_i^2-F_i^2)
+ +(2L_i-1)\beta_iA_i^2
+ +2NA_if_i e_i(\beta_i).
+\end{align*}
+Since \(L_i\ge1\), this is at least
\begin{equation}
- W_{\mathrm{new}}\le E^{0.26}(E+P)^{0.75}.
- \label{eq:entropy-phase-volume}
+ \mathcal B_i:=
+ \beta_i^2f_i^2+2\epsilon a\beta_i f_i^2+\beta_i A_i^2
+ -2\epsilon a\beta_iF_i^2
+ -2NA_if_i|d\beta_i|_{h_{i-1}}.
+ \label{eq:sweep-beta-correction}
\end{equation}
-The same assertion holds if the initial data have parameter $E_0$,
-surplus $P_0$, and $W\le E_0^{0.32}(E_0+P_0)^{0.75}$, provided
-$E\ge E_0$ is sufficiently large dimensionally and
-$E_0^{0.32}\le\tfrac12E^{0.26}$.
-\end{lemma}
-\begin{proof}[Proof sketch]
-Iterate \eqref{eq:entropy-increment}. The energy coefficients
-multiply to $\prod(1+l(\epsilon_p))=E/\epsilon_{\mathrm{in}}$; the
-volume additions sum to $W_{\mathrm{new}}-W$; the surplus sums to
-$P-P_{\mathrm{in}}$; and the scalar constant grows by
-$C_n\sum l(\epsilon_p)^2$.
+On the uncut region the actual floor has nonnegative
+differentiated contribution by
+\eqref{eq:graph-floor-derivative}, and its square loses at
+most the quantity in \eqref{eq:graph-floor-square}.
+On a cutoff transition \(F_i=0\) and \(d\beta_i=0\).
+There \eqref{eq:graph-cutoff-small} makes the actual scalar
+change small. It also makes every term on the right of the
+single-step expansion just obtained small:
+\(\beta_i A_i^2=(\beta_i f_i)t_i\),
+the longitudinal terms are controlled by \(N|\II_i|\) and
+\(t_i\), and the other terms are controlled by
+\(Z_i,\beta_i f_i,A_i\). The first term in
+\eqref{eq:graph-ideal-derivative} is zero. Thus this
+expanded lower bound still holds on a cutoff transition with
+an arbitrarily small additive error in these units. It is
+also valid off the bumps, where all its terms vanish.
+Choose the floor and transition errors for each color so their
+sum is at most \(\delta_0\) at every point. Telescoping
+\eqref{eq:graph-identity} over the colors gives
+\begin{align}
+ G\ge \sum_i\bigl[&
+ Z_i^2-2NU_i(Z_i)+N^2|\II_i|^2
+ +N^2|d\log L_i|_{h_i}^2
+ +\epsilon N^2|dF_i|_{h_0}^2+\mathcal B_i\bigr]
+ \notag\\
+ &-\epsilon N^2e_{h_+}(Y)-\delta_0 .
+ \label{eq:sweep-raw}
+\end{align}
+This argument uses the full differentiated floor and cutoff
+estimates, rather than differentiating the value error in
+\eqref{eq:sweep-prescription-error}.
-\emph{Summability.} Put $t_p=\log(1+\epsilon_p)$. Since
-$\epsilon_p\ge\epsilon_*>0$ and $l_*$ is fixed small,
+The input energy is already included in
+\eqref{eq:graph-ideal-bound}. For the output, normalization is an
+orthogonal projection in the target:
\[
- t_{p+1}-t_p=\log\Bigl(1+l(\epsilon_p)\frac{\epsilon_p}{1+\epsilon_p}\Bigr)
- \asymp_n l(\epsilon_p)=\frac{l_*}{(1+t_p)^2}.
+ dY=r^{-1/2}(I-Y\otimes Y)\,df .
\]
-Comparing with the integrals of $(1+t)^2$ and $(1+t)^{-2}$ gives
-both bounds in \eqref{eq:entropy-summability}.
-
-\emph{Volume bound.} Let $k$ be the number of sweeps up to $E$. By
-Lemma~\ref{lem:entropy-surplus}, since $\epsilon_p\le E$ and
-$P_p\le P$,
+The rank-one inverse identity and metric monotonicity give
\[
- \sum_pW_p\le C_nk E^{1/4}(E+P)^{3/4}.
+ N^2|df_i|_{h_i}^2=A_i^2,\qquad
+ N^2|df_i|_{h_+}^2=A_i^2(1-C_i)\le A_i^2 .
\]
-For large $E$, $C_nk\le\tfrac12E^{0.01}$ by
-\eqref{eq:entropy-summability}, giving \eqref{eq:entropy-phase-volume}
-when $W=0$. For the initial-data case, the term
-$E_0^{0.32}(E+P_0)^{0.75}$ is at most
-$\tfrac12E^{0.26}(E+P)^{0.75}$.
-\end{proof}
-
-The exponents in \eqref{eq:entropy-phase-volume} are chosen so that
-each subsequent phase can absorb the outgoing volume from the previous
-one. Specifically, we require
+Consequently
\begin{equation}
- 0.26=\tfrac14+0.01,\qquad
- 0.26+0.04<\frac{1+0.32}{4},\qquad
- 0.26+\tfrac34(0.04)<0.32,\qquad
- 0.02>\tfrac1{100}.
- \label{eq:entropy-exponent-arithmetic}
-\end{equation}
-These leave room for (respectively): the number of sweeps, the
-conformal scalar cost, the outgoing volume estimate, and the final
-gradient target.
-\subsection{Phase exit}
-\label{sec:entropy-phase-exit}
-
-The accumulation lemma produces the entropy inequality at the target
-parameter $E$, but does not control the absolute size of $e_h(F)$.
-Without that bound, the label overlaps could grow beyond the fixed
-graph as the sweeps accumulate. The phase exit reduces the gradient
-at the cost of a bounded volume increase.
-
-\begin{lemma}[Phase exit]
-\label{lem:entropy-phase-exit}
-There is a dimensional constant $c_{\mathrm{exit}}>0$ with the
-following property. Suppose at parameter $E\ge1$ that
-\begin{equation}
- \wscal(h_0,\phi_0)\ge-C+2W_0-n\log E+E\,e_{h_0}(Y)+P_0,
- \label{eq:entropy-phase-exit-in}
+ N^2e_{h_+}(Y)
+ \le\frac1r\sum_i A_i^2(1-C_i).
+ \label{eq:sweep-output-energy}
\end{equation}
-with $W_0\ge0$, $P_0\ge0$, and
-\begin{equation}
- e_{h_0}(Y)^{1/2}\le K_0E^{-1/100}
- \label{eq:entropy-phase-exit-grad}
+
+Put
+\[
+ K_i=(X_i-t_i)^2+N^2|\II_i^{\mathrm{rem}}|^2
+ +N^2|d\log L_i|_{h_i}^2,\qquad
+ g_i=a(\sigma_i-r)+u_i/2 .
+\]
+The definitions imply
+\[
+ Z_i=\frac{\epsilon f_i}{r}g_i,\quad
+ \frac r\epsilon\sum_iZ_i^2=\epsilon\E_\nu g_i^2,\quad
+ \frac r\epsilon(L_i-1)t_i^2=(L_i-1)A_i^2u_i .
+\]
+Use these identities, \eqref{eq:graph-ideal-bound}, and
+\eqref{eq:sweep-output-energy} in \eqref{eq:sweep-raw}.
+After multiplication by \(r/\epsilon\), the result is
+\begin{align}
+ Q\ge{}&T_0+\frac r\epsilon\sum_iK_i
+ +\epsilon\E_\nu g_i^2
+ +\sum_i A_i^2\bigl[(2a-4a^2)\sigma_i+2ar-1\bigr]
+ +\frac r\epsilon\left(\sum_i\mathcal B_i-\delta_0\right).
+ \label{eq:sweep-expanded}
+\end{align}
+Indeed \(rF_i^2/f_i^2=\sigma_i\). The terms with \(B_i\) and
+\(L_i-1\) in \eqref{eq:graph-ideal-bound} become the corresponding
+terms of \(T_0\), while the output contributes
+\(-A_i^2+A_i^2C_i\). The remaining height terms give the displayed
+quadratic expression.
+
+The remaining quadratic expression has an exact mean and
+variance decomposition. Use \(\E_\nu\sigma_i=1\) and
+\(\E_\nu u_i=\bar u\) from \eqref{eq:sweep-statistics}, and write
+\(w_i=u_i-\bar u\). Dividing the quadratic expression in
+\eqref{eq:sweep-expanded} by \(\epsilon\), its contribution
+from the means is
+\begin{align*}
+ [a(1-r)+\bar u/2]^2+\bar u(2a-4a^2+2ar-1)
+ &= [a(1-r)-\bar u/2]^2-(2a-1)^2\bar u\\
+ &=M_0^2-4\theta^2\bar u .
+\end{align*}
+The coefficient of
+\(\E_\nu[w_i(\sigma_i-1)]\) in the centered expression is
+\[
+ a+(2a-4a^2)=3a-4a^2=2a\lambda .
+\]
+Completing its square therefore gives the identity
+\begin{align}
+ \epsilon\E_\nu g_i^2
+ &+\sum_i A_i^2\bigl[(2a-4a^2)\sigma_i+2ar-1\bigr] \notag\\
+ &=
+ \epsilon\left(M_0^2+\E_\nu V_i^2+
+ (\tfrac14-\lambda^2)\Var_\nu u\right)
+ -4\theta^2m .
+ \label{eq:sweep-mean-identity}
+\end{align}
+Here
+\(\tfrac14-\lambda^2=2\theta-4\theta^2\ge7\theta/4\).
+This supplies a reserve for all three squares in the
+proposition. We finish by paying the terms in
+\(\mathcal B_i\) from fixed fractions of this reserve and of
+the nonnegative \(\beta_i\) terms.
+
+Write \(s_i=\beta_{s,i}\) and \(b_i=\epsilon\psi(R_i)\), so
+\(\beta_i=s_i+b_i\) with both summands nonnegative.
+In particular \(\beta_i^2\ge s_i^2+b_i^2\).
+The negative \(F_i^2\) term in
+\eqref{eq:sweep-beta-correction} has no \(s_i\) part, because
+\(s_i\) vanishes near the support of \(F_i\).
+The spatial derivative term satisfies
+\begin{align}
+ 2NA_if_i|ds_i|
+ &\le2c_0 A_if_i(\theta\epsilon^{3/2}+s_i^{3/2}) \notag\\
+ &\le c_0(s_i^2f_i^2+s_iA_i^2)
+ +Cc_0\theta(\epsilon^2f_i^2+A_i^4/f_i^2).
+ \label{eq:sweep-spatial-cost}
+\end{align}
+For the first pair of terms use \(2xy\le x^2+y^2\).
+For the other pair, divide by \(\epsilon^2f_i^2\) and use
+\(t\le1+t^4\) with \(t=A_i/(\sqrt\epsilon f_i)\).
+This derivative cost occurs only in rows with \(F_i=0\), hence
+\(\sigma_i=0\). In those rows,
+\[
+ V_i=-a+\lambda w_i,\qquad
+ ar+\bar u/2=a-M_0=\lambda w_i-V_i-M_0 .
+\]
+Since \(a\ge1/2\) and \(|\lambda|\le1/2\), these equations give
+\[
+ r+u_i\le2(|M_0|+|V_i|+|w_i|),\qquad
+ r^2+u_i^2\le12(M_0^2+V_i^2+w_i^2).
+\]
+After multiplication by \(r/\epsilon\), the last cost in
+\eqref{eq:sweep-spatial-cost}, summed over these rows, is
+\[
+ Cc_0\epsilon\theta
+ \sum_{\{\sigma_i=0\}}\nu_i(r^2+u_i^2).
+\]
+The preceding bound shows that it is a small fraction of the
+three squares in \eqref{eq:sweep-mean-identity} when \(c_0\)
+is a sufficiently small numerical constant. The first cost
+in \eqref{eq:sweep-spatial-cost} is paid by a small fraction
+of \(\beta_i^2f_i^2+\beta_iA_i^2\).
+This proves the required absorption when \(\psi=0\), leaving
+a fixed fraction of every asserted square.
+
+Suppose now that \(\psi=\kappa\Psi\) and
+\(\epsilon\ge\epsilon_*\). Metric addition over earlier
+colors implies, as quadratic forms,
+\[
+ h_{i-1}\ge N^2\sum_{j<i}df_j\otimes df_j .
+\]
+For a unit vector in \(h_{i-1}\), the vector
+\((Ndf_j)_{j<i}\) consequently has Euclidean norm at most one.
+Cauchy--Schwarz applied to
+\(dR_i=2\sum_{j<i}f_jdf_j\) proves
+\begin{equation}
+ |dR_i|_{h_{i-1}}\le2\sqrt{R_i}/N .
+ \label{eq:sweep-prefix-gradient}
\end{equation}
-for some $K_0\ge0$. For every sufficiently large dimensional
-$E_1\ge E$ one can construct $(h_1,\phi_1,Z_1)$ and $(h_2,\phi_2,Z_2)$
-with:
-\begin{itemize}
-\item $Z_1,Z_2$ root probability maps with the same label orbits as $Y$;
-\item $\supp Z_1,\supp Z_2$ in a uniform neighborhood of $\supp Y$;
-\item $e_{h_j}(Z_j)^{1/2}\le E_1^{-1/100}$ for $j=1,2$;
-\item for $j=1,2$,
- \begin{equation}
- \wscal(h_j,\phi_j)\ge-C+2W_j-n\log E_1+E_1e_{h_j}(Z_j)+P_j,
- \label{eq:entropy-after-exit}
- \end{equation}
- with $W_j\ge0$, $P_j\ge c_{\mathrm{exit}}P_0$, and
- $W_j\le E^{0.32}(E+P_j')^{0.75}$;
-\item the density $\rho$ is unchanged from the input; equivalently
- $\rho\le dV_g$ persists.
-\end{itemize}
+For any fixed small \(\zeta>0\), the derivative bound in
+\eqref{eq:sweep-boost-profile} and Young's inequality give
+\[
+ 2NA_if_i|db_i|
+ \le4\epsilon A_if_i\sqrt{R_i}\,|\psi'(R_i)|
+ \le\zeta\epsilon\psi(R_i)A_i^2
+ +C_\zeta\epsilon\kappa f_i^2R_i .
+\]
+The first term is a small fraction of \(\beta_iA_i^2\).
+For the other costs, use
+\[
+ \sum_i f_i^2R_i
+ =\tfrac12\left(r^2-\sum_i f_i^4\right)\le r^2/2,
+ \qquad
+ \sum_i\psi(R_i)F_i^2\le C\kappa r^2\sum_iF_i^2=C\kappa r^2.
+\]
+The profile and its derivative vanish up to \(8\). Thus the
+sum of all remaining boost costs in the units of \(G\) is at
+most
+\begin{equation}
+ C(\epsilon+\epsilon^2)\kappa r^2\indic_{\{r\ge8\}}.
+ \label{eq:sweep-boost-cost}
+\end{equation}
+
+Two positive terms cover this cost. First, for \(r\ge8\),
+\[
+ -M_0=a(r-1)+\bar u/2\ge7r/16,
+ \qquad
+ \frac{\epsilon^2}{r}M_0^2\ge c\epsilon^2r .
+\]
+The latter is the mean square in
+\eqref{eq:sweep-mean-identity} expressed in the units of \(G\).
+Second, the increments \(f_i^2\) in \(R_i\) are at most \(4\).
+For \(r\) beyond a fixed threshold, the rows with
+\(R_i\ge r/2\) carry total increment at least
+\(r/2-4\ge r/4\). In fact the first prefix crossing \(r/2\)
+overshoots it by at most \(4\). Once \(r/2\ge R_\Psi\), the
+lower profile bound on each of these rows gives
+\begin{equation}
+ \sum_i f_i^2 b_i^2
+ =\epsilon^2\sum_i f_i^2\psi(R_i)^2
+ \ge c\epsilon^2\kappa^2 r^5
+ \qquad(r\ge C_*).
+ \label{eq:sweep-boost-square}
+\end{equation}
+
+To make the parameter choice explicit, let
+\(K_*=C(1+\epsilon_*^{-1})\). The cost
+\eqref{eq:sweep-boost-cost} is at most
+\(K_*\epsilon^2\kappa r^2\) on \(r\ge8\).
+Choose \(\kappa\) so small that
+\(\kappa^{-1/2}\ge C_*\) and \(K_*\sqrt\kappa\) is smaller
+than the fixed fractions of the preceding two reserves
+available for this purpose. If
+\(8\le r\le\kappa^{-1/2}\), this cost is at most
+\(K_*\sqrt\kappa\,\epsilon^2r\). If
+\(r\ge\kappa^{-1/2}\), it is at most
+\(K_*\sqrt\kappa\,\epsilon^2\kappa^2r^5\).
+It is therefore covered in both ranges. Fixed fractions of
+the mean square and of \eqref{eq:sweep-boost-square} remain.
+One can simultaneously retain a fixed fraction of
+\(\sum_i\beta_i^2f_i^2\) for \(\mathcal X\): reserve, for
+example, one half of this sum at the outset and use
+\(\beta_i^2\ge s_i^2+b_i^2\) on the other half for the
+spatial and boost estimates. Take \(c_0,\zeta,\kappa\) small
+enough for these fixed allocations.
+
+Insert the absorptions into \eqref{eq:sweep-expanded} and
+\eqref{eq:sweep-mean-identity}. A fixed fraction of
+\(\sum_iK_i+\sum_i\beta_i^2f_i^2=\mathcal X\) and of the
+three mean and variance squares survives. The remaining
+part of \eqref{eq:sweep-boost-square}, after multiplication
+by \(r/\epsilon\), is
+\(c_*\epsilon r^6\indic_{\{r\ge C_*\}}\), with
+\(c_*\) a positive multiple of \(\kappa^2\).
+The terms \(T_0\) and \(-4\theta^2m\) have not been spent.
+Choose the constant \(c\) in \eqref{eq:sweep-positive-reserve}
+below the fixed numerical fractions that survive for \(\mathcal X\)
+and the mean and variance squares. These fractions are independent
+of \(\epsilon_*\), because \(\kappa\) was chosen to fit the fixed
+allocations.
+This proves \eqref{eq:sweep-inequality}.
+
+Finally, the choices of errors impose no estimate on the
+growth of analytic constants from one color to the next.
+For a fixed color, the reference buffers are fixed for the
+pass, and the smaller analytic boxes and all previous
+geometry bounds are fixed before its \(H_*\) is chosen.
+There are finitely many colors, so the raw scalar error can
+be made at most \(\delta_0/k\) per color and the squared
+value error at most \(\eta_0/k\) per color. Later choices
+of \(H_*\) may be arbitrarily larger. The summed errors
+are therefore precisely those asserted in
+\eqref{eq:sweep-prescription-error} and
+\eqref{eq:sweep-inequality}.
+\end{proof}
+\section{Entropy metrics with controlled labels}\label{sec:entropy}
+
+This section turns a root probability map for a metric of scalar curvature
+\(-1\) into a map with a large energy coefficient in a weighted scalar
+inequality. The accompanying volume term is the logarithm of the actual
+volume expansion from a density no larger than the original volume density.
+We will also make the gradient absolutely small and control all label
+overlaps by one graph. The last two properties allow the energy coefficient
+to be chosen larger than the combinatorial quantities used later.
+
+Write \(\exp^{[0]}(t)=t\) and
+\(\exp^{[j+1]}(t)=\exp(\exp^{[j]}(t))\). We call a function
+\(\mathcal B:[1,\infty)\to[0,\infty)\) \emph{elementary} if, for some fixed
+\(H\in\N\cup\{0\}\) and \(A,B>0\),
+\[
+ \mathcal B(q)\leq \exp^{[H]}\bigl(B(1+q)^A\bigr)\qquad(q\geq1).
+\]
+When the function is prescribed in a fixed dimension, its parameters
+\(H,A,B\) are allowed to depend on that dimension.
+
+\begin{theorem}[Entropy metrics]\label{thm:entropy}
+Let \((M^n,g)\) be a closed connected Riemannian manifold, where \(n\geq3\)
+and \(\Scal_g=-1\), and put \(\Gamma=\pi_1(M)\). Let
+\((\Omega_0,\mu_0)\) be a standard Borel probability space with a
+measure-preserving \(\Gamma\)-action; the one-point space is allowed.
+Suppose \(\mathcal L_0\) is a countable set with finitely many free
+\(\Gamma\)-orbits and
+\[
+ F^{\mathrm{in}}:\widetilde M\times\Omega_0\longrightarrow
+ \ell^2(\mathcal L_0),\qquad
+ F^{\mathrm{in}}_\ell\geq0,\qquad
+ \sum_{\ell\in\mathcal L_0}(F^{\mathrm{in}}_\ell)^2=1
+\]
+is equivariant, Borel in the transverse variable, and smooth along
+\(\widetilde M\). Assume the following uniformity for this input. In
+charts lifted from a fixed finite atlas on \(M\), with fixed inner buffers,
+the supports are locally finite with a uniform bound on their number,
+and all leafwise derivatives of the coordinate functions have uniform
+bounds. For each orbit representative \(\ell\) there is a compact set
+\(K_\ell\subset\widetilde M\) such that
+\(\supp F^{\mathrm{in}}_{\gamma\ell}(\,\cdot\,,\omega)
+\subset\gamma K_\ell\) for all \(\gamma\in\Gamma\) and almost every
+\(\omega\). The bounds and compact sets in these assumptions may depend
+on the input.
+
+For every prescribed elementary function \(\mathcal B\) and every finite
+number \(E_{\min}\), there are a standard Borel probability
+\(\Gamma\)-space \((\Omega,\mu)\) extending \((\Omega_0,\mu_0)\), a
+countable set \(\mathcal L\) with finitely many free \(\Gamma\)-orbits, and
+equivariant leafwise data on \(\widetilde M\times\Omega\): a metric \(h\),
+a function \(\phi\), a positive density \(\rho\), and a root probability
+map \(F\) with labels \(\mathcal L\). There are also \(E>E_{\min}\), an
+integer \(q\geq n+1\), and a constant \(C=C(n,\mathcal B)<\infty\) such
+that, on an invariant conull set of transverse parameters,
+\begin{equation}\label{eq:entropy-contract}
+\begin{split}
+ &\rho\leq dV_g,\qquad
+ W:=\log\frac{dV_h}{\rho}\geq0,\\
+ &\wscal(h,\phi)\geq
+ -C+2\left(W-\frac n2\log E\right)+E e_h(F),
+ \qquad e_h(F)^{1/2}\leq E^{-1/100},\\
+ &E>\mathcal B(q).
+\end{split}
+\end{equation}
+Here the extension means that there is an equivariant measure-preserving
+map \(\Omega\to\Omega_0\). All output objects are Borel transversely and
+smooth along leaves. In the same lifted charts, the coefficients of
+\(h\), their derivatives of every order, and the coefficients of \(h^{-1}\)
+have uniform bounds; the function \(\phi\), the density ratio
+\(\rho/dV_g\), its reciprocal, and their derivatives of every order have
+uniform bounds. The coordinates of \(F\) have the same derivative and
+uniform local finiteness properties as the input. There are equivariant
+compact sets containing their supports, uniformly in the transverse
+parameter. All these output bounds may depend on the input and on the
+chosen finite construction.
+
+There is, in addition, a fixed \(\Gamma\)-invariant graph \(\mathcal G\)
+on \(\mathcal L\), independent of the transverse parameter, of finite
+maximum degree \(\Delta\). It contains an edge between every pair of
+distinct labels that are simultaneously nonzero at any point of any leaf.
+We may take
+\[
+ q\geq \max\{n+1,\#(\Gamma\backslash\mathcal L),\Delta+1\}
+\]
+and a Borel equivariant coloring
+\(\chi:\mathcal L\times\Omega\to\{1,\ldots,q\}\) such that labels of
+graph distance one or two have different colors. The set \(\mathcal L\)
+can be taken to consist of finitely many copies of \(\mathcal L_0\).
+The constant \(C\) is independent of the manifold, metric, input family,
+all the uniform bounds just specified, and \(E_{\min}\).
+\end{theorem}
+
+We prove the theorem using Proposition~\ref{prop:sweep}. The bootstrap
+will fix the density \(\rho\), which then remains fixed through all later
+sweeps and conformal changes. For
+\(W=\log(dV_h/\rho)\), the balance propagated after the bootstrap is
+\[
+ \wscal(h,\phi)-\epsilon e_h(F)-2W+n\log\epsilon\geq-C+P,
+ \qquad W\geq0,\quad P\geq0.
+\]
+We first raise an initially small energy coefficient to a dimensional
+positive value. Subsequent sweeps have a summable scalar loss and
+retain the nonnegative surplus \(P\). At the end of a group of sweeps,
+that surplus pays for a conformal change which makes the gradient
+absolutely small. The small gradient then controls how fast the fixed
+graph of label overlaps can grow in the next group of sweeps.
+The inverse-metric telescope and the assembly of logarithmic shortening
+also occur in \cite[Lemma~6.1 and Theorem~6.3]{OpenAIScalarInessentiality2026};
+the proof below establishes the forcing and entropy estimates needed here.
+
+\subsection{Colorings and a conformal bootstrap}
+
+We first supply the colors required for the sweeps. Let \(\mathcal H\)
+be any fixed invariant graph of maximum degree
+\(\Delta_{\mathcal H}<\infty\) on a countable label set \(\mathcal L\).
+Adjoin to the transverse space independent uniform priorities in
+\([0,1]^{\mathcal L}\), with the \(\Gamma\)-action permuting coordinates.
+This is a standard Borel probability extension. Direct every edge
+from the larger priority to the smaller, after discarding the invariant
+null set on which two priorities coincide. This set is null because the
+label set is countable and the priorities have continuous laws. For a
+fixed vertex the probability of a strictly descending path of length
+\(k\) is at most \(\Delta_{\mathcal H}^k/(k+1)!\): there are at most
+\(\Delta_{\mathcal H}^k\) paths, and the priorities of the \(k+1\)
+distinct vertices in a given descending path have the required order
+with probability \(1/(k+1)!\). The probability of arbitrarily long
+such paths is therefore zero. Since the label set is countable, on
+an invariant conull set every vertex has finite descending depth.
+Greedily assign to a vertex the least color in
+\(\{1,\ldots,\Delta_{\mathcal H}+1\}\) unused by its lower-priority
+neighbors, recursively in that finite depth. This defines a proper
+coloring. It is Borel, since each vertex's color is determined in
+some finite descending neighborhood, and it is equivariant, since
+the rule only uses the invariant graph and priorities. Applying this
+to the graph square separates vertices of distance at most two.
+
+For the initial map and throughout the bootstrap, graphs of the
+required finite degree exist with input-dependent bounds. To see
+this, choose a compact set in \(\widetilde M\) whose translates cover
+the universal cover. Each label support lies in one of finitely many
+fixed compact sets translated by its group label. A support
+enlargement by a bounded distance in any one of the current uniformly
+controlled metrics is contained in an input-dependent compact
+enlargement in the lifted metric \(g\). Properness of the deck action
+implies that only finitely many translates of any of these compact
+sets can meet another. Joining all such possible pairs, over all
+transverse parameters, gives one invariant finite-degree graph for
+that step. The color construction just proved supplies the colors
+of Proposition~\ref{prop:sweep}. The graph and color choices are
+made before the analytic errors for the corresponding sweep are
+chosen. This proves availability at each step of the finite
+bootstrap, with input-dependent degree. The phase argument below
+will control the degree relative to the eventual energy coefficient.
+
+Here and below all the notation of a single sweep is that of
+Proposition~\ref{prop:sweep}. In particular, its input is
+\((h_0,\phi_0,F)\), its output is \((h_+,\phi_+,Y)\), and
+\[
+\begin{gathered}
+ N^2=l\epsilon,\qquad a=\tfrac12+\theta,\qquad
+ \lambda=\tfrac12-2\theta,\\
+ r=\sum_i f_i^2>0,\qquad Y=f/\sqrt r,\qquad
+ W_p=\sum_i\log L_i,\qquad m=\sum_i A_i^2.
+\end{gathered}
+\]
+The weights \(\nu_i=f_i^2/r\) sum to one. We recall the probability
+statistics and their centered combinations from
+\eqref{eq:sweep-statistics}:
+\[
+\begin{gathered}
+ u_i=\frac{A_i^2}{\epsilon\nu_i},\qquad
+ \sigma_i=\frac{F_i^2}{\nu_i},\qquad
+ \bar u=\frac m\epsilon,\\
+ M_0=a(1-r)-\frac{\bar u}{2},\qquad
+ V_i=a(\sigma_i-1)+\lambda(u_i-\bar u),\qquad
+ t_i=\frac{A_i^2}{f_i}.
+\end{gathered}
+\]
+Indices of vanishing bumps are omitted. In particular
+\(\E_\nu u_i=\bar u\) and \(\E_\nu\sigma_i=1\); the ratios \(\sigma_i\)
+need not be at most one. We write \(\E\) and \(\Var\) for
+this finite weighted mean and variance in the pointwise calculations.
+The graph operations give
+\begin{equation}\label{eq:entropy-sweep-density}
+ \frac{dV_{h_+}}{dV_{h_0}}=e^{W_p},\qquad
+ \phi_+=\phi_0-W_p,\qquad
+ N^2 e_{h_+}(Y)\leq \frac mr.
+\end{equation}
+The added graph tensors are nonnegative. Thus, as quadratic forms,
+\(h_0\leq h_{i-1}\leq h_i\leq h_+\); the inverse forms reverse these
+inequalities. In particular, for every covector \(\alpha\),
+\[
+ |\alpha|_{h_+}\leq|\alpha|_{h_i}
+ \leq|\alpha|_{h_{i-1}}\leq|\alpha|_{h_0}.
+\]
+The positive reserve \(\Pi_p\) is defined in
+\eqref{eq:sweep-positive-reserve}, with coefficient exactly one on
+\(T_0\). In a sweep without boost its final term is omitted; in a
+boosted sweep its constants \(c_*,C_*\) may depend on the fixed lower
+threshold for \(\epsilon\). The sweep inequality reads
+\begin{equation}\label{eq:entropy-sweep-form}
+\begin{split}
+ Q&:=\frac{rN^2}{\epsilon}
+ \bigl[\wscal(h_+,\phi_+)-\wscal(h_0,\phi_0)
+ +\epsilon e_{h_0}(F)-\epsilon e_{h_+}(Y)\bigr]\\
+ &\geq \Pi_p-4\theta^2m-\frac r\epsilon\delta_0 .
+\end{split}
+\end{equation}
+The positive errors \(\delta_0,\eta_0\) can be prescribed independently
+before each sweep. The construction realizes those errors by choosing
+the analytic parameters for each color after the geometry of the
+preceding colors has been fixed.
+
+\begin{lemma}[Conformal bootstrap]\label{lem:entropy-bootstrap}
+There are constants \(\epsilon_*>0\) and \(C_{\mathrm{boot}}<4\),
+depending only on \(n\), with the following property. Starting with
+the input of Theorem~\ref{thm:entropy}, finitely many sweeps without
+boost, followed individually by conformal changes, produce a metric
+\(h_*\), a root probability map \(F_*\), and
+\(\epsilon_0\in[\epsilon_*,2\epsilon_*]\) such that
+\begin{equation}\label{eq:entropy-bootstrap-output}
+ \rho:=dV_{h_*}\leq dV_g,\qquad
+ \Scal_{h_*}\geq-C_{\mathrm{boot}}+\epsilon_0e_{h_*}(F_*).
+\end{equation}
+The metric and map have the uniformity and support properties in
+Theorem~\ref{thm:entropy}. The number of sweeps may depend on the input.
+\end{lemma}
+
+\begin{proof}
+We use sweeps with \(\phi_0=0\), no boost, and \(l=\theta\), where \(l>0\)
+will be a sufficiently small fixed dimensional number. Then
+\(\phi_+=-W_p\). After a sweep set
+\[
+ k=e^{-2W_p/(n-1)}h_+ .
+\]
+The scalar conformal formula and the definition of \(\wscal\) give
+\begin{equation}\label{eq:entropy-bootstrap-reset}
+ \Scal_k=e^{2W_p/(n-1)}
+ \left(\wscal(h_+,\phi_+)+\frac{|dW_p|_{h_+}^2}{n-1}\right).
+\end{equation}
+The same reset contracts the volume density:
+\begin{equation}\label{eq:entropy-bootstrap-density}
+ dV_k=e^{-nW_p/(n-1)}dV_{h_+}
+ =e^{-W_p/(n-1)}dV_{h_0}\leq dV_{h_0}.
+\end{equation}
+We claim that there are dimensional \(C_b\) and \(\epsilon_b>0\) such
+that, whenever \(2\leq C_0\leq4\), \(0<\epsilon\leq\epsilon_b\), and
+\(\Scal_{h_0}\geq-C_0+\epsilon e_{h_0}(F)\), one can choose the errors
+in the sweep so that
+\begin{equation}\label{eq:entropy-bootstrap-step}
+ \Scal_k\geq-(C_0+C_b l\epsilon)+(1+l)\epsilon e_k(Y).
+\end{equation}
+
+We first derive the budget needed for this step. The reset gives
+\(e_k(Y)=e^{2W_p/(n-1)}e_{h_+}(Y)\). Substituting the definition of
+\(Q\) and the input scalar inequality into
+\eqref{eq:entropy-bootstrap-reset}, and dropping its nonnegative
+gradient term, shows that it suffices to have
+\[
+ \frac{\epsilon Q}{rN^2}\geq
+ l\epsilon e_{h_+}(Y)+C_0(1-e^{-2W_p/(n-1)})
+ -C_b l\epsilon e^{-2W_p/(n-1)}.
+\]
+Multiply by \(rN^2\), and use
+\(N^2 e_{h_+}(Y)\leq m/r\) and \(N^2=l\epsilon\). A sufficient budget
+for \eqref{eq:entropy-bootstrap-step} is therefore
+\begin{equation}\label{eq:entropy-bootstrap-budget}
+ \epsilon Q\geq
+ l\epsilon m+C_0l\epsilon r(1-e^{-2W_p/(n-1)})
+ -C_b l^2\epsilon^2r e^{-2W_p/(n-1)}.
+\end{equation}
+
+To pay this budget when \(\epsilon\) is small, we need control of both
+\(m\) and the deviation of \(r\) from one. The sweep inequality implies
+\begin{equation}\label{eq:entropy-bootstrap-coercivity}
+ \epsilon Q\geq
+ c_n\bigl(\epsilon^2(1-r)^2+m^2\bigr)
+ -C_n\epsilon l^2m-r\delta'_0 ,
+\end{equation}
+where \(\delta'_0>0\) can be chosen arbitrarily small. Here is a proof.
+Fix a sufficiently small dimensional \(m_b>0\). If \(m\geq m_b\) and
+\(\epsilon_b\) is small enough, then \(\bar u=m/\epsilon\geq4a\). Since
+\[
+ -M_0=ar+\frac{\bar u}{2}-a,
+\]
+the square \(\epsilon^2M_0^2\) controls
+\(\epsilon^2(1-r)^2+m^2\). This is one of the positive terms in
+\(\epsilon\Pi_p\).
+
+Suppose next that \(m<m_b\). The prescription term \(D_i^0\) of
+Proposition~\ref{prop:sweep} satisfies
+\[
+ ND_i^0+t_i=\frac{\epsilon f_i}{r}
+ \left[V_i+M_0+(\lambda+\tfrac12)\bar u+
+ (\tfrac32-\lambda)u_i\right].
+\]
+Both displayed coefficients of \(\bar u\) and \(u_i\) are positive and
+bounded below. Moving \(V_i+M_0\) to the other side, taking positive
+parts, and summing squares therefore yields
+\begin{equation}\label{eq:entropy-bootstrap-t}
+ \sum_i t_i^2\leq
+ \frac{C\epsilon^2}{r}\bigl(\E V_i^2+M_0^2\bigr)
+ +C\sum_i(ND_i^0+t_i)_+^2 .
+\end{equation}
+The prescription error relation in Proposition~\ref{prop:sweep} is
+\(ND_i=ND_i^0-\beta_i f_i+\mathrm{err}_i\), with
+\(\sum_i|\mathrm{err}_i|^2\) arbitrarily small. Moreover, the
+longitudinal component of \(\II_i\) is \(-X_i/N\), so
+\[
+ ND_i+t_i=-(X_i-t_i)
+ +N\bigl(\tr\II_i^{\mathrm{rem}}-U_i(\phi_{i-1})\bigr).
+\]
+The terms \((X_i-t_i)^2\), \(N^2|\II_i^{\mathrm{rem}}|^2\), and
+\(\beta_i^2f_i^2\) occur in \(\mathcal X\).
+For the remaining term, choose \(m_b<1/4\). Then
+\(L_i\leq C\) and \(|d\log L_i|_{h_i}\leq C A_i|\II_i|\).
+Because the metrics increase during the sweep and
+\(\phi_{i-1}=-\sum_{j<i}\log L_j\), Cauchy--Schwarz gives
+\[
+\begin{split}
+ \sum_iN^2U_i(\phi_{i-1})^2
+ &\leq C m^2\sum_jN^2|\II_j|^2\\
+ &\leq C' m^2\left(\mathcal X+\sum_jt_j^2\right).
+\end{split}
+\]
+For example, the first line follows by bounding
+\(|d\phi_{i-1}|_{h_{i-1}}\) by
+\(C\sum_{j<i}A_j|\II_j|\), squaring with
+\(\sum_jA_j^2=m\), and then summing the factor \(A_i^2\).
+Choosing \(m_b\) small absorbs the last sum of \(t_j^2\) in
+\eqref{eq:entropy-bootstrap-t}. Since
+\[
+ m^2=\left(\sum_i f_i t_i\right)^2\leq r\sum_i t_i^2,
+ \qquad
+ \epsilon^2(1-r)^2\leq C\bigl(\epsilon^2M_0^2+m^2\bigr),
+\]
+fractions of the \(M_0^2,\E V_i^2,\mathcal X\) terms in
+\(\epsilon\Pi_p\) control both required squares. The arbitrarily
+small errors in the prescription contribute \(r\delta'_0\).
+The term \(-4\theta^2m\) in \eqref{eq:entropy-sweep-form} is
+\(-4l^2m\). This proves \eqref{eq:entropy-bootstrap-coercivity}.
+
+We now compare the right side of
+\eqref{eq:entropy-bootstrap-coercivity} with the required budget.
+When \(m\geq m_b\), the first two terms on the right of
+\eqref{eq:entropy-bootstrap-budget} are at most
+\(l\epsilon(m+4r)\). The squares
+\(m^2+\epsilon^2(1-r)^2\) dominate this expression with a fixed
+fraction to spare after \(l\) is chosen small and then \(\epsilon_b\)
+is decreased if needed. One may check this separately for \(r\leq2\),
+where \(m\geq m_b\) controls the constant part, and for \(r>2\), where
+Young's inequality controls \(\epsilon r\) by
+\(\epsilon^2(r-1)^2\) and a sufficiently small fixed part of \(m^2\).
+The term \(C_n\epsilon l^2m\) is covered at the same time.
+
+When \(m<m_b\), the identity
+\(L_i=(1-A_i^2)^{-1/2}\) gives \(W_p\leq C m\).
+For \(r\notin(1/2,2)\), Young's inequality applied to
+\(l\epsilon m(1+r)\) again gives the required domination by the two
+squares, with \(l\) small. For \(1/2<r<2\), the positive terms on the
+right of \eqref{eq:entropy-bootstrap-budget}, together with the term
+\(C_n\epsilon l^2m\) from the coercivity estimate, are at most
+\(C_nl\epsilon m\). Here
+\[
+ C_nl\epsilon m\leq \frac{c_n}{2}m^2+C'_n l^2\epsilon^2.
+\]
+On this region \(r e^{-2W_p/(n-1)}\) has a dimensional positive lower
+bound. We may therefore choose \(C_b\) dimensionally to cover the
+last term. These comparisons can be made strict by increasing \(C_b\)
+and decreasing \(l,\epsilon_b\). They leave a positive multiple of
+\(l^2\epsilon^2r\) in the middle region, and positive square margins
+outside it; dividing the latter margins by \(r\) gives a positive
+lower bound for fixed \(l,\epsilon\). We can consequently choose
+\(\delta'_0\) small enough for the sweep. This proves
+\eqref{eq:entropy-bootstrap-step}.
+
+The input map has a finite uniform energy bound in \(g\). Thus for
+some positive, possibly input-dependent \(\epsilon_{\mathrm{in}}\)
+small enough,
+\(\Scal_g=-1\geq-2+\epsilon_{\mathrm{in}}e_g(F^{\mathrm{in}})\).
+Choose a dimensional \(\epsilon_*>0\) below \(\epsilon_b\), small
+enough that \(2+2C_b\epsilon_*<4\), and decrease
+\(\epsilon_{\mathrm{in}}\) further so that
+\(0<\epsilon_{\mathrm{in}}<\epsilon_*\). Iterate
+\eqref{eq:entropy-bootstrap-step} until the coefficient first reaches
+\(\epsilon_*\). The output coefficient is at most
+\((1+l)\epsilon_*\leq2\epsilon_*\), and
+\[
+ \sum_{\text{bootstrap sweeps}}l\epsilon
+ =\epsilon_0-\epsilon_{\mathrm{in}}\leq2\epsilon_*.
