==== TABLE OF CONTENTS ====
The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law//figures/comparison-path.tex
The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law//figures/fixed-marginal.tex
The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law//main.tex
The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law//sections/01-introduction.tex
The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law//sections/02-foundations.tex
The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law//sections/03-paths.tex
The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law//sections/04-reduction.tex
The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law//sections/05-inverse.tex
The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law//sections/06-global.tex
The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law//sections/07-budgets.tex
The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law//sections/08-statistics.tex
The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law//sections/09-expansion.tex
The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law//sections/10-parameters.tex
The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law//sections/11-companion-interface.tex
The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law//sections/12-notation.tex
The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law//sections/consequences.tex
The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law//references.bib
===========================
==== BEGIN The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law//figures/comparison-path.tex ====
\begin{figure}[htbp]
\centering
\begin{tikzpicture}[>=Latex,node distance=14mm,
box/.style={draw=black!60,rounded corners=2pt,align=center,inner sep=7pt,text width=37mm,font=\small},
lab/.style={font=\footnotesize,align=center,fill=white,inner sep=3pt}]
\node[box] (a) at (0,3.5) {Completed SK system at $N$\\$Y_N$};
\node[box] (b) at (8,3.5) {Completed SK system at $N'$\\$Y_{N'}$};
\node[box] (c) at (0,0) {Smoothed auxiliary\\system at $N$};
\node[box] (d) at (8,0) {Smoothed auxiliary\\system at $N'$};
\draw[->,thick] (a)--(c) node[midway,right,lab,text width=32mm] {Vary strength\\and fields;\\smooth the path};
\draw[->,thick] (c.south)--(0,-.9)--(8,-.9)--(d.south);
\node[lab,text width=105mm] at (4,-1.4) {Vary the clocks and compare neighboring sizes\\$\ell=e^{\theta/6}$};
\draw[->,thick] (d)--(b) node[midway,left,lab,text width=32mm] {Undo the smoothing,\\strength and field changes};
\draw[->,dashed] (a.north)--(0,4.5)--(8,4.5)--(b.north);
\node[lab,text width=105mm] at (4,4.9) {Resulting comparison of centered log Laplace transforms};
\end{tikzpicture}
\caption{The route from $N$ to an arbitrary $N'\in[N,2N]$. The displayed endpoints are the centered, rescaled bare systems. The proof bounds the changes along the route and the errors from the small regularizations at its ends. One parameter vector is selected for each finite route and each Laplace test.}
\label{fig:comparison-path}
\end{figure}
==== END The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law//figures/comparison-path.tex ====
==== BEGIN The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law//figures/fixed-marginal.tex ====
\begin{figure}[htbp]
\centering
\begin{tikzpicture}[>=Latex,x=1cm,y=1cm,every node/.style={font=\small},dot/.style={circle,fill,inner sep=1.6pt}]
\node[dot,label=above:$u$] (u) at (0,2.6) {};
\node[dot,label=left:$v$] (v) at (-.9,1.3) {};
\coordinate (a) at (-1.7,0);
\coordinate (b) at (1.3,0);
\coordinate (q) at (-.1,0);
\draw[thick] (u)--(v)--(a) node[below] {$a$};
\draw[thick] (u)--(b) node[below] {$b$};
\draw[dashed] (v)--(q) node[below,align=center] {extra\\branch};
\node at (0,-1.15) {Before marginalization};
\draw[->] (2.1,1.4)--(3.7,1.4) node[midway,above,align=center,font=\footnotesize] {replace aliases;\\prune};
\node[dot,label=above:$u$] (u2) at (5.5,2.6) {};
\coordinate (a2) at (3.8,0);
\coordinate (b2) at (6.8,0);
\draw[thick] (u2)--(a2) node[below] {$a$};
\draw[thick] (u2)--(b2) node[below] {$b$};
\node[draw,circle,inner sep=2pt,label=left:$v$] at ($(u2)!.5!(a2)$) {};
\node at (5.5,-1.15) {The retained law has the fixed split $u$};
\end{tikzpicture}
\caption{The earlier-pair configuration in the centered integration argument. The pair $(a,b)$ splits at the fixed mass $u1$, we prove that the zero-field Gaussian Ising Sherrington--Kirkpatrick free energy, centered by its expectation and divided by its standard deviation, converges in distribution to a nondegenerate law as the system size tends to infinity through all integers. We also prove that its variance divided by $n^{1/3}$ converges to a finite positive constant.
\end{abstract}
\setcounter{tocdepth}{1}
\tableofcontents
\clearpage
\input{sections/01-introduction}
\input{sections/02-foundations}
\input{sections/03-paths}
\input{sections/04-reduction}
\input{sections/05-inverse}
\input{sections/06-global}
\input{sections/07-budgets}
\input{sections/08-statistics}
\input{sections/09-expansion}
\input{sections/10-parameters}
\input{sections/consequences}
\appendix
\input{sections/11-companion-interface}
\input{sections/12-notation}
\begingroup
\raggedright
\bibliographystyle{plain}
\bibliography{references}
\endgroup
\end{document}
==== END The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law//main.tex ====
==== BEGIN The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law//sections/01-introduction.tex ====
\section{Introduction}\label{sec:introduction}
At a fixed temperature below the transition, how large are the
sample-to-sample fluctuations of the Sherrington--Kirkpatrick free
energy, and do they approach a law after centering and rescaling?
We prove a limiting law for every fixed inverse temperature $\beta>1$
in the zero-field Gaussian Ising model. The result includes a positive
variance prefactor at scale $n^{1/3}$ and a characterization of the
law by finite-system moments.
Let $(g_{ij})_{1\le i1$, put
\begin{equation}\label{eq:model}
H_n(\sigma)=\frac1{\sqrt n}\sum_{i1$, Aronow and Lopatto prove
\[
c_\beta n^{4/15}\le \Var F_n(\beta)\le C_\beta n^{7/15}
\]
in \cite[Theorem~1.4]{AL2026}, and quantitative tail estimates in a
separate deviation window in \cite[Theorem~1.5]{AL2026}.
These bounds allow, but do not establish, variance of order $n^{1/3}$.
Recent critical-window limits~\cite[Theorem~2.1]{ChengLiuShaoXu2026CriticalWindow}
concern $\beta_n\to1$, while the all-temperature results of
Chen, Dey, and Panchenko require a nonzero field~\cite{ChenDeyPanchenko2017}.
Neither hypothesis covers the fixed zero-field model here.
Our companion \emph{The low-temperature Sherrington--Kirkpatrick
fluctuation scale}~\cite{PE}, abbreviated \emph{PE}, supplies both its
finite-tree comparison toolkit and the fixed-temperature exponent
\begin{equation}\label{eq:PE-exponent}
s_n(\beta)=n^{1/6+o(1)}.
\end{equation}
The exponent alone does not identify a limiting distribution or prove
convergence of the variance divided by $n^{1/3}$. The additional work
here is a quantitative comparison of different sizes, with a summable
error. Appendix~\ref{sec:companion-interface} lists the precise PE
inputs and their hypotheses.
\subsection{The limiting law}
\begin{theorem}[Limiting law]\label{thm:main}
For every fixed real $\beta>1$, there is a nondegenerate Borel probability measure $\nu_\beta$ such that
\[
X_n(\beta)\ \Longrightarrow\ \nu_\beta
\qquad\text{as }n\to\infty\text{ through all integers}.
\]
The measure $\nu_\beta$ has mean zero, variance one, and a finite exponential moment in a neighborhood of the origin. It is the unique probability measure with the moments specified in Proposition~\ref{def:law} below.
\end{theorem}
The centering and variance are the exact finite-size quantities. The
formula below specifies the moments of $\nu_\beta$; no identification with a named
distribution is made. The quantitative statement
below proves that the variance has a positive limiting prefactor and
provides the rate needed for the moment characterization.
For the quantitative result, we use a Gaussian completion convenient for interpolation. Let $\zeta_n$ be a centered Gaussian of variance $\beta^2/2$, independent of the couplings, and define
\begin{equation}\label{eq:completed-free-energy}
F_n^c=F_n(\beta)-n\log2+\zeta_n,
\qquad Y_n=n^{-1/6}(F_n^c-\mathbb E F_n^c).
\end{equation}
Different copies below include independent couplings and independent completion variables. The subtraction of $n\log2$ changes the counting spin prior into the uniform probability prior. The completion changes the covariance into the homogeneous quadratic-overlap covariance used later.
\begin{theorem}[Moment convergence at the fluctuation scale]\label{thm:moments}
For each fixed $\beta>1$, there are real numbers $a_k=a_k(\beta)$, $k\ge1$, with the following property. For every integer $k\ge1$, there are constants $c_k>0$ and $C_k<\infty$ such that
\begin{equation}\label{eq:moment-rate}
\big|\mathbb E Y_n^k-a_k\big|\le C_k n^{-c_k},\qquad n\ge2.
\end{equation}
There are also constants $c>0$ and $C<\infty$, depending on $\beta$, for which
\begin{equation}\label{eq:exponential-moments}
\sup_{n\ge2}\mathbb E e^{c|Y_n|}\le C.
\end{equation}
The limits satisfy $a_1=0$ and $00,
\]
and the centered completion vanishes in $L^2$ after multiplication by $n^{-1/6}$. The exact finite-size standardization is then justified by Slutsky's theorem. We give the complete deduction, including nondegeneracy, in Section~\ref{sec:reduction}.
For the physical total free energy $-\beta^{-1}F_n$, the variance
constant is $a_2/\beta^2$ and its standardized limit is the reflection
of $\nu_\beta$. For its density $-(\beta n)^{-1}F_n$, the variance is
asymptotic to $(a_2/\beta^2)n^{-5/3}$.
\subsection{A finite-system characterization of the law}
The power rate in Theorem~\ref{thm:moments} allows each limiting
moment to be recovered from a convergent series of finite-size averages.
The following random-size experiment expresses this characterization
using independent copies of the finite SK model.
\begin{proposition}[Moment specification]\label{def:law}
Fix $\beta>1$ and let $C=\sum_{l=2}^{\infty}l^{-2}$. For an integer $k\ge1$, draw an integer $J\ge2$ with
\[
\mathbb P(J=l)=\frac{l^{-2}}C.
\]
Conditional on $J$, draw $k+1$ independent copies $F_J^{c,0},\ldots,F_J^{c,k}$ of the completed free energy in \eqref{eq:completed-free-energy}. Set
\[
t=\log J,
\qquad P_k=\prod_{i=1}^k J^{-1/6}(F_J^{c,0}-F_J^{c,i}).
\]
Conditional averaging over the independent comparison copies gives
$\mathbb E[P_k\mid J=l]=\mathbb E Y_l^k$.
The constants in Theorem~\ref{thm:moments} satisfy
\begin{equation}\label{eq:law-moments}
a_k=C\sum_{h=0}^{\infty}
\mathbb E\!\left[P_k\left(\frac{t^h}{h!}
-\mathbf1_{\{h\ge1\}}\frac{t^{h-1}}{(h-1)!}\right)\right],
\end{equation}
where the second term is absent for $h=0$. Each expectation is taken before the series is summed. The series of these scalar expectations is absolutely convergent. We have $a_2>0$, and the law in Theorem~\ref{thm:main} is uniquely determined by
\[
\int z^k\,\nu_\beta(dz)=\frac{a_k}{a_2^{k/2}},\qquad k\ge1.
\]
\end{proposition}
Section~\ref{sec:reduction} proves the formula and its absolute convergence from the moment rate in Theorem~\ref{thm:moments}.
This characterization involves only a fixed prior on finite system sizes and finite Gaussian experiments. It does not require the choice of a subsequence or knowledge of an asymptotic variance. The order in \eqref{eq:law-moments} is essential: it is not permissible to interchange the unaveraged pointwise series with expectation. Section~\ref{sec:reduction} also realizes each $a_k$ as the expectation of a single integrable random variable obtained from an almost surely finite experiment.
\subsection{The size comparison and its main ingredients}
The new estimate compares centered log Laplace transforms at different
system sizes with a summable error. For a large dyadic size $N$, put
$x=N^{-1/6}$. The comparison gives, for every integer $N'\in[N,2N]$,
\[
\left|\log\mathbb E e^{q x^\varepsilon Y_{N'}}
-\log\mathbb E e^{q x^\varepsilon Y_N}\right|
\le C_q x^{\varepsilon+c'},
\]
where $c'>0$, $q>0$ is fixed, and $\varepsilon>0$ is sufficiently
small and fixed. The saving $c'$ is independent of sufficiently small
$\varepsilon$. Estimates for positive Laplace tests, together with
preliminary exponential-tail bounds, recover comparisons of each fixed
moment. Their errors can then be summed over dyadic intervals. This is the reduction
proved in Section~\ref{sec:reduction}.
To obtain the transform estimate, we interpolate through hierarchical
Gaussian systems. At each level, a normalized transition kernel
specifies the law of the next Gaussian increments; paths can share an
initial history and then branch independently. Overlaps are normalized
scalar products of the terminal spin configurations. The hierarchy is
indexed by an averaging parameter, called mass; its matrix and field
clocks record the cumulative variances along this index. A one-site
scalar model built from the Parisi minimizer supplies reference overlaps
along these clocks.
The scalar input has its own development. Auffinger and Chen proved
structural properties of Parisi measures and uniqueness of the Parisi minimizer
\cite{AuffingerChenProperties2015,AuffingerChenUniqueness2015};
Jagannath and Tobasco gave variational support conditions
\cite{JagannathTobasco2017}. Zhou established interval support near the
critical temperature~\cite{Zhou2025}. We use Lopatto's full-support
result for every fixed $\beta>1$~\cite[Theorem~1.1 and Lemma~2.2]{Lopatto2026},
in the form developed in PE. The positive initial density needed here
is proved in that companion from the scalar identities and full
support.
The direct antecedent for our pinned-field and optimized-path
comparisons is Aronow and Lopatto~\cite{AL2026}. Hierarchical positive
laws, interpolation, and cavity calculations also have an established
lineage through Ruelle's cascade construction~\cite{Ruelle1987} and
the cavity variational principle of Aizenman, Sims, and
Starr~\cite{AizenmanSimsStarr2003}. Here we work with finite normalized
transition kernels and prove the quantitative larger-tree and
higher-moment estimates required by the size comparison.
These comparisons use normalized probability laws. An extra future
branch can be integrated out because its conditional law has mass one;
conditioning on an observation from that branch can instead change the
law retained in the calculation. PE's appendix ``Companion interfaces''
proves the polynomial-observation bound and permits an auxiliary common
temperature to vary with size; we recall these arguments when verifying
their use here. The changed-constraint estimates require the additional
local arguments developed in this article.
Appendix~\ref{sec:companion-interface} lists the imported estimates
and the hypotheses retained in their use.
Along a hierarchy we retain the initial disorder as a random input
and average over the later Gaussian increments. Ordinary averaging of
the resulting continuation gives one pressure; exponential averaging
with a small positive parameter $s$ gives the other. Their difference,
multiplied by $s$ times the number of sites, is the centered log Laplace
transform. We vary the covariance clocks and take finite differences
in the number of spins. Proposition~\ref{prop:increment} bounds their
combined effect with a power saving beyond the per-site scale $x^5$.
With $s=q x^{1+\varepsilon}$ and $N=x^{-6}$, multiplication by $Ns$
converts this into the saving in the displayed transform estimate.
The route from $N$ to $N'$ first changes the covariance data at fixed
size, then varies the size, and finally reverses the initial changes.
Writing $n=N e^\theta$ and $\ell=e^{\theta/6}$ keeps the endpoint
normalization equal to $n^{-1/6}$. Figure~\ref{fig:comparison-path}
shows these three stages.
\input{figures/comparison-path}
The comparison rests on four estimates.
\begin{enumerate}
\item We solve the linear equations for covariances of overlap errors.
The inverse bound remains uniform when the hierarchy is refined or its
masses become small. Summation by parts cancels the small-mass
denominators. Contributions from branching at an already specified
vertex are retained in the bound.
\item We vary the matrix and field covariances to turn pressure
comparisons into bounds on mean squared overlap errors. The constrained
minimization requires estimates in both the mass coordinate and the
quantile's value. A derivative of the minimum controls every minimizer,
so the argument does not need to select a minimizer measurably.
\item We obtain higher moments for the actual comparison paths by
projecting spin observations onto their revealed Gaussian histories and
integrating squared martingale increments. Before integrating a centered
quantity over a moving branch point, we show that the paths retained in
the calculation have a law independent of that point.
\item We sum the signed pressure coefficients before taking absolute
values. Summing the scalar contributions above a fixed cut, combining
the quadratic correction, and rescaling size cancel the leading terms
of degrees two, three, and four. The remaining increments are summable.
\end{enumerate}
\subsection{Consequences}
Theorem~\ref{thm:main} and the variance asymptotic above instantiate
the ferromagnetic comparison of Dey and Kang
\cite[Proposition~1.5]{DeyKang2026}. The comparison does not establish
the zero-field law, but it gives the following consequence once that
law and its positive variance prefactor are available.
\begin{corollary}[SK--Curie--Weiss fluctuations below unit coupling]
\label{cor:skcw}
Fix $\beta>1$ and $0\le\gamma<1$. Using the same independent standard
Gaussian couplings as in \eqref{eq:model}, define the exponent and log
partition function
\[
\begin{gathered}
\mathcal H_n^{\mathrm{CW}}(\sigma;\beta,\gamma)
=
\frac{\beta}{\sqrt n}\sum_{1\le i0$ is the constant $a_2$ above at the same $\beta$.
The centering and standard deviation are the perturbed model's own
exact finite-size quantities.
\end{corollary}
The comparison controls the added free energy in $L^2$ uniformly in
$n$. Since the unperturbed standard deviation grows like $n^{1/6}$,
this error disappears after either model's exact standardization.
The full argument is in Section~\ref{sec:consequences}.
This consequence is confined to the displayed zero-field ordered
Gaussian model at fixed $\beta>1$ and $0\le\gamma<1$; it does not cover
$\gamma=1$ or a coupling varying with $n$.
For the original zero-field model, the variance asymptotic also gives
the following consequence of Chatterjee's general Gaussian and
replacement inequalities.
\begin{corollary}[Upper disorder-chaos scales in zero-field SK]
\label{cor:disorder-chaos}
Fix $\beta>1$ and $n\ge2$. Put $E_n=\{(i,j):1\le i0$, set $g^t=e^{-t}g+\sqrt{1-e^{-2t}}\,g'$. For an integer
$0\le k\le M$, let $A$ be independent of $g,g'$ and uniform over the
$k$-element subsets of $E_n$. Set $g^A_{ij}=g'_{ij}$ on $A$ and
$g^A_{ij}=g_{ij}$ otherwise. Then
\begin{align}
\mathbb E\langle R^2\rangle_{g,g^t}
&\le \frac1n+\frac{2\Var F_n(\beta)}
{\beta^2n(1-e^{-t})},\label{eq:chaos-ou}\\
\mathbb E\langle R^2\rangle_{g,g^A}
&\le \frac1n+\frac{2(M+1)\Var F_n(\beta)}
{\beta^2n(k+1)}+C_\beta n^{-1/2},\label{eq:chaos-replacement}
\end{align}
for a finite constant $C_\beta$ depending only on $\beta$. As $n\to\infty$, for
$t_n\ge0$ and integers $0\le k_n\le M$, these bounds imply
\[
\begin{aligned}
t_n n^{2/3}\to\infty
&\quad\Longrightarrow\quad
\mathbb E\langle R^2\rangle_{g,g^{t_n}}\to0,\\
(k_n/M)n^{2/3}\to\infty
&\quad\Longrightarrow\quad
\mathbb E\langle R^2\rangle_{g,g^{A_n}}\to0,
\end{aligned}
\]
where $A_n$ has size $k_n$ as above.
\end{corollary}
Chatterjee's variance identity and replacement inequality bound the
mean squared overlap by the free-energy variance. Substitution of the
$n^{1/3}$ variance asymptotic gives the displayed sufficient scales;
Section~\ref{sec:consequences} checks both normalizations.
These are sufficient annealed perturbation scales, not a sharp
transition or a quenched statement. No overlap conclusion is asserted
for the SK--Curie--Weiss model in Corollary~\ref{cor:skcw}.
\subsection{Organization and order of limits}
Sections~\ref{sec:foundations}--\ref{sec:paths} develop the finite-tree
calculus and admissible comparison paths. The pair inverse in
Section~\ref{sec:inverse} first supplies the global trial accuracy of
Section~\ref{sec:global}; that accuracy and the local auxiliary comparisons
give the genuine-path budgets of Section~\ref{sec:budgets}. The analytic
inputs for the modified optimizers are proved within that section before
their use. Section~\ref{sec:statistics} derives genuine-path statistics
from the budgets and uses the pair inverse again in its high-mass
covariance equation. Scalar compression in Section~\ref{sec:expansion}
and the common parameter order in Section~\ref{sec:parameters} complete
Proposition~\ref{prop:increment}. The conditional deduction in
Section~\ref{sec:reduction} then gives full-sequence moment convergence,
positive limiting variance, moment determinacy, the finite-system law
specification, and exact finite-size standardization. Section~\ref{sec:consequences}
contains the proofs of the two corollaries stated above.
All conclusions are for a fixed $\beta>1$. Moment orders, expansion orders, and propagation depths are finite and fixed before $N\to\infty$. Their values may depend on the desired estimate. The exponent used for the small Laplace test is chosen only after the power saving in the increment estimate is established.
==== END The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law//sections/01-introduction.tex ====
==== BEGIN The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law//sections/02-foundations.tex ====
\section{Hierarchical Gaussian calculus}
\label{sec:foundations}
The size comparison uses two kinds of calculation: expectations under
hierarchical probability laws, and signed sums of such expectations produced
by covariance differentiation. We define these separately. The resulting
finite-order cavity and size expansions express pressure changes in terms
of overlap errors relative to a one-site scalar model.
\subsection{Normalization and positive tree laws}
We refer to \cite{PE} as PE; references explicitly marked PE concern that
paper. Fix \(T=\beta^2>1\). As in \eqref{eq:completed-free-energy}, we
add an independent spin-constant Gaussian of variance \(T/2\) and use the
uniform probability prior on spins. The resulting matrix covariance is
\(mTQ^2/2\), where \(Q\) is the overlap defined below. The added Gaussian
has bounded moments, and replacing the counting prior by the probability
prior shifts only the mean.
We work in dyads \(N\le m\lesssim 2N\) (with small enlargements), \(m\) the integer number of sites, and put \(x=N^{-1/6}\). All exponents below, including any order of a Taylor expansion, are fixed as \(N\to\infty\); constants can depend on these choices and on \(T\). We use ``with subpower loss'' or \(\lesssim_\circ\) to allow \(C_\epsilon x^{-\epsilon}\) for \emph{every} sufficiently small fixed \(\epsilon>0\). ``Power small'' means \(O(x^c)\) for some fixed \(c>0\); for assertions in every fixed \(L^p\) a power saving not indexed by \(p\) means the exponent is independent of the fixed finite moment order (constant and threshold need not be). Power slack choices in estimates with an indicated hierarchy of exponents are made before using subpower losses.
Estimates in this dyadic notation apply with \(N/2\le m\le3N\); changing comparison scales by bounded factors from this convention is immaterial. Several pieces have local variables (including error majorants and truncation orders) redefined in each calculation. In particular the path martingale \(X_v\) below is unrelated to the notation for the standardized free energy, and in Section~\ref{sec:budgets} the letter \(X\) without a path subscript is also used for a scale ratio.
Fix a terminal mass \(a\in[1,4]\). The coordinate \(v\in(0,a)\)
is the replica mass, and the deterministic nonnegative nondecreasing
functions \(b,h\) are the matrix and spin-field clocks. Jumps are allowed.
Their cumulative Gaussian covariance at mass \(v\) is
\[
m(b(v)Q^2/2+h(v)Q),\qquad
Q=m^{-1}\sum_{i\le m}\sigma_i\sigma'_i.
\]
For step clocks, construct independent Gaussian increments with these
covariance increases, including any initial increment at mass zero.
If \(H\) is their sum, the terminal continuation is
\(a^{-1}\log\int e^{aH}\), with the integral taken against the uniform
probability prior. Compute earlier continuations backwards: for an
increment at mass \(v>0\), set
\[
F_{\rm before}=v^{-1}\log\mathbf E\exp(vF_{\rm after}),
\]
where this expectation averages only that increment. At mass zero use
ordinary expectation. Divide the resulting deterministic value by \(m\)
and subtract \(a(b(a-)/4+h(a-)/2)\); this defines the self-corrected
pressure \(f_m(b,h)\). Its interval length \(a\) will be implicit.
The same continuations define a forward probability law. Reveal each
Gaussian increment with its original Gaussian density multiplied by
\(\exp(v(F_{\rm after}-F_{\rm before}))\), and then draw the terminal
spin from the Gibbs law of exponent \(a\). The backward recursion makes
each transition integrate to one. On a fixed finite rooted tree, branches
share the increments before their fork and use conditionally independent
forward transitions afterwards. We call this the \emph{positive tree law}
and write \(\nu[P]\) for its expectation of \(P\), including both fields
and terminal spins. The tree topology and clocks are fixed in this
expectation. Bounded monotone clocks are obtained by the fixed-site step
limits from PE stated below.
A specified observation may lie on either side of a jump or at an
intermediate clock value. At a simultaneous jump we use the joint filling
\[
(b_t,h_t)=(b_-,h_-)+t(\Delta b,\Delta h),\qquad 0\le t\le1,
\]
with mass held fixed, and preserve this filling in step approximations.
Endpoint tree laws and mass-integrated identities do not depend on the
order of revelation within a same-mass jump. The separate matrix and
field contributions to quadratic variation, and observations at
intermediate cuts, do use the prescribed filling.
For a single path let \(y=\sigma/\sqrt m\),
\[
\begin{gathered}
X_v=\mathbf E_v y,\qquad C_v=\mathbf E_v(y-X_v)(y-X_v)^{\mathsf T},\\
S_v=\mathbf E|X_v|^2,\qquad
B_v=\mathbf E\|\mathbf E_v yy^{\mathsf T}\|_{\rm HS}^2,\qquad D_v^2=B_v-S_v^2.
\end{gathered}
\]
Here \(\mathbf E_v\) is conditional expectation given the fields revealed
through the specified cut, and \(\mathbf E\) averages the full single-path
positive law. Thus \(X_v,C_v\) are functions of the random prefix, whereas
\(S_v,B_v,D_v\) are deterministic statistics of that law. Write \(H_v\)
for the Hessian of the continuation with respect to a source coupled to
\(y\) in the downstream Hamiltonian, with the revealed prefix held fixed. Put
\(L_v=y\cdot X_v-|X_v|^2/2\) and
\(W_v=(y-X_v)^{\mathsf T}H_v(y-X_v)\). These last two observables also
retain the terminal spin.
\subsection{Pinned moments and projection}
The next estimates distinguish a variation of the deterministic clocks
from movement along one fixed revealed path. In the pressure formula,
\(db(v),dh(v)\) are clock variations in an external parameter. In
the susceptibility inequality, \(dS_v\) and \(dh\) are instead
Stieltjes increments along the fixed path, including its prescribed
filling at a jump. The source Hessian \(H_v\) always keeps the
revealed prefix fixed.
\begin{lemma}[Pressure, source, and single-path estimates]
\label{found:estimates}
For the positive laws just defined, the following identities and
inequalities hold at every fixed size. The moment bounds in
\eqref{eq:A1}--\eqref{eq:A2} hold for every fixed real \(2\le p<\infty\),
with deterministic \(k\in L^2(0,a)\) and zero deterministic external
field.
\[
\begin{split}
df_m&=-\tfrac14\int B_v\,db(v)\,dv-\tfrac12\int S_v\,dh(v)\,dv,\\
vC_v\preceq H_v\preceq a C_v,\qquad
d S_v&\ge m\,\mathbf E\operatorname{tr}H_v^2\,dh,\qquad
\mathbf EW_v=\mathbf E\operatorname{tr}C_v H_v,\\
\left\|\int k_v(L_v-\mathbf EL_v)dv\right\|_p^2&\le c_p\left\|\int k_v^2 W_v\,dv\right\|_{p/2}.
\end{split} \tag{A1}\label{eq:A1}
\]
The last line is a moment estimate under the full single-path law, with
deterministic weights \(k_v\) and zero deterministic external field.
More generally, for
polynomials \(P_v\) of fixed degree with bounded deterministic coefficients,
\[
\left\|\int k_v(P_v(L_v)-\mathbf E P_v(L_v))dv\right\|_p^2
\le c_p\left\|\int k_v^2 P'_v(L_v)^2 W_v\,dv\right\|_{p/2}.
\tag{A2}\label{eq:A2}
\]
For the projection bound only, assume that the total clocks are at most
a fixed power of \(\log N\), and that the positive mass and field
length are bounded below by inverse powers of \(N\). If a branch
has traversed a pure or mixed clock interval with field length
\(\Delta h>0\) after separating from the data determining an anchor
vector \(V\), and its mass is at least \(v>0\) throughout that
interval, then for every fixed even \(p\ge2\),
\[
\|V\cdot(y-X_{\rm end})\|_p
\lesssim_p \frac{\log^2 N}{v\sqrt{m\Delta h}}\,\|V\|_p \tag{A3}\label{eq:A3}
\]
where the vector norm inside \(L^p\) is Euclidean. The branch must
retain its normalized future kernel conditional on the prefix and outside
data; in particular, the bound applies to conditionally independent
resamplings.
\end{lemma}
\begin{proof}
Equations \eqref{eq:A1} and \eqref{eq:A3} are the pressure, source,
pinned-moment, and projection estimates of \cite[Lemmas~2.1--2.3
and~3.5, equations~(1) and~(13)]{PE}. To match the terminal-mass-one
normalization there, put
\[
u=v/a,\qquad \widetilde b(u)=a^2b(au),\qquad
\widetilde h(u)=a^2h(au),\qquad \widetilde F=aF.
\]
The conditional spin means and covariances are unchanged, while
\(H_{au}=a\widetilde H_u\). The projection denominator is unchanged
as well, since \((v/a)\sqrt{m a^2\Delta h}=v\sqrt{m\Delta h}\).
Appendix~\ref{sec:companion-interface} records the pressure and moment
conversions. The fixed-size path-limit result is PE, Proposition~2.14.
For \eqref{eq:A2}, we include the polynomial argument of PE,
Appendix, paragraph ``Source and pressure calculus; the scope of pinned
moments.'' Pin the terminal spin and write the Gaussian increments in
terms of standard Gaussian drivers, as in the proof of PE, Lemma~2.3. Let \(\mathsf A_v\) be the Hessian of the continuation
in these drivers and \(\mathsf B_v\) its mixed Hessian with the
normalized spin source. After the transition densities telescope, the
negative logarithm of the pinned driver density is the Gaussian
quadratic potential minus \(aH_\sigma\), which is linear, plus
\(\int_0^a F_v\,dv\), where \(F_v\) is the continuation at the cut
\(v\). Consequently
\[
\begin{pmatrix}\mathsf A_v&\mathsf B_v\\
\mathsf B_v^{\mathsf T}&H_v\end{pmatrix}\succeq0,
\qquad
\mathsf Q=I+\int_0^a\mathsf A_v\,dv,
\]
where \(\mathsf Q\) is the Hessian of the negative logarithm of the
pinned driver density. The driver gradient of \(L_v\) is
\(g_v=\mathsf B_v(y-X_v)\). Thus, for every driver direction \(z\),
\( |z\cdot g_v|^2\le(z^{\mathsf T}\mathsf A_vz)W_v\).
For the polynomial observable set
\[
G_P=\int_0^a k_vP'_v(L_v)g_v\,dv,
\qquad R_P=\int_0^a k_v^2P'_v(L_v)^2W_v\,dv.
\]
Cauchy--Schwarz in \(v\) gives
\( |z\cdot G_P|^2\le(z^{\mathsf T}\mathsf Qz)R_P\), hence
\(G_P^{\mathsf T}\mathsf Q^{-1}G_P\le R_P\).
The inverse-Hessian Brascamp--Lieb inequality~\cite[Theorem~4.1]{BL1976}
therefore gives \eqref{eq:A2} for \(p=2\). Applying the same variance
inequality to the \(p/2\)-th absolute power of the centered integral,
as in PE, Lemma~2.3, gives the assertion for every fixed \(p>2\).
The zero-field spin gauge transformations act transitively on terminal
spins and preserve both \(L_v\) and \(W_v\). Because the coefficients of \(P_v\) and the weights \(k_v\)
are deterministic, they also preserve the polynomial integral and
\(R_P\). Their conditional laws are consequently independent of the
pinned spin, including the mean used for centering. Averaging over that
spin yields the full single-path estimate. Bounded-weight approximation
and then the fixed-\(m\) path limit give the stated range of clocks
and weights.
Clocks are bounded for each \(N\), possibly by a power of \(\log N\); \eqref{eq:A1} is clock independent. For \eqref{eq:A3}, we apply PE, equation~(13), only in this bounded or polylogarithmic clock regime; together with the polynomial mass and field-length bounds, this gives the polynomial coefficient-ratio bounds needed in its moment iteration, as detailed in Appendix~\ref{sec:companion-interface}. Our estimates can always be passed from affine finite-step interpolations, or from inserted fixed cuts, by the fixed-\(m\) limit assertion, keeping chosen revelation sides and the joint filling at observed intermediate jump cuts, and using the corresponding integrated inequalities. In particular no discretization \emph{rate} in \(m\) is used.
The pinned inequalities in this lemma concern the full single-path law. When a tree contains additional branches, we first marginalize the branches not used by the test. Random factors retained on other branches are subsequently handled by unconditional H\"older inequalities; they are not made into random weights in \eqref{eq:A2}.
\end{proof}
\subsection{Signed allocation of replica labels}
Covariance derivatives introduce additional replica labels. For a simple
example, start with one observed leaf above a cut \(v\), and divide its
edge at a level \(u\in(v,a)\). An extra label may coincide with that
leaf, with coefficient \(a\), or split from it along either edge
interval, with coefficient minus the interval length. On a constant
summand the total is \(a-(a-u)-(u-v)=v\). In a larger tree, attachment
to a fork also contributes an atom. We sum all these placements by the
signed rule of PE, Section~2,
Proposition~2.5, equation~(2), and Lemmas~2.6, 2.8, and~2.9. A new
label coinciding with an observed leaf has coefficient \(a\); a new
split on an occupied edge has coefficient \(-dv\); attachment at an
existing mass-\(v\) fork with \(d\) children has coefficient
\(-(d-1)v\). These coefficients multiply expectations under the
positive laws just defined. They are not probabilities.
The row-sum rule sums one free label whose summand is constant over all
its attachments in a block above a cut \(v\). The result is that constant
times \(v\), as in the example. A row whose summand depends
on its attachment cannot be collapsed this way. Empty levels can be
inserted continuously without changing the calculation. Different
infinitesimal fork variables specify different tree vertices even when
the covariance clocks have the same value at those vertices.
A sum \emph{per root mass}, denoted by \((\cdot)_\#\), begins without
specified old labels. Its first label is an ordinary observation with
weight 1, and all subsequent labels use the signed rule. This is the
limit of normalized summation as the incoming mass tends to zero. Sums
without \(\#\) extend whatever old labels are already present.
Products of symbolic slot polynomials concatenate all their labels;
a coefficient involving only some labels is evaluated on the restriction
of the final topology to those labels. The labeled product rules in PE,
Proposition~2.5 make this convention associative, commutative, and
independent of allocation order. In a per-root product we concatenate first and apply the root
normalization only once.
Constant diagonal covariance changes contribute only to the self
correction. Thus pressure insertions after that correction use ordered
off-diagonal pairs and a factor \(1/2\) per insertion. The first
insertion gives \eqref{eq:A1}; for subsequent insertions the constant
diagonal terms cancel by the row-sum rule in PE, equation~(2). The same
rule applies to any affine positive semidefinite interpolation between
finite feature models with constant diagonal and fixed \(a\), after
subtracting \(a/2\) times the terminal diagonal. For total pressures,
the inserted covariance change is the total covariance change, rather
than its per-site value. Deterministic additive terms common to the two
ensembles cancel when we take their pressure difference.
Enlarging a positive-law tree preserves its original marginals. For any
fixed number of inserted labels, the total variation of the signed
allocation coefficients is uniformly bounded. Taking absolute values
in placement order leaves finitely many integrals over new fork masses,
existing-fork atoms with bounded multiplicities times their masses,
and terminal coincidences. We may therefore reorder the exact signed
sum with its topology restrictions first, and then estimate with this
absolute allocation measure. This order of operations is essential
when a later argument uses cancellation.
\subsection{Scalar references and Laplace tests}
The signed calculus will compare the matrix model with independent
one-site scalar systems. For the reference model take one spin and only a deterministic spin clock
\(K(v)\). Its continuation can be parametrized by cumulative variance
\(t\) instead of mass. The mass already reached by time \(t\) is
\(\alpha(t)=|\{v\in(0,a):K(v)\le t\}|\). Write \(V(t,z)\) for
the scalar continuation at field \(z\), and \(Z_t\) for its field under
the forward positive law. They satisfy
\[
V_t=-\tfrac12(V_{zz}+\alpha V_z^2),\qquad dZ_t=\alpha V_z(t,Z_t)dt+dB_t ,
\qquad V_{\rm term}=a^{-1}\log\cosh(a z).
\]
Set \(\Gamma(t)=\mathbf E V_z(t,Z_t)^2\), with zero starting field
unless otherwise stated. A terminal mass-\(a\) extension in scalar time
changes earlier continuation values only by a spatial constant. In
particular it leaves their derivatives in \(z\) unchanged. On a fixed
tree let \(\tau_i\) be the scalar spin at leaf \(i\), and let
\(v_{ij}\) be the mass where distinct leaves \(i,j\) separate.
Their positive-law moment is
\(\nu_K[\tau_i\tau_j]=\Gamma(K(v_{ij}))\). Common covariance insertions and source derivatives of these moments again use the tree calculus. In particular any fixed spatial derivative of positive order of \(V\) is bounded uniformly (even with long but finite clocks). Below put
\[
q(v)=\Gamma(K(v)),\qquad h_*(v)=K(v)-b(v)q(v),\qquad d(v)=h(v)-h_*(v).
\]
The physical paths used below have \(b(v)=b(0+)\) and \(h(v)=0\)
for \(v0\) sufficiently small
also relative to \(1/2-s_*\), and \(p_*>0\) sufficiently small.
Later small exponents may be decreased subject to their stated order.
Logarithms and constants are absorbed with powers much smaller than
\(p_*\) times the available gaps between exponents of \(\lambda\).
\subsection{The fixed-temperature scalar reference}
For a one-site clock \(K\) of terminal mass 1, write
\(f_{\rm spin}(K)=f_1(0,K)\) for its self-corrected pressure. Let
\(z:(0,1)\to[0,1]\) be the deterministic nondecreasing quantile
minimizing the fixed-temperature scalar objective
\[
f_{\rm spin}(Tz)+\frac T4\int_0^1z(w)^2\,dw.
\]
The corresponding variance-time quantile is \(Tz(w)\). Its distribution
function is
\(A_0(\xi)=|\{w\in(0,1):Tz(w)\le\xi\}|\), and \(\Gamma_0\)
is the scalar response of this clock as defined in
Section~\ref{sec:foundations}. These are the fixed base objects;
the reference clocks below will perturb them.
\begin{lemma}[Regular scalar profile and a strict top plateau]
\label{paths:base-profile}
At fixed \(T>1\), the scalar minimizing distribution has support \([0,\xi_*]\), satisfies \(\Gamma_0(\xi)=\xi/T\) there, and has a positive initial density. Its active-side distribution function has bounded one-sided derivatives of every fixed order. Moreover \(p_{\rm top}=A_0(\xi_*-)\) belongs to \((0,1)\).
\end{lemma}
\begin{proof}
We use PE, Lemma~5.8, whose full-support input is \cite[Theorem~1.1 and Lemma~2.2]{Lopatto2026}. Its measure has support \([0,\xi_*]\), \(\xi_*>0\); on this support \(\Gamma_0(\xi)=\xi/T\). Its hypotheses are precisely fixed \(T>1\), zero field, and the Ising
two-spin scalar objective above, with terminal mass 1. The lemma gives \(A_0(\xi)\sim \kappa_T\xi/T\) at 0, \(\kappa_T>0\), and continuity of the quantile on the open mass interval. The measure therefore has support down to 0 with no interior gap or atom at 0.
At the minimizing \(z\), vary \(z\) to \(z+tP_0(z)\) with smooth \(P_0\) supported in \((0,1)\) (both signs admissible for small \(t\)). By scalar \eqref{eq:A1} the derivative of the objective is \(T\int (z-\Gamma_0(Tz))P_0(z)/2=0\), giving self-consistency by continuity throughout the support. On \([0,\xi_*]\) viewed from the active side, \(A_0\) is smooth with bounded one-sided derivatives of all orders, \(A'_0\ge c>0\) near zero. Indeed \(\Gamma_0''=0\) in the interior gives
\(A_0=\mathbf E (V_{zzz})^2/(2\mathbf E(V_{zz})^3)\) almost everywhere by PE, Section~5, Lemma~5.3 (in variance time).
The numerator and denominator are continuous moment functions. The
positive-Hessian argument of the same lemma, restricted to a bounded
field interval of uniformly positive probability, bounds the denominator
away from zero throughout the compact active interval. Their ratio therefore
gives a continuous representative of \(A_0\), extending Lipschitzly
to both endpoints. Differentiation with the scalar PDE and It\^o
generator improves this representative one derivative at a time;
the positive denominator allows the iteration to every fixed order.
The positive initial slope is the initial-density conclusion of PE.
In fact \(p_{\rm top}=A_0(\xi_*-)\) belongs to \((0,1)\). Here is a useful argument. Extend by a mass-one annealed clock, and let \(\rho\) denote the law of \(Z_{\xi_*}\). The finite measure \(w(dz)=\rho(dz)/\cosh z\) is even and log-concave.
To see the sign of the potential, take a step approximation ending at
\(\xi_*\) and telescope its transition weights. After division by
\(\cosh z\), the joint density of the Gaussian increments is their
Gaussian density times the exponential of negative convex continuation
potentials. Each coefficient is a nonnegative mass increment; the
unused terminal mass similarly multiplies \(-\log\cosh z\).
