diff ee77827..626631e simplicial_volume_scalar_bound_v2.txt
diff --git a/simplicial_volume_scalar_bound_v2.txt b/simplicial_volume_scalar_bound_v2.txt
index 6b1149b..94b9b12 100644
--- a/simplicial_volume_scalar_bound_v2.txt
+++ b/simplicial_volume_scalar_bound_v2.txt
@@ -11,7 +11,6 @@
\hypersetup{
pdftitle={An integral scalar curvature bound for real simplicial volume},
- pdfauthor={OpenAI}
}
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
@@ -42,14 +41,9 @@
\newcommand{\cl}{\operatorname{cl}}
\newcommand{\indic}{\mathbf 1}
-\title{An integral scalar curvature bound for real simplicial volume}
-\author{OpenAI}
-\date{October 5, 2026}
\begin{document}
-\maketitle
-\begin{abstract}
For every integer $n\ge3$ there is a constant $a_n>0$, depending only on
$n$, such that every closed connected oriented smooth $n$-manifold $M$
and every smooth Riemannian metric $g$ on $M$ satisfy
@@ -68,17 +62,12 @@ large simplicial volume.
The proof has two inputs. Qualitatively, a closed manifold admitting a
metric of nonnegative scalar curvature has zero real simplicial volume;
-this vanishing theorem is proved in the companion paper
-\cite{OpenAIScalarInessentiality2026} and is used here once, in
-\S\ref{sec:reduction-detector}. Quantitatively, we deform metrics by a
-prescribed-curvature graph flow, iterate it to build an entropy metric
-with a large energy coefficient, and convert the bounded-cohomology
-detector of $\|M\|$ into measured comparison degrees. Those degrees are
+for spin universal covers this vanishing theorem is the result of \cite{MaWangXieYuZhu2026}, and it is used here once. Quantitatively, a prescribed-curvature graph flow deforms the metric and is iterated to build an entropy metric with a large energy coefficient, and it converts the bounded-cohomology detector of $\|M\|$ into measured comparison degrees. Those degrees are
bounded by a weighted integral that, after the deformations, is
comparable to $\Vol_g(M)$. All constants depend only on $n$.
-\end{abstract}
-\section{Introduction}
-\label{sec:introduction}
+
+
+\par\noindent{\footnotesize\ttfamily src-ma2026.pdf}\par
The \emph{real simplicial volume} of a closed connected oriented
$n$-manifold $M$ is the $\ell^1$-seminorm of its real fundamental class:
@@ -103,6 +92,9 @@ is invariant under a constant rescaling $g\mapsto\lambda^2g$, so it is
dimensionless; and it records both how negative the scalar curvature
becomes and over how much volume it does so.
+
+\par\noindent{\footnotesize\url{https://doi.org/10.1007/BFb0075649}}\par
+
\begin{theorem}[Main theorem]
\label{thm:main}
For every integer $n\ge3$ there is a constant $a_n>0$, depending only on
@@ -137,7 +129,9 @@ all such manifolds and \eqref{eq:pointwise-form} for all metrics on all
such manifolds are equivalent by a Yamabe rescaling; see
\cite[Lemma~2.1]{MinZhengZhu2026}.
-\subsection*{Prior results}
+
+\par\noindent{\footnotesize\ttfamily src-min2026.pdf}\par
+
Several related statements were known before the general case.
@@ -147,24 +141,26 @@ Several related statements were known before the general case.
uniformly bounded above; their constant depends also on that bound.
\item Ma, Wang, Xie, Yu and Zhu \cite{MaWangXieYuZhu2026} prove that
$\|M\|=0$ when the universal cover is spin and $\Scal_g\ge0$. This is
- the qualitative companion result used below.
+ the qualitative vanishing result used below.
\item Min, Zheng and Zhu \cite{MinZhengZhu2026} prove the integral
estimate with coefficient $27/2$ for every metric on every closed
K\"ahler surface, and for an infinite family of non-K\"ahler
symplectic four-manifolds.
\end{itemize}
-The one external input used in this paper is the qualitative vanishing
-theorem of the companion manuscript
-\cite[Corollary~1.5]{OpenAIScalarInessentiality2026}, stated precisely as
-Theorem~\ref{thm:vanishing-input} in \S\ref{sec:reduction-detector}: a
+
+\par\noindent{\footnotesize\ttfamily src-braunsauer2021.pdf}\par
+
+The one external input is the qualitative vanishing
+theorem of
+\cite{MaWangXieYuZhu2026}, stated precisely as
+Theorem~\ref{thm:vanishing-input}: a
closed connected oriented manifold of positive dimension admitting a
metric of nonnegative scalar curvature has zero real simplicial volume.
Everything else is proved here. The work is the passage from that
qualitative statement to a quantitative estimate whose constant is
uniform in the topology and in the metric.
-\subsection*{The logical spine}
The proof has five stages. Each is stated as a theorem or proposition
whose hypotheses are no stronger than what the previous stage delivers,
@@ -173,7 +169,7 @@ and the last stage composes them.
\begin{enumerate}
\item \emph{Conformal reduction to $\Scal_g=-1$}
- (\S\ref{sec:reduction-detector}, Proposition~\ref{prop:yamabe-reduction}).
+ Proposition~\ref{prop:yamabe-reduction}.
In every conformal class the Yamabe minimizer has constant scalar
curvature. If $\|M\|>0$, the vanishing theorem forces that constant to
be negative, and a constant rescaling normalizes it to $-1$.
@@ -187,7 +183,7 @@ and the last stage composes them.
which is Theorem~\ref{thm:scalar-volume}.
\item \emph{A detector for $\|M\|$.}
- (\S\ref{sec:reduction-detector}, Proposition~\ref{prop:detector}).
+ Proposition~\ref{prop:detector}.
Duality between the $\ell^1$-seminorm on homology and the supremum
seminorm on bounded cohomology, together with the Mapping Theorem,
produces a bounded group cocycle $c$ of norm at most one whose value
@@ -195,10 +191,10 @@ and the last stage composes them.
becomes a closed $n$-form $\eta$ on the simplex of probability vectors,
and an equivariant probability map $F$ from the universal cover pulls
$\eta$ back with total integral $I$. So \eqref{eq:normalized-target}
- follows once we bound $I$ by $C_n\Vol_g(M)$.
+ follows once one bounds $I$ by $C_n\Vol_g(M)$.
\item \emph{The graph engine.}
- (\S\ref{sec:graphs}). Replace a metric $h$ by the metric induced on the
+ Replace a metric $h$ by the metric induced on the
graph of a function $f$, and shift the weight $\phi$ so that the
weighted volume $e^\phi dV_h$ does not change. The weighted graph
identity, Lemma~\ref{lem:graph-identity}, writes the change of weighted
@@ -206,13 +202,13 @@ and the last stage composes them.
nonnegative errors. Since $D$ can be prescribed, a solvability theorem
for the prescribed weighted mean curvature equation
(Lemma~\ref{lem:graph-prescription}, localized in
- Lemma~\ref{lem:graph-localization}) lets us make that square
+ Lemma~\ref{lem:graph-localization}) permits making that square
systematically positive. Sweeping the coordinates of $F$ in order
yields Proposition~\ref{prop:sweep}: one pass improves the energy of $F$
at a cost bounded by an explicit nonnegative reserve $\Pi_p$.
\item \emph{Iteration to an entropy metric.}
- (\S\ref{sec:entropy}). A conformal bootstrap raises the energy
+ A conformal bootstrap raises the energy
coefficient from a very small value to a fixed dimensional one
(Lemma~\ref{lem:entropy-bootstrap}). Phases of sweeps then raise it to
an arbitrary prescribed $E$ while keeping the accumulated volume
@@ -227,11 +223,11 @@ and the last stage composes them.
(Theorem~\ref{thm:entropy}).
\item \emph{Degree and transfer.}
- (\S\ref{sec:degree} and \S\ref{sec:transfer}). The measured degree
+ The measured degree
estimate, Theorem~\ref{thm:degree}, bounds the signed degree of a map
from a measured family of $d$-manifolds to a $d$-cube by an integral of
$\exp(-\wscal/2)$ times a penalty for each coordinate gradient. The
- transfer of \S\ref{sec:transfer} builds such a map whose degree is the
+ transfer builds such a map whose degree is the
detector pairing $I$ and whose tangential and normal lengths are
$b_*\sqrt E$ and $b_*\sqrt{qE}$. The factor $E^{n/2}$ produced by the
degree bound cancels the $E^{-n/2}$ from the product volume, the
@@ -247,17 +243,16 @@ $L_T^{-m}L_N^{-n}$ of the degree denominator, and the entropy inequality
contributes $E^{n/2}e^{-W}$; the three factors multiply to
$b_*^{-n}q^{-n/2}\rho$, in which no power of $E$ appears.
-\subsection*{How to read this paper}
-The three technical engines are Sections~\ref{sec:graphs},
-\ref{sec:entropy} and \ref{sec:degree}. They are independent of each
-other in the following sense: \S\ref{sec:graphs} uses only Riemannian
-geometry and elliptic PDE; \S\ref{sec:entropy} uses only
-\S\ref{sec:graphs}; and \S\ref{sec:degree} uses only geometric measure
-theory, with no reference to simplicial volume or bounded cohomology.
-Only the final \S\ref{sec:transfer} uses all of them. A reader willing
+The three technical engines are the graph engine, the entropy
+iteration and the degree estimate. They are independent:
+the graph engine uses only Riemannian geometry and elliptic PDE,
+the entropy iteration uses only the graph engine, and the degree
+estimate uses only geometric measure theory, with no reference to
+simplicial volume or bounded cohomology.
+Only the transfer uses all of them. A reader willing
to accept the three engines as black boxes can read
-\S\ref{sec:reduction-detector}, \S\ref{sec:transfer} and the statements
+the reduction, the transfer and the statements
of Theorem~\ref{thm:entropy} and Theorem~\ref{thm:degree} for a complete
proof of Theorem~\ref{thm:main}.
@@ -266,9 +261,8 @@ $C_n,c_n,C$ are positive and depend only on the dimension unless a
subscript or an explicit dependence is indicated; their values may change
from line to line, and no effort is made to keep them sharp. Integrals
over measured families of leaves always include the transverse measure;
-the construction is given in \S\ref{sec:reduction-detector}.
+the construction is given with the detector.