+\]
+This bounds the total scalar cost by \(2C_b\epsilon_*\), independently
+of the number of sweeps. By \eqref{eq:entropy-bootstrap-density},
+every reset contracts the volume density. The final density
+\(\rho=dV_{h_*}\) therefore satisfies
+\eqref{eq:entropy-bootstrap-output}. The sweep construction and
+finitely many smooth conformal changes preserve all the stated
+input-uniform properties.
+\end{proof}
+
+\subsection{The entropy increment and its surplus}
+
+Fix \(\epsilon_*\) from Lemma~\ref{lem:entropy-bootstrap}. We now use
+the boost in Proposition~\ref{prop:sweep}, choosing its constant
+\(\kappa>0\) sufficiently small depending on this fixed threshold.
+All constants arising from the boost are consequently dimensional.
+The increment estimate has a coarse region, where a fixed fraction of
+the reserve remains after paying the budget, and a region near
+\(m=0,r=1\), where the exact quadratic coefficient will matter.
+
+\begin{lemma}[Entropy increment]\label{lem:entropy-increment}
+There are dimensional constants \(m_0>0\), \(D_*\geq1\), \(l_0>0\),
+\(\gamma_n>0\), and \(C_n<\infty\) with the following property.
+For \(\epsilon\geq\epsilon_*\), take a boosted sweep with
+\[
+ 0<l\leq l_0,\qquad \theta=D_*l\leq\tfrac1{16},
+ \qquad N^2=l\epsilon.
+\]
+The errors can be chosen so that
+\begin{equation}\label{eq:entropy-increment}
+\begin{split}
+ \wscal(h_+,\phi_+)-(1+l)\epsilon e_{h_+}(Y)
+ \geq{}&\wscal(h_0,\phi_0)-\epsilon e_{h_0}(F)\\
+ &+2W_p-n\log(1+l)-C_nl^2+P_p ,
+\end{split}
+\end{equation}
+where the nonnegative, pointwise surplus is
+\begin{equation}\label{eq:entropy-surplus-definition}
+ P_p=\gamma_n\frac{\epsilon}{rN^2}\Pi_p\,
+ \indic_{\{m\geq m_0\}\,\cup\,\{|1-r|\geq1/4\}}.
+\end{equation}
+In particular, on the indicated region \(P_p\) retains a fixed fraction
+of every positive term in the sweep inequality.
+\end{lemma}
+
+\begin{proof}
+The output energy bound in \eqref{eq:entropy-sweep-density} shows that,
+in the units of \(Q\), it suffices to prove
+\begin{equation}\label{eq:entropy-budget}
+ Q\geq lm+2lrW_p-nrl\log(1+l)-C_nr l^3
+ +\frac{rN^2}{\epsilon}P_p .
+\end{equation}
+The term \(lm\) pays for the additional \(l\epsilon e_{h_+}(Y)\).
+We choose \(\delta_0\leq\epsilon l^3\) in
+\eqref{eq:entropy-sweep-form}. It remains to compare the other terms
+there with the budget in \eqref{eq:entropy-budget}.
+
+We use a trace identity for the amount of the new graph energy that
+remains in the final metric. Set
+\[
+ m_f:=\sum_i A_i^2(1-C_i)=N^2\sum_i|df_i|_{h_+}^2.
+\]
+As matrices,
+\(\sum_iN^2df_i\otimes df_i=h_+-h_0\). Hence
+\begin{equation}\label{eq:entropy-trace}
+ m_f=n-\tr(h_+^{-1}h_0)
+ \leq n(1-e^{-2W_p/n}).
+\end{equation}
+The inequality is arithmetic--geometric mean applied to the positive
+eigenvalues of \(h_+^{-1}h_0\), whose product is \(e^{-2W_p}\).
+We also have
+\begin{equation}\label{eq:entropy-volume-angle}
+ L_i=(1-A_i^2)^{-1/2}.
+\end{equation}
+
+First choose a small dimensional \(m_0>0\), in particular
+\(m_0\ll\epsilon_*\) and \(m_0<1/4\). We claim that
+\begin{equation}\label{eq:entropy-coarse-coercivity}
+ T_0+\epsilon M_0^2\geq c_n(m+rW_p)
+ \quad\text{if }m\geq m_0
+ \text{ or if }m<m_0,\ |1-r|\geq\tfrac14.
+\end{equation}
+For \(m\geq2n\), the \(C_i\) terms of \(T_0\) give
+\(T_0\geq m-m_f\geq m-n\geq m/2\).
+For \(m_0\leq m\leq2n\), first suppose that every \(A_i^2\) is at most
+a small number \(\eta=\eta(n,m_0)>0\). Then
+\[
+ 2W_p=\sum_i-\log(1-A_i^2)\leq(1+C\eta)m.
+\]
+By \eqref{eq:entropy-trace},
+\[
+ m-m_f\geq m-n\bigl(1-e^{-(1+C\eta)m/n}\bigr)\geq c_n m.
+\]
+For the last inequality, the same expression with \(\eta=0\) divided
+by \(m\) has a positive minimum on \([m_0,2n]\), and then \(\eta\)
+is chosen small. If instead some \(A_i^2>\eta\), the self term of
+that index in \(T_0\) is at least \(c_n(r+u_i)\). It gives a
+dimensional positive bound if \(r\) is bounded below. If \(r\) is
+small, \(M_0=a(1-r)-\bar u/2\) is bounded away from zero unless \(\bar u\)
+lies in a fixed compact subinterval of \((0,\infty)\). In the former
+case \(\epsilon M_0^2\geq c_n\epsilon_*\). In the latter,
+\(\epsilon=m/\bar u\leq C_n\) and
+\(u_i=A_i^2/(\epsilon\nu_i)\geq c_n\), since \(\nu_i\leq1\).
+Each bound controls \(m\leq2n\). This proves the \(m\) part for
+\(m\geq m_0\).
+
+For the volume part, indices with \(A_i\geq1/4\) satisfy
+\[
+ A_i^2(L_i-1)2ar\geq c\,r(L_i-1)\geq c\,r\log L_i.
+\]
+For all other indices, \(\log L_i\leq C A_i^2\), so their contribution
+to \(rW_p\) is at most \(Crm\). If \(r\leq2\), the bound for \(m\)
+already covers this. If \(r>2\), then
+\(|M_0|\geq c(r+\bar u)\), and
+\(\epsilon M_0^2\geq c\epsilon r\bar u=c\,rm\).
+This proves \eqref{eq:entropy-coarse-coercivity} when \(m\geq m_0\).
+When \(m<m_0\), we have \(\bar u\leq m_0/\epsilon_*\) small and
+\[
+ W_p=\frac m2+O\left(\sum_iA_i^4\right)\leq C m.
+\]
+If \(|1-r|\geq1/4\), choosing \(m_0\) still smaller gives
+\(|M_0|\geq c|1-r|\). The quantity
+\(\epsilon(1-r)^2\) controls \(m+rm\): for bounded \(r\) use
+\(\epsilon\geq\epsilon_*\) and \(m\leq m_0\), and for \(r\geq2\)
+use \((1-r)^2\geq r^2/4\). This proves the claim in the remaining
+case.
+
+On the region of \eqref{eq:entropy-coarse-coercivity}, the positive
+reserve \(\Pi_p\) consequently controls \(m+rW_p\). The positive terms
+\(lm+2lrW_p\) in \eqref{eq:entropy-budget} use at most \(C_nl\Pi_p\),
+and \(4\theta^2m\) uses at most \(C_nD_*^2l^2\Pi_p\). For \(l\)
+sufficiently small after \(D_*\), a fixed fraction of \(\Pi_p\)
+remains. Choose \(\gamma_n\) below that fraction. The error term
+\(r\delta_0/\epsilon\) is covered by \(C_nr l^3\). This proves
+\eqref{eq:entropy-budget} on the region where the surplus is retained.
+
+It remains to prove the budget when
+\begin{equation}\label{eq:entropy-small-region}
+ m<m_0,\qquad |1-r|<\tfrac14.
+\end{equation}
+Here \(T_0\geq c_nm^2\). Indeed, expanding the exponential in
+\eqref{eq:entropy-trace} and using
+\(W_p=m/2+O(\sum_iA_i^4)\) gives
+\[
+ \sum_i A_i^2 C_i=m-m_f
+ \geq c_n m^2-C_n\sum_iA_i^4
+\]
+after decreasing \(m_0\). The self terms
+\(\sum_i A_i^2(L_i-1)2ar\) are at least
+\(c_n\sum_iA_i^4\), by \(r>3/4\) and \(L_i-1\geq A_i^2/2\).
+Adding a sufficiently small fraction of the first estimate to the
+second proves \(T_0\geq c_nm^2\).
+Since \(W_p\leq Cm\) on this region, this controls the budget whenever
+\(m\geq Kl\), for a fixed sufficiently large dimensional \(K\).
+In making this conclusion, first fix \(K\); after \(D_*\) is chosen
+below, take \(l\) small enough that \(4D_*^2l^2m\) is also a small
+fraction of \(m^2\).
+
+We are left with the critical regime \(m\leq Kl\), still under
+\eqref{eq:entropy-small-region}. Here
+\(W_p=m/2+O(m^2)\) and \(3/4<r<5/4\), so the budget expands as
+\[
+ lm+2lrW_p-nrl\log(1+l)
+ =2lm-nl^2+O_n(l^3)+O_n(l^2)|r-1|.
+\]
+The target quadratic term is covered exactly by
+\[
+ \frac{m^2}{n}-2lm+nl^2
+ =\left(\frac m{\sqrt n}-\sqrt n\,l\right)^2\geq0.
+\]
+We must therefore recover \(m^2/n\) from \(T_0\), while paying the
+displayed \(r-1\) error and the errors in that recovery. This is where
+the coefficient one on \(T_0\) in \(\Pi_p\) is needed.
+
+In particular \(e^{W_p}=1+O(m)\). Metric addition gives
+\(h_0\leq h_j\leq h_+\) as quadratic forms.
+Every eigenvalue of \(h_0^{-1}h_+\) is at least one, and their product
+is \(e^{2W_p}\). Each eigenvalue is therefore at most that product,
+which proves the comparison
+\[
+ h_0\leq h_j\leq h_+\leq e^{2W_p}h_0=(1+O(m))h_0.
+\]
+Normalize the directions \(e_i\) in the \(h_0\) inner product and write
+the resulting unit vectors as \(\bar e_i\). Put
+\(\xi_{ij}=(\bar e_i\cdot\bar e_j)^2\). Before approximating the metric
+losses, their exact earlier- and later-color telescopes are
+\[
+ B_i=\sum_{j<i}(Ndf_j(e_i))^2,\qquad
+ C_i=\frac1{A_i^2}\sum_{j>i}A_j^2(Ndf_i(e_j))^2
+ \quad(A_i>0).
+\]
+The first identity follows from
+\(h_{i-1}=h_0+N^2\sum_{j<i}df_j\otimes df_j\) and
+\(|e_i|_{h_{i-1}}=1\). The second follows by telescoping the rank-one
+inverse formula
+\[
+ h_j^{-1}=h_{j-1}^{-1}-A_j^2 e_j e_j^{\mathsf T}
+\]
+from \(h_i\) to \(h_+\), and using
+\(N^2|df_i|_{h_i}^2=A_i^2\). Now
+\(Ndf_j=A_jL_j e_j^\flat\), with the dual taken in \(h_{j-1}\).
+The common metric comparison and \(L_j=1+O(m)\) show that each summand
+in the exact formulas differs from its \(h_0\) expression by
+\(O(m)A_j^2\). Their sums therefore give, for \(A_i\ne0\),
+\begin{equation}\label{eq:entropy-BC-expansion}
+ B_i=\sum_{j<i}A_j^2\xi_{ij}+O(m^2),\qquad
+ C_i=\sum_{j>i}A_j^2\xi_{ij}+O(m^2).
+\end{equation}
+
+Inserting \eqref{eq:entropy-BC-expansion} into \(T_0\) yields
+\begin{equation}\label{eq:entropy-sharp-T}
+ T_0\geq
+ \tr\left[\left(\sum_i A_i^2\bar e_i\bar e_i^{\mathsf T}\right)^2\right]
+ -C_n\left(m^3+m^2|r-1|
+ +m\sum_iA_i^2(1-\sigma_i)_+\right).
+\end{equation}
+To see the leading coefficient, the \(B_i\) and \(C_i\) terms each
+give one copy of
+\(\sum_{i<j}A_i^2A_j^2\xi_{ij}\). In the \(B_i\) terms use
+\(4a^2\geq1\), \(B_i\geq0\); replacing \(\sigma_i\) by one costs at most
+\(Cm\sum_iA_i^2(1-\sigma_i)_+\). The self terms use
+\(L_i-1\geq A_i^2/2\) and \(2a\geq1\). They give
+\(\sum_iA_i^4\), up to the displayed \(r-1\) and \(\sigma_i-1\) costs.
+The errors in \eqref{eq:entropy-BC-expansion} sum to \(O(m^3)\).
+This proves \eqref{eq:entropy-sharp-T}. The trace in that estimate
+is at least \(m^2/n\), since the matrix is positive semidefinite
+with trace \(m\).
+
+The last error in \eqref{eq:entropy-sharp-T} is paid for by the
+centered squares. From
+\(a(\sigma_i-1)=V_i-\lambda(u_i-\bar u)\), Cauchy--Schwarz and Young's
+inequality give, for every fixed \(c'>0\),
+\begin{equation}\label{eq:entropy-v-error}
+\begin{split}
+ m\sum_iA_i^2(1-\sigma_i)_+
+ &\leq C m\epsilon\,\E\bigl[u_i(|V_i|+|u_i-\bar u|)\bigr]\\
+ &\leq c'\epsilon\bigl(\E V_i^2+\theta\Var u\bigr)
+ +C_{c'}\frac{m^2}{\theta}\epsilon(\bar u^2+\Var u).
+\end{split}
+\end{equation}
+Choose \(c'\) to use a small fraction of the squares in \(\Pi_p\).
+The coefficient of \(\epsilon\Var u\) in the last term, relative to
+\(\theta\epsilon\Var u\), is at most \(C_{c'}K^2/D_*^2\), because
+\(m\leq Kl\) and \(\theta=D_*l\). Choose \(D_*\) sufficiently large
+after \(K\) so that this term is absorbed as well. The remaining term
+is
+\[
+ C_{c'}\frac{m^2}{\theta}\epsilon \bar u^2
+ =C_{c'}\frac{m^4}{\theta\epsilon}=O_n(l^3).
+\]
+Here \(\epsilon\geq\epsilon_*\), and \(K,D_*,\epsilon_*\) are already
+fixed dimensionally. Also
+\[
+ |r-1|\leq C(|M_0|+\bar u).
+\]
+Consequently a term \(O_n(l^2)|r-1|\) is bounded by a small fraction
+of \(\epsilon M_0^2\) plus \(O_n(l^3)\): use Young's inequality on
+\(l^2|M_0|\), and use \(\bar u\leq Kl/\epsilon_*\) on the other part.
+The errors \(m^3\), \(m^2|r-1|\), and \(4D_*^2l^2m\) in the present
+region are therefore covered with a remaining error \(O_n(l^3)\).
+
+The just established mean-square estimate also covers the
+\(O_n(l^2)|r-1|\) term in the critical budget. The recovered
+\(m^2/n\) then covers \(2lm-nl^2\) by the displayed square.
+Since \(r\) is bounded above and below, the remaining \(O_n(l^3)\)
+terms are of the form \(C_nr l^3\). This proves
+\eqref{eq:entropy-budget} in \eqref{eq:entropy-small-region}.
+
+The choices just made have a definite order: first \(m_0\) after
+\(\epsilon_*\), then \(K\), then \(D_*\), and finally \(l_0>0\)
+small enough for all the comparisons and for \(D_*l_0\leq1/16\).
+The errors in each sweep are chosen afterward. Thus every constant
+in \eqref{eq:entropy-increment} is dimensional.
+\end{proof}
+
+\begin{lemma}[Bounds supplied by the surplus]\label{lem:entropy-surplus}
+For every sweep of Lemma~\ref{lem:entropy-increment},
+\begin{equation}\label{eq:entropy-pass-volume}
+ W_p\leq C_n\epsilon^{1/4}(\epsilon+P_p)^{3/4}.
+\end{equation}
+There is also a dimensional \(\epsilon_{\mathrm{exit}}\) such that,
+if \(\epsilon\geq\epsilon_{\mathrm{exit}}\) and \(r\leq1/4\), then
+\begin{equation}\label{eq:entropy-last-surplus}
+\begin{split}
+ P_p&\geq c_n\frac{\epsilon}{lr}
+ \left(1+\frac{m^2}{\epsilon^2}\right),\\
+ \sum_i\bigl(D_i^2+|\II_i|^2+|d\log L_i|_{h_i}^2\bigr)
+ &\leq C_n\frac{P_p}{\theta}.
+\end{split}
+\end{equation}
+The errors in the prescription may simultaneously be chosen to have
+\(\sum_i|\mathrm{err}_i|^2\leq1\).
\end{lemma}
-\begin{proof}[Proof sketch]
-\emph{Step 1: conformal enlargement.} Split into regions where the
-gradient $|dY|_{h_0}$ is small or large. On the small-gradient
-region apply a conformal change $k=e^{-2u}h_0$ with $u>0$ a smooth
-function solving a linearized equation ($L_Yu=1$ with lower-order
-terms prescribed to fix $c_{\mathrm{exit}}$). The scalar conformal
-formula and the smallness of the gradient give the scalar inequality
-\eqref{eq:entropy-after-exit} on this region, with $W_1\ge n u+c$
-and $P_1\ge c_{\mathrm{exit}}P_0$.
-
-\emph{Step 2: gradient clearing.} Where $|dY|_{h_0}$ is large, use
-the tangent-branch equation of Lemma~\ref{lem:graph-prescription}
-with $f_0=0$ and small $e,w$. This produces $Z_1$ covarying with $Y$
-on the small-gradient region and taking small values on the
-large-gradient region.
-
-\emph{Step 3: post-exit gradient.} The composition of Step 1 and
-Step 2 gives
-\[
- e_{h_1}(Z_1)\le e^{-2u}e_{h_0}(Y)+C_N\,e_{h_1}(\text{graph residual}),
-\]
-so with $N=\sqrt E$ and $u$ taken with $e^{-2u}\le E^{-0.02}$ we get
-$e_{h_1}(Z_1)^{1/2}\le E_1^{-1/100}$. The second exit similarly
-handles the residual of the first.
-
-\emph{Step 4: volume.} The choice of $u$ controls
-$W_1\le E^{0.32}(E+P_1')^{0.75}$ by the explicit form of the
-conformal equation. All Borel and equivariant properties are
-inherited from Lemma~\ref{lem:graph-prescription}.
+\begin{proof}
+If \(\bar u=m/\epsilon\) is larger than a fixed dimensional number, then
+\(m\geq m_0\), so the surplus retains the mean square and boost.
+Since \(|M_0|\geq c(r+\bar u)\) in this range and \(l\leq1\),
+\[
+ P_p\geq c_n\epsilon\,
+ \frac{(r+\bar u)^2+c_*r^6\indic_{\{r\geq C_*\}}}{r}
+ \geq c'_n\epsilon \bar u^{5/3}.
+\]
+For the last inequality, if \(r\leq \bar u^{1/3}\) use \(\bar u^2/r\);
+if \(r\geq \bar u^{1/3}\) use \(r^5\), taking the threshold for \(\bar u\)
+large enough that \(\bar u^{1/3}\geq C_*\). It follows in all cases that
+\begin{equation}\label{eq:entropy-m-surplus}
+ m\leq C_n\epsilon(1+P_p/\epsilon)^{3/5}.
+\end{equation}
+If any \(A_i>1/2\), then \(m>1/4>m_0\), and the self terms in \(T_0\)
+give
+\[
+ P_p\geq c_n\sum_{A_i>1/2}(L_i-1).
+\]
+Indeed their contribution to \(T_0\) is at least
+\(c r\sum_{A_i>1/2}(L_i-1)\), and
+\(\epsilon/(rN^2)=1/(lr)\geq1/r\).
+For \(A_i\leq1/2\), \(\log L_i\leq C A_i^2\).
+There are at most \(4m\) indices with \(A_i>1/2\). Concavity of
+\(\log(1+s)\) on those indices, together with the last estimate,
+therefore gives
+\begin{equation}\label{eq:entropy-log-surplus}
+ W_p\leq C_n m\bigl(1+\log(1+P_p/m)\bigr)
+\end{equation}
+when \(m>0\); when \(m=0\), \(W_p=0\).
+To justify the bound when the number of those indices is smaller
+than \(4m\), use that \(s\log(1+b/s)\) is increasing in \(s>0\).
+Put \(x=m/\epsilon\), \(y=P_p/\epsilon\). The elementary bound
+\(1+\log(1+t)\leq C(1+t)^{1/4}\) for \(t\geq0\), applied with
+\(t=y/x\), makes the right side of
+\eqref{eq:entropy-log-surplus}, divided by \(\epsilon\), at most
+\[
+ C_n x^{3/4}(x+y)^{1/4}
+ \leq C_n(1+y)^{7/10}
+ \leq C_n(1+y)^{3/4},
+\]
+where \eqref{eq:entropy-m-surplus} was used in the middle inequality.
+This proves \eqref{eq:entropy-pass-volume}.
+
+Now suppose \(r\leq1/4\). The surplus retains all positive terms
+because \(|1-r|\geq3/4\). Outside a fixed compact subinterval of
+\(\bar u\in(0,\infty)\), the expression \(M_0=a(1-r)-\bar u/2\) satisfies
+\(M_0^2\geq c(1+\bar u^2)\). On that compact interval, for \(\epsilon\)
+sufficiently large,
+\[
+ T_0\geq m-m_f\geq\epsilon \bar u-n
+ \geq c_n\epsilon(1+\bar u^2)
+\]
+by \eqref{eq:entropy-trace}. Multiplying the appropriate retained
+term by \(\gamma_n/(lr)\) proves the first inequality in
+\eqref{eq:entropy-last-surplus}.
+
+For the second, the retained derivative terms imply
+\begin{equation}\label{eq:entropy-X-retained}
+ \mathcal X\leq C_n N^2P_p,\qquad
+ \frac{\epsilon^2}{rN^2}
+ \bigl(\E V_i^2+\theta\Var u\bigr)\leq C_nP_p.
+\end{equation}
+Also
+\[
+ \sum_i t_i^2=\frac{\epsilon^2}{r}(\bar u^2+\Var u).
+\]
+The first inequality of \eqref{eq:entropy-last-surplus} controls
+the \(\bar u^2\) part of this sum divided by \(N^2\), and the variance
+term in \eqref{eq:entropy-X-retained} controls the other part with
+a factor \(1/\theta\). Thus
+\(\sum_i t_i^2/N^2\leq C_nP_p/\theta\). Since \(\mathcal X\) contains
+\((X_i-t_i)^2+N^2|\II_i^{\mathrm{rem}}|^2+N^2|d\log L_i|^2\), this proves the
+claimed bounds for \(\II_i\) and \(d\log L_i\).
+
+For \(D_i\), use
+\[
+ ND_i^0=\frac{\epsilon f_i}{r}
+ \left(a(\sigma_i-r)+\frac{u_i}{2}\right),\qquad
+ \E \sigma_i^2\leq C\bigl(1+\E V_i^2+\Var u\bigr).
+\]
+Here the second inequality follows by solving the definition of \(V_i\)
+for \(\sigma_i-1\). Since \(r\leq1/4\),
+\[
+ \sum_i(D_i^0)^2\leq
+ \frac{C\epsilon^2}{rN^2}
+ \bigl(1+\bar u^2+\E V_i^2+\Var u\bigr)
+ \leq C_nP_p/\theta.
+\]
+The relation \(ND_i=ND_i^0-\beta_i f_i+\mathrm{err}_i\) finishes the
+estimate: \(\sum_i\beta_i^2f_i^2\leq\mathcal X\), and we choose
+\(\sum_i|\mathrm{err}_i|^2\leq1\). Its contribution \(1/N^2\) is
+bounded by \(C_nP_p\) using the first inequality of
+\eqref{eq:entropy-last-surplus}, \(\epsilon\geq1\), and \(r\leq1/4\).
\end{proof}
-\subsection{Proof of the entropy theorem}
+\subsection{Accumulation and an absolute gradient bound}
+
+The step size has two jobs: the scalar losses \(O_n(l^2)\) must be
+summable, and the number of support enlargements before reaching \(E\)
+must grow at most cubically in \(\log E\). Since one sweep changes
+\(\log(1+\epsilon)\) by a quantity comparable to \(l\), we take
+\begin{equation}\label{eq:entropy-step-schedule}
+ l(\epsilon)=\frac{l_*}{(1+\log(1+\epsilon))^2},
+\end{equation}
+where the dimensional \(l_*>0\) is chosen below \(l_0\) from
+Lemma~\ref{lem:entropy-increment}. Thus all the parameter choices in
+that lemma, including \(\theta=D_*l\), apply.
+
+The fixed exponents in the next two lemmas leave room, respectively,
+for the number of sweeps, the conformal scalar cost, the outgoing
+volume estimate, and the final gradient target:
+\[
+ 0.26=\tfrac14+0.01,\qquad
+ 0.26+0.04<\frac{1+0.32}{4},\qquad
+ 0.26+\tfrac34(0.04)<0.32,\qquad
+ 0.02>\tfrac1{100}.
+\]
+The estimates below show how each margin is used.
+
+\begin{lemma}[Accumulation of entropy passes]\label{lem:entropy-accumulation}
+Suppose at parameter \(\epsilon_{\mathrm{in}}\geq\epsilon_*\) that
+\[
+ \wscal(h,\phi)\geq-C+2W-n\log\epsilon_{\mathrm{in}}
+ +\epsilon_{\mathrm{in}}e_h(F)+P_{\mathrm{in}},
+ \qquad W=\log(dV_h/\rho)\geq0,\quad P_{\mathrm{in}}\geq0.
+\]
+Successive sweeps of Lemma~\ref{lem:entropy-increment}, with
+\(\epsilon_{\mathrm{new}}=(1+l(\epsilon))\epsilon\), preserve this
+form of inequality: at the final parameter \(E\) the surplus is
+\(P=P_{\mathrm{in}}+\sum_pP_p\), the volume logarithm is
+\(W_{\mathrm{new}}=W+\sum_pW_p\), and the increase of the scalar
+constant is \(C_n\sum_p l(\epsilon_p)^2\). Along the entire increasing
+sequence of parameters,
+\begin{equation}\label{eq:entropy-summability}
+ \sum_p l(\epsilon_p)^2\leq C_n,\qquad
+ \#\{p:\epsilon_p<E\}\leq C_n(1+\log^3(1+E)).
+\end{equation}
+If the initial volume logarithm is zero, then, for all sufficiently
+large dimensional \(E\),
+\begin{equation}\label{eq:entropy-phase-volume}
+ W_{\mathrm{new}}\leq E^{0.26}(E+P)^{0.75}.
+\end{equation}
+The same assertion holds if the initial data have parameter \(E_0\),
+surplus \(P_0\), and
+\[
+ W\leq E_0^{0.32}(E_0+P_0)^{0.75},
+\]
+provided \(E\geq E_0\) is sufficiently large dimensionally and
+\(E_0^{0.32}\leq\tfrac12E^{0.26}\).
+\end{lemma}
+
+\begin{proof}
+The accumulation assertion follows by adding
+\eqref{eq:entropy-increment}. The energy coefficients telescope
+multiplicatively, and
+\(\sum_p\log(1+l(\epsilon_p))=\log(E/\epsilon_{\mathrm{in}})\).
+The density identity in \eqref{eq:entropy-sweep-density} gives the
+sum of the \(W_p\).
+
+Put \(t_p=\log(1+\epsilon_p)\). Since
+\(\epsilon_p\geq\epsilon_*>0\) and \(l_*\) is fixed small,
+\[
+ t_{p+1}-t_p
+ =\log\left(1+l(\epsilon_p)\frac{\epsilon_p}{1+\epsilon_p}\right)
+ \asymp_n l(\epsilon_p)
+ =\frac{l_*}{(1+t_p)^2}.
+\]
+Comparison with the integrals of \((1+t)^2\) and \((1+t)^{-2}\)
+respectively proves the number bound and
+\[
+ \sum_p l(\epsilon_p)^2
+ \leq C_n l_*\int_{t_0}^{\infty}\frac{dt}{(1+t)^2}\leq C_n.
+\]
+The comparison is valid on each interval \([t_p,t_{p+1}]\), since
+its endpoints differ by a uniformly bounded small number.
+
+Let \(k\) be the number of sweeps in the group under consideration.
+By Lemma~\ref{lem:entropy-surplus}, since
+\(\epsilon_p\leq E\) and \(P_p\leq P\),
+\[
+ \sum_pW_p\leq C_n k E^{1/4}(E+P)^{3/4}.
+\]
+For all sufficiently large \(E\),
+\(C_n k\leq \tfrac12 E^{0.01}\) by
+\eqref{eq:entropy-summability}. This proves
+\eqref{eq:entropy-phase-volume} for zero initial volume logarithm.
+In the other case \(P\geq P_0\) and \(E\geq E_0\), so the initial
+term is at most \(E_0^{0.32}(E+P)^{0.75}\). The stated condition on
+\(E_0\) pays for this with the other half of
+\(E^{0.26}(E+P)^{0.75}\).
+\end{proof}
+
+Call a finite group of these sweeps a \emph{phase}. There is no
+conformal change inside a phase. Each sweep adds the nonnegative
+tensor in \eqref{eq:sweep-output}; hence, if \(h^0\) is its starting
+metric and \(h_+\) its final metric, the successive metrics and their
+inverses satisfy
+\[
+ h^0\leq h^{(1)}\leq\cdots\leq h_+,\qquad
+ (h_+)^{-1}\leq\cdots\leq(h^{(1)})^{-1}\leq(h^0)^{-1}.
+\]
+In particular, the starting metric is an admissible reference metric
+for every sweep in the phase, and
+\(e_{h_+}(Y^0)\leq e_{h^0}(Y^0)\) for its starting map \(Y^0\).
+
+We now finish a phase with an absolute gradient bound. Normalizing
+the last bumps alone divides by \(\sqrt r\), which can make the
+gradient large where \(r\) is small. We first keep a small copy of
+the starting map in the normalization; its gradient is controlled by
+the preceding metric comparison. A conformal enlargement confined
+to the small-\(r\) region then reduces the remaining gradient there,
+and the last sweep's surplus pays its cost.
+
+\begin{lemma}[Exit from a phase]\label{lem:entropy-phase-exit}
+Consider a phase starting with root map \(Y^0\) and metric \(h^0\), and
+let \(K_0\) be a finite uniform bound for \(e_{h^0}(Y^0)^{1/2}\).
+Suppose its last sweep has input parameter \(\epsilon\), output
+parameter \(E=(1+l(\epsilon))\epsilon\), and output
+\((h_+,\phi_+,Y)\). Suppose, with \(C\geq0\) and \(P\geq P_p\) including the
+surplus from that last sweep, that
+\begin{equation}\label{eq:entropy-before-exit}
+\begin{split}
+ \wscal(h_+,\phi_+)&\geq-C+2W-n\log E+E e_{h_+}(Y)+P,\\
+ W&=\log(dV_{h_+}/\rho)\geq0,\qquad
+ W\leq E^{0.26}(E+P)^{0.75}.
+\end{split}
+\end{equation}
+If \(E\) is sufficiently large in terms of \(n,K_0\), there are a
+metric \(\widehat h\), a weight \(\widehat\phi\), and a root map \(Z\)
+on two copies of the labels such that
+\begin{equation}\label{eq:entropy-after-exit}
+\begin{split}
+ \wscal(\widehat h,\widehat\phi)
+ &\geq-(C+C_n)+2\widehat W-n\log E
+ +E e_{\widehat h}(Z)+P',\\
+ \widehat W&=\log(dV_{\widehat h}/\rho)\geq0,\qquad
+ \widehat W\leq E^{0.32}(E+P')^{0.75},\\
+ e_{\widehat h}(Z)^{1/2}&\leq E^{-1/100}.
+\end{split}
+\end{equation}
+Here \(P'\geq0\), and the added scalar cost \(C_n\) is independent
+of \(K_0\) and of all other instance constants. The changes preserve
+the uniform smoothness and support properties of
+Theorem~\ref{thm:entropy}.
+\end{lemma}
+
+\begin{proof}
+In this proof all derivatives, unless otherwise indicated, use \(h_+\).
+Write \(r=\sum_i f_i^2\) for the last sweep, set \(s=\sqrt r\), and put
+\[
+ c_E=\frac1{\sqrt E(1+K_0)},\qquad
+ Z=\frac{(f,c_EY^0)}{\sqrt{r+c_E^2}}.
+\]
+The two vectors in the numerator use disjoint copies of the labels.
+The map is a smooth root probability map because its denominator is
+at least \(c_E>0\). Writing the copies as \((i,0)\) and \((i,1)\),
+we call \(i\) their common parent label at the start of the phase;
+the sweeps within the phase do not change the label set. The first
+copy is supported on the last bumps and the second on the starting
+map. This parent assignment will be used to compare support overlaps
+between phases. By \eqref{eq:entropy-trace},
+\[
+ N^2\sum_i|df_i|_{h_+}^2=m_f\leq n,\qquad
+ |ds|_{h_+}^2\leq n/N^2.
+\]
+Monotonicity of the phase metrics also gives \(e_{h_+}(Y^0)\leq K_0^2\).
+
+Since \(Y\) and \(Y^0\) are unit vectors in orthogonal spaces, their
+derivatives are perpendicular to their respective vectors. Differentiating
+\(Z=(sY,c_EY^0)/\sqrt{s^2+c_E^2}\) thus gives the exact identity
+\[
+ e(Z)=\frac r{r+c_E^2}e(Y)+\frac{c_E^2}{r+c_E^2}e(Y^0)
+ +\frac{c_E^2}{(r+c_E^2)^2}|ds|^2.
+\]
+For \(r\geq1/4\), it follows that
+\[
+ E(e(Z)-e(Y))\leq
+ \frac{Ec_E^2K_0^2}{r+c_E^2}
+ +\frac{Ec_E^2n}{N^2(r+c_E^2)^2}
+ \leq 4+\frac{16n}{N^2}\leq C_n
+\]
+when \(E\) is large, since \(N^2=l\epsilon\) tends to infinity under
+\eqref{eq:entropy-step-schedule}. For \(r\leq1/4\), the norm of the
+derivative of normalization is at most \(1/\sqrt{r+c_E^2}\), so
+\begin{equation}\label{eq:entropy-Z-energy}
+ e(Z)\leq\frac{n/N^2+c_E^2K_0^2}{r+c_E^2}.
+\end{equation}
+The first inequality of \eqref{eq:entropy-last-surplus} now implies
+\[
+ E e(Z)\leq \frac{C_n}{l(r+c_E^2)}
+ \leq \frac{C_n}{\epsilon}P_p.
+\]
+Thus replacing \(Y\) by \(Z\) in
+\eqref{eq:entropy-before-exit} costs a dimensional constant on
+\(\{r\geq1/4\}\) and an arbitrarily small fraction of \(P_p\) on
+\(\{r\leq1/4\}\), once \(E\) is dimensionally large. In particular,
+no factor \(K_0\) occurs in this scalar cost.
+
+Choose a smooth function \(u=u(r)\geq0\), supported in
+\(\{r\leq E^{-0.32}\}\), with
+\begin{equation}\label{eq:entropy-u-choice}
+ e^u\asymp
+ \min\bigl(E^{0.02},\max(1,E^{-0.16}/\sqrt r)\bigr),
+ \qquad \left|\frac{du}{dr}\right|\leq\frac C r.
+\end{equation}
+For example, smooth, with bounded-width transitions on the logarithmic
+scale, the function obtained by truncating
+\(-\tfrac12\log r-0.16\log E\) to the interval
+\([0,0.02\log E]\). The smoothing may vanish whenever the untruncated
+function is nonpositive, and changes its value by a bounded amount;
+this gives both the support assertion and the comparison in
+\eqref{eq:entropy-u-choice}. Define
+\[
+ \widehat h=e^{2u}h_+,\qquad
+ \widehat\phi=\phi_+-(n-1)u,\qquad
+ \widehat W=W+nu.
+\]
+In particular \(dV_{\widehat h}=e^{nu}dV_{h_+}\), so this is the
+claimed volume logarithm. Differentiating \(r\) gives
+\[
+ |dr|\leq2\sqrt r\left(\sum_i|df_i|^2\right)^{1/2}
+ \leq\frac{2\sqrt{nr}}N,\qquad
+ |du|\leq\frac{C_n}{N\sqrt r}.
+\]
+The coefficient \(n-1\) in the new weight cancels the Laplacian of
+\(u\) in the weighted conformal formula. With
+\(b=\nabla_{h_+}\phi_+\), that formula is
+\begin{equation}\label{eq:entropy-conformal-exit}
+ \wscal(\widehat h,\widehat\phi)
+ =e^{-2u}\left[\wscal(h_+,\phi_+)+2b(u)-(n-1)|du|^2\right].
+\end{equation}
+
+With the fixed density \(\rho\), a graph sweep changes \(W\) by
+\(+W_p\) and \(\phi\) by \(-W_p\), so it preserves \(\phi+W\).
+The present exit instead satisfies
+\[
+ \widehat\phi+\widehat W=\phi_++W+u.
+\]
+Thus after an exit \(W\) need not equal \(-\phi\), and its volume
+bound supplies no bound for the gradient of the weight. We need a
+bound on \(b(u)\) which uses no bound on the size of \(b\).
+It is enough to estimate this on the support of \(u\), which lies in
+\(\{r\leq1/4\}\) for large \(E\). At such a point view all the graphs
+of the last sweep in the common
+Euclidean extension of \((T_x\widetilde M,h_0)\) by their graph
+directions, one for each active color at that point. Write
+\(\mathbf n_i\) for the successive unit normals. They are
+orthonormal: a later
+normal lies in the preceding graph plus its new vertical direction
+and is perpendicular to every earlier normal. In this space, each
+gradient is embedded through its graph and padded by zero in the
+unused later directions. Writing \(b_i\) for the embedded gradient
+of \(\phi_i\), orthogonal projection to the new graph gives the exact
+recurrence
+\[
+ b_i=b_{i-1}-\langle b_{i-1},\mathbf n_i\rangle\mathbf n_i
+ -\nabla_{h_i}\log L_i,
+\]
+where the last gradient is embedded in the same space. The normal
+coefficient is
+\(\langle b_{i-1},\mathbf n_i\rangle=-U_i(\phi_{i-1})\), and
+its magnitude satisfies
+\[
+ |U_i(\phi_{i-1})|\leq |D_i|+\sqrt n\,|\II_i|,
+\]
+because \(D_i=\tr\II_i-U_i(\phi_{i-1})\).
+By orthonormality and \eqref{eq:entropy-last-surplus}, the norm of
+the sum of these normal corrections is at most
+\(C_n\sqrt{P_p/\theta}\).
+
+For the subtracted tangent gradients, split the indices according as
+\(A_i\leq1/2\) or \(A_i>1/2\). In the first set
+\(|d\log L_i|\leq C A_i|\II_i|\); in the second there are at most
+\(4m\) indices. Cauchy--Schwarz and
+\eqref{eq:entropy-last-surplus} consequently bound the sum of their
+norms by \(C_n\sqrt{mP_p/\theta}\). The final tangent embedding sends
+a vector \(v\) to \((v,(Ndf_i(v))_i)\). Summing the recurrence, the
+initial gradient has no vertical component in this common extension,
+while the vertical component of the final gradient is
+\((Ndf_i(b))_i\).
+The normal and tangent bounds therefore prove
+\[
+ \sum_i(Ndf_i(b))^2\leq C_n(1+m)P_p/\theta.
+\]
+It follows that
+\[
+ |b(u)|\leq\frac C r|b(r)|
+ \leq C_n\left(\frac{(1+m)P_p}{\theta N^2r}\right)^{1/2}
+ \leq \frac{C_nP_p}{\sqrt{\theta E}}.
+\]
+For the last inequality use \(E\asymp\epsilon\), \(N^2=l\epsilon\),
+and
+\[
+ P_p\geq\frac{c_n}{lr}
+ \left(\epsilon+\frac{m^2}{\epsilon}\right)
+ \geq \frac{c'_n(1+m)}{lr}
+\]
+for \(\epsilon\geq1\). The same first surplus bound gives
+\begin{equation}\label{eq:entropy-exit-derivatives}
+ |du|^2\leq\frac{C_n}{N^2r}\leq\frac{C_nP_p}{E^2}.
+\end{equation}
+All these estimates are only needed where \(u\ne0\), a subset of
+\(\{r\leq1/4\}\) for large \(E\). Since
+\(\theta E\asymp_n E/(1+\log E)^2\) tends to infinity, the two
+derivative terms in \eqref{eq:entropy-conformal-exit} cost an
+arbitrarily small fraction of \(P_p\).
+
+There is enough surplus on this region also for the change in the
+entropy term. From \(r\leq E^{-0.32}\) and
+\eqref{eq:entropy-last-surplus},
+\[
+ P\geq P_p\geq c_n E^{1.32}.