Marginalizing the increments at fixed terminal field preserves
log-concavity~\cite{Prekopa1973}. Fixed-clock approximation, as in PE,
Proposition~2.14, passes it to \(w\). The limiting field law has a
density because its diffusion has bounded drift over positive time. At subsequent time \(\xi_*+d\), the derivative \(\mathbf E\chi^2\), \(\chi=V_{zz}\), is
\[
e^{-d/2}\int\operatorname{sech}^3 z\,(w*G_d)(dz)
\le e^{-d/2}\Gamma'_0(\xi_*),
\]
with \(G_d\) the centered Gaussian kernel. The inequality follows by even unimodality (translate overlaps of upper level intervals). At \(d=0\) the two sides agree, so the inequality makes the right
second derivative strictly negative. The left second derivative is
zero by \(\Gamma_0(\xi)=\xi/T\). The spatial moments in
\(\Gamma''=\mathbf E(V_{zzz}^2-2A_0\chi^3)\) agree across the
endpoint; only the mass changes. Consequently
\[
\Gamma_0''(\xi_*+)-\Gamma_0''(\xi_*-)
=-2(1-p_{\rm top})\mathbf E\chi^3<0,
\]
which gives \(p_{\rm top}<1\). Its positivity follows from the
positive initial density and monotonicity of \(A_0\). Thus the base
quantile is constant on a nonempty terminal mass interval.
\end{proof}
\subsection{Creating a positive scalar gap}
The minimizing scalar reference is marginal: \(\Gamma'_0=1/T\)
on its active support, so in these normalized coordinates the choice
\(b=T\) leaves no margin for \(-q\,db\). We perturb its density
near zero to create a gap of order \(\lambda t^2\). The matrix
construction will add a second margin of order \(\lambda U^2\).
Choose \(z_0>0\) small fixed with \(A'_0\ge c>0\) up to \(2z_0\), and let \(P(\xi)\) be smooth nonnegative, bounded by \(\xi\), equal to \(\xi\) on \([0,z_0/2]\) and zero after \(z_0\). Adding \(\lambda P\) gives the distribution
\(A_\lambda=A_0+\lambda P\), which is admissible for sufficiently
small \(\lambda\). Its direct physical clock is
\(K(v)=t(v)=\ell^{-2}A_\lambda^{-1}(v/\ell)\); it is constant
on the top plateau \(v/\ell\in[p_{\rm top},1]\).
For the smoothed clock, fix the final strength. Spread the jump
\(1-p_{\rm top}\) linearly over \([\xi_*,\xi_*+\omega]\), add
\(\omega\xi\) to this mass profile for \(0\le\xi0\)
such that
\[
\sup_{0\le\xi\le R,\ z\in\mathbb R}
\left|\partial_j\partial_z^r\widehat V_j(\xi,z)\right|
\le C_r e^{-c_rM}.
\]
For a fixed sufficiently small \(\xi_0>0\), this implies the more
precise cancellation
\[
\left|\partial_j\bigl(
\widehat\Gamma'_j(\xi)-\widehat\Gamma'_j(0)\bigr)\right|
\le C e^{-cM}\xi^2,
\qquad 0\le\xi\le\xi_0. \tag{P2a}\label{eq:P2a}
\]
For a fixed sufficiently small mass \(v_0>0\), the smoothed reference
also satisfies, for each fixed \(r\ge0\),
\[
\sup_{0\le v\le v_0}
\bigl(|\partial_v^r t|+|\partial_v^r q|
+|\partial_\theta\partial_v^r t|
+|\partial_\theta\partial_v^r q|\bigr)\le C_r,
\]
with one-sided derivatives at zero. The physical initial spatial
derivatives obey the rescaling bound, for each fixed \(r\ge1\),
\[
\left|\partial_\theta\left(
\ell^{1-r}\partial_z^r V(0,0)\right)\right|
\le C_r e^{-c_rM}.
\]
Over the full mass interval \((0,4)\), the smoothed clocks \(t,q\)
are absolutely continuous in \(\theta\), with
\[
\int_0^4\bigl(|\partial_\theta t(v)|
+|\partial_\theta q(v)|\bigr)\,dv
\le C M^C. \tag{P2b}\label{eq:P2b}
\]
The constants in this integrated bound contain no inverse power of
\(\omega\).
\end{lemma}
\begin{proof}
For the direct reference, \(A_0'(0)>0\) gives the inverse bounds.
The perturbation of its inverse is confined to the initial interval,
and differentiating its inverse relation gives the stated
\(O(\lambda)\) bounds. The linearized PDE and coupled diffusion in
the proof of Lemma~\ref{paths:lift} give the bound on
\(\partial_{\log\lambda}q\). That proof also differentiates the
scalar gap only after subtracting its value at zero, which preserves
the factor \(\xi^2\).
For size variations, the normalized smoothed profile below \(M\) is
independent of \(\ell\); only its terminal datum changes through
\(j=4/\ell\). The diffusion representation of
\(\partial_j\widehat V_j\) has terminal datum
\(\partial_j[j^{-1}\log\cosh(jz)]\). It differs from the constant
\(\log2/j^2\) by at most
\(C\sqrt{1-\tanh^2(jz)}\). The conditional martingale estimate in
the preceding proof therefore makes its expectation exponentially
small, uniformly in the starting field on a fixed compact time interval.
Higher spatial derivatives are uniformly bounded by the step power rules with one
bounded smooth terminal insertion and its bounded derivatives; the
remaining slots are bounded spin derivatives controlled by PE,
Corollary~2.10. If the uniform value error is \(\varepsilon\), global
finite-difference interpolation between orders zero and any fixed
\(k>r\) bounds the derivative of order \(r\) by
\(C_{r,k}\varepsilon^{1-r/k}\). This proves the asserted exponential
bound uniformly over all starting fields. The calculation differentiates
only the terminal datum while the mass profile below \(M\) stays fixed.
To obtain \eqref{eq:P2a}, differentiate the scalar second-derivative
identity at fixed normalized time. The spatial derivatives just bounded
are exponentially small, \(\widehat V_z\) and
\(\widehat V_{zzz}\) are odd, and the normalized mass is \(O(\xi)\)
near zero. Differentiating the Brownian coupling gives a field change
\(O_p(e^{-cM}\sqrt\xi)\). Hence
\(\partial_j\widehat\Gamma''_j(\xi)=O(e^{-cM}\xi)\), and
integration from zero proves \eqref{eq:P2a}.
Repeated PDE and generator differentiation bounds the response time
derivatives near zero, where the normalized mass profile is smooth with
bounded derivatives. Applying the same calculation to the equations
linearized in \(j\) bounds the mixed derivatives
\(\partial_j\partial_\xi^r\widehat\Gamma_j\) there. Since
\(A_0'(0)>0\), composition with the
normalized inverse and the exact rescalings
\(\xi=\ell^2t\), \(\widehat A(\xi)=v/\ell\) gives the stated
one-sided mass derivatives and their size derivatives. Finally,
\(\ell^{1-r}\partial_z^r V(0,0)=\partial_z^r\widehat V_j(0,0)\),
and \(\partial_\theta j=-j/6\). Thus treating the normalized initial
spatial derivatives as constants during a size differentiation incurs
only exponentially small errors.
It remains to control these derivatives after integration over all
masses. Off the terminal plateau, the inverse relation is
\(\widehat A(\xi)=v/\ell\), with \(\xi=\ell^2t(v)\). Since
\(\partial_\theta\ell=\ell/6\), differentiation at fixed \(v\) gives
\[
\partial_\theta t(v)
=-\frac{t(v)}3-
\frac{v}{6\ell^3\widehat A'(\ell^2t(v))}.
\]
The second term can be large pointwise, but under
\(dv=\ell\widehat A'(\xi)\,d\xi\) its inverse density cancels.
Both resulting integrals are \(O(M)\). On the terminal plateau,
\(t=M/\ell^2\) and \(\partial_\theta t=-t/3\). Its moving boundary
creates no jump term because the clock is continuous there. In
particular \(\int|\partial_\theta t(v)|\,dv\lesssim M^C\), with
no inverse power of \(\omega\).
For \(q(v)=\widehat\Gamma_j(\xi)\), there is also dependence on the
terminal mass \(j=4/\ell\). We bound it at fixed normalized time
\(\xi\) by representing \(\widehat\Gamma_j(\xi)\) as the expectation
of the product of two terminal spins whose branches separate at that
time. This test is bounded by 1 and does not depend on \(j\). In a
step approximation, differentiate the logarithm of its positive tree
density, telescoped as in PE, Proposition~2.14. Below \(M\) all
normalized masses are fixed. The derivative
\(\partial_j[j^{-1}\log\cosh(jz)]\) is uniformly bounded, and the
diffusion representation gives the same bound for interior continuation
derivatives. Their coefficients have total variation bounded by the
number of observed leaves times the terminal mass, independently of
the number of steps. The remaining terminal terms are bounded by
\(C(1+|Z_M^1|+|Z_M^2|)\). Hence the logarithmic density derivative
has \(L^1\) norm at most \(C(1+M)\), because bounded scalar drift
gives \(\mathbf E|Z_M^i|\lesssim1+M\). It follows that
\[
|\partial_j\widehat\Gamma_j(\xi)|\le C(1+M),
\qquad 0\le\xi\le M,
\]
uniformly in the step resolution and in \(\omega\). The resulting
uniform Lipschitz bound passes to continuous clocks by PE,
Proposition~2.14; it supplies absolute continuity and the same derivative
bound almost everywhere without requiring convergence of derivatives.
Off the plateau, the chain rule now reads
\[
\partial_\theta q(v)
=-\frac{v\,\widehat\Gamma_j'(\xi)}
{6\ell\widehat A'(\xi)}
-\frac j6\partial_j\widehat\Gamma_j(\xi).
\]
After integration in \(v\), the first term again loses its inverse
density. Its integral is bounded by
\(C\int_0^M\widehat\Gamma_j'(\xi)\,d\xi\le C\), since
\(\widehat\Gamma_j\) is increasing and lies in \([0,1]\). The
second term has integral \(O(1+M)\). On the plateau only this second
term remains. Thus \(q\), like \(t\), has an absolutely integrable
size derivative of norm \(O(M^C)\).
\end{proof}
\subsection{Constructing admissible matrix and field clocks}
We now use the scalar gap to allow a varying matrix clock. The positive
term from the lift pays for its increase, while a negative constant
offset supplies a further margin at the smallest masses. We then add
field increments, checking separately the single ramp that can decrease.
Choose a small fixed mass \(v_b>0\), such that inverse densities are regular with bounded derivatives and bounded above and below up to \(8v_b\). Take a smooth nondecreasing \(G(v)\) equal to zero on \([0,x]\), \(v^2\) on \([8x,v_b]\) and constant positive above \(2v_b\), \(0\le G'\le C v\). Use dyadic masses \(u=2^{-h}\) in \([16x,v_b/16]\), a fixed smooth nonnegative \(\phi\) supported on \([1/2,4]\), positive on \([1,2]\), and parameters \(\gamma=(\gamma_0,\gamma_u)\). These cutoffs are common across path segments on the dyad and independent of \(\ell\). Take
\[
b(v)=b_\circ+c_b\lambda\left(G(v)-U^2-c_1\gamma_0(G(v)+U^2)
-c_1\sum_u\gamma_u u^2\phi(v/u)\right), \tag{P3}\label{eq:P3}
\]
with sufficiently small fixed \(c_1,c_b>0\). The parameters \(\gamma\) may range in \([-4,4]\). They are held fixed
when taking a positive-law expectation; later we average their coordinates
independently over \([1/3,2/3]\). By bounded overlap, \(0\le b'(v)\le Cc_b\lambda v\), \(b\asymp1\), \(c_g=-\partial_{\gamma_0}b\asymp\lambda(v^2+U^2)\) throughout this bounded mass interval. Also \(h_*=K-bq\) is nonnegative nondecreasing, with
\[
dh_*\ \ge c\lambda(v^2+U^2)\,dt,\qquad
h_*(v)\lesssim\lambda(v^3+vU^2)\quad\text{for small }v . \tag{P4}\label{eq:P4}
\]
Indeed, along the inverse clock, including constant pieces,
\[
dh_*=(1-b\Gamma'(t))\,dt-q\,db.
\]
The lift \eqref{eq:P2} supplies the \(\lambda v^2\) gap near zero,
and the negative \(U^2\) offset in \eqref{eq:P3} supplies
\(\lambda U^2\). The term \(q\,db\) occurs only at small active
mass, where \(q\asymp v\) and the inverse density is regular; choosing
\(c_b\) small absorbs it in the positive gap. Outside that interval
\(db=0\). At large \(t\), \(\Gamma'\) is small and the gap is
bounded below by a constant. This proves the first bound in
\eqref{eq:P4}; integrating its small-time upper counterpart from zero
gives the second. We also have for every \(z\ge0\) (with either weight \(w(y)=1\) or \(1-y\))
\[
\int_0^1 w(y)\,[1-b(v)\Gamma'(t(v)+y b(v)(z-q(v)))]\,dy
\ge c\lambda(v^2+U^2). \tag{P5}\label{eq:P5}
\]
Indeed the first half of the secant lies at time at least \(t(v)/2\) since \(h_*\ge0\), hence the base lift there dominates \(c\lambda v^2\). The subtracted \(U^2\) offset in \eqref{eq:P3} provides the \(U^2\) term unless \(\Gamma'\) itself is small; everywhere the loss relative to these bounds from \eqref{eq:P3} is at most \(C c_b\lambda v^2\).
Set \(h(v)=0\) for \(v0\), \(\widetilde U=\min(U_0,U)\), a smooth taper \(\Upsilon(U)\) equal to 1 on \([0,U_0/2]\), zero on \([U_0,1]\), with values in \([0,1]\). Define \(E_b>0\) by
\[
p_{\rm top}-A_0(\xi_*-E_b)=\widetilde U^4/E_b^2=d_-,\qquad d_+=\widetilde U^4.
\]
It exists uniquely near zero by strict increase, with \(|dE_b/d\log\lambda|\lesssim E_b\). On the plateau \(w=v/\ell\in[p_{\rm top},1]\) set
\[
H_e=\lambda_1\Upsilon(U)\,\widetilde U^2
\left((1-w+d_+)^{-1/2}-(w-p_{\rm top}+d_-)^{-1/2}\right).
\tag{P7}\label{eq:P7}
\]
On the active part ramp linearly in \(\xi=\ell^2 t(v)\) from zero at \(\xi_*-E_b\) to its value at the plateau; below the ramp set \(H_e=0\). Thus on each dyadic strip of distance \(\asymp d'\) from an edge of the plateau up to its midpoint (using also one capped strip of width \(\asymp d_\pm\) adjacent to each edge), the clock increase per unit mass due to \(H_e\), for small \(U\), is comparable to \(\lambda_1 U^2/(d')^{3/2}\). On the active-side ramp its derivative in \(t\) may be negative,
but its magnitude is \(O(\lambda_1)\). This ramp lies at a fixed
positive mass, where \eqref{eq:P4} gives an increase at least
\(c\lambda\,dt\). Since \(\lambda_1/\lambda=\lambda^\eta\),
the negative part is absorbed for large \(N\). Thus the total field
clock remains nonnegative and nondecreasing even where the added defect
is negative; all paths are admissible. Both \(H_e\) and its \(\log\lambda\) derivative are bounded, on the ramp by \(C\lambda_1 E_b\), on the plateau by \(C\lambda_1\widetilde U^2/\sqrt{d'}\), vanishing away from \(U\le U_0\). In differentiating, \(\ell\) is then fixed and the inverse reference near the ramp does not vary.
\subsection{Defect bounds, endpoint tests, and parameter monotonicity}
In particular the field defect \(d=h-h_*\) satisfies
\[
\int d^2/c_g\,dv\ \lesssim_\circ\
\lambda^{1+2\eta}U^5+\chi_e^2\lambda^{1+2\eta}U^4+\chi_{\rm lin}^2 x^6/\lambda. \tag{P8}\label{eq:P8}
\]
To verify \eqref{eq:P8}, separate the part below the cut from the
three added field terms. Below \(x\), \(d=-h_*\), and
\eqref{eq:P4} gives
\[
\int_0^x\frac{d^2}{c_g}\,dv
\lesssim\lambda U^2x^3
\lesssim\lambda^{1+2\eta}U^5.
\]
The last inequality uses \(x/U\lesssim\lambda^{1/12}\) and sufficiently
small \(\eta\). For \(H_{\rm lo}\), integration of
\((v^2+U^2)^{-1}\) contributes \(O(U^{-1})\); after squaring its
amplitude, the bound is \(\lambda^{1+2\eta}U^5\) with logarithmic
loss. On the active ramp for \(H_e\), the squared amplitude is at
most \(C\lambda_1^2E_b^2\), its mass length is \(\ell d_-\), and
\(c_g\asymp\lambda\). The identity
\(E_b^2d_-=\widetilde U^4\) therefore gives
\(O(\lambda^{1+2\eta}U^4)\). On the plateau, the squares of the two
reciprocal-square-root terms integrate with only logarithmic loss and
give the same bound. Finally, \(H_{\rm lin}\) is supported at mass
bounded away from zero, where \(c_g\gtrsim\lambda\), so its contribution
is \(O(x^6/\lambda)\). The inequality for the square of a sum combines
these contributions with the displayed multipliers.
\paragraph{Strength derivatives.}
During a direct strength sweep, \(\ell,\gamma\) are fixed,
\((\chi_{\rm lin},\chi_e)=(0,1)\), and a dot means
\(\partial_{\log\lambda}\). The differentiated defects obey the
same localized bounds. On the mask they are
\(O(\lambda(v^3+vU^2))\): differentiate the scalar gap only after
subtracting its value at zero. Above the mask, differentiating
\eqref{eq:P6}--\eqref{eq:P7} gives the claimed ramp and plateau bounds.
In this regime all dotted clocks, \(\dot q\), and \(\dot b\) are
bounded by \(O_\circ(\lambda)\).
\paragraph{Size derivatives.}
On the smoothed size segment, the strength is final,
\((\chi_{\rm lin},\chi_e)=(1,0)\), and a dot means
\(\partial_\theta\), with \(\gamma\) fixed. Exact scalar rescaling
and Lemma~\ref{paths:parameter-regularity} give the same small-mass
defect bounds for \(d\) and \(\dot d\). In particular,
\eqref{eq:P2a} retains the factor \(\xi^2\) after differentiating
the scalar gap relative to its time-zero value. Above small
mass, \(d,\dot d=O_\circ(x^3)\). The other dotted clocks are
absolutely integrable with \(O_\circ(1)\) norms by the global
parameter bounds proved above.
\paragraph{Direct comparison endpoints.}
Rescale a direct path to \([0,1]\) by \(w=v/\ell\). The PE comparison
hypotheses at every lower scale \(w\gtrsim x\), with
\(T_m=\ell^2b_\circ\to T\), take the form
\[
\ell^2 b(\ell w)=T_m+o(w^2),\qquad \ell^2 h(\ell w)=o(w^3),\qquad w\gtrsim x,
\tag{P9}\label{eq:P9}
\]
because \(\lambda U^j\le x^j\lambda^{1-js_*/2}\) for \(j=2,3\).
The gaps \(1-js_*/2>0\) make these errors vanish relative to the
required powers at the lower comparison scales.
In varying \(\gamma\) it is useful to subtract \(\int b q^2/4\) from \(f_m^0\). Its increasing derivative along any matrix decrease \(c=-\partial_\gamma b\ge0\) accompanied by the path prescription satisfies
\[
\partial_\gamma(f_m^0-\int bq^2/4)
\ \ge\ \tfrac14\int c\,(D_v^2+(S_v-q)^2)\,dv, \tag{P10}\label{eq:P10}
\]
To check the sign, note that \(q\) is fixed under the \(\gamma\)
variation. Above the mask the prescribed field satisfies
\(\partial_\gamma h=cq\), so \eqref{eq:A1} gives the integrand
\[
\frac c4(B_v-2qS_v+q^2)
=\frac c4\big(D_v^2+(S_v-q)^2\big).
\]
On the mask, \(h=0\) remains fixed and the actual integrand is
\(c(B_v+q^2)/4\). It exceeds the displayed square by
\(cqS_v/2\ge0\), proving \eqref{eq:P10}.
\subsection{Interchanging the direct and smoothed paths}
\begin{proposition}[Comparison of the two regularizations]
\label{paths:switch}
At final strength \(\lambda=x^{p_*}\), with
\((\chi_{\rm lin},\chi_e)=(1,0)\), the direct and smoothed systems
satisfy \eqref{eq:P11}, uniformly with the same parameters
\(\gamma,\ell\). Their scalar trial gaps
\[
f_{\rm spin}(K)+\frac14\int b^{-1}(K-h)^2\,dv-f_m^0,
\]
where \(f_{\rm spin}\) is the self-corrected one-site pressure at
the corresponding terminal mass, also differ by \(O(x^{5+c})\).
The same fixed \(c>0\) works for these three comparisons, independently
of the site count.
\end{proposition}
\begin{proof}
Fix \(\lambda=x^{p_*}\), \(\chi_{\rm lin}=1\), and \(\chi_e=0\),
and keep \(\gamma,\ell\) the same in both systems. First extend the
direct system to length 4 by a single field jump at mass \(\ell\).
For \(v>\ell\), put \(K=K_{\rm post}=M/\ell^2,\ q=1,\ b=b_{\rm post}=b(\ell-),\ h=h_{\rm post}=K_{\rm post}-b_{\rm post}+H_{\rm lo}(\ell)+H_{\rm lin}(\ell)\). On this extension \(q=1\) is a deterministic comparison function,
rather than a claim about the finite-time scalar response. We first
compare these four extended functions to their smoothed counterparts.
For fixed \(0<\epsilon<1/21\), put
\[
E_\epsilon=\omega M^2+\omega^{1-\epsilon}+e^{-c'M}.
\]
Then
\[
\begin{split}
&\|K_{\rm hard}-K_{\rm smoothed}\|_1
+\|q_{\rm hard}-q_{\rm smoothed}\|_1\\
&\qquad+\|b_{\rm hard}-b_{\rm smoothed}\|_1
+\|h_{\rm hard}-h_{\rm smoothed}\|_1
\lesssim E_\epsilon.
\end{split}
\]
Here and below the norms integrate over \(v\in(0,4)\).
For the inverse clocks,
the area between monotone quantiles equals the area between their
mass profiles; the spread jump and density floor contribute
\(O(\omega M^2)\). Below mass \(\ell\), both scalar times remain
in a fixed compact interval, so the earlier
\(O_\epsilon(\omega^{1-\epsilon})\) moment and initial-derivative
comparisons control the response at a common scalar time and
\(b_\circ\). The remaining change in \(q\), caused by evaluating
the response at different scalar times, is bounded by
\(\sup|\Gamma'|\,|K_{\rm hard}-K_{\rm smoothed}|\). Its integral
is therefore controlled by the inverse-clock estimate.
Above \(\ell\), the portion before the smoothed terminal plateau has
mass length \(O(\omega M)\). Outside that strip, the martingale bound
gives \(q=1-O(e^{-c'M})\). Equations \eqref{eq:P3} and \eqref{eq:P6}
then give the asserted \(L^1\) estimates for \(b,h\) as well.
Since \(\omega=x^{5+1/4}\), the choice \(\epsilon<1/21\) gives
\((5+1/4)(1-\epsilon)>5\). With \(M=\log^3(1/x)\), this proves
\(E_\epsilon=O(x^{5+c})\) for a fixed \(c>0\).
We next compute the pressure of this extension, denoted by
\(f_{\rm hard}\). Condition on arbitrary earlier Hamiltonian values
\((H_\sigma)_{\sigma\in\{-1,1\}^m}\), and set
\(D=h_{\rm post}-h(\ell-)\), \(p=\ell/4\). If
\(z\) has independent standard Gaussian coordinates and \(\Phi\)
is the standard Gaussian distribution function, then
\[
\begin{split}
&\Phi(\ell\sqrt D)^m e^{m\ell^2D/2}
\sum_\sigma e^{\ell H_\sigma}\\
&\qquad\le
\mathbf E_z\left(\sum_\sigma
e^{4H_\sigma+4\sqrt D\,z\cdot\sigma}\right)^p
\le e^{m\ell^2D/2}\sum_\sigma e^{\ell H_\sigma}.
\end{split}
\]
The upper bound is subadditivity for \(00\) with the following property. For every sufficiently small fixed
\(\varepsilon>0\), every fixed \(q>0\), and
\(s=q x^{1+\varepsilon}\), the following bounds hold for every
sufficiently large fixed stencil order:
\begin{enumerate}
\item On each direct segment,
\[
\int \mathbf E_\gamma\!\left[
{\bf1}_{\{\mathcal K_m(s)\le3x^\varepsilon\}}
|\dot\psi_m(s)|\right]\,|d\vartheta|
\lesssim x^{5+c},
\]
uniformly for \(N/2\le m\le3N\), where \(\vartheta\) is that
segment's parameter.
\item On the size segment the corresponding bound holds for
\begin{equation}
\left|\dot\psi_m(s)+\sum_{j=0}^r
c_j(m+j)\psi_{m+j}(s)\right|.\tag{I}\label{eq:I}
\end{equation}
It also holds when \(m\) is a measurable function of \(\theta\) with
values in this integer range, or when the observables are averaged
with nonnegative probability weights on these integers depending
measurably on \(\theta\).
\end{enumerate}
All integrations use absolute path-parameter length. The saving \(c\)
can be chosen independently of sufficiently small \(\varepsilon\).
Constants may depend on the fixed test coefficient \(q\), stencil order,
and the fixed temperature.
\end{proposition}
\subsection{Exponential augmentation and the stopped change of measure}
To transfer moment estimates to the tilted law, we need fixed
\(L^p\) bounds for its density, which involve Laplace transforms at
larger fixed multiples of \(s\). Adding independent exponential
noise produces a log-concave law whose centered positive Laplace
transform controls its variance and both tails.
At fixed site count, path point, and parameters \(\gamma\), the
transform \(\mathcal K_m(s)\) is deterministic because its defining
disorder expectation has already been taken. Thus the stop
\(\mathcal K_m(s)\le3x^\varepsilon\) restricts path data. All density
norms below use the untilted initial Gaussian law.
\begin{lemma}[Augmentation along the physical paths]\label{red:lem-augmentation}
Let \(G_m\) be the continuation in \eqref{eq:A4}, and let
\(\mathcal E_{x/2}\) be an independent exponential random variable of
rate \(x/2\). Along every physical path used above,
\(G_m+\mathcal E_{x/2}\) has a log-concave density. On
\(\mathcal K_m(s)\le3x^\varepsilon\), the Radon--Nikodym density
\[
R_m=\frac{e^{sG_m}}{\mathbf E e^{sG_m}}
\]
has bounded \(L^p\) norm for every fixed finite \(p\), uniformly in
the path and the allowed site counts for all sufficiently large \(N\).
\end{lemma}
\begin{proof}
First take a finite hierarchy and include all future Gaussian drivers
under the normalized positive law. Write the positive revelation masses
as \(v_1<\cdots0\) such that \(\mathbf P(Z\ge c_1d_G)\ge c_1\).
On the stopped set,
\[
\log\mathbf E e^{sZ}
=\mathcal K_m(s)-\log(1-2s/x)-2s/x
\le3x^\varepsilon+O(x^{2\varepsilon}).
\]
The function \(e^t-1-t\) is nonnegative, and on the displayed
right-hand event it is at least \((s c_1d_G)^2/2\). Since
\(\mathbf EZ=0\), it follows that
\[
c_1(s c_1d_G)^2/2
\le\mathbf E(e^{sZ}-1-sZ)
=e^{\log\mathbf E e^{sZ}}-1
\lesssim x^\varepsilon.
\]
Consequently \(s d_G\lesssim x^{\varepsilon/2}\). The two-sided
exponential tails now bound \(\mathbf E e^{p sZ}\) for every fixed
\(p\). Conditional Jensen over the independent centered exponential
transfers this bound to \(G_m-\mathbf E G_m\). Since
\(\mathcal K_m(s)\ge0\), this proves the asserted bound for \(R_m\).
\end{proof}
The active-component law in \eqref{eq:A4} may therefore be estimated by
H\"older from its untilted marginal, with an arbitrarily small power
loss when the required moment orders are fixed sufficiently large.
For a product involving several active components, apply these marginal
bounds and H\"older to the finite collection of density factors. Their
roots remain independent; no common untilted root is introduced.
Outside the stopped set no derivative estimate of this accuracy will
be used.
\subsection{Mollification in the number of sites}
The size path changes both the clocks and the number of sites.
At fixed integer \(m\), differentiation of
\(\mathcal K_m(s)=sm\psi_m(s)\) accounts only for the clock change.
We average over nearby integers to obtain a differentiable function
of a real size center. Differentiating the averaging weights then
produces the finite-difference term in \eqref{eq:I}; a neighboring-site
bound will let us remove the averaging at the endpoints.
Let \(\rho\) be a fixed nonnegative smooth compactly supported
probability density, and set
\[
D=x^{-1/2},\qquad n=Ne^\theta,\qquad
w_m(n)=D^{-1}\rho\bigl((m-n)/D\bigr).
\]
The real center \(n\) is frozen on direct segments. Smooth
sum--integral comparison gives, for every fixed \(k\),
\[
\sum_m w_m(n)=1+O_k(D^{-k}).
\]
For example, repeated integration by parts in the Fourier series of
the periodization proves the estimate uniformly in \(n\).
\begin{lemma}[Stencil and neighboring-site errors]\label{red:lem-site-smoothing}
For any sequence \(Z_m\), supported or restricted to the relevant
neighborhood of the center,
\[
\frac{d}{dn}\sum_m w_m(n)Z_m
=\sum_m w_m(n)\sum_{j=0}^r c_j Z_{m+j}
+O_r\!\left(D^{-r-1}\sup_m|Z_m|\right).
\]
At the same clock data, uniformly along the paths under consideration,
\begin{equation}\label{red:eq-neighbor}
|\mathcal K_{m+1}(s)-\mathcal K_m(s)|\le Cs(1+M).
\end{equation}
\end{lemma}
\begin{proof}
Move the stencil from \(Z\) to the weights. The coefficient of
\(Z_m\) becomes \(\sum_jc_jw_{m-j}\). Taylor expansion of the shifted
weights, together with the defining polynomial identities of the
stencil, leaves the derivative in \(n\). Its remainder has size
\(O_r(D^{-r-2})\) for each weight and is summed over \(O(D)\) sites.
For \eqref{red:eq-neighbor}, append a zero-input independent spin and
interpolate to the completed \((m+1)\)-site model with the adjusted
matrix normalization. The change in total cumulative covariance is
\(O(1+M)\), and the affine interpolation is positive semidefinite.
Apply pressure differentiation to both tests in \eqref{eq:A4}.
The spin prior is a probability measure, so the unused spin contributes
no additional constant. This gives the displayed bound.
\end{proof}
\begin{proposition}[Dyadic transform comparison]\label{red:prop-dyadic}
Assume Proposition~\ref{prop:increment}. There is \(c'>0\), independent
of sufficiently small fixed \(\varepsilon\), such that for every fixed
positive test coefficient \(q\), every integer \(N'\in[N,2N]\), and
\(s=q x^{1+\varepsilon}\),
\[
\left|\log\mathbb E e^{(s/x)Y_{N'}}
-\log\mathbb E e^{(s/x)Y_N}\right|
\lesssim x^{\varepsilon+c'},\qquad
Y_m=m^{-1/6}(F_m^c-\mathbb EF_m^c).
\]
The constants are uniform in the integer \(N'\).
\end{proposition}
\begin{proof}
Use a direct sweep at center \(N\), the switch \eqref{eq:P11}, the size
sweep to \(\theta=\log(N'/N)\), the reverse switch, and the reverse
direct sweep at center \(N'\). At both ends the regularization strength
is \(x^{12}\); the clocks differ from the bare data
\((T/\ell^2,0)\) by \(O_\circ(x^{12})\). The self corrections cancel
in the centered transform.
Apply Fubini to the finite sum of the nonnegative stopped derivative
integrals, with the site weights above. Their total expectation in
\(\gamma\) is \(O(x^{5+c})\), so one choice of \(\gamma\) has this
bound up to a fixed constant. The weights can be normalized by their
sum, with an error smaller than any fixed power by increasing the
sum--integral order. The weighted form of \eqref{eq:I} expressly
permits their dependence on \(\theta\). The choice is made for this
one dyad comparison and one test \(s\); different tests may use
different paths because the bare endpoint laws do not depend on
\(\gamma\).
Write
\[
\overline{\mathcal K}(\theta)=\sum_mw_m(n)\mathcal K_m(s).
\]
By \eqref{red:eq-neighbor}, its active site values have oscillation
\(O(sD(1+M))=x^{\varepsilon+1/2-o(1)}\). Since
\(\mathcal K_m\ge0\) and \(\sum_mw_m=1+O_k(D^{-k})\), whenever
\(\overline{\mathcal K}\le x^\varepsilon\), all contributing site
counts, and the boundedly many neighboring stencil counts, lie in
their individual stops \(\mathcal K_m\le3x^\varepsilon\).
On a size segment use \(Z_m=m\psi_m(s)\) in
Lemma~\ref{red:lem-site-smoothing}. The derivative is
\begin{align*}
\frac{d}{d\theta}\overline{\mathcal K}
={}&sn\sum_mw_m
\left(\dot\psi_m+\sum_{j=0}^r c_j(m+j)\psi_{m+j}\right)\\
&+s\sum_mw_m(m-n)\dot\psi_m
+O_\circ\bigl(sN^2D^{-r-1}\bigr).
\end{align*}
Indeed \(Z_m=O_\circ(N)\), by the pressure bounds in
\eqref{eq:A1}. The dotted-clock \(L^1\) bounds and
\eqref{eq:A1} make the middle term \(O_\circ(sD)\), after integration
as well. The two auxiliary errors have sizes
\[
sD=x^{\varepsilon+1/2},\qquad
sN^2D^{-r-1}=x^{\varepsilon-11+(r+1)/2}
\]
up to subpower losses. Choose the fixed stencil order large enough to
leave a positive saving. Multiplying the increment bounds by
\(sn=O(x^{-5+\varepsilon})\), and including the switches
\eqref{eq:P11}, gives total drift
\(O(x^{\varepsilon+c'})\). The bare endpoint approximation costs
\(O_\circ(sN x^{12})=O_\circ(x^{7+\varepsilon})\).
The savings can be decreased independently of \(\varepsilon\).
At the bare initial endpoint the fixed-temperature exponent theorem
of \cite[Theorem~1.1]{PE}, together with the augmented
log-concave estimate, gives
\(\mathcal K_m(s)=o(x^\varepsilon)\) uniformly in the active window.
Indeed the variance scale is \(x^{-2-o(1)}\) before multiplication
by \(x\), and the small-positive-parameter bound gives
\(O(x^{2\varepsilon-o(1)})\). Continuity on each segment, and the
small switch errors, exclude a first violation of
\(\overline{\mathcal K}\le x^\varepsilon\). Only stopped derivative
estimates have entered this argument.
Remove the endpoint weights using \eqref{red:eq-neighbor}; the cost
is again \(O(sD(1+M))\). At the final bare endpoint the terminal mass
is \(\ell=(N'/N)^{1/6}\), while the covariance is \(T/\ell^2\).
The terminal continuation is therefore distributed as
\(F_{N'}^c/\ell\), with the same physical inverse temperature.
Since \(x/\ell=N'^{-1/6}\), its \(x\)-scaled centered law is exactly
\(Y_{N'}\). The initial law is \(Y_N\). This proves the claim,
including its uniformity throughout the dyad.
\end{proof}
\subsection{Moments, positive variance, and full-sequence convergence}
\begin{proposition}[Consequences of the increment bound]\label{red:prop-moments}
Assume Proposition~\ref{prop:increment}. For every fixed integer
\(k\ge1\), the moments \(\mathbb EY_n^k\) converge at a positive
power rate. Their limiting second moment is finite and strictly
positive, and the centered variables \(Y_n\) have uniformly bounded
exponential moments on some constant neighborhood of the origin.
They converge weakly to a moment-determinate law. Removing the
completion and standardizing by the exact finite-size variance gives
the convergence asserted in Theorem~\ref{thm:main}.
\end{proposition}
\begin{proof}
Fix \(k\). The dyadic comparison gives values of the log Laplace
transform at positive arguments. We recover its derivatives at zero
by polynomial interpolation, choosing \(\varepsilon\) after \(k\)
so that the interpolation loss is smaller than the comparison saving.
At bare endpoints, the exponent theorem, the augmented tail bound,
and conditional Jensen give a scale \(L_n=x^{-o(1)}\), uniform in
the dyad, with \(\mathbb E e^{|Y_n|/L_n}\le C\). Hence, for a
sufficiently small constant \(a>0\),
\(M_n(z)=\mathbb E e^{zY_n}\) satisfies
\(|M_n(z)-1|\le1/2\) on
\(|z|\le r_n^{\mathrm{an}}:=a/L_n\), where its logarithm is analytic
and bounded by an absolute constant.
Cauchy's estimate on this disk gives
\[
\left|\log M_n(z)-
\sum_{j=0}^{k}\frac{(\log M_n)^{(j)}(0)}{j!}z^j\right|
\le C_k(L_n|z|)^{k+1},
\qquad |z|\le r_n^{\mathrm{an}}/2,
\]
which is \(x^{(k+1)\varepsilon-o(1)}\) at each fixed multiple of \(x^\varepsilon\) and uses
only the preliminary subpower scale.
Interpolate at the \(k+1\) nonnegative nodes
\(0,x^\varepsilon,\ldots,kx^\varepsilon\).
Proposition~\ref{red:prop-dyadic} bounds each nonzero-node difference;
at zero the difference is zero. Division by the interpolation scale
loses at most \(x^{-k\varepsilon}\). Thus every cumulant through
order \(k\) has dyadic difference bounded by
\[
O\bigl(x^{c'-(k-1)\varepsilon}
+x^{\varepsilon-o(1)}\bigr).
\]
Take \(\varepsilon\) sufficiently small after fixing \(k\), for
example with \(k\varepsilon0\). The lower bound in the PE
exponent theorem gives \(v_n\ge n^{-2\eta}\) eventually for every
fixed \(\eta>0\), after a harmless adjustment of the exponent
notation. Choosing \(2\eta0\).
The bounded variances improve the tail scale to a uniform constant.
To see the normalization explicitly, put \(r_n=n^{-1/6}\) at a bare
system of size \(n\), and let \(E_n\) be independent exponential
noise of rate \(r_n/2\). The bare version of the pinning calculation
above gives a log-concave density for \(F_n^c+E_n\). At a bare path
the negative continuation coefficient in that calculation is the
terminal mass one, which exceeds \(r_n/2\). Thus
\[
\widetilde Y_n=Y_n+r_n(E_n-\mathbb E E_n)
\quad\hbox{has variance}\quad v_n+4.
\]
The log-concave tail estimate applies with a common constant scale.
Conditional Jensen, using the convex function
\(z\mapsto e^{t|z|}\), removes the centered independent noise.
Consequently, for some \(t_0>0\),
\[
\sup_{n\ge2}\mathbb E e^{t_0|Y_n|}<\infty.
\]
Finitely many small sizes can be included by decreasing \(t_0\).
These bounds give tightness and uniform integrability of every fixed
power. They also imply moment determinacy, for example through the
resulting factorial moment bounds. Every weak subsequential limit
therefore has the same moments and is the same probability law.
Thus weak convergence holds along the full sequence.
The probability spin prior changes the original free energy only by
the deterministic \(n\log2\). The completion is an independent
centered Gaussian of variance \(T/2\), whose scaled \(L^2\) norm is
\(O(n^{-1/6})\) and whose scaled variance is \(T/(2n^{1/3})\).
It follows that the uncompleted scaled variance has the same positive
limit \(v\). Removing that Gaussian and dividing by the exact
finite-size standard deviation is now justified by Slutsky's theorem.
\end{proof}
\subsection{Identification by finite random-size experiments}
\begin{proposition}[Validity of the law specification]\label{red:prop-identification}
Under Proposition~\ref{prop:increment}, the constants in
Proposition~\ref{def:law} are well defined by an absolutely convergent
series after the indicated expectations. They are precisely the
limiting centered moments of \(Y_n\), with \(a_2>0\).
The standardized moments \(a_k/a_2^{k/2}\) determine one Borel
probability measure uniquely. Each \(a_k\) is also the expectation of
a single integrable random variable constructed from fixed priors and
finitely many Gaussian systems almost surely.
\end{proposition}
\begin{proof}
Fix \(k\), let \(m_k(l)=\mathbb EY_l^k\), and denote its power-rate
limit by \(m_k\). Recall
\(C=\sum_{l\ge2}l^{-2}\),
\(\mathbb P(J=l)=l^{-2}/C\), and
\[
P_k=\prod_{i=1}^k J^{-1/6}(F^{c,0}-F^{c,i}),
\qquad t=\log J,
\]
where the completed systems are conditionally independent given
\(J\). Conditional on \(J=l\) and copy zero, the other \(k\)
copies average the product to the \(k\)-th centered power of copy
zero. Thus
\[
\mathbb E[P_k\mid J=l]=m_k(l).
\]
The uniform moment bounds proved above also control every fixed
absolute moment of \(P_k\) conditional on \(J=l\), uniformly in \(l\).
For \(\operatorname{Re}z>0\), put
\[
B_k(z)=Cz\,\mathbb E[P_k e^{(1-z)t}]
=z\sum_{l\ge2}m_k(l)l^{-1-z}.
\]
If \(|m_k(l)-m_k|\le C_k l^{-a}\) for some \(a>0\), the difference
series is holomorphic on \(\operatorname{Re}z>-a\), locally
absolutely convergent there. The constant part also extends across
zero. Indeed, for \(\operatorname{Re}z>0\), the difference between
the sum and its integral is
\[
\sum_{l\ge2}\left(l^{-1-z}-\int_l^{l+1}r^{-1-z}\,dr\right)
-\int_1^2 r^{-1-z}\,dr.
\]
On a compact subset of \(\operatorname{Re}z>-1\), each summand is
bounded by a constant times \(l^{-2-\operatorname{Re}z}\).
The discrepancy therefore converges locally uniformly and is
holomorphic there. Since
\(z\int_1^\infty r^{-1-z}\,dr=1\), multiplication by \(z\)
removes the integral's pole and gives value one at zero. Combining
this with the difference series, \(B_k\) extends to
\(\operatorname{Re}z>-c_k\) for some \(c_k>0\), with
\(B_k(0)=m_k\).
Its Taylor series about \(z=1\) has radius strictly greater than one.
To evaluate it at zero write \(z=1-w\). Near \(w=0\), differentiation
under the expectation is legitimate: for every fixed \(h\) and
\(0<\delta<1\), the conditional absolute-moment bound on \(P_k\)
and \(\sum_{l\ge2}l^{-2+\delta}(\log l)^h<\infty\) give a
dominating integrable function on \(|w|\le\delta\).