-\subsection*{The toolkit, in one page}
Four techniques carry the proof, and the logical spine above says where
each is used. Yamabe theory supplies, in every conformal class, a metric
@@ -290,20 +284,16 @@ Geometric measure theory supplies the measured degree and its bound by a
weighted integral with gradient penalties, obtained by slicing, weighted
perimeter minimization, coarea with the correct lapse, and varifold
compactness with Allard regularity.
-\section{Reduction and the bounded cohomology detector}
-\label{sec:reduction-detector}
-We first separate the conformal reduction from the quantitative part of
-the proof. We then construct a differential form on a simplex of
+First separate the conformal reduction from the quantitative part of
+the proof. Then construct a differential form on a simplex of
probability vectors that detects simplicial volume. This form will remain fixed while
the metric and the probability map are deformed.
-\subsection{The conformal reduction}
-We use the following result, stated here in the precise form needed.
+Use the following result, stated here in the precise form needed.
-\begin{theorem}[OpenAI, Corollary 1.5 of
- \cite{OpenAIScalarInessentiality2026}]
+\begin{theorem}
\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
@@ -311,12 +301,12 @@ scalar curvature, then its real simplicial volume is zero.
\end{theorem}
In particular, the hypothesis permits scalar curvature to vanish
-identically. Theorem~\ref{thm:vanishing-input} is the only result from
-the companion manuscript invoked below without proof.
+identically. Theorem~\ref{thm:vanishing-input} is the only external result
+invoked below without proof.
The Yamabe reformulation of the scalar curvature inequality is
standard; \cite[Lemma 2.1]{MinZhengZhu2026} records the equivalence of
-its uniform pointwise and integral forms. We include the short
+its uniform pointwise and integral forms. Include the short
classwise reduction that identifies the use of the vanishing input.
\begin{proposition}
@@ -376,14 +366,19 @@ Consequently \(\Vol_h(M)\le\int_M(\Scal_g^-)^{n/2}\,dV_g\), and
\eqref{eq:yamabe-transfer}.
\end{proof}
-For the rest of the proof we fix \(M^n\) and \(g\) with
-\(\Scal_g=-1\), and prove \eqref{eq:scalar-minus-one-target}. We may
-continue to assume \(\|M\|>0\).
-\subsection{A bounded cocycle and its probability form}
+\par\noindent{\footnotesize J. Math. Pures Appl. (9) 55 (1976), 269--296}\par
+\par\noindent{\footnotesize\url{https://doi.org/10.4310/jdg/1214439291}}\par
+\par\noindent{\footnotesize\ttfamily src-trudinger1968.pdf}\par
+\par\noindent{\footnotesize\ttfamily src-yamabe1960.pdf}\par
+\par\noindent{\footnotesize\ttfamily src-leeparker1987.pdf}\par
+
+For the rest of the proof one fixes \(M^n\) and \(g\) with
+\(\Scal_g=-1\), and prove \eqref{eq:scalar-minus-one-target}. Continue to assume \(\|M\|>0\).
+
Let \(\Gamma=\pi_1(M)\). This group is countable because \(M\) has a
-finite triangulation. We use two standard facts about bounded
+finite triangulation. Use two standard facts about bounded
cohomology with trivial real coefficients. First, duality between the
\(\ell^1\)-seminorm on homology and the supremum seminorm on bounded
cohomology gives
@@ -397,11 +392,15 @@ See \cite[Section 1.1]{Gromov1982}.
Second, the isometric Mapping Theorem identifies \(H_b^k(M;\R)\) with
\(H_b^k(\Gamma;\R)\) for every \(k\ge2\), through the classifying map.
The version used here applies to connected countable CW spaces and
-preserves the seminorms. We use the Mapping Theorem of Gromov
+preserves the seminorms. Use the Mapping Theorem of Gromov
\cite[Section 3.1]{Gromov1982} and its canonical group formulation in
\cite[Theorems 8.3 and 8.6]{IvanovNotes}.
-We use the homogeneous model for group cochains. Thus an \(n\)-cochain
+
+\par\noindent{\footnotesize\ttfamily src-gromov1982.pdf}\par
+\par\noindent{\footnotesize\ttfamily src-ivanov2017.pdf}\par
+
+Use the homogeneous model for group cochains. Thus an \(n\)-cochain
is a bounded function on \(\Gamma^{n+1}\), invariant under simultaneous
left translation, and its coboundary is the alternating sum obtained by
omitting one argument. Alternation is a cochain projection of norm at
@@ -425,7 +424,10 @@ group representative of norm at most \(1+\varepsilon\), alternate it,
and divide by \(1+\varepsilon\), with \(0<\varepsilon\le1/2\). Replacing
the cocycle by its negative first, if necessary, fixes the sign.
-We allow a finite number of copies of every group label. More
+
+\par\noindent{\footnotesize\ttfamily src-frigerio2016.pdf}\par
+
+Allow a finite number of copies of every group label. More
precisely, let \(\mathcal L=\Gamma\times\{1,\ldots,s\}\), with the free
left \(\Gamma\)-action on the first factor, and evaluate \(c\) on
\(\mathcal L^{n+1}\) by projection to \(\Gamma^{n+1}\). The same
@@ -446,7 +448,7 @@ On each finite-dimensional face define the \(n\)-form
\end{equation}
These definitions agree on common faces.
-We call a map \(F=(F_\ell)\) with nonnegative coordinates and
+Call a map \(F=(F_\ell)\) with nonnegative coordinates and
\(\sum_\ell F_\ell^2=1\) a \emph{root probability map}. For the maps
used here, whose coordinate supports are locally finite, the squared
coordinates \(p_\ell=F_\ell^2\) give the associated map to
@@ -519,10 +521,8 @@ such labelings give cochain-homotopic maps by the simplex prism
construction.
\end{proof}
-\subsection{Measured equivariant families}
-The deformations below use auxiliary probability choices. We record
-their integration convention so that the detector value remains
+The deformations below use auxiliary probability choices. Record their integration convention so that the detector value remains
unambiguous.
Let \((\Omega,\mu)\) be a standard Borel probability space with a
@@ -540,7 +540,7 @@ proper in the \(\widetilde M\) factor.
Related equivariant measured bundles are central to the macroscopic
volume argument of Braun and Sauer \cite[Sections 1.2 and 4]{BraunSauer2021}.
-Here we need only Borel transverse data and use the following direct
+Here only Borel transverse data are needed, and the following direct
integration convention.
Choose a finite evenly covered
@@ -556,7 +556,7 @@ integrable equivariant leafwise \(n\)-form \(\alpha\), set
\end{equation}
Equivariance and invariance of \(\mu\) show that this value is
independent of the lifts and partition. The same convention applies
-to nonnegative densities. We write \(dV_h\) for the resulting measured
+to nonnegative densities. Write \(dV_h\) for the resulting measured
Riemannian density when \(h=h_\omega\) is a leafwise metric.
The data used in the first deformation will have bounds in a fixed
@@ -582,17 +582,15 @@ same properties. Stokes' theorem on the measured product with
\]
Thus every probability map constructed below continues to detect the
same number \(I\). Enlarging \(\Omega\) by an independent probability
-space also preserves the integral by Fubini. We may discard invariant
+space also preserves the integral by Fubini. Discard invariant
null sets whenever needed. More general measured spaces of leaves
-will enter the degree estimate in Section~\ref{sec:degree}; their chart and
+will enter the degree estimate in the degree estimate below; their chart and
integration conventions will be specified there.
-\section{Weighted graph deformations and a sweep}
-\label{sec:graphs}
This section develops a deformation that changes the metric while
preserving a weighted volume density. Its input is a smooth function
on a Riemannian manifold; its output is the metric induced on the
-graph of that function. We first compute the change of weighted
+graph of that function. First compute the change of weighted
scalar curvature, then solve graph equations prescribing weighted mean
curvature to control this change. A localized version of the construction will give an
estimate for one sweep through the coordinates of a root probability
@@ -611,10 +609,12 @@ is Perelman's \(R^m\), with his \(f=-\phi\)
\cite[Section~1.3]{Perelman2002}. The estimates below concern its
behavior under particular graph deformations. For antecedents of
the graph and vertical Jacobi identities, see
-\cite[Proposition~4.1]{OpenAIScalarInessentiality2026}; we include the
+\cite{MaWangXieYuZhu2026}; one includes the
calculation with the conventions needed here.
-\subsection{The graph identity}
+
+\par\noindent{\footnotesize\ttfamily src-perelman2002.pdf}\par
+
\begin{lemma}[Weighted graph identity]
\label{lem:graph-identity}
@@ -808,9 +808,8 @@ floor, and localization introduces controlled cutoff errors. A sweep
will also have to account for normalization of the output map and the
interaction of the ordered graph metrics.
-\subsection{Prescribed graphs on complete leaves}
-We specify the uniformity required for the analytic construction.
+Specify the uniformity required for the analytic construction.
A manifold has \emph{bounded geometry} here if it has coordinate
charts of a common positive size whose smaller concentric domains
cover the manifold, with a common positive buffer to each chart
@@ -825,13 +824,13 @@ obtained by taking the supremum of the corresponding Hölder norms in
these charts, after a fixed shrinkage within the buffers.
The use of complete prescribed graphs has precedents in
-\cite[Theorem~5.2]{OpenAIScalarInessentiality2026}. The
+\cite{MaWangXieYuZhu2026}. The
following statement records the precise uniform and measurable
conclusions used here and in the hypersurface argument. Its positive
branch realizes the designed prescription with a positive floor and
will be localized for the sweep. Its tangent branch supplies the value
approximation and the intrinsic gradient bound used in
-Section~\ref{sec:degree}.
+the degree estimate.
\begin{lemma}[Prescribed graphs on a complete manifold]
\label{lem:graph-prescription}
@@ -921,7 +920,7 @@ any equivariant identifications of the complete problems.
\end{lemma}
\begin{proof}
-We give existence and the estimates together, since the estimates
+Give existence and the estimates together, since the estimates
also give uniqueness and measurable dependence.
\emph{Height barriers and comparison.}
@@ -980,8 +979,7 @@ For a solution, \eqref{eq:graph-translation} and the chain rule give
+p_z(1-v_z^2)+p_vT(v_z).
\label{eq:graph-angle-equation}
\end{equation}
-Here \(p_x\) differentiates with \(z,v\) fixed, and we used
-\(T(z)=1-v_z^2\). The ambient curvature and weight bounds imply
+Here \(p_x\) differentiates with \(z,v\) fixed, and use was made of \(T(z)=1-v_z^2\). The ambient curvature and weight bounds imply
\(P\ge-C\), since the remaining term \(|\II|^2\) is nonnegative.