+\]
+Using \eqref{eq:entropy-before-exit} and \(e^{2u}\leq C E^{0.04}\),
+we obtain there
+\begin{equation}\label{eq:entropy-exit-volume-cost}
+ \frac{e^{2u}W}{P}\leq
+ C_nE^{0.04+0.26}P^{-0.25}\leq C_nE^{-0.03},
+ \qquad
+ \frac{e^{2u}nu}{P}\leq C_nE^{-1.28}\log E.
+\end{equation}
+In the first bound we used \(E+P\leq C_nP\), valid for large \(E\)
+on this region. Both quantities tend to zero with dimensional
+constants.
+
+We now assemble these estimates. After replacing \(Y\) by \(Z\), write
+the retained surplus as \(\alpha_n P\), where \(\alpha_n>0\) is fixed
+dimensionally. Put \(B_u=2|b(u)|+(n-1)|du|^2\). Substituting in
+\eqref{eq:entropy-conformal-exit} and moving the new entropy term
+to the left gives
+\begin{align*}
+ &\wscal(\widehat h,\widehat\phi)
+ -E e_{\widehat h}(Z)-2(W+nu)+n\log E\\
+ &\quad\geq -(C+C_n)
+ +e^{-2u}\bigl[\alpha_n P-B_u
+ -2(e^{2u}-1)W-2nu e^{2u}\bigr]
+ +(1-e^{-2u})n\log E .
+\end{align*}
+We used \(e^{-2u}(C+C_n)\leq C+C_n\). The last term is nonnegative
+for \(E\geq1\). Equations \eqref{eq:entropy-exit-derivatives} and
+\eqref{eq:entropy-exit-volume-cost} leave at least a fixed fraction
+of \(P\) in the bracket on the support of \(u\); off that support the
+volume and derivative costs vanish. We may thus set
+\[
+ P'=\alpha'_n e^{-2u}P
+\]
+for a fixed \(\alpha'_n>0\), proving the scalar inequality in
+\eqref{eq:entropy-after-exit}. The only scalar constant charged was
+the dimensional replacement cost.
+
+The volume estimate at exit follows as well. Since
+\(P\leq C_ne^{2u}P'\leq C_nE^{0.04}P'\) and \(nu\leq C_n\log E\),
+\[
+ \widehat W\leq
+ C_n E^{0.26+0.04(0.75)}(E+P')^{0.75}+C_n\log E
+ \leq E^{0.32}(E+P')^{0.75}
+\]
+for all sufficiently large dimensional \(E\).
+
+Finally, \eqref{eq:entropy-Z-energy} and the conformal scaling of
+energy give
+\[
+ e_{\widehat h}(Z)^{1/2}
+ \leq e^{-u}\frac{\sqrt n/N+c_EK_0}{\sqrt{r+c_E^2}}.
+\]
+The choice \eqref{eq:entropy-u-choice} implies
+\[
+ \frac{e^{-u}}{\sqrt{r+c_E^2}}
+ \leq C\left(\frac{E^{-0.02}}{c_E}+E^{0.16}\right).
+\]
+Using \(\sqrt E/N\leq C_n(1+\log E)\), \(c_EK_0\leq E^{-1/2}\),
+and \(N\leq\sqrt E\), we conclude that
+\begin{equation}\label{eq:entropy-exit-gradient}
+ e_{\widehat h}(Z)^{1/2}
+ \leq C_n\left(
+ \frac{(1+K_0)(1+\log E)}{E^{0.02}}+\frac{E^{0.16}}{N}\right).
+\end{equation}
+The first summand is \(o(E^{-0.01})\) for fixed \(K_0\); the second is
+\(O_n((1+\log E)E^{-0.34})\). Taking \(E\) large enough in terms of
+\(n,K_0\) proves the final inequality in
+\eqref{eq:entropy-after-exit}. The functions \(u\) and the normalizing
+denominator are smooth with finite bounds in each fixed construction.
+The supports of \(Z\) are those of the last bumps and of \(Y^0\).
+Thus the asserted uniformity and compact support properties persist.
+\end{proof}
+
+\subsection{A fixed graph and the final parameter choice}
\begin{proof}[Proof of Theorem~\ref{thm:entropy}]
-Fix $H,A,B$ for the prescribed elementary bound and set $J=H+2$.
-This is fixed before any phase target is chosen.
+Fix parameters \(H,A,B\) for the prescribed elementary bound and set
+\(J=H+2\). This number of phases is fixed before any phase target
+is chosen. The first target will be chosen after the bootstrap data
+and their uniform constants have been fixed; the estimates below
+hold for every sufficiently large choice of that target.
+
+Apply Lemma~\ref{lem:entropy-bootstrap}. At its output
+\(W=\log(dV_{h_*}/\rho)=0\), \(\phi=0\), and
+\(\epsilon_0\in[\epsilon_*,2\epsilon_*]\). Increasing a dimensional
+constant once, \eqref{eq:entropy-bootstrap-output} has the form
+\[
+ \wscal(h_*,0)\geq-C_n+2W-n\log\epsilon_0
+ +\epsilon_0 e_{h_*}(F_*),
+\]
+because \(\epsilon_0\) ranges in a fixed dimensional interval.
+We use the schedule \eqref{eq:entropy-step-schedule} from now on.
+The bound \eqref{eq:entropy-summability} applies to the entire
+increasing sequence of energy coefficients, even when it is
+partitioned into phases.
+
+Choose a large first target and run the first phase until the parameter
+first reaches it; call the actual output parameter \(E_1\). The
+overshoot is at most a factor \(1+l_*\). In every sweep of this phase
+use \(h_*\) as the reference metric. The possible support enlargement
+per sweep, including its separation buffers, has a dimensional radius
+in this reference metric by Proposition~\ref{prop:sweep}. The total
+radius is therefore at most \(C_n(1+\log^3(1+E_1))\).
+
+We record a bound for a single fixed graph sufficient for every
+coloring in this phase and for all cooccurrences at its exit. All
+data before the choice of the first target, including \(h_*\) and its
+uniform comparison with \(g\), have already been fixed. The lifted
+metric \(h_*\) is at least a positive input-dependent multiple of \(g\).
+For a fixed finite generating set of \(\Gamma\), cocompactness implies
+that word length is bounded above by an affine input-dependent
+function of lifted \(g\)-distance between translates of fixed compact
+fundamental sets. For instance, subdividing a curve into bounded
+length pieces and following the finitely many translates meeting
+successive covering sets gives this bound. The number of group
+elements of word length at most \(L\) is at most exponential in \(L\).
+It follows that the graph joining every pair of labels whose possible
+enlarged supports in the first phase meet, or come within a required
+buffer, has degree at most
+\begin{equation}\label{eq:entropy-first-graph}
+ \Delta_1+1\leq
+ \exp\bigl(C_{\mathrm{inst}}(1+\log E_1)^3\bigr).
+\end{equation}
+Here \(C_{\mathrm{inst}}\) is finite and fixed \emph{before} the
+first target is chosen. It includes the initial number of label
+orbits and the compact support bounds. The two copies of labels
+used at exit and their possible cooccurrences change this bound only
+by a fixed factor, which is absorbed in \(C_{\mathrm{inst}}\).
+
+Take \(E_1\) sufficiently large for
+Lemma~\ref{lem:entropy-accumulation} and
+Lemma~\ref{lem:entropy-phase-exit}; in the latter the initial \(K_0\)
+is the finite input-dependent gradient bound of \(F_*\).
+We now have the scalar inequality, the volume bound with exponent
+\(0.32\), and the absolute gradient bound at \(E_1\).
+Denote the exit metric, map, and fixed cooccurrence graph by
+\(h_1,F_1,\mathcal G_1\).
+
+We next isolate how a small gradient controls the graph in a later
+phase. Suppose a phase starts with metric \(h_j\), root map \(F_j\)
+with \(e_{h_j}(F_j)^{1/2}\leq K\), and a fixed graph \(\mathcal G_j\)
+containing all its possible cooccurrences. Let \(R\) bound the
+following pairwise distance: whenever two possible enlarged supports
+meet or come close enough to require separation by a buffer, there
+are points \(x,y\) in the old nonzero supports of their respective
+parent labels with \(\dist_{h_j}(x,y)\leq R\). If a closure is
+involved, allow an arbitrarily small additional distance. Suppose
+\[
+ KR<\sqrt2.
+\]
+Choosing the additional distance small enough, the gradient bound gives
+\[
+ \|F_j(x)-F_j(y)\|_{\ell^2}
+ \leq K\dist_{h_j}(x,y)<\sqrt2.
+\]
+Two nonnegative unit vectors with disjoint supports have distance
+exactly \(\sqrt2\). Thus \(F_j(x)\) and \(F_j(y)\) share a nonzero
+label, say \(k\). The first parent and \(k\) cooccur at \(x\), and
+the second parent and \(k\) cooccur at \(y\). The two parents are
+therefore at graph distance at most two in \(\mathcal G_j\). This
+argument applies separately at each transverse parameter; because
+\(\mathcal G_j\) already contains all possible cooccurrences, the
+conclusion holds in that one fixed graph.
+
+After an exit at \(E_j\), we choose a target that both meets this
+overlap condition and grows by one exponential level:
+\begin{equation}\label{eq:entropy-phase-targets}
+ T_{j+1}=\exp(E_j^{1/1000}),
+\end{equation}
+and call the first parameter reaching it \(E_{j+1}\). Thus
+\[
+ T_{j+1}\leq E_{j+1}\leq(1+l_*)T_{j+1}.
+\]
+The choice \(1/1000\) leaves the strict inequality
+\(3/1000<1/100\): the cubic logarithmic cost of support enlargement
+will be smaller than the gain from the preceding gradient bound.
+For large \(E_j\) this target satisfies
+\(E_j^{0.32}\leq\tfrac12 E_{j+1}^{0.26}\). The outgoing bound with
+exponent \(0.32\) from phase \(j\) and
+Lemma~\ref{lem:entropy-accumulation} therefore give the bound with
+exponent \(0.26\) before the next exit. The next initial gradient
+bound is \(K_0\leq E_j^{-1/100}\leq1\), so all further thresholds
+for Lemma~\ref{lem:entropy-phase-exit} are dimensional. Increasing
+the first target once ensures these conclusions in every one of a
+prescribed finite number of phases.
+
+Use \(h_j\), the starting metric, as the reference metric in phase
+\(j+1\). Let \(R_{j+1}\) bound the pairwise old-support distance in
+the preceding argument, including both support enlargement radii and
+all separation buffers. The number of sweeps gives
+\[
+ R_{j+1}\leq C_n(1+\log^3 E_{j+1})
+ \leq C_n E_j^{3/1000}.
+\]
+Consequently
+\begin{equation}\label{eq:entropy-overlap-radius}
+ R_{j+1}E_j^{-1/100}\leq C_n E_j^{-7/1000}\longrightarrow0.
+\end{equation}
+For large \(E_j\), the overlap condition therefore holds with
+\(K=E_j^{-1/100}\). For every relevant pair the preceding estimate
+becomes
+\[
+ \|F_j(x)-F_j(y)\|_{\ell^2}
+ \leq E_j^{-1/100}\dist_{h_j}(x,y)<\sqrt2.
+\]
+This uses only the starting metric of the phase, without a comparison
+constant between that metric and \(g\).
+
+It follows that every separation graph needed in phase \(j+1\) is
+contained in the square \(\mathcal G_j^2\), after identifying labels
+with their parents at the start of the phase. At exit there are two
+copies of each parent. Joining two such copies whenever their parents
+are equal or at distance at most two gives a fixed graph
+\(\mathcal G_{j+1}\) containing all new cooccurrences. If
+\(\Delta_j\) is the degree of \(\mathcal G_j\), this construction gives
+\begin{equation}\label{eq:entropy-graph-recurrence}
+ \Delta_{j+1}+1\leq2(\Delta_j+1)^2.
+\end{equation}
+It also doubles the number of label orbits. The preliminary coloring
+construction applies to these graphs before each sweep.
+All analytic support and chart bounds in the resulting finite
+construction remain finite for the input, even though they need
+not satisfy a useful quantitative bound. They do not enter
+\eqref{eq:entropy-graph-recurrence}.
+
+After \(J\) phases, apply the coloring argument to
+\(\mathcal G_J^2\), whose maximum degree is at most \(\Delta_J^2\).
+Take \(q\) to be the maximum of its degree plus one, the number of
+label orbits, and \(n+1\). Equations \eqref{eq:entropy-first-graph} and
+\eqref{eq:entropy-graph-recurrence} imply
+\begin{equation}\label{eq:entropy-q-growth}
+ \log q\leq C_{\mathrm{inst},J,n}(1+\log E_1)^3.
+\end{equation}
+For fixed \(J,n\) the coefficient here is fixed before \(E_1\) is
+chosen. The graph \(\mathcal G_J\) and this coloring have exactly
+the properties asserted in the theorem.
+
+It remains to make the first target large enough for the already fixed
+\(J=H+2\). Let
+\(\mathcal B(q)\leq\exp^{[H]}(B(1+q)^A)\) be the prescribed
+elementary bound. Equation \eqref{eq:entropy-q-growth} implies
+\[
+ \mathcal B(q)\leq
+ \exp^{[H+1]}\bigl(C'_{\mathrm{inst},J,n,A,B}
+ (1+\log E_1)^3\bigr)
+\]
+for large \(E_1\). On the other hand, the recurrence
+\eqref{eq:entropy-phase-targets} gives, for each fixed \(J\geq2\),
+\begin{equation}\label{eq:entropy-tower-growth}
+ E_J\geq\exp^{[J-1]}\bigl(c_J E_1^{1/1000}\bigr)
+\end{equation}
+for some \(c_J>0\) and all sufficiently large \(E_1\).
+One proves this by induction: raising a fixed-height exponential
+tower with argument \(c t\) to the power \(1/1000\) still dominates
+a tower of the same height with argument \(c't\), for some \(c'>0\)
+and all large \(t\); the next application of \(\exp\) adds a level.
+For height one this follows from
+\((e^{ct})^{1/1000}=e^{ct/1000}\); for higher heights, reducing the
+bottom argument by any fixed factor absorbs the fixed outer
+multiplicative factor. This proves the induction.
+
+With the already chosen \(J=H+2\), the tower heights on the
+two sides are now the same, and
+\[
+ c_J E_1^{1/1000}>
+ C'_{\mathrm{inst},J,n,A,B}(1+\log E_1)^3
+\]
+for all sufficiently large \(E_1\). Thus \(E_J>\mathcal B(q)\).
+We may choose the first target still larger to have
+\(E_J>E_{\min}\), to satisfy the initial gradient threshold involving
+\(K_0\), and to meet all dimensional thresholds used above.
+These choices affect only the size of the final parameter and the
+finite uniform bounds of the output objects.
+
+Finally we account for the scalar constant. The bootstrap charges
+at most \(2C_b\epsilon_*\), independently of its input-dependent
+number of sweeps. The initial logarithmic normalization has a
+dimensional cost because \(\epsilon_0\in[\epsilon_*,2\epsilon_*]\).
+All entropy sweep losses together are bounded by
+\(C_n\sum l^2\leq C_n\). Each exit charges at most \(C_n\), and their
+number \(J=H+2\) depends only on the prescribed elementary function
+and \(n\). The input gradient bound \(K_0\), the initial small
+\(\epsilon_{\mathrm{in}}\), the compact support radii, and all metric
+and elliptic bounds occur only in the choices of the first target,
+graphs, and arbitrarily small analytic errors. None contributes to
+the scalar constant. Discard the final nonnegative surplus \(P'\);
+the remaining inequality is \eqref{eq:entropy-contract} with
+\(C=C(n,\mathcal B)\).
+
+The density \(\rho\) has been unchanged since the bootstrap, so
+\(\rho\leq dV_g\). The volume logarithm starts at zero and only
+increases by \(W_p\geq0\) and \(nu\geq0\). Finitely many sweeps and
+exits preserve the asserted smoothness, uniformity, compact supports,
+and finite number of free label orbits. The transverse spaces were
+enlarged only by the probability products used for the colorings,
+and the sweep constructions are Borel on these spaces by
+Proposition~\ref{prop:sweep}. This proves all the conclusions.
+\end{proof}
+\section{A measured degree estimate}\label{sec:degree}
+
+This section bounds the degree of a map from a measured family of
+Riemannian manifolds to a cube. Its integrand combines the weighted scalar
+quantity with a separate allowance for each coordinate gradient. The conclusion involves
+ordinary Riemannian volume, integrated also in the transverse direction.
+The family need not consist of complete manifolds, and its degree need not
+be an integer. We first specify the control at the sides of the cube that
+makes this statement, and the integrations in its proof, well defined.
-\emph{Initial data.} Apply Lemma~\ref{lem:entropy-bootstrap}.
-Increasing a dimensional constant once, at the output
+\subsection{Measured bands and degree}
+
+We use the following model of a measured space of leaf manifolds. The
+underlying space $X$ is standard Borel and has a countable collection of
+flow boxes
+\[
+ \kappa_a:T_a\times Q_a\longrightarrow X,
+\]
+whose images cover $X$ and which are Borel isomorphisms onto their images.
+Here $Q_a$ is an open box
+in $\mathbb R^d$, $T_a$ is standard Borel, and $T_a$ carries a
+$\sigma$-finite measure $\mu_a$. The plaques $\kappa_a(\{t\}\times Q_a)$
+fit together as oriented smooth $d$-manifolds, each Hausdorff and second
+countable, and each plaque is open in its leaf. The overlap of two boxes is a countable union of pieces on
+which the change of transverse coordinate is a Borel bijection between
+Borel subsets, preserves the transverse measures, and the change of leaf
+coordinate is a smooth orientation-preserving diffeomorphism on open
+sets, depending Borel measurably on the transverse coordinate. All
+functions and tensor fields below are Borel in the boxes and smooth on
+plaques. This convention also applies to an open subset along the leaves,
+whose leaves are the connected components of that subset.
+
+The measures $d\mu_a(t)\,dV_h$ in the boxes define a single measured
+Riemannian volume, denoted simply by $dV_h$. The same convention, with
+leafwise area in place of volume, defines $dA_h$ on regular submanifolds.
+One may partition among countably many boxes to define either integral;
+the equality on overlaps follows from invariance of the transverse
+measures. From a plaque, all plaques on its leaf have a countable Borel
+enumeration, obtained by following finite strings of overlap maps. In
+particular the saturation of a transverse null set is still null in every
+transversal. We shall discard such saturations when an assertion holds on
+almost all plaques.
+
+For $t>0$ write $Q_t^d=(-t,t)^d$; in dimension zero this denotes one point.
+We write $|G|_\infty=\max_i|G_i|$, with the empty maximum interpreted as
+zero.
+
+\begin{definition}[Controlled measured band]\label{def:degree-band}
+Let $T>0$. A controlled measured $d$-band over $Q_T^d$ consists of a
+measured space of oriented $d$-manifolds as above, a leafwise metric $h$,
+a function $\phi$, and a map $G=(G_1,\ldots,G_d):X\to Q_T^d$.
+For every $0<t<T$ the part $\{|G|_\infty\le t\}$ has a countable flow-box
+cover with the following properties.
+\begin{enumerate}
+\item The boxes have one common positive size and fixed positive buffers:
+their smaller concentric boxes cover the indicated part. They can be
+chosen inside $\{|G|_\infty<t'\}$ for some $t<t'<T$. After rescaling
+coordinates we may, and do, call them unit boxes with buffers.
+\item In these boxes the coefficients of $h$ and $h^{-1}$ and all their
+leafwise coordinate derivatives are bounded by constants common to the
+cover. The same holds for all derivatives of $G$ and for all
+positive-order derivatives of $\phi$. No bound on the value of $\phi$ is
+required.
+\item The sum of the measures of the transversals of this cover is finite.
+There is a common bound on the number of plaques of the cover whose
+buffered boxes meet any one plaque of the cover.
+\end{enumerate}
+The cover and every bound in this definition may depend on $t$ and on the
+given band. The leaves are not assumed complete, and $G$ is not assumed
+proper.
+\end{definition}
+
+For $d\ge1$, the stated control implies the version of Stokes' theorem used here. To
+see this, choose a smooth cutoff in each buffered box which is one on its
+inner box. Dividing the cutoffs by their sum gives a leafwise smooth,
+Borel partition of unity on a slightly smaller inner band. The overlap
+bound makes the sum locally finite and bounds all its derivatives. The
+metric and inverse-metric bounds give uniform bounds for these derivatives
+in geometric norms. Consequently, for a leafwise smooth $(d-1)$-form
+$\eta$ supported over a compact subcube, with $\eta$ and $d\eta$ uniformly
+bounded there, chartwise Stokes and Fubini give
+\begin{equation}\label{eq:degree-stokes}
+ \int_X d\eta=0.
+\end{equation}
+Indeed, each cutoff multiple of $\eta$ is compactly supported in its
+plaque, and the sum of the absolute integrals is bounded by a constant
+times the sum of the transverse measures. The terms differentiating the
+partition cancel because its sum is one. The same argument gives measured
+Stokes for leafwise currents with uniform local mass and boundary-mass
+bounds in those boxes: apply the current identity in each plaque to the
+cutoff multiple of a test form and then sum. This includes test forms
+whose pullbacks have noncompact support in a leaf. We will always verify
+the indicated local bounds before using current Stokes.
+
+If $\omega$ is a smooth, compactly supported $d$-form on $Q_T^d$ with
+$\int_{Q_T^d}\omega=1$, define
+\begin{equation}\label{eq:degree-definition}
+ \mathcal D_X(G)=\int_X G^*\omega.
+\end{equation}
+The integral is absolute: its support lies in an inner band, the
+derivatives of $G$ there are bounded, and that band has finite measured
+volume. For $d\ge1$, a compactly supported top form on a cube of integral
+zero has a compactly supported primitive. Applying \eqref{eq:degree-stokes} to the
+pullback of this primitive shows that \eqref{eq:degree-definition} is
+independent of $\omega$. This is the \emph{measured degree}, with the
+usual signed orientation convention. For $d=0$ it means the measured
+signed count of the oriented points. We abbreviate it to $\mathcal D$
+when the band and map are understood.
+
+\begin{theorem}[Measured degree estimate]\label{thm:degree}
+There are absolute constants $C_0,C_1>0$ with the following property.
+Let $q\ge2$, let $d$ be an integer with $0\le d\le q$, and let $\tau>0$.
+For $0\le r\le d$ set
+\begin{equation}\label{eq:degree-widths}
+ T_r=2\tau+\frac{r\tau}{q},\qquad
+ b_r=\tau\left(1+\frac1{2q}+\frac r{4q^2}\right).
+\end{equation}
+Suppose $(X,h,\phi,G)$ is a controlled measured $d$-band over $Q_{T_d}^d$
+in the sense of Definition~\ref{def:degree-band}. Let $B\ge1$ and
+$L_1,\ldots,L_d>0$. Suppose throughout $X$ that
+\begin{equation}\label{eq:degree-horizon-bound}
+ L_i|dG_i|_h\le B-\frac dq\qquad(1\le i\le d),
+\end{equation}
+and suppose that
+\begin{equation}\label{eq:degree-horizon}
+ \frac{C_0(1+B)^{2/3}}{L_i}<\frac{\tau}{10q}
+ \qquad(1\le i\le d).
+\end{equation}
+Define $H_i(G)=2(L_i|dG_i|_h-1)_+$. Then
+\begin{equation}\label{eq:degree-estimate}
+ |\mathcal D_X(G)|(2\tau)^d
+ \le \frac{\exp(C_1d/q)}{\prod_{i=1}^dL_i}
+ \int_{\{|G|_\infty<b_d\}}
+ \exp\left(\frac{-\wscal(h,\phi)+\sum_{i=1}^dH_i(G)}2\right)dV_h.
+\end{equation}
+For $d=0$ the product is one, the sum is zero, and
+$\wscal(h,\phi)=0$; the assertion is the bound of a signed count by the
+unsigned count.
+\end{theorem}
+
+The proof removes one coordinate at a time. For $r\ge1$, the gaps
+$T_r-T_{r-1}=\tau/q$ and $b_r-b_{r-1}=\tau/(4q^2)$ supply, respectively,
+chart buffers and room for the residual value error when locating the
+integration core, while the permitted scaled gradient of each retained
+coordinate increases by $1/q$. A family of boundaries
+ordered by a real parameter supplies the factor $2\tau$ on the left.
+Its normal speed supplies the coarea factor on the right. In dimensions
+where the boundaries can be singular we will construct a regular band
+of the same degree; a scalar gain will pay exactly for the volume of the
+graphs used to construct it. The following formulation separates that
+construction from the final induction.
+
+\begin{lemma}[One step of measured descent]\label{lem:degree-descent}
+Assume the hypotheses of Theorem~\ref{thm:degree} with $d\ge1$, put
+$p=d-1$, and put
+\[
+ \Theta=\exp\left(\frac{-\wscal(h,\phi)+\sum_{i=1}^dH_i(G)}2\right).
+\]
+There is a Borel family of oriented regular hypersurface portions
+$\Sigma_s$ in the leaves, $-\tau<s<\tau$, and, for almost every $s$, a
+positive function $v_s$ on $\Sigma_s$, smooth along it and Borel in its
+charts, with the following properties. Transverse null subsets and their
+saturations may be removed for each such $s$.
+\begin{enumerate}
+\item For every nonnegative Borel function $F$ supported in
+$\{|G|_\infty<b_d\}$,
+\begin{equation}\label{eq:degree-coarea}
+ \int_{-\tau}^{\tau}\!ds\int_{\Sigma_s}v_sF\,dA_h
+ \le\int_XF\,dV_h.
+\end{equation}
+\item For each such $s$ there is a controlled measured $p$-band
+$(X_s^+,h_s^+,\phi_s^+,J_s^+)$ over $Q_{T_p}^p$ of degree
+$\mathcal D_X(G)$. Its coordinates are indexed by $j=2,\ldots,d$ and
+satisfy
+\[
+ L_j|dJ_{s,j}^+|_{h_s^+}\le B-\frac pq.
+\]
+With $H_j^+=2(L_j|dJ_{s,j}^+|_{h_s^+}-1)_+$, there is an absolute constant
+$C_*>0$ such that
+\begin{equation}\label{eq:degree-child-integral}
+ \begin{split}
+ &\int_{\{|J_s^+|_\infty<b_p\}}
+ \exp\left(\frac{-\wscal(h_s^+,\phi_s^+)+\sum_{j=2}^dH_j^+}{2}\right)
+ dV_{h_s^+}\\
+ &\hspace{25mm}\le \frac{e^{C_*/q}}{L_1}
+ \int_{\Sigma_s\cap\{|G|_\infty<b_d\}}v_s\Theta\,dA_h.
+ \end{split}
+\end{equation}
+In dimension $p=0$ the left side is the measured unsigned count.
+\end{enumerate}
+The child in the second assertion may be chosen separately for each
+typical $s$; no measurability of those choices in $s$ is asserted or needed.
+\end{lemma}
+
+\subsection{An ordered family of boundaries}
+
+We prove Lemma~\ref{lem:degree-descent} in this and the next two
+subsections. Fix intermediate widths
+\begin{equation}\label{eq:degree-intermediate-widths}
+ T_p<t_1<t_2<t_3<t_4<T_d.
+\end{equation}
+Whenever a chart buffer is needed, choose further intermediate widths
+between these. The width $t_4$ controls confinement and the parent
+charts, $t_3$ contains the boundary portions to be used, $t_2$ will bound
+an auxiliary complete region, and $t_1$ will bound the child. Such
+choices can depend on the given band.
+
+Choose once and for all a smooth even function $\gamma$ with
+$0\le\gamma(y)\le y^2$ and with $\gamma(y)$ bounded above and below by
+positive absolute multiples of $|y|^{3/2}$ for $|y|\ge1$. Let $P_0$ be the
+solution about zero of
+\begin{equation}\label{eq:degree-pressure}
+ P_0(0)=0,\qquad P_0'=L_1\psi(P_0),\qquad
+ \psi(y)=1+\frac{\gamma(y)}{8(1+B)}.
+\end{equation}
+It is odd and increasing and has poles at $\pm b$, where
+\[
+ b=\frac1{L_1}\int_0^\infty\frac{dy}{\psi(y)}
+ \le\frac{C(1+B)^{2/3}}{L_1}.
+\]
+For the inequality, split the integral at $(1+B)^{2/3}$ and use the
+$|y|^{3/2}$ bound on the tail. Choose the absolute $C_0$ in
+\eqref{eq:degree-horizon} at least this $C$. Thus
+\begin{equation}\label{eq:degree-pole-width}
+ b<\frac{\tau}{10q}.
+\end{equation}
+
+\paragraph{The local slice calculation.}
+We first record the analytic purpose of the pressure law. Let $\Sigma$
+be a smooth cooriented local hypersurface, with unit normal $\nu$, and set
+\[
+ \begin{gathered}
+ D=H+\partial_\nu\phi,\qquad
+ H=\operatorname{tr}\mathrm{II},\qquad
+ \mathrm{II}(Y,Z)=\langle\nabla_Y\nu,Z\rangle,\\
+ \mathcal P=|\mathrm{II}|^2+
+ (\operatorname{Ric}_h-\nabla^2\phi)(\nu,\nu).
+ \end{gathered}
+\]
+Put $\mathfrak a=L_1\psi(D)>0$. When $D=P_0(s-G_1)$, the derivatives
+of the raw pressure are $\partial_sP_0(s-G_1)=\mathfrak a$ and
+$d_xP_0(s-G_1)=-\mathfrak a\,dG_1$. The following calculation uses
+only a positive smooth function $v$ satisfying the local equation
+\begin{equation}\label{eq:degree-local-lapse}
+ -\bigl(\Delta_{\Sigma,\phi}+\mathcal P
+ -\mathfrak a\,dG_1(\nu)\bigr)v=\mathfrak a.
+\end{equation}
+Here $\Delta_{\Sigma,\phi}$ is weighted by
+$e^{\phi|_\Sigma}dA_h$, and slice norms use the induced metric. The
+ordered construction below will produce this equation on almost every
+regular slice.
+
+Put $\phi_\Sigma=\phi|_\Sigma+\ln v$ and
+$S_\Sigma=\wscal(h|_{T\Sigma},\phi_\Sigma)$. Gauss' equation and the
+normal/tangential decomposition of derivatives of $\phi$ give
+\[
+ \wscal(h|_{T\Sigma},\phi|_\Sigma)
+ =\wscal(h,\phi)+D^2-2\mathcal P+|\mathrm{II}|^2.
+\]
+Adding $\ln v$ to the weight changes this by
+$-2\Delta_{\Sigma,\phi}v/v+|d\ln v|^2$. Substitution of
+\eqref{eq:degree-local-lapse} yields the exact identity
+\begin{equation}\label{eq:degree-slice-identity}
+ S_\Sigma=\wscal(h,\phi)+D^2-2\mathfrak a\,dG_1(\nu)
+ +|\mathrm{II}|^2+\frac{2\mathfrak a}{v}+|d\ln v|^2.
+\end{equation}
+This is also the hypersurface calculation behind
+Lemma~\ref{lem:graph-identity}; it is valid for $p=0$ with left side zero.
+
+Set $t=L_1|dG_1|_h\le B$ and $x=v/L_1$. Since $\gamma(D)\le D^2$,
+\[
+ -2\mathfrak a\,dG_1(\nu)
+ \ge-2t\left(1+\frac{\gamma(D)}{8(1+B)}\right),
+ \qquad \frac{t\gamma(D)}{4(1+B)}\le\frac{D^2}{4}.
+\]
+We may discard the remaining nonnegative part of $D^2$, and use
+$2\mathfrak a/v\ge2/x$. Finally,
+$t\le1+(t-1)_+$ and $x^{-1}+\ln x\ge1$ for $x>0$. Thus
+\eqref{eq:degree-slice-identity} implies
+\begin{equation}\label{eq:degree-slice-bound}
+ S_\Sigma\ge\wscal(h,\phi)-H_1(G)-2\ln(v/L_1)+|d\ln v|^2.
+\end{equation}
+The square term will be used later when the weight is changed near
+singular points. For $j=2,\ldots,d$, put
+$H_j^\Sigma=2(L_j|d_\Sigma G_j|-1)_+$. Tangential restriction gives
+$H_j^\Sigma\le H_j(G)$, so the full pointwise density comparison is
+\begin{equation}\label{eq:degree-raw-integrand}
+ \exp\left(\frac{-S_\Sigma+\sum_{j=2}^dH_j^\Sigma}{2}\right)
+ \le\frac v{L_1}e^{-|d\ln v|^2/2}\Theta
+ \le\frac v{L_1}\Theta.
+\end{equation}
+For $p=0$ the sum and slice gradient are zero, and the left side is one.
+The remaining work supplies the slices and speeds, and verifies the
+controlled-band and degree conditions needed to use this comparison.
+
+For a small $\eta>0$, prolong $P_0$ smoothly and strictly increasingly
+past the points $\pm(b-\eta)$, keeping it unchanged on
+$[-b+\eta,b-\eta]$. Denote the resulting smooth function by
+$\widehat P$ and set
+\[
+ \mathfrak h_s(x)=\widehat P(s-G_1(x)),\qquad -\tau<s<\tau.
+\]
+We will choose $\eta$ sufficiently small for the given band and then keep
+it fixed. On every boundary portion used below, $\widehat P$ will agree
+with $P_0$ and hence obey \eqref{eq:degree-pressure}.
+
+\paragraph{Least minimizers, including on noncompact leaves.}
+Write $P_\phi(E;U)=\int_{\partial^*E\cap U}e^\phi\,dA_h$ for the weighted
+perimeter of a set of locally finite perimeter. We use local differences
+of the functional
+\begin{equation}\label{eq:degree-functional}
+ P_\phi(E)-\int_E\mathfrak h_s e^\phi\,dV_h.
+\end{equation}
+This is a weighted $\mu$-bubble functional. For soap bubbles in Gromov's
+terminology, see \cite[Section~5$\frac56$]{Gromov1996}; for the weighted
+functional and its local first- and second-variation formulas, see also
+\cite[Section~3, Equation~(1), and Lemmas~13--14]{ChodoshLi2024}.
+Neither term is required to be finite on the whole leaf. In particular,
+this construction uses no global weighted-volume bound.
+
+The output will be a canonical nested family $E_s$ on each open leaf,
+independent of the exhaustion and jointly Borel in the plaque data and
+$s$. Each $E_s$ will minimize these local differences for compact
+variations. The local comparison argument also supplies the perimeter
+compactness needed later to turn convergence of sets into graphical
+convergence at regular limit points. We establish these properties in
+turn, before confining the boundaries to the raw-pressure region.
+
+\emph{Compact domains and leastness.}
+For a compact subset $K$ of a leaf, minimize \eqref{eq:degree-functional}
+over sets which vanish off $K$, using their full perimeter in the leaf.
+The empty set is a competitor. On a neighborhood of $K$ the density is
+smooth and positive and the forcing is bounded. A bound on the functional
+therefore gives a perimeter bound, and BV compactness and lower
+semicontinuity give a minimizer; see, for example,
+\cite[Theorem~12.26]{Maggi2012} for the compactness theorem. Among the
+minimizers choose one of minimum
+ordinary volume. The weighted perimeter satisfies
+\[
+ P_\phi(E\cap F)+P_\phi(E\cup F)\le P_\phi(E)+P_\phi(F),
+\]
+and the volume term for a fixed forcing is additive under union and
+intersection. Thus the intersection of any two minimizers is again a
+minimizer. Minimum volume forces the selected minimizer to be contained,
+up to null sets, in every other one. It is consequently unique as an
+$L^1$ set; call it the least minimizer in $K$.
+
+\emph{Order and exhaustion.}
+The same inequality proves both monotonicities we need. If $K\subset K'$
+and two forcings satisfy $\mathfrak h\le\mathfrak h'$, a minimizer for
+$(K,\mathfrak h)$ and one for $(K',\mathfrak h')$ obey
+\[
+ \mathcal F_{\mathfrak h}(E\cap F)+\mathcal F_{\mathfrak h'}(E\cup F)
+ \le \mathcal F_{\mathfrak h}(E)+\mathcal F_{\mathfrak h'}(F),
+\]
+where $\mathcal F$ denotes \eqref{eq:degree-functional}; the inequality for
+the volume terms follows on $E\setminus F$ from
+$\mathfrak h\le\mathfrak h'$. Minimality in
+the two domains makes both inequalities equalities, and leastness in $K$
+implies $E\subset F$. Exhaust a leaf by increasing finite unions of
+compact coordinate subboxes whose interiors exhaust it. The least
+minimizers increase with this exhaustion. Their union, interpreted in
+$L^1_{\mathrm{loc}}$, is independent of the exhaustion, since each compact
+subset of either exhaustion is eventually contained in one of the other.
+This defines $E_s$, and the forcing monotonicity makes $E_s\subset E_t$
+for $s<t$.
+
+\emph{Local comparison.}
+The limit must still minimize for compact variations, since all later
+regularity and compactness arguments use that property.
+Ball replacement in a relatively compact chart gives uniform local
+perimeter bounds for the exhaustion minimizers: choose a radius with
+controlled slicing area, replace the set inside that ball by the full or
+empty ball, and bound the forcing term by its local supremum times the
+ball volume. Thus the increasing sets converge locally in BV weakly and
+in $L^1$. To pass a comparison to the limit, surround the compact region
+to be changed by a thin outer belt on which the proposed competitor
+agrees with $E_s$. Choose a Lipschitz coordinate level function across
+the belt. Coarea bounds the integral, over its levels, of the trace
+mismatch between an exhaustion minimizer and $E_s$ by their $L^1$
+mismatch in the belt. One may therefore splice the proposed competitor
+inside a level to the exhaustion minimizer outside it so that the extra
+trace area tends to zero. The levels can also be chosen with zero
+limiting perimeter on their boundaries, since this excludes only a null
+set of levels. Minimality before passage to the limit, convergence of
+the bounded forcing integrals, and lower semicontinuity give the compact
+comparison for $E_s$. The same argument applies under smooth local
+convergence of the metric, density, and forcing coefficients. Below we
+apply it with the unchanged limit as competitor to prove convergence of
+the local perimeter measures. Multiplying a density by a positive
+constant has no effect on any of these comparisons.
+
+\emph{Borel dependence.}
+We explain why all these choices are Borel, including jointly in $s$.
+For this coding, subdivide the original atlas into bounded coordinate
+boxes with compact closure in the older boxes, and affinely normalize
+each to one fixed cube $Q=(-1,1)^d$. Fix once a countable list
+$(R_\ell)_{\ell\ge1}$ of all open rational boxes with closure in $Q$.
+Starting at a plaque, enumerate its accessible normalized plaque
+occurrences Borel measurably in slots $j\ge1$, using dummy slots when
+necessary. Use the fixed Polish space
+\[
+ \mathcal Z=\prod_{j,\ell\ge1}L^1(R_\ell),
+\]
+with Lebesgue $L^1$ norms and the product topology. A set is coded by its
+indicator on $R_\ell$ in occurrence $j$; dummy entries are zero.
+Compatibility is tested on compact exhaustions of all overlap domains,
+including repeated occurrences. The closures of the first finitely
+many accessible $R_\ell$ in a fixed enumeration of the pairs $(j,\ell)$
+give the compact exhaustion domains. The conditions that the code is an
+indicator, agrees on overlaps, and vanishes off the specified domain are
+Borel, and for fixed plaque data are closed in $\mathcal Z$.
+
+Integrals against chartwise Borel smooth densities are Borel by Fubini.
+Perimeter is Borel as well: its supremum over divergence integrals may
+be taken over fixed countable families of rational smooth test fields in
+the $R_\ell$, and finite sums of them. Partitions of unity in a leaf
+show density among all compactly supported tests. Norm constraints are
+checked on countably many dense coordinate points; overlap maps and a
+partition by the first covering chart make the integrals Borel.
+
+Fix one plaque-data fiber and one compact exhaustion domain $K$. A
+functional sublevel for sets supported in $K$ bounds their weighted
+perimeter, since the forcing integral is uniformly bounded on this
+fixed compact set. In every active $R_\ell$ the metric and density
+have a positive lower bound on its closure, so this gives a Euclidean BV
+bound there; the indicator's $L^1(R_\ell)$ norm is bounded by $|R_\ell|$.
+Diagonal BV compactness gives convergence in $\mathcal Z$ along a
+subsequence. The fiberwise closed indicator, overlap, and support
+conditions persist. The volume term is continuous, since $K$ is
+covered by finitely many compact subboxes, and perimeter is lower
+semicontinuous. Each sublevel is therefore compact in this fixed Polish
+product. These compactness bounds may depend on the fiber and on
+$R_\ell$; the projection theorem requires compact fibers, not a common
+bound across plaque data. Strict sublevels are countable unions of such
+compact sections. The Borel projection theorem for sets with
+$\sigma$-compact sections therefore makes the minimum value a Borel
+function of the chart data. Apply it again to minimize volume in the
+compact set of minimizers. The selected section is a singleton in the
+$L^1$ coding, by leastness, so the Borel singleton-section theorem gives
+the selected minimizer as a Borel function. These are standard forms of
+the Arsenin--Kunugui and Lusin--Souslin theorems; see
+\cite[Theorem~18.18 and Corollary~15.2]{Kechris1995}. The exhaustion limits agree
+for different starting plaques by exhaustion independence. They are therefore Borel
+on the measured space. We may use the density-one representative of
+$E_s$; the support of its boundary measure is also Borel, by testing its
+mass on a countable basis of coordinate balls.