The coefficient of \(w^h\)
in \((1-w)e^{wt}\) is
\[
\frac{t^h}{h!}
-{\bf1}_{\{h\ge1\}}\frac{t^{h-1}}{(h-1)!},
\]
with the second term omitted at \(h=0\). Hence
\[
B_k(0)=C\sum_{h=0}^\infty
\mathbb E\!\left[P_k\left(
\frac{t^h}{h!}
-{\bf1}_{\{h\ge1\}}\frac{t^{h-1}}{(h-1)!}
\right)\right]=a_k,
\]
and the scalar series is absolutely convergent. The expectation is
taken before the infinite sum. Indeed, at each fixed \(J\) the
bracket series equals \(e^t-e^t=0\); averaging that pointwise sum
would therefore fail to recover the positive second moment.
The limiting law from Proposition~\ref{red:prop-moments} proves
existence of the measure with these moments, its positive variance
proves \(a_2>0\), and its exponential moment bound proves uniqueness.
The final construction uses randomized single-term unbiased estimation
with replication conditional on the sampled level. For this method, see
\cite[Section~2, equation~(10), and Section~4]{RheeGlynn2015}.
Here the levels index Taylor coefficients; the integrability and
almost-sure finiteness proved below do not assert finite expected work
or import the SDE complexity guarantees of that reference.
For the final assertion, draw an independent integer \(H\ge0\) with
\(\mathbb P(H=h)=2^{-h-1}\). Conditional on \(H=h\), repeat the whole
random-size experiment independently \(256^h\) times, and let
\(\overline A_{k,h}\) be the sample average of
\[
A_{k,h}=CP_k\left(
\frac{t^h}{h!}
-{\bf1}_{\{h\ge1\}}\frac{t^{h-1}}{(h-1)!}\right).
\]
Return \(W_k=2^{H+1}\overline A_{k,H}\).
The estimate \(t^h/h!\le4^h e^{t/4}\), the prior proportional to
\(l^{-2}\), and the uniform conditional moments of \(P_k\) give
\(\|A_{k,h}\|_2\le C_k4^h\), including the shifted term.
Independence of the repetitions yields
\[
\mathbb E\left|
\overline A_{k,h}-\mathbb EA_{k,h}\right|
\le\frac{C_k4^h}{\sqrt{256^h}}=C_k4^{-h}.
\]
The means \(\mathbb EA_{k,h}\) are absolutely summable by the Taylor
argument. Consequently
\[
\mathbb E|W_k|
=\sum_{h\ge0}\mathbb E|\overline A_{k,h}|<\infty,
\qquad
\mathbb EW_k=\sum_{h\ge0}\mathbb EA_{k,h}=a_k.
\]
All priors are fixed and every experiment uses finitely many finite-size
Gaussian systems almost surely. Neither a limiting random input nor a
choice of subsequence enters the specification.
\end{proof}
==== END The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law//sections/04-reduction.tex ====
==== BEGIN The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law//sections/05-inverse.tex ====
\section{Inverting the scalar pair kernel}
\label{sec:inverse}
The cavity equations couple errors carried by different pairs of
leaves. To solve them, we need an inverse whose bound survives refinement
of the hierarchy and the small masses used in the comparison. The
proposition below provides that bound for arbitrary fixed old trees;
its proof keeps the signed cancellations before taking absolute values.
A pair function assigns a deterministic number to each pair and each
allowed extension of the fixed old-label tree. In the applications,
that number is a positive-law expectation of an overlap test. The
operator sums below vary the extended topology using signed allocation
coefficients; the scalar expectations in its kernel use the positive
law on each such topology.
For a finite resolution with terminal mass one, choose mass cuts
\(00,\qquad \chi_r=V_{zz}(K_r,Z_{K_r}).
\]
Bounds called polynomial in this section are in \(1+K_{\rm top}+\|b\|_\infty+\operatorname{var}b+e^{-1}\), independent of the number of steps. On the retained levels
\[
u_{ab}\ \longmapsto\ u_{ab}-\sum_{cd\ {\rm retained}}{\bf M}_{ab,cd} b_{cd}u_{cd}
\tag{H1}\label{eq:H1}
\]
has an inverse on each finite step resolution. Each output is an integral of inputs over allocations of up to two more labels (some arguments can also be reused), with weights dominated by a polynomial times the ordinary \emph{absolute} unrestricted attachment rules to the old tree including \(ab\). Restrictions and bounded weights can depend on the topology. This formulation in particular controls new split densities, allowing coincidences and ties to existing forks. It applies with \(ab\) using old labels too. We only need symmetric arrays (on antisymmetric ones the subtracted sum is zero).
\end{proposition}
The proof separates the scalar calculation from the construction of the
inverse. We first resolve pair arrays into two sectors and state the
scalar response identities they require. With those identities in hand,
we construct the inverse and prove its attachment bound. The detailed
scalar verification follows afterward.
\subsection{Pair resolutions and scalar response}
Initially keep the incoming root power \(p_0>0\) and the clocks
strictly ordered. We will remove both restrictions after obtaining
bounds independent of the resolution.
The scalar time profile is \(\alpha=p_i\) on \((K_{i-1},K_i)\), where
\(K_{-1}=0\). Add a mass-one terminal interval ending at
\(K_{R+1}\); it leaves all earlier field derivatives unchanged.
For a function of a leaf, \(P_k\) replaces that leaf by a signed
allocation within its \(p_k\)-block and divides the result by \(p_k\).
The signed row-sum rule therefore gives \(P_k1=1\), and
\(P_{R+1}=I\). Put
\(E_0=P_0\) and \(E_k=P_k-P_{k-1}\) for \(k\ge1\).
These differences separate successive block resolutions. When a pair
lies on sheet \(t\), resolve each endpoint only within its own child:
the coarsest operator is then
\(\widetilde E^t_{t+1}=P_{t+1}\), followed by
\(\widetilde E^t_i=E_i\) for \(i>t+1\).
\paragraph{The additive and row-zero sectors}
For the counting calculation alone, treat the symbols \(p_i\) as integer
block sizes with sufficiently large integer ratios \(p_i/p_{i+1}\).
Thus this temporary integer tree has decreasing block sizes; the
increasing real masses are substituted only after the counting
identities have been proved. On the integer tree use symmetric
off-diagonal pair arrays, with inner product summed over unordered
pairs.
Nested block averages on this tree satisfy
\[
P_iP_j=P_{\min(i,j)},\qquad
E_kE_l=\mathbf1_{\{k=l\}}E_k,\qquad \sum_{k=0}^{R+1}E_k=I.
\]
The coefficient argument below transfers these identities to real masses.
Fix a leaf function \(\epsilon\) satisfying \(E_k\epsilon=\epsilon\).
A \emph{sheet-additive} array has value
\(f_t(\epsilon_a+\epsilon_b)\) when \(ab\) lies on sheet \(t\).
The coefficients \(f_t\) may vary with the sheet. Summing over the
second leaf in a row on sheet \(t\) gives
\(d_t^k f_t\epsilon_a\), where
\[
d_t^k=p_t-p_{t+1}+p_t{\bf1}_{k\le t}-p_{t+1}{\bf1}_{k\le t+1},\qquad \mu_t^k=-d_t^k .
\]
On this integer tree, its squared norm divided by
\(\sum_a\epsilon_a^2\) is \(\sum_t d_t^k f_t^2\).
The kernel \({\bf M}\) preserves the arrays of this form and acts on
the level coefficients \(f_t\) by a matrix independent of
\(\epsilon\). To see this, fix the output pair \(ab\). Tree symmetry
makes the sum of a one-leaf input constant on each block away from the
paths to \(a\) and \(b\). The resulting leaf operators are block
projections along those two paths. On the range of \(E_k\), each such
projection is either zero or the identity, so the answer is a multiple
of \(\epsilon_a+\epsilon_b\).
For a general pair array, write
\((R_tu)_a=\sum_{b:\,ab\ {\rm on\ sheet}\ t}u_{ab}\).
The additive projection is explicitly
\[
(\Pi_{\rm add}u)_{ab}\big|_t
=\sum_k\frac{(E_kR_tu)_a+(E_kR_tu)_b}{d_t^k}.
\]
Subtracting this projection leaves an array whose row sum on each
sheet is zero. We call this the \emph{row-zero} part. For a
row based at a leaf in a \(p_t\)-block, the sum is over all other
children of that block together; it need not vanish separately in each
child.
\paragraph{Real masses and arbitrary old-label topologies}
The transfer to real masses concerns coefficients of moment expressions,
not Gaussian expectations as functions of an integer replica count.
Introduce a formal variable for each pair or four-leaf moment, recording
all leaf coincidences and overlap levels. Include products of pair
moments as separate expressions. Also leave every input value as a
formal variable indexed by its complete genealogy with the old labels.
After expanding a block projection or kernel action, the coefficient of
each such expression is a rational function of the \(p_i\).
The counts in these coefficients are falling powers: when adding a leaf,
one either uses an occupied child or chooses a new child. These are
exactly the coefficients in the real-power rule of PE, Section~2.
To verify a coefficient identity, clear denominators and regard the
branching ratios as independent variables. Integer counting shows that
the resulting polynomial vanishes for every sufficiently large integer
choice of these variables, also if even ratios are required. Applying
the one-variable polynomial identity successively in each variable
shows that the polynomial is identically zero. We may therefore
substitute the increasing real masses.
In particular, the additive projection is obtained by taking a row sum
on sheet \(t\), applying \(E_k\), dividing by \(d_t^k\), and adding
the result at the two endpoints. The denominators are nonzero for
strictly increasing real masses. Subtracting these projections gives
the stated row-zero remainder. Because the input variables retain
their entire old-label genealogy, these identities apply to arbitrary
pair functions in the proposition, not just sheet-constant functions.
The signed real-mass calculation uses these coefficient identities;
it makes no assertion about a vector space of negative dimension.
The decomposition reduces inversion to two scalar calculations. On the
row-zero sector, the kernel is diagonal in the two child resolutions.
On the additive sector, it is a diagonal operator minus a positive
Hessian kernel integrated against sheet masses. The next lemma gives
both actions and the estimates that will cancel block denominators.
\begin{lemma}[Scalar response on the two sectors]
\label{inv:sector-response}
Use the finite resolution above, and put
\(m_t=V_z(K_t,Z_{K_t})\), \(\chi_t=V_{zz}(K_t,Z_{K_t})\).
Let \(\mathbf E_t\) condition on the scalar prefix through \(K_t\).
On the row-zero sector, different sheets do not mix. On sheet \(r\),
the multiplier in the child resolutions
\(\widetilde E_i^r\otimes\widetilde E_l^r\), \(i,l\ge r+1\), is
\[
g_r^{il}=\mathbf E[(\mathbf E_r\chi_{i-1})
(\mathbf E_r\chi_{l-1})].
\]
On the additive sector associated with \(E_k\), the action on
sheet coefficients is
\[
({\bf M}^k f)_r=g_r^k f_r-\sum_t B^k_{rt}\mu_t^k f_t,\qquad
g_t^k=\mathbf E[\chi_t\chi_{\max(k-1,t)}].
\tag{H2}\label{eq:H2}
\]
Here \(A=K_{k-1}\), with \(A=0\) for \(k=0\), and
\[
B^k_{rt}=\mathbf E\left[
\operatorname{Hess}_{z,\alpha}V(A,Z_A)[\beta_r,\beta_t]\right],
\qquad
\beta_t=\begin{cases}
(m_t,0),&t0\), the first derivatives from the two
opposite colors cancel pointwise before the fork. The second derivative
of the normalized root value is consequently
\[
\mathbf E[\ddot V+2\mathcal Z\dot V_z+\mathcal Z^2\chi] .
\]
For \(k=0\) it is the root continuation derivative with \(\mathcal Z=0\).
Set \(\eta=2\sum_{t\ge k}s_tf_t\delta_{K_t}\), \(J=D_\alpha V_A[\eta]=\sum_{t\ge k}s_tf_t\mathbf E_A m_t^2\). Differentiating the moved times gives (at fixed starting field)
\[
\begin{split}
\dot V&= f_{R+1}-F(\chi+p u^2)-J,\\
\ddot V&=G_{R+1}-G_{k-1}(\chi+p u^2)-\sum_{t\ge k}s_t G_t\mathbf E_A m_t^2
-2\sum_{t\ge k}s_t f_t^2\mathbf E_A\chi_t^2\\
&\qquad+4F^2\partial_{AA} V-4F\partial_A J+
\operatorname{Hess}_{\alpha\alpha}V_A[\eta,\eta].
\end{split}
\]
To account for the terms, moving a step location changes the profile by
minus its jump times the displacement. Its first response is
\(\mathbf E_A m_t^2/2\), whose time derivative is
\(\mathbf E_A\chi_t^2/2\). These give \(-J\) in the first line and
the two sums over \(t\ge k\) in the second. The remaining terms
differentiate the starting time, the annealed terminal constant, and
the profile twice.
The impulses can also be obtained by integrating over the short
intervals swept out by the moving steps. The Hessian kernels below are
continuous when the impulse times meet. At fixed bounded clocks, the
continuation and its needed derivatives are continuous under
\(L^1\) profile perturbations, by their bounded-coefficient diffusion
equations. This justifies the impulse limit. Extend the initial
constant-mass interval slightly when differentiating the starting time;
strict clock separation keeps all other steps outside that interval.
The prefix is sampled under its normalized positive law, rather than
under the original Gaussian measure. Multiplying its successive density
factors gives a telescoping exponential. Gaussian integration by parts
therefore differentiates both the endpoint test and this density. For
\(Y\) depending on the endpoint field, the first two identities are
\[
\begin{split}
\mathbf E[\mathcal ZY]&=\mathbf E(FY_z+HY),\\
\mathbf E[(\mathcal Z^2-G_{k-1})Y]&=\mathbf E[F^2Y_{zz}+2FHY_z+(H^2+N')Y],\\
H&=pF u-\sum_{t1\) and the exponent choices of Section~\ref{sec:paths}.
For \(m\) in the stated dyad, the following bound is uniform over the
allowed \(\lambda,\ell,\gamma\). There is a saving \(c_G>0\)
independent of the allowed \(p_*\) and of \(s_*\) sufficiently close
to \(1/2\), such that for all sufficiently large \(N\),
\[
0\le F_*-f_m^0\le x^{4+c_G},\qquad
F_*= f_{\rm spin}(t)+\tfrac14\int bq^2\,dv . \tag{G1}\label{eq:G1}
\]
Here \(f_m^0=f_m(b,h_*)\), and \(f_{\rm spin}(t)\) is the
one-site scalar pressure with the same terminal mass \(\ell\).
The size threshold may depend on the fixed exponent choices.
\end{proposition}
\begin{proof}
Replacing the matrix by the field \(bq\) gives a nonnegative
square-error interpolation, which proves the first inequality.
Throughout the proof normalize \(\ell\) to 1 as in \eqref{eq:A1}:
multiply the pressure by \(\ell\), the clocks by \(\ell^2\), and use
mass \(v/\ell\). The size rescaling is held fixed; below \(\theta\)
denotes a new interpolation parameter. We will remove the matrix in a
constrained interpolation. The next two inputs control its endpoints:
PE bounds the cost of an initial small removal, while scalar coercivity
bounds the endpoint at which the matrix has been removed completely.
\paragraph{The finite-propagation comparison used below}
We use PE, Proposition~6.6, in its ordinary all-mass case
\(I=[0,1]\) (PE, Definition~6.1 and equations~(3)--(6)). Its
parameters are a small fixed power slack \(m_{\rm PE}>0\),
\(\delta_{\rm PE}=m_{\rm PE}/10000\), and scales satisfying
\[
01\), with \(T_m\) in a fixed compact subset
of \((1,\infty)\) and held fixed throughout each parent/child tree
at size \(m\). The scale restrictions and propagation slack stay
the same. This sequential extension is proved in
\cite[Appendix: Companion interfaces, paragraph
``A common prelimit temperature'']{PE}.
We recall its argument to specify how it applies here without a rate
of convergence for \(T_m\).
First, the constraints and conditional moment estimates in PE,
Section~3 use bounded clocks and fixed comparability constants;
its endpoint Taylor hypothesis enters only the subsequent profile
argument. At the regular calibration cut in PE, Proposition~4.1,
use \(b_*=T_m+O(W^2)\), \(K_*=T_mS_*+O(W^3)\), and set
\(J_0=T_m\chi^2-1\) in PE~(26)--(27). The proof divides by
\(\chi K_*\asymp W\) and absorbs
\(\eta_n|J_0|x_{\rm test}^2/W^2\). These operations use the actual
\(T_m\), so they introduce no \(T_m-T\) error; the conclusion
PE~(22) likewise uses the actual scalar clock.
In the small-scale proof in PE, Section~5 take time \(K/T_m\), subtract \(K/T_m\) in the scalar compactness and uniform scalar linearization lemmas, and put \(((b_0-T_m)S+h_0)/x_{\rm test}^3\) for the endpoint remainder at test scale \(x_{\rm test}\). All prelimit identities and bounds there are then identical (scalar derivative and strict curvature bounds use only compact temperature and total time bounds); in the rescaled ODE/potential limits \(T_m\) converges to \(T\).
In the macroscopic direct result (PE, Section~5, Proposition~5.9) the actual endpoint convergence and limiting variational problem are at \(T\), so its fixed-temperature scalar-minimizer input needs no uniform version. This gives the same positive upper/lower bounds on limiting slopes for the profile contradictions.
Child data, covariance margins (choose with room at the limit), and moving barriers in PE, Section~6 then use PE~(4) with that common \(T_m\), and the fixed finite depth subsequence proof applies unchanged. Thus the parameter replacement invokes no theorem uniform up to the critical temperature or change of the fixed propagation slack.
\paragraph{Scalar coercivity}
We first establish a coercive comparison. If \(K'=h_*+b z\) is an admissible scalar path with \(0\le z\le C\), then
\[
f_{\rm spin}(K')+\tfrac14\int b z^2
\ \ge F_*+c\int_0^{Q_0}\left(\int_s^{Q_0}(\alpha_{K'}-\alpha_t)\right)^2 ds ,
\tag{G2}\label{eq:G2}
\]
where \(Q_0\) is a fixed clock bound, after annealed extension of
the time distributions. Use the base optimal feedback martingale
\(u_s=V_z(s,Z_s)\) as control in both scalar problems, as verified
in PE, Section~5. This gives the lower first variation
\(\frac12\int\Gamma(s)(\alpha_{K'}-\alpha_t)(s)ds\), together
with the terminal \(\log\cosh\) convexity remainder.
Write \(P(s)=\int_s^{Q_0}(\alpha_{K'}-\alpha_t)\) and
\(A=\int_0^{Q_0}P(s)^2ds\). The terminal displacement is
\(Y=\int(\alpha_{K'}-\alpha_t)u=\int P(s)\chi_s\,dB_s\),
where \(\chi_s=V_{zz}(s,Z_s)\). On bounded clocks,
\(\mathbf E\chi_s^2\ge c\), while \(\chi_s\) is bounded, so
\(\mathbf EY^2\ge cA\) and \(\mathbf EY^4\le CA^2\).
Also \(Y\) is deterministically bounded from its original integral.
Choose a fixed compact set for the base terminal argument with
complement probability at most \(c^2/(4C)\). Cauchy--Schwarz leaves
at least \(cA/2\) of \(\mathbf EY^2\) on this set. The second
derivative of \(\log\cosh\) is bounded below along every displaced
segment there. Its convexity remainder is therefore at least \(c'A\),
as required in \eqref{eq:G2}.
The first variation plus penalty change cancels linearly, leaving
\(\tfrac12\int dv\, b(z-q)^2\int_0^1(1-y)[1-b\Gamma'(t+y b(z-q))]dy\ge0\) by \eqref{eq:P5}, without needing monotonicity of \(z\). The lower bound on \(\mathbf E\chi_s^2\) here follows also by comparison to the base \(T\) scalar path on a fixed compact including its annealed extension as in \eqref{eq:P2}.
\paragraph{A constrained interpolation}
Choose \(e_0=x^{d_g}\), where the small exponent \(d_g>0\) will
be fixed below, and set \(d_{\rm PE}=e_0^{1/4}\). Apply PE with
\(d=d_{\rm PE}\), strength \(e_0\), and error parameter
\(E=x^{4-d_g/16}d_{\rm PE}=x^{4+3d_g/16}\). Taking its fixed
power slack sufficiently small relative to \(d_g\) satisfies
PE~(3) and makes \(m^{20\delta_{\rm PE}}E\le x^{4+c}\).
The endpoint scale is \(a_{\rm PE}\asymp x^{1-d_g/16}\gtrsim x\).
Also \(e_0/d_{\rm PE}^3\to0\), so the taper is identically 1:
this comparison removes the constant \(e_0\) and inserts \(e_0r\).
Equations~\eqref{eq:P2}--\eqref{eq:P4} and \eqref{eq:P9} verify
the endpoint and admissibility hypotheses with the common base
\(T_m\to T\) just discussed. The optimized value, including
\(\int e_0r^2/4\), is consequently within \(x^{4+c}\) of the
bare pressure.
We can pass to onto inverse paths \(\alpha:[0,2]\to[0,1]\) with constant bounds
\(\kappa=x^{4+1/4}\le\alpha'\le M_c=x^{-10}\), at upper-cost loss \(O(\kappa+1/M_c)\). Indeed any radius law on \([0,1]\) can be approximated in first transport moment to that accuracy, by a slight bounded-density spreading within the enlarged interval and mixing a uniform floor. Cost follows by \eqref{eq:A1}. Keep these latter constraints fixed and minimize
\[
\Phi(\theta)=\min_r\{f_m(b-e,h_*+e r)+\textstyle\int e r^2/4\},\qquad
e=(1-\theta)e_0+\theta b,\quad 0\le\theta\le1 .
\]
The density box is compact at fixed size, so a minimizer exists.
Equation~\eqref{eq:A1} gives monotonicity and Lipschitz continuity
of \(\Phi\), as well as its envelope upper derivative at a
minimizer. All paths are admissible. At the two endpoints,
\(\Phi(0)\le f_m(b,h_*)+x^{4+c}\) by the PE comparison and
\(\Phi(1)\ge F_*\) by \eqref{eq:G2}. At every intermediate
\(\theta\), trying a constrained approximation to \(q\) and
replacing the remaining matrix by the field \((b-e)q\) gives
\(\Phi(\theta)\le F_*+O(\kappa+M_c^{-1})\).
Fix \(\theta\) and a minimizing \(r\). The objective is an averaged
pressure, so this choice, its inverse \(\alpha\), and all intervals
chosen from them are deterministic before taking the positive-law
expectation. Evaluate \(S,B,D\) at this changed path. A subscript in
radius \(s\) means the cut at mass \(\alpha(s)\). Put
\[
\widetilde b=b-e,\qquad
K^{\rm emp}=h_*+er+\widetilde bS.
\]
This empirical scalar clock replaces each remaining matrix overlap by
its mean \(S\). It is nonnegative and nondecreasing.
It remains to show that \(\Phi\) increases by at most
\(O(x^{4+c})\). Its envelope derivative is bounded by the integral
of \(D^2+(S-r)^2\). We first use the minimizing density constraints
to control \(S-r\), then compare \(K^{\rm emp}\) to \(t\).
Deterministic radius buffers will give high moments of \(Q-S\);
the pair inverse will improve these to the required integrated bound
on \(D^2\). The final envelope estimate combines these four steps.
First variation gives a potential \(H\) with
\(H'(s)=e(\alpha(s))(s-S_{\alpha(s)})\). Indeed the variation against
\(\delta\alpha\) is \(-\int H'\delta\alpha/2\), by changing variables
in \eqref{eq:A1}. The inverse is bi-Lipschitz at fixed size, and the
required path moments are continuous under these perturbations by PE,
Section~2. After integration by parts this is a linear functional of
the density variation. Exchanging density between two locations, while
keeping total mass fixed, gives a constant shift of \(H\) for which
\(\alpha'=\kappa\) on \(H>0\) and \(\alpha'=M_c\) on \(H<0\).
A component of \(H<0\) has radius width at most \(M_c^{-1}\).
Its endpoint signs and monotonicity put \(S\) between its two radius
endpoints; at exterior boundaries use \(0\le S\le1\). On \(H=0\),
\(S=s\) almost everywhere. We call the complement of the positive
components \emph{nonfloor}. Thus \(|S-s|\le M_c^{-1}\) there, while
the floor has total mass at most \(2\kappa\). Integrating \(H'\) on
a negative component also gives \(H\ge-C(e_0+\theta)M_c^{-2}\).
We will use the following local consequence. Within distance \(O(h)\)
of a nonfloor point \(s_0\), where \(M_c^{-1}\ll h\),
\(S=s_0+O(h)\), provided one further interval of length \(h\) fits
on each required side. For if \(S\) exceeded this range, monotonicity
would keep \(s-S\) of one sign and of size \(\gtrsim h\) on such
an interval. Since \(e\asymp e_0+\theta\), integrating \(H'\) to
or from \(s_0\), where \(H(s_0)\le0\), would force a negative
value of size \(\gtrsim(e_0+\theta)h^2\). This contradicts the
lower bound for \(H\).
\paragraph{A quantitative coarse clock comparison}
The empirical scalar clock is a full mass path. Its time distribution
will be denoted by \(\alpha_{K^{\rm emp}}\), with field time as its
argument. We first need the coarse estimate
\[
\|\alpha_{K^{\rm emp}}-\alpha_t\|_1\le x^{c_1}, \tag{G3}\label{eq:G3}
\]
where \(c_1>0\) is independent of sufficiently small \(d_g\). Here and below these are scalar time profiles, not optimizer inverse paths. There is a monotone \(0\le z\le1\) with
\[
f_{\rm spin}(h_*+e r+\widetilde b z)+\int\widetilde b z^2/4
\le f_m(\widetilde b,h_*+e r)+x^{c_0}
\]
for some absolute \(c_0>0\). If \(\sup\widetilde b\lesssim x^{1/10}\),
take \(z=0\) and use the pressure Lipschitz bound in \eqref{eq:A1}.
Otherwise \(\widetilde b=(1-\theta)(b-e_0)\) has
\(b_*:=\inf\widetilde b\gtrsim x^{1/10}\), since \(b-e_0\asymp1\).
We use the quantitative bounds in the proof of PE, Section~5,
Lemma~5.7, as follows.
In that scalar-trial interpolation, denote its interpolation time by
\(\tau\), its inverse floor and cap by \(\eta_m=m^{-1/8}\) and
\(Q_m=m^{1/8}\), and its optimizer by \(z_\tau\). In this paragraph
the observables refer to this inner optimizer, with clocks
\(((1-\tau)\widetilde b,h_*+er+\tau\widetilde b z_\tau)\).
The mismatch
satisfies \(\int(S-z_\tau)^2dv\le\eta_m+Q_m^{-2}\). For
\(\tau\ge\varepsilon\), after also discarding masses below
\(\varepsilon\), the covariance trace satisfies
\[
\int_\varepsilon^1\mathbf E\operatorname{tr}C_v^2dv
\le R:=\frac{Q_m}{m b_*\varepsilon^3},
\]
and the variance of an unnormalized \(L\)-interval integral is at
most \(C Q_m/(m b_*\varepsilon^2)\). Partition the mass interval
where \(S\) varies by at most \(\Delta\), and discard end strips
of width \(\rho\). There are at most \(C(1+\Delta^{-1})\) strips.
The omitted masses and times therefore cost
\(O(\varepsilon+\rho(1+\Delta^{-1}))\). On every retained mass,
the earlier and later \(L\)-averages give
\[
\bigl\||X_v|^2-S_v\bigr\|_2
\le C\Delta+\frac C\rho
\left(\frac{Q_m}{m b_*\varepsilon^2}\right)^{1/2}=:A.
\]
Finally use
\(D_v^2=\operatorname{Var}(|X_v|^2)+\mathbf E\operatorname{tr}C_v^2
+2\mathbf E X_v^{\mathsf T}C_vX_v\), with the last term bounded
in integrated mass by \(2R^{1/2}\). The integrated envelope derivative
is consequently at most
\[
C\{\eta_m+Q_m^{-2}+\varepsilon+\rho(1+\Delta^{-1})
+A^2+R+R^{1/2}\}.
\]
With \(\varepsilon=\Delta=x^{1/100}\) and \(\rho=x^{2/100}\),
each term saves a fixed power of \(x\): for example,
\(R\lesssim x^{128/25}\) and
\(A\lesssim x^{1/100}+x^{509/200}\).
These estimates use only bounded monotone actual clocks and the lower
bound on \(b_*\), so they are uniform over our minimizing \(r\).
The scalar endpoint of this interpolation supplies the asserted trial.
We now return to the observables of the outer minimizing \(r\).
Put \(K'=h_*+er+\widetilde b z\) and
\(\bar z=(er+\widetilde b z)/b\). The exact penalty decomposition is
\[
er^2+\widetilde b z^2
=b\bar z^2+\frac{e\widetilde b}{b}(r-z)^2.
\]
Apply \eqref{eq:G2} to \(K'=h_*+b\bar z\). The trial inequality and
\(\Phi(\theta)\le F_*+O(\kappa+M_c^{-1})\) then bound both
the squared primitive in \eqref{eq:G2} and
\(\int e\widetilde b(r-z)^2dv\) by \(O(x^{c_0})\), after
decreasing \(c_0\).
To pass from the primitive to the profiles, write
\(g=\alpha_{K'}-\alpha_t\) and \(P(s)=\int_s^{Q_0}g\).
The total variation of \(g\) is bounded. Averaging over intervals
of length \(a\) gives
\(\|g\|_1\le C(a+a^{-1}\|P\|_1)\); boundary intervals have
the same \(O(a)\) cost. Since \(\|P\|_1\lesssim x^{c_0/2}\),
choosing \(a=x^{c_0/4}\) gives a power-small profile error.
Moreover the cap/floor mismatch and Cauchy--Schwarz give
\[
\int\widetilde b|S-z|dv
\lesssim\kappa+M_c^{-1}+e_0^{-1/2}x^{c_0/2}.
\]
For monotone clocks, the \(L^1\) distance between their quantiles
equals that between their time distributions. Thus this is also the
cost of replacing \(K'\) by \(K^{\rm emp}\). Choose \(d_g\)
sufficiently small relative to \(c_0\); a fixed smaller exponent
\(c_1>0\) now proves \eqref{eq:G3} independently of \(d_g\).
Decrease \(c_1\) once more, if needed, to cover the bounded-clock
scalar derivative comparison used at the end of the proof; all
remaining exponents are chosen after this choice.
\paragraph{Integrated pinned moments and rough concentration}
Choose \(\delta=x^{a_g}\) with \(02a_g\).
Thus these neighborhoods and all later cuts have mass
\(\gtrsim\delta\). They have room for the required buffers, since
\(s_0\le1+M_c^{-1}\), while the radius interval extends to 2.
For fixed \(p\ge2\), on a radius window \(I\) there (within \([0,5/4]\)) we have
\[
\left\|\int_I W\,ds\right\|_{p/2}\lesssim_{p,\circ} x^6/(e_0\delta^2). \tag{G4}\label{eq:G4}
\]
\paragraph{A reusable residual estimate.}
\label{par:global-block-estimates}
We isolate the deterministic blocking calculation also used in
Sections~\ref{sec:budgets} and~\ref{sec:statistics}. It applies to the full single-path
positive law and does not require an optimizing radius.
\begin{lemma}[Residual blocks and integrated increments]
\label{global:residual-blocks}
Fix a positive path law with terminal mass \(a\in[1,4]\), and
parametrize a deterministic increasing family of revealed cuts by \(s\). At a cut \(s\), let branches \(\mathsf a,\mathsf b\)
be conditionally independent continuations with their normalized positive
kernels, given the common prefix. Partition their futures into the same
\(k\) deterministic blocks, the last ending at the terminal spin.
Write \(\Delta_j\) for the martingale increments, so that
\(y-X_s=\sum_{j=1}^k\Delta_j\), and let \(\mu'_j\le a\)
be the largest mass in block \(j\). Evaluating \(W_s\) on branch
\(\mathsf a\), we have
\begin{equation}
W_s\le k\sum_{i,j}\mu'_j\,
\mathbf E_{\mathsf b}[(\Delta_i^{\mathsf a}\cdot\Delta_j^{\mathsf b})^2\mid\mathsf a].
\label{eq:global-block-decomposition}
\end{equation}
Conditioning on \(\mathsf a\) retains its whole branch and the common prefix;
only continuation \(\mathsf b\) is averaged. For any deterministic interval
\(J\), any increment length \(l>0\), and \(q\ge1\), provided all
indicated cuts exist, put \(s_-=\inf J\) and \(s_+=\sup J+2l\). Then
\begin{equation}
\left\|\int_J|X_{s+2l}-X_{s+l}|^2ds\right\|_q
\le C_q l\,\bigl\||X_{s_+}-X_{s_-}|^2\bigr\|_q .
\label{eq:global-grid}
\end{equation}
The same estimate holds with increment \(X_{s+l}-X_s\).
For any nonnegative process \(V_s\), not necessarily adapted,
\begin{equation}
\left\|\int\mathbf E_sV_s\,ds\right\|_q
\le C_q\left\|\int V_s\,ds\right\|_q .
\label{eq:global-dual-doob}
\end{equation}
\end{lemma}
\begin{proof}[Proof of the lemma]
The step Hessian rule writes \(H_s\) as the conditional sum of future
increment covariances weighted by their masses; the last term is the
terminal Gibbs covariance multiplied by \(a\). Conditional orthogonality
gives
\[
H_s\preceq\sum_j\mu'_j\mathbf E_s
\Delta_j\Delta_j^{\mathsf T}.
\]
Expanding only branch \(\mathsf a\)'s residual and applying Cauchy--Schwarz
gives \eqref{eq:global-block-decomposition}. In particular
\(H_s\preceq aC_s\). The fixed-size approximation in
\eqref{eq:A1} passes this statement to filled clocks.
For the integrated estimate, the shifted-grid identity is
\[
\int_J|X_{s+2l}-X_{s+l}|^2ds
=\int_0^l\sum_{j:\,u+jl\in J}
|X_{u+(j+2)l}-X_{u+(j+1)l}|^2du.
\]
At each fixed \(u\) the increments are disjoint. Apply the martingale
square-function and Doob inequalities to \(X-X_{s_-}\), and integrate
in \(u\), to obtain \eqref{eq:global-grid}; for \(q=1\) martingale
orthogonality suffices. The earlier increment has the same proof.
For \eqref{eq:global-dual-doob} with \(q>1\), pair with a nonnegative
\(Z\) of unit \(L^{q'}\) norm. Move the conditional expectation onto
\(Z\), bound \(\mathbf E_sZ\) by its supremum, and apply Doob and
H\"older. For \(q=1\), the two expectations are equal.
\end{proof}
\paragraph{Application to the integrated pinned moment.}
We return to the normalized terminal mass \(a=1\) and prove
\eqref{eq:G4}. Here \(H_s\), the source Hessian, is distinct from
the optimizer potential \(H(s)\). The lemma also applies after an
increment on branch \(\mathsf b\) is averaged back to its fork, by
\eqref{eq:global-dual-doob}. We use \(\mu'_j\le1\), or directly
\(H_s\preceq C_s\), giving
\[
W_s\le\mathbf E_{\mathsf b}[((y^{\mathsf a}-X_s)\cdot(y^{\mathsf b}-X_s))^2\mid\mathsf a].
\]
For the residual decomposition in \eqref{eq:G4}, fix a small constant
\(L_*>0\) with \(s+L_*<2\) for \(s\in I\). Choose
\(l_0=2^{-J_*}L_*\) of arbitrarily small inverse-polynomial size.
Use the initial block \(X_{s+l_0}-X_s\), the blocks
\(X_{s+2l}-X_{s+l}\) for \(l=l_0,2l_0,\ldots,L_*/2\), and
the terminal residual \(y-X_{s+L_*}\). There are
\(k=O(\log(1/x))\) blocks and their sum is exactly \(y-X_s\).
For any pair other than two initial blocks, project the block with
the later starting distance, denoted \(l_{\max}\). The anchor
\(A\) is the other branch's increment. Write
\(A=|A|\widehat A\), with \(\widehat A=0\) when \(A=0\),
and apply \eqref{eq:A3} to \(\widehat A\). Conditional on the
common prefix, the whole other branch is exterior data, so fixing
this anchor leaves the projected branch's normalized future kernel
unchanged. Equal starting distances may be resolved in either order.
The gap after the fork has field length at least \(e_0l_{\max}\)
and mass at least \(c\delta\). For a finite block, conditional
Jensen converts the residual estimate at its start into the same
estimate for its increment.
Fix a small \(\epsilon>0\). At a sufficiently high fixed even
moment, Markov's inequality gives
\[
\mathbf P\left\{
|\widehat A\cdot\Delta_{\rm later}|^2
>\frac{x^{6-\epsilon}}{e_0\delta^2l_{\max}}\right\}
=O(x^K)
\]
for any prescribed fixed \(K\). Indeed \eqref{eq:A3} gives the
same square scale with \(x^6\log^4N\); taking the moment order
large relative to \(K/\epsilon\) supplies this tail bound. The
factor \(|A|^2\) remains in the subsequent integral. On the
exceptional event the unnormalized squared product is bounded,
since \(|y|,|X|\le1\). Jensen before and after conditional averaging
shows that its integral has \(L^{p/2}\) norm \(O(x^{2K/p})\).
Choose \(K\) large enough to make this negligible.
By \eqref{eq:global-grid}--\eqref{eq:global-dual-doob}, the earlier
squared anchor integrated over \(I\) has \(L^{p/2}\) norm at most
\(C_p l_{\max}\). If both blocks are terminal, boundedness gives
the same assertion since \(l_{\max}=L_*\) is fixed. Thus each
projected pair contributes at most
\(C_p x^{6-\epsilon}/(e_0\delta^2)\). For two initial blocks,
bound one by a constant and integrate the other's square; the cost
is \(O_p(l_0)\), negligible by the choice of \(l_0\).
Summing the logarithmically many pairs in
\eqref{eq:global-block-decomposition} proves \eqref{eq:G4}.
\paragraph{Earlier and later buffers.}\label{par:global-brackets}
We next turn the integrated pinned estimate into control at individual
cuts. The following bracket argument also records what is needed when
the later applications use different radius windows.
\begin{lemma}[Buffer brackets and a common past]
\label{global:buffer-brackets}
Fix a full single-path positive law with terminal mass \(a\in[1,4]\)
and deterministic revealed cuts. Let \(w\) be one such cut, and let
\(\rho_-\), \(\rho_+\) be deterministic probability measures supported
on cuts before and after \(w\), respectively. Write
\[
A_-=\int L_v\,\rho_-(dv),\qquad
A_+=\int L_v\,\rho_+(dv).
\]
Suppose that, for a deterministic number \(\bar S\),
\[
|S_v-\bar S|\le\eta\quad
\text{on the two buffers},\qquad
\|A_\pm-\mathbf EA_\pm\|_p\le\varepsilon_p,
\qquad 2\le p<\infty.
\]
The bound on \(S_v\) is required only almost everywhere for the two
buffer measures. Then
\[
\bigl\||X_w|^2-\bar S\bigr\|_p+
\left\|\int|X_w-X_v|^2\rho_-(dv)\right\|_p
\le C(\eta+\varepsilon_p).
\]
If two cuts \(w,w'\) use the same earlier measure \(\rho_-\), supported
before both, and each has a later buffer satisfying the same hypotheses
with the same \(\bar S,\eta,\varepsilon_p\), then
\[
\bigl\||X_w-X_{w'}|^2\bigr\|_p\le C(\eta+\varepsilon_p).
\]
\end{lemma}
\begin{proof}[Proof of the lemma]
Since \(2\mathbf EL_v=S_v\), the buffer assumptions give
\(|2\mathbf EA_\pm-\bar S|\le\eta\). The martingale identities are
\[
\begin{split}
2\mathbf E_w A_-&=|X_w|^2-
\int|X_w-X_v|^2\rho_-(dv),\\
2\mathbf E_w A_+&=\int\mathbf E_w|X_v|^2\rho_+(dv)
\ge |X_w|^2.
\end{split}
\]
Thus \(|X_w|^2\) lies between \(2\mathbf E_wA_-\) and
\(2\mathbf E_wA_+\), each of which is within
\(\eta+2\varepsilon_p\) of \(\bar S\) in \(L^p\), by conditional
Jensen. The first identity then also bounds its nonnegative past
squared-increment integral. For two cuts, integrate
\[
|X_w-X_{w'}|^2\le
2|X_w-X_v|^2+2|X_{w'}-X_v|^2
\]
against their common earlier measure and use the two preceding bounds.
\end{proof}
Apply the lemma with normalized radius averages over buffers of length
comparable to \(h\), before and after cuts near \(s_0\). Their measures
are deterministic because the optimizer was fixed before sampling.
For an interval of radius length \(h\), the mass weight in
\eqref{eq:A1} is \(k(v)=1/[h\alpha'(r(v))]\). Consequently
\eqref{eq:G4} gives
\[
\|A_\pm-\mathbf EA_\pm\|_p
\lesssim_{p,\circ}\frac{x^3}{\sqrt{e_0}\delta h\sqrt\kappa}
=x^{7/8-d_g/2-3a_g}\,O_{p,\circ}(1)=O_p(h).
\]
The last estimate holds by taking the fixed exponents small before
choosing the subpower loss. The local potential bound gives
\(S_v=s_0+O(h)\) throughout these buffers. Lemma~\ref{global:buffer-brackets}
therefore applies with \(\bar S=s_0\), \(\eta=O(h)\), and
\(\varepsilon_p=O_p(h)\). It bounds both
\(\||X_w|^2-s_0\|_p\) and the averaged past squared increments by
\(O_p(h)\). Using the same earlier buffer for two nearby cuts gives
\(\||X_w-X_{w'}|^2\|_p=O_p(h)\).
Now take two branches splitting at \(s_0\), and project their
terminal spins to cuts a distance comparable to \(h\) later.
Their squared norms are \(s_0+O_{L^p}(h)\), and their increments
from the common ancestor have squared norms \(O_{L^p}(h)\).
Polarization therefore puts their projected inner product within
\(O_{L^p}(h)\) of \(s_0\). The two projection errors in
\eqref{eq:A3} are at most
\(O_{p,\circ}(x^3/(\delta\sqrt{e_0h}))\), so in particular
\(\|Q-S\|_p=O_p(\delta)\) at this nonfloor split.
For splits below radius \(\delta\), choose instead a nonfloor cut
at radius comparable to a sufficiently large fixed multiple of
\(\delta\). To find it, take a mass band of that scale.