At an infimum of \(v_z\), taken by the same recentering procedure,
\(\Delta_{\Sigma,\phi}v_z\ge0\) and \(T(v_z)=0\). Consequently, if
@@ -1096,12 +1094,15 @@ to a solution produces a solution of the identified problem, so
uniqueness proves equivariance.
\end{proof}
-\subsection{Localization of the positive prescription}
+
+\par\noindent{\footnotesize\url{https://doi.org/10.1007/978-3-642-61798-0}}\par
+\par\noindent{\footnotesize\url{https://doi.org/10.1007/978-1-4612-4190-4}}\par
+
The positive solution in Lemma~\ref{lem:graph-prescription} is
defined on the complete manifold and need not have compact support.
-For a probability coordinate we need a graph supported near that
-coordinate. We obtain one by forcing the complete solution close
+For a probability coordinate one needs a graph supported near that
+coordinate. Obtain one by forcing the complete solution close
to a small constant on an outer region, and cutting it off there.
The following lemma states the quantitative features of this
procedure that enter the sweep.
@@ -1196,7 +1197,7 @@ operation.
\end{lemma}
\begin{proof}
-We first construct the outer coefficient while keeping its
+First construct the outer coefficient while keeping its
transition width bounded. Starting at a fixed positive distance
from \(K\), a scalar profile \(b\) can increase from zero to
\(H_*\) at speed at most
@@ -1277,7 +1278,7 @@ end. This gives \(|d\chi|^2\le C\chi\) and the other asserted
cutoff properties. If some outer regions are empty, the
corresponding assertions are vacuous.
-We next prove that the complete solution can be cut on this
+Next prove that the complete solution can be cut on this
plateau. All constants in the rest of the proof may use the
fixed data and parameters, but not \(H_*\). Write
\[
@@ -1442,7 +1443,7 @@ Indeed the usual finite-difference interpolation of a
derivative norm gives an exponential factor times a power of
\(c_H^{-1}\), which is smaller than every power of \(H_*^{-1}\).
-We can now cut off. On the transition region,
+Now cut off. On the transition region,
\(\widetilde f\asymp c_H\), its positive-order derivatives are
smaller than every power of \(H_*^{-1}\), and the derivatives
of \(\chi\) are fixed. On \(\{\chi>0\}\),
@@ -1517,12 +1518,10 @@ properties for the complete solution. This proves the final
assertions.
\end{proof}
-\subsection{A sweep of the root probabilities}
-We now apply the localized graph to each coordinate of a root
+Now apply the localized graph to each coordinate of a root
probability map. The order of the operations matters: a graph
-enlarges the metric available to all subsequent coordinates. We
-use finitely many colors so that graphs in the same color have
+enlarges the metric available to all subsequent coordinates. Use finitely many colors so that graphs in the same color have
disjoint supports and can be performed simultaneously.
For a root probability map, its energy is
\[
@@ -1541,7 +1540,7 @@ where \(R_\Psi\) and the constants are numerical. Such a
function can be obtained as the square of a smooth
nondecreasing function that is zero up to \(8\), has bounded
derivative, and equals \(s-8\) from a fixed larger value on.
-We use either \(\psi=0\), or \(\psi=\kappa\Psi\) for a small
+Use either \(\psi=0\), or \(\psi=\kappa\Psi\) for a small
constant \(\kappa>0\). In the latter case,
\[
\psi\le C\kappa s^2,\qquad
@@ -1550,7 +1549,7 @@ constant \(\kappa>0\). In the latter case,
\]
\paragraph{Notation for an ordered sweep.}
-We introduce the pointwise quantities for any candidate sequence of
+Introduce the pointwise quantities for any candidate sequence of
graphs before asserting that the sequence can be constructed. The
pass parameters are
\begin{equation}
@@ -1657,7 +1656,7 @@ Finally, define two nonnegative quantities:
\end{align}
The quantity \(T_0\) records the gains from the metric comparisons
and from each graph's stretching. Its full coefficient will yield
-the sharp quadratic term \(m^2/n\) in Section~\ref{sec:entropy}.
+the sharp quadratic term \(m^2/n\) in the entropy iteration.
The quantity \(\mathcal X\) retains the derivative squares and the
applied coefficients. Together with the mean and variance squares
below, it controls graph curvature; those moment squares also control
@@ -1785,7 +1784,7 @@ is at most \(\eta_0/k\), the last conclusion of
Lemma~\ref{lem:graph-localization} gives
\eqref{eq:sweep-prescription-error}.
-We prove the curvature estimate in the units before
+Prove the curvature estimate in the units before
multiplication by \(r/\epsilon\). Write
\[
G=N^2\bigl[\wscal(h_+,\phi_+)-\wscal(h_0,\phi_0)
@@ -1794,7 +1793,7 @@ multiplication by \(r/\epsilon\). Write
\]
so \(Q=(r/\epsilon)G\). For each \(i\), the one-coordinate calculation
applies with \(h=h_{i-1}\), \(h'=h_i\), and
-\(h_{\mathrm{in}}=h_0\). We first account for the difference between
+\(h_{\mathrm{in}}=h_0\). First account for the difference between
the ideal target \(Z_i\) for \(ND_i\) and the realized prescription.
For the moment omit the floor in
@@ -1926,7 +1925,7 @@ Completing its square therefore gives the identity
Here
\(\tfrac14-\lambda^2=2\theta-4\theta^2\ge7\theta/4\).
This supplies a reserve for all three squares in the
-proposition. We finish by paying the terms in
+proposition. Finish by paying the terms in
\(\mathcal B_i\) from fixed fractions of this reserve and of
the nonnegative \(\beta_i\) terms.
@@ -2082,18 +2081,17 @@ are therefore precisely those asserted in
\eqref{eq:sweep-prescription-error} and
\eqref{eq:sweep-inequality}.
\end{proof}
-\section{Entropy metrics with controlled labels}\label{sec:entropy}
This section turns a root probability map for a metric of scalar curvature
\(-1\) into a map with a large energy coefficient in a weighted scalar
inequality. The accompanying volume term is the logarithm of the actual
volume expansion from a density no larger than the original volume density.
-We will also make the gradient absolutely small and control all label
+Also make the gradient absolutely small and control all label
overlaps by one graph. The last two properties allow the energy coefficient
to be chosen larger than the combinatorial quantities used later.
Write \(\exp^{[0]}(t)=t\) and
-\(\exp^{[j+1]}(t)=\exp(\exp^{[j]}(t))\). We call a function
+\(\exp^{[j+1]}(t)=\exp(\exp^{[j]}(t))\). Call a function
\(\mathcal B:[1,\infty)\to[0,\infty)\) \emph{elementary} if, for some fixed
\(H\in\N\cup\{0\}\) and \(A,B>0\),
\[
@@ -2162,7 +2160,7 @@ There is, in addition, a fixed \(\Gamma\)-invariant graph \(\mathcal G\)
on \(\mathcal L\), independent of the transverse parameter, of finite
maximum degree \(\Delta\). It contains an edge between every pair of
distinct labels that are simultaneously nonzero at any point of any leaf.
-We may take
+Take
\[
q\geq \max\{n+1,\#(\Gamma\backslash\mathcal L),\Delta+1\}
\]
@@ -2174,7 +2172,7 @@ The constant \(C\) is independent of the manifold, metric, input family,
all the uniform bounds just specified, and \(E_{\min}\).
\end{theorem}
-We prove the theorem using Proposition~\ref{prop:sweep}. The bootstrap
+Prove the theorem using Proposition~\ref{prop:sweep}. The bootstrap
will fix the density \(\rho\), which then remains fixed through all later
sweeps and conformal changes. For
\(W=\log(dV_h/\rho)\), the balance propagated after the bootstrap is
@@ -2182,19 +2180,18 @@ sweeps and conformal changes. For
\wscal(h,\phi)-\epsilon e_h(F)-2W+n\log\epsilon\geq-C+P,
\qquad W\geq0,\quad P\geq0.
\]
-We first raise an initially small energy coefficient to a dimensional
+First raise an initially small energy coefficient to a dimensional
positive value. Subsequent sweeps have a summable scalar loss and
retain the nonnegative surplus \(P\). At the end of a group of sweeps,
that surplus pays for a conformal change which makes the gradient
absolutely small. The small gradient then controls how fast the fixed
graph of label overlaps can grow in the next group of sweeps.
The inverse-metric telescope and the assembly of logarithmic shortening
-also occur in \cite[Lemma~6.1 and Theorem~6.3]{OpenAIScalarInessentiality2026};
+also occur in \cite{MaWangXieYuZhu2026};
the proof below establishes the forcing and entropy estimates needed here.
-\subsection{Colorings and a conformal bootstrap}
-We first supply the colors required for the sweeps. Let \(\mathcal H\)
+First supply the colors required for the sweeps. Let \(\mathcal H\)
be any fixed invariant graph of maximum degree
\(\Delta_{\mathcal H}<\infty\) on a countable label set \(\mathcal L\).
Adjoin to the transverse space independent uniform priorities in
@@ -2247,7 +2244,7 @@ Proposition~\ref{prop:sweep}. In particular, its input is
W_p=\sum_i\log L_i,\qquad m=\sum_i A_i^2.
\end{gathered}
\]
-The weights \(\nu_i=f_i^2/r\) sum to one. We recall the probability
+The weights \(\nu_i=f_i^2/r\) sum to one. Recall the probability
statistics and their centered combinations from
\eqref{eq:sweep-statistics}:
\[
@@ -2262,7 +2259,7 @@ statistics and their centered combinations from
\]
Indices of vanishing bumps are omitted. In particular
\(\E_\nu u_i=\bar u\) and \(\E_\nu\sigma_i=1\); the ratios \(\sigma_i\)
-need not be at most one. We write \(\E\) and \(\Var\) for
+need not be at most one. Write \(\E\) and \(\Var\) for
this finite weighted mean and variance in the pointwise calculations.
The graph operations give
\begin{equation}\label{eq:entropy-sweep-density}
@@ -2311,7 +2308,7 @@ Theorem~\ref{thm:entropy}. The number of sweeps may depend on the input.