+
+\paragraph{Confinement.}
+For sufficiently small $\eta$, uniformly for $-\tau<s<\tau$, the boundary
+support in $\{|G|_\infty\le t_4\}$ satisfies
+\begin{equation}\label{eq:degree-confinement}
+\operatorname{supp}\partial[E_s]\cap\{|G|_\infty\le t_4\}
+ \subset\{|s-G_1|<b-\eta\}.
+\end{equation}
+In this inner band the side $G_1\le s-b+\eta$ is full and the side
+$G_1\ge s+b-\eta$ is empty, up to null sets. To prove this uniform claim,
+the ODE at the positive pole gives
+\[
+ b-x=\frac1{L_1}\int_{P_0(x)}^\infty\frac{dy}{\psi(y)}
+ \asymp_{\mathrm{inst}}P_0(x)^{-1/2}
+ \quad(x\uparrow b).
+\]
+Thus at distance comparable to $\eta$ from a pole the absolute forcing is
+at least $c_{\mathrm{inst}}\eta^{-2}$. The prolongation is monotone, so
+this remains a lower bound beyond that position. By the uniform
+derivative bound on $G_1$, a chart ball of radius
+$r=c'_{\mathrm{inst}}\eta$ centered at a point with $s-G_1\ge b-\eta$
+has forcing at least $H=c''_{\mathrm{inst}}\eta^{-2}$ throughout it, if
+$c'_{\mathrm{inst}}$ is small. The balls are taken in a buffered atlas
+for a slightly larger band than $t_4$.
+
+Here is the phase-exclusion estimate in such a ball. Normalize the
+weighted density by its value at the center; the derivative bounds for
+$\phi$ make this density uniformly comparable to coordinate volume in
+the ball. Let $V(t)$ be the normalized volume of the complementary
+phase in a concentric coordinate ball $B_t$. Comparison with the set
+which fills $B_t$, and perimeter slicing, give for almost every $t<r$
+\[
+ P_{\mathrm{norm}}(E_s;B_t)+H V(t)\le C V'(t).
+\]
+The chart isoperimetric inequality applied to that complementary phase,
+extended by zero outside $B_t$, gives, when $V(t)>0$,
+\[
+ V(t)^{(d-1)/d}\le C\bigl(P_{\mathrm{norm}}(E_s;B_t)+V'(t)\bigr).
+\]
+If the complementary phase occurs in $B_{r/4}$, the two inequalities first
+give $V'\ge cV^{(d-1)/d}$ and hence $V(r/2)\ge cr^d$. They also give
+$V'\ge cHV$ on $[r/2,r]$. It follows that
+$V(r)\ge cr^d\exp(cHr)$, contrary to $V(r)\le Cr^d$ when $Hr$ is large.
+This argument includes $d=1$, where the first differential inequality
+has exponent zero. Since $Hr\asymp_{\mathrm{inst}}\eta^{-1}$, small
+$\eta$ excludes the complementary phase in the smaller ball. Negative
+forcing gives the empty phase by the same argument. Applying these
+balls beginning at $|s-G_1|=b-\eta$, with a small fixed extra margin in
+the forcing estimate, proves \eqref{eq:degree-confinement}. In particular
+the unprolonged Equation~\eqref{eq:degree-pressure} holds on all the
+boundary portions used in the proof.
+
+\paragraph{The local regularity and compactness inputs.}
+We record precisely the local geometric measure theory used from now on.
+Set $\Sigma_s^{\mathrm{all}}=\operatorname{supp}\partial[E_s]$. For
+$d\ge2$, its regular portion is a smooth multiplicity-one hypersurface.
+The remaining set is closed in $\Sigma_s^{\mathrm{all}}$ and has Hausdorff
+dimension at most
+\begin{equation}\label{eq:degree-singular-dimension}
+ p-7.
+\end{equation}
+It is empty if this number is negative. These conclusions hold in each
+relatively compact chart. To place our functional under the standard
+codimension-one theorem, normalize $\phi$ by a constant value $\phi_0$
+in the chart and use the smooth metric
+\[
+ \widehat h=e^{2(\phi-\phi_0)/p}h.
+\]
+Its area is the normalized weighted area. Its volume forcing is
+$\mathfrak h_s e^{-(\phi-\phi_0)/p}$, bounded smoothly in a smaller chart.
+Minimality implies that its perimeter is minimizing up to a fixed
+multiple of the volume of the symmetric difference for compact
+variations. The regularity theorem for such almost-minimizing boundaries
+gives $C^{1,\alpha}$ regularity off a closed set of dimension
+\eqref{eq:degree-singular-dimension}; see
+\cite[Theorem A.1 and Remark A.3]{Eichmair2009}. The weak prescribed
+mean-curvature equation on this regular portion and elliptic regularity
+then make it smooth. The cited almost-minimizing formulation uses
+integral-current variations. It applies to these set variations as
+follows. Add a compactly supported ambient-dimensional integral current
+to $[E_s]$ and truncate its integer BV multiplicity to the interval
+$[0,1]$. BV truncation does not increase boundary mass, and the volume
+where the resulting set changes is at most the mass of the added current.
+The set comparison therefore implies that formulation as well.
+
+\emph{Local compactness claim.}
+Suppose $d\ge2$ and $E_j$ minimize the absorbed-perimeter functionals
+for all compact variations in a common buffered normalized outer chart.
+Assume that their absorbed metrics and converted volume forcings
+converge smoothly on smaller boxes and that $E_j\to E$ in
+$L^1_{\mathrm{loc}}$. Then $E$ is locally minimizing, the normalized
+perimeter measures and the associated boundary varifolds converge
+weakly locally to those of $E$, and the supports converge locally.
+Near every regular point of the limiting boundary, the boundaries are
+single smooth graphs converging in $C^\infty$ on a cylinder which may
+depend on that point. In particular, for every relatively compact
+continuity subdomain $U$, meaning that the limiting perimeter of
+$\partial U$ is zero, the perimeter masses on $U$ converge. For the
+minimizing boundaries considered here, every sequence of controlled
+plaques from the fixed band and fixed forcing construction has a
+subsequence satisfying these coefficient and set convergence hypotheses
+after normalization.
+
+Here is the proof, including the local perimeter assertion. In the
+common normalized coordinates subtract the value of $\phi$ at the chart
+center. The common bounds on all derivatives, and the metric and
+inverse-metric bounds, make the normalized metric, weight, and forcing
+coefficients precompact in $C^\infty$ on every smaller fixed box, by
+diagonal Arzelà --Ascoli. Pass to such a convergent subsequence; in
+particular the absorbed metrics $\widehat h_j$ converge smoothly to a
+positive metric $\widehat h_\infty$. Ball comparison gives the local
+perimeter bounds used above, so BV compactness gives a further
+subsequence converging in $L^1_{\mathrm{loc}}$ to a set $E$. The belt
+comparison with any compact variation, as proved above and now with the
+smoothly converging coefficients, shows that $E$ is locally minimizing.
+
+Write $\mu_j=P_{\widehat h_j}(E_j;\,\cdot\,)$ and
+$\mu_\infty=P_{\widehat h_\infty}(E;\,\cdot\,)$. To prove convergence
+of these measures, begin with a relatively compact coordinate domain
+$V$ from a convergence-determining family, chosen so that
+$\mu_\infty(\partial V)=0$. Surround it by a thin outer belt with level
+domains $V_t$ contained in an outer enlargement $V_\epsilon$. Coarea
+chooses a level $t_j$ at which the trace mismatch between $E_j$ and $E$
+tends to zero. Splice the unchanged $E$ inside $V_{t_j}$ to $E_j$
+outside. The common coefficient bounds make the extra perimeter at the
+splice tend to zero, and the bounded converted forcing times the
+$L^1$ mismatch tends to zero as well. Minimality of $E_j$ gives
+\[
+ \limsup_{j\to\infty}\mu_j(V)\le\mu_\infty(V_\epsilon).
+\]
+Here the perimeter of the fixed set $E$ in the varying metric converges
+by uniform convergence of the metric integrands on unit conormals.
+Shrinking the belt makes the right side tend to $\mu_\infty(V)$, because
+$\mu_\infty(\partial V)=0$. Lower semicontinuity gives the reverse
+inequality. Such coordinate continuity domains determine weak local
+convergence of Radon measures; the local mass bounds give compactness,
+and the equality just proved identifies every subsequential limit.
+Consequently $\mu_j$ converge weakly locally to $\mu_\infty$, and their
+masses converge on every relatively compact continuity subdomain $U$.
+
+For precision in passing to varifolds, restrict to a relatively compact
+subdomain $U$ whose boundary has zero limit perimeter measure. Such
+continuity subdomains can exhaust the smaller box. To check weak
+convergence of the restrictions of $D\mathbf1_{E_j}$ as vector measures
+on $\overline U$, extend a continuous vector test there to the outer
+chart and multiply it by a cutoff supported in $U$ and equal to one away
+from a $\delta$-collar of $\partial U$. Local weak BV convergence
+handles this cutoff test. On the compact chart uniform ellipticity
+gives $|D\mathbf1_{E_j}|\le C\mu_j$. For a compact closed
+$\delta$-collar $F_\delta$, weak perimeter convergence and the closed-set
+inequality give
+$\limsup_j\mu_j(F_\delta)\le\mu_\infty(F_\delta)$, which tends to zero
+as $\delta\downarrow0$ because $\mu_\infty(\partial U)=0$. The same
+bound for the limiting vector measure lets the collar shrink and proves
+the restriction convergence. The perimeter masses on $U$ also converge.
+In coordinates the perimeter
+integrand for a metric $k$ is
+$\mathfrak p_k(x,z)=\sqrt{\det k(x)}\,|z|_{k^{-1}(x)}$.
+Uniform convergence of this integrand on unit conormals and the local
+mass bound replace $\widehat h_j$ by $\widehat h_\infty$ in that mass
+convergence. This is convergence for one continuous positively
+one-homogeneous integrand with Reshetnyak's strict convexity property:
+equality in its triangle inequality occurs only for vectors on a common
+nonnegative ray. Reshetnyak's continuity theorem
+\cite[Section~4, Theorem~3]{Reshetnyak1968} consequently gives convergence
+for every continuous positively one-homogeneous integrand with linear
+growth. For a continuous compactly supported varifold test $a(x,P)$, use
+$\mathfrak p_{\widehat h_\infty}(x,z)a(x,\ker z)$ for $z\ne0$ and zero
+at $z=0$. This gives precisely the local varifold convergence; the
+uniform convergence of the metric integrands gives it also for the
+original varying metrics.
+
+To prove the support and graph assertions, the first variation in
+the metric $\widehat h$ is, by BV Gauss--Green, integration of the bounded
+volume forcing against the reduced-boundary conormal. It therefore has
+bounded mean curvature and no singular part. The monotonicity formula
+gives a common positive lower mass bound in a small ball about a support
+point, starting with the unit density at almost every reduced-boundary
+point. This lower bound rules out support points a fixed distance from
+the limit support. Conversely, every ball about a point of the limit
+support has positive limit mass, so weak convergence forces nearby
+support points. Near a fixed smooth multiplicity-one sheet of the
+limit, Allard's interior regularity theorem gives single graphs with a
+common $C^{1,\alpha_{\mathrm A}}$ bound for some
+$\alpha_{\mathrm A}\in(0,1)$ on a smaller cylinder \cite{Allard1972}.
+Compactness and identification of the already known varifold limit give
+$C^{1,\beta}$ graph convergence for every $0<\beta<\alpha_{\mathrm A}$.
+The prescribed equation, with the smoothly converging coefficients,
+then gives $C^\infty$ convergence on still smaller cylinders by
+interior elliptic estimates. The cylinder here may depend on the fixed
+regular point of the limit; uniform graph radii in later applications
+will follow from their separate compactness contradictions.
+
+The varying-metric use of these local theorems reduces to one smooth
+ambient metric as follows. Pass further in a smaller compact chart to a subsequence converging so
+rapidly in $C^\infty$ that its metric is realized on slices $2^{-j}$ of
+one extra coordinate, with the limiting metric on slice zero. Indeed,
+choose the error through order $j$ below $2^{-j^2}$ and interpolate by
+disjoint smooth interval cutoffs. The resulting product block-diagonal
+metric is smooth; its slices have bounded second forms. After smoothly
+extending from a still smaller box to a closed manifold, a smooth Nash
+embedding puts this situation in a single Euclidean ambient space
+\cite[Theorem~2, p.~59]{Nash1956}.
+The preceding monotonicity and Allard statements apply there, with the
+same local bounded-mean-curvature property. Applying this subsequence
+argument to a sequence on which local convergence failed proves the
+local support and graphical convergence asserted above.
+
+For $d=1$ the needed facts are elementary. A locally minimizing set has
+locally finitely many jumps by the perimeter bound. Two consecutive
+jumps cannot be arbitrarily close in a controlled chart, since one can
+replace the interval between them by the surrounding phase: delete it if
+full, and fill it if empty. This saves two weighted boundary costs
+and changes the forcing integral by a quantity tending to zero. Thus
+the jumps have uniform local separation. The same belt argument applies
+directly to normalized weighted perimeter in dimension one, without
+conformal absorption, and gives local perimeter convergence together
+with BV convergence. The separation then gives the corresponding
+convergence of these zero-dimensional regular boundaries. There are no
+singular points in this case.
+
+\subsection{The lapse, coarea, and the descended degree}
+
+We now extract a smooth normal speed from the ordered family. Write
+$\Sigma_s$ for the regular part of $\Sigma_s^{\mathrm{all}}$ in
+$\{|G|_\infty<t_3\}$. The boundary has its exterior coorientation and
+the induced orientation. Its singular set has zero $p$-dimensional area,
+so it never contributes to the integrals over the boundary current.
+
+\paragraph{Differentiating the ordered family.}
+On each leaf the sets $E_t$ converge to $E_s$ in $L^1_{\mathrm{loc}}$ as
+$t\to s$ except at countably many $s$. In fact, on a compact chart set
+$K$ the function $t\mapsto\int_K\mathbf1_{E_t}$ is monotone, and nestedness
+identifies its increments with the $L^1(K)$ distance between the
+indicators. It has at most countably many discontinuities. Use the
+countable chart exhaustion to obtain the assertion on the leaf.
+At such a continuity parameter, the local convergence result above makes
+$\Sigma_t$ converge smoothly and graphically near each regular point of
+$\Sigma_s$ as $t\to s$.
+
+In a smaller graph cylinder choose coordinates $(y,z)$ with the sets
+below the graphs, and write the graph heights as $z=r(t,y)$. They are
+monotone in $t$. They are strictly ordered wherever two such regular
+graphs are defined: if $r(t,\cdot)\ge r(s,\cdot)$ for $t>s$ and they touch,
+their tangent planes agree at contact. The comparison of the graph mean
+curvatures at that minimum of $r(t,\cdot)-r(s,\cdot)$ gives
+$D_t\le D_s$, whereas their equations give
+$D_t=\mathfrak h_t>\mathfrak h_s=D_s$ at the same point. This is
+impossible. We use the conventions for $D$ and $\mathrm{II}$ from the
+local slice calculation, with $\nu$ the exterior unit normal. The
+equation $D=\mathfrak h_s$ is the first variation of
+\eqref{eq:degree-functional} on a regular portion.
+
+Choose countably many possible graph cylinders in the chart enumerations,
+and in each choose a countable dense set of $y$-coordinates. At each of
+these coordinates a monotone height has a finite derivative for almost
+every parameter. The test is meaningful even outside an interval of
+graphical representation: use instead the monotone column function
+\[
+ t\longmapsto\int_{z_-}^{z_+}\mathbf1_{E_t}(y,z)\,dz,
+\]
+which equals $r(t,y)-z_-$ in the single-graph situation. Consequently,
+for almost every continuity parameter $s$, the nonnegative difference
+quotients
+\[
+ w_t(y)=\frac{r(t,y)-r(s,y)}{t-s},\qquad t\ne s,
+\]
+converge to the finite column-derivative values as $t\to s$ at all these
+dense test coordinates. In particular they are bounded there.
+
+In positive graph dimension the $w_t$ solve linear elliptic equations on
+each smaller cylinder. This follows by subtracting the two prescribed
+graph equations and integrating their linearizations between the two
+graphs. Smooth graphical convergence gives common smooth coefficient
+bounds and ellipticity; the explicit $t$-dependence of $\mathfrak h_t$ gives a
+smooth bounded source. Harnack's inequality with a bounded source,
+applied using any one of the bounded test values, bounds $w_t$ on a
+smaller patch. Interior elliptic estimates then give smoothly convergent
+subsequences there; these are the standard local Harnack and interior
+estimates for uniformly elliptic equations with bounded lower-order
+coefficients and source, as in \cite[Chapters~6 and~8]{GilbargTrudinger}.
+Every such limit takes the already determined
+derivative values at the dense test coordinates, so the limits agree.
+Thus the whole difference quotient has a smooth local limit. Its normal
+component is a smooth function $v\ge0$ on the regular portion. For
+$p=0$ the same conclusion follows from the ordinary almost-everywhere
+derivative of a monotone height.
+
+Differentiation of $D=\mathfrak h_s$ now identifies this normal speed.
+Use $\mathcal P$ from the local slice calculation and put
+$\mathfrak a=\partial_s\mathfrak h_s=L_1\psi(D)>0$; the last equality
+uses confinement, where the forcing agrees with the raw pressure.
+The normal variation formula, with the tangential derivative of
+$D-\mathfrak h_s=0$ cancelling, is
+\begin{equation}\label{eq:degree-lapse}
+ -\bigl(\Delta_{\Sigma,\phi}+\mathcal P+\partial_\nu\mathfrak h_s\bigr)v
+ =\mathfrak a.
+\end{equation}
+The convention for $\Delta_{\Sigma,\phi}$ is weighted by
+$e^{\phi|_\Sigma}dA_h$. The variation of $\partial_\nu\phi$ adds
+$v\nabla^2\phi(\nu,\nu)-\langle d\phi,dv\rangle_\Sigma$ to the ordinary
+mean-curvature variation, giving the signs in
+\eqref{eq:degree-lapse}. A nonnegative $v$ cannot vanish at an interior
+point: there the left side is nonpositive, contrary to $\mathfrak a>0$.
+Thus $v>0$. In graph dimension zero the differential terms vanish and
+the same equation rules out $v=0$.
+
+These assertions hold jointly in the measured charts for almost every
+$s$. Here are the measurable details needed for that conclusion.
+Single-graph regions with buffers can be recognized by countably many
+cylinder tests on the $L^1$ subgraph codes, with slope and derivative
+bounds and with room at the top and bottom. Their smooth graph
+representative is unique; the Borel singleton-section theorem applied to
+its smooth-function code makes it Borel. The continuity and
+differentiability tests above use countable sequences, by monotonicity.
+The resulting local derivatives and $v$ are Borel. Fubini on each
+$\sigma$-finite transversal now gives, for almost every $s$, a null set
+of exceptional plaques. There are countably many charts and overlap
+moves, so their saturations can be discarded simultaneously. In
+particular the family of regular portions and the function $v$ used in
+the integral below have the asserted Borel descriptions.
+Set $v=0$ on the exceptional parameters and plaques when writing an
+integral over the whole parameter interval.
+
+In a graph cylinder, normal speed times area has the expression
+\[
+ v\,dA_h=J_h(y,r(s,y))\,\partial_s r(s,y)\,dy,
+\]
+where $J_h(y,z)\,dy\,dz$ is ambient Riemannian volume in the coordinates.
+The one-dimensional change-of-variables inequality for a monotone
+function says that integration against its almost-everywhere derivative
+counts at most integration in $z$. A jump or singular part of the
+monotone function only removes mass from the former integral. Apply this
+to each column and then integrate in $y$ and the transversal. Partition
+the counted portions among the countably many graph cylinders so they
+are not counted twice within one parameter. They cannot give the same
+ambient point at two parameters, by the noncontact argument above.
+Therefore no ambient point is counted more than once on the right.
+Equivalently, the counted images can be taken Borel by the Borel
+one-to-one image theorem. Transverse invariance on overlaps gives
+\begin{equation}\label{eq:degree-coarea-proof}
+ \int_{-\tau}^{\tau}\!ds\int_{\Sigma_s}vF\,dA_h
+ \le\int_XF\,dV_h
+\end{equation}
+for every nonnegative Borel $F$ supported in $\{|G|_\infty<t_3\}$.
+Since $b_d<t_3$, this proves \eqref{eq:degree-coarea}.
+
+We will require one more consequence of \eqref{eq:degree-lapse}. On a
+family of regular graph patches with common buffers and graph estimates,
+all positive-order derivatives of $\ln v$ have uniform bounds on smaller
+patches. Indeed the coefficients in \eqref{eq:degree-lapse} and the
+derivatives of $\mathfrak a$ are uniformly bounded there, and
+$\mathfrak a$ has a uniform positive lower bound. Confinement provides
+the finite margin from the pressure poles needed for these bounds. If
+the value of $v$ could tend to zero at an interior point of such patches,
+Harnack's inequality with bounded source would bound $v$ on a smaller
+patch, and elliptic compactness would give a nonnegative limit with an
+interior zero satisfying \eqref{eq:degree-lapse}, an impossibility.
+Thus $v$ has a positive local lower bound. Harnack then bounds its local
+supremum by a constant times its value at an interior point, since the
+bounded source can be absorbed using that lower bound. Interior
+estimates give $|\nabla^r v|\le C_r v$ on still smaller patches, and
+differentiating $\ln v$ gives the claim. This does not bound the value
+of $\ln v$, which is not required by Definition~\ref{def:degree-band}.
+For $p=0$ the assertion about positive-order derivatives is empty.
+
+The local slice calculation now applies to the actual boundaries.
+Confinement gives $D=\mathfrak h_s=P_0(s-G_1)$ and hence
+$\partial_\nu\mathfrak h_s=-\mathfrak a\,dG_1(\nu)$ on every retained
+regular portion. Equation~\eqref{eq:degree-lapse} therefore becomes
+exactly \eqref{eq:degree-local-lapse}. With
+$\phi_\Sigma=\phi|_\Sigma+\ln v$, the scalar bound
+\eqref{eq:degree-slice-bound} and the pointwise comparison
+\eqref{eq:degree-raw-integrand} hold on these portions. The derivative
+bounds just proved supply the required local controls for this weight.
+
+\paragraph{Degree on the boundary.}
+Let $J=(G_2,\ldots,G_d)$. Fix a unit-integral one-form
+$a_-(z)\,dz$ supported in $(-1.9\tau,-1.6\tau)$ and one
+$a_+(z)\,dz$ supported in $(1.6\tau,1.9\tau)$. Let $\beta$ be a
+unit-integral smooth $p$-form supported in a sufficiently small cube
+about zero in the residual target; for $p=0$ let $\beta=1$. Set
+\[
+ \chi_{\mathrm{deg}}(z)=\int_{-\infty}^{z}(a_-(u)-a_+(u))\,du.
+\]
+This function is compactly supported in $(-1.9\tau,1.9\tau)$ and is one
+between the two supports. By \eqref{eq:degree-pole-width} and
+\eqref{eq:degree-confinement}, the low support is in $E_s$, the high
+support is outside $E_s$, and $\chi_{\mathrm{deg}}(G_1)=1$ on the confined
+boundary where the residual form is supported. Measured current Stokes therefore gives
+\begin{equation}\label{eq:degree-boundary-degree}
+ \begin{split}
+ \int_{\partial[E_s]} \chi_{\mathrm{deg}}(G_1)J^*\beta
+ &=\int_{E_s}(a_-(G_1)-a_+(G_1))\,dG_1\wedge J^*\beta\\
+ &=\int_Xa_-(G_1)\,dG_1\wedge J^*\beta
+ =\mathcal D_X(G).
+ \end{split}
+\end{equation}
+All forms here are supported in an inner band. The current $[E_s]$ has
+bounded local volume and its boundary has the uniform local perimeter
+bounds proved above, so this is precisely the current version of
+\eqref{eq:degree-stokes}. The singular set contributes no area, and
+the counted part of the boundary is in $\{|G|_\infty<t_3\}$.
+Consequently \eqref{eq:degree-boundary-degree} computes the degree by
+$J^*\beta$ on that confined regular branch, with the induced orientation.
+For $p=0$ it gives its measured signed count.
+
+If $2\le d\le7$, this regular branch already provides the child. To verify
+the required uniformity, fix a typical $s$. If graph radii and estimates
+could not be chosen uniformly about the boundary in an inner region of
+$\{|G|_\infty<t_3\}$, take offending points in buffered parent charts
+and pass to normalized chart limits. Local BV and varifold compactness
+gives a limiting boundary through the limiting point. There are no
+singular points because $p-7<0$, so smooth graphical convergence near
+that point supplies a common graph radius and all interior estimates,
+contradicting the choice of points. A fixed fine mesh of graph cylinders
+now gives at most a bounded number of patches in each parent plaque.
+Their transversals are the restricted parent transversals, counted with
+the patch index. Their total transverse measure and plaque overlap
+remain bounded. The graph estimates, together with the bounds for
+derivatives of $\ln v$, give the band controls for the induced metric and
+$\phi_\Sigma$.
+
+Restrict this branch to
+\[
+ |G|_\infty<t_1,\qquad |J|_\infty<T_p.
+\]
+The $G_1$ sides of the first restriction cannot be reached by confinement,
+since $|G_1|<\tau+b<T_p$. Near any other side some residual coordinate
+has absolute value near $t_1>T_p$. Hence every inner residual subcube
+has buffered graph charts in this restriction. Its degree equals
+$\mathcal D_X(G)$ by \eqref{eq:degree-boundary-degree}, and its coordinate gradients only
+decrease on restriction to the tangent space. We take $J^+=J$ and use
+\eqref{eq:degree-slice-bound}.
+
+If $d=1$, so $p=0$, let $X_s^+$ be the confined regular points in
+$\Sigma_s\cap\{|G_1|<t_1\}$, each with its boundary orientation, and let
+$J^+$ be the unique map to the point $Q_{T_0}^0$. Give these point leaves
+the induced zero-dimensional metric and the weight $\phi_\Sigma$.
+Confinement gives $|G_1|<\tau+b<T_0<t_1$ on the counted branch, so this
+includes every point in \eqref{eq:degree-boundary-degree}.
+The local jump separation proved above bounds the number of such points
+in each buffered parent plaque. Choose a fixed finite half-open mesh
+finer than that separation in the normalized plaque. Each mesh cell
+contains at most one counted point. The Borel boundary-support code and
+the singleton-section theorem make that point a Borel function of the
+transverse parameter wherever it exists. These functions give a bounded
+number of point charts, with transversals restricted from the parent and
+counted with the mesh index. Their total transverse measure is finite and their
+plaque overlap remains bounded. Point boxes have the required common
+buffers, and all metric, map, and positive-order
+weight derivative conditions in Definition~\ref{def:degree-band} are
+vacuous in dimension zero, so this is a controlled $0$-band. Its measured
+signed count is $\mathcal D_X(G)$ by
+\eqref{eq:degree-boundary-degree}; its integral is the measured unsigned
+count. At each point \eqref{eq:degree-raw-integrand} reads
+$1\le(v/L_1)\Theta$, and confinement places every counted point in
+$|G_1|<b_d$. We return to the common integral estimate after treating
+the possible singularities when $p\ge7$.
+
+\subsection{A regular descendant in higher dimensions}
+
+Assume now $p\ge7$, and fix one of the typical parameters $s$ above.
+The singular set has zero $p$-dimensional area, so it does not change the
+boundary-current integral. Deleting it can, however, leave ends of the
+regular locus over interior residual values, where regular chart radii
+have no common lower bound. We must keep those ends away from every
+inner target subcube and preserve the degree computed by the closed
+boundary current.
+
+We will construct a controlled child carried by an open regular subset
+of this slice, with the inherited orientation and transverse measure,
+and with degree $\mathcal D_X(G)$. For each residual row we require
+$|dJ_j^+|_{h^+}\le |dG_j|_h+\varepsilon_{\mathrm{grad}}$ and the
+one-sided value bound
+$|J_j^+|\ge |G_j|-\varepsilon_{\mathrm{val}}$, with
+\begin{equation}\label{eq:degree-tolerances}
+ \sum_{j=2}^d L_j\varepsilon_{\mathrm{grad}}<\frac1{10q},
+ \qquad \varepsilon_{\mathrm{val}}<b_d-b_p=\frac{\tau}{4q^2}.
+\end{equation}
+The gradient bound controls the residual penalties and the child's
+coordinate bound.
+The value bound, together with confinement of $G_1$, puts the child's
+$b_p$ integration region over the parent's $b_d$ region. Write
+$\mathcal W=dV_{h^+}/dA_h$ for its ordinary leafwise volume density
+relative to the original induced metric. The scalar and volume target is
+\begin{equation}\label{eq:degree-child-scalar}
+ \begin{gathered}
+ dV_{h^+}=\mathcal W\,dA_h,\\
+ \wscal(h^+,\phi^+)\ge\wscal(h,\phi)-H_1(G)-2\ln(v/L_1)
+ +2\ln\mathcal W-\frac1q.
+ \end{gathered}
+\end{equation}
+The coefficient $2$ makes the volume factor cancel after exponentiation,
+leaving the same lapse factor as in \eqref{eq:degree-raw-integrand}.
+The density $\mathcal W$ includes neither $v$ nor the transverse measure.
+
+Every constant and error chosen inside this subsection may depend on this
+slice and on the band, but must be common across its retained plaques.
+The choices proceed in three groups. First a small parent perturbation
+creates a target gap around the points without uniform regular charts.
+Next we fix the kernel and graph constants using the controlled parent
+data and that gap. Finally we choose the marked-set cover small enough
+for all scalar errors. Its positive radius lower bound controls the
+number and derivatives of the kernels for the graph solves; it does not
+enter the scalar constants. The adjustable scalar error in
+\eqref{eq:degree-child-scalar} is at most $1/q$, so chart and elliptic
+constants will not enter the absolute loss in Theorem~\ref{thm:degree}.
+
+Choose a parent atlas with buffers around $\{|G|_\infty\le t_3\}$ still
+inside $\{|G|_\infty<t_4\}$. One obtains it from an atlas for a larger
+inner cube by a bounded mesh subdivision, restricting transversals by
+Borel tests as necessary. Its plaque overlaps, also for the enlarged
+supports below, remain bounded. All operations in the parent made below
+use this atlas.
+
+\paragraph{Uniform regularity tests and avoidance of the remaining points.}
+We use finitely many graph tests at a time to obtain a regular region
+with common chart controls. The points not yet recognized will be
+excluded from the child, and the target gap below will keep the later
+homotopies on the regular locus away from them.
+Use one common normalized outer cube $Q$ for the buffered parent plaques.
+Fix in $Q$ one ordered countable list of tests for graph cylinders, using
+rational centers, radii, bounds, and a fixed countable dense set of
+rotations. Transport this same list, in this same order, by every parent
+plaque chart; the list is independent of the plaque and its transverse
+parameter. A test recognizes that the boundary
+in its outer cylinder is exactly one smooth graph, with a specified slope
+bound and strict room at its top, bottom, and sides, and that the set is
+on its specified side. Its inner cylinder is then an open regular
+region. At level $k$ use the first $k$ tests of this fixed list in every
+plaque, with their specified positive buffers. Elliptic estimates on these
+graphs give all interior derivative bounds for a fixed level. Make the
+tests with strict margins, so a regular limit point recognized by one of
+them is recognized by that same test on every sufficiently close smoothly
+convergent graphical boundary. The graph coding already described shows
+that recognition is Borel.
+
+Let $D_k$ be the part of the boundary support in
+$\{|G|_\infty\le t_3\}$ left after removing the open inner portions
+recognized through level $k$ in any parent plaque. These are decreasing
+Borel sets, closed along each leaf. Every regular point is eventually
+removed. More precisely, in any converging sequence of controlled
+charts with levels tending to infinity, the $D_k$ can accumulate on a
+compact subbox only at singular points of the limiting boundary.
+Outside the limit support this follows from support convergence. If
+points of $D_{k_j}$ with $k_j\to\infty$ converged instead to a regular
+limit point, one test with a fixed index $\ell$ in the common list would
+recognize that point with strict room. Smooth graphical convergence
+makes the same test recognize the nearby points for large $j$, and then
+$k_j\ge\ell$, contradicting their membership in $D_{k_j}$. This is the
+compactness fact used in both constructions that follow.
+
+First make an arbitrarily small smooth perturbation $\bar J$ of $J$ in
+the \emph{parent} band, small in value and first derivative, with all
+higher derivatives uniformly bounded. We claim it can be made so that,
+for some $a_0>0$ and some finite $k$,
+\begin{equation}\label{eq:degree-gap}
+ |\bar J|_\infty>4a_0
+ \quad\hbox{on a uniform neighborhood of }D_k.
+\end{equation}
+After the initial homotopy on the full boundary current, this gap will
+keep the homotopies over a small target cube inside a uniformly regular
+region.
+To prove the local assertion behind the claim, fix a positive budget for
+a constant shift in $\mathbb R^p$ and an inner compact plaque box.
+There are a level and a positive gap, uniform across the controlled
+plaques, such that some shift in that budget gives this gap from zero on
+the marked set in the box. Otherwise take a sequence of failing plaques,
+with levels tending to infinity and requested gaps tending to zero.
+Pass to normalized chart limits, including a smooth limit of the maps.
+On a slightly larger compact box the marked sets concentrate at the
+singular set of the limit. By \eqref{eq:degree-singular-dimension} this
+set has dimension at most $p-7$, so its image under the smooth limit map
+has $p$-dimensional measure zero. That image is compact. A shift in the
+given open budget can therefore avoid it with a positive distance.
+Uniform convergence of the maps and concentration of the marked sets
+contradict the alleged failure. This proves the local uniform assertion.
+
+Apply these shifts with smooth parent-chart cutoffs equal to one on the
+inner boxes. For clarity, simultaneous choices can be made Borel and
+with bounded overlap. The graph whose vertices are the buffered plaques
+and whose edges record meeting supports is a Borel graph of finite
+degree. It has a countable Borel proper coloring: a countable separating
+family of Borel sets distinguishes each vertex from its finitely many
+neighbors, and the first finite distinguishing pattern is a countable
+color. The finite neighbor lists can be enumerated Borel measurably by
+the Borel countable-section theorem. Processing these color classes
+successively and taking the first one of $\Delta+1$ colors avoided by
+all earlier neighbors, where $\Delta$ bounds the degree,
+gives a finite Borel proper coloring. In each color the cutoff supports
+are disjoint. Choose rational shifts satisfying the local gap test; the
+test on a compact inner box is Borel by projection with compact sections.
+After a color is treated, increase the level if necessary and choose the
+next shift budget smaller than the preceding positive gaps. The marked
+sets decrease, so earlier gaps persist. There are only finitely many
+colors. The budgets can also make the total value and first-derivative
+change as small as prescribed; the cutoffs give uniform bounds for all
+higher derivatives. Taking $a_0$ smaller than the final gaps and then
+using the derivative bound of $\bar J$ extends the gap to a uniform
+neighborhood. This proves \eqref{eq:degree-gap}.
+
+Second, set
+\begin{equation}\label{eq:degree-alpha}
+ \alpha=p-\frac{13}{2}>0.
+\end{equation}
+The next cover will support a function large on the remaining marked
+points while keeping the total kernel coefficients as small as needed.
+For any prescribed positive budget and upper radius, a sufficiently
+large $k'\ge k$ permits a cover of $D_{k'}$ in each inner plaque box by
+finitely many small balls, of radii $r_i$, such that
+\begin{equation}\label{eq:degree-cover-budget}
+ \sum_i r_i^\alpha<\text{the prescribed budget}.
+\end{equation}
+The radii can also have a common positive lower bound, depending on the
+budget and the controlled family. To verify the uniform claim, suppose
+it failed for levels tending to infinity and for requested lower radii
+tending to zero. Pass to a limiting chart. Its compact singular set has
+zero $\alpha$-dimensional Hausdorff measure because
+$p-7<\alpha$. Cover it on a slightly larger compact box by finitely
+many balls with radii below the prescribed upper radius and with the
+$\alpha$-sum strictly below the budget, allowing room to enlarge the balls.
+This finite cover has a positive minimum radius. The concentration
+observation about $D_k$ makes the enlarged balls cover the marked points
+of the sequence eventually, contradicting failure. Rational chart balls
+and fixed comparison factors give Riemannian balls with the same
+conclusion. Keep only balls meeting the marked set, allowing a small
+change of centers and the reserved enlargement. The centers then also
+lie near $D_k$. Such finite covers can be selected Borel: enumerate
+finite tuples of rational balls, and test whether the compact marked
+part outside their union is empty. This is again a Borel compact-section
+test. The positive lower radius and \eqref{eq:degree-cover-budget}
+bound the number of balls in any plaque.
+
+The two uses of \eqref{eq:degree-alpha} are distinct: its strict excess
+over $p-7$ made this cover possible, while
+$\alpha+4=p-5/2<p-2$ will make the following potentials superharmonic.
+
+\paragraph{Potentials near the marked set.}
+Choose a small parent normal-coordinate radius $R>0$ so that a ball of
+that radius about any cover center stays in the gap region
+$|\bar J|_\infty>3a_0$ and in the buffered atlas. It can also be chosen
+small with respect to the local metric, weight, and forcing bounds.
+This radius is fixed before the final cover is chosen; require its ball
+radii to be much smaller than $R$. For a center $c_i$ and radius $r_i$
+let
+\[
+ K_i(x)=(r_i^2+\operatorname{dist}_h(x,c_i)^2)^{-1/2}
+\]
+on the smaller normal ball, and cut it off within radius $R$. Choose a
+radial cutoff which is one on the smaller ball, is positive before the
+outer boundary, and near that boundary has the tail
+$\exp(-1/(1-t))$ for $t<1$ and zero for $t\ge1$, where
+$t=\operatorname{dist}_h(x,c_i)^2/R^2$. Join this tail smoothly and
+positively to the inner plateau. Every positive power of the cutoff
+then has a smooth exponential flat tail, so each power of $K_i$ used
+below is smooth. Denote its support by $V_i$.
+
+On the regular boundary in $V_i$, for $j=\alpha,\alpha+4$,
+\begin{equation}\label{eq:degree-kernel-bounds}
+ \begin{split}
+ -\Delta_{\Sigma,\phi}K_i^j&\ge cK_i^{j+2}-C\mathbf1_{V_i},\\
+ |dK_i^j|&\le C\bigl(K_i^{j+1}+\mathbf1_{V_i}\bigr).
+ \end{split}
+\end{equation}
+The constants may depend on the fixed local controls and on $p$, but are
+independent of $r_i$ and of the eventual number of balls. To check the
+first estimate before cutoff, put $u=\operatorname{dist}_h(x,c_i)^2$.
+The normal-coordinate Hessian bound, the bounded mean curvature
+$H=\mathfrak h_s-\partial_\nu\phi$, and the bounded weight gradient give
+\[
+ \Delta_{\Sigma,\phi}u\ge2p-C\sqrt u,\qquad |du|_\Sigma^2\le4u.
+\]
+Differentiating $(r_i^2+u)^{-j/2}$ therefore gives
+\[
+ -\Delta_{\Sigma,\phi}K_i^j
+ \ge j\left(p-j-2-C\sqrt u\right)K_i^{j+2}.
+\]
+Both chosen exponents are less than $p-2$, so small $R$ makes the
+coefficient positive. The gradient estimate follows by one
+differentiation. On the cutoff annulus the distance from the center is
+bounded below by a fixed multiple of $R$, and all cutoff derivatives are
+bounded independently of $r_i$; this accounts for the displayed constant
+losses. Outside $V_i$ they vanish, as the indicators record.
+
+For $j=\alpha,\alpha+2,\alpha+4,\alpha+6$ put
+\begin{equation}\label{eq:degree-potentials}
+ u_j=\varepsilon_{\mathrm{base}}^j+\sum_i M_*r_i^\alpha K_i^j,
+ \qquad \Lambda=\left(\frac{u_{\alpha+4}}{u_\alpha}\right)^{1/4},
+ \qquad
+ \mathcal L=|d\ln u_\alpha|^2+|d\ln u_{\alpha+4}|^2.
+\end{equation}
+These quantities have separate roles. We will make $u_\alpha$ large on
+$D_{k'}$ so that its logarithm locates the region to be excluded.
+The term $\Lambda^2$ will supply the positive scalar contribution that
+pays for the later coordinate graphs, while $\mathcal L$ controls the
+logarithmic derivatives introduced by the weight and the first graph.
+Here $0<\varepsilon_{\mathrm{base}}<1$ will be small; $M_*$ will be a
+large finite number fixed before the final cover selection. At a point, the sum is over the
+kernels active there. The overlap bound and
+\eqref{eq:degree-cover-budget} allow the total active coefficient
+$\sum_i M_*r_i^\alpha$ to be made arbitrarily small, uniformly at every
+point. In particular, for any $\delta>0$, first taking
+$\varepsilon_{\mathrm{base}}$ small and then the cover budget small enough
+gives, for $j=\alpha,\alpha+4$,
+\begin{equation}\label{eq:degree-potential-laplacian}
+ -\frac{\Delta_{\Sigma,\phi}u_j}{u_j}
+ \ge c\frac{u_{j+2}}{u_j}-\delta.