Equation~\eqref{eq:G3} and monotonicity put \(K^{\rm emp}\)
between comparable multiples of \(\delta\) on an interior band;
the discarded floor has mass at most \(2\kappa\ll\delta\).
At a nonfloor point of this band, \(h_*=o(\delta)\) and
\(r=S+O(M_c^{-1})\), so
\(K^{\rm emp}=br+o(\delta)\) and \(r\asymp\delta\), as needed.
The same buffers give \(\||X|^2\|_p=O_p(\delta)\) there.
Project both branches to this cut, using its last comparable
subinterval for \eqref{eq:A3}; Cauchy--Schwarz bounds their projected
product by \(O_p(\delta)\). Since \(S\lesssim\delta\) at the
original cut, we obtain, after weakening the exponent,
\(\|Q-S\|_p\lesssim_p x^{a_g/2}\) at almost every nonfloor split,
for every fixed moment order.
\paragraph{Centered inversion and absorption}
At the sampled clock \(K^{\rm emp}\), the matrix multiplier
\(\widetilde b\) has the gap required in \eqref{eq:H1}, of size
at least \(ce_0\). To check this, \eqref{eq:G3} and the bounded-clock
PDE estimates compare scalar derivatives at a common time with error
\(O(x^{c_1})\).
For \(v\ge\delta\), monotonicity and the same area argument give
\(K^{\rm emp}(v)\gtrsim v\). The positive part of
\(b(v)-b_\circ\) is at most \(Cc_b\lambda v^2\), which is
absorbed by \eqref{eq:P2} when \(c_b\) is small. Below \(\delta\)
this positive part is \(o(e_0)\). Finally, replacing \(b\) by
\(b-e\) gains \(e\Gamma'\): wherever this is not comparable to
\(e_0\), \(\Gamma'\) is small and the original gap is bounded
below by a constant. The common-time error is \(o(e_0)\), so the
gap persists in both cases.
Fix the old pair \(ij\) and use \eqref{eq:A7} to first order about
\(K^{\rm emp}\), with test \(T_{ij}=Q_{ij}-S_{ij}\).
The row mismatch is exactly
\[
J=(h_*+er)+\widetilde b Q-K^{\rm emp}
=\widetilde b(Q-S).
\]
The test and the deterministic centers \(S\) remain fixed throughout
this cavity interpolation; the actual row and the independent scalar
row at \(K^{\rm emp}\) are its positive-semidefinite endpoints.
The unknown pair function is the deterministic, topology-indexed
expectation
\(u_{ab}=\nu[T_{ij}(Q_{ab}-S_{ab})]\).
It is unchanged on adding unused labels, by positive-law
marginalization. The scalar degree-zero term, including any
deterministic offset on a given input topology, vanishes because
\(\nu[T_{ij}]=0\). The linear terms give \eqref{eq:H1} with
multiplier \(\widetilde b\). Its errors are \(O(x^6)\) and
products of \(T_{ij}\) with at least two \(Q-S\) insertions.
Apply the inverse, evaluate at \(ab=ij\), and then integrate the
old fork mass. Absolute attachment domination bounds new fork
densities by a polynomial in \(e_0^{-1}\); a fork reused at \(ij\)
is now integrated as well. Truncate one error insertion at
\(x^{a_g/4}\). Its higher fixed moments, just proved off the floor,
make the exceptional probability \(O(x^6)\) by Markov's inequality,
and all factors are bounded there. To charge a floor placement,
follow its fork to the first time that vertex was introduced. It is
either the original \(ij\) fork, whose mass is now integrated, or a
fresh fork with bounded density. Later reuses retain the ordinary
bounded atom weights and introduce no new fixed mass. The floor mass
is \(O(\kappa)\), so the finite union of these placements also costs
\(O(\kappa)\) under the dominated allocation measure.
On the retained event this leaves \(x^{a_g/4}|T_{ij}(Q_{ab}-S_{ab})|\).
Cauchy--Schwarz bounds its expectation by
\(x^{a_g/4}D_{v_{ij}}D_{v_{ab}}\); the bounded fork measures and
\(2D_vD_w\le D_v^2+D_w^2\) give the integrated estimate
\[
\int D_v^2dv\le O(e_0^{-C})\big(\kappa+x^6+x^{a_g/4}\int D_v^2dv\big).
\]
For nonstep paths, perform this inversion first at fixed \(m\) on
step approximations, using each approximation's own \(S\) and
\(K^{\rm emp}\) in the cavity formula. Their moments converge at
almost every placement and in mass \(L^1\). The common-time scalar
comparison and monotone lower clock bound persist, so the sheet gap
remains positive when \(b,e\) are sampled from their paths.
Pass to the limit inside the dominated integrated error bounds, as
in the \hyperref[par:inverse-limits]{fixed-resolution limit argument}
of Proposition~\ref{inv:pair-inverse}.
Now choose \(d_g\) so small that \(Cd_g0\) independent of sufficiently small \(p_*\).
\end{proposition}\begin{proof}[Reduction to localized comparisons]
We first obtain the size-path bounds from the global comparison. Fix a
direct endpoint and an unseeded, unmasked path whose global parameter
is smaller by a fixed \(\delta>0\). To reach the endpoint, remove
\(\delta c_g\) from the matrix clock and add \(\delta c_gq+d\) to
the field clock. Both paths are admissible. Completing the square
along their affine interpolation and using \eqref{eq:G1} bounds
\[
R=f_m^0-\int bq^2/4+\int_{v\ge x}d q/2
\]
below by \(f_{\rm spin}(K)-O_\delta({\cal B})\); the corresponding upper bound follows by the trial replacing the remaining matrix to reach \(K\). Here, with subpower losses harmless,
\[
{\cal B}\lesssim_\circ x^{4+c_G}+
\lambda^{1+2\eta}U^5+\chi_e^2\lambda^{1+2\eta}U^4+\chi_{\rm lin}^2 x^6/\lambda.
\]
The lower comparison costs \(\int d^2/(\delta c_g)\), and the upper
comparison costs \(\int d^2/b\). Omitting the mask costs
\(O(\lambda U^2x^3)\). The linear restoration is independent of
\(\gamma\), so \eqref{eq:P10} can be integrated between the global
parameter endpoints. For some fixed \(b_s>0\), this gives
\[
\int(v^2+U^2)\,\mathbf E_{\gamma_0} A_v\,dv
\lesssim x^4\lambda^{-2s_0+b_s}
\]
also when the global interval of averaging is any fixed interior interval leaving crossing room. We can choose \(b_s,\tau_0\) sufficiently small relative to \(\eta\), with further room to absorb errors \(x^4u\lambda^{k_c/3-1-o(1)}\) on the right. At final strength with edge off this argument and \eqref{eq:P11} give both parts of \eqref{eq:C2}, with savings independent of further decreases in \(p_*\): even the differentiated estimate uses only \(O_\circ(x^{4+c_G}/\lambda+\lambda^{2\eta}U^5+x^6/\lambda^2)\) plus the switch cost from \eqref{eq:P11} divided by \(\lambda\).
For \eqref{eq:C1}, the remaining task is a direct-path dyad \(u\) of
\eqref{eq:P3} with \(U\lesssim u\le\lambda^{b_\ell}\), where
\(b_\ell>0\) is sufficiently small relative to \(b_s\). Bounded
overlaps already suffice when \(U\) is large. Fix all parameters
other than \(\gamma_0,\gamma_u\), and start at \(\gamma_u=0\).
At this endpoint we minimize over an auxiliary quantile \(r\): the
matrix removal is \(C(r)>0\), the field addition is \(C(r)r\),
and the penalty is \(\int C(r)r^2/4\). We will specify its density
constraints below. Writing \(X=x/u\), it is enough to prove the
following two assertions for the final constraints:
\begin{enumerate}[label=(\roman*)]
\item averaged excess of this optimized value over \(f_m^0\) at most \(O(x^4 u\lambda^{1-2s_0})\);
\item for the used optimizers, on \(\operatorname{supp}(c_u)\), \(c_u=-\partial_{\gamma_u}b\), one has \(C(r)\gg c_u\) and
\[
\mathbf E_{\gamma_0}\left[\sup |r-q|^2\right]
\lesssim (u X^2\lambda^{-s_0})^2 . \tag{C3}\label{eq:C3}
\]
\end{enumerate}
To see this, compare a minimizing auxiliary path to the actual endpoint
\(\gamma_u=1\). Starting at that actual path, the square interpolation
removes \(C(r)-c_u\) and adds \(C(r)r-c_uq\). The increase of the
restored pressure \(R\) between the two endpoints is therefore at most
the optimizer excess plus
\(\int C(r)c_u(r-q)^2/[4(C(r)-c_u)]\).
Averaging this bound controls the nonnegative derivative in
\eqref{eq:P10}; summing over dyads costs at most a logarithm.
Before differentiating on a smoothed size path, transfer the same
pressure difference by \eqref{eq:P11}. Its cost is \(O(x^{5+c})\),
since only the restoration \(-\int bq^2/4\) affects the difference.
Fixed rescalings of mass change these bounds only by bounded factors.
The second requirement is an estimate for every optimizer outside one
common null set of the global parameter. It is enough to construct a
measurable scalar majorant for all these optimizers; no choice of an
optimizer needs to be integrated. The remaining subsections establish
these two local assertions and hence complete \eqref{eq:C1}. Both
parts of \eqref{eq:C2}, with a saving independent of the local
strengthening depth, have already been proved above.
\end{proof}
\subsection{Local auxiliary comparisons and coarse profiles}
For the local argument normalize each direct path to length 1, reusing notation for its matrix and field coefficients multiplied by \(\ell^2\), the scaled scalar clock and moments, and writing \(v\) for normalized mass. Thus \(q(v)\) here is the old \(q(\ell v)\), and \(q\) and \(q^{-1}\) have bounded derivatives and positive slopes on fixed initial intervals.
All pressure bounds (with their corrections) change by bounded normalization factors. The scale \(u\) of \eqref{eq:C3} need not be rescaled: its relevant mass range is still contained in a fixed compact positive band of scale \(u\). Fix such a dyad, with \(\gamma_u=0\). In particular
\[
x\lambda^{-s_*/2}\lesssim u\le\lambda^{b_\ell},\quad X=x/u,\quad
\Lambda=\lambda^{k_c},\quad A=X^2\Lambda^{-1/3},\quad a_{\rm goal}=X^2\lambda^{-s_0}.
\tag{C4}\label{eq:C4}
\]
Thus \(U=x\lambda^{-s_*/2}\) for large sizes here. Use an arbitrarily small fixed \(\zeta>0\) (depending on previously required power margins including \(p_*,b_\ell\), decreased as needed). Apply \cite[Proposition~6.6]{PE} with
\[
d_{\rm PE}=c_d u\Lambda^{1/3},\qquad E=x^{4-\zeta}d_{\rm PE},\qquad
C(r)=C_0 f(C_0 r/d_{\rm PE}^3),\qquad C_0=u^2\Lambda ,
\]
i.e. at strength \(C_0\); \(f\) here is the PE taper of (6), constant 1 near 0. Choose the fixed \(c_d\) sufficiently large to have this constant profile on \([0,c_I u]\) for an arbitrarily large fixed \(c_I\), in particular well beyond the compact range of scale \(u\) where \eqref{eq:C3} will be tested.
PE power slack (its small parameter in (3), and \(\delta\)) is chosen sufficiently small relative to \(\zeta\). Write \(k(r)\) for the PE floor (5), \(v_*=(mE)^{-1}\asymp x^\zeta u A\). These parameters satisfy (3) since \(X\lesssim\lambda^{s_*/2}\), and \(C_0\le d_{\rm PE}^2=o(1)\). Conditions (4)--(6) with base \(T_m\to T>1\) hold by \eqref{eq:P3}--\eqref{eq:P9}; all masses are changed, with no radius wall, and a fixed positive covariance margin throughout. Use the common-base version explained at \eqref{eq:G1}. The optimized excess over the actual untilted pressure is thus \(\lesssim x^{4-2\zeta}u\Lambda^{1/3}\).
The modified optimizations below, at this same strength and with the same objective \(f_m(b-C(r),h+C(r)r)+\int C(r)r^2/4\), use inverse constraints on \([0,1]\)
\[
\bar k(r)\le\alpha'(r)\le M_j,\qquad
\bar k(r)=k(r)+\kappa\,{\bf1}_{r\in I},\qquad
0\le\kappa\lesssim\lambda^{2s_0}\Lambda^{-2/3}. \tag{C5}\label{eq:C5}
\]
Here \(I/u\) is a fixed closed subinterval of \((0,c_I)\), and
\(M_j=2^{j+1}x^{-100}\), with only finitely many fixed stages
\(j\ge-1\). The inverse distribution \(\alpha\) maps onto \([0,1]\);
its inverse \(r=\alpha^{-1}\) is the forward quantile. The initial
choice \(\kappa=0,\ j=-1\) contains the original PE class.
All these constraints are deterministic and independent of
\(\gamma_0\). Compactness and the segment continuity in
\eqref{eq:A1} give a minimizer. In the local argument, the symbols
\(S,B,D,X_v,W_v\) refer to the changed path of any such minimizer,
unless stated otherwise.
\begin{lemma}[Modified optimizer geometry]\label{bud:geometry}
For every minimizer in \eqref{eq:C5}, the first-variation potential can be normalized so that its derivative is
\[
{\cal H}'=C(r)\big[(2-p)r-2(1-p)S-p B/r\big],\qquad p=-r C'/C .
\tag{C6}\label{eq:C6}
\]
The minimizing density equals the cap $M_j$ where $\mathcal H<0$
and the floor $\bar k$ where $\mathcal H>0$; the remaining points are
contacts. Each cap has radius width at most $M_j^{-1}$. Outside the
floor, apart from a null set of contact exceptions,
$|S-r|\lesssim_\circ D+O(x^{90})$. At contacts where $C$ is constant, $S=r$; on every cap contained
in that constant region, $|S-r|\le M_j^{-1}$.
These conclusions are the parts of the geometry of
\cite[Lemma~3.3, equations~(9)--(10)]{PE} that will be used below.
\end{lemma}\begin{proof}
The first variation minimizes a linear functional of \(\alpha'\),
subject to its prescribed lower and upper bounds and total mass one.
The multiplier for total mass adds a constant to the primitive
\({\cal H}\). With this choice, the minimizing density equals
\(M_j\) where \({\cal H}<0\), and \(\bar k\) where
\({\cal H}>0\). These are the cap and floor regions, respectively;
the remaining points are contacts. Differentiation of inverse monotone
maps is justified as at \eqref{eq:G1}; the radii at which a jump is
crossed form a null set.
The monotonicities of \(S,B,p\), together with the zero initial wall,
give the endpoint signs, almost-everywhere contact relations, and the
cap bounds in PE (9)--(10). Each cap has radius width at most
\(M_j^{-1}\). We call cap and contact points \emph{nonfloor points},
discarding the null set of contact exceptions. There
\(|S-r|\lesssim_\circ D+O(x^{90})\); on a component in the constant
part of \(C\), the stronger bound is \(|S-r|\le M_j^{-1}\).
Also \(S\le r\) at almost every contact. On a negative cap, \(S\)
cannot exceed its right endpoint radius: otherwise \(B\ge S^2\)
and monotonicity make \({\cal H}'<0\) on a terminal portion of the
cap, contradicting its endpoint sign. At the outer endpoint the same
argument uses \(S\le1\).
\end{proof}
Put
\[
y_c=x^{-10\zeta}\big(X\Lambda^{-1/6}+X^{3/4}\lambda^{0.12}\big),\qquad w_c=y_c u .
\]
Here and in this local argument \(\delta\) on a PE power denotes \(\delta_{\rm PE}\) as recalled at \eqref{eq:G1}. In particular \(a_{\rm PE}\asymp x^{1-\zeta}\); \(w_c/u\ll\sqrt\Lambda\), \(x/w_c\) is power small, and
\(w_c\ge m^\delta\sqrt m E/\sqrt{C_0}\). Indeed \(X\lesssim\lambda^{s_*/2},\ \lambda\le x^{p_*}\) allow the prefactor \(x^{-10\zeta}\) in the first assertion if \(\zeta\) is small enough. Also \(C(w_c)=C_0,\ C_0 w_c^2\gtrsim x^{-18\zeta}mE^2\).
\Needspace{4\baselineskip}
\begin{proposition}[Comparability and the calibrated initial slope]\label{bud:coarse-profile}
For every optimizer in \eqref{eq:C5}, uniformly,
\[
r(v)\asymp v\ (w_c\le v\le1),\qquad
\sup_{c u\le v\le C u}|r(v)-q(v)|=o(u) \tag{C7}\label{eq:C7}
\]
for each fixed \(00\) and \(|X_t|\le1\).
We will bound it uniformly, without assuming an upper bound on the
mass of a cap. On a regular small radius band \(J\asymp v\), we claim
\[
\left\|\int_J W_{\alpha(r)}\,dr\right\|_l
\lesssim_{l,\circ} Q_{\rm raw}^2 x^6/C(v). \tag{C9}\label{eq:C9}
\]
Fork an independent continuation at each radius \(r\). Decompose both
residuals into future martingale increments, using the weighted Hessian
representation in Lemma~\ref{global:residual-blocks}. Retaining
its mass weight is necessary here. First use dyadic distances from a
sufficiently small inverse power of size up to a fixed fraction of
\(v\). Then use doubling scales up to a fixed positive radius and the
terminal layer. For a pair of layers with largest scale \(z\gtrsim v\),
the Hessian weight is \(O(z)\); for two short layers it is \(O(v)\).
For short layers, let \(h'\) be the starting distance of the later one.
Unless both are the first increment, \eqref{eq:A3} bounds the squared
product by the earlier squared anchor times
\(x^{6-\epsilon}/(v^2C(v)h')\), apart from a negligible tail.
Here one first uses arbitrarily high fixed moments for the normalized
anchor, then truncates; boundedness of the actual increments controls
the discarded tail. The shifted-grid and dual conditional-expectation
estimates used for \eqref{eq:G4} bound the integral of the earlier
squared increment by \(O_l(h'Q_{\rm raw}^2v)\) in \(L^l\).
Thus the three factors in a short-layer contribution are
\[
\underbrace{v}_{\text{Hessian weight}}\,
\underbrace{\frac{x^{6-\epsilon}}{v^2C(v)h'}}_{\text{projection}}\,
\underbrace{h'Q_{\rm raw}^2v}_{\text{integrated anchor}}
=\frac{x^{6-\epsilon}Q_{\rm raw}^2}{C(v)}.
\]
Two first increments are negligible by the square-sum estimate and the
choice of their inverse-power length.
For a larger layer of scale \(z\), there is a preceding regular
interval of length and mass comparable to \(z\). Projection costs
\(x^{6-\epsilon}/(z^3C(z))\), and the squared opposite anchor has
\(L^l\) norm \(O_l(Q_{\rm raw}^2z)\). Its integrated contribution is
therefore at most
\[
z\,\frac{x^{6-\epsilon}}{z^3C(z)}\,
Q_{\rm raw}^2z\,|J|
=\frac{x^{6-\epsilon}Q_{\rm raw}^2|J|}{zC(z)}
\lesssim\frac{x^{6-\epsilon}Q_{\rm raw}^2}{C(v)}.
\]
The last step uses \(|J|=O(v)\) and \(zC(z)\gtrsim vC(v)\),
also at the fixed outer cutoff. Summing the logarithmically many layer
pairs proves \eqref{eq:C9}.
By \eqref{eq:A1} and the floor lower bound, the centered average of
\(L\) over a future radius window of length comparable to \(v\)
has \(L^{2l}\) norm at most \(\varepsilon_m vQ_{\rm raw}\), where
\(\varepsilon_m\to0\) with a power saving. At scale \(v\), its bound
is \(O_\circ([mC(v)k(v)v^4]^{-1/2})\); the polynomial divergence
proved above absorbs the subpower loss. Conditional expectation
back to an earlier cut bounds \(|X|^2/2\) by this average. Its
deterministic mean is \(O(v)\). For a cut inside the nonregular region,
choose the window at a fixed larger regular scale comparable to \(w\);
for a regular small cut, choose it at a fixed multiple of that cut's
scale. Macroscopic cuts use \(|X|\le1\). Taking the supremum over
these deterministic cuts gives
\[
Q_{\rm raw}^2\le C_l+\varepsilon_m Q_{\rm raw}.
\]
Finiteness of \(Q_{\rm raw}\) now allows absorption and proves its
uniform bound. For a small fork, project both branches into the same
type of later window and telescope the remaining synchronized layers,
as in PE (16). If \(v\) is the window scale, the residual is
\(O_{p,\circ}(x^3/(v\sqrt{C(v)}))\ll v\). This proves
\eqref{eq:C8}, and the PE (16) postprojection bounds whenever the
required preceding regular intervals fit.
\end{proof}
\Needspace{4\baselineskip}
\begin{lemma}[Second moments with an enhanced floor]\label{bud:outer-second}
Under the same outer-regularity hypotheses, on regular mass dyads, with
$B_f=X^{3/2}\lambda^{s_0/2}\Lambda^{-5/12}$,
\[
(\operatorname{avg}D^2)^{1/2}
\lesssim_\circ (x^6/(C(v)v))^{1/3}+x^3/(v\sqrt{C(v)})
+m^{\delta/2}v_*+{\bf1}_{v\asymp u}\,u B_f . \tag{C10}\label{eq:C10}
\]
The average is normalized by the length of the mass dyad. The indicator
denotes a fixed band covering the enhanced floor, when present. The
constants are uniform over optimizers with the stated outer bounds.
\end{lemma}
\begin{proof}
\leavevmode\par
\paragraph{Second moments and the enhanced floor.}
For the nonfloor contribution to \eqref{eq:C10}, use the whole-cap trace and bracket estimates PE (20)--(21), with cap width \(\le M_j^{-1}\), \eqref{eq:C6} giving the same \(\Delta S\) and contact hypotheses, and the admissible changed clocks giving \eqref{eq:A1}. Enlarge bands to include complete caps with comparable masses by regularity and the small radius width. This gives the first two terms. Floor mass from \(k\) uses simply \eqref{eq:C8}.
For a nonzero extra floor, use sliding radius windows on \(I\), where
\(\alpha'\ge\kappa\) and \(S\lesssim u\). Equation
\eqref{eq:A1}, integrated in radius, gives
\[
\int\mathbb EW\,dr\lesssim x^6/C_0,
\qquad
\int\mathbb E\operatorname{tr}C_v^2\,dr
\lesssim x^6/(C_0u).
\]
Here \(C_v\) denotes the spin covariance and the masses are comparable
to \(u\). Let \(\overline L^{\,+}_{r,h}\) and
\(\overline L^{\,-}_{r,h}\) be normalized averages on the following
and preceding radius windows of length \(h=t'u\). Applied in mass
coordinates, their deterministic weight is \(1/(h\alpha')\).
Let \(I_h\) omit radius strips of width \(h\) at the ends of \(I\),
so both windows remain where \(\alpha'\ge\kappa\).
Equation \eqref{eq:A1} and Fubini give
\[
\int_{I_h}\operatorname{Var}(\overline L^{\,\pm}_{r,h})\,dr
\lesssim\frac1{\kappa h}\int_I\mathbb EW\,dr.
\]
The single inverse power of \(h\) comes from integrating the moving
window, whose multiplicity is at most \(h\).
For two ordered cuts \(i0$ independent of $p$.
The cut is chosen from deterministic second moments and geometry before
spin sampling; it is the same cut for all these moment estimates.
\end{lemma}
\begin{proof}
\leavevmode\par
\paragraph{Inner mass and radius estimates.}
For \(v\gtrsim W\) the right side of \eqref{eq:C10} is polynomially smaller than \(w^2/W\) (\(v C(v)\gtrsim W^3\)); for the extra floor this uses
\[
u B_f W/w_c^2\lesssim\lambda^{s_0/2+k_c/12-0.24}.
\]
On regular bands between a large fixed multiple of \(w\) and scale
\(W\), \eqref{eq:C10} saves a power relative to
\(w^2/(W^2v)^{1/3}\). The extra floor is absent there.
We also need
\[
\int D^2=o(w^3)
\]
on bands with mass between fixed positive multiples of \(w\) and
radius \(O(w)\), separately in mass and radius. In radius, apply the
sliding estimate with \(S=O(w)\), field slope \(C_0\), and floor
lower bound \(k(O(w))\). Its errors vanish because
\(mC_0k(O(w))w^4\gg1\); narrow domains and boundary strips use
\eqref{eq:C8}.
In mass, the floor contributes \(o(w^3)\), since its mass is \(o(w)\)
and the raw scale is \(O(w)\). On caps and contact, use PE (20)--(21)
with that same raw scale. If a cap crosses the lower cutoff \(cw\)
from below \(cw/2\), use its portion above \(cw/2\). This portion
is a fixed fraction of its total mass, because outer comparability
puts its upper endpoint at \(O(w)\). The whole-cap bound on
\(\Delta S\) and the constant cap density give the trace estimate
on that portion. Thus the trace bound is always used with a positive
lower mass cutoff. The condition \(mC_0w^4\gg1\) finishes the
mass estimate. Summing these bands and the regular intermediate
estimates, and using \eqref{eq:C8} below them, gives
\(\int_0^{C'W}D\,dv=o(w^2)\) for every fixed \(C'\).
\paragraph{A good calibration cut.}
Choose a regular band of scale \(W\), above any required fixed multiple
of \(w\). The constant removal, \eqref{eq:C6}, and \eqref{eq:C10}
give a nonfloor cut \(*\) there with \(S_*\asymp W\) and
\(D_*\) polynomially smaller than \(w^2/W\). This selection uses
deterministic moments of the fixed optimizer and parameters. It is made
before sampling spins and independently of a moment order.
At this same cut, \(\|Q_*-S_*\|_p\ll W\) for every fixed \(p\),
with a saving exponent independent of \(p\). To prove this, take earlier
and later radius buffers of length \(h'=x^{c_\theta}W\), with fixed
\(c_\theta>0\) sufficiently small. Integrating \eqref{eq:C6} to or
from \(*\) shows that \(S\) varies by \(O(h')\) on fixed
enlargements: use \({\cal H}(*)\le0\), the local lower bound
\({\cal H}\ge-O(C_0/M_j^2)\), and monotonicity of \(S\).
Equations \eqref{eq:C9} and \eqref{eq:A1} make the centered averages
on these buffers polynomially smaller than \(W\). The past/future brackets and common-past comparison in
Lemma~\ref{global:buffer-brackets} then give the same improvement for
\(\||X|^2-S_*\|_p\) and for squared path increments in \(L^p\).
Finally project both leaves to a central cut a fixed fraction of
\(h'\) beyond their fork. Equations \eqref{eq:A3} and \eqref{eq:C8},
with synchronized residual layers, bound its residual by
\(O_{p,\circ}(x^3\sqrt{W/h'}/(W\sqrt{C_0}))\).
The two normalized errors just used are bounded, up to subpower losses,
by
\[
[mC_0k(W)W^4]^{-1/2}(W/h'),\qquad
[mC_0W^4]^{-1/2}(W/h')^{1/2},
\]
respectively. Both save a power when \(c_\theta\) is small enough,
independently of the fixed moment order. This radius-window argument
uses the floor lower bound \(mC(v)k(v)v^4\gg1\); it does not
require the upper-cap bound used in PE's original mass-window proof.
\end{proof}
Set
\[
\widehat K=h+C(r)r+(b-C(r))S,
\]
and let \(\Gamma_{\widehat K}\) be the scalar moment curve for this
entire clock path.
\begin{proposition}[Closure for the modified constraints]\label{bud:modified-closure}
Along any fixed sequence of optimizers with the strong-removal hypotheses
above and fixed outer-comparability constants, the following estimate holds
for targets \(i\) with
\(cw\le v_i\le Cw\) and \(r_i\le Cw\),
\begin{equation}\label{eq:bud-coarse-closure}
\frac{|S_i-\Gamma_{\widehat K}(\widehat K_i)|}{w^3}
\le C'\{\varepsilon_m+(D_i/w)^\vartheta\},
\qquad \varepsilon_m\longrightarrow0,\quad\vartheta>0.
\end{equation}
The constants and modulus are common to the fixed target band. Thus
this difference is \(O(w^3)\) throughout the band, and is
\(o(w^3)\) along any moving target with \(D_i=o(w)\).
Since the target-band integrals of \(D^2\) are \(o(w^3)\), the same
modulus gives, separately in the two coordinates,
\begin{equation}\label{eq:bud-coarse-closure-integrals}
\frac1w\int
\frac{|S_i-\Gamma_{\widehat K}(\widehat K_i)|}{w^3}\,dv_i=o(1),
\qquad
\frac1w\int
\frac{|S_{\alpha(r)}-\Gamma_{\widehat K}(\widehat K_{\alpha(r)})|}
{w^3}\,dr=o(1).
\end{equation}
Each integral is restricted to the target band just described.
\end{proposition}
\begin{proof}
The constraints \eqref{eq:C5} differ from those in the statement of
PE's closure proposition. We therefore use its proof, and verify the
replacement inputs below. A \emph{block} here consists of leaves sharing
through a fixed continuity cut of mass comparable to \(W\). A boundary
edge joins two such blocks. The tests used in that proof are the fixed finite products of centered
overlap factors and projected factors in \cite[Definition~4.2]{PE};
their natural raw size is denoted by \(p_0\).
The constants may depend on that fixed degree. All centers are the
actual moments \(S\), held fixed during removal of one site.
\begin{enumerate}[label=(\roman*)]
\item \emph{Projection estimates and row changes.}
Use the selected star \(*\) and a regular continuity cut at a large
fixed multiple of \(W\) to form the blocks. For each projection at
scale \(v\), outer comparability supplies a preceding radius interval
of length comparable to \(v\), after the low fork, with masses
comparable to \(v\). Its field length is at least
\(C(v)v/O(\log m)\). Above a sufficiently large fixed multiple of
\(W\), \(vC(v)\gtrsim W^3\). Define
\(z_v=w^3/(W^{3/2}\sqrt v)\). The postprojection bound and the
star-anchor version of \eqref{eq:A3} then read
\[
\frac{x^3}{v\sqrt{C(v)}}
\lesssim (x/w)^3 z_v,
\qquad
\frac{x^3\sqrt W}{\sqrt{v^3C(v)}}
\lesssim (x/w)^3\frac{w^3}{Wv}.
\]
The factor \((x/w)^3\) is power small. These are the two boundary
estimates in PE (13), (16), and its Section~4, ``Analytic input and
finite-degree tests''. They also hold for conditional replacements.
The raw bounds \eqref{eq:C8}, the good-star bound, and the second-moment
estimates above supply its remaining inputs (12), (17)--(19), with
\(m,w,W\) in place of \(n,x,W\).
The clock \(\widehat K\) is monotone, bounded, and \(O(v+w)\).
Above the regularity threshold this follows from outer comparability,
\eqref{eq:C8}, and \eqref{eq:P9}; below it use monotonicity.
Consequently the scalar row sum \(\chi\) in that proof is bounded
above and below by positive constants, and its two-spin coefficient
at distinct high mass \(v\) is \(O(v)\).
The bounded-row comparison of \cite[Lemma~4.3]{PE} applies: all
clocks remain bounded and genuine, and the ordinary field increments
on the remaining sites do not change. Their raw projection norms also
follow by conditional Jensen from positive overlap tests.
\item \emph{The finite-degree expansion and calibration.}
The row interpolation replaces the cavity-spin kernel by
\[
h+C(r)r+(b-C(r))Q^{-},
\]
where \(Q^{-}\) is the overlap with the deleted site omitted but with
the original normalization. Its difference from \(\widehat K\)
is \((b-C(r))(Q^{-}-S)\), up to the \(O(m^{-1})\) common-noise
convention. Every row interpolation preserves positive covariance.
Thus the row expansion in \cite[Lemmas~4.5, 4.7 and 4.8]{PE},
(23)--(25), has the same algebra and the same centers.
Here is how its errors are bounded after substitution.
A high error integrated with bounded split density, including repeated
vertices in one dyad, costs a power-small multiple of \(w^2/W\).
The error at \(*\) has the same bound. Ordinary low factors cost
\(O_p(W)\), or \(O_p(w)\) if their split is at the target scale.
A varying low error costs \(\int_0^{C'W}D=o(w^2)\); an error
at the old target costs \(D_i+O(m^{-1})\), with positive powers of
\(D_i/w\) after moment interpolation. Deleting the row costs
\((mw)^{-1}=o(w^3)\). The observed-block truncation step uses
variations power-smaller than \(Wz_v\) and \(p_0z_v/w\), which
were supplied in (i). These estimates justify the truncation before
any signed sums are bounded absolutely.
For completeness, suppress the common raw test factor \(p_0\).
In the terms with three low insertions, an integrated error is
\(o(w^2)\) and its other low factors cost \(O(W^2)\) or
\(O(W^3)\). If the error is at the old target and \(w\ll W\),
one additional factor has scale \(w\). When necessary, this factor
is obtained by turning off revelation below a smaller block cut of
scale \(w\), then restoring it once; the zero-field sign symmetry
cancels the term before restoration. High remainders contain either
one high error and two low factors, or two high errors and one low
factor. Their bounds are power-small multiples of \(Ww^2\) and
\(w^4/W\), respectively. Moments slightly above two suffice for
the integrated errors; their interpolation loss is paid by the strict
power saving.
If a third block is already observed, first project the fresh endpoint
in the other target block to the star and collapse its scalar row to
\(\chi\). Next replace the factor at the second leaf of the third
block by its value at the first leaf. The postprojection estimate in
(i) gives a power-saved \(w^3/(Wv)\) for this replacement. Its
scalar coefficient is \(O(v)\), and the other low factor costs
\(O(W)\), so the total is \(o(w^3)\le o(Ww^2)\). After this
replacement that leaf's
scalar row also collapses to \(\chi\). These are the two row
reductions in \cite[Lemma~4.7, observed-third-block case]{PE}.
Let \(v_{\rm blk}\asymp W\) denote the continuity cut defining the
blocks. The tests have no overlap factor internal to an old block;
its old internal relationships occur only in fixed scalar coefficients
and the prescribed topology.
Only the unweighted attachment of the first leaf in the third
block remains. At its fresh-at-cut value the summand is independent of
the deeper attachment, so the signed row-sum rule, including all old-fork atoms and coincidences,
gives the factor
\(v_{\rm blk}=O(W)\). At the star the baseline product difference
is \(O(WD_*)\); its contribution is therefore
\[
O(v_{\rm blk}WD_*)=O(W^2D_*)=o(Ww^2).
\]
Restoring the deeper attachment uses the boundary comparison of
\cite[Lemma~4.5]{PE}; its high-layer bound is
\[
\varepsilon_m v(Wz_v)(z_v/w)
=\varepsilon_m w^5/W^2\le\varepsilon_m Ww^2,
\qquad \varepsilon_m\to0,
\]
where \(\varepsilon_m\) has the fixed power saving supplied by (i). If the third block is unobserved,
its varying low error costs \(W\int_0^{O(W)}D=o(Ww^2)\);
a star attachment has the same entry factor \(O(W)\).
At the old target the two low factors have scale \(w\), with the
positive-power modulus in \(D_i/w\) described above. Thus both
observed and unobserved third blocks obey the required remainder bound.
For the leading scalar-row replacement, we spell out the test
difference in \cite[Lemma~4.8, proof of the first replacement]{PE}.
By linearity take an admissible monomial
\[
P_a=H Y_a^k E_{oa},\qquad P_c=H Y_c^k E_{oc},\qquad
Y_d=X_*\cdot y_d-S_*,\qquad E_{od}=Q_{od}-S_j,
\]
where \(H\) uses only exterior leaves, the incoming edge has split
\(j\in\{i,*\}\), and the edge factor is omitted if absent.
The leaf \(c\) is a fresh relative of \(a\) in its block.
Let \(\mathcal F\) be the complete target-trunk prefix through
\(*\), excluding exterior continuations and subsequent target
innovations. Define \(\Delta P_{ac}\) by averaging \(P_a-P_c\)
over the entire exterior joint continuation, conditional on
\(\mathcal F\) and the joint future of \(a,c\).
The signed exchange identity is
\[
\sum_c\gamma_{ac}\nu[(Y_c-Y_a)P_a]
=\frac12\sum_c\gamma_{ac}\nu[(Y_c-Y_a)\Delta P_{ac}],
\]
where \(\gamma_{ac}\) is the scalar two-spin coefficient with
revelation below the block cut moved to that cut, and the sum uses the
signed attachment rule within the block. This is the source's first
symmetrization, followed by conditional expectation.
Put \(\mathfrak s_i=w\), \(\mathfrak s_*=W\), and let \(p_H\) be the raw scale
of \(H\). Then \(p_0=p_HW^k\mathfrak s_j\) with the incoming edge
and \(p_0=p_HW^k\) without it. The exterior anchor
\(V=\mathbb E[Hy_o\mid\mathcal F]\) obeys
\[
\|V\|_p\le\|H\|_{2p}\|Q_j\|_p^{1/2}
\lesssim_p p_H\sqrt{\mathfrak s_j}.
\]
This is the conditional Cauchy--Schwarz estimate in that same source
proof. It uses the ordinary split-\(j\) marginal; no exterior
terminal spin is conditioned upon. The target projection powers stay
outside this averaging and outside the anchor.
For a regular innovation interval before a high split of scale \(v\),
write the unit-anchor bound as
\[
\pi_v=O_{p,\circ}\left(\frac{x^3}{\sqrt{v^3C(v)}}\right)
\le\varepsilon_m\frac{w^3}{W^{3/2}v},
\]
with \(\varepsilon_m\) power small. Then
\(\|Y_c-Y_a\|_p\lesssim\sqrt W\pi_v\).
Telescoping \(P_a-P_c\), a changed projection power costs
\(p_0\pi_v/\sqrt W\); the changed incoming edge, after the exterior
average, costs \(p_0\pi_v/\sqrt{\mathfrak s_j}\). Apply projection before
H\"older with the remaining powers, as in PE's test-difference
estimate. Thus
\[
\|\Delta P_{ac}\|_p
\lesssim\frac{\varepsilon_m p_0}{v}
\left(\frac{w^3}{W^2}+\frac{w^{5/2}}{W^{3/2}}\right).
\]
The norm is of the exterior-averaged difference, not of the original
incoming-overlap difference. At a distinct split, the attachment
variation per dyad and \(\gamma_{ac}\) each contribute \(O(v)\).
Consequently the symmetrized error per dyad is at most
\[
C v^2\left(\frac{\varepsilon_m w^3}{Wv}\right)
\frac{\varepsilon_m p_0}{v}
\left(\frac{w^3}{W^2}+\frac{w^{5/2}}{W^{3/2}}\right)
\lesssim\varepsilon_m^2p_0Ww^2,
\]
since \(w\le W\). The saving pays the dyadic sum.
For a connected high insertion, one \(O(v)\) placement factor
suffices, because its integrated high error supplies the additional
saved \(w^2/W\); this remains true when the vertices share a dyad.
This verifies the remainder estimates in PE (24)--(25).
No variation of the optimizer occurs in this finite-tree argument.
We spell out how the finite recursion preserves their uniform target
modulus. For a block $\mathscr A$ at the chosen continuity cut
$v_{\rm blk}$, let $\nu^{\rm sc}$ be the scalar law with revelation
below $v_{\rm blk}$ moved to that cut, and
put $\gamma_{ac}=\nu^{\rm sc}[\tau_a\tau_c]$. The connected insertion
of PE (23), paired with a test $P$, is
\[
\mathcal L(P,a)=\frac12
\sum_{\substack{c,e,f\in\mathscr A\\ e\ne f}}
\bigl(\nu^{\rm sc}[\tau_a\tau_c\tau_e\tau_f]
-\gamma_{ac}\gamma_{ef}\bigr)
(b-C)_{ef}\,\nu[P(Q_{ef}-S_{ef})].
\]
Here $S_{ef}$ is the actual moment at the split of $e,f$, and all
three added labels use the signed within-block attachment rule.
The coefficients and expectations are evaluated on the same final
topology. This formula defines the insertion through its pairing;
no positive probability interpretation of the signed sum is used.
At the star, $\widehat K_*=T_mS_*+O(W^3)$ and the changed matrix is
$T_m+O(W^2)$. Put $J_0=T_m\chi^2-1$. The recursion of
\cite[Lemma~4.9]{PE} divides by $\chi\widehat K_*\asymp W$.
The remaining pairing of the same degree at the partner root is removed
by applying the row identity to two fresh interchangeable roots absent
from $P$. Their equal leading pairings have total coefficient
$2\chi\widehat K_*$; every remaining term adds a star factor at cost
$O(W^{-1})$.
At depth $k$, accumulated coefficients are $O_k(W^{-k})$ and every
unresolved term has $k$ new centered star factors. Lemma~\ref{bud:common-star}
therefore bounds its terminal contribution by $O_k(p_0x^{kc_*})$.
Choose $k$ fixed so that $kc_*>2$, using $w\ge w_c\gg x$;
then this is $o(w^2p_0)$. Only after choosing $k$ do we fix the finite
degrees and H\"older orders. The intermediate errors give
\[
|\mathcal L(P,a)|\le Cw^2p_0
\left\{\varepsilon_m+(D_i/w)^\vartheta
+\varepsilon_m|J_0|/W^2\right\},
\qquad \varepsilon_m\to0,
\]
with the target term absent for star-only tests. There are only finitely
many target exponents, so their positive minimum gives one $\vartheta$
for the whole fixed target band.
Return to the finite-degree row identity with $P=1$, as in
\cite[Lemma~4.10]{PE}. Its surviving high corrections are pairings
against one star factor, of raw size $p_0=W$. Apply the preceding
bound to those pairings and use
$\Gamma_{\widehat K}(\widehat K_*)=\chi^2\widehat K_*+O(W^3)$.
Indeed, the second scalar coefficient is a two-edge path through an
unobserved third block: its $O(W)$ entry factor multiplies two low
factors of size $O(W)$, and bounded finite-label variation controls
the third-order remainder.
This gives
\[
S_*=\chi^2T_mS_*+O(W^3)
+O\!\left(\varepsilon_m Ww^2(1+|J_0|/W^2)\right).
\]
Dividing by $S_*\asymp W$ gives
\[
|J_0|\le CW^2+\varepsilon_m w^2
+\varepsilon_m|J_0|w^2/W^2.
\]
Since $w\le W$, the last term is absorbed. Hence $J_0=O(W^2)$,
and the calibrated bound is
\[
|\mathcal L(P,a)|\le Cw^2p_0
\{\varepsilon_m+(D_i/w)^\vartheta\}.
\]
The common base $T_m$ has remained fixed throughout the finite tree;
no rate for $T_m-T$ enters these estimates.