\end{lemma}
\begin{proof}
-We use sweeps with \(\phi_0=0\), no boost, and \(l=\theta\), where \(l>0\)
+Use sweeps with \(\phi_0=0\), no boost, and \(l=\theta\), where \(l>0\)
will be a sufficiently small fixed dimensional number. Then
\(\phi_+=-W_p\). After a sweep set
\[
@@ -2327,7 +2324,7 @@ The same reset contracts the volume density:
dV_k=e^{-nW_p/(n-1)}dV_{h_+}
=e^{-W_p/(n-1)}dV_{h_0}\leq dV_{h_0}.
\end{equation}
-We claim that there are dimensional \(C_b\) and \(\epsilon_b>0\) such
+Claim that there are dimensional \(C_b\) and \(\epsilon_b>0\) such
that, whenever \(2\leq C_0\leq4\), \(0<\epsilon\leq\epsilon_b\), and
\(\Scal_{h_0}\geq-C_0+\epsilon e_{h_0}(F)\), one can choose the errors
in the sweep so that
@@ -2335,7 +2332,7 @@ in the sweep so that
\Scal_k\geq-(C_0+C_b l\epsilon)+(1+l)\epsilon e_k(Y).
\end{equation}
-We first derive the budget needed for this step. The reset gives
+First derive the budget needed for this step. The reset gives
\(e_k(Y)=e^{2W_p/(n-1)}e_{h_+}(Y)\). Substituting the definition of
\(Q\) and the input scalar inequality into
\eqref{eq:entropy-bootstrap-reset}, and dropping its nonnegative
@@ -2354,7 +2351,7 @@ for \eqref{eq:entropy-bootstrap-step} is therefore
-C_b l^2\epsilon^2r e^{-2W_p/(n-1)}.
\end{equation}
-To pay this budget when \(\epsilon\) is small, we need control of both
+To pay this budget when \(\epsilon\) is small, one needs control of both
\(m\) and the deviation of \(r\) from one. The sweep inequality implies
\begin{equation}\label{eq:entropy-bootstrap-coercivity}
\epsilon Q\geq
@@ -2424,7 +2421,7 @@ small errors in the prescription contribute \(r\delta'_0\).
The term \(-4\theta^2m\) in \eqref{eq:entropy-sweep-form} is
\(-4l^2m\). This proves \eqref{eq:entropy-bootstrap-coercivity}.
-We now compare the right side of
+Now compare the right side of
\eqref{eq:entropy-bootstrap-coercivity} with the required budget.
When \(m\geq m_b\), the first two terms on the right of
\eqref{eq:entropy-bootstrap-budget} are at most
@@ -2449,12 +2446,12 @@ right of \eqref{eq:entropy-bootstrap-budget}, together with the term
C_nl\epsilon m\leq \frac{c_n}{2}m^2+C'_n l^2\epsilon^2.
\]
On this region \(r e^{-2W_p/(n-1)}\) has a dimensional positive lower
-bound. We may therefore choose \(C_b\) dimensionally to cover the
+bound. Therefore choose \(C_b\) dimensionally to cover the
last term. These comparisons can be made strict by increasing \(C_b\)
and decreasing \(l,\epsilon_b\). They leave a positive multiple of
\(l^2\epsilon^2r\) in the middle region, and positive square margins
outside it; dividing the latter margins by \(r\) gives a positive
-lower bound for fixed \(l,\epsilon\). We can consequently choose
+lower bound for fixed \(l,\epsilon\). Consequently choose
\(\delta'_0\) small enough for the sweep. This proves
\eqref{eq:entropy-bootstrap-step}.
@@ -2482,9 +2479,8 @@ finitely many smooth conformal changes preserve all the stated
input-uniform properties.
\end{proof}
-\subsection{The entropy increment and its surplus}
-Fix \(\epsilon_*\) from Lemma~\ref{lem:entropy-bootstrap}. We now use
+Fix \(\epsilon_*\) from Lemma~\ref{lem:entropy-bootstrap}. Now use
the boost in Proposition~\ref{prop:sweep}, choosing its constant
\(\kappa>0\) sufficiently small depending on this fixed threshold.
All constants arising from the boost are consequently dimensional.
@@ -2525,11 +2521,11 @@ in the units of \(Q\), it suffices to prove
+\frac{rN^2}{\epsilon}P_p .
\end{equation}
The term \(lm\) pays for the additional \(l\epsilon e_{h_+}(Y)\).
-We choose \(\delta_0\leq\epsilon l^3\) in
+Choose \(\delta_0\leq\epsilon l^3\) in
\eqref{eq:entropy-sweep-form}. It remains to compare the other terms
there with the budget in \eqref{eq:entropy-budget}.
-We use a trace identity for the amount of the new graph energy that
+Use a trace identity for the amount of the new graph energy that
remains in the final metric. Set
\[
m_f:=\sum_i A_i^2(1-C_i)=N^2\sum_i|df_i|_{h_+}^2.
@@ -2542,13 +2538,13 @@ As matrices,
\end{equation}
The inequality is arithmetic--geometric mean applied to the positive
eigenvalues of \(h_+^{-1}h_0\), whose product is \(e^{-2W_p}\).
-We also have
+There also holds
\begin{equation}\label{eq:entropy-volume-angle}
L_i=(1-A_i^2)^{-1/2}.
\end{equation}
First choose a small dimensional \(m_0>0\), in particular
-\(m_0\ll\epsilon_*\) and \(m_0<1/4\). We claim that
+\(m_0\ll\epsilon_*\) and \(m_0<1/4\). Claim that
\begin{equation}\label{eq:entropy-coarse-coercivity}
T_0+\epsilon M_0^2\geq c_n(m+rW_p)
\quad\text{if }m\geq m_0
@@ -2588,7 +2584,7 @@ already covers this. If \(r>2\), then
\(|M_0|\geq c(r+\bar u)\), and
\(\epsilon M_0^2\geq c\epsilon r\bar u=c\,rm\).
This proves \eqref{eq:entropy-coarse-coercivity} when \(m\geq m_0\).
-When \(m<m_0\), we have \(\bar u\leq m_0/\epsilon_*\) small and
+When \(m<m_0\), one has \(\bar u\leq m_0/\epsilon_*\) small and
\[
W_p=\frac m2+O\left(\sum_iA_i^4\right)\leq C m.
\]
@@ -2630,7 +2626,7 @@ In making this conclusion, first fix \(K\); after \(D_*\) is chosen
below, take \(l\) small enough that \(4D_*^2l^2m\) is also a small
fraction of \(m^2\).
-We are left with the critical regime \(m\leq Kl\), still under
+What remains is the critical regime \(m\leq Kl\), still under
\eqref{eq:entropy-small-region}. Here
\(W_p=m/2+O(m^2)\) and \(3/4<r<5/4\), so the budget expands as
\[
@@ -2642,7 +2638,7 @@ The target quadratic term is covered exactly by
\frac{m^2}{n}-2lm+nl^2
=\left(\frac m{\sqrt n}-\sqrt n\,l\right)^2\geq0.
\]
-We must therefore recover \(m^2/n\) from \(T_0\), while paying the
+It remains to recover \(m^2/n\) from \(T_0\), while paying the
displayed \(r-1\) error and the errors in that recovery. This is where
the coefficient one on \(T_0\) in \(\Pi_p\) is needed.
@@ -2857,19 +2853,16 @@ for \(\sigma_i-1\). Since \(r\leq1/4\),
\leq C_nP_p/\theta.
\]
The relation \(ND_i=ND_i^0-\beta_i f_i+\mathrm{err}_i\) finishes the
-estimate: \(\sum_i\beta_i^2f_i^2\leq\mathcal X\), and we choose
-\(\sum_i|\mathrm{err}_i|^2\leq1\). Its contribution \(1/N^2\) is
+estimate: \(\sum_i\beta_i^2f_i^2\leq\mathcal X\), and one chooses \(\sum_i|\mathrm{err}_i|^2\leq1\). Its contribution \(1/N^2\) is
bounded by \(C_nP_p\) using the first inequality of
\eqref{eq:entropy-last-surplus}, \(\epsilon\geq1\), and \(r\leq1/4\).
\end{proof}
-\subsection{Accumulation and an absolute gradient bound}
The step size has two jobs: the scalar losses \(O_n(l^2)\) must be
summable, and the number of support enlargements before reaching \(E\)
must grow at most cubically in \(\log E\). Since one sweep changes
-\(\log(1+\epsilon)\) by a quantity comparable to \(l\), we take
-\begin{equation}\label{eq:entropy-step-schedule}
+\(\log(1+\epsilon)\) by a quantity comparable to \(l\), one takes \begin{equation}\label{eq:entropy-step-schedule}
l(\epsilon)=\frac{l_*}{(1+\log(1+\epsilon))^2},
\end{equation}
where the dimensional \(l_*>0\) is chosen below \(l_0\) from
@@ -2973,9 +2966,9 @@ In particular, the starting metric is an admissible reference metric
for every sweep in the phase, and
\(e_{h_+}(Y^0)\leq e_{h^0}(Y^0)\) for its starting map \(Y^0\).
-We now finish a phase with an absolute gradient bound. Normalizing
+Now finish a phase with an absolute gradient bound. Normalizing
the last bumps alone divides by \(\sqrt r\), which can make the
-gradient large where \(r\) is small. We first keep a small copy of
+gradient large where \(r\) is small. First keep a small copy of
the starting map in the normalization; its gradient is controlled by
the preceding metric comparison. A conformal enlargement confined
to the small-\(r\) region then reduces the remaining gradient there,
@@ -3024,7 +3017,7 @@ Write \(r=\sum_i f_i^2\) for the last sweep, set \(s=\sqrt r\), and put
The two vectors in the numerator use disjoint copies of the labels.
The map is a smooth root probability map because its denominator is
at least \(c_E>0\). Writing the copies as \((i,0)\) and \((i,1)\),
-we call \(i\) their common parent label at the start of the phase;
+one calls \(i\) their common parent label at the start of the phase;
the sweeps within the phase do not change the label set. The first
copy is supported on the last bumps and the second on the starting
map. This parent assignment will be used to compare support overlaps
@@ -3107,7 +3100,7 @@ The present exit instead satisfies
\widehat\phi+\widehat W=\phi_++W+u.
\]
Thus after an exit \(W\) need not equal \(-\phi\), and its volume
-bound supplies no bound for the gradient of the weight. We need a
+bound supplies no bound for the gradient of the weight. Need a
bound on \(b(u)\) which uses no bound on the size of \(b\).