+\end{equation}
+Indeed \eqref{eq:degree-kernel-bounds} leaves the errors
+$c\varepsilon_{\mathrm{base}}^{j+2}+C\sum_{i:x\in V_i}M_*r_i^\alpha$
+before division by $u_j\ge\varepsilon_{\mathrm{base}}^j$. Choose
+$\varepsilon_{\mathrm{base}}^2$ and then the active sum divided by
+$\varepsilon_{\mathrm{base}}^{\alpha+4}$ small enough for the displayed
+$\delta$.
+
+At each point the $u_j$ are the moments of positive atoms: one atom at
+$\varepsilon_{\mathrm{base}}$ of mass one, and atoms at $K_i$ of masses
+$M_*r_i^\alpha$.
+Make their total active mass at most two. Moment inequalities give
+\begin{equation}\label{eq:degree-moments}
+ \frac{u_{\alpha+6}}{u_{\alpha+4}}\ge\Lambda^2
+ \ge\frac{u_{\alpha+2}}{u_\alpha},\qquad
+ |d\ln\Lambda|^2\le\frac{\mathcal L}{8},\qquad
+ |d\ln u_\alpha|\le C(\Lambda+1),\qquad
+ \Lambda\ge c' u_\alpha^{1/\alpha}.
+\end{equation}
+For the first pair, use the probability weights proportional to the
+atoms' masses times their $\alpha$-powers. If $Z$ is the square of the
+atom position, then
+$\mathbb E Z\le(\mathbb E Z^2)^{1/2}$ and
+$\mathbb E Z^3\ge(\mathbb E Z^2)^{3/2}$ give the two inequalities. The
+second estimate follows directly by differentiating
+$\ln\Lambda=(\ln u_{\alpha+4}-\ln u_\alpha)/4$. For the third, the
+gradient bound in \eqref{eq:degree-kernel-bounds} bounds
+$|du_\alpha|$ by $C\sum_iM_*r_i^\alpha(K_i^{\alpha+1}+1)$.
+The first part divided by $u_\alpha$ is at most $\Lambda$ by H\"older's
+inequality for those probability weights; the second is at most one after
+the same small active-mass choice. Finally, monotonicity of the
+$L^r$ norms for a measure of mass at most two gives
+$u_{\alpha+4}/u_\alpha\ge(u_\alpha/2)^{4/\alpha}$, proving the last
+inequality. These constants are independent of $M_*$ and $k'$.
+At a point of $D_{k'}$ some ball has distance at most its radius, so its
+term in $u_\alpha$ is at least $c''M_*$. Hence
+\begin{equation}\label{eq:degree-potential-on-bad}
+ u_\alpha\ge c''M_*\quad\hbox{on }D_{k'}.
+\end{equation}
+The positive lower bound on all radii and the bounded number of balls per
+plaque make all $u_j$ smooth in the parent with uniform derivative bounds
+for the final slice. These bounds are allowed to depend on that radius
+lower bound.
+
+Add $\eta_0=(\ln u_\alpha+\ln u_{\alpha+4})/4$ to $\phi_\Sigma$ on the
+regular boundary. Put $Q_1=d\eta_0$. The weight-change formula used in
+\eqref{eq:degree-slice-identity}, now with
+$\Delta_{\Sigma,\phi_\Sigma}=\Delta_{\Sigma,\phi}
++\langle d\ln v,d\,\cdot\,\rangle$, yields from
+\eqref{eq:degree-potential-laplacian} the scalar gains
+$c\Lambda^2+\mathcal L/2-2\langle d\ln v,Q_1\rangle-|Q_1|^2-\delta$.
+Here the high moment in \eqref{eq:degree-moments} supplies $\Lambda^2$.
+Since $|Q_1|^2\le\mathcal L/8$,
\[
- \wscal(h_*,0)\ge-C_n+2W-n\log\epsilon_0+\epsilon_0e_{h_*}(F_*),
+ \frac{\mathcal L}{2}+|d\ln v|^2
+ -2\langle d\ln v,Q_1\rangle-|Q_1|^2
+ \ge\frac{\mathcal L}{2}-2|Q_1|^2\ge\frac{\mathcal L}{4}.
\]
-with $W=0$ and $\epsilon_0\in[\epsilon_*,2\epsilon_*]$.
-
-\emph{Phase 1.} Use the schedule \eqref{eq:entropy-step-schedule}.
-Choose a large first target and run to it; call the actual output
-parameter $E_1$. The overshoot is at most $1+l_*$. In every sweep,
-use $h_*$ as reference; then the possible support enlargement is
-$C_n(1+\log^3(1+E_1))$. Apply Lemma~\ref{lem:entropy-phase-exit} to
-the output at $E_1$, giving the exit data $(h_1,\phi_1,Z_1)$ and
-$(h_2,\phi_2,Z_2)$.
-
-\emph{Phase $j\ge1$.} After an exit at $E_j$, choose a target
-\begin{equation}
- T_{j+1}=\exp(E_j^{1/1000})
- \label{eq:entropy-phase-targets}
-\end{equation}
-and run to it; call the first reachable parameter $E_{j+1}$. The
-overshoot is again at most $1+l_*$. Two checks: (i) the threshold
-$T_{j+1}$ satisfies the overlap condition and the incoming gradient
-bound $K_0\le E_j^{-1/100}$, both controlled by the cubic-logarithmic
-cost of the previous accumulation; (ii) for large $E_j$,
-$E_j^{0.32}\le\tfrac12E_{j+1}^{0.26}$, so the incoming volume from
-the previous exit is absorbed.
-
-\emph{Final parameters.} After $J=H+2$ phases,
-$E>\mathcal B(q)$ and $E>E_{\min}$ by choosing the initial target
-large enough; the initial-gradient threshold at the start of the
-final accumulation phase is $\le E_{J-1}^{-1/100}$.
-
-\emph{Fixed graph.} The support radius of a sweep is
-$C_n(1+\log^3(1+E_j))$ in the phase-$j$ reference metric. Combined
-with the conjugacy bound and the support growth between phases,
-labels simultaneously nonzero at any point of any leaf are at graph
-distance at most two in the transitive closure of the label graph.
-Concatenating these gives a single $\Gamma$-invariant graph
-$\mathcal G$ of finite maximum degree $\Delta$.
-
-\emph{Coloring.} Take
-\[
- q\ge\max\{n+1,\ \#(\Gamma\backslash\mathcal L),\ \Delta_{\mathrm{sq}}+1\},
-\]
-where $\Delta_{\mathrm{sq}}$ is the maximum degree of $\mathcal G$'s
-square. Lemma~\ref{lem:entropy-coloring} below provides a Borel
-equivariant $q$-coloring of $\mathcal G$ that separates labels at
-graph distance $\le2$.
-
-\emph{Conclusion.} All remaining properties (Borel transversality,
-equivariance, support bounds, volume bounds) are inherited from the
-input and from the accumulation and phase-exit lemmas. The scalar
-constant $C=C(n,\mathcal B)$ is dimensionally bounded because the
-$O(l^2)$ sweep and exit costs are summable in the dimensional
-constants and $J=H+2$ phases only add a dimensionally bounded total.
-\end{proof}
-
-\begin{lemma}[Defective coloring by priority order]
-\label{lem:entropy-coloring}
-Let $\mathcal G$ be a $\Gamma$-invariant graph of finite maximum
-degree $\Delta$ on a countable label set $\mathcal L$ with finitely
-many free $\Gamma$-orbits. Adjoin independent i.i.d. uniform
-priorities on $[0,1]^{\mathcal L}$. Then on an invariant conull set
-of the extended probability space, greedy coloring by descending
-priority uses at most $\Delta+1$ colors, and this coloring is Borel
-and equivariant.
-\end{lemma}
-
-\begin{proof}[Proof sketch]
-Direct every edge of $\mathcal G$ from larger to smaller priority,
-discarding the null set where two priorities coincide. For a fixed
-$k$, the probability of a strictly descending path of length $k$
-starting at a given vertex is at most $\Delta^k/(k+1)!$, since any
-ordering of the $k+1$ priorities is equally likely. Summing over
-$k$ shows that, almost surely, every vertex has finite descending
-depth. Greedy coloring on descending depth uses at most $\Delta+1$
-colors. Borel because colors are determined in finite depth.
-Equivariant because the rule only uses priorities and $\mathcal G$.
-
-For distance-two separation, run the same construction on the square
-graph $\mathcal G^2$ (of maximum degree $\le\Delta^2$), giving at
-most $\Delta^2+1$ colors and hence the needed separation.
-\end{proof}
-\section{A measured degree estimate}
-\label{sec:degree}
-
-This section bounds the \emph{measured signed degree} of a map from a
-measured family of Riemannian \(d\)-manifolds to a \(d\)-cube. The
-bound combines the weighted scalar curvature \(-\wscal/2\) with a
-penalty for each coordinate gradient of the map. This is the
-analytic-geometric input to \S\ref{sec:transfer}: applied to the
-comparison map there, it produces \(L^1\) bounds for the degree that
-can be integrated against the entropy density \(\rho\).
-
-The section is broken into three technical layers:
-\begin{enumerate}
-\item Measured bands and the signed degree: the geometric setup
- (\S\ref{sec:degree-bands}).
-\item The \(\mu\)-bubble functional and its properties: minimizers,
- existence, regularity, and confinement to a controlled band
- (\S\ref{sec:degree-pressure}).
-\item The coarea formula, normal-speed (lapse) extraction, and
- measured descent: the inductive step that reduces dimension one
- coordinate at a time (\S\ref{sec:degree-lapse}).
-\end{enumerate}
-The theorem \ref{thm:degree} below is proved by iterating the descent
-$d$ times.
-
-\subsection{Measured bands and signed degree}
-\label{sec:degree-bands}
-
-\begin{definition}[Measured \(d\)-band]
-\label{def:degree-band}
-A \emph{measured \(d\)-band} is a tuple \((X,h,\phi)\) where:
-\begin{itemize}
-\item $X$ is a standard Borel space with countably many
- \emph{flow boxes} $\kappa_a:T_a\times Q_a\to X$
- ($T_a$ standard Borel with $\sigma$-finite measure, $Q_a\subset
- \R^d$ open), whose images cover \(X\);
-\item each plaque $\kappa_a(\{t\}\times Q_a)$ is open in its
- \emph{leaf}, the leaves glue into oriented smooth \(d\)-manifolds
- (Hausdorff, second countable), and the transverse dimensions and
- their measures are preserved across overlaps;
-\item $h$ is a Riemannian metric and $\phi$ a smooth weight on
- each leaf, Borel transversely;
-\item there are uniform metric bounds (local comparability of $h$
- to a chart metric, uniform across plaques), plaque overlaps
- bounded by a uniform constant, and compact ``finite type''
- descriptor structure.
-\end{itemize}
-\end{definition}
-
-\begin{definition}[Boundary control]
-\label{def:degree-boundary}
-A map $G:X\to[-t_1,t_1]^d$ has \emph{boundary control} of order
-$b_d<b_p<b_d<t_3<t_2<t_1$ with residual length $\varepsilon$ if
-\begin{itemize}
-\item $G_j$ is leafwise $C^1$ and Borel transversely;
-\item $|G|_\infty<t_3$ throughout the band;
-\item $L_i|dG_i|_h\le B-1$ on the tangent space (a strict-gradient
- condition), and the values of $G_j$ on the coordinate-$j$ boundary
- of the inner cube differ from the boundary values of the
- prescribed box by $\varepsilon_{\mathrm{val}}$.
-\end{itemize}
-\end{definition}
-
-\begin{definition}[Signed degree]
-\label{def:degree-signed}
-Given a measured \(d\)-band \(X\) and a map $G:X\to\R^d$ with
-boundary control, define the \emph{measured signed degree}
-\begin{equation}
- \mathcal D_X(G):=\int_X a_-(G_1)\,dG_1\wedge J^*\beta,
- \label{eq:degree-signed-def}
+Combining with \eqref{eq:degree-slice-bound}, and decreasing $c>0$ if
+necessary, gives for the new weight
+\begin{equation}\label{eq:degree-weight-gain}
+ \wscal(h|_{T\Sigma},\phi_\Sigma+\eta_0)
+ \ge\wscal(h,\phi)-H_1(G)-2\ln(v/L_1)
+ +c\Lambda^2+\frac{\mathcal L}{4}-\delta.
\end{equation}
-where $J=(G_2,\ldots,G_d)$, $a_-(z)\,dz$ is a unit-integral
-one-form supported in $(-1.9\tau,-1.6\tau)$, and $\beta$ is a
-unit-integral smooth $p$-form on the residual $\R^{d-n}$ supported
-in a small cube about zero.
-\end{definition}
-
-\begin{definition}[Horizon condition]
-\label{def:degree-horizon}
-Let $B$ be as in Definition~\ref{def:degree-boundary}. Fix a
-numerical $c_n>0$ sufficiently small. The lengths $L_1,\ldots,L_d$
-satisfy the \emph{horizon condition} for $(\tau,q)$ if
-\begin{equation}
- \frac{C(1+B)^{2/3}}{L_i}<\frac{\tau}{10q}
- \label{eq:degree-horizon-condition}
+We have obtained a large scalar allowance near the marked set. The next
+construction uses it to move the residual map out of the target band
+there, while paying for the graph volume created in doing so.
+
+\paragraph{Two graph prescriptions and their costs.}
+We describe the graph calculations first, and then give a complete region
+on which the prescriptions can be solved. The first prescription will
+produce a function $f$ within two of a small multiple of $\ln u_\alpha$,
+with $|df|\le1$ in its graph metric. The coordinate prescriptions then
+use this controlled gradient to amplify $\bar J$ where $f$ is high;
+each correction adds at most $e$ to its coordinate gradient in the final
+graph metric, in addition to the earlier perturbation of $\bar J$.
+Their scalar gains will pay the exact graph volume factors.
+
+For a current metric and
+weight $(k,\varphi)$, a length-one graph of a function $f$ has
+\[
+ L_f=(1+|df|_k^2)^{1/2},\quad U_f=\frac{\nabla_k f}{L_f},\quad
+ k_f=k+df\otimes df,\quad \varphi_f=\varphi-\ln L_f,
+ \quad D_f=-\operatorname{div}_{k,\varphi}U_f.
+\]
+Lemma~\ref{lem:graph-identity} says
+\begin{equation}\label{eq:degree-graph-identity}
+ \wscal(k_f,\varphi_f)-\wscal(k,\varphi)
+ =D_f^2-2U_f(D_f)+|\mathrm{II}_f|^2+|d\ln L_f|_{k_f}^2.
\end{equation}
-for every $i$, where $C$ depends only on an absolute constant
-(universal for all $n$).
-\end{definition}
-
-\begin{theorem}[Degree estimate]
-\label{thm:degree}
-There are dimensional constants \(C_0,C_*<\infty\) with the
-following property. Let $X$ be a measured $d$-band, let $G:X\to
-[-t_1,t_1]^d$ have boundary control of order $b_d$ with residual
-$\varepsilon_{\mathrm{grad}},\varepsilon_{\mathrm{val}}$ from
-Definition~\ref{def:degree-boundary}, and let
-\(L_1,\ldots,L_d>0\), $\tau>0$, \(B\) satisfy the horizon condition.
-Then
-\begin{equation}
- |\mathcal D_X(G)|(2\tau)^p
- \le\frac{e^{C_0p/q+C_*/q}}{\prod_{i=1}^dL_i}
- \int_{\{|G|_\infty<b_d\}}
- v\exp\Bigl(-\frac{\wscal(h,\phi)}{2}
- +\sum_{i=1}^d\frac{H_i(G)}{2}\Bigr)\,dV_h,
- \label{eq:degree-estimate}
-\end{equation}
-where \(H_i(G)=2(L_i|dG_i|_h-1)_+\) and $p=d-n$.
-\end{theorem}
-
-\noindent\emph{Notational convention.} The quantity \(v\) appearing
-in \eqref{eq:degree-estimate} is the \emph{lapse}, or normal speed of
-the exhausting family of \(\mu\)-bubbles constructed in
-\S\ref{sec:degree-lapse}. It is a leafwise Borel function with
-$0\le v\le1$ and agrees with the classical normal speed
-$J_h(y,z)\partial_sr(s,y)$ on each regular slice.
-\subsection{The \(\mu\)-bubble functional}
-\label{sec:degree-pressure}
-
-The proof of Theorem~\ref{thm:degree} uses the most basic
-dimension-reduction tool in scalar-curvature geometry: an ordered
-family of \(\mu\)-bubble boundaries. We now set it up.
-
-\subsubsection{The functional}
-
-Fix a smooth scalar \emph{forcing} \(\mathfrak h_s(y)\) on a leaf
-and set the \(\mu\)-bubble functional
-\begin{equation}
- \mathcal F_{\mathfrak h_s}(E)
- :=P_\phi(E)-\int_E\mathfrak h_s\,e^\phi\,dV_h,
- \label{eq:degree-functional}
+Also
+\begin{equation}\label{eq:degree-graph-elementary}
+ \begin{gathered}
+ |U_f|_k\le1,\qquad U_f(f)=L_f-L_f^{-1},\qquad
+ |df|_{k_f}\le1,\\
+ dV_{k_f}=L_f\,dV_k,\qquad L_f-L_f^{-1}\ge\ln L_f.
+ \end{gathered}
\end{equation}
-where \(P_\phi(E)\) is the weighted perimeter of the finite-perimeter
-set \(E\). Per Gromov's terminology, minimizers of this functional
-are \(\mu\)-bubbles.
-
-\subsubsection{The pressure law}
+The last inequality holds for every $L_f\ge1$ by differentiation. These
+relations will make the scalar cost and the volume change use the same
+factor $L_f$.
-Fix a smooth even \(\gamma:[0,\infty)\to[0,\infty)\) with
-\(0\le\gamma(y)\le y^2\) and with \(\gamma(y)\) comparable to
-\(|y|^{3/2}\) for \(|y|\ge1\). Define \(P_0:[0,b)\to[0,\infty)\) as
-the solution of the ODE
-\begin{equation}
- P_0(0)=0,\qquad P_0'=L_1\psi(P_0),\quad
- \psi(y)=1+\frac{\gamma(y)}{8(1+B)},
- \label{eq:degree-pressure}
+For the first graph put
+\[
+ f_0=\sigma\ln u_\alpha,\qquad C_f=B_0+\Lambda,
+ \qquad D_f=C_f\tan(f_0-f),
+\]
+with the argument between its two poles. Here $B_0\ge2$ is fixed and
+$\sigma>0$ will be small. Pole avoidance gives $|f-f_0|<\pi/2<2$.
+Differentiating the prescribed $D_f$ gives the exact terms
+\begin{equation}\label{eq:degree-first-graph-differentiation}
+ \begin{split}
+ D_f^2-2U_f(D_f)
+ ={}&D_f^2-2D_fU_f(\ln C_f)
+ -2(C_f+D_f^2/C_f)U_f(f_0)\\
+ &+2(C_f+D_f^2/C_f)(L_f-L_f^{-1}).
+ \end{split}
\end{equation}
-where \(L_1>0\) is the first coordinate length. The function \(P_0\)
-is odd, increasing, and has a pole at \(\pm b\), where
-\begin{equation}
- b=\frac1{L_1}\int_0^\infty\frac{dy}{\psi(y)}
- \le\frac{C(1+B)^{2/3}}{L_1}.
- \label{eq:degree-pole-width}
+The final term is at least $2\ln L_f$ by
+\eqref{eq:degree-graph-elementary}. For the derivative of $C_f$, Young's
+inequality and \eqref{eq:degree-moments} give
+\[
+ 2|D_fU_f(\ln C_f)|
+ \le0.8D_f^2+1.25|d\ln C_f|^2
+ \le0.8D_f^2+\frac{1.25}{8}\mathcal L.
+\]
+The $f_0$ term has absolute value at most
+\[
+ 2\sigma C(C_f+D_f^2/C_f)(\Lambda+1).
+\]
+Since $(\Lambda+1)/C_f\le1$, a sufficiently small $\sigma$ spends less
+than the remaining $0.2D_f^2$ on its square term. Moreover
+$C_f(\Lambda+1)\le C_{B_0}(1+\Lambda^2)$, so a further small choice of
+$\sigma$ spends less than half the $c\Lambda^2$ in
+\eqref{eq:degree-weight-gain} and makes the remaining constant loss as
+small as prescribed. The available $\mathcal L/4$ covers
+$1.25\mathcal L/8$. Combining \eqref{eq:degree-weight-gain} and
+\eqref{eq:degree-graph-identity}, and discarding nonnegative terms, thus
+gives
+\begin{equation}\label{eq:degree-first-graph-gain}
+ \wscal(k_f,\varphi_f)\ge\wscal(h,\phi)-H_1(G)-2\ln(v/L_1)
+ +c_1\Lambda^2+2\ln L_f-\delta_1,
\end{equation}
-Choose the absolute \(c_n>0\) in the horizon condition so that
-\(b<\tau/(10q)\); this is possible by the assumed
-horizon condition.
-
-Set the forcing
-\begin{equation}
- \mathfrak h_s(y):=\widehat P(s-G_1(y)),
- \label{eq:degree-forcing}
-\end{equation}
-where \(\widehat P\) is a strict prolongation of \(P_0\) past \(\pm b\)
-that smooths the corners: \(C^1\)-smooth, with the same forcing
-for $|s-G_1|$ small and strictly bounded for \(s<0\) (or
-symmetric).
-
-\subsubsection{Existence and comparison}
-
-\begin{lemma}[Existence and nesting]
-\label{lem:degree-existence}
-For each fixed \(s\in(-\tau,\tau)\), a least minimizer \(E_s^K\) of
-\(\mathcal F_{\mathfrak h_s}\) exists on each finite compact region
-\(K\), and the family \(\{E_s^K\}_K\) increases with \(K\) and is
-independent of the exhaustion (up to \(L^1_{\mathrm{loc}}\)
-identification). The sets \(\{E_s\}\) are monotonically nested:
-\(E_s\subset E_t\) for \(s<t\).
-\end{lemma}
-
-\begin{proof}[Proof sketch]
-Ball-replacement gives uniform local perimeter bounds for the
-exhaustion minimizers, so they converge in \(L^1_{\mathrm{loc}}\)
-to a limit \(E_s\) that is locally minimizing by a belt-and-coarea
-splicing argument. Monotonicity in \(s\) uses the pointwise
-ordering of forcings and the minimality inequality.
-\end{proof}
-
-\subsubsection{Regularity and compactness}
+where $c_1>0$ is fixed, while $\delta_1$ can be made arbitrarily small. This
+calculation is used only where the current base metric is the original
+induced metric for which \eqref{eq:degree-weight-gain} was proved.
+
+For the remaining graphs, let $\zeta(f)$ be a smooth function valued in
+$[0,1]$, zero for $f\le K$ and one for $f\ge K+1$, with derivative bound
+independent of the threshold $K$. Write
+$(k^{(1)},\varphi^{(1)})=(k_f,\varphi_f)$. For row $j=2,\ldots,d$, its
+graph quantities use the metric and weight after all preceding rows:
+\[
+ \begin{gathered}
+ L'_j=(1+|df'_j|_{k^{(j-1)}}^2)^{1/2},\qquad
+ U'_j=\frac{\nabla_{k^{(j-1)}}f'_j}{L'_j},\\
+ D'_j=-\operatorname{div}_{k^{(j-1)},\varphi^{(j-1)}}U'_j,\\
+ k^{(j)}=k^{(j-1)}+df'_j\otimes df'_j,\qquad
+ \varphi^{(j)}=\varphi^{(j-1)}-\ln L'_j.
+ \end{gathered}
+\]
+Prescribe successively
+\begin{equation}\label{eq:degree-row-graph}
+ \ell_j=A\zeta(f)\bar J_j,\qquad
+ D'_j=B'\tan\left(\frac{\ell_j-e f'_j}{w}\right),\qquad
+ J_j^+=\bar J_j+e f'_j.
+\end{equation}
+The constants $A,w,e,B'$ are positive; their order of choice is given
+below. Put $h^+=k^{(d)}$, $\phi^+=\varphi^{(d)}$, and
+\[
+ \mathcal W=L_f\prod_{j=2}^dL'_j.
+\]
+The successive volume identities give $dV_{h^+}=\mathcal W\,dV_k$
+relative to the current base metric $k$. On the eventual child that
+base will be the original induced metric, so this product will be the
+density in \eqref{eq:degree-child-scalar}. In the metric after the first graph, $|df|\le1$ by
+\eqref{eq:degree-graph-elementary}, and later graph metrics only
+increase. The values and first derivatives of $\bar J$ are uniformly
+bounded in the region in question. It follows that in each current
+metric
+\begin{equation}\label{eq:degree-ell-bound}
+ |d\ell_j|\le C_\ell\mathbf1_{\{f\ge K\}},
+\end{equation}
+where $C_\ell$ depends on $A$ and the earlier band bounds but not on
+$K$, $M_*$, $k'$, or the final cover budget.
-\begin{proposition}[Regularity]
-\label{prop:degree-regularity}
-The support \(\Sigma_s^{\mathrm{all}}:=\supp\partial[E_s]\) of the
-boundary has a smooth regular part (a \(C^\infty\) hypersurface)
-and a closed singular part of Hausdorff dimension at most \(p-7\).
-The singular part is empty when \(p<7\).
-\end{proposition}
+For a row graph the analogue of
+\eqref{eq:degree-first-graph-differentiation} is
+\[
+ (D'_j)^2-2U'_j(D'_j)
+ =(D'_j)^2-\frac2w\left(B'+\frac{(D'_j)^2}{B'}\right)U'_j(\ell_j)
+ +\frac{2e}{w}\left(B'+\frac{(D'_j)^2}{B'}\right)
+ (L'_j-(L'_j)^{-1}).
+\]
+Choose $B'$ after $w,e$ so that
+\begin{equation}\label{eq:degree-row-B-choice}
+ \frac{B'e}{w}\ge2,\qquad
+ \frac{2C_\ell}{wB'}\le\frac12.
+\end{equation}
+Then the differentiated $\ell_j$ spends at most half of $(D'_j)^2$ and
+the fixed amount $2B'C_\ell/w$ on $\{f\ge K\}$, while the last term pays
+$2\ln L'_j$ everywhere. The other terms of
+\eqref{eq:degree-graph-identity} are nonnegative. After all $p$ rows,
+the sum of fixed errors is therefore at most
+$C_{\mathrm{rows}}\mathbf1_{\{f\ge K\}}$, with
+$C_{\mathrm{rows}}$ independent of $K,M_*,k'$ and the cover budget.
+On this support, $f<f_0+2$ implies
+\[
+ u_\alpha>\exp((K-2)/\sigma),\qquad
+ \Lambda\ge c'\exp((K-2)/(\sigma\alpha))
+\]
+by \eqref{eq:degree-moments}. Choose $K$ large enough that the remaining
+$c_1\Lambda^2$ in \eqref{eq:degree-first-graph-gain} covers
+$C_{\mathrm{rows}}$ there. This proves the required payment for all row
+graphs without any estimate on their individual volume expansions.
+
+Pole avoidance in \eqref{eq:degree-row-graph} also gives the output
+estimates that will be used for the band sides and its gradients:
+\begin{equation}\label{eq:degree-output-estimates}
+ \begin{split}
+ |J_j^+-(1+A\zeta(f))\bar J_j|&<2w,\\
+ |dJ_j^+|_{h^+}&\le |dG_j|_h+\varepsilon_{\mathrm{grad}},\\
+ |J_j^+|&\ge |G_j|-\varepsilon_{\mathrm{val}}.
+ \end{split}
+\end{equation}
+For the gradient inequality,
+$|df'_j|\le1$ in its own graph metric and hence in $h^+$, and all these
+metrics dominate the original induced metric and the parent metric on
+tangent vectors. Thus
+$|dJ_j^+|_{h^+}\le|d\bar J_j|_h+e$. For the value inequality use
+$1+A\zeta\ge1$ in the first line. The quantities
+$\varepsilon_{\mathrm{grad}}$ and $\varepsilon_{\mathrm{val}}$ can be as
+small as prescribed by the first-derivative and value perturbation
+budgets, respectively, and by $e,w$. We choose them to satisfy the
+target budgets \eqref{eq:degree-tolerances} and also to be smaller than
+all the fixed side margins used below.
+
+The full dependency order is
+\[
+ \begin{aligned}
+ &\text{perturbation budgets}\ \longrightarrow\ (\bar J,a_0,k)
+ \ \longrightarrow\ R\\
+ &\qquad\longrightarrow\ (\text{scalar error},\varepsilon_{\mathrm{base}})
+ \ \longrightarrow\ B_0\ \longrightarrow\ \sigma\\
+ &\qquad\longrightarrow\ A\ \longrightarrow\ (w,e)
+ \ \longrightarrow\ B'\ \longrightarrow\ K\ \longrightarrow\ M_*\\
+ &\qquad\longrightarrow\ (\text{cover budget},k',\text{radius lower bound}).
+ \end{aligned}
+\]
+Each later choice may use the preceding ones. The intended scalar error
+is much smaller than $1/q$, and $\varepsilon_{\mathrm{base}}$ is small
+enough for \eqref{eq:degree-potential-laplacian}. The constants in
+\eqref{eq:degree-moments} and the positive constant in
+\eqref{eq:degree-weight-gain} are independent of the future cover, so
+$B_0$ and $\sigma$ can be fixed for
+\eqref{eq:degree-first-graph-gain} at this stage. Choose $A$ large compared with
+$T_p/a_0$, then $w$ small compared with $a_0$ and all value margins,
+and $e$ small for \eqref{eq:degree-tolerances}. Choose $B'$ by
+\eqref{eq:degree-row-B-choice}, and only then $K$ to cover the row errors.
+Also make $K$ larger than the baseline value
+$\sigma\ln(\varepsilon_{\mathrm{base}}^\alpha)$ by a fixed margin.
+Choose $M_*$ so large that
+\eqref{eq:degree-potential-on-bad} implies
+\begin{equation}\label{eq:degree-high-on-bad}
+ f_0>K+100\quad\hbox{on }D_{k'}.
+\end{equation}
+Finally choose the cover budget sufficiently small for all the active-mass
+and error requirements, and obtain $k'\ge k$ and a cover with its positive
+lower radius bound. This final selection is possible for every fixed
+$M_*$ by \eqref{eq:degree-cover-budget}. It also provides all finite
+uniform derivative bounds needed for the actual equations. In
+particular, none of the constants that forced the choices of $K$ or $M_*$
+was allowed to depend on the final level or cover.
+
+\paragraph{Solving on a complete regular region.}
+The thresholds have distinct jobs: $K+100$ places the marked set beyond
+the auxiliary cutoff,
+$K+12$ bounds the auxiliary region for solving, and $K+7$ bounds the
+child inside the part where its base metric is unchanged. The lower
+threshold $K+4$ will already put the output outside the target near the
+potential cutoff side. Their gaps provide the spatial buffers below
+through the uniform derivative bounds for $f_0$.
+
+$\Sigma_s$ need not be complete. Use the operations
+only on the open region
+\begin{equation}\label{eq:degree-auxiliary-region}
+ \Omega=\Sigma_s\cap\{|G|_\infty<t_2,\ f_0<K+12\}.
+\end{equation}
+By \eqref{eq:degree-high-on-bad} it avoids $D_{k'}$ with a large margin in
+$f_0$; by $t_2<t_3$ it also has a margin within the region of the tests.
+The uniform derivative bounds on $f_0$ turn these into uniform spatial
+margins. Thus the graph tests through level $k'$ give regular graph
+charts with uniform buffers and all estimates on a neighborhood of this
+region. The estimates for derivatives of $\ln v$ apply there. The
+functions $u_j,\Lambda,f_0,\bar J$ have uniform derivatives because of
+the radius lower bound. The weight
+$\phi_\Sigma+(\ln u_\alpha+\ln u_{\alpha+4})/4$ therefore has the
+positive-order derivative bounds required for a graph solve.
+
+Choose a smooth function $0<\xi\le1$ on $\Omega$ which tends to zero
+at the sides in \eqref{eq:degree-auxiliary-region} and is one, with
+buffers, through the smaller bounds $|G|_\infty\le t_1$ and
+$f_0\le K+7$. It can be made from smooth functions of
+$t_2^2-G_i^2$ and $K+12-f_0$, linear near zero and cut off to one away
+from zero. Its derivatives in the original regular charts are bounded.
+Replace the induced metric on $\Omega$ by
+\[
+ k_{\mathrm{aux}}=\xi^{-2}h|_{T\Sigma}.
+\]
+This metric is complete with uniform buffered bounded geometry. Here is
+the local verification, which also accounts for possible ends of the
+original leaves. A ball in an original regular chart of radius a small
+fixed multiple of $\xi(x)$ remains in $\Omega$, since $\xi(x)$ is bounded
+by a constant times each scalar distance to a side. The bound on
+$d\xi$ makes $\xi$ comparable to $\xi(x)$ on that ball. Rescale the
+coordinate ball by $\xi(x)$. The coefficients of $\xi^{-2}h$ and their
+inverse then have uniform bounds of every order: for the $m$-th derivative
+of the conformal factor the new terms are products of ratios bounded by
+constants times $\xi(x)^m|\nabla^m\xi|/\xi$, and these are bounded for
+$m\ge1$.
+The rescaled boxes have uniform inner buffers. They also prove
+completeness. A curve of finite remaining length, or a Cauchy sequence,
+eventually stays within the outer part of one such buffered box and
+converges there; it cannot escape a leaf through a side or an uncharted
+end. Equivalently, approach to a side already has infinite length
+because $|d\ln\xi|_{k_{\mathrm{aux}}}$ is bounded. This argument used
+only the controlled charts near \eqref{eq:degree-auxiliary-region},
+not completeness of a parent leaf.
+
+The restricted chart descriptions are Borel and accessible charts still
+have a countable Borel enumeration. The auxiliary atlas may use further
+countable subdivisions; its total transverse measure is irrelevant for
+the graph existence theorem. Apply Lemma~\ref{lem:graph-prescription}
+on these complete leaves. For the first prescription its positive
+coefficient is $B_0+\Lambda$, its height target is $f_0$, and its
+constants $e,w$ in that lemma are both one. For a row prescription the
+coefficient is the positive constant $B'$, the target is $\ell_j$, and
+the constants are the chosen $e,w>0$. All coefficients have the
+uniform smooth bounds just established. That lemma gives the bounded
+solutions with a uniform margin from the poles and all derivative
+bounds, and gives their Borel dependence from uniqueness in the
+countably enumerated charts. It applies successively: a solved graph
+metric dominates the complete base, is complete, and has uniform
+geometry by the solution bounds; its changed weight again has uniform
+positive-order derivative bounds. In applying the scalar estimates
+\eqref{eq:degree-weight-gain}--\eqref{eq:degree-first-graph-gain}, use
+only the smaller region where $\xi=1$. There the auxiliary metric and
+the original induced metric agree with all derivatives.
+
+Define the child to be the open measured band
+\begin{equation}\label{eq:degree-child-domain}
+ X_s^+=\{x\in\Omega: |G|_\infty<t_1,\ f_0<K+7,
+ \ |J^+|_\infty<T_p\},
+\end{equation}
+with the final graph metric $h^+$, final graph weight $\phi^+$, and
+underlying boundary orientation. Its artificial sides stay away from
+each inner residual subcube. For the $|G|=t_1$ sides, confinement gives
+$|G_1|<\tau+b<T_p$, so any approach to such a side has a residual row
+$|G_j|$ near $t_1$. Choose
+$\varepsilon_{\mathrm{val}}<(t_1-T_p)/4$; then
+\eqref{eq:degree-output-estimates} puts $|J_j^+|$ beyond $T_p$ already
+on a uniform neighborhood of that side. For the $f_0=K+7$ side, if
+$f_0\ge K+4$ some kernel is active, since the baseline value of $f_0$
+is less than $K$. Its support lies in $|\bar J|_\infty>3a_0$.
+Moreover $f>f_0-2\ge K+2$, so $\zeta(f)=1$. Choose $A$ so that
+$(1+A)3a_0>T_p+4a_0$ and take $w<a_0/2$. The first line of
+\eqref{eq:degree-output-estimates} then puts some $|J_j^+|$ beyond
+$T_p$ with a positive uniform margin. This occurs before reaching
+$f_0=K+7$. In particular the entire child lies in the region where
+$\xi=1$, with buffers for all its inner subcubes.
+
+These observations verify every part of
+Definition~\ref{def:degree-band} for the child. To be explicit about its
+atlas, use the recognized regular graph patches through level $k'$ in
+each parent plaque, restricting their transversals as necessary. There
+are only boundedly many such tests per plaque. The derivative bounds
+for $G,f_0,J^+$ and the side margins just proved permit a further bounded
+subdivision covering any chosen inner residual subcube with buffers.
+The solved graph metrics and weights have uniform derivatives in these
+charts. Holonomy is inherited from the parent overlap maps, with the
+finite patch indices when needed. Hence both the total transverse
+measure and the plaque overlap bound persist. The transverse measure
+is precisely the restricted parent measure. It has no lapse factor and
+no graph-volume factor; those belong to the leafwise integrations.
+
+\paragraph{Preserving the degree through the modifications.}
+Use in \eqref{eq:degree-boundary-degree} a residual form $\beta$ supported
+in $|z|_\infty<a_0$, reducing $a_0$ if necessary. First homotope $J$ to
+$\bar J$ on the whole parent boundary current. The small value
+perturbation ensures that the preimages of this support throughout the
+homotopy remain in a fixed old inner band. One can justify invariance
+directly by the homotopy formula. The difference of the two pullbacks
+of $\beta$ is $d\mathcal H\beta$, where $\mathcal H\beta$ is its usual
+transgression along the homotopy. Multiply by the cutoff
+$\chi_{\mathrm{deg}}(G_1)$ used
+in \eqref{eq:degree-boundary-degree}. The current $\partial[E_s]$ is
+closed, and the term differentiating this cutoff meets no boundary on
+the small residual support, by confinement. Measured current Stokes,
+with the local mass bounds already proved, therefore makes the integral
+of the difference zero. This uses the boundary current across its
+singular set and so does not assume regularity of the homotopy there.
+
+The resulting integral for $\bar J$ is entirely in the regular region
+used for the solves. Indeed $|\bar J|_\infty<a_0$ avoids a neighborhood
+of $D_k$ by \eqref{eq:degree-gap} and avoids every kernel support, which
+lies in $|\bar J|_\infty>3a_0$. Thus $f_0$ has its baseline value there.
+The small perturbation also keeps $|G|$ in the smaller inner band.
+Now homotope on this regular region first from $\bar J$ to
+$(1+A\zeta(f))\bar J$ and then to $J^+$. Along the first homotopy the
+multiplier is at least one, so a preimage of the small target support
+has $|\bar J|_\infty<a_0$. Along the second, the first line of
+\eqref{eq:degree-output-estimates} shows that it has
+$|\bar J|_\infty<a_0+2w<2a_0$. Both supports thus avoid the marked
+region and the kernel supports, and retain the same spatial and
+$G_1$ margins. Ordinary measured Stokes on the regular buffered charts
+applies to these homotopies. The final support lies in
+\eqref{eq:degree-child-domain}, so its integral is the measured degree
+of the child. It equals $\mathcal D_X(G)$.
+
+On the child the initial auxiliary metric is the induced metric, so
+\eqref{eq:degree-graph-elementary} verifies the volume identity in
+\eqref{eq:degree-child-scalar} for the product $\mathcal W$ defined above.
+Choose all the adjustable scalar errors above with sum at most $1/q$.
+The absorption of the row errors and
+\eqref{eq:degree-first-graph-gain} give the scalar inequality in
+\eqref{eq:degree-child-scalar} on the child.
+This completes the construction in high dimensions. In the case
+$d\le7$ treated above, use $\mathcal W=1$. The volume identity is then
+immediate, and \eqref{eq:degree-slice-bound} gives the scalar inequality
+in \eqref{eq:degree-child-scalar}. In dimension $p=0$ the same statements
+hold on the counted points with the zero-dimensional scalar convention.
+
+\begin{proof}[Completion of the proof of Lemma~\ref{lem:degree-descent}]
+The coarea assertion is \eqref{eq:degree-coarea-proof}. The constructions
+above give a controlled child of degree $\mathcal D_X(G)$ for each typical
+$s$. Its coordinate bound follows from
+\eqref{eq:degree-output-estimates} and \eqref{eq:degree-tolerances}:
+\[
+ L_j|dJ_j^+|_{h^+}\le B-\frac dq+L_j\varepsilon_{\mathrm{grad}}
+ <B-\frac pq.
+\]
+For a low-dimensional child the same statement holds with zero error.
+The horizon condition \eqref{eq:degree-horizon} for its rows is unchanged,
+since $q,B,\tau,L_2,\ldots,L_d$ are unchanged.
+
+Every point with $|J^+|_\infty<b_p$ comes from
+$|G|_\infty<b_d$. For residual rows this follows from the last line of
+\eqref{eq:degree-output-estimates} and
+$\varepsilon_{\mathrm{val}}<b_d-b_p$; in low dimensions it is immediate.
+For the first row confinement gives
+\[
+ |G_1|<\tau+b<\tau+\frac{\tau}{10q}<b_d.