\item \emph{Return from the calibration scale to the target.}
Repeat the mean expansion with the block cut at a large fixed multiple
of \(w\). The two regular-dyad bounds make high errors power-smaller
than \(w\), and all low factors now have raw scale \(w\).
An old-target error contributes a positive power of \(D_i/w\),
while the remaining errors tend to zero uniformly over the target band.
The surviving mean equation at this smaller block cut is
\[
S_i-\Gamma_{\widehat K}(\widehat K_i)
=2(b_i-C(r_i))\chi_w\,
\mathcal L_w(Y_a^{(i)},a)+\operatorname{Rem}_i,
\qquad Y_a^{(i)}=X_i\cdot y_a-S_i.
\]
Here $\mathcal L_w$ is the same connected pairing with block cut of
scale $w$, and $\chi_w$ its scalar row sum. The remainder is bounded
by $Cw^3\{\varepsilon_m+(D_i/w)^\vartheta\}$, uniformly in the
fixed target band. The changed matrix coefficient is $b_i-C(r_i)$.
To use the calibrated insertion, first remove insertion edges between
this cut and the calibration block cut $v_{\rm blk}$. Their cost is
\[
O(w)\int_0^{C'W}D=o(w^3).
\]
Next change the high scalar kernels; above a sufficiently large fixed
\(c'W\), the cost is
\[
O(wW)\int_{c'W}^1D=o(w^3).
\]
These are precisely the two cutoff changes in PE's ``Return to the
target cut''. Allocate the error pair first relative to the sole old
root $a$ in its block, and then the summed leaf $c$; this gives the
bounded split density used in these integrals without a high-fork
atom. After the scalar-kernel change, independent block flips force
$c$ into $a$'s block, while the error endpoints share a block; if that
block is different, the connected scalar covariance vanishes.
The remaining pairing is therefore $\mathcal L(Y_a^{(i)},a)$ at the
calibration-scale block cut $v_{\rm blk}$. The test $Y_a^{(i)}$ has raw scale $p_0=w$ and is admissible by
its fresh-branch representation. Its calibrated bound is therefore
$Cw^3\{\varepsilon_m+(D_i/w)^\vartheta\}$; $b-C$ and $\chi_w$
are bounded. This proves \eqref{eq:bud-coarse-closure}. The already proved target-band
\(D^2\) bounds then give both integrals in
\eqref{eq:bud-coarse-closure-integrals}.
\end{enumerate}
For example, when \(0<\vartheta\le2\), H\"older bounds its discrepancy
term by a constant times
\( (w^{-3}\int D^2)^{\vartheta/2}\); a larger exponent may first
be decreased using the raw bound \(D=O(w)\).
\end{proof}
\begin{proof}[Proof of Proposition~\ref{bud:coarse-profile}]
We first exclude a failure of comparability, using a PE child for weak
removal and Proposition~\ref{bud:modified-closure} for strong removal.
The final step identifies the initial slope by a macroscopic transfer.
Profile arguments allow subselection. If loose fixed comparability
fails, choose an offending mass $w\ge w_c$ comparable to the supremum
of offending masses. This supplies the outer regularity above a fixed
multiple of $w$. Choose the loose constants strictly outside the bounds
of the PE direct profile limits. The scale estimates displayed before
Lemma~\ref{bud:outer-moments} follow from $w\ge w_c$.
\paragraph{Step 1: weak removal and feasible barriers.}
First suppose \(C(w)=o(w^2)\). Apply PE finite propagation to a child acting from the changed path of this optimizer, with the data \(d',E'\) of \cite[Lemma~6.2]{PE} at \(w\), and ordinary PE constraints. The data verification in that lemma uses (3) and the parent strength, cutoff and outer bound (not the parent's optimization constraints); hence (3)--(4) hold in the child. It has nonnegative monotone bare paths with bounded clocks and a fixed positive all-mass matrix margin, choosing its fixed \(d'/w\) small. Its final error is at most \(m^{20\delta}E'\); write \(z\) for a final minimizing quantile and \(\mathcal D(z)=d'^2 f(z/d')\). The movement inequality of \cite[Lemma~6.3]{PE} applies with parent-removal \(C\) to any trial satisfying \eqref{eq:C5}: its proof only uses feasibility, optimality, the slope bound on \(C\) and the square interpolation. We explain the corresponding barriers to handle the modified floor.
Here \(d'=c_0 w,\ E'=\min(m^{-1/2}d'^2,m^{-100\delta}C(w)w^3)\), \(c_0>0\) sufficiently small fixed. In particular the parent changed endpoint still tests against the common \(T_m\) by the estimates of \cite[Lemma~6.2]{PE}, since \(r=O(y)\) on comparable mass bands of scale \(y\gtrsim w\), \(r=O(w)\) below that.
Suppose that \(r\) and \(z\) are separated by fixed thresholds whose
gap is comparable to \(y\), on a mass interval of length comparable
to \(y\), where \(y\gtrsim w\). Macro intervals lie strictly inside
\((0,1)\). Failure of agreement in normalized measure produces such
an interval by monotonicity, and outer regularity gives \(r\le C'y\)
there. Here agreement means that, on each fixed comparable mass band,
the mass of \(\{|r-z|>\epsilon y\}\), divided by \(y\), tends to
zero for every fixed \(\epsilon>0\).
The forward constraints are
\(l\le r'\le1/\bar k(r)\), with \(l=M_j^{-1}\). We have
\(\sup\bar k=o(1)\) and \(\int_0^1\bar k=o(w)\), the latter
using \(w\gtrsim u\sqrt\Lambda\). Reserve fixed short portions
at the two ends of the separation interval for the following barriers.
If \(z\) is above \(r\), choose an intermediate threshold \(H\)
and an early subinterval \([a,b]\), with \(b-a\asymp y\), and set
\[
b_\uparrow(v)=lv+
\begin{cases}
0,&v\le a,\\
H(v-a)/(b-a),&at\), whether \(s(t)\)
is near that endpoint or farther into the gap. For infinite strength,
the conclusion follows from the derivative \(-2Tg(t)\).
Both contradict nonnegativity and the zero endpoint value. Reflection
excludes a positive derivative at the right endpoint as well.
But on a gap of length \(L\), the curvature is
\(g''(t)=At-\mathrm{const}\) and the endpoint values are zero,
so the sum of endpoint derivatives is \(AL^2/6>0\).
No gap is possible.
Above the support infimum \(l\), we thus have \(g=0\) and
\(U_1(t)=At/B\). If \(l>0\), then \(g''=At\) below \(l\)
and \(g(l)=g'(l)=0\), contradicting \(g(0)=0\).
Consequently \(U_1(t)=At/B\) throughout. This is the linear quantile
limit PE (28), with the same fixed slope bounds. It contradicts our
choice of loose comparability constants and proves the first part of
\eqref{eq:C7}.
The first part of \eqref{eq:C7} has now been proved for every
optimizer. In particular Lemmas~\ref{bud:outer-moments}
and~\ref{bud:outer-second} can henceforth be used with $w=w_c$ and fixed uniform comparability constants. This is the
version of \eqref{eq:C8}--\eqref{eq:C10} used in the strengthening
argument below; it does not invoke the strong-removal closure again.
\paragraph{Step 3: identification of the initial slope.}
Repeat the weak-removal transfer at \(w=u\), now without assuming
a violation. We identify the child's linear slope by its macroscopic
clock limit. First,
\(S_{\rm child}-z\to0\) in mass, by PE (18), geometry, and
regularity on each fixed interior band. The required error estimates
are
\[
m\mathcal D(v)\gtrsim mu^3,\qquad
E'/d'^3\lesssim m^{-1/2}/u,\qquad
m^{\delta/2}/(mE')\longrightarrow0.
\]
Compare \(z\) to a further ordinary PE minimizer at final strength,
with fixed small removal parameter \(d''>0\) and
\(E''=m^{-1/2}d''^2\). Its macroscopic endpoint hypothesis PE (4)
follows from \eqref{eq:P9} and the vanishing preceding removals;
all-mass admissibility and a positive covariance margin remain.
Finite propagation gives error \(m^{20\delta}E''\ll u^3\), and
\cite[Proposition~5.9]{PE} gives the further minimizer's limiting
quantile \(q_T\).
The ordinary PE barriers transfer this limit to \(z\). At a fixed
interior mass scale, full clipping is feasible with the child's minimum
slope \(1/M(v)\); both forced lengths are \(o(1)\).
The movement fraction holds because \(\sup\mathcal D=o(1)\),
whereas the newest removal is bounded below. A fixed threshold
separation would cost \(\gtrsim\inf\mathcal D\gtrsim u^3\),
contradicting the propagation error. Thus
\(h_{\rm child}+b_{\rm child}S_{\rm child}\to Tq_T\) in
macroscopic \(L^1\).
Bounded-clock PDE stability, as at \eqref{eq:G3}, now gives convergence
of the scalar spatial derivatives at zero. Equivalently one can use
drift comparison after \(L^1\) convergence of the inverse-clock
distributions. The slope formula \(A/B\) in PE (29), together with
\cite[Lemma~5.8]{PE}, therefore identifies the child's inverse-profile slope as
\(\kappa_T\). Agreement transfers its linear quantile limit to
\(r\); meanwhile \(q(v)/v\to1/\kappa_T\) by \eqref{eq:P2}
and the normalized inverse construction. This proves the second part
of \eqref{eq:C7}. On compact positive radius bands of scale \(u\)
inside the constant-removal region, it also gives \(S-r=o(u)\)
uniformly, by \eqref{eq:C6}, monotonicity, and the local \(o(u)\)
floor mass.
\end{proof}
\subsection{Finite strengthening and common parameter envelopes}
Use the notation
\[
K_*=h_*+bq,\quad d=h-h_*,\quad K_s=h+C(r)r+(b-C(r))S,\quad
\Gamma_*=\Gamma_{K_*},\quad \Gamma_s=\Gamma_{K_s}
\]
in the normalized local argument (\(s\) on \(K_s,\Gamma_s\) here denotes the sampled path, not a tilt). Let \(\beta_0(r)=q^{-1}(r)\) locally near the patch; it and \(q\) have slopes between fixed positive constants there. Choose finitely many nested compact intervals scaled by \(u\), strictly interior to the constant profile and at positive radii of order \(u\), each containing with room the required image of the mass band by \eqref{eq:C7}. Their fixed spacings can be as small as needed for the number of stages.
Take
\[
a_{-1}=1,\quad
a_j=\max(a_{\rm goal},\lambda^{-0.12} A^{1/3} a_{j-1}^{\,2/3}),\quad
\kappa_j=(A/a_{j-1})^2\quad (j\ge0). \tag{C11}\label{eq:C11}
\]
These scales are nonincreasing, \(a_0=o(1)\), and reach \(a_{\rm goal}\) in a bounded fixed number of steps: the iterated term has fixed point \(A\lambda^{-0.36}\ll a_{\rm goal}\), while \(A\gtrsim\lambda^{2/p_*}\). Use \eqref{eq:C5} with the indicated floor on the next nested patch \(I_j\), dropping the previous extra floor.
\begin{lemma}[Finite strengthening]\label{bud:strengthening}
On a slightly smaller interval interior to \(I_j\), at each stage,
\[
|\alpha-\beta_0|\lesssim u a_j Z \tag{C12}\label{eq:C12}
\]
for almost every global parameter, simultaneously for all its optimizers, where \(Z\ge1\) has bounded averaged square in that parameter. For each fixed choice of the other path data, the exceptional set is independent of the optimizer and may be taken common to the finitely many stages. At successive stages that parameter may be restricted to slightly smaller fixed interior intervals, always leaving room beyond \([1/3,2/3]\); parameters not explicitly varied need no restriction beyond those of \eqref{eq:C3}. At stage \(-1\) the analogous assertion holds by \eqref{eq:C7} with \(Z=1\).
\end{lemma}
We prove the lemma by induction. First we show that the preceding-stage
bound pays the new optimization and supplies a common global norm bound.
We then rule out a violation at the new stage by a buffered contradiction.
The comparison cost therefore depends only on the already proved accuracy.
\begin{lemma}[Stage cost and an envelope common to all minimizers]\label{bud:envelope}
Suppose the preceding stage satisfies \eqref{eq:C12}. At the new stage
the average optimized excess is at most
\(O(x^{4-2\zeta}u\Lambda^{1/3})\). For almost every global parameter,
every minimizer simultaneously satisfies \eqref{eq:C13} below,
with a common measurable majorant \(Z\) whose
averaged square is bounded.
\end{lemma}
\begin{proof}
Let the old optimizer have radius law \(k\,dr+\nu\). On a fixed
enlargement of \(I_j\), convolve the fraction \(c'\kappa_j\) of
\(\nu\) with the symmetric uniform kernel of radius
\(\sigma=\min(c_{\rm small}u,C_{\rm big}u a_{j-1}Z)\).
Choose the fixed constants, depending on the stage, so that all motion
stays inside the preceding controlled band and the constant part of
\(C\). Every radius interval of length \(2\sigma\) used here carries
mass \(\gtrsim\sigma\). This follows from \eqref{eq:C12} and the
positive lower slope of \(\beta_0\); if the fixed cutoff determines
\(\sigma\), use \eqref{eq:C7} instead. Subtracting the density \(k\)
does not affect this lower bound. Taking \(c'\) sufficiently large
therefore supplies density at least \(\kappa_j\) throughout \(I_j\).
The convolved fraction is \(o(1)\), so the new upper density bound is
at most twice the old one.
Write \(r_{\rm old},r_{\rm new}\) for the two quantiles. Conditional
Jensen shows that the new radius law dominates the old in convex order
and has the same mean. For every \(a'\in[0,1]\), the upper-tail
quantile formula
\[
\int_{a'}^1r(v)\,dv
=\inf_{s'}\{(1-a')s'+\mathbb E(r-s')_+\}
\]
therefore gives a nonnegative upper-tail integral of
\(r_{\rm new}-r_{\rm old}\). Integration by parts against the
nondecreasing moment \(S_t\) at each forward interpolation time yields
\[
\int_0^1 S_t(v)(r_{\rm new}(v)-r_{\rm old}(v))\,dv\ge0.
\]
Since all motion occurs where \(C=C_0\), \eqref{eq:A1} makes the
pressure change nonpositive. Only the penalty must be paid; it is
\(O(C_0\kappa_ju\sigma^2)\lesssim x^4u\Lambda^{1/3}Z^2\).
Averaging and starting from the bound before \eqref{eq:C5} gives the
asserted cost at every stage.
We next obtain a norm bound common to every new minimizer. Write
\(g=\gamma_0\), let \(F_j(g,\alpha)\) be the objective with the
inverse profile fixed, and put \(V_j(g)=\min_\alpha F_j(g,\alpha)\).
The admissible density box is independent of \(g\).
Equation~\eqref{eq:A1}, followed by its fixed-size path limit, makes
each frozen objective differentiable and bounds its derivative uniformly.
The minimum \(V_j\) is therefore Lipschitz.
At a differentiability point of \(V_j\), fix any minimizing
\(\alpha\). The function
\(g'\mapsto F_j(g',\alpha)-V_j(g')\) is nonnegative and vanishes
at \(g\). Its two one-sided derivatives give
\[
\partial_gF_j(g,\alpha)=V_j'(g).
\]
The exceptional set is the nondifferentiability set of the single
function \(V_j\); it is independent of the minimizer.
Restore the scalar terms by setting
\[
\mathcal R_j(g)=V_j(g)-\frac14\int bq^2\,dv
+\frac12\int_{v\ge x}d(v)q(v)\,dv .
\]
For each minimizer define
\[
\mathcal N^2=\int(v^2+U^2)
[D_v^2+(S_v-q)^2]\,dv.
\]
The aligned matrix and field variation in \eqref{eq:P10} gives
\(\mathcal R_j'(g)\ge c\lambda\mathcal N^2\).
Above the mask this is the completed square
\(c_g[D^2+(S-q)^2]/4\); below it the field stays fixed and the
extra term is nonnegative. Thus the same derivative bounds
\(\mathcal N\) for every minimizer at \(g\).
Choose an upper parameter endpoint outside the common null set in a fixed positive-length margin
beyond the next averaging interval, where the optimized excess is at
most a fixed multiple of its average on that margin. At the lower
endpoint use the nonnegativity of the excess. Integrating
\(\mathcal R_j'\) and using the direct pressure comparison gives
\[
c\lambda\,\mathbf E_{\gamma_0}\mathcal N^2
\le\mathbf E_{\gamma_0}\mathcal R_j'
\lesssim\mathcal B+x^{4-2\zeta}u\Lambda^{1/3}.
\]
This notation denotes a bound by the common measurable derivative,
not integration of a chosen optimizer. The endpoint depends only on
the stage and the fixed path data.
Choose \(00\) is fixed and sufficiently small relative to the
power gaps below. To obtain this buffer, whenever its discrepancy bound
fails, move to a point where the discrepancy has at least doubled and
repeat. There are only logarithmically many moves. The sum of the
\(u\sqrt L\) travel bounds is dominated by the last one, which is
\(o(u)\) by \eqref{eq:C7}; the sum of the
\(u\lambda^\rho\) bounds is also \(o(u)\).
Thus all selected points stay in the interior of the patch.
Put
\[
h_1=u\max(L,\sqrt\lambda),\qquad
G_1=\min(\max(L^2,\lambda),\Lambda),\qquad P_1=LG_1.
\]
The buffer contains unboundedly many lengths \(h_1\). On each smaller
fixed fraction, every cut has nonfloor points within \(O(d_0)\) on
both sides: otherwise the small floor density contradicts the positive
lower slope of \(\beta_0\) and the discrepancy bound.
Equation \eqref{eq:C6} and monotonicity then give
\(|S-r|=O(d_0)\), while the oscillations of \(S\) and \(\alpha\)
on a radius interval of length \(l\) are \(O(d_0+l)\).
At nonfloor points, \(|S-r|\le M_j^{-1}\).
Every later buffer restriction is by a fixed proportion; the finite
number needed in the moment iteration is specified there.
To compare scalar curvatures, define in this buffer
\[
k_*(r)=\Gamma_*^{-1}(r),\qquad
\widetilde r(v)=\Gamma_*(K_s(v)),\qquad
\widetilde\alpha(r)=|\{v:\widetilde r(v)\le r\}|.
\]
Near the tested band the defect \(d\) consists only of the logarithmic
seed, so
\[
|d|\lesssim_\circ u^3 X^3\lambda^{1+\eta-3s_*/2}=o(u^3P_1).
\]
The comparison uses \(G_1\ge\sqrt\lambda L\) and
\(L\ge X^2\lambda^{-s_0}\). Split the latter lower bound at
\(\sqrt\lambda\) and \(\sqrt\Lambda\); in each range the ratio
has a power saving. At the \(\sqrt\lambda\) breakpoint the saving
is \(1/4+O(\tau+\eta)\) in \(\lambda\).
Equation \eqref{eq:P2} and monotone bracketing now give
\(\widetilde r(\alpha(r))=r+O(d_0)\) and
\(\widetilde\alpha-\beta_0=O(d_0)\).
Extend the scalar clocks annealed to a common bounded time interval,
and write their distribution difference in mass length as
\(\delta\alpha_t=|\{K_s\le t\}|-|\{K_*\le t\}|\). We claim
\[
\int \min(u,t)|\delta\alpha_t|dt=o(d_0).
\]
The one-dimensional inverse-transport bound, together with
\eqref{eq:C7}, \eqref{eq:C8}, and \eqref{eq:P4}--\eqref{eq:P7},
bounds the left side by
\(\lesssim\int\min(u,v+w_c)|K_s-K_*|\,dv\).
Here the low clock is controlled because
\(\lambda_1U^3\log(1/x)\lesssim w_c\).
Masses \(v\le O(w_c)\) contribute \(O(w_c^3)\ll d_0\).
Above them, the terms \(b(S-q)\) and \(d\) contribute
\(O(\sqrt u\,x^2\lambda^{-s_0}Z)\), by \eqref{eq:C13} and
\eqref{eq:P8}.
It remains to bound \(C(r)(r-S)\). Off the floor, use
\eqref{eq:C6} and the same norm. The original floor contributes
\(O(uC_0\int k)\) by boundedness. On the extra floor, at
\(r\asymp u\), the \(S-q\) part is again controlled by the norm.
Fubini and \eqref{eq:C7} bound the remaining \(|r-q|\) integral by
\[
O(\kappa_j)\int_{I_j}|\alpha(r)-\beta_0(r)|dr
\lesssim \kappa_j\int_{v\asymp u}|r(v)-q(v)|\,dv ,
\]
on a comparable enlargement. Its extra-floor portion can be absorbed,
since \(\kappa_j=o(1)\); its other portions have just been bounded.
Multiplication by \(uC_0\) proves the claimed transport estimate,
with a power margin for subpower losses.
\Needspace{4\baselineskip}
\begin{lemma}[Scalar curvature in a buffer]\label{bud:curvature}
By the scalar second derivative identity we now have uniformly a.e. in the inner buffer
\[
\frac{d^2}{dr^2}\big[\Gamma_s(k_*(r))-r\big]
=-A_* (\widetilde\alpha(r)-\beta_0(r))+o(d_0), \tag{C14}\label{eq:C14}
\]
where \(A_*\) converges to a positive constant (\(2(V_{*,zz}(0,0))^3 k_*'(0)^2\)). \end{lemma}
\begin{proof}
Let \(Q_0\) be a fixed bound for the extended clocks. At time
\(t\asymp u\), PDE drift comparison bounds the difference of
continuation derivatives in sup norm by
\(O(\int_t^{Q_0}|\delta\alpha_v|\,dv)\); odd derivatives gain
an additional factor \(|z|\). For a bounded smooth even observable,
changing the forward diffusion costs at most
\[
C\int_0^t v\left(|\delta\alpha_v|
+\int_v^{Q_0}|\delta\alpha_{v'}|\,dv'\right)\,dv.
\]
Indeed the drift difference is odd and bounded by the indicated
coefficient times \(|Z_v|\). The propagated spatial gradient is also
odd with bounded derivative, and \(\mathbb EZ_v^2=O(v)\).
These comparisons require only bounded spatial drift derivatives on
the bounded time interval.
Now use
\(\Gamma''=\mathbb EV_{zzz}^2-2\alpha_t\mathbb EV_{zz}^3\).
The two masses are \(O(u)\), and their difference is \(O(d_0)\).
Oddness in the first term and the preceding transport estimates leave,
to order \(d_0\), only the mass-distribution difference in the second
term. Multiplying by \(k_*'^2\) gives the displayed coefficient
\(A_*\). The chain-rule term involving \(k_*''=O(\lambda u)\)
uses the same comparison for \(\Gamma_s'-\Gamma_*'\) and is
\(o(d_0)\). This proves \eqref{eq:C14}.
\end{proof}
\subsection{Buffered moments, free forks, and power margins}
Write \(\delta_i=Q_i-S_i\) at a fork. For a radius interval
\(J=[a,b]\) in the inner buffer with \(|J|\asymp d_0\), define
the two box integrals by
\[
\operatorname{avg}_{J,r}F
=\frac1{d_0}\int_a^b F_{\alpha(r)}\,dr,
\qquad
\operatorname{avg}_{J,v}F
=\frac1{d_0}\int_{\alpha(a)}^{\alpha(b)}F_v\,dv.
\]
Endpoints have zero measure in these integrals. Both use the scale
\(d_0\) as denominator. In particular the second is not normalized by
the mass of the interval: that mass is \(O(d_0)\), but can be
arbitrarily small. In box assertions, \(\operatorname{avg}\) denotes
either operation, or their sum, uniformly over moving intervals of this
kind. An explicitly subscripted dyadic average such as
\(\operatorname{avg}_{v\sim w}\) instead divides the mass integral
by the length of that dyadic band.
\begin{lemma}[Moments on moving boxes]\label{bud:box-moments}
Uniformly inside the buffer and for these averages, for every fixed
moment order, we have
\[
\|\delta_i\|_p\lesssim_p d_0,\qquad
\operatorname{avg}\|\delta_i\|_2^2\lesssim d_0^2\lambda^{0.14},\qquad
\operatorname{avg}\|Q_i-|X_i|^2\|_2^2
\lesssim_\circ \lambda^{-\rho}u^2 A^3. \tag{C15}\label{eq:C15}
\]
\end{lemma}\begin{proof}
Fix the optimizer and the moment order \(p\). Choose a finite sequence
of nested buffers \(B_0\supset B_1\supset\cdots\supset B_{n_{\rm buf}}\),
with successive boundaries separated by fixed fractions of
\(b_{\rm buf}\). Their number will be fixed below, before taking the
size limit. Define
\[
{\cal Q}_k=\max\left\{1,\sup_{t,t'\,\text{in }B_k}
\frac{\||X_t-X_{t'}|^2\|_p}
{d_0+|r(t)-r(t')|}\right\}.
\]
The cuts in this supremum are deterministic and lie on one path.
Boundedness gives finiteness at fixed size, and \eqref{eq:C8} gives
\({\cal Q}_0=O_p(L^{-1})\). To estimate cuts in \(B_{k+1}\),
use only windows lying in \(B_k\). If such a window \(J\) has
radius length \(\Delta\gtrsim d_0\), the weighted-layer calculation
for \eqref{eq:C9} gives
\[
\left\|\int_J W\,dr\right\|_{p/2}
\lesssim_{p,\circ} u^3 A^3\Delta({\cal Q}_k+\lambda^{-\rho}).
\]
For layers ending within radius distance \(O(u)\) and still at
comparable scales, projection after distance \(t'\) costs
\(O_\circ(x^6/(C_0ut'))\) times the earlier squared increment,
including the Hessian weight. The normalized-anchor truncation in
Paragraph~\ref{par:global-block-estimates} retains that increment;
it requires no higher moment of \({\cal Q}_k\). When
\(t'\lesssim\Delta\), \eqref{eq:global-grid} bounds its integral by
\(O_p(t'{\cal Q}_k\Delta)\), using terminal increments on an
interval of length \(O(\Delta)\) inside \(B_k\). When
\(t'\gtrsim\Delta\), its pointwise norm is
\(O_p(t'{\cal Q}_k)\), giving the same integrated bound.
Layers leaving \(B_k\) start at distance
\(t'\gtrsim u\lambda^\rho\) and use \eqref{eq:C8}; this accounts
for \(\lambda^{-\rho}\). The first inverse-power increments are
negligible by the square-sum estimate. Outer layers at scale
\(v\gtrsim u\) cost \(O_{p,\circ}(x^6/(vC(v)))\) per unit
radius length, which also fits the displayed bound.
Using \(\alpha'\ge\kappa_j\), normalized \(L\) averages on such intervals have centered norm relative to \(\Delta\) at most
\(O_{p,\circ}((A/L)^{3/2}\kappa_j^{-1/2})({\cal Q}_k+\lambda^{-\rho})^{1/2}\).
The prefactor, before subpower losses, is at most
\(\lambda^{0.18}\) by \eqref{eq:C11}. For two cuts in
\(B_{k+1}\), take earlier and later windows of length comparable to
\(\Delta=\max(d_0,\text{their radius separation})\).
The brackets in Lemma~\ref{global:buffer-brackets} then control
both centered square norms and their squared increment. The
deterministic oscillation of \(S\) costs \(O(\Delta)\). After
absorbing subpower losses and the \(\lambda^{-\rho}\) term, this is
the recurrence
\[
{\cal Q}_{k+1}\le C_p\bigl(1+\lambda^{0.1}\sqrt{{\cal Q}_k}\bigr).
\]
There is a fixed \(K\) with \({\cal Q}_0\lesssim_p\lambda^{-K}\).
After \(n_{\rm buf}\) iterations its potentially growing term is bounded by a
constant times
\(\lambda^{0.2(1-2^{-n_{\rm buf}})-K2^{-n_{\rm buf}}}\).
Choose the fixed \(n_{\rm buf}\)
so that this exponent is positive. This proves bounded ratios on
\(B_{n_{\rm buf}}\), and the same brackets give
\(\||X_i|^2-S_i\|_p\lesssim_p d_0\).
Project both leaves to radius distance comparable to \(d_0\) after
their fork. The synchronized later layers cost
\(\lesssim_{p,\circ}uA^{3/2}/\sqrt L\ll d_0\).
Polarization controls the projected inner product, since the squared
distance of the two branches through their common prefix is
\(O_p(d_0)\). This proves the first assertion of \eqref{eq:C15}.
\paragraph{Second moments in radius and in mass.}
For the second bound, over each box we have
\[
\int\mathbf EW\,dr\lesssim u^3 A^3d_0,\qquad
\int\mathbf E\operatorname{tr}C_v^2\,dr\lesssim u^2 A^3d_0.
\]
These estimates follow from \eqref{eq:A1}, the \(S\) oscillation
bound, and constant removal strength. Use earlier and later windows of
length \(h=t'd_0\). The sliding variance estimate proved at
\eqref{eq:C10} is now
\[
\int_{J_h}\operatorname{Var}(\overline L^{\,\pm}_{r,h})\,dr
\lesssim\frac1{\kappa_jh}\int_{J^+}\mathbb EW\,dr.
\]
Here \(J_h\) omits endpoint strips of width \(h\), so the windows
stay inside \(J\); \(J^+\) merely provides the available integral
upper bound. Discard those strips with the pointwise moment bound.
Apply the absorption calculation at \eqref{eq:C10} on height scale
\(d_0\), now with trace term \({\cal T}\lesssim A^3/L^2\) and
variance term \((A/L)^3/(\kappa_jt')\). It gives
\[
\int D^2dr/d_0^3\lesssim t'+(A/L)^3/(\kappa_j t')+A^3/L^2.
\]
Take \(t'=(A/L)^{3/2}/\sqrt{\kappa_j}\le\lambda^{0.18}\).
This proves the radius estimate and also the mass estimate on the floor,
whose density is bounded above. To treat the remaining mass, enlarge
endpoint caps to whole caps. Total mass and \(S\) oscillation remain
\(O(d_0)\). Equation \eqref{eq:C6} and PE (20), with \(p=0\),
give the same integrated \(W\) and trace bounds on caps and contact.
Order this measure by the deterministic coordinate
\(s(v)=\int^v\mathbf1_{\rm nonfloor}(t)\,dt\), and take windows
of \(s\)-length \(h=t'd_0\). The optimizer and its floor set were
fixed before sampling; hence the window indicators are deterministic
weights in \eqref{eq:A1}. Fubini now bounds the integrated centered
variance by \(h^{-1}\int\mathbb EW\,dv\), with no floor-density
denominator. The ordered identity and monotonicity control the noise
and mean changes as in radius. If the whole nonfloor measure has length
less than two windows, the endpoint-strip bound alone suffices. This
proves the second assertion of \eqref{eq:C15} in both measures.
\paragraph{The alias error.}
For the final bound of \eqref{eq:C15}, it remains by \eqref{eq:A1}'s covariance convention to bound \(\mathbf E(X_i\cdot(y-X_i))^2\). Telescope \(X_i\) into past layers ending at dyadic distances from the current radius, down to \(x^{40}\). For an increment ending at distance \(t'\) after starting at distance \(\asymp t'\), with \(x^{40}\lesssim t'\ll u\), \eqref{eq:A3} costs for its squared product with \(y-X_i\) at most \(O_\circ(x^6/(u^2 C_0 t'))\) times the layer's squared norm in expectation. That norm has box average \(O(t'\lambda^{-\rho})\). In radius this follows for \(t'\le d_0\) by sliding the \(S\) variation, for \(d_0\le t'\ll b_{\rm buf}\) pointwise, and otherwise by \eqref{eq:C8}.
In mass, the floor uses the radius estimate, and around a nonfloor evaluation the \(S\) oscillation at distances \(O(t')\) within the buffered patch is \(O(t')\), by \eqref{eq:C6} and the lower potential bound \(-O(C_0/M_j^2)\), as at the good cut. The unprojected last past increment thus has negligible squared norm on average by the same facts, and the initial anchor, projected from fixed comparable distance within constant strength, costs \(O_\circ(u^2 A^3)\). Summing proves the last assertion.
\end{proof}
Here are needed integrated estimates on free forks. Choose a sufficiently large fixed \(c_H\) so \(v\ge c_Hu\) in mass lies beyond the extra floor, and away from the observed buffer with room for prior comparable projection intervals. \begin{lemma}[Integrated errors at free forks]\label{bud:free-forks}
Uniformly,
\[
\begin{split}
\int_0^{c_Hu}D_v dv&\lesssim_\circ u^2 T_1,\qquad T_1=A+B_f+y_c^2,\\
\int_0^{c_Hu}D_v^2dv&\lesssim_\circ u^3 T_2,\quad
T_2=A^2+B_f^2+A^{3/2}y_c+A y_c^{5/3},\\
(\operatorname{avg}_{v\sim w}D_v^2)^{1/2}&\lesssim_\circ u A
\quad (w\ge c_Hu).
\end{split} \tag{C16}\label{eq:C16}
\]
\end{lemma}\begin{proof}
Regular bands use \eqref{eq:C10}, with \(y_c\gtrsim\sqrt A\); the inner region \(v\lesssim w_c\) uses \eqref{eq:C8} for the first line, also on floor (of mass \(O(u A)\)) for the second.
For contact and comparable-endpoint caps in inner dyads \(v/u\asymp t'\le O(y_c)\), (20)--(21) with constant field slope lower bound \(C_0\) and raw scale \(u y_c\) give for the dyadic mass average of \(D^2/u^2\) at most
\[
O_\circ((A^3 y_c/t'^2)^{2/3}+A^3/t'^2)
\]
apart from negligible cap-width and strip errors. Clip also by \(O(y_c^2)\); summation weighted by \(t'\) costs \(O_\circ(A^{3/2}y_c)\).
For noncomparable-endpoint caps meeting the region the upper masses are \(O(u y_c)\) by \eqref{eq:C7} and the width bound. Slice each such cap geometrically into units at masses comparable to their lengths, of scale \(u t'\). On one unit the whole-cap \(\Delta S\le M_j^{-1}\) trace proof and constant cap density cost an additional factor \(O(y_c/t')\) in both trace bounds of (20). Use one height bin there: in (21)'s proof the squared mean-oscillation term costs negligible since \(S\) varies by at most \(M_j^{-1}\), including for reverse ordering (its fourth moment estimate uses \eqref{eq:C8}). Thus the same average is bounded by \(O_\circ(A^3 y_c/t'^3)\), again clipped by \(O(y_c^2)\). There are boundedly many such units per mass dyad by disjointness; summation gives \(O_\circ(A y_c^{5/3})\). All band calculations can stop at \(v\sim x^{10}\) with negligible cost. This proves \eqref{eq:C16}.
For completeness, the additional broad-cap cost can be seen directly.
Writing \(y=y_c\), its normalized mass sum is bounded by
\[
\sum_{t'\,\text{dyadic}}t'\min\{y^2,A^3y/(t')^3\}.
\]
The two terms meet at \(t'_0=A y^{-1/3}\). Below this scale the
geometric sum is at most \(Cy^2t'_0\); above it the sum is at most
\(CA^3y/(t'_0)^2\). Both equal \(O(Ay^{5/3})\).
If the crossover is outside the finite summation range, the corresponding
one-sided geometric bound is only smaller. The original-floor raw
contribution is at most \(O(Ay^2)\), absorbed in this same term since
\(y=o(1)\). The analogous crossover for comparable caps is
\(t'_0=A^{3/2}/y\), giving \(O(A^{3/2}y)\).
The trace conversion on a sliced cap uses the lower mass bound of that
slice. The whole-cap numerator is divided by its smaller slice mass,
which is precisely the factor \(y/t'\) already paid above. No trace
estimate is applied below its lower mass cutoff. These are mass
estimates; the radius estimates used in Lemma~\ref{bud:box-moments}
and in the coarse profile proof were proved separately by radius slides.
\end{proof}
\begin{lemma}[High projection and centered primitives]\label{bud:high-primitive}
Work in the buffered violation constructed above, with
\(d_0=uL\), \(L\ge a_{\rm goal}Z\),
\(h_1=u\max(L,\sqrt\lambda)\), and
\(G_1=\min(\max(L^2,\lambda),\Lambda)\).
Here \(Z\ge1\) is the deterministic majorant in the global parameter
from Lemma~\ref{bud:envelope}; it is not a random variable under the
positive spin law. For a high dyad \(v\sim w\ge c_Hu\), replace an
overlap at the observed buffer fork by the inner product of the two
branches projected to a fixed cut of order \(w\), after their
separation and strictly before the dyad. In each fixed moment the cost
is at most
\[
O_{p,\circ}(z_w),\qquad z_w=x^3\min\{(w\sqrt{C(w)})^{-1},\lambda^{-1/2}w^{-2}\}.
\tag{C17}\label{eq:C17}
\]
The integral of \(L_v-\mathbb EL_v\) over a high dyad, against any
deterministic weight bounded by one, has \(L^2\) norm
\(o(u^2G_1)\), with a fixed power saving. This includes the plateau
for every edge multiplier used on the comparison path.
\end{lemma}
\begin{proof}
The first option in \eqref{eq:C17} is the postprojection bound obtained
after \eqref{eq:C9}. The actual field gives the second option:
\(dh\gtrsim\lambda w^2\,dv\) on comparable active intervals by
\eqref{eq:P4} and bounded active density; the edge ramp loss is
absorbed there. Even at the plateau or a macroscopic band, a preceding
active interval is available at a comparable scale. Telescope both
branches in regular layers to obtain the stated minimum.
We next estimate the centered primitive. On the active part,
\eqref{eq:A1} gives
\[
\left\|\int k_v(L_v-\mathbb EL_v)\,dv\right\|_2
\lesssim_\circ x^3\min\{C(w)^{-1/2},\lambda^{-1/2}/w\}
\lesssim_\circ u^2X^3\Lambda^{-1/3}\lambda^{-1/6}.
\]
For the first bound integrate \(\mathbb EW\) using PE (20) on
comparable caps and contact, and in radius on the floor. Complete a
partial cap before estimating it. This option also holds on the
plateau. The second option uses the actual-field slope just given;
balancing the two in \(w/u\) gives the last expression. Its ratio to
\(u^2G_1\) is power small by \eqref{eq:C18} below.
For the plateau, first suppose \(\chi_e>0\). From \eqref{eq:C13},
\({\cal N}\le x^2\lambda^{-s_0}Z\le U^2Z\).
On a plateau strip of width \(d'\), Cauchy--Schwarz and monotonicity
compare \(S\) to the constant reference \(q\) on neighboring
intervals of length comparable to \(d'\). The denominator is
\(\sqrt{d'}\), so the resulting oscillation bound is
\[
\operatorname{osc}S\lesssim {\cal N}/\sqrt{d'}
\lesssim ZU^2/\sqrt{d'}.
\]
On the capped left strip, use also the preceding active interval of
mass comparable to \(d_-\). Its reference varies by at most
\(O(E_b)=O(U^2/\sqrt{d_-})\), which fits the same bound. On the
capped right strip, boundedness suffices, because its width is
\(d_+=U^4\) and \(U^2/\sqrt{d_+}=1\).
The edge slope on this strip is
\(dh/dv\gtrsim\lambda_1\chi_e U^2/(d')^{3/2}\).
Thus \eqref{eq:A1} bounds the squared primitive norm there by
\[
O_\circ\left(
\frac{x^6(d')^{3/2}}{\lambda_1\chi_e U^2}
\frac{ZU^2}{\sqrt{d'}}\right)
=O_\circ\left(\frac{x^6Zd'}{\lambda_1\chi_e}\right).
\]
Summing the dyadic strips gives the edge bound
\(O_\circ(x^3\lambda_1^{-1/2}\chi_e^{-1/2}\sqrt Z)\).
When \(\chi_e=1\), the auxiliary-removal bound at \(w\asymp1\)
is \(O_\circ(x^3u^{-3/2}\Lambda^{-1/2})\). Taking the smaller
bound, and using \(\min(a,b)\le a^{2/3}b^{1/3}\), gives
\[
O_\circ(u^2X^3\Lambda^{-1/3}\lambda_1^{-1/6}Z^{1/6}).
\]
Equation \eqref{eq:C18} again makes its ratio to \(u^2G_1\)
power small, including the dependence on \(Z\).
Other values of \(\chi_e\) occur only at
\(\lambda=x^{p_*}\), with \(\chi_{\rm lin}=1\).
The violation bound supplies the useful lower bound
\[
u^2G_1\ge u^2\sqrt\lambda L
\ge x^2\lambda^{1/2-s_0}Z.
\]
There are three cases.
\begin{enumerate}[label=(\alph*)]
\item If \(\chi_e\ge x^{h_e}\), the edge bound divided by this
lower bound is at most
\[
O_\circ\left(
x^{1-h_e/2}\lambda_1^{-1/2}\lambda^{s_0-1/2}/\sqrt Z\right)=o(1).
\]
\item If \(\chi_e\frac14.
\]
The minimum occurs at \(\xi=s_0/2+1/4\). Thus a large majorant
only improves this ratio; no pointwise upper bound on \(Z\) was used.
\end{proof}
\subsection{Quantitative one-row closure}
\begin{proposition}[Buffered one-row closure]\label{bud:closure}
In any of the boxes of \eqref{eq:C15} we claim
\[
\operatorname{avg}|S_i-\Gamma_s(K_s(i))|=o(u^3 P_1). \tag{C19}\label{eq:C19}
\]
\end{proposition}\begin{proof}
Apply \eqref{eq:A7} through degree two at \(K_s\), with the
untilted law, the two observed labels \(a,b\), and test \(1\).
Each mismatch insertion is \((b-C)\delta\). Linear terms vanish
termwise after averaging spins under the actual law.
For the remainders, expand the cavity-spin test to order two, then each
quotient term of degree \(j\le2\) to order \(2-j\), from the
independent-row endpoint to the actual row. Taylor's integral remainder
has three mismatch factors on a fixed finite extension of the tree.
It retains \(\tau_a\tau_b\), either in the interpolated test or in
one deterministic scalar coefficient; every additional cavity spin
occurs in an inserted pair.
After deleting the row, nonnegative spin-only bounds transfer by the
bounded-covariance Gr\"onwall rule, also during the bounded positive
semidefinite row changes below. Deletion itself costs
\(O(m^{-1})\ll u^3P_1\), since
\(P_1\ge\sqrt\lambda\,a_{\rm goal}^2\).
Keep deleted overlaps during sign arguments, so the other mismatch
factors use only the remaining rows. Changing this convention costs
another \(O(m^{-1})\). Moving a revelation cutoff means an affine
positive semidefinite interpolation of the row before applying PE (2).
It does not change the conditioning in a term already being estimated.
The allocation rule after \eqref{eq:A1} supplies all fork measures.
Distinct free vertices away from the observed fork have bounded joint
densities in their mass coordinates, by sequential attachment.