It is enough to estimate this on the support of \(u\), which lies in
\(\{r\leq1/4\}\) for large \(E\). At such a point view all the graphs
@@ -3182,18 +3175,18 @@ entropy term. From \(r\leq E^{-0.32}\) and
P\geq P_p\geq c_n E^{1.32}.
\]
Using \eqref{eq:entropy-before-exit} and \(e^{2u}\leq C E^{0.04}\),
-we obtain there
+one obtains there
\begin{equation}\label{eq:entropy-exit-volume-cost}
\frac{e^{2u}W}{P}\leq
C_nE^{0.04+0.26}P^{-0.25}\leq C_nE^{-0.03},
\qquad
\frac{e^{2u}nu}{P}\leq C_nE^{-1.28}\log E.
\end{equation}
-In the first bound we used \(E+P\leq C_nP\), valid for large \(E\)
+In the first bound use was made of \(E+P\leq C_nP\), valid for large \(E\)
on this region. Both quantities tend to zero with dimensional
constants.
-We now assemble these estimates. After replacing \(Y\) by \(Z\), write
+Now assemble these estimates. After replacing \(Y\) by \(Z\), write
the retained surplus as \(\alpha_n P\), where \(\alpha_n>0\) is fixed
dimensionally. Put \(B_u=2|b(u)|+(n-1)|du|^2\). Substituting in
\eqref{eq:entropy-conformal-exit} and moving the new entropy term
@@ -3206,11 +3199,11 @@ to the left gives
-2(e^{2u}-1)W-2nu e^{2u}\bigr]
+(1-e^{-2u})n\log E .
\end{align*}
-We used \(e^{-2u}(C+C_n)\leq C+C_n\). The last term is nonnegative
+Use was made of \(e^{-2u}(C+C_n)\leq C+C_n\). The last term is nonnegative
for \(E\geq1\). Equations \eqref{eq:entropy-exit-derivatives} and
\eqref{eq:entropy-exit-volume-cost} leave at least a fixed fraction
of \(P\) in the bracket on the support of \(u\); off that support the
-volume and derivative costs vanish. We may thus set
+volume and derivative costs vanish. Thus set
\[
P'=\alpha'_n e^{-2u}P
\]
@@ -3239,7 +3232,7 @@ The choice \eqref{eq:entropy-u-choice} implies
\leq C\left(\frac{E^{-0.02}}{c_E}+E^{0.16}\right).
\]
Using \(\sqrt E/N\leq C_n(1+\log E)\), \(c_EK_0\leq E^{-1/2}\),
-and \(N\leq\sqrt E\), we conclude that
+and \(N\leq\sqrt E\), one concludes that
\begin{equation}\label{eq:entropy-exit-gradient}
e_{\widehat h}(Z)^{1/2}
\leq C_n\left(
@@ -3254,7 +3247,6 @@ The supports of \(Z\) are those of the last bumps and of \(Y^0\).
Thus the asserted uniformity and compact support properties persist.
\end{proof}
-\subsection{A fixed graph and the final parameter choice}
\begin{proof}[Proof of Theorem~\ref{thm:entropy}]
Fix parameters \(H,A,B\) for the prescribed elementary bound and set
@@ -3272,7 +3264,7 @@ constant once, \eqref{eq:entropy-bootstrap-output} has the form
+\epsilon_0 e_{h_*}(F_*),
\]
because \(\epsilon_0\) ranges in a fixed dimensional interval.
-We use the schedule \eqref{eq:entropy-step-schedule} from now on.
+Use the schedule \eqref{eq:entropy-step-schedule} from now on.
The bound \eqref{eq:entropy-summability} applies to the entire
increasing sequence of energy coefficients, even when it is
partitioned into phases.
@@ -3285,7 +3277,7 @@ per sweep, including its separation buffers, has a dimensional radius
in this reference metric by Proposition~\ref{prop:sweep}. The total
radius is therefore at most \(C_n(1+\log^3(1+E_1))\).
-We record a bound for a single fixed graph sufficient for every
+Record a bound for a single fixed graph sufficient for every
coloring in this phase and for all cooccurrences at its exit. All
data before the choice of the first target, including \(h_*\) and its
uniform comparison with \(g\), have already been fixed. The lifted
@@ -3314,12 +3306,12 @@ Take \(E_1\) sufficiently large for
Lemma~\ref{lem:entropy-accumulation} and
Lemma~\ref{lem:entropy-phase-exit}; in the latter the initial \(K_0\)
is the finite input-dependent gradient bound of \(F_*\).
-We now have the scalar inequality, the volume bound with exponent
+Now there holds the scalar inequality, the volume bound with exponent
\(0.32\), and the absolute gradient bound at \(E_1\).
Denote the exit metric, map, and fixed cooccurrence graph by
\(h_1,F_1,\mathcal G_1\).
-We next isolate how a small gradient controls the graph in a later
+Next isolate how a small gradient controls the graph in a later
phase. Suppose a phase starts with metric \(h_j\), root map \(F_j\)
with \(e_{h_j}(F_j)^{1/2}\leq K\), and a fixed graph \(\mathcal G_j\)
containing all its possible cooccurrences. Let \(R\) bound the
@@ -3345,7 +3337,7 @@ argument applies separately at each transverse parameter; because
\(\mathcal G_j\) already contains all possible cooccurrences, the
conclusion holds in that one fixed graph.
-After an exit at \(E_j\), we choose a target that both meets this
+After an exit at \(E_j\), one chooses a target that both meets this
overlap condition and grows by one exponential level:
\begin{equation}\label{eq:entropy-phase-targets}
T_{j+1}=\exp(E_j^{1/1000}),
@@ -3449,13 +3441,13 @@ two sides are now the same, and
C'_{\mathrm{inst},J,n,A,B}(1+\log E_1)^3
\]
for all sufficiently large \(E_1\). Thus \(E_J>\mathcal B(q)\).
-We may choose the first target still larger to have
+Choose the first target still larger to have
\(E_J>E_{\min}\), to satisfy the initial gradient threshold involving
\(K_0\), and to meet all dimensional thresholds used above.
These choices affect only the size of the final parameter and the
finite uniform bounds of the output objects.
-Finally we account for the scalar constant. The bootstrap charges
+Finally one accounts for the scalar constant. The bootstrap charges
at most \(2C_b\epsilon_*\), independently of its input-dependent
number of sweeps. The initial logarithmic normalization has a
dimensional cost because \(\epsilon_0\in[\epsilon_*,2\epsilon_*]\).
@@ -3479,19 +3471,17 @@ enlarged only by the probability products used for the colorings,
and the sweep constructions are Borel on these spaces by
Proposition~\ref{prop:sweep}. This proves all the conclusions.
\end{proof}
-\section{A measured degree estimate}\label{sec:degree}
This section bounds the degree of a map from a measured family of
Riemannian manifolds to a cube. Its integrand combines the weighted scalar
quantity with a separate allowance for each coordinate gradient. The conclusion involves
ordinary Riemannian volume, integrated also in the transverse direction.
The family need not consist of complete manifolds, and its degree need not
-be an integer. We first specify the control at the sides of the cube that
+be an integer. First specify the control at the sides of the cube that
makes this statement, and the integrations in its proof, well defined.
-\subsection{Measured bands and degree}
-We use the following model of a measured space of leaf manifolds. The
+Use the following model of a measured space of leaf manifolds. The
underlying space $X$ is standard Borel and has a countable collection of
flow boxes
\[
@@ -3519,11 +3509,11 @@ the equality on overlaps follows from invariance of the transverse
measures. From a plaque, all plaques on its leaf have a countable Borel
enumeration, obtained by following finite strings of overlap maps. In
particular the saturation of a transverse null set is still null in every
-transversal. We shall discard such saturations when an assertion holds on
+transversal. Discard such saturations when an assertion holds on
almost all plaques.
For $t>0$ write $Q_t^d=(-t,t)^d$; in dimension zero this denotes one point.
-We write $|G|_\infty=\max_i|G_i|$, with the empty maximum interpreted as
+Write $|G|_\infty=\max_i|G_i|$, with the empty maximum interpreted as
zero.
\begin{definition}[Controlled measured band]\label{def:degree-band}
@@ -3536,7 +3526,7 @@ cover with the following properties.
\item The boxes have one common positive size and fixed positive buffers:
their smaller concentric boxes cover the indicated part. They can be
chosen inside $\{|G|_\infty<t'\}$ for some $t<t'<T$. After rescaling
-coordinates we may, and do, call them unit boxes with buffers.
+coordinates one may, and does, call them unit boxes with buffers.
\item In these boxes the coefficients of $h$ and $h^{-1}$ and all their
leafwise coordinate derivatives are bounded by constants common to the
cover. The same holds for all derivatives of $G$ and for all
@@ -3570,7 +3560,7 @@ partition cancel because its sum is one. The same argument gives measured
Stokes for leafwise currents with uniform local mass and boundary-mass
bounds in those boxes: apply the current identity in each plaque to the
cutoff multiple of a test form and then sum. This includes test forms
-whose pullbacks have noncompact support in a leaf. We will always verify
+whose pullbacks have noncompact support in a leaf. Always verify
the indicated local bounds before using current Stokes.
If $\omega$ is a smooth, compactly supported $d$-form on $Q_T^d$ with
@@ -3585,7 +3575,7 @@ zero has a compactly supported primitive. Applying \eqref{eq:degree-stokes} to
pullback of this primitive shows that \eqref{eq:degree-definition} is
independent of $\omega$. This is the \emph{measured degree}, with the
usual signed orientation convention. For $d=0$ it means the measured
-signed count of the oriented points. We abbreviate it to $\mathcal D$
+signed count of the oriented points. Abbreviate it to $\mathcal D$
when the band and map are understood.
\begin{theorem}[Measured degree estimate]\label{thm:degree}
@@ -3626,7 +3616,7 @@ integration core, while the permitted scaled gradient of each retained
coordinate increases by $1/q$. A family of boundaries
ordered by a real parameter supplies the factor $2\tau$ on the left.
Its normal speed supplies the coarea factor on the right. In dimensions
-where the boundaries can be singular we will construct a regular band
+where the boundaries can be singular one will construct a regular band
of the same degree; a scalar gain will pay exactly for the volume of the
graphs used to construct it. The following formulation separates that
construction from the final induction.
