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\hypersetup{
pdftitle={An integral scalar curvature bound for real simplicial volume},
pdfauthor={OpenAI}
}
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\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
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\DeclareMathOperator{\Scal}{Scal}
\DeclareMathOperator{\Ric}{Ric}
\DeclareMathOperator{\Vol}{Vol}
\DeclareMathOperator{\supp}{supp}
\DeclareMathOperator{\tr}{tr}
\DeclareMathOperator{\diver}{div}
\DeclareMathOperator{\Var}{Var}
\DeclareMathOperator{\dist}{dist}
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\newcommand{\N}{\mathbb N}
\newcommand{\Z}{\mathbb Z}
\newcommand{\E}{\mathbb E}
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\newcommand{\II}{\mathrm{II}}
\newcommand{\HS}{\mathrm{HS}}
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\title{An integral scalar curvature bound for real simplicial volume}
\author{OpenAI}
\date{October 5, 2026}
\begin{document}
\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
\[
\int_M (\Scal_g^-)^{n/2}\,dV_g \;\ge\; a_n\,\|M\|.
\]
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.
\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:
\[
\|M\| = \inf \Bigl\{ \sum_j |a_j| : \sum_j a_j\sigma_j \in
C_n^{\mathrm{sing}}(M;\R) \text{ represents } [M]_{\R} \Bigr\}.
\]
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
\[
\int_M (\Scal_g^-)^{n/2}\,dV_g
\]
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.
\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
\begin{equation}
\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
\[
\Vol_g(M) \ge a_n\,[n(n-1)]^{-n/2}\,\|M\|.
\]
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 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}.
\subsection*{Context and prior results}
Several cases of the conjecture were already known or partial.
\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.
\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
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.
\subsection*{Overview of the argument}
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.
\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}.
\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}.
\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.
\subsection{The conformal reduction}
\begin{proposition}[Yamabe reduction]
\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
\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
\begin{equation}
\|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 $g$:
\[
Y(M,[g])=\inf_{u\in C^\infty(M),\,u>0}
\frac{\displaystyle\int_M
\Bigl(\tfrac{4(n-1)}{n-2}|du|_g^2+\Scal_g u^2\Bigr)dV_g}
{\displaystyle\Bigl(\int_M u^{2n/(n-2)}\,dV_g\Bigr)^{(n-2)/n}}.
\]
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
\[
\int_M \Scal_g u^2\,dV_g \ge
-\left(\int_M (\Scal_g^-)^{n/2}\,dV_g\right)^{2/n}
\left(\int_M u^{2n/(n-2)}\,dV_g\right)^{(n-2)/n}.
\]
The gradient term in the Yamabe quotient is nonnegative, so taking the
infimum yields
\[
-Y(M,[g]) \le \left(\int_M (\Scal_g^-)^{n/2}\,dV_g\right)^{2/n}.
\]
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}.
\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}
Let $\Gamma=\pi_1(M)$.
\medskip\noindent\textbf{Bounded cohomology duality.}
For every integer $n\ge1$ and every closed oriented $n$-manifold $M$,
\[
\|M\| = \sup\Bigl\{\langle[\alpha],[M]_{\R}\rangle :
[\alpha]\in H_b^n(\Gamma;\R), \ \|\alpha\|_\infty\le1\Bigr\},
\]
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 $\tfrac12\|M\|$:
\begin{equation}
I:=\langle[c],[M]_{\R}\rangle\ge\tfrac12\|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:
\[
\Delta_{\mathcal L}=\Bigl\{p=(p_\ell)_{\ell\in\mathcal L} :
p_\ell\ge0,\ \sum_\ell p_\ell=1,\ p\text{ finitely supported}\Bigr\}.
\]
On each finite 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}
\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
\begin{equation}
\wscal(h,\phi)\ge -C_{\mathrm{ent}}+2W-n\log E+E\,e_h(F),
\label{eq:entropy-overview}
\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 $n$.
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}
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{The graph identity}
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:
\begin{equation}
e^{\phi_f}dV_{h_f}=e^{\phi}dV_h.
\label{eq:graph-volume-preservation}
\end{equation}
\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}
\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}
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}
\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{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}
\end{equation}
For every sufficiently large $H_*>0$ there is a smooth solution with
\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}
\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
$z$:
\[
\partial_z p=-\frac{\epsilon a(F^2+\delta^2)+(1-v^2)/2}{z^2}
-\frac{\epsilon a+\alpha}{N^2}\le-\frac{\epsilon a}{N^2}
\]
in case (i), and
\[
\partial_z p=-\frac{e(\epsilon a+\alpha)}{w}
\sec^2\bigl((f_0-ez)/w\bigr)\le-c_0
\]
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
\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}
\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.
\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
\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}
\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}
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
\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}
\end{equation}
The comparison errors from the sweep metrics are
\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}
\end{equation}
and $0\le B_i,C_i\le1$ by metric monotonicity.
\subsubsection{Sweep statistics}
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}
\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}
\end{equation}
Finally, define two nonnegative quantities
\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}
\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
\begin{equation}
\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 $\beta_{s,i}\ge0$ vanishes near $\supp F_i$ and satisfies the
spatial cutoff conditions of Lemma~\ref{lem:graph-localization} in
$h_{i-1}$. Writing
\begin{equation}
ND_i^0=\epsilon a(F_i^2/f_i-f_i)+A_i^2/(2f_i)
\label{eq:sweep-ND0}
\end{equation}
on $\{f_i>0\}$, extended by zero off the bump, one has
\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}
\end{equation}
\end{proposition}
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
\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}
\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 $i$:
\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}
\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
\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
\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}
\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
$\epsilon^2M_0^2/r\ge c\epsilon^2r$.