Repeated attachments at one vertex use its single variable.
Ordering restrictions are deterministic and may be retained or dropped
when taking bounds. An inserted pair is \emph{pinned} if its two leaves
split at the observed fork. These give the only split atoms, since
diagonal pairs are omitted. Every other pair is called low or high
according as its split mass is below or above \(c_Hu\).
\begin{enumerate}[label=(\roman*)]
\item Terms with no pinned pair: two low errors cost \(O_\circ(\int_0^{c_Hu}D^2)\) by marginalization and Cauchy--Schwarz, whether at the same vertex or not. One low and one high cost \(O_\circ(u^3 T_1 A)\) using distinct-vertex densities. Any extras here can be dropped.
If all errors are high one gains an extra \(O_p(u)\) low covariance. Indeed turn off the relevant cavity-row revelation below \(c_Hu\), retaining its later cumulative covariance by a jump there. This includes the scalar row law containing \(\tau_a\tau_b\) if that factor occurs in a deterministic coefficient: restoring that law costs \(O(u)\) by PE (2) since \(K_s=O(u)\) there. Products can be treated factor by factor. With low revelation off, the term containing those two spins vanishes by sign flips of a whole branch component (negating the attached row Gaussians if interacting); they are in distinct components and all other pairs are internal.
In an interpolated interacting row this is an admissible PSD interpolation (move the cumulative row-specific field and rest-row couplings, or their affine mixture with the scalar row, to the jump). Spin-only tests on remaining rows are invariant. The new pair for restoring this row costs \(O_p(u)\) at each low split by \eqref{eq:C8} and bounded row transfer, since its coefficient, apart from \(O(1/m)\), is bounded by \(O(|Q_v^{\rm deleted}|+u)\). Take large fixed moments here and slight higher-than-second moments on the two high errors, costing arbitrarily small power losses by boundedness and \eqref{eq:C16}. Old topology and weights remain fixed during restoration. Thus all-high terms cost \(O_\circ(u^3 A^2)\).
\item At degree at least three with pinned errors: for three pinned factors use one \eqref{eq:C15} averaged RMS saving and pointwise higher orders on two, paying \(O(\lambda^{0.07}d_0^3)\) (or with arbitrarily small extra losses); \(\lambda^{0.07} L^2\ll G_1\).
For two pinned and one free factor use the pointwise costs \(d_0^2\) times \(O_\circ(u A+u^2 T_1)\). With one pinned and two free, use a high fixed moment for the pinned error and second moments with slight power loss for the others. The latter two integrated cost at most \(O_\circ(u^2 A^2+u^3 T_2+u^3 A T_1)\). These bounds and the preceding ones are sufficient by \eqref{eq:C18}.
\item At degree two with one pinned error, a low free error costs
\(O_\circ(d_0\lambda^{0.07}u^2T_1)\), using second moments and the
average at the observed fork. For a high free error, fix its dyad
\(v\asymp w\) and project the pinned overlap as in
\eqref{eq:C17}. The error, including the free-vertex mass length, is
\[
O_\circ(wz_w\,uA),\qquad
wz_w\lesssim u^2X^3\Lambda^{-1/3}\lambda^{-1/6}.
\]
It is \(o(u^3P_1)\) by \eqref{eq:C18}.
For the projected term we must keep its joint law fixed as \(v\)
varies. Fix the placement topology and all other free vertex
coordinates. Use a projection cut fixed throughout the dyad, prune the
pinned branches at that cut, and retain one path through the high
vertex to its terminal spin. The sharing of these retained data is
independent of \(v\): any sharing through the high fork already
extends beyond the projection cut. All discarded branches have
normalized continuation kernels. Let \(F\) be the projected centered
pinned factor; it is measurable with respect to the retained earlier
data. Conditional on those data and the high prefix, the high pair has
independent continuations with conditional mean \(X_v\). Hence
\[
\mathbb E[F(Q_v-S_v)]
=\mathbb E[F(|X_v|^2-S_v)]
=2\mathbb E[F(L_v-\mathbb EL_v)].
\]
The second equality conditions the retained terminal spin on its prefix.
The scalar coefficients and allocation restrictions are bounded
deterministic weights \(k_v\), and \(\|F\|_2=O(d_0)\).
Lemma~\ref{bud:high-primitive} therefore gives
\[
\left|\mathbb E\left[F\int k_v\,2(L_v-\mathbb EL_v)\,dv\right]\right|
\le2\|F\|_2
\left\|\int k_v(L_v-\mathbb EL_v)\,dv\right\|_2
=o(d_0u^2G_1)=o(u^3P_1).
\]
This uses unconditional Cauchy--Schwarz on the fixed marginal law.
If other slots attach at the same high vertex, there is still only one
integration in its mass. Fubini and the power margins permit summation
over topologies and dyads.
\end{enumerate}
\paragraph{Two insertions pinned at the observed fork.}
It remains to treat both degree-two pairs pinned. Write each centered overlap as \(m_i+\eta_{\rm pair}\) with \(m_i=|X_i|^2-S_i\) on the common prefix and \(\eta_{\rm pair}=Q_i-|X_i|^2\). Single \(\eta\) terms vanish by conditioning.
For two \(\eta\)'s the coefficients individually gain \(O(u)\): either a third child has entered at the fork (coefficient \(O(u)\)), or both pairs run across the two occupied children. In the latter case the total number of scalar spins, including old spins, in each child is odd; at least one scalar moment factor has a sign cancellation on turning off scalar revelation through this cut (keep later cumulative clock via a jump after the observation, or take its immediate right limit). Indeed the two children then have independent sign symmetry. If using cuts immediately to the right for a fixed placement with the same children, they separate from all further distinct fork vertices eventually; the restoring bound by PE (2) is uniform \(O(u)\). Thus \eqref{eq:C15}'s final bound and \eqref{eq:C18} suffice.
The factor \(\mathbb E m_i^2\) is common to all pinned topologies,
so extract it before summing coefficients. We claim that the remaining
signed scalar coefficient, before the common \((b-C)^2\), is
\(\Gamma_s''(K_s(i))/2\) at almost every buffered cut.
To identify it, compress a small neighborhood of the observed mass
into a clock jump at that mass, retaining the observation at its
original intermediate clock value. At a continuity cut there is room
to move this value in both directions: \(K_s\) is strictly increasing
locally because its field contains \(C_0r\). On an already extended
old tree, increase the common subincrement at the distinguished vertex
by variance \(\epsilon\), and decrease the following subincrement
in every occupied child by the same amount. The covariance direction
is then one for pairs entering different children of that vertex and
zero for same-child pairs, diagonals, and pairs outside its subtree.
Total leaf variances and all other split covariances stay fixed.
Thus this is exactly the pinned direction, even when earlier
insertions have occupied more than two children.
In the quotient expansion of \eqref{eq:A5}, the degree-two scalar
terms are
\[
N_2-N_1\Psi_1+N_0(\Psi_1^2-\Psi_2).
\]
In every term with positive denominator degree, allocate one whole
such \(\Psi\) moment last. Previously placed factors pull back from
the induced old genealogy, and the extracted prefix expectation is
independent of placement. The complete signed sum of that denominator
moment is therefore a covariance derivative of the constant-one
observation on this genealogy, hence zero. This cancellation requires
the full signed sum; it is not a row-sum identity for a nonconstant
restricted test. Coincidences within the permitted children remain
part of that sum.
Only \(N_2\) remains. Its coefficient includes the Taylor factor
\(1/2!\). The two subincrements at the same positive mass compose
into the original scalar transition, so moving their intervening split
changes only the evaluation time of \(\Gamma_s\). The coefficient
is consequently \(\Gamma_s''/2\). This calculation introduces no
additional split atom: a new attachment at a same-mass subdivision
without old divergence has zero coefficient, because that edge has
zero length and one continuing child. A different vertex with the
same numerical mass likewise creates no atom at the observed fork.
Finally shrink the compressed neighborhood. Bounded-clock path limits
with the cut retained pass the aggregated coefficients to the original
path. Other freely varying fork masses in the shrinking interval have
vanishing absolute weight. The scalar identity
\(\Gamma_s''=\mathbb EV_{zzz}^2-2\alpha_t\mathbb EV_{zz}^3\)
uses exactly the fork mass at the intermediate observation. At the
limit this remains its mass, since \(K_s\) grows strictly locally
and the spatial derivatives and path moments converge. Null contact
and jump exceptions do not affect either box integral. This proves
the claimed coefficient identity.
By the drift estimates at \eqref{eq:C14}, \(\Gamma_s''(K_s(i))=O(d_0+\lambda u)\). The \(m_i^2\) average by \eqref{eq:C15} is at most \(O(d_0^2\lambda^{0.14})\); their product suffices since \(\lambda^{0.14}L(L+\lambda)\ll G_1\). This completes \eqref{eq:C19}.
In particular the error budget remains relative to the removal strength:
\[
\frac{u^3P_1}{d_0h_1^2}
=\frac{G_1}{\max(L^2,\lambda)}
\asymp\min(1,R_n)
\]
with the notation used next. This observation will be needed when
\(R_n\) tends to zero; an absolute \(o(1)\) error would not suffice.
\end{proof}
\subsection{Buffered rigidity and completion of the budgets}
\begin{proof}[Proof of Lemma~\ref{bud:strengthening}]
The ratio \(R_n\) defined below compares the removal strength with the
curvature scale of the rescaled box. We first treat \(R_n\to\infty\),
with a second rescaling if the box shrinks, and then bounded \(R_n\),
including \(R_n\to0\). The closure error is
\(o(\min(1,R_n))\); its relative form is essential when calibration
at support points requires division by \(R_n\).
Use \(r=r_0+h_1 z,\ \theta=d_0/h_1\) (local notation, unrelated here to size-sweep time), and indices \(n\) along the growing systems. Put
\[
\begin{aligned}
m_n(z)&=(\alpha-\beta_0)(r)/d_0,\qquad
\widetilde m_n(z)=(\widetilde\alpha-\beta_0)(r)/d_0,\\
Y_n(z)&=(S-r)/d_0,\qquad
\varphi_n(z)=(\widetilde r(\alpha(r))-r_0)/h_1,\\
g_n(z)&=[\Gamma_s(k_*(r))-r]/(d_0 h_1^2).
\end{aligned}
\]
The first three functions are bounded on compact sets with a common bound, and we can take \(\beta_0'\to c_\beta>0,\ c_q=1/c_\beta\) in these local ranges. We have on compacts
\[
g_n''=-A_*\widetilde m_n+o(1),\qquad \varphi_n=z+\theta(Y_n+o(1)),
\]
the curvature statement a.e. (also distributionally, since the scalar first derivatives of \(\Gamma\) are absolutely continuous) with \(A_*\) now its positive limiting value. Here \(\varphi_n\) is a nondecreasing map. Indeed Taylor expand \(\widetilde r(v)\) at \(K_*(v),\ v=\alpha(r)\); the argument increment is \(b(S-q)-C_0(S-r)+d\) and \(\Gamma_*''=O(\lambda u)\), so the quadratic error is \(O(\lambda u d_0^2)=o(u^3P_1)\). Also \(q(v)-r=d_0(c_qm_n+o(1))\). Hence \eqref{eq:C19}, with \(d=o(u^3P_1)\), gives
\[
\begin{aligned}
g_n(\varphi_n)&=R_n Y_n + D_n(Y_n-c_qm_n)+e_n,\\
R_n&=\Gamma_*'(K_*(v)) C_0/h_1^2,\qquad
D_n=[1-b(v)\Gamma_*'(K_*(v))]/h_1^2 .
\end{aligned}
\tag{C20}\label{eq:C20}
\]
where \(e_n=o(\min(1,R_n))\) in box-averaged absolute value (both measures as before) on all moving boxes of length comparable to \(\theta\) in compact \(z\)-ranges. By \eqref{eq:P2}--\eqref{eq:P5},
\(D_n\asymp\lambda u^2/h_1^2\lesssim1,\ D_n=o(R_n)\); the pointwise lift pays the small \(v^2\) offset in \eqref{eq:P3}, and \(U\lesssim u\). On compacts \(D_n\) varies by \(o(1)\), \(R_n\) by \(o(R_n)\), by the active derivative bounds (\(b'=O(\lambda u),\ \Gamma_*''=O(\lambda u)\)). We subselect repeatedly.
We have uniform limiting bounds on \(g_n,g_n'\) independent of compact interval. Indeed intervals of length a sufficiently large fixed multiple of \(\theta\) contain nonfloor physical mass \(\gtrsim d_0\) by the buffer bounds on \(m_n\); there \(R_n|Y_n|\le O(R_n/(d_0 M_j))=o(1)\). Thus \eqref{eq:C20} calibrates bounded values at bounded spacings, sufficient by the curvature bound. Take a \(C^1_{\rm loc}\) limit \(g\). Recall \({\cal H}'=2C_0(r-S)\) in radius and
\[
{\cal H}(\alpha'-\beta_0')\le0,\qquad {\cal H}\ge-O(C_0/M_j^2)
\]
locally since \(\beta_0'\) lies strictly between floor and cap. On nonfloor \({\cal H}\le0\).
\paragraph{Unbounded rescaled removal.}
Suppose first \(R_n\to\infty\). Equation \eqref{eq:C20} gives \(Y_n=o(1)\) in box-averaged radius, hence uniformly on compacts (a violation persists over a fraction of a box on one side, by monotonicity of \(S\)). Then \(\widetilde m_n-m_n\to0\) in local \(L^1\): \(\widetilde r(\alpha(r))-r=o(d_0)\), and by Fubini the area between distributions on each such interval of length \(O(h_1)\) costs \(o(d_0)\) times \(O(h_1)\) mass (crossings are local by monotonicity). Write
\[
J_n(z)=\Gamma_*'(K_*(\alpha(r_0)))\,{\cal H}(r)/(2d_0 h_1^3),\qquad
\ell_1=\lim c_q D_n .
\]
The variational sign gives \(J_nm_n'\le0\). Equation
\eqref{eq:C20} gives
\(J_n'=-R_n(0)Y_n=-g_n-\ell_1m_n+o(1)\), where the error tends
to zero in the moving radius-box integrals. Nearby nonfloor
calibrations give \(J_n=O(\theta)\), with a common local bound;
the normalized \(M_j^{-2}\) potential error is \(o(\theta)\).
Take a locally uniform limit \(J\) and a weak-star limit
\(m_n\rightharpoonup m_\infty\). Then
\(J'=-g-\ell_1m_\infty\) and \(g''=-A_*m_\infty\).
For clarity, the energy estimate follows by two integrations by parts.
On a fixed interval, integrate \(J_nm_n'\le0\), substitute the
equation for \(J_n'\), and use
\(m_n=-g_n''/A_*+o(1)\). This gives
\[
0\ge[J_nm_n]+\int g_nm_n+\ell_1\int m_n^2+o(1),
\qquad
\int g_nm_n
=-[g_ng_n'/A_*]+\int g_n'^2/A_*+o(1).
\]
Consequently
\[
\int (g_n'^2/A_*+\ell_1m_n^2)\le [g_n g_n'/A_* -J_n m_n]_{\rm left}^{\rm right}+o(1).
\]
All boundary factors are uniformly bounded, independently of the fixed
interval. Passing to the limit and then enlarging the interval first
shows that \(g'\in L^2(\mathbb R)\). Bounded curvature makes
\(g'\) uniformly continuous, hence it tends to zero at both ends.
If \(\theta\to0\), then \(J=0\), so both boundary terms vanish
as the endpoints tend to infinity. If \(\theta\) has positive
limit, the measure
\(d(c_\beta z+(\lim\theta)m_\infty)\) has support unbounded in
both directions. Nonfloor approximants give \(J=0\) on that support,
because the floor density vanishes. Send the endpoints to infinity
through its support; again both boundary terms vanish. The energy
estimate now gives \(g'=0\), and \(g''=-A_*m_\infty\) gives
\(m_\infty=0\). Finally \(J'=-g\) and boundedness of \(J\)
give \(g=0\). For positive limiting \(\theta\), monotonicity of
\(\alpha\) then gives the pointwise contradiction
\(|m_n(0)|=1\).
\paragraph{Unbounded removal: the shrinking-box subcase.}
If \(\theta\to0\), we have \(\ell_1>0\) since \(h_1^2=\lambda u^2\). Put \(K_n(t)=J_n(\theta t)/\theta,\ M_n(t)=m_n(\theta t)\). They are bounded on compacts with uniform limiting bounds, \(K_n'=-\ell_1 M_n+o_{L^1_{\rm loc}}(1)\) by the moving box estimate and \(g=0\), \(K_n M_n'\le0\). \(M_n\) converge on subselection in local \(L^1\) and at continuity points by monotonicity with the deterministic trend added. Now \(\ell_1\int M_n^2\le-[K_n M_n]+o(1)\) on fixed intervals. The limit thus has finite global squared norm; sending endpoints to infinities through continuity points where it tends to zero makes that norm zero. Again monotonicity gives the pointwise contradiction.
\paragraph{Bounded rescaled removal, including a vanishing limit.}
Suppose now that \(R_n\) is bounded. Then \(\theta=1\).
After adding their deterministic trends, \(m_n,Y_n\) are monotone;
subselection therefore gives almost-everywhere and local \(L^1\)
limits. The transport estimate also gives
\(\widetilde m_n-m_n\to0\) in local \(L^1\). Indeed on nonfloor
points the displacement \(\widetilde r(\alpha(r))-r\) is
\(o(d_0)\); floor crossings travel only \(O(d_0)\) and have
density \(o(1)\). Fubini bounds the area between the inverse
distributions on a compact interval by \(o(d_0h_1)\).
It follows that \(g''=-A_*m_\infty\).
The normalized potential
\(\mathcal H(r)/(2C_0d_0h_1)\) has derivative \(-Y_n\).
Its lower bound and nearby nonfloor references give a bounded
nonnegative limiting function \(\bar J\).
We next calibrate at the support of
\(\mu=d(c_\beta z+m_\infty)\). The measures
\(\mu_n=d\alpha/h_1\) in this coordinate converge vaguely to
\(\mu\). Every fixed neighborhood of a support point has positive
limiting mass. Its floor mass is \(o(1)\), while the mass integral
of \(|e_n|/R_n\) tends to zero. Hence it contains a nonfloor point
with \(e_n/R_n\to0\). At this point \(Y_n=o(1)\) and
\(D_n/R_n\to0\), so \eqref{eq:C20} gives
\(g_n(\varphi_n)=o(R_n)\). Shrink the fixed neighborhoods and
choose a diagonal subsequence. This produces approximants to each
support endpoint used below. At a contact the potential is zero; in a
cap its nearest zero is within \(M_j^{-1}\), whose normalized error
vanishes. We conclude that \(\bar J=g=0\) on
\(\operatorname{supp}\mu\), even if \(R_n\to0\).
A floor interval can still have positive rescaled radius length; only
its mass has vanished. We therefore retain the separate radius estimate
for \(e_n/R_n\) on any such support gap. After a further subsequence,
for almost every radius coordinate \(z\),
\[
s(z)=\lim\varphi_n(z)=z+Y(z),\qquad
g(s(z))=(\lim R_n)Y(z),\qquad \bar J'=-Y.
\]
The function \(s\) is nondecreasing. On a support gap \((a',b')\),
monotonicity and the endpoint approximants give \(s\in[a',b']\).
The gap is bounded, and \(g''\) is affine there with slope
\(A_*c_\beta>0\).
Suppose \(g'(a')>0\). If \(\lim R_n>0\), then
\(g(s)=(\lim R_n)(s-z)\) forces \(s(z)>z\) almost everywhere
just to the right of \(a'\): this holds if \(s\) is near
\(a'\), where \(g\) is positive to its right, and also if \(s\)
is farther inside the gap. Thus \(\bar J'=-Y<0\), contradicting
\(\bar J(a')=0\) and \(\bar J\ge0\).
If \(R_n\to0\), then \(g(s)=0\), and the sign requires the
relative error. Fix an almost-everywhere point \(z>a'\) where the
radius error divided by \(R_n\) tends to zero. If \(s(z)=a'\),
take nonfloor approximants \(z_n^a\to a'\) as above. Eventually
\(z_n^a0\). Since the value at
the approximant is \(o(R_n)\) and \(R_n\) is asymptotically
constant on compact intervals,
\[
\liminf_n\frac{g_n(\varphi_n(z))}{R_n(z)}\ge0.
\]
Divide \eqref{eq:C20} by \(R_n\). It gives \(Y(z)\ge0\),
contrary to \(Y(z)=a'-z<0\). Every other zero of \(g\) is
bounded away from \(a'\) in this neighborhood, and hence gives
\(s(z)>z\). Again \(Y>0\) almost everywhere just to the right,
contradicting the potential sign. Reflection excludes
\(g'(b')>0\) in both cases.
But \(g(a')=g(b')=0\) and the affine curvature imply
\[
g'(a')+g'(b')=A_*c_\beta(b'-a')^2/6>0.
\]
At least one endpoint derivative must be positive. Thus no support gap
exists.
Finally, the boundedness of \(m_\infty\) makes the support of
\(d(c_\beta z+m_\infty)\) unbounded in both directions. With no gaps
it is all of \(\mathbb R\), so \(g=0\) and \(m_\infty=0\).
In each regime where the normalized discrepancies converge locally in
\(L^1\) to zero, the contradiction at the originally selected point is
pointwise: monotonicity of the inverse distribution, after adding back
its deterministic trend, gives for fixed continuity points \(\pm h\)
\[
-C h+o(1)\le m_n(0)\le C h+o(1).
\]
Here one may take \(C=c_\beta/\theta_\infty\) when
\(\theta\to\theta_\infty>0\), and \(C=c_\beta\) in the bounded
removal case \(\theta=1\).
Letting \(h\downarrow0\) contradicts \(|m_n(0)|=1\).
For the second-rescaling sequence the identical argument is applied to
\(M_n\), whose deterministic trend has slope tending to \(c_\beta\).
No pointwise relative error at the selected point is used.
This proves \eqref{eq:C12}, completing the finite induction. \end{proof}
\begin{proof}[Completion of Proposition~\ref{bud:targets}]
At its last stage the radius assertion converts by \eqref{eq:C7} to \eqref{eq:C3}; the averaged excess \(O(x^{4-2\zeta}u\Lambda^{1/3})\) suffices there. This completes the local comparison required for \eqref{eq:C1}.
Here are the final exponent checks. The local pressure derivative has
scale \(\lambda u\) when it is used to bound
\(u\int_{v\asymp u}A_v\,dv\). The optimized excess therefore contributes
\[
x^{4-2\zeta}\lambda^{k_c/3-1}
=x^4\lambda^{-2s_0}
\bigl(x^{-2\zeta}\lambda^{k_c/3-1+2s_0}\bigr).
\]
The parenthesized factor is power small because
\(k_c/3>1-2s_0\) and \(\zeta\) is chosen after \(p_*\).
The discrepancy term in \eqref{eq:C3} contributes
\(u^4a_{\rm goal}^2=x^4\lambda^{-2s_0}\).
Since \(U^4=x^4\lambda^{-2s_*}\), the remaining factor
\(\lambda^{2\tau_0}\) pays dyadic logarithms and the permitted small
losses. At the smallest dyad \(u\asymp U\), the switch error from
\eqref{eq:P11}, divided by \(\lambda u\) and then by \(U^4\), is at most
\[
x^{c'}\lambda^{5s_*/2-1},
\]
which has a strict saving. Larger dyads only improve this estimate.
The global budget handles the remaining dyads
\(u\ge\lambda^{b_\ell}\), as in the initial reduction.
Both parts of \eqref{eq:C2} were already obtained from the global
comparison, with a final-strength power independent of subsequent
sufficiently small choices of \(p_*\). Thus the local induction does
not enter the choice of the later size Taylor order.
\end{proof}
==== END The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law//sections/07-budgets.tex ====
==== BEGIN The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law//sections/08-statistics.tex ====
\section{Statistics for the genuine paths}\label{sec:statistics}
The cavity equations require moments of the random overlap $Q$ and
of its centered version $Q-S_v$. We derive them from two outputs of
the comparison argument. The mass-averaged budget \eqref{eq:C1}
first gives deterministic averaging windows for the single-path
observable $L_v$. A useful window must contain enough field increase
for projection while keeping the variation of $S_v$ small. Balancing
these requirements gives the pointwise and integrated low-mass
estimates below, including the direct-path estimates needed during
the strength comparison.
On size paths, the pointwise scalar-trial excess in \eqref{eq:C2}
has a separate role: it gives uniform coarse accuracy in every fixed
moment. That argument uses the single-path inequalities and the scalar
gap, and does not use the refined low-mass estimates. Its power saving
therefore fixes the Taylor order before the final choice of $p_*$.
We then combine the refined low-mass estimates with the pair inverse
to close the high-mass covariance equation. Throughout, $S_v$ is
the untilted deterministic mean, including when a tilted law is tested.
\subsection{Scales and the pointwise estimates}
Fix the path parameters and the site count $m$. Unless stated
otherwise, expectations and $L^p$ norms in this section are then under
the genuine untilted positive law. Strength variations use the direct
path with multipliers $(0,1)$. The comparison budget \eqref{eq:C1}
allows us to choose a deterministic number $1\le Z\le x^{-C}$, depending
on these fixed data, such that $\mathbf E_\gamma Z^4\lesssim_\circ1$ and,
with $Y=UZ$,
\begin{equation}
\int_{v\asymp r} (S_v-q(v))^2\,dv/r\le C Y^4/r^2\qquad(r\gtrsim U). \tag{D1}\label{eq:D1}
\end{equation}
For example, let $B$ be the maximum dyadic budget appearing in
\eqref{eq:C1} and take $Z=1+C(B/U^4)^{1/4}$. This explains both the
fourth parameter moment and the pointwise bound \eqref{eq:D1}.
The same estimate holds on fixed comparable enlargements of the dyads.
Since $(S_v-q(v))^2\le A_v$, the budget indeed controls the mean error.
The number $Z$ is random only when the parameters $\gamma$ are averaged;
it is fixed under every positive-law expectation below.
We first assume $Y=o(1)$, with $U=x\lambda^{-s_*/2}$. Large $Y$ and
multiplier changes are treated separately. We will also use a
\textbf{low-region variant}: the path is a size path,
$Y=UZ=o(1)$ satisfies \eqref{eq:D1} at small masses, and refined estimates
are requested only below a sufficiently small fixed mass. Put
\begin{equation*}
z_r^0=\frac{x^3}{\sqrt{\lambda_1}\,r^3},\qquad
g_0=\lambda^{(3s_*-1-\eta)/2},\qquad g_B=g_0/Z^{3/2}.
\end{equation*}
Thus $z_r^0=g_0(U/r)^3$. These are dimensionless projection scales,
not scalar fields.
In envelope estimates with arbitrarily small power losses, the envelope itself may be chosen for the desired loss: more precisely we use a geometric slack \(b_\epsilon=x^\epsilon\) with \(\epsilon>0\) arbitrarily small, and lose fixed powers of \(b_\epsilon\), as well as subpower factors. Power losses in the resulting applications, involving any fixed moment/expansion orders, can thus all be taken as small as desired. We write \(\lesssim_\diamond\) with this convention. Exponents in such estimates below do not cost an extra power of \(Z\) unless explicitly shown.
\begin{proposition}[Pointwise moments]\label{prop:statistics-pointwise}
Assume the direct-path hypotheses above and $Y=o(1)$, or their low-region
size-path variant. For every fixed $1\le p<\infty$ the following estimates hold:
\begin{equation}
\begin{array}{ll}
\||X_v|^2\|_p+\|L_v\|_p+\|Q_{\rm split\ at\ }v\|_p\lesssim_p v+Y;\\
\|Q_{\rm split\ at\ }v-q(v)\|_p+
\|2L_v-q(v)\|_p+\||X_v|^2-q(v)\|_p\lesssim_{p,\diamond} e(v)
& (v\gtrsim Y).
\end{array} \tag{D2}\label{eq:D2}
\end{equation}
In the low-region variant the second line is only low. \(e\) is deterministic conditional on the parameters, with, on each such dyad \(v\asymp r\),
\begin{equation*}
x^C\le e(v)\lesssim r,\qquad
\int e(v)^2\,dv/r\lesssim_\diamond m_r^2,\qquad m_r=Y^2/r.
\end{equation*}
On those regions \(|d(v)|+|\dot d(v)|\lesssim_\diamond\lambda r^2 e(v)\), by the corresponding path estimates. At \(v\lesssim Y\) use \(r=e=Y\) in this last assertion and \eqref{eq:D2}.
\end{proposition}
The envelope $e$ depends on the deterministic path data and on the
comparison majorant $Z$, not on a sample from the positive path law.
We first construct it, including the windows needed in the proof.
\subsection{Deterministic widths and averaging windows}
All window lengths in this construction are measured in $z$;
mass integrals are converted by $dv=G_z\,dz$. On the active interval
of a direct path, set $z=\xi\in[0,z_*]$, where $z_*=\xi_*$.
Extend this coordinate increasingly over the plateau by
\begin{equation*}
dz=U^2[(1-w+d_+)^{-3/2}+(w-p_{\rm top}+d_-)^{-3/2}]\,dw,
\qquad w=v/\ell,
\end{equation*}
and write $G_z=dv/dz$. The function $q(v(z))$ is Lipschitz. On the
active interval $G$ has bounded derivatives of every fixed order and
is bounded above and below near zero, although it may vanish elsewhere.
On the plateau put
\[
\mathfrak h(z)=\max\left(
\frac{U^2}{\sqrt{w-p_{\rm top}+d_-}},
\frac{U^2}{\sqrt{1-w+d_+}}\right).
\]
The explicit coordinate derivative makes this width Lipschitz and
gives $G_z\mathfrak h^3\asymp U^4$. At the two ends of the plateau
its width is comparable respectively to $E_b$ and to one.
The extended $z$ interval has bounded length. Partition it into dyadic
bands $z\asymp r$ near zero and finitely many macroscopic intervals
above, placing a boundary at the active endpoint. On each band its
length is comparable to $r$, and
$r\asymp\min(z,1)\asymp v$. Fixed comparable enlargements may be
used for buffers. We work on bands $r\gtrsim Y$.
On their active parts \eqref{eq:P4} gives a lower field slope, with
the negative ramp slope absorbed as in \eqref{eq:P7}. On their plateau
parts the formulas for $dz/dw$ and $H_e$ give $dh/dz\gtrsim\lambda_1$;
these masses are macroscopic. Thus throughout the construction,
\begin{equation}
dh\ge c\lambda_1r^2\,dz. \tag{D3}\label{eq:D3}
\end{equation}
Write $f(v)=S_v-q(v)$ here; this $f$ is not a pressure.
Monotonicity, \eqref{eq:D1}, and $q(v)\lesssim v$ imply
$S_v\lesssim v+Y$.
The active width has three requirements: enough mass for projection,
enough room for the mean error, and compatibility with the endpoint
ramp. The next four functions impose these requirements. Put \(r(z)=\min(z,1)\). Reflect the active \(G_z\) evenly at the endpoints for defining integrals outside, and add an artificial constant \(x^K\), \(K\) arbitrarily large fixed. Use
\begin{equation*}
H_0(z)=\min(b_\epsilon r(z),\,l_0),\qquad
l_0^2\int_{z-l_0}^{z+l_0}(G_{s,\mathrm{refl}}+x^K)ds
=U^6/r(z)^3.
\end{equation*}
Add the cone envelope
\begin{equation*}
H_f(z)=\sup_{s\ {\rm active}}
[\min(b_\epsilon r(s), |f(v(s))|/Z^2)-A|s-z|]_+
\end{equation*}
with a sufficiently large constant \(A\). Near the upper active endpoint, writing \(d_*=z_*-z,\ E=E_b\), add also the two functions
\begin{equation*}
(E/2-d_*)_+,\qquad
[\,2\min(E,b_\epsilon)-(d_*-E)_+\,]_+ .
\end{equation*}
Take the maximum of the four functions as \(\mathfrak h(z)\). We only need this width at \(z\gtrsim Y\). Finally set \(e(v(z))=\min(r(z),Z^2\mathfrak h(z))\).
We will repeatedly compare the mass of neighboring intervals in the
$z$ coordinate. On the active side, if $J$ and $J'$ have comparable
length $l\le Cb_\epsilon$, lie within $O(l)$ of one another, and stay
inside a fixed comparable band, then
\[
\sup_J G\le C\left(\frac1{|J'|}\int_{J'}G\,dz+x^K\right).
\]
Here and below $K$ may be increased to make additive errors negligible.
To prove this, expand the smooth nonnegative function $G$ to a fixed
order at the scale $l$. Choose that order so that the Taylor remainder
is at most $x^K$. On the rescaled bounded intervals, polynomial norm
equivalence bounds the supremum by the average on any comparable
subinterval; nonnegativity permits replacing the polynomial's absolute
average by that of $G$, with the same remainder. The constants depend
on the fixed Taylor order, not on the interval. For a reflected
interval meeting an endpoint, restrict to its part on the original
side, whose length is comparable. This gives the same comparison.
\begin{lemma}[Width construction]\label{lem:statistics-widths}
The preceding construction gives a deterministic envelope with the
size, integral, and defect bounds stated in
Proposition~\ref{prop:statistics-pointwise}. The random moment bounds
\eqref{eq:D2} will be proved in the next subsection. The widths and
windows have the following additional properties.
\begin{enumerate}[label=\textup{(\roman*)},leftmargin=*]
\item At comparable points, \(\mathfrak h(z')\lesssim\mathfrak h(z)+|z'-z|\), with comparability of widths when \(|z'-z|\) is a sufficiently small fraction of \(\mathfrak h(z)\). Also \(x^C\le\mathfrak h\lesssim r\), \(e\gtrsim b_\epsilon Y^2/r\), \(|f|\lesssim b_\epsilon^{-1}e\), and \(\int_{\rm band}G_z(\mathfrak h/r)^2 dz/r\lesssim_\diamond(U/r)^4\).
\item Unless \(Z^2\mathfrak h\gtrsim b_\epsilon r\), there are past and future windows of \(z\)-length comparable to \(\mathfrak h\), adjacent or at that distance, each with average \(G_*\) there satisfying
\begin{equation}
\sup G_z\lesssim G_*,\qquad G_*(\mathfrak h/r)^3\gtrsim (U/r)^6 . \tag{D4}\label{eq:D4}
\end{equation}
They can be taken small fixed fractions of the indicated scales. From such a nontrivial point there is also remaining forward \(z\)-length at least a small fixed macroscopic constant (for taking longer layers up to a macroscopic mass), with \eqref{eq:D3} holding using each successive scale.
\item Partition any full band as above into consecutive tiles of lengths comparable to \(\mathfrak h\), with this width and \(e\) comparable over each tile. Writing \(\mu_I=\int_I G_z dz/r\),
\begin{equation}
\sup_I G_z\lesssim b_\epsilon^{-C}(r\mu_I/|I|+x^K),\quad
\Delta_I S\lesssim b_\epsilon^{-1}\min(r,Z^2 |I|),\quad
\sum_I \sup_I(e/r)^2\lesssim Z^2 . \tag{D5}\label{eq:D5}
\end{equation}
The variation assertion holds also on comparable-scale enlargements of these tiles within the path.
\end{enumerate}
\end{lemma}
\Needspace{6\baselineskip}
\begin{proof}
\leavevmode\par
\paragraph{The intrinsic width.}
The map
$l\mapsto l^2\int_{z-l}^{z+l}(G_{s,\mathrm{refl}}+x^K)ds$
is strictly increasing. Ball inclusion therefore compares its solution
at neighboring centers. Together with the cap $b_\epsilon r$, this
proves the stated quasi-Lipschitz bound for $H_0$ and comparability
within a small fraction of $H_0$. Boundedness of $G$ gives
\[
H_0\gtrsim\min(b_\epsilon r,U^2/r).
\]
The adjacent-interval comparison at this radius also gives
$G_zH_0^3\lesssim U^6/r^3$, up to an arbitrarily small additive error.
Since $G$ is bounded, the interpolation
\[
G_zH_0^2=(G_zH_0^3)^{2/3}G_z^{1/3}\lesssim U^4/r^2
\]
proves the squared width budget after integration over a band of
$z$-length $O(r)$ and division by $r^3$.
\paragraph{The deviation cone.}
For each source point $s$, the function
$[\min(b_\epsilon r(s),|f(v(s))|/Z^2)-A|s-z|]_+$
is $A$-Lipschitz. Their supremum is therefore Lipschitz too. At its
source, a height of order $l$ implies $|f|\gtrsim Z^2l$.
If $f$ is positive, monotonicity of $S$ and Lipschitz continuity of
$q(v(z))$ preserve half this deviation on a forward interval of
length $cl$; for a negative deviation use the backward interval.
Thus each source of height $l$ supplies a one-sided interval $J_s$ with
\[
Z^4l^2\int_{J_s}G_z\,dz
\lesssim \int_{J_s}f(v(z))^2G_z\,dz.
\]
Its cone is supported within $O(l)$ of $s$. The adjacent-interval
comparison bounds the mass of that support by the mass of $J_s$, with
negligible additive error. At each dyadic height $l\ge x^K$, choose
separated source points to cover the cone superlevel set. The
corresponding intervals have bounded overlap at that height. Summing
the preceding inequality, and then the logarithmically many heights,
gives
\[
\int_{\rm band}G_zH_f(z)^2\,dz
\lesssim_\diamond Z^{-4}\int_{\rm enlarged\ band}f(v)^2\,dv
\lesssim_\diamond U^4/r.
\]
Sources of height below $x^K$ contribute negligibly. The lower boundary
causes no difficulty because $l\le b_\epsilon r$. If a positive
source's forward interval reaches the upper active endpoint, its
deviation persists through the plateau. The macroscopic version of
\eqref{eq:D1} then gives $Z^4l^2\lesssim Y^4$, or $l\lesssim U^2$;
these sources already obey the same macroscopic squared width budget.
\paragraph{The plateau and its active edge.}
The reference $q$ is constant on the plateau. A positive
deviation of $S-q$ persists towards its upper end; a negative deviation
persists towards its lower end. Integrate \eqref{eq:D1} over the
corresponding endpoint strip, whose mass is comparable to the distance
to that endpoint. Away from the capped strips, the result is
$|f|\lesssim Z^2\mathfrak h$. Inside the upper $d_+$ strip,
$\mathfrak h\asymp1$ and the bound follows from boundedness of $S,q$.
For a negative deviation inside the lower $d_-$ strip, the backward
interval must instead use
the active side. The last active $z$-interval of length $E=E_b$ has
mass comparable to $d_-=U^4/E^2$. Either $|f|=O(Z^2E)$ already, or
the Lipschitz variation of $q$ over that interval is too small to
remove half the negative deviation. In the latter case \eqref{eq:D1}
gives
\[
|f|^2\frac{U^4}{E^2}\lesssim Y^4,
\quad\hbox{hence}\quad |f|\lesssim Z^2E,
\]
which is the required lower-edge bound.
The explicit plateau formula also gives
$\int_{\rm plateau}\mathfrak h^2\,dv\lesssim_\circ U^4$ by
integrating $U^4/(d+d_\pm)$ from each endpoint. The first extra active
function has height $O(E)$ on a region of mass $O(U^4/E^2)$, so its
squared cost is $O(U^4)$. The second has height at most
$2\min(E,b_\epsilon)$; its part inside the last $E$ costs the same.
For its additional layer of length $2\min(E,b_\epsilon)$, compare
adjacent active intervals at that length. Their mass is controlled by
the preceding last-$E$ region, giving again $O_\diamond(U^4)$.
These two functions also majorize the active ramp defect and its
parameter derivative divided by $\lambda$, with at most a
$b_\epsilon^{-1}$ loss. The bounds in \eqref{eq:P6}--\eqref{eq:P8}
then give the claimed defect estimates. Below $Y$, those estimates
use only the low and mask bounds from \eqref{eq:P6}, \eqref{eq:P8}.
\paragraph{Matching the two coordinates at the join.}
Suppose first that $E\lesssim b_\epsilon$, and let $d_*=z_*-z$ on
the active side. We claim, for $d_*\le Cb_\epsilon$, that
\[
|f(v(z))|/Z^2\lesssim E+d_*.
\]
For $d_*\lesssim E$, use the last active interval of length $E$ or
the adjacent plateau, in the direction in which the deviation
persists. The preceding mass calculation proves the claim.
For $d_*\gtrsim E$, an interval of length comparable to $d_*$ fits
on the active side in either direction. Unless the claimed bound
already holds, the deviation persists on that interval. Adjacent
interval comparison bounds below its mass by that of the last active
length $E$, up to negligible error. Applying \eqref{eq:D1} as above
would then give $|f|\lesssim Z^2E$, proving the claim in this case too.
With $A$ sufficiently large, sources of the cone obeying this linear
bound imply $H_f(z)\lesssim E+d_*$. The same bound holds for $H_0$:
a ball of radius $C(E+d_*)$ contains the last active length $E$, and
its squared radius times that mass is at least a constant times $U^4$,
which exceeds the intrinsic target $U^6/r^3$ on this macroscopic band.
When $E>b_\epsilon$, the corresponding upper bounds follow from the
caps in $H_0,H_f$. The explicit endpoint functions provide the lower
bound near the join. Thus at distances much smaller than $E$, both
the active and plateau widths are comparable to $E$. Their separate
quasi-Lipschitz bounds consequently remain valid across the join.
\paragraph{Windows and tiles.}
Consider first an active point with $d_*\ge E$ for which
$Z^2\mathfrak h0$. At each
fixed fork, declare an exception when the squared projection exceeds
$x^{-\varepsilon}$ times its squared scale in \eqref{eq:A3}.
Choose the projection moment sufficiently high after $\varepsilon$
and the required negligible exponent $K$ are fixed. Markov's
inequality then makes this exception have probability $O(x^K)$.
The factor $x^{-\varepsilon}$ is included in the permitted losses
below. Bounded spins,
integration over the fork coordinate, and conditional Jensen make its
contribution negligible in the integral norm. This does not require an
event simultaneous over all real forks, or a higher-moment bound for
$|A|$ proportional to the bootstrap constant. These operations can
first be performed with inserted deterministic cuts and then passed to
the fixed-size path limit.
\paragraph{Raw-moment bootstrap.}
Fix a large even exponent $p$ and set
\[
B_0=\max\left(1,\sup_v\frac{\||X_v|^2\|_p}{v+Y}\right).
\]
This number is finite at each fixed size because $|X_v|\le1$ and
$Y>0$. At a small scale $r\gtrsim Y$ in the regular inverse region,
let $J$ be a fork window of $z$-length comparable to $r$. We claim
\begin{equation}
\left\|\int_J W_z\,dz/r\right\|_{p/2}
\lesssim_{p,\circ}(z_r^0)^2r^3B_0.