@@ -3673,9 +3663,8 @@ The child in the second assertion may be chosen separately for each
typical $s$; no measurability of those choices in $s$ is asserted or needed.
\end{lemma}
-\subsection{An ordered family of boundaries}
-We prove Lemma~\ref{lem:degree-descent} in this and the next two
+Prove Lemma~\ref{lem:degree-descent} in this and the next two
subsections. Fix intermediate widths
\begin{equation}\label{eq:degree-intermediate-widths}
T_p<t_1<t_2<t_3<t_4<T_d.
@@ -3707,7 +3696,7 @@ $|y|^{3/2}$ bound on the tail. Choose the absolute $C_0$ in
\end{equation}
\paragraph{The local slice calculation.}
-We first record the analytic purpose of the pressure law. Let $\Sigma$
+First record the analytic purpose of the pressure law. Let $\Sigma$
be a smooth cooriented local hypersurface, with unit normal $\nu$, and set
\[
\begin{gathered}
@@ -3754,7 +3743,7 @@ Set $t=L_1|dG_1|_h\le B$ and $x=v/L_1$. Since $\gamma(D)\le D^2$,
\ge-2t\left(1+\frac{\gamma(D)}{8(1+B)}\right),
\qquad \frac{t\gamma(D)}{4(1+B)}\le\frac{D^2}{4}.
\]
-We may discard the remaining nonnegative part of $D^2$, and use
+Discard the remaining nonnegative part of $D^2$, and use
$2\mathfrak a/v\ge2/x$. Finally,
$t\le1+(t-1)_+$ and $x^{-1}+\ln x\ge1$ for $x>0$. Thus
\eqref{eq:degree-slice-identity} implies
@@ -3781,13 +3770,13 @@ $\widehat P$ and set
\[
\mathfrak h_s(x)=\widehat P(s-G_1(x)),\qquad -\tau<s<\tau.
\]
-We will choose $\eta$ sufficiently small for the given band and then keep
+Choose $\eta$ sufficiently small for the given band and then keep
it fixed. On every boundary portion used below, $\widehat P$ will agree
with $P_0$ and hence obey \eqref{eq:degree-pressure}.
\paragraph{Least minimizers, including on noncompact leaves.}
Write $P_\phi(E;U)=\int_{\partial^*E\cap U}e^\phi\,dA_h$ for the weighted
-perimeter of a set of locally finite perimeter. We use local differences
+perimeter of a set of locally finite perimeter. Use local differences
of the functional
\begin{equation}\label{eq:degree-functional}
P_\phi(E)-\int_E\mathfrak h_s e^\phi\,dV_h.
@@ -3799,12 +3788,16 @@ functional and its local first- and second-variation formulas, see also
Neither term is required to be finite on the whole leaf. In particular,
this construction uses no global weighted-volume bound.
+
+\par\noindent{\footnotesize\url{https://doi.org/10.1007/978-1-4612-4098-3_1}}\par
+\par\noindent{\footnotesize\ttfamily src-chodoshli2024.pdf}\par
+
The output will be a canonical nested family $E_s$ on each open leaf,
independent of the exhaustion and jointly Borel in the plaque data and
$s$. Each $E_s$ will minimize these local differences for compact
variations. The local comparison argument also supplies the perimeter
compactness needed later to turn convergence of sets into graphical
-convergence at regular limit points. We establish these properties in
+convergence at regular limit points. Establish these properties in
turn, before confining the boundaries to the raw-pressure region.
\emph{Compact domains and leastness.}
@@ -3826,8 +3819,11 @@ minimizer. Minimum volume forces the selected minimizer to be contained,
up to null sets, in every other one. It is consequently unique as an
$L^1$ set; call it the least minimizer in $K$.
+
+\par\noindent{\footnotesize\url{https://doi.org/10.1017/CBO9781139108133}}\par
+
\emph{Order and exhaustion.}
-The same inequality proves both monotonicities we need. If $K\subset K'$
+The same inequality proves both monotonicities one needs. If $K\subset K'$
and two forcings satisfy $\mathfrak h\le\mathfrak h'$, a minimizer for
$(K,\mathfrak h)$ and one for $(K',\mathfrak h')$ obey
\[
@@ -3866,13 +3862,12 @@ limiting perimeter on their boundaries, since this excludes only a null
set of levels. Minimality before passage to the limit, convergence of
the bounded forcing integrals, and lower semicontinuity give the compact
comparison for $E_s$. The same argument applies under smooth local
-convergence of the metric, density, and forcing coefficients. Below we
-apply it with the unchanged limit as competitor to prove convergence of
+convergence of the metric, density, and forcing coefficients. Below, apply it with the unchanged limit as competitor to prove convergence of
the local perimeter measures. Multiplying a density by a positive
constant has no effect on any of these comparisons.
\emph{Borel dependence.}
-We explain why all these choices are Borel, including jointly in $s$.
+Explain why all these choices are Borel, including jointly in $s$.
For this coding, subdivide the original atlas into bounded coordinate
boxes with compact closure in the older boxes, and affinely normalize
each to one fixed cube $Q=(-1,1)^d$. Fix once a countable list
@@ -3923,7 +3918,7 @@ the selected minimizer as a Borel function. These are standard forms of
the Arsenin--Kunugui and Lusin--Souslin theorems; see
\cite[Theorem~18.18 and Corollary~15.2]{Kechris1995}. The exhaustion limits agree
for different starting plaques by exhaustion independence. They are therefore Borel
-on the measured space. We may use the density-one representative of
+on the measured space. Use the density-one representative of
$E_s$; the support of its boundary measure is also Borel, by testing its
mass on a countable basis of coordinate balls.
@@ -3979,7 +3974,7 @@ the unprolonged Equation~\eqref{eq:degree-pressure} holds on all the
boundary portions used in the proof.
\paragraph{The local regularity and compactness inputs.}
-We record precisely the local geometric measure theory used from now on.
+Record precisely the local geometric measure theory used from now on.
Set $\Sigma_s^{\mathrm{all}}=\operatorname{supp}\partial[E_s]$. For
$d\ge2$, its regular portion is a smooth multiplicity-one hypersurface.
The remaining set is closed in $\Sigma_s^{\mathrm{all}}$ and has Hausdorff
@@ -3988,7 +3983,7 @@ dimension at most
p-7.
\end{equation}
It is empty if this number is negative. These conclusions hold in each
-relatively compact chart. To place our functional under the standard
+relatively compact chart. To place the functional under the standard
codimension-one theorem, normalize $\phi$ by a constant value $\phi_0$
in the chart and use the smooth metric
\[
@@ -4011,6 +4006,9 @@ $[0,1]$. BV truncation does not increase boundary mass, and the volume
where the resulting set changes is at most the mass of the added current.
The set comparison therefore implies that formulation as well.
+
+\par\noindent{\footnotesize\ttfamily src-eichmair2009.pdf}\par
+
\emph{Local compactness claim.}
Suppose $d\ge2$ and $E_j$ minimize the absorbed-perimeter functionals
for all compact variations in a common buffered normalized outer chart.
@@ -4098,6 +4096,9 @@ at $z=0$. This gives precisely the local varifold convergence; the
uniform convergence of the metric integrands gives it also for the
original varying metrics.
+
+\par\noindent{\footnotesize\url{https://doi.org/10.1007/BF02196453}}\par
+
To prove the support and graph assertions, the first variation in
the metric $\widehat h$ is, by BV Gauss--Green, integration of the bounded
volume forcing against the reduced-boundary conormal. It therefore has
@@ -4119,6 +4120,9 @@ interior elliptic estimates. The cylinder here may depend on the fixed
regular point of the limit; uniform graph radii in later applications
will follow from their separate compactness contradictions.
+
+\par\noindent{\footnotesize\url{https://doi.org/10.2307/1970868}}\par
+
The varying-metric use of these local theorems reduces to one smooth
ambient metric as follows. Pass further in a smaller compact chart to a subsequence converging so
rapidly in $C^\infty$ that its metric is realized on slices $2^{-j}$ of
@@ -4134,6 +4138,9 @@ same local bounded-mean-curvature property. Applying this subsequence
argument to a sequence on which local convergence failed proves the
local support and graphical convergence asserted above.
+
+\par\noindent{\footnotesize\url{https://doi.org/10.2307/1969989}}\par
+
For $d=1$ the needed facts are elementary. A locally minimizing set has
locally finitely many jumps by the perimeter bound. Two consecutive
jumps cannot be arbitrarily close in a controlled chart, since one can
@@ -4147,9 +4154,8 @@ with BV convergence. The separation then gives the corresponding
convergence of these zero-dimensional regular boundaries. There are no
singular points in this case.
-\subsection{The lapse, coarea, and the descended degree}
-We now extract a smooth normal speed from the ordered family. Write
+Now extract a smooth normal speed from the ordered family. Write
$\Sigma_s$ for the regular part of $\Sigma_s^{\mathrm{all}}$ in
$\{|G|_\infty<t_3\}$. The boundary has its exterior coorientation and
the induced orientation. Its singular set has zero $p$-dimensional area,
@@ -4174,7 +4180,7 @@ their tangent planes agree at contact. The comparison of the graph mean
curvatures at that minimum of $r(t,\cdot)-r(s,\cdot)$ gives
$D_t\le D_s$, whereas their equations give
$D_t=\mathfrak h_t>\mathfrak h_s=D_s$ at the same point. This is
-impossible. We use the conventions for $D$ and $\mathrm{II}$ from the
+impossible. Use the conventions for $D$ and $\mathrm{II}$ from the
local slice calculation, with $\nu$ the exterior unit normal. The
equation $D=\mathfrak h_s$ is the first variation of
\eqref{eq:degree-functional} on a regular portion.
@@ -4273,7 +4279,7 @@ one-to-one image theorem. Transverse invariance on overlaps gives
for every nonnegative Borel $F$ supported in $\{|G|_\infty<t_3\}$.
Since $b_d<t_3$, this proves \eqref{eq:degree-coarea}.
-We will require one more consequence of \eqref{eq:degree-lapse}. On a
+Require one more consequence of \eqref{eq:degree-lapse}. On a
family of regular graph patches with common buffers and graph estimates,
all positive-order derivatives of $\ln v$ have uniform bounds on smaller
patches. Indeed the coefficients in \eqref{eq:degree-lapse} and the
@@ -4358,7 +4364,7 @@ since $|G_1|<\tau+b<T_p$. Near any other side some residual coordinate
has absolute value near $t_1>T_p$. Hence every inner residual subcube
has buffered graph charts in this restriction. Its degree equals
$\mathcal D_X(G)$ by \eqref{eq:degree-boundary-degree}, and its coordinate gradients only
-decrease on restriction to the tangent space. We take $J^+=J$ and use
+decrease on restriction to the tangent space. Take $J^+=J$ and use
\eqref{eq:degree-slice-bound}.