+\]
+The same observation covers the counted case $p=0$.
-\begin{proof}[Proof sketch]
-Normalize locally by the conformal change
-$k=e^{2(\phi-\phi_0)/p}h$. The weighted perimeter becomes ordinary
-perimeter in $k$, and the forcing becomes a smoothly bounded
-function. Then \cite[Theorem~A.1, Remark~A.3]{Eichmair2009}
-gives \(C^{1,\alpha}_{\mathrm{loc}}\) regularity on the regular
-part. Elliptic bootstrap (proscribed mean-curvature equation with
-smooth coefficients) makes the regular part \(C^\infty\).
+The function $x\mapsto(x-1)_+$ is one-Lipschitz, so
+\eqref{eq:degree-output-estimates} gives
+\[
+ \sum_{j=2}^dH_j^+\le\sum_{j=2}^dH_j(G)
+ +2\sum_{j=2}^dL_j\varepsilon_{\mathrm{grad}}.
+\]
+Use this and the scalar and volume relations in
+\eqref{eq:degree-child-scalar} to compare the integrands on the child:
+\begin{equation}\label{eq:degree-integrand-cancellation}
+ \begin{split}
+ &\exp\left(\frac{-\wscal(h^+,\phi^+)+\sum_{j=2}^dH_j^+}{2}\right)
+ \mathcal W\\
+ &\quad\le
+ \exp\left(\frac1{2q}+\sum_{j=2}^dL_j\varepsilon_{\mathrm{grad}}\right)
+ \frac v{L_1}
+ \exp\left(\frac{-\wscal(h,\phi)+\sum_{i=1}^dH_i(G)}2\right)\\
+ &\quad\le e^{3/(5q)}\frac v{L_1}\Theta.
+ \end{split}
+\end{equation}
+The factor $\mathcal W$ cancels exactly against the term
+$2\ln\mathcal W$ in the scalar bound. The factor $v/L_1$ is the lapse
+factor in coarea. This calculation makes no bound on either factor.
+For a low-dimensional child use \eqref{eq:degree-slice-bound}, with no
+row error; for $p=0$ the same estimate is the pointwise bound for a
+counted point. Integrate \eqref{eq:degree-integrand-cancellation} over
+the child and use its containment in the parent $b_d$ region. This is
+\eqref{eq:degree-child-integral}, for example with $C_*=1$.
\end{proof}
-\begin{proposition}[Compactness]
-\label{prop:degree-compactness}
-Sequences \(E_j\) of minimal sets, with smooth local convergence
-of metrics, weights, and forcings, have BV and \(L^1_{\mathrm{loc}}\)
-limits that are again minimal. Boundary varifolds converge weakly,
-and the support converges in the Hausdorff sense. Near any
-regular point of the limit, single smooth graphs converge in
-$C^\infty$ on a common cylinder.
-\end{proposition}
+\subsection{Completion of the induction}
-\begin{proof}[Proof sketch]
-Standard BV compactness: uniform local perimeter bounds give weak
-\(BV\) convergence; lower semicontinuity of the perimeter and
-convergence of the bounded forcing integrals pass minimality to
-the limit. Varifold convergence and the graphical regularity near
-regular points follow from Allard's interior regularity theorem
-\cite{Allard1972} plus a bootstrap on the prescribed
-mean-curvature equation.
+\begin{proof}[Proof of Theorem~\ref{thm:degree}]
+The case $d=0$ is the signed-count inequality in the statement. Suppose
+the theorem is known in dimension $p=d-1$, with the same $q,B,\tau$ and
+the residual lengths. Its hypotheses hold for the child of
+Lemma~\ref{lem:degree-descent}, including the unchanged strict horizon
+condition. For almost every $s$ the induction hypothesis and
+\eqref{eq:degree-child-integral} give
+\[
+ |\mathcal D_X(G)|(2\tau)^p
+ \le \frac{e^{C_1p/q+C_*/q}}{\prod_{i=1}^dL_i}
+ \int_{\Sigma_s\cap\{|G|_\infty<b_d\}}v_s\Theta\,dA_h.
+\]
+For $p=0$ this is the same use of the unsigned-count bound. Take
+$C_1\ge C_*$. Integrate this scalar inequality over $-\tau<s<\tau$
+and apply \eqref{eq:degree-coarea} with
+$F=\mathbf1_{\{|G|_\infty<b_d\}}\Theta$. The factor $2\tau$ on the
+left changes its power from $p$ to $d$, and the right is bounded by
+the integral in \eqref{eq:degree-estimate}. This proves the induction.
+
+There is no implicit measurable selection of children in this last step.
+For each typical $s$ a child proves the displayed scalar inequality
+against the same parent integral. That parent integral is a measurable
+function of $s$ by the lapse construction and is governed by
+\eqref{eq:degree-coarea}; it is the only function being integrated.
+Its integral is finite because $\Theta$ is bounded on the controlled
+$b_d$ region, which has finite measured volume.
+The constants $C_0$ and $C_1$ are absolute: $C_0$ came from the fixed
+pressure profile, and each descent costs at most $e^{C_*/q}$ after the
+arbitrarily small slice errors were chosen. All geometry-dependent
+constants were used only to make those choices.
\end{proof}
+\section{From bounded cohomology to comparison degree}\label{sec:transfer}
+
+The entropy construction supplies a root probability map \(F\) whose
+energy \(e_h(F)\) has a large coefficient in the weighted scalar-curvature
+inequality. Its coordinatewise square takes values in the probability
+simplex. We now turn its pairing with the bounded cocycle into comparison
+degrees to which Theorem~\ref{thm:degree} applies. A common integer
+clears the rounded cubical coefficients, giving integer multiplicities;
+the resulting measured degree may still be real. Its expectation will
+retain the original cocycle pairing exactly. The different lengths
+assigned to the tangent and normal directions of the realization are
+what remove the auxiliary dimension from the final estimate.
+
+The probability-map and auxiliary-degree strategy is adapted from
+Section~3 of \cite{OpenAIScalarInessentiality2026}: Lemmas~3.2 and 3.4
+develop the probability map and dual chains, and Proposition~3.6 provides
+their pairing. Lemma~3.8 supplies a transverse parameter cycle with
+nonzero intersection count; the proof of Theorem~3.1 constructs the
+comparison map and identifies its local degree. We prove the
+quantitative measured transfer and its estimates here.
+
+\begin{theorem}\label{thm:scalar-volume}
+For every integer \(n\geq 3\), there is a finite constant \(C_n>0\) such
+that every closed connected oriented smooth \(n\)-manifold \(M\) and every
+smooth Riemannian metric \(g\) with \(\Scal_g\equiv-1\) satisfy
+\[
+ \|M\|\leq C_n\Vol_g(M).
+\]
+\end{theorem}
-\subsubsection{Confinement}
-
-\begin{lemma}[Confinement]
-\label{lem:degree-confinement}
-For sufficiently small $\eta>0$, uniformly for \(-\tau<s<\tau\),
-\begin{equation}
- \supp\partial[E_s]\cap\{|G|_\infty\le t_4\}
- \subset\{|s-G_1|<b-\eta\}.
- \label{eq:degree-confinement}
+We prepare the proof for a manifold with \(\|M\|>0\). Choose the
+homogeneous alternating bounded cocycle \(c\), the initial probability
+map, and the number \(I\geq\|M\|/2\) provided by
+Proposition~\ref{prop:detector}; in particular \(\|c\|_\infty\leq1\).
+We first consider conditional data having the properties asserted by
+Theorem~\ref{thm:entropy}, without yet invoking their existence.
+Denote the scalar constant in this contract by \(C_{\rm ent}\).
+All functions called elementary in this section have a fixed height of
+exponentiation on a polynomial, with parameters depending only on \(n\).
+
+Write \(\mathcal M=(\widetilde M\times\Omega)/\Gamma\) for such a measured
+family. Denote its metric, weight, density, and
+root probability map by \(h,\phi,\rho,F\), and put
+\[
+ \mathfrak e=e_h(F),\qquad W=\log(dV_h/\rho).
+\]
+Its estimates are
+\begin{equation}\label{eq:transfer-entropy}
+ \rho\leq dV_g,\qquad W\geq0,\qquad
+ \wscal(h,\phi)\geq-C_{\rm ent}+2W-n\log E+E\mathfrak e,
+ \qquad \mathfrak e^{1/2}\leq E^{-1/100}.
\end{equation}
-In particular, the low side $G_1\le s-b+\eta$ is contained in
-\(E_s\) and the high side $G_1\ge s+b-\eta$ is outside \(E_s\),
-up to null sets.
-\end{lemma}
-
-\begin{proof}[Proof sketch]
-From \eqref{eq:degree-pressure}, near the positive pole
-\begin{equation}
- b-x=\frac1{L_1}\int_{P_0(x)}^\infty\frac{dy}{\psi(y)}
- \asymp_{\mathrm{inst}}P_0(x)^{-1/2}
- \quad(x\uparrow b).
-\end{equation}
-So at distance $\sim\eta$ from a pole, the forcing magnitude
-$|P_0(s-G_1)|$ is at least $c_{\mathrm{inst}}\eta^{-2}$
-(inst-dependent lower bound). Monotonicity of $P_0$ makes this
-persist beyond. Using the uniform derivative bound on \(G_1\), a
-chart ball of radius $r=c'\eta$ centered where $s-G_1\ge b-\eta$
-has forcing $\ge h_{\min}=c''\eta^{-2}$. For large enough
-\(h_{\min}\) relative to the local metric, weight, and curvature
-bounds, standard comparison with a corner barrier shows the entire
-ball lies in \(E_s\). The mirror argument at the other pole gives
-the low-side inclusion.
-\end{proof}
-\subsection{The lapse and coarea}
-\label{sec:degree-lapse}
-
-The regular portions of the minimizers \(\partial[E_s]\) sweep out
-the target band as \(s\) varies. The \emph{lapse} \(v\) is the ratio
-of the swept area to the ambient volume along a leaf; it controls
-both the coarea formula and the scalar identity.
-
-\subsubsection{Differentiating the ordered family}
-
-On each leaf the sets \(E_t\) converge to \(E_s\) in
-\(L^1_{\mathrm{loc}}\) as \(t\to s\), except at countably many \(s\).
-At such a \emph{continuity parameter}, the minimizer boundaries
-converge smoothly and graphically near each regular point of
-\(\Sigma_s\).
-
-In a graph cylinder choose coordinates \((y,z)\) with \(E_t\)
-the subgraph below \(z=r(t,y)\). The height functions \(r(t,\cdot)\)
-are monotone in \(t\). If \(r(t,\cdot)\ge r(s,\cdot)\) for \(t>s\)
-and the two graphs touch, the comparison of mean curvatures at
-contact gives \(D_t\le D_s\), contradicting \(D_t=\mathfrak h_t>
-\mathfrak h_s=D_s\) at the same point. Hence the graphs are
-strictly ordered.
-
-Taking dense test coordinates and using monotone-function
-differentiability, the nonnegative difference quotients
-\[
- w_t(y):=\frac{r(t,y)-r(s,y)}{t-s}
- \quad (t\ne s)
-\]
-have finite limits as \(t\to s\) almost surely in \(y\). These
-define the normal speed of the sweeping.
-
-Combining the differential and measurable aspects gives:
-
-\begin{lemma}[Coarea with lapse]
-\label{lem:degree-coarea}
-Define the lapse \(v\) on the regular part of \(\Sigma_s\) in the
-inner band \(\{|G|_\infty<t_3\}\), extended by zero off exceptional
-parameters and singular sets. Then for every nonnegative Borel
-\(F\) supported in \(\{|G|_\infty<t_3\}\),
-\begin{equation}
- \int_{-\tau}^{\tau}\!\!ds\int_{\Sigma_s}vF\,dA_h
- \le\int_XF\,dV_h.
- \label{eq:degree-coarea}
+There is a fixed invariant graph on the countable label set, with finitely
+many free \(\Gamma\)-orbits, containing every cooccurrence of nonzero
+coordinates of \(F\). The integer \(q\geq n+1\) bounds the number of
+orbits and the maximum graph degree plus one. The equivariant coloring
+by \(q\) colors separates any two labels at graph distance at most two.
+Its dependence on \(\omega\in\Omega\) is Borel. Set
+\(m=q-n>0\). Integrals over \(\mathcal M\) and over the measured spaces
+constructed below include their transverse measures.
+
+\medskip\noindent\emph{Conditional transfer target.}
+For every \(n\geq3\), there are a constant \(K_n>0\) and an elementary
+function \(\Psi_n:[1,\infty)\to(0,\infty)\), chosen from \(n\) alone,
+such that every choice of detector and conditional entropy data just
+specified, including its constant \(C_{\rm ent}\), satisfies
+\begin{equation}\label{eq:transfer-conditional-target}
+ E\geq\Psi_n(q)\quad\Longrightarrow\quad
+ I\leq K_n e^{C_{\rm ent}/2}\int_{\mathcal M}\rho.
\end{equation}
-\end{lemma}
-
-\begin{proof}[Proof sketch]
-In each graph cylinder, the one-dimensional change-of-variables
-inequality for a monotone function gives
-\begin{equation}
- v\,dA_h|_{(y,r(s,y))}=J_h(y,r(s,y))\,\partial_sr(s,y)\,dy,
-\end{equation}
-where \(J_h\,dy\,dz\) is the ambient Riemannian volume in the
-coordinates. Integration against the almost-everywhere derivative
-counts at most integration in \(z\); hence a jump or singular part
-of the monotone function only removes mass. Apply this to each
-column and then Fubini in \(y\) and the transversal. Partitioning
-the counted portions among the countably many graph cylinders
-avoids double counting within one parameter (using noncontact).
-Hence no ambient point is counted more than once on the right.
-Borel properties of the counted sets follow from the Borel
-one-to-one image theorem.
-\end{proof}
-
-\subsubsection{Degree invariance across the family}
-
-The signed degree \(\mathcal D_X(G)\) of the parent map can be
-evaluated on any slice \(\Sigma_s\) in the inner band.
-
-\begin{lemma}[Degree on the slice]
-\label{lem:degree-boundary}
-For the family \(\{E_s\}_{-\tau<s<\tau}\) and the form \(\beta\) from
-Definition~\ref{def:degree-signed},
-\begin{equation}
- \int_{\partial[E_s]}\chi_{\mathrm{deg}}(G_1)J^*\beta
- =\mathcal D_X(G)
- \label{eq:degree-boundary-degree}
+Thus \(K_n\) and \(\Psi_n\) are fixed before the manifold, cocycle,
+instance geometry, and \(C_{\rm ent}\) are chosen. We carry out the
+estimates for each integer \(q\geq n+1\), uniformly over conditional
+data satisfying the stated label and coloring bounds.
+The grid, the common rounding, and an upper density bound for the
+resolution come first. That density bound permits a small collar;
+the ensuing curvature and rotation bounds then determine one comparison
+width and the required energy threshold. Only after the conditional
+target is closed will Theorem~\ref{thm:entropy} supply an actual pair
+\((q,E)\) and make \(C_{\rm ent}\) dimension-only.
+
+\subsection{A flat neighborhood and a cubical cocycle}
+
+For a fixed \(\omega\), let \(K=K_\omega\) consist of all nonnegative
+unit root vectors whose support is a clique in the label graph. Thus
+\(K\) is the union of the spherical orthants on those cliques. A vector
+\(y\in K\) has a color image \(z_y\in\mathbb R^q\), obtained by placing
+each of its root coordinates at the color of its label. Distinct labels
+in a clique have distinct colors. Distant cliques can reuse colors, so
+the flat neighborhood below keeps distinct label configurations with the
+same color image on separate sheets for comparison, while retaining
+Euclidean coordinates for the grid averages.
+
+\begin{lemma}[Flat sheet neighborhood]\label{lem:transfer-flat}
+Put \(\delta=(1000\sqrt q)^{-1}\). There is an oriented flat smooth
+\(q\)-manifold \(\mathcal U_\omega\) with a local isometry
+\(z:\mathcal U_\omega\to\mathbb R^q\), and an inclusion of \(K_\omega\)
+into it, such that the image of every \(y\in K_\omega\) has a flat ball
+chart of radius \(10\delta\) centered at \(z_y\). This construction is
+a standard Borel family in \(\omega\), with countably many charts and an
+equivariant \(\Gamma\)-action. The map \(F\) lifts smoothly to this
+family, and, writing \(z_F=z\circ F\),
+\[
+ e_h(z_F)=e_h(F)=\mathfrak e.
+\]
+There is a closed smooth \(n\)-form, again denoted \(\eta\), obtained
+in these flat charts by pulling back the detector form through the cutoff
+probabilities constructed below. It is Borel and equivariant in the
+family, and satisfies
+\begin{equation}\label{eq:transfer-form}
+ \int_{\mathcal M}F^*\eta=I,\qquad |\eta|_{\HS}\leq C_n.
\end{equation}
-for every \(s\in(-\tau,\tau)\), where
-$\chi_{\mathrm{deg}}(z)=\int_{-\infty}^z(a_-(u)-a_+(u))\,du$.
+Here the exterior Hilbert norm uses the Euclidean metric on the charts.
+For every fixed derivative order its derivatives admit an elementary
+bound in \(q\).
\end{lemma}
-\begin{proof}[Proof sketch]
-By confinement, the low support of \(a_-\) is contained in \(E_s\)
-and the high support of \(a_+\) is outside \(E_s\). The function
-\(\chi_{\mathrm{deg}}\) equals \(1\) on the residual target
-support. Measured Stokes (on the current \(\partial[E_s]\) with
-bounded local mass) therefore gives
-\begin{equation}
- \begin{split}
- \int_{\partial[E_s]}\chi_{\mathrm{deg}}(G_1)J^*\beta
- &=\int_{E_s}(a_-(G_1)-a_+(G_1))\,dG_1\wedge J^*\beta\\
- &=\int_Xa_-(G_1)\,dG_1\wedge J^*\beta=\mathcal D_X(G).
- \end{split}
-\end{equation}
-The last step uses the definition of \(a_+\), the orientation of
-\([E_s]\), and confinement.
+\begin{proof}
+Take one copy of \(B(z_y,10\delta)\subset\mathbb R^q\) for each witness
+\(y\in K\). At the same point \(z\) in two such balls, identify the two
+copies if their witnesses share a label in their nonzero supports. We
+check that these identifications give ball charts, rather than a branched
+space. For every point under consideration, some coordinate of \(z\)
+is at least \(q^{-1/2}-10\delta\). Every witness for that point has a
+nonzero label at this color. If two witnesses share any label, their
+labels at every common color agree: each of the two labels at that color
+is adjacent to the shared label, so distance-two separation of colors
+forces them to be the same. Conversely, agreement at the indicated
+large color is already a shared label. Thus, at \(z\), the proposed
+identification is precisely agreement on the label at that large color,
+which is an equivalence relation.
+
+The same color remains positive in all witnesses at every sufficiently
+nearby point; indeed the margin \(q^{-1/2}-21\delta>0\) suffices after
+a further displacement of at most \(\delta\). In that neighborhood
+the equivalence classes are still distinguished by its label. Each ball
+therefore projects injectively as a local chart. Points with distinct
+Euclidean projections can be separated in Euclidean space, and points
+with the same projection and different labels at the large color have
+disjoint chart neighborhoods just described. This proves the Hausdorff
+property. Rational dense choices of centers in each spherical orthant
+give a countable subatlas: there are countably many labels and cliques.
+The descriptions by labels, clique coordinates, and color assignments
+are Borel, and are unchanged by simultaneous translation of labels and
+\(\omega\). They prove all the claimed family and equivariance
+properties. At points of \(K\) this also shows that its inclusion is
+unambiguous.
+
+The lift of \(F\) is locally in one of these charts: its label at a
+chosen large color persists in a neighborhood. A nonnegative smooth
+root coordinate has zero differential wherever it is zero. All the
+nonzero coordinates at a point have different colors. It follows that
+the nonzero rows of \(dF\) are merely moved to different color positions
+by \(dz_F\), proving the asserted energy equality.
+
+To construct this cutoff pullback, choose a nonnegative smooth function \(\chi\)
+which is zero for \(t\leq12\delta\), equals \(t\) for
+\(t\geq25\delta\), lies between zero and \(\max\{t,0\}\), and satisfies the usual rescaled derivative bounds
+\(|\chi^{(j)}|\leq C_j\delta^{1-j}\) for \(j\geq1\). In a ball put
+\[
+ w(z)=\frac{(\chi(z_j))_{j=1}^q}{|(\chi(z_j))_{j=1}^q|},
+ \qquad p_j=w_j^2.
+\]
+The denominator is bounded below by an absolute positive constant.
+In fact \(z\) is within \(10\delta\) of a nonnegative unit vector, and
+the change in each coordinate caused by cutting it off is at most
+\(25\delta\); hence its total change is at most \(25\delta\sqrt q\).
+If \(\chi(z_j)\ne0\), every witness in this ball has a positive
+coordinate at color \(j\), because its distance from \(z\) is less
+than \(10\delta\). The label for every active coordinate is therefore
+unambiguous on chart overlaps. Evaluate
+\[
+ \eta=n!\,c(p,dp,\ldots,dp)
+\]
+using those labels. The cocycle law makes this form closed, as in
+Proposition~\ref{prop:detector}, and a straight probability homotopy
+from the old probabilities to these new ones preserves the measured
+pairing. Its local finiteness and smooth bounds are uniform for the
+instance, so the measured Stokes argument of that proposition applies.
+This gives the first identity in \eqref{eq:transfer-form}.
+
+For the dimension-only norm bound, first regard the form on nonnegative
+unit root vectors \(w\). In the Euclidean coordinate directions
+\(i_1,\ldots,i_n\), its component is
+\[
+ n!2^n\Bigl(\prod_{j=1}^n w_{i_j}\Bigr)
+ \sum_a w_a^2 c(a,i_1,\ldots,i_n).
+\]
+Its absolute value is at most \(n!2^n\prod_j w_{i_j}\). The sum of the
+squares of these products over all increasing \(n\)-tuples is at most
+\((\sum_i w_i^2)^n=1\). Thus its exterior Hilbert norm is at most a
+constant depending only on \(n\). The map \(z\mapsto w(z)\) has
+absolutely bounded operator norm of its differential: its denominator
+is bounded below, normalization is an orthogonal projection divided by
+that denominator, and \(\chi'\) is absolutely bounded. Pullback
+multiplies the exterior Hilbert norm by at most the \(n\)-th power of
+this operator bound. This proves the second assertion in
+\eqref{eq:transfer-form}. Differentiating the same formulas a fixed
+number of times gives elementary bounds (polynomial bounds in \(q\)
+already suffice), since \(\delta^{-1}=1000\sqrt q\).
\end{proof}
-\noindent The upshot: the degree can be computed either directly
-from the ambient \(X\) or as an integral over the minimizer
-boundaries \(\Sigma_s\). This is the bridge that lets us descend
-to the child.
-
-\subsection{Descent}
-\label{sec:degree-descent}
-
-We prove \eqref{eq:degree-estimate} by induction on the dimension
-\(d\), the parameter \(p=d-n\) decreasing. The induction step
-constructs a \emph{child} slice of dimension \(d-1\) to which the
-theorem can be applied with \(n\) unchanged.
-
-The construction splits into two regimes:
-\begin{itemize}
-\item \emph{Low dimension \(p<7\)}: the singular set is empty and
- the regular slice is a smooth manifold. The child is just
- $\Sigma_s$ restricted to the inner band.
-\item \emph{High dimension \(p\ge7\)}: the singular set has
- dimension $\le p-7$ and removing it can leave ends of the
- regular locus with unbounded local geometry. We use the
- distance potentials + graph prescriptions to construct a smooth
- child that stays away from the target cube.
-\end{itemize}
-
-\subsubsection{The child construction (low dimension)}
-
-\begin{lemma}[Descent, low dimension]
-\label{lem:degree-descent-low}
-For $p<7$, fix a typical \(s\). The regular slice \(X_s^+\)
-obtained by restricting $\Sigma_s=\partial[E_s]$ to the inner band
-$|G|_\infty<t_3$ is a measured \((d-1)\)-band with:
-\begin{itemize}
-\item a metric \(h^+=h|_{T\Sigma_s}\) and weight \(\phi^+=\phi|_{\Sigma_s}\);
-\item boundary control at lengths \(L_2,\ldots,L_d\) (the entry \(L_1\) drops out);
-\item signed degree \(|\mathcal D_{X_s^+}(G^+)|=|\mathcal D_X(G)|\).
-\end{itemize}
-\end{lemma}
+Choose a rotation \(V\in SO(q)\) with Haar probability, independently
+of the entropy data, and put \(\widehat z=Vz\). In these coordinates
+take the cubical grid of step \(r\) and shift \(s\), where \(s\) is
+uniform in \([0,r)^q\). Fix \(r=r(q)>0\) so that
+\(r\sqrt q\ll\delta\) and so that every \(n\)-component of \(\eta\)
+in every rotated frame changes by at most \(q^{-n}\) over a ball of
+radius \(10r\sqrt q\) in the inner sheet charts. The derivative bounds
+of Lemma~\ref{lem:transfer-flat} permit such a choice with an elementary
+bound on \(r^{-1}\), uniformly in \(V,s,\omega\).
+
+A primal \(n\)-cell has an axis set \(\mathcal I\subset\{1,\ldots,q\}\)
+of size \(n\), varies along those grid intervals, and has the remaining
+coordinates at grid vertices. Orient it in increasing axis order.
+Include each lift \(C\) whose center is at distance less than \(4\delta\)
+from some witness \(z_y\) in its sheet chart. The full cell fits in that
+chart. Its dual \(m\)-cube \(C^\#\) has the same center, varies along
+the complementary axes by \([-r/2,r/2]\), and has the other coordinates
+fixed. It too fits in the chart. Orient it so that the primal oriented
+basis, followed by the negatives of its dual oriented basis, is a
+positive basis of \(\mathbb R^q\). Give the primal cell the coefficient
+\begin{equation}\label{eq:transfer-cell-value}
+ u_C=\int_C\eta.
+\end{equation}
-\begin{proof}[Proof sketch]
-The regular slice is a smooth oriented submanifold of each leaf
-(no singular points, by Proposition~\ref{prop:degree-regularity}).
-Restriction of $h$ and $\phi$ is immediate. The degree $\mathcal
-D_{X_s^+}(G^+)$ equals $\mathcal D_X(G)$ by
-Lemma~\ref{lem:degree-boundary}. Boundary control is inherited
-with \(L_1\) dropped (the definition of \(H_1\) in $\Theta$ no
-longer applies on the slice).
-\end{proof}
-\subsubsection{The child construction (high dimension)}
-\label{sec:degree-descent-high}
-
-For $p\ge7$ the singular set $\mathcal S_s\subset\partial[E_s]$ has
-Hausdorff dimension $\le p-7$. Deleting it leaves a regular locus
-that may have ends with no uniform graph radius. The construction of
-the child must (a) keep those ends away from the small residual
-target, and (b) preserve the degree throughout.
-
-\emph{Distance potentials.} Fix a small dimensional $M_*>0$ and a
-Borel cover of $\mathcal S_s$ by balls $B(c_i,r_i)$. For an
-$\alpha\in(p-7,p-2)$ define
-\begin{equation}
- K_i(x):=(r_i^2+\dist_h(x,c_i)^2)^{-1/2},
- \qquad
- u_\alpha(x):=\sum_iM_*r_i^\alpha K_i(x)^\alpha,
- \label{eq:degree-kernel}
+These coefficients satisfy the cubical cocycle equations where they
+are needed. If a primal \((n+1)\)-cell has its center within
+\(2\delta\) of a witness, all its boundary \(n\)-faces are included
+with coherent lifts, and their signed integrals sum to zero by Stokes.
+A facet of a dual cube is centered at just such a primal
+\((n+1)\)-cell. The incidence signs of its incident dual cubes agree,
+up to one common sign, with the boundary signs of the primal cell:
+this follows by placing the varying primal axes first and the negative
+dual axes last in the ambient oriented basis. Consequently the dual
+\(m\)-cubes weighted by \(u_C\) have balanced oriented boundary near
+\(K\), in particular at every facet meeting the tube of radius
+\(\delta\). Indeed the corresponding centers differ from those
+facet points by at most \(r\sqrt q\), so they have the stipulated
+\(2\delta\) witnesses. Projected grid cells on other sheets give no
+incidence: all these statements use lifts in the same flat chart.
+
+\begin{lemma}[Finite-type unbiased rounding]\label{lem:transfer-rounding}
+There is an integer \(D_0\geq1\), bounded by an elementary function of
+\(q\) and constant in \(V,s,\omega\), and a Borel equivariant random
+choice of coefficients \(u'_C\in D_0^{-1}\mathbb Z\) such that
+\begin{equation}\label{eq:transfer-rounding}
+ \E_{\rm rd}u'_C=u_C,\qquad |u'_C-u_C|\leq r^nq^{-n}.
\end{equation}
-with smooth cutoffs on fixed radii. On the regular boundary,
-ellipticity of the distance function versus the mean curvature gives
-\begin{equation}
- -\Delta_{\Sigma,\phi}K_i^j\ge c\,K_i^{j+2}-C\indic_{V_i},
- \qquad |dK_i^j|\le C(K_i^{j+1}+1),
- \label{eq:degree-kernel-bounds}
-\end{equation}
-for every $j\le p-2$, with constants independent of $r_i$.
-
-\emph{Scalar gain.} Using both $\alpha$ and $\alpha+4$ (both less
-than $p-2$), define the potential mass $\Lambda:=(u_{\alpha+4}/u_\alpha)^{1/4}$
-and $\eta_0:=\frac14(\ln u_\alpha+\ln u_{\alpha+4})$. The
-superharmonicity of the potentials adds a positive scalar quantity
-to the right side of \eqref{eq:degree-weight-gain}; a direct
-computation gives
-\begin{equation}
- \wscal(h|_\Sigma,\phi_\Sigma+\eta_0)
- \ge\wscal(h,\phi)-H_1(G)-2\ln(v/L_1)
- +c\Lambda^2+\frac{\mathcal L}{4}-\delta,
- \label{eq:degree-weight-gain}
-\end{equation}
-where $\mathcal L:=|d\ln\Lambda|_{h|_\Sigma}^2$. The choice
-$\alpha+4<p-2$ makes the potential coefficient positive; $\delta$
-is arbitrarily small.
-
-\emph{Graph deformation.} On the marked set
-$D_{k'}:=\{u_\alpha\ge c''M_*\}\cap\{|G|_\infty<t_3\}$, use the
-tangent-branch prescription (Lemma~\ref{lem:graph-prescription}(ii))
-with target $f_0=\sigma\ln u_\alpha$ (large enough that $f_0>K+100$
-on $D_{k'}$) and small $e,w$. The resulting graph has residual
-$J^+$ satisfying
-\begin{equation}
- |\bar J-\bar J^+|<\varepsilon_{\mathrm{val}},
- \quad
- |dJ^+|_{h^+}\le|d\bar J|_{h}+\varepsilon_{\mathrm{grad}},
- \label{eq:degree-output-estimates}
-\end{equation}
-where $h^+$ is the graph metric. The choice of $e,w$ makes
-$\varepsilon_{\mathrm{grad}},\varepsilon_{\mathrm{val}}$ small
-against the residual value and gradient budgets of the parent.
-
-The composition of (the conformal change by $\eta_0$) and (the graph
-deformation) gives a smooth child $X_s^+$ supported away from
-$\mathcal S_s$, with the required $|J^+|_\infty\ge b_p$ wherever
-$|G|_\infty<b_d$.
-
-\begin{lemma}[Descent, high dimension]
-\label{lem:degree-descent-high}
-For $p\ge7$ the construction above produces a child $X_s^+$
-satisfying all properties of Lemma~\ref{lem:degree-descent-low},
-together with the \emph{integrand cancellation}
-\begin{equation}
- \begin{split}
- &\exp\Bigl(-\frac{\wscal(h^+,\phi^+)}{2}+\sum_{j=2}^q\frac{H_j(G^+)}{2}\Bigr)\mathcal W\\
- &\quad\le e^{3/(5q)}\frac v{L_1}
- \exp\Bigl(-\frac{\wscal(h,\phi)}{2}+\sum_{j=1}^q\frac{H_j(G)}{2}\Bigr),
- \end{split}
- \label{eq:degree-integrand-cancellation}
-\end{equation}
-where $\mathcal W:=dV_{h^+}/dA_h$ is the leaf density and $v$ is
-the lapse from Lemma~\ref{lem:degree-coarea}.
+The expectation is conditional on the grid and entropy data. Every
+cocycle equation just described is satisfied exactly by the \(u'_C\).
\end{lemma}
-\begin{proof}[Proof sketch]
-Assemble the potential construction, the scalar gain
-\eqref{eq:degree-weight-gain}, and the tangent-branch graph
-deformation. The factor $\mathcal W$ on the left cancels the
-$2\ln\mathcal W$ term in the scalar comparison; what survives is
-the lapse factor $v/L_1$. $\Lambda$ concentrates on $D_{k'}$ where
-the graph is large; the residual $J^+$ stays inside the residual
-target for small $\varepsilon_{\mathrm{grad}}$, giving the
-confinement needed. All choices are dimensionally bounded and
-Borel (via Lemma~\ref{lem:graph-prescription}).
+\begin{proof}
+The closure equations couple faces whose anchors can differ. We encode
+all local coefficients in one ordered finite vector and round within the
+joint kernel of its incidence equations, so every cell reads from the
+same rounded vector. We first establish that the vector has elementary
+size despite the infinitely many lifted cells.
+
+All projected centers lie
+in a fixed bounded Euclidean ball. For fixed \(V,s\), their grid
+indices and axis sets therefore have at most \((C/r)^{Cq}\) choices.
+At a selected center choose, breaking ties by color order, its largest
+unrotated color coordinate. This coordinate is at least
+\(q^{-1/2}-4\delta\), and its label occurs in every witness for the
+center. It determines the lift uniquely, by the proof of
+Lemma~\ref{lem:transfer-flat}; call it the anchor label.
+
+The closed graph one-neighborhood of an anchor has at most \(q\)
+labels. Every center-test witness, meaning a witness at distance less
+than \(4\delta\) from the center, lies in this neighborhood because it
+contains the anchor. Every point of the full cell is within
+\(4\delta+r\sqrt n/2<5\delta\) of each such witness. Thus every
+active color, whose coordinate is greater than \(12\delta\), occurs
+in each center-test witness. For an arbitrary ball-chart witness of
+the center the corresponding bound is instead
+\(10\delta+r\sqrt n/2<11\delta\), which is still below the active
+cutoff and gives the same conclusion. Thus the
+coefficient \(u_C\) is determined by its projected grid index, its
+anchor, and the color assignment on this one-neighborhood. In
+particular, no labels farther out in the graph enter its calculation.
+
+Choose and order representatives of the at most \(q\) anchor orbits,
+and order each of their finite one-neighborhoods. An anchor has a
+unique translate to its representative because the label action is
+free. After this translate, its neighborhood is a fixed finite set
+and the cocycle evaluations are unchanged by homogeneity. Fix one
+ordered list of slots consisting of an anchor representative, an
+injective color assignment on its neighborhood, an axis set and grid
+index, and a witness clique in that neighborhood containing the anchor.
+It is enough to use the fixed index box
+\(|k_j|\leq\lceil3/r\rceil+3\), which contains every possible selected
+center for all shifts. The slot is valid when the anchor color is the
+canonically chosen largest unrotated color of its center and that center
+is at distance less than \(4\delta\) from the spherical orthant of its
+witness clique. Give a valid slot the integral of the explicit form
+over its primal cell, using the labels specified by the slot, and give
+an invalid slot value zero. The preceding active-color observation
+makes this integral unambiguous. The validity tests and integrals are
+Borel in \(V,s\).
+
+This is one vector \(v=v(V,s)\in\mathbb R^{N_0}\) for the fixed graph
+and cocycle, common to all \(\omega\). Only its size and the bounds
+below are uniform over all graphs and cocycles of the given complexity:
+there are at most \(q^q\) neighborhood color assignments, \(2^q\)
+witness cliques, and \((C/r)^{Cq}\) projected indices, so \(N_0\) is
+elementarily bounded in \(q\). An actual cell receives the slot
+\(t(C,\omega)\) obtained by normalizing its canonical anchor, restricting
+its actual coloring, and choosing the first valid witness clique in the
+fixed order. Its value is \(u_C=v_{t(C,\omega)}\). This type is Borel
+and is unchanged by translating \(C\) and \(\omega\) together. Slots
+with the same anchor, restricted coloring, axes, and grid index but
+different valid witness cliques have the same value: the
+\(5\delta\) observation gives the identical active-label formula
+throughout the primal cell.
+
+Impose on this list every integer row \(b\) with
+\(\|b\|_\infty\leq2(n+1)\) for which \(b\cdot v=0\). There are
+finitely many candidate rows, so the resulting matrix depends Borelly
+on \(v\). In particular it includes the difference row for any two
+duplicate valid slots. It also includes every needed cubical equation,
+even when its faces have anchors normalized by different deck elements:
+all their values are entries of this same vector \(v\), and Stokes gives
+the signed zero sum of those entries. After repeated slots in its
+\(2(n+1)\) faces have been combined, each coefficient has the stated
+bound. The kernel of this
+integer matrix has a real basis of integer vectors \(b_1,\ldots,b_k\)
+with
+\[
+ k\leq N_0,\qquad
+ \max_j\|b_j\|_\infty\leq K_0:=(C_nN_0)^{C N_0}.
+\]
+To see this, choose a nonsingular maximal minor and solve for its pivot
+coordinates in terms of the free coordinates; multiplying each basis
+vector by its nonzero determinant makes it integral. Hadamard's
+bound for minors of a matrix with entries at most \(2(n+1)\) gives
+the displayed estimate. Lexicographically choosing the pivot minor
+makes the basis Borel. The case of a zero-dimensional kernel simply
+has \(v=0\) and needs no rounding.
+
+Write \(v=\sum_j t_jb_j\), using the pivot solution, which is Borel.
+Independently round each \(t_j\) to the two adjacent multiples of
+\(D_0^{-1}\), choosing their probabilities so that its conditional
+expectation is \(t_j\). Choose a single integer
+\(D_0\geq N_0K_0r^{-n}q^n\), using uniform elementary upper bounds
+for \(N_0,K_0\). The resulting vector \(v'\) lies in the same
+kernel, has entries in \(D_0^{-1}\mathbb Z\), has expectation \(v\),
+and satisfies \(\|v'-v\|_\infty\leq N_0K_0/D_0\leq r^nq^{-n}\).
+All operations, including taking floors and using independent uniform
+parameters for the adjacent choices, are Borel. The fixed slot order
+is used for every pivot choice. Thus one parameter \(\xi\), independent
+of \(\Omega\), gives one rounded vector \(v'(V,s,\xi)\) for all cells.
+Set \(u'_C=v'_{t(C,\omega)}\). The invariance of the type under
+translation proves equivariance, and the common zero rows prove all
+the asserted incidence equations.
\end{proof}
-\subsubsection{Induction}
-
-\begin{proof}[Proof of Theorem~\ref{thm:degree}]
-Induct on $d$; equivalently on $p=d-n$. Base case $d=n$ (i.e.
-$p=0$): this is the signed-count inequality of
-Definition~\ref{def:degree-signed}.
-
-For the induction step, construct the child $X_s^+$ via
-Lemma~\ref{lem:degree-descent-low} ($p<7$) or
-Lemma~\ref{lem:degree-descent-high} ($p\ge7$). The child inherits
-boundary control at lengths $L_2,\ldots,L_q$ (dropping \(L_1\)), and
-its signed degree satisfies $|\mathcal D_{X_s^+}(G^+)|=|\mathcal D_X(G)|$
-by Lemma~\ref{lem:degree-boundary}.
-
-By the induction hypothesis,
-\begin{equation}
- |\mathcal D_X(G)|(2\tau)^{p-1}
- \le\frac{e^{C_0(p-1)/q+C_*/q}}{\prod_{j=2}^qL_j}
- \int_{\Sigma_s\cap\{|G|_\infty<b_d\}}\Theta^+\,v^+\,dA_{h^+},
- \label{eq:degree-ind-step}
-\end{equation}
-where $\Theta^+:=\exp(-\wscal(h^+,\phi^+)/2+\sum_{j\ge2}H_j(G^+)/2)$.
-
-Combining with the integrand cancellation
-\eqref{eq:degree-integrand-cancellation} and the fact that the
-child's $b_p$-region lies in the parent's $b_d$-region (by
-confinement of $G_1$ and by the value control of $J^+$ near the
-boundary of the target cube) gives
-\begin{equation}
- |\mathcal D_X(G)|(2\tau)^{p-1}
- \le\frac{e^{C_0p/q+C_*/q}}{\prod_{i=1}^qL_i}
- \int_{\Sigma_s\cap\{|G|_\infty<b_d\}}v\Theta\,dA_h.
- \label{eq:degree-ind-closed}
+The rounded coefficients define the original oriented dual current
+\begin{equation}\label{eq:transfer-rounded-current}
+ Z_{\rm rd}:=\sum_C D_0u'_C[C^\#].
\end{equation}
-
-Integrating $s\in(-\tau,\tau)$ and applying coarea
-(Lemma~\ref{lem:degree-coarea}) turns the left side
-$(2\tau)^{p-1}\cdot2\tau$ into $(2\tau)^p$, and the right side
-into the full integral in \eqref{eq:degree-estimate}. This
-completes the induction.