\tag{D6}\label{eq:D6}
\end{equation}
Use an initial block of arbitrarily small inverse-polynomial length,
then blocks starting at doubled distances from the fork, and finally
a terminal residual. Stop the doubling at a fixed small macroscopic
distance, with a full preceding interval still inside the regular
region. Consider a block pair other than two initial blocks, and let
$lr$ be its later starting distance. If $l\lesssim1$, the projection
gap has mass at least $cr$ and field length at least
$c\lambda_1r^3l$. Its squared projection coefficient, before the mass
weight, is consequently
$O_\circ(x^6/(\lambda_1r^5l))$.
The earlier squared anchor has the integral bound
\[
\left\|\int_J|A_z|^2\,dz/r\right\|_{p/2}
\le C_p lB_0r.
\]
Indeed, \eqref{eq:global-grid} integrates translated grids of disjoint
increments, and the raw bootstrap bounds the terminal squared norm on
the enlarged window by $CB_0r$. If that anchor belongs to the
independently resampled continuation, apply
\eqref{eq:global-dual-doob} to its nonnegative squared increment to
average it back to the fork. Thus the contribution from this block
pair is at most
\[
\underbrace{Cr}_{\text{mass weight}}\,
\underbrace{C_{p,\circ}\frac{x^6}{\lambda_1r^5l}}
_{\text{squared projection coefficient}}\,
\underbrace{C_p lB_0r}_{\text{integrated squared anchor}}
\lesssim_{p,\circ}(z_r^0)^2r^3B_0.
\]
The two initial blocks instead use the sliding estimate without
projection. Choose their length so small that the resulting term is
negligible after the polynomial prefactors.
For a later starting distance comparable to $r'\gtrsim r$, use a
full gap at masses comparable to $r'$, with field length at least
$c\lambda_1(r')^3$. This includes a macroscopic $r'$ for the
terminal block. The mass weight, squared projection coefficient, and
raw squared anchor now cost respectively
\[
Cr',\qquad
C_{p,\circ}\frac{x^6}{\lambda_1(r')^5},\qquad
C_pB_0r'.
\]
Their product is
$O_{p,\circ}((z_{r'}^0)^2(r')^3B_0)$, no larger than the
right side of \eqref{eq:D6}. The normalized fork window has bounded
length. Summing the logarithmically many block pairs in
\eqref{eq:global-block-decomposition} proves \eqref{eq:D6}. The
sliding and conditional integration bounds are uniform in the inserted
grid sizes, so this argument uses no size-uniform path approximation
rate and no derivative of $S$.
Let $L_J=|J|^{-1}\int_JL_u\,du$ for a future mass window
$J$ of width comparable to $r$. The future bracket identity in
Lemma~\ref{global:buffer-brackets} gives
$|X_v|^2\le2\mathbf E_vL_J$, while $\mathbf EL_J=O(r)$ by the
deterministic mean bound. Equations \eqref{eq:A1} and \eqref{eq:D6}
give
\[
\|L_J-\mathbf EL_J\|_p^2
\lesssim r^{-1}\left\|\int_JW_u\,du/r\right\|_{p/2}
\lesssim_{p,\circ}(z_r^0)^2r^2B_0.
\]
Conditional Jensen therefore yields
$\||X_v|^2\|_p\lesssim_pr(1+z_r^0\sqrt{B_0})$.
For $v\lesssim Y$ use a future window at scale $Y$; at macroscopic
masses use $|X_v|\le1$. Taking the supremum in the definition of $B_0$
now gives $B_0\le C_p(1+g_0\sqrt{B_0})$, including the harmless
subpower factors. The fixed power saving in $g_0$ absorbs these
factors, and this quadratic inequality bounds $B_0$ uniformly.
For \(L,Q\), project endpoints into a future interval at comparable scale \(v+Y\) and then telescope both branches in future doubled layers with terminal block, as above. Each tail product uses \eqref{eq:A3} on the later layer (with comparable preceding clock fitting strictly after the fork); the opposite anchor has norm \(O_p(\sqrt{r'})\) at worst at scale \(r'\), so these tails cost \(O_{p,\circ}(z_{r'}^0 r')\). For \(L\) the anchor \(X_v\) already precedes the gap. This gives the first line of \eqref{eq:D2}. It suffices here too to telescope at small scales.
\paragraph{Refined-moment bootstrap.}
For this argument write $Z_h=Z$ on direct paths and $Z_h=1$ in the
low-region size-path variant. Let $B_1\ge1$ be the least constant for
which
\begin{equation*}
\||X_{z'}-X_z|^2\|_p
\le B_1Z_h^2(|z-z'|+\mathfrak h(z))
\end{equation*}
holds for comparable nearby cuts in the bands $r\gtrsim Y$ and their
buffers. The supremum defining $B_1$ is finite at fixed size because
$\mathfrak h\ge x^C$; take it simultaneously over the bands to avoid
artificial interior boundaries. At $z\asymp Y$ the raw bound already
suffices with a $b_\epsilon^{-1}$ loss, as it does on the large-width
auxiliary ramp. Cuts whose separation is not a small fraction of their
scale can likewise use the raw bound.
Let $J$ be one of the windows in \eqref{eq:D4} near $z_0$, and put
$h_0=\mathfrak h(z_0)/r$, so $|J|\asymp rh_0$. We claim
\[
\left\|\int_J W_z\,dz/r\right\|_{p/2}
\lesssim_{p,\diamond}(z_r^0)^2r^3h_0B_1Z_h^2.
\]
For a block pair whose later starting distance is $lr$ with
$l\ll1$, the same mass and projection factors as above apply. The
integrated earlier anchor satisfies, in both ranges of $l$,
\[
\left\|\int_J|A_z|^2\,dz/r\right\|_{p/2}
\le C_p lB_1Z_h^2rh_0.
\]
When $l\ge h_0$, the bootstrap and quasi-Lipschitz width bound give
$\||A_z|^2\|_{p/2}\le CB_1Z_h^2lr$ pointwise; integration contributes
$|J|/r\asymp h_0$. When $l0\), on all the bands in \eqref{eq:D7} together,
\begin{equation}
\int_{\{e(v)\ge x^{\delta'}\}} D_v^2\,dv
\lesssim_\diamond x^{-C_0\delta'} z_1^0 Y^2 Z^2 \tag{D8}\label{eq:D8}
\end{equation}
with an absolute finite loss exponent \(C_0\) independent of \(\delta'\). \end{lemma}
\begin{proof}
Retain every tile meeting $\{e\ge x^{\delta'}\}$. Such a tile
has $r\gtrsim x^{\delta'}$, and width comparability makes $e$ at least
a constant times $x^{\delta'}$ throughout it. Write
$a_I=\sup_I e/r$ and recall that its mass is $r\mu_I$.
Bracket within a retained tile using mass windows of width $h$.
The pointwise error bound in \eqref{eq:D2} gives squared accuracy
$O_\diamond(r^2a_I^2)$ near the tile edges. Since their mass is
$O(h)$, this costs $O_\diamond(ra_I^2h)$ in $dv/r$.
Monotonicity and $\Delta_IS\lesssim_\diamond ra_I$ give that same
bound for the interior mean errors. The tile estimate for $W$ in the
preceding proof gives centered cost
$O_\diamond((z_r^0)^2r^3Z^2\mu_I/h)$. Balance these two terms by taking
\[
h=\frac{rz_r^0Z\sqrt{\mu_I}}{a_I},\qquad
ra_I^2h+\frac{(z_r^0)^2r^3Z^2\mu_I}{h}
\lesssim r^2z_r^0Z a_I\sqrt{\mu_I}.
\]
If $h$ does not fit in the tile, use the raw squared error on the whole
tile instead. Its cost $r^2a_I^2\mu_I$ is no larger than the displayed
bound when $h\gtrsim r\mu_I$. This also handles overlapping buffers.
Tiles of negligible mass may first be discarded so the additive
$x^K$ errors in the $W$ bound do not enter the denominator.
The trace contribution is bounded by the tile $W$ estimate divided
by $r$. For the mixed term use Cauchy--Schwarz and the raw fourth
moment of $X$ on the tile. These two contributions together cost at
most $O_\diamond(r^2z_r^0Z\mu_I)$. To sum them and the variance
bound, the squared envelope budget gives on the retained tiles
\[
\sum_I\mu_I
\lesssim_\diamond x^{-2\delta'}\frac{Y^4}{r^2}
\lesssim_\diamond x^{-C\delta'}Y^4,
\qquad
\sum_I a_I\sqrt{\mu_I}
\le\Big(\sum_Ia_I^2\Big)^{1/2}
\Big(\sum_I\mu_I\Big)^{1/2}
\lesssim_\diamond Zx^{-C\delta'}Y^2.
\]
The second inequality uses \eqref{eq:D5}. Converting $dv/r$ back to
$dv$ multiplies a band bound by $r$, and $r^3z_r^0=z_1^0$.
Since $Y=o(1)$ and $Z\ge1$, the sum of the mixed and trace terms is
also bounded by $x^{-C\delta'}z_1^0Y^2Z^2$. Summing the logarithmically
many retained bands gives \eqref{eq:D8}, with a fixed exponent $C_0$
independent of $\delta'$.
\end{proof}
\begin{lemma}[The lower band]\label{lem:statistics-lower-band}
At \(v\le C Y\) we will discard \(v0$, then a finite moment order large
enough that \eqref{eq:D2} makes the tail negligible; $W$ is bounded.
Choose the geometric slack in \eqref{eq:D2} last, small enough for that
moment order. The same truncation of the other factors at a fork,
followed by Cauchy--Schwarz in $dv/r$, bounds its alias errors by the
same expression.
At \(x\le v\le C Y\) a centered \textbf{linear} integral of \(2L_v-S_v\) with bounded coefficients instead costs \(O_\circ(E_0)\) before any mass normalization, by \eqref{eq:D9} and \eqref{eq:A1}. When there are no other random factors in the expectation of that linear term, no alias error is required even with an initial density tilt: conditional averaging reduces the overlap to \(|X_v|^2\) and then \(2L_v\) exactly.
\end{proof}
\paragraph{Stopping, forests, and the order of truncation.}\label{par:statistics-tilt-transfer}
For a tilted active path on the stop, \eqref{eq:A4} represents
its expectation as an untilted expectation multiplied by a density
having every fixed moment required here. H\"older therefore transfers
the integrated second moments and squared errors in
\eqref{eq:D7}--\eqref{eq:D10}. The needed norm is slightly above
$L^2$. Interpolation between its $L^2$ bound and its deterministic
bound loses an arbitrarily small power, since the target scale is
polynomial in $x$ and the path observables are bounded.
The same observation applies to the low linear centered integral.
In a finite product, first truncate the other factors using their
fixed moments from \eqref{eq:D2}; their deterministic envelope bounds
then cost only the chosen small power loss, and their discarded tails
are negligible. These operations concern active subtrees or bounded
deterministic values at masked pairs, as specified in \eqref{eq:A4}.
They do not by themselves give a common marginal law when the topology
varies. Such a law must first be constructed for the centered integral
and its retained tests.
For a forest, apply the change of measure separately on each active
component. A bounded number of component densities has every required
fixed moment on the stop, so H\"older handles their product. Neither
this transfer nor \eqref{eq:A2} is an assertion about a common untilted
root for the whole forest. When other random path factors remain,
first realize the centered integral and those factors under one fixed
marginal law, apply the unconditional centered estimate, and only then
apply H\"older. The existence of that marginal when a fork coordinate
varies is proved explicitly in Lemma~\ref{lem:statistics-low-sources}.
\begin{corollary}[Projection to a common low cut]\label{cor:statistics-low-projection}
One more consequence of \eqref{eq:A3}, \eqref{eq:D2} is the following: for \(Y\lesssim c'\) up to a sufficiently small fixed scale, a pair split well before \(c'\) (say mass \(\le c'/8\)) changes in overlap by at most \(O_{p,\circ}(z_{c'}^0 c')\) in every fixed moment upon projecting down from later cuts to mass \(c'\). \end{corollary}
\begin{proof}
Use future doubling layers up to a small macroscopic mass before the long terminal block, hitting the later layer in each product via \eqref{eq:A3} with a full gap after the fork (also for projected rather than terminal ends). This is the two-branch raw argument of \eqref{eq:D6}.
\end{proof}
\subsection{Coarse paths and multiplier changes}
\begin{lemma}[Coarse and switch contributions]\label{lem:statistics-coarse-switch}
For sufficiently small fixed $\kappa>0$, the direct-strength contribution
from $Y>x^\kappa$ satisfies the direct part of
Proposition~\ref{prop:increment}. The two final-strength multiplier
changes also satisfy its required power-saving bound.
\end{lemma}
\begin{proof}
\leavevmode\par
\paragraph{The large-$Y$ event.}
Choose $\kappa>0$ small relative to the power gains below.
Markov's inequality and $\mathbf E_\gamma Z^4\lesssim_\circ1$ give
\[
\mathbf P_\gamma(Y>x^\kappa)\lesssim_\circ U^4x^{-4\kappa}.
\]
We first bound the contribution at each parameter value on this event.
Here $U$ retains its cap $\min(1,x\lambda^{-s_*/2})$. Since
$v\,dh/dv\gtrsim\lambda_1U^3$ for $v\ge x$, the source estimate
and $0\le S\le1$ give
\begin{equation*}
\int_{v\ge x}\mathbf EW\,dv\lesssim e_1^2,
\qquad e_1=\frac{x^3}{\sqrt{\lambda_1}U^{3/2}}.
\end{equation*}
With unit raw bounds, mass slides of width $e_1$ then cost
$O(e_1)$ in both their mean and centered parts. The trace term on
$v\ge\max(e_1,x)$ costs $O(e_1)$ by division by $v$; below $e_1$
use the unit raw bound. Cauchy--Schwarz on dyads handles the mixed
covariance term. Hence
\[
\int_{v\ge x}D_v^2\,dv\lesssim_\circ e_1.
\]
The dotted clocks in the strength derivative are $O_\circ(\lambda)$.
In the pressure difference from \eqref{eq:A1}, the portion $vx^\kappa\). The deterministic width properties there only use \eqref{eq:D1}, \(S,q\) bounded, and do not require \(Y=o(1)\). The plateau lower slope in \eqref{eq:D3} comes from $\chi_eH_e$;
for $\chi_e>0$ its use therefore costs $\chi_e^{-1}$. The active
part has at least the same weaker bound. Consequently for \(H=H_{\rm lin}\) or \(H_e\), respectively,
\begin{equation*}
\mathbf E\int H^2 W\,dv\ \lesssim_\diamond\
\chi_e^{-1}x^6\lambda_1^{-1} Z^2
\begin{cases}
x^6,\\ \lambda_1^2 U^4.
\end{cases}
\end{equation*}
Indeed both are supported at masses bounded below, the first kernel is uniformly \(O(x^3)\), the second bounded by \(O_\diamond(\lambda_1\mathfrak h)\); apply the tile expectation bound of \eqref{eq:D7} with that same loss \(\chi_e^{-1}\) and the integrated squared width estimate. By \eqref{eq:A1}, \eqref{eq:A4} the derivative difference along each multiplier costs the square root of this bound on the stopped set (again exact linear conditional averaging); \(\chi_e^{-1/2}\) is integrable. Both \(x^6/\sqrt{\lambda_1}\) at final strength and \(x^3\sqrt{\lambda_1} U^2\) allow power saving beyond \(x^5\). This handles the multiplier changes.
\end{proof}
\subsection{Uniform coarse accuracy on size paths}
The pointwise trial excess in \eqref{eq:C2} provides an independent
starting estimate on size paths. We turn it into an integrated bound
for $|S-q|$ by a convex variation of the mass coordinate, then use the
scalar gap to make that bound uniform. The final moment upgrade uses
the same block and buffer identities as above with unit raw bounds.
Its purpose is to supply one positive exponent for every fixed moment
order before choosing a Taylor degree.
\Needspace{4\baselineskip}
\begin{proposition}[Uniform all-moment accuracy]\label{prop:statistics-size-accuracy}
Here \(\lambda=x^{p_*}\); assume \(p_*\) sufficiently small. Uniformly at every parameter value on smoothed size paths in the untilted system there is a fixed \(\delta_*>0\), available independently of further decreases in \(p_*\), such that
\begin{equation}
\sup_v\|Q-q(v)\|_p\le C_p x^{\delta_*} \tag{D11}\label{eq:D11}
\end{equation}
for a pair split at \(v\), for \textbf{every} fixed \(p\). \end{proposition}
\begin{proof}
Use both coordinates \(v,t=t(v)\). Up to \(t_M=M/\ell^2\), before the terminal constant piece beginning at \(v_M\), both coordinate derivatives are bounded by \(C/\omega\); the two are inverse, with \(v_M=\ell+O(\omega M)\). For \(v>x\) we have \(dh/dt\gtrsim\lambda v^2\) by \eqref{eq:P4}.
\paragraph{Freezing beyond the wall.}
At \(t\ge t_M/2\) and thereafter \(1-q\) and \(1-S\) are smaller than every power of \(x\). In fact for the scalar spin the conditional mean martingale above a fixed macroscopic time has diffusion coefficient between \(c(1-V_z^2)\) and \(C(1-V_z^2)\), giving exponential decay of \(\Gamma'\) as after \eqref{eq:P2}. Thus \(dh/dt\ge c>0\) above a sufficiently large fixed time (with \(b\) constant there). For each actual spin, the conditional mean \(m_i=\sqrt m (X_v)_i\) has martingale bracket at least \(c(1-m_i^2)^2dh\) there, by the \(i\)-th diagonal source-Hessian bound. Indeed the field contributes diffusion matrix \(m H_v\) to all these coordinates (the source-Hessian and clock identities in~\cite[Section~2]{PE}); matrix innovations are independent. More explicitly, since the masses here are bounded below,
$H_{ii}\ge c(1-m_i^2)/m$. The field contribution to quadratic
variation is at least
$m^2H_{ii}^2dh\ge c(1-m_i^2)^2dh$; the independent matrix
innovations only increase it. With $dh/dt\ge c$, localized It\^o
calculus for $F(u)=\sqrt{1-u^2}$ gives
\[
\frac d{dt}\mathbf EF(m_i(t))
\le -c\mathbf E
\frac{(1-m_i(t)^2)^2}{(1-m_i(t)^2)^{3/2}}
=-c\mathbf EF(m_i(t)).
\]
Thus its expectation decays exponentially, uniformly in $i,m$. The calculation works first across filled finite increments at masses bounded below, then by approximation. Averaging proves the assertion; since \(Q\le1,\ \nu Q=S\), it gives \eqref{eq:D11} in this region already.
\paragraph{Integrated fluctuation control.}
Let \(\mathfrak a=1/200,\ V_0'=x^{\mathfrak a}\), with \(p_*<\mathfrak a/100\). On \([V_0',v_M]\), \eqref{eq:A1} gives
\begin{equation*}
\int \mathbf E W\,dt\lesssim x^6/(\lambda (V_0')^3).
\end{equation*}
Let $R=x^6/(\omega\lambda(V_0')^3)$. The bounds on the two
coordinate derivatives give
\[
\int\mathbf EW\,dv
=\int\mathbf EW\frac{dv}{dt}\,dt\lesssim R,
\qquad
\int\mathbf EW\Big(\frac{dt}{dv}\Big)^2dv
=\int\mathbf EW\frac{dt}{dv}\,dt\lesssim R.
\]
The second integral is the one required by \eqref{eq:A1} for a
$t$-average, since that average has weight $dt/dv$ when expressed in
mass. Mass slides and $t$-slides can therefore each use width
$R^{1/2}$, with unit raw bounds. They give
\begin{equation}
\int_{V_0'}^{v_M} D_v^2 dv+
\int_{t(V_0')}^{t_M} D_{v(t)}^2 dt\le x^{1/4}. \tag{D12}\label{eq:D12}
\end{equation}
Use the bracket identities of Lemma~\ref{global:buffer-brackets}, with raw unit bounds, monotone \(S\), and width \((x^6/(\omega\lambda (V_0')^3))^{1/2}\) in each coordinate, using \(dt/dv\) in the \(t\)-averages in \eqref{eq:A1}. The \(\operatorname{tr} C^2\) terms cost at most \(C x^6/(\omega\lambda(V_0')^4)\) and Cauchy suffices for \(X^{\mathsf T}CX\) (coordinate lengths at most \(O_\circ(1)\)). All these bounds have the indicated slack.
\paragraph{A convex mass variation.}
We use the trial gap in \eqref{eq:C2} to control the integral
of $|S-q|$ in mass. Let $\phi$ be a smooth bump of either sign,
supported between $V_0'$ and $\ell$, and set
\[
\frac1{m_\theta(v)}=\frac1v-\theta\frac{\phi(v)}{v^2},
\qquad 0\le\theta\le1.
\]
The bumps used below have small slope and $|\phi|/v\ll1$, making
$m_\theta$ an increasing bi-Lipschitz change of mass. Reassign each
clock increment formerly at $v$ to $m_\theta(v)$, keeping the increment
itself, the terminal length, and the terminal self correction fixed.
Let $F(\theta)$ be the resulting pressure.
The exponential variational formula expresses each step power mean
as a conditional supremum of expectation minus relative entropy
divided by the step mass. After iteration, the complete pressure is
a supremum over the choices of transition laws; the entropy
coefficients are the reciprocal masses $1/m_\theta(v)$, which are
affine in $\theta$. Thus $F$ is convex. The initial mass-zero
expectations are unchanged. Passing from step clocks to the fixed-size
limit preserves convexity. The segment formula \eqref{eq:A1} gives
\[
F'(0)=\frac14\int(Bb'+2Sh')\phi\,dv.
\]
The clocks are Lipschitz on the bump at fixed size, so this chain rule
also follows directly by approximating with continuous observation cuts.
In the new mass coordinate $w=m_\theta(v)$, compare to the fixed scalar
trial $K(w)$. Square completion along the affine interpolation to that
trial gives
\[
F(\theta)\le f_{\rm spin}(K)+P(\theta),\qquad
P(\theta)=\int\frac{(K(w)-h(v))^2}{4b(v)}\,dw,
\quad v=m_\theta^{-1}(w).
\]
At $\theta=0$, \eqref{eq:C2} bounds this trial excess by $x^{4+c}$.
Convexity therefore yields
\[
F'(0)\le F(1)-F(0)
\le x^{4+c}+P(1)-P(0).
\]
Differentiating the penalty and returning to the old coordinate gives
\begin{equation*}
P'(\theta)=\tfrac14\int(z_\theta^2b'+2z_\theta h')
\partial_\theta m_\theta(v)\,dv,
\qquad z_\theta=\frac{K(m_\theta(v))-h(v)}{b(v)}.
\end{equation*}
If the oscillation of $K$ on the swept support is at most
$x^{3\mathfrak a}$, then $z_\theta-z_0=O(x^{3\mathfrak a})$.
Also
$\partial_\theta m_\theta=\phi(1-\theta\phi/v)^{-2}
=\phi[1+O(\sup|\phi/v|)]$.
These are the two errors in comparing $P'(\theta)$ to $P'(0)$.
Integrating that comparison in $\theta$ gives
\begin{equation}
\int [(B-z_0^2)b'+2(S-z_0)h']\phi
\le 4x^{4+c}+C(x^{3\mathfrak a}+\sup|\phi/v|)\int(b'+h')|\phi|. \tag{D13}\label{eq:D13}
\end{equation}
Here \(z_0=q-d/b\) is bounded and \(|d|\lesssim_\circ x^3\) throughout, by \eqref{eq:P6}.
Partition $[V_0',\ell]$ into cells of mass width
$l=x^{5\mathfrak a}$. Give each cell a fixed enlargement, with
uniformly bounded overlap. Discard the boundary cells and every cell
whose enlargement has $K$-oscillation greater than $x^{3\mathfrak a}$.
Since $K$ is nondecreasing and its total increase is $O(M)$,
\[
\#\{\hbox{discarded interior cells}\}\,x^{3\mathfrak a}
\lesssim M,
\qquad
|\hbox{discarded mass}|\lesssim l+Ml/x^{3\mathfrak a}.
\]
On each retained enlargement, Lipschitz continuity of $\Gamma$ gives
$q$-oscillation $O(x^{3\mathfrak a})=o(x^{\mathfrak a})$.
If $|S-q|\ge Cx^{\mathfrak a}$ somewhere in its cell, monotonicity
of $S$ preserves that sign, at half the threshold, on a one-sided
interval of length comparable to $l$ in the enlargement: use the
forward side for a positive deviation and the backward side for a
negative one.
Choose a smooth bump $\phi$ on this one-sided interval, with that sign,
height $cl$, slopes bounded by a sufficiently small fixed constant,
and $\int|\phi|\asymp l^2$. Its swept support remains in the good
enlargement. Moreover $|\phi|/v\lesssim l/V_0'=x^{4\mathfrak a}$;
the small slope and this bound make all $m_\theta$ increasing and
bi-Lipschitz. Since $|d|\lesssim_\circ x^3$, $b\asymp1$, and
$q\gtrsim V_0'$ on this support, $z_0=q-d/b\ge0$ and
\[
(S-z_0)\phi\gtrsim x^{\mathfrak a}|\phi|.
\]
Put $I_\phi=\int h'|\phi|$. The error on the right of \eqref{eq:D13}
is at most
\[
C(x^{3\mathfrak a}+x^{4\mathfrak a})(1+x^{-\mathfrak a})I_\phi
=O(x^{2\mathfrak a})I_\phi
=o(x^{\mathfrak a}I_\phi),
\]
because $b'\lesssim h'/V_0'$.
To estimate the remaining term on its left, use the exact identity
\[
B-z_0^2=(S-z_0)(S+z_0)+D_v^2.
\]
The first summand has favorable sign after multiplication by
$b'\phi$. The second can have unfavorable sign only for a negative
bump. If $b'$ meets the bump, its support lies in the regular low
region, where $b'\lesssim\lambda$ and
$h'\gtrsim\lambda(V_0')^2$. By \eqref{eq:D12},
\[
\int D_v^2b'|\phi|\lesssim\lambda l x^{1/4},\qquad
x^{\mathfrak a}I_\phi
\gtrsim\lambda x^{3\mathfrak a}l^2.
\]
Their ratio is $O(x^{1/4-8\mathfrak a})=o(1)$. If the bump lies
outside that low region, $b'=0$ and there is no variance error.
Finally $h'\gtrsim x^3$ everywhere on the bump: use the regular low
inverse before $4v_b$, and $H_{\rm lin}$ thereafter. Thus
\[
x^{\mathfrak a}I_\phi\gtrsim x^{3+11\mathfrak a}
\gg x^{4+c},
\]
contradicting \eqref{eq:D13}. Every retained cell therefore has
$|S-q|\lesssim x^{\mathfrak a}$. Since $S,q$ are bounded, the low
interval of length $V_0'$ and the discarded cells contribute
$O(x^{\mathfrak a})$ and $O(l+Ml/x^{3\mathfrak a})$, respectively.
The strip between $\ell$ and $v_M$ has mass $O(\omega M)$, so its
contribution is $O(\omega M)=o(x^{\mathfrak a})$, including its part
before the frozen region. The remaining tail is controlled by the
freezing estimate for $t\ge t_M/2$. This proves
\begin{equation}
\int |S-q|\,dv\lesssim x^{\mathfrak a}. \tag{D14}\label{eq:D14}
\end{equation}
\paragraph{From integrated means to pointwise means.}
Set \(K_S=h+b S\), nondecreasing, and write \(\Gamma_S\) for its scalar response (time variable), extending both profiles as needed by terminal annealed clocks. Use one common terminal extension for the two monotone clocks.
The area between their graphs can be integrated first in mass or first
in time, giving
\[
\int|\alpha_{K_S}(t)-\alpha_K(t)|\,dt
=\int|K_S(v)-K(v)|\,dv
\lesssim x^{\mathfrak a}
\]
by \eqref{eq:D14} and $K_S-K=bf+d$. Thus
\begin{equation*}
\sup_t|\Gamma_S(t)-\Gamma(t)|\lesssim (1+M)^C x^{\mathfrak a}
\end{equation*}
on the needed times. To see the polynomial constant, interpolate inverse profiles with evaluation time fixed; \(\Gamma(y)/2\) is the root impulse derivative in the profile. Its derivative uses the \hyperref[par:inverse-kernel-estimates]{post/post Hessian kernel} in the proof of Lemma~\ref{inv:sector-response}, which is uniformly \(O(1+M)\) for mixtures as well: the spatial derivative \(L_z^{\,y}(s)\) of the conditional scalar pair moment divided by two is bounded by the labeled source derivative rule (also directly by telescoping the conditional transition log-density when shifting the field, with bounded gradient of each scalar continuation). Impulse formulas pass from step profiles as there. Extend to one common terminal time \(O(M+1)\).
At reference $K_S$, the returning mismatch is exactly
$h+bQ-K_S=b(Q-S)$. Apply \eqref{eq:A7} with $P=1,k=1$.
Its absolute attachment measure has only one prescribed fork: the
tested pair's split at $v$. An insertion tied there costs $D_v$;
a freely placed insertion has bounded mass density and costs
$\int D_u\,du$. The row-deletion contact contributes $O(m^{-1})$.
Consequently
\begin{equation*}
|S_v-\Gamma_S(K_S(v))|\lesssim D_v+\int D_u du+O(m^{-1}).
\end{equation*}
All row changes are bounded uniformly. Integrate over \(00,C_H<\infty\), with \(b_H\gg C_Ha_1,\ a_1\gg C_Ha_2\), for which
\begin{equation}
\sup_{v\ge V_1}|f(v)|\le x^{2+b_H}(Z^2+Z_g),\qquad
D_{H,i}:=\sup_{v\ge V_i}D_v\le x^{3-C_H a_i}Z^2\quad(i=1,2). \tag{D15}\label{eq:D15}
\end{equation}
\end{proposition}
We first isolate the centered low-source estimate needed to invert the
high equations. Its topology assumptions are part of the statement.
\begin{lemma}[Low sources with a retained high pair]\label{lem:statistics-low-sources}
Consider an allocation with bounded degree and bounded topology
weights. The old prescribed topology is either absent or consists of
one pair splitting at mass at least $V_i$; in particular it contains no
prescribed low fork. Call designated factors at $vv$, its projected endpoints are
fixed-time observables of these same retained paths. The centered
inequality remains unconditional, followed by H\"older with the earlier
random factor. Figure~\ref{fig:fixed-marginal} records this configuration.
\end{remark}
\input{figures/fixed-marginal}
\Needspace{6\baselineskip}
\begin{proof}[Proof of Proposition~\ref{prop:statistics-high}]
\leavevmode\par
\paragraph{The covariance equation.}
Fix an old pair $ab$ splitting at or above $V_i$, where
$i\in\{1,2\}$, and put $P=Q_{ab}-S_{ab}$. For every tested pair
$jk$ at or above the same threshold, define
\[
u_{jk}=\nu[P(Q_{jk}-S_{jk})].
\]
These are pair functions on extensions of the old labels $ab$; unused
labels are ignored by marginalization. Apply \eqref{eq:A7} at the
reference $K$, expanding $J=b\delta+bf+d$, where
$\delta=Q-S$. Choose a fixed truncation degree large enough that
\eqref{eq:D11} makes the true Taylor remainder $O(x^K)$ for any needed
fixed $K$. The row-deletion error remains $O(m^{-1})=O(x^6)$.
Every all-deterministic term vanishes against $P$.
Retain the high stochastic linear term on the left. The equations
have the form
\[
u_{jk}-\sum_{cd:\ v_{cd}\ge V_i}{\bf M}_{jk,cd}b_{cd}u_{cd}
=R_{jk}^{\rm low}+R_{jk}^{\rm nonlinear}
+O(x^K)+O(x^6).
\]
The first source includes the stochastic linear term below $V_i$;
the second contains at least two insertions. This left side is exactly
\eqref{eq:H1}, because the positive-law expectation of each inserted
pair with $P$ depends only on the indicated labels. By \eqref{eq:P5},
its gap on retained levels is at least $c\lambda V_i^2$. Proposition
\ref{inv:pair-inverse} therefore inverts it with cost $O(x^{-Ca_i})$,
once $p_*\ll a_i$ is chosen. Evaluate the result at $jk=ab$.
The inverse adds only a bounded number of freely attached labels.
Their attachment densities and existing-fork weights are bounded after
removing the inverse cost; they do not turn a new low fork into a
prescribed atom.
As explained in the \hyperref[par:inverse-limits]{fixed-resolution limit argument}, these operations can first be made on resolutions keeping the tested cut and threshold. At fixed system size gaps persist on fine approximations; inverse domination with block restrictions allows replacing path expectations and coefficients outside that inverse by continuous-path ones with errors tending to zero (empty step intervals can be refined for attachment). Our bounds apply to bounded measurable topology weights from that inverse after its cost factor is removed.
Consider a nonlinear term containing a high stochastic
factor $\delta_{cd}$. Keep $P$ and that factor untruncated. There is
at least one further small factor: truncate it at $x^{\delta_*/2}$,
and bound all other factors uniformly. Equation \eqref{eq:D11}
controls the discarded tails in arbitrarily high fixed moments.
Cauchy--Schwarz on the two untruncated factors gives
\[
|\nu[P\delta_{cd}\,\hbox{remaining factors}]|
\lesssim x^{\delta_*/2}\|P\|_2\|\delta_{cd}\|_2+O(x^K)
\le x^{\delta_*/2}D_{ab}D_{H,i}+O(x^K).
\]
This preserves the exact second-moment factors; no higher norm
proportional to $D_{ab}$ is assumed. Every remaining source contains
only low stochastic factors, so Lemma~\ref{lem:statistics-low-sources}
applies, including to the low linear term.
\paragraph{The coefficient of a sole low factor.}
If just one low insertion occurs at \(v\), its scalar coefficient indeed costs \(O(v)\): in every term of the finite quotient it belongs to a spin pair moment (possibly with other pairs), and all other pairs in that moment, including the tested pair if present, are high. Immediately across the low branching its two ends thus give two different groups of odd counts. Conditional on prefixes up to that cut, group expectations on independent continuations are odd smooth bounded functions (for the odd groups) with bounded gradients by the scalar source rules. Their product costs \(O(K(v))=O(v)\) using Brownian moments and bounded drift. More explicitly, the two odd group functions $F_1(z),F_2(z)$
vanish at zero and have uniformly bounded derivatives, so
$|F_1(z)F_2(z)|\le C|z|^2$. The shared scalar prefix field at time
$K(v)$ has second moment $O(K(v))$ by its Brownian representation
with bounded drift. Hence this scalar coefficient is $O(K(v))=O(v)$.
Additional labels and coincidences within the groups do not change
the odd parity or the derivative bound.
Combining these estimates after inversion gives
\begin{equation*}
D_{ab}^2\lesssim_\diamond x^{-Ca_i}
\bigl(x^{\delta_*/2}D_{ab}D_{H,i}
+x^3\lambda^{-C}Z^2D_{ab}+x^6\bigr).
\end{equation*}
The final $x^6$ is the row-deletion contact error. Increasing the
Taylor degree does not remove it. Take the supremum over $ab$ and
write $D=D_{H,i}$. Choose the power losses so that
$Cx^{\delta_*/2-Ca_i}<1/2$. Absorption then leaves
\[
D^2\lesssim_\diamond x^{3-Ca_i}\lambda^{-C}Z^2D
+x^{6-Ca_i}.
\]
If $D^2\le AD+B$ with $D\ge0$, then $D\le A+\sqrt B$ up to a fixed
constant. Since $Z\ge1$ and $p_*\ll a_i$, the factors
$\lambda^{-C}$ and the permitted small losses can be included in a
fixed $C_Ha_i$. This proves
$D_{H,i}\le x^{3-C_Ha_i}Z^2$ for $i=1,2$.
\paragraph{The longitudinal equation.}
For $r\ge V_1$, apply \eqref{eq:A7} without a random test.
The longitudinal identity \eqref{eq:H2} separates the linear term
into its diagonal and its integral part. The resulting scalar equation
is
\[
[1-b(r)\Gamma'(K(r))]f(r)
=\Gamma'(K(r))d(r)+R_{\rm int}(r)+R_{\rm nonlinear}(r)+O(x^6).
\]
The diagonal is in field units; returning from normalized mass changes
only bounded constants. Its lower bound is
\[
1-b(r)\Gamma'(K(r))\ge c\lambda V_1^2.
\]
For the integral term, the post kernel bound in Section~\ref{sec:inverse}
gains the earlier time $K(u)\lesssim u$ when $u$ is small. It gives
\[
|R_{\rm int}(r)|\lesssim(1+M)^C\left(
\int|d|\,du+\int_{u0\), after taking the allowed losses smaller
than these gains.
\end{lemma}
\begin{proof}
Set \(a_s=r_s/Y\), \(A=w/Y\), and \(m_{r_s}=Y/a_s\). The ratios
\(a_s\) and \(A\) are at least one. At a vertex carrying \(k\) factors,
\eqref{eq:D2} and the integrated envelope bound give
\[
P_k=m_{r_s}^{\min(2,k)}r_s^{k-\min(2,k)}
\]
after integration in \(dv_s/r_s\). Thus one or two factors use the
first or second envelope moment, and each further factor costs \(r_s\).
Changing \(dv_s/w\) to \((r_s/w)dv_s/r_s\) in \eqref{eq:E1} shows that
the multiplier relative to \(\lambda Y^5\) is
\[
F=A^{5-j-l}\prod a_s^{1+k_s-2\min(2,k_s)}
\]
with the additional factor \(\min_s a_s/A\) when \(j=l=2\).
The exponent of an individual \(a_s\) is \(0\) for a single factor,
\(-1\) for a double factor, and \(k_s-3\) for \(k_s\ge3\). Consequently
\[
\begin{array}{c|c}
\text{multiplicities}&F\text{ or an upper bound for }F\\ \hline
l=1&A\\
(1,2)&1/a_{\rm duplicate}\\
(1,1)&\min(a_1,a_2)\\
\text{all other cases with }l\ge2&A^{-1}.
\end{array}
\]
If an assigned vertex lies below \(x\), its integration uses only the
fraction \(x/Y\) of the lower band. In this case at least one \(a_s=1\),
and every entry in the table is at most a constant. Raw bounds in the
two ensembles therefore give the first contribution in \eqref{eq:E2}.
We may now assume that all assigned vertices are at least \(x\).
Fully deterministic products cancel, so there is a stochastic factor.
Suppose first that all stochastic factors occur at one vertex \(s\).
At that vertex, replace the overlaps by the common ancestor norm
squared and then by \(2L_v\) on a path through the fork. The alias
errors are controlled by \eqref{eq:D7}. Center the resulting polynomial
at its untilted expectation and apply \eqref{eq:D10}, treating the
other times and restrictions as deterministic coefficients. After
truncating the other factors at their envelopes, the cost is
\(O_\diamond(z_{r_s}^0r_sZ)\) times \(1\) for \(k_s=1\), or times
\(m_{r_s}r_s^{k_s-2}\) for \(k_s\ge2\). Relative to \(P_{k_s}\), this
gives at least the gain \(g_B/a_s\). On the lower band the same gain
follows from conditional averaging and the linear bound after
\eqref{eq:D9} when there is one stochastic power, and from the
integrated \(D_v^2\) bound when there are two or more. Multiplying by
the appropriate entry for \(F\) gives \(O_\diamond(g_B)\).
If stochastic factors occur at several vertices, select two of them
at scales \(aY,bY\). Their integrated second moments about \(S\), from
\eqref{eq:D7} and \eqref{eq:D9}, are bounded by
\(O_\diamond(g_BY^2/a)\) and \(O_\diamond(g_BY^2/b)\).
First apply unconditional Cauchy--Schwarz to the two selected stochastic
factors at fixed masses, and truncate the remaining factors at their
deterministic envelopes. At a selected vertex of scale \(r=aY\)
and multiplicity \(k\ge2\), Cauchy--Schwarz in mass then gives
\[
\int e(v)^{k-1}D_v\,\frac{dv}{r}
\lesssim r^{k-2}
\left(\int e(v)^2\frac{dv}{r}\right)^{1/2}
\left(\int D_v^2\frac{dv}{r}\right)^{1/2}
\lesssim_\diamond r^{k-2}m_r
(g_BY^2/a)^{1/2}.
\]
Dividing by the raw budget \(P_k=m_r^2r^{k-2}\) gives the factor
\(O_\diamond(\sqrt{g_Ba})\); for \(k=1\), division of
\((\int D_v^2\,dv/r)^{1/2}\) by \(P_1=m_r\) gives the same
factor. Thus the two selected vertices improve the raw product by
\(O_\diamond(g_B\sqrt{ab})\). Since \(\sqrt{ab}\le A\),
this proves the claim whenever \(F\lesssim A^{-1}\).
There are two exceptional multiplicity patterns. For \((1,2)\), let
\(a_d\) be the duplicate scale and \(a_s\) the singleton scale. If both
factors at the duplicate are stochastic, selecting them uses
\(\int D_v^2\,dv/r\) at that vertex. Relative to its raw budget
\(m_r^2\), this gains \(O_\diamond(g_Ba_d)\), so
\(F g_Ba_d=g_B\). If only one is stochastic, selecting it together
with the singleton gives
\[
Fg_B\sqrt{a_sa_d}=g_B\sqrt{a_s/a_d},
\]
which suffices when \(a_s\lesssim a_d\). In the remaining case the
later singleton is linear and stochastic. Integrating it as a centered
path integral gives \(g_B/A\), so the product with \(F=1/a_d\) is
again at most \(g_B\). For \((1,1)\), both factors must be stochastic.
The lower-band case has \(a_1=a_2=1\) and follows directly from
Cauchy--Schwarz. Otherwise the later linear factor has scale \(A\)
and gives \(g_B/A\); multiplying by
\(F=\min(a_1,a_2)\) gives at most \(g_B\).
For completeness, the last centered integration has a fixed marginal
law even though its fork moves. There is only one earlier stochastic
pair to retain. At most one of its leaves descends through the later
fork, so choose that leaf for the \(L\)-path when it is present. After
the alias replacement the other child of the moving fork has no
observable and integrates to one. The retained pair always splits at
its fixed earlier time. A prior projection merely evaluates these
same two paths at fixed later cuts. If neither endpoint descends
through the moving fork, choose a continuation whose separation from
the retained tree is at another fixed vertex. In either case the
moving time occurs only in the \(L\)-observable and in deterministic
kernels. Apply \eqref{eq:A2}, or its consequence \eqref{eq:D10},
unconditionally under this marginal and then apply H\"older with the
earlier random factor. This is the construction in
Remark~\ref{rem:statistics-earlier-pair}; it does not require an
inequality conditioned on that factor. Deterministic ties belong to
the kernels, and stochastic ties form one polynomial power. The
truncation and tilted-law transfers follow the conventions after
\eqref{eq:D10}. This proves \eqref{eq:E2}.