If $d=1$, so $p=0$, let $X_s^+$ be the confined regular points in
@@ -4383,23 +4389,21 @@ signed count is $\mathcal D_X(G)$ by
\eqref{eq:degree-boundary-degree}; its integral is the measured unsigned
count. At each point \eqref{eq:degree-raw-integrand} reads
$1\le(v/L_1)\Theta$, and confinement places every counted point in
-$|G_1|<b_d$. We return to the common integral estimate after treating
+$|G_1|<b_d$. Return to the common integral estimate after treating
the possible singularities when $p\ge7$.
-\subsection{A regular descendant in higher dimensions}
Assume now $p\ge7$, and fix one of the typical parameters $s$ above.
The singular set has zero $p$-dimensional area, so it does not change the
boundary-current integral. Deleting it can, however, leave ends of the
regular locus over interior residual values, where regular chart radii
-have no common lower bound. We must keep those ends away from every
+have no common lower bound. Those ends must be kept away from every
inner target subcube and preserve the degree computed by the closed
boundary current.
-We will construct a controlled child carried by an open regular subset
+Construct a controlled child carried by an open regular subset
of this slice, with the inherited orientation and transverse measure,
-and with degree $\mathcal D_X(G)$. For each residual row we require
-$|dJ_j^+|_{h^+}\le |dG_j|_h+\varepsilon_{\mathrm{grad}}$ and the
+and with degree $\mathcal D_X(G)$. For each residual row one requires $|dJ_j^+|_{h^+}\le |dG_j|_h+\varepsilon_{\mathrm{grad}}$ and the
one-sided value bound
$|J_j^+|\ge |G_j|-\varepsilon_{\mathrm{val}}$, with
\begin{equation}\label{eq:degree-tolerances}
@@ -4427,8 +4431,8 @@ Every constant and error chosen inside this subsection may depend on this
slice and on the band, but must be common across its retained plaques.
The choices proceed in three groups. First a small parent perturbation
creates a target gap around the points without uniform regular charts.
-Next we fix the kernel and graph constants using the controlled parent
-data and that gap. Finally we choose the marked-set cover small enough
+Next one fixes the kernel and graph constants using the controlled parent
+data and that gap. Finally one chooses the marked-set cover small enough
for all scalar errors. Its positive radius lower bound controls the
number and derivatives of the kernels for the graph solves; it does not
enter the scalar constants. The adjustable scalar error in
@@ -4443,7 +4447,7 @@ supports below, remain bounded. All operations in the parent made below
use this atlas.
\paragraph{Uniform regularity tests and avoidance of the remaining points.}
-We use finitely many graph tests at a time to obtain a regular region
+Use finitely many graph tests at a time to obtain a regular region
with common chart controls. The points not yet recognized will be
excluded from the child, and the target gap below will keep the later
homotopies on the regular locus away from them.
@@ -4481,7 +4485,7 @@ compactness fact used in both constructions that follow.
First make an arbitrarily small smooth perturbation $\bar J$ of $J$ in
the \emph{parent} band, small in value and first derivative, with all
-higher derivatives uniformly bounded. We claim it can be made so that,
+higher derivatives uniformly bounded. Claim it can be made so that,
for some $a_0>0$ and some finite $k$,
\begin{equation}\label{eq:degree-gap}
|\bar J|_\infty>4a_0
@@ -4618,7 +4622,7 @@ For $j=\alpha,\alpha+2,\alpha+4,\alpha+6$ put
\qquad
\mathcal L=|d\ln u_\alpha|^2+|d\ln u_{\alpha+4}|^2.
\end{equation}
-These quantities have separate roles. We will make $u_\alpha$ large on
+These quantities have separate roles. Make $u_\alpha$ large on
$D_{k'}$ so that its logarithm locates the region to be excluded.
The term $\Lambda^2$ will supply the positive scalar contribution that
pays for the later coordinate graphs, while $\mathcal L$ controls the
@@ -4699,12 +4703,12 @@ necessary, gives for the new weight
\ge\wscal(h,\phi)-H_1(G)-2\ln(v/L_1)
+c\Lambda^2+\frac{\mathcal L}{4}-\delta.
\end{equation}
-We have obtained a large scalar allowance near the marked set. The next
+A large scalar allowance has been obtained near the marked set. The next
construction uses it to move the residual map out of the target band
there, while paying for the graph volume created in doing so.
\paragraph{Two graph prescriptions and their costs.}
-We describe the graph calculations first, and then give a complete region
+Describe the graph calculations first, and then give a complete region
on which the prescriptions can be solved. The first prescription will
produce a function $f$ within two of a small multiple of $\ln u_\alpha$,
with $|df|\le1$ in its graph metric. The coordinate prescriptions then
@@ -4868,7 +4872,7 @@ $|dJ_j^+|_{h^+}\le|d\bar J_j|_h+e$. For the value inequality use
$1+A\zeta\ge1$ in the first line. The quantities
$\varepsilon_{\mathrm{grad}}$ and $\varepsilon_{\mathrm{val}}$ can be as
small as prescribed by the first-derivative and value perturbation
-budgets, respectively, and by $e,w$. We choose them to satisfy the
+budgets, respectively, and by $e,w$. Choose them to satisfy the
target budgets \eqref{eq:degree-tolerances} and also to be smaller than
all the fixed side margins used below.
@@ -5119,7 +5123,6 @@ the child and use its containment in the parent $b_d$ region. This is
\eqref{eq:degree-child-integral}, for example with $C_*=1$.
\end{proof}
-\subsection{Completion of the induction}
\begin{proof}[Proof of Theorem~\ref{thm:degree}]
The case $d=0$ is the signed-count inequality in the statement. Suppose
@@ -5152,12 +5155,11 @@ pressure profile, and each descent costs at most $e^{C_*/q}$ after the
arbitrarily small slice errors were chosen. All geometry-dependent
constants were used only to make those choices.
\end{proof}
-\section{From bounded cohomology to comparison degree}\label{sec:transfer}
The entropy construction supplies a root probability map \(F\) whose
energy \(e_h(F)\) has a large coefficient in the weighted scalar-curvature
inequality. Its coordinatewise square takes values in the probability
-simplex. We now turn its pairing with the bounded cocycle into comparison
+simplex. Now turn its pairing with the bounded cocycle into comparison
degrees to which Theorem~\ref{thm:degree} applies. A common integer
clears the rounded cubical coefficients, giving integer multiplicities;
the resulting measured degree may still be real. Its expectation will
@@ -5165,12 +5167,8 @@ retain the original cocycle pairing exactly. The different lengths
assigned to the tangent and normal directions of the realization are
what remove the auxiliary dimension from the final estimate.
-The probability-map and auxiliary-degree strategy is adapted from
-Section~3 of \cite{OpenAIScalarInessentiality2026}: Lemmas~3.2 and 3.4
-develop the probability map and dual chains, and Proposition~3.6 provides
-their pairing. Lemma~3.8 supplies a transverse parameter cycle with
-nonzero intersection count; the proof of Theorem~3.1 constructs the
-comparison map and identifies its local degree. We prove the
+The probability-map and auxiliary-degree strategy has antecedents in
+\cite{MaWangXieYuZhu2026}. Prove the
quantitative measured transfer and its estimates here.
\begin{theorem}\label{thm:scalar-volume}
@@ -5182,11 +5180,11 @@ smooth Riemannian metric \(g\) with \(\Scal_g\equiv-1\) satisfy
\]
\end{theorem}
-We prepare the proof for a manifold with \(\|M\|>0\). Choose the
+Prepare the proof for a manifold with \(\|M\|>0\). Choose the
homogeneous alternating bounded cocycle \(c\), the initial probability
map, and the number \(I\geq\|M\|/2\) provided by
Proposition~\ref{prop:detector}; in particular \(\|c\|_\infty\leq1\).
-We first consider conditional data having the properties asserted by
+First consider conditional data having the properties asserted by
Theorem~\ref{thm:entropy}, without yet invoking their existence.
Denote the scalar constant in this contract by \(C_{\rm ent}\).
All functions called elementary in this section have a fixed height of
@@ -5223,7 +5221,7 @@ specified, including its constant \(C_{\rm ent}\), satisfies
I\leq K_n e^{C_{\rm ent}/2}\int_{\mathcal M}\rho.
\end{equation}
Thus \(K_n\) and \(\Psi_n\) are fixed before the manifold, cocycle,
-instance geometry, and \(C_{\rm ent}\) are chosen. We carry out the
+instance geometry, and \(C_{\rm ent}\) are chosen. Carry out the
estimates for each integer \(q\geq n+1\), uniformly over conditional
data satisfying the stated label and coloring bounds.
The grid, the common rounding, and an upper density bound for the
@@ -5233,7 +5231,6 @@ width and the required energy threshold. Only after the conditional
target is closed will Theorem~\ref{thm:entropy} supply an actual pair
\((q,E)\) and make \(C_{\rm ent}\) dimension-only.
-\subsection{A flat neighborhood and a cubical cocycle}
For a fixed \(\omega\), let \(K=K_\omega\) consist of all nonnegative
unit root vectors whose support is a clique in the label graph. Thus
@@ -5272,8 +5269,7 @@ bound in \(q\).
\begin{proof}
Take one copy of \(B(z_y,10\delta)\subset\mathbb R^q\) for each witness
\(y\in K\). At the same point \(z\) in two such balls, identify the two
-copies if their witnesses share a label in their nonzero supports. We
-check that these identifications give ball charts, rather than a branched
+copies if their witnesses share a label in their nonzero supports. Check that these identifications give ball charts, rather than a branched
space. For every point under consideration, some coordinate of \(z\)
is at least \(q^{-1/2}-10\delta\). Every witness for that point has a
nonzero label at this color. If two witnesses share any label, their
@@ -5406,10 +5402,9 @@ cocycle equation just described is satisfied exactly by the \(u'_C\).