\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{equation}
Q\ge\Pi_p-4\theta^2m-\frac r\epsilon\delta_0,
\label{eq:sweep-inequality}
\end{equation}
where the \emph{positive reserve} is
\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,
\begin{equation}
\rho\le dV_g,\qquad
W:=\log(dV_h/\rho)\ge0,
\label{eq:entropy-density}
\end{equation}
\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}
\end{equation}
\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
\[
q\ge\max\{n+1,\ \#(\Gamma\backslash\mathcal L),\ \Delta+1\},
\]
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}
The proof of Theorem~\ref{thm:entropy} is the content of the rest of
this section.
\subsection{The bootstrap}
The first stage produces an initial datum with a small but positive
energy coefficient.
\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
\begin{equation}
\rho:=dV_{h_*}\le dV_g,\qquad
\Scal_{h_*}\ge-C_{\mathrm{boot}}+\epsilon_0e_{h_*}(F_*).
\label{eq:entropy-bootstrap-output}
\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}
\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
\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}
\end{equation}
and the volume changes as
\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
\[
\mathcal F(\text{output})\ge
\mathcal F(\text{input})+2W_p-n\log(1+l)-C_nl^2+P_p,
\]
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
\[
0<l\le l_0,\qquad \theta=D_*l\le\tfrac1{16},\qquad N^2=l\epsilon.
\]
The errors $(\delta_0,\eta_0)$ in Proposition~\ref{prop:sweep} can be
chosen so that
\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}
\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}
\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
\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}
\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
$m_0<1/4$).
\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{equation}
l(\epsilon)=\frac{l_*}{(1+\log(1+\epsilon))^2},
\label{eq:entropy-step-schedule}
\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 $E$
\begin{equation}
\wscal(h',\phi')\ge-C+2W_{\mathrm{new}}-n\log E
+E e_{h'}(F')+P,
\label{eq:entropy-phase-output}
\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,
\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}
\end{equation}
If the initial volume logarithm is zero, then for sufficiently large
dimensional $E$,
\begin{equation}
W_{\mathrm{new}}\le E^{0.26}(E+P)^{0.75}.
\label{eq:entropy-phase-volume}
\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
$C_n\sum l(\epsilon_p)^2$.
\emph{Summability.} Put $t_p=\log(1+\epsilon_p)$. Since
$\epsilon_p\ge\epsilon_*>0$ and $l_*$ is fixed small,
\[
t_{p+1}-t_p=\log\Bigl(1+l(\epsilon_p)\frac{\epsilon_p}{1+\epsilon_p}\Bigr)
\asymp_n l(\epsilon_p)=\frac{l_*}{(1+t_p)^2}.
\]
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
$P_p\le P$,
\[
\sum_pW_p\le C_nk E^{1/4}(E+P)^{3/4}.
\]
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
\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}
\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}
\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}
\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}.
\end{proof}
\subsection{Proof of the entropy theorem}
\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.
\emph{Initial data.} Apply Lemma~\ref{lem:entropy-bootstrap}.
Increasing a dimensional constant once, at the output
\[
\wscal(h_*,0)\ge-C_n+2W-n\log\epsilon_0+\epsilon_0e_{h_*}(F_*),
\]
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}
\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}
\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}
\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}
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}
\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}
\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}
\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}
\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 \(C^\infty\).
\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}
\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.
\end{proof}
\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}
\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}
\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}
\end{equation}
for every \(s\in(-\tau,\tau)\), where
$\chi_{\mathrm{deg}}(z)=\int_{-\infty}^z(a_-(u)-a_+(u))\,du$.
\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.
\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}
\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}
\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}.
\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}).
\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}
\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}
\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}
\end{equation}
and all fixed-order derivatives of $\eta$ admit an elementary bound
in $q$.
\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.
\end{proof}
\subsection{The cubical chain complex}
\label{sec:transfer-cells}
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}.
\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.
\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}
\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}
\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.
\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
\[
G:=V(z\circ X)
\]
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.
\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}
\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}
\end{equation}
The horizon condition (Definition~\ref{def:degree-horizon}) holds
because
\[
\frac{C(1+B)^{2/3}}{L_T}
\le\frac{C_nq^{1/3}}{b_*\sqrt E}<\frac{\tau}{10q}
\]
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
\[
H_i^{\mathrm{tangent}}=2(L_T^{-1}+|A_i|_h-1)_+
\le L_T^2\mathfrak e
\]
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}
\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,
\[
\sum_iH_i\le 2L_N\sqrt{q\mathfrak e}+4qL_N/L_T
\le 2b_*q\sqrt{E\mathfrak e}+4q^{3/2}.
\]
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}
\end{equation}
\emph{Scalar comparison.} The weight is $\phi$ (constant in tiles),
so
\[
\wscal(h_{\mathrm{tot}},\phi)=\wscal(h,\phi)+L_T^{-2}\Scal_k.
\]
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}
\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}
\end{equation}
At straight points, \eqref{eq:transfer-straight-bound} gives
\[
e^{(\sum_iH_i-E\mathfrak e)/2}\le e^{c_nU_{\mathcal I}^2},
\]
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}
\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}
\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}
\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}
\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
\[
\|M\|/2\le I
\le 2C_{\mathrm{out}}C_{\mathrm{str},n}
e^{C_{\mathrm{ent}}/2}\Vol_g(M).
\]
Setting $a_n^{-1}:=4C_{\mathrm{out}}C_{\mathrm{str},n}
e^{C_{\mathrm{ent}}/2}$ proves the normalized case.
If $\|M\|=0$, the inequality is vacuous.
The general case follows from Yamabe reduction.
\end{proof}
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