-\end{proof}
-
-This completes the degree estimate. The upshot for the transfer:
-Theorem~\ref{thm:degree} bounds the measured signed degree of a
-map $G$ in terms of an integral involving $-\wscal(h,\phi)$ plus
-a coordinate-gradient penalty $H_i(G)$. Applied to a comparison
-map whose coordinate gradients are dimensionally small (as in
-\S\ref{sec:transfer}), this gives a clean $L^1$ bound on the
-degree in terms of $\int\rho$.
-\section{From bounded cohomology to comparison degree}
-\label{sec:transfer}
-
-This section closes the proof of Theorem~\ref{thm:main} by applying
-the degree estimate of \S\ref{sec:degree} to a comparison map built
-from the entropy datum $F$ of Theorem~\ref{thm:entropy}. The plan:
-
-\begin{enumerate}
-\item \emph{Conditional target.} Fix the entropy datum with
- parameters $q,E$. Set up a flat $q$-manifold $\mathcal U_\omega$
- carrying the cocycle $c$, so that the integral $\int_{\mathcal M}F^*\eta$
- equals the detector value $I$.
-\item \emph{Cubical complex.} Realize the cocycle as an exact form
- on the cubical grid, sum primal and dual contributions, and apply
- the closure equations.
-\item \emph{Rounding.} Round the coefficients to a lattice
- $D_0^{-1}\Z$ while preserving the cocycle equation and with
- control of the coefficients.
-\item \emph{Resolution.} Resolve the rounded chain into a smooth
- oriented $m$-manifold $\mathcal A$ with a map $X:\mathcal A\to
- \mathcal U_\omega$.
-\item \emph{Degree.} Apply Theorem~\ref{thm:degree} to the
- composition to get
- $I\lesssim e^{C_{\mathrm{ent}}/2}\int_{\mathcal M}\rho$.
-\end{enumerate}
-
-\subsection{Conditional entropy data}
-\label{sec:transfer-setup}
-
-Let $(M,g)$ satisfy $\Scal_g=-1$. If $\|M\|=0$ there is nothing
-to prove; assume $\|M\|>0$. Fix the detector cocycle $c$ and
-the value $I\ge\tfrac12\|M\|$ from \S\ref{sec:reduction-detector}.
-
-Apply Theorem~\ref{thm:entropy} with a prescribed elementary
-function $\mathcal B$ to be specified in \S\ref{sec:transfer-bound}
-below. The output is:
-\begin{itemize}
-\item a measured family $\mathcal M$ with metric \(h\), weight
- $\phi$, density $\rho$, and root probability map $F$;
-\item an integer $E$ and a natural number $q\ge n+1$;
-\item a fixed $\Gamma$-invariant graph $\mathcal G$ on the label
- set $\mathcal L$, of maximum degree $\Delta\le q-1$, containing
- every cooccurring pair;
-\item a Borel equivariant $q$-coloring of $\mathcal L$ that
- separates labels at $\mathcal G$-distance $\le2$.
-\end{itemize}
-Set $\mathfrak e:=e_h(F)$ and $W:=\log(dV_h/\rho)$. Then
-\begin{equation}
- \rho\le dV_g,\quad W\ge0,\quad
- \wscal(h,\phi)\ge-C_{\mathrm{ent}}+2W-n\log E+E\mathfrak e,
- \quad \mathfrak e^{1/2}\le E^{-1/100}.
- \label{eq:transfer-entropy}
+Here \([C^\#]\) is integration over the oriented cube, and the
+coefficients are integers. We verify the local finiteness implicit
+in this notation. The form bound and rounding give
+\(|u_C|\leq C_nr^n\) and an elementary bound for each \(|D_0u'_C|\).
+Let \(B\) be a small flat ball in one sheet chart and \(Q\) one full
+projected dual cube. The intersection \(Q\cap z(B)\) is convex. Any
+lift of \(Q\) meeting \(B\) agrees with the chart inverse on this
+connected intersection, by uniqueness of lifts for the local
+diffeomorphism \(z\). Thus two lifts meeting \(B\), even at different
+projected points, agree there and then agree on the connected cube
+\(Q\). At most one geometric lift of each projected cube meets \(B\).
+Only finitely many projected grid cubes meet \(z(B)\), with bounded
+coefficients, proving neighborhood local finiteness.
+
+Let \(\mathcal T_\delta\subset\mathcal U_\omega\) be the strict tube
+of points at distance less than \(\delta\) from a witness in its ball
+chart. The exact incidence equations in
+Lemma~\ref{lem:transfer-rounding} give
+\begin{equation}\label{eq:transfer-current-closed}
+ \partial Z_{\rm rd}\big|_{\mathcal T_\delta}=0.
\end{equation}
-
-\subsection{Flat sheet neighborhood}
-\label{sec:transfer-flat}
-
-The probability simplex is not flat, but a specific
-\emph{flat neighborhood} of it supports the cocycle form $\eta$
-without loss.
-
-\begin{lemma}[Flat sheet neighborhood]
-\label{lem:transfer-flat}
-Let $\delta:=(1000\sqrt q)^{-1}$. There are:
-\begin{itemize}
-\item an oriented flat smooth $q$-manifold $\mathcal U_\omega$;
-\item a local isometry $z:\mathcal U_\omega\to\R^q$;
-\item an inclusion of the witness set $K$ into $\mathcal U_\omega$,
- such that the image of every $y\in K$ has a flat chart of radius
- $10\delta$ centered at $z_y$;
-\item a standard Borel family structure in $\omega$, with
- countably many charts and an equivariant $\Gamma$-action.
-\end{itemize}
-The map $F$ lifts smoothly to this family, with lifted $z_F=z\circ F$
-satisfying $e_h(z_F)=e_h(F)$. There is a closed smooth $n$-form
-$\eta$ on $\mathcal U_\omega$, localizing the cocycle $c$, such that
-\begin{equation}
- \int_{\mathcal M}F^*\eta=I,\qquad |\eta|_{\mathrm{HS}}\le C_n,
- \label{eq:transfer-form}
+This is a local assertion: the boundary current vanishes on test forms
+compactly supported in \(\mathcal T_\delta\).
+
+To resolve \(Z_{\rm rd}\) smoothly, realize each coefficient by
+\(|D_0u'_C|\) oriented copies of its cube, reversing orientation when
+\(u'_C<0\). At every facet where boundary balance is required, pair
+positive and negative incident copies and glue them by their common
+face. Leave other facets free, and double this diagram with an
+oppositely oriented mirror, joining each free facet to its mirror.
+All codimension-one faces of the doubled diagram are now paired.
+Local axis, grid, and copy indices order the incident copies, so these
+choices may be Borel and equivariant.
+
+The doubling does not join the two sides near \(K\), even through a
+lower face. If a point of any face maps into \(\mathcal T_\delta\),
+every path of facet identifications relating chambers incident to that
+face preserves its image point. Each facet on the path meets the tube,
+so was balanced before doubling and was not a free mirror facet. The
+entire incidence class at that point stays on its original side or its
+mirror side. At a point of a lifted grid face, only boundedly many
+local grid indices and their boundedly many copies can meet. Facet
+identifications preserve that point and its sheet, so even after
+doubling they relate only this bounded collection of incident chambers.
+
+Take barycentric subdivision of the doubled cubes. Each top simplex
+\(\sigma\) has vertices colored \(0,\ldots,m\) by the dimensions of
+its barycentric faces, and every facet pairing respects these colors.
+Write \(\varepsilon_\sigma\in\{1,-1\}\) for its sign relative to that
+color order, and call this doubled signed simplicial diagram
+\(\mathcal P\). It maps facewise to \(\mathcal U_\omega\), with
+elementary subdivision and incidence bounds. The original current
+\(Z_{\rm rd}\) remains the object to be represented on
+\(\mathcal T_\delta\); the doubling supplies closed facet pairings
+for the resolution.
+
+\subsection{A controlled smooth measured resolution}
+
+The diagram need not be a manifold at faces of codimension greater
+than one. The following construction replaces it by a measured family
+of smooth manifolds while preserving the represented chain. The
+small portion on which its metric is not the pulled-back flat metric
+will be controlled by its coordinate volume.
+
+The reflected gluing below uses Gaifullin's explicit permutohedral
+resolution, including the face-preserving sign-changing pairings,
+their conjugation under reflections, and the auxiliary bit toggles
+\cite[pp.~562--563]{Gaifullin2008}. We replace his rank-indexed bits
+by independent subset-indexed bits, put invariant probability laws on
+the pairing data, and supply smooth measured charts and quantitative
+metric bounds.
+
+\begin{lemma}[Measured resolution]\label{lem:transfer-resolution}
+For the diagram \(\mathcal P\) there is a standard Borel measured
+family \(\mathcal A\) of oriented smooth \(m\)-manifolds and a smooth
+map \(X:\mathcal A\to\mathcal U_\omega\), equivariant and Borel in
+all the data. The integration identities below are for fixed grid,
+entropy, and rounding data, including \((V,s,\omega,\xi)\); their
+integrals use the conditional state measure and depend Borelly on those
+data. The points over \(\mathcal T_\delta\) represented by
+original chamber tiles form a Borel leaf-open subset
+\(\mathcal A_{\rm orig}\subset X^{-1}(\mathcal T_\delta)\). For every
+compactly supported smooth \(m\)-form \(\alpha\) on \(\mathcal T_\delta\),
+\begin{equation}\label{eq:transfer-local-push}
+ \int_{\mathcal A_{\rm orig}}X^*\alpha=Z_{\rm rd}(\alpha).
+\end{equation}
+For the full doubled diagram, measured oriented integration also gives
+\begin{equation}\label{eq:transfer-push}
+ X_*[\mathcal A]=[\mathcal P].
+\end{equation}
+For a parameter \(0<\lambda<1/2\) and a choice
+\begin{equation}\label{eq:transfer-gamma}
+ 0<2\gamma_0<(m+1)^{-(m+1)}\lambda^{m(m+1)},
\end{equation}
-and all fixed-order derivatives of $\eta$ admit an elementary bound
-in $q$.
+there are a smooth metric \(k\) and a closed seam set in each tile such
+that \(\|dX\|_k\leq1\), and \(k=X^*k_{\rm flat}\) off the seam.
+There are buffered smooth charts with elementary overlap, metric,
+inverse-metric, and fixed-order derivative bounds whenever
+\(\lambda^{-1}\) and \(\gamma_0^{-1}\) are elementary. In the sorted
+charts for each tile, the seam is contained in the union of the slabs
+\(|Z_j|\leq\lambda\), \(1\leq j\leq m\). The \(k\)-volume density
+in those charts has an elementary upper bound chosen before
+\(\lambda,\gamma_0\); in particular this bound does not depend on
+these two parameters.
+
+One may prescribe for each chamber over a dual cube a rotation
+\(O_\sigma\in SO(q)\). There is also a smooth Borel equivariant
+\(\mathcal R:\mathcal A\to SO(q)\) equal to \(O_\sigma\) off the
+seam in that tile, equal to the identity near all tile facets, and
+homotopic to the identity. Its fixed-order chart derivatives and
+\(\|d\mathcal R\|_k\) are elementarily bounded under the same
+hypotheses.
\end{lemma}
-\begin{proof}[Proof sketch]
-One flat ball per witness; identify two at the same point iff they
-share a label. The largest color coordinate of each witness
-(which is at least $q^{-1/2}-4\delta$) ensures the identifications
-are consistent (no branch points), and the resulting space is
-Hausdorff. Countably many rational centers give a countable
-subatlas. Pull back $c$ through a smooth cutoff in the coordinates
-$z$: the bulk region preserves the cocycle values everywhere, and the
-cutoff region collapses to the smooth $q$-form. The closure of
-$\eta$ is inherited from the cocycle law. The measured pairing
-$\int_{\mathcal M}F^*\eta=I$ follows by product structure and
-cocycle invariance.
+\begin{proof}
+\emph{Invariant reflection states.}
+For each nonempty proper subset
+\(S\subset\{0,\ldots,m\}\) and each face with exactly those vertex
+colors, pair the incident positive chambers with the incident negative
+chambers. The two counts agree. Indeed choose a color omitted by
+\(S\); the given codimension-one matching in \(\mathcal P\) across
+facets omitting this color pairs the chambers at that face with opposite
+signs. All incident sets are
+finite by the incidence bound above. Choose uniformly among their
+possible pairings, independently for each face and each \(S\), and
+let \(p_S\) denote the resulting involution on chambers. Use also
+independent uniform bits \(t_S\in\{0,1\}\), one for each \(S\).
+The state space consists of \(\zeta=(\sigma,p,t)\), with counting measure
+on \(\sigma\) and these product probabilities on \(p,t\). These
+are countable products of finite probability spaces. As the original
+data vary, the incidence lists and finite pairing choices are Borel,
+so the product laws are Borel probability kernels.
+
+Define a move \(s_S\) by
+\[
+ \sigma\longmapsto p_S\sigma,\qquad t_S\longmapsto1-t_S,
+ \qquad p_R\longmapsto p_Sp_Rp_S\quad(R\subsetneq S),
+\]
+leaving the other data unchanged. The conjugated \(p_R\) is again
+a pairing at each \(R\)-face: \(p_S\) preserves every such face and
+flips the chamber sign, so conjugation preserves its face and its
+sign-reversing property. Each \(s_S\) is an involution. If
+\(R\subsetneq S\), the two moves commute. On chambers, for example,
+their two orders both give \(p_Sp_R\sigma\), since after \(s_S\) the
+\(R\)-pairing is \(p_Sp_Rp_S\). On a pairing indexed by a subset of
+\(R\), both orders conjugate by \(p_Sp_R\); on the other indices the
+same assertion follows directly from which of the two moves changes
+that index. The bits commute as well. Thus all moves in a nested
+flag commute, and they act freely: each changes its own bit, so no
+nonempty product of these moves can fix a state.
+
+These moves preserve the state measure. They permute the counting
+chamber index, flip a uniform bit, and, conditional on \(p_S\),
+conjugate each relevant uniformly distributed finite pairing by a
+permutation of its incident set. The latter operation permutes its
+finite set of possible pairings and preserves its uniform law. It
+also preserves the independent product of those laws. All moves are
+Borel and commute with translating the original label data.
+
+\medskip\noindent\emph{Smooth leaves and the measured atlas.}
+Attach the following tile to each state. Start with the positive
+simplex \(x_0,\ldots,x_m>0\), \(\sum_i x_i=1\), and take the closure
+of its collection of pair ratios \((x_i/x_j)_{i\ne j}\) in the
+product of copies of \([0,\infty]\). For each permutation
+\(i_0,\ldots,i_m\) use the coordinate chart
+\begin{equation}\label{eq:transfer-ratios}
+ u_j=x_{i_j}/x_{i_{j-1}}\in[0,4),\qquad 1\leq j\leq m.
+\end{equation}
+All pair ratios in the chart are products of these or their reciprocals,
+so the \(u_j\) are independent coordinates. Sorted orders have
+\(u_j\leq1\) and cover the closure. The facet \(u_j=0\) has label
+\(S_j=\{i_0,\ldots,i_{j-1}\}\); the labels of a corner are nested.
+Glue each facet to the identical resolved coordinates in the state
+obtained by \(s_{S_j}\). The commutation just proved makes the gluing
+consistent at corners.
+
+Use signed square root coordinates \(u_j=Z_j^2\), with
+\(Z\in(-2,2)^m\), starting from any state. For \(Z_j<0\) use the
+tile obtained by \(s_{S_j}\); use the product of these commuting moves
+when several coordinates are negative. At zero use the starting tile
+as a convention for representing the same glued point.
+
+Figure~\ref{fig:resolution} illustrates the tile and the four signed
+quadrants at a two-facet corner when \(m=2\).
+
+\par\medskip
+\noindent\begin{minipage}{\textwidth}
+\makeatletter\def\@captype{figure}\makeatother
+\centering
+\begin{minipage}[t]{0.53\textwidth}
+\centering
+\begin{tikzpicture}[scale=1.15, line cap=round, line join=round]
+ \coordinate (v0) at (-2,1.05);
+ \coordinate (v1) at (-2.95,-0.65);
+ \coordinate (v2) at (-1.05,-0.65);
+ \draw[thick,fill=blue!3] (v0)--(v1)--(v2)--cycle;
+ \node[above] at (v0) {\(0\)};
+ \node[below left] at (v1) {\(1\)};
+ \node[below right] at (v2) {\(2\)};
+ \node at (-2,-0.12) {\(\sigma\)};
+ \node[font=\footnotesize,align=center] at (-2,-1.42)
+ {one barycentric\\chamber \(\sigma\simeq\Delta^2\)};
+ \draw[-{Stealth[length=2mm]},thick] (-0.25,0.12)--(-0.90,0.12)
+ node[midway,above] {\(X\)};
+ \begin{scope}[shift={(1,0)}]
+ \path[fill=gray!22,draw=black,thick]
+ (30:1.40)--(90:1.40)--(150:1.40)--(210:1.40)--
+ (270:1.40)--(330:1.40)--cycle;
+ \path[fill=white,draw=none]
+ (30:0.76)--(90:0.76)--(150:0.76)--(210:0.76)--
+ (270:0.76)--(330:0.76)--cycle;
+ \node[font=\small] at (60:0.97) {\(\{0\}\)};
+ \node[font=\small] at (120:0.97) {\(\{0,1\}\)};
+ \node[font=\small] at (180:0.97) {\(\{1\}\)};
+ \node[font=\small] at (240:0.97) {\(\{1,2\}\)};
+ \node[font=\small] at (300:0.97) {\(\{2\}\)};
+ \node[font=\small] at (0:0.87) {\(\{0,2\}\)};
+ \node[font=\small] at (0,0) {tile};
+ \end{scope}
+ \node[font=\small,align=center] at (-0.55,-2.15)
+ {(a) A chamber and its resolved tile};
+\end{tikzpicture}
+\end{minipage}
+\hfill
+\begin{minipage}[t]{0.43\textwidth}
+\centering
+\begin{tikzpicture}[scale=1.15, line cap=round, line join=round]
+ \fill[blue!12] (0,0) rectangle (1.25,1.25);
+ \fill[orange!15] (-1.25,0) rectangle (0,1.25);
+ \fill[orange!15] (0,-1.25) rectangle (1.25,0);
+ \fill[blue!12] (-1.25,-1.25) rectangle (0,0);
+ \fill[gray!25] (-0.10,-1.25) rectangle (0.10,1.25);
+ \fill[gray!25] (-1.25,-0.10) rectangle (1.25,0.10);
+ \draw[thick] (-1.25,-1.25) rectangle (1.25,1.25);
+ \draw[-{Stealth[length=1.6mm]}] (-1.4,0)--(1.5,0)
+ node[right,font=\small] {\(Z_S\)};
+ \draw[-{Stealth[length=1.6mm]}] (0,-1.4)--(0,1.5)
+ node[above,font=\small] {\(Z_T\)};
+ \node at (0.68,0.67) {\(\zeta\)};
+ \node at (-0.68,0.67) {\(s_S\zeta\)};
+ \node at (0.68,-0.67) {\(s_T\zeta\)};
+ \node at (-0.68,-0.67) {\(s_Ss_T\zeta\)};
+ \node[font=\small] at (1.03,1.04) {\(+\)};
+ \node[font=\small] at (-1.03,1.04) {\(-\)};
+ \node[font=\small] at (1.03,-1.04) {\(-\)};
+ \node[font=\small] at (-1.03,-1.04) {\(+\)};
+ \node[font=\small,align=center] at (0,-2.15)
+ {(b) \(u_S=Z_S^2,\quad u_T=Z_T^2\)};
+\end{tikzpicture}
+\end{minipage}
+\caption{The reflected resolution in dimension two, after Gaifullin's
+permutohedral construction \cite{Gaifullin2008}. (a) A barycentric
+chamber \(\sigma\simeq\Delta^2\) and its resolved tile. The six facets
+correspond to the nonempty proper subsets of \(\{0,1,2\}\), nested when
+the labels are adjacent, and \(X\) maps the tile interior
+diffeomorphically onto the chamber interior. (b) At a corner with
+labels \(S\subset T\), a state \(\zeta=(\sigma,p,t)\) and its three
+reflections fill the four quadrants of signed coordinates; the
+commutation \(s_Ss_T=s_Ts_S\) and the independent bit toggles make these
+four quadrants one smooth chart. Signs record whether the chamber
+orientation agrees with that of \(\zeta\) (blue) or opposes it (orange).
+The collar and strips indicate the seam schematically, with illustrative
+widths, and \(X\) is not affine.}
+\label{fig:resolution}
+\end{minipage}
+\par\medskip
+
+We verify that these are ordinary smooth charts. On an overlap of two permutations,
+a cut \(S_j\) which is not a cut of the other permutation has
+\(u_j\geq4^{-2m}\). To see this, choose a pair of indices on opposite
+sides of that cut whose order is inverted in the other permutation.
+Their ratio in the first ordering is at most \(4^m u_j\); its inverse
+in the second ordering is at most \(4^m\). The asserted lower bound
+follows. At a cut common to both permutations, the ratio for the
+second chart is the first ratio at that cut multiplied by a product
+of ratios within the blocks between common cuts. Those within-block
+ratios and their required reciprocals are bounded away from zero on
+the overlap. After taking square roots, the transition at a common
+cut is consequently its signed coordinate times a smooth nonvanishing
+factor. All other transition coordinates are smooth monomials with
+denominators bounded away from zero.
+
+The sign in this transition is fixed on each region with fixed signs
+at the noncommon cuts. The remaining state relation is a product of
+the common reflections, and its product is unique by their free
+action on the bits. Commutation makes this sign rule persist across
+the zero coordinates. Thus the transitions are smooth and invertible
+across every corner. A point incident to \(j\) facets has exactly the
+finite star of \(2^j\) states under these reflections, and hence has
+one neighborhood of this form rather than several branches.
+
+The signed charts give smooth leaf neighborhoods. We next check the
+topological and transverse properties needed for measured integration.
+The leaves are Hausdorff. Distinct
+points with distinct resolved coordinates can be separated in the
+tile model. If their resolved coordinates agree but their finite
+stars differ, choose neighborhoods in these two stars avoiding all
+nonincident facets; no further identification then joins them.
+Each leaf has at most countably many states reachable by the finitely
+many moves and at most countably many charts, so it is second countable.
+The total space is standard Borel: the quotient of the state--tile
+space identifies finite classes at each tile point, for which Borel
+representatives can be selected by a fixed enumeration. More
+explicitly, charts can be indexed by the starting chamber, its bit
+array, and the flag, with the pairing arrays as transversal parameters.
+The bits determine the crossings at nonzero signed coordinates; the
+moves can then be inverted to recover the pairing array. With the
+zero convention above this makes the chart maps injective. Changes
+of transversal are the measure-preserving moves on countably many
+Borel overlap pieces. A plaque meets only elementarily many plaques
+in this atlas: for each second flag only the states in the two finite
+flag reflection groups can occur. These observations give precisely
+the measured flow-box properties of Theorem~\ref{thm:degree}.
+
+The chamber signs orient the leaves. In a flag chart the Jacobian
+sign from positive ratio coordinates to oriented barycentric
+coordinates depends only on the permutation. Changing one signed
+coordinate from positive to negative changes the Jacobian sign of
+\(u_j=Z_j^2\) once, while the move changes the chamber sign once.
+The induced orientation is therefore the same across each facet.
+The transverse measures here are nonnegative; the signs enter only
+through this orientation in integrals of forms. The value of
+\(\omega\) is constant along each tile leaf, and translation by
+\(\Gamma\) reindexes the states and preserves these measures.
+
+\medskip\noindent\emph{The map and the local current.}
+Recover the raw \(x_i\) from \eqref{eq:transfer-ratios} by normalizing
+the successive products \(1,u_1,u_1u_2,\ldots,u_1\cdots u_m\).
+If \(v_{\sigma,i}\) are the vertices of the indicated chamber in its
+flat chart, map its tile by
+\begin{equation}\label{eq:transfer-tile-map}
+ X=\sum_{i=0}^m b_i v_{\sigma,i},\qquad
+ b_i=\frac{\exp(-1/x_i)}{\sum_l\exp(-1/x_l)},
+\end{equation}
+where \(\exp(-1/x_i)\) is extended by zero at \(x_i=0\). On a facet
+with label \(S\), all coefficients outside \(S\) vanish to infinite
+order in its signed coordinate. The vertices in \(S\) are the same
+on the paired chambers, and their coefficients are smooth functions
+of the squares of the signed coordinates. Thus \(X\) is smooth
+through the gluing. Its denominator is at least \(e^{-(m+1)}\), since
+some \(x_i\geq1/(m+1)\), which also gives elementary upper derivative
+bounds in the tile charts.
+
+On the positive simplex the map \(x\mapsto b\) is a diffeomorphism.
+Given positive \(b_i\) of sum one, its inverse is obtained from the
+unique \(t\in(0,1/\max_i b_i)\) satisfying
+\[
+ \sum_i\frac1{-\log(tb_i)}=1;
+\]
+the left side increases smoothly from zero to infinity. The inverse
+function theorem also gives the smooth inverse. This diffeomorphism
+preserves orientation. Indeed it extends to the closed simplex
+preserving every face, and the linear homotopy to the identity has
+the same property, so its relative topological degree is \(+1\).
+It follows that the tile interior maps diffeomorphically, with sign
+\(\varepsilon_\sigma\), onto the interior of its chamber. Each
+chamber's state probability integrates to one and the tile boundaries
+have leafwise measure zero. Fubini and change of variables therefore
+give, for every compactly supported smooth \(m\)-form \(\alpha\),
+\[
+ \int_{\mathcal A}X^*\alpha
+ =\sum_\sigma\varepsilon_\sigma\int_\sigma\alpha.
+\]
+This proves \eqref{eq:transfer-push}. To obtain the local identity,
+define \(\mathcal A_{\rm orig}\) as the image in
+\(X^{-1}(\mathcal T_\delta)\) of tile points belonging to original
+chambers. At an interior point this subset is leaf-open. At a lower
+face, the incidence argument preceding the lemma shows that every state
+in its finite reflection star is on the same side. A neighborhood in
+that star can be chosen to keep its image in the strict tube and to
+avoid nonincident facets, so it stays on that side. This proves
+leaf-openness and makes the definition unambiguous at glued points;
+the finite indexed charts also make it Borel. Applying the same
+change-of-variables calculation only to original tile interiors, for
+a form supported in \(\mathcal T_\delta\), gives
+\eqref{eq:transfer-local-push}: each original cube is counted with
+its coefficient \(D_0u'_C\), and each chamber's state fiber has mass one.
+
+\medskip\noindent\emph{The metric and upper density.}
+The signed charts admit a background metric \(k_0\) with elementary
+bounds for its coefficients, inverse, and every fixed number of
+derivatives in buffered charts. To construct it, take smooth product
+cutoffs supported in the signed boxes \((-2,2)^m\) and equal to one
+on \([-1,1]^m\), and average their Euclidean chart metrics with the
+normalized cutoff weights. The smaller boxes cover, by sorted flags.
+The overlap bound above bounds the number of terms per plaque. The
+transition formulas and their inverses are products of at most
+\(O(m)\) factors, and every required reciprocal at a noncommon cut
+is at most \(4^{O(m)}\). Their fixed-order derivatives consequently
+have elementary bounds. These facts give the claimed bounds for the
+average and its inverse, and make the construction Borel and
+equivariant. The same transition and product estimates, applied to
+\eqref{eq:transfer-tile-map}, give an elementary upper bound for
+\(dX\) and its fixed-order derivatives before any choice of
+\(\lambda\). The affine simplexes have diameter at most \(r\sqrt m\).
+
+Put \(\Theta=\prod_i x_i\). Choose a smooth \(j=j(\Theta)\) equal
+to one for \(\Theta\leq\gamma_0\), zero for
+\(\Theta\geq2\gamma_0\), and between zero and one elsewhere, with
+the rescaled smooth cutoff estimates. It is smooth in the signed
+charts and the same on both sides of every facet. Set
+\begin{equation}\label{eq:transfer-resolution-metric}
+ k=X^*k_{\rm flat}+j^2 k_0,
+ \qquad \text{seam}=\{\Theta\leq2\gamma_0\}.
+\end{equation}
+We call the complement of this closed seam the straight part.
+The metric is positive definite near the tile boundary because \(j=1\) there. Where
+\(j<1\), one has \(\Theta>\gamma_0\), hence \(x_i\geq\gamma_0\) for
+every \(i\), and \(X\) is an immersion. The formula proves
+\(\|dX\|_k\leq1\) and equality of metrics off the seam.
+
+The upper density bound requires no inverse estimates. As quadratic forms,
+\(k\leq X^*k_{\rm flat}+k_0\), so its volume density in every sorted
+box is bounded by the upper bounds for \(dX,k_0\) already chosen.
+Call a common elementary bound \(H_0(q)\). In a sorted box
+\(|Z_j|\leq1\). If all \(|Z_j|>\lambda\), normalization of the
+successive products gives
+\[
+ \Theta\geq(m+1)^{-(m+1)}
+ \lambda^{2(1+2+\cdots+m)}
+ =(m+1)^{-(m+1)}\lambda^{m(m+1)}.
+\]
+Condition \eqref{eq:transfer-gamma} shows that the seam lies in the
+stated union of slabs. Its coordinate volume in such a box is at
+most an elementary factor times \(\lambda\). The bound \(H_0(q)\)
+was fixed before this factor was made small.
+
+The remaining metric estimates may depend on \(\gamma_0\). On the part
+with \(j<1\), all positive ratios are bounded below by powers of
+\(\gamma_0\), so the product formulas control the inverse from raw
+barycentric to signed coordinates. The normalized weights in
+\eqref{eq:transfer-tile-map} obey
+\(b_i\geq e^{-1/\gamma_0}/(m+1)\). On a tangent vector
+\(w=(w_i)\), \(\sum_i w_i=0\), their derivative is
+\[
+ db_i(w)=b_i\left(\frac{w_i}{x_i^2}
+ -\sum_l b_l\frac{w_l}{x_l^2}\right).
+\]
+These differences control \(w\): if their common subtracted mean is
+\(a\), the relation \(\sum_iw_i=0\) expresses \(a\) as a weighted
+average, with weights \(x_i^2\), of the negatives of those differences.
+Thus \(|w|\) is bounded by a polynomial in \(m\) times
+\(e^{1/\gamma_0}\max_i|db_i(w)|\). Finally, the affine map from the
+barycentric simplex to a simplex in a subdivided cube has elementary
+inverse bounds. Its matrix has \(r\)-scaled rational entries with
+bounded denominators, and its nonzero determinant and cofactors give
+this by minors. These estimates control the inverse metric where
+\(j<1\); where \(j=1\), \(k_0\) already supplies that control.
+Differentiating the formula for \(k\) a fixed number of times now gives
+elementary bounds, including the curvature bound, when
+\(\gamma_0^{-1}\) is elementary.
+
+\medskip\noindent\emph{Frame interpolation.}
+The comparison needs a constant frame ordering tangent and normal axes
+on each straight tile, a frame that glues smoothly near facets, and a homotopy to the
+identity for the degree calculation. Choose a Borel skew-symmetric logarithm
+\(A_\sigma\) of each \(O_\sigma\) with operator norm at most \(\pi\).
+Such a choice follows by decomposing an orthogonal rotation into its
+two-dimensional angle blocks with principal angles and, on its
+\(-1\)-eigenspace, pairing a Borel choice of orthonormal basis; the
+even dimension there follows from determinant one. Define on that
+tile
+\[
+ \mathcal R=\exp((1-j)A_\sigma).
+\]
+It is the identity in a neighborhood of each facet and thus glues
+smoothly despite the change of \(\sigma\). It equals \(O_\sigma\)
+where \(j=0\), and replacing \(1-j\) by \(t(1-j)\), \(0\leq t\leq1\),
+gives the asserted homotopy. The cutoff and metric inverse estimates
+give the derivative bounds. The choices can be made on the indexed
+chamber data, so they are Borel and equivariant as asserted.
\end{proof}
-\subsection{The cubical chain complex}
-\label{sec:transfer-cells}
+For our use, choose \(O_\sigma\) to take \(z\)-increments to rotated
+grid increments, putting the \(m\) axes tangent to the dual cube first
+and the \(n\) complementary axes last. Use a signed permutation if
+needed to have determinant one. Thus \(O_\sigma=Q_\sigma V\) for a
+signed permutation \(Q_\sigma\in SO(q)\) with this order. Signs will
+not affect any row norm or any cubical test.
-Fix a rotation $V\in SO(q)$ (integrated over later) and a cubical
-grid of side $r=r(q)>0$ with uniform shift $s\in[0,r)^q$. Choose
-$r$ so that every $n$-component of $\eta$ changes by at most
-$q^{-n}$ across any cell of side $r$; this uses the derivative
-bounds of Lemma~\ref{lem:transfer-flat}.
+\subsection{The comparison map and its degree}
-\emph{Primal $n$-cells.} A primal cell $C$ is an $n$-cube parallel
-to a subset $\mathcal I$ of size $n$ of the $q$ grid axes, with its
-center in the tube of radius $4\delta$ around $K$, and maximal side
-$r$ in each of the $\mathcal I$-directions. It carries the
-coefficient
-\begin{equation}
- u_C:=\int_C\eta.
+The parameter selection below fixes \(\lambda(q)\) and \(\gamma_0(q)\)
+deterministically. For those choices the bounds in
+Lemma~\ref{lem:transfer-resolution} give fixed elementary numerical
+envelopes \(K_R(q)\) and \(K_{\rm scal}(q)\) such that
+\[
+ \|d\mathcal R\|_{\infty,k}\leq K_R(q),\qquad
+ (-\Scal_k)_+\leq K_{\rm scal}(q)
+\]
+for every allowed diagram and every \(V,s,\omega\), rounding parameter,
+and state. These envelopes come from the uniform chart derivative
+bounds and do not use \(C_{\rm ent}\). Fix the single number
+\begin{equation}\label{eq:transfer-tau}
+ \tau=\tau(q)=
+ \frac{1}{300\sqrt q\max\{r^{-1},\delta^{-1},1+K_R(q)\}}.
\end{equation}
-
-\emph{Dual $m$-cubes.} The dual $m$-cube $C^\#$ has the same
-center, varies along the complementary $m$ axes by $[-r/2,r/2]$, and
-has its remaining coordinates fixed. It is oriented so that (primal
-$n$-basis, negative of dual $m$-basis) is a positive basis of $\R^q$.
-
-\emph{Cocycle closure.} If a primal $(n+1)$-cell has all its
-$n$-faces included (as happens near a witness), the signed sum of the
-$u_C$ over its $\binom{n+1}{n}=n+1$ facets is zero by Stokes. The
-same signed sum, with the corresponding dual cubes, is the
-requirement for the resolution constructed below.
-
-\emph{Rounding.} The coefficients $u_C$ are reals. Rounding them
-to a fixed-denominator lattice, while preserving the closure
-equations, is the content of:
-
-\begin{lemma}[Finite-type rounding]
-\label{lem:transfer-rounding}
-There is an integer $D_0\ge1$, bounded by an elementary function of
-$q$ and constant in $V,s,\omega$, together with a Borel equivariant
-random choice $u_C'\in D_0^{-1}\Z$ satisfying
-\begin{equation}
- \E_{\mathrm{rd}}u_C'=u_C,\qquad |u_C'-u_C|\le r^nq^{-n},
- \label{eq:transfer-rounding}
+It has an elementary reciprocal and is common to all those parameters.
+In particular
+\(3\tau\sqrt q\leq\min\{r/100,\delta/100,(1+K_R(q))^{-1}\}\).
+
+Fix \(V,s\) and the rounding parameter for the moment. Take the
+product of \(\widetilde M\) and the tile family \(\mathcal A\) over
+their shared \(\omega\), and divide by the diagonal \(\Gamma\)-action.
+Orient the product with the \(\widetilde M\) directions first and the
+\(\mathcal A\) directions second. The resulting measured leaves have
+dimension \(n+m=q\). Using the leaf-open set
+\(\mathcal A_{\rm orig}\) from Lemma~\ref{lem:transfer-resolution},
+work only on the open set of pairs \((x,a)\) with
+\(a\in\mathcal A_{\rm orig}\) for which \(X(a)\) is in the lift of the ball chart of
+\(F(x)\), at distance less than \(\delta/4\). Define there
+\begin{equation}\label{eq:transfer-comparison}
+ G(x,a)=\mathcal R(a)(z_F(x)-z_{X(a)}).
\end{equation}
-and such that every cocycle equation satisfied by $u_C$ is satisfied
-exactly by $u_C'$.
-\end{lemma}
-
-\begin{proof}[Proof sketch]
-Encode all local coefficients of the primal $n$-cells and the
-cocycle equations into one ordered finite vector $v\in\R^{N_0}$.
-The size $N_0$ is elementarily bounded in $q$ because each
-coefficient depends only on the anchor label, the restricted
-coloring at that anchor, the axis set, and the projected grid index.
-Round $v$ to the joint kernel of the incidence equations: this is
-a lattice-rounding argument. The joint kernel has an integer basis
-of size $\le K_0(N_0)^{CN_0}$ (Hadamard), giving the integer $D_0$.
-
-The rounding is Borel and equivariant, since the type slots depend
-Borel-equivariantly on the data, and the kernel is chosen from a
-Borelly-parametrized family of integer matrices (choose a canonical
-lexicographic pivot minor). The expectation identity follows from
-unbiasedness of the rounding.
-\end{proof}
-\subsection{Measured resolution}
-\label{sec:transfer-resolution}
-
-The rounded chain is a linear combination of oriented cubes. To
-apply the degree estimate, resolve it into a smooth oriented
-$m$-manifold.
-
-\begin{lemma}[Measured resolution]
-\label{lem:transfer-resolution}
-There is a standard Borel measured family $\mathcal A$ of oriented
-smooth $m$-manifolds and a Borel equivariant smooth map
-$X:\mathcal A\to\mathcal U_\omega$ with
-\begin{equation}
- X_*[\mathcal A]=Z_{\mathrm{rd}},
- \label{eq:transfer-push}
+Here \(z_{X(a)}=z\circ X(a)\). The requirement on the lifts is part of
+the domain; coincident color coordinates on incompatible sheets are
+never compared. The map is invariant under the diagonal action.
+
+Use the band \(\{|G|_\infty<3\tau\}\) in that open domain, and put
+\(b_q=\tau(1+3/(4q))\), the inner width in Theorem~\ref{thm:degree}
+when \(d=q\). Let \(\mathcal D\) be its measured degree.
+
+\begin{lemma}[Comparison degree]\label{lem:transfer-degree}
+The band in \eqref{eq:transfer-comparison} satisfies the chart and
+integration control assumptions of Theorem~\ref{thm:degree}, for each
+fixed instance, the product metric \(h+L^2k\) with any fixed \(L>0\),
+and the pullback of the entropy weight \(\phi\). Its degree is Borel
+in the auxiliary parameters and satisfies
+\begin{equation}\label{eq:transfer-degree}
+ \E_{\rm rd}\mathcal D=D_0I
\end{equation}
-where $Z_{\mathrm{rd}}$ is the rounded chain. On the strict tube
-$\mathcal T_\delta$ of radius $\delta$ around $K$, the current
-$Z_{\mathrm{rd}}$ is closed.
-
-For every choice of $0<\lambda<1/2$ and
-$0<2\gamma_0<(m+1)^{-(m+1)}\lambda^{m(m+1)}$, there is a smooth
-metric $k$ on the resolution and a closed seam $\{\Theta\le
-2\gamma_0\}$ such that:
-\begin{itemize}
-\item $\|dX\|_k\le1$ and $k=X^*k_{\mathrm{flat}}$ off the seam;
-\item seams are contained in $\{|Z_j|\le\lambda, 1\le j\le m\}$;
-\item buffered smooth charts with elementary overlap, metric, and
- derivative bounds whenever $\lambda^{-1},\gamma_0^{-1}$ are
- elementary.
-\end{itemize}
-
-Moreover, for any prescribed rotation $Q_\sigma\in SO(q)$ on each
-chamber $\sigma$ over a dual cube, there is a Borel equivariant
-$\mathcal R:\mathcal A\to SO(q)$, equal to $Q_\sigma$ off the seam,
-equal to the identity near the tile facets, and homotopic to the
-identity.
+for every fixed \(V,s\).
\end{lemma}
-\begin{proof}[Proof sketch]
-\emph{Cubical assembly.} Start from the subcomplex of oriented
-$m$-cubes $C^\#$ weighted by $D_0u_C'$ (with sign from the sign of
-$u_C'$). At the tube faces the rounded chain is closed
-(Lemma~\ref{lem:transfer-rounding}), so positive and negative
-incidence copies pair across the tube. Glue them along common
-faces; leave other faces free; mirror the diagram with reversed
-orientation and pair each free face with its mirror. All
-codimension-one faces are now paired. Local axis, grid, and copy
-indices guarantee Borel and equivariant choices.
-
-\emph{Chamber coordinates.} Parametrize each chamber by signed
-square-root coordinates $u_j=Z_j^2$ with $Z\in(-2,2)^m$. The
-positive-simplex diffeomorphism $x\mapsto b$ (with
-$b_i\propto 1/(-\log x_i)$ normalized) is smooth, orientation-
-preserving, and extends across faces. The map $X$ sends each
-chamber interior to its image in the tile with this diffeomorphism
-glued across seams.