\end{lemma}
\begin{proof}
-The closure equations couple faces whose anchors can differ. We encode
-all local coefficients in one ordered finite vector and round within the
+The closure equations couple faces whose anchors can differ. Encode all local coefficients in one ordered finite vector and round within the
joint kernel of its incidence equations, so every cell reads from the
-same rounded vector. We first establish that the vector has elementary
+same rounded vector. First establish that the vector has elementary
size despite the infinitely many lifted cells.
All projected centers lie
@@ -5515,7 +5510,7 @@ The rounded coefficients define the original oriented dual current
Z_{\rm rd}:=\sum_C D_0u'_C[C^\#].
\end{equation}
Here \([C^\#]\) is integration over the oriented cube, and the
-coefficients are integers. We verify the local finiteness implicit
+coefficients are integers. Verify the local finiteness implicit
in this notation. The form bound and rounding give
\(|u_C|\leq C_nr^n\) and an elementary bound for each \(|D_0u'_C|\).
Let \(B\) be a small flat ball in one sheet chart and \(Q\) one full
@@ -5570,7 +5565,6 @@ elementary subdivision and incidence bounds. The original current
\(\mathcal T_\delta\); the doubling supplies closed facet pairings
for the resolution.
-\subsection{A controlled smooth measured resolution}
The diagram need not be a manifold at faces of codimension greater
than one. The following construction replaces it by a measured family
@@ -5581,11 +5575,14 @@ will be controlled by its coordinate volume.
The reflected gluing below uses Gaifullin's explicit permutohedral
resolution, including the face-preserving sign-changing pairings,
their conjugation under reflections, and the auxiliary bit toggles
-\cite[pp.~562--563]{Gaifullin2008}. We replace his rank-indexed bits
+\cite[pp.~562--563]{Gaifullin2008}. Replace his rank-indexed bits
by independent subset-indexed bits, put invariant probability laws on
the pairing data, and supply smooth measured charts and quantitative
metric bounds.
+
+\par\noindent{\footnotesize\ttfamily src-gaifullin2008.pdf}\par
+
\begin{lemma}[Measured resolution]\label{lem:transfer-resolution}
For the diagram \(\mathcal P\) there is a standard Borel measured
family \(\mathcal A\) of oriented smooth \(m\)-manifolds and a smooth
@@ -5781,7 +5778,7 @@ widths, and \(X\) is not affine.}
\end{minipage}
\par\medskip
-We verify that these are ordinary smooth charts. On an overlap of two permutations,
+Verify that these are ordinary smooth charts. On an overlap of two permutations,
a cut \(S_j\) which is not a cut of the other permutation has
\(u_j\geq4^{-2m}\). To see this, choose a pair of indices on opposite
sides of that cut whose order is inverted in the other permutation.
@@ -5805,7 +5802,7 @@ across every corner. A point incident to \(j\) facets has exactly the
finite star of \(2^j\) states under these reflections, and hence has
one neighborhood of this form rather than several branches.
-The signed charts give smooth leaf neighborhoods. We next check the
+The signed charts give smooth leaf neighborhoods. Next check the
topological and transverse properties needed for measured integration.
The leaves are Hausdorff. Distinct
points with distinct resolved coordinates can be separated in the
@@ -5919,7 +5916,7 @@ charts and the same on both sides of every facet. Set
k=X^*k_{\rm flat}+j^2 k_0,
\qquad \text{seam}=\{\Theta\leq2\gamma_0\}.
\end{equation}
-We call the complement of this closed seam the straight part.
+Call the complement of this closed seam the straight part.
The metric is positive definite near the tile boundary because \(j=1\) there. Where
\(j<1\), one has \(\Theta>\gamma_0\), hence \(x_i\geq\gamma_0\) for
every \(i\), and \(X\) is an immersion. The formula proves
@@ -5987,14 +5984,13 @@ give the derivative bounds. The choices can be made on the indexed
chamber data, so they are Borel and equivariant as asserted.
\end{proof}
-For our use, choose \(O_\sigma\) to take \(z\)-increments to rotated
+For the present purpose, choose \(O_\sigma\) to take \(z\)-increments to rotated
grid increments, putting the \(m\) axes tangent to the dual cube first
and the \(n\) complementary axes last. Use a signed permutation if
needed to have determinant one. Thus \(O_\sigma=Q_\sigma V\) for a
signed permutation \(Q_\sigma\in SO(q)\) with this order. Signs will
not affect any row norm or any cubical test.
-\subsection{The comparison map and its degree}
The parameter selection below fixes \(\lambda(q)\) and \(\gamma_0(q)\)
deterministically. For those choices the bounds in
@@ -6053,7 +6049,7 @@ The local pushforward \eqref{eq:transfer-local-push} will express the
degree using a smoothed form of \(Z_{\rm rd}\). Its primal-cell periods
are the integer coefficients \(D_0u'_C\). Their expectations match the periods of
\(D_0\eta\); a compatible piecewise cubical primitive and mollification
-then give the same measured pairing with \(F\). We first verify the
+then give the same measured pairing with \(F\). First verify the
band control that makes these integrations valid.
\emph{The controlled comparison band.}
@@ -6149,7 +6145,7 @@ multiplicity one. Our fiber integration convention is
\int_{C\times C^\#}\mathsf d^*\beta\) with the base-first product
orientation. The differential of \(\mathsf d(z,w)=z-w\) sends this
ordered frame to the primal basis followed by the negative dual basis,
-so its orientation is positive by our choice of the dual orientation.
+so its orientation is positive by the choice of the dual orientation.
Its integral is therefore
one, proving \eqref{eq:transfer-convolution-cells}. The small support
and the chart buffers ensure that only the indicated lifts contribute.
@@ -6159,7 +6155,7 @@ Set \(\alpha=\E_{\rm rd}\alpha_{\rm rd}-D_0\eta\). It is a smooth
closed form on that tube, and \eqref{eq:transfer-rounding},
\eqref{eq:transfer-cell-value}, and
\eqref{eq:transfer-convolution-cells} give \(\int_C\alpha=0\) on all
-the primal \(n\)-cells there. We explain why its measured pairing
+the primal \(n\)-cells there. Explain why its measured pairing
with \(F\) vanishes, including when \(F\) meets grid faces.
Use all lifted full \(q\)-cells needed to cover the tube of radius
@@ -6207,15 +6203,14 @@ Using \eqref{eq:transfer-form} in
\eqref{eq:transfer-degree}.
\end{proof}
-\subsection{The two comparison lengths and the final average}
-We finish by estimating these degrees. We will see separately how
+Finish by estimating these degrees. It will be seen separately how
the straight part uses the dimension-only Hilbert norm of \(\eta\)
and how the small coordinate volume of the seam pays for its rougher
bound.
\begin{proof}[Proof of Theorem~\ref{thm:scalar-volume}]
-The case \(\|M\|=0\) is immediate. Otherwise we first prove the
+The case \(\|M\|=0\) is immediate. Otherwise, first prove the
conditional target for the data above, choosing \(\Psi_n\) from the
requirements on \(E\) derived below, and then apply it to the entropy
output. Choose a positive constant \(b_*\), depending only on \(n\),
@@ -6271,7 +6266,7 @@ and the dimensional choice of \(b_*\), and
\(13/75=1/2-(2/3)(1/2-1/100)>0\). Thus the displayed strict horizon
inequality follows if
\(E\geq(20c_{{\rm hor},n}q^{4/3}/\tau)^{75/13}\).
-We may therefore apply Theorem~\ref{thm:degree}
+Therefore apply Theorem~\ref{thm:degree}
with \(d=q\).
Write \(H_i=2(L_i|dG_i|-1)_+\), as in that theorem. At a straight
@@ -6413,8 +6408,7 @@ cells.
The form bound holds in every rotated frame. Using Cauchy--Schwarz
jointly in \(V\) and the \(\binom qn\) axis sets, and then
-\eqref{eq:transfer-sphere}, we obtain
-\begin{equation}\label{eq:transfer-form-average}
+\eqref{eq:transfer-sphere}, one obtains \begin{equation}\label{eq:transfer-form-average}
\begin{split}
\E_V\sum_{|\mathcal I|=n}
(|\eta_{\mathcal I}(y)|+2q^{-n})e^{c_nU_{\mathcal I}^2}
@@ -6456,7 +6450,7 @@ projected grid index the uniqueness proved before
lift. Apply the rough copy bound
\(D_0|u'_C|\leq C_{\eta,n}D_0r^n\) from that equation, where
\(C_{\eta,n}\) depends only on the form norm, and use the uniform
-density bound. Only after these bounds may we drop the restrictions
+density bound. Only after these bounds may the restrictions be dropped
on available sheets, roundings, and chambers and sum over all projected
centers and chamber shapes.
@@ -6516,7 +6510,7 @@ so far, this gives
\int_{\mathcal M}\rho.
\end{equation}
-We now fix these bounds before invoking the entropy theorem. For each
+Now fix these bounds before invoking the entropy theorem. For each
integer \(q\geq n+1\), the form derivative bound first gives a
deterministic \(r(q)\) with elementary reciprocal. The size bound for
the universal type list, \(D_0\), incidence counts, and the bounds for
@@ -6640,12 +6634,6 @@ Russian Mathematical Surveys \textbf{63} (2008), no. 3, 562--564.
{doi:10.1070/RM2008v063n03ABEH004540};
\href{https://arxiv.org/abs/0806.3580}{arXiv:0806.3580}.
-\bibitem{Gaifullin2013}
-A. Gaifullin, \emph{Universal realisators for homology classes},
-Geometry \& Topology \textbf{17} (2013), no. 3, 1745--1772.
-\href{https://doi.org/10.2140/gt.2013.17.1745}
-{doi:10.2140/gt.2013.17.1745}.
-
\bibitem{GilbargTrudinger}
D. Gilbarg and N. S. Trudinger,
\emph{Elliptic Partial Differential Equations of Second Order},
@@ -6720,11 +6708,6 @@ J. Nash, \emph{The imbedding problem for Riemannian manifolds},
Annals of Mathematics (2) \textbf{63} (1956), no. 1, 20--63.
\href{https://doi.org/10.2307/1969989}{doi:10.2307/1969989}.
-\bibitem{OpenAIScalarInessentiality2026}
-OpenAI, \emph{Positive scalar curvature forces rational inessentiality},
-OpenAI Math Release preprint
-\href{https://github.com/openai/math/blob/main/preprints/Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026/paper.pdf}{OAI:Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026},
-2026.
\bibitem{Perelman2002}
G. Perelman, \emph{The entropy formula for the Ricci flow and its