\end{proof}
\subsection{Taylor remainders and separation of the masses}
On size paths, \eqref{eq:D11} makes each insertion small in every fixed
moment. A sufficiently high fixed degree therefore bounds the Taylor
remainders using the absolute allocation measure. The differentiated
clock weights in \eqref{eq:A9} have absolute integrated norm
\(O_\circ(1)\). Their placements have bounded densities, including when
the added pair attaches to an existing integration variable; there is
no prescribed fork in this pressure calculation. Thus they cost only
a subpower factor.
The same weighted accounting applies to row contacts and normalization
errors. Since \(m\asymp N=x^{-6}\), their contribution is bounded by
\[
O(m^{-1})\left(1+
\|\dot b\|_{L^1}+\|\dot h\|_{L^1}
+\|\dot K\|_{L^1}+\|\dot q\|_{L^1}\right)
=O_\circ(x^6).
\]
Here the norms denote the absolute mass integrals with the
bounded-density allocation weights from \eqref{eq:A9}. The stencil
has finitely many fixed coefficients. Moreover,
Lemma~\ref{found:size-increment} gives a total-pressure remainder, so
no further factor \(m\) multiplies these errors. This proves that
\eqref{eq:A7}--\eqref{eq:A9} and the stencil reduce the size increment
to \eqref{eq:E3} with the required saving. On direct paths the
differentiated weights carry the additional factor
\(O_\circ(\lambda)\).
For a direct variation, start with the monotone reference
\(K_S=h+bS\). Apply \eqref{eq:A7} with insertions \(b\delta\), expanding
the pressure gradient of \eqref{eq:A9} through order \(J_0-1\).
By \eqref{eq:P8}, the test multiplying the cavity moment has degree
at most one and weight \(O_\circ(\lambda)\). In the true Taylor
remainder distinguish two cases. If at most one insertion has
\(e(v)\ge x^{\delta'}\), at least \(J_0-2\) insertions have smaller
envelope. Their moment bounds from \eqref{eq:D2} supply an arbitrarily
large fixed power of \(x\) once \(\delta'>0\) is fixed and \(J_0\) is
large enough. Choose the truncation losses first so that they do not
use up this gain.
If two insertions have \(e(v)\ge x^{\delta'}\), apply Cauchy--Schwarz
and \eqref{eq:D8} to those insertions, and bound the others uniformly.
The selected vertices have bounded absolute density even if the
insertions are tied. The one-row interpolated laws are comparable to
the genuine laws by \eqref{eq:A7}; the transfer to tilted active
components uses \eqref{eq:A4} and the stopping conventions after
Lemma~\ref{lem:statistics-centered}. Before averaging over \(\gamma\),
the cost is
\[
O_\diamond\!\left(\lambda x^{-C_0\delta'}z_1^0Y^2Z^2\right)
=O_\diamond\!\left(
x^{5-C_0\delta'}\lambda^{(1-\eta)/2-s_*}Z^4\right).
\]
The fourth moment of \(Z\) therefore gives the averaged bound
\[
O_\diamond(x^{5-C_0\delta'}\lambda^{(1-\eta)/2-s_*}),
\]
which saves a power when
\(C_0\delta'0\}.
\]
Thus \(D\) counts blocks in the full skeleton, whereas \(D_\pi\)
counts a block again each time it is used by another part. Neither
counts endpoint multiplicities.
On the fixed skeleton, let \(\zeta^A_u\), \(0\le u\le c\), be the
scalar prefix fields under the positive scalar law. They share their
Brownian driver until their skeleton separation, and thereafter use
independent transitions. Each part \(P\) uses a separate copy of this
law restricted to its blocks. The normalized source sum for \(k\)
slots inside one block is
\[
s_k(z)=c^{-1}e^{-cV(t(c),z)}\partial_z^k e^{cV(t(c),z)}.
\]
With the explicit edge weights of \eqref{eq:E3} kept outside, the
coefficient multiplying the skeleton measure is
\[
\sum_{\pi}(-1)^{|\pi|-1}(|\pi|-1)!\ c^{D_\pi}
\prod_{P\in\pi}\mathbf E_{\rm low}
\prod_{A\ {\rm in}\ P} s_{k_{A,P}}(\zeta^A_c).
\tag{E5}\label{eq:E5}
\]
The expectation \(\mathbf E_{\rm low}\) is probabilistic; the preceding
grouping of allocation coefficients by skeleton is a signed sum.
For example, take two copies of the edge \(AB\) between cut blocks.
Their original terminal labels may differ; projection makes their
spin-model pair tests identical. Abbreviate
\(s_1(\zeta^A_c)=\mathsf M_A\),
\(s_2(\zeta^A_c)=\mathsf P_A\). The one-part partition uses two
blocks, whereas the two-part partition uses each block twice. Their
combined internal coefficient is therefore
\[
c^2\mathbf E_{\rm low}(\mathsf P_A\mathsf P_B)
-c^4\bigl(\mathbf E_{\rm low}(\mathsf M_A\mathsf M_B)\bigr)^2.
\]
The skeleton measure contributes \(c^{-1}\), giving net powers
\(c\) and \(c^3\). There are two orientations of the second edge
for a fixed first edge, so its factor in \eqref{eq:E3} is
\(2/(2^2 2!)=1/4\). The square counterterm uses this same skeleton,
with internal coefficient \(c^2\) and weight \(-b/4\). These
normalizations will give the quadratic cancellation below.
\paragraph{The finite signed identity.}
First expand each cumulant into products of positive scalar moments
on its indicated edge sets. Once the skeleton assignments are fixed,
there is no restriction on the internal placements within a block.
A part observes only its own induced genealogy, so the internal
signed sums for different parts can be performed separately, even
when their actual terminal leaves coincide.
This factorization can be checked directly at finitely many levels
including \(c\). In the labeled product rule of
\cite[Proposition~2.5]{PE}, the coefficients choosing distinct children
are falling factorials of incoming-to-child mass ratios, and every
terminal source slot carries its terminal mass factor. Keep the
induced genealogies as discrete marks and first take the successive
ratios, including an auxiliary root ratio, to be sufficiently large
positive integers. The coefficients then count child identities.
After distinct block identities have been assigned at \(c\), the
choices for disjoint sets of slots inside a block are independent
counts, with terminal slots chosen with replacement. Their counts
therefore multiply. The resulting rational coefficient identity
extends by polynomial identity to the real masses, keeping terminal
mass fixed and dividing by the root mass before its zero limit.
This argument concerns only the coefficients; it does not extend
spin tests or probability laws from integer ratios.
\paragraph{Evaluation of the internal source sums.}
For one part, the positive scalar prefix law on its restricted
skeleton is independent of the internal placements after \(c\).
Conditional on these prefix fields, the source product rule sums
\(k\) internal slots in a block to
\(e^{-cV}\partial_z^k e^{cV}=c s_k\). Different blocks use independent
future kernels. Thus every used block contributes one factor \(c\),
giving \(c^{D_\pi}\), and the cumulant partition formula gives
\eqref{eq:E5}. Signed path limits and bounded fixed-order derivatives,
as supplied by PE, pass this finite-level identity to the continuous
paths. Equivalently, one may fix the prefix fields and approximate
only the future clocks within the blocks. There are no prescribed
internal split times to preserve in that approximation.
\paragraph{Differentiation at a fixed skeleton.}
The cut and skeleton measure stay fixed. Differentiating the scalar
moments adds one covariance pair with weight \(\dot K\). By
\eqref{eq:P8}, its absolute integrated weight is finite; its
non-atomic placements have bounded density for fixed cut and label
count. The same bound holds along affine comparison chords of the
clocks, so the finite signed differentiation rule and Fubini apply
to the internal sums.
To pass from chords to ordinary derivatives, fix \(N\). The
clock difference quotients are uniformly bounded in mass: for size
paths this follows from the smoothing floor and clipping, and for
direct paths from the localized smooth inverse perturbation. For
almost every parameter they converge at every mass in the low
region and at almost every other mass. The scalar kernels converge
by fixed-size path continuity. Dominated convergence therefore
passes the divided covariance insertions to the dotted insertions,
also for almost every fixed skeleton. A reused split is either one
of these integration coordinates or a prescribed low skeleton mass,
where convergence is pointwise; constant diagonals still cancel.
This proves that internal summation and a.e. coefficient
differentiation commute. We may choose the low-region partition
separately at each parameter when estimating the result: the
coefficient has already been differentiated, so no moving indicator
or cut is being differentiated.
\paragraph{Scalar jets and common-prefix estimates.}
Write
\(\chi=V_{zz}(0,0)\), \(g=V_{zzzz}(0,0)\),
\(k_0=t'(0+)>0\), and \(G(u,z)=V(t(u),z)\).
The notation \(O_*(A)\) controls both the value and the parameter
derivative: on size paths the derivative is \(\partial_\theta\),
and on direct paths it is \(\lambda^{-1}\partial_{\log\lambda}\).
The same convention applies to fixed \(L^p\) norms when fields are
coupled by common Brownian drivers in mass coordinates. Derivatives
of the explicit weights \(b,d\) will be handled separately.
By \eqref{eq:P1}--\eqref{eq:P2}, the inverse \(t\) is smooth here,
with \(t'\) bounded above and below and with the required higher
derivatives bounded. The positive-order spatial derivatives of \(G\)
also have bounded mass and spatial derivatives of every required
fixed order, allowing polynomial growth in \(z\) after parameter
differentiation. To see the mass derivatives, use
\[
\partial_uG=-\tfrac12t'(G_{zz}+uG_z^2).
\]
For direct derivatives, differentiate the scalar PDE in \(\lambda\)
at fixed normalized time, as in the lift proof, and then compose with
the regular inverse. For size derivatives, use the rescaled scalar
jets, including the terminal-mass derivative already estimated in
Section~\ref{sec:paths}. These operations preserve evenness in the
field. In particular, on size paths,
\[
\partial_\theta k_0=-3 k_0/6,\qquad
\partial_\theta\chi=\chi/6+O(e^{-\Omega(M)}),\qquad
\partial_\theta g=3g/6+O(e^{-\Omega(M)}). \tag{E6}\label{eq:E6}
\]
The normalized inverse slope and initial spatial derivatives are
constant under this rescaling, up to the exponentially small terminal
effect. This gives \eqref{eq:E6}. Moreover, \(\chi\) is uniformly
bounded above and away from zero. At every designated split,
\[
b=b_\circ+O(\lambda c^2),\qquad b_\circ=\chi^{-2},
\]
and the error has the same bound under unscaled parameter
differentiation. By \eqref{eq:D2}, each \(d\) or \(\dot d\) can
likewise contribute \(O_\diamond(\lambda c^2)\) to the coefficient,
leaving its ordinary envelope factor.
The low prefix equation is
\[
d\zeta=\sqrt{t'}\,dB+u t'G_z(u,\zeta)du.
\]
Starting from zero, this equation gives
\(\zeta_u=O_*(\sqrt u)\) and
\(\zeta_u-\sqrt{k_0}B_u=O_*(u^{3/2})\) in every fixed moment.
For a conditional evolution starting at \(z\), the field and its
parameter derivative have norms at most
\(C_p(|z|+\sqrt c)(1+|z|)^C\). Its spatial flow derivatives have
bounded moments, allowing polynomial growth in \(z\). These bounds
follow by differentiating the additive-noise equation: the drift is
smooth with bounded spatial derivatives, the differentiated
coefficients have polynomial growth, and each odd differentiated
drift vanishes at zero.
Evaluate the block observables at their fields at \(c\), unless
another time is specified. Taylor expansion, together with even/odd
parity and the definition of \(s_k\), gives
\[
\begin{split}
\mathsf M_A := s_1(\zeta^A_c)&=G_z(c,\zeta^A_c)=\chi\zeta^A_c+O_*(c^{3/2}),\\
\mathsf P_A:=s_2(\zeta^A_c)&=G_{zz}(c,\zeta^A_c)+c(G_z(c,\zeta^A_c))^2
=\chi+O_*(c),\\
s_3(\zeta^A_c)&=g\zeta^A_c+O_*(c^{3/2}),\qquad
s_4(\zeta^A_c)=g+O_*(c).
\end{split} \tag{E7}\label{eq:E7}
\]
For blocks \(A,B\) splitting at \(v_{AB}1\) and the neighborhoods, tapers, and constants
\(c_1,c_b\) in \eqref{eq:P1}--\eqref{eq:P7}, leaving strict room in
the admissibility and derivative bounds. The global saving \(c_G\)
in \eqref{eq:G1} is available before choosing \(s_*\) near \(1/2\)
or the ultimate small \(p_*\). In its proof, the coarse clock estimate
\eqref{eq:G3} supplies a fixed saving first. Choose \(a_g\) below
that saving, then \(d_g\) small enough to absorb the fixed
inverse-kernel loss in the covariance estimate following
\eqref{eq:G4}, and finally the PE power slack
for this global comparison small relative to \(d_g\). The endpoint
condition \eqref{eq:P9} uses the common base temperature as explained
in Section~\ref{sec:global}. It requires \(\lambda\to0\), without
requiring \(\lambda\le x^{a_g}\). Thus later decreases of \(p_*\)
do not decrease \(c_G\).
\item Set \(k_c=0.03\). Choose
\(\tau\ll\min(10^{-4},c_G)\), put \(s_*=1/2-\tau\), and then
choose \(\eta\ll\tau\) and \(\tau_0\ll\eta\), with
\(s_0=s_*-\tau_0\), as in \eqref{eq:C1}--\eqref{eq:C2}.
Choose \(b_s>0\) so that
\[
2(\tau+\tau_0)+b_s<\min(c_G/12,k_c/3),\qquad
b_s<2(\eta-\tau_0),
\]
and then \(b_\ell\ll b_s\). These inequalities leave room for the
local comparison errors throughout \(\lambda\ge x^{12}\).
Choose also \(h_e\ll\min(1,c_G)\).
Both final-strength estimates in \eqref{eq:C2} now have a fixed
saving \(c>0\) for every sufficiently small \(p_*\). Indeed, with
\(\lambda=x^{p_*}\), \(U=x\lambda^{-s_*/2}\), and the edge seed
off, the differentiated bound in its proof has powers
\[
4+c_G-p_*,\qquad 5+p_*(2\eta-5s_*/2),\qquad 6-2p_*.
\]
All remain separated above \(4\) as \(p_*\) decreases. The
smoothing error \(O(x^{5+c'})\) in \eqref{eq:P11}, divided by
\(\lambda\), has exponent \(5+c'-p_*>4\) as well.
Likewise \eqref{eq:D11} supplies a fixed
\(\delta_*>0\), for example \((1/200)/50\), under the preliminary
restriction \(p_*<(1/200)/100\). Neither saving depends on the
remaining choice of \(p_*\).
\item Fix the size row and pressure Taylor degrees using
\(\delta_*\). In the covariance equation proving \eqref{eq:D15},
choose the row degree so that the Taylor remainder before inversion,
apart from the \(O(m^{-1})\) contacts, is smaller than \(x^6\) by
a fixed power. This is possible by \eqref{eq:D11} and requires no
further information about \(p_*\). For these fixed degrees the
inverse has fixed polynomial losses in
\((\lambda V_i^2)^{-1}\) and \(1+M\); the low sources have a fixed
loss \(\lambda^{-C}\). Fix constants bounding these losses before
choosing the cutoffs, large enough to cover all the estimates below.
Now choose \(a_0\) small relative to \(c,\delta_*\), take \(b_H\)
to be a sufficiently small fixed fraction of \(a_0\), then choose
\(a_1\) small relative to \(a_0,b_H\), and \(a_2\) small relative
to \(a_1\). Finally choose \(p_*\) small relative to these gaps and
to \(h_e\), including the restriction needed for the cap on \(Y\)
after \eqref{eq:D11}. With \(V_i=x^{a_i}\), the high-mass terms
of \eqref{eq:E3} then have the following positive margins over
\(x^5\):
\[
\begin{gathered}
1-2C_Ha_1>0,\qquad b_H-C_Ha_1>0,\qquad b_H-Cp_*>0,\\
a_1-C_Ha_2-p_*s_*>0,\qquad
\delta_*/2-C_Ha_2-p_*s_*>0.
\end{gathered}
\]
The first two margins cover two high stochastic factors and a high
stochastic factor paired with a high deterministic factor. The third
covers a high deterministic factor times the low-source estimate
\eqref{eq:D16}. The last two cover a sole high stochastic factor,
using respectively the parity-weighted integral \(V_1Y^2\) or a
further factor \(x^{\delta_*/2}Y^2\). In both cases
\(Y^2=x^2\lambda^{-s_*}Z^2\). The size projection errors
\(x^6\lambda^{-C}=x^{6-Cp_*}\) also have the positive margin
\(1-Cp_*\). No subsequent step decreases \(p_*\).
\item Fix the number of strengthening stages in \eqref{eq:C11}.
Choose \(\zeta\) small relative to the power margins in
Section~\ref{sec:budgets}; in particular, it is smaller than \(p_*\)
times each required gap in a \(\lambda\)-exponent, including the gaps
involving \(b_\ell\) and \(\tau_0\). Then choose the local PE
power slacks small relative to \(\zeta\), and the buffer exponent
\(\rho\) small enough for \eqref{eq:C18}. The remaining finite
profile and moment iterations, and their subpower losses, can be
chosen with these parameters fixed.
The extra optimized excess
\(O(x^{4-2\zeta}u\Lambda^{1/3})\) fits the budget at
\eqref{eq:C3}: after division by \(\lambda u\), its ratio to
\(x^4\lambda^{-2s_0}\) is
\(x^{-2\zeta}\lambda^{k_c/3-1+2s_0}\), which is power small by
these choices. The final-strength switch error
\(O(x^{5+c'})\) from \eqref{eq:P11} also fits after division by
\(\lambda u\). At the smallest dyad \(u\asymp U\), its ratio
to \(U^4\) is at most
\(x^{c'}\lambda^{5s_*/2-1}\); larger dyads improve this bound.
Both exponents are positive, so the switch leaves room for the
remaining losses.
\item For the direct paths, choose the coarse cutoff exponent
\(\kappa>0\) small relative to the gains in
Lemma~\ref{lem:statistics-coarse-switch}. More precisely, its
large-\(Y\) event costs \(x^{-4\kappa}\), so choose
\[
4\kappa<\min\left\{p_*(1-2s_*),\,
(1-\eta)/s_*-2\right\}.
\]
Both quantities on the right are positive by the choices of
\(s_*,\eta\). The gains used in the
remaining direct expansion are
\[
1-2s_*=2\tau>0,\qquad
(1-\eta)/2-s_*=\tau-\eta/2>0,\qquad
s_*/2-\eta>0.
\]
A factor \(\lambda^g\) with any of these exponents \(g>0\) is at
most \(x^{p_*g}\). Choose \(\delta'>0\) smaller than \(\kappa\)
and small enough that
\[
C_0\delta'0\).
Choose the fixed moment and change-of-measure orders large enough
that truncation and transfer lose only small fractions of the margins
just retained. Then choose the geometric loss in \eqref{eq:D1} and
\eqref{eq:D10}, and the subpower losses, still smaller. The smooth
density comparisons on intervals of length \(O(b_\epsilon)\) may
require higher fixed orders after this geometric choice; they change
constants only. Finally take the stencil order large enough for
both the fixed size pressure polynomial and the site-weight smoothing
in Section~\ref{sec:reduction}. Enlarging this order does not require
enlarging the size Taylor degree.
\end{enumerate}
These choices give a common positive saving in the stopped increment
bounds. For a direct strength sweep, Lemma~\ref{lem:statistics-coarse-switch}
handles \(Y>x^\kappa\); on the complement apply the allocation
and compression estimates \eqref{eq:E2}, \eqref{eq:E4} to the
pressure polynomial \eqref{eq:E3}. The two multiplier sweeps use the
separate switch bounds of the same lemma. The size sweep uses the
capped low statistics, \eqref{eq:D15}, and the size versions of these
polynomial estimates. Integrating the direct strength
parameter costs only \(O(\log(1/x))\). The size segment has bounded
length, and the multiplier bound has at most the integrable endpoint
factor \(\chi_e^{-1/2}\). The fixed-degree dyadic and projection-cut
sums cost subpower factors. Decreasing the common saving absorbs
these integration and summation costs.
We spell out the uniformity needed when the site count varies. At
each fixed size-path point \(\theta\), every estimate above is
applied to the particular integer \(m\) being tested, uniformly for
\(N/2\le m\le3N\). In particular, the averaging in
\eqref{eq:C1}--\eqref{eq:C2} is over \(\gamma\) at this fixed
\((\theta,m)\). The physical clocks are common to these site
counts. Their absolute dotted-clock norms are bounded up to subpower
loss as explained after \eqref{eq:P8}; integrating any remaining
dotted weight in mass therefore gives a common integrable envelope
in \(\theta\). Thus the bound is uniform in \(m\) before
integrating \(\theta\).
Let \(w_m(\theta)\ge0\) be measurable probability weights on the
allowed integers. Multiplying the parameter-averaged bound at
\((\theta,m)\) by \(w_m(\theta)\) and summing preserves its
common envelope, since \(\sum_m w_m(\theta)=1\). A measurable
choice of a single \(m(\theta)\) is the special case of a point
mass. Fixed-size path approximation makes the pressure tests and
statistics measurable, and absolute continuity supplies their a.e.
path derivatives. Fubini first removes a null set of \(\theta\)
for each parameter-averaged estimate. At fixed \(N\) there are only
finitely many allowed \(m\), so their exceptional sets can then be
united into one null set before weighting and integrating.
For each summand the change of measure is used only on the central
stop \(\mathcal K_m(s)\le3x^\varepsilon\). The comparisons with
finitely many added rows in \eqref{eq:A6}--\eqref{eq:A9} have bounded
covariance changes and estimate their remainders with moments of the
original \(m\)-site system. They therefore require this same stop,
without a new stopping condition for the added-row systems. Fix the
common saving before taking the small Laplace exponent
\(\varepsilon\) in Section~\ref{sec:reduction}; this exponent is
distinct from the geometric slack. The density bounds on the stop hold
for every fixed finite order, so they cover all transfer orders chosen
above, for every sufficiently small fixed \(\varepsilon>0\) and each
fixed test coefficient \(q>0\) in \(s=qx^{1+\varepsilon}\).
For one such test, the same parameter bounds cover the finitely many
segments in the comparison from \(N\) to any \(N'\in[N,2N]\).
Summing their stopped bounds and then choosing \(\gamma\) gives the
single route used in Section~\ref{sec:reduction}. Different tests may
choose different routes: their bare endpoint laws are independent of
\(\gamma\). Thus the later interpolation at finitely many Laplace
tests requires no parameter choice uniform over an infinite test family.
This proves Proposition~\ref{prop:increment}, including its version
with a measurably varying site count or site weights.
Section~\ref{sec:reduction} then proves Theorem~\ref{thm:moments}
and proves Proposition~\ref{def:law}. Removing the independent
Gaussian completion and dividing by the exact finite-size standard
deviation, as done there, proves Theorem~\ref{thm:main} for every
fixed \(\beta>1\).
==== END The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law//sections/10-parameters.tex ====
==== BEGIN The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law//sections/11-companion-interface.tex ====
\section{Inputs from the fluctuation-scale companion}
\label{sec:companion-interface}
We collect the inputs from \cite{PE}, with their conventions and their
uses in this paper. The moment and comparison estimates are separate inputs
from the exponent theorem. The target temperature is fixed in $(1,\infty)$; the auxiliary common
bases $T_m\to T>1$ are treated below. All replica tests have a fixed finite number of labels, and all moment,
Taylor and recursion orders are fixed before the system size tends to infinity.
\subsection{Covariance and mass normalization}
The Hamiltonian in \eqref{eq:model}, multiplied by $\beta$, has covariance
$mTQ^2/2-T/2$. Adding the independent spin-constant Gaussian of variance
$T/2$ in \eqref{eq:completed-free-energy} gives $mTQ^2/2$, exactly the
covariance of \cite{PE}. The probability spin prior subtracts $m\log2$.
Consequently the centered completed variable in the companion is our
$F_m^c-\mathbf E F_m^c$.
Here is the conversion from our terminal mass $a$ to terminal mass one.
For $u=v/a$, put
\[
\widetilde b(u)=a^2b(au),\qquad
\widetilde h(u)=a^2h(au),\qquad
\widetilde F=aF.
\]
The terminal expression becomes $\log\int e^{aH}$; at an intermediate
mass $v=au$, multiplying the continuation by $a$ changes its power mean
to the one of exponent $u$. Thus, including self corrections,
\[
f_m^{\rm PE}(\widetilde b,\widetilde h)=a f_m(b,h),\qquad
\widetilde X_u=X_{au},\quad \widetilde C_u=C_{au},\quad
H_{au}=a\widetilde H_u,\quad W_{au}=a\widetilde W_u.
\]
In particular $vC_v\preceq H_v\preceq aC_v$. The projection denominator
is unchanged:
\[
(v/a)\sqrt{m(a^2\Delta h)}=v\sqrt{m\Delta h}.
\]
The same change of variables passes the deterministic-weight pinned
inequality to $dv$ with precisely the right side in \eqref{eq:A1}.
At a bare size endpoint $a=\ell$, $b=T/\ell^2$, the total continuation
is $F_m^c/\ell$, so $x/\ell=m^{-1/6}$ when $m=N e^\theta$.
\subsection{Finite-tree calculus and its probability laws}
The following list specifies both the source statement and the additional
conditions retained when it is used.
\begin{enumerate}
\item \cite[Lemmas~2.1--2.3, equation~(1)]{PE} give the source Hessian,
pressure variation, clock inequality, and single-spin pinned moments in
Lemma~\ref{found:estimates}. The pinned estimate requires zero deterministic
external field, deterministic weights, and the full single-path law. If
other branches are present, their normalized kernels are first marginalized;
remaining random factors are treated afterwards by unconditional H\"older.
The polynomial form \eqref{eq:A2} is stated and proved in
\cite[Appendix, ``Source and pressure calculus; the scope of pinned
moments'']{PE}. Section~\ref{sec:foundations} recalls its proof in our
normalization: for fixed terminal spin the gradient of $P_v(L_v)$ is
$P'_v(L_v)$ times the original gradient. The same block-Hessian
inequality and Brascamp--Lieb argument apply, and gauge symmetry removes
the pin without changing the centering. The coefficients of $P_v$ are
deterministic under the positive path law. This estimate is used in
Lemma~\ref{lem:statistics-centered}.
\item \cite[Lemma~3.5, equation~(13)]{PE} supplies \eqref{eq:A3}.
The anchor is determined by the prefix and branches exterior to the
future being projected. Conditional on those data, that future must
retain its normalized positive kernel. The traversed field length and
its positive lower mass are bounded below by inverse powers of size.
In every application here, the total clocks are at most a fixed power
of $\log m$, as stipulated in Section~\ref{sec:foundations}; hence the
coefficient ratios in the finite projection iteration are polynomial
in the size. The lemma itself does not require the optimized constraints
of PE.
The layer estimates in Sections~\ref{sec:global}--\ref{sec:statistics}
use this version with unit-normalized anchors and then integrate the
anchor magnitude. Their Hessian blocking is the calculation in
\cite[proof of Proposition~3.4, preceding equation~(15)]{PE}.
\item \cite[Proposition~2.5, Lemmas~2.6, 2.8 and~2.9, equation~(2)]{PE}
give the real-power allocation rule, row sums, bounded split densities,
and fixed-order Taylor estimates. Each independent Gaussian increment
covariance remains positive semidefinite throughout each interpolation.
Spin tests and their scalar centers
are held fixed during these interpolations; a varying test requires its
own product-rule derivatives. These are the inputs to
Lemmas~\ref{found:cavity-quotient}--\ref{found:size-increment}, the
coefficient algebra in Proposition~\ref{inv:pair-inverse}, and the
compression identity \eqref{eq:E5}. A row collapses only after its
summand is independent of the new label's detailed attachment.
Allocation order is changed before absolute values are taken. A reused
fork retains its atom; a new fork has a continuous mass variable.
\item \cite[Corollary~2.10]{PE} bounds fixed-order source derivatives.
The terminal-datum derivatives in Lemma~\ref{paths:parameter-regularity} are treated
by its diffusion representation with one bounded smooth insertion;
the source bounds control the remaining fixed derivative slots.
The scalar time identities are those of \cite[Lemma~5.3]{PE}, converted
from time $K/T$ to variance time $K$. They are also used for the
single-coordinate freezing estimate in \eqref{eq:D11}.
\item \cite[Proposition~2.14]{PE} is a limit at each fixed system size,
with fixed finite observed topology. Terminal clocks and every prescribed
cut or revelation side are retained. No rate uniform in system size is
inferred. At a simultaneous jump we choose the joint linear filling
\[
(b_-,h_-)+t(\Delta b,\Delta h),\qquad 0\le t\le1.
\]
If an intermediate filled revelation is observed, approximations preserve
this filling and its specified observations. Integrated clock identities
refer to that chosen filling; only endpoint laws are independent of it.
Nonnegative contributions of filled jumps can be discarded in trace
upper bounds. Fixed-size limits are used in Lemma~\ref{found:estimates},
the scalar path construction, Lemma~\ref{red:lem-augmentation}, and the
finite-resolution passage in Proposition~\ref{inv:pair-inverse}.
The latter first uses the inverse at a finite resolution with its
uniform absolute attachment bound; it does not require convergence of
the inverse kernels themselves.
\end{enumerate}
\Needspace{4\baselineskip}
For the centered-coupling argument of \cite[Lemmas~2.11--2.13 and~4.5]{PE},
the conditioning sigma-field is generated by the common prefix, the
unaffected exterior branches, and the fixed topology. It contains no
observation of the affected future. This preserves the two separate
conditional marginals. The weighted exterior anchor in
\cite[Lemma~4.8]{PE} averages the exterior joint law rather than
conditioning on its terminal spins. The applications here retain these
conditions. In particular Lemma~\ref{lem:statistics-low-sources} and
Remark~\ref{rem:statistics-earlier-pair} explicitly construct a marginal
law independent of the moving fork before using a centered integral.
\subsection{Optimized comparisons and changed constraints}
The all-mass calls in Sections~\ref{sec:global} and~\ref{sec:budgets}
use \cite[Definition~6.1 and Proposition~6.6]{PE}, with data (3)--(6)
there. Writing $m$ for the site count and $\epsilon_{\rm PE}>0$ for
its fixed power slack, the error parameter obeys
\[
m^{-2/3+\epsilon_{\rm PE}}d_{\rm PE}\lesssim E
\le m^{-1/2}d_{\rm PE}^2.
\]
The bare paths and all changed competitors are admissible, total clocks
are uniformly bounded, and a fixed positive matrix margin remains after
the comparison. The upper bound for $d_{\rm PE}^2$ has a fixed positive
gap below the limiting $T$. The endpoint condition is (4) of PE on
every required comparable mass band, with the common base $T_m$ of
\eqref{eq:P9}. The common-base extension is stated and proved in
\cite[Appendix, ``A common prelimit temperature'']{PE}.
Section~\ref{sec:global} verifies its use here and recalls the
substitutions $J_0=T_m\chi^2-1$ in PE (26)--(27) and time $K/T_m$
in PE Lemmas~5.4--5.5. This base stays fixed through each finite
parent/child tree. The macroscopic limit is at the fixed
$T>1$, so \cite[Lemma~5.8 and Proposition~5.9]{PE} apply at that limit.
No uniform assertion near the critical temperature is required.
For clarity, the further source inputs to the modified local optimization
are as follows.
\begin{enumerate}
\item The geometry of \cite[Lemma~3.3, equations~(9)--(10)]{PE} is
rederived for the prescribed density box in Lemma~\ref{bud:geometry}.
The first variation has no state-dependent integrating factor in this
box. The original PE density cap is not assumed for its minimizers.
\item Under the stated outer-regularity hypotheses,
Lemmas~\ref{bud:outer-moments} and~\ref{bud:outer-second} establish
the replacement raw and projection estimates \eqref{eq:C8}--\eqref{eq:C9}
and second moments \eqref{eq:C10}, including the enhanced floor.
Under the additional strong-removal hypotheses, Lemma~\ref{bud:common-star}
gives separate mass and radius estimates and one good calibration cut
for all fixed moment orders. For cap/contact
units these proofs use \cite[Lemmas~3.11--3.12, equations~(20)--(21)]{PE}:
whole-cap trace bounds, controlled mean oscillation, deterministic
weights, and comparable masses. Broad caps are sliced with their
additional trace cost retained. These same inputs are used in
Lemmas~\ref{bud:box-moments}, \ref{bud:free-forks} and
\ref{bud:high-primitive}. The enhanced floor is treated separately by
radius slides; small physical floor mass is never used to discard
radius intervals.
\item The weak-removal argument uses the child-data proof of
\cite[Lemma~6.2]{PE}, the movement inequality of
\cite[Lemma~6.3]{PE}, and feasible-barrier arguments of
\cite[Lemma~6.4]{PE}. The child has the ordinary PE constraints.
The parent movements are proved feasible for the new density box
before using movement. Child success gives agreement and then outer
regularity for the child, as in \cite[Lemma~6.5]{PE}; only then are
\cite[Theorem~5.1 and Proposition~5.9]{PE} used.
\item Proposition~\ref{bud:modified-closure}, used in the strong-removal
argument, reuses the proof of
\cite[Proposition~4.1]{PE}, rather than applying its original
optimization hypotheses to the new density box. The preceding
analytic estimates replace PE (12)--(13), (16)--(19). Fixed-degree
tests remain those of \cite[Definition~4.2]{PE}; bounded row changes,
observed-block truncation, boundary covariance, test-family closure,
and projected replacements are \cite[Lemmas~4.3--4.8]{PE}.
The finite-depth cancellation and calibration are
\cite[Lemmas~4.9--4.10, equations~(23)--(27)]{PE}.
The division is by $\chi K_*\asymp W$, not by a stability gap.
One common good cut is used for the fixed finite recursion and its
required moment orders; these orders are fixed before the size limit.
The resulting moving-target estimate is stated explicitly in
\eqref{eq:bud-coarse-closure}; \eqref{eq:bud-coarse-closure-integrals}
gives its two integrated forms, with the mass and radius normalizations
kept separate. These are the versions of PE (22) used here.
The profile proof is then repeated using \cite[Lemmas~5.3--5.6]{PE}
and PE (28)--(29). Identification of its slope uses
\cite[Lemma~5.8 and Proposition~5.9]{PE} and the macroscopic transfer
in Proposition~\ref{bud:coarse-profile}.
\end{enumerate}
The coarse scalar trial used for \eqref{eq:G3} comes from the bounds
in the proof of \cite[Lemma~5.7]{PE}, with the explicit floor, cap,
and cutoff choices given in Section~\ref{sec:global}. Its proof is
used quantitatively; the limiting variational statement alone would
not provide the needed power saving. The scalar control representation
used for coercivity and pair-kernel positivity is proved in
\cite[proof of Lemma~5.8]{PE} by completing the square in It\^o's
formula.
\subsection{The exponent and the one-dimensional input}
The fixed-temperature exponent theorem \cite[Theorem~1.1]{PE} is
used twice in Section~\ref{sec:reduction}: to initialize the stopped
small-positive-Laplace comparison, and to rule out zero limiting
variance after a power rate of variance convergence has been obtained.
The first use combines the exponent with
\cite[Lemmas~7.6--7.7]{PE}; along the nonbare physical paths the
appropriate exponential augmentation is instead proved in
Lemma~\ref{red:lem-augmentation} from the telescoped pinned density.
That proof uses the full joint positive law and then gauge symmetry.
The one-dimensional tail, positive-tail-mass and small-Laplace bounds
are exactly \cite[Lemma~7.7]{PE}.
The exponent supplies neither a variance prefactor nor a limiting law.
Those conclusions use the summable increment estimate of this paper.
Finally $\operatorname{Var}F_m^c=\operatorname{Var}F_m+T/2$; the common
Gaussian therefore disappears after division by $m^{1/6}$. The theorem
is stated for log partition functions. For the physical total free
energy $-F_m/\beta$, the variance constant is divided by $\beta^2$
and the standardized limiting law is reflected.
==== END The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law//sections/11-companion-interface.tex ====
==== BEGIN The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law//sections/12-notation.tex ====
\section{Guide to shared notation}
\label{sec:notation-guide}
This table collects notation used across several sections. Local scale
ratios, optimizer profiles, and rescaled discrepancy functions are defined
in the arguments where they occur.
\begingroup
\small
\renewcommand{\arraystretch}{1.12}
\begin{longtable}{@{}p{0.23\linewidth}p{0.71\linewidth}@{}}
\toprule
Notation & Meaning and defining location \\
\midrule
\endhead
\(F_n^c,Y_n,a_k\)
& Completed free energy, scaled centered free energy, and limiting
moments; \eqref{eq:completed-free-energy} and Proposition~\ref{def:law}. \\
\(N,m,n,x\)
& Dyadic reference size \(N\), integer size indices \(m,n\), and
\(x=N^{-1/6}\). Section~\ref{sec:reduction} temporarily also uses
\(n=Ne^\theta\) as a real mollification center. \\
\(\theta,\ell,a\)
& Size coordinate, \(\ell=e^{\theta/6}\), and terminal mass:
\(a=\ell\) on direct paths and \(a=4\) on smoothed paths;
Section~\ref{sec:paths}. \\
\(v,t=K(v),\xi\)
& Replica mass, scalar variance time, and normalized scalar time
\(\xi=\ell^2t\); Sections~\ref{sec:foundations}--\ref{sec:paths}. \\
\(b,h,K,q\)
& Matrix and field clocks, scalar reference clock, and response
\(q(v)=\Gamma(K(v))\); Section~\ref{sec:foundations}. \\
\(h_*,d,J\)
& Reference field \(h_*=K-bq\), field defect \(d=h-h_*\),
and row mismatch \(J=b\Delta+d\); Section~\ref{sec:foundations}. \\
\(Q,Q^*,\Delta,\delta\)
& Overlap, tilt-masked overlap, and error \(\Delta=Q^*-q\).
Section~\ref{sec:expansion} also uses \(\delta=Q^*-S_v\), so
\(\Delta=\delta+(S_v-q)\);
\eqref{eq:A4}. \\
\(X_v,C_v,H_v\)
& Posterior spin mean, conditional covariance, and source Hessian;
Section~\ref{sec:foundations}. \\
\(S_v,B_v,D_v,L_v,W_v\)
& Deterministic overlap statistics and single-path fluctuation
observables; Section~\ref{sec:foundations} and \eqref{eq:A1}. \\
\(\psi_m(s),\mathcal K_m(s)\)
& Difference of pressure tests and centered log Laplace transform
\(\mathcal K_m(s)=sm\psi_m(s)\); \eqref{eq:A4}. \\
\(\lambda,U,\gamma\)
& Regularization strength, low scale, and deterministic path parameters;
\eqref{eq:P1} and \eqref{eq:P3}. \\
\(Z,Z_g,e,Y=UZ\)
& Parameter-dependent majorants, deterministic error envelope, and
derived scale; \eqref{eq:D1} and Propositions~\ref{prop:statistics-pointwise}
and~\ref{prop:statistics-high}. \\
\(\nu,\mathbf E_v,\mathbf E_\gamma\)
& Positive-tree expectation, conditioning on the revealed prefix, and
averaging over path parameters; Sections~\ref{sec:foundations}
and~\ref{sec:reduction}. \\
\((\cdot)_\#\)
& Signed allocation per root mass; Section~\ref{sec:foundations}. \\
\(\lesssim_\circ,\lesssim_\diamond\)
& Subpower losses and the stated arbitrarily small geometric power
losses; Sections~\ref{sec:foundations} and~\ref{sec:statistics}. \\
\bottomrule
\end{longtable}
\endgroup
==== END The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law//sections/12-notation.tex ====
==== BEGIN The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law//sections/consequences.tex ====
\section{Proofs of the fluctuation and disorder-chaos corollaries}
\label{sec:consequences}
The two consequences in the introduction now follow from the main
variance asymptotic and existing comparison inequalities. We keep the
finite-size normalizations explicit.
\subsection{The SK--Curie--Weiss comparison}
\begin{proof}[Proof of Corollary~\ref{cor:skcw}]
Abbreviate $F_n^{\mathrm{CW}}=F_n^{\mathrm{CW}}(\beta,\gamma)$ and
couple it with $F_n(\beta)$ using the same disorder. For
$\Delta_n=F_n^{\mathrm{CW}}-F_n(\beta)$, \cite[Proposition~1.5]{DeyKang2026} gives
$\Delta_n\ge0$ and, for every $u\ge0$ and $n$,
\[
\mathbb P(\Delta_n\ge u)\le C_\gamma e^{-u},
\qquad C_\gamma=(1-\gamma)^{-1/2}.
\]
Integrating the tail yields the uniform bounds
\[
\begin{gathered}
\mathbb E\Delta_n^2
=2\int_0^\infty u\,\mathbb P(\Delta_n>u)\,du
\le2C_\gamma,\\
\|\Delta_n-\mathbb E\Delta_n\|_2
\le K_\gamma:=\sqrt{2C_\gamma}.
\end{gathered}
\]
In particular the perturbed mean and variance are finite. Let
$t_n=\sqrt{\Var F_n^{\mathrm{CW}}}$. The reverse triangle inequality
in $L^2$ gives
\[
|t_n-s_n(\beta)|
\le\|\Delta_n-\mathbb E\Delta_n\|_2\le K_\gamma.
\]
Since $s_n(\beta)\sim\sqrt{a_2(\beta)}n^{1/6}\to\infty$, we have
$t_n/s_n(\beta)\to1$, which proves the variance asymptotic and makes
$t_n$ positive for all sufficiently large $n$. For those $n$, using
$\|X_n(\beta)\|_2=1$,
\[
\left\|\frac{F_n^{\mathrm{CW}}-\mathbb E F_n^{\mathrm{CW}}}{t_n}
-X_n(\beta)\right\|_2
\le\frac{K_\gamma}{t_n}
+\left|\frac{s_n(\beta)}{t_n}-1\right|
\le\frac{2K_\gamma}{t_n}\longrightarrow0.
\]
Theorem~\ref{thm:main} and Slutsky's theorem now give the asserted law.
\end{proof}
\subsection{Disorder chaos in the zero-field model}
\begin{proof}[Proof of Corollary~\ref{cor:disorder-chaos}]
For the Gaussian field in \eqref{eq:model},
\[
\rho(\sigma,\tau):=\Cov(H_n(\sigma),H_n(\tau))
=\frac1n\sum_{i