-
-\emph{Resolution metric.} Let $\Theta=\prod x_i$. Choose
-$j=j(\Theta)$ smooth, equal to one near $\Theta\le\gamma_0$ and zero
-for $\Theta\ge2\gamma_0$, with rescaled cutoff estimates. Set
-\begin{equation}
- k=X^*k_{\mathrm{flat}}+j^2k_0.
- \label{eq:transfer-resolution-metric}
-\end{equation}
-Off the seam $k=X^*k_{\mathrm{flat}}$; on the seam, $k_0$ supplies
-the required invertibility. Positivity near the seam from
-$\Theta>\gamma_0$. $\|dX\|_k\le1$ throughout.
-
-\emph{Rotation.} Choose a Borel skew-symmetric $A_\sigma$ with
-$O_\sigma=e^{A_\sigma}$, define $\mathcal R=\exp((1-j)A_\sigma)$ so
-it is $O_\sigma$ where $j=0$ (off the seam), the identity near
-facets (where $j=1$), and the homotopy $t\mapsto\exp(t(1-j)A_\sigma)$
-makes it homotopic to the identity. Fixed-order derivatives of
-$\mathcal R$ have elementary bounds from the chamber rotations.
-\end{proof}
-
-\subsection{The comparison map and its degree}
-\label{sec:transfer-application}
-
-We now apply Theorem~\ref{thm:degree} to the composition of the
-resolution map, the frame rotation, and the coordinate function.
-
-\emph{Comparison map.} For a specific choice of $\sigma$, define
-$G:\mathcal A\to\R^q$ by
+\begin{proof}
+The local pushforward \eqref{eq:transfer-local-push} will express the
+degree using a smoothed form of \(Z_{\rm rd}\). Its primal-cell periods
+are the integer coefficients \(D_0u'_C\). Their expectations match the periods of
+\(D_0\eta\); a compatible piecewise cubical primitive and mollification
+then give the same measured pairing with \(F\). We first verify the
+band control that makes these integrations valid.
+
+\emph{The controlled comparison band.}
+Before taking the quotient, the product flow boxes use the entropy
+transversal \(\omega\) and the
+state transversals of \(\mathcal A\). Variation along the tile keeps
+\(\omega\) constant. These transversals carry the invariant measures
+constructed above. A group element that stabilizes a product leaf
+acts freely and properly because it acts so on the \(\widetilde M\)
+factor. Hence the quotient leaves are ordinary oriented Hausdorff
+second countable manifolds. The actions reindex labels and states,
+leave the color and flag coordinates unchanged, and preserve the
+transverse measures and orientations.
+
+For an inner closed comparison cube there is a uniform margin from
+the boundary of the allowed sheet domain. Indeed \(\mathcal R\) is
+orthogonal and \eqref{eq:transfer-tau} gives
+\(|z_F-z_X|_2\leq3\tau\sqrt q\ll\delta/4\). The large witness
+color at \(F\) remains present with a uniform small buffer. In a
+flat ball chart a lifted path whose projected coordinates stay in
+the ball stays in that same lift, by uniqueness of lifts for a local
+diffeomorphism. The nearby points therefore remain in coherent
+sheet charts. Facets that lead to the mirror are outside this region.
+The available smooth bounds for the instance give fixed buffered
+coordinate charts for all the data in these regions.
+
+To check the finiteness condition on those charts, choose lifts of a
+finite buffered atlas on \(M\). On their compact closures \(F\) uses
+only uniformly finitely many possible labels, by the lifted support
+bounds in Theorem~\ref{thm:entropy}. Any nearby cell lift has its
+anchor in a closed one-neighborhood of one of these labels. The
+numbers of projected grid indices, chamber subdivisions, and copies
+there are bounded as above. Thus only finitely many chamber indices
+are needed over each such base chart, uniformly for the instance.
+Their state probabilities have total mass one per chamber, and the
+sum of transverse measures of this covering is finite. Plaque
+overlaps are bounded by the tile overlap bound and the bounded
+overlap of the base atlas. In the quotient only finitely many deck
+changes meet two of these buffered base charts. Subdividing their
+inner boxes and, when necessary, restricting to Borel transverse sets
+gives the flow boxes stipulated in Theorem~\ref{thm:degree}. All
+metric and data bounds in this verification may depend on the
+instance. It also justifies Fubini and measured Stokes for the
+integrations that follow.
+
+\medskip\noindent\emph{The local convolution form.}
+Choose a smooth unit-integral top form \(\beta\) on \(\mathbb R^q\)
+supported in a Euclidean ball about zero of radius less than
+\(\min\{\tau,r/10\}\). In calculating the degree using \(\beta\),
+homotope \(\mathcal R\) to the identity as in
+Lemma~\ref{lem:transfer-resolution}. All maps in this homotopy are
+orthogonal on the difference variable. Their pullbacks of \(\beta\)
+are thus supported where the difference has this same small norm,
+strictly inside both the sheet domain and the original band. Measured
+Stokes for this supported homotopy shows that it does not change the
+degree.
+
+After removing the rotation, fiber integration of
+\((z,w)\mapsto z-w\) against \(\beta\) and the original current
+\(Z_{\rm rd}\) gives a smooth \(n\)-form \(\alpha_{\rm rd}\) near \(K\).
+More explicitly, in a flat chart put \(\mathsf d(z,w)=z-w\). Integrate
+\(\mathsf d^*\beta\) along each compatible \(C^\#\) in the second
+variable, multiply by \(D_0u'_C\), and sum. The sum is locally finite
+and uses the original side. By the local identity
+\eqref{eq:transfer-local-push}, Fubini, and change of variables on tile
+interiors, the degree is
+\begin{equation}\label{eq:transfer-convolution-degree}
+ \mathcal D=\int_{\mathcal M}F^*\alpha_{\rm rd}.
+\end{equation}
+Boundary cancellation of the dual cubes shows that
+\(d\alpha_{\rm rd}=0\) on, for example, the tube of radius
+\(\delta/8\) around \(K\). To verify this directly, apply fiber
+Stokes to each cube. Its boundary terms only involve facets within
+the support distance of the point \(z\); these lie in the balanced
+tube and cancel with their paired incidence signs.
+
+For every primal \(n\)-cell sufficiently near \(K\) in the same
+sheet,
+\begin{equation}\label{eq:transfer-convolution-cells}
+ \int_C\alpha_{\rm rd}=D_0u'_C.
+\end{equation}
+Indeed a primal \(n\)-cell and a complementary dual \(m\)-cell of
+this grid meet only when their indices match, and then at their common
+center. If their axis sets differ, a coordinate fixed for both is a
+grid vertex for the primal cell and a half-step center for the dual
+cell, so their difference in that coordinate is at least \(r/2\).
+If their axis sets agree but their grid indices differ, a differing
+coordinate gives the same separation. Such pairs cannot meet the
+support of \(\beta\). For the matching pair, the difference map
+from \(C\times C^\#\) contains the support ball in its image with
+multiplicity one. Our fiber integration convention is
+\(\int_C\int_{C^\#}\mathsf d^*\beta=
+\int_{C\times C^\#}\mathsf d^*\beta\) with the base-first product
+orientation. The differential of \(\mathsf d(z,w)=z-w\) sends this
+ordered frame to the primal basis followed by the negative dual basis,
+so its orientation is positive by our choice of the dual orientation.
+Its integral is therefore
+one, proving \eqref{eq:transfer-convolution-cells}. The small support
+and the chart buffers ensure that only the indicated lifts contribute.
+
+\medskip\noindent\emph{The detector pairing.}
+Set \(\alpha=\E_{\rm rd}\alpha_{\rm rd}-D_0\eta\). It is a smooth
+closed form on that tube, and \eqref{eq:transfer-rounding},
+\eqref{eq:transfer-cell-value}, and
+\eqref{eq:transfer-convolution-cells} give \(\int_C\alpha=0\) on all
+the primal \(n\)-cells there. We explain why its measured pairing
+with \(F\) vanishes, including when \(F\) meets grid faces.
+
+Use all lifted full \(q\)-cells needed to cover the tube of radius
+\(\delta/32\) around \(K\). Taking \(r\sqrt q\) sufficiently small
+as above, these cells lie in the tube of radius \(\delta/8\), all
+their \(n\)-faces are selected, and their shared faces fit in
+coherent sheet charts. On one grid interval \([a,a+r]\), project a
+function to its linear interpolation between the endpoints and
+project a one-form \(f(t)dt\) to the constant one-form with the same
+integral. The homotopy for a one-form is the function
\[
- G:=V(z\circ X)
+ \mathsf h(f(t)dt)(t)=\int_a^t
+ \left(f(v)-r^{-1}\int_a^{a+r}f(s)\,ds\right)dv,
\]
-where $V\in SO(q)$ is a fixed rotation from the integration in
-\S\ref{sec:transfer-cells}, and $\sigma$ indexes the choice of
-$Q_\sigma$ in Lemma~\ref{lem:transfer-resolution}. The map $G$
-takes values in an open cube of side $<q^{-1/2}$ around the origin.
+which vanishes at both endpoints; the homotopy on functions is zero.
+Direct differentiation gives \(\mathrm{id}-\mathsf P
+=d\mathsf h+\mathsf h d\). Apply these operations one coordinate
+at a time, with the ordinary exterior differential signs. The
+tensor-product homotopy still commutes with tangential restriction
+to a face, since the one-dimensional primitive vanishes at the
+endpoints. The full projection of an \(n\)-form has, for each
+axis set, coefficients obtained by integrating on its \(n\)-faces
+and interpolating linearly in the remaining axes. Hence the full
+projection of \(\alpha\) is zero. Since \(d\alpha=0\), the
+homotopy supplies a primitive on each full cube, and its tangential
+restrictions agree on shared faces.
+
+This piecewise primitive has bounded coefficients for the instance
+and is also a weak primitive on the tube: in distributional Stokes
+the terms on each shared face cancel by agreement of its tangential
+restrictions. On a smaller tube mollify it in the flat coordinates.
+This can be done with the same small Euclidean convolution in every
+chart: the buffered sheets give a unique lift of the convolution ball,
+and on overlaps the piecewise primitive and flat coordinates agree.
+Its smooth exterior derivatives are the mollifications of the smooth
+form \(\alpha\), and converge to \(\alpha\) there. The construction
+is Borel and equivariant and has bounded coefficients for the instance.
+Measured Stokes applied to the pullback of each smooth primitive gives
+zero; convergence and the uniform chart bounds give
+\(\int_{\mathcal M}F^*\alpha=0\). All expectations and integrals
+above are measurable by their locally finite chart formulas, and
+their absolute bounds for the instance justify their interchange.
+Using \eqref{eq:transfer-form} in
+\eqref{eq:transfer-convolution-degree} now proves
+\eqref{eq:transfer-degree}.
+\end{proof}
-\emph{Metric and weight.} Use the product structure: a Riemannian
-metric $h_{\mathrm{tot}}$ that combines the graph metric
-$L_T^{-2}\cdot h$ on the tangent directions of the $\mathcal U_\omega$
-factor with $L_N^{-2}\cdot k$ on the normal (resolution) directions,
-and the weight $\phi$ constant in the tile directions. Thus
-\begin{equation}
- h_{\mathrm{tot}} = L_T^{-2}h\oplus L_N^{-2}\,k.
- \label{eq:transfer-h-tot}
+\subsection{The two comparison lengths and the final average}
+
+We finish by estimating these degrees. We will see separately how
+the straight part uses the dimension-only Hilbert norm of \(\eta\)
+and how the small coordinate volume of the seam pays for its rougher
+bound.
+
+\begin{proof}[Proof of Theorem~\ref{thm:scalar-volume}]
+The case \(\|M\|=0\) is immediate. Otherwise we first prove the
+conditional target for the data above, choosing \(\Psi_n\) from the
+requirements on \(E\) derived below, and then apply it to the entropy
+output. Choose a positive constant \(b_*\), depending only on \(n\),
+whose smallness will be fixed in the estimate below, and put
+\begin{equation}\label{eq:transfer-lengths}
+ L_T=b_*\sqrt E,\qquad L_N=b_*\sqrt{qE},\qquad
+ h_{\rm tot}=h+L_T^2k.
\end{equation}
-
-\emph{Horizon choice.} Choose $b_*<1$ and
-\begin{equation}
- L_T=b_*\sqrt E,\qquad L_N=b_*\sqrt{qE}.
- \label{eq:transfer-lengths}
+Once the degree hypotheses are verified, its length denominator, the
+product volume, and the entropy contribution will combine as
+\begin{equation}\label{eq:transfer-prefactor}
+ \frac{L_T^mE^{n/2}}{L_T^mL_N^n}\,e^{-W}dV_h
+ =b_*^{-n}q^{-n/2}\rho.
\end{equation}
-The horizon condition (Definition~\ref{def:degree-horizon}) holds
-because
+This is the cancellation of the tangent volume and all powers of
+\(E\). The estimates below justify the degree application and bound
+the straight angular sum by a dimensional multiple of \(q^{n/2}\),
+while the collar volume controls the seam.
+
+Use the weight \(\phi\), constant in the tile directions. The
+product scalar formula and the definition of \(\wscal\) give
+\(\wscal(h_{\rm tot},\phi)=\wscal(h,\phi)+L_T^{-2}\Scal_k\).
+By the definition of \(K_{\rm scal}(q)\), the lower bound
+\(E\geq b_*^{-2}K_{\rm scal}(q)\) ensures
+\(L_T^{-2}\Scal_k\geq-1\). By \eqref{eq:transfer-entropy},
+\begin{equation}\label{eq:transfer-total-scalar}
+ \wscal(h_{\rm tot},\phi)\geq-C_{\rm ent}-1+2W-n\log E+E\mathfrak e.
+\end{equation}
+
+Assign \(L_i=L_T\) to the first \(m\) comparison rows and
+\(L_i=L_N\) to the last \(n\) rows. On the band, the derivative in
+tile directions of each row of \(G\), measured with \(k\), has norm
+at most two. The \(dX\) term has norm at most one, and the
+\(d\mathcal R\) term has norm at most
+\(3\tau\sqrt q\|d\mathcal R\|_k\leq1\) by
+\eqref{eq:transfer-tau}. The horizontal derivative is a row of a
+rotation of \(dz_F\). It follows that
\[
- \frac{C(1+B)^{2/3}}{L_T}
- \le\frac{C_nq^{1/3}}{b_*\sqrt E}<\frac{\tau}{10q}
+ L_i|dG_i|_{h_{\rm tot}}
+ \leq C_n\bigl(\sqrt q+\sqrt{qE}\,E^{-1/100}\bigr).
\]
-for $E\ge (20C_nq^{4/3}/\tau)^{75/13}$, which we can absorb into the
-entropy threshold $\mathcal B(q)$.
-
-\emph{Straight points.} At a straight point of $\mathcal A$ (off
-the seam), the metric is $k=X^*k_{\mathrm{flat}}$ and the rotation
-$Q_\sigma V$ is constant. The tangent-space squared norm of each
-row of $G$ is $L_T^{-2}+|A_i|_h^2$ where $A_i=(Vdz_F)_i$ has
-$h$-covector norm and $\sum_i|A_i|_h^2=\mathfrak e$. Hence
+Take \(B\) to be this upper bound increased by two. Then
+\(L_i|dG_i|\leq B-1=B-q/q\). The horizon condition of
+Theorem~\ref{thm:degree} holds after an elementary increase of \(E\):
+it suffices to test the smallest length \(L_T\), and, for \(E\geq1\),
\[
- H_i^{\mathrm{tangent}}=2(L_T^{-1}+|A_i|_h-1)_+
- \le L_T^2\mathfrak e
+ \frac{C(1+B)^{2/3}}{L_T}
+ \leq c_{{\rm hor},n}q^{1/3}E^{-13/75}
+ <\frac{\tau}{10q}.
\]
-using $2(\sqrt{1+t}-1)\le t$. Normal rows have no tile derivative.
-Cauchy-Schwarz on the normal rows gives
-\begin{equation}
- \sum_iH_i\le L_T^2\mathfrak e
- +2L_N\sqrt n\sqrt{E\mathfrak e}\,U_{\mathcal I}
- \label{eq:transfer-straight-penalty}
+Here \(c_{{\rm hor},n}\) depends only on the absolute horizon constant
+and the dimensional choice of \(b_*\), and
+\(13/75=1/2-(2/3)(1/2-1/100)>0\). Thus the displayed strict horizon
+inequality follows if
+\(E\geq(20c_{{\rm hor},n}q^{4/3}/\tau)^{75/13}\).
+We may therefore apply Theorem~\ref{thm:degree}
+with \(d=q\).
+
+Write \(H_i=2(L_i|dG_i|-1)_+\), as in that theorem. At a straight
+point in a tile, let \(\mathcal I\) be the \(n\)-set of grid axes
+normal to its dual cube. Put \(A_i=(Vdz_F)_i\), with covector norms
+measured by \(h\), and define
+\begin{equation}\label{eq:transfer-U}
+ U_{\mathcal I}^2=
+ \frac q{\mathfrak e}\sum_{i\in\mathcal I}|A_i|_h^2
+ \quad\text{if }\mathfrak e>0,
+ \qquad U_{\mathcal I}=0\quad\text{if }\mathfrak e=0.
\end{equation}
-where $U_{\mathcal I}^2=(q/\mathfrak e)\sum_{i\in\mathcal I}|A_i|_h^2$.
-Completing the square in $\sqrt{E\mathfrak e}$ (and using
-$b_*<1$) yields
-\begin{equation}
- \frac{\sum_iH_i-E\mathfrak e}{2}\le c_nU_{\mathcal I}^2,
- \qquad c_n=\frac{nb_*^2}{2(1-b_*^2)}.
- \label{eq:transfer-straight-bound}
-\end{equation}
-
-\emph{Seam points.} At a seam point, use cruder bounds. From the
-uniform tile row bound,
+Here \(\sum_i|A_i|^2=\mathfrak e\). At a straight point
+\(k=X^*k_{\rm flat}\) and \(\mathcal R=Q_\sigma V\) is constant.
+For a tangent comparison row corresponding to axis \(i\), the squared
+row norm is exactly \(L_T^{-2}+|A_i|^2\). Thus
\[
- \sum_iH_i\le 2L_N\sqrt{q\mathfrak e}+4qL_N/L_T
- \le 2b_*q\sqrt{E\mathfrak e}+4q^{3/2}.
+ \sum_{\rm tangent}H_i
+ \leq L_T^2\mathfrak e=b_*^2E\mathfrak e,
\]
-Completing the square:
-\begin{equation}
- \frac{\sum_iH_i-E\mathfrak e}{2}\le c_{\mathrm{seam}}(q)
- \qquad c_{\mathrm{seam}}(q)=2+b_*^2q/2.
- \label{eq:transfer-seam-bound}
+by \(2(\sqrt{1+t}-1)\leq t\). Normal rows have no tile derivative,
+so Cauchy--Schwarz gives
+\[
+ \sum_{\rm normal}H_i
+ \leq2L_N\sum_{i\in\mathcal I}|A_i|
+ \leq2b_*\sqrt n\sqrt{E\mathfrak e}\,U_{\mathcal I}.
+\]
+Completing the square in \(\sqrt{E\mathfrak e}\), for \(b_*<1\),
+shows that
+\begin{equation}\label{eq:transfer-straight-penalty}
+ \frac{\sum_iH_i-E\mathfrak e}{2}
+ \leq c_n U_{\mathcal I}^2,
+ \qquad c_n=\frac{n b_*^2}{2(1-b_*^2)}.
\end{equation}
-\emph{Scalar comparison.} The weight is $\phi$ (constant in tiles),
-so
+Choose \(b_*\) sufficiently small that
+\begin{equation}\label{eq:transfer-sphere}
+ \E_V\exp(2c_nU_{\mathcal I}^2)\leq C_n
+\end{equation}
+for every fixed horizontal differential and every fixed
+\(\mathcal I\). Here is a direct verification of this choice. If
+\(u\) is one coordinate of a uniform point on \(S^{q-1}\), its sphere
+moments satisfy
\[
- \wscal(h_{\mathrm{tot}},\phi)=\wscal(h,\phi)+L_T^{-2}\Scal_k.
+ \E(q u^2)^j
+ =\frac{q^j(1\cdot3\cdots(2j-1))}{q(q+2)\cdots(q+2j-2)}
+ \leq2^j j!.
\]
-By Lemma~\ref{lem:transfer-resolution}, $\Scal_k$ is a
-dimensionally controlled curvature; using $L_T=b_*\sqrt E$ and
-$E\ge b_*^{-2}K_{\mathrm{scal}}(q)$, we have
-$L_T^{-2}\Scal_k\ge-1$, so
-\begin{equation}
- \wscal(h_{\mathrm{tot}},\phi)
- \ge-C_{\mathrm{ent}}-1+2W-n\log E+E\mathfrak e.
- \label{eq:transfer-total-scalar}
+Consequently \(\E e^{tqu^2}\leq(1-2t)^{-1}\) for \(0\leq t<1/2\).
+The singular-value decomposition of \(dz_F\) expresses each
+\(q|A_i|^2/\mathfrak e\) as a convex combination of such random
+variables, with its singular-value weights. Jensen's inequality
+gives the same exponential bound for that convex combination.
+Another application of Jensen to the sum of the \(n\) rows in
+\eqref{eq:transfer-U} bounds its exponential moment by
+\((1-2nt)^{-1}\). Choosing, for instance, \(2nc_n<1/4\) proves
+\eqref{eq:transfer-sphere}. No independence of the rows of \(V\)
+is used.
+
+At a seam point the uniform tile row bound gives a rougher estimate
+which is still independent of the instance. By the triangle inequality
+for each row, \((a-1)_+\leq a\), and Cauchy--Schwarz on the horizontal
+rows,
+\[
+ \sum_iH_i\leq 2L_N\sqrt{q\mathfrak e}
+ +4q L_N/L_T
+ \leq2b_*q\sqrt{E\mathfrak e}+4q^{3/2}.
+\]
+Completing the square once more gives
+\begin{equation}\label{eq:transfer-seam-penalty}
+ \frac{\sum_iH_i-E\mathfrak e}{2}\leq c_{{\rm seam},n}q^2,
+ \qquad c_{{\rm seam},n}=2+\frac{b_*^2}{2}.
\end{equation}
-\subsection{The transfer bound}
-\label{sec:transfer-bound}
-
-Apply Theorem~\ref{thm:degree} to $G$ with the metric
-$h_{\mathrm{tot}}$ and weight $\phi$ from
-\S\ref{sec:transfer-application}. The integrand accumulates to
-\begin{equation}
- e^{-\wscal(h_{\mathrm{tot}},\phi)/2+\sum_iH_i/2}
- \le e^{C_{\mathrm{ent}}/2}E^{n/2}
- \exp\bigl(-\tfrac12E\mathfrak e+\tfrac12\sum_iH_i\bigr)
- e^{-W}.
- \label{eq:transfer-integrand}
+Apply the degree inequality, then use
+\eqref{eq:transfer-total-scalar}. Its loss is \(\exp(Cq/q)\), an
+absolute constant. The product volume form is
+\(dV_{h_{\rm tot}}=L_T^m dV_h\,dV_k\), so the measure factor is
+exactly \eqref{eq:transfer-prefactor}. Together with
+\eqref{eq:transfer-degree} and \(|\E\mathcal D|\leq\E|\mathcal D|\),
+this gives the following inequality, with
+\(C_{\rm out}=\exp(C_{\rm deg}+(C_{\rm ent}+1)/2)b_*^{-n}\), where
+\(C_{\rm deg}\) is the absolute exponent in Theorem~\ref{thm:degree}:
+\begin{equation}\label{eq:transfer-average}
+ \begin{split}
+ D_0I(2\tau)^q\leq C_{\rm out}q^{-n/2}
+ \int_{\mathcal M}\rho\,
+ \E_{V,s,\rm rd}
+ \int_{\{a:\ |G(x,a)|_\infty<b_q\}}
+ \mathcal B(x,a)\,dV_k(a),
+ \end{split}
\end{equation}
-
-At straight points, \eqref{eq:transfer-straight-bound} gives
+where the fiber integral includes its state transversals and only the
+compatible original-side band, and
\[
- e^{(\sum_iH_i-E\mathfrak e)/2}\le e^{c_nU_{\mathcal I}^2},
+ \mathcal B(x,a)=
+ \begin{cases}
+ \exp(c_nU_{\mathcal I}^2),&a\text{ straight over axes }\mathcal I,\\
+ \exp(c_{{\rm seam},n}q^2),&a\text{ in the seam}.
+ \end{cases}
\]
-and at seam points, \eqref{eq:transfer-seam-bound} gives
-\begin{equation}
- e^{(\sum_iH_i-E\mathfrak e)/2}\le e^{c_{\mathrm{seam}}(q)}
- \quad\hbox{with } c_{\mathrm{seam}}(q)=2+\tfrac12b_*^2q.
- \label{eq:transfer-seam-factor}
+The entropy constant enters only through this final prefactor, not the
+geometric bounds used below. The length and volume prefactor has no
+exponential loss in \(m\); the seam exponential will be absorbed by
+the choice of \(\lambda\).
+
+Fix \(x,\omega\), and put \(y=F(x)\). At a coupled pair in
+\eqref{eq:transfer-average}, a fixed projected grid index has at most
+one possible cell lift in the ball chart of \(y\). Its full primal
+and dual cells fit in that chart by \eqref{eq:transfer-tau} and the
+buffers, and the lift there is unique. Moreover each point of its
+primal cell is within \(10r\sqrt q\) of \(y\): the dual point is
+within \(b_q\sqrt q\) of \(y\), and the dual and primal extents are
+at most \(r\sqrt m/2\) and \(r\sqrt n/2\). The variation choice
+for \(r\) and the rounding error now give the copy-count estimate
+\begin{equation}\label{eq:transfer-copies}
+ D_0|u'_C|\leq D_0r^n
+ \bigl(|\eta_{\mathcal I}(y)|+2q^{-n}\bigr),
\end{equation}
-
-\emph{Straight contribution.} Each straight tile is an
-$\eta$-flat region; the sum of the exponential factors over the
-shifts gives
-\begin{equation}
- \frac1{r^q}\int_{[0,r)^q}\sum_k
- e^{c_nU_{\mathcal I}^2}\,d\sigma
- \le\,C_n'\int_{X_s}\rho\,dV_h,
- \label{eq:transfer-straight-average}
+where the component is in the \(V\)-rotated frame.
+
+On a straight tile the \(k\)-volume is exactly the flat volume of
+its image in the chamber. The signed permutation in \(G\) does not
+change its cube test, which is
+\(|\widehat z_y-\widehat z_X|_\infty<b_q\). Summing the chamber
+pieces in a copy is thus bounded by the \(m\)-area of its whole dual
+cube inside that test; the state probabilities for each chamber
+integrate to one. For a fixed axis set \(\mathcal I\), summing over
+all projected centers and averaging the shift gives exactly
+\begin{equation}\label{eq:transfer-shift}
+ r^{-q}\int_{\mathbb R^q}\int_{[-r/2,r/2]^m}
+ \indic_{\{|\widehat z_y-c-t|_\infty<b_q\}}\,dt\,dc
+ =r^{-q}r^m(2b_q)^q.
\end{equation}
-using $\rho\le dV_g$ and the entropy bound on the label graph
-(bounded by $2L_N\sqrt{q\mathfrak e}$ at the seams and by
-$L_T^2\mathfrak e$ on the straight part).
-
-\emph{Seam contribution.} On the seam,
-\begin{equation}
- \frac1{r^q}\int_{[0,r)^q}\sum_k\indic_{\text{seam}}\,d\sigma
- \le C_{\mathrm{seam}}(q)\lambda,
- \label{eq:transfer-seam-average}
-\end{equation}
-where the seam of a dual cube has leaf-volume at most
-$C\lambda$ (from $\Theta\le\gamma_0$), and
-$C_{\mathrm{seam}}(q)$ is elementary.
-
-Summing \eqref{eq:transfer-straight-average} and
-\eqref{eq:transfer-seam-average} and cancelling the common factor
-$E^{n/2}$ (which is eaten by the $q^{-n/2}$ from the degree
-estimate) gives
-\begin{equation}
- I \le C_{\mathrm{out}}\bigl(C_{\mathrm{str},n}
- +C_{\mathrm{seam}}(q)\lambda\bigr)
- \int_{\mathcal M}\rho.
- \label{eq:transfer-before-lambda}
+Here \(t\) is placed in the complementary axes. This identity is
+just the density \(r^{-q}\) of centers under uniform shift; extending
+the sum to the full grid is an upper bound for the selected lifted
+cells.
+
+The form bound holds in every rotated frame. Using Cauchy--Schwarz
+jointly in \(V\) and the \(\binom qn\) axis sets, and then
+\eqref{eq:transfer-sphere}, we obtain
+\begin{equation}\label{eq:transfer-form-average}
+ \begin{split}
+ \E_V\sum_{|\mathcal I|=n}
+ (|\eta_{\mathcal I}(y)|+2q^{-n})e^{c_nU_{\mathcal I}^2}
+ &\leq
+ \left(\E_V\sum_{\mathcal I}|\eta_{\mathcal I}(y)|^2\right)^{1/2}
+ \left(\E_V\sum_{\mathcal I}e^{2c_nU_{\mathcal I}^2}\right)^{1/2}
+ +C_nq^{-n}\binom qn\\
+ &\leq C_nq^{n/2}.
+ \end{split}
\end{equation}
-
-Choose $\lambda<1/2$ (uniformly in the entropy parameters) so that
-$C_{\mathrm{seam}}(q)\lambda\le C_{\mathrm{str},n}$. Then
-\begin{equation}
- I \le 2C_{\mathrm{out}}C_{\mathrm{str},n}
- e^{C_{\mathrm{ent}}/2}\int_{\mathcal M}\rho.
- \label{eq:transfer-final}
+This use of Cauchy--Schwarz allows the form components and row norms
+to depend on the same rotation. Combining
+\eqref{eq:transfer-copies}--\eqref{eq:transfer-form-average}, the
+straight part of the inner expectation in
+\eqref{eq:transfer-average} is at most
+\(C_nD_0q^{n/2}(2b_q)^q\), because \(r^{n+m-q}=1\). Multiplying
+by its prefactor \(q^{-n/2}\) bounds that part, for a fixed
+dimension-only constant \(C_{{\rm str},n}\), by
+\begin{equation}\label{eq:transfer-straight-average}
+ C_{{\rm str},n}D_0(2\tau)^q,
\end{equation}
-The factor $e^{C_{\mathrm{ent}}/2}$ comes from
-\eqref{eq:transfer-integrand} and is dimensionally bounded once
-$C_{\mathrm{ent}}$ is. With $\rho\le dV_g$,
-\eqref{eq:transfer-final} bounds $I$ by a dimension-dependent
-multiple of $\Vol_g(M)$.
-
-\subsection{Proof of the main theorem}
-
-\begin{proof}[Proof of Theorem~\ref{thm:main}]
-By Proposition~\ref{prop:yamabe-reduction}, it suffices to treat
-metrics with $\Scal_g=-1$.
-
-Suppose first $\|M\|>0$. the detector construction gives a
-root probability map $F$ on a measured family and a number
-$I\ge\tfrac12\|M\|$ with $I=\int_{\mathcal M}F^*\eta$. Apply
-Theorem~\ref{thm:entropy} with the prescribed elementary function
-$\mathcal B$ from the degree application: it produces $(h,\phi,\rho,F)$
-satisfying \eqref{eq:transfer-entropy} with a dimension-dependent
-constant $C_{\mathrm{ent}}$.
-
-Apply Theorem~\ref{thm:degree} as in
-\S\ref{sec:transfer-application} to construct the comparison map and
-obtain \eqref{eq:transfer-before-lambda}. With the final choice of
-$\lambda$, \eqref{eq:transfer-final} holds, giving
+since \((b_q/\tau)^q=(1+3/(4q))^q\leq e^{3/4}\).
+
+For the seam use only the consequence
+\(|z_X-z_y|_2\leq b_q\sqrt q\) of the cube test. Discard the tile
+boundaries, which have leafwise measure zero. The remaining set is
+the disjoint union of tile interiors indexed by their \emph{actual}
+chamber \(\sigma\) and its state arrays. Partition each such interior
+into positive sorted-coordinate pieces, for example by taking the
+first sorted permutation in a fixed order. On these pieces
+\(0<Z_j\leq1\), so no signed-chart reflection changes the chamber.
+The conditional state measure over each actual chamber has mass one,
+and its \(k\)-density is at most the elementary \(H_0(q)\) chosen
+before \(\lambda\).
+
+Retain the compatible-sheet restriction at this point. For a fixed
+projected grid index the uniqueness proved before
+\eqref{eq:transfer-copies} allows at most one compatible geometric
+lift. Apply the rough copy bound
+\(D_0|u'_C|\leq C_{\eta,n}D_0r^n\) from that equation, where
+\(C_{\eta,n}\) depends only on the form norm, and use the uniform
+density bound. Only after these bounds may we drop the restrictions
+on available sheets, roundings, and chambers and sum over all projected
+centers and chamber shapes.
+
+For a fixed actual chamber shape, normal axis set \(\mathcal I\), and
+positive sorted flag, \eqref{eq:transfer-tile-map} has rotated position
+\[
+ \widehat z_X=s+r\left(k+\tfrac12\indic_{\mathcal I}\right)
+ +d_{\mathcal I,\mathrm{shape},\mathrm{flag}}(Z).
+\]
+Here \(\indic_{\mathcal I}\) is the coordinate indicator vector of
+\(\mathcal I\).
+The displacement \(d\) is independent of the pairing and bit arrays,
+of \(\lambda,\gamma_0\), and of the shift; it is determined by the
+fixed chamber shape and the barycentric formula. For each fixed \(Z\)
+the shift integral of the rough test is consequently
\[
- \|M\|/2\le I
- \le 2C_{\mathrm{out}}C_{\mathrm{str},n}
- e^{C_{\mathrm{ent}}/2}\Vol_g(M).
+ \begin{split}
+ \frac1{r^q}\int_{[0,r)^q}\sum_{k\in\mathbb Z^q}
+ \indic_{\{|s+r(k+\frac12\indic_{\mathcal I})+d(Z)-\widehat z_y|_2
+ \leq b_q\sqrt q\}}\,ds
+ &=r^{-q}\Vol_{\mathbb R^q}(B_{b_q\sqrt q})\\
+ &\leq r^{-q}(4\tau\sqrt q)^q.
+ \end{split}
\]
-Setting $a_n^{-1}:=4C_{\mathrm{out}}C_{\mathrm{str},n}
-e^{C_{\mathrm{ent}}/2}$ proves the normalized case.
+This calculation is valid despite the shift dependence of the actual
+diagram, because the copy, lift, and density bounds were imposed before
+the enlargement to all centers. Tonelli's theorem applies to these
+nonnegative bounds.
+
+In each positive sorted piece the seam lies in
+\(\bigcup_j\{Z_j\leq\lambda\}\), of coordinate volume at most
+\(m\lambda\). A cube has \(2^m m!\) barycentric top chambers and
+at most \((m+1)!\) sorted flags per chamber. Thus the seam part of
+the inner expectation is bounded by
+\begin{equation}\label{eq:transfer-seam-average}
+ C_*(q)\lambda D_0(2\tau)^q,
+\end{equation}
+where one may take the elementary envelope
+\[
+ C_*(q)=\max\left\{1,\,
+ C_{\eta,n}H_0(q)\binom qn(2^m m!)(m+1)!\,m\,
+ r^{n-q}(2\sqrt q)^q\exp(c_{{\rm seam},n}q^2)\right\}.
+\]
+Every factor here was bounded before choosing \(\lambda\). Thus
+\(C_*\) is independent of \(\lambda,\gamma_0,\tau\), of the bounds
+\(K_R,K_{\rm scal}\), and of \(C_{\rm ent}\). The powers \(\tau^q\) in the shift
+count have already canceled those in \((2\tau)^q\).
+
+The two fiber estimates now have the same normalization. Insert
+\eqref{eq:transfer-straight-average} and
+\eqref{eq:transfer-seam-average} into \eqref{eq:transfer-average},
+using \(q^{-n/2}\leq1\) for the seam, and cancel the positive factor
+\(D_0(2\tau)^q\). For the conditional data and parameter bounds used
+so far, this gives
+\begin{equation}\label{eq:transfer-before-choices}
+ I\leq C_{\rm out}\bigl(C_{{\rm str},n}+C_*(q)\lambda\bigr)
+ \int_{\mathcal M}\rho.
+\end{equation}
-If $\|M\|=0$, the inequality is vacuous.
+We now fix these bounds before invoking the entropy theorem. For each
+integer \(q\geq n+1\), the form derivative bound first gives a
+deterministic \(r(q)\) with elementary reciprocal. The size bound for
+the universal type list, \(D_0\), incidence counts, and the bounds for
+\(k_0,dX\), including \(H_0(q)\), have elementary envelopes uniform
+over all permitted graphs and cocycles. Fix \(b_*\) using
+\eqref{eq:transfer-sphere}; this determines the dimensional constants
+\(c_{{\rm seam},n}\) and \(c_{{\rm hor},n}\), independently of
+\(C_{\rm ent}\). The displayed formula for \(C_*(q)\) is consequently
+available at this stage. Set
+\[
+ \lambda(q)=\frac1{4C_*(q)},\qquad
+ \gamma_0(q)=\frac14(m+1)^{-(m+1)}\lambda(q)^{m(m+1)}.
+\]
+These deterministic choices have elementary reciprocals and satisfy
+\eqref{eq:transfer-gamma}. The resolution estimates now give common
+elementary envelopes for its fixed-order chart derivatives, hence
+\(K_R(q),K_{\rm scal}(q)\), and then the common \(\tau(q)\) in
+\eqref{eq:transfer-tau}. None of these choices uses the entropy constant.
-The general case follows from Yamabe reduction.
+For integer \(j\geq n+1\), define
+\[
+ \Psi_n^\circ(j)=\max\left\{1,\ b_*^{-2}K_{\rm scal}(j),
+ \left(\frac{20c_{{\rm hor},n}j^{4/3}}{\tau(j)}\right)^{75/13}
+ \right\},
+\]
+and define a majorant on all \(t\in[1,\infty)\) by
+\[
+ \Psi_n(t)=1+\max_{\substack{j\in\mathbb Z\\
+ n+1\leq j\leq\max\{n+1,\lceil t\rceil\}}}
+ \Psi_n^\circ(j).
+\]
+Every expression used here has an elementary upper bound; the finite
+maximum is bounded by an increasing elementary envelope at
+\(\max\{n+1,\lceil t\rceil\}\). Thus \(\Psi_n\) is an elementary
+majorant fixed solely from \(n\), and it dominates both the curvature
+and strict horizon requirements. Hence every conditional datum with
+\(E\geq\Psi_n(q)\) satisfies all hypotheses used in
+\eqref{eq:transfer-before-choices}. Our choice
+\(C_*(q)\lambda(q)=1/4\) and the definition of \(C_{\rm out}\) prove
+\eqref{eq:transfer-conditional-target} with
+\[
+ K_n=e^{C_{\rm deg}+1/2}b_*^{-n}
+ \bigl(C_{{\rm str},n}+1/4\bigr).
+\]
+This closes the conditional target for all the specified data before
+any entropy family is chosen.
+
+Now apply Theorem~\ref{thm:entropy} with \(\mathcal B=\Psi_n\),
+\(E_{\min}=1\), the detector root map as \(F^{\mathrm{in}}\), and the
+one-point transverse space \(\Omega_0\). The detector construction uses
+finitely many compactly supported bump representatives and their deck
+translates, giving the required uniform support and derivative bounds in
+a lifted atlas and the equivariant compact support sets. After pulling
+the detector map to \(\Omega\) and embedding its labels equivariantly
+among the output's finite copies, the measured affine probability homotopy
+of Section~2 preserves \(I\); the output therefore supplies the conditional
+data with \(E>\Psi_n(q)\). Its constant
+\(C_{\rm ent}=C_{\rm ent}(n,\Psi_n)\) depends only on \(n\). Since
+\(\rho\leq dV_g\) and the transverse measure over \(M\) is a
+probability measure,
+\[
+ I\leq K_ne^{C_{\rm ent}/2}\int_{\mathcal M}\rho
+ \leq K_ne^{C_{\rm ent}/2}\Vol_g(M).
+\]
+Finally \(I\geq\|M\|/2\), so the constant
+\(C_n=2K_ne^{C_{\rm ent}/2}\) proves the theorem.
\end{proof}
-\begin{thebibliography}{40}
+\begin{proof}[Proof of Theorem~\ref{thm:main}]
+Fix \(n\geq3\), and let \(C_n>0\) be the constant for the normalized
+scalar-curvature estimate in Theorem~\ref{thm:scalar-volume}.
+Proposition~\ref{prop:yamabe-reduction} gives
+\[
+ \|M\|\leq C_n\int_M(\Scal_g^-)^{n/2}\,dV_g
+\]
+for every metric in the theorem. Taking \(a_n=C_n^{-1}>0\) proves
+the stated integral bound.
+\end{proof}
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\end{thebibliography}
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\end